Research Article Design and Implementation of a Chaotic ...downloads.hindawi.com/journals/jcnc/2016/5976282.pdf · An enhancement technique is suggested by using single reference

Research ArticleDesign and Implementation of a Chaotic Scheme inAdditive White Gaussian Noise Channel

Nizar Al Bassam and Oday Jerew

Middle East College, Knowledge Oasis Muscat, PB No. 79, Al Rusayl, 124 Muscat, Oman

Correspondence should be addressed to Nizar Al Bassam; [email protected]

Received 30 January 2016; Accepted 3 May 2016

Academic Editor: Gianluigi Ferrari

Copyright © 2016 N. Al Bassam and O. Jerew. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

A new chaotic scheme named Flipped Chaotic On-Off Keying (FCOOK) is proposed for binary transmission. In FCOOK, the lowcorrelation value between the stationary signal and its mirrored version is utilized. Transmitted signal for binary 1 is a chaoticsegment added to its time flipped (mirrored) version within one bit duration, while in binary 0, no transmission takes placewithin the same bit duration.The proposed scheme is compared with the standard chaotic systems: Differential Chaos Shift Keying(DCSK) and CorrelationDelay Shift Keying (CDSK).The Bit Error Rate (BER) of FCOOK is studied analytically based onGaussianapproximation method. Results show that the BER performance of FCOOK outperforms DCSK and CDSK in AWGN channelenvironment and with various 𝐸𝑏/𝑁𝑜 levels. Additionally, FCOOK offers a double bit rate compared with the standard DCSK.

1. Introduction

In recent years, chaotic signals become natural candidatesfor spreading narrow band information due to their wide-band characteristic. Thus, using chaotic signals to encodeinformation, the resulting signals are spread spectrum signalshaving larger bandwidths and lower power spectral densities.Chaotic signals enjoy all the benefits of spread spectrumsignals such as a difficulty of uninformed detection, mitiga-tion of multipath fading, and antijamming. Moreover, a largenumber of spreading waveforms can be produced easily as aconsequence of the sensitive dependence upon initial condi-tions and parameters variations.This provides a low cost andsimple means for spread spectrum communications [1].

A number of modulation and demodulation schemeshave been proposed for digital chaos based communications[2–5]. In most proposed methods, the basic principle is tomap the digital symbols to nonperiodic chaotic basic signals.For instance, Chaos Shift Keying (CSK) maps differentsymbols to different chaotic basic signals, which are producedfrom various dynamical systems. If synchronized copies ofthe chaotic basis signals are available at the receiver, detectioncan be conducted by evaluating synchronization error [6] orbased on a conventional correlator type [7, 8]. This class of

detection is known as a coherent detection. An enhancedtechnique to improve BER is suggested in [9]. If synchro-nized copies of the chaotic basis signal are not availableat the receiver, detection has to be done by noncoherentdetection as in noncoherent CSK and Chaotic On-OffKeying(COOK). Detection depends on estimation of transmittedsignal energy. However, determining an optimum thresholdbetween signal elements is the major drawback due to noisepower contribution [6, 7] and the noise performance can beincreased only by increasing the distance between signal ele-ments.This requires the transmitter to consumemore energyfor each bit to increase the distance between signal elementsand the receiver. Noise performance in coherent system issuperior to the noncoherent systems. On the other hand,accuracy in synchronization is required to be achieved incoherent systems, which is difficult task [8].

Another widely used technique for modulation is knownas differential coherent systems where Differential ChaosShift Keying (DCSK) and Correlation Delay Shift Keying(CDSK) [10] are the basic schemes. Each information bit istransmitted by the generation of twin chaotic segments; firstsignal is called reference and the second, which is modulatedby the information bit, is called information bearing signal.Information detection is performed by correlating reference

Hindawi Publishing CorporationJournal of Computer Networks and CommunicationsVolume 2016, Article ID 5976282, 7 pageshttp://dx.doi.org/10.1155/2016/5976282

2 Journal of Computer Networks and Communications

with the information signal over last half of bit duration.Therefore, half of the average bit energy is wasted withrespect to noncoherent systems. An enhancement techniqueis suggested by using single reference signal for multiple bitstransmission [11]. A generalized Correlation Delay Shift Key-ing scheme is suggested in [12] by producing several delayedversions of a chaotic signal; selected version is modulatedwith the information bits. Although the system requires bankof transmitter, it can achieve similar performance comparedto that of DCSK at reasonable 𝐸𝑏/𝑁𝑜 range. Additionally,sending the reference signal and the information signal on thesame time slot is discussed in [13–18]. Data rate and BER per-formance are increased, but schemes are complex and requireadditional synchronization circuits. Multicarrier modulationforDCSK (MC-DCSK) signal is explored in [13]. In spite of anefficient utilization of the transmitted energy, system imple-mentation in (MC-DCSK) includes bank of narrow bandmodulator that needs a high degree of accuracy in the designto maintain subcarrier synchronization.

In this paper, we introduce new version of chaotic on-offscheme named (FCOOK). In FCOOK the stationary signaland its flipped version are added togetherwithin one bit dura-tion to represent binary 1. for binary 0, no transmission takesplace for the same bit duration. Based on the fact that thechaotic signal and its time flipped version are stationary, lowcross correlation can be achieved [10]. This helps to designa new signal for the binary information transmission. Thedesign allows the receiver to correlate each first half of incom-ing signal with the flipped version of the second half. Sinceboth halves contain noncorrelated noise segments, this can beused to overcome the threshold shifting problem.

The remainder of this paper is organized as follows.Standard differentially coherent systems are described inSection 2. In Section 3, the transmitter and receiver modelsof FCOOK are described. Theoretical estimation of BER isderived in Section 4. Finally, simulation results are discussedin Section 5.

2. Chaos Based Systems: Revisited

2.1. DCSK. The transmitter structure of the DCSK is shownin Figure 1. Each information bit 𝑏, where 𝑏 ∈ {1, −1}, is trans-mitted by sending two chaotic segments into successive iden-tical time slots. Each slot is occupied by𝑀 samples, where 2𝑀represent the spreading factor. First time slot contains a refer-ence signal and second slot contains a delayed version of thereference signal multiplied by the information bit 𝑏. There-fore, the 𝑖th transmitted signal 𝑠𝑖 for a single bit can be writtenas

𝑠𝑖 =

{

{

{

𝑥𝑖, 0 < 𝑖 ≤ 𝑀,

𝑏𝑥𝑖−𝑀, 𝑀 < 𝑖 ≤ 2𝑀.

(1)

Hence, average bit energy 𝐸𝑏 = 2𝑀𝑉(𝑥𝑖), where 𝑉(⋅) is thevariance operator and the expected value of 𝐸(𝑋) = 0.

At the receiver, each received signal 𝑟𝑖 is multiplied byits delayed version 𝑟𝑖−𝑀 and averaged over half bit duration.Assume that the channel is AWGN, and then the received

Chaosgenerator

Data b

Delay Mbxi−M

xi bxi−M · · · xi

Figure 1: DCSK transmitter.

Threshold

Delay Mri−M

riri−MZDCSK =

M

∑i=1

ri⌣b

Figure 2: DCSK receiver.

signal 𝑟𝑖 = 𝑠𝑖 + 𝜁𝑖, where 𝜁 is stationary random process with𝐸(𝜁) = 0 and the two-sided noise power spectral density isgiven by 𝑉(𝜁) = 𝑁𝑜/2. The correlator output 𝑍DCSK at theend of bit duration can be written as

𝑍DCSK =𝑀

∑

𝑖=1

𝑟𝑖𝑟𝑖−𝑀 =

𝑀

∑

𝑖=1

(𝑠𝑖 + 𝜁𝑖) (𝑠𝑖−𝑀 + 𝜁𝑖−𝑀)

= 𝑏

𝑀

∑

𝑖=1

𝑥2

𝑖+

𝑀

∑

𝑖=1

𝑥𝑖 (𝜁𝑖−𝑀 + 𝑏𝜁𝑖) +

𝑀

∑

𝑖=1

𝜁𝑖𝜁𝑖−𝑀.

(2)

The DCSK receiver structure is shown in Figure 2. Informa-tion bit is decoded using the following rule:

�� =

{

{

{

1, 𝑍DCSK ≥ 0,

−1, 𝑍DCSK < 0.(3)

Under the standard assumptions that 𝑥 is stationary and𝑥𝑖 is statistically independent from 𝜁𝑗 at any (𝑖, 𝑗), correlatoroutput𝑍DCSK tends to haveGaussian distribution at sufficientvalue of 𝑀 [10]. Therefore, theoretical evaluation can beobtained by calculating the mean and variance of conditionalProbability Density Function (PDF) of𝑍DCSK at ±1. If Pr(0) =Pr(1) = 0.5, then BER can be calculated as [10]

BERDCSK

=1

2erfc(√

𝐸𝑏

4𝑁𝑜

(1 +2

5𝑀

𝐸𝑏

𝑁𝑜

+𝑁𝑜

2𝐸𝑏

𝑀)

−1

) ,

(4)

where

erfc (𝑦) = 2

√𝜋∫

∞

𝑦

𝑒−𝑡2/2𝑑𝑡. (5)

Journal of Computer Networks and Communications 3

Chaos source

Delay M

xi

xi−Mblxi−M

si = xi + blxi−M

bl ∈ {−1, 1}

Figure 3: CDSK transmitter.

2.2. CDSK. Unlike DCSK, each information bit in CDSK issent by transmitting a signal which is the sum of a chaoticsequence 𝑥𝑖 and of the delayed chaotic sequence multipliedby the information signal 𝑏ℓ𝑥𝑖−𝐿, where ℓ is the bit counter.Thus, transmitted signal of CDSK at any instant 𝑖 is given by

𝑠𝑖 = 𝑥𝑖 + 𝑏ℓ𝑥𝑖−𝐿, (ℓ − 1)𝑀 < 𝑖 ≤ ℓ𝑀, (6)

where 𝐿 ≥ 𝑀 and 𝐸𝑏 = 2𝑀𝑉(𝑥).As shown in Figure 3, the CDSK transmitter differs from

DCSK in that the switch in the transmitter is now replacedby an adder and the transmitted signal is never repeated.Therefore, data rate is doubled compared with that in DCSK[10]. Putting delay 𝐿 = 𝑀, then the receiver of CDSK issimilar to that of DCSK, and each segment is correlated withthe previous one. Correlator output 𝑍CDSK can be calculatedas

𝑍CDSK =𝑀

∑

𝑖=1

𝑟𝑖𝑟𝑖−𝑀

=

𝑀

∑

𝑖=1

(𝑥𝑖 + 𝑏ℓ𝑥𝑖−𝑀 + 𝜁𝑖) (𝑥𝑖−𝑀 + 𝑏ℓ−𝑀𝑥𝑖−2𝑀 + 𝜁𝑖−𝑀)

= 𝑏ℓ

𝑀

∑

𝑖=1

𝑥2

𝑖−𝑀+

𝑀

∑

𝑖=1

𝑥𝑖𝑥𝑖−𝑀 + 𝑏ℓ−𝑀

𝑀

∑

𝑖=1

𝑥𝑖𝑥𝑖−2𝑀

+

𝑀

∑

𝑖=1

𝑥𝑖𝜁𝑖−𝑀 + 𝑏ℓ𝑏ℓ−𝑀

𝑀

∑

𝑖=1

𝑥𝑖−𝑀𝑥𝑖−2𝑀

+ 𝑏ℓ

𝑀

∑

𝑖=1

𝑥𝑖−𝑀𝜁𝑖−𝑀 +

𝑀

∑

𝑖=1

𝑥𝑖−𝑀𝜁𝑖 + 𝑏ℓ−𝑀

𝑀

∑

𝑖=1

𝑥𝑖−2𝑀𝜁𝑖

+

𝑀

∑

𝑖=1

𝜁𝑖𝜁𝑖−𝑀.

(7)

It can be clearly observed that the correlator output𝑍CDSK contains more intrasignal and noise terms comparedto DCSK. Thus, BER performance is decreased based on theprevious assumptions in DCSK and given that 𝐸(𝑥𝑖𝑥𝑖+𝑘) = 0,where 𝑘 is an integer and 𝐸(⋅) is the average value. It canbe easily shown that 𝑍CDSK has a Gaussian distribution andtheoretical value of BER can be found by calculating themean and variance of conditional PDF of𝑍CDSK at +1 and −1

Chaotic signal

Mirrored chaotic signal

Binary 1 Binary 0

Transmitted signal

i

i

i

Information

+

i =M

2

Figure 4: Flipped Chaotic On-Off Keying signal format for binary1 and binary 0.

Chaos source

Datacontrolled

switchSignal flipping

Chaotic segment

Mirrored version of thechaotic segment

Delay Mxi

√Ps

over MxM−i+1 b ∈ {1, 0}

si

Figure 5: Flipped COOK transmitter.

transmission. Decoding is performed according to the samerule in (3). Thus, BER is given by [10]

BERCDSK

=1

2erfc(√

𝐸𝑏

8𝑁𝑜

(1 +19

20𝑀

𝐸𝑏

𝑁𝑜

+𝑁𝑜

4𝐸𝑏

𝑀)

−1

) .

(8)

3. Proposed Scheme: Flipped Chaotic On-OffKeying (FCOOK)

3.1. Signal Format. In our scheme, binary 1 is transmittedby sending the sum of adding identical, mirrored chaoticsegments with length of𝑀. Thereby, the resultant signal forbinary 1 will be symmetric around 𝑖 = 𝑀/2 as shown in Fig-ure 4. On the other hand, the receiver keeps silence for binary0 duration and no signal will be transmitted.

3.2. Transmitter Description. In order to generate theFCOOK signal in Figure 4, the FCOOK transmitter setupis proposed as shown in Figure 5. A flipped signal 𝑥𝑀−𝑖+1 isgenerated by receiving a copy of𝑀 samples and flipping them


over one bit duration.This process requires the forward signalto be delayed for one bit duration till the end of one segmentgeneration. To achieve time synchronization between theemitted segment and its flipped version, a delay element isinserted before the adder. However, this has no effect onsystem notation or analysis because we assume that bothsegments are ready to be added at any instant 𝑖. Then, thesymmetric transmitted signal for single bit duration can bewritten as

𝑠𝑖 = 𝑏√𝑃𝑠 (𝑥𝑖 + 𝑥𝑀−𝑖+1) , 0 < 𝑖 ≤ 𝑀, (9)

where √𝑃𝑠 is signal level and 𝑏 ∈ {1, 0}. Compared with (3),the FCOOK system transmits all the signal components only

within𝑀 samples. This offers double bit rate compared withDCSK system.

3.3. Receiver Description. At the receiver, each incomingsignal, 𝑟𝑖, is multiplied with its mirrored version 𝑟𝑀−𝑖+1 overduration of 𝑀/2 only to avoid having duplicated terms inthe correlator output. Signal is mirrored by sampling andreverting the received signal over one bit duration as shown inFigure 6.The delay element is added to compensate the delayincurred in the time reversal of the received signal.

Assume that both transmitter and receiver are fullysynchronized and the channel is AWGN; then the correlatoroutput 𝑍 at the end of single bit duration can be given as

𝑍 =

𝑀/2

∑

𝑖=1

𝑟𝑖𝑟𝑀−𝑖+1 = (𝑏√𝑃𝑠 (𝑥𝑖 + 𝑥𝑀−𝑖+1) + 𝜁𝑖)(𝑏√𝑃𝑠 (𝑥𝑀−𝑖+1 + 𝑥𝑖) + 𝜁𝑀−𝑖+1)

= 𝑏𝑃𝑠

𝑀/2

∑

𝑖=1

(𝑥2

𝑖+ 𝑥2

𝑀−𝑖+1)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

𝐴

+ 2𝑏𝑃𝑠

𝑀/2

∑

𝑖=1

(𝑥𝑖 ⋅ 𝑥𝑀−𝑖+1)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

𝐵

+ 𝑏√𝑃𝑠

𝑀/2

∑

𝑖=1

(𝑥𝑖 ⋅ 𝜁𝑀−𝑖+1) + 𝑏√𝑃𝑠

𝑀/2

∑

𝑖=1

(𝑥𝑀−𝑖+1 ⋅ 𝜁𝑀)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

𝐶

+ 𝑏√𝑃𝑠

𝑀/2

∑

𝑖=1

(𝑥𝑖 ⋅ 𝜁𝑖) + 𝑏√𝑃𝑠

𝑀/2

∑

𝑖=1

(𝑥𝑀−𝑖+1 ⋅ 𝜁𝑀−𝑖+1)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

𝐷

𝑀/2

∑

𝑖=1

(𝜁𝑖 ⋅ 𝜁𝑀−𝑖+1)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

𝐸

.

(10)

Detector input 𝑍 consists of the required signal (𝐴),intrasignal (𝐵) term, signal-noise terms (𝐶,𝐷), and noise-noise term (𝐸); all these terms are contributed only if binary1 is transmitted. In binary 0 transmission, except for (𝐸), allthe terms are disappeared because 𝑏 = 0 in (10).

For the sake of simplicity, we set the threshold 𝛼th value atthemiddle between signal elements (0, 𝐸𝑏).Thus, the receivedsignal is decoded according to the following rule:

𝑏 =

{

{

{

1, 𝑍 ≥ 𝛼th,

0, 𝑍 < 𝛼th.(11)

4. Performance Evaluation

A baseband model is used to evaluate the BER performanceof FCOOK. Our performance computation is based onGaus-sian approximation method that provides good estimation ofBER at large spreading factor [10, 15]. For small spreadingfactor values, bit energy distribution is required [19, 20]. Westart our evaluation by revisiting some properties of chaoticsignals and few standard assumptions.

Symmetric Tent Map, defined by the equation 𝑥𝑛+1 = 1 −2|𝑥𝑛|, is applied to generate the discrete chaotic signal. Signalis stationary and is uniformly distributed between (−1, 1)with zero mean and computed variances 𝑉(𝑥) = 1/3 and𝑉(𝑥2) = 4/45 = 4/5 ∗ 𝑉

2(𝑥) [6].

Recall that a chaotic signal, as any spreading spectrumsignals, has an impulse like autocorrelation value such that

𝐸 (𝑥𝑖𝑥𝑗) =

{

{

{

𝑉(𝑥𝑖) , 𝑖 = 𝑗,

0, 𝑖 = 𝑗.

(12)

Therefore, it is easy to show that, for sufficient value of𝑀,𝐸(𝑥𝑘𝑥𝑀−𝑖+1) = 0, where 𝑖 ≤ 𝑀/2 and

𝑉 (𝑥𝑖𝑥𝑀−𝑖+1) = 𝑉 (𝑥𝑖) + 𝑉 (𝑥𝑀−𝑖+1)

− 2 cov (𝑥𝑖𝑥𝑀−𝑖+1)

≈ 𝑉 (𝑥𝑘) + 𝑉 (𝑥𝑀−𝑘+1) .

(13)

It is logically to assume that 𝜁𝑖 is statistically independentfrom 𝑥𝑖 at any (𝑖, 𝑗) and 𝜁𝑗 for any 𝑖 = 𝑗. Based on the centrallimit theorem and with previous assumptions, correlatoroutput in (10) can be regarded as a Gaussian distributed[10]; thereby the performance can be fully characterized bycalculating themean and variances for the PDF of𝑍 at binary1 and 0 transmission. Hence,

𝐸 (𝑍 | 𝑏 = 1) = 𝐸 (𝐴) + 𝐸 (𝐵) + 𝐸 (𝐶) + 𝐸 (𝐷)

+ 𝐸 (𝐹) = 𝑃𝑠

𝑀/2

∑

𝑖=1

𝐸 (𝑥2

𝑖+ 𝑥2

𝑀−𝑖+1)

= 𝑃𝑠𝑀𝑉(𝑥) = 𝐸𝑏,


Threshold

Signal flipping

Received signal with noisefor binary “1”

Received signal with noise forbinary “1” (mirrored version)

Delay M

over M

ri

rM−i+1

Z =

M/2

∑i=1

rirM−i+1

⌣b

Figure 6: Flipped COOK receiver.

𝑉 (𝑍 | 𝑏 = 1) = 𝑉 (𝐴) + 𝑉 (𝐵) + 𝑉 (𝐶) + 𝑉 (𝐷)

+ 𝑉 (𝐸) ,

𝑉 (𝐴) =

𝑀/2

∑

𝑖=1

𝑉(𝑃𝑠𝑥2

𝑖+ 𝑃𝑠𝑥2

𝑀−𝑖+1)

=4

5𝑃2

𝑠𝑀𝑉(𝑥)𝑉 (𝑥) =

4

5

𝐸2

𝑏

𝑀,

𝑉 (𝐵) = 𝑉(2

𝑀/2

∑

𝑖=1

𝑃𝑠𝑥𝑖𝑥𝑀−𝑖+1)

= 4𝑃2

𝑠𝑉(

𝑀/2

∑

𝑖=1

𝑥𝑖𝑥𝑀−𝑖+1) = 2𝐸2

𝑏

𝑀,

𝑉 (𝐶) = 𝑉(√𝑃𝑠

𝑀/2

∑

𝑖=1

(𝑥𝑖 ⋅ 𝜁𝑀−𝑖+1))

+ 𝑉(√𝑃𝑠

𝑀/2

∑

𝑖=1

(𝑥𝑀−𝑖+1 ⋅ 𝜁𝑀))

= 𝑀𝑃𝑠𝑉 (𝑥)𝑉 (𝜁) =𝐸𝑏𝑁𝑜

2.

(14)

Similarly 𝑉(𝐷) = 𝐸𝑏𝑁𝑜/2. Consider

𝑉 (𝐸) = 𝑉(

𝑀/2

∑

𝑖=1

𝜁𝑖𝜁𝑀−𝑖+1) =

𝑀/2

∑

𝑖=1

𝑉 (𝜁𝑖𝜁𝑀−𝑖+1)

=𝑀𝑁2

𝑜

8,

𝑉 (𝑍 | 𝑏 = 1) =4

5

𝐸2

𝑏

𝑀+ 2𝐸2

𝑏

𝑀+ 𝐸𝑏𝑁𝑜 +

𝑀𝑁2

𝑜

8

=14

5

𝐸2

𝑏

𝑀+ 𝐸𝑏𝑁𝑜 +

𝑀𝑁2

𝑜

8,

𝐸 (𝑍 | 𝑏 = 0) − 0,

𝑉 (𝑍 | 𝑏 = 0) = 𝑉 (𝐸) =𝑀𝑁2

𝑜

8.

(15)

FCOOK theory (Eb/No = 16dB)FCOOK simulation (Eb/No = 16dB)

10−3

10−2

10−1

100

BER

200 300 400 500 600 700 800100Spreading factor M

Figure 7: BER performance versus spreading factor𝑀.

If the input to the decision device is less than the threshold𝛼th, a received bit 1 can be detected as bit 0, while bit 0 can bedetected as bit 1 if the metric is more than the threshold andassumes that 𝑃𝑟(0) = 𝑃𝑟(1) = 0.5.

Then the average probability of error,𝑃𝑒, can be expressedas

𝑃𝑒 =1

2(

1

√2𝜋𝑉 (𝑍 | 𝑏 = 1)

∫

𝛼th

−∞

𝑒−(𝑍−𝐸

𝑏)2/2𝑉(𝑍|𝑏=1)

𝑑𝑍

+1

√2𝜋𝑉 (𝑍 | 𝑏 = 0)

∫

∞

𝛼th

𝑒−𝑍2/2𝑉(𝑍|𝑏=0)

𝑑𝑍) ,

(16)

𝑃𝑒 =1

2(𝑄(√

𝐸𝑏

𝑁𝑜

(14𝐸𝑏

𝑀𝑁𝑜

+ 1 +𝑀

8

𝑁𝑜

𝐸𝑏

)

−1

)

+ 𝑄(𝐸𝑏

𝑁𝑜

√2

𝑀)) .

(17)

5. Simulation Results and Discussion

In this section, the BER performance of DCSK, CDSK, andFCOOK is tested with the wide range of spreading factors𝑀values and at various 𝐸𝑏/𝑁𝑜 levels. To have fair comparison,signal level in the proposed scheme is set to √2. Henceaverage energy 𝐸𝑏 for each bit is given, (4𝑀𝑉(𝑥) + 0)/2 =2𝑀𝑉(𝑥). This is exactly equal to the average bit energy usedin DCSK and CDSK [10].

Apparently, small disagreement between theoretical esti-mation and simulation results appears at relatively small𝑀in Figure 7. This is one of Gaussian approximation methoddrawback as mentioned in Section 4. However, clear agree-ment between analytical estimate and the simulation resultscan be observed at𝑀 > 100, which positively supports ourclosed form expression in (17). Additionally, the suboptimum


FCOOK (simulation)FCOOK (theory)DCSK (simulation)

DCSK (theory)CDSK (simulation)CDSK (theory)

10−3

10−2

10−1

100

BER

5 10 15 200Eb/No (dB)

Figure 8: Relation between BER and 𝐸𝑏/𝑁𝑜 for DCSK, CDSK, andFCOOK by simulations and theoretical estimation at𝑀 = 100.

FCOOKDCSK

CDSK

10−4

10−3

10−2

10−1

100

BER

5 10 15 20 250Eb/No (dB)

Figure 9: Simulated BERs versus 𝐸𝑏/𝑁𝑜 for FCOOK, DCSK, andCDSK at𝑀 = 500.

performance for our prosed system can be noticed between𝑀 = 100 and𝑀 = 200, but this decreases after𝑀 = 200. Thereason behind this performance is that when𝑀 increases, thenoise-noise contribution grows faster than the signal energyin (10).

The theoretical estimations for BER, which are obtainedby Gaussian approximation method in [10, 15], and sim-ulation results for DCSK and CDSK are compared withFCOOK at 𝑀 = 100 as shown in Figure 8. In addition tothe matching between theoretical estimation and simulationresults, FCOOK can always pass the performance of CDSKand outperform DCSK in considerable range of 𝐸𝑏/𝑁𝑜.

FCOOKDCSK

CDSK

10−4

10−3

10−2

10−1

100

BER

5 10 15 20 250Eb/No (dB)


FCOOKDCSK

CDSK

10−4

10−3

10−2

10−1

100

BER

5 10 15 20 250Eb/No (dB)


Furthermore, CDSK seems always to be 2 dB worse thanFCOOK in Figures 9, 10, and 11, respectively; the reason ofwhich is that there are much more intrasignal interferenceand noise interference components in the correlator outputof CDSK. The proposed system performs almost 1 dB betterthan DCSK with𝑀 = 600 and 700 at 𝐸𝑏/𝑁𝑜 less than 18 dB.This can be explained by the fact that, compared toDCSK, thecontribution of noise terms in (2) is halved.This contributionis the reason of nonlinearity at large value of𝑀 which causesall the systems to have low BER performances at very largespreading factors.


6. Conclusion

A new chaotic communication scheme named FCOOK isdeveloped for binary transmission. The system makes anadvantage of the low correlation between the chaotic seg-ments and its mirrored twin. The BER performance of thesystem is derived analytically using Gaussian approximationmethod. Simulation results illustrate the capability of theexpression to predict BER at different spreading factor.More-over, performance of the proposed scheme is compared withDCSK and CDSK systems by plotting BER with 𝐸𝑏/𝑁𝑜 at thesame spreading factor. Results show that FCOOK has advan-tage of 2 dB compared with CDSK and 1 dB compared withthe DCSK.

Competing Interests

The authors declare that they have no competing interests.

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