Application of the Fractal Market Hypothesis

ISAST Transactions on No. 1, Vol. 2, 2008 (ISSN 1797-2329)

Electronics and Signal Processing Regular Papers

Hamid Z. Fardi: Modeling and Characterization of 4H-SiC Bipolar Transistors ...................1 Yun Zhang, Mei Yu and Gangyi Jiang: Evaluation of Typical Prediction Structures for Multi-view Video Coding 7 Adriana Serban, Magnus Karlsson and Shaofang Gong: Microstrip Bias Networks for Ultra-Wideband Systems............................16 Jonathan M. Blackledge: Multi-algorithmic Cryptography using Deterministic Chaos with Applications to Mobile Communications...................................................21 Magnus Karlsson and Shaofang Gong: Monopole and Dipole Antennas for UWB Radio Utilizing a Flex-rigid Structure......................................................................................................59 Magnus Karlsson and Shaofang Gong: Monofilar spiral antennas for multi-band UWB system with and without air core ........................................................................................................64 Jean C. Chedjou, Kyandoghere Kyamakya, Van Duc Nguyen, Ildoko Moussa and Jacques Kengne: Performance Evaluation of Analog Systems Simulation Methods for the Analysis of Nonlinear and Chaotic Modules in Communications .............71 Adriana Serban, Magnus Karlsson and Shaofang Gong: A Frequency-Triplexed RF Front-End for Ultra-Wideband Systems 3.1-4.8 GHz.................................................................................................83 Jonathan M. Blackledge: Application of the Fractal Market Hypothesis for Macroeconomic Time Series Analysis............................................................................................89 Pär Håkansson, Duxiang Wang and Shaofang Gong: An Ultra-Wideband Six-port I/Q Demodulator Covering from 3.1 to 4.8 GHz ...........................................................................................................111

Greetings from ISAST Dear Reader, You have the first ISAST Transactions on Electronics and Signal Processing on your hands. It consists of ten original contributed scientific articles at the various fields of intelligent systems. Every article has gone through peer-review process. ISAST - International Society for Advanced Science and Technology – was founded in 2006 for the purpose of promote science and technology, mainly electronics, signal processing, communications, networking, intelligent systems, computer science, scientific computing, and software engineering, as well as the areas near to those, not forgetting emerging technologies and applications. To show, how large the diversity of computers and software engineering field is today, we shortly summarize the contents of this Transactions Journal: Hamid Z. Fardi has a research paper about a new model for design and optimization of 4H-SiC bipolar transistors. Yun Zhang, Mei Yu and Gangyi Jiang have analyzed and evaluated different typical multi-view video coding schemes in their research paper. On the other hand Adriana Serban, Magnus Karlsson and Shaofang Gong have a study about optimizing broadband microstrip bias networks to reduce resonance in broadband RF circuits. Jonathan M. Blackledge has an extended paper about multi-algorithmic cryptography using deterministic chaos. Magnus Karlsson and Shaofang Gong have presented two research papers about ultra wideband antenna design. The first paper discusses circular monopole and dipole antennas utilizing a flex-rigid structure and the second paper discusses spiral antennas with and without air core structure. In the paper of Jean C. Chedjou, Kyandoghere Kyamakya, Van Duc Nguyen, Ildoko Moussa and Jacques Kengne have evaluated different analog systems simulation methods for their performance. They have used these methods to investigate nonlinear and chaotic dynamics in communication systems. Adriana Serban, Magnus Karlsson and Shaofang Gong have a novel design of a RF front-end for multiband and ultra wideband systems that has fully integrated filter, triplexer network and a flat gain low noise amplifier. In his second extended paper Jonathan M. Blackledge introduces an idea about using fractal market hypothesis in macroeconomic time series analysis. Finally Pär Håkansson, Duxiang Wang and Shaofang Gong study a six-port I/Q demodulator that covers ultra wideband spectrum. We are happy to see how much we have obtained manuscripts with ambitious and impressive ideas. We hope that you will inform of the existence of our Society to your colleagues to all over the academic, engineering, and industrial world. Best Regards, Professor Timo Hämäläinen, University of Jyväskylä, FINLAND, Editor-in-Chief Professor Jyrki Joutsensalo, University of Jyväskylä, FINLAND, Vice Editor-in-Chief M.Sc. Simo Lintunen, University of Jyväskylä, FINLAND, Co-editor

Abstract—PISCES-IIB two dimensional device simulation

program is used to model the behavior of 4H-SiC bipolar junction transistors. The physical material parameters in PISCES such as carrier’s mobility and lifetime, temperature dependent bandgap, and the density of states are modified to accurately represent 4H-SiC. The simulation results are compared with the measured experimental data obtained by others. The comparisons made with the experimental data are for two different devices that are of interest in power electronics and RF applications. The simulation results predict a dc current gain of about 25 for power device and a gain of about 20 for RF device in agreement with the experimental data. The comparisons confirm the accuracy of the modeling employed. The simulated current-voltage characteristics indicate that higher gain may be achieved for 4H-SiC transistors if the leakage current is reduced. The simulation work discussed in this paper complements the current research in the design and characterization of 4H-SiC bipolar transistors. The model presented will aid in interpreting experimental data at a wide range of temperatures. This paper reports on a new model that provides insight into the device behavior and shows the trend in the dc gain performance important for the design and optimization of 4H-SiC bipolar transistors operating at or above the room temperature.

Index Terms—4H-SiC, BJT, dc gain, base resistance.

I. INTRODUCTION

There has been considerable interest in Silicon Carbide (SiC) bipolar transistors for high power high temperature and high frequency applications due to its superiority in physical and electrical properties such as wide bandgap, high saturation velocity, high breakdown voltage, and high thermal conductivity. Recently high power and high performance high frequency 4H-SiC bipolar transistors have been fabricated with demonstrated differential dc gain levels of about 20 to 25 at room temperature [1,2,3]. Researchers and investigators have fabricated and used 4H-SiC in a variety of device applications such as high performance bipolar transistors [4, 5]. In order to develop and design devices

H. Z. Fardi is with the Electrical Engineering Department University of Colorado at Denver and Health Sciences Center, Campus Box 110, P.O. Box 173364, Denver, CO 80217-3364, email: [email protected] , phone: 303-556-4938, fax: 303-556-2383.

based on the electronic properties of 4H-SiC, a thorough knowledge of their transport properties is needed. While some of this information is already available through device measurements and characterization, device modeling investigates an in-depth analysis of dc gain in 4H-SiC bipolar transistor operating at and above room temperature. By means of device modeling the device structure and material parameters can be related directly to measurements and physical parameters; the model is hence valuable in interpreting experimental data at a wide range of temperatures. The simulation work discussed in this paper complements the current research in the design and characterization of 4H-SiC bipolar transistors [6, 7, 8].

Advances in the processing and characterization of SiC devices further demonstrate the need for device optimization and design through the use of accurate device modeling. Some of these efforts are already established. For example, field-dependent carrier mobility model as a function of temperature and concentration is presented in [9], where a simplified analytical expression can model accurately the field-velocity relationship in SiC devices similar to the progress made in Silicon materials and devices.

II. DEVICE STRUCTURE

Two device structures are investigated in this study: device A, designed for high frequency application; device B designed for power. On the RF side, device A [2] has an active emitter area of 5x50 m2. The emitter base spacing is 5 m. The 4H-SiC emitter is 0.2-micron thick with 2x1019 cm–3 doped n-type. The base is 0.1-micron thick with 2x1018 cm–3 p-type. The n-layer 4H-SiC sub-collector with concentration of 2x1016cm–3 is 3-micron thick. Device B designed for high power high temperature electronics [5] has an active area of 1.2 mm2. It contains 43 emitter fingers, each having a dimension of 1186 m x 14 m doped n-type with a doping density of about 2 x 1019 cm-3. The emitter is 0.8m thick. The base emitter spacing is 3 m thick. The base is p-typed doped with a density of 4.1 x 1017cm-3. The sub-collector doping density is about 8.5 x 1015cm-3 and 12m thick. The schematic cross sectional view for these two devices is similar and is shown in Figure 1. A summary of the device geometry

Modeling and Characterization of 4H-SiC Bipolar Transistors

H. Z. Fardi, Senior Member, IEEE

Regular Paper Original Contribution

ISAST Transactions on Electronics and Signal Prosessing, No. 1, Vol. 2, 2008 Fardi H.Z.: Modeling and Characterization of 4H-SiC Bipolar Transistors

1

and the dimensions, including the doping levels for the two devices A and B is given in Table I.

Fig. 1. Schematic of 4H-SiC bipolar transistors represents device A taken from [1] and device B taken from [4]. The geometry and doping data and data related to the structure and dimensions are given in Table I.

TABLE I DEVICE GEOMETY AND DOPING LEVELS FOR THE TWO DEVICES STUDIED

Parameter Device A [2] Device B [5]

Emitter active area

5 x 50μm2

1.2 mm2

Emitter Width

5 μm

7μm

Emitter base spacing

5 μm

3μm

Emitter thickness (μm)

0.2 μm

0.8μm

Emitter doping ( cm -3)

2 x 10 19

2 x 10 19

Base thickness

0.1 μm

1 μm

Base contact density ( cm -3)

1 x 10 19

8 x 10 19

Base doping density( cm -3)

2 x 10 18

4.1 x 10 17

Sub-Collector length

3 μm

12 μm

Collector doping density ( cm -3)

2 x 10 16

8.5 x 10 15

III. DEVICE MODELING The two-dimensional (2D) device simulator PISCES-IIB is used for modeling 4H-SiC bipolar transistor [10]. This

device simulator solves the drift- diffusion partial differential equations and Poisson’s equation self-consistently for the electric potential, and the electron and hole concentrations. The simulation program was designed primarily for Silicon devices; however, material parameters can be modified to incorporate other semiconductor materials. In our case, libraries of important known material parameters for 4H-SiC were assembled from the literature [3,6,11]. Physical models incorporated in the simulation include Shockley-Read-Hall (SRH) for modeling the leakage current, surface generation-recombination mechanisms, mobility models, and ionized and neutral impurity effects as well as velocity-field relationship. The room temperature effective density of states in the conduction band and in the valance band are calculated based on the results obtained in reference [6].

The mobility parameters for 4H-SiC are similar to those of Si material taking into the consideration the doping dependency, high field and saturation velocity at a given temperature [6,9,11]. The room temperature (T0) doping level dependency of 4H-SiC is based on the following equation [12]:

KTTii

igii

i

i

NN

NB 300min,max,

,max,min,

0

)()( ==−

+=

μμ

μμ γ

(1)

where i is for electrons (n) and holes (p). The values of the constants imax,μ , imin,μ igN , , and iγ are presented in Table

II, taken from the literature approximated for the 4H-SiC material [13-16]. In relationship with Equation (1), the temperature dependence mobility is modeled by:

ii

i

TTNB

TTNB

TTNi

i

iiβα

β

μμ++

−=)).((1

)).(()(),(

0

00max, (2)

Where T0 is the room temperature and ),( TNiμ is the doping (N) dependent carrier mobility. The values of the constant iα and iβ are given in Table II.

TABLE II MOBILITY DATA USED IN 4H-SIC TRANSISTORSA

Parameters Electrons Holes

imax,μ (cm2 V-1s-1 ) 880 950

117

imin,μ (cm2 V-1s-1) 30 40

33

igN , (cm-3) 2 x 1017 2 x 1017

1 x 1019

iγ 0.67, 0.76 0.5

iα 2.6 -

iβ 0.5 -

a Data are taken from [6,7,13,14].

PISCESII-B is a drift-diffusion device modeling program, which uses both empirical formulation as well as real physical


2

equations for the temperature dependent material parameters. The general expression adopted for temperature dependent bandgap energy (Eg) for Si devices is based on the experimental values of the energy bandgap using the following empirical fitted relationship [17]:

bTaTETE gg +

−=2

)0()( , (3)

Where is the energy gap at zero Kelvin. Since no reports exist on 4H-SiC bandgap temperature dependence, it is assumed that 4H-SiC has the same temperature dependent as that of 6H-SiC , where constants Eg (0) = 3.359 eV, a=3.3 x 10 -4

eV/K and b=0 are assumed. This assumption has shown to be useful in characterization of 4H-SiC BJT [8]. The temperature dependent carrier lifetimes ( pn,τ ) and the

effective density of states( vcN , ) are modeled by the following

theoretical equations assuming that the phonon scattering is the dominant process at or above the room temperature[18]:

v

TTTpnpn ⎥

⎦

⎤⎢⎣

⎡=

0)0(,, ττ (4)

σ⎥⎦

⎤⎢⎣

⎡=

0)0(,, T

TTvcNvcN (5)

where the constants ν = -0.5 and σ = +1.5 are assumed. The relationship nτ = pτ is used. The above equations (4)

and (5) are empirical mathematical derivations used to simplify the numerical modeling [19, 20]. The ionization energy for the n-type layers using Nitrogen is about 100 meV and for the p-type layers using Aluminum is about 220 meV [1,2,3]. A SRH recombination carrier lifetime of 50 ns at room temperature is used in the simulation which fits the experimental measured gain of about 20 [19,21,22,23]. It should be noted that recently the device performance and processing of 4H-SiC have been enhanced considerably and much higher dc gain is achieved [24].

The material and device parameters used in the simulation are summarized in Table III. The two devices have been modeled using the same set of input parameters given in Tables II and III.

TABLEI II

PARAMETERS FOR NPN 4H-SIC TRANSISTORS AT ROOM TEMPERATURE

The focus of this work is not to extract any physical device

parameters from the numerical simulation, but rather to gain insight into the device behavior and to show the trend in the dc gain performance important for the design and optimization of 4H-SiC bipolar transistors operating at or above the room temperature.

IV. RESULTS AND DISCUSSIONS

In Figure 2, the maximum dc gain as a function of temperature is simulated for the device A and is compared with the measured data [1]. The room temperature dc gain is about 20. The gain decreases as the temperature increases as less current is available at collector region. The discrepancy at temperatures above 500K is attributed to both the accuracy of physical parameters and the series resistance at the p-type base contact that prevented graphing the absolute values rather than the normalized. A simplified analytical transport model of bipolar transistor predicts a similar gain-temperature relationship [6].

0.00.20.40.60.81.01.2

1.0 1.5 2.0 2.5 3.0 3.5

Temperature 1000/T (k-1)

Gai

n (N

orm

aliz

ed) Simulated Measured: Device A

Fig. 2. The simulated maximum current gain versus 1000/T

compared with the measured experimental device A [1]. The differential base resistance is an important parameter

for bipolar transistors at high frequency application. In bipolar junction transistors, the emitter prevents the diffusion of majority carriers from base (holes) into the emitter, which results in high electron injection efficiency. The differential base resistance is obtained from the inverse slope of the base current (Ib) at a given emitter-base bias (VBE) and temperature in low injection region, Figure 3. The high doping density of the base allows the base resistance to decrease without significantly sacrificing the emitter efficiency. As shown in Fig. 3, a similar trend is observed experimentally [1,2,3]. The measured base resistance in device A remains relatively constant at high temperature. The comparison shows the overall strength of the device modeling in predicting the

Parameters Value

Ref.

Saturation velocity Nc (300 K) Nv (300 K) SRH lifetime Surface-recom. velocity

3 x 107 cm/s 1.64 x1019cm

3.22 x1019cm–3 50 ns

1x105 cm/s 1x104 cm/s

[6] [6] [6] [23] [21] [22]


3

device behavior, making it a useful tool for optimization and design reducing the costly iterative experimental procedure.

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4

Temperature 1000/T (k-1)

Bas

e R

esis

tanc

e (N

orm

aliz

ed)

Simulation Measured-Device A

Fig. 3. The simulated base resistance versus 1000/T compared with the measured experimental device A [1].

Figure 4 shows the measured dc current gain over a range of emitter-base voltage for device A [1]. The collector-base bias was kept at a constant voltage of 10 V (The base-emitter bias was probed so that the differential base resistance can be measured directly from Ib-VBE data). As shown in Fig. 4, the maximum current gain simulated for the device A is about 20 which is comparable with the measured data.

0

4

8

12

16

20

0 5 10 15 20Base Emitter Voltage (V)

Gai

n

Measured: Device A Simulated

Figure 4. The measured current gain for the device A [1,2,3] is compared with the simulated results (solid lines) as a function of the emitter-base bias, T=300 K. The collector-base voltage was kept at a constant voltage of 10 V.

The measured differential dc gain as a function of collector current density for the power device B is shown in Figure 5

[5]. Both the room temperature and the data for T=423 K are shown. The maximum current gain simulated for the device B is about 25 at room temperature and about 15 at T=423 K compared with the measured data. The maximum dc gain simulated is in close agreement with the experimental result. The measured gain is somewhat lower than the simulated result at low collector current (low injection) indicating leakage current in measured data.

0

5

10

15

20

25

30

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Collector Current (A)

Gai

n

Simulated T=300 K Simulated T=423 KMeasured: Device B T=300 K Measured: Device B T=423 K

Fig. 5. The simulated current gain for the device B is compared with the measured data [4]. The junction becomes more non-ideal as the temperature increases. The maximum current gain of 25 at room temperature and about 15 at T=423 K are in agreement with the simulated results.

The leakage current is shown by simulating the base current for Device B using SRH recombination model. The base current as a function of base-emitter bias for device B at two different temperatures is shown in Figure 6. These results predict that the SRH recombination current at low-level injection may result in the degradation of the transistor’s gain (shown in Figure 5). That is, in the base region, SRH recombination rate would be high, reducing the number of carriers that would otherwise be available at the collector region, resulting in lower gain values.

Also shown in Figure 6 is the simulated turn-on voltage compared with the experimental data at two different temperatures for the device B. The measured data is obtained by other investigators [3,5]. The turn-on voltage decreases as the temperature increases. This can be explained by the diode property of the junction. The drift-diffusion model employed predicts accurately the junction current voltage characteristic at high injection.


4

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Base Emitter Voltage (V)

Cur

rent

Ib (

A)

Simulated T=300 K

Measured: Device B T=300 K

Simulated: T=423 K

Measured:Device B T=423 K

Fig. 6. The turn-on voltage for the device B at two different temperature T=300 K and T=423 K. The measured data are also shown [4].

The simulated room temperature base current and collector current (Ic) are shown in Figure 7 as compared with the measured experimental data for device A [1] The transistor starts to show gain at base emitter voltage of about 3 Volts . The simulation model predicts lower base current and at higher base emitter voltage than the experimental data. The discrepancy in base current at low base emitter bias is attributed by Perez et al. to the leakage current caused by the incomplete removal of the emitter epitaxial layer on top of the extrinsic base as reported in detail in the original experimental work [1] In a later work, they report on enhancing the device fabrication process that resulted in better device performance [24]. The physical drift-diffusion model applied in the simulation is relatively accurate for most of the device data at the high injection that is dominated by high series resistance. In this regime, the physical model normally tends to fail to predict accurately the current-voltage relationship since the series resistance dominates and the model tends to fail to predict the current-voltage characteristics. At low injection, the current-voltage characteristics are modeled by a relatively high SRH recombination rate with a carrier lifetime of 50 ns that may be at the high end [23]. The simulation results indicate that higher dc gain can be achieved if the leakage current is reduced.

1.E-10

1.E-09

1.E-08

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

0 1 2 3 4 5 6 7 8 9 10

Base Emitter Voltage (V)

Cur

rent

(A)

Ib Measured: Device A Ic Measured: Device A

Ib Simulated Ic Simulated

Fig. 7. The simulated room temperature base current and collector current for the device A as compared with the measured data [1,2,3]. The measured base current shows leakage current. A maximum dc gain of about 20 is obtained for this device at room temperature. High level injection is dominated by series effect.

V. CONCLUSION A 2D drift-diffusion simulation program is utilized to model 4H-SiC bipolar transistors used in both power electronics and high frequency high temperature applications. The 4H-SiC model parameters are applied to an existing device simulator, PISCES IIB, developed primarily for Silicon transistors. The simulated results are obtained for the base resistance, differential dc gain, and the current –voltage characteristics over a wide range bias and temperature. The comparison made with the measured experimental data confirms the accuracy of the modeling established over the range of temperature investigated. The simulation results predict a maximum dc gain of about 20 for device A and about 25 for device B in agreement with the measured data. The modeling results show that a higher dc gain can be achieved if leakage current is to be minimized for 4H-SiC BJT’s.

REFERENCES [1] Perez-Wurfl, I. et al. “4H-SiC bipolar junction transistor with high

current and power density,” Solid State Electronics, vol. 47, pp. 229-231, 2003.

[2] Perez-Wurfl, I., Konstantinov,A., Torvik, J., and Van Zeghbroeck, B., “RF 4H-SiC Bipolar Transistors, “Proceedings of the Leste rEastman Conference, Newark, DE, Aug. 6-8, 2002.

[3] Perez-Wurfl, I., Torvik, J., and Van Zeghbroeck, B., “4H-SiC RF Bipolar Junction Transistors, “ Proceedings of DRC, pp. 27-28, 2003.

[4] Huang C.-F., Cooper, J. A. Jr., “high performance power BJTs in 4H-SiC,” IEEE 7803-7447, 2002.

[5] Zhao, J. H., et al., “A high voltage (1750V) and High Current gain (β=24.8) 4H-SiC Bipolar Transistor using a thin (12 μm) drift Layer,


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ICSRM manuscript no. 470, session ThP3-14, Lyon, France, Oct. 5-10, 2003.

[6] Danialsson, E., “Processing and electrical characterization of GaN/SiC heterojunctions and SiC bipolar transistors”, ISRN KTH/EKT/FR-01/1-SE, KTH, Royal Institute of Technology, Stockholm, 2001.

[7] Danielsson, E., et al., Solid State electronics, vol. 47 , p. 639, 2003. [8] Li, X, et al., “on the temp. coefficient of 4H-SiC BJT current gain,”

Solid State Electronics, vol. 47, pp. 233-239, 2003. [9] Mnatsakanov,T.T., et al., “ carrier mobility model for simulation of

SiC-based electronic devices, Electronic Journal: Semiconductor Science and Technology, 17, No. 9, pp. 974-977, 2002.

[10] Pinto, M. R., Rafferty, C. S., Dutton, R. W., PISCES II: Poisson and Continuity Equation Solver, Integrated Circuits laboratory, EE Department, Stanford University, CA., 1984.

[11] Raghunathan, R., and Baliga, B. J., “p-type 4H and 6H-SiC High-Voltage Schottky Barrier Diodes, “ IEEE EDL, vol. 19, no. 3, pp. 71-73, 1988.

[12] Caughey, D. M., and Thomas, R.E., Proceedings of IEEE, vol. 55, p. 2192, 1967.

[13] Mnatsakanov T T, Pomortseva L I and Yurkov S. N., Semiconductors 35 394, 2001.

[14] Roschke M and Schwierz F., IEEE Trans. Electron Devices 48 1442, 2001.

[15] Morkoc, H., et al., “Large-band-gap SiC III-V nitride, and II-VI ZnSe-based semiconductor,” J. Appl. Phys. 76 (3), 1, pp. 1363-1398, 1994.

[16] Joshi, R. P., “ Monte Carlo calculation of the temperature- and field-dependent electron transport parameters for 4H-SiC,” J. Appl. Phys. 78(9), pp. 5518-5521, 1995.

[17] Thurmond, C. D.,”The standard thermodynamic for the formation of electrons and holes in Ge, GaAs, GaP’s,” J. Electrochem., Soc., vol. 122, pp. 1133-1141, 1975.

[18] Sze, S. M., Physics of Semiconductor Devices, John Wiley & Sons, New York, 1981.

[19] Arora, N. D., Hauser, J. R., and Roulston, D.J., IEEE Trans. ED, vol. 29, p. 292, 1982.

[20] Ayalew, Tesfaye ,SiC semiconductor Devices technology, Modeling, and Simulation, PhD Thesis, Technischen Universitat Wien, 2004.

[21] Galeckas, A., et al., Appl. Phys. Lett., 79, 365, 2001. [22] Ivanov, P. A., et al., “Factors, limiting the current gain in high-voltage

4H-SiC npn BJTs, Solid State Electronics, vol. 46, pp. 567-572, 2002. [23] Fardi, H. Z., “Modeling the dc gain of 4H-SiC Bipolar Transistors as a

function of surface recombination velocity”, Solid State Electronics, 2005.

[24] Van Zeghbroeck Bart, Perez, I., Zhao, F., and Torvik, J., “Technology development of 4H-SiC BJTs with 5GHx fMAX,” CS MANTECH Conference, pp. 223-226 ,Vancouver, Canada, April 24-27, 2006.

H. Z. Fardi received the B.S. degree in Physics from Tehran University, Iran in 1978, and the M.S. and Ph.D. degrees in electrical engineering from the University of Colorado at Boulder in 1982 and 1986, respectively. He joined the University of Colorado at Denver in 1992 where he is now a professor of electrical engineering. Prior to this, he was with the National Science Foundation Center for Millimeter-Microwave Computer Aided Design of the University of Colorado at Boulder. He has worked on numerous research projects related to physics and modeling of novel semiconductor devices and solid state electronics. His research work on physics and characterization of indium phosphate photovoltaic devices, gallium arsenide field effect transistors, pnpn thyristor light emitting diodes, and silicon sarbide bipolar transistors are published in 42 refereed national and international technical journals and proceedings. He received the Western Universities Fellowship awards thru National Renewable Energy Laboratory in 1996 and 1997 working on multi-quantum well superlattices. Dr. Fardi was

the recipient of the Researcher of the Year Award form the College of Engineering of University of Colorado at Denver in 1997.

Dr. Fardi is a senior member of the Institute of Electrical and Electronics Engineers (IEEE) and the Treasurer of Electron Device Society-Denver Chapter.


6

Evaluation of Typical Prediction Structures for Multi-view Video Coding

Yun Zhang, Mei Yu, and Gangyi Jiang

Abstract The main requirements of the prediction

structure for multi-view video coding (MVC) are that they have to provide high compression efficiency, operate with reasonable complexity and memory requirements, and facilitate random access. Up to now, many MVC prediction structures have been proposed mainly for obtaining high coding gain, however there are more and more importance is attached to MVC schemes functionalities, such as random accessibility, coding or decoding complexity, and scalability. It is desirable to make a comprehensive evaluation of the schemes that contributes to further understanding over MVC prediction techniques and the corresponding MVC schemes. In this paper, we analyze nine typical MVC prediction structures in six categories at length, and then experimentally compare their performances, including complexity, random accessibility, and scalability as well as compression efficiency. Finally, we discuss the trade-off between the compression related requirements for further researches on MVC. Generally, MVC with hierarchical B picture is the most efficient scheme in compression, but results in high complexity and low random accessibility. And simulcast prediction structure is most suitable for the applications without strong storage limitations but with low-delay or real-time requirements. Other schemes have a trade-off between compression efficiency and functionalities by adopting multiple intra frames, sequential prediction, multiple reference prediction and intra frame centered structure etc1.

Index Terms Multi-view Video Coding, Prediction Structure, Random Access, Coding and decoding Complexity, Scalability.

I. INTRODUCTION Multi-view video capturing, analysis, coding and display

have attracted a lot of attention in recent years since three

Manuscript received on August 1, 2007. This work was supported by Natural Science Foundation of China (grant 60472100, 60672073), the Program for New Century Excellent Talents in University (NCET-06-0537), the Key Project of Chinese Ministry of Education (grant 206059), and the Natural Science Foundation of Ningbo China (grant 2007A610037).

Yun Zhang was with the Faculty of Information Science and Engineering, Ningbo University, Ningbo, China, he is now with the Institute of Computing Technology, Chinese Academic of Science and Graduate School of Chinese Academic of Science, Bejing, 100080, China (email: zhangyun_8851@163. com)

Mei Yu is with the Faculty of Information Science and Engineering, Ningbo University, Ningbo, 315211, China (e-mail: [email protected]). She is corresponding author of the manuscript.

Gangyi Jiang is with the Faculty of Information Science and Engineering, Ningbo University, Ningbo, 315211, China (e-mail: [email protected]).

dimension television is expected to be the next killer application of digital media technologies in the coming decade. However, there are still quite a lot of technological difficulties needed to be overcome. Due to the massive mount of data of the multi-view video, data transmission and processing requires much more bandwidth and computational power than that of mono-video. Thus, Multi-view video coding (MVC) [1]-[3] has become one of the key techniques for multi-view video applications.

Many MVC schemes have been proposed, including MVC with simulcast[4], sequential view prediction structure (SVPS)[1], group-of-GOP coding structure (GoGOP)[5][6], MVC with multi-directional picture (M-picture) [7][8] and MVC with hierarchical B picture (HBP)[9][10] etc.. SVPS adopts view-by-view prediction in order to achieve relatively high coding efficiency by relieving occlusion and exposure. MVC with M-picture prediction structure introduces a new type of picture, M-picture, which supports 21 coding modes, for high coding gain. MVC with HBP prediction structure introduces HBP picture which significantly improves compression efficiency. These researches on MVC mainly focused on rate-distortion (RD) performance, because high compression efficiency is demanded by strong transmission/storage limitations of available networks and storage devices.

On the other hand, nowadays, there are more and more importance attached to MVC schemes functionalities, such as random access (RA), coding or decoding complexity, and scalability. Interactive multimedia applications, such as free viewpoint video communication, will let the user freely change viewing position and direction while downloading and streaming a video content. Therefore, fast RA in view and time dimension is a key performance of MVC[11]-[13]. GoGOP coding structure is mainly proposed to improve RA by adopting multiple intra frames in 2D GOP. In addition, MVC schemes are also required to provide additional compression related requirements for functionalities, such as low delay encoding-decoding, spatial/temporal/SNR/ view scalability low memory and complexity requirement[2][3]. These requirements should be jointly taken into account to evaluate MVC schemes.

The rest of the paper is organized as follows, in section II and III, structure related requirements and nine typical MVC coding schemes in six categories are discussed, respectively, and then, comparisons among the performances of the typical

ISAST Transactions on Electronics and Signal Prosessing, No. 1, Vol. 2, 2008 Yun Zhang et al.: Evaluation of Typical Prediction Structures for Multi-view Video Coding

Regular Paper Original Contribution 7

MVC prediction structures are presented and analyzed in section IV. Finally, a conclusion is given.

II. REQUIREMENTS FOR MVC In addition to high compression efficiency, an excellent

prediction structure for MVC should operate with reasonable complexity, memory requirements and fast RA, and provide scalability and so on.

A. Random Accessibility MVC schemes should support RA in view and time

dimension and RA to a spatial area in a picture[3]. RA directly affects the system capabilities that support view switching, view sweeping and frozen moment[14] etc. Two cost parameters, average and maximum path lengths of random viewpoint access, can be used to describe random accessibility of MVC scheme [10][15]. The smaller the costs are, the better RA a MVC scheme supports. Fig.1(a) shows the simulcast prediction structure, in which each view is coded independently. The group of multi-view video can be thought as a two-dimensional (2-D) coding cell or a matrix, in which each element is a picture. The horizontal axis represents views, and the vertical is time axis. Each rectangle represents a frame and the arrow point to the compulsory reference picture.

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Let xi be the number of frames that must be pre-decoded

before the ith frame being decoded in a coding cell with Mview views and Ntime time-steps. Let pi be the probability for the ith frame to be accessed by viewer. The average path length of random viewpoint access [14], Fav, is defined by

∑×

=

=viewtime MN

iii pxXE

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)( (1)

And the maximum path length of random viewpoint access is defined as follows

max max 0i view viewF x i M N= < ≤ × (2)

B. Computational Complexity In H.264/AVC based MVC coding platform, a good

portion of the gain comes from high precise disparity compensation prediction (DCP) and motion compensation prediction (MCP), however, at the cost of high coding complexity and memory consumption. MCP or DCP occupies most of coding time, so in this paper, we estimate the computational complexity of a MVC prediction structure by using the minimum number of reference frames of a coding cell, PNmin. (Note that the coding structures could expand the number of referenced frames for high compression efficiency.) The lager PNmin is, the more complicated the encoder is. For example, simulcast scheme, as illustrated in Fig.1(a), adopts P frames that reference at least one frame in the encoding procedure. Therefore, its PNmin is 30.

C. Memory Requirement General memory requirements stem from AVC, since all

frames from all views are reordered and treated as one single video stream by the coder. And the decoded picture buffer (DPB) in H.264/AVC is used to store the reference frames. Assume that each scheme adopts the optimal coding order to minimize the DPB size, represented by DPBmin here. For the simulcast prediction structure, since P frames are single-referenced and are coded view-by-view, only one frame has to be stored in DPB for next frame, i.e. DPBmin equals to 1.

D. View Scalability View scalability enables the video to be displayed on a

multitude of different displayers, such as multi-view displayer, stereo displayer or HDTV, and terminals over networks with varying conditions[3]. Therefore, the decoder should be able to selectively decode the appropriate number of views according to the type of display modes with view scalability. In this paper, we define two cost variables, FSV, FDV, to represent the average number of compulsorily decoded frames in a coding cell when single view or double views are displayed, respectively.

Let On be a set of frames in a coding cell and Xi,j be a set of compulsory decoded frames when the frame at (i,j) position in a coding cell is displayed. Thus i , j nX O⊆ . Suppose ρj is the probability for jth view to be chosen to display by the user, and ρj,k is the probability for both jth view and kth view to be accessed. FSV and FDV are defined as

1 1

timeview NM

SV i, j jj i

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= ⋅ ∑ U (3)

( )1 1 1

timeview view NM M

DV i, j i ,k j ,kj k j i

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= ⋅ ∑ ∑ UU (4)

where Card is cardinality of a set. On the other hand, the sender is just required to send the compulsorily decoded


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frames so as to save transportation stream bandwidth as well as decoders computation power. The smaller the cost variables are, the better view scalability the decoder supports.

Besides the requirements discussed above, there are other compression related requirements and system related requirements should be supported by the multi-view video system. More details are available in [3].

III. TYPICAL MVC PREDICTION STRUCTURES A. Simulcast As a reference of evaluating MVC schemes compression

efficiency, simulcast coding structure was analyzed based on H.264/AVC coding platform, in which each view in multi-view video is coded independently [4]. The first frame of each view is coded as an intra frame (I-frame), the remaining frames are coded as P-frames, as shown in Fig.1(a). Simulcast coding can be achieved by independently coding the coding cell column by column. The simulcast coding structure is a direct expansion of mono-video coding scheme.

On account of the deficiency of simulcasts RD performance, Sun et al have proposed a MVC scheme based on DCP, denoted as IPPP, as shown in Fig.1(b). It is clear that the key frame of the center view in each group of GOP is coded as I-frame, while other key frames are coded using inter-view DCP from center to both sides. The structure enhances coding efficiency by eliminating inter-view redundancy of the first time of a coding cell, while maintains relatively fast RA.

B. Sequential View Prediction Structure [1] Fig.2(a) illustrates a sequential prediction structure for

MVC, where the video sequence from the first camera video is encoded normally using temporal prediction. Then, each frame in the second viewpoint is predicted using the corresponding frame in the first viewpoint, as well as with temporal prediction from other frames in the second viewpoint. Next, the third viewpoint sequence is predicted using the second viewpoint sequence, and so on. There may be several variations of this sequential approach. For example, as illustrated in Fig.2(b), bi-prediction may be used to increase coding efficiency. In this case, bi-predicted pictures in certain views would be stored so that neighboring pictures in different views could use these pictures as a reference for prediction.

In Fig.2(b), there exist five types of frames, they are I (pure intra coded), P' (single view prediction, use only I or other P' as reference), P (single temporal prediction, use only I or other P as reference), B' (bi-predicted temporally and spatially, uses at most two references of P, P' or other B' frames), and B (also bi-predicted temporally and spatially, but uses three references including stored B from other preceding view).

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Fig. 3(a) GoGOP AB Fig. 3(b) GoGOP SR

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C. Group-of-GOP Coding Structure NTT Corp. proposed a MVC scheme with the GoGOP

structure, which is capable of decoding a view in low delay [5][6]. The GoGOP is the extended structure from GOP and provides the low-delay RA in view dimension as well as in time dimension. In GoGOP, all GOPs are categorized into two, Base-GOP and Inter-GOP. A picture in a Base-GOP may reference decoded pictures only in the current GOP, however, a picture in an Inter-GOP may use decoded pictures in other GOPs as well as in the current GOP.

The method to code all pictures as Base-GOP, is called GoGOP AB (All in Base-GOP), as illustrated in Fig.3(a). It is proposed to achieve lowest delay though deficient in compression efficiency. To improve coding efficiency, other two advanced schemes are proposed. One is the Single Reference (SR) Inter GOP coding method, denoted as GoGOP SR. In GoGOP SR, a picture in an Inter-GOP refers to previous and current pictures in Base-GOPs and previous pictures in the same GOP, as illustrated in Fig.3(b). Extending the structure of GoGOP SR, the Multiple Reference (MR) Inter GOP coding method, denoted as GoGOP MR, was proposed. In GoGOP MR, a picture in an


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Inter GOP refers to previous and current pictures in other GOPs (Base-GOP and Inter-GOP) and previous pictures in the current GOP, as illustrated in Fig.3(c).

D. Lims MVC Scheme Lim et al proposed a multi-view sequence CODEC with

flexibility and view scalability[15]. The encoder generates two types of bitstreams, a main bitstream and an auxiliary one. The main bitstream is coded using temporal prediction. The auxiliary bitstream contains information concerning the remaining multi-view sequences except for the reference sequences. Fig.4 illustrates one kind of Lims prediction structure with one I-frame, named One-I. In the prediction structure five kinds of frames are defined, they are I-frame, Pt frame and Bt frame for removing temporal redundancy using MCP, Ps frame and Bs frame for view redundancy using DCP, and Bs,t frame predicted from temporal and spatial dimension simultaneously for removing redundancy in both temporal and spatial domain. The I-frame is centered in the coding cell for fast RA. In Fig.4, the middle view of the matrix is encoded into the main bitstream, and the rest views are coded into auxiliary bitstream. The codec are selectively determined at the receiver according to the type of display devices and display modes. The viewers can choose an arbitrary number of views by checking the information so that only the views selected are decoded and displayed. For higher compression efficiency, we implemented Lims prediction structure on H.264/AVC platform for experimental comparison.

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… … … … … Fig. 5 MVC with M-picture Fig. 6 MVC with HBP

E. MVC with Multi-directional Picture Oka and Fujii proposed a new picture type for coding

dynamic ray-space named multidirectional picture, which utilizes inter-image prediction in both temporal and spatial domains [8]. M-picture has 21 coding modes classified into five categories for high compression efficiency. Fig.5 illustrates MVC with M-picture prediction structure in a coding cell with five views and seven temporal intervals. Rectangles labeled M denote the multidirectional pictures. Each M-picture are multi-referencing at least four surrounding B or P pictures.

The introduction of M-picture improves compression efficiency because multidirectional prediction enables the coder to encode pictures with fewer residual errors and bits. Furthermore, a new cost function has been proposed for effective mode selection that maximizes the coding efficiency of the coder.

F. MVC with Hierarchical B Picture HBP significantly improves RD performance when

quantization parameters (QP) for the various pictures are assigned appropriately [17]. Additionally, HBP provides hierarchical temporal scalability. Based on the statistical correlation analysis of multi-view video, MVC with HBP is proposed for the case of a 1D camera arrangement (linear or arc). Later, it is extended to support 2D camera arrangement (cross or 2D-array) [10]. Fig.6 shows the prediction structure for 1D camera arrangement. In the figure, rectangles labeled Blevel are the HBPs with different levels, where level indicates the different coding level of B picture. And the higher-level B pictures are allowed to reference from the lower-level pictures only.

The inter-view/temporal prediction structure in Fig.6 applies HBP in temporal and inter-view dimension. For H.264/AVC compatible encoder, the multi-view video sequences are combined into one single uncompressed video stream using a specific scan[9]. This is a pure encoder optimization, and the only change to the encoder is the increase of the decoded picture buffer size to 2 × Ntime + Mview to store all necessary images, and a potentially larger number of output pictures per second than it is currently allowed in H.264/MPEG4-AVC.

The MVC with HBP outperforms other approaches in compression efficiency significantly and provides multi-level temporal scalability. In addition, the multi-view video data are reorganized into a single uncompressed video stream that is fed into standard H.264/AVC encoder, thus the resulting bitstream is H.264/AVC compatible.

IV. EXPERIMENTAL RESULTS AND ANALYSES

A. Compression Efficiency Comparison In order to evaluate compression efficiency of the nine

MVC schemes, the MVC prediction structures presented in section 3 are realized using an H.264/AVC encoder with extended memory capabilities. And multi-view video sequences, from KDDI, MERL and Tanimoto Lab, with varying content, different spatial-temporal density and different resolutions, such as flamenco1, race2, golf2, vassar, Aquarium and xmas with 30mm camera interval, have been coded. Information of the sequences is specified in TABLE I. Encoding parameters are illustrated in TABLE II. And it is important to point out that, in our experiments, I, P, B and M pictures adopt a consistent QP in a coding


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process and the coders use the reference frames as few as possible for low-complexity.

Fig.7 illustrates the PSNR-Y over bit-rate averaged over all views of a data set. Here, HBP depicts MVC with HBP coding structure, and BSVP and PSVP denote SVPS using P and B pictures, respectively. GoGOP SR and GoGOP MR depicts the two GoGOP coding structure. Jeong represents the MVC schemes proposed by Lim et al, and Mpicture depicts the MVC with M-picture coding scheme proposed by Fujii et al. Additionally, simulcast coding scheme and the IPPP coding structure proposed by Sun et al is represented by Simulcast and IPPP, respectively.

TABLE I

TEST SEQUENCES FOR MULTI-VIEW VIDEO

Data Set Video characteristics Resolution Camera distance

flamenco1 Slow movement, varying chroma race2 Cars move fast golf2 Cameras move horizontally, slow

320×240 20cm

vassar Big static background 640×480 19.5cm xmas Dense linear camera arrangement 640×480 30mm

aquarium Arc camera arrangement 320×240 ≈3cm

TABLE II CODING PARAMETER

Platform JM8.5 main profile RDO Yes

Entropy Coding Loop filter, CABAC Search range ±32 for CIF/VGA

Motion/Disparity Compensation 1/4Pixel Accuracy, Full SearchQP 18, 24, 30, 36

0 0.1 0.2 0.3 0.4 0.5 0.6 0.732

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Fig. 7 PSNR results for the five data set

In Fig.7, it is obvious that the HBP outperforms other

schemes significantly about 1dB at all bit-rates for tested multi-view sequences, (note that the curves are linked from 4 specified points). BSVP, PSVP, Mpicture and Jeong schemes win the second place in coding efficiency and outperform simulcast from 1dB to 4dB through spatial and temporal prediction. GoGOP and IPPP curves nearly overlap simulcast for all sparse sequences, but for the dense xmas sequence, they outperform simulcast 1dB to 2dB. Obviously, the compression efficiency does not only depend on the parameters of the sequence such as camera distance and frame rate, but also on the content of the sequence itself. From Fig.7(b), Mpicture scheme is inferior to IPPP scheme in RD performance because M-pictures are not efficient enough when the chroma of flamenco1 varies in temporal domain. SVPS, Jeong, HBP and Mpicture obtain significant compression efficiency from spatial prediction. HBP also obtains a good portion of the gain that comes from the HBPs in temporal dimension.

For most multi-view sequences, the majority of correlation is mainly in temporally preceding frames. For Aquarium, only a smaller gain, no more than 1dB, could be achieved by spatial and inter-view prediction because the percentage of spatial correlation is low. However, there is also a significant amount of dependencies that are better predicted from the spatial direction [18]. For the sequences flamenco1, golf2, race2, this amount is rather high that leads to the assumption that for these sequences a significant coding gain could be achieved by a proper multi-view prediction scheme. Multiple reference prediction reduces temporal and inter-view dependencies to improve compression efficiency. But for the sequence of which correlations are concentrated on temporal domain, a multi-reference prediction scheme is unable to achieve more gain compared to simulcast coding. So to achieve both high compression efficiency and low complexity, its reasonable to predict from spatial domain, temporal

domain or spatial-temporal jointly according to the spatial-temporal correlation.

TABLE II PERFORMANCE COMPARISON TABLE

Random Access Cost

View Scalablity

Coding Structure

Fav Fmax PNmin

DPBmin FSV FDV

Simulcast 3.0 6 30 1 7 14 GoGOP SR 3.6 9 111 16 12.6 21.7 GoGOP MR 4.6 14 114 16 15.4 25.2

PSVP 11.0 34 58 7 21 28 BSVP 7.5 19 83 7 21 28

MVC with Mpicture 6.0 20 97 16 16 22.4

MVC with HBP* 7.6 16 96# 21 15.2 26.1

Lims scheme 3.1 5 62 8 12.6 18.2

IPPP 4.2 8 34 3 8.2 15.2 Note * represents the corresponding performance value in 5×8 coding structure # represents more complexity due to recursive MCP and DCP

B. Random Accessibility Comparison In MVC with HBP scheme, the maximum number of

reference frames Fmax is necessary for those B frames with highest level Lmax and highest view number Mview within a GOP. Thus, Fmax can be calculated as Fmax=3×Lmax+2× (Mview -1)/2. Applying this to the coding structure in Fig.6, a total of Fmax=16 referencing frames have to be decoded, namely. In TABLE III, it is obvious that Fmax is generally coinciding with Fav for the compared schemes.

Simulcast scheme is the best one in Fav owing to single reference prediction and multiple I-frames in a coding cell. However, it is inferior to Lims scheme a bit in Fmax because simulcast sequentially predicts in temporal domain of each view. Lims scheme is excellent in RA owning to its I-frame centered structure, although it may cost more buffers. IPPP and GoGOP schemes are in second place in random accessibility. Compared with simulcast, IPPP structure enhances compression efficiency by eliminating inter-view redundancy of the first time-step of a coding cell. It reduces I-frames and extends path length of sequential prediction. GoGOP maintains relative fast RA thanks to multiple I-frames and its small Base-GOPs size. But frames in Inter-GOPs are multi-referencing, especially for frames in MR Inter-GOPs. They not only increase complexity but also extend the number of pre-decoded frames that causes more delay. SVPS is the worst of all in RA. It is clear that PSVP has to pre-decode 11 frames on average when one frame is randomly selected to be decoded. In case of the bottom-right frame is selected, the decoder has to pre-decode 34 frames. Introduction of B pictures in SVPS cuts down the path length of sequential prediction and improves random accessibility. MVC with HBP and MVC with M-picture are not good in RA due to multiple reference prediction.


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C. Computational Complexity Comparison Column PNmin in TABLE III shows the computational

complexity of the MVC schemes. From the table, GoGOP is the most time-consuming scheme of which PNmin is more than 110, because most frames in Inter-GOPs reference from 6 to 8 frames. MVC with M-picture, MVC with HBP and BSVP are in the second place in complexity. Most HBPs and all M-pictures reference from four frames, and most B frames in BSVP reference from three frames. Lims MVC scheme and PSVP are in the third place. PSVP mainly adopts P frames, most of which are two-frame referencing, either temporally or spatially according to RD cost. In Lims MVC scheme, most frames are also two-frame referencing. Simulcast and IPPP mainly adopts single reference P frames that contribute to their lowest complexity.

D. Memory Requirement Comparison Column DPBmin in TABLE III shows the memory

requirements of the compared MVC schemes. For MVC with HBP, in the case of a coding cell with five views and seven time-steps, DPBmin would be 21 whatever the encoding order is. GoGOP and MVC with M-picture are also memory consuming for DPBmin equals to 16. We attribute the three schemes high memory requirement to multi-reference prediction structure. For instance, all I, B and P frames have to be stored in MVC with M-picture scheme while encoding M-pictures. SVPS and Lims scheme are in the middle class in memory requirement because of sequential prediction and I-frame centered structure, respectively. Simulcast and IPPP are the best thanks to single-reference prediction. From the DPB size and structure analysis, we can obtain that single-reference prediction, sequential prediction and I-frame centered structure may contribute to low memory requirement.

E. View Scalability Comparison Assume that the view switching among the views is a

average probability event, that is ρj=0.2 and ρj,k=0.1. Columns FSV, FDV in TABLE III show the number of decoded frames for displaying single view and double views, respectively. As we can see in the table, simulcast is the best one and there is no additional decoded frame in this scheme. IPPP needs one or two additional frames because the frames in first time step are coded using DCP from center to both sides, but it is still a good structure in view scalability. The view scalability cost of Lims scheme and GoGOP SR is generally 1.5 times of the displayed frames which means 1.5 times computational power of decoder and bandwidth. GoGOP MR, MVC with M-picture and MVC with HBP require nearly two times computational power. SVPS is the worst of all. It expenses three times cost for single view displaying and two times decoding frames for double view displaying. Generally, view-by-view sequential prediction and multiple reference prediction may have burden on view scalability of the multi-view video system.

From the results above, MVC with HBP provides extremely

outstanding compression efficiency as well as temporal scalability that come from HBP and distinctive prediction structure (one I frame is used by two coding cells). However, this coding structure requires large memory (more than a GOP.), which is disadvantageous especially for real-time applications. Though not superior in RA, resource consumption and view scalability, it is a MVC scheme with high compression ratio and really good for storage-faced applications.

Simulcast and IPPP schemes, contrary to MVC with HBP, provide fast RA, low complexity, low memory requirements and perfect view scalability, but their common fatal disadvantage is that they are not good enough in RD performance, though IPPP improves a bit when compared with simulcast. As a result, simulcast and IPPP are recommended in the applications without strong bandwidth limitation but with low-delay requirements, such as multi-view video system in LAN or receivers without enough computational power and memory.

Lims MVC scheme jointly adopts MCP and DCP to eliminate temporal and inter-view redundancies. The improvements in better random accessibility and lower complexity can be significantly achieved by placing the I-frame at the center of the 2D GOP. However, I-frame centered structure will introduce some initial delay and require more DPB size on the receiver side, so dose MVC with HBP. Lims scheme is a well-balanced scheme among compression efficiency, RA and complexity etc.

Though M-pictures provide high compression efficiency in MVC with M-picture, bottom-right and up-right frames in a coding cell are coded in low efficiency due to long-distant inter-view prediction. It may work well for high-density camera multi-view video sequences of which the correlations of both domains are not so different. On the other hand, the coder is very time-consuming owing to multiple reference prediction and rate-distortion optimization so that it is difficult to be applied in real-time applications. Additionally, MVC with M-picture is poor in RA and view scalability that are essential to multi-view video applications, such as free viewpoint video.

SVPS takes advantage of multiple reference prediction to eliminate inter-view and temporal redundancy. Furthermore it relieves exposure and occlusion through view-by-view prediction. However, the sequential prediction may cause error propagation and poor performance in RA and view scalability. Fortunately, the introduction of B picture may improve RA a bit.

GoGOP structure was once proposed for providing low-delay RA. Even if Inter-GOPs are not decoded, all views can be obtained by decoding Base-GOPs and interpolation. However, its RA is still not good enough, especially for GoGOP MR. GoGOP is inferior to simulcast and Lims


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scheme. From analysis over a larger set of multi-view video sequences, Fecker and Mueller et al have once drawn conclusion that temporal prediction is the most efficient prediction mode because temporal correlation is much more than inter-view correlation generally, though there are differences between the data sets in scene complexity, frame rate etc. In GoGOP, it is efficientless for Inter-GOPs predicting from frames at different views and different time-steps relative to the current frame. Multiple reference prediction in Inter-GOP improves RD performance slightly but at the expense of huge computational power and RA performance.

V. CONCLUSION Nine typical MVC prediction structures with six categories

were analyzed in detail. And comparisons on compression efficiency, RA, view scalability and complexity have been made. From the experimental results, we can obtain that MVC with HBP is the most efficient in compression, and simulcast is most suitable for the applications without strong bandwidth limitations but with low-delay requirements. Other schemes have a trade-off between compression efficiency and functionalities by adopting multiple intra frames, sequential prediction, multiple reference prediction and I-frame centered structure etc.

For most multi-view video, the majority of correlation is mainly in temporally preceding frames. However, there is also a significant amount of dependencies that are better predicted from the spatial direction. Multiple reference prediction may reduce prediction residue and improve coding efficiency, but inordinate and fulsome multiple reference predictions not only increase complexity and memory requirements, but also make against view scalability and random accessibility. Multiple reference prediction should be used more effectively according to the relationship between temporal and inter-view dependencies. Additionally, it is obvious that view-by-view sequential prediction relieves exposure and occlusion problems but goes against with RA performance, view scalability. Fortunately, using B picture in sequential prediction structure may improve RA a bit. And improvements on RA and scalability can be achieved by placing I-frame in the center of the 2D GOP. RA can be improved by adopting multiple I-frames in a 2D GOP while at the cost of compression efficiency.

In future work, we will do further researches to propose some smart prediction structures that predict from spatial domain, temporal domain or spatial-temporal joint domain according to the spatial-temporal correlation. And the structure will provide high coding efficiency, low complexity, reasonable memory requirements and fast RA.

ACKNOWLEDGMENT Mitsubishi Electric Research Laboratories, KDDI R&D

Laboratories, and Tanimoto Laboratory at Nagoya University have kindly provided multi-view video test sequences. HHI have kindly provided joint model software.

REFERENCES [1] ISO/IEC JTC1/SC29/WG11 N6909, Survey of Algorithms used for

MVC, Hong Kong, Jan. 2005. [2] M.Tanimoto, T.Fujii, H.Kimata and S.Sakazawa, Proposal on

Requirements for FTV, the 23th JVT Meeting, Doc.JVT-W127, California, USA, Apr. 2007.

[3] ISO/IEC JTC1/SC29/WG11 N8218, Requirements on Multi-view Video Coding v.7, Klagenfurt, Austria, July 2006.

[4] U. Fecker, and A. Kaup, H.264/AVC-Compatible Coding of Dynamic Light Fields Using Transposed Picture Ordering, in Proc 13th European Signal Processing Conference, Antalya, Turkey, 2005.

[5] Kimata H, Kitahara M, Kamikura K, et al, Low-delay multiview video coding for free-viewpoint video communication, Systems and Computers in Japan, vol.38, no.5, pp.14-29, May 2007.

[6] H. Kimata, M. Kitahara, and K. Kamikura, Multi-view video coding using reference picture selection for free-viewpoint video communication, in Proc. of the Picture Coding Symposium-2004, pp.499-502, San Francisco, 2004.

[7] S. Oka, P. N. Bangchang, and T. Fujii, Dynamic Ray-Space Coding for FTV (in Japanese), in Proc. 3D Image Conference, pp.139-142, Tokyo, Japan, Jun. 2004.

[8] S. Oka, T. Endo, and T. Fujii, Dynamic Ray-Space Coding using Multi-directional Picture, IEICE Technical Report, vol.104(493), pp.15-20, Dec. 2004.

[9] P. Merkle, A. Smolic, K. Müller and T. Wiegand, Efficient Prediction Structures for Multi-view Video Coding, to be published in IEEE Transactions on Circuits and Systems for Video Technology, 2007.

[10] ISO/IEC JTC1/SC29/WG11 W8019, Description of Core Experiments in MVC, Montreux, Switzerland, Apr. 2006.

[11] Y. Liu, Q. Huang, X. Ji, D. Zhao, and W. Gao, Multi-view Video Coding with Flexible View-Temporal Prediction Structure for Fast Random Access, Lecture Notes in Computer Science, vol.4261, pp.564-571, 2006.

[12] Y. Liu, Q.Huang, D. Zhao, and W. Gao, Low-delay View Random Access for Multi-view Video Coding, in Proc. ISCAS-2007, pp.997-1000, May 2007.

[13] X. Tong, and R.M. Gray, Interactive rendering from compressed light fields, IEEE Transactions on Circuits and Systems for Video Technology, vol.13, no.11, pp.1080-1091, Nov. 2003.

[14] J.G. Lou, H. Cai, and J. Li, A RealTime Interactive MultiView Video System, in Proc. 13th ACM International Conference on Multimedia, Singapore, Nov. 2005.

[15] G. Jiang, M.Yu, Y. Zhou, and Q. Xu, A New Multi-View Video Coding Scheme for 3DAV Systems, in Proc. Picture Coding Symposium-2006, Beijing, China, Apr. 2006.

[16] J.E. Lim, K. N. Ngan, and W.Yang, A multiview sequence CODEC with view scalability, Signal Processing: Image Communication, vol.19, pp.239-256, 2004.

[17] H. Schwarz, D. Marpe, and T. Wiegand, Hierarchical B pictures, presented at the 16th JVT Meeting (JVT-P014), Poznan, PL, Jul. 2005.

[18] U. Fecker, and A. Kaup, Statistical Analysis of Multi- Reference Block Matching for Dynamic Light Field Coding, ISO/IEC JTC1/SC29/WG11, Doc.M11546, Hong Kong, China, Jan. 2005.

Yun Zhang received his B.S. and M.S degrees in information and electronic engineering from Faculty of Information Science and Engineering, Ning Bo University, China, in 2004 and 2007. He is now pursing a doctoral degree in Institute of Computing Technology, Chinese Academy of Sciences of China. His research interests mainly include digital video compression and communications, SoC design and embedded system for

consumer electronics.

Mei Yu received her M.S degree from Hangzhou Institute of Electronics Engineering, China, in 1993, and Ph.D degree from Ajou University, Korea, in 2000. She is now a professor at Faculty of Information Science and


14

Engineering, Ningbo University, China. Her research interests include image and video coding, and visual perception.

Gangyi Jiang received his M.S degree from Hangzhou University, China, in 1992, and received his Ph.D degree from Ajou University, Korea, in 2000. He is now a professor at Faculty of Information Science and Engineering, Ningbo University, China. His research interests mainly include digital video compression and communications, multi-view video coding, image based rendering, and image processing.


15

Abstract—Bias networks with radio frequency (RF) chokes can

be implemented using different microstrip elements. They can have different advantages in terms of bandwidth and occupied area. However, sharp discontinuities of the transfer functions have been observed in these types of bias networks. In this paper they are explained by resonances generated within the DC path of the bias network. As the resonance behavior degrades the performance of broadband RF circuits, the robustness of different bias networks against resonance was investigated. Different bias networks were fabricated and measured. Both simulation and experimental results show that broadband microstrip bias networks can be optimized to avoid or reduce the resonance phenomena.

Index Terms— Bias network, broadband amplifier, butterfly stub, low-noise amplifier, microstrip components, radial stub, RF-choke.

I. INTRODUCTION IAS NETWORKS using microstrip transmission lines are widely used in radio frequency (RF) and microwave

circuits because of their simple structures, easy processing, and good affinity with the rest of active circuits, e.g., amplifier and mixer [1]. Due to rapid evolution of the mobile communication systems towards ultra-wideband (UWB) systems, the performances of every circuit in a radio chain must be optimized over large frequency bandwidths. The general problem of wideband low-noise amplifier (LNA) design was extensively discussed in last years. Different methods were developed to simultaneously minimize noise figure and optimize power gain over wide frequency band [2]-[5]. Nevertheless, the problem of bias-network design for typical UWB applications has not been explicitly addressed. For example, the LNA in a multi-band UWB transceiver in Band Group 1 defined by [6] must operate over the frequency interval 3.1-4.8 GHz. For such an amplifier the bias network must correctly operate at least over the same frequency bands. However, even if the bandwidth specification is achieved for the bias network itself, single or multiple, sharp-discontinuities (notches) of the transfer function have been observed when the complete UWB LNA module was

Manuscript received October 8, 2007. Ericsson AB in Sweden is acknowledged for financial support of this work.

Adriana Serban; email: [email protected], Magnus Karlsson; email: [email protected], and Shaofang Gong are with Linköping University, Sweden.

simulated [3]. Furthermore, resonances have been reported in other RF circuits where band-limited RF chokes are present [7]-[8]. In the case of LNA, these notches can critically degrade the performance, as both the noise figure and the flatness of the power gain are affected. Moreover, simulation results indicated the possibility of unstable operation of the amplifier.

At RF frequencies, using bias circuits implemented with distributed components in the form of microstrip transmission lines, the bandwidth limitation of bias circuits implemented with discrete components can be avoided [1]. Classical bias-networks have the characteristic of bandpass filters using impedance transforming properties of quarter-wave transmission lines to generate a virtual RF ground. They provide good isolation between the RF and DC ports and are referred usually as RF chokes. Different types of microstrip stubs can be used to implement the RF choke, such as straight microstrip stub, radial stub and butterfly stub. Analytical models and accurate algorithms for modeling and characterizing of radial stubs have been developed for a long time [9]-[15]. Their properties and performances have been compared to those of straight stubs when used for impedance tuning, bias networks, lowpass and bandpass filters. Butterfly stubs models have also been developed and verified in different wideband filter applications [16]-[17]. Recent investigations [18]-[23] supported by the extraordinary development of electronic design automation (EDA) simulation tools [24] have confirmed their applicability in different type of modern high-frequency circuits. Radial stub have been used in order to improve the spurious response of bandpass filters [18], while butterfly stubs have been innovatively used as Photonic Bandgap (PBG) cells [19] and in dual-mode ring bandpass filters [20]-[21] with increased bandwidth and good rejection properties. For further increased bandwidth and more compact area, novel RF components based on the butterfly stub were discussed in [22]-[23].

In this paper, bias networks implemented with different distributed components are studied with focus on bandwidth and their robustness to secondary effects. Simulation results and experimental measurements are compared with each other.

II. MICROSTRIP BIAS NETWORKS WITH RF CHOKE Classically, an RF amplifier can be described as a three-

Microstrip Bias Networks for Ultra-Wideband Systems

Adriana Serban, Magnus Karlsson, and Shaofang Gong, Member, Member, IEEE

B


ISAST Transactions on Electronics and Signal Prosessing, No. 1, Vol. 2, 2008 Serban A. et al.: Microstrip Bias Networks for Ultra-Wideband Systems

16

stage circuit, see Fig. 1. It consists of an active device, symbolized by the transistor, the input matching network (IMN) and the output matching network (OMN).

IMN

OMN

IMN

OMN

Fig. 1. Simplified schematic of an amplifier including bias network (dashed line) with RF choke.

In Fig. 1 Cin and Cout are DC-block capacitors and C is the decoupling capacitor for suppressing the power supply noise. Design techniques for broadband RF amplifiers focus on the amplifier topology or on special techniques providing wideband input- and output matching networks [3]-[5]. However, the DC bias network, indicated by the dashed line in Fig. 1, must also operate correctly over at least the same frequency band. The main component in a bias network is the RF choke. As RF chokes using discrete inductors may not have sufficient bandwidth, microstrip RF chokes are instead used [1]. As seen in Fig.1, the microstrip RF choke is composed of two quarter wavelength (λ/4) transformers: a λ/4 open-circuited stub and a λ/4 connecting line, where λ is the wavelength.

The open-circuited stub can be implemented using different microstrip elements, as shown in Figs. 2a-2c. It is known that the radial stub can provide broader bandwidth than a straight microstrip stub and can be used successfully in some broadband bias networks [9]-[15]. In this paper, a third implementation of the RF choke is considered and analyzed, i.e., the bias network with RF choke using a butterfly stub shown in Fig. 2c.

λ/4 A

B

λ/4

Port 3

λ/4A

B

A λ/4

lA

RF choke

rO,RS

α

rO,BS

(a) (b) (c) Port 3

Port 1 Port 2 Port 1 Port 2

Port 3

B

Port 1 Port 2

α

Fig. 2. Bias network with RF choke using (a) microstrip stub, (b) radial stub, and (c) butterfly stub.

The connection of the bias network to an active circuit, e.g., the RF amplifier shown in Fig. 1, should not disturb the RF signal. A low-loss RF-route from Port 1 to Port 2 shown in Fig. 2 is achieved by using the RF choke which, in theory, provides infinite impedance towards Port 3, i.e., the DC bias port. At the center frequency, ideally, the open-circuited stub provides a low impedance level – RF short – at point B. The λ/4 transmission line transforms the RF short at point B to an RF open, i.e., infinite impedance at junction point A. For wideband applications it is important that these requirements are met not only at the center frequency but also over the entire bandwidth.

III. SIMULATIONS AND MEASUTEMENT SET-UPS The three bias networks presented in Fig. 2 were designed

and simulated using Advanced Design System (ADS) 2005A from Agilent Technologies Inc. Simulations were done on both schematic and layout levels. Layout level simulations were done with the electromagnetic (EM) simulator Momentum in ADS. The bias networks were manufactured using a two-layer printed circuit board (PCB) with the Rogers material RO4350B. The PCB parameters are listed in Table I

Fig. 3. Photograph of the three manufactured bias networks shown in Fig. 2, and centimeter scaled ruler.

The 50 Ω transmission line has the width w = 0.542 mm.

The length of the λ/4 transmission line at 4 GHz is l = 11.5 mm. Referring to Figs. 2b and 2c, the radial stub has the optimized radius rO,RS = 6.8 mm while the butterfly stub has the radius rO,BS = 8.3 mm. The angle for both radial and butterfly stubs is α = 60. A photograph of manufactured bias networks is shown in Fig. 3.

Measurements were done with a Rohde&Schwartz ZVM

RFin

Cin

Cout

RFout

C

VCC

λ/4 λ/4

A

B

TABLE I PCB PROCESS PARAMETERS

Material RO4350B

Dielectric thickness 0.254 mm Dielectric constant 3.48 ± 0.05 Dissipation factor 0.0037 Metal thickness 0.035 mm Metal conductivity 5.8 x 107 S/m Surface roughness 0.001 mm


17

vector network analyzer. All bias networks were designed for the 3.1 – 4.8 GHz frequency band, i.e., the Band Group 1 in a multiband UWB system [6].

As the DC supply-line is usually shunt-connected with several decoupling capacitors to reduce noise from the DC-source, the load impedance at Port 3 shown in Fig. 2 is more close to an RF short than a 50 Ω. In simulations and measurements Port 3 is terminated with an RF short to emulate decoupling capacitors.

IV. SIMULATIONS AND MEASUREMENT RESULTS Use Figs. 4a-4c show the transmission coefficient, |S21|, of

the bias networks implemented with a microstrip stub, a radial stub, and a butterfly stub, respectively. The solid line curves represent the measured data, while the dashed line curves simulated data.

1 2 3 4 5 6 7-40

-30

-20

-10

0

Forw

ard

trans

mis

sion

(dB

)

Frequency (GHz)

Simulated

Measured

(a) Microstrip stub, layout length lA = 28 mm and λ/4=11.5 mm, see Fig 2a.

1 2 3 4 5 6 7-40

-30

-20

-10

0

Forw

ard

trans

mis

sion

(dB

)

Frequency (GHz)

Simulated

Measured

(b) Radial stub, layout length lA = 28 mm and rO,RS = 6.8 mm, see Fig 2b.

1 2 3 4 5 6 7-40

-30

-20

-10

0

Forw

ard

trans

mis

sion

(dB)

Frequency (GHz)

Measured

Simulated

(c) Butterfly stub, layout length lA = 28 mm and rO,BS = 8.3 mm, see Fig 2c.

Fig. 4. Simulated and measured transmission coefficients of the bias networks with (a) microstrip stub, (b) radial stub, and (c) butterfly stub.

By comparing the EM simulated transmission coefficients to the measured ones, it can be seen that a good agreement has been obtained for all three bias networks except some deviation above 5.5 GHz. As expected, the bias network using the radial stub has wider bandwidth than that using the straight line stub. It is also important to notice that the bias network using the butterfly stub not only has the widest bandwidth but also is most robust against the sharp discontinuities of the transmission function. Fewer notches with smaller amplitude can be identified in Fig. 4c, as compared to Figs. 4a and 4b.

V. DISCUSSION These discontinuities shown in Fig. 4 can be explained by

resonances in the DC-path of the bias network. To verify the resonance phenomena along the DC path, the input impedance amplitude at point A (see Fig. 1) when looking into the RF-choke, |ZinA|, is analyzed. Based on the simulated input reflection coefficient S11, |ZinA| was calculated for the microstrip stub bias network.

In Fig. 5, comparing |ZinA| to the transmission coefficient, it can be seen that the discontinuities of the |S21| appear at the same frequencies where |ZinA| ≈ 0. Thus, it can be concluded that discontinuities of the transmission coefficient are caused by equivalent series resonance in the DC path. The DC path from junction point A to Port 3 (see Fig. 2) results in a short-circuited λ/2 transmission line resonator.

1 2 3 4 5 6 7-40

-30

-20

-10

0

1

10

100

1000

10000

Forw

ard

trans

mis

sion

(dB

)

Frequency (GHz)

|ZinA | (Ω

)

|ZinA

|

|S21

|

Fig. 5. Microstrip stub bias network. |S21| and the input impedance at point A (see Fig. 2 ), |ZinA|.

The origin of such resonance phenomena can be explained

by (a) leakage of the RF signal into the DC-route and (b) near zero termination of the DC port, i.e., Port 3 in Fig. 2.

Since the length lA from point B to Port 3 (see Fig.2) is a layout-dependent and arbitrary length, it is interesting to analyze its influence on the transmission coefficient. Simulation results presented in Fig. 6 show that, if lA is increased to lA = 100 mm, more resonances within the frequency band degrade the performance of the three bias networks as compared to those shown in Fig. 4 where lA = 28 mm. The worst is the case from the microstrip stub bias network, while the butterfly stub bias network is not only more broad-banded, but also less affected by the resonance phenomenon.


18

1 2 3 4 5 6 7-30-20-10

0

Radial stub

Forw

ard

trans

mis

sion

(dB

)

Frequency (GHz)

Butterfly stub

2 3 4 5 6 7-30-20-10

0 2 3 4 5 6 7-30-20-10

0Microstrip stub

Fig. 6. Simulated transmission coefficient when lA=100 mm. Fig. 7 shows simulation results when the termination

resistance at Port 3 is 10, 50 and 90 Ω, respectively. It is important to notice that the RF choke using the butterfly stub is the one least influenced by different termination conditions.

1 2 3 4 5 6 7-20-15-10-50

Radial stub

10 Ω 50 Ω 90 Ω

Forw

ard

trans

mis

sion

(dB

)

Frequency (GHz)

Butterfly stub

2 3 4 5 6 7-20-15-10-50 2 3 4 5 6 7

-20-15-10-50

Microstrip stub

Fig. 7. Simulated transmission coefficient when the DC port termination resistance is 10, 50 and 90 Ω, respectively.

Depending on applications, a circuit layout can be more

complex than the basic configurations shown in Figs. 2a-2c. Stand alone characterization of components like RF-chokes is not enough, their surroundings, i.e., in particular neighboring components in the system affects their performance. By including other necessary components, e.g., decoupling capacitors, stabilization resistors and matching networks, the transmission coefficient of the bias network can be disturbed by resonance phenomena whenever a non-optimized RF choke is used. As a consequence, key parameters of an active circuit, e.g., power gain and noise figure, could be seriously affected. Moreover, as the bandwidth of today’s RF circuits increases, resonance phenomena can appear at different frequencies in the band. In order to identify their presence within the defined operation frequency band, it is important to perform layout level simulations of the complete RF circuit, including the bias network, all passive components, via holes, pads and terminations. Using EM simulation tools, the broadband bias network using band-limited RF chokes can be optimized to avoid the appearance of resonance phenomena.

Choosing the appropriate microstrip components, e.g., a butterfly stub instead of a radial stub can also minimize the effect of such resonances on the performance of broadband RF circuits.

VI. CONCLUSION Undesired, single or multiple, sharp discontinuities of the

broadband bias network transfer function have been observed. They can be explained by equivalent series resonances generated within the DC path of the bias network. The origin of such resonance phenomena can be explained by (a) leakage of the RF signal into the DC-route and (b) near zero termination of the DC port. In our experiments the DC path, from junction point A to Port 3 shown in Fig.2, results in a short-circuited λ/2 transmission line resonator.

The microstrip structure of the RF choke using straight line, radial or butterfly stub determines the blocking frequency range and therefore the robustness of the RF choke. The butterfly stub is best suited for broadband RF choke applications. The RF choke using the butterfly stub gives not only the broadest band characteristic but also the most robust bias network towards (a) different layout geometries connecting the RF choke to the DC port, and (b) load impedance variation of the DC port.

REFERENCES [1] S. Hong and M. J. Lancaster, “Microstrip Filters for RF/Microwave

Applications,” John Wiley & Sons, Inc., 2001, Chapter 6, pp. 188-190. [2] S.-L.S. Yang, Q. Xue; K.-M. Luk, ”A Wideband Low-Noise Amplifiers

Design Incorporating the CMRC Circuitry,” IEEE Microwave and Wireless Components Letters, vol. 15, pp. 315 – 317, May 2005.

[3] A. Serban and S. Gong, “Ultra-wideband Low-Noise Amplifier Design for 3.1-4.8 GHz,” in Proc. GigaHertz 2005 Conf., Uppsala, Sweden, 8-9 Nov. 2005, pp. 291-294.

[4] R. Ludwig and P. Bretchko, “RF Circuit Design. Theory and Applications,” Prentice Hall 2000.

[5] G. Gonzalez, “Microwave Transistor Amplifier Design. Analysis and Design,” Prentice Hall 1997.

[6] “First report order, revision of part 15 of commission’s rules regarding ultra-wideband transmission systems,” FCC., Washington, 2002.

[7] L. Jiang and S. C. Shi, “Investigation on the Resonance Observed in the Embedding-Impedance Response of an 850-GHz Waveguide HEB Mixer,” IEEE Microwave and Wireless Components Letters, vol. 15, No. 4, pp. 196-198, Apr. 2005.

[8] Wenhua Yu and Raj Mittra, “Accurate modeling of planar microwave circuit using FDTD algorithm,” IEEE Electronics Letters, vol. 36. No. 7, pp. 618-619, Mar. 2000.

[9] B. A. Syrett, “A Broad-Band Element for Microstrip Bias or Tuning Circuits,” IEEE Trans. Microwave Theory and Tech., vol. MTT-28. No. 8, pp. 925-927, Aug. 1980.

[10] H. A. Atwater, “Microstrip Reactive Circuit Elements,” IEEE Tran. Microwave Theory and Tech., Vol. 83, pp.488 – 491, Jun. 1983.

[11] F. Giannini, R. Sorrentino and J. Vrba, “Planar Circuit Analysis of Microstrip Radial Stub,”, IEEE Trans. Microwave Theory and Tech., vol. 32, pp.1652-1655, Dec. 1984.

[12] F. Giannini, M. Ruggieri, J. Vrba, “Shunted-Connected Microstrip Radial Stub”, IEEE Trans. Microwave Theory Tech., vol. 34, no. 3, pp.3632-366, Mar. 1986.

[13] S. L. March, “Analyzing Lossy Radial-Line Stubs,” IEEE Trans. Microwave Theory and Tech., vol. 33, pp. 269 – 271, Mar. 1985.

[14] R. Sorrentino, L. Roselli, “A New Simple and Accurate Formula for Microstrip Radial Stub”, IEEE Trans. Microwave Theory and Tech., vol. 2, no. 12, pp.480-482, Dec. 1992.


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[15] I. Sakagami, Y. Hao, M. Mohemaiti, A. Tokunou, “On a transmission-line Butterworth lowpass filter using radial stubs,” IEEE International Symposium on Circuits and Systems, 2002, ISCAS 2002, vol. 3, pp. 867 – 870.

[16] F. Giannini, C. Paoloni, M. Ruggieri, “CAD-Oriented Lossy Models for Radial Stubs,” IEEE Tran. Microwave Theory and Tech., vol. 36, pp. 305 – 313, Feb. 1988.

[17] F. Giannini, M Salerno, R. Sorrentino, “Two-Octave Stopband Microstrip Low-Pass Filter Design using Butterfly Stubs,” 16th European Microwave Conference, 1986, pp. 292 – 297.

[18] J. Zhu, Z. Feng, “Enhancement of Stopband Rejection of Microstrip Bandpass Filters by Radial Stubs,” International Conference on Microwave and Millimeter Wave Technology, 2007, ICMMT '07, April 2007, pp. 1 – 3.

[19] B. T. Tan, J. J. Yu, S. J. Koh, S. T. Chew, “Investigation into Broadband PBG Using a Butterfly-Radial Slot (BRS),” Microwave Symposium Digest, 2003 IEEE MTT-S International, vol.2, 8-13 Jun. 2003, pp. 1107 – 1110.

[20] B. T. Tan, J. J. Yu, S. T. Chew M.-S. Leong, B.-L. Ooi, “A Miniaturized Dual-Mode Ring Bandpass Filter with a New perturbation,” IEEE Tran. Microwave Theory and Tech., vol. 53, pp. 343 – 348, Jan. 2005.

[21] R.-J. Mao, X.-H. Tang, F. Xiao, “Miniaturized Dual-Mode Ring Bandpass Filters With Patterned Ground Plane,” IEEE Tran. Microwave Theory and Tech., vol. 55, pp.1539 – 1547, Jul. 2007.

[22] R. K. Joshi, A. R. Harish, “Characteristics of a Rotated Butterfly Radial Stub”, Microwave Symposium Digest, 2006. IEEE MTT-S International, Jun. 2006, pp.1165 – 1168.

[23] R. Dehbashi, K. Forooraghi, Z. Atlasbaf, N. Amiri, “A Novel Broad-Band Band-Stop Resonator with Compact Size,” Asia-Pacific Conference on Applied Electromagnetics, 2005, APACE 2005, Dec. 2005.

[24] Agilent Technologies Inc., “Advanced Design System (ADS),” http://eesof.tm.agilent.com/.

Adriana Serban received the M.Sc degree in electronic engineering from Politehnica University, Bucharest, Romania. From 1981 to 1990 she was with Microelectronica Institute, Bucharest as a Principal Engineer where she was involved in mixed integrated circuits design. From 1992 to 2002 she was with Siemens AG, Munich, Germany and with Sicon AB, Linkoping,

Sweden as analog and mixed signal integrated circuits Senior Design Engineer. Since 2002 she is a Lecturer at Linkoping University teaching in analog/digital system design and RF circuit design. She works towards her Ph.D degree in Communication Electronics. Her main research interest has been RF circuit design and high-speed integrated circuit design.

Magnus Karlsson was born in Västervik, Sweden in 1977. He received his M.Sc. and Licentiate of Engineering from Linköping University in Sweden, in 2002 and 2005, respectively.

In 2003 he started his Ph.D. study in the Communication Electronics research group at Linköping University. His main work involves wideband antenna-

techniques and wireless communications.

Shaofang Gong was born in Shanghai, China, in 1960. He received his B.Sc. degree from Fudan University in Shanghai in 1982, and the Licentiate of Engineering and Ph.D. degrees from Linköping University in Sweden, in 1988 and 1990, respectively.

Between 1991 and 1999 he was a senior researcher at the microelectronic institute – Acreo in Sweden. From

2000 to 2001 he was the CTO at a spin-off company from the institute. Since 2002 he has been full professor in communication electronics at Linköping University, Sweden. His main research interest has been communication electronics including RF design, wireless communications and high-speed data transmissions.


20

Multi-algorithmic Cryptography using DeterministicChaos with Applications to Mobile Communications

Jonathan M Blackledge, Fellow, IET, Fellow, BCS, Fellow, IMA, Fellow, RSS

Abstract— In this extended paper, we present an overviewof the principal issues associated with cryptography, providinghistorically significant examples for illustrative purposes as partof a short tutorial for readers that are not familiar with thesubject matter. This is used to introduce the role that nonlineardynamics and chaos play in the design of encryption engineswhich utilise different types of Iteration Function Systems (IFS).The design of such encryption engines requires that they conformto the principles associated with diffusion and confusion forgenerating ciphers that are of a maximum entropy type. Forthis reason, the role of confusion and diffusion in cryptography isdiscussed giving a design guide to the construction of ciphers thatare based on the use of an IFS. We then present the backgroundand operating framework associated with a new product -CrypsticTM - which is based on the use of multi-algorithmic IFSto design an encryption engine that is mounted on a USB memorystick and uses both disinformation and obfuscation to ‘hide’a forensically inert application. The protocols and proceduresassociated with the use of this product are also discussed.

Index Terms— Cryptography, Nonlinear Dynamics, IterationFunction Systems, Chaos, Multi-algorithmicity

I. I NTRODUCTION

T HE quest for inventing innovative techniques which onlyallow authorized users to transfer information that is

impervious to attack by others has, and continues to be, anessential requirement in the communications industry (e.g.[1], [2], [3]). This requirement is based on the importanceof keeping certain information secure, obvious examples beingmilitary communications and financial transactions, the formerexample, being a common theme in the history and develop-ment of Cryptology [4], [5], [6].

Cryptography is the study of mathematical and computa-tional techniques related to aspects of information security(e.g. [7]-[9]). The word is derived from the GreekKryp-tos, meaning hidden, and is related to disciplines such asCryptanalysis and Cryptology. Cryptanalysis is the art ofbreaking cryptosystems by developing techniques for the re-trieval of information from encrypted data without havingapriori knowledge of the required decryption process (typically,knowledge of the key) [10]. Cryptology is the science thatunderpins Cryptography and Cryptanalysis and can include

Manuscript received November 1, 2007. The work reported in this paperhas been supported by the Council for Science and Technology, Managementand Personnel Services Limited and Crypstic Limited

Jonathan Blackledge is Professor of Information and CommunicationsTechnology, Applied Signal Processing Research Group, Department ofElectronic and Electrical Engineering, Loughborough University, England andProfessor of Computer Science, Department of Computer Science, Univer-sity of the Western Cape, Cape Town, Republic of South Africa (e-mail:[email protected]).

a broad range of mathematical concepts, computational al-gorithms and technologies. In other words, Cryptology isa multi-disciplinary subject that covers a wide spectrum ofdifferent disciplines and increasingly involves using a range ofengineering concepts and technologies through the innovationassociated with term ‘technology transfer’. These include areassuch as Synergetics, which is an interdisciplinary scienceexplaining the formation and self-organization of patterns andstructures in non-equilibrium open systems and Semiotics,which is the study of both individual and grouped signs andsymbols, including the study of how meaning is constructedand understood [11].

Cryptology is often concerned with the application of formalmathematical techniques to design a cryptosystem and toestimate its theoretical security. This can include the use offormal methods for the design of security software whichshould ideally be a ‘safety critical’ [12]. However, althoughthe mathematically defined and provable strength of a cryp-tographic algorithm or cryptosystem is necessary, it is not asufficient requirement for a system to be acceptably secure.This is because it is difficult to estimate the security of acryptosystem in any formal sense when it is implementedunder operational conditions that can not always be pre-dicted and thus, simulated. The security associated with acryptosystem can be checked only by means of proving itsresistance to various kinds of known attack that are likelyto be implemented. However, in practice, this does not meanthat the system is secure since other attacks may exist thatare not included in simulated or test conditions. The reasonfor this is that humans possess a broad range of abilities fromunbelievable ineptitude to astonishing brilliance which can notbe formalised in a mathematical sense or on a case by casebasis.

The practical realities associated with Cryptology are in-dicative of the fact that ‘security a process, not a product’[13]. Whatever the sophistication of the security product(e.g. the encryption and/or key exchange algorithm(s), forexample), unless the user adheres strictly to the procedures andprotocols designed for its use, the ‘product’ can be severelycompromised. A good example of this is the use of the Enigma[14] cipher by Germany during the Second World War. It wasnot just the ‘intelligence’ of the ‘code breakers’ at BletchleyPark in England that allowed the allies to break many of theEnigma codes but the ‘irresponsibility’ and, in many cases,the sheer stupidity of the way in which the system was usedby the German armed and intelligence services at the time.

The basic mechanism for the Enigma cipher, which hadbeen developed as early as 1923 by Artur Schubius for se-

ISAST Transactions on Electronics and Signal Prosessing, No. 1, Vol. 2, 2008 Blackledge J.M.: Multi-algorithmic Cryptography using Deterministic Chaos with Applications to Mobile Communications


curing financial transactions, was well known to the allies dueprimarily to the efforts of the Polish Cipher Office at Poznan inthe 1930s. The distribution of some 10000 similar machines tothe German army, navy and air force was therefore a ‘problemwaiting to happen’. The solution would have been to designa brand new encryption engine or better still, a range ofdifferent encryption engines given the technology of the time,and use the Enigma machine to propagate disinformation.Indeed, some of the new encryption engines introduced bythe Germans towards the end of the Second World War werenot broken by the allies.

These historically intriguing insights are easy to contem-plate in hindsight, but they can also help to focus on themethodologies associated with developing new technologiesfor knowledge management which is a focus of the materialconsidered in this work. Here, we explore the use of deter-ministic chaos for designing ciphers that are composed ofmany different pseudo chaotic number generating algorithms- Meta-encryption-engines. This multi-algorithmic or Meta-engine approach provides a way of designing an unlimitedclass of encryption engines as opposed to designing a singleencryption engine that is operated by changing the key(s)- which for some systems, public key system in particular,involves the use of prime numbers. There are of course anumber of disadvantages to this approach which are discussedlater on but it is worth stating at this point, that the principalpurpose for exploring the application of deterministic chaos incryptography is:

• the non-reliance of such systems on the use of primenumbers which place certain limits on the characteristicsand arithmetic associated with an encryption algorithm;

• the unlimited number of chaos based algorithms that canbe, quite literally, invented to produce a meta-encryptionengine.

II. I NFORMATION AND KNOWLEDGE MANAGEMENT

With regard to information security and the managementof information in general, there are some basic concepts thatare easy to grasp and sometimes tend to get lost in the detail.The first of these is that the recipient of any encrypted messagemust have some form ofa priori knowledge on the method (thealgorithm, for example) and the operational conditions (e.g.the key) used to encrypt a message. Otherwise, the recipientis in no better ‘state of preparation’ than the potential attacker.The conventional approach is to keep thisa priori informationto a minimum but in such a way that it is critical to thedecryption process. Another important reality is that in anattack, if the information transmitted is not deciphered in goodtime, then it may become redundant. Coupled with the factthat an attack usually has to focus on a particular criterion(such as a specific algorithm), one way to enhance the securityof a communications channel is to continually change theencryption algorithm and/or process offered by the technologyavailable.

Another approach to information management is to disguiseor camouflage the encrypted message in what would appearto be ‘innocent’ or ‘insignificant’ data such as a digital

photograph, a music file or both, for example1. This is knownas Steganography[15]-[17]. Further, the information securityproducts themselves should be introduced and ‘organised’in such a way as to reflect their apparent insignificance interms of both public awareness and financial reward whichhelps to combat the growing ability to ‘hack and crack’ usingincreasingly sophisticated software that is readily available,e.g. [18]. This is of course contrary to the disseminationof many encryption systems, a process that is commonlyperceived as being necessary for business development throughthe establishment of a commercial organisation, internationalpatents, distribution of marketing material, elaborate and so-phisticated Web sites, authoritative statements on the strengthof a system to impress customers, publications and so on.Thus, a relatively simple but often effective way of maintainingsecurity with regard to the use of an encryption system is tonot tell anyone about it. The effect of this can be enhancedby publishing other systems and products that are designedto mislead the potential attacker. In this sense, informationmanagement and Information and Communication Technology(ICT) security products should be treated in the same way asmany organisations treat a breach of security, i.e. not to publishthe breach in order to avoid embarrassment and loss of faithby the client base.

A. Secrets and Ultra-secrets

A classic mistake (of historical importance) of not ‘keep-ing it quiet’, in particular, not maintaining ‘silent warfare’[19], was made by Winston Churchill when he publishedhis analysis of World War I. In his book,The World Crisis1911-1918, published in 1923, he stated that the British haddeciphered the German Naval codes for much of the waras a result of the Russians salvaging a code book from thesmall cruiserMagdeburgthat had ran aground off Estonia onAugust 27, 1914. The code book was passed on to Churchillwho was, at the time, the First Sea Lord. This helped theBritish maintain their defences with regard to the Germannavy before and after the Battle of Jutland in May, 1916. TheGerman navy became impotent which forced Germany into apolicy of unrestricted submarine warfare. In turn, this led to anevent (the sinking on May 7, 1915 of the Lusitania, torpedoedby a German submarine, the U-20) that galvanized Americanopinion against Germany and played a key role in the UnitedStates’ later entry into the Fisrt World War on April 17, 1917and the defeat of Germany [20], [21].

Churchill’s publication did not go un-noticed by the Ger-man military between the First and Second World Wars.Consequently, significant efforts were made to develop newencryption devices for military communications. This resultedin the famous Enigma machine, named after Sir EdwardElgar’s masterpiece, theEnigma Variations[22]. Enigma wasan electro-mechanical machine about the size of a portabletypewriter which, through application of both electrical (‘plug-board’) and mechanical (‘rotor’) settings offered∼ 2 × 1020

permutations for establishing a ‘key’. The machine could be

1By encoding the encrypted message in the least significant bit or bit-pairof the host data, for example.

ISAST Transactions on Electronics and Signal Prosessing, No. 1, Vol. 2, 2008 Blackledge J.M.: Multi-algorithmic Cryptography using Deterministic Chaos with Applications to Mobile Communications22

used without difficulty by semi-skilled operators under themost extreme battle conditions. The keys could be changeddaily or several times a day according to the number ofmessages transmitted.

The interest in cryptology by Germany that was undoubt-edly stimulated by Churchill’s indiscretions included establish-ing a specialist cipher school in Berlin. Ironically, it was at thisSchool that some of the Polish mathematicians were trainedwho later worked for the Polish Cipher Office, opened in ut-most secrecy at Poznan in 1930 [23], [24]. In January 1929, theDean of the Department of Mathematics, Professor ZdzislawKrygowski from the University of Poznan, provided a list ofhis best graduates to start working at this office. One of thesegraduates was the brilliant young logician, Marian Rejewskiwho pioneered the design of theBomba kryptologiczna, anelectro-mechanical device used for eliminating combinationsthat had not been used to encrypt a message with the Enigmacipher [25]. However, the design of theBomba kryptologicznawas only made possible through the Poles gaining access to theEnigma machine and obtaining knowledge of its mechanismwithout alerting the Germans to their activities. In modernterms, this is equivalent to obtaining information on the typeof encryption algorithm used in a cryptosystem.

The Bomba kryptologicznahelped the Poles to deciphersome 100,000 Enigma messages from as early January 1933 toSeptember 1939 including details associated with the remili-tarization of the Rhine Province, Anschluss of Austria andseizure of the Sudetenland. It was Rejewski’s original workthat formed the basis for designing the advanced electro-mechanical and later, the electronic decipher machines (in-cluding ‘Colossus’ - the world’s first programmable computer)constructed and utilized at Bletchley Park between 1943 and1945 [26], [27].

After the Second World War, Winston Churchill made surethat he did not repeat his mistake, and what he referred toas his ‘Ultra-secret’ - the code breaking activities undertakenat Station X in Bletchley Park, England - was ordered byhim to be closed down and the technology destroyed soonafter the end of the war. Further, Churchill never referredto his Ultra-secret in any of his publications after the war.Those personnel who worked at Bletchley Park were requiredto maintain their silence for some fifty years afterwards andsome of the activities at Bletchley Park remain classified tothis day. Bletchley Park is now a museum which includesa reconstruction of ‘Colossus’ undertaken in the mid-1990s.However, the type of work undertaken there in the early 1940scontinues in many organisations throughout the world suchas the Government Communications Head Quarters (GCHQ)based at Cheltenham in England [28] where a range of‘code making’ and ‘code breaking’ activities continue to bedeveloped.

The historical example given above clearly illustrates theimportance of maintaining a level of secrecy when undertakingcryptographic activities. It also demonstrates the importanceof not publishing new algorithms, a principle that is at oddswith the academic community; namely, that the security ofa cryptosystem should not depend upon algorithm secrecy.However, this has to be balanced with regard to the dissem-

ination of information in order to advance a concept throughpeer review, national and international collaboration. Taken toan extreme, the secrecy factor can produce a psychologicalimbalance that is detrimental to progress. Some individualslike to use confidential information to enhance their status.In business, this often leads to issues over the signing ofNon-Disclosure Agreements or NDAs, for example, leadingto delays that are of little value, especially when it turns outthat there is nothing worth disclosing. Thus, the whole issueof ‘keeping it quiet’ has to be implemented in a way that isbalanced, such that confidentiality does not lead to stagnationin the technical development of a cryptosystem. However, usedcorrectly and through the appropriate personality, issues overconfidentiality coupled with the ‘feel important’ factor can beused to good effect in the dissemination of disinformation.

B. Home-Spun Systems Development

The development and public use of information securitytechnology is one of the most interesting challenges for statecontrol over the ‘information society’. As more and moremembers of the younger generation become increasingly ITliterate, it is inevitable that a larger body of perfectly ableminds will become aware of the fact that cryptology isnot as difficult as they may have been led to believe. Aswith information itself, the days when cryptology was in thehands of a select few with impressive academic credentialsand/or luxury civil service careers are over and cryptosystemscan now be developed by those with a diverse portfolio ofbackgrounds which does not necessarily include a Universityeducation. This is reflected in the fact that after the Cold War,the UK Ministry of Defense, for example, developed a strategyfor developing products driven by commercially availablesystems. This Commercial-Off-The-Shelf or COTS approachto defence technology has led directly to the downsizing ofthe UK Scientific Civil Service which, during the Cold War,was a major source of scientific and technical innovation.

The average graduate of today can rapidly develop theability to write an encryption system which, although relativelysimple, possibly trivial and ill-informed, can, by the very na-ture of its non-compliance to international standards, providesurprisingly good security. This can lead to problems withthe control and management of information when increasinglymore individuals, groups, companies, agencies and nationstates decide that they can ‘go it alone’ and do it themselves.While each home grown encryption system may be relativelyweak, compared to those that have had expert developmentover many years, have been well financed and been testedagainst the very best of attack strategies, the proliferation ofsuch systems is itself a source of significant difficulty for anyauthority whose role is to monitor communications traffic in away that is timely and cost effective. This is why governmentsworld-wide are constantly attempting to control the use andexploitation of new encryption methods in the commercialsector2. It also explains the importance of international en-cryption standards in terms of both public perception and

2For example, the introduction of legislation in mainland UK concerningthe decryption of massages by a company client through enforcement of theRegulation of Investigatory Powers (RIP) Act, 2000.


free market exploitation. Government and other controllingauthorities like to preside over a situation in which everybodyelse is confidently reliant for their information security onproducts that have been developed by the very authoritiesthat encourage their use, a use that is covertly ‘diffused’ intothe ‘information society’ through various legitimate businessventures coupled with all the usual commercial sophisticationand investment portfolios. Analysis of this type can lead toa range of unsubstantiated conspiracy theories, but it only bythinking through such possible scenarios, that new concepts ininformation management, some of which may be of practicalvalue, are evolved. The proliferation of stand-alone encryptionsystems that are designed and used by informed individualsis not only possible but inevitable, an inevitability that isguided by the principle that if you want to know what you areeating then you should cook it yourself. Security issues of thistype have become the single most important agenda for futuregovernment policy on information technology, especially whensuch systems have been ‘home spun’ by those who havelearned to fully respect that they should, in the words ofShakespeare, ‘Neither a borrower, nor a lender be’3.

C. Disinformation

Disinformation is used to tempt the ‘enemy’ into believingcertain kinds of information. The information may not be trueor contain aspects that are designed to cause the enemy toreact in an identifiable way that provides a strategic advantage[29], [30]. Camouflage, for example, is a simple example ofdisinformation [31]. This includes techniques for transformingencrypted data into forms that resemble the environmentsthrough which an encrypted message is to be sent [32], [33].At a more sophisticated level, disinformation can includeencrypted messages that are created with the sole purpose ofbeing broken in order to reveal information that the enemywill react to by design.

Disinformation includes arranging events and processesthat are composed to protect against an enemy acquiringknowledge of a successful encryption technology and/or asuccessful attack strategy. A historically significant example ofthis involved the Battle of Crete which began on the morningof 20 May 1941 when Nazi Germany launched an airborneinvasion of Crete under the code-name Unternehmen Merkur(Operation Mercury) [34]. During the next day, through mis-communication and the failure of Allied commanders to graspthe situation, the Maleme airfield in western Crete fell to theGermans which enabled them to fly in heavy reinforcementsand overwhelm the Allied forces. This battle was unique intwo respects: it was the first airborne invasion in history4;it was the first time the Allies made significant use of theirability to read Enigma codes. The British had known for someweeks prior to the invasion of Crete that an invasion waslikely because of the work being undertaken at Bletchley Park.They faced a problem because of this. If Crete was reinforcedin order to repel the invasion then Germany would suspect

3From William Shakespeare’s play, Hamlet.4Illustrating the potential of paratroopers and so initiating the Allied

development of their own airborne divisions.

that their encrypted communications were being compromised.But this would also be the case if the British and otherAllied troops stationed on Crete were evacuated. The decisionwas, therefore, taken by Churchill to let the German invasionproceed with success but not without giving the invadersa ‘bloody nose’. Indeed, in light of the heavy casualtiessuffered by the parachutists, Hitler forbade further airborneoperations and Crete was dubbed ‘the graveyard of the Germanparachutists’. The graveyard for German, British, Greek andAllied soldiers alike was not a product of a fight over desirableand strategically important territory (at least for the British). Itwas a product of the need to secure Churchill’s ‘Ultra-secret’.In other words, the Allied efforts to repulse the Germaninvasion of Crete was, in reality, a form of disinformation,designed to secure a secret that was, in the bigger picture,more important than the estimated 16,800 dead and woundedthat the battle cost.

D. Plausible Deniability

Deniable encryption allows an encrypted message to bedecrypted in such a way that different and plausible plaintextscan be obtained using different keys [35]. The idea is to makeit impossible for an attacker to prove the existence of the realmessage, a message that requires a specific key. This approachprovides the user with a solution to the ‘gun to the headproblem’ as it allows the sender to have plausible deniabilityif compelled to give up the encryption key.

There are a range of different methods that can be designedto implement such a scheme. For example, a single ciphertextcan be generated that is composed of randomised segmentsor blocks of data which correlate to blocks of differentplaintexts encrypted using different keys. A further key isthen required to assemble the appropriate blocks in order togenerate the desired decrypt. This approach, however, leads tociphertext files that are significantly larger than the plaintextsthey contain. On the other hand, a ciphertext file should notnecessarily be the same size as the plaintext file and paddingout the plaintext before encryption can be used to increase theEntropy of the ciphertext (as discussed in Section VIII).

Other methods used for deniable encryption involve estab-lishing a number of abstract ‘layers’ that are decrypted to yielddifferent plaintexts for different keys. Some of these layers aredesigned to include so-called ‘chaff layers’. These are layersthat are composed of random data which provide the owner ofthe data to have plausible deniability of the existence of layerscontaining the real ciphertext data. The user can store ‘decoyfiles’ on one or more layers while denying the existence ofothers, identifying the existence of chaff layers as required.The layers are based on file systems that are typically storedin a single directory consisting of files with filenames thatare either randomized (in the case where they belong to chafflayers), or are based on strings that identify cryptographic data,the timestamps of all files being randomized throughout.

E. Obfuscation

In a standard computing (windows) environment, a simpleform of camouflage can be implemented by renaming files to


be of a different type; for example, storing an encrypted datafile as a.exeor .dll file. Some cryptosystems output files withidentifiable extensions such as .enc which can then be simplyfiltered by a firewall. Another example includes renaming filesin order to access data and/or execute an encryption engine.For example, by storing an executable file as a.dll (dynamiclink library) file (which has a similar structure to a.exefile) ina directory full of real.dll files associated with some complexapplications package, the encryption engine can be obfuscated,especially if it has a name that is similar to the environmentof files in which it is placed. By renaming the file back to its‘former self’, execution of a cryptosystem can be undertakenin the usual way. However, this requires that the executable fileis forensically inert, i.e. it does not contain data that reflectsits purpose. A simple way of implementing this requirement isto ensure that the source code (prior to compilation) is devoidof any arrays, comments etc. that include references (throughuse of named variables, for example) to the type of application(e.g. comments suchencrypt the dataor named arrays suchasdecryptarray[i]).

F. Steganographic Encryption

It is arguable that disinformation should, where possible,be used in conjunction with the exchange of encrypted in-formation which has been camouflaged using steganographictechniques for hiding the ciphertext. For example, supposethat it had been known by Germany that the Enigma cipherswere being compromised by the British during the SecondWorld War. Clearly, it would have then been strategicallyadvantageous for Germany to propagate disinformation usingEnigma. If, in addition, ‘real information’ had been encrypteddifferently and the ciphertexts camouflaged using broadcaststhrough the German home radio service, for example, thenthe outcome of the war could have been very different. Theuse of new encryption methods coupled with camouflage anddisinformation, all of which are dynamic processes, provides amodel that, while not always of practical value, is strategicallycomprehensive and has only rarely been fully realised. Never-theless, some of the techniques that have been developed andare reported in this work are the result of an attempt to realisethis model.

III. B ASIC CONCEPTS

Irrespective of the wealth of computational techniques thatcan be invented to encrypt data, there are some basic con-cepts that are a common theme in modern cryptography.The application of these concepts typically involves the useof random number generators and/or the use of algorithmsthat originally evolved for the generation of random numberstreams, algorithms that are dominated by two fundamentaland interrelated themes [4]-[6]: (i) the use of modular arith-metic; (ii) the application of prime numbers. The applicationof prime numbers is absolutely fundamental to a large rangeof encryption processes and international standards such asPKI (Public Key Infrastructure) details of which are discussedlater.

Fig. 1. Alice and Bob can place a message in a box which can be securedusing a combination lock and sent via a public network - the postal service,for example.

Using a traditional paradigm, we consider the problem ofhow Alice (A) and Bob (B) can pass a message to and fromeach other without it being compromised or ‘attacked’ by anintercept. As illustrated in Figure 1, we consider a simple boxand combination lock scenario. Alice and Bob can write amessage, place it in the box, lock the box and then send itthrough an open ‘channel’ - the postal services, for example.In cryptography, the strength of the box is analogous to thestrength of the cipher. If the box is ‘weak’ enough to beopened by brute force, then the strength of the lock is relativelyinsignificant. This is analogous to a cipher whose statisticalproperties are poor, for example, i.e. whose Probability Den-sity Function (PDF) is narrow and whose information Entropyis relatively low, with a similar value to the plaintext. Thestrength of the lock is analogous to the strength of the keyin a real cryptographic system. This includes the size of thecombination number which is equivalent to the length of thekey that is used. Clearly a four rotor combination lock asillustrated in Figure 1 represents a very weak key since thenumber of ordered combinations required to attempt a bruteforce attack to open the lock are relatively low, i.e. for a 4-digit combination lock where each rotor has ten digits 0-9, thenumber of possible combinations is 10000 (including 0000).However, the box-and-lock paradigm being used here is forillustrative purposes only.

A. Symmetric Encryption

Symmetric encryption is the simplest and most obviousapproach to Alice and Bob sending their messages. Alice andBob agree on a combination numbera priori. Alice writes amessage, puts it in the box, locks it and sends it off. Uponreceipt, Bob unlocks the box using the combination numberthat has been agreed and recovers the message. Similarly,Bob can send a message to Alice using exactly the sameapproach or ‘protocol’. Since this protocol is exactly the samefor Alice and Bob it has a symmetry and thus, encryptionmethods that adopt this protocol are referred to as symmetricencryption methods. Given that the box and the lock havebeen designed to be strong, the principal weakness associatedwith this method is its vulnerability to attack if a third partyobtains the combination number at the point when Alice and


Bob invent it and agree upon it. Thus, the principal problemin symmetric encryption is how Alice and Bob exchange thekey. Irrespective of how strong the cipher and key are, unlessthe key exchange problem can be solved in an appropriate anda practicable way, symmetric encryption always suffers fromthe same fundamental problem - key exchange!

If E denotes the encryption algorithm that is used whichdepends upon a keyK to encrypt plaintextP , then we canconsider the ciphertextC to be given by

C = EK(P ).

Decryption can then be denoted by the equation

P = EK(C).

Note that it is possible to encrypt a number of times usingdifferent keysK1,K2, ... with the same encryption algorithmto give a double encrypted cipher text

C = EK2(EK1(P ))

or a triple encrypted ciphertext

C = EK3(EK2(EK1(P ))).

Decryption, is then undertaken using the same keys in thereverse order to which they have been applied, i.e.

P = EK1(EK2(EK3(C))).

Symmetric encryption systems, which are also referred toas shared secret systems or private key systems, are usuallysignificantly easier to use than systems that employ differentprotocols (such as asymmetric encryption). However, the re-quirements and methods associated with key exchange some-times make symmetric system difficult to use. Examples ofsymmetric encryption systems include the Digital EncryptionStandard DES and DES3 (essentially, but not literally, theDigital Encryption Standard with triple encryption) and theAdvanced Encryption Standard (AES). Symmetric systems arecommonly used in many banking and other financial institutesand in some military applications. A well known historicalexample of a symmetric encryption engine, originally designedfor securing financial transactions, and later used for militarycommunications, was theEnigma.

B. Asymmetric Ciphers

Instead of Alice and Bob agreeing on a combination numbera priori, suppose that Alice sets her lock to be open with acombination number known only to her. If Bob then wishes tosend Alice a message, he can make a request for her to sendhim an open lock. Bob can then write his message, place itin the box which is then locked and sent on to Alice. Alicecan then unlock the box and recover the message using thecombination number known only to her. The point here isthat Bob does not need to know the combination number, heonly needs to receive an open lock from Alice. Of courseBob can undertake exactly the same procedure in order toreceive a message from Alice. Clearly, the processes that areundertaken by Alice and Bob in order to send and receive asingle message are not the same. The protocol is asymmetric

and we refer to encryption systems that use this protocol asbeing asymmetric. Note that Alice could use this protocol toreceive messages from any number of senders provided theycan get access to one of her open locks. This can be achievedby Alice distributing many such locks as required.

One of the principal weaknesses of this approach relatesto the lock being obtained by a third party whose interestis in sending bogus or disinformation to Alice. The problemfor Alice is to find a way of validating that a message sentfrom Bob (or anyone else who is entitled to send messagesto her) is genuine, i.e. that the message is authentic. Thus,data authentication becomes of particular importance whenimplementing asymmetric encryption systems.

Asymmetric encryption relies on both parties having twokeys. The first key (the public key) is shared publicly. Thesecond key is private, and is kept secret. When workingwith asymmetric cryptography, the message is encrypted usingthe recipients’ public key. The recipient then decrypts themessage using the private key. Because asymmetric cipherstend to be computationally intensive (compared to symmet-ric encryption), they are usually used in combination withsymmetric systems to implement public key cryptography.Asymmetric encryption is often used to transfer a session keyrather than information proper - plaintext. This session key isthen used to encrypt information using a symmetric encryptionsystem. This gives the key exchange benefits of asymmetricencryption with the speed of symmetric encryption. A wellknown example of asymmetric encryption - also known aspublic key cryptography - is the RSA algorithm which isdiscussed later. This algorithm uses specific prime numbers(from which the private and public keys are composed) inorder to realize the protocol.

In order to provide users with appropriate prime numbers, aninfrastructure needs to be established by a third party whose‘business’ is to distribute the public/private key pairs. Thisinfrastructure is known as the Public Key Infrastructure orPKI. The use of a public key is convenient for those who wishto communicate with more than one individual and is thus, amany-to-one protocol that avoids multiple key-exchange. Onthe other hand, a public key provides a basis for cryptanalysis.Given thatC = EK(P ) whereK is the public key, the analystcan guessP and check the answer by comparingC with theintercepted ciphertext, a guess that is made easier if it is basedon a known Crib - i.e. information that can be assumed tobe a likely component of the plaintext. Public key algorithmsare therefore often designed to resist chosen-plaintext attack.Nevertheless, analysis of public key and asymmetric systemsin general, reveals that the level of security is not as significantas that which can be achieved using a well-designed symmetricsystem. One obvious and fundamental issue relates to the thirdparty responsible for the PKI and how much trust shouldbe assumed, especially with regard to legislation concerningissues associated with the use of encrypted material.

C. Three-Way Pass Protocol

The three-way pass protocol, at first sight, provides asolution to the weaknesses associated with symmetric and


asymmetric encryption. Suppose that Alice writes a message,puts it in the box, locks the box with a lock whose combinationnumber is known only to her and sends it onto Bob. Uponreceipt Bob cannot open the box, so Bob locks the box withanother lock whose combination number is known only tohimself and sends it back to Alice. Upon receipt, Alice canremove her lock and send the box back to Bob (secured withhis lock only) who is then able to remove his lock and recoverthe message. Note that by using this protocol, Alice and Bobdo not need to agree upon a combination number; this avoidsthe weakness of symmetric encryption. Further, Alice and Bobdo not need to send each other open locks which is a weaknessof asymmetric encryption.

The problem with this protocol relates to the fact that itrequires the message (secured in the locked box) to be ex-changed three times. To explain this, suppose we have plaintextin the form of an American Standard Code for InformationInterchange or ASCII-value arrayp[i] say. Alice generates aciphern1[i] using some appropriate strength random numbergenerator and an initial condition based on some long integer- the key. Let the ciphertextc[i] be generated by adding thecipher to the plaintext, i.e.

c1[i] = p[i] + n1[i]

which is transmitted to Bob. This is a substitution-basedencryption process and is equivalent to Alice securing themessage in the box with her lock - the first pass. Bob generatesa new ciphern2[i] using the same (or possibly a different)random number generator with a different key and generatesthe ciphertext

c2[i] = c1[i] + n2[i] = p[i] + n1[i] + n2[i]

which is transmitted back to Alice - the second pass. Alicenow uses her cipher to generate

c3[i] = c2[i]− n1[i] = p[i] + n2[i]

which is equivalent to her taking off her lock from the boxand sending the result back to Bob - the third pass. Bob thenuses his cipher to recover the message, i.e.

c3[i]− n2[i] = p[i].

However, suppose that the cipher textsc1, c2 and c3 areintercepted, then the plaintext array can be recovered since

p[i] = c3[i] + c1[i]− c2[i].

This is the case for any encryption process that is commutativeand associative. For example, if the arrays are considered tobe bit streams and the encryption process undertaken usingthe XOR process (denoted by⊕), then

c1 = n1 ⊕ p,

c2 = n2 ⊕ c1 = n2 ⊕ n1 ⊕ p,

c3 = n1 ⊕ c2 = n2 ⊕ p

andc1 ⊕ c2 ⊕ c3 = p.

This is because for any bit streama, b andc

a⊕ a⊕ b = b

and because the XOR operation is both commutative andassociative i.e.

a⊕ b = b⊕ a

anda⊕ (b⊕ c) = (a⊕ b)⊕ c.

These properties are equivalent to the fact that when Alicereceives the box at the second pass with both locks on it, shecan, in principle, remove the locks in any order. If, however,she had to remove Bob’s lock before her own, then the protocolwould become redundant.

D. Private Key Encryption

One of the principal goals in private key cryptography isto design Pseudo Random Number Generators (PRNGs) thatprovide outputs (random number streams) where no elementcan be predicted from the preceding elements given completeknowledge of the algorithm. Another important feature is toproduce generators that have long cycle lengths. A furtherimportant feature, is to ensure that the Entropy of the randomnumber sequence is a maximum, i.e. that the histogram of thenumber stream is uniform.

The use of modular integer arithmetic coupled with the useof prime numbers in the development of encryption algorithmstends to provide functions which are not invertible. They areone-way functions that can only be used to reproduce a specific(random) sequence of numbers from the same initial condition.

The basic idea in stream ciphers - as used for private key(symmetric) cryptography - is to convert a plaintext into aciphertext using a key that is used as a seed for the PRNG. Aplaintext file is converted to a stream of integer numbers usingASCII conversion. For example, suppose we wish to encryptthe authors surnameBlackledgefor which the ASCII5 decimalinteger stream or vector is

p = (66, 108, 97, 99, 107, 108, 101, 100, 103, 101).

Suppose we now use the linear congruential PRNG definedby6

ni+1 = animodP

where a = 13, P = 131 and let the seed be 250659, i.e.n0 = 250659. The output of this iteration is

n = (73, 32, 23, 37, 88, 96, 69, 111, 2, 26).

If we now add the two vectors together, we generate the cipherstream

c = f+n = (139, 140, 120, 136, 195, 204, 170, 211, 105, 127).

Clearly, provided the recipient of this number stream hasaccess to the same algorithm (including the values of theparametersa and P ) and crucially, to the same seedn0,

5Any code can be used.6Such a PRNG is not suitable for cryptography and is being used for

illustrative purposes only.


the vectorn can be regenerated andp obtained fromc bysubtractingn from c. However, in most cryptographic systems,this process is usually accomplished using binary streamswhere the binary stream representation of the plaintextp andthat of the random number stream or ciphern are used togenerate the ciphertext binary streamc via the process

c = n⊕ f .

Restoration of the plaintext is then accomplished via theoperation

f = n⊕ c = n⊕ n⊕ f .

The processes discussed above are examples of digitalconfusion in which the information contained in the fieldf(the plaintext) is ‘confused’ using a stochastic functionc (thecipher) via addition (decimal integer process) or with an XORoperator (binary process). Here, the seed plays the part of a keythat it utilized for the process of encryption and decryption.This is an example of symmetric encryption in which the keyis a private key known only to the sender and recipient of theencrypted message.

Given that the algorithm used to generate the randomnumber stream has public access (together with the parametersit uses which are typically ‘hard-wired’ in order to provide arandom field pattern with a long cycle length), the problemis how to securely exchange the key to the recipient of theencrypted message so that decryption can take place. If thekey is particular to a specific communication and is used onceand once only for this communication (other communicationsbeing encrypted using different keys), then the process isknown as a one-time pad, because the key is only used once.Simple though it is, this process is not open to attack. Inother words, no form of cryptanalysis will provide a way ofdeciphering the encrypted message. The problem is how toexchange the keys in a way that is secure and thus, solutionsto the key exchange problem are paramount in symmetricencryption,

The illustration of stream cipher encryption given abovehighlights the problem of key exchange, i.e. providing thevalue of n0 to both sender and receiver. In addition todeveloping the technology for symmetric encryption (e.g. thealgorithm or algorithms), it is imperative to develop appro-priate protocols and procedures for using it effectively withthe aim of reducing inevitable human error, the underlyingprinciples being: (i) the elimination of any form of temporalcorrelation in the used algorithm; (ii) the generation of a keythat is non-intuitive and at best random; (iii) the exchange ofthe key once it has been established.

E. Public-Private Key Encryption

Public-Private Key Encryption [36]-[40] is fundamentallyasymmetric and in terms of the box and combination-lockparadigm is based on considering a lock which has twocombinations, one to open the lock and another to lock it. Thesecond constraint is the essential feature because one of thebasic assumptions in the use of combination locks is that theycan be locked irrespective of the rotor positions. Thus, afterwriting a message, Alice uses one of Bobs specially designed

locks to lock the box using a combination number that isunique to Bob but is openly accessible to Alice and otherswho want to send Bob a message. This combination numberis equivalent to the public key. Upon reception, Bob can openthe lock using a combination number that is known only tohimself - equivalent to a private key. However, to design sucha lock, there must be some mechanical ‘property’ linkingthe combination numbers required to first lock it and thenunlock it. It is this property that is the principal vulnerabilityassociated with public/private key encryption, a property thatis concerned with certain precise and exact relationships thatare unique to the use of prime numbers and their applicationswith regard to generating pseudo random number streams andstochastic functions in general [41].

The most common example of a public-private key encryp-tion algorithm is the RSA algorithm [39] which gets its nameafter the three inventors, Rivest, Shamir and Adleman whodeveloped the generator in the mid 1970s7. It has since with-stood years of extensive cryptanalysis. To date, cryptanalysishas neither proved nor disproved the security of the algorithmin a complete and self-consistent form which suggests a highconfidence level in the algorithm.

The basic generator is given by

ni+1 = nei mod(pq)

wherep, q ande are prime numbers ande < pq. Although thisgenerator can be used to compute a pseudo random numberstreamni, the real value of the algorithm lies in its use fortransforming plaintextPi (taken to be a decimal integer arraybased on ASCII 7-bit code, for example) to ciphertextCi

directly via the equation

Ci = P ei mod(pq).

We then consider the decryption process to be based on thesame type of transform, i.e.

Pi = Cdi mod(pq).

The problem is then to findd given e, p and q. The ‘key’ tosolving this problem is to note that ifed − 1 is divisible by(p− 1)(q − 1), i.e. d is given by the solution of

de = mod[(p− 1)(q − 1)]

thenCd

i mod(pq) = P edi mod(pq) = Pimod(pq)

usingFermat’s Little Theorem, i.e. for any integera and primenumberp

ap = amodp.

Note that this result is strictly dependent on the fact thated−1is divisible by (p − 1)(q − 1) making e a relative prime of(p− 1)(q − 1) so thate and(p− 1)(q − 1) have no commonfactors except for 1. This algorithm, is the basis for manypublic/private or asymmetric encryption methods. Here, the

7There are some claims that the method was first developed at GCHQ inEngland and then re-invented (or otherwise) by Rivest, Shamir and Adlemanin the USA; the method was not published openly by GCHQ - such are therealities of ‘keeping it quiet’.


public key is given by the numbere and the productpq whichare unique to a given recipient and in the public domain (likean individuals telephone number). Note that the prime numbersp and q and the numbere < pq must be distributed to Aliceand Bob in such a way that they are unique to Alice and Bobon the condition thatd exists! This requires an appropriateinfrastructure to be established by a trusted third party whos‘business’ is to distribute values ofe, pq and d to its clientsa Public Key Infrastructure. A PKI is required in order todistribute public keys, i.e. different but appropriate values ofe andpq for use in public key cryptography (RSA algorithm).This requires the establishment of appropriate authorities anddirectory services for the generation, management and certifi-cation of public keys.

Recovering the plaintext from the public key and the ciphertext can be conjectured to be equivalent to factoring theproduct of the two primes. The success of the system, which isone of the oldest and most popular public key cryptosystems, isbased on the difficulty of factoring. The principal vulnerabilityof the RSA algorithm with regard to an attack is thate andpqare known and thatp andq must be prime numbers - elementsof a large but (assumed) known set. To attack the cipher,dmust be found. But it is known thatd is the solution of

de = mod[(p− 1)(q − 1)]

which is only solvable ife < pq is a relative prime of(p−1)(q−1). An attack can therefore be launched by searchingthrough prime numbers whose magnitudes are consistent withthe productpq (which provides a search domain) until therelative prime condition is established for factorsp and q.However, factoringpq to calculated given e is not trivial. Itis possible to attack an RSA cipher by guessing the value of(p − 1)(q − 1) but this is no easier than factoringpq whichis the most obvious means of attack. It is also possible fora cryptanalyst to try every possibled but this brute forceapproach is less efficient than trying to factorpq.

In general, RSA cryptanalysis (see Section IV) has shownthat the attacks discovered to date illustrate the pitfalls to beavoided when implementing RSA. Thus, even though RSAciphers can be attacked, the algorithm can still be consideredsecure when used properly. In order to ensure the continuedstrength of the cipher, RSA run factoring challenges on theirwebsites. As with all PKI and other cryptographic products,this algorithm is possibly most vulnerable to authorities (atleast those operating in the UK) having to conform to theRegulation of Investigatory Powers Act 2000, Section 49.

IV. CRYPTANALYSIS

Any cryptographic system must be able to withstand crypt-analysis [42]. Cryptanalysis methods depend critically on theencryption techniques which have been developed and are,therefore, subject to delays in publication. Cryptanalysts workon ‘attacks’ to try and break a cryptosystem. In many cases,the cryptanalysts are aware of the algorithm used and willattempt to break the algorithm in order to compromise thekeys or gain access to the actual plaintext. It is worth notingthat even though a number of algorithms are freely published,

this does not in any way mean that they are the most secure.Most government institutions and the military do not revealthe type of algorithm used in the design of a cryptosystem.The rationale for this is that, if we find it difficult to break acode with knowledge of the algorithm, then how much moredifficult is it to break a code if the algorithm is unknown?On the other hand, within the academic community, securityin terms of algorithm secrecy is not considered to be ofhigh merit and publication of the algorithm(s) is alwaysrecommended. It remains to be understood whether this isa misconception within the academic world (due in part tothe innocence associated with academic culture) or a covertlyinduced government policy (by those who are less innocent!).For example, in 2003, it was reported that the Americans hadbroken ciphers used by the Iranian intelligence services. Whatwas not mentioned, was the fact that the Iranian ciphers werebased on systems purchased indirectly from the USA and thus,based on USA designed algorithms [45].

The ‘known algorithm’ approach originally comes from thework of Auguste Kerchhoff. Kerchhoff’s Principle states that:A cryptosystem should be secure even if everything about thesystem, except the key, is public knowledge. This principlewas reformulated by Claude Shannon asthe enemy knowsthe systemand is widely embraced by cryptographers worldwide. In accordance with the Kerchhoff-Shannon principle,the majority of civilian cryptosystems make use of publiclyknown algorithms. The principle is that of ‘security throughtransparency’ in which open-source software is considered tobe inherently more secure than closed source software. Onthis basis there are several methods by which a system can beattacked where, in each case, it is assumed that the cryptanalysthas full knowledge of the algorithm(s) used.

A. Basic Attacks

We provide a brief overview of the basic attack strategiesassociated with cryptanalysis.

Ciphertext-only attack is where the cryptanalyst has aciphertext of several messages at their disposal. All messagesare assumed to have been encrypted using the same algorithm.The challenge for the cryptanalyst is to try and recover theplaintext from these messages. Clearly a cryptanalyst will bein a valuable position if they can recover the actual keys usedfor encryption.

Known-plaintext attack makes the task of the cryptanalysissimpler because, in this case, access is available to both theplaintext and the corresponding ciphertext. It is then necessaryto deduce the key used for encrypting the messages, or designan algorithm to decrypt any new messages encrypted withthe same key.

Chosen-plaintext attack involves the cryptanalyst possessingboth the plaintext and the ciphertext. In addition, the analysthas the ability to encrypt plaintext and see the ciphertextproduced. This provides a powerful tool from which the keyscan be deduced.


Adaptive-chosen-plaintext attack is an improved version ofthe chosen-plaintext attack. In this version, the cryptanalysthas the ability to modify the results based on the previousencryption. This version allows the cryptanalyst to choose asmaller block for encryption.

Chosen-ciphertext attack can be applied when thecryptanalyst has access to several decrypted texts. In addition,the cryptanalyst is able to use the text and pass it though a‘black box’ for an attempted decrypt. The cryptanalyst has toguess the keys in order to use this method which is performedon an iterative basis (for different keys), until a decrypt isobtained.

Chosen-key attack is based on some knowledge onthe relationship between different keys and is not of practicalsignificance except in special circumstances.

Rubber-hose cryptanalysis is based on the use of humanfactors such as blackmail and physical threat for example.It is often a very powerful attack and sometimes very effective.

Differential cryptanalysis is a more general form ofcryptanalysis. It is the study of how differences in an inputcan affect differences in the output. This method of attack isusually based on a chosen plaintext, meaning that the attackermust be able to obtain encrypted ciphertexts for some setof plaintexts of their own choosing. This typically involvesacquiring a Crib of some type as discussed in the followingsection.

Linear cryptanalysis is a known plaintext attack whichuses linear relations between inputs and outputs of anencryption algorithm which holds with a certain probability.This approximation can be used to assign probabilities to thepossible keys and locate the one that is most probable.

B. Cribs

The problem with any form of chosen-plaintext attack is,of course, how to obtain part or all of the plaintext in thefirst place. One method that can be used is to obtain a Crib.A Crib, a term that originated at Bletchley Park during theSecond World War, is a plaintext which is known or suspectedof being part of a ciphertext. If it is possible to comparepart of the ciphertext that is known to correspond with theplaintext then, with the encryption algorithm known, one canattempt to identify which key has been used to generate thecipherext as a whole and thus decrypt an entire message. Buthow is it possible to obtain any plaintext on the assumptionthat all plaintexts are encrypted in their entirety? One wayis to analyse whether or not there is any bad practice beingundertaken by the user, e.g. sending stereotyped (encrypted)messages. Analysing any repetitive features that can be ex-pected is another way of obtaining a Crib. For example,suppose that a user was writing letters using Microsoft word,for example, having established an electronic letter template

with his/her name, address, company reference number etc.Suppose we assume that each time a new letter is written, theentire document is encrypted using a known algorithm. If itis possible to obtain the letter template then a Crib has beenfound. Assuming that the user is not prepared to share theelectronic template (which would be a strange thing to askfor), a simple way of obtaining the Crib could be to write tothe user in hardcopy and ask that the response from the sameuser is of the same type, pleading ignorance of all forms ofICT or some other excuse. This is typical of methods thatare designed to seed a response that includes a useful Crib.Further, there are a number of passive cribs with regard toletter writing that can be assumed, the use ofDear andYourssincerely, for example.

During the Second World War, when passive cribs suchas daily weather reports became rare through improvementsin the protocols associated with the use of Enigma and/oroperators who took their work seriously, Bletchley Park wouldask the Royal Air Force to create some ‘trouble’ that was oflittle military value. This included seeding a particular areain the North Sea with mines, dropping some bombs on theFrench coast or, for a more rapid response, asking fighter pilotsto go over to France and ‘shoot up’ targets of opportunity8,processes that came to be known as ‘gardening’. The Enigmaencrypted ciphertexts that were used to report the ‘trouble’could then be assumed to contain information such as thename of the area where the mines had been dropped and/or theharbour(s) threatened by the mines. It is worth noting that theability to obtain cribs by gardening was made relatively easybecause of the war in which ‘trouble’ was to be expected andto be reported. Coupled with the efficiency of the German warmachine with regard to its emphasis on accurate and timelyreports, the British were in a privileged position in which theycould create cribs at will and have some fun doing it!

When a captured and interrogated German stated thatEnigma operators had been instructed to encode numbers byspelling them out, Alan Turing reviewed decrypted messages,and determined that the number ‘eins’ appeared in 90% ofthe messages. He automated the crib process, creating an‘Eins Catalogue’, which assumed that ‘eins’ was encoded atall positions in the plaintext. The catalogue included everypossible key setting which provided a very simple and effectiveway of recovering the key and is a good example of how thestatistics (of a word or phrase) can be used in cryptanalysis.

The use of Enigma by the German naval forces (in par-ticular, the U-boat fleet) was, compared to the German armyand air force, made secure by using a password from oneday to the next. This was based on a code book providedto the operator prior to departure from base. No transmissionof the daily passwords was required, passive cribs were rareand seeding activities were difficult to arrange. Thus, if notfor a lucky break, in which one of these code books (whichwere printed in ink that disappeared if they were dropped inseawater) was recovered in tact by a British destroyer (HMSBulldog) from a damaged U-boat (U-110) on May 9, 1941,

8Using targets of opportunity became very popular towards the end of thewar. Fighter pilots were encouraged to, in the words of General J Dolittle,‘get them in the air, get them on the ground, just get them’.


breaking the Enigma naval transmissions under their time-variant code-book protocol would have been very difficult. ABritish Naval message dated May 10, 1941 reads: ‘1. Captureof U Boat 110 is to be referred to as operation Primrose;2. Operation Primrose is to be treated with greatest secrecyand as few people allowed to know as possible...’ Clearly,and for obvious reasons, the British were anxious to makesure that the Germans did not find out that U-110 and itscodebooks had been captured and all the sailors who took partin the operation were sworn to secrecy. On HMS Bulldog’sarrival back in Britain a representative from Bletchley Park,photographed every page of every book. The ‘interesting pieceof equipment’ turned out to be an Enigma machine, and thebooks contained the Enigma codes being used by the Germannavy.

The U-boat losses that increased significantly through thedecryption of U-boat Enigma ciphers led Admiral Carl Doenitzto suspect that his communications protocol had been com-promised. He had no firm evidence, just a ‘gut feeling’ thatsomething was wrong. His mistake was not to do anythingabout it9, an attitude that was typical of the German HighCommand who were certifiable with regard to their confidencein the Enigma system. However, they were not uniquely cer-tifiable. For example, on April 18, 1943, Admiral Yamamoto(the victor of Pearl Harbour) was killed when his plane wasshot down while he was attempting to visit his forces in theSolomon Islands. Notification of his visit from Rabaul to theSolomon’s was broadcast as Morse coded ciphertext over theradio, information that was being routinely decrypted by theAmericans. At this point in the Pacific War, the Japanesewere using a code book protocol similar to that used by theGerman Navy, in which the keys were changed on a dailybasis, keys that the Americans had ‘generated’ copies of. Someweeks before his visit, Yamamoto had been given the optionof ordering a new set of code books to be issued. He hadrefused to give the order on the grounds that the logisticsassociated with transferring new code books over Japaneseheld territory was incompatible with the time scale of his visitand the possible breach of security that could arise through anew code book being delivered into the hands of the enemy.This decision cost him his life. However, it is a decisionthat reflects the problems associated with the distribution ofkeys for symmetric cryptosystems especially when a multi-user protocol needs to be established for execution over a widecommunications area. In light of this problem, Yamamoto’sdecision was entirely rational but, nevertheless, a decisionbased on the assumption that the cryptosystem had not alreadybeen compromised. Perhaps it was his ‘faith in the system’ andthereby his refusal to think the ‘unthinkable’ that cost him hislife!

The principles associated with cryptanalysis that have beenbriefly introduced here illustrate the importance of using adynamic approach to cryptology. Any feature of a securityinfrastructure that has any degree of consistency is vulnerableto attack. This can include plaintexts that have routine phrasessuch as those used in letters, the key(s) used to encrypt the

9An instinct can be worth a thousand ciphers, ten-thousand if you like.

plaintext and the algorithm(s) used for encryption. One of theprincipal advantages of using chaoticity for designing ciphersis that it provides the cryptographer with a limitless and dy-namic resource for producing different encryption algorithms.These algorithms can be randomly selected and permuted toproduce, in principle, an unlimited number of Meta encryptionengines that operate on random length blocks of plaintext. Theuse of block cipher encryption is both necessary in order toaccommodate the relatively low cycle length of chaotic ciphersand desirable in order to increase the strength of the cipherby implementing a multi-algorithmic approach. Whereas inconventional cryptography, emphasis focuses on the number ofpermutations associated with the keys used to ‘seed’ or ‘drive’an algorithm, chaos-based encryption can focus on the numberof permutations associated with the algorithms that are used,algorithms that can, with care and understanding, be quiteliterally ‘invented on the fly’. Since cryptanalysis requires thatthe algorithm is known and concentrates on trying to findthe key, this approach, coupled with other important detailsthat are discussed later on in this paper, provides a methodthat can significantly enhance the cryptographic strength ofthe ciphertext. Further, in order to satisfy the ‘innocence’ ofthe academic community, it is, of course, possible to openlypublish such algorithms (as in this paper, for example), but inthe knowledge that many more can be invented and publishedor otherwise. This provides the potential for generating ahost of ‘home-spun’ ciphers which can be designed andimplemented by anyone who wishes to by-pass establishedpractices and ‘cook it themselves’.

V. D IFFUSION AND CONFUSION

In terms of plaintexts, diffusion is concerned with the issuethat, at least on a statistical basis, similar plaintexts shouldresult in completely different ciphertexts even when encryptedwith the same key. This requires that any element of theinput block influences every element of the output block inan irregular fashion. In terms of a key, diffusion ensures thatsimilar keys result in completely different ciphertexts evenwhen used for encrypting the same block of plaintext. Thisrequires that any element of the input should influence everyelement of the output in an irregular way. This property mustalso be valid for the decryption process because otherwise anintruder may be able to recover parts of the input from anobserved output by a partly correct guess of the key used forencryption. The diffusion process is a function of sensitivityto initial conditions conditions that a cryptographic systemshould have and further, the inherent topological transitivitythat the system should also exhibit causing the plaintext to bemixed through the action of the encryption process.

Confusion ensures that the (statistical) properties of plain-text blocks are not reflected in the corresponding ciphertextblocks. Instead every ciphertext must have a random ap-pearance to any observer and be quantifiable through appro-priate statistical tests. However, diffusion and confusion areprocesses that are of fundamental importance in the design andanalysis of cryptological systems, not only for the encryptionof plaintexts but for data transformation in general.


A. The Diffusion Equation

In a variety of physical problems, the process of diffusioncan be modelled in terms of certain solutions to the diffusionequation whose basic (linear) form is given by10 [44]-[47]

∇2u(r, t)− σ∂

∂tu(r, t) = S(r, t),

∇2 =∂2

∂x2+

∂2

∂y2+

∂2

∂z2,

u(r, 0) = u0(r), σ =1D

,

where D is the ‘Diffusivity’, S is a source term anduis a function which describes physical properties such astemperature, light, particle concentration and so on with initialvalueu0.

The diffusion equation describes fieldsu that are the resultof an ensemble of incoherent random walk processes, i.e.walks whose direction changes arbitrarily from one step to thenext and where the most likely position after a timet is pro-portional to

√t. Further, the diffusion equation differentiates

between past and future, i.e.±t . This is because the diffusingfield u represents the behaviour of some average property ofan ensemble of many ‘agents’ which cannot in general go backto their original state. This fundamental property of diffusiveprocesses has a synergy with the use of one-way functionsin cryptology, i.e. functions that, given an input, produce anoutput that is not reversible - an output from which it is notpossible to compute the input.

Consider the process of diffusion in which a source ofmaterial diffuses into a surrounding homogeneous medium,the material being described by some initial conditionu(r, 0).Physically, it is to be expected that the material will increas-ingly ‘spread out’ as time evolves and that the concentrationof the material decreases further away from the source. Thegeneral solution to the diffusion equation yields a result inwhich the spatial concentration of material is given by theconvolution of the initial condition with a Gaussian function.This solution is determined by considering how the process ofdiffusion responds to a single point source which yields theGreen’s function (in this case, a Gaussian function) given by[47]-[49],

G(r, t) =1σ

( σ

4πt

)n2

exp[−(

σr2

4t

)], t > 0, r =| r | .

which is the solution to(∇2 − σ

∂

∂t

)u(r, t) = δn(r)δ(t)

whereδ denotes the Dirac delta function [50], [51] andn =1, 2, 3 determines the dimension of the solution.

In the infinite domain, the general solution to the diffusionequation can be written in the form [47]

u(r, t) = G(r, t)⊗r ⊗tS(r, t) + σG(r, t)⊗r u(r, 0)

which requires that the spatial extent of the source function isinfinite but can include functions that are localised provided

10r = xx + yy + zz denotes the spatial vector andt denotes time

Fig. 2. Image of an optical source (left), the same source imaged throughsteam (centre) and a simulation of this effect obtained by convolving thesource image with a Gaussian Point Spread Function (right).

that S → 0 as r → ∞ - a Gaussian function for example.The solution is composed of two terms. The first term isthe convolution (in space and time, denoted by⊗r and⊗t

respectively) of the source function with the Green’s functionand the second term is the convolution (in space only) of theinitial conditionu(r, 0) with the same Green’s function where

G(r, t)⊗r⊗tS(r, t) =∫ ∫

G(| r−r′ |, t−τ)S(r′, τ)d3r′dτ.

Thus, for example, in two-dimensions, for the case whenS =0, and ignoring scaling byσ/(4πt)), the solution foru is

u(x0, y0, t) = exp[− σ

4t(x2 + y2)

]⊗⊗u0(x, y)

where we have introduced⊗⊗ to denote the two-dimensionalconvolution integral. Here, the field at timet > 0 is given bythe field at timet = 0 convolved with the two-dimensionalGaussian functionexp[−σ(x2 + y2)/(4t)]. This result can,for example, be used to model the diffusion of light throughan optical diffuser. An example of such an effect is given inFigure 2 which shows a light source (the ceiling light of asteam room) imaged through air and through steam togetherwith a simulation. Steam, as composed of a complex of smallwater droplets, effects light by scattering it a large number oftimes. The high degree of multiple scattering that takes placeallows us to model the transmission of light in terms of adiffusive rather than a propagative processe where the functionu is taken to denote the intensity of light. The initial conditionu0 is taken to denote the initial image which is, in effect, theimage of the light source recorded through air. As observedin Figure 2, the details associated with the light source areblurred through a mixing process which is determined by theGaussian function that is characteristic of diffusion processesin general. In imaging science, functions of this type determinehow a point of light is affected by the convolution process11

and is thus referred to as the Point Spread Function or PSF[52]. The PSF is a particularly important characteristic of anyimaging system in general, a characteristic that is related tothe physical processes through which light is transforms fromthe object plane(input) to theimage plane(output).

If we record a diffused fieldu after some timet = T , isit possible to reconstruct the field at timet = 0, i.e. to solvethe inverse problem or de-diffuse the field measured? We can

11Convolution is sometimes referred to by its German equivalent, i.e. bythe word ‘Faltung’ which means ‘mixing’ or ‘diffusing’.


expressu(r, 0) in terms ofu(r, T ) using the Taylor series

u0(r) ≡ u(r, 0) = u(r, T ) +∞∑

n=1

(−1)n

n!Tn

[∂n

∂tnu(r, t)

]t=T

.

From the diffusion equation

∂2u

∂t2= D∇2 ∂u

∂t= D2∇4u,

∂3u

∂t3= D∇2 ∂2u

∂t2= D3∇6u

and so on. Thus, in general we can write[∂n

∂tnu(r, t)

]t=T

= Dn∇2nu(x, y, T ).

and after substituting this result into the series foru0 givenabove, we obtain

u0(r) = u(r, T ) +∞∑

n=1

(−1)n

n!(DT )n∇2nu(r, T )

∼ u−DT∇2u, DT << 1.

B. Diffusion of a Stochastic Source

For the case when(∇2 − σ

∂

∂t

)u(r, t) = −S(r, t), u(r, 0) = 0

the solution is

u(r, t) = G(r, t)⊗r ⊗tS(r, t), t > 0.

If a source is introduced in terms of an impulse att = 0, thenthe ‘system’ will react accordingly and diffuse fort > 0. Thisis equivalent to introducing a source function of the form

S(r, t) = s(r)δ(t).

The solution is then given by

u(r, t) = G(r, t)⊗r s(r), t > 0.

Observe that this solution is of the same form as the homoge-neous case with initial conditionu(r, 0) = u0(r) and that thesolution for initial conditionu(r, 0) = u0(r) is given by

u(r, t) = G(r, t)⊗r [s(r)+u0(r)] = G(r, t)⊗u0(r)+n(r, t)

wheren(r, t) = G(r, t)⊗r s(r), t > 0.

Note that if s is a stochastic function (i.e. a random depen-dent variable characterised, at least, by a Probability DensityFunction (PDF) denoted byPr[s(r)]), then n will also bea stochastic function. Also note that for a time-independentsource functionS(r),

u0(r) = u(r, T )

+∞∑

n=1

(−1)n

n![(DT )n∇2nu(r, T ) + D−1∇2n−2S(r)]

and that ifS is a stochastic function, then the fieldu can notbe de-diffused (since it is not possible to evaluateu0 exactly

Fig. 3. Progressive diffusion and confusion of an image (top-left) - fromleft to right and from top to bottom - for uniform distributed noise. Theconvolution is undertaken using the convolution theorem and a Fast FourierTransform (FFT)

givenPr[S(r)]). In other words, any error or ‘noise’ associatedwith diffusion leads to the process being irreversible - a ‘one-way’ process. This, however, depends on the magnitude ofthe diffusivity D which for large values cancels out the effectof any noise, thus making the process potentially reversible.In cryptography, it is therefore important that the process ofdiffusion applied (in order that a key affects every bit of theplaintext irrespective of the encryption algorithm that is used)has a low diffusivity.

The inclusion of a stochastic source function provides uswith a self-consistent introduction to another important con-cept in cryptology, namely ‘confusion’. Taking, for example,the two-dimensional case, the fieldu is given by

u(x, y) =σ

4πtexp

[− σ

4t(x2 + y2)

]⊗⊗u0(x, y) + n(x, y).

We thus arrive at a basic model for the process of diffusionand confusion, namely

Output=Diffusion+Confusion.

Here, diffusion involves the ‘mixing’ of the initial conditionwith a Gaussian function andconfusion is compounded inthe addition of a stochastic or noise function to the diffusedoutput. The relative magnitudes of the two terms determinesthe dominating effect. As the noise functionn increases inamplitude relative to the diffusion term, the output will becomeincreasingly determined by the effect of confusion alone. Inthe equation above, this will occur ast increases since themagnitude of the diffusion term depends of the scaling factor1/t. This is illustrated in Figure 3 which shows the combinedeffect of diffusion and confusion for an image of the phrase

Confusion+

Diffusion

as it is (from left to right and from top to bottom) progressivelydiffused (increasing values oft) and increasingly confused fora stochastic functionn that is uniformly distributed. Clearly,the longer the time taken for the process of diffusion to occur,the more the output is confusion dominated. This is consistentwith all cases when the level of confusion is high and whenthe stochastic field used to generate this level of confusion


is unknown (other than possible knowledge of its PDF).However, if the stochastic function has been synthesized12 andis thus knowna priori, then we can compute

u(x, y)− n(x, y) =1

4πtexp

[− σ

4t(x2 + y2)

]⊗⊗u0(x, y)

from which u0 may be computed approximately via applica-tion of a deconvolution algorithm [52].

VI. STOCHASTIC FIELDS

By considering the diffusion equation for a stochasticsource, we have derived a basic model for the ‘solutionfield’ or ‘output’ u(r, t) in terms of the initial condition orinput u0(r). We now consider the principal properties ofstochastic fields, considering the case where the fields arerandom variables that are functions of timet.

A. Independent Random Variables

Two random variablesf1(t) and f2(t) are independent iftheir cross-correlation function is zero, i.e.

∞∫−∞

f1(t + τ)f2(τ)dτ = f1(t) f2(t) = 0

where is used to denote the correlation integral above. Fromthe correlation theorem [53], [54], it then follows that

F ∗1 (ω)F2(ω) = 0

where

F1(ω) =

∞∫−∞

f1(t) exp(−iωt)dt

and

F2(ω) =

∞∫−∞

f2(t) exp(−iωt)dt.

If each function has a PDFPr[f1(t)] and Pr[f2(t)] respec-tively, the PDF of the functionf(t) that is the sum off1(t) andf2(t) is given by the convolution ofPr[f1(t)] and Pr[f2(t)],i.e. the PDF of the function

f(t) = f1(t) + f2(t)

is given by [55], [56]

Pr[f(t)] = Pr[f1(t)]⊗ Pr[f2(t)].

Further, for a number of statistically independentstochastic functionsf1(t), f2(t), ..., each with a PDFPr[f1(t)],Pr[f2(t)], ..., the PDF of the sum of thesefunctions, i.e.

f(t) = f1(t) + f2(t) + f3(t) + ...

is given by

Pr[f(t)] = Pr[f1(t)]⊗ Pr[f2(t)]⊗ Pr[f1(t)]⊗ ...

12The synthesis of stochastic functions is a principal issue in cryptology.

These results can derived using the Characteristic Function[57]. For a strictly continuous random variablef(t) with dis-tribution functionPf (x) = Pr[f(t)] we define the expectationas

E(f) =

∞∫−∞

xPf (x)dx,

which computes the mean value of the random variable, theMoment Generating Function as

E[exp(−kf)] =

∞∫−∞

exp(−kx)Pf (x)dx

which may not always exist and the Characteristic Functionas

E[exp(−ikf)] =

∞∫−∞

exp(−ikx)Pf (x)dx

which will always exist. Observe that the moment generatingfunction is the (two-sided) Laplace transform [58] ofPf andthe Characteristic Function is the Fourier transform ofPf .Thus, if f(t) is a stochastic function which is the sum ofN independent random variablesf1(t), f2(t), ..., fN (t) withdistributionsPf1(x), Pf2(x), ..., PfN

(x), then

f(t) = f1(t) + f2(t) + ... + fN (t)

and

E[exp(−ikf)] = E[exp[−ik(f1 + f2 + ... + fN )]

= E[exp(−ikf1)]E[exp(−ikf2)]...E[exp(−ikfN )]

= F1[Pf1 ]F1[Pf2 ]...F1[PfN]

whereF1 is the one-dimensional Fourier transform operartordefined as

F ≡∞∫

−∞

dx exp(−ikx).

In other words, the Characteristic Function of the randomvariablef(t) is the product of the Characteristic Functions forall random variables whose sum iff(t). Using the convolutiontheorem for Fourier transforms, we then obtain

Pf (x) =N∏

i=1

⊗ Pfi(x) = Pf1(x)⊗ Pf2(x)⊗ ...⊗ PfN(x).

Further, we note that iff1, f2, ..., fN are all identicallydistributed then

E[exp[−ik(f1 + f2 + ... + fN )] = (F [Pf1 ])N

and

Pf (x) = Pf1(x)⊗ Pf1(x)⊗ ...


B. The Central Limit Theorem

The Central Limit Theorem stems from the result thatthe convolution of two functions generally yields a functionwhich is smoother than either of the functions that are beingconvolved. Moreover, if the convolution operation is repeated,then the result starts to look more and more like a Gaussianfunction - a normal distribution - at least in an approximatesense [59], [60]. For example, suppose we have a number ofindependent random variables each of which is characterisedby a distribution that is uniform. As we add more and more ofthese functions together, the resulting distribution is then givenby convolving more and more of these (uniform) distributions.As the number of convolutions increases, the result tends toa Gaussian distribution. If we consider the effect of applyingmultiple convolutions of the uniform distribution

P (x) =

1X , | x |≤ X/2;0, otherwise

then be considering the effect of multiple convolutions inFourier space (through application of the convolution theorem)and working with a series representation of the result, it canbe shown that (see Appendix I)

N∏i=1

⊗ Pi(x) ≡ P1(x)⊗ P2(x)⊗ ...⊗ PN (t)

'√

6πN

exp(−6x2/XN)

wherePi(x) = P (x), ∀i andN is large. Figure 4 illustratesthe effect of successively adding uniformly distributed butindependent random times series (each consisting of 500elements) and plotting the resulting histograms (using 32 bins),i.e. given the discrete times seriesf1[i], f2[i], f3[i], f4[i] fori=1 to 500, Figure 4 shows the time series

s1[i] = f1[i],

s2[i] = f1[i] + f2[i],

s3[i] = f1[i] + f2[i] + f3[i],

s4[i] = f1[i] + f2[i] + f3[i] + f4[i]

and the corresponding 32-bin histograms of the signalssj , j =1, 2, 3, 4. Clearly asj increases, the histogram starts to ‘look’increasing normally distributed. Here, the uniformly distrib-uted discrete time seriesfi, i = 1, 2, 3, 4 have been computedusing the uniform pseudo random number generator

fi+1 = afimodP

wherea = 77 andP = 232 − 1 is a Mersenne prime number,by using four different seedsf0 in order to provide time seriesthat are ‘independent’.

The Central Limit Theorem has been considered specificallyfor the case of uniformly distributed independent randomvariables. However, in general, it is approximately applicablefor all independent random variables, irrespective of their

Fig. 4. Illustration of the Central Limit Theorem. The top-left image showsplots of a 500 element uniformly distributed time series and its histogramusing 32 bins. The top-right image shows the result of adding two uniformlydistributed and independent time series together and the 32 bin histogram.The bottom-left image is the result after adding three uniformly distributedtimes series and the bottom-right image is the result of adding four uniformlydistributed times series.

distribution. In particular, we note that for a standard normal(Gaussian) distribution given by

Gauss(x;σ, µ) =1√2πσ

exp

[−1

2

(x− µ

σ

)2]

where ∞∫−∞

Gauss(x)dx = 1

and∞∫

−∞

Gauss(x) exp(−ikx)dx = exp(ikµ) exp(−σ2k2

2

),

then, since

Gauss(x) ⇐⇒ exp(ikµ) exp(−σ2k2

2

),

N∏j=1

⊗ Gauss(x) ⇐⇒ exp(ikNµ) exp(−Nσ2k2

2

)so that

N∏j=1

⊗ Gauss(x) =(

12πNσ2

)exp

[− 1

2N

(x− µ

σ

)2]

where ⇐⇒ denotes transformation form ‘real’ to ‘Fourierspace’. In other words, the addition of Gaussian distributedfields produces a Gaussian distributed field.


VII. STOCHASTIC DIFFUSION

Given the classical diffusion/confusion model of the type

u(r) = p(r)⊗r u0(r) + n(r)

discussed above, we note that both the operator and thefunctional form of p are derived from solving a physicalproblem (using a Green’s function solution) compounded ina particular PDE - diffusion equation. This is an exampleof ‘Gaussian diffusion’ since the characteristic Point SpreadFunction is a Gaussian function. However, we can use thisbasic model and consider a variety of PSFs as required.Although arbitrary changes to the PSF are inconsistent withclassical diffusion, in cryptology we can, in principal, chooseany PSF that is of value in ‘diffusing’ the data. For example,in Fresnel optics [61], [62], the PSF is of the same Gaussianform but with a complex exponential. Iff(x, y) is the ‘objectfunction’ describing the ‘object plane’ andu(x, y) is the imageplane wave function, then [63], [64]

u(x, y) = p(x, y)⊗⊗f(x, y)

where the PSFp is given by (ignoring scaling) [52]

p(x, y) = exp[iα(x2 + y2)]; | x |≤ X, | y |≤ Y

whereα = π/(zλ), λ being the wavelength andz the distancebetween the object and image planes, and whereX and Ydetermine the spatial support of the PSF.

Stochastic diffusion involves interchanging the roles ofpandn, i.e. replacingp(r) - a deterministic PSF - withn(r) -a stochastic function. Thus, noise diffusion is compounded inthe result

u(r) = n(r)⊗r u0(r) + p(r)

wherep can be any function or

u(r) = n1(r)⊗r u0(r) + n2(r)

where bothn1 andn2 are stochastic function which may be ofthe same type (i.e. have the same PDFs) or of different types(with different PDFs). This form of diffusion is not ‘physical’in the sense that it does not conform to a physical model asdefined by the diffusion equation, for example. Heren(r) canbe any stochastic function (synthesized or otherwise).

The simplest form of noise diffusion is

u(r) = n(r)⊗r u0(r).

There are two approaches to solving the inverse problem:Given u and n, obtain u0. We can invert or deconvolve byusing the convolution theorem giving (for dimensionn =1, 2, 3)

u0(r) = F−1n

[U(k)N∗(k)| N(k) |2

]whereN is the Fourier transform ofn and U is the Fouriertransform ofu. However, this approach requires regularisationin order to eliminate any singularities when| N |2= 0 throughapplication of a constrained deconvolution filter such as theWiener filter [52]. Alternatively, ifn is the result of some

random number generating algorithm, we can construct thestochastic field

m(r) = F−1n

[N∗(k)| N(k) |2

]where| N(k) |2> 0, the diffused field now being given by

u(r) = m(r)⊗r u0(r).

The inverse problem is then solved by correlatingu with n,since

n(r)r u(r) ⇐⇒ N∗(k)U(k)

andN∗(k)U(k) = N∗(k)M(k)U0(k)

= N∗(k)N∗(k)| N(k) |2

U0(k) = U0(k)

so thatu0(r) = n(r)r u(r).

The condition that| N(k) |2> 0 is simply achieved byimplementing the following process:∀k, if | N(k) |2= 0,then | N(k) |2= 1. This result can be used to ‘embed’ onedata field in another.

Consider the case when we have two independent imagesI1(x, y) ≥ 0∀x, y and I2(x, y) ≥ 0∀x, y and we considerthe case of embeddingI1 with I2. We construct a noise fieldm(x, y) ≥ 0∀x, y a priori and consider the equation

u(x, y) = Rm(x, y)⊗⊗I1(x, y) + I2(x, y)

where

‖m(x, y)⊗⊗I1(x, y)‖∞ = 1 and ‖I2(x, y)‖∞ = 1.

By normalising the terms in this way, the coefficient0 ≤ R ≤1, can be used to adjust the relative magnitudes of the termssuch that the diffused imageI1 is a perturbation of the ‘hostimage’ I2. This provides us with a way of watermarking [65]one image with another,R being referred to as the watermark-ing ratio13. This approach could of course be implementedusing a Fresnel diffuser. However, for applications in imagewatermarking, the diffusion of an image with a noise fieldprovides a superior result because: (i) a noise field providesmore uniform diffusion; (ii) noise fields can be generated usingrandom number generators that depend on a single initial valueor seed (i.e. a private key). An example of this approach isshown in Figure 5. Here an imageI2 (the ‘host image’) iswatermarked by another imageI1 (the ‘watermark image’) andbecauseR = 0.1, the outputu is ‘dominated’ by the imageI2. The noise fieldn, is computed using a uniform randomnumber generator in which the output arrayn is normalizedso that‖n‖∞ = 1 and used to generaten(xi, yi) on a row-by-row basis. Here, the seed is any integer such as1873...which can be based on the application of a PIN (PersonalIdentity Number) or a password (e.g. ‘Enigma’, which in termsof an ASCII string - using binary to decimal conversion - is‘216257556149’). Recovery of the watermark image requires

13Equivalent, in this application, to the standard term ‘Signal-to-Noise’ orSNR ratio as used in signal and image analysis.


Fig. 5. Example of watermarking an image with another image using noisebased diffusion. The ‘host image’I2 (top-left ) is watermarked with the‘watermark image’I1 (top-centre) using the diffuser (top-right) given by auniform noise fieldn whose pixel-by-pixel values depend upon the seed used(the private key). The result of computingm⊗⊗I1 (bottom-left) is added tothe host image forR = 0.1 to generate the watermarked imageu (bottom-centre). Recovery of the watermark imageI1 (bottom-right) is accomplishedby subtracting the host image from the watermarked image and correlatingthe result with the noise fieldn.

knowledge of the PIN or Password and the host imageI2

The effect of adding the diffused watermark image to thehost image yields a different, slightly brighter image becauseof the perturbation ofI2 by Rm ⊗ ⊗I1. This effect can beminimized by introducing a smaller watermarking ratio suchthat the perturbation is still recoverable by subtracting the hostimage from the watermarked image.

The expected statistical distribution associated with theoutput of a noise diffusion process is Gaussian. This can beshown if we consideru0 to be a strictly deterministic functiondescribed by a sum of delta functions, equivalent to a binarystream in 1D or a binary image in 2D (discrete cases), forexample. Thus if

u0(r) =∑

i

δn(r− ri)

then

u(r) = n(r)⊗r u0(r) =N∑

i=1

n(r− ri).

Now, each functionn(r − ri) is just n(r) shifted byri andwill thus be identically distributed. Hence

Pr[u(r)] = Pr

[N∑

i=1

n(r− ri)

]=

N∏i=1

⊗ Pr[n(r)]

and from the Central Limit Theorem, we can expectPr[u(r)]to be normally distributed for largeN . This is illustrated inFigure 6 which shows the statistical distributions associatedwith a binary image, a uniformly distributed noise field andthe output obtained by convolving the two fields together.

Given the equation

u(r) = p(r)⊗r u0(r) + n(r),

if the diffusion by noise is based on interchangingp andn, then the diffusion of noise is based on interchangingu0

Fig. 6. Binary image (top-left), uniformly distributed 2D noise field (top-centre), convolution (top-right) and associated 64-bin histograms (bottom-left,-centre and -right respectively).

and n. In effect, this means that we consider the initial fieldu0 to be a stochastic function. Note that the solution tothe inhomogeneous diffusion equation for a stochastic sourceS(r, t) = s(r)δ(t) is

n(r, t) = G(r, t)⊗r s(r)

and thus,n can be considered to be diffused noise. If weconsider the model

u(r) = p(r)⊗r n(r),

then for the classical diffusion equation, the PSF is a Gaussianfunction. In general, given the convolution operation,p canbe regarded as only one of a number of PSFs that can beconsidered in the ‘production’ of different stochastic fieldsu.This includes PSFs that define self-affine stochastic fields orrandom scaling fractals [66]-[68] that are based on fractionaldiffusion processes.

A. Print Authentication

The method discussed above refer to electronic-to-electronictype communications in which there is no loss of information.Steganography and watermarking techniques can be developedfor hardcopy data which has a range of applications. Thesetechniques have to be robust to the significant distortionsgenerated by the printing and/or scanning process. A simpleapproach is to add information to a printed page that is difficultto see. For example, some modern colour laser printers,including those manufactured by HP and Xerox, print tinyyellow dots which are added to each page. The dots are barelyvisible and contain encoded printer serial numbers, date andtime stamps. This facility provides a useful forensics tool fortracking the origins of a printed document which has onlyrelatively recently been disclosed.

If the watermarked image is printed and scanned back intoelectronic form, then the print/scan process will yield an arrayof pixels that will be significantly different from the originalelectronic image even though it might ‘look’ the same. Thesedifferences can include the size of the image, its orientation,brightness, contrast and so on. Of all the processes involvedin the recovery of the watermark, the subtraction of the host


image from the watermarked image is critical. If this processis not accurate on a pixel-by-pixel basis and deregistered forany of many reasons, then recovery of the watermark bycorrelation will not be effective. However, if we make useof the diffusion process alone, then the watermark can berecovered via a print/scan because of the compatibility of theprocesses involved. However, in this case, the ‘watermark’ isnot covert but overt.

Depending on the printing process applied, a number of dis-tortions will occur which diffuse the information being printed.Thus, in general, we can consider the printing process tointroduce an effect that can be represented by the convolutionequation

uprint = pprint ⊗⊗u.

whereu is the original electronic form of a diffused image(i.e. u = n⊗⊗u0) andpprint is the point spread function ofthe printer. An incoherent image of the data, obtained using aflat bed scanner for example (or any other incoherent opticalimaging system) will also have a characteristic point spreadfunction pscan. Thus, we can consider a scanned image to begiven by

uscan = pscan ⊗⊗uprint

whereuscan is taken to be the digital image obtain from thescan. Now, because convolution is commutative, we can write

uscan = pscan⊗⊗pprint⊗⊗p⊗⊗u0 = p⊗⊗pscan/print⊗⊗u0

wherepscan/print = pscan ⊗⊗pprint

which is the print/scan point spread function associated withthe processing cycle of printing the image and then scanningit. By applying the method discussed earlier, we can obtain areconstruction of the watermark whose fidelity is determinedby the scan/print point spread function. However, in practice,the scanned image needs to be re-sized to that of the original.This is due to the scaling relationship (for a functionf withFourier transformF )

f(αx, βy) ⇐⇒ 1αβ

F

(kx

α,ky

β

).

The size of any image captured by a scanner or other devicewill depend on the resolution used. The size of the imageobtained will inevitably be different from the original becauseof the resolution and window size used to print the diffusedimage u and the resolution used to scan the image. Sincescaling in the spatial domain causes inverse scaling in theFourier domain, the scaling effect must be ‘inverted’ before thewatermark can be recovered by correlation since correlationis not a scale invariant process. Re-sizing the image (using anappropriate interpolation scheme such as the bi-cubic method,for example) requires a set of two numbersn and m (i.e.the n×m array used to generate the noise field and executethe diffusion process) that, along with the seed required toregenerate the noise field, provides the ‘private keys’ neededto recover the data from the diffused image. An example ofthis approach is given in Figure 7 which shows the result of re-constructing four different images (a photograph, finger-print,

Fig. 7. Example of the application of ‘diffusion only’ watermarking. In thisexample, four images of a face, finger-print, signature and text have beendiffused using the same noise fieldm and printed on the front (top-left) andback (bottom-left) of an impersonalized identity card using a 600 dpi printer.The reconstructions (top-right and bottom-right, respectively) are obtainedusing a conventional flat-bed scanner based on a 300 dpi grey-level scan.

signature and text) used in the design of an impersonalizeddebit/credit card. The use of ‘diffusion only’ watermarkingfor print security can be undertaken in colour by applyingexactly the same diffusion/reconstruction methods to the red,green and blue components independently. This provides twoadditional advantages: (i) the effect of using colour tendsto yield better quality reconstructions because of the colourcombination process; (ii) for each colour component, it ispossible to apply a noise field with a different seed. In thiscase, three keys are required to recover the watermark.

Because this method is based on convolution alone and since

uscan = pscan/print ⊗⊗u0

as discussed earlier, the recovery of thef will not be negatedby the distortion of the point spread function associatedwith the print/scan process, just limited or otherwise by itscharacteristics. Thus, if an image is obtained of the printed datafield p⊗⊗u0 which is out of focus due to the characteristicsof pscan/print, then the reconstruction ofu0 will be out offocus to the same degree. Decryption of images with thischaracteristic is only possible using an encryption scheme thatis based a diffusion only approach. Figure 8 illustrates therecovery of a diffused image printed onto a personal identitycard obtained using a flat bed scanner and then captured usingmobile phone camera. In the latter case, the reconstruction isnot in focus because of the wide-field nature of the lens used.However, the fact that recovery of the watermark is possiblewith a mobile phone means that the scrambled data can betransmitted securely and the card holders image (as in thisexample) recovered remotely and transmitted back to the samephone for authentication. This provides the necessary physicalsecurity needed to implement such a scheme in practice andmeans that specialist image capture devices are not requiredon site.

The diffusion process can be carried out using a varietyof different noise fields other than the uniform noise field


Fig. 8. Original image (top-left), diffused image (top-right), reconstructionusing a flatbed scanner (bottom-left) and reconstruction using a mobile phone(bottom-right). These images have been scanned in grey scale from theoriginal colour versions printed on to a personalised identity card at 600dpistamp-size (i.e. 2cm×1.5cm).

considered here. Changing the noise field can be of valuein two respects: first, it allows a system to be designed that,in addition to specific keys, is based on specific algorithmswhich must be knowna priori. These algorithms can bebased on different pseudo uniform random number generatorsand/or different pseudo chaotic number generators that arepost-processed to provide a uniform distribution of numbers.Second, the diffusion field depends on both the characteristicsof the watermark image and the noise field. By utilizingdifferent noise fields (e.g. Gaussian noise, Poisson noise,fractal noise and so on), the texture of the output field canbe changed. The use of different noise fields is of valuewhen different textures are required that are aestheticallypleasing and can be used to create a background that is printedover the entire document. In this sense, variable noise baseddiffusion fields can be used to replace complex print securityfeatures with the added advantage that, by de-diffusing them,information can be recovered. Further, these fields are veryrobust to data degradation created by soiling, for example. Inthe case of binary watermark images, data redundancy allowsreconstructions to be generated from a binary output, i.e. afterbinarizing the diffusion field (with a threshold of 50% forexample). This allows the output to be transmitted in a formthat can tolerate low resolution and low contrast copying, e.g.a fax.

The tolerance of this method to printing and scanning isexcellent provided the output is cropped accurately (to within

Fig. 9. Example of the diffusion of composite images with the inclusion ofa reference frame for enhancing and automating the processes of copping andorientation. In each case the data fields have been printed and scanned at 300dpi.

a few pixels) and oriented correctly. The processes of croppingand oreintion can be enhanced and automated by providing areference frame in which the diffused image is inserted. Thisis illustrated in Figure 9 which, in addition shows the effectof diffusing a combination of images. This has the effect ofproducing a diffused field that is very similar but neverthelessconveys entirely different information.

B. Covert Watermarking

Watermarking is usually considered to be a method in whichthe watermark is embedded into a host image in an unobtrusiveway. Another approach is to consider the host image to bea data field that, when processed with another data field,generates new information.

Consider two imagesi1 and i2. Suppose we construct thefollowing function

n = F2

(I1

| I1 |2I2

)where I1 = F2[i1] and I2 = F2[i2]. If we now correlatenwith i1, then from the correlation theorem

i1 n ⇐⇒ I∗1I1

| I1 |2I2 ⇐⇒ i2.

In other words, we can recoveri2 from i1 with a knowledge ofn. Because this process is based on convolution and correlationalone, it is compatible and robust to printing and scanning,i.e. incoherent optical imaging. An example of this is given inFigure 10. In this scheme, the noise fieldn is the private keyrequired to reconstruct the watermark and the host image canbe considered to be a public key.

C. Application to Encryption

One of the principal components associated with the de-velopment of methods and algorithms to ‘break’ cyphertext isthe analysis of the output generated by an attempted decryptand its evaluation in terms of an expected type. The output


Fig. 10. Example of a covert watermarking scheme.i1 (top-left) is‘processed’ withi2 (top-middle) to produce the noise field (top-right).i2 isprinted at 600 dpi, scanned at 300 dpi and then re-sampled back to its originalsize (bottom-left). Correlating this image with the noise field generates thereconstruction (bottom-centre). The reconstruction depends on just the hostimage and noise field. If the noise field and/or the host image are differentor corrupted, then a reconstruction is not achieved, as in the example given(bottom-right).

type is normally assumed to be plain text, i.e. the output isassumed to be in the form of characters, words and phrasesassociated with a natural language such as English or German,for example. If a plain text document is converted into animage file then the method described in the previous Sectionon ‘covert watermarking’ can be used to diffuse the plain textimagei2 using any other imagei1 to produce the fieldn. Ifboth i1 andn are then encrypted, any attack on these data willnot be able to make use of an ‘analysis cycle’ which is basedon the assumption that the decrypted output is plaintext. Thisapproach provides the user with a relatively simple method of‘confusing’ the cryptanalyst and invalidates attack strategiesthat have been designed and developed on the assumption thatthe encrypted data have been derived from plaintext alone.

VIII. E NTROPY CONSCIOUSCONFUSION AND DIFFUSION

Consider a simple linear array such as a deck of eight cardswhich contains the ace of diamonds for example and wherewe are allowed to ask a series of sequential questions as towhere in the array the card is. The first question we couldask is in which half of the array does the card occur whichreduces the number of cards to four. The second question is inwhich half of the remaining four cards is the ace of diamondsto be found leaving just two cards and the final question iswhich card is it. Each successive question is the same butapplied to successive subdivisions of the deck and in this waywe obtain the result in three steps regardless of where thecard happens to be in the deck. Each question is a binarychoice and in this example, 3 is the minimum number of binarychoices which represents the amount of information requiredto locate the card in a particular arrangement. This is the sameas taking the binary logarithm of the number of possibilities,since log2 8 = 3. Another way of appreciating this result, isto consider a binary representation of the array of cards, i.e.000,001,010,011,100,101,110,111, which requires three digits

or bits to describe any one card. If the deck contained 16cards, the information would be 4 bits and if it contained 32cards, the information would be 5 bits and so on. Thus, ingeneral, for any number of possibilitiesN , the informationIfor specifying a member in such a linear array, is given by

I = − log2 N = log2

1N

where the negative sign is introduced to denote that informa-tion has to be acquired in order to make the correct choice,i.e. I is negative for all values ofN larger than 1. We can nowgeneralize further by considering the case where the numberof choicesN are subdivided into subsets of uniform sizeni. Inthis case, the information needed to specify the membershipof a subset is given not byN but by N/ni and hence, theinformation is given by

Ii = log2 Pi

where Pi = ni/N which is the proportion of the subsets.Finally, if we consider the most general case, where the subsetsare non-uniform in size, then the information will no longerbe the same for all subsets. In this case, we can consider themean information given by

I =N∑

i=1

Pi log2 Pi

which is the Shannon Entropy measure established in hisclassic works on information theory in the 1940s [69]. Infor-mation, as defined here, is a dimensionless quantity. However,its partner entity in physics has a dimension called ‘Entropy’which was first introduced by Ludwig Boltzmann as a measureof the dispersal of energy, in a sense, a measure of disorder,just as information is a measure of order. In fact, Boltzmann’sEntropy concept has the same mathematical roots as Shannon’sinformation concept in terms of computing the probabilities ofsorting objects into bins (a set of N into subsets of sizeni) andin statistical mechanics the Entropy is defined as [70], [71]

E = −k∑

i

Pi lnPi

wherek is Boltzmann’s constant. Shannon’s and Boltzmann’sequations are similar.E and I have opposite signs, butotherwise differ only by their scaling factors and they convertto one another byE = −(k ln 2)I. Thus, an Entropy unitis equal to−k ln 2 of a bit. In Boltzmann’s equation, theprobabilitiesPi refer to internal energy levels. In Shannon’sequationsPi are nota priori assigned such specific roles andthe expression can be applied to any physical system to providea measure of order. Thus, information becomes a conceptequivalent to Entropy and any system can be described interms of one or the other. An increase in Entropy impliesa decrease of information and vise versa. This gives rise tothe fundamental conservation law:The sum of (macroscopic)information change and Entropy change in a given system iszero.

From the point of view of designing an appropriate substitu-tion cipher, the discussion above clearly dictates that the ciphern[i] should be such that the Entropy of the ciphertextu[i] is a


Fig. 11. A 3000 element uniformly distributed random number stream (topleft) and its 64-bin discrete PDF (top right) withI = 4.1825 and a 3000element Gaussian distributed random number stream (bottom left) and its 64-bin discrete PDF (bottom right) withI = 3.2678.

maximum. This requires that a PRNG algorithm be designedthat outputs a number stream whose Entropy is maximum -as large as is possible in practice. The stream should have aPDF Pi that yields the largest possible values forI. Figure11 shows a uniformly distributed and a Gaussian distributedrandom number stream consisting of 3000 elements and thecharacteristic discrete PDFs using 64-bins (i.e. forN = 64).The Information Entropy, which is computed directly from thePDFs using the expression forI given above, is always greaterfor the uniformly distributed field. This is to be expectedbecause, for a uniformly distributed field, there is no biasassociated with any particular numerical range and hence, nolikelihood can be associated with a particular state. Hence,one of the underlying principles associated with the design ofa ciphern[i] is that it should output a uniformly distributedsequence of random numbers. However, this does not meanthat the ciphertext itself will be uniformly distributed since if

u(r) = u0(r) + n(r)

thenPr[u(r)] = Pr[u0(r)]⊗ Pr[n(r)].

This is illustrated in Figure 12 which shows 128-bin his-tograms for a 7-bit ASCII plaintext (the LaTeX file associatedwith this paper)u0[i], a stream of uniformly distributed inte-gersn[i], 0 ≤ n ≤ 127 and the ciphertextu[i] = u0[i]+n[i].The spike associate with the plaintext histogram reflects the‘character’ that is most likely to occur in the plaintext of anatural Indo-European language, i.e. a space with ASCII value32. Although the distribution of the ciphertext is broader thanthe plaintext it is not as broad as the cipher and certainly

Fig. 12. 128-bin histograms for an 7-bit ASCII plaintextu0[i] (left), astream of uniformly distributed integers between 0 and 127n[i] (centre) andthe substitution cipheru[i] (right).

not uniform. Thus, the Entropy of the ciphertext, althoughlarger than the plaintext (in this exampleIu0 = 3.4491 andIu = 5.3200), the Entropy of the ciphertext is still less thatthen that of the cipher (in this exampleIn = 5.5302). Thereare two ways in which this problem can be solved. The firstmethod is to construct a ciphern with a PDF such that

Pn(x)⊗ Pu0(x) = U(x)

whereU(x) = 1, ∀x. Then

Pn(x) = U(x)⊗Q(x)

where

Q(x) = F−11

(1

F [Pu0(x)]

).

But this requires that the cipher is generated in such a waythat its output conforms to an arbitrary PDF as determinedby the plaintext to be encrypted. The second method isbased on assuming that the PDF of all plaintexts will be ofthe form given in Figure 12 with a characteristic dominantspike associated with the number of spaces that occur in theplaintext14. Noting that

Pn(x)⊗ δ(x) = Pn(x)

then as the amplitude of the spike increases, the output increas-ingly approximates a uniform distribution; the Entropy of theciphertext increases as the Entropy of the plaintext decreases.One simple way to implement this result is to pad-out theplaintext with a single character. Padding out a plaintextfile with any character provides a ciphertext with a broaderdistribution, the character ? (with an ASCII decimal integer of63) providing a symmetric result. The statistical effect of thisis illustrated in Figure 13 whereIu0 = 1.1615, In = 5.5308andIu = 5.2537.

IX. STATISTICAL PROPERTIES OF ACIPHER

Diffusion has been considered via the properties associatedwith the homogeneous (classical) diffusion equation and the

14This is only possible provided the plaintext is an Indo-European alpha-numeric array and is not some other language or file format - a compressedimage file, for example.


Fig. 13. 127-bin histograms for an 7-bit ASCII plaintextu0[i] (left) afterspace-character padding, a stream of uniformly distributed integers between0 and 255n[i] (centre) and the substitution cipheru[i] (right).

general Green’s function solution. Confusion has been consid-ered through the application of the inhomogeneous diffusionequation with a stochastic source function and it has beenshown that

u(r) = p(r)⊗r u0(r) + n(r)

where p is a Gaussian Point Spread Function andn is astochastic function.

Diffusion of noise involves the case whenu0 is a stochasticfunction. Diffusion by noise involves the use of a PSFp thatis a stochastic function. Ifu0 is taken to be deterministicinformation, then we can consider the processes of noisediffusion and confusion to be compounded in terms of thefollowing:

Diffusionu(r) = n(r)⊗r u0(r).

Confusionu(r) = u0(r) + n(r).

Diffusion and Confusion

u(r) = n1(r)⊗r u0(r) + n2(r).

The principal effects of diffusion and confusion have beenillustrated using various test images. This has been undertakenfor visual purposes only but on the understanding that such‘effects’ apply to fields in different dimensions in a similarway.

The statistical properties associated with independent ran-dom variables has also bee considered. One of the mostsignificant results associated with random variable theory iscompounded in the Central Limit Theorem. When data isrecorded, the stochastic termn, is often the result of manyindependent sources of noise due to a variety of physical,electronic and measuring errors. Each of these sources mayhave a well-defined PDF but ifn is the result of the additionof each of them, then the PDF ofn tends to be Gaussian dis-tributed. Thus, Gaussian distributed noise tends to be common

in the large majority of applications in whichu is a record ofa physical quantity.

In cryptology, the diffusion/confusion model is used ina variety of applications that are based on diffusion only,confusion only and combined diffusion/confusion models. Onesuch example of the combined model is illustrated in Figure 5which shows how one data field can be embedded in anotherfield (i.e. how one image can be used to watermark anotherimage using noise diffusion). In standard cryptography, oneof the most conventional methods of encrypting informationis through application of a confusion only model. This isequivalent to implementing a model where it is assumed thatthe PSF is a delta function so that

u(r) = u0(r) + n(r).

If we consider the discrete case in one-dimension, then

u[i] = u0[i] + n[i]

whereu0[i] is the plaintext array or just ‘plaintext’ (a streamof integer numbers, each element representing a symbol as-sociated with some natural language, for example),n[i] isthe ‘cipher’ andu[i] is the ‘ciphertext’. Methods are thenconsidered for the generation of stochastic functionsn[i] thatare best suited for the generation of the ciphertext. This isthe basis for the majority of substitution ciphers where eachvalue of each element ofu0[i] is substituted for another valuethrough the addition of a stochastic functionn[i], a functionthat should: (i) include outputs that are zero in order that thespectrum of random numbers is complete15; (ii) have a uniformPDF. The conventional approach to doing this is to designappropriate PRNGs or, as discussed later in this work, pseudochaotic ciphers. In either case, a cipher should be generatedwith maximum Entropy which is equivalent to ensuring thatthe cipher is a uniformly distributed stochastic field. However,it is important to appreciate that the statistics of a plaintextare not the same as those of the cipher when encryption isundertaken using a confusion only model; instead the statisticsare determined by the convolution of the PDF of the plaintextwith the PDF of the cipher. Thus, if

u(r) = u0(r) + n(r)

thenPr[u(r)] = Pr[n(r)]⊗r Pr[u0(r)].

One way of maximising the Entropy ofu is to constructu0

such thatPr[u0(r)] = δ(r). A simple and practical methodof doing this is to pad the datau0 with a single element thatincrease the data size but does not intrude on the legibility ofthe plaintext.

Assuming that the encryption of a plaintextu0 is undertakenusing a confusion only model, there exist the possibility ofencrypting the ciphertext again. This is an example of doubleencryption, a process that can be repeated an arbitrary number

15The Enigma cipher, for example, suffered from a design fault with regardto this issue in that a letter could not reproduce its self -u[i] 6= u0[i]∀i. Thisprovided a small statistical bias which was nevertheless significant in thedecryption of Enigma ciphers.


of times to give triple and quadruple encrypted outputs.However, multiple encryption procedures in which

u(r) = u0(r) + n1(r) + n2(r) + ...

wheren1, n2,... are different ciphers, each consisting of uni-formly distributed noise, suffer from the fact that the resultantcipher is normally distributed because, from the Central LimitTheorem

Pr[n1 + n2 + ...] ∼ Gauss(x).

For this reason, multiple encryption systems are generally notpreferable to single encryption systems. A notable exampleis the triple DES (Data Encryption Standard) or DES3 system[72] that is based on a form of triple encryption and originallyintroduced to increase the key length associated with thegeneration of a single ciphern1. DES3 was endorsed by theNational Institute of Standards and Technology (NIST) as atemporary standard to be used until the Advanced EncryptionStandard (AES) was completed in 2001 [73].

The statistics of an encrypted field formed by the diffusionof u0 (assumed to be a binary field) with noise produces anoutput that is Gaussian distributed, i.e. if

u(r) = n(r)⊗r u0(r)

then

Pr[u(r)] = Pr[n(r)⊗r u0(r)] ∼ Gauss(x).

Thus, the diffusion ofu0 produces an output whose statisticsare not uniform but normally distributed. The Entropy of adiffused field using uniformly distributed noise is thereforeless than the Entropy of a confused field. It is for this reason,that a process of diffusion should ideally be accompanied bya process of confusion when such processes are applied tocryptology in general.

The application of noise diffusion for embedding or water-marking one information field in another is an approach thathas a range of applications in covert ciphertext transmission.However, since the diffusion of noise by a deterministicPSF produces an output whose statistics tend to be normallydistributed, such fields are not best suited for encryption.However, this process is important in the design of stochasticfields that have important properties for the camouflage ofencrypted data.

X. I TERATED FUNCTION SYSTEMS AND CHAOS

In cryptography, the design of specialized random numbergenerators with idealized properties forms the basis of many ofthe algorithms that are applied. Although the type of randomnumber generators considered so far are of value in the gen-eration of noise fields, the properties of these algorithms arenot well suited for cryptography especially if the cryptosystemis based on a public domain algorithm. This is because it isrelatively easy to apply brute force attacks in order to recoverthe parameters used to ‘drive’ a known algorithm especiallywhen there is a known set of rules required to optimise thealgorithm in terms of parameter specifications. In generalstream ciphers typically use an iteration of the type

xi+1 = f(xi, p1, p2, ...)

wherepi is some parameter set (e.g. prime numbers) andx0 isthe key. The cipherx, which is usually of decimal integer type,is then written in binary form (typically using ASCII 7-bitcode) and the resulting bit stream used to encrypt the plaintext(after conversion to a bit stream with the same code) using anXOR operation. The output bit stream can then be convertedback to ASCII ciphertext form as required. Decryption isthen undertaken by generating the same cipher (for the samekey) and applying an XOR operation to the ciphertext (binarystream). The encryption/decryption procedure is thus of thesame type and attention is focused on the characteristics ofthe algorithm that is used for computing the cipher. How-ever, whatever algorithm is designed and irrespective of its‘strength’ and the length of the key that is used, in all cases,symmetric systems require the users to exchange the key. Thisrequires the use of certainkey exchange algorithms. Streamciphers are essential Vernam type ciphers which encrypt bitstreams on a bit by bit basis. By comparison block ciphersoperate of blocks of the stream and may apply permutationsand shifts to the data which depend on the key used. In thissection we provide the foundations for the use of IFS forgenerating Vernam ciphers that are constructed from randomlength blocks of data that are based on the application differentIFS.

A. Background to Chaos

The word Chaos appeared in early Greek writings anddenoted either the primeval emptiness of the universe beforethings came into being or the abyss of the underworld. Bothconcepts occur in the Theogony of Hesiod16. This concept tiedin with other early notions that saw in Chaos the darkness ofthe underworld. In later Greek works, Chaos was taken todescribe the original state of things, irrespective of the waythey were conceived. The modern meaning of the word isderived from Ovid (Publius Ovidius Naso - known to theEnglish speaking world asOvid), a Roman poet (43BC -17AD) and a major influence in early Latin literature, whosaw Chaos as the original disordered and formless mass, fromwhich the ordered universe was derived.

The modern notion of chaos - apart from being a term todescribe a mess - is connected with the behaviour of dynamicalsystems that appear to exhibit erratic and non-predictablebehaviour but, on closer ‘inspection’, reveal properties thathave definable ‘structures’. Thus, compared with the originalGreek concept of chaos, chaotic systems can reveal order,bounded forms and determinism, a principal feature being theirself-organisation and characterisation in terms of self-affinestructures. This aspect of chaos immediately suggests thatchaotic systems are not suitable for applications to cryptogra-phy which requires ciphers that have no predictable dynamicbehaviour or structure of any type, e.g. pseudo random numberstreams that are uniformly distributed with maximum entropy.However, by applying appropriate post-conditioning criterionto a pseudo chaotic number stream, a cipher can be designedthat has the desired properties.

16Hesiod, 700 BC, one of the earliest Greek poets. His epic ‘Theogony’describes the myths of the gods.


The idea that a simple nonlinear but entirely deterministicsystems can behave in an apparently unpredictable and chaoticmanner was first noticed by the great French mathematicianHenri Poincare in the late Nineteenth Century. In spite ofthis, the importance of chaos was not fully appreciated untilthe widespread availability of digital computers for numericalsimulations and the demonstration of chaos in various physicalsystems. In the early 1960s, the American mathematician,Edward Lorenz re-discovered Poincare’s observations whileinvestigating the numerical solution of a system of non-linearequations used to model atmospheric turbulence, equationsthat are now known as the Lorenz equations.

A primary feature of chaotic systems is that they exhibitself-affine structures when visualised and analysed in an ap-propriate way, i.e. an appropriate phase space. In this sense, thegeometry of a chaotic system may be considered to be fractal.This is the principal feature that provides a link betweenchaotic dynamics and fractal geometry.

A key feature of chaotic behavior in different systems isthe sensitivity to initial conditions. Thus,‘It may happen thatsmall differences in the initial conditions produce very greatones in the final phenomena. A small error in the formerwill produce an enormous error in the future. Predictionbecomes impossible’(Edward Lorenz17). This aspect of achaotic system is ideal for encryption in terms of the diffusionrequirement discussed earlier, i.e. that a cryptographic systemshould be sensitivity to the initial conditions (i.e. the key)that is applied. However, in a more general context, thesensitivity to initial conditions of chaotic systems theory is animportant aspect of using the theory to develop a mathematicaldescription of complex phenomena such a Brownian andfractaional Brownian process, weather changes in meteorologyor population fluctuations in biology. The relative success ofchaos theory for modelling complex phenomena has caused animportant paradigm shift that has provided the first ‘scientific’explanation for the coexistence of such concepts as law anddisorder, determinism and unpredictability.

Formally, chaos theory can be defined as the study ofcomplex nonlinear dynamic systems. The word ‘complex’ isrelated to the recursive and nonlinear characteristics of thealgorithms involved, and word ‘dynamic’ implies the non-constant and non-periodic nature of such systems. Chaoticsystems are commonly based on recursive processes, either inthe form of single or coupled algebraic equations or a set of(single or coupled) differential equations modeling a physicalor virtual system.

Chaos is often but incorrectly associated with noise inthat it is taken to represent a field which is unpredictable.Although this is the case, a field generated by a chaotic systemgenerally has more structure if analysed in an appropriateway, a ‘structure’ that may exhibits features that are similarat different scales. Thus, chaotic field are not the same asnoise fields either in terms of their behaviour or the way inwhich they are generated. Simple chaotic fields are typicallythe product of an iteration of the formxi+1 = f(xi) where thefunction f is some nonlinear map which depends on a single

17Cambel, A B, ‘Applied Chaos Theory’, Gorman, 2000.

or a set of parameters. The chaotic behaviour ofxi dependscritically of the value of the parameter(s). The iteration processmay not necessarily be a single nonlinear mapping but consistof a set of nonlinear coupled equations of the form

x(1)i+1 = f1(x

(1)i , x

(2)i , ..., x

(N)i ),

x(2)i+1 = f2(x

(1)i , x

(2)i , ..., x

(N)i ),

...

x(N)i+1 = fN (x(1)

i , x(2)i , ..., x

(N)i )

where the functionsf1, f2, ..., fN may all be nonlinear ornonlinear and linear. In turn, such a coupled system can bethe result of many different physical models covering a widerange of applications in science and engineering.

B. Vurhulst Processes and the Logistic Map

Suppose there is a fixed population ofN individuals livingon an island (with no one leaving or entering) and a fataldisease (for which there is no cure) is introduced, whichis spread through personal contact causing an epidemic tobreak out. The rate of growth of the disease will normally beproportional to the number of carriersc say. Suppose we letx = c/N be the proportion of individuals with the disease sothat100x is the percentage of the population with the disease.Then, the equation describing the rate of growth of the diseaseis

dx

dt= kx

whose solution is

x(t) = x0 exp(kt)

where x0 is the proportion or the population carrying thedisease att = 0 (i.e. when the disease first ‘strikes’) andk is a constant of proportionality defining the growth rate.The problem with this conventional growth rate model, isthat whenx = 1, there can be no further growth of thedisease because the island population no longer exists andso we must impose the condition that0 < x(t) ≤ 1, ∀t.Alternatively, suppose we include the fact that the rate ofgrowth must also be proportional to the number of individuals1 − x who do not become carriers, due to isolation of theiractivities and/or genetic disposition, for example. Then, ourrate equation becomes

dx

dt= kx(1− x)

and if x = 1, the epidemic is extinguished. This equationcan be used to model a range of situations similar to thatintroduced above associated with predator-prey type processes.(In the example given above, the prey is the human and thepredator could be a virus or bacterium, for example). Finitedifferencing over a time interval∆t, we have

xi+1 − xi

∆t= kxi(1− xi)

orxi+1 = xi + k∆txi(1− xi)


orxi+1 = rxi(1− xi)

wherer = 1+k∆t. This is a simple quadratic iterator knownas the logistic map and has a range of characteristics dependingon the value ofr. This is illustrated in Figure 14 whichshows the output (for just 30 elements) from this iteratorfor r = 1, r = 2, r = 3 and r = 4 and for an initialvalue of 0.118. For r = 1 and r = 2, convergent behaviour

Fig. 14. Output (30 elements) of the logistic map for values ofr = 1 (topleft), r = 2 (top right),r = 3 (bottom left) andr = 4 (bottom right) and aninitial value of 0.1.

takes place; forr = 3 the output is oscillatory and forr = 4 the behaviour is chaotic. The transition from monotonicconvergence to oscillatory behaviour is known as a bifurcationand is better illustrated using a so called Fiegenbaum mapor diagram which is a plot of the output of the iterator interms of the values produced (after iterating enough timesto produce a consistent output) for different values ofr. Anexample of this for the logistic map is given in Figure 15for 0 < r ≤ 4 and shows convergent behaviour for valuesof r from 0 to approximately 3, bifurcations for values ofrbetween approximately 3 and just beyond 3.5 and then a regionof chaotic behaviour, achieving ‘full chaos’ atr = 4 where,in each case, the output consists of values between 0 and 1.However, closer inspection of this data representation revealsrepeating patterns, an example being given in Figure 16 whichis a Fiegenbaum diagram of the output for values ofr between3.840 and 3.855 and values ofx between 0.44 and 0.52. Asbefore, we observe a region of convergence, bifurcation andthen chaos. Moreover, from Figure 16 we observe anotherregion of this map (for values ofr around 3.854) in which thissame behaviour occurs. The interesting feature about this mapis that the convergence→bifurcation→chaos characteristics arerepeated albeit at smaller scales. In other words, there isa similarity of behaviour at smaller scales, i.e. the pattern

18The initial value, which is taken to be any value between 0 and 1, changesthe ‘signature’ of the output but not its characteristics, at least, in the idealcase.

Fig. 15. Feigenbaum diagram of the logistic map for0 < r < 4 and0 < x < 1.

Fig. 16. Feigenbaum diagram of the logistic map for3.840 < r < 3.855and0.44 < x < 0.52.

of behaviour. Further, this complex behaviour comes from aremarkably simple iterator, i.e. the mapx → rx(1− x).

C. Examples of Chaotic Systems

In addition to the logistic map, which has been used inthe previous section to introduce a simple IFS that givesa chaotic output, there are a wide variety of other mapswhich yield signals that exhibit the same basic propertiesas the logistic map (convergence→bifurcation→chaos) withsimilar structures at different scales at specific regions ofthe Feigenbaum diagram. Examples, include the ‘maps’ givenbelow:

1) Linear functions:The sawtooth map

xi+1 = 5ximod4.

The tent map

xi+1 = r(1− | 2xi − 1 |).


The generalized tent map

xi+1 = r(1− | 2xi − 1 |m), m = 1, 2, 3, ...

2) Nonlinear functions:The sin map

xi+1 =| sin(πrxi | .

The tangent feedback map

xi+1 = rxi[1− tan(xi/2)].

The logarithmic feedback map

xi+1 = rxi − [1− log(1 + xi)].

Further, there are a number of ‘variations on a theme’ that areof value, an example being the ‘delayed logistic map’

xi+1 = rxi(1− xi−1)

which arises in certain problems to population dynamics.Moreover, coupled iterative maps occur from the developmentof physical models leading to nonlinear coupled differentialequations, a famous and historically important example beingthe Lorenz equations given by

dx1

dt= a(x2−x1),

dx2

dt= (b−x3)x1−x2,

dx3

dt= x1x2−cx3

wherea, b andc are constants. These equations were originallyderived by Lorenz from the fluid equations of motion (theNavier Stokes equation, the equation for thermal conductivityand the continuity equation) used to model heat convection inthe atmosphere and were studied in an attempt to explore thetransition to turbulence where a fluid layer in a gravitationalfield is heated from below. By finite differencing, these equa-tions, we convert the functionsxi, i = 1, 2, 3 into discreteform xn

i , i = 1, 2, 3 giving (using forward differencing)

x(n+1)1 = x

(n)1 + ∆ta(x(n)

2 − x(n)1 ),

x(n+1)2 = x

(n)2 + ∆t[(b− x

(n)3 )x(n)

1 − x(n)2 ],

x(n+1)3 = x

(n)3 + ∆t[x(n)

1 x(n)2 − cx

(n)3 ].

For specific values ofa, b and c (e.g. a = 10, b = 28 andc = 8/3) and a step length∆t, the digital signalsx(n)

1 , x(n)2

and x(n)3 exhibit chaotic behaviour which can be analysed

quantitatively in the three dimension phase space(x1, x2, x3)or variations on this theme, e.g. a three dimensional plot withaxes(x1 +x2, x3, x1−x3) or as a two dimensional projectionwith axes(x1+x2, x3) an example of which is shown in Figure17. Here, we see that the path is confined to two domainswhich are connected. The path is attracted to one domain andthen to another but this connection (the point at which the pathchanges form one domain to the next) occurs in an erratic way- an example of a ‘strange attractor’.

As with the simple iterative maps discussed previously,there are a number of nonlinear differential equations (coupledor otherwise) that exhibit chaos whose behaviour can bequantified using an appropriate phase space. These in include:

The Rossler equations

dx1

dt= −x2 − x3,

dx2

dt= x3 + ax2,

dx3

dt= b + x3(x1 + c).

Fig. 17. Two dimensional phase space analysis of the Lorenz equationsillustrating the ‘strange attractor’.

The Henon-Heiles equations

dx1

dt= px,

dpx

dt= −x−2xy;

dx2

dt= py,

dpy

dt= −y−x2+y2.

The Hill’s equations

d2

dt2x(t) + Ω2(t)x(t) = 0,

a special case being the Mathieu equation when

Ω2(t) = ω20(1 + λ cos ωt),

ω0 andλ being constants. The Duffing Oscillator

dx

dt= v,

dv

dt= av + x + bx3 + cos t

where a and b are constants. The non-linear Schrodingerequation

d2

dt2x(t) + ω2x(t) =| x(t) |2 x(t).

In each case, the chaotic nature of the output to these systemsdepends on the values of the constants.

For iterative processes where stable convergent behaviour isexpected, an output that is characterised by exponential growthcan be taken to be due to unacceptable numerical instability.However, with IFS that exhibit intrinsic instability, in that theoutput does not converge to a specific value, the Lyapunovexponent is used to quantify the characteristics of the output.This exponent or ‘Dimension’ provides a principal measure of‘chaoticity’ and is derived in Appendix II.

XI. ENCRYPTION USINGDETERMINISTIC CHAOS

The use of chaos in cryptology was first considered in theearly 1950s by the American electrical engineer Claude Shan-non and the Russian mathematician Vladimir AlexandrovichKotelnikov who laid the theoretical foundations for moderninformation theory and cryptography. It was Shannon whofirst explicitly mentioned the basic stretch-and-fold mechanism


associated with chaos for the purpose of encryption:Good mix-ing transformations are often formed by repeated products oftwo simple non-commuting operations[11]. Hopf19 consideredthe mixing of dough by such a sequence of non-commutingoperations. The dough is first rolled out into a thin slab, thenfolded over, then rolled, and then folded again and so on. Thesame principle is used in the making of a Japanese sword,the aim being to produce a material that is a highly diffusedversion of the original material structure.

The use of chaos in cryptography was not fully appre-ciated until the late 1980s when the simulation of chaoticdynamical systems became common place and when the roleof cryptography in IT became increasingly important. Sincethe start of the 1990s, an increasing number of publicationshave considered the use of chaos in cryptography, e.g. [74]-[78]. These have included schemes based on synchronizedchaotic (analogue) circuits, for example, which belong tothe field of steganography and secure radio communication[79]. Over the 1990s cryptography started to attract a varietyof scientists and engineers from diverse fields who startedexploiting dynamical systems theory for the purpose of en-cryption. This included the use of discrete chaotic systemssuch as the cellular automata, Kolmogorov flows and discreteaffine transformations in general to provide more efficientencryption schemes [80]-[83]. Since 2000, the potential ofchaos-based communications, especially with regard to spreadspectrum modulation, has been recognized. Many authors havedescribed chaotic modulations and suggested a variety ofelectronics based implementations, e.g. [76]-[79]. However,the emphasis has been on information coding and informationhiding and embedding. Much of this published work has beenof theoretical and some technological interest with work beingundertaken in both an academic and industrial research context(e.g. [84]-[90]). However, it is only relatively recently thatthe application of chaos-based ciphers have been implementedin software and introduced to the market. One example ofthis is the basis of the authors own company - CrypsticTM

Limited - in which the principle of multi-algorithmicity usingchaos-based ciphers [11], [91] has been use to produce meta-encryption engines that are mounted on a single, a pair or agroup of flash (USB - Universal Serial Bus) memory sticks.Some of these memory sticks have been designed to includea hidden memory accessible through a covert procedure (suchas the renaming - by delation - of an existing file or folder)from which the encryption engine(s) can be executed.

Consider an algorithm that outputs a number stream whichcan be ordered, chaotic or random. In the case of an orderednumber stream (those generated from a discretized piecewisecontinuous functions for example), the complexity of the fieldis clearly low. Moreover, the information and specifically theinformation entropy (the lack of information we have about theexact state of the number stream) is low as is the informationcontent that can be conveyed by such a number stream.

A random number stream (taken to have a uniform PDF,

19Hopf, Eberhard F. F, (1902-1983), an Austrian mathematician who madesignificant contributions in topology and ergodic theory and studied the mixingin compact spaces, e.g.On Causality, Statistics and Probability, Journal ofMathematics and Physics, 13, 51-102, 1934.

for example) will provide a sequence from which, under idealcircumstances, it is not possible to predict any number in thesequence from the previous values. All we can say is thatthe probability of any number occurring between a specifiedrange is equally likely. In this case, the information entropyis high. However, the complexity of the field, in terms erratictransitions from one type of localized behaviour to another,is low. Thus, in comparison to a random field, a chaoticfield is high in complexity but its information entropy, whilenaturally higher than an ordered field is lower than that of arandom field, e.g. chaotic fields which exhibit uniform numberdistributions are rare. Such fields therefore need to be post-processed in order that the output conforms to a uniformdistribution.

We consider a dynamic continuous-state continuous-timesystemS = 〈X,K, f〉 as follows:

dx

dt= f (x, k) , x ∈ X, k ∈ K

wheref is a a smooth function,X is a state space andK is aparameter space. The equation is taken to satisfy the conditionsof the existence and uniqueness of solutionsx(x0, t) with theinitial condition x0 = x (x0, 0), the solution curveϕt (x0, t)being the trajectory.

For cryptography, we focus on dynamic discrete-time sys-tems which can be written in the following form:

xi+1 = f (xi, k) , xi ∈ X, k ∈ K, i = 0, 1, 2, . . .

where xi is a discrete state of the system. The trajectoryϕ (xi, x0) is defined by the sequencex0, x1, x2, . . .. Thisequation is similar to the cryptographic iterated functionsused for pseudo random number generation, block ciphers andother constructions such as the DES, RSA nd AES ciphers.Consequently, in both nonlinear dynamics and cryptographywe deal with an iterated key-dependent transformation ofinformation. There are several sufficient conditions satisfiedby a dynamic system to guarantee chaos; the sensitivity toinitial conditions and topological transitivity being the mostcommon.

A chaotic continuous-state discrete-time system is a dy-namic systemS = 〈X, f〉 with two properties [?]: (i) given ametric spaceX and a mappingf : X → X, we say thatf istopologically transitive onX if, for any two open setsU, V ⊂X, there isn ≥ 0 such thatfn(U)∩V 6= ∅; (ii) the mapf issaid to be sensitive to initial conditions if there isδ > 0, n ≥ 0given that for anyx ∈ X and for any neighborhoodHx ofx there isy ∈ Hx, such that|fn(x)− fn(y)| > δ. Theseproperties can be interpreted as follows: a dynamic systemis chaotic if all trajectories are bounded (by the attractor)and nearby trajectories diverge exponentially at every pointof the phase space. The trajectories are continuous and belongto a two-dimensional system that is said to be chaotic. Thisyields to a natural synergy between chaotic and cryptographicsystems that can be described in terms of the following: (i)topological transitivity which ensures that the system outputcovers all the state space, e.g. any plaintext can be encryptedinto any ciphertext; (ii) sensitivity to initial condition which


corresponds to Shannon’s original requirements for an encryp-tion system in the late 1940s. In both chaos and cryptographywe are dealing with systems in which a small variation of anyvariable changes the outputs considerably.

A. Stream Cipher Encryption

The use of discrete chaotic fields for encrypting data canfollow the same basic approach as used with regard to theapplication of pseudo random number generating algorithmsfor stream ciphers. Pseudo chaotic numbers are in principle,ideal for cryptography because they produce number streamsthat are ultra-sensitive to the initial value (the key). However,instead of using iterative based maps using modular arithmeticwith integer operations, here, we require the application ofnonlinear maps using floating point arithmetic. Thus, the firstdrawback concerning the application of deterministic chaosfor encryption concerns the processing speed, i.e. pseudo ran-dom number generators typically output integer streams usinginteger arithmetic whereas pseudo chaotic number generatorsproduce floating point streams using floating point arithmetic.Another drawback of chaos based cryptography is that thecycle length (i.e. the period over which the number streamrepeats itself) is relatively short and not easily quantifiablewhen compared to the cycle length available using conven-tional PRNGs, e.g. additive generators, which commence byinitialising an arrayxi with random numbers (not all of whichare even) so that we can consider the initial state of thegenerator to bex1, x2, x3, ... to which we then apply

xi = (xi−a + xi−b + ... + xi−m)mod2n

wherea, b, ...,m andn are assigned integers20, have very longcycle lengths of the order of2f (255 − 1) where0 ≤ f ≤ nand linear feedback shift registers with the form

xn = (c1xn−1 + c2xn−2 + cmxn−m)mod2k

which, for specific values ofc1, c2, ...cm have cycle lengthsof 2k.

The application of deterministic chaos to encryption has twodistinct disadvantages relative to the application of PRNGs.Another feature of IFS is that the regions over which chaoticbehaviour can be generated may be limited. However, thislimitation can be overcome by designing IFS with the specificaim of increasing the range of chaos. One method is to usewell known maps and modify them to extend the region ofchaos. For example, the Matthews cipher is a modification ofthe logistic map to [92]

xi+1 = (1 + r)(

1 +1r

)r

xi(1− xi)r, r ∈ (0, 4].

The effect of this generalization is seen in Figure 18 whichshows the Feigenbaum diagram for values ofr between 1 and4. Compared to the conventional logistic mapxi+1 = rxi(1−xi), r ∈ (0, 4] which yields full chaos atr = 4, the chaoticbehaviour of the Matthews map is clearly more extensiveproviding full chaos for the majority (but not all) of values of

20A well known example is the ‘Fish generator’xi = (xi−55 +xi−24)mod232

Fig. 18. Feigenbaum map of the Matthews cipher

r between approximately0.5 and4. In the conventional case,the key is the value ofx0 (the initial condition). In addition,because there is a wide range of chaotic behaviour for theMatthews map, the value ofr itself can be used as a primary(or secondary) key.

The approach to using deterministic chaos for encryptionhas to date, been based on using conventional and otherwell known chaotic models of the type discussed above withmodifications such as the Matthew map as required. However,in cryptography, the physical model from which a chaoticmap has been derived is not important; only the fact thatthe map provides a cipher that is ‘good’ at scrambling theplaintext in terms of diffusion and confusion. This point leadsto an approach which exploits two basic features of chaoticmaps: (i) they increase the complexity of the cipher; (ii)there are an unlimited number of maps of the formxi+1 =f(xi), for example, that can be literally ‘invented’ and thentested for chaoticity to produce a data base of algorithms.However, it is important to stress that such ciphers, once‘invented’, needs to be post-processed to ensure that the cipherstream is uniformly distributed which, in turn, requires furthercomputational overheads and, as discussed in the followingsection, may include significant cipher redundancy.

The low cycle lengths associated with chaotic iterators canbe overcome by designing block ciphers where the iteratorproduces a cipher stream only over a block of data whoselength is significantly less than that of the cycle length ofthe iterator, each block being encrypted using a differentkey and/or algorithm. The use of different algorithms forencrypting different blocks of data provides an approach thatis ‘multi-algorithmic’.

B. Block Cipher Encryption and Multi-algorithmicity

Instead of using a single algorithm (such as a Matthewscipher) to encrypt data over a series of blocks using different(block) keys, we can use different algorithms, i.e. chaoticmaps. Two maps can be used to generate the length of each


block and the maps that are used to encrypt the plaintextover each block. Thus, suppose we have designed a database consisting of 100 chaotic maps, say, consisting of IFSf1, f2, f3, ..., f100, each of which generates a floating pointnumber steam through the operation

xi+1 = fm(xi, p1, p2, ...)

where the parametersp1, p2, ... are pre-set or ‘hard-wired’to produce chaos for any initial valuex0 ∈ (0, 1) say. An‘algorithm selection key’ is then introduced in which twoalgorithms (or possibly the same algorithm) are chosen to‘drive’ the block cipher -f50 and f29 say, the key in thiscase being(50, 29). Here, we shall consider the case wheremap f50 determines the algorithm selection and mapf29

determines the block size. Mapf50 is then initiated with thekey 0.26735625 say and mapf29 with the key 0.65376301 say.The output from these maps (floating point number streams)are then normalized, multiplied by 100 and 1000, respectively,for example, and then rounded to produce integer streams withvalues ranging from 1 to 100 and 1 to 1000, respectively. Letus suppose that the first few values of these integer streamsare28, 58, 3, 61 and202, 38, 785, 426, respectively. The blockencryption starts by using map 28 to encrypt 202 elements ofthe plaintext using the key 0.78654876 say. The second blockof 38 elements is then encrypted using map 58 (the initialvalue being the last floating point value produced by algorithm28) and the third block of 785 elements is encrypted usingalgorithm 3 (the initial value being the last floating point valueproduced by algorithm 58) and so on. The process continuesuntil the plaintext has been fully encrypted with the ‘sessionkey’ (50,29,0.26735625,0.65376301,0.78654876).

Encryption is typically undertaken using a binary repre-sentation of the plaintext and applying an XOR operationusing a binary representation of the cipher stream. This can beconstructed using a variety of ways. For example, one couldextract the last significant bits from the floating point formatof xi. Another approach, is to divide the floating point rangeof the cipher into two compact regions and apply a suitablethreshold. For example, suppose that the outputxi from a mapoperating over a given block consists of floating point valuebetween0 and1, then, with the application of a threshold of0.5, we can consider generating the bit stream

b(xi) =

1, xi ∈ (0.5, 1];0, xi ∈ [0, 0.5).

However, in applying such a scheme, we are assuming thatthe distribution ofxi is uniform and this is rarely the casewith chaotic maps. Figure 19 shows the PDF for the logisticmap xi+1 = 4xi(1 − xi) which reveals a non-uniformdistribution with a bias for floating point numbers approaching0 and 1. However, the mid range (i.e. forxi ∈ [0.3, 0.7]) isrelatively flat indicating that the probability for the occurrenceof different numbers generated by the logistic map in the midrange is the same. In order to apply the threshold partitioningmethod discussed above in a way that provides an output thatis uniformly distributed for a any chaotic map, it is necessaryto introduce appropriate conditions and modify the above to

Fig. 19. Probability density function (with 100 bins) of the output from thelogistic map for 10000 iterations.

Fig. 20. Illustration of the effect of using multiple algorithms for generatinga stream cipher on the computationalEnergyrequired to attempt a brute forceattack.

the form

b(xi) =

1, xi ∈ [T, T + ∆+);0, xi ∈ [T −∆−, T );−1, otherwise.

whereT is the threshold and∆+ and ∆− are those valueswhich yield an output stream that characterizes (to a goodapproximation) a uniform distribution. For example, in thecase of the logistic mapT = 0.5 and ∆+ = ∆− =0.2. This aspect of the application of deterministic chaos tocryptography, together with the search for a parameter or setof parameters that provides full chaos for an ‘invented’ map,determines the overall suitability of the function that has been‘invented’ for this application.

The ‘filtering’ of a chaotic field to generate a uniformly dis-tributed output is equivalent to maximizing the entropy of thecipher stream (i.e. generating a cipher stream with a uniformPDF) which is an essential condition in cryptography. In termsof cryptanalysis and attack, the multi-algorithmic approach todesigning a block cipher introduces a new ‘dimension’ to theproblem. The conventional problem associated with an attackon a symmetric cipher is to search for the private key(s) givenknowledge of the algorithm. Here, the problem is to searchnot only for the session key(s), but the algorithms they ‘drive’as illustrated in Figure 20.

One over-riding issue concerning cryptology in general, isthat algorithm secrecy is weak. In other words, a cryptographicsystem should not rely of the secrecy of its algorithms and


all such algorithms should be openly published21. The systemdescribed here is multi-algorithmic, relying on many differentchaotic maps to encrypt the data. Here, publication of thealgorithms can be done in the knowledge that many more mapscan be invented as required (subject to appropriate conditionsin terms of generating a fully chaotic field with a uniform PDF)by a programmer, or possibly with appropriate ‘training’ of adigital computer.

XII. C RYPSTICTM

CrypsticTM is the trade mark for a USB based product thatcurrently uses three approaches for providing secure mobileinformation exchange: (i) obfuscation; (ii) disinformation; (iii)multi-algorithmic encryption using chaos22. The product hasbeen designed for floating point computations with 32-bitprecision operating on PC platforms with an XP or Vistaenvironment.

A. Obfuscation and Disinformation

Obfuscation is undertaken by embedding the application(the .exe file that performs the encryption/decryption) in anenvironment (i.e. the USB memory) that contains a wealthof data (files and folders etc.) that is ideally designed toreflet the users portfolio. This can includes areas that arepassword protected and other public domain encryption sys-tems with encrypted files as required that may be brokenand even generate apparently valuable information (given asuccessful attack) but are in fact provided purely as a formof disinformation. This environment is designed in order toprovide a potential attacker, who has gained access to ausers CrypsticTM through theft, for example, with a ‘targetrich’ environment. The rationale associated with the use of aCrypsticTM as a mobile encryption/decryption device followsthat associated with a users management of a key ring. Inother words, it is assumed that the user will maintain andimplement the CrypsticTM in the same way as a conventionalset of keys are used. However, in the case of loss or theft, a newCrypsticTM must be issued which includes a new encryptionengine and under no circumstances is the original CrypsticTM

re-issued. Management of the encryption engines and theirdistribution is of course undertaken by CrypsticTM Limitedwhich maintains a data base of current users and the encryptionengines provided to them in compliance with the RIP Act,2000, Section 49, which deals with the power of disclosure, i.e.for CrypsticTM Limited to provide the appropriate encryptionengine for the decryption of any encrypted data that is underinvestigation by an appropriate authority.

B. Encryption Engine Design

The encryption engine itself can be based on any numberof algorithms, each algorithm having been ‘designed’ with

21Except for some algorithms developed by certain government agencies -perhaps they have something to hide!

22Incorporated under the Companies Act 1985 for England and Wales asa Private Company that is Limited on 19th January, 2005; Company Number5337521.

respect to the required (maximum entropy) performance con-ditions through implementation of the appropriate thresholdparametersT and ∆±. The design is based on applying thefollowing basic steps:

Step 1: Invent a (non-linear) functionf and apply the IFSxi+1 = f(xi, p1, p2, ...)

Step 2: Normalise the output of the IFS so thatx∞ = 1.

Step 3: Graph the outputxi and adjust parametersp1, p2, ...until the output ‘looks’ chaotic.

Step 4:Graph the histogram of the output and observe if thereis a significant region of the histogram over which it is ‘flat’.

Step 5: Set the values of the thresholdsT and∆± based on‘observations’ made in Step 4.

Anlysing of the ISF using a Faigenbaum diagram can also beundertaken but this can be computationally intensive. Further,each ISF can be categorised in terms of paremeters such as theLaypunov Dimension (Appendix II) and information entropy,for example. However, in practice, such parmeters yield littlein terms of the design of an IFS and are primarily ‘academic’.Indeed, the invention and design of such algorithms has acertain ‘art’ to it which improves with experience. It shouldbe noted that many such inventions fail to be of value in thatthe statistics may not be suitable (e.g. the PDF may not beflat enough or is flat only over a vary limited portion of thePDF), chaoticity may not be guaranteed for all values of theseedx0 between 0 and 1 and the numerical performance ofthe algorithm may be poor. The aim is to obtain a simpleIFS that is numerically relatively trivial to compute, has abroad statistical distribution and is valid for all floating pointvalues ofx0 between 0 and 1. Examples of the IFS used forCrypsticTM are given in the following table where the valuesof T , ∆+ and ∆− apply to the normalised output streamgenerated by each function.

Functionf(x) r T ∆+ ∆−rx(1− tan(x/2)) 3.3725 0.5 0.3 0.3rx[1− x(1 + x2)] 3.17 0.5 0.25 0.35

rx[1− x log(1 + x)] 2.816 0.6 0.3 0.2r(1− | 2x− 1 |1.456) 0.9999 0.5 0.3 0.3| sin(πrx1.09778) | 0.9990 0.6 0.25 0.25

The functions given in the table above produce outputs thathave a relatively broad and smooth histogram which can bemade flat by application of the values ofT and ∆± goven.Some functions, however, produce poor characteristic in thisrespect. For example, the function

f(x) = r | 1− tan(sinx) |, r = 1.5

has a highly irregular histogram (see Figure 21) which is notsuitable in terms of applying values ofT and∆± and, as such,is not an appropriate IFS a crypstic application

C. Graphical User Interface

In conventional encryption systems, it is typical to providea Graphical User Interface (GUI) with fields for inputting


Fig. 21. The first 1000 elements forxi+1 = r | 1 − tan(sin xi) |, r =1.5, 0 < x0 < 1 (above) and associated histogram (below).

the plaintext and outputting the ciphertext where the nameof the output (including file extension) is supplied by the user.CrypsticTM outputs the ciphertext by overwriting the inputfile. This allows the file name, including the extension, to beused to ‘seed’ the encryption engine and thus requires that thename of the file remains unchanged in order to decrypt. Theseed is used to initiate the session key discussed in SectionXI(B). The file name is converted to an ASCII 7-bit decimalinteger stream which is then concatenated and the resultingdecimal integer used to seed a hash function whose output isof the form (d, d, f, f, f) whered is a decimal integer andfis a 32-bit precision floating point number between 0 and 1.

The executable file is camouflaged as a.dll file which isembedded in a folder containing many such.dll files. Thereason for this is that the structure a.dll file is close tothat of a .exefile. Nevertheless, this requires that the sourcecode must be written in such a way that all references toits application are void as discussed in Section II(E). Thisincludes all references to the nature of the data processinginvolved including words such asEncrypt and Decrypt, forexample23, so that the compiled file, although camouflaged as a.dll file, is forensically inert to attacks undertaken with systemssuch a WinHEX [93]. In other words, the source code shouldbe written in a way that is ‘incomprehensible’, a conditionthat is consistent with the skills of many software engineers!This must include the development of a run time help facil-ity. Clearly, such criteria are at odds with the ‘conventionalwisdom’ associated with the development of applications butthe purpose of this approach is to develop a forensically inertexecutable file that is obfuscated by the environment in whichit is placed. An example of the GUI is given in Figure 22.

D. Procedure

The approach to loading the application to encrypt/decrypta file is based on renaming the.dll file to an .exefile with agiven name as well as the correct extension. Simply renaminga .dll file in this way can lead to a possible breach of securityby a potential attacker using a key logging system [94]. In oderto avoid such an attack, CrypsticTM uses an approach in whichthe name of the.dll file can be renamed to a.exefile by using

23Words that can be replaced by E and D respectively in a GUI.

Fig. 22. GUI of CrypsticTM encryption application.

a ‘deletion dominant’ procedure. For example, suppose theapplication is calledenigma.exe, then by generating a.dll filecalled enginegmaxindex.dll, renaming can be accomplishedby deleting (in the order given)lld. followed bydni x followedby en followed by g and then inserting a . betweenae andincludinge afterex. A further application is required such thatupon closing the application, the.exefile is renamed back toits original .dll form. This includes ensuring that the time anddate stamps associated with the file are not updated.

The procedure described above is an attampt to obfuscatethe use of passwords which are increasingly open to attack[18]. For example, the Russian based company ElcomsoftLimited recently filed a US patent for a password crackingtechnique that relies on the parallel processing capabilitiesof modern graphics processors. The technique increases thespeed of password cracking by a factor of 25 using a GeForce8800 Ultra graphics card from Nvidia. ‘Cracking times can bereduced from days or hours to minutes in some instances andthere are plans to introduce the technique into password crack-ing products’ (http://techreport.com/discussions.x/13460).

E. Protocol

CrypsticTM is a symmetric encryption system that relies onthe user working with a USB memory stick and maintaininga protocol that is consistent with the use of a conventionalset of keys, typically located on a key ring. The simplestuse of CrysticTM is for a single user to be issued with aCrypsticTM which incorporates an encryption engine that isunique (through the utilisation of a unique set of algorithmswhich is registered with CrypsticTM Limited for a given user).The user can then use the CrypsticTM to encrypt/decrypt filesand/or folders (after application of a compression algorithmsuch aspkzip, for example) on a PC before closure of asession. In this way, the user maintains a secure environmentusing a unique encryption engine with a ‘key’ that includes acovert access route, coupled with appropriate disinformationas discussed in previous sections. Different encryption enginescan be incorporated that are used to encrypt disinformation inorder to provide a solution to the ‘gun to the head problem’as required.

In the case of communications between ‘Alice and Bob’,both users are issued with Crystics that have encryptionengines unique to Alice and Bob, each of whom can use thefacility for personal data security as above, and, in addition,can encrypt files for email communications. If any stick, by


any party, is lost, then a new pair of Crypstics are issued withnew encryption engines unique to both parties. In addition toa two-party user system, Crypstics can be issued to groups ofusers in a way that provides an appropriate access hierarchyas required.

XIII. D ISCUSSION

The material discussed in this paper has covered some ofthe basic principles associated with cryptography in general,including the role of diffusion and confusion for designingciphers that have no statistical bias. This has been used as aguide in the design of ciphers that are based on the applicationof IFS exhibiting chaotic behaviour. The use of IFS allow forthe design of encryption engines that are multi-algorithmic,each algorithm being based on an IFS that is invented, subjectto the condition, that the output stream has a uniform PDF.The principle of multi-algorithmicity has been used to developa new product - CrypsticTM - that is based on the following:(i) a multi-algorithmic block encryption engine consisting ofa unique set of IFS; (ii) maximum entropy conversion to a bitstream cipher; (iii) a key that is determined by the file name tobe encrypted/decrypted. The approach has passed all statisticaltests [11] recommended by National Institute of Standards andTechnology (NIST) [95].

Access and use of the encryption engine is based onutilizing an commercial-off-the-shelf USB flash memory viaa combination of camouflage, obfuscation and disinformationin order to elude any potential attacker. The approach hasbeen based on respecting the following issues: (i) security is aprocess not a product; (ii) never underestimate the enemy; (iii)the longer that any cryptosystem, or part thereof, remains ofthe same type with the same function, the more vulnerablethe system becomes to a successful attack. Point (iii) is asingularly important feature which CrypsticTM overcomes byutilizing a dynamic approach to the design and distribution ofencryption engines.

A. Chaos Theory and Cryptography

We have discussed cryptography in the context of chaostheory and there is clearly a fundamental relationship betweencryptography and chaos. In both cases, the object of study isa dynamic system that performs an iterative nonlinear trans-formation of information in an apparently unpredictable butdeterministic manner. In terms of chaos theory, the sensitivityto the initial conditions together with the mixing propertyensures cryptographic confusion and diffusion, as originallysuggested by Shannon. However, there are also a numberof conceptual differences: (i) chaos theory studies dynamicsystems defined on an infinite state space (e.g. vectors ofreal numbers or infinite binary strings), whereas cryptographyrelies on a finite-state machine and all chaos models imple-mented on a computer are approximations, i. e. pseudo-chaos;(ii) chaos theory studies the asymptotic behaviour of a nonlin-ear system (the behaviour as the number of iteration approachinfinity when the Lyapunov dimension can be quantified - seeAppendix II in which the definition of the Lyapunov dimensionis based onN → ∞), whereas cryptography focuses on the

effect of a small number of iterations; (iii) chaos theory is notconcerned with the algorithmic complexity of the IFS, whilein cryptography, complexity is the key issue; in other words,the concepts of cryptographic security and efficiency have nocounterparts in chaos theory; (iv) classical chaotic systemshave visually recognizable attractors where as in cryptography,we attempt to hide any visible structure; (v) chaos theory isoften associated with the mathematical model used to quantifya physically significant problem, whereas in cryptography,the physical model is of no importance; (vi) unlike chaos ingeneral, cryptographic systems use a combination of all inde-pendent variables to provide an output that is unpredictableto an observer. The following table provides a comparisonbetween chaos theory and cryptography in terms of thoseaspects of the two subjects that have been considered in hispaper.

Chaos Theory CryptographyChaotic system Pseudo-chaotic systemNonlinear transform Nonlinear transformInfinite number of states Finite statesInfinite number of iterations Finite iterationsInitial state PlaintextFinal state CiphertextInitial condition(s) keyand/or parameter(s)Asyptotic independenece of Confusioninitial and final statesSensitivity to initial Diffusioncondition(s) andparameter(s) mixing

Chaotic systems are algorithmically random and thus cannotbe predicted by a deterministic Turing machine even withinfinite power. However, chaotic systems are predictable bya probabilistic Turing machine and thus, finding probabilisti-cally unpredictable chaotic systems is a central problem forchaos based cryptography. In this paper, the generation of anunpredictable cipher has been undertaken by filtering thosenumbers that belong to a uniform partition of the PDF. Thisapproach comes at the expense of numerical performance sincea relatively large percentage of the floating point numbers thatare computed are discarded.

Chaos theory is not related to number theory in the sameway as conventional cryptographic algorithms nor is chaostheory related to the computational complexity analysis thatunderpins digital cryptography. Hence, neither chaos, norpseudo-chaos can guarantee pseudo-randomness and resistanceto different kinds of cryptanalysis based on conventional senar-ios. The use of floating-point arithmetic is the most obvioussolution of approximating continuous chaos on a finite-statemachine. However, there is no straightforward applicationto pseudo-random number generation and cipher generation.Critical problems can include: (i) growing rounding-off errors;(ii) structural instability, i.e. different initial conditions andparameters yield different cryptographic properties, such asvery short cycles, weak plaintexts or weak keys.


Chaotic systems based on smooth nonlinear functions (e.g.x2, sin(x), tan(x) and log(x)) produce sequences with ahighly non-uniform distribution and can therefore be predictedby a probabilistic machine. By applying a partitioning strategyto generate a uniform output, a bit stream cipher with uniformstatistical properties can be obtained which passes all pseudo-randomness tests. Some piecewise-linear maps generate se-quences, which have theoretically flat distributions. However,in practice, these maps are less suitable than nonlinear mapsbecause of the overall effect of linearity, rounding and iterativetransformations and may be characterised by highly non-uniform statistics. The need to post-process the outputs formchaotic iterators in order to provide bit-streams with no statis-tical bias leads to a cryptosystem that is relatively inefficientwhen compared to conventional PRNGs. Further, the lackof any fundamental theoretical framework with regard to thepseudo-random properties of IFS leads to a basic incompati-bility with modern cryptography. However, this is off-set bythe flexibility associated with the use multi-algorithmicity forgenerating numerous and, theoretically, an unlimited numberof unique encryption engines.

All conventional cryptographic systems (encryptionschemes, pseudo-random generators, hash functions) canbe considered to be binary pseudo-chaotic systems, basedon bit stream encryption defined over a finite space. Suchsystems are periodic but have a limited sensitivity to theinitial conditions, i.e. the Lyapunov exponents are positiveonly if measured at the beginning of the process (before onecan see the cycles). The mixing property leads to pseudo-randomness. Pseudo-chaotic systems typically have manyorbits of different length. Measuring the minimal, averageand maximal length of a system is not a trivial problem, butclearly, ideal cryptographic systems have a single orbit thatincludes all the possible states.

Iterative block ciphers can be viewed as a combination oftwo linked pseudo-chaotic systems; data and round-key sys-tems. The iterated functions of such system includes nonlinearsubstitutions, row shifts, column mixing etc. The round-keysystem is a pseudo-random generator providing a sensitivitydependence of the ciphertext on the key. Technically, mostpseudo-random generators are based on the stretch-and-foldtransformation: first, the state is stretched over a large space(e.g, multiplying or raising in power), then folded into theoriginal state space (using a periodic function such asmodandsin). In mathematical chaos, the stretch-and-fold transfor-mation forms the basis of the majority of iterated functions.

In the design on any chaos based cryptosystem, it is ofparamount important to have a structurally stable cryptosys-tem, i.e. a system that has (almost) the same cycle lengthand Lyapunov exponents for all initial conditions and a givencontrol parameter set. Many pseudo-chaotic systems do notpossess this quality. Approximations to chaos are usually basedon fixed precision computations. However, it is possible toincrease the precision or resolution (e.g. the length of a binarystate string) in each iteration, a precision that can, accordingto a set of rules, be used to estimate any error impact. Aone-way transformation forms the basis of a PRNG, whereasa key-dependent invertible transformation is the essence of

a cipher or encryption scheme. Most chaos based cipherscan be extended to include invertible transformations suchas XOR, cyclic shifts and other permutations and the lattertransformations can also be considered as pseudo-chaoticmaps. Further, asymmetric cryptographic systems and basedon trapdoor functions, i.e. functions that have the one-wayproperty unless a secret parameter (trapdoor) is known. Nocounterpart of a trapdoor transformation is known in chaostheory and thus it is not currently possible to produce anequivalent to the RSA algorithm using an IFS. However, it isnoted that asymmetric encryption algorithms such as the RSAalgorithm can be used to transfer a database of algorithmsused for the multi-algorithmic symmetric encryption schemeconsidered in this paper.

B. Covertext and Stegotext

One of the principal weaknesses of all encryption systems isthat the form of the output data (the ciphertext), if intercepted,alerts the intruder to the fact that the information beingtransmitted may have some importance and that it is thereforeworth attacking and attempting to decrypt it. In Figure 1,for example, if a postal worker observed some sophisticated‘strong box’ with an impressive lock passing through the postoffice, it would be natural for them to wonder what might beinside. It would also be natural to assume that the contents ofthe box would have a value in proportion with the strengthof the box/lock. These aspects of ciphertext transmission canbe used to propagate disinformation, achieved by encryptinginformation that is specifically designed to be intercepted anddecrypted. In this case, we assume that the intercept will beattacked, decrypted and the information retrieved. The key tothis approach is to make sure that the ciphertext is relativelystrong (but not too strong!) and that the information extractedis of high quality in terms of providing the attacker with‘intelligence’ that is perceived to be valuable and compatiblewith their expectations, i.e. information that reflects the con-cerns/interests of the individual(s) and/or organisation(s) thatencrypted the data. This approach provides the interceptor witha ‘honey pot’ designed to maximize their confidence especiallywhen they have had to put a significant amount of work in to‘extracting it’. The trick is to make sure that this process isnot too hard or too easy. ‘Too hard’ will defeat the object ofthe exercise as the attacker might give up; ‘too easy’, and theattacker will suspect a set-up!

In addition to providing an attacker with a honey-pot forthe dissemination of disinformation it is of significant valueif a method can be found that allows the real information tobe transmitted by embedding it in non-sensitive informationafter (or otherwise) it has been encrypted, e.g. camouflagingthe ciphertext using methods ofSteganography. This providesa significant advantage over cryptography alone in that en-crypted messages do not attract attention to themselves. Nomatter how well plaintext is encrypted (i.e. how unbreakableit is), by default, a ciphertext will arouse suspicion and mayin itself be incriminating, as in some countries encryption isillegal. With reference to Figure 1, Steganography is equivalentto transforming the ‘strong box’ into some other object that


will pass through without being noticed - a ‘chocolate-box’,for example.

The word ‘Steganography’ is of Greek origin and means‘covered’, or ‘hidden writing’. In general, a steganographicmessage appears as something else orCovertext. The conver-sion of a ciphertext to another plaintext form is calledSte-gotextconversion and is based on the use of covertext. Somecovertext must first be invented and the ciphertext embeddedor mapped on to it in some way to produce the stegotext. Thebasic principle is given in the following schematic diagram:

Data → Covertext↓

Plaintext → Ciphertext → Stegotext↓

Transmission

Note that this approach does not necessarily require the useof plaintext to ciphertext conversion as illustrated above andthat plaintext can be converted into stegotext directly. Forexample, a simple approach to this is to use a mask to deleteall characters in a message except those that are to be read bythe recipient of the message. Apart from establishing a methodof exchanging the mask, which is equivalent to the key incryptography, the principal problem with this approach is thatdifferent messages have to be continuously ‘invented’ in orderto accommodate hidden messages and that these ‘inventions’must appear to be legitimate. However, the wealth of datathat is generated and transmitted in todays environment andthe wide variety of formats that are used, means that there ismuch greater potential for exploiting steganographic methodsthan were previously available. In other words, the wealth ofinformation now available has generated a camouflage richenvironment in which to operate and one can attempt tohide plaintext or ciphertext (or both) in a host of data types,including audio and video files and digital images. More-over, by understanding the characteristics of a transmissionenvironment, it is possible to conceive techniques in whichinformation can be embedded in the transmission noise, i.e.where natural transmission noise is the covertext. There areof course a range of counter measures - steganalysis - thatcan be implemented in order to detect stegotext. However, thetechniques usually requires access to the covertext which isthen compared with the stegotext to see if any modificationshave been introduced. The problem is to find ways of obtainingthe original stegotext which is equivalent to a plaintext attack.

C. Hiding Data in Images

The relatively large amount of data contained in digitalimages makes them a good medium for undertaking steganog-raphy. Consequently digital images can be used to hidemessages in other images. A colour image typically has 8bits to represent the red, green and blue components. Eachcolour component is composed of 256 colour values and themodification of some of these values in order to hide otherdata is undetectable by the human eye. This modification isoften undertaken by changing the least significant bit in thebinary representation of a colour or grey level value (for greylevel digital images). For example, the grey level value 128

has the binary representation 10000000. If we change the leastsignificant bit to give 10000001 (which corresponds to a greylevel value of 129) then the difference in the output image,in terms of a single pixel, will not be discernable. Hence,the least significant bit can be used to encode informationthrough modification of pixel intensity. Further, if this is donefor each colour component then a letter of ASCII text can berepresented for every three pixels. The larger the host imagecompared with the hidden ‘image’, the more difficult it is todetect the message. Further, it is possible to hide an imagein another image for which there are a number of approachesavailable.

CrypsticTM explicitly uses the method discussed in SectionVII on Stochastic Diffusion for steganographic applications.The plaintext (which, in the case of written material, is limitedin this application to an image of a single text page) is firstconverted into an image file which is then diffused with a noisefield that is generated by CrypsticTM. The host image (whichis embedded in an environment of different digital images) isdistributed with each CrypsticTM depending on the protocoland user network associated with its application. Note that thehost image represents, quite literally, the key to recovering thehidden image. The additive process applied is equivalent to theprocess of confusion that is the basis for a substitution cipher.Rather than the key being used to generate a random numberstream using a pre-defined algorithm from which the streamcan be re-generated (for the same key), the digital image is, ineffect, being used as the cipher. By diffusing the image witha noise field, it is possible to hide the output in a host imagewithout having to resort to quantization. In the case of largeplaintext documents, each page is converted into an image fileand the image stream embedded in a host video.

D. Hiding Data in Noise

The ‘art’ of steganography is to use what ever covertext isreadily available to make the detection of plaintext or, ideally,the ciphertext as difficult as possible. This means that theembedding method used to introduce the plaintext/cipherextinto the covertext should produce a stegotext that is indis-tinguishable from the covertext in terms of its statisticalcharacteristics and/or the information it conveys. From aninformation theoretic point of view, the covertext should havesignificantly more capacity than the cipheretext, i.e. there mustbe a high level of redundancy. Utilising noisy environmentsoften provides an effective solution to this problem. There arethree approaches that can be considered: (i) embedding theciphertext in real noise; (ii) transforming the ciphertext intonoise that is then added to data; (iii) replacing real noise withciphertext that has been transformed in to synthetic noise withexactly the same properties as the real noise.

In the first case, we can make use of noise sources suchas thermal noise, flicker noise, and shot noise associated withelectronics that digitize an analogue signal. In digital imagingthis may be noise from the imaging Charge Couple Device(CCD) element; for digital audio, it may be noise associatedwith the recording techniques used or amplification equipment.Natural noise generated in electronic equipment usually pro-vides enough variation in the captured digital information that


it can be exploited as a noise source to ‘cover’ hidden data.Because such noise is usually a linear combination of differentnoise types generated by different physical mechanisms, it isusually characterised by a normal or Gaussian distribution asa result of the Central Limit Theorem (see Appendix I).

In the second case, the ciphertext is transformed into noisewhose properties are consistent with the noise that is to beexpected in certain data fields. For example, lossy compressionschemes (such as JPEG - Joint Photographic Expert Group)always introduce some error (numerical error) into the de-compressed data and this can be exploited for steganographicpurposes. By taking a clean image and adding ciphertext noiseto it, information can be transmitted covertly providing allusers of the image assume that it is the output of a JPEG orsome other lossy compressor. Of course, if such an image isJPEG compressed, then the covert information may be badlycorrupted.

In the third case, we are required to analyse real noise andderive an algorithm for its synthesis. Here, the noise has tobe carefully synthesized because it may be readily observableas it represents the data stream in its entirety rather thandata that is ‘cloaked’ in natural noise. This technique alsorequires that the reconstruction/decryption method is robustin the presence of real noise that we should assume will beadded to the synthesized noise during a transmission phase.In this case, random fractal models are of value becausethe spectral properties of many noise types found in naturesignify fractal properties to a good approximation [52], [68].This includes transmission noise over a range of radio andmicrowave spectra, for example, and Internet traffic noise[33]. With regard to Internet traffic noise, the time series datarepresenting packet size and inter-arrival times shows welldefined random fractal properties with a fractal dimensionthat varies over a 24 hour cycle. This can be used to submitemails by fracturing files into byte sizes that characterise thepacket size time series and submitting each fractured file attime intervals that characterise the inter-arrival times at thepoint of submission[96], [97]. In both cases, the principal‘characteristic’ is the fractal dimension computed from liveInternet data.

APPENDIX ICENTRAL L IMIT THEOREM FOR AUNIFORM

DISTRIBUTION

We study the effect of applying multiple convolutions of theuniform distribution

P (x) =

1X , | x |≤ X/2;0, otherwise

and show thatN∏

i=1

⊗ Pi(x) ≡ P1(x)⊗ P2(x)⊗ ...⊗ PN (t)

'√

6πN

exp(−6x2/XN)

where Pi(x) = P (x), ∀n and N is large. by consideringthe effect of multiple convolutions in Fourier space (through

application of the convolution theorem) and then working witha series representation of the result.

The Fourier transform ofP (x) is given by

P (k) =

∞∫−∞

P (x) exp(−ikx)dx

=

X/2∫−X/2

1X

exp(−ikx)dx = sinc(kX/2)

wheresinc(x) = sin(x)/x - the ‘sinc’ function. Thus,

P (x) ⇐⇒ sinc(kX/2)

where⇐⇒ denotes transformation into Fourier space, andfrom the convolution theorem in follows that

Q(x) =N∏

i=1

⊗ Pi(x) ⇐⇒ sincN(kX/2).

Using the series expansion of the sin function for an arbitraryconstantα,

sinc(αk) =1

αk

(αk− 1

3!(αk)3 +

15!

(αk)5 − 17!

(αk)7 + . . .

)= 1− 1

3!(αk)2 +

15!

(αk)4 − 17!

(αk)6 + . . .

The N th power of sinc(αk) can be written in terms of abinomial expansion giving

sincN(αk) =(

1− 13!

(αk)2 +15!

(αk)4 − 17!

(αk)6 + . . .

)N

= 1−N

(13!

(αk)2 − 15!

(αk)4 +17!

(αk)6 − . . .

)

+N(N − 1)

2!

(13!

(αk)2 − 15!

(αk)4 +17!

(αk)6 − . . .

)2

−N(N − 1)(N − 2)3!

((αk)2

3!− (αk)4

5!+

(αk)6

7!− . . .

)3

+ . . .

= 1−Nα2k2

3!+ N

α4k4

5!− k

α6k6

7!

− . . . +N(N − 1)

2!

(α4k4

(3!)2− 2

α6k6

3!5!+ . . .

)−N(N − 1)(N − 2)

3!

(α6k6

(3!)3+ . . .

)+ ...

= 1− N

3!α2k2 +

(N

5!α4 +

N(N − 1)2!(3!)2

α4

)k4

−(

N

7!α6 +

N(N − 1)3!5!

α6 +N(N − 1)(N − 2)

3!(3!)3α6

)k6 + . . .

Now the series representation of the exponential (for anarbitrary positive constantc) is

exp(−ck2) = 1− ck2 +12!

c2k4 − 13!

c3k6 + . . .


Equating terms involvingk2, k4 and k6 it is clear that(evaluating the factorials),

c =16Nα2,

12c2 =

(1

120N +

172

N(N − 1))

α4

or

c2 =(

136

N2 − 190

N

)α4,

and

16c3 =

(N

5040+

N(N − 1)720

+N(N − 1)(N − 2)

1296

)α6

or

c3 =(

1216

N3 − 11080

N2 +1

2835N

)α6.

Thus, by deduction, we can conclude that

cn =(

16N

)n

α2n + O(Nn−1α2n).

Now, for large N , the first term in the equation abovedominates to give the following approximation for the constantc,

c ' 16Nα2.

We have therefore shown that theN th power of thesinc(αk)function approximates to a Gaussian function (for largeN ),i.e.

sincN(αk) ' exp(−1

6Nα2k2

).

Thus, if α = X2 , then

Q(x) ⇐⇒ exp(−X

24Nk2

)approximately. The final part of the proof is therefore toFourier invert the functionexp(−XNk2/24), i.e. to computethe integral

I =12π

∞∫−∞

exp(− 1

24XNk2

)exp(ikx)dk

=12π

∞∫−∞

exp

−(√XN

24k −

√24

XN

ix

2

)2

+6x2

XN

dk

=1π

√6

XNe−

6x2XN

∞+ix√

6XN∫

−∞+ix√

6XN

e−y2dy

after making the substitution

y =

√XN

6k

2− ix

√6

XN.

By Cauchy’s theorem

I =1π

√6

XNe−

6x2XN

∞∫−∞

e−z2dz =

√6

πXNe−

6x2XN

where we have use the result∞∫

−∞

exp(−y2)dy =√

π.

Thus, we can write

Q(x) =N∏

i=1

⊗ Pi(x) '√

6πXN

exp[−6x2/(XN)]

for largeN .

APPENDIX IITHE LYAPUNOV DIMENSION

Consider the iterative system

fn+1 = F (fn) = f + εn

where εn is a perturbation to the value off at an iteraten which is independent of the value off0. If the systemconverges tof as n → ∞ then εn → 0 as n → ∞ andthe system is stable. If this is not the case, then the systemmay be divergent or chaotic. Suppose we modelεn in termsof an exponential growth (σ >0) or decay (σ <0) so that

εn+1 = c exp(nσ)

wherec is an arbitrary constant. Thenε1 = c, ε2 = ε1 exp(σ),ε3 = ε1 exp(2σ) = ε2 exp(σ) and thus, in general, we canwrite

εn+1 = εn exp(σ).

Noting that

ln(

εn+1

εn

)= σ

we can writeN∑

n=1

ln(

εn+1

εn

)= Nσ.

Thus, we can defineσ as

σ = limN→∞

1N

N∑n=1

ln(

εn+1

εn

).

The constantσ is known as the Lyapunov exponent. Since wecan write

σ = limN→∞

1N

N∑n=1

(ln εn+1 − ln εn)

and noting that (using forward differencing)

d

dxln ε ' ln εn+1 − ln εn

δx= ln εn+1 − ln εn, δx = 1

we see thatσ is, in effect, given by the mean value of thederivatives of the natural logarithm ofε. Note that, if the valueof σ is negative, then the iteration is stable and will approachf since we can expect that asN → ∞, εn+1/εn < 1 and,thus, ln(εn+1/εn) < 0. If σ is positive, then the iterationwill not converge tof but will diverge or, depending onthe characteristics of the mapping functionF , will exhibitchaotic behaviour. The Lyapunov exponent is a parameter that


is a characterization of the ‘chaoticity’ of the signalfn. Inparticular, if we computeσN usingN elements of the signalfn and then computeσM usingM elements of the same signal,we can define the Lyapunov dimension as

DL =

1− σN

σM, σM > σN ;

1− σM

σN, σM < σN .

where

σN = limN→∞

1N

N∑n=1

ln∣∣∣∣εn+1

εn

∣∣∣∣ .ACKNOWLEDGMENT

Some aspects of this paper are based on the research andPhD Theses of the following former research students of theauthor: Dr S Mikhailov, Dr N Ptitsyn, Dr D Dubovitski, DrK Mahmoud, Dr N Al-Ismaili and Dr R Marie. The author isgrateful to the following for their contributions to CrypsticTM

Limited: Mr Bruce Murray, Mr Dean Attew, Major GeneralJohn Holmes, Mr William Kidd, Mr David Bentata and MrAlan Evans.

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[77] T. Caroll and L. M. Pecora, 1990,Synchronization in Chaotic Systems,Phys. Rev. Letters,64(8), 821-824, 1990.

[78] T. Caroll and L. M. Pecora,Driving Systems with Chaotic Signals, Phys.Rev. A44(4), 2374-2383, 1991.

[79] T. Caroll and L. M. Pecora,A Circuit for Studying the Synchronizationof Chaotic Systems, Journal of Bifurcation and Chaos,2(3), 659-667,1992.

[80] J. M. Carroll, J. Verhagen and P. T. Wong,Chaos in Cryptography: TheEscape From the Strange Attractor, Cryptologia,16(1), 52-72, 1992.

[81] M. S. Baptista,Cryptography with Chaos, Physics Letters A,240(1-2),50-54, 1998.

[82] E. Alvarez, A. Fernandez, P. Garcia, J. Jimenez and A. Marcano,NewApproach to Chaotic Encryption, Physics Letters A,263(4-6), 373-375,1999.

[83] L. Cappelletti, An FPGA Implementation of a Chaotic EncryptionAlgorithm, Bachelor Thesis. Universita Degli Studi di Padova, 2000http://www.lcappelletti.f2s.com/Didattica/thesis.pdf

[84] L. Kocarev,Chaos-based Cryptography: a Brief Overview, Journal ofCircuits and Systems,1(3), 6-21, 2001.

[85] Y. H. Chu and S. Chang,Dynamic cryptography based on synchronizedchaotic systems, Electronic Letters,35(12), 1999.

[86] Y. H. Chu and S. Chang,Dynamic data encryption system based onsynchronized chaotic systems, Electronic Letters,35(4), 1999.

[87] F. Dachselt, K. Kelber and W. Schwarz,Chaotic Coding and Crypt-analysis, 1997, http://citeseer.nj.nec.com/355232.html

[88] J. Fridrich,Secure Image Ciphering Based on Chaos, Final TechnicalReport. USAF, Rome Laboratory, New York, 1997.

[89] J. B. Gallagher and J. Goldstein,Sensitive dependence cryptography,Technical Report, 1996, http://www.navigo.com/sdc/

[90] Gao,Gao’s Chaos Cryptosystem Overview, Technical Report, 1996,http://www.iisi.co.jp/ppt/enggcc/

[91] N. V. Ptitsyn, J. M. Blackledge and V. M. Chernenky,DeterministicChaos in Digital Cryptography, Proceedings of the First IMA Con-ference on Fractal Geometry: Mathematical Methods, Algorithms andApplications (Eds. J M Blackledge, A K Evans and M Turner), HorwoodPublishing Series in Mathematics and Applications, 189-222, 2002.

[92] R. Matthews,On the derivation of a chaotic encryption algorithm,Cryptologia, (13): 2942, 1989.

[93] http://www.x-ways.net/winhex/[94] http://www.wellresearchedreviews.com/computer-monitoring/[95] http://www.nist.gov/[96] R. Marie, Fractal-Based Models for Internet Traffic and their Ap-

plication to Secure Data Transmission, PhD Thesis, LoughboroughUniversity, 2007.

[97] R. Marie, J. M. Blackledge and H. Bez,Characterisation of InternetTraffic using a Fractal Model, Proc. 4th IASTED Int. Conf. on SignalProcessing, Pattern Recognition and Applications, Innsbruck, 2007, 487-501.

Jonathan Blackledge received a BSc in Physicsfrom Imperial College, London University in 1980,a Diploma of Imperial College in Plasma Physicsin 1981 and a PhD in Theoretical Physics fromKings College, London University in 1983. As a Re-search Fellow of Physics at Kings College (LondonUniversity) from 1984 to 1988, he specialized ininformation systems engineering undertaking workprimarily for the defence industry. This was followedby academic appointments at the Universities ofCranfield (Senior Lecturer in Applied Mathematics)

and De Montfort (Professor in Applied Mathematics and Computing) wherehe established new post-graduate MSc/PhD programmes and research groupsin computer aided engineering and informatics. In 1994, he co-foundedManagement and Personnel Services Limited where he is currently ExecutiveDirector. His work for Microsharp (Director of R & D, 1998-2002) includedthe development of manufacturing processes now being used for digitalinformation display units. In 2002, he co-founded a group of companiesspecializing in information security and cryptology for the defence andintelligence communities, actively creating partnerships between industry andacademia. He currently holds academic posts in the United Kingdom andSouth Africa, and in 2007 was awarded Fellowships of the City and GuildsLondon Institute and the Institute of Leadership and Management togetherwith Freedom of the City of London for his role in the development of theHigher Level Qualification programmes in Engineering, ICT and BusinessAdministration, most recently, for the nuclear industry, security and financialsectors respectively.


Abstract—Fully integrated monopole and dipole antennas for ultra-wideband (UWB) radio utilizing flexible and rigid printed circuit boards are presented in this paper. A circular monopole antenna for the entire UWB frequency band 3.1-10.6 GHz is presented. A circular dipole antenna with an integrated balun for the frequency band 3.1-4.8 GHz is also presented. The balun utilizes broadside-coupled microstrips, integrated in the rigid part of the printed circuit board. Furthermore, an omnidirectional radiation pattern and high radiation efficiency are predicted by simulations.

Index Terms—Broadside-coupled, circular, dipole antenna,

monopole antenna, UWB

I. INTRODUCTION ltra-wideband (UWB) radio has gained popularity in recent years [1]-[8]. The entire UWB frequency-band

used for short range high speed communications have been defined between 3.1-10.6 GHz [1]-[5]. Ever since the effort to achieve one sole UWB standard halted in early 2006, two dominating UWB specifications have remained as top competitors [8]. One is based on the direct sequence spread spectrum technique [4], [7]-[8]. The other is based on the multi-band orthogonal frequency division multiplexing technique (Also known as “Wimedia UWB”, supported by Wimedia alliance) [5]-[6], [9]-[10]. The multi-band specification divides the frequency spectrum into 500 MHz sub-bands (528 MHz including guard carriers and 480 MHz without guard carriers, i.e., 100 data carriers and 10 guard carriers). The three first sub-bands centered at 3.432, 3.960, and 4.488 GHz form the so-called Mode 1 band group (3.1-4.8 GHz) [4]-[8], [10].

All the research efforts that have been made during the era of UWB antenna development have resulted in many ideas for good wideband antennas [11]-[15]. However, the general focus has so far been on the antenna element but not so much on how the antenna can be integrated in a UWB system. Utilizing a flexible substrate the antenna can be bent and placed in many ways without a major distortion of the antenna performance [16]. In this paper the concept of utilizing a flexible and rigid (flex-rigid) substrate is presented. Using this flex-rigid concept the antenna is made on the flexible part of

Manuscript received Oct. 23, 2007. Ericsson AB in Sweden is

acknowledged for financial support of this work. Magnus Karlsson; email: [email protected], and Shaofang Gong are with

Linköping University, Sweden.

the flex-rigid structure. In the rigid part other circuitries are designed and placed as with any other regular multi-layer printed circuit board. For instance, in this paper the dipole antenna balun is placed in the rigid part.

II. OVERVIEW OF THE SYSTEM As shown in Fig. 1 all prototypes were manufactured using

a four metal layer flex-rigid printed circuit board. Two dual-layer NH9326 boards were processed together with a polyimide-based flexible substrate. The rigid and the flexible substrates are bonded together in a printed circuit board bonding process. The laminates are made of sheet material (e.g., glass fabric) impregnated with a resin cured to an intermediate stage, ready for multi-layer printed circuit board bonding.

Table 1. Printed circuit board parameters Parameter (Polyimide) Dimension Dielectric height 0.1524 mm Dielectric constant 3.4±0.05 Dissipation factor 0.002 Parameter (NH9326) Dimension Dielectric height 0.254 mm Dielectric constant 3.26±0.1 Dissipation factor 0.0025 Parameter (Metal, common) Dimension Metal thickness, layer 1, 4 0.035 mm Metal thickness, layer 2, 3 0.025 mm Metal conductivity 5.8x107 S/m (Copper)Surface roughness 0.001 mm

Table 1 lists the printed circuit board parameters, with the

stack of the printed circuit board layers shown in Fig. 1a. Metal layers 1 and 4 are thicker than metal layers 2 and 3 because the surface layers are plated twice (the embedded metal layers 2 and 3 are plated once). Fig. 1b shows the advantage using the flex-rigid concept, i.e., the bendable property of the flex-rigid substrate.

Metal 1 Metal 4

RigidFlex Rigid

Metal 2: antenna Metal 3: ground

(a) Substrate cross-section.

Monopole and Dipole Antennas for UWB Radio Utilizing a Flex-rigid Structure

Magnus Karlsson, and Shaofang Gong, Member, IEEE

U


ISAST Transactions on Electronics and Signal Prosessing, No. 1, Vol. 2, 2008 Karlsson M. and Gong S.: Monopole and Dipole Antennas for UWB Radio Utilizing a Flex-rigid Structure59

Rigid Flex Rigid

(b) Bendable property.

Fig. 1. Printed circuit board structure: (a) detailed cross-section, and (b) bendable property.

A. Monopole antenna Fig. 2 shows a circular monopole antenna integrated in the

flex-rigid substrate. As shown in Fig. 2, the ground plane is integrated in the rigid part. The radiating antenna element is placed entirely on the flex part of the substrate. The circular antenna geometry provides good omni-directionality [11].

Circular monopole antenna

Single-ended feed-line

Ground plane (rigid part)

Polyimide foil

30 mm

Fig. 2. Monopole antenna.

B. Dipole antenna Fig. 3 shows a circular dipole antenna integrated in the flex-

rigid substrate. The radiating antenna element is placed entirely on the flex part of the substrate. Furthermore, the balun is integrated in the rigid part of the substrate. The balun utilizes broadside-coupled microstrips [13].

Single ended feed-line

38 mm

Circular dipole antenna

Polyimide foil

Rigid part

Fig. 3. Circular dipole antenna.

C. Balun used with the dipole antennas Fig. 4a shows the illustration of the broadside-coupled

microstrips. Fig. 4b shows the proposed balun structure. The balun is used together with the dipole antennas and built with the broadside-coupled microstrips. By implementing the balun in a multilayer structure a more compact design is achieved.

Metal 1Metal 2Ground

Microstrip lines

(a) Broadside-coupled microstrips.

Single ended feed-line (Port 1, Metal 1)

Differential feed-line

Metal 2 connected to ground

λ/2

Port 2, Metal 2

Port 3, Metal 2

(b) Broadside-coupled balun.

Fig. 4. Balun: (a) broadside-coupling, and (b) a broadside-coupled balun.

III. SIMULATED RESULTS Design and simulation were done with ADS2006A from

Agilent Technologies Inc. Electromagnetic simulations were done with Momentum, a built-in 2.5D method of moment field solver.

A. Monopole antenna Fig. 5 shows voltage standing wave ratio (VSWR)

simulation of a circular monopole antenna on the flex-rigid substrate. It is seen that the designed circular monopole antenna shown in Fig. 2 has a wide impedance bandwidth using the proposed flex-rigid structure. It covers the entire UWB frequency-band 3.1-10.6 GHz at VSWR<2.

1 2 3 4 5 6 7 8 9 10 111

2

3

4

5

VS

WR

Frequency (GHz)

Fig. 5. VSWR simulation of the monopole antenna.


B. Dipole antenna Fig. 6 shows VSWR simulation of a circular dipole antenna

on the flex-rigid substrate. Fig. 6a shows the VSWR simulation result without the balun, and Fig. 6b shows the simulation result with the balun. It is seen that the circular dipole antenna has a wide impedance bandwidth using the suggested flex-rigid structure. Furthermore, it is seen that the balun is the component limiting the bandwidth.

1 2 3 4 5 6 7 8 9 101

2

3

4

5

VS

WR

Frequency (GHz)

(a) VSWR simulation without balun.

1 2 3 4 5 6 7 8 9 101

2

3

4

5

VS

WR

Frequency (GHz)

(b) VSWR simulation with balun.

Fig. 6. Dipole antenna: (a) VSWR simulation without balun, and (b) VSWR simulation with balun.

Fig. 7 shows radiation simulation of the circular dipole

antenna. The radiation patterns are similar in the three sub-bands of 3.432, 3.960, and 4.488 GHz, as seen in Figs 7a-7c. The pattern becomes slightly more focused at higher frequencies, which is expected since the physical size is larger compared to the wavelength at the higher frequencies [17].

-30

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00

30

60

90

120

150180

210

240

270

300

330

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-5

0

φ=0º

(a) Normalized Eθ radiation pattern at 3.432 GHz, φ=0º.

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-5

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30

60

90

120

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210

240

270

300

330

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0

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(b) Normalized Eθ radiation pattern at 3.960 GHz, φ=0º.

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300

330

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(c) Normalized Eθ radiation pattern at 4.488 GHz, φ=0º.

Fig. 7. Dipole antenna simulations, normalized radiation pattern: (a) radiation pattern at 3.432 GHz, φ=0, (b) radiation pattern at 3.960 GHz, φ=0, and (c) radiation pattern at 4.488 GHz, φ=0.

Table 2 lists simulated gain and radiation efficiency. It is

seen that both the monopole and the dipole antennas have high radiation efficiency when implemented using the flex-rigid structure. The simulated gain of the dipole antenna is slightly


above 2 dBi as expected. The monopole has even higher gain but this is due to the 2.5D simulation that uses an infinitively large ground-plane [11], [17]. However, the VSWR still ought to be correct since the minimum distance to the ground-plane is correct [17].

Table 2. Simulated maximum co-polarization antenna gain and efficiency

Frequency (GHz) 3.432 3.960 4.488 Gain, Monopole (dBi) 4.139 3.572 4.854 Gain, Dipole (dBi) 2.272 2.459 2.581 Efficiency, Monopole (%) 99.68 98.78 99.99 Efficiency, Dipole (%) 95.17 94.65 93.82

C. Balun used with the dipole antenna Fig. 8 shows simulations of the proposed broadside-coupled

balun shown in Fig. 4. It is seen in Fig. 8a that the balun has an insertion loss (IL) less than 0.8 dB in the Mode 1 UWB frequency-band. Fig. 8b shows a rather symmetric performance of the two signal paths. S21 and S31 are single-ended forward transmissions, from Port 1 to Port 2 and 3, respectively. However, it is seen that above the Mode 1 UWB frequency bandwidth (3.1-4.8 GHz) the IL increases. Fig. 8c shows simulated phase balance (single-ended S21 and S31). It is seen that the phase shift is linear, and the phase difference is close to 90º between 2.0 to 7.1 GHz. It is noticed that a small notch is seen at 3.05 GHz. It occurs when the total length from the antenna feed-point (Port 2 and Port 3 in Fig. 4b) to the grounded end of each path of the balun is equal to one quarter wavelength. In this work, the length of the differential antenna feed-line was optimized so that the small notch fell below the UWB frequency-band.

2 3 4 5 6 7 8-50

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0

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trans

mis

sion

(dB)

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(a) Simulation of forward transmission.

2 3 4 5 6 7 8-50

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trans

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)

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S21

S31

(b) Simulation of forward transmission.

2 3 4 5 6 7 8

-200-150-100-50

050

100150200

Pha

se (°

)

Frequency (GHz)

S21

S31

(c) Simulation of phase balance.

Fig. 8. Balun: (a) simulated forward transmission, single-differential port, (b) simulated forward transmission, single-single port, and (c) simulated phase.

IV. DISCUSSION Simulation results show that the monopole antenna has a

wide operational frequency band. The monopole antenna is smaller than the dipole antenna but needs a quite large ground plane. On the contrary, the dipole antenna only requires a small ground plane beneath the balun, but the total size is larger. The simulations of the circular dipole antenna indicate that the antenna has a typical radiation pattern as expected from a common dipole antenna. It is also observed (Fig. 6) that the balun limits the bandwidth if the dipole antenna is used with a single-ended port. However, this antenna with the balun (Fig. 6b) has a good bandpass characteristic, covering the frequency band 3.1-4.8 GHz required by the Mode 1 UWB specification.

V. CONCLUSION Simulations show that a circular monopole antenna can be

implemented on a flex-rigid substrate with VSWR<2.0 over the entire UWB frequency bandwidth (3.1-10.6 GHz). Moreover, a circular dipole antenna implemented using the flex-rigid substrate can cover the Mode 1 UWB frequency-bandwidth (3.1-4.8) at VSWR<1.54 without a balun and VSWR<1.68 with a balun.


REFERENCES [1] “First report order, revision of part 15 of commission’s rules regarding

ultra-wideband transmission systems” FCC., Washington, 2002. [2] G. R. Aiello and G. D. Rogerson, “Ultra Wideband Wireless Systems,”

IEEE Microwave Magazine, vol. 4, no. 2, pp. 36-47, Jun. 2003. [3] M. Karlsson and S. Gong, “Wideband patch antenna array for multi-

band UWB,” Proc. IEEE 11th Symp. on Communications and Vehicular Tech., Ghent, Belgium, Nov. 2004.

[4] L. Yang and G. B. Giannakis, “Ultra-Wideband Communications, an Idea Whose Time has Come,” IEEE Signal Processing Magazine, pp. 26-54, Nov. 2004.

[5] M. Karlsson and S. Gong, “An integrated spiral antenna system for UWB,” Proc. IEEE 35th European Microwave Conf., Paris, France, Oct. 2005, pp 2007-2010.

[6] J. Balakrishnan, A. Batra, and A. Dabak, “A multi-band OFDM system for UWB communication,” Proc. Conf. Ultra-Wideband Systems and Technologies, Reston, VA, 2003, pp.354–358.

[7] W. D. Jones, "Ultrawide gap on ultrawideband," IEEE Spectrum, vol. 41, no. 1, pp. 30, Jan. 2004.

[8] D. Geer, "UWB standardization effort ends in controversy," Computer, vol. 39, no. 7, pp. 13-16, July 2006.

[9] S. Chakraborty, N. R. Belk, A. Batra, M. Goel, A. Dabak, "Towards fully integrated wideband transceivers: fundamental challenges, solutions and future," Proc. IEEE Radio-Frequency Integration Technology: Integrated Circuits for Wideband Communication and Wireless Sensor Networks 2005, pp. 26-29, 2 Dec. 2005.

[10] Geer, D., "UWB standardization effort ends in controversy," Computer, vol.39, no.7, pp. 13-16, July 2006.

[11] H. Schantz, "The Art and Science of Ultrawideband Antennas," Artech House Inc., ISBN: 1-58053-888-6, 2005.

[12] M. Karlsson, P. Håkansson, A. Huynh, and S. Gong, “Frequency-multiplexed Inverted-F Antennas for Multi-band UWB,” IEEE Wireless and Microwave Conf. 2006, pp. 2.1-2.3, 2006.

[13] M. Karlsson, and S. Gong, "A Frequency-Triplexed Inverted-F Antenna System for Ultra-wide Multi-band Systems 3.1-4.8 GHz," Accepted for publication in ISAST Transactions on Electronics and Signal Processing, 2007.

[14] Z. N. Chen, M. J. Ammann, X. Qing; X. H. Wu, T. S. P. See, A. Cai, "Planar antennas," Microwave Magazine, IEEE, vol. 7, no. 6, pp. 63-73, Dec. 2006.

[15] W. S. Lee, D. Z. Kim, K. J. Kim; K. S. Son, W. G. Lim, J. W. Yu, "Multiple frequency notched planar monopole antenna for multi-band wireless systems," Proc. IEEE 35th European Microwave Conf., Paris, France, Oct. 2005, pp 535-537.

[16] B. Kim, S. Nikolaou, G. E. Ponchak, Y.-S. Kim, J. Papapolymerou, M. M. Tentzeris, "A curvature CPW-fed ultra-wideband monopole antenna on liquid crystal polymer substrate using flexible characteristic," IEEE Antennas and Propagation Society Int. Symp. 2006, pp. 1667-1670, 9-14 July 2006.

[17] V. F. Fusco, Foundations of Antenna Theory and Techniques, Edinburgh Gate, Harlow, Essex, England, Pearson Education Limited, pp. 45, 2005.


In 2003 he started his Ph.D. study in the Communication Electronics research group at

Linköping University. His main work involves wideband antenna-techniques, wideband transceiver front-ends, and wireless communications.


Between 1991 and 1999 he was a senior researcher at the microelectronic institute – Acreo in Sweden. From 2000 to 2001 he was the CTO at a spin-off company from the institute. Since 2002 he has been full professor in communication electronics at Linköping University, Sweden. His main research interest has been communication electronics including RF design, wireless communications and high-speed data transmissions.


Monofilar spiral antennas for multi-band UWB system with and without air core

Magnus Karlsson and Shaofang Gong

Linköping University, Department of Science and Technology - ITN, LiU Norrköping, SE-601 74 Norrköping, Sweden, Phone +46 11363491

Abstract — One of the trends in wireless communication

is that systems require more and more frequency spectrum. Consequently, the demand for wideband antennas increases as well. For instance, the ultra wideband radio (UWB) utilizes the frequency band of 3.1-10.6 GHz. Such a bandwidth is more than what is normally utilized with a single low-profile antenna. Low profile antennas are popular because they are integratable on a printed circuit board. However, the fractional bandwidth is usually an issue for low profile antennas because of the limited substrate height. The monofilar spiral antenna on the other hand has higher fractional bandwidth, and at GHz frequencies the physical dimensions of the spiral is reasonable. Furthermore, a study of how spiral dimensions impact on antenna gain and standing wave ratio (SWR) was conducted and presented. Simulated results were compared with measurements.

I. INTRODUCTION Wireless communication systems require more and

more spectrums, while single module solutions grow in popularity. An integratable low-profile antenna structure limits design options, regarding various performance aspects. For instance, the fractional bandwidth is limited. A monofilar spiral antenna is known to have wide bandwidth, i.e., good fractional bandwidth compared to other planar antennas [1-4]. One known drawback is the physical size of a plane monofilar spiral antenna, i.e., the single arm spiral is physically large in terms of its diameter [3, 4]. Furthermore it is known that even though the spiral may have very wide impedance bandwidth, dispersion may cause problems for using spiral antennas in wideband impulse-systems [1, 2, 5]. However, in multi-band systems the antennas need only to transmit and receive a signal with limited bandwidth, i.e., the width of a single multi-band pulse determines the maximum frequency spectrum used simultaneously [6]. Flat antenna performance is desired in UWB systems [7]. A terminating resistor at the arm end may be used to reduce reflections in order to flatten the antenna input impedance, but it causes signal loss [5]. Our approach is instead to optimize the physical properties of the antenna so that the input impedance becomes acceptable close to a 50 Ω real load. The antennas are intended to be integrated on a multilayer printed circuit board (PCB). A concept to reduce the total size of the complete design is to place components on a layer below the antenna as proposed in our previous publications [3, 6]. This requires that the antenna structure is compatible with an overall module technology. Ultra wideband radio (UWB)

has been specified in the frequency range 3.1 - 10.6 GHz [4].

II. OVERVIEW OF THE ANTENNA

A. Monofilar Spiral Antenna Fig. 1 shows a photo of one of the monofilar spiral

antenna prototypes presented in this paper. No matter which antenna technique used, most antennas should be connected to a 50 Ω port. The soldered 50 Ω feed point is seen in the center of the spiral. Table I lists antenna specifications.

Fig. 1. Photo of a spiral antenna prototype. Table I. Antenna specifications

Antenna Turn distance (∆s) Radius (r) 1.1 4.5 mm 75 mm1.2 5.5 mm 75 mm 1.3 6.5 mm 75 mm 2.1 4.5 mm 50 mm 2.2 5.5 mm 50 mm 2.3 6.5 mm 50 mm 3.1 4.5 mm 30 mm 3.2 5.5 mm 30 mm 3.3 6.5 mm 30 mm

B. Material and PCB Structures Table II. Four layer PCB parameters

Parameter Layer Dimension Dielectric top RCC Height, h=0.05 mmDielectric core RO4350B (1x) h=1.524 mm Dielectric bottom RCC h=0.05 mm Dielectric constant All layers 3.48±0.05Dissipation factor All layers 0.004 Metal thickness Top and bottom 0.025 mm Metal thickness RO4350B (core) 0.018 mm Metal conductivity All layers 5.8x107 S/mSurface roughness All layers 0.001 mm

Table II. lists material parameters used for simulations,

and Fig. 2 illustrates cross-sections of the two PCB-structures. Fig. 2a shows the PCB-structure used for the antennas without an air core, and Fig. 2b shows the PCB-structure used for all antennas with an air core. Rogers RO4350B has been used since it is suitable for radio

ISAST Transactions on Electronics and Signal Prosessing, No. 1, Vol. 2, 2008 Karlsson M. and Gong S.: Monofilar spiral antennas for multi-band UWB system with and without air core


frequency (RF) modules [6, 8]. The proposed RF module with an air core is built with two PCBs that are separated with an additional air gap. Separating the antenna element from the ground plane with an air gap improves radiation characteristics [3, 9]. The antenna PCB and the component PCB heights are chosen separately, and the number of layers of the component PCB can be chosen freely. The antenna is fed with one via, i.e., a via through all layers to the antenna. Since the module is manufactured as two separate PCBs the via connecting the antenna feed point is soldered afterwards.

Antenna PCB

Component PCB

Air core Component layer Gnd

Top

Via

Component layer

Gnd Antenna layer

RO4350B

Antenna Via layer

hair

Bottom

(a) Cross-section. (b) Cross-section.

Fig. 2. Cross-sections of substrate: (a) four layer PCB, and (b) air core substrate structure.

C. Principle of the Monofilar Spiral As shown in Fig. 3, the circumference of the radiation

zone determines the radiation frequency. The circumference is one λ for the first radiating zone, where λ is the wavelength. When the circumference is >2λ the antenna will radiate a tilted beam [10]. The tilted beam consists of one component for each wavelength the circumference reaches, i.e., for circumference 2-3λ the beam consists of two radiating zones and so on. In general, when one half circumference is equal to n times λ/2. Two travelling waves exist on the spiral, forward and backward. Each wave will radiate when passing through an appropriate radiation zone [2]. The net result is the far-field composed of positive and negative modes coming from these two oppositely directed waves. Existence of the two waves will determine the antenna performance in a certain direction at a specific frequency [10, 11]. A stable wave flow and a smooth standing wave ratio (SWR) are closely related to each other, if the substrate loss is low. The antenna has circular polarization in the frequency range where the total length of the spiral arm is electrically large when compared to the wavelength [4, 5].

Feeding to the spiral can be done either from the centre or from the outside. In the prototypes shown in this paper the feeding is done by a via to the centre of the spiral, see Fig. 3. The input impedance depends on the line width together with the distance to the ground plane, because the characteristic impedance of the spiral arm is dependant of the line width as in the case of a microstrip line. The real part of the input impedance can be controlled by the line width, while the imaginary part is more difficult to control. If implemented in a narrow band system the spiral antenna can be matched using a classical RF matching technique for optimal performance in that frequency region. However, if a wideband

operation is required the issue must be solved with other techniques.

Antenna

Open end

Feed point

Turn distance, Δs Radiating zone

Radius, r

n∗λ/2

X

Z

Y

phi (φ)

theta (θ)

Fig. 3. Layout of a monofilar spiral antenna, oriented in the XY plane, φ=0°.

D. Methods Design and simulation were done using the ADS

Momentum, which is a 2.5D planar electromagnetic (EM) simulator. The antennas simulated are of planar structure therefore the Momentum-engine is a reliable choice, despite the lack of surface roughness consideration [12]. The simulations were done with an infinite ground plane, imposed by ADS.

Voltage standing wave ratio (VSWR) measurements were done with a Rhode&Schwartz ZVM vector network analyzer (NWA). Radiation measurements were done in an anechoic antenna chamber. An HP 8510C vector-NWA together with an HP 8517B S-parameter test set and the MiDAS 2.01g software was used to measure the antennas. Linearly polarized horn antennas with known antenna gain were used as reference antennas.

III. SIMULATION RESULTS

A. Monofilar Spiral Antennas of 50 Ω Fig. 4 shows the layout and VSWR simulation of a

monofilar spiral antenna. To optimize performance, the real part of the characteristic impedance was calculated to be 50 Ω, i.e., the line width was calculated to be 3.43 mm with a substrate thickness of 1.5 mm. Three radii with three different turn distances for each size of the antenna were designed and simulated. It is shown that in all three simulations in Figs. 4a-4c, a more dense turn distance (Δs) provides a high fractional bandwidth. As mentioned in the introduction, in theory the infinite number of turns gives the infinite fractional bandwidth. However, owing to the chosen line width and the need of spacing between turns and limited number of turns, the bandwidth is limited in our case as seen in Figs. 4a-4c. Fig. 4d shows a rather good match between simulated and measured data.

3 4 5 6 7 8 9 102 11

2

3

4

1

5

Frequency (GHz)

VS

WR

(a) Simulation, r=30 mm.

Δs=6.5 mm

Δs=4.5 mm Δs=5.5 mm 2

ISAST Transactions on Electronics and Signal Prosessing, No. 1, Vol. 2, 2008 Karlsson M. and Gong S.: Monofilar spiral antennas for multi-band UWB system with and without air core65

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Fig. 4. VSWR simulation of Monofilar spiral antennas of 50 Ω input impedance: (a) r=30 mm, (b) r=50 mm, (c) r=75 mm, and (d) simulation and measurement comparison, r=75 mm, Δs=4.5 mm.

B. Monofilar Spiral Antenna Gain Considerations Fig. 5 shows antenna gain simulations for various

monofilar spiral antennas. Simulations in 5a-5c are done with substrate definitions exactly displayed in Table II and Fig. 2a, also called 1x. The simulation marked with 2x has the same material but with a double core layer height. Thus, the height from the antenna plane to the ground plane is 1.5 and 3.0 mm, respectively. Figs. 5a-5c shows a series of simulations about how the distance between turns affects the antenna gain at various antenna radii. The optimal density of turns varies with the radius. Fig. 5d shows how an additional air-gap improves the antenna gain. Fig. 5e shows how the dielectric loss and the substrate thickness impact on the antenna gain. The substrate marked with 2x-air is the 1x substrate with a 1.5 mm air distance added between the ground plane and the PCB, as seen in Fig. 2b.

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Fig. 5. Antenna gain simulations: (a) antenna gain, r=30 mm, (b) antenna gain, r=50 mm, (c) antenna gain, r=75 mm, (d) antenna gain, air core, r=75 mm, Δs=4.5 mm, and (e) antenna gain vs. different substrates and heights.

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IV. MEASURED RESULTS

A. Monofilar Spiral Antenna⎯r=30 mm Figs. 6a shows VSWR measurements of a spiral

antenna with a radius of 30 mm. The turn distance is 4.5 mm for the antennas measured in Fig. 6a. It is seen that the VSWR performance is limited in the lower frequency-band, due to the limited radius. The limitation was slightly less than expected from the simulations shown in Fig. 4a but the VSWR response was less flat. The antenna with the largest turn distance has a more stable gain performance in the measured spectrum, see Fig. 6b.

(a) VSWR measurement, r=30 mm, Δs=4.5 mm.

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Fig. 6. VSWR and antenna gain measurement of two spiral antennas with a radius of 30 mm: (a) turn distance 4.5 mm, and (b) maximum antenna gain measured from three θ sweeps with φ=0°, 45°, and 90°, respectively.

B. Monofilar Spiral Antenna⎯r=50 mm Figs. 7a shows VSWR measurement of a spiral

antenna with a radius of 50 mm. The turn distance is 6.5 mm, for the antenna measured in Fig. 7a. The radius of 50 mm is large enough to provide a VSWR<3 through the entire UWB frequency-band. The magnitude of the resonances is smaller than that from the antennas with a 30-mm radius because of less reflections owing to more turns. It is seen that the VSWR is slightly lower than that from the simulations shown in Fig. 4b. The antenna with the largest turn distance has highest gain in the majority of the measured spectrum, see Fig. 7b.

(a) Measured VSWR, r=50 mm, Δs=6.5 mm.

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Fig. 7. VSWR and antenna gain measurements of two spiral antennas with a radius of 50 mm: (a) turn distance 6.5 mm, and (b) maximum antenna gain measured from three θ sweeps with φ=0°, 45°, and 90°, respectively.

C. Monofilar Spiral Antenna⎯r=75 mm Figs. 8a-8c show VSWR measurements of three spiral

antennas with a radius of 75 mm. The turn distance is 4.5, 5.5 and 6.5 mm, for the antenna measured in Figs. 8a, 8b and 8c, respectively. It is seen that the antenna 1.1 reaches VSWR<2 at 2.1 GHz. Larger fractional bandwidth is achieved as the turn distance decreases. The simulated VSWR deviates more from the measured as the frequency increases, see Figs. 4c, 8a-8c. However, the antenna with largest turn-distance has the smoothest VSWR performance. Fig. 8d shows that the radiation pattern is dependant of the frequency. It is seen in Fig. 8e that the 4.5 mm turn distance is the most optimal at low frequencies, but as the frequency increases the optimal turn distance increases.

(a) Measured VSWR, r=75 mm, Δs=4.5 mm.

(b) Measured VSWR, r=75 mm, Δs=5.5 mm.

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Fig. 8. VSWR and antenna gain measurements of three spiral antennas with a radius of 75 mm: (a) turn distance 4.5 mm, (b) turn distance 5.5 mm, (c) turn distance 6.5 mm, (d) measured radiation characteristics of the antenna with 4.5-mm turn distance, φ=90°, and (e) maximum antenna gain measured from three θ sweeps with φ=0°, 45°, and 90°, respectively.

D. Air Core Spiral Antennas Fig. 9 shows measurements of two air core spiral

antennas (see Fig. 2b) with r=75 mm. It is seen in Figs. 9a and 9b that the substrate with an air gap gives a more stable VSWR response compared to the antennas in Figs 8a-8c, and the VSWR is kept low for a wide frequency range. It is shown in Fig. 9a that with an air core substrate (1.5 mm RO4350B + 1.5 mm air), a VSWR<2 can be achieved for the entire UWB frequency band. Besides low VSWR the air core substrate provides good impedance stability, which results in an almost resonance free VSWR response. Figs 9d-9e show a much higher antenna gain compared to the equally sized antennas show in Fig. 8. This is due to a high height between the antenna and the ground plane and low loss tangent of the air layer.

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Fig. 9. Air core antenna measurements, r=75 mm: (a) VSWR, 1.5 mm RO4350B core + 1.5 mm air, (b) VSWR, 0.8 mm RO4350B core + 2.2 mm air, (c) measured E-field of the 1.5 mm air core antenna, φ=90°, (d) measured antenna gain, 0.8 mm RO4350B + 2.2 mm air, and (e) maximum antenna gain measured from three θ sweeps with φ=0°, 45°, and 90°, respectively.

V. DISCUSSIONS A study of monofilar spiral antennas for UWB is

presented in this paper. Antennas covering the entire UWB frequency band of 3.1-10.6 GHz can be realized with various SWR numbers, but the physical size limits the fractional bandwidth and determines the centre frequency. The turn distance was found to be a crucial factor for the antenna gain, and the optimal turn distance varies with the radius of the antenna. However, increasing the turn distance decreases the fractional bandwidth so a trade-off must be made. Some missmatching between simulated and measured results is seen, especially regarding to fractional bandwidth and the center frequency, see Fig. 4. The difference may come from the fact that the simulations are done with an infinite ground-plane while the prototype ground-plane is only 30 mm larger than the spiral. Moreover the simulations are done with a more limited number of points. Momentum is as mentioned not a full 3D simulator. More advanced EM simulators like Ansoft HFSS or CST Microwave studio may give more accurate results.

A single arm spiral antenna can have good performance over a wide frequency range. However, the radiation pattern is dependent of the frequency so that the performance in a certain direction varies throughout the frequency-range (see Figs. 8d and 9c). The performance varies owing to the fact that the beam moves around the z-axis (see Fig 3.) such that several modes interact. Furthermore, the high frequency ripple in VSWR affects performance and causes the antenna gain to vary throughout the frequency-range. These variations are due to the physical properties of the single arm spiral that introduces non-real input impedance. Moreover, some inductance might be introduced by the SMA connector, i.e., the feeding via through the PCB is not impedance controlled. A more uniform and high gain can be achieved when an air core is used (see Fig. 9). This is due to the fact that with a high gain less signal power reaches the arm end, i.e., the backward travelling wave gets suppressed relative to the forward travelling wave. Furthermore, the air core effectively reduces the high frequency variation in VSWR. In general an additional

air gap affects positively the antenna performance in two ways. Firstly, the input impedance is improved. Secondly, an air gap increases the maximum gain. The increase in gain is similar to what is reported with electromagnetic bandgap structures (EBG) [13, 14]. However, the radiation pattern of the air core monofilar-spiral antenna is not as smooth as for instance what is reported for the EBG curl antenna [14].

VI. CONCLUSIONS A study of monofilar spiral antenna for UWB has been

conducted. How the spiral radius, turn distance, and substrate dissipation factor and thickness affect the antenna gain were shown. Moreover, antenna gain improvement with an air gap between the ground plane and the antenna plane has been shown.

Monofilar spiral antenna solutions for UWB optimized for an input source impedance of 50 Ω was designed, simulated and measured. It is shown that a planar monofilar spiral antenna implemented on a PCB is suitable for RF module designs.

Monofilar spiral antennas with a radius of 30, 50 and 75 mm were designed, simulated and measured. One monofilar spiral antenna of these sizes covers the entire UWB frequency band (3.1-10.6 GHz) at a VSWR<4, <3, <2, respectively.

Monofilar spiral antennas with a radius of 75 mm consisting of an air core were also designed, simulated and measured. One monofilar spiral antenna with the air core covers the entire UWB frequency band at VSWR<1.8. The VSWR response is also more stable. The antenna gain is higher than that from any of the monofilar antennas without an air core.

REFERENCES [1] R. G. Corzine and I. A. Mosko, “Four-arm spiral antennas,” Wood

MA, Anech House, 1990. [2] R. H. DuHamel and J. P. Scherer, “Frequency-independent

antennas," Antenna Engineering Handbook, 3rd ed., Johnson R. C. Ed., McGraw Hill, New York, 1993, Ch. 14, pp. 53-82.

[3] M. Karlsson and S. Gong, “An integrated spiral antenna system for UWB,” Proc. IEEE 35th European Microwave Conf., Paris, France, Oct. 2005, pp 2007-2010.

[4] E. Gschwendtner, D. Löffler, W. Wiesbeck, “Spiral antenna with external feeding for planar applications,” IEEE Africon, vol. 2, Sep. 1999, pp. 1011-1014.

[5] C. Kinezos and V. Ungvichian, “Ultra-wideband circular polarized microstrip archimedean spiral antenna loaded with chip-resistor,” IEEE Antennas and Propagation Society International Symposium, vol 3, Jun. 2003, pp. 612-615.

[6] M. Karlsson and S. Gong, “Wideband patch antenna array for multi-band UWB,” Proc. IEEE 11th Symp. on Communications and Vehicular Tech., Ghent, Belgium, Nov. 2004.

[7] G. R. Aiello and G. D. Rogerson, “Ultra Wideband Wireless Systems,” IEEE Microwave Magazine, vol. 4, no. 2, pp. 36-47, Jun. 2003.

[8] S. Gong, M. Karlsson, and A. Serban, ”Design of a Radio Front End at 5 GHz,” Proc. IEEE 6th Circuits and Systems Symp. on Emerging Tech., Shanghai, China, Jun. 2004, vol. 1, pp. 241-244.

[9] D. Guha and J. Y. Siddiqui, “Resonant frequency of equilateral triangular microstrip antenna with and without air gap,” IEEE Trans. on Antennas and Propagation, vol. 52, no. 8, Aug. 2004.

[10] H. Nakano, Y. Okabe, H. Mimaki, J. Yamauchi, “A Monofilar Spiral Antenna Excited Through a Helical Wire,” IEEE Trans. on Antennas and Propagation, vol. 51, no. 3, Mar. 2003.


[11] H. Nakano, J. Eto, Y. Okabe, J. Yamauchi, “Tilted- and Axial-Beam Formation by a Single-Arm Rectangular Spiral Antenna With Compact Dielectric Substrate and Conducting Plane,” IEEE Trans. on Antennas and Propagation, vol. 50, no. 1, Jan. 2002.

[12] A. P. Jenkins, A. M. Street, D. Abbott, “Filter design using CAD. II. 2.5-D simulation, Effective Microwave CAD,” IEE Colloquium, no. 1997/377, pp. B1/1-B1/5, Dec. 1997.

[13] P. de Maagt, R. Gonzalo, Y.C. Vardaxoglou, J.-M. Baracco, "Electromagnetic bandgap antennas and components for microwave and (Sub)millimeter wave applications," IEEE Trans. on Antennas and Propagation, vol.51, no.10, pp. 2667- 2677, Oct. 2003.

[14] J.-M. Baracco, M. Paquay, P. de Maagt, "An electromagnetic bandgap curl antenna for phased array applications," IEEE Trans. on Antennas and Propagation, vol.53, no.1, pp. 173- 180, Jan. 2005.

Magnus Karlsson was born in Västervik, Sweden in 1977. He received the M.Sc. and Licentiate of Engineering from Linköping University in Sweden, in 2002 and 2005, respectively.

In 2003 he started his Ph.D. study in the Communication Electronics research group at Linköping University. His main work involves

wideband antenna-techniques and wireless communications.

Shaofang Gong was born in Shanghai, China, in 1960. He received the B.Sc. degree from Fudan University in Shanghai in 1982, and the Licentiate of Engineering and Ph.D. degree from Linköping University in Sweden, in 1988 and 1990, respectively.

Between 1991 and 1999 he was a senior researcher at the microelectronic institute – Acreo in Sweden. From

2000 to 2001 he was the CTO at a spin-off company from the institute. Since 2002 he has been full professor in communication electronics at Linköping University, Sweden. His main research interest has been communication electronics including RF design, wireless communications and high-speed data transmissions.


Performance Evaluation of Analog Systems Simulation Methods for the Analysis of Nonlinear

and Chaotic Modules in Communications

J. C. Chedjou*, K. Kyamakya

Institute for Smart-Systems Technologies,

University of Klagenfurt Klagenfurt, Austria

e-mails: [email protected] , Kyandoghere.Kyamakya@uni-

klu.ac.at

Van Duc Nguyen

Faculty of Electronics and Telecommunications,

C9-P403 Hanoi University of Technology

e-mail: [email protected]

Ildoko Moussa, J. Kengne

Doctoral School of Electronics, Information Technology, and

Experimental Mechanics (UDETIME),

University of Dschang Dschang, Cameroon

e-mail: [email protected]

Abstract— The revolutionary idea of setting Analog Cellular Computers based on Cellular Neural Networks systems (CNNs) to change the way analog signals are processed is a proof of the high importance devoted to the analog simulation methods. This paper provides basics of the methods that can be exploited for the analog simulation of very complex systems (an implementation on chip using CNN technology is possible even on FPGA). We evaluate the performance of analog systems simulation methods. These methods are applied for the investigation of the nonlinear and chaotic dynamics in some modules of communication systems. We list some problems encountered when using this approach and propose appropriate techniques to tackle them. The overall motivation is to encourage the research community to use analog methods for systems’ analysis, despite the strong focus on the numerical approaches that kept analog simulation alternatives a bit in the dark during the past decades. Both advantages and limitations of the analog modelling schemes are discussed versus those of their numerical counterpart. To illustrate the concepts, a communication module consisting of a shunt type Colpitts oscillator is considered. The electrical structure of the oscillator is addressed and the modeling process is performed to derive the equations of motion. The numerical analysis is carried out to obtain various bifurcation diagrams showing scenarios leading to chaos. Both PSPICE based simulations and laboratory experimental realizations (with analog circuits) are considered to validate the modeling and to confirm the numerical results. Near-sinusoidal oscillations, sub-harmonics and chaos are observed. The bifurcation study reveals that the system moves from near-sinusoidal regime to chaos via the usual paths of period doubling and sudden transitions. One of the interests of this work (amongst many others) is to prove that the analog systems simulation approach is more suitable than its numerical counterpart for the analysis of the striking and complex dynamics of non-linear parts/module of communication systems.

Keywords- Communication Systems; Shunt Colpitts Oscillator; Bifurcation; Chaos; Analog Systems Simulation Methods

I. INTRODUCTION The last decade has witnessed a tremendous attention on

the effects of nonlinearity in sinusoidal oscillators [1-23]. The interest devoted to these effects is explained by the rich and complex behaviour the oscillators can exhibited in their nonlinear states and also by the various technological and fundamental applications of such oscillators. Indeed, in their nonlinear states these oscillators can be exploited in many applications such as measurement, instrumentation and telecommunications. In their regular states, the oscillators can be used for instrumentation and measurements while the chaotic behaviour (irregular state) exhibited by the oscillators can be used in chaotic secure communication [24] just to name a few.

Concerning either shunt [1-6] or non-shunt [7, 8] structures of the Colpitts oscillators, some interesting works have been carried out. Reference [1] does consider an analytical approach based on asymptotic method to analyse the dynamics of the Colpitts oscillator. Bifurcation scenarios are obtained numerically to confirm the richness of the modes exhibited by the Colpitts oscillator. The extreme sensitivity of this oscillator to tiny changes in its parameters is shown. Reference [2] deals with the observation of chaos in the Colpitts oscillator. A model (set of equations) describing the autonomous states of the oscillator is proposed. A piecewise-linear circuit model is considered. Chaotic behaviour is observed numerically and experimentally. The authors of reference [3] focussed on the relationship between chaotic Colpitts oscillator and Chua’s circuit. They showed that the Colpitts oscillator might be


ISAST Transactions on Electronics and Signal Prosessing, No. 1, Vol. 2, 2008 Chedjou J.C. et al.: Performance Evaluation of Analog Systems Simulation Methods for the Analysis of Nonlinear and Chaotic Modules in Communications71

mailto:[email protected]





mapped to the Chua’s oscillator with an asymmetric nonlinearity. Some bifurcation structures of the oscillators are obtained and the structure under consideration exhibits striking phenomena amongst which period doubling scenarios to chaos are observed. Reference [4] develops a methodological approach to the analysis and design of a Colpitts oscillator. A nonlinear approach for the two quasi sinusoidal and chaotic operating modes was considered. In particular, the generation technique of regular and irregular (chaotic) oscillations in terms of the circuit parameter was shown. Reference [5] considers non-smooth bifurcations in piecewise-linear model of the Colpitts oscillator. An approximate 1D map was proposed for predicting border collision bifurcation (common in power electronics) of the Colpitts oscillators. In reference [6] one considers the modelling of chaotic generators using microwave transistors. The transition from a simplified mathematical model to a model of RF chaotic source or microwave band is discussed. Reference [7] exploits a Lur’e system form to clarify the occurrence of chaos in the Colpitts oscillator. Components of the autonomous chaotic Colpitts oscillator causing the variation of equilibrium points are identified. This study was extended by the same authors to the case of the forced (non-autonomous) Colpitts oscillator. Both amplitude and frequency of the external excitation were used for chaos control in the oscillator. The dynamic maps of locking, transition, and normal areas, with their related frequencies and output powers, were depicted by measurements. Simulation and experimental results of injection-locked behaviour were discussed and presented.

The works summarised above show the Colpitts oscillator (either a shunt or non-shunt structure) as a chaotic generator. It appears that the non shunt structure of the Colpitts oscillator has been intensively considered while the literature is very poor concerning information related to the shunt structure of such an oscillator. It has been shown that tiny changes (imperfection or instability in the shunt structure of the oscillator) in the parameters values can generate the chaotic behaviour of the oscillator. One of the advantages of the shunt Colpitts oscillator (amongst many others) can be found in practical realisation. Indeed, this structure is simple to be realised. Moreover, the good stability of the fundamental characteristics (that is the amplitude and phase) of the waveforms generated by such a structure is due to the fact that the biasing current is not flowing in the oscillatory network as observed in the non-shunt structure. These are some advantages of the shunt structure of the Colpitts oscillator.

This paper considers the shunt structure of the Colpitts oscillator. To the best of our knowledge approximate analytical results available in the literature concerning such a structure are obtained from analysis tools based on Lur’e system forms. These models were used to study the stability conditions of regular oscillations and the possible appearance of chaos in the shunt structure of the Colpitts oscillator. Nevertheless the literature does not propose a direct modelling of the shunt structure of the Colpitts and does not analyse the chaoticity (degree of chaos) of the oscillator from the real model of the

oscillator. Our aim in this paper is to list some difficulties that can be faced when performing analog experimental simulations and propose appropriated methods to tackle them. We also contribute to the general understanding of the behaviour of the shunt structure of Colpitts oscillator and complete the results obtained so far by a) carrying out a systematic and methodical analysis of its nonlinear dynamics; b) providing both theoretical and experimental (analog) tools, which will be of precious use for design and control engineers since they can be used to get full insight of the nonlinear dynamical behaviour of the oscillator; c) pointing out some of the unknown and striking behaviour the shunt Colpitts oscillator.

One of the traditional properties of sinusoidal oscillators is the possibility of adjusting the frequencies of the waveforms generated starting from RC or LC resonators components. Nevertheless such an operation becomes very delicate or obsolete when the effects of nonlinearity are taken into account, since a rigorous analysis shows the strong dependence upon nonlinearity of the fundamental characteristics (that is both amplitude and frequency) of the waveforms generated. This can clearly be demonstrated by performing a rigorous analysis to express the fundamental characteristics of the waveforms generated. in terms of the parameters of the bipolar junction transistor responsible of the nonlinearity phenomena observed in the structural behaviour of such oscillators.

The structure of the paper is as follows. Section 2 presents the theoretical methods versus analogue methods. An evaluation of some difficulties currently faced when performing each of these methods is carried out. Some appropriate solutions are proposed to tackle these difficulties. In section 3 we use a simpler model of the bipolar junction transistor for modelling. The state equations of the circuit designed corresponding to the shunt structure of the Colpitts oscillator are obtained. The numerical simulation is carried out and various bifurcation diagrams associated to their corresponding graphs of largest one dimensional (1D) Lyapunov exponent are obtained showing both complex and striking scenarios to chaos. Section 4 exploits the Pspice software to simulate the dynamics of the shunt structure of the Colpitts oscillator using the “trial and error” approach. The chaotic behaviour of the oscillator is observed. In section 5, experimental measurements on a real circuit are performed to confirm the results from both numerical and Pspice simulation tools. Section 6 deals with conclusions and proposals for further works.

II. EFFICIENCY OF THE THEORETICAL METHODS VERSUS THE ANALOG METHODS: EVALUATION AND SOME PROPOSALS

A. Theoretical methods Vs. Analog methods We briefly describe and compare both analog and

numerical methods. Invariably the question arises - Which is better, analog or numerical method? Our wish is to present both limits and advantages of each of the methods in order to maintain opened the answer to the above question. We present


and prescribe some practical advises when dealing with analog implementation techniques. Our aim is to encourage engineers to use these techniques for the analysis of nonlinear models despite some practical difficulties faced (when performing these techniques) such as saturation and offset phenomena of the discrete components (diodes, transistors, operational amplifiers, and multipliers) of the electronic circuits. In addition to these difficulties is the dependence of the accuracy of the analog techniques upon the precision and stability of the electronic components. Hopefully, by proposing some techniques to tackle the difficulties encountered during analog implementations this will encourage interested researchers to use analog systems simulation techniques for the analysis of nonlinear problems.

Theoretical approaches (analytical and numerical methods) are commonly used to investigate the dynamics of nonlinear systems. However, the problems faced when performing numerical simulation methods are well-known: a) lack of method to choose the appropriate numerical integration step size, b) lack of method to determine the duration of the transient phase of a numerical simulation, and c) numerical simulation of complex dynamical systems is very time consuming while compared to its analog counterparts that are very fast [24]. Though the analog implementation is always limited by the saturation and offset phenomena of analogue devices such as operational amplifiers (LM741 and LF351) and multipliers (AD-633JN), it does however offers good ways to tackle the above difficulties faced by the numerical analysis. Analytical methods can provide only approximate solutions of nonlinear dynamical models while analog methods give exact solutions. These are some major reasons for the increasing interest devoted to this type of simulation for the analysis of nonlinear and chaotic physical systems [24 – 28]. In fact, a properly designed circuit can provide sufficiently good real-time results faster than a numerical simulation on a fast computer [24]. Such a circuit must use high precision resistors and capacitors. In addition, the offset voltage of the operational amplifiers and multipliers must be well controlled.

The analog techniques do not take into account the notion of “algorithm” and there is no need to translate quantities into appropriate symbolic forms. For these techniques, variables are represented by physical quantities on which the operations are performed. The simulation is carried out by some physical systems that obey the same mathematical relations that control the physical or technical phenomenon under investigation [29]. This procedure is in some sense more natural to both physicists and engineers [30]. A virtue of the analog techniques is that their basic design concepts are usually easy to recognize. What goes on inside is understandable since it is an analog of the real system whereas the numerical type simulator is a product of pure logic. It cannot be described as similar to something with which we are familiar [31, 32].

Therefore, although the numerical analysis had long ago superseded the analog techniques due, to a large extent, to the spectacular development of digital technology, we still believe

that analog techniques might bring some fresh air to the theory of computation in the field of non linear dynamics.

B. Practical problems and advices Concerning the effects of nonlinearity in an unspecified

sinusoidal oscillator, we mention that they come mainly from the discrete components (bipolar junction transistor (BJT), operational amplifier (Opamp), or analog circuits multipliers) constituting the gain elements. Thus, the effects of nonlinearity will be differently perceived depending upon the type of gain element. Some recent scientific contributions [1-4] have presented analytical approaches to explain the nonlinear behavior exhibited by classical oscillators using bipolar junction transistors. The exponential dependence of the current flowing through the collector of the BJT with respect to the voltage drop between its base-emitter regions was presented as the origin of nonlinearity in the oscillators. Bifurcation structures were presented, showing the coexistence between the regular states (limit cycles) and the irregular states (chaotic) of the oscillators.

The difficulties faced by theoretical (analytical and numerical) analyses methods are avoided when performing analog implementation techniques. Nevertheless various practical problems are currently encountered during the implementation of analog circuits. Among these problems are some that automatically induce errors in analog calculations. The development below lists some practical problems encountered and proposes solutions to tackle each of them.

• Offset phenomenon:

This phenomenon is the presence of a static voltage at inputs of analog devices (such as operational amplifiers (UA741), circuits multipliers (AD633JN), ...…) when they are biased by a DC voltage source. The offset phenomenon that occurs at outputs of the analog devices usually[39]. We have demonstrated the cancellation technique of this phenomenon in Ref. [24]. Indeed, such phenomenon can be compensated by using a compensation array that consists of monitoring a precise potentiometer to reduce the effect of the phenomenon on the dynamical behavior of the analog devices [24, 39]. The steps to perform offset cancellation are threefold. We use a potentiometer (P) having three points amongst which the middle point is movable between the two others that are fix points. We connect the fix points of (P) to the pin numbers 1 and 5 of the Opamp (for example: UA741 or LF 351). The second step is the connection of the movable point of (P) to a DC voltage source (biasing for instance). By monitoring a potentiometer (P), we measure the evolution of the DC voltage both at inputs and output of the Opamp. The situation where the movable point is very close to the pin numbers 1 and 5 should be avoided. This can lead to the destruction of the device (Opamp) due to a simultaneous presence of the entire value of the bias both at pins 1 and 5. The potentiometer is monitored to transform the magnitude of the input voltages of Opamps into almost the same order. The offset cancellation becomes very complex when the electronic circuit is of a self-


sustained type. In this case, the voltages at inputs of the analog device can be a direct consequence of the self-sustained character of the circuit. When the value of the self-sustained voltage at inputs of analog devices can be predicted (by a circuit theory analysis or by taking into account the expected performances of the circuit that may be defined before the realization of the circuit) the offset cancellation method can be used to fix the predicted values.

• Saturation phenomenon:

The dynamics of analog circuits is limited by the value(s) of the DC voltage(s) source(s) used for biasing. The saturation phenomenon occurs when a signal of magnitude greater than the value(s) of the DC voltage(s) source(s) used for biasing is found at a given point in the electronic circuit. To overcome the saturation problem, the scaling factor process is applied by rescaling the state voltages at different points of the electronic circuit in order to fit within the biasing range. We use a “Static Check” to verify if the system has been wired correctly. By tracing through the system we can calculate what the output voltage of each component should be. If it is determined that all outputs are of correct magnitude and sign (when measuring them), it can be safely assumed that the system is wired correctly.

• Power transfer:

When the electric current is flowing from one electronic network (transmitter) to another (receiver), the power is transferred in the same direction. A problem may arise where the power is not transferred. This can be explained by the fact that the dynamical resistor at the output of the transmitter is not of the order of that at the input of the receiver. Such a problem can be solved by adapting the total dynamical resistor (impedance) between the two electronic networks. This is generally achieved by adding, in parallel, a dynamical resistor at the output of the first electronic network (transmitter) or at the input of the second electronic network (receiver). Also, the power may not be transferred because the connection between the transmitter and the receiver is open. This problem can be detected by measuring the voltage at each point of the analog circuit.

• Defective electronic devices:

Damage of analog devices is generally caused by their wrong supply (biasing) or by a complete involuntary shunt performed between their inputs. When defective, the temperature within the analog devices may be very high. This may be a usual checking for some analog devices such as Opamps and analog multipliers AD633JN. Modules for a direct test of defective components are available. Concerning Opamps, they can be tested as voltage device followers. Analog circuit multipliers can also be tested by loading their inputs by well-known electrical signals. Nevertheless, a situation may occur where despite the fact that the analog devices (Opamps and analog multipliers) are not defective there is no signal at their outputs or the signals at outputs are not those expected. This is a classical problem related to the

bandwidth of the analog devices used that may be controlled when performing analog experimentation.

• Time scaling:

It is well-known that the state accuracy of electronic circuits depends on the accuracy of their electronic components (Opamps, analog multipliers, resistors, capacitors, …) [24, 39]. Yet, the dynamics of electronic circuits is limited by the frequency bandwidth of the analog devices (Opamps, multipliers,…). When an analog device operates within a range of frequency not included in its bandwidth, this affects the behavior of the electronic circuit containing this device and, consequently the results obtain are not correct. The time scaling process offers to the analog devices (e.g. Opamps, Circuit multipliers, …) the possibility to operate under their bandwidth. This process is currently used to restrict the high frequencies into low frequencies and inversely, this depending upon the frequency bandwidth of the analog devices in order to expect their good functioning. The time scaling process is also of high importance while performing analog simulation. It offers the possibility to simulate the behavior of the system at very high frequencies by performing an appropriate time scaling that consists of expressing the real time variable t versus the analog simulation time variable τ ( )τa−=10te.g. , allowing the simulation frequency to be

times less than the real frequency. Here, a is positive integer depending on the values of the resistors and capacitors used in the analog simulator. One of the advantages of time scaling amongst many others is the possibility it offers to the integrators to manage both high and low frequencies signals. Time scaling also allow the simulation of either high frequency or large broadband phenomena using analog devices (Opams, analog multipliers,…) that operate in a restricted frequency bandwidth [39].

a+10

III. IMPORTANT CONTRIBUTION Some interesting proposals were presented to tackle

problems encountered by numerical simulation. Concerning the problem due to the integration discontinuities related to the choice of the numerical integration step size, Thomas Rübner-Petersen [33] proposed an efficient algorithm using backward time-scale differences for solving stiff differential-algebraic systems. The proposed approach has computational advantages in simplicity and flexibility with respect to variations of the integration order. In fact, this algorithm allows the order within each step to be changed in an optimal way between k and

1+k . The implementation of the algorithm is described as part of a nonlinear analysis program, which has proved to be quite efficient for simulations of electronic networks. This program provides parameters in the DC analysis mode to be varied with automatic control of the step size. We have found that though the proposed method is very interesting in solving the numerical convergence problem since it varies automatically the step size to obtain an appropriate converging one, it requires very long integration time. Moreover, the


integration duration may become even much larger because it increases with increasing nonlinearity in the system under investigation.

The community providing usable technical solutions for computer based design (or CAD) has proposed the possibility of using GEAR algorithm [34] in Spice to overcome the divergence problem due to an inappropriate choice of the integration step size [35]. The proposed method, though quite interesting, is limited by the fact that the simulation using Spice is still a theoretical analysis because the characteristics (or the internal parameters) of the analog components (Diodes, Transistors, Operational Amplifiers, and Multipliers) are chosen to be ideal (that is are transportable to real compnents which are generally far from being ideal). In addition to this, Spice is emulation and the calculations it performs are done through algorithmic processes on a computing platform of the Von Neuman type.

Mention that Pspice and Simulink are calculation tools that are currently used for analog analysis rather than a real physical implementation (see the subsection below). These simulation tools are purely theoretical and still rely on some form of numerical computation in the background. Furter, the analog components they use are generally considered in the states where their characters are ideal. This would have been an advantage since the results obtained are of very good accuracy. Unfortunately the simulation of complex dynamical systems using these simulation tools is very time consuming (due to the still numerical computations in the background). Nevertheless, it is clear and sufficiently convincing that analog systems simulation (eitheir analog simulation of the circuits or the direct implementation of the circuits) is more suitable than its other counterparts for the analysis of complex nonlinear phenomena. It is a very precious tool for reliably detecting some strange phenomena such as chaos, modulation, demodulation and also synchronization, to name a few.

IV. SAMPLE RESULTS TO ILLUSTRATE THE CONCEPTS

We consider the shunt type structure of the Colpitts oscillator. The interest devoted to this oscillator is its possibility to behave chaotically both at low and high frequencies. The stability of the shunt type oscillator is also of high importance since it allows an efficient exploitation of such a structure in instrumentation, measurement and telecommunication. Our aim is to propose a shunt type structure of the Colpitts oscillator of practical interest to enrich the literature concerning nonlinear oscillators. The proposed structure might fulfill the above requirements. We show that the proposed structure can be realized experimentally. We also show that analog systems simulation (that is both analog simulation design and direct implementation with analog electronic components) are very suitable to get full insight of the behavior of the oscillator.

A. Circuit description Fig. 1 is the design of the shunt structure of the Colpitts

oscillator under investigation. The bipolar junction transistor used in the common-base-configuration, plays the role of

nonlinear gain element. The feedback network consists of the inductor

1Q

L and the capacitors and . These capacitors

act as voltage divider. is a coupling capacitor which may be of very low impedance within the frequency bandwidth in which the oscillator operates. The biasing is provided by the DC voltage source . is an ideal current source. The difference between the series type and the shunt type Colpitts oscillators is that the biasing current doesn’t flow through the feedback network in the latter type.

1C 2C

3C

CCV 0I

Figure 1. Circuit diagram of the shunt type Colpitts oscillator

The fundamental frequency of a shunt type Colpitts

oscillator can be estimated as follows:

10

1 2

12

C CfLC Cπ+

= 2 (1)

In the structure of Fig. 1, the nonlinear device is the bipolar junction transistor , which nonlinear character is responsible of the chaotic behaviour exhibited by the electronic circuit. The transistor Q2N3904 is chosen for the investigations. The choice of this type of transistor is motivated by the fact that the dynamical input impedance

1Q

121

11

+=

hh

Zinput (2)

is expressed in terms of the hybrid parameters which values give an appropriate value of the dynamical input impedance that allows a good power transfer. Another interest on the Q2N3904 is its availability in Pspice simulation package. Therefore the results from Pspice can be compared with experimental results. Three steps are considered for the investigation of the dynamical behaviour of the shunt type Colpitts oscillator: (1) modelling of the oscillator and analysis of chaotic behaviour exhibited by the oscillator; (2) Pspice simulation of the oscillator using the ‘trial and error’


approach; (3) direct implementation of the oscillator. These steps allow the confirmation or validation of the results obtained concerning the behaviour of the oscillator.

B. Circuit modelling and chaotic behavior • Circuit modeling

The BJT operates in just two regimes namely either the forward conducting or the non-conducting one. Therefore, for theoretical analysis, a simplified model [1] consisting of one current-controlled source and a single diode with exponential characteristic is convenient. Under these assumptions, the emitter and collector currents are defined as follows:

exp 1BEE S

T

VI IV

⎛ ⎞⎛ ⎞= ⎜ ⎜ ⎟

⎝ ⎠⎝ ⎠− ⎟

E

(3a)

C FI Iα= (3b)

where is the saturation current of the base-emitter junction

and is the value of the thermal voltage at room temperature.

SImVVT 26=

Fα denotes the common-base short-circuit forward current gain of the BJT.

We emphasize that the idea is to select the simplest possible model which maintains the essential features [16] exhibited by the real circuit. If we denote by the current

flowing through the inductor LI

L , and the

voltages across the capacitors C , the Kirchhoff Current Laws (KCL) can be applied to the circuit of Fig. 1 to obtain the following set of differential equations describing the evolution of the voltages .within the electronic circuit:

( )3,2,1=iVi

i

iV

21 VVdt

dIL L += (4a)

(4b)

3211

1

LEF

CC

RIIR

VVVVdt

dVRC

−−

+−−−=

α

( )[ (4c) 1 0

3212

2

IIIR

VVVVdt

dVRC

LEF

CC

−−−−

+−−−=

α ]

(4d) 3213

3 EFCC IRVVVVdt

dVRC α−−−−=

Assuming 1=Fα (this is well justified for most transistors

with 200100 ≤≤ Fβ ) and introducing the following dimensionless quantities:

T

L

Vi

xρ

= (5a)

TV

Vy 1= (5b)

TV

Vz 2= (5c)

TV

V3=ϑ (5d)

where 2LC=τ , τ

θ t= ,

2CL

=ρ , 1

21 C

C=ε ,

3

22 C

C=ε ,

T

S

VIρ

γ = , Rρδ = ,

T

CC

VV

=σ , and

TVI 0ρ

ζ = , Eqs. (4) can be transformed into the following

first order differential equations:

zyddx

+=θ

(6a)

( )[ ])(.1 zfzyxddy

−−−−+−= ϑσδεθ

(6b)

( ) ζϑσδθ

−−−−−= xzyddz . (6c)

( )[ ])(..2 zfzydd

−−−−= ϑσδεθϑ

(6d)

where ( )f z is the exponential function derived from Eqs. (3) and expressed as follows: ( )( ) exp( ) 1)f z zγ= − − (6e) The model described by Eqs. (6) is the one we propose for the investigation of the dynamical behaviour of the structure under consideration of the shunt type Colpitts oscillator.

• Chaotic behavior

Eqs. (6) are solved numerically to define routes to chaos. We use the fourth-order Runge-Kutta algorithm [36, 37] for the sets of the parameters used in this work, the time step is always 005.0≤Δt and the calculations are performed using real variables and constants in extended mode. The integration time is always . Here, the types of motion are identified using two indicators. The first indicator is the bifurcation diagram, the second being the largest 1D numerical Lyapunov exponent denoted by

610≥T

[ ]⎥⎦⎤

⎢⎣⎡=

∞→ ttd

t

)(lnlimmaxλ (7a)

where


( ) ( ) ( ) ( )2222)( δϑδδδ +++= zyxtd (7b) and computed from the variational equations obtained by perturbing the solutions of Eqs. (6) as follows: xxx δ+→ ,

yyy δ+→ , zzz δ+→ , and δϑϑϑ +→ . is the distance between neighbouring trajectories [38]. Asymptotically,

)(td

( ttd .exp)( max )λ= . Thus, if 0max >λ , neighbouring trajectories diverge and the state of the oscillator is chaotic. 0max <λ , these trajectories converge and the state

of the oscillator in non-chaotic. 0max =λ for the torus state of the oscillator [38]. Setting the values of the components of Fig. 1 ( , , VVCC 12= mVVT 26= nFC 1002 = , FC μ223 = ,

FL μ470= , , , and

) we analyze the effects of the capacitor (or

the parameter

fAI S 734.6= mAI 50 =Ω= 318R 1C

1ε ) on the behaviour of the oscillator. Therefore, a scanning process is performed to investigate the sensitivity of the oscillator to tiny changes in (1C 1ε ). The investigations are carried out in the following windows:

FCF μμ 155 1 ≤≤ , and FCF μμ 2515 1 ≤≤ .

Figure 2. a)- Bifurcation diagram of the current flowing through the

inductor

LiL in terms of the feedback divider capacitor for

1CFCF μμ 155 1 ≤≤

Considering the effects of the capacitor , it appears that the structure of the oscillator in Fig. 1 leads to complex dynamical behaviour, such as torus, multi-periodic, quasi-periodic, and chaotic states. We observe various routes to chaos (such as sudden transition, period adding, period-doubling, torus breakdown, or quasi-periodic routes) with

several kinds of periodic and multi-periodic windows. Fig. 2 provides some sample results, showing the bifurcation diagram

1C

[ ])(),(1 mAinFC L when FCF μμ 155 1 ≤≤ . Period-doubling routes to chaos are shown both with increasing and decreasing . Also shown is a period 2 sudden transition route to chaos.

1C

Figure 3. a)- Bifurcation diagram of the current flowing through the

inductor

LiL in terms of the feedback divider capacitor for

1CFCF μμ 2515 1 ≤≤

Various tiny windows of chaotic states of the oscillator are shown alternating with windows of regular motion. The weak chaoticity (degree of chaos) of the oscillator is shown. This is clearly demonstrated by the small values of the largest 1D numerical Lyapunov exponent that were always less than 0.0185 for FCF μμ 155 1 ≤≤ . Fig. 3 shows the bifurcation

diagram [ ])(),(1 mAinFC L for FCF μμ 2515 1 ≤≤ . A period-adding scenario to chaos (period 5→period 7→chaos) is shown. Also shown is the period 4 sudden transition route to chaos that occurs in a tiny window of found between 17.5nF and 18nF. The weak chaoticity of the oscillatior is also shown.

1C

We have drown in Fig. 4 some phase portraits of the current flowing in the inductor Li L for sample values . This figure confirms the period-doubling scenario to chaos shown by the bifurcation diagrams. The following transition: period 1→ period 2→period 4→perid 8→chaos is clearly shown. The attractor of period-8 is not shown because of its high instability due to the tiny window within which it co-exits both with period 4 and chaotic attractors situated respectively at left and at right of the value

1C

nFC 91 = as clearly shown in Fig. 2a.


Figure 4. Numerical phase portraits of : a)- Period-1 or limit cycle,

, b)- Period-2 ( , c)- period-4

, and d)- chaos ( )

Li( 4.7nF1 =C ) )

)nFC 0.81 =

( nFC 5.81 = nFC 0.181 =

Different routes to chaos observed in the shunt structure of the Colpitts oscillator are commonly observed in nonlinear systems, such as forced systems, coupled autonomous systems, and coupled forced systems [38], to name a few. This serves to justify the richness of the bifurcations in the shunt Colpitts oscillator and also the striking phenomena exhibited by such oscillators. The model proposed for the shunt colpitts oscillator has been computed numerically to have full insight of the behaviour of the oscillator. The simulation on Pspice is performed to verify the numerical results obtained and also to validate the proposed model for the shunt type Colpitts oscillator.

C. Pspice simulation of the oscillator We use the same model of the bipolar junction transistor

defined in the preceding section namely the Q2N3904. The circuit of Fig. 1 is implemented in Pspice. Here, the “trial and error” approach [12-13] is substantially exploited. The following values of the circuit components are defined in Pspice to obtain some phase portraits showing the evolution of the typical phase-space trajectories of the current flowing

through the inductor: , ,

,

LiVVCC 12= mVVT 26=

nFC 1002 = FC μ223 = , FL μ470= , 210=Fβ ,

, and . The numerical phase portraits mAI 50 = Ω= 600R

Figure 5. Phase portraits of in Pspice: a)- Period-1 or limit cycle, Li( )nFC 7.41 = , b)- Period-2 ( )nFC 0.221 = , c)- period-4

( )nFC 251 = , and d)- chaos ( )nFC 471 =

shown in Fig. 5 are obtained from the Pspice simulation. These phases portraits are qualitatively similar to those obtained numerically. Moreover the sequence of bifurcation shown numerically (period 1→ period 2→period 4→perid 8→chaos) is confirmed by the Pspice simulation. The results from Psipce simulation were generally in very good agreement with those from the numerical analysis despite the divergence observe in the values of the bifurcation points (values ). This divergence can be explained by the (real) characteristics of the bipolar transistor used for/in Pspice simulation. It is well-known that the characteristics of the transistor used (that is the Q2N3904) are predefined and stored in Pspice simulation package some of them being considered as ideal. Moreover, in order to understand the operation mode of the BJT in the oscillator, further simulations were performed using two models of the BJT: a)- the Ebers-Moll model and b)- the even simpler transistor model consisting of a simple diode and a current controlled source. The results obtained in both cases were similar to the one previously obtained using Pspice own model for the BJT. Thus the simpler model is adequate for investigating the essential behaviour of the system. This makes it possible later to adopt relatively simple state equations to describe the oscillator. The divergence between both numerical and Pspice simulation results were explained by the non-real characteristics of the BJT used. This justifies the interest devoted to the real physical implementation of the shunt Colpitts oscillator since this method uses real electronic components and consequently the characteristics of the electronic components are real.

1C

D. Real physical implementation of the oscillator According to the previous results, the shunt type Colpitts

oscillator can exhibit complex and striking bifurcation scenarios leading to chaos, when the feedback divider capacitor is monitored. The study here is focussed on both 1C


design and analogue experimentation of the shunt type Colpitts oscillator. The experimental results obtained from a real implementation of the oscillator are compared with the results obtain by both numerical and Pspice simulation methods.

Figure 6. Experimental setup for measurements on the Shunt Colpitts oscillator.

Figure 6 is the proposed experimental setup for measurements on the shunt type Colpitts oscillator. This circuit is built on a breadboard. Fig. 6 shows the basic scheme of the shunt type Colpitts oscillator shown in Fig.1 with the following values of the circuit components: Ω= 600R ,

, nFC 1002 = FC μ223 = , FL μ470= , and

. The network consisting of the operational amplifier U

VVCC 12=1 with related resistors is an implementation of the

ideal current generator. If the following condition is fulfilled: 1 3

2 4

R R5R R R

=+

(8a)

The current pulled from the load is given by : 0I2

01 5

iR VI

R R= (8b)

where is the output voltage of the network using the operational amplifier U

iV2 with related resistors which electronic

function is an inverting amplifier. Therefore, with the values of the components in Fig. 6, the relationship between the control voltage and the current is : iV 0I

0 1000iVI = (8c)

Thus, is supposed to vary between 0 and 12mA as 0I10R varies between 0 and 100 since the inverting input of

UkΩ

2 is connected to -12V.

Figure 7. Experimental phase portraits of : a)- Period-1 or limit cycle, Li

)/1:.,/5:(7.41 divVYanddivmAXnFC = , b)- Period-2

)/2:.,/5:(221 divVYanddivmAXnFC = , c)- period-4

)/2:.,/5:(3.251 divVYanddivmAXnFC = , and d)- chaos

)/1:.,/5:(471 divVYanddivmAXnFC =

In order to investigate how the feedback ratio affects the dynamics of the circuit, is chosen as a control parameter.

The variation of is performed by connecting in parallel standard capacitor components to obtain the desired value. The value of the biasing current is set to 5mA (as in the

above Pspice simulations) using the resistor . A 1

1C

1C

0I

10R Ω

resistor is added in series to the inductor L to sensor it’s current . The experimental results are obtained by observing as a function of time the voltage across the inductor and by plotting phase-space trajectories ( , ) using the oscilloscope in the XY mode.

Li

Li Lv

As in the case of Pspice simulations the dynamical

behaviour of the oscillator changes substantially as is monitored. This is clearly demonstrated by the experimental pictures in Fig. 7 showing the real behaviour of the shunt type Colpitts oscillator proposed in this paper. As it appears in Fig. 7 the real circuit shows the same bifurcation scenarios as observed using both numerical and Pspice simulation methods. Figure 7 shows an evolution of starting from normal near-sinusoidal oscillations to chaos via a period doubling sequence when is increased. This evolution shows identical bifurcation scenarios (period 1→ period 2→period 4→perid 8→chaos) with those form the preceding simulation methods. Note that pictures in Fig. 7 are very close to the numerical phase portraits. This can be considered to validate the proposed model (Eqs. 6) for the investigation of the dynamical behaviour of the shunt type Colpitts oscillator. During our experimental investigations, we have also found

1C

Li

1C


period-adding and sudden transition scenarios to chaos exhibited by the system. These scenarios were also reported using both numerical and Pspice simulation methods. The experimental results were generally very close those obtained from these methods. A very good agreement is obtained while comparing the experimental values of the bifurcation parameter with the values from Pspice simulation. 1C

V. CONCLUSION This paper was motivated by the wish of encouraging

engineers to deal with analogue simulation. Nowadays, the revival of this method is encouraged due to the technological exploitation of analogue systems simulation in various fields namely telecommunication, biocomputing, traffic management, electronics (instrumentation and measurements), to name a few. The advantages and limits of analogue simulation were discussed and compared with those of its numerical counterpart. Some proposals to tackle some problems faced during the experimental realisation were presented. The concepts were illustrated by proposing a structure of the shunt type Colpitts oscillator. The choice of this type of oscillator was motivated by our wish to enrich the literature by showing the capability of the proposed oscillator to exhibiting very complex and striking phenomena. Three methods were considered during our investigations: the numerical, the Pspice simulations, and finally the real physical implementation of the proposed oscillator. These methods were compared to validate the results obtained. The KCL theorem was used to derive a model describing the dynamical behaviour of the oscillator. Taking one feedback divider capacitor as control parameter, bifurcation diagrams associated to their corresponding graphs of largest 1D numerical Lyapunov exponent were plotted to summarise the scenario leading to chaos. The studies revealed that the proposed configuration of the Colpitts oscillator can exhibit near-sinusoidal oscillations, quasi-periodic, multi-periodic, and chaotic oscillations. Very complex bifurcation structures were obtained: torus, period- adding, period- doubling, and sudden transition scenarios to chaos. The results from different methods were compared and a very good agreement was observed.

1C

An interesting question under investigation is that of finding the relationship between the loop gain and the dynamics of the oscillator. Another problem under consideration is that of coupling two identical chaotic oscillators of this type and searching for the synchronization threshold. Such an investigation is of high importance in many application areas such as chaotic secure communications where chaos synchronisation is being exploited in wave coding processes. It is also of particular interest to consider the implementation of analog methods exploiting the CNN technology for the analog simulation of very complex systems especially on VLSI chip implementations (for example on FPGA). This is particularly necessary when the number of analog nodes

(needed for simulating a given very complex system) is very high (many orders of magnitude)).

ACKNOWLEDGMENT J. C. Chedjou would like to acknowledge the financial

support from both the Swedish International Development Cooperation Agency (SIDA) through the International Centre for Theoretical Physics (ICTP), Trieste, Italy. Further, he does express his profound gratitude to the Institute for Smart-systems Technologies (IST), Faculty of engineering, University of Klagenfurt, Austria.

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[26] A. Azzouz, R. Duhr and M. Hasler, IEEE Trans. Circuits Syst., Vol. 30, pp.913-914, 1983.

[27] T. S. Parker and L. O. Chua, Proc. IEEE, Vol. 75, pp. 9821008, 1987.

[28] I. Jonshon. ‘’Analog Computer Techniques’’, New York: McGraw-Hill, 1963.

[29] S. Puchta, IEEE Annals of the History of Computing, Vol.18, pp.49-59, 1996.

[30] W. S. McCulloch and W. Pitts, Bulletin of the Mathematical Biophysics, Vol.5, pp.115-133, 1943.

[31] General Electric Management Consultant Services Division. The Next Step in Management and Aprisal of Cybernetics, 1952.

[32] L. Owens, IEEE Annals of the History of Computing, Vol.18, pp.34-41, 1996.

[33] T. Rübner-Petersen: An Efficient Algorithm Using Backward Time-Scaled Differences For Solving Stiff Differential-Algebraic System, Report 16/5 73, Institute of Circuit Theory and Telecommunication, Technical University of Denmark.

[34] C. W. Gear, “Simultaneous numerical solution of differential equations,” IEEE Trans. CT-18. No. 1, pp. 89-94, January 1972.

[35] J. Pierce, “The advantages of a Front-to-Back Flow for Windows-Based PCB Design,” Cadence Design Systems, Inc. pp. 1-6, 2002.

[36] J. S. Vandergraft, “Introduction to numerical computation,” Academic, New York, 1978

[37] J. C. Chedjou, K. Kyamakya, I. Moussa, H. -P. Kuchenbecker, W. Mathis, “Behavior of a Self-Sustained Electromechanical Transducer and Routes to Chaos,” Journal of Vibration and Acoustics, ASME transactions, Vol. 128, pp. 282-293, 2006.

[38] J. C. Chedjou, L. K. Kana, I. Moussa, K. Kyamakya, and A. Laurent, "Dynamics of a Quasi-periodically Forced Rayleigh Oscillator," Journal of Dynamic Systems Measurement and Control, Transactions on the ASME, vol. 128, pp. 600-608, 2006.

[39] J. C. Chedjou, K. Kyamakya, W. Mathis, I. Moussa, A. Fomethe, A. V. Fono, "Chaotic Synchronization in Ultra Wide Band Communication and Positioning Systems," Journal of Vibration and Acoustics, Transactions on the ASME (In Press, to appear in 2007).


Jean Chambe

Moussa Ildoko

High School Teacher (

rlain Chedjou received in 2004 his doctorate in Electrical Engineering at the Leibniz University of Hanover, Germany. He has been a DAAD (Germany) scholar and also an AUF research Fellow (Postdoc.). From 2000 to date he has been a Junior Associate researcher in the Condensed Matter section of the ICTP (Abdus Salam International Centre for Theoretical Physics) Trieste, Italy.

Currently, he is a senior researcher at the Institute for Smart Systems Technologies of the Alpen-Adria University of Klagenfurt in Austria. His research interests include Electronics Circuits Engineering, Chaos Theory, Analog Systems Simulation, Cellular Neural Networks, Nonlinear Dynamics, Synchronization and related Applications in Engineering. He has authored and co-authored 2 books and more than 22 journals and conference papers.

Kyandoghere Kyamakya obtained the M.S. in Electrical

Engineering in 1990 at the University of Kinshasa. In 1999 he received his Doctorate in Electrical Engineering at the University of Hagen in Germany. He then worked three years as post-doctorate researcher at the Leibniz University of Hannover in the field of Mobility Management in Wireless Networks. From 2002 to 2005 he was junior professor for Positioning Location

Based Services at Leibniz University of Hannover. Since 2005 he is full Professor for Transportation Informatics and Director of the Institute for Smart Systems Technologies at the University of Klagenfurt in Austria. Van Duc Nguyen received the Bachelor and Master of

Engineering degrees in Electronics and Communications from the Hanoi University of Technology, Vietnam, in 1995 and 1997, respectively, and the Doctorate degree in Communications Engineering from the University of Hannover, Germany in 2003. From 1995 to 1998, he worked for the

Technical University of Hanoi as an Assistant Researcher. In 1996, he participated in the student exchange program between the Technical University of Hanoi and the Munich University of Applied Sciences for one term. From 1998 to 2003, he was with the Institute of Communications Engineering, University of Hannover, first as a DAAD scholarship holder and then as a member of the scientific staff. From 2003 to 2004, he was employed with Agder University College in Grimstad, Norway, as a Postdoctoral Researcher. He was with International University of Bremen as a Postdoctoral

Fellow. In 2007, he spent 2 months at the Sungkyungkwan University, Korea, as a Research Professor. His current research interests include Mobile Radio Communications, especially MIMO-OFDM systems, and radio resource management, channel coding for wireless networks.

holds an MSc. in control and signal processing and a Doctorate degree in Electronics from the University of “Valenciennes et du Hainaut-Cambrésis” in France, respectively in 1982 and 1985. He is currently an Associate researcher at UDETIME (Doctorate School of Electronics, Information Technology, and Experimental Mechanics) at the

University of Dschang, Cameroon. Besides, he is a Senior Lecturer at the University of Yaoundé 1, Cameroon. He research interests are related to Nonlinear Dynamics, Analog Circuits Design, Chaos based Secure Communications. Kengne Jacques He obtained the diploma of Technical

DIPET II) from the Department Electrical Engineering (ENSET/ University of Douala / Cameroon) in 1995 and the Master of Science (M.Sc) degree from the faculty of Sciences/University of Dschang in 2007 both in Electronics. From 1995 up to now he has been working as a Technical high school teacher. He is currently an associate researcher at UDETIME (Doctorate

School of Electronics, Information Technology, and Experimental Mechanics) at University of Dschang. Mr. Kengne is a doctorate student in Electrical Engineering in the field of Non linear dynamics and its application in Communications.


Abstract— A multi-band and ultra-wideband (UWB) 3.1-4.8 GHz receiver front-end consisting of a fully integrated filter and triplexer network, and a flat gain low-noise amplifier (LNA) is presented in this paper. The front-end utilizes a microstrip network and three combined broadside- and edge-coupled bandpass filters to connect the three sub-bands. The LNA design employs dual-section input and output microstrip matching networks for wideband operation with a flat power gain and a low noise figure. The system is fully integrated in a four-metal-layer printed circuit board. The measured power gain is 10 dB and the noise figure of the front-end is 6 dB at each center frequency of the three sub-bands. The minimum isolation between the sub-bands is -27 dB and the isolation between the non-neighboring alternate sub-bands is -52 dB. The out-of-band interferer attenuation is below -30 dB.

Index Terms—Bandpass filter, broadside coupled, edge

coupled, frequency multiplexing, low-noise amplifier, matching network, triplexer, multi-band OFDM system, ultra-wideband, UWB.

I. INTRODUCTION he ultra-wideband (UWB) technology for short-range communication applications in the 3.1-10.6 GHz range

has been a target for intensive research in recent years [1]-[6]. The general interest in UWB technology from academia and industry has started in 2002 as the unlicensed UWB operation was permitted by the Federal Communication Commission (FCC) [1]. The FCC specifications included the spectral mask, and the bandwidth limitations of a UWB device, but not the type of modulation scheme or signal. As a result, to exploit the 7.5 GHz of spectrum different approaches have been proposed [2]-[3]. Currently, there are two dominating and technically very different versions to the UWB technology. One approach, known as “WiMedia UWB”, [4]-[5] is based on the multi-band orthogonal frequency-division multiplexing (OFDM) modulation technique. The other one is a single-band impulse-based or Direct Sequence UWB (DS-UWB) radio, as described in [2]-[3], [6].

The multi-band OFDM specification divides the frequency spectrum into 500 MHz sub-bands (528 MHz including guard carriers and 480 MHz without guard carriers). The first three sub-bands, known as the Band Group 1, cover the spectrum

Manuscript received Nov. 5, 2007. Ericsson AB in Sweden is

acknowledged for financial support of this work. Adriana Serban; email: [email protected], Magnus Karlsson; email:

[email protected], and Shaofang Gong are with Linköping University, Sweden.

from 3.1 to 4.8 GHz and are centered at 3.432, 3.960, and 4.488 GHz, respectively.

Due to the characteristics of the UWB signals, i.e., very low radiated power (-41.3 dBm/MHz) and large bandwidth (the minimum bandwidth is 500 MHz), a multi-band OFDM UWB receiver requires better receiver sensitivity and a lower noise figure than, for example, an IEEE 802.11a receiver [7]. The expected receiver sensitivity is around -70 dBm [8], and it can be achieved by an optimal design of the low-noise amplifier (LNA) in terms of wideband, near-to-minimum noise figure and reasonable power gain. Furthermore, an optimal integration of the entire RF front-end including the antenna can also contribute to better receiver sensitivity by minimizing losses. Another challenge of the UWB front-end design is caused by the problem of narrowband interferers, e.g., out-of-band, in-band, or other unintentional radiation of electronic devices [9]. In particular, services around 2.4 and 5 GHz (IEEE 802.15.1, IEEE 802.11a) can hinder the UWB communication and must be taken into consideration in UWB front-end implementation. One solution is to filter interferences at radio frequencies (RF) before or within the first amplification stage, e.g., LNA [9]. Two different techniques can be employed to achieve selective operation in different band groups or, more restrictive, within each frequency band group, i.e., in different sub-bands. In [10], LNAs with selective gain-frequency characteristics employ multi-resonance load networks which shape the LNA transfer function. These techniques require area consuming and complex LC (inductor and capacitor) load networks. Sometimes, they also need load center frequency control mechanisms by means of noisy switching pulses.

An alternative approach with a multi-band LNA covering the multi-band UWB is presented in this paper. It is a frequency-triplexed RF front-end using one 3.1-4.8 GHz LNA for Band Group 1 UWB systems. The proposed solution combines a multi-band pre-selecting filter function with the frequency multiplexing function to connect the three different RF inputs to only one LNA. The LNA is optimized for a near-to-minimum noise figure and flat gain response. The RF front-end is completely integrated into a four-metal layer printed circuit board and is dedicated to a complete integration of UWB antenna-LNA system on the same RF module.

A Frequency-Triplexed RF Front-End for Ultra-Wideband Systems 3.1-4.8 GHz

Adriana Serban, Magnus Karlsson, and Shaofang Gong, Member, IEEE

T

ISAST Transactions on Electronics and Signal Prosessing, No. 1, Vol. 2, 2008 Serban A.: A Frequency-Triplexed RF Front-End for Ultra-Wideband Systems 3.1-4.8 GHz


II. OVERVIEW OF THE UWB FRONT-END The proposed front-end shown in Fig. 1 consists of three

FMN

Matchingnetwork LNA Matching

network

UWB LNATriplexer

Metal 4Metal 1-2

FMN

Matchingnetwork LNA Matching

network

UWB LNATriplexer

Metal 4Metal 1-2

Fig. 1. Block diagram of the proposed multi-band UWB front-end. RF inputs for connecting antennas, a frequency multiplexing network (FMN) [11]-[12] and a 3.1-4.8 GHz flat-gain LNA. A selective multi-band operation is automatically achieved within the FMN block. The antenna system and the triplexer have also been studied, and the results are presented in [11]-[12].

A. Triplexer network The frequency multiplexer, i.e., the triplexer in this case, is

used between the three RF inputs and the UWB LNA to simultaneously filter the potential out-of-band and in-band interferers and for the multiplexing purpose. Fig. 2 shows the schematic of the proposed triplexer network realized with the microstrip technology. The triplexer consists of three bandpass filters, three transmission lines for filter tuning, and three series quarter-wavelength (λ/4) transmission lines.

The bandpass filters for multi-band UWB applications require 500 MHz bandwidth at each center frequency of 3.432, 3.960 and 4.488 GHz. They are implemented as fifth order broadside- and edge-coupled filters. The filter tuning lines optimize the stop band impedance of each filter to provides a high stop band impedance in the neighboring bands. The three series λ/4 transmission lines provide a high impedance at the respective frequency band.

Port 1

Port 2

Port 3

BPF

BPF

BPF

Triplexer output (to LNA)

sub-band #1

sub-band #2

sub-band #3

λ/4 @ 3.432 GHz

λ/4 @ 3.960 GHz

λ/4 @ 3.488 GHz

Stopband tuningtransmission line

Port 1

Port 2

Port 3

BPF

BPF

BPF

Triplexer output (to LNA)

sub-band #1

sub-band #2

sub-band #3

λ/4 @ 3.432 GHz

λ/4 @ 3.960 GHz

λ/4 @ 3.488 GHz

Stopband tuningtransmission line

Fig. 2. Principle of the triplexer.

Fig. 3 shows the principle of broadside- and edge-coupling techniques and the filter structure used in the UWB front-end. The start and the stop segments are placed on metal layer 1, while the rest of the filter is placed on metal layer 2.

Metal 1 Metal 2 Ground

Broadside coupling Edge coupling

Fig. 3. Filter structure: combined broadside- and edge-coupled filter.

The λ/4 network and the bandpass filters are optimized simultaneously for uniform passband performance within the sub-bands. Furthermore, since the sub-bands are so close with each other, a sharp bandpass transfer function was prioritized over low reflection for optimal filtering of potential interference at radio frequencies.

B. UWB LNA Optimally, wideband LNA design methodologies should

provide improved receiver sensitivity and thus accurate low-level signal processing. The UWB LNA design handles trade-offs among LNA topology selection, wideband matching for near-to-minimum noise figure, flat power gain, and wideband bias network design [13]. In addition, as any loss that occurs before the LNA in the system will substantially degrade the noise figure of the front-end, the LNA and the antenna system should be designed simultaneously and preferably integrated on the same substrate.

The presented UWB LNA is designed for a noise figure below 4 dB and a flat transfer function over the 3.1-4.8 GHz bandwidth. The wideband amplifier topology relies on reactive matching networks of low nodal quality factor [14]. Other classical broadband amplifier topologies, such as amplifiers using negative-feedback or distributed amplifiers result in increased noise figure and/or increased consumption of power and area. Using matching networks implemented with microstrip lines, the large variation of noise figure and power gain due to tolerances of the discrete components can be avoided. The area occupied by distributed matching networks is in this case not critical as the front-end module area is dominated by the triplexer area. The simplified schematic of the UWB LNA is presented in Fig. 4.

S2PRF In RF Out

Rstab

Data File component

Input matching network

Microstrip stubs

Output matching network

S2PRF In RF Out

Rstab

Data File component

Input matching network

Microstrip stubs

Output matching network

Fig. 4. Microstrip UWB LNA, simplified schematic.


84

The active device (MAX2649) of the amplifier is

represented as a two-port network (*.s2p) containing the measured noise and S-parameters provided by MAXIM Inc.

III. UWB FRONT-END MANUFACTURING AND EVALUATION

A. UWB front-end manufacturing The manufactured front-end includes a triplexer network

and a UWB LNA. The triplexer photograph is shown in Fig. 5a. Three SMA connectors mounted from the side, Ports 1-3, connect the three sub-band RF inputs. Port 4 is soldered at the output of the front-end. The photograph of the LNA, including the broadband bias network using butterfly radial stub [13] is presented in Fig. 5b. The UWB LNA is integrated on the back side of the four-layer board. The LNA input is connected to the triplexer output by a via hole.

The prototype has a size of 90 x 58 mm, but note that the actual design only partially fills the printed circuit board. In addition, separate triplexer and UWB LNA modules were fabricated.

Port 1

Port 2

Port 3

Port 4Broadbandbias network

Port 1

Port 2

Port 3

Port 4Broadbandbias network

(a) Photo of the front-side. (b) Photo of the back-side.

Fig. 5. Frequency triplex UWB front-end: (a) photo of the triplexer implementation, front side, and (b) photo of the UWB LNA, back side.

The three-dimensional (3-D) wideband integration using a four-layer PCB is a challenging task since the electrical performance can be degraded by parasitic and radiation losses. To realize it, electromagnetic (EM) simulations were performed. All prototypes were manufactured using a four metal layer printed circuit board. Two dual-layer RO4350B boards were processed together with a RO4450 prepreg, as shown in Fig. 6.

S-parameter measurements were done with a Rhode&Schwartz ZVM vector network analyzer. Agilent’s N8974A Noise Figure Analyzer is used to measure the noise figure of the LNA and of the UWB front-end.

The RO4450 prepreg is made of a sheet material (e.g., glass fabric) impregnated with a resin cured to an intermediate stage, ready for multi-layer printed circuit board bonding.

RO4350B RO4450B RO4350B

Metal 1: Triplexer Metal 2: Triplexer Metal 3: Ground Metal 4: LNA

RO4350BRO4450BRO4350B

Fig. 6. Printed circuit board structure.

Table 1. Printed circuit board parameters Parameter (Rogers 4350B) Dimension Dielectric height 0.254 mm Dielectric constant 3.48±0.05 Dissipation factor 0.004 Parameter (Rogers 4450B) Dimension Dielectric height 0.200 mm Dielectric constant 3.54±0.05 Dissipation factor 0.004 Parameter (Metal, common) Dimension Metal thickness, layer 1, 4 0.035 mm Metal thickness, layer 2, 3 0.025 mm Metal conductivity 5.8x107 S/m (Copper)Surface roughness 0.001 mm

Table 1 lists the printed circuit board parameters, and Fig. 2

illustrates the stack of the printed circuit board layers. Metal layers 1 and 4 are thicker than metal layers 2 and 3 because the surface layers are plated twice while the embedded metal layers 2 and 3 are plated once.

B. The triplexer Fig. 7a shows forward transmission |S21| measurement of

the triplexer in the antenna system. A rather flat response is seen. The transmission line network is optimized together with the filters to achieve a high blocking of neighboring bands. The measured total insertion loss is 3.0-3.5 dB for the three sub-bands. All sub-bands have at least 500 MHz bandwidth at the -3 dB criterion, i.e., less than 3dB variation within the desired frequency spectrum. Fig. 7b shows the isolation between the multiplexed ports. It is seen that the minimum isolation is -23 dB. The minimum isolation occurs between the neighboring sub-bands, and in the remaining spectrum the isolation is better than -23 dB. Furthermore, the isolation between the non-neighboring sub-bands, |S31|, is –51 dB.

2 3 4 5 6-80-70-60-50-40-30-20-10

0

Forw

ard

trans

mis

sion

(dB

)

Frequency (GHz)

Sub-band #1 Sub-band #2 Sub-band #3

(a) Measured forward transmission.


85

2 3 4 5 6-80-70-60-50-40-30-20-10

0

Forw

ard

trans

mis

sion

(dB

)

Frequency (GHz)

S21 (dB) S32 (dB) S31 (dB)

(b) Measured isolation.

Fig. 7. Measured performance of the triplexer: (a) Forward transmission, and (b) Isolation.

C. The UWB LNA Fig. 8a and 8b show the measured performances of the LNA. The selected topology was optimized for a near-to-minimum noise figure and the maximum flat power gain. The measured forward transfer coefficient |S21| is greater than 13 dB with a ± 0.6 dB variation. Measured noise figure is smaller 4 dB over the three sub-bands, between 3.1 and almost 4.8 GHz. The measured noise figure follows the minimum noise figure of the simulated device with some deviation at the upper frequency edge. The supply voltage is 3 V and the consumed current is 13 mA.

2 3 4 5 60

5

10

15 Front-end NF LNA NF LNA NF-min

Noi

se fi

gure

(dB)

Frequency (GHz)

(a) Measured noise figure.

2 3 4 5 60

5

10

15

Forw

ard

trans

mis

sion

(dB)

Frequency (GHz)

(b) Measured forward transmission

Fig. 8. Measured performance of the LNA: (a) measured noise figure, (b) measured forward transmission.

D. UWB Front-End Evaluation Fig. 9a shows the forward transmission simulation results

of the UWB front-end, and Fig. 9b shows the corresponding measurement results. It is seen that all three sub-bands have at least 500 MHz bandwidth at -3 dB from the top, i.e., the maximum forward gain within respective sub-band. The measured overall gain is 3.1-3.5 dB lower than the simulated gain for the three sub-bands. This is mostly due to slightly higher insertion loss in the triplexer network than estimated by the simulations [12] and lower measured LNA gain compared to the simulated. Figs 9a and 9b also show how the proposed front-end effectively attenuates the out-of-band signals, e.g., the narrowband 2.4 GHz interferers, while it parallely connects and amplifies the three RF inputs.

2 3 4 5 6-70-60-50-40-30-20-10

01020

Forw

ard

trans

mis

sion

(dB)

Frequency (GHz)


(a) Simulation of forward transmission of the front-end.

2 3 4 5 6-70-60-50-40-30-20-10

01020

Forw

ard

trans

mis

sion

(dB

)

Frequency (GHz)


(b) Measured forward transmission of the front-end.

Fig. 9. Triplex LNA system, forward transmission: (a) simulation, and (b) measurement.

Fig. 10a and 10b show the front-end noise figure simulation

and measurement results, respectively. The LNA noise figure and the simulated minimum noise figure are also shown. It is seen that the measured noise figure for the entire system within each 500 MHz sub-band is kept below 6 dB at the sub-band center frequency. However, the measured noise figure of the front-end is larger than the simulated noise figure. The larger values of the noise figure compared to the simulated values can be explained by (a) larger insertion loss in the triplexer, (b) lower amplifier gain.


86

2 3 4 5 60

5

10

15

20

Noi

se fi

gure

(dB

)

Frequency (GHz)

Front-end NF LNA NF LNA NF-min

(a) Simulation of noise figure.

2 3 4 5 60

5

10

15

20

Noi

se fi

gure

(dB

)

Frequency (GHz)

Front-end NF LNA NF

(b) Measurement of noise figure.

Fig. 10. Triplex LNA system, noise figure: (a) simulation, and (b) measured.

2 3 4 5 6-80-70-60-50-40-30-20-10

0

Forw

ard

trans

mis

sion

(dB

)

Frequency (GHz)

S21 (dB) S32 (dB) S31 (dB)

(a) Simulation of isolation.

2 3 4 5 6-80-70-60-50-40-30-20-10

0

Forw

ard

trans

mis

sion

(dB

)

Frequency (GHz)

S21 (dB) S32 (dB) S31 (dB)

(b) Measurement of isolation.

Fig. 11. Triplex LNA system, isolation: (a) simulation, and (b) measurement.

Fig. 11a shows isolation simulation, between Ports 2-4 of the UWB front-end, and Fig. 11b shows the corresponding measurement. It is seen that the minimum measured isolation is -27 dB. The minimum isolation occurs at the boundary of the neighboring sub-bands, so in the three passbands the isolation is better than -27 dB. The isolation between the non-neighboring alternate sub-bands is -52 dB.

IV. DISCUSSION The measured forward gain was approximately 3.1 dB

lower than the simulated value. This is mostly due to slightly lower LNA gain and higher insertion loss in the triplexer network than predicted by the simulations. A small shift in frequency for all designs is also seen, i.e., approximately a rather static error of 2.5 %. This is due to the fact that the simulated electrical length differs from the measured one, i.e., the simulated phase velocity is higher than the measured one. To filter interference at radio frequencies was one of the main targets of this project. Consequently, better isolation between the three sub-bands and good attenuation of out-of band signals was prioritized over noise figure. However, the minimum noise figure values at the center frequency of each sub-band are below 6 dB.

V. CONCLUSION In this paper, a new multi-band UWB 3.1-4.8 GHz front-

end was presented. It consists of a fully integrated pre-selective filter and triplexer network and a wideband and flat gain low-noise amplifier. The UWB front-end is dedicated to a complete integration of the UWB antenna-LNA system on the same RF module. Using a microstrip network and three combined broadside- and edge-coupled bandpass filters, a multi-band transfer function and RF frequency multiplexing are achieved simultaneously. The low-noise amplifier has been designed for a near-to-minimum noise figure and a flat power gain over the 3.1-4.8 GHz frequency band. The measured LNA noise figure is below 4 dB while the measured overall front-end noise figure is below 6 dB. The attenuation of potential narrowband interference and good isolation between the three sub-channels are achieved.

REFERENCES [1] “First report order, revision of part 15 of commission’s rules regarding

ultra-wideband transmission systems” FCC., Washington, 2002. [2] G. R. Aiello and G. D. Rogerson, “Ultra wideband wireless systems,”

IEEE Microwave Magazine, vol. 4, no. 2, pp. 36-47, Jun. 2003. [3] L. Yang and G. B. Giannakis, “Ultra-wideband communications, an idea

whose time has come,” IEEE Signal Processing Magazine, pp. 26-54, Nov. 2004.

[4] A. Batra, J. Balakrishnan, G. R. Aiello, J. R. Foerster, A. Dabak, “Design of a multiband OFDM system for realistic UWB channel environments,” IEEE Tran. Microwave Theory and Tech., Vol. 52, Issue 9, Part 1, pp. 2123 – 2138, Sep. 2004.


87

[5] S. Chakraborty, N. R. Belk, A. Batra, M. Goel, A. Dabak, "Towards fully integrated wideband transceivers: fundamental challenges, solutions and future," Proc. IEEE Radio-Frequency Integration Technology: Integrated Circuits for Wideband Communication and Wireless Sensor Networks 2005, pp. 26-29, Dec. 2005.

[6] M. Z. Win and R. A. Scholtz, “Ultra-Wide Bandwidth Time-Hopping Spread-Spectrum Impulse Radio for Wireless Multiple-Access Communications,” IEEE Transactions on Communications, vol. 48, pp. 679-691, Apr. 2000.

[7] B. Razavi, T. Aytur, C. Lam, F. R. Yang, K. Y. Li, R. H. Yan, H. C. Hang, C. C. Hsu, C. C. Lee, “A UWB CMOS Transceiver,” IEEE Journal of Solid-State Circuits, vol. 40, Issue 12, Dec. 2005, pp. 2555-2562.

[8] Standard ECMA-368, “High Rata Ultra Wideband PHY and Mac Standard”, 1st Edition – December 2005, www.ecma-international.org/publications/files/ECMA-ST/ECMA-368.pdf”.

[9] T. W. Fischer, B. Kelleci, K. Shi, A. I. Karşilayan, E. Serpedin, "An analog approach to suppressing in-band narrow-band interference in UWB receivers," IEEE Transactions on Circuits and Systems, Vol. 54, No. 5, pp. 941-950, May. 2007.

[10] G. Cusmai, M. Brandolini, P. Rossi, F. Svelto, “A 0.18-µm CMOS Selective Receiver Front-End for UWB Applications,” IEEE Journal of Solid-State Circuits, vol. 41, Issue 8, Aug. 2006, pp. 1764-1771.

[11] M. Karlsson, and S. Gong, "A frequency-triplexed inverted-F antenna system for ultra-wide multi-band Systems 3.1-4.8 GHz," to be published in ISAST Transactions on Electronics and Signal Processing, 2007.

[12] M. Karlsson, P. Hakansson, S. Gong,” A frequency triplexer for ultra-wideband systems utilizing combined broadside- and edge-coupled filters,” submitted for publication in IEEE Transactions on Advanced Packaging, 2007.

[13] A. Serban, M. Karlsson and S. Gong, “Microstrip bias networks for ultra-wideband systems,” submitted for publication in ISAST Transactions on Electronics and Signal Processing, 2007.

[14] G. Gonzalez, “Microwave Transistor Amplifier Design. Analysis and Design,” Prentice Hall 1997, pp. 344-345.

Adriana Serban received the M.Sc degree in electronic engineering from Politehnica University, Bucharest, Romania.

From 1981 to 1990 she was with Microelectronica Institute, Bucharest as a Principal Engineer where she was involved in mixed integrated circuits (ICs) design. From 1992 to 2002 she was with Siemens AG, Munich,

Germany and with Sicon AB, Linkoping, Sweden as analog and mixed signal ICs Senior Design Engineer. Since 2002 she is a Lecturer at Linkoping University teaching in analog/digital system design and RF circuit design. She works towards her Ph.D degree in Communication Electronics. Her main research interest has been RF circuit design and high-speed integrated circuit design.


In 2003 he started his Ph.D. study in the Communication Electronics research group at Linköping University. His main work involves wideband antenna-

techniques, wideband transceiver front-ends, and wireless communications.


Between 1991 and 1999 he was a senior researcher at the microelectronic institute – Acreo in Sweden. From 2000 to 2001 he was the CTO at a spin-off company from the institute. Since 2002 he has been full professor in communication electronics at Linköping University, Sweden. His main research interest has been communication electronics including RF design, wireless communications and high-speed data transmissions.


88

Application of the Fractal Market Hypothesis forMacroeconomic Time Series Analysis

Jonathan M Blackledge, Fellow, IET, Fellow, IoP, Fellow, IMA, Fellow, RSS

Abstract— This paper explores the conceptual background tofinancial time series analysis and financial signal processingin terms of the Efficient Market Hypothesis. By revisiting theprincipal conventional approaches to market analysis and thereasoning associated with them, we develop a Fractal MarketHypothesis that is based on the application of non-stationaryfractional dynamics using an operator of the type

∂2

∂x2− σq(t) ∂q(t)

∂tq(t)

where σ−1 is the fractional diffusivity and q is the Fourierdimension which, for the topology considered, (i.e. the one-dimensional case) is related to the Fractal Dimension1 < DF < 2by q = 1−DF + 3/2.

We consider an approach that is based on the signalq(t) andits interpretation, including its use as a macroeconomic volatilityindex. In practice, this is based on the application of a movingwindow data processor that utilises Orthogonal Linear Regres-sion to compute q from the power spectrum of the windoweddata. This is applied to FTSE close-of-day data between 1980 and2007 which reveals plausible correlations between the behaviourof this market over the period considered and the amplitudefluctuations of q(t) in terms of a macroeconomic model that iscompounded in the operator above.

Index Terms— Fractional Diffusion Equation, Time SeriesAnalysis, Macroeconomic Modelling, Volatility Index

I. I NTRODUCTION

T HE application of statistical techniques for analysingfinancial time series is a well established practice. This

includes a wide range of stochastic modelling methods andthe use of certain partial differential equations for describ-ing financial systems (e.g. the Black-Scholes equation forfinancial derivatives). Attempts to develop stochastic modelsfor financial time series, which are essentially digital signalscomposed of ‘tick data’1 [1], [2] can be traced back to theearly Twentieth Century when Louis Bachelier [3] proposedthat fluctuations in the prices of stocks and shares (whichappeared to be yesterday’s price plus some random change)could be viewed in terms of random walks in which price

Manuscript received December 1, 2007. The work reported inthis paper was supported by Management and Personnel ServicesLimited (http://www.mapstraining.co.uk) and by the Schneider Group(http://schneidertrading.com).

Jonathan Blackledge ([email protected]) is Visiting Professor,Department of Electronic and Electrical Engineering, Loughborough Uni-versity, England (http://www.lboro.ac.uk/departments/el/staff/blackledge.html)and Extraordinary Professor, Department of Computer Science, Uni-versity of the Western Cape, Cape Town, Republic of South Africa(http://www.cs.uwc.ac.za).

1Data that provides traders with daily tick-by-tick data - time and sales - oftrade price, trade time, and volume traded, for example, at different samplingrates as required.

changes were entirely independent of each other. Thus, one ofthe simplest models for price variation is based on the sum ofindependent random numbers. This is the basis for Brownianmotion (i.e. the random walk motion first observed by theScottish Botanist, Robert Brown [4], who, in 1827, noted thatpollen grains suspended in water appear to undergo continuousjittery motion - a result of the random impacts on the pollengrains by water molecules) in which the random numbers areconsidered to conform to a normal distribution.

With macroeconomic financial systems, the magnitude of achange in pricedu tends to depend on the priceu itself. Wetherefore need to modify the Brownian random walk modelto include this observation. In this case, the logarithm of theprice change as a function of timet (which is also assumedto conform to a normal distribution) is modelled according tothe equation

du

u= αdv + βdt or

d

dtlnu = β + α

dv

dt(1)

where α is the volatility, dv is a sample from a normaldistribution andβ is a drift term which reflects the averagerate of growth of an asset2. Here, the relative price change ofan asset is equal to a random value plus an underlying trendcomponent - a ‘log-normal random walk’, e.g [5] - [8].

Brownian motion models have the following basic prop-erties: (i) statistical stationarity of price increments in whichsamples of Brownian motion taken over equal time incrementscan be superimposed onto each other in a statistical sense;(ii) scaling of price where samples of Brownian motion corre-sponding to different time increments can be suitably re-scaledsuch that they too, can be superimposed onto each other in astatistical sense. Such models fail to predict extreme behaviourin financial time series because of the intrinsic assumptionthat such time series conform to a normal distribution, i.e.Gaussian processes that are stationary in which the statistics -the standard deviation, for example - do not change with time.

Random walk models, which underpin the so called Effi-cient Market Hypothesis (EMH) [9]-[12], have been the basisfor financial time series analysis since the work of Bachelierin the late Nineteenth Century. Although the Black-Scholesequation [13], developed in the 1970s for valuing options, isdeterministic (one of the first financial models to achieve deter-minism), it is still based on the EMH, i.e. stationary Gaussianstatistics. The EMH is based on the principle that the currentprice of an asset fully reflects all available information relevantto it and that new information is immediately incorporatedinto the price. Thus, in an efficient market, the modelling

2Note that bothα andβ may very with timet


ISAST Transactions on Electronics and Signal Prosessing, No. 1, Vol. 2, 2008 Blackledge J.M.: Application of the Fractal Market Hypothesis for Macroeconomic Time Series Analysis89

of asset prices is concerned with modelling the arrival ofnew information. New information must be independent andrandom, otherwise it would have been anticipated and wouldnot be new. The arrival of new information can send ‘shocks’through the market (depending on the significance of theinformation) as people react to it and then to each other’sreactions. The EMH assumes that there is a rational andunique way to use the available information and that all agentspossess this knowledge. Further, the EMH assumes that this‘chain reaction’ happens effectively instantaneously. Theseassumptions are clearly questionable at any and all levels ofa complex financial system.

The EMH implies independence of price increments and istypically characterised by a normal of Gaussian ProbabilityDensity Function (PDF) which is chosen because most pricemovements are presumed to be an aggregation of smallerones, the sums of independent random contributions having aGaussian PDF. However, it has long been known that financialtime series do not follow random walks. An illustration ofthis is given in Figure 1 which shows a (discrete) financialsignalu(t) (data obtained from [14]), the log derivative of thissignald log u(t)/dt and a Gaussian distributed random signal.The log derivative is considered in order to: (i) eliminate thecharacteristic long term exponential growth of the signal; (ii)obtain a signal on the daily price differences3 in accord withthe left hand side term of equation (1). Clearly, there is amarked difference in the characteristics of a real financialsignal and a random Gaussian signal. This simple comparisonindicates a failure of the statistical independence assumptionwhich underpins the EMH.

The shortcomings of the EMH model (as illustrated inFigure 1) include: failure of the independence and Gaussiandistribution of increments assumption, clustering, apparentnon-stationarity and failure to explain momentous financialevents such as ‘crashes’ leading to recession and, in someextreme cases, depression. These limitations have prompted anew class of methods for investigating time series obtainedfrom a range of disciplines. For example, Re-scaled RangeAnalysis (RSRA), e.g. [15], [16], which is essentially basedon computing the Hurst exponent [17], is a useful tool forrevealing some well disguised properties of stochastic time se-ries such as persistence (and anti-persistence) characterized bynon-periodic cycles. Non-periodic cycles correspond to trendsthat persist for irregular periods but with a degree of statisticalregularity often associated with non-linear dynamical systems.RSRA is particularly valuable because of its robustness in thepresence of noise. The principal assumption associated withRSRA is concerned with the self-affine or fractal nature of thestatistical character of a time-series rather than the statistical‘signature’ itself. Ralph Elliott first reported on the fractalproperties of financial data in 1938 (e.g. [18] and referencetherein). He was the first to observe that segments of financialtime series data of different sizes could be scaled in such a waythat they were statistically the same producing so called Elliotwaves. Since then, many different self-affine models for pricevariation have been developed, often based on (dynamical)

3The gradient is computed using forward differencing.

Fig. 1. Financial time series for the FTSE value (close-of-day) from 02-04-1984 to 12-12-2007 (top), the log derivative of the same time series (centre)and a Gaussian distributed random signal (bottom).

Iterated Function Systems (IFS). These models can capturemany properties of a financial time series but are not basedon any underlying causal theory of the type attempted in thispaper.

A good stochastic financial model should ideally considerall the observable behaviour of the financial system it isattempting to model. It should therefore be able to providesome predictions on the immediate future behaviour of thesystem within an appropriate confidence level. Predicting themarkets has become (for obvious reasons) one of the mostimportant problems in financial engineering. Although, at leastin principle, it might be possible to model the behaviour ofeach individual agent operating in a financial market, onecan never be sure of obtaining all the necessary informationrequired on the agents themselves and their modus operandi.This principle plays an increasingly important role as thescale of the financial system, for which a model is required,increases. Thus, while quasi-deterministic models can be ofvalue in the understanding of micro-economic systems (withknown ‘operational conditions’), in an ever increasing globaleconomy (in which the operational conditions associated withthe fiscal policies of a given nation state are increasingly open),we can take advantage of the scale of the system to describeits behaviour in terms of functions of random variables.

II. M ARKET ANALYSIS

The stochastic nature of financial time series is well knownfrom the values of the stock market major indices such asthe FTSE (Financial Times Stock Exchange) in the UK, the


Fig. 2. Evolution of the 1987, 1997 and 2007 financial crashes. Normalisedplots (i.e. where the data has been rescaled to values between 0 and 1inclusively) of the daily FTSE value (close-of-day) for 02-04-1984 to 24-12-1987 (top), 05-04-1994 to 24-12-1997 (centre) and 02-04-2004 to 24-09-2007(bottom)

Dow Jones in the US which are frequently quoted. A principalaim of investors is to attempt to obtain information that canprovide some confidence in the immediate future of the stockmarkets often based on patterns of the past, patterns that areultimately based on the interplay between greed and fear. Oneof the principle components of this aim is based on the obser-vation that there are ‘waves within waves’ and ‘events withinevents’ that appear to permeate financial signals when studiedwith sufficient detail and imagination. It is these repeatingpatterns that occupy both the financial investor and the systemsmodeller alike and it is clear that although economies haveundergone many changes in the last one hundred years, thedynamics of market data do not appear to change significantly(ignoring scale). For example, Figure 2 shows the build up tothree different ‘crashes’, the one of 1987 and that of 1997(both after approximately 900 days) and what may turn outto be a crash of 2007 (at the time of writing this paper).The similarity in behaviour of these signals is remarkable andis indicative of the quest to understand economic signals interms of some universal phenomenon from which appropriate(macro) economic models can be generated. In an efficientmarket, only the revelation of some dramatic information cancause a crash, yet post-mortem analysis of crashes typicallyfail to (convincingly) tell us what this information must havebeen.

In modern economies, the distribution of stock returnsand anomalies like market crashes emerge as a result ofconsiderable complex interaction. In the analysis of financialtime series, it is inevitable that assumptions need to be madeto make the derivation of a model possible. This is the most

vulnerable stage of the process. Over simplifying assumptionslead to unrealistic models. There are two main approachesto financial modelling: The first approach is to look at thestatistics of market data and to derive a model based on aneducated guess of the ‘mechanics’ of the market. The modelcan then be tested using real data. The idea is that this processof trial and error helps to develop the right theory of marketdynamics. The alternative is to ‘reduce’ the problem and try toformulate a microscopic model such that the desired behaviour‘emerges’, again, by guessing agents’ strategic rules. Thisoffers a natural framework for interpretation; the problem isthat this knowledge may not help to make statements about thefuture unless some methods for describing the behaviour canbe derived from it. Although individual elements of a systemcannot be modelled with any certainty, global behaviour cansometimes be modelled in a statistical sense provided thesystem is complex enough in terms of its network of inter-connection and interacting components.

In complex systems, the elements adapt to the aggregatepattern they co-create. As the components react, the aggregatechanges, as the aggregate changes the components react anew.Barring the reaching of some asymptotic state or equilibrium,complex systems keep evolving, producing seemingly stochas-tic or chaotic behaviour. Such systems arise naturally in theeconomy. Economic agents, be they banks, firms, or investors,continually adjust their market strategies to the macroscopiceconomy which their collective market strategies create. Itis important to appreciate that there is an added layer ofcomplexity within the economic community: Unlike manyphysical systems, economic elements (human agents) reactwith strategy and foresight by considering the implicationsof their actions (some of the time!). Although we can notbe certain whether this fact changes the resulting behaviour,we can be sure that it introduces feedback which is thevery essence of both complex systems and chaotic dynamicalsystems that produce fractal structures.

The link between dynamical systems, chaos and the econ-omy is an important one because it is dynamical systems thatillustrate that local randomness and global determinism canco-exist. Global determinism can be considered, at least ina qualitative sense, in terms of broad social issues and thereaction of distinct groups to changing social attitudes, partic-ularly in economies that have traditionally been enhanced byan open and often pro-active policy towards the immigrationof peoples from diverse cultural backgrounds. For example, in1656, Cromwell permitted an open door policy to immigrationfrom continental Europe, partly in an attempt to enhance theeconomy of England that had been severely compromised bythe English Civil wars of 1642-46 and 1648-49 [19]. The longterm effect of this was to provide a new financial infrastructurethat laid the foundations for future economic development.It is arguable that Cromwell’s policy is the principal reasonwhy the ‘English revolution’ of the Eighteenth Century wasprimarily an industrial one. Issues concerning the current andfuture economic welfare of England may then be appreciatedin terms of the attitudes and values associated with newwaves of immigrants and the policy of appeasement adoptedat government level.


Complex systems can be split into two categories: equi-librium and non-equilibrium. Equilibrium complex systems,undergoing a phase transition, can lead to ‘critical states’ thatoften exhibit random fractal structures in which the statistics ofthe ‘field’ are scale invariant. For example, when ferromagnetsare heated, as the temperature rises, the spins of the electronswhich contribute to the magnetic field gain energy and beginto change in direction. At some critical temperature, the spinsform a random vector field with a zero mean and a phasetransition occurs in which the magnetic field averages to zero.But the field is not just random, it is a self-affine random fieldwhose statistical distribution is the same at different scales,irrespective of the characteristics of the distribution. Non-equilibrium complex systems or ‘driven’ systems give rise to‘self organised critical states’, an example is the growing ofsand piles. If sand is continuously supplied from above, thesand starts to pile up. Eventually, little avalanches will occuras the sand pile inevitably spreads outwards under the force ofgravity. The temporal and spatial statistics of these avalanchesare scale invariant.

Financial markets can be considered to be non-equilibriumsystems because they are constantly driven by transactions thatoccur as the result of new fundamental information about firmsand businesses. They are complex systems because the marketalso responds to itself, often in a highly non-linear fashion, andwould carry on doing so (at least for some time) in the absenceof new information. The ‘price change field’ is highly non-linear and very sensitive to exogenous shocks and it is probablethat all shocks have a long term effect. Market transactionsgenerally occur globally at the rate of hundreds of thousandsper second. It is the frequency and nature of these transactionsthat dictate stock market indices, just as it is the frequency andnature of the sand particles that dictates the statistics of theavalanches in a sand pile. These are all examples of randomscaling fractals [20]-[28].

III. D OES A MACROECONOMY HAVE MEMORY?

When faced with a complex process of unknown origin, itis usual to select an independent process such as Brownianmotion as a working hypothesis where the statistics and prob-abilities can be estimated with great accuracy. However, usingtraditional statistics to model the markets assumes that they aregames of chance. For this reason, investment in securities isoften equated with gambling. In most games of chance, manydegrees of freedom are employed to ensure that outcomes arerandom. In the case of a simple dice, a coin or roulette wheel,for example, no matter how hard you may try, it is physicallyimpossible to master your roll or throw such that you cancontrol outcomes. There are too many non-repeatable elements(speeds, angles and so on) and non-linearly compoundingerrors involved. Although these systems have a limited numberof degrees of freedom, each outcome is independent of theprevious one. However, there are some games of chance thatinvolve memory. In Blackjack, for example, two cards are dealtto each player and the object is to get as close as possible to21 by twisting (taking another card) or sticking. In a ‘bust’(over 21), the player loses; the winner is the player that stays

closest to 21. Here, memory is introduced because the cardsare not replaced once they are taken. By keeping track ofthe cards used, one can assess the shifting probabilities asplay progresses. This game illustrates that not all gamblingis governed by Gaussian statistics. There are processes thathave long-term memory, even though they are probabilisticin the short term. This leads directly to the question, doesthe economy have memory? A system has memory if whathappens today will affect what happens in the future.

Memory can be tested by observing correlations in thedata. If the system today has no affect on the system at anyfuture time, then the data produced by the system will beindependently distributed and there will be no correlations. Afunction that characterises the expected correlations betweendifferent time periods of a financial signalu(t) is the Auto-Correlation Function (ACF) defined by

A(t) = u(t) u(t) =∫ ∞

−∞u(τ)u(τ − t)dτ.

where denotes that the correlation operation. This functioncan be computed either directly (evaluation of the aboveintegral) or via application of the power spectrum using thecorrelation theorem

u(t) u(t) ⇐⇒| U(ω) |2

where⇐⇒ denotes transformation from real spacet to Fourierspaceω (the angular frequency), i.e.

U(ω) = F [u(t)] =

∞∫−∞

u(t) exp(−iωt)dt

whereF denotes the Fourier transform operator. The powerspectrum| U(ω) |2 characterises the amplitude distribution ofthe correlation function from which we can estimate the timespan of memory effects. This also offers a convenient way tocalculate the correlation function (by taking the inverse Fouriertransform of | U(ω) |2). If the power spectrum has morepower at low frequencies, then there are long time correlationsand therefore long-term memory effects. Inversely, if there isgreater power at the high frequency end of the spectrum, thenthere are short-term time correlations and evidence of short-term memory. White noise, which characterises a time serieswith no correlations over any scale, has a uniformly distributedpower spectrum.

Since prices movements themselves are a non-stationaryprocess, there is no ACF as such. However, if we calculatethe ACF of the price incrementsdu/dt, then we can observehow much of what happens today is correlated with whathappens in the future. According to the EMH, the economyhas no memory and there will therefore be no correlations,except for today with itself. We should therefore expect thepower spectrum to be effectively constant and the ACF to bea delta function. The power spectra and the ACFs of log pricechangesd log u/dt and their absolute value| d log u/dt | forthe FTSE 100 index (daily close) from 02-04-1984 to 24-09-2007 is given in Figure 3. The power spectra of the data isnot constant with rogue spikes (or groups of spikes) at theintermediate and high frequency portions of the spectrum. For


Fig. 3. Log-power spectra and ACFs of log price changes and absolutelog price changes for FTSE 100 index (daily close) from 02-04-1984 to 24-09-2007. Top-left: log price changes; top-right: absolute value of log pricechanges; middle: log power spectra; bottom: ACFs.

the absolute log price increments, there is evidence of a powerlaw at the low frequency end, indicating that there is additionalcorrelation in the signs of the data.

The ACF of the log price changes is relatively featureless,indicating that the excess of low frequency power within thesignal has a fairly subtle effect on the correlation function.However, the ACF of the absolute log price changes containsa number of interesting features. It shows that there area large number of short range correlations followed by anirregular decline up to approximately 1500 days after whichthe correlations start to develop again, peaking at about 2225days. The system governing the magnitudes of the log pricemovements clearly has a better long-term memory than itshould. The data used in this analysis contains 5932 daily pricemovements and it is therefore improbable that these results arecoincidental and correlations of this, or any similar type, whatever the time scale, effectively invalidates the independenceassumption of the EMH.

IV. STOCHASTIC MODELLING OF MACROECONOMICDATA

Developing mathematical models to simulate stochasticprocesses has an important role in financial analysis andinformation systems in general where it should be noted thatinformation systems are now one of the most important aspectsin terms of regulating financial systems, e.g. [29]-[32]. A goodstochastic model is one that accurately predicts the statisticswe observe in reality, and one that is based upon some welldefined rationale. Thus, the model should not only describethe data, but also help to explain and understand the system.

There are two principal criteria used to define the charac-teristics of a stochastic field: (i) The PDF or the Characteristic

Function (i.e. the Fourier transform of the PDF); the PowerSpectral Density Function (PSDF). The PSDF is the functionthat describes the envelope or shape of the power spectrum ofa signal. In this sense, the PSDF is a measure of the fieldcorrelations. The PDF and the PSDF are two of the mostfundamental properties of any stochastic field and variousterms are used to convey these properties. For example, theterm ‘zero-mean white Gaussian noise’ refers to a stochasticfield characterized by a PSDF that is effectively constant overall frequencies (hence the term ‘white’ as in ‘white light’) andhas a PDF with a Gaussian profile whose mean is zero.

Stochastic fields can of course be characterized using trans-forms other than the Fourier transform (from which the PSDFis obtained) but the conventional PDF-PSDF approach servesmany purposes in stochastic systems theory. However, ingeneral, there is no general connectivity between the PSDFand the PDF either in terms of theoretical prediction and/orexperimental determination. It is not generally possible tocompute the PSDF of a stochastic field from knowledge ofthe PDF or the PDF from the PSDF. Hence, in general, thePDF and PSDF are fundamental but non-related propertiesof a stochastic field. However, for some specific statisticalprocesses, relationships between the PDF and PSDF canbe found, for example, between Gaussian and non-Gaussianfractal processes [33] and for differentiable Gaussian processes[34].

There are two conventional approaches to simulating astochastic field. The first of these is based on predicting thePDF (or the Characteristic Function) theoretically (if possible).A pseudo random number generator is then designed whoseoutput provides a discrete stochastic field that is characteristicof the predicted PDF. The second approach is based onconsidering the PSDF of a field which, like the PDF, is ideallyderived theoretically. The stochastic field is then typicallysimulated by filtering white noise. A ‘good’ stochastic modelis one that accurately predicts both the PDF and the PSDFof the data. It should take into account the fact that, ingeneral, stochastic processes are non-stationary. In addition, itshould, if appropriate, model rare but extreme events in whichsignificant deviations from the norm occur.

New market phenomenon result from either a strong the-oretical reasoning or from compelling experimental evidenceor both. In econometrics, the processes that create time seriessuch as the FTSE have many component parts and the inter-action of those components is so complex that a deterministicdescription is simply not possible. As in all complex systemstheory, we are usually required to restrict the problem tomodelling the statistics of the data rather than the data itself,i.e. to develop stochastic models. When creating models ofcomplex systems, there is a trade-off between simplifyingand deriving the statistics we want to compare with realityand simulating the behaviour through an emergent statisticalbehaviour. Stochastic simulation allows us to investigate theeffect of various traders’ behavioural rules on the globalstatistics of the market, an approach that provides for a naturalinterpretation and an understanding of how the amalgamationof certain concepts leads to these statistics.

One cause of correlations in market price changes (and


volatility) is mimetic behaviour, known as herding. In general,market crashes happen when large numbers of agents place sellorders simultaneously creating an imbalance to the extent thatmarket makers are unable to absorb the other side withoutlowering prices substantially. Most of these agents do notcommunicate with each other, nor do they take orders froma leader. In fact, most of the time they are in disagreement,and submit roughly the same amount of buy and sell orders.This is a healthy non-crash situation; it is a diffusive (random-walk) process which underlies the EMH and financial portfoliorationalization.

One explanation for crashes involves a replacement for theEMH by the Fractal Market Hypothesis (FMH) which is thebasis of the model considered in this paper. The FMH proposesthe following: (i) The market is stable when it consists ofinvestors covering a large number of investment horizonswhich ensures that there is ample liquidity for traders; (ii)information is more related to market sentiment and technicalfactors in the short term than in the long term - as investmenthorizons increase and longer term fundamental informationdominates; (iii) if an event occurs that puts the validityof fundamental information in question, long-term investorseither withdraw completely or invest on shorter terms (i.e.when the overall investment horizon of the market shrinksto a uniform level, the market becomes unstable); (iv) pricesreflect a combination of short-term technical and long-termfundamental valuation and thus, short-term price movementsare likely to be more volatile than long-term trades - they aremore likely to be the result of crowd behaviour; (v) if a securityhas no tie to the economic cycle, then there will be no long-term trend and short-term technical information will dominate.Unlike the EMH, the FMH states that information is valuedaccording to the investment horizon of the investor. Becausethe different investment horizons value information differently,the diffusion of information will also be uneven. Unlike mostcomplex physical systems, the agents of the economy, andperhaps to some extent the economy itself, have an extraingredient, an extra degree of complexity. This ingredient isconsciousness.

V. RANDOM WALK PROCESSES

The purpose of revisiting random walk processes is thatit provides a useful conceptual reference to the model that isintroduced later on in this paper and in particular, appreciationof the use of the fractional diffusion equation for describingself-affine stochastic fields, an equation that arises through theunification of coherent and incoherent random walks. We shallconsider a random walk in the plane where the amplituderemains constant but where the phase changes, first by aconstant factor and then by a random value between0 and2π.

A. Coherent (Constant) Phase Walks

Consider a walk in the (real) plane where the length fromone step to another is constant - the amplitudea - and wherethe direction that is taken after each step is the same. In thissimple case, the ‘walker’ continues in a straight line and after

n steps the total length of the path the walker has taken willbe justan. We define this value as the resultant amplitudeA- the total length of the walk - which will change only byaccount of the number of steps taken. Thus,

A = an.

If each step takes a set period of timet to complete, then itis clear that

A(t) = at.

This scenario is limited by the fact that we are assuming thateach step is of precisely the same length and takes precisely thesame period of time to accomplish. In general, we considera to be the mean value of all the step lengths andt to bethe cumulative time associated with the average time taken toperform all steps. A walk of this type has a coherence fromone step or cluster of steps to the next, is entirely predictableand correlated in time.

If the same walk takes place in the complex plane then thephaseθ from one step to the next is the same. Thus, the resultis given by

A exp(iθ) =∑

n

a exp(iθ) = na exp(iθ).

The resultant amplitude is given byna as before and the totalphase value isθ. We can also define the intensity which isgiven by

I =| A exp(iθ) |2= A2

Thus, as a function of time, the intensity associated with thiscoherent phase walk is given by

I(t) = a2t2.

Suppose we make the walk slightly more complicated andconsider the case where the phase increases by a small constantfactor of θ at each step. Aftern steps, the result will be givenby the sum of all the steps taken, i.e.

A exp(iΘ) =∑

n

a exp(inθ)

= a[1 + exp(iθ) + exp(2iθ) + ... + exp[i(n− 1)θ]

= a[1− exp(inθ)][1− exp(iθ)]

= aexp(inθ/2)[exp(−inθ/2)− exp(inθ/2)]

exp(iθ/2)[exp(−iθ/2)− exp(iθ/2)]

= a exp[i(n− 1)θ/2)]sin(nθ/2)sin θ/2

.

Now, after many steps, whenn is large,

α = (n− 1)θ/2 ' nθ/2

and when the phase changeθ is small,

sin(θ/2) ' θ

2' α

nand we obtain the result

A exp(iΘ) = na exp[i((n− 1)/2)θ]sincα, sincα =sinα

α.

For very small changes in the phaseθ << 1, sincα ' 1 andthe resultant amplitudeA is, as before, given byan or as afunction of time, byat.


B. Incoherent (Random) Phase Walk

Incoherent or random phase walks are the basis of modellingmany kinds of statistical fluctuations. It is also the principlephysical model associated with the stochastic behaviour ofan ensemble of particles that collectively exhibit the processof diffusion. The first quantitative description of Brownianmotion was undertaken by Albert Einstein and published in1905 [35]. The basic idea is to consider a random walk inwhich the mean value of each step isa but where there is nocorrelation in the direction of the walk from one step to thenext. That is, the direction taken by the walker from one stepto next can be in any direction described by an angle between0 and 360 degrees or0 and 2π radians - for a walk in theplane. The angle that is taken at each step is entirely randomand all angles are taken to be equally likely. Thus, the PDFof angles between0 and2π is given by

Pr[θ] =

12π , 0 ≤ θ ≤ 2π;0, otherwise.

If we consider the random walk to take place in the complexplane, then aftern steps, the position of the walker will bedetermined by a resultant amplitudeA and phase angleΘgiven by the sum of all the steps taken, i.e.

A exp(iΘ) = a exp(iθ1) + a exp(iθ2) + ... + a exp(iθn)

= a

n∑m=1

exp(iθm).

The problem is to obtain a scaling relationship betweenAandn. Clearly we should not expectA to be proportional tothe number of stepsn as is the case with a coherent walk.The trick to finding this relationship is to analyse the resultof taking the square modulus ofA exp(iΘ). This provides anexpression for the intensityI given by

I = a2

∣∣∣∣∣n∑

m=1

exp(iθm)

∣∣∣∣∣2

= a2n∑

m=1

exp(iθm)n∑

m=1

exp(−iθm)

= a2

n +n∑

j=1,j 6=k

exp(iθj)n∑

k=1

exp(−iθk)

.

Now, in a typical term

exp(iθj) exp(−iθk) = cos(θj − θk) + i sin(θj − θk)

of the double summation, the functionscos(θj − θk) andsin(θj − θk) have random values between±1. Consequently,as n becomes larger and larger, the double sum will reducesto zero since more and more of these terms cancel each otherout. This insight is the basis for stating that forn >> 1

I = a2n

and the resulting amplitude is therefore given by

A = a√

n.

In this case,A is proportional to the square root of the numberof steps taken and if each step is taken over a mean timeperiod, then we obtain the result

A(t) = a√

t.

With a coherent walk we can state that the resulting amplitudeafter a time t will be at. This is a deterministic result.However, with an incoherent random walk, the interpretationof the above result is thata

√t is the amplitude associated with

the most likely position that the random walker will be aftertime t. If we imagine many random walkers, each starting outon their ‘journey’ from the origin of the (complex) plane att = 0, record the distances from the origin of this plane after aset period of timet, then the PDF ofA will have a maximumvalue - the ‘mode’ of the distribution - that occurs ata

√t. In

the case of a perfectly coherent walk, the PDF will consist ofa unit spike that occurs atat.

Figure 4 shows coherent and a incoherent phase walks in theplane. Each position of the walk(xj , yj), j = 1, 2, 3, ..., Nhas been computed using (fora = 1)

xj =j∑

i=1

cos(θi)

yj =j∑

i=1

sin(θi)

where θi ∈ [0, 2π] is uniformly distributed and computedusing the standard linear congruential pseudo random numbergenerator

xi+1 = aximodP, i = 1, 2, ..., N (2)

with a = 77 andP = 231 − 1 and an arbitrary value ofx0 -the ‘seed’. For the coherent phase walk

θi =2π

16xi

‖x‖∞which limits the angle to a small range between 0 andπ/8radians4. For the incoherent phase walk, the range of valuesis between 0 and2π radians, i.e.

θi = 2πxi

‖x‖∞

VI. PHYSICAL INTERPRETATION

In the (classical) kinetic theory of matter (including gases,liquids, plasmas and some solids), we considera to be theaverage distance a particle travels before it randomly collidesand scatters from another particle. The scattering process istaken to be entirely elastic, i.e. the interaction does not affectthe particle in any way other than to change the direction inwhich it travels. Thus,a represents themean free pathof aparticle. The mean free path is a measure how far a particle cantravel before scattering with another particle which in turn, isrelated to the number of particle per unit volume - the density

4‖x‖∞ denote the uniform norm, equivalent to the maximum value of thearray vectorx.


Fig. 4. Examples of a coherent (top) and incoherent (bottom) random walkin the plane forN = 100.

of a gas for example. If we imagine a particle ‘diffusing’through an ensemble of particles, then the mean free pathis a measure of the ‘diffusivity’ of the medium in whichthe process of diffusion takes place. This is a feature of allclassical diffusion processes which can be formulated in termsof the diffusion equation with diffusivityD. The dimensions ofdiffusivity are length2/time and may be interpreted in termsof the characteristic distance of a random walk process whichvaries with the square root of time.

If we consider a wavefront travelling through space andscattering from a site that changes the direction of propagation,then the mean free path can be taken to be the average numberof wavelengths taken by the wavefront to propagate from oneinteraction to another. After scattering from many sites, thewavefront can be considered to have diffused through the‘diffuser’. Here, the mean free path is a measure of the densityof scattering sites, which in turn, is a measure of the diffusivityof the material - an optical diffuser, for example.

We can use the random walk model associated with awavefield to interpret the flow of information through acomplex network of ‘sites’ that are responsible for passingon the information from one site to the next. If a packet ofinformation (e.g. a stream of bits of arbitrary length) travelsdirectly from A to B, then, in terms of the random walk modelsdiscussed above, the model associated with this informationexchange is ‘propagative’; it is a coherent process which iscorrelated in time and its principal physical characteristic isdetermined by the speed at which the information flows fromA to B. On the other hand, suppose that this information packetis transferred from A to B via information interchange sites C,D,...,Z,... In this case the flow of information is diffusive and ischaracterised by the diffusivity of the information interchange

‘system’. To a first order approximation, the diffusivity willdepend on the number of sites that are required to manage thereception and transmission of the information packet. As thenumber of sites decreases the flow of information becomesmore propagative and less diffusive. Thus, we can considerthe Internet, for example (albeit a good one) to be a sourceof information diffusion, not in terms of the diffusion ofthe information it coveys but in terms of the way in whichinformation packets ‘walk through’ the network. Further, wecan think of the internet itself as being an active mediumfor the propagation of financial information from one site toanother.

A. The Classical Diffusion Equation

The homogeneous diffusion equation is given by (for theone-dimension casex) [36](

∂2

∂x2− σ

∂

∂t

)u(x, t) = 0

for a diffusivity D = σ−1. The fieldu(x, t) represents a mea-surable quantity whose space-time dependence is determinedby the random walk of a large ensemble of particles or amultiple scattered wavefield or information flowing through acomplex network. We consider an initial value for this fielddenoted byu0 ≡ u(x, 0) = u(x, t) at t = 0. For example,ucould be the temperature of a material that starts ‘radiating’heat at timet = 0 from a point in spacex due to a massof thermally energised particles, each of which undertakesa random walk from the source of heat in which the mostlikely position of any particle after a timet is proportional to√

t. In optical diffusion, for example,u denotes the intensityof light. The light wavefield is taken to be composed of anensemble of wavefronts or rays, each of which undergoesmultiple scattering as it propagates through the diffuser. For asingle wavefront element, multiple scattering is equivalent toa random walk of that element.

The relationship between a random walk model and thediffusion equation can also be attributed to Einstein [35]who derived the diffusion equation using a random particlemodel system assuming that the movements of the particlesare independent of the movements of all other particles andthat the motion of a single particle at some interval of time isindependent of its motion at all other times. The derivation isas follows: Letτ be a small interval of time in which a particlemoves some distance betweenλ andλ+dλ with a probabilityP (λ) whereτ is long enough to assume that the movementsof the particle in two separate periods ofτ are independent. Ifn is the total number of particles and we assume thatP (λ) isconstant betweenλ andλ + dλ, then the number of particleswhich will travel a distance betweenλ and λ + dλ in τ isgiven by

dn = nP (λ)dλ.

If u(x, t) is the concentration (number of particles per unitvolume) then the concentration at timet + τ is described bythe integral of the concentration of particles which have been


displaced byλ in time τ , as described by the equation above,over all possibleλ, i.e.

u(x, t + τ) =

∞∫−∞

u(x + λ, t)P (λ)dλ.

Since,τ is assumed to be small, we can approximateu(x, t+τ)using the Taylor series and write

u(x, t + τ) ' u(x, t) + τ∂

∂tu(x, t).

Similarly, using a Taylor series expansion ofu(x + λ, t), wehave

u(x + λ, t) ' u(x, t) + λ∂

∂xu(x, t) +

λ2

2!∂2

∂x2u(x, t)

where the higher order terms are neglected under the assump-tion that if τ is small, then the distance travelled,λ, must alsobe small. We can then write

u(x, t) + τ∂

∂tu(x, t) = u(x, t)

∞∫−∞

P (λ)dλ

+∂

∂xu(x, t)

∞∫−∞

λP (λ)dλ +12

∂2

∂x2u(x, t)

∞∫−∞

λ2P (λ)dλ.

For isotropic diffusion,P (λ) = P (−λ) and soP is an evenfunction with usual normalization condition

∞∫−∞

P (λ)dλ = 1.

As λ is an odd function, the productλP (λ) is also an oddfunction which, if integrated over all values ofλ, equates tozero. Thus we can write

u(x, t) + τ∂

∂tu(x, t) = u(x, t) +

12

∂2

∂x2u(x, t)

∞∫−∞

λ2P (λ)dλ

so that

∂

∂tu(x, t) =

∂2

∂x2u(x, t)

∞∫−∞

λ2

2τP (λ)dλ.

Finally, defining the diffusivity as

D =

∞∫−∞

λ2

2τP (λ)dλ

we obtain the diffusion equation

∂

∂tu(x, t) = D

∂2

∂x2u(x, t).

B. The Classical Wave Equation

The wave equation (homogeneous form) is given by (forthe one-dimension case) [36](

∂2

∂x2− 1

c2

∂2

∂t2

)u(x, t) = 0

wherec is the wave speed andu denotes the amplitude of thewavefield. A possible solution to this equation is

u(x, t) = p(x− ct)

which describes a wave with distributionp moving alongx atvelocity c. For the initial value problem where

u(x, 0) = v(x),∂

∂tu(x, 0) = w(x)

the (d’Alembert) general solution is given by [36]

u(x, t) =12[v(x− ct) + v(x + ct)] +

12c

x+ct∫x−ct

w(ξ)dξ.

This solution is of limited use in that the range ofx isunbounded and only applies to the case on an ‘infinite string’.For the case whenw = 0, the solution can be taken to describetwo identical waves with amplitude distributionv(x) travellingaway from each other. Neither wave is taken to undergo anyinteraction as it travels along a straight path and thus, aftertime t the distance travelled will bect. This is analogousto a walker undertaking a perfectly coherent walk with anaverage step length ofc and after a period of timet reachinga positionct. The point here, is that we can relate the diffusionequation and the wave equation to two types of processes. Thediffusion equation describes a field generated by incoherentrandom processes with no time correlations whereas the waveequation describes a field generated by coherent processes thatare correlated in time. One of the aims of this paper is toformulate an equation that models the intermediate case - thefractional diffusion equation - in which random walk processhave a directional bias.

VII. H URST PROCESSES

For a walk in the plane,A(t) = at for a coherent walk andA(t) = a

√t for an incoherent walk. However, what would

be the result if the walk was neither coherent or incoherentbut partial coherent/incoherent? In other words, suppose therandom walk exhibited a bias with regard to the distributionof angles used to change the direction. What would be theeffect on the scaling law

√t? Intuitively, one expects that

as the distribution of angles reduces, the corresponding walkbecomes more and more coherent, exhibiting longer and longertime correlations until the process conforms to a fully coherentwalk. A simulation of such an effect is given in Figure 5 whichshows a random walk in the (real) plane as the (uniform)distribution of angles decreases. The walk becomes less andless random as the width of the distribution is reduced.

The equivalent effect for a random phase walk in three-dimensions is given in Figure 6. Each position of the walk

(xj , yj , zj), j = 1, 2, 3, ..., N


has been computed using

xj =j∑

i=1

cos(θi) cos(φi)

yj =j∑

i=1

sin(θi) cos(φi)

zj =j∑

i=1

sin(φi)

for N = 500. The uniform random number generator usedto computeθi and φi is the same - equation (2) - but withdifferent seeds. Conceptually, scaling models associated withthe intermediate case(s) should be based on a generalisation ofthe scaling laws

√t andt to the formtH where0.5 ≤ H < 1.

This reasoning is the basis for generalising the random walkprocesses considered so far, the exponentH being known asthe Hurst exponent or ‘dimension’.

Fig. 5. Random phase walks in the plane for a uniform distribution of anglesθi ∈ [0, 2π] (top left), θi ∈ [0, 1.9π] (top right),θi ∈ [0, 1.8π] (bottom left)andθi ∈ [0, 1.2π] (bottom right).

H E Hurst (1900-1978) was an English civil engineer whodesigned dams and worked on the Nile river dam projects inthe 1920s and 1930s. He studied the Nile so extensively thatsome Egyptians reportedly nicknamed him ‘the father of theNile’. The Nile river posed an interesting problem for Hurstas a hydrologist. When designing a dam, hydrologists needto estimate the necessary storage capacity of the resultingreservoir. An influx of water occurs through various naturalsources (rainfall, river overflows etc.) and a regulated amountneeds to be released for primarily agricultural purposes, forexample, the storage capacity of a reservoir being based onthe net water flow. Hydrologists usually begin by assumingthat the water influx is random, a perfectly reasonable as-sumption when dealing with a complex ecosystem. Hurst,

Fig. 6. Three dimensional random phase walks for a uniform distribution ofangles(θi, φi) ∈ ([0, 2π], [0, 2π]) (top left),(θi, φi) ∈ ([0, 1.6π], [0, 1.6π])(top right), (θi, φi) ∈ ([0, 1.3π], [0, 1.3π]) (bottom left) and(θi, φi) ∈([0, π], [0, π]) (bottom right).

however, had studied the 847-year record that the Egyptianshad kept of the Nile river overflows, from 622 to 1469. Henoticed that large overflows tended to be followed by largeoverflows until abruptly, the system would then change to lowoverflows, which also tended to be followed by low overflows.There appeared to be cycles, but with no predictable period.Standard statistical analysis of the day revealed no significantcorrelations between observations, so Hurst, who was awareof Einstein’s work on Brownian motion, developed his ownmethodology [37] lead to the scaling lawtH . This scaling lawmakes no prior assumptions about any underlying distribu-tions. It simply tells us how the system is scaling with respectto time. So how do we interpret the Hurst exponent? We knowthat H = 0.5 is consistent with an independently distributedsystem. The range0.5 < H ≤ 1, implies a persistent timeseries, and a persistent time series is characterized by positivecorrelations. Theoretically, what happens today will ultimatelyhave a lasting effect on the future. The range0 < H ≤ 0.5indicates anti-persistence which means that the time seriescovers less ground than a random process. In other words,there are negative correlations. For a system to cover lessdistance, it must reverse itself more often than a randomprocess.

VIII. L EVY PROCESSES

The generalisation of Einstein’s equationA(t) = a√

t byHurst to the formA(t) = atH , 0 < H ≤ 1 was necessary inorder for Hurst to analyse the apparent random behaviour ofthe annual rise and fall of the Nile river for which Einstein’smodel was inadequate. In considering this generalisation,Hurst paved the way for an appreciation that most naturalstochastic phenomena which, at first site, appear random, havecertain trends that can be identified over a given period of


time. In other words, many natural random patterns have abias to them that leads to time correlations in their stochasticbehaviour, a behaviour that is not an inherent characteristic ofa random walk model and fully diffusive processes in general.This aspect of stochastic field theory was taken up in the late1930s by the French mathematician Paul Levy (1886-1971)[38].

Levy processes are random walks whose distribution hasinfinite moments. The statistics of (conventional) physicalsystems are usually concerned with stochastic fields that havePDFs where (at least) the first two moments (the meanand variance) are well defined and finite. Levy statistics isconcerned with statistical systems where all the moments(starting with the mean) are infinite.

Many distributions exist where the mean and variance arefinite but are not representative of the process, e.g. the tail ofthe distribution is significant, where rare but extreme eventsoccur. These distributions include Levy distributions. Levy’soriginal approach5 to deriving such distributions is based onthe following question: Under what circumstances does thedistribution associated with a random walk of a few stepslook the same as the distribution after many steps (exceptfor scaling)? This question is effectively the same as askingunder what circumstances do we obtain a random walk thatis statistically self-affine. The characteristic function (i.e. theFourier transform)P (k) of such a distributionp(x) was firstshown by Levy to be given by (for symmetric distributionsonly)

P (k) = exp(−a | k |q), 0 < q ≤ 2

wherea is a (positive) constant. Ifq = 0,

p(x) =12π

∞∫−∞

exp(−a) exp(ikx)dk = exp(−a)δ(x)

and the distribution is concentrated solely at the origin asdescribed by the delta functionδ(x). Whenq = 1, the Cauchydistribution

p(x) =12π

∞∫−∞

exp(−a | k |) exp(ikx)dk =1π

a

a2 + x2

is obtained and whenq = 2, p(x) is characterized by theGaussian distribution

p(x) =12π

∞∫−∞

exp(−ak2) exp(ikx)dk

=12π

√π

aexp[−x2/(4a)],

whose first and second moments are finite. The Cauchy distri-bution has a relatively long tail compared with the Gaussiandistribution and a stochastic field described by a Cauchydistribution is likely to have more extreme variations whencompared to a Gaussian distributed field. For values ofq

5P Levy was the research supervisor of B Mandelbrot, the ‘inventor’ of‘fractal geometry’.

between 0 and 2, Levy’s characteristic function correspondsto a PDF of the form

p(x) ∼ 1x1+q

, x →∞.

This can be shown as follows6: For 0 < q < 1 and since thecharacteristic function is symmetric, we have

p(x) = Re[f(x)]

where

f(x) =1π

∞∫0

eikxe−kq

dk

=1π

[1ix

eikxe−kq

]∞k=0

− 1ix

∞∫0

eikx(−qkq−1e−kq

)dk

=

q

2πix

∞∫−∞

dkH(k)kq−1e−kq

eikx, x →∞

where

H(k) =

1, k > 00, k < 0

For 0 < q < 1, f(x) is singular atk = 0 and the greatestcontribution to this integral is the inverse Fourier transform ofH(k)kq−1. Noting that [27]

F−1

[1

(ik)q

]∼ 1

x1−q

whereF−1 denotes the inverse Fourier transform, and that

H(k) ⇐⇒ δ(x) +i

πx∼ δ(x), x →∞

then, using the convolution theorem, we have

f(x) ∼ q

iπx

i1−q

xq

and thus

p(x) ∼ 1x1+q

, x →∞

For 1 < q < 2, we can integrate by parts twice to obtain

f(x) =q

iπx

∞∫0

dkkq−1e−kq

eikx

=q

iπx

[1ix

kq−1e−kq

eikx

]∞k=0

+q

πx2

∞∫0

dkeikx[(q − 1)kq−2e−kq

− q(kq−1)2e−kq

]

=q

πx2

∞∫0

dkeikx[(q−1)kq−2e−kq

−q(kq−1)2e−kq

], x →∞.

6The author acknowledges Dr K I Hopcraft, School of MathematicalSciences, Nottingham University, England, for his help in deriving this result.


The first term of this result is singular and therefore providesthe greatest contribution and thus we can write,

f(x) ' q(q − 1)2πx2

∞∫−∞

H(k)eikx(kq−2e−kq

)dk.

In this case, for1 < q < 2, the greatest contribution to thisintegral is the inverse Fourier transform ofkq−2 and hence,

f(x) ∼ q(q − 1)πx2

i2−q

xq−1

so thatp(x) ∼ 1

x1+q, x →∞

which maps onto the previous asymptotic asq → 1 from theabove.

For q ≥ 2, the second moment of the Levy distributionexists and the sums of large numbers of independent trials areGaussian distributed. For example, if the result were a randomwalk with a step length distribution governed byp(x), q > 2,then the result would be normal (Gaussian) diffusion, i.e. aBrownian process. Forq < 2 the second moment of this PDF(the mean square), diverges and the characteristic scale of thewalk is lost. This type of random walk is called a Levy flight.

IX. T HE FRACTIONAL DIFFUSION EQUATION

We can consider a Hurst process to be a form of fractionalBrownian motion based on the generalization

A(t) = atH , H ∈ (0, 1].

Given that incoherent random walks describe processes whosemacroscopic behaviour is characterised by the diffusion equa-tion, then, by induction, Hurst processes should be charac-terised by generalizing the diffusion operator

∂2

∂x2− σ

∂

∂t

to the fractional form

∂2

∂x2− σq ∂q

∂tq

where q ∈ (0, 2] and D = 1/σ is the fractional diffusivity.Fractional diffusive processes can therefore be interpretedas intermediate between classical diffusive (random phasewalks with H = 0.5; diffusive processes withq = 1) and‘propagative process’ (coherent phase walks forH = 1;propagative processes withq = 2), e.g. [39], [40] and [38]- references therein. Fractional diffusion equations can alsobe used to model Levy distributions [41] and fractal timerandom walks [42], [43]. However, it should be noted that thefractional diffusion operator given above is the result of a phe-nomenology. It is no more (and no less) than a generalisationof a well known differential operator to fractional form whichfollows from a physical analysis of a fully incoherent randomprocess and it generalisation to fractional form in terms of theHurst exponent. Note that the diffusion and wave equations canbe derived rigorously from a range of fundamental physicallaws (conservation of mass, the continuity equation, Fourier’slaw of thermal conduction, Newton’s laws of motion and so

on) and that, in comparison, our approach to introducing afractional differential operator is based on postulation alone.It is therefore similar to certain other differential operators, anotable example being Schrodinger’s operator.

The fractional diffusion operator given above is appropriatefor modelling fractional diffusive processes that are stationary.For non-stationary fractional diffusion, we could consider thecase where the diffusivity is time variant as defined by thefunction σ(t). However, a more interesting case arises whenthe characteristics of the diffusion processes change over timebecoming less or more diffusive. This is illustrated in termsof the random walk in the plane given in Figure 7. Here, thewalk starts off being fully diffusive (i.e.H = 0.5 andq = 1),changes to being fractionally diffusive (0.5< H < 1 and1 < q < 2) and then changes back to being fully diffusive. Theresult given in Figure 7 shows a transition from two episodesthat are fully diffusive which has been generated using uniformphase distributions whose width changes from2π to 1.8π andback to2π. In terms of fractional diffusion, this is equivalentto having an operator

∂2

∂x2− σq ∂q

∂tq

where q = 1, t ∈ (0, T1]; q > 1, t ∈ (T1, T2]; q = 1, t ∈(T2, T3] where T3 > T2 > T1. If we want to generalisesuch processes over arbitrary periods of time, then we shouldconsiderq to be a function of time. We can then introduce anon-stationary fractional diffusion operator given by

∂2

∂x2− σq(t) ∂q(t)

∂tq(t).

This operator is the theoretical basis for the Fractal MarketHypothesis considered in this paper.

Fig. 7. Non-stationary random phase walk in the plane.

X. FRACTIONAL DYNAMIC MODEL

We consider an inhomogeneous non-stationary fractionaldiffusion equation of the form[

∂2

∂x2− σq(t) ∂q(t)

∂tq(t)

]u(x, t) = F (x, t)

whereF is a stochastic source term with some PDF anduis the stochastic field whose solution we require. Specifying


q to be in the range0 ≤ q ≤ 2, leads to control over thebasic physical characteristics of the equation so that we candefine an anti-persistent fieldu(x, t) when q < 1, a diffusivefield whenq = 1 and a propagative field whenq = 2. In thiscase, non-stationarity is introduced through the use of a timevarying fractional derivative whose values modify the physicalcharacteristics of the equation.

The range of values ofq is based on deriving an equationthat is a generalisation of both diffusive and propagativeprocesses using, what is fundamentally, a phenomenology.When q = 0 ∀t, the time dependent behaviour is determinedby the source function alone; whenq = 1 ∀t, u describesa diffusive process whereD = σ−1 is the ‘diffusivity’; whenq = 2 we have a propagative process whereσ is the ‘slowness’(the inverse of the wave speed). The latter process shouldbe expected to ‘propagate information’ more rapidly than adiffusive process leading to transients or ‘flights’ of some type.We refer toq as the ‘Fourier Dimension’ which is related tothe Hurst Exponent byq = H + DT /2 where DT is theTopological Dimension and to the Fractal DimensionDF byq = 1−DF + 3DT /2 as shown in Appendix I.

Since q(t) ‘drives’ the non-stationary behaviour ofu, theway in which we modelq(t) is crucial. It is arguable thatthe changes in the statistical characteristics ofu which leadto its non-stationary behaviour should also be random. Thus,suppose that we let the Fourier dimension at a timet bechosen randomly, a randomness that is determined by somePDF. In this case, the non-stationary characteristics ofu willbe determined by the PDF (and associated parameters) alone.Also, sinceq is a dimension, we can consider our model to bebased on the ‘statistics of dimension’. There are a variety ofPDFs that can be applied which will in turn affect the range ofq. By varying the exact nature of the distribution considered,we can ‘drive’ the non-stationary behaviour ofu in differentways. However, in order to apply different statistical modelsfor the Fourier dimension, the range ofq can not be restrictedto any particular range, especially in the case of a normaldistribution. We therefore generalize further and consider theequation[

∂2

∂x2− σq(t) ∂q(t)

∂tq(t)

]u(x, t) = F (x, t),−∞ < q(t) < ∞,∀t.

which allows us to apply different PDFs forq coveringarbitrary ranges. For example, suppose we consider a systemwhich is assumed to be primarily diffusive; then a ‘normal’PDF of the type

Pr[q(t)] =1

σ√

2πexp[−(q − 1)2/2σ2], −∞ < q < ∞

whereσ is the standard deviation, will ensure thatu is entirelydiffusive whenσ → 0. However, asσ is increased in value,the likelihood of q = 2 (and q = 0) becomes larger. Inother words, the standard deviation provides control over thelikelihood of the process becoming propagative.

Irrespective of the type of distribution that is considered,the equation[

∂2

∂x2− σq(t) ∂q(t)

∂tq(t)

]u(x, t) = F (x, t)

poses a fundamental problem which is how to define and workwith the term

∂q(t)

∂tq(t)u(x, t).

Given the result (for constantq)

∂q

∂tqu(x, t) =

12π

∞∫−∞

(iω)qU(x, ω) exp(iωt)dω

we might generalize as follows:

∂q(τ)

∂tq(τ)u(x, t) =

12π

∞∫−∞

(iω)q(τ)U(x, ω) exp(iωt)dω.

However, if we consider the case where the Fourier dimensionis a relatively slowly varying function of time, then we canlegitimately considerq(t) to be composed of a sequence ofdifferent statesqi = q(ti). This approach allows us to developa stationary solution for a fixedq over a fixed period of time.Non-stationary behaviour can then be introduced by using thesame solution for different values ofq over fixed (or varying)periods of time and concatenating the solutions for allq.

XI. GREEN’ S FUNCTION SOLUTION

We consider a Green’s function solution to the equation

(∂2

∂x2− σq ∂q

∂tq

)u(x, t) = F (x, t), −∞ < q < ∞

when F (x, t) = f(x)n(t) where f(x) and n(t) are bothstochastic functions. Applying a separation of variables hereis not strictly necessary. However, it yields a solution inwhich the terms affecting the temporal behaviour ofu(x, t)are clearly identifiable. Thus, we require a general solution tothe equation(

∂2

∂x2− σq ∂q

∂tq

)u(x, t) = f(x)n(t).

Let

u(x, t) =12π

∞∫−∞

U(x, ω) exp(iωt)dω

and

n(t) =12π

∞∫−∞

N(ω) exp(iωt)dω.

Then, using the result

∂q

∂tqu(x, t) =

12π

∞∫−∞

U(x, ω)(iω)q exp(iωt)dω

we can transform the fractional diffusion equation to the form(∂2

∂x2+ Ω2

q

)U(x, ω) = f(x)N(ω)


where we shall take

Ωq = i(iωσ)q2

and ignore the case forΩq = −i(iωσ)q2 . Defining the Green’s

function g to be the solution of [44], [45](∂2

∂x2+ Ω2

q

)g(x | x0, ω) = δ(x− x0)

whereδ is the delta function, we obtain the following solution:

U(x0, ω) = N(ω)

∞∫−∞

g(x | x0, ω)f(x)dx (3)

where [36]

g(x | x0, k) =i

2Ωqexp(iΩq | x− x0 |)

under the assumption thatu and ∂u/∂x → 0 as x → ±∞.This result reduces to conventional solutions for cases whenq = 1 (diffusion equation) andq = 2 (wave equation) as shallnow be shown.

A. Wave Equation Solution

Whenq = 2, the Green’s function defined above provides asolution for the outgoing Green’s function. Thus, withΩ2 =−ωσ, we have

U(x0, ω) =N(ω)2iωσ

∞∫−∞

exp(−iωσ | x− x0 |)f(x)dx.

Fourier inverting and using the convolution theorem for theFourier transform, we get

u(x0, t) =12σ

∞∫−∞

dxf(x)...

12π

∞∫−∞

N(ω)iω

exp(−iωσ | x− x0 |) exp(iωt)dω

=12σ

∞∫−∞

dxf(x)

t∫−∞

n(t− σ | x− x0 |)dt

which describes the propagation of a wave travelling at veloc-ity 1/σ subject to variations in space and time as defined byf(x) and n(t) respectively. For example, whenf and n areboth delta functions,

u(x0, t) =12σ

H(t− σ | x− x0 |).

This is a d’Alembertian type solution to the wave equationwhere the wavefront occurs att = σ | x − x0 | in the causalcase.

B. Diffusion Equation Solution

Whenq = 1 andΩ1 = i√

iωσ,

u(x0, t) =12

∞∫−∞

dxf(x)...

12π

∞∫−∞

exp(−√

iωσ | x− x0 |)√iωσ

N(ω) exp(iωt)dω.

For p = iω, we can write this result in terms of a Bromwichintegral (i.e. an inverse Laplace transform) and using theconvolution theorem for Laplace transforms with the result

c+i∞∫c−i∞

exp(−a√

p)√

pexp(pt)dp =

1√πt

exp[−a2/(4t)],

we obtain

u(x0, t) =

12√

σ

∞∫−∞

dxf(x)

t∫0

exp[−σ(x0 − x)2/(4t0)]√πt0

n(t− t0)dt0.

Now, if for example, we consider the case whenn is a deltafunction, the result reduces to

u(x0, t) =

12√

πσt

∞∫−∞

f(x) exp[−σ(x0 − x)2/(4t)]dx, t > 0

which describes classical diffusion in terms of the convolutionof an initial sourcef(x) (introduced at timet = 0) with aGaussian function.

C. General Series Solution

The evaluation ofu(x0, t) via direct Fourier inversion forarbitrary values ofq is not possible due to the irrationalnature of the exponential functionexp(iΩq | x − x0 |) withrespect toω. To obtain a general solution, we use the seriesrepresentation of the exponential function and write

U(x0, ω) =iM0N(ω)

2Ωq

[1 +

∞∑m=1

(iΩq)m

m!Mm(x0)

M0

](4)

where

Mm(x0) =

∞∫−∞

f(x) | x− x0 |m dx.

We can now Fourier invert term by term to develop a seriessolution. Given that we consider−∞ < q < ∞, this requiresus to consider three distinct cases.


1) Solution forq = 0: Evaluation ofu(x0, t) in this caseis trivial since, from equation (3)

U(x0, ω) =M(x0)

2N(ω) or u(x0, t) =

M(x0)2

n(t)

where

M(x0) =

∞∫−∞

exp(− | x− x0 |)f(x)dx.

2) Solution forq > 0: Fourier inverting, the first term inequation (4) becomes

12π

∞∫−∞

iN(ω)M0

2Ωqexp(iωt)dω =

M0

2σq2

12π

∞∫−∞

N(ω)(iω)

q2

exp(iωt)dω

=M0

2σq2

1(2i)q

√π

Γ(

1−q2

)Γ

(q2

) ∞∫−∞

n(ξ)(t− ξ)1−(q/2)

dξ.

The second term is

−M1

212π

∞∫−∞

N(ω) exp(iωt)dω = −M1

2n(t).

The third term is

− iM2

2.2!12π

∞∫−∞

N(ω)i(iωσ)q2 exp(iωt)dω =

M2σq2

2.2!d

q2

dtq2n(t)

and the fourth and fifth terms become

M3

2.3!12π

∞∫−∞

N(ω)i2(iωσ)q exp(iωt)dω = −M3σq

2.3!dq

dtqn(t)

and

iM4

2.4!12π

∞∫−∞

N(ω)i3(iωσ)3q2 exp(iωt)dω =

M4σ3q2

2.4!d

3q2

dt3q2

n(t)

respectively with similar results for all other terms. Thus,through induction, we can writeu(x0, t) as a series of theform

u(x0, t) =

M0(x0)2σq/2

1(2i)q

√π

Γ(

1−q2

)Γ

(q2

) ∞∫−∞

n(ξ)(t− ξ)1−(q/2)

dξ

−M1(x0)2

n(t) +12

∞∑k=1

(−1)k+1

(k + 1)!Mk+1(x0)σkq/2 dkq/2

dtkq/2n(t).

Observe that the first term involves a fractional integral (theRiemann-Liouville integral), the second term is composedof the source functionn(t) alone (apart from scaling) and

the third term is an infinite series composed of fractionaldifferentials of increasing orderkq/2. Also note that the firstterm is scaled by a factor involvingσ−q/2 whereas the thirdterm is scaled by a factor that includesσkq/2.

3) Solution forq < 0: In this case, the first term becomes

12π

∞∫−∞

iN(ω)M0

2Ωqexp(iωt)dω

=M0

2σ

q2

12π

∞∫−∞

N(ω)(iω)q2 exp(iωt)dω =

M0

2σ

q2

dq2

dtq2n(t).

The second term is the same is in the previous case (forq > 0)and the third term is

− iM2

2.2!12π

∞∫−∞

N(ω)i(iωσ)

q2

exp(iωt)dω

=M2

2.2!1

σq/2

1(2i)q

√π

Γ(

1−q2

)Γ

(q2

) ∞∫−∞

n(ξ)(t− ξ)1−(q/2)

dξ.

Evaluating the other terms, by induction we obtain

u(x0, t) =M0(x0)σq/2

2dq/2

dtq/2n(t)− M1(x0)

2n(t)

+12

∞∑k=1

(−1)k+1

(k + 1)!Mk+1(x0)

σkq/2

1(2i)kq

√π

Γ(

1−kq2

)Γ

(kq2

) ...

∞∫−∞

n(ξ)(t− ξ)1−(kq/2)

dξ

where q ≡| q |, q < 0. Here, the solution is composed ofthree terms: a fractional differential, the source term and aninfinite series of fractional integrals of orderkq/2. Thus, theroles of fractional differentiation and fractional integration arereversed asq changes from being greater than to less thanzero. All fractional differential operators associated with theequations above and hence forth should be considered in termsof the definition for a fractional differential given by

Dqf(t) =dn

dtn[In−qf(t)], n− q > 0

where I is the fractional integral operator (the Riemann-Liouville transform),

Ipf(t) =1

Γ(p)

t∫−∞

f(ξ)(t− ξ)1−p

dξ, p > 0 (5)

The reason for this is that direct fractional differentiationcan lead to divergent integrals. However, there is a deeperinterpretation of this result that has a synergy with the issueover whether a macroeconomic system has ‘memory’ andis based on observing that the evaluation of a fractionaldifferential operator depends on the history of the function inquestion. Thus, unlike an integer differential operator of ordern, a fractional differential operator of orderq has ‘memory’


because the value ofIq−nf(t) at a time t depends on thebehaviour off(t) from −∞ to t via the convolution witht(n−q)−1/Γ(n − q). The convolution process is of coursedependent on the history of a functionf(t) for a given kerneland thus, in this context, we can consider a fractional derivativedefined via the result above to have memory. In this sense, theoperator

∂2

∂x2− σq(t) ∂q(t)

∂tq(t)

decribes a process, compounded in a fieldu(x, t), that has anon-stationary memory association with the temporal charac-teristics of the system it is attempting to model. This is notan intrinsic charcteristic of systems that are purely diffusiveq = 1 or propagativeq = 2.

D. Asymptotic Solutions for an Impulse

We consider a special case in which the source functionf(x) is an impulse so that

Mm(x0) =

∞∫−∞

δ(x) | x− x0 |m dx =| x0 |m .

This result immediately suggests a study of the asymptoticsolution

u(t) = limx0→0

u(x0, t) (6)

=

1

2σq/21

(2i)q√

π

Γ( 1−q2 )

Γ( q2 )

∞∫−∞

n(ξ)(t−ξ)1−(q/2) dξ, q > 0;

n(t)2 , q = 0;

σq/2

2dq/2

dtq/2 n(t), q < 0.

The solution for the time variations of the stochastic fieldufor q > 0 are then given by a fractional integral alone andfor q < 0 by a fractional differential alone. In particular, forq > 0, we see that the solution is based on the convolutionintegral (ignoring scaling)

u(t) =1

t1−q/2⊗ n(t), q > 0

where⊗ denotes convolution and inω-space (ignoring scaling)

U(ω) =N(ω)

(iω)q/2.

This result is the conventional random fractal noise model forFourier dimensionq. Table I quantifies the results for differentvalues ofq with conventional name associations7. The fielduhas the following fundamental property forq ∈ (0, 2):

λq/2Pr[u(t)] = Pr[u(λt)].

This property describes the statistical self-affinity ofu. Thus,the asymptotic solution considered here, yields a result thatdescribes a random scaling fractal field characterized by aPSDF of the form1/ | ω |q which is a measure of the timecorrelations in the signal.

7Note that Brown noise conventionally refers to the integration of whitenoise but that Brownian motion is a form of pink noise because it classifiesdiffusive processes identified by the case whenq = 1.

q-value t-space ω-space (PSDF) Name

q = 0 1t⊗ n(t) 1 White noise

q = 1 1√t⊗ n(t) 1

|ω| Pink noise

q = 2tRn(t)dt 1

ω2 Brown noise

q > 2 t(q/2)−1 ⊗ n(t) 1|ω|q Black noise

TABLE I

NOISE CHARACTERISTICS FOR DIFFERENT VALUES OFq. NOTE THAT THE

RESULTS GIVEN ABOVE IGNORE SCALING FACTORS.

Note thatq = 0 defines the Hilbert transform ofn(t) whosespectral properties in the positive half space are identical ton(t) because

1t⊗ n(t) ⇐⇒ −iπsign(ω)N(ω)

where

sign(ω) =

1, ω > 0;−1, ω < 0.

The statistical properties of the Hilbert transform ofn(t) aretherefore the same asn(t) so that

Pr[t−1 ⊗ n(t)] = Pr[n(t)].

Hence, asq → 0, the statistical properties ofu(t) will ‘reflect’those ofn, i.e.

Pr[

1t1−q/2

⊗ n(t)]

= Pr[n(t)], q → 0.

However, asq → 2 we can expect the statistical properties ofu(t) to be such that the width of the PDF ofu(t) is reduced.This reflects the greater level of coherence (persistence in time)associated with the stochastic fieldu(t) for q → 2.

E. Other Asymptotic Solutions

A similar result to the asymptotic solution forx0 → 0 isobtained when the diffusivity is large, i.e.

limσ→0

u(x0, t)

=M0(x0)2σq/2

1(2i)q

√π

Γ(

1−q2

)Γ

(q2

) ∞∫−∞

n(ξ)(t− ξ)1−(q/2)

dξ

−M1(x0)2

n(t), q > 0. (7)

Here, the solution is the sum of fractal noise and white noise.Further, by relaxing the conditionσ → 0 we can consider theapproximation

u(x0, t) 'M0(x0)2σq/2

1(2i)q

√π

Γ(

1−q2

)Γ

(q2

) ∞∫−∞

n(ξ)(t− ξ)1−(q/2)

dξ


−M1(x0)2

n(t) +M2(x0)

2.2!σq/2 dq/2

dtq/2n(t), q > 0, σ << 1

(8)in which the solution is expressed in terms of the sum offractal noise, white noise and the fractional differentiation8 ofwhite noise.

F. Equivalence with a Wavelet Transform

The wavelet transform is defined in terms of projections off(t) onto a family of functions that are all normalized dilationsand translations of a prototype ‘wavelet’ functionw [47], i.e.

W[f(t)] = FL(t) =

∞∫−∞

f(τ)wL(τ, t)dτ

where

wL(τ, t) =1√L

w

(τ − t

L

), L > 0.

The independent variablesL andt are continuous dilation andtranslation parameters respectively. The wavelet transforma-tion is essentially a convolution transform wherewL(t) is theconvolution kernel with dilation variableL. The introductionof this factor provides dilation and translation properties intothe convolution integral that gives it the ability to analysesignals in a multi-resolution role (the convolution integral isnow a function ofL), i.e.

FL(t) = wL(t)⊗ f(t), L > 0.

In this sense, the asymptotic solution (ignoring scaling)

u(t) =1

t1−q/2⊗ n(t), q > 0 x → 0

is compatible with the case of a wavelet transform where

w1(t) =1

t1−q/2

for the stationary case and where, for the non-stationary case,

w1(t, τ) =1

t1−q(τ)/2.

XII. FTSE ANALYSIS USING OLR

We consider the basic model for a financial signal to begiven by

u(t) =1

t1−q/2⊗ n(t), q > 0

which has characteristic spectrum

U(ω) =N(ω)

(iω)q/2

and is a solution to the fractional diffusion equation(∂2

∂x2− σq ∂q

∂tq

)u(x, t) = δ(x)n(t), x → 0

The PSDF is thus characterised byω−q, ω ≥ 0 and ourproblem is thus, to computeq from the dataP (ω) =| U(ω) |2, ω ≥ 0. For this data, we consider the PSDF

P (ω) =c

ωq

8As defined by equation (5).

orln P (ω) = C + q lnω

where C = ln c. The problem is therefore reduced to im-plementing an appropriate method to computeq (and C) byfinding a best fit of the lineln P (ω) to the datalnP (ω).Application of the least squares method for computingq,which is based on minimizing the error

e(q, C) = ‖ lnP (ω)− ln P (ω, q, C)‖22with regard toq and C, leads to errors in the estimates forq which are not compatible with market data analysis. Thereason for this is that relative errors at the start and endof the datalnP may vary significantly especially becauseany errors inherent in the dataP will be ‘amplified’ throughapplication of the logarithmic transform required to linearisethe problem. In general, application of a least squares approachis very sensitive to statistical heterogeneity [48] and in thisapplication, may provide values ofq that are not compatiblewith the rationale associated with the FMH (i.e. values of1 <q < 2 that are intermediate between diffusive and propagativeprocesses). For this reason, an alternative approach must beconsidered which, in this paper, is based on Orthogonal LinearRegression (OLR).

Applying a standard moving window,q(t) is computed byrepeated application of OLR based on the m-code availablefrom [49]. Sinceq is, in effect, a statistic, its computationis only as good as the quantity (and quality) of data thatis available for its computation. For this reason, a relativelylarge window is required whose length is compatible with:(i) the number of samples available; (ii) the autocorrelationfunction and long-term memory effects as discussed in SectionIII. An example of theq(t) signal obtained using a 1000element window is given in Figure 8 which includesq(t) afterit has been smoothed using a Gaussian low-pass filtered toreveal the underlying trends inq. Inspection of the data (i.e.closer inspection of the time series than is shown in Figure 8)clearly illustrates a qualitative relationship between trends inthe financial data andq(t) in accordance with the theoreticalmodel considered. In particular, over periods of time in whichq increases in value, the amplitude of the financial signalu(t) decreases. Moreover, and more importantly, an upwardtrend in q appears to be a pre-cursur to a downward trend inu(t). A more detailed example of this behviour is shown inFigure 9 for close of day FTSE data over a smaller period oftime (i.e. from 1994 to 1997), a correlation that is compatiblewith the idea that a rise in the value ofq relates to the‘system’ becoming more propagative, which in stock marketterms, indicates the likelihood for the markets becoming ‘bear’dominant in the future.

The results of using the method discussed above not onlyprovides for a general appraisal of different macroeconomicfinancial time series, but, with regard to the size of selectedwindow used, an analysis of data at any point in time.The output can be interpreted in terms of ‘persistence’ and


Fig. 8. Application of OLR using a 1000 element window for analysingfinancial time series composed of FTSE values (close-of-day) from 02-04-1984 to 13-02-2008. The plot shows the time varying Fourier Dimensionq(t) (green) onto which is superimposed a Gaussian low-pass filtered versionof the signal (red) and the FTSE time series after normalisation.

Fig. 9. Application of OLR using a 1000 element window for analysingfinancial time series composed of FTSE values (close-of-day) from 05-04-1994 to 24-12-1997. The plot shows the time varying Fourier Dimensionq(t) (green) onto which is superimposed a Gaussian low-pass filtered versionof the signal (red) and the FTSE time series after normalisation.

‘anti-persistence’ and in terms of the existence or absenceof after-effects (macroeconomic memory effects). For thoseperiods in time whenq(t) is relatively constant, the existingmarket tendencies usually remain. Changes in the existingtrends tend to occur just after relatively sharp changes inq(t) have developed. This behaviour indicates the possibilityof using the time seriesq(t) for identifying the behaviourof a macroeconomic financial system in terms of both inter-market and between-market analysis. These results support thepossibility of usingq(t) as an independent macroeconomicvolatility predictor. It is noted that, at the time of writingthis paper, the value ofq(t) associated with those days afterapproximately day 4800 in Figure 8 (representing the latterhalf of 2007) indicate the growth of propagative behaviour andthus the macroeconomic instability compounded in the term‘Credit Crunch’. This is not surprising if it is assumed that thedownward trend from approximately day 3000 to day 3700shown in Figure 8 is a natural consequence of the effect of ahigher inflationary global economy resulting from the end ofthe cold war and that the upward trend from approximately day3700 to 5000 is a consequence of credit policies adopted bybanks in an attempt to compensate for this natural inflationarypressure. Under this assumption, the ‘Credit Crunch’ of 2007represents a transition that is compounded in a reappraisal ofthe definition of poverty, namely, that poverty is not a measureof how little one has but a measure of how much one owes.

XIII. D ISCUSSION

This paper is concerned with the introduction and theoreticalanalysis (in terms of general a solution) associated with thenon-stationary fractional diffusion operator

∂2

∂x2− σq(t) ∂q(t)

∂tq(t)

in the context of a macroeconomic model. By considering asource function of the typeδ(x)n(t) wheren(t) is white noise,we have shown that, forx → 0, the fractional diffusive fieldu(t) at timeτ is given by (ignoring scaling)

u(t, τ) =1

t1−q(τ)/2⊗ n(t)

which has Power Spectral Density Function characterised by| ω |−q(τ)/2 - a random scaling fractal. It should be noted,that the data analysis reported in this paper is based on anasymptotic solution (i.e.x → 0) used to obtain equation (6)and is thus, limited in the extent to which it ‘reflects’ thephysical principles upon which the model has been established.However, it is noted that the computation ofq(t) in thepresence of additive white noise is equivalent to the inversionof equation (7) forq (and for arbitrary values ofx0) whenσ → 0. In this sense, the power spectrum method used tocomputeq(t) is valid under the assumption that a fractionaldiffusive process occurs with high diffusivity and a highsignal-to-noise ratio (i.e.‖M1(x0)‖ → 0). For the case whenσ << 1, the inversion of equation (8) to computeq from umight be possible using an iterative approach which can beextended to solve the general case as required.

The non-stationary nature of this model is taken to ac-count for stochastic processes that can vary in time and areintermediate between diffusive and propagative or persistentbehaviour. Application of Orthogonal Linear Regression tomacroeconomic time series data provides an accurate androbust method to computeq(t) when compared to other statis-tical estimation techniques such as the least squares method.As a result of the physical interpretation associated with thefractional diffusion equation and the ‘meaning’ ofq(t), wecan, in principal, use the signalq(t) as a predictive measure inthe sense that as the value ofq(t) continues to increases, thereis a greater likelihood for volatile behaviour of the markets.This is reflected in the data analysis that is compounded inFigure 8 for the FTSE close-of-day between 1980 to 2007and in other financial data, the results of which lie beyond thescope of this paper9. It should be noted that because financialtime series data is assumed to be self-affine, the computationof q(t) can be applied over any time scale, and that the FTSEclose-of-day is only one example that has been used in thispaper as an illustrative case study.

In a statistical sense,q(t) is just another measure that may,or otherwise, be of value to market traders. In comparisonwith other statistical measures, this can only be assessedthrough its practical application in a live trading environment.However, in terms of its relationship to a stochastic modelfor macroeconomic data,q(t) does provide a measure that

9Similar results being observed for other major stock markets.


is consistent with the physical principles associated with arandom walk that includes a directional bias, i.e. fractionalBrownian motion. The model considered, and the signalprocessing algorithm proposed, has a close association withre-scaled range analysis for computing the Hurst exponentHsince forDT = 1, q = H + 1/2 (see Appendix I) [48]. Inthis sense, the principal contribution of this paper has been toconsider a model that is quantified in terms of a physicallysignificant (but phenomenological) model that is compoundedin a specific (fractional) partial differential equation. As withother financial time series, their derivatives, transforms etc., arange of statistical measures can be used to characteriseq(t),an example being given in Figure 8 and Figure 9 whereq(t)has been smoothed to provide a measure of the underlyingtrends.

In terms of the non-stationary fractional diffusive modelconsidered in this work, the time varying Fourier dimensionq(t) can be interpreted in terms of a ‘gauge’ on the charac-teristics of a dynamical system. This includes the manage-ment processes from which all modern economies may beassumed to be derived. In this sense, the FMH is based onthree principal considerations: (i) the non-stationary behaviourassociated with any system undergoing continuous change thatis driven by a management infrastructure; (ii) the cause andeffect that is inherent at all scales (i.e. all levels of managementhierarchy); (iii) the self-affine nature of outcomes relating topoints (i) and (ii). In a modern economy, the principal issueassociated with any form of financial management is based onthe flow of information and the assessment of this informationat different points connecting a large network. In this sense,a macroeconomy can be assessed in terms of its informationnetwork which consists of a distribution of nodes from whichinformation can flow in and out. The ‘efficiency’ of the systemis determined by the level of randomness associated with thedirection of flow of information to and from each node. Thenodes of the system are taken to be individuals or smallgroups of individuals whose assessment of the informationthey acquire together with their remit, responsibilities andinitiative, determines the direction of the information flowfrom one node to the next. The determination of the efficiencyof a system in terms of randomness is the most critical in termsof the model developed. It suggests that the performance ofa business is related to how well information flows throughan organisation. If the information flow is entirely random,then we might surmise that the decisions made which ‘drive’the direction of the ‘system’ are also entirely random. Theprincipal point here is that the flow of information has a directrelationship on the management decisions that are made onbehalf of an organisation.

The non-stationary but statistically self-affine nature of themarkets leads directly to the use of the Fourier dimension asa measure for quantifying their ‘state of coherence’. Just asthis parameter can be used as a market index for managing afinancial portfolio, so, it may be of value in quantifying the‘state’ of any organisation undergoing change (management).The conceptual basis associated with the Fourier dimensionand the system behaviour that it reflects leads directly to anapproach to management where the principles of openness and

Fractal type Fractal DimensionFractal Dust 0 < DF < 1Fractal Curve 1 < DF < 2Fractal Surface 2 < DF < 3Fractal Volume 3 < DF < 4Fractal Time 4 < DF < 5Hyper-fractals 5 < DF < 6...

...

TABLE II

FRACTAL TYPES AND CORRESPONDING FRACTAL DIMENSIONS

transparency articulate the degree of coherence of informationflow through an organisation from one level to another. Ineffect, the sustained organisational approach to managingcontinuous change is the basis for a portfolio in whichq(t) > 1and increases with time.

The FMH and the self-affine nature of organisations ingeneral provides a model in which the work-force at any onelevel (i.e. department/section/group etc.) of an organisationcan empathise with all other levels by cultivating an under-standing in which each level is a reflection of their own, e.g.problems/solutions at middle management are a reflection ofthe same type of problems/solutions at executive level. This‘empathy’ is a two-way entity which differs only in termsof its scale. Sustained organisational change and the examplemethods of implementing it is a self-affine process and shouldthus be introduced with this aspect in mind [50]. In tacklingproblems at any level within an organisation, one is, in effect,taking consideration of such problems above and below thatsame level in terms of the dynamic behaviour of the ‘system’as a whole, a macroeconomy being the antithesis of such a‘system’.

APPENDIX IRELATIONSHIP BETWEEN THEHURST EXPONENT AND THE

TOPOLOGICAL, FRACTAL AND FOURIER DIMENSIONS

Suppose we cut up some simple one-, two- and three-dimensional Euclidean objects (a line, a square surface anda cube, for example), make exact copies of them and thenkeep on repeating the copying process. LetN be the numberof copies that we make at each stage and letr be the lengthof each of the copies, i.e. the scaling ratio. Then we have

NrDT = 1, DT = 1, 2, 3, ...

where DT is the topological dimension. The similarity orfractal dimension is that value ofDF which is usually (but notalways) a non-integer dimension ‘greater’ that its topologicaldimension (i.e. 0,1,2,3,... where 0 is the dimension of a pointon a line) and is given by

DF = − log(N)log(r)

.

The fractal dimension is that value that is strictly greaterthan the topological dimension as given in Table II. In eachcase, as the value of the fractal dimension increases, the fractalbecomes increasingly ‘space-filling’ in terms of the topologicaldimension which the fractal dimension is approaching. In each


case, the fractal exhibits structures that are self-similar. Aself-similar deterministic fractal is one where a change in thescale of a functionf(x) (which may be a multi-dimensionalfunction) by a scaling factorλ produces a smaller version,reduced in size byλ, i.e.

f(λx) = λf(x).

A self-affine deterministic fractal is one where a change inthe scale of a functionf(x) by a factorλ produces a smallerversion reduced in size by a factorλq, q > 0, i.e.

f(λx) = λqf(x).

For stochastic fields, the expression

Pr[f(λx)] = λqPr[f(x)]

describes a statistically self-affine field - a random scalingfractal. As we zoom into the fractal, the shape changes, butthe distribution of lengths remains the same.

There is no unique method for computing the fractal di-mension. The methods available are broadly categorized intotwo families: (i) Size-measure relationships, based on recursivelength or area measurements of a curve or surface usingdifferent measuring scales; (ii) application of relationshipsbased on approximating or fitting a curve or surface to a knownfractal function or statistical property, such as the variance.

Consider a simple Euclidean straight line` of lengthL(`)over which we ‘walk’ a shorter ‘ruler’ of lengthδ. The numberof steps taken to cover the lineN [L(`), δ] is thenL/δ whichis not always an integer for arbitraryL andδ. Since

N [L(`), δ] =L(`)

δ= L(`)δ−1,

⇒ 1 =lnL(`)− lnN [L(`), δ]

ln δ= −

(lnN [L(`), δ]− lnL(`)

ln δ

)which expresses the topological dimensionDT = 1 of theline. In this case,L(`) is the Lebesgue measure of the lineand if we normalize by settingL(`) = 1, the latter equationcan then be written as

1 = − limδ→0

[lnN(δ)

ln δ

]since there is less error in countingN(δ) asδ becomes smaller.We also then haveN(δ) = δ−1. For extension to a fractalcurvef , the essential point is that the fractal dimension shouldsatisfy an equation of the form

N [F (f), δ] = F (f)δ−DF

whereN [F (f), δ] is ‘read’ as the number of rulers of sizeδneeded to cover a fractal setf whose measure isF (f) whichcan be any valid suitable measure of the curve. Again we maynormalize, which amounts to defining a new measureF ′ assome constant multiplied by the old measure to get

DF = − limδ→0

[lnN(δ)

ln δ

]whereN(δ) is taken to beN [F ′(f), δ] for notational conve-nience. Thus a piecewise continuous field has precise fractal

properties over all scales. However, for the discrete (sampled)field

D = −⟨

lnN(δ)ln δ

⟩where we choose valuesδ1 and δ2 (i.e. the upper and lowerbounds) satisfyingδ1 < δ < δ2 over which we applyan averaging processes denoted by〈〉. The most commonapproach is to utilise a bi-logarithmic plot oflnN(δ) againstln δ, choose valuesδ1 and δ2 over which the plot is uniformand apply an appropriate data fitting algorithm (e.g. a leastsquares estimation method or, as used in this paper, OrthogonalLinear Regression) within these limits.

The relationship between the Fourier dimensionq and thefractal dimensionDF can be determined by considering thismethod for analysing a statistically self-affine field. For afractional Brownian process (with unit step length)

A(t) = tH , H ∈ (0, 1]

where H is the Hurst dimension. Consider a fractal curvecovering a time period∆t = 1 which is divided up intoN =1/∆t equal intervals. The amplitude increments∆A are thengiven by

∆A = ∆tH =1

NH= N−H .

The number of lengthsδ = N−1 required to cover eachinterval is

∆A∆t =N−H

N−1= N1−H

so thatN(δ) = NN1−H = N2−H .

Now, since

N(δ) =1

δDF, δ → 0,

then, by inspection,

DF = 2−H.

Thus, a Brownian process, whereH = 1/2, has a fractaldimension of 1.5. For higher topological dimensionsDT

DF = DT + 1−H.

This algebraic equation provides the relationship between thefractal dimensionDF , the topological dimensionDT and theHurst dimensionH. We can now determine the relationshipbetween the Fourier dimensionq and the fractal dimensionDF .

Consider a fractal signalf(x) over an infinite support witha finite samplefX(x), given by

fX(x) =

f(x), 0 < x < X;0, otherwise.

A finite sample is essential as otherwise the power spectrumdiverges. Moreover, iff(x) is a random function then for anyexperiment or computer simulation we must necessarily takea finite sample. LetFX(k) be the Fourier transform offX(x),PX(k) be the power spectrum andP (k) be the power spectrumof f(x). Then


fX(x) =12π

∫ ∞

−∞FX(k) exp(ikx)dk,

PX(k) =1X|FX(k)|2

andP (k) = lim

X→∞PX(k).

The power spectrum gives an expression for the power of asignal for particular harmonics.P (k)dk gives the power inthe rangek to k + dk. Consider a functiong(x), obtainedfrom f(x) by scaling thex-coordinate by somea > 0, thef -coordinate by1/aH and then taking a finite sample as before,i.e.

gX(x) =

g(x) = 1aH f(ax), 0 < x < X;

0, otherwise.

Let GX(k) and P ′X(k) be the Fourier transform and powerspectrum ofgX(x), respectively. We then obtain an expressionfor GX in terms ofFX ,

GX(k) =∫ X

0

gX(x) exp(−ikx)dx =

1aH+1

∫ X

0

f(s) exp(− iks

a

)ds

wheres = ax. Hence

GX(k) =1

aH+1FX

(k

a

)and the power spectrum ofgX(x) is

P ′X(k) =1

a2H+1

1aX

∣∣∣∣FX

(k

a

)∣∣∣∣2and, asX →∞,

P ′(k) =1

a2H+1P

(k

a

).

Sinceg(x) is a scaled version off(x), their power spectra areequal, and so

P (k) = P ′(k) =1

a2H+1P

(k

a

).

If we now setk = 1 and then replace1/a by k we get

P (k) ∝ 1k2H+1

=1kβ

.

Now sinceβ = 2H + 1 andDF = 2−H, we have

DF = 2− β − 12

=5− β

2.

The fractal dimension of a fractal signal can be calculateddirectly fromβ using the above relationship. This method alsogeneralizes to higher topological dimensions giving

β = 2H + DT .

Thus, sinceDF = DT + 1−H,

thenβ = 5− 2DF for a fractal signal andβ = 8− 2DF fora fractal surface so that, in general,

β = 2(DT + 1−DF ) + DT = 3DT + 2− 2DF

and

DF = DT + 1−H = DT + 1− β −DT

2=

3DT + 2− β

2,

the Fourier dimension being given byq = β/2.

ACKNOWLEDGMENT

Some of the material presented in this paper is based onthe PhD Theses of two former research students of the author,Dr Mark London and Dr Irena Lvova. The author is gratefulto Mr Bruce Murray (Lexicon Data Limited) for his helpin the assessment of the financial data analysis carried outby the author and to Ms Mariam Fardoost (Merrill Lynch,London) for her advice with regard to global economics, macrooverviews and market forecasts.

REFERENCES

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Thomson Business Press, 1996.[9] E. Fama,The Behavior of Stock Market Prices, Journal of Business Vol.

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Work, Journal of Finance Vol. 25, 383-417, 1970.[12] G. M. Burton, Efficient Market Hypothesis, The New Palgrave: A

Dictionary of Economics, Vol. 2, 120-23, 1987.[13] F. Black and M. Scholes,The Pricing of Options and Corporate

Liabilities, Journal of Political Economy, Vol. 81(3), 637-659, 1973.[14] http://uk.finance.yahoo.com/q/hp?s=%5EFTSE[15] B. B. Mandelbrot and J. R. Wallis,Robustness of the Rescaled Range

R/S in the Measurement of Noncyclic Long Run Statistical Dependence,Water Resources Research, Vol. 5(5), 967-988, 1969.

[16] B. B. Mandelbrot,Statistical Methodology for Non-periodic Cycles:From the Covariance to R/S Analysis, Annals of Economic and SocialMeasurement, Vol. 1(3), 259-290, 1972.

[17] E. H. Hurst, A Short Account of the Nile Basin, Cairo, GovernmentPress, 1944.

[18] http://en.wikipedia.org/wiki/Elliottwaveprinciple[19] http://www.olivercromwell.org/jews.htm[20] B. B. Mandelbrot,The Fractal Geometry of Nature, Freeman, 1983.[21] J. Feder,Fractals, Plenum Press, 1988.[22] K. J. Falconer,Fractal Geometry, Wiley, 1990.[23] H. O. Peitgen, H. Jurgens and D. Saupe D,Chaos and Fractals: New

Frontiers of Science, Springer, 1992.[24] P. Bak,How Nature Works, Oxford University Press, 1997.[25] M. J. Turner, J. M. Blackledge and P. Andrews,Fractal Geometry in

Digital Imaging, Academic Press, 1997.[26] N. Lam and L. De Cola L,Fractal in Geography, Prentice-Hall, 1993.[27] J. M. Blackledge,Digital Image Processing, Horwood, 2006.


[28] H. O. Peitgen and D. Saupe (Eds.),The Science of Fractal Images,Springer, 1988.

[29] A. J. Lichtenberg and M. A. Lieberman,Regular and Stochastic Motion:Applied Mathe-matical Sciences, Springer-Verlag, 1983.

[30] J. J. Murphy,Intermarket Technical Analysis: Trading Strategies for theGlobal Stock, Bond, Commodity and Currency Market, Wiley FinanceEditions, Wiley, 1991.

[31] J. J. Murphy,Technical Analysis of the Futures Markets: A Comprehen-sive Guide to Trad-ing Methods and Applications, New York Instituteof Finance, Prentice-Hall, 1999.

[32] T. R. DeMark,The New Science of Technical Analysis, Wiley, 1994.[33] J. O. Matthews, K. I. Hopcraft, E. Jakeman and G. B. Siviour,Accuracy

Analysis of Measurements on a Stable Power-law Distributed Series ofEvents, J. Phys. A: Math. Gen. 39, 1396713982, 2006.

[34] W. H. Lee, K. I. Hopcraft, and E. Jakeman,Continuous and DiscreteStable Processes, Phys. Rev. E 77, American Physical Society, 011109,1-4.

[35] A. Einstein,On the Motion of Small Particles Suspended in Liquids atRest Required by the Molecular-Kinetic Theory of Heat, Annalen derPhysik, Vol. 17, 549-560, 1905.

[36] J. M. Blackledge, G. A. Evans and P. Yardley,Analytical Solutions toPartial Differential Equations, Springer, 1999.

[37] H. Hurst, Long-term Storage Capacity of Reservoirs, Transactions ofAmerican Society of Civil Engineers, Vol. 116, 770-808, 1951.

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[47] S. Mallat,A Wavelet Tour of Signal Processing, Academic Press, ISBN:0-12-466606-X, 1999.

[48] I. Lvova, Application of Statistical Fractional Methods for the Analysisof Time Series of Currency Exchange Rates, PhD Thesis, De MontfortUniversity, 2006.

[49] http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=6716&objectType=File

[50] C. Davies, Sustained Organisational Change: A Hearts and MindsApproach, PhD Thesis, Loughborough University, 2007.

Jonathan Blackledge received a BSc in Physicsfrom Imperial College, London University in 1980,a Diploma of Imperial College in Plasma Physicsin 1981 and a PhD in Theoretical Physics fromKings College, London University in 1983. As a Re-search Fellow of Physics at Kings College (LondonUniversity) from 1984 to 1988, he specialized ininformation systems engineering undertaking workprimarily for the defence industry. This was followedby academic appointments at the Universities ofCranfield (Senior Lecturer in Applied Mathematics)

and De Montfort (Professor in Applied Mathematics and Computing) wherehe established new post-graduate MSc/PhD programmes and research groupsin computer aided engineering and informatics. In 1994, he co-foundedManagement and Personnel Services Limited where he is currently ExecutiveDirector. His work for Microsharp (Director of R & D, 1998-2002) includedthe development of manufacturing processes now being used for digitalinformation display units. In 2002, he co-founded a group of companiesspecializing in information security and cryptology for the defence andintelligence communities, actively creating partnerships between industryand academia. He currently holds academic posts in the United Kingdomand South Africa, and in 2007 was awarded Fellowships of the City andGuilds London Institute and the Institute of Leadership and Managementtogether with Freedom of the City of London for his role in the developmentof the Higher Level Qualification programmes in Engineering, ICT andBusiness Administration, most recently, for the nuclear industry, securityand financial sectors respectively. Professor Blackledge has published overone hundred scientific and engineering research papers and technical reportsfor industry, six industrial software systems, fifteen patents, ten books andhas been supervisor to sixty research (PhD) graduates. He lectures widelyto a variety of audiences composed of mathematicians, computer scientists,engineers and technologists in areas that include cryptology, communicationstechnology and the use of artificial intelligence in process engineering,financial analysis and risk management. His current research interests includecomputational geometry and computer graphics, image analysis, nonlineardynamical systems modelling and computer network security, working in bothan academic and commercial context. He holds Fellowships with England’sleading scientific and engineering Institutes and Societies including theInstitute of Physics, the Institute of Mathematics and its Applications, theInstitution of Electrical Engineers, the Institution of Mechanical Engineers,the British Computer Society, the Royal Statistical Society and the Instituteof Directors. He is a Chartered Physicist, Chartered Mathematician, CharteredElectrical Engineer, Chartered Mechanical Engineer, Chartered Statisticianand a Chartered Information Technology Professional.


Abstract—This paper presents an ultra-wideband I/Q

demodulator based on the six-port technique. The six-port I/Q demodulator covers the frequency spectrum from 3.1 to 4.8 GHz, i.e., it covers the lower band of the UWB spectrum. The demodulator has thus an relative bandwidth of 43%. This six-port circuit utilizes three ultra-wideband broadband 3-dB 90º branch couplers and one 3-dB 0º Wilkinson power divider. It is manufactured utilizing microstrips on a printed circuit board. Simulation and measurement results of this new six-port and the traditional six-port correlators are compared. The designed six-port correlator shows good phase and amplitude balances. The I/Q demodulator with this new correlator also shows good demodulation results in the frequency range without any calibration.

Index Terms— directional coupler, broadband correlator, quadrature hybrid, receivers, six-port receiver, UWB systems

I. INTRODUCTION

Modern communication systems require high data-rate,

wide bandwidth, small size and low cost. A trend in communications is towards reconfigurable radio terminals, i.e., software defined radio (SDR). SDR requires receivers with wideband capabilities to support as many different services as possible in a very wide band of frequencies. The homodyne topology offers advantages in reducing complicity and cost of a radio receiver. Therefore, it is of particular interest and much research effort has been done on homodyne receivers. However, conventional homodyne receivers are traditionally of narrowband. To overcome this problem, the six-port receiver, with a wideband property, is a promising architecture [1]. The six-port as a communication receiver was first introduced in 1994 by Ji Li, R. G. Bosisio and Ke Wu [2]. The six-port receiver is a wideband solution but still with limited bandwidth. Today, ultra-wideband (UWB) (>20% relative bandwidth) is needed to archive high speed for short range wireless-communication. There are two dominating UWB solutions for high data rate and short range wireless communication. One is based on the direct sequence spread spectrum technique and the other is based on the multi-band

Manuscript received 2007-10-18. Vinnova, a Swedish funding organization, is acknowledged for financial support of this study.

Pär Håkansson and Shaofang Gong is with Linköping University, Department of Science and Technology, SE-60174 Norrköping, Sweden. (phone: +46-11-363368, Fax: +46-11-363270, E-mail: [email protected]) Duxiang Wang is with Electronic Equipment Institute, P.O.Box 1610, Nanjing, 210007 Jiangsu, P.R.China.

orthogonal frequency division multiplexing technique [3]. The multi-band specification divides the frequency spectrum into 500 MHz sub-bands. Three sub-bands are mandatory, centered at 3.432, 3.960, and 4.488 GHz, respectively [3]. Here, we present a new ultra-wideband solution of an I/Q demodulator based on the six-port principle covering the frequency range 3.1 – 4.8 GHz.

Another advantage with the six-port receiver is the receiver sensitivity, which is higher compared to a standard homodyne receiver [4]. This makes it a good candidate for tomorrow’s high frequency and high data rate receivers as well as SDR receivers. The six-port I/Q demodulator presented in this paper utilizes an ultra-wideband correlator implemented in a standard printed circuit board. The designed ultra-wideband correlator shows good phase and amplitude balances. This makes it possible to use the I/Q demodulator over a wide frequency band without utilizing any calibration technique. Previous publications have shown a 23-31 GHz receiver implemented with a microwave monolithic integrated circuit (MMIC), i.e., a relative bandwidth of 30% without utilizing any calibration [5-7]. There are several previous showing wide operating bandwidth utilizing calibration methods [8 - 9]. In general, to increase the bandwidth calibration methods can be utilized for six-port receivers [8]. However, if a six-port receiver does not use any calibration procedure the data rate can be increased since it reduces the signal processing requirement [7]. This paper presents a six-port I/Q demodulator without any calibration but with good demodulation results in the UWB spectrum 3.1 to 4.8 GHz, i.e., a relative bandwidth of 43%, which has never been reported before.

II. DESIGN OF I/Q DEMODULATOR

A. Block diagram of the I/Q Demodulator Fig. 1 shows the block diagram of a six-port homodyne

receiver. It consists of a low noise amplifier (LNA), a six-port correlator, four radio frequency diodes, four low-pass filters and a judgment circuit. Fig. 2 shows two different kinds of judgment circuits, the analog and the digital judgment circuit, respectively. The analog judgment circuit can be implemented using an instrumentation amplifier [7]. If the received signal is modulated using quadrature phase shift keying (QPSK) a simple comparator can be used as the demodulator [4]. The benefit of using a digital solution is that calibration and compensation techniques can be utilized to compensate for hardware phase and amplitude errors. Thus, increase the

Pär Håkansson, Duxiang Wang, Shaofang Gong

An Ultra-Wideband Six-port I/Q Demodulator Covering from 3.1 to 4.8 GHz


ISAST Transactions on Electronics and Signal Prosessing, No. 1, Vol. 2, 2008 Pär Håkanson et al.: An Ultra-Wideband Six-port I/Q Demodulator Covering from 3.1 to 4.8 GHz111

bandwidth of the receiver. The principle of the six-port circuit as a communication

receiver is well explained in [10] by Hentschel T. If input signal is assumed to be

( ))cos()()sin()()cos()( tttxttxttxs rfrfBBrfQrfIrf ϕϖϖϖ +⋅=⋅+⋅= (1)

where xBB is the amplitude, ωrf is the anglular frequency, and φrf is the phase of the RF input signal. The local oscillator (LO) signal is assumed to be

)cos( lolololo tAs ϕϖ +⋅= (2) where Alo is the amplitude of the LO and φlo the phase of the LO-signal. Assuming wrf=wlo and φlo=0, i.e., the so-called coherent reception, and assuming an ideal six-port correlator i.e., output voltages expressed in Fig. 3, diodes operating in the square-law region and an ideal low pass filters. Then the following I- and Q- output voltages are produced at the output ports w3 –w6 utilizing an analog judgment circuit in Fig. 2a: (3)

)(5.0 2165 txAKKww Qlo=− (4)

where K1 is the transfer function of diodes and K2 is the transfer function of the low pass filter. Hence, as seen in Eq. 1 the I- and Q-data are produced. However, this requires a good amplitude balance and phase balance between the output ports, i.e., w3 to w6 in Fig. 1. To archive the w3-w4 and w5-w6 operations shown in Fig. 2a two instrumentation amplifiers are utilized, being the analog judgment circuit. The schematic of the instrumentation amplifier utilized in the designed judgment circuit is shown in Fig. 4. The purpose of the instrumentation amplifier is to amplify the difference between the input ports i.e., w3-w4 and w5-w6, and have a high common mode rejection ratio.

B. Prototype of the ultra-wideband I/Q demodulator

The ultra-wideband six-port I/Q demodulator is implemented utilizing microstrips on a two-layer printed circuit board. The parameters of the substrate are listed in TABLE I. The six-port correlator consists of three 90° branch couplers and a Wilkinson power divider. A Wilkinson power divider can reach 40% relative bandwidth which meets the specification of the designed correlator. However, the conventional branch coupler has only a relative bandwidth of 10% [11]. This paper utilizes our modified microstrip branch coupler [12], with wideband matching networks in order to maintain low insertion loss, small amplitude imbalance and small phase imbalance within the operating frequency range of 3.1 to 4.8 GHz. Detailed design information can be found in [12].

Fig. 5 shows a photo of the designed I/Q demodulator. Note that in Fig. 5 the judgment circuit is not on the board. It is connected to the output ports, i.e., P1 to P4. The amplifiers used in the instrumentation amplifier are OPA3691 from Texas Instrument Inc. The instrumentation amplifier has a gain of approximately 10.8 dB and a -3 dB bandwidth of 110 MHz. The diodes used are zero-biased schottkey diodes BAT 15-07LRF from Infineon Technologies Inc., operating in the square-law region. The low pass filter is designed to have a bandwidth of 500 MHz.

2a 2b

Fig. 2a and 2b. Judgment circuits (a) analog and (b) digital for six-port recivers.

+

+-

-I-data

Q-data

w3

w4

w5

w6

DSP

ADC

ADC

ADC

ADC

I-data

Q-data

w3

w4

w5

w6

Fig. 3. Block diagram of the ideal six-port correlator.

Power divider

slo 50 Ω

srf

0.7 slo

0.7s

rf

j0.7

s rf

j0.5 slo + j0.5 slo

j0.5 slo + 0.5 srf

0.7 slo 0.5 slo + j0.5 srf

0.5 slo - 0.5 srf

Port 3 Port 4

Port 5 Port 6

90° coupler

270 Ω

+ -

+ -

- +

560 Ω

330 Ω 330 Ω

75 Ω

270 Ω

56

0 Ω

Vout

Vin-

Vin+

Fig. 4. The simplified schematic of the instrumentation amplifier used as an analog judgement circuit.

I/Q demodulator

I-data

Judg

emen

t circ

uit

AG

Six- Port LNA

RF Diode

LP Filter

RFin Q-data

LO

w3

w4

w5

w6

Fig. 1. Block diagram of a six-port receiver with the I/Q demodulator in the dashed line block.

Cor

rela

tor

)(5.0 2143 txAKKww Ilo=−


Fig. 5. Photo of the designed six-port I/Q demodulator.

C. Test set-up

The test set-up for the proposed six-port I/Q demodulator is shown in Fig. 6. The vector signal generator used in the test set-up is an SMIQ 06 B from Rode Schwartz. To generate the LO signal a continuous sinusoidal wave from a ZVM from Agilent Technologies Inc. is utilized. To analyze the demodulated signals an wideband oscilloscope and a spectrum analyzer from Agilent Technologies are used. The vector signal analyzer software package VSA 89600 from Agilent Technologies Inc. is used to further analyze the I/Q vector signals.

III. EXPERIMENTAL RESULTS

A. The ultra-wideband correlator The key component in the proposed wideband I/Q

demodulator is the ultra-wideband correlator. Fig. 7 shows the layout of the designed six-port correlator where P1 is the LO-input port, P6 is the RF input port and P3-P5 are the output ports. Port P7 is terminated into a 50Ω load. All the design and

optimization of the correlator are done using Momentum in Advanced Design System from Agilent Technology Inc. Figs. 8 and 9 show the scattering parameters (s-parameters) from the LO-input and the RF-input to the outputs of the six-port, respectively. Fig. 10 shows the measured s-parameter from the RF-port to LO-port i.e., from P1 to P6. It is seen that the LO-RF leakage is lower than -22.5 dB within the frequency range 3.1-4.8 GHz. The theoretical phase and amplitude difference from the RF and LO inputs to the adjacent output ports should be 0 dB and 90°, respectively, as shown in Fig. 3. In Fig. 11 phase differences from the LO and RF input to adjacent output ports, i.e., from P1 and P6 to P2-P3 and P4-P5 are seen. Fig. 12 shows the amplitude difference from P1 and P6 to P2-P3 and P4-P5. Table II summarizes the key parameters of the correlator from 3.1 to 4.8 GHz.

Fig. 7. Layout of the designed ultra-wideband six-port correlator.

P1

P2

P3

P6

P4

P5

P7

Power divider

21 mm

90° branch coupler

3.3 mm

12 mm

Matching networks

I/Q demodulator

Vector Signal

Generator

LO

Oscilloscope

Ref clock

I

Q

Vector Signal

Analyzer

Instrumentation amplifier

Fig. 6. The set-up used for measurement of the I/Q demodulator.

TABLE I. SUBSTRATE PROPERTIES Dielectric thickness 0.254 mm Relative dielectric constant 3.48

Loss factor 0.004 Metal thickness 25 µm

Metal conductivity 58 MS/m Surface roughness 1 µm

3 3.5 4 4.5 5-25

-20

-15

-10

-5

0

Frequency (GHz)

Forw

ard

trans

fer f

unct

ion

(dB

)

S21S31S41S51

Fig. 8. Measured S-parameters from the LO-input (P1) to the output ports (P2-P5).

Dio

des

LP fi

lter

LO

RF input

Wid

e ba

nd 9

0°

Bra

nch

coup

lers

Wilkinson power divider

P1

P2

P3

P4


B. The I/Q demodulator

To evaluate the complete ultra-wideband I/Q demodulator shown in Fig 5 an instrumentation amplifier shown in Fig. 4 is connected to the output ports of the I/Q demodulator.

The measured I/Q constellation diagram for a 64-QAM signal at the center frequency, i.e,. 3.96 GHz is shown in Fig. 13. The error vector magnitude (EVM), magnitude error, phase error and gain imbalance in the center frequencies of the three sub-channels of the lower band of the Multiband UWB proposal (3.1 – 4.8 GHz) are shown in Table III. The data rate during this measurement is 2 Mbps with an LO signal 0 dBm and the RF signal -20 dBm.

Figs. 14 and 15 show the measured and simulated I- and Q-output data with an LO signal at 3.8 GHz and 5 dBm, and an QPSK modulated RF of -10 dBm at 2 Mbps. Fig. 16a shows the measured eye-diagram of a 30 Mbps demodulated QPSK signal, when the LO-signal is 0 dBm and the RF input signal is 15 dBm. It is seen that the eye opening is approximately 700 mV. In Fig. 16b the RF input is decreased to -30 dBm and the eye opening reduces to 10 mV. Accordingly, at the detectable signal level down to 10 mV a dynamic range of more than 45 dB is measured at the center frequency. The dynamic range in all the three bands i.e., 3.432, 3.960 and 4.488 GHz, is above 40 dBm with an LO power of 0 dBm.

TABLE II SIMULATED AND MEASURED RESULTS OF THE DESIGNED ULTRA-

WIDEBAND SIX-PORT CORRELATOR Simulated Measured Maximum phase error (P1 to P2-P3 and P4-P5)

< 6.0° < 7.1°

Maximum phase error (P6 to P2-P3 and P4-P5)

< 7.8° < 7.0°

Maximum amplitude imbalance (P6 to P2-P3 and P4-P5)

<1.0 dB <1.2 dB

Maximum amplitude imbalance (P1 to P2-P3 and P4-P5)

<1.1 dB <1.4 dB

Maximum loss (P1 to P2-P5)

< -7.8 dB < -9.1 dB

Maximum loss (P6 to P2-P5)

< -9.1 dB < -10.4 dB

3 3.5 4 4.5 5-25

-20

-15

-10

-5

0

Frequency (GHz)

Forw

ard

trans

fer f

unct

ion

(dB

)

S26S36S46S56

Fig. 9. Measured S-parameters from the RF-input port (P6) to the output ports (P2-P5).

3 3.5 4 4.5 5-40

-35

-30

-25

-20

-15

-10

-5

0

Frequency (GHz)

LO-R

F le

akag

e (d

B)

S61

Fig. 10. Measured leakage from the RF input port to the LO input port.

3 3.5 4 4.5 580

82

84

86

88

90

92

94

96

98

100

Frequency (GHz)

Pha

se d

iffer

ece

(deg

rees

)

phase(S21)-phase(S31)phase(S41)-phase(S51)phase(S26)-phase(S36)phase(S46)-phase(S56)

Fig. 11. Measured phase difference between the RF and LO input adjacent output ports.

3 3.5 4 4.5 5-3

-2

-1

0

1

2

3

Frequency (GHz)

Am

biltu

de d

iffer

ence

(dB

)

S21-S31S41-S51S26-S36S46-S56

Fig. 12. Measured amplitude difference between the RF and LO inputs and adjacent output ports.


IV. DISCUSSION It is shown in [12] that a conventional six-port correlator has

a relative bandwidth of 10 % and a phase imbalance of is 25° within the band 3.1-4.8 GHz. To overcome the limited bandwidth problem of conventional six-port I/Q demodulators, calibration techniques to compensate for amplitude and phase imbalances can be utilized [8]. However, this requires more complex baseband digital processing. The I/Q demodulator presented in this paper does not require these types of calibration procedures. This can increase the data rate of the I/Q demodulator since it reduces the signal processing requirements [7]. The use of an analog instrumentation amplifier to recover the I/Q outputs avoids the use of four ADCs in the receiver [7]. This paper also presents higher order QAM demodulation results, i.e., 64-QAM compared to previous publications, e.q., 16-QAM [7]. The receiver in [7] shows the capability to demodulated OPSK signals within a relative bandwidth of 30% [7]. However, QPSK modulated signals can be demodulated at a much higher EVM then presented in this paper i.e., 5.7% in the three lower subbands of UWB.

The bandwidth of the I/Q demodulator presented in this paper covers from 3.1 to 4.8 GHz. However, the -3dB bandwidth of the instrumentation amplifier used in this work is 120 MHz. Thus, the used instrumentation amplifier limits the achievable maximum data rate. The dynamic range of the I/Q demodulator is more than 40 dB in all three sub-bands with an LO signal of 0 dBm. To increase the dynamic range an LNA for the RF signal with an automatic gain control can be utilized.

The size (14x6.5 cm) of the manufactured six-port I/Q demodulator is relatively large. Since the lowpass filters utilizing microstrips occupy a large portion of the printed circuit board, an implementation of the lowpass filters utilizing discrete components will reduce the size significantly. Furthermore, a multi-layer printed circuit board design can be used to reduce the size significantly.

In wireless communications the received signal is distorted

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.5

0

0.5

Symbol

Q-o

utpu

t (V

)

Eye opening (a)

TABLE III. MEASURED PROPERTIES OF THE I/Q DEMODULATOR flo = frf 3.432 GHz 3.96 GHz 4.488 GHz Error Vector Mag (%RMS)

5.7 2.2 5.0

Mag. Error (%RMS)

4.2 1.3 2.3

Phase error (°)

3.1 1.7 5.6

Gain imbalance (dB)

-0.9 -0.3 -0.5

Fig. 15. Simulated I and Q output with a QPSK-modulated signal at a data rate of 2 Mbps.

0 2 4 6 8 10 12 14 16 18 20

-0.1

-0.05

0

0.05

0.1

Time (us)

I-out

put (

V)

0 2 4 6 8 10 12 14 16 18 20

-0.1

-0.05

0

0.05

0.1

Time (us)

Q-o

utpu

t (V)

0 2 4 6 8 10 12 14 16 18 20

-0.1

-0.05

0

0.05

0.1

Time (us)

I-out

put (

V)

0 2 4 6 8 10 12 14 16 18 20

-0.1

-0.05

0

0.05

0.1

Time (us)

Q-o

utpu

t (V

)

Fig. 14. Measured I and Q output with a QPSK-modulated signal a data rate of 2 Mbps.

-0,2 -0,1 0,0 0,1 0,2

-0,2

-0,1

0,0

0,1

0,2

I

QFig. 13. Measured I and Q constellation diagram of a 64-QAM signal at 3.96 GHz.

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.01

-0.005

0

0.005

0.01

Symbol

Q-o

utpu

t (V

)

Eye opening (b)

Fig. 16. Measured eye-diagram of the Q-output with QPSK modulated signal at 3.96 GHz, with a 0 dBm LO signal and a RF of (a) 15 dBm and (b) -30 dBm.


by channel imperfections such as multi-path reflections. Therefore, in coherent reception the carrier must be recovered locally [13]. There have been several studies covering carrier recovery methods such as De Costas Loop and the Reverse-Modulation Loop for six-port receivers [13-15].

V. CONCLUSIONS An ultra-wideband six-port I/Q demodulator covering the

UWB spectrum from 3.1 to 4.8 GHz is presented. The used ultra-wideband correlator is implemented in a standard printed circuit board showing a relative bandwidth of 43%. The small phase and amplitude imbalances of the wideband correlator, i.e., 7° and 1.4 dB, make it possible to produce a high quality RF signal receiver without using any calibration technique in the UWB frequency band 3.1 – 4.8 GHz. The complete demodulator shows an EVM lower than 5.7% RMS in the three UWB sub-bands between 3.1 – 4.8 GHz.

ACKNOWLEDGMENT The authors would like to express their gratitude to Magnus

Karlsson, Allan Huynh and Adriana Serban at the Department of Science and Technology of Linköping University, Sweden, for their assistance during the course of this work.

REFERENCES [1] T.Eireiner, T. Schnurr, and T.Müller, “Integration of a six-port receiver

for mm-wave communication”, IEEE MELECON 2006, Benalmádena (Málaga), Spain, pp.371- 376.

[2] Li J., R.G. Bosisio, K. Wu ”Computer and Measurment Simulation of a New Digital Receiver Operating Directly at Millimeter-Wave Frequencies”, IEEE Trans. Microw. Theory Tech., Vol 43,No 12, pp.2766-2772, December 1995.

[3] D. Geer, "UWB standardization effort ends in controversy," Computer, vol. 39, no. 7, pp. 13-16, July 2006.

[4] J.-C. Schiel, S. O. Tatu, K.Wu, and R. G. Bosisio, “Six-port direct digital receiver (SPDR) and standard direct receiver (SDR) results for QPSK modulation at high speeds,” in IEEE MTT-S Int. Microwave Symp. Dig., 2002, pp. 931–934.

[5] S. O.Tatu, E.Moldovan, K.Wu, and R.G.Bosisio, “A new direct millimeter-wave six-port receiver”, IEEE Trans. Microwave Theory Tech., vol. MTT-49, no.12, pp.2517-2522, Dec. 2001.

[6] S. O. Tatu, E. Moldovan, G.Brehm, K. Wu, and R.G.Bosisio, “Ka-band direct digital receiver”, IEEE Trans. Microwave Theory Tech., vol.MTT-50, no.11, pp.2436-2442, Nov. 2002.

[7] S. O. Tatu, E. Moldovan, K. Wu, R.G.Bosisio and Tayeb A. Denidni “Ka-band analog Front-end for Software Defined Direct Conversion Receiver”, IEEE Trans. Microwave Theory Tech., vol-53, no. 9, pp.2768-2776, Sept, 2005

[8] F.R de Sausa and B. Huyart, ”1.8 – 5.5 Ghz Integrated Five-Port Front-End for Wideband transceivers”, 7th European Conference on Wireless Technology, 2004, pp. 67 – 69.

[9] T. Mack, A. Honold and J-F. Luy, “An Extreamly Broadband Software Configurable Six-Port Receiver Platform” Proc. Of 33rd European Microwave conference”, Munich 2003, pp. 623 – 626.

[10] Hentschel Tim. “The Six-Port as a Communication Receiver” IEEE Trans. Microw. Theory Tech., vol 53, no. 3, pp 1039-1047, March 2005

[11] G. P. Riblet, “A directional coupler with very flat coupling”, IEEE Trans. Microwave Theory and Tech., vol.26, no.2, pp.70-74, 1978.

[12] Duxiang Wang, Allan Huynh, Pär Håkansson, Ming Li and Shaofang Gong, "Study of Wideband Microstrip Correlators for Ultra-wideband Communication Systems," Proc. Of Asia Pacific Microwave Conf. 2007, Bangkok, Accepted for publication.

[13] F.R de Sausa and B. Huyart, “Reconfigurable Carrier Recovery Loop”, Microw. And Optical Tech. Letters., Vol 43, No5, pp. 406-408, Dec. 2004.

[14] E. Marsan, J.-C. Schiel, K. Wu Gailon Brehm and R. G. Bosisio, „High-Speed Carrier Recovery Circuit Suitable for Direct Digital QPSK Transceivers“ Proc. of RAWCON 2002, pp. 103 – 106.

[15] F.R de Sausa and B. Huyart, ”Carrier Recovery in five-port receivers”, Proc. Euro Conf. Wireless Technology., 2003, pp. 419-421.

Pär Håkansson was born in Karlshamn, Sweden in 1979. He received his M.Sc. degree from Linköping University in Sweden in 2003. From 2004 to 2005 he worked as a research engineer in the research group of Communication Electronics at Linköping University, Sweden. In 2005 he started his Ph.D. study in the research group. His main work involves both wireless and wired high-speed data communications.

Duxiang Wang was born in Zhenjiang City, Jiangsu, China, in 1965. He received his BEE from Nanjing University of Aeronautics and Astronautics, China, in 1982, and his MSEE from Shanghai University, China, in 1990. After graduating, he joined Nanjing Electronic Equipment Institute, China, where he contributed to microwave circuit and microwave receiver and system design. Since 2001 he has been a professor in Nanjing Electronic Equipment. In 2006 he had been a researcher for six months in communication electronics at Linköping University, Sweden, as a senior visiting scholar. Duxiang was the recipient of special award from China State Council in 2002 and is presently a senior member of China Institute of electronics (CIE)

Shaofang Gong was born in Shanghai, China, in 1960. He received his B.Sc. degree from Fudan University in Shanghai in 1982, and the Licentiate of Engineering and Ph.D. degree from Linköping University in Sweden, in 1988 and 1990, respectively. Between 1991 and 1999 he was a senior researcher at the microelectronic institute – Acreo in Sweden. From 2000 to 2001 he was the CTO at a spin-off company from the institute. Since 2002 he has been full professor in

communication electronics at Linköping University, Sweden. His main research interest has been communication electronics including RF design, wireless communications and high-speed data transmissions.


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