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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 50, NO. 1, JANUARY 2003 3

Spectral Properties of Chaos-Based FM Signals:Theory and Simulation Results

Sergio Callegari, Member, IEEE, Riccardo Rovatti, Senior Member, IEEE, and Gianluca Setti, Senior Member, IEEE

Abstract—This paper addresses the problem of generatingconstant-envelope wideband (CEW) signals, for which applica-tions are emerging both in telecommunications (as informationcarriers) and in digital/power electronics (to aid the synthesis oftiming signals which favor electromagnetic compliance). A flexiblegeneration technique consists in driving a frequency modulation(FM) modulator with random or chaotic sequences. Mathematicaltools for predicting some spectral properties of random-FM andchaotic-FM CEW signals are herein introduced by commenting onrecent results and presenting novel ones in a coherent framework.

Index Terms—Chaos, frequency modulation (FM), randomness,spread spectrum.

I. INTRODUCTION

SPREAD-SPECTRUM signal-processing techniques areencountering a growing interest thanks to the constantly

increasing number of applications where they pair or outclassthe performance of more conventional approaches. Forinstance, most of the next-generation telecommunication sys-tems will be based on spread-spectrum schemes [1]–[3] whichachieve both good robustness to noise and low co-channelinterference [3]. Furthermore, by no means spread-spectrummethodologies be confined to data transmission. Conversely,they can be fruitfully applied to other fields, where someof those characteristic features exploited in communicationsare still of capital importance. Preeminently, the ability tooperate with low electromagnetic interference to narrowbandneighboring equipment is appealing to those designingdc-dc converters, power actuators, clock-signals distributionnetworks, etc. [4], [5].

The focus of this paper is on a particular aspect ofspread-spectrum signal processing, namely the generation ofconstant-envelope wideband (CEW) signals. Indeed, manyspread-spectrum signal processing schemes base their op-eration on the availability of large-band constant-envelope“carriers”. In telecommunications, such signals are properlyused as information carriers. For example, in FM-DCSK [6], arunning CEW signal is segmented in order to obtain referencechunks for the principal differential shift keying modulationprocess. In other cases, CEW signals are used as the basis for

Manuscript received May 4, 2001; revised February 8, 2002. This paper wasrecommended by Associate Editor T. Saito.

S. Callegari and R. Rovatti are with the Department of Electronic, ComputerScience, and Systems (DEIS), and the Advanced Research Center for ElectronicSystems-ARCES, University of Bologna, I-40136 Bologna, Italy. (e-mail: [email protected]; [email protected]).

G. Setti is is with the Department of Engineering, University of Ferrara,I-44100 Ferrara, Italy, and is also with the Advanced Research Center forElectronic Systems-ARCES, University of Bologna, I-40136 Bologna, Italy(e-mail: [email protected]).

Digital Object Identifier 10.1109/TCSI.2002.807510

the generation ofON-OFFsequences for timing or power control[4], [7]. As a matter of fact, many applications require CEWsignals that conforms to some spectral properties. The aim ofthis paper is to present mathematical tools for dealing with thepower density spectrum (PDS) of a broad subclass of CEWsignals, i.e., those generated via the frequency modulation(FM) of chaotic sequences.

From the very beginning, it is worth noticing that control overthe spectrum of a constant-envelope signal cannot be achieveda posteriori in a trivial way. For instance, any attempt to use(linear) filters on some reference signal does always result inamplitude modulation effects that hinder the constant-envelopeproperty. This is why it is convenient to directly generate con-stant-envelope signals whicha priori have the desired spectralfeatures, making FM a natural choice [6].

The idea is to drive a modulator with a low-pass widebandsignal, thus reducing the generation problem to the productionof a suitable broad-band low-pass modulating waveform.Herein, we refer to this scheme, employing periodically sam-pled modulating sequences, as depicted in Fig. 1. Dependingon how the modulating sequence is obtained, we distinguishamongrandom-FMandchaotic-FMCEW signals. In the firstcase, we suppose to have access to a perfectly random signalsource; in the second we exploit a chaotic dynamical system.

In both cases, the goal is to formalize a design path so that,once a target PDS is given, we can suitably set the modulatorparameters and design an appropriate source for the modulatingsequence in order to obtain a CEW waveform approximating thetarget spectrum as well as possible. Such a design path requirestwo logical steps. The first consists in identifyinganalysis toolsto pass from a description of the modulating signal source (andthe modulator parameters) to the PDS of the modulated wave-form. The second consists in inverting such analysis tools intoa synthesis methodology.

For what concerns analysis, the random-FM case is signifi-cantly easier than the chaotic-FM one. As a matter of fact, onecan safely rely on well-acceded methodologies developed forcommunications, where the determination of the spectrum ofFM signals is an already tackled problem [3], [8]. Simply, indigital communications one considers numeric inputs, so that afew techniques need to be extended to continuous-value cases[9]. In chaos-based modulations, on the contrary, such method-ologies cannot be trivially applied: in communications the inputinformation passes through acoding stage, so that it can beassumed to be eventually made ofindependentsamples (i.e.,random like). Conversely, a chaotic signal isfully deterministicand the assumption of independence does simply not apply. Inthis paper, we review the theory about random based modula-tions and we introduce novel methodologies for dealing withthe chaotic ones, at least in notable cases. Note that the theory

1057-7122/03$17.00 © 2003 IEEE

4 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 50, NO. 1, JANUARY 2003

Fig. 1. Reference architecture.

about random based modulations is revisited with regard to [3],[8] taking the unconventional approach of deriving our theoremproofs relative to random modulations as particular cases ofchaos-based modulations.

For what concerns synthesis, we show notable cases wherethe analysis methodologies mentioned above can be inverted.When such an inversion cannot be applied in a straightforwardanalytic fashion, we show how the analysis tools can be appliedin iterative design methodologies to still identify a synthesispath. An important aspect is that we can guarantee that itis always possible to synthesize a chaos-based system whichperforms no worse than an equivalent random based CEWgenerator. Since chaotic generator are generally regarded to beimplementatively much cheaper thanaperiodic pseudonoise(PN) generators [10], this result is important in that it assuresthat the adoption of chaos-based techniques can deliver costsavings at no performance loss. As an aside, note that someresults already exist proving that chaos-based CEW modulatorscan surpass random based ones inperformanceand not justin cost [11]. However, they are still too tied to specific tasksfor reporting in detail.

This paper is organized as follows.

• In Section II, preliminary concepts and notation are intro-duced. With this, a logical framework can be created forboth estabilshed and new results.

• Section III proposes and comments theorems to be used asanalysis tools for both random-FM and chaotic-FM CEWsignals.

• In Section IV, cases in which the analysis formulas canbe reversed to provide a synthesis path are discussed andesemplified.

• Proofs of the original theorems are reported in theAppendix.

II. PRELIMINARY CONCEPTS AND ACLASSIFICATION OF

SIMPLIFYING ASSUMPTIONS

A. Notation

In the following, the CEW signal is indicated as . Sinceit is the result of an FM modulation, we can write

f t f

where indicates the carrier frequency, the frequencydeviation, and the input signal which is discrete time anddefined as

(2)

given that is a unit pulse of duration and that the valuesconstitute the modulating sequence. To better

express the modulation characteristics, it is convenient tointroduce a modulation index . It is also convenientto strip the carrier information from by considering its

low-pass equivalentf

. Within theinterval , with , can be cast as

(3)

hence, by setting

(4)

a more convenient general expression for can be derived

(5)

The object of this work, as mentioned in the introduction, is toprovide tools for determining the PDS of which we indicateas . Obviously, it makes sense to speak about spectra onlyif is wide sense stationary. Fortunately, it can be proven thatif is ergodic, then is cyclo stationary [3], [9]. With this,we can compute the averaged autocorrelation function

(6)

where the asterisk denotes complex conjugation, and in-dicates an expectation with regard to the process responsible forthe modulating sequence. The PDS of can be ob-tained as its Fourier transform. The relation among and

, is then [3]

(7)

CALLEGARI et al.: SPECTRAL PROPERTIES OF CHAOS-BASED FM SIGNALS: THEORY AND SIMULATION RESULTS 5

Fig. 2. Simplifying assumptions.

which, considering only positive frequencies, can be approxi-mated by

(8)

provided that is sufficiently concentrated around ,i.e. that the CEW signal bandwidth is sufficiently smaller thanthe carrier frequency.

As obvious from (6), the PDS of depends on the statisticsof . It will be shown that the most important dependence ison the probability density function (PDF), that will be indicatedas .

B. Simplifying Assumptions

Since we want to predict , it is valuable to identifycases which ease its derivation process. The following twomajor simplifying assumptions exist.

1) The modulation index may be very large. As it will beshown thelarge-massumption is a very important one,which favors the introduction of synthesis tools.

2) The modulating sequence may be made of sampleswhich are statistically independent of each other. This isthe idealrandom-FMcase, well distinguished from thechaotic-FMone to which the largest part of this work isdevoted. Remember that a chaotic sequence canbe by no meansindependentsince a single term setsdeterministicallyall the following , with .

These assumptions enable us to break the analysis space in fourregions, as shown in Fig. 2. Note that, at the logical level, itis important to distinguish the assumption of a largefromthat of aquasi-stationarymodulation. The latter is obtainedwhen and is immediately perceivable observingwhich, under this assumption, becomes a sequence oflong si-nusoidal chunks at different frequencies, i.e. a slowlyfrequencyhopping tone. On the contrary, the assumption of a largemeans that is sufficiently large with regard to . Obvi-ously, the modulation stationarity plays no role in determiningthe shape of (and in fact we can propose mathematicalderivations based on equivalent low-pass models, whereisabsent), while the modulation index does. In practice, the mod-ulation index and the modulation stationarity often become con-fused, since most application set bothand , leaving asthe sole free parameter. This is why, following an establishedstandard [12], in Fig. 2, we speak aboutslow modulations(large

) and fast modulations(small ), while in fact, it would bemore proper to distinguish among large and small modulation

indexes. Also, throughout this paper, theorems about slow mod-ulations express this condition as , while it could alsobe expressed as .

III. A NALYSIS TOOLS

In this section, we provide analysis tools for predictingthe PDS of random-FM and chaotic-FM signals. Note thatpreviously available results [9] involve an exploration of thetwo top-most regions of diagram 2, i.e., they consider only therandom-FM case.

A. Random-FM CEW Signals

Our knowledge about the PDS of random-FM signals can besummarized in three theorems.

Theorem 1 (Random-FM Spectrum):Assumptions: is a CEW signal obtained by an FM

modulator with frequency deviation , driven by a sequencehaving a pulse period and made of independent samples

whose PDF is .Statement:

xK2 x f

x K3 x f(9)

where

(10)

Theorem 2 (Slow-Modulation random-FM Spectrum):Assumptions:Same as in Theorem 1.Statement:

(11)

Theorem 3 (Random-FM Spectrum Symmetry):Assumptions:Same as in Theorem 1.Statement:

(12)

Commentary:Theorem 1 is a general analysis tool where (9)can conveniently take an integral form since the statistical inde-pendence of implies that . Theexcellent conformance to experimental results can be remarkedby an example. Suppose that a random modulating sequence,whose PDF is as depicted on the top in Fig. 3, is fed to an FMmodulator with MHz and MHz at a samplerate of Msample/s. The bottom of the figure shows the pre-dicted spectrum and the experimental one, almost completelysuperimposed.

Theorem 2 offers a specialized and simplified analysis tool,for the slow modulation case. As a rule of thumb, if issmooth, it is sufficient to have larger than a few units to obtain

. The advantages of specializationlay in this much easier formulation of as a function of

, which implies that, for slow modulations, the CEW signalPDS isshapedas . The simplified formulation is also easily


Fig. 3. Application example for Theorem 1. Top: modulating sequence PDF.Bottom: correspondent CEW signal spectrum.

reversible, permitting to determine in order to deliver aCEW signal conforming to a given spectrum. In other words,this theorem is also a synthesis tool. An example in this sense isprovided in Section IV.

Finally, Theorem 3 states that even in the fast modulationcase it is possible to exactly determine some features of therandom-FM CEW signal PDS. Namely, the even symmetry ofthe modulating sequence PDF isalwaystransformed in an even-symmetry of the random-FM CEW signal PDS.

B. Chaotic-FM CEW Signals

A notable nonrandom-FM case is offered by chaotic-FMCEW signals obtained from modulating sequences generated as

(13)

where is a suitable nonlinear function.Equation (13) identifies a class of chaotic systems for whichelectronic implementations have already been shown to be fea-sible and convenient in the literature [10], [13]–[15]. We shallassume that the map satisfies particular constraints in orderto bemixing[16] and thus ergodic, so that its features univocallydetermine the PDF of the samples .

An important characteristic parameter of systems of this sortis therate of mixing [17], corresponding to the modulus ofthe second largest eigenvalue of thePerron–Frobenius Operator[16], a linear operator associated to the map and used to achievea statistical description of discrete-time chaotic systems. In thepresent context, is useful for expressing the rate of decayof correlations relative to the sequence [17]–[19], [20].Formally, it is possible to define a function norm (suchas, for instance, the bounded variation norm [16]) satisfying

(14)

Fig. 4. Comparison of random-FM and chaotic-FM CEW signals spectra.

so that for any two functions and for which andare finite

(15)

The convergence rate is thus quantified since inde-pendent from and so that

(16)

In other words, for mixing maps there is always an exponen-tial decay of correlations, such that the rate of decay is set by

, the higher , the slower the decay [17]. Theimportance of in the present context is easily sketched.The lower , the less correlated the sequences produced bya one-dimensional (1-D) discrete-time chaotic system. Thoughthis cannot prevent the sequences samples from being determin-istically tied to each other, it makes themmore similarto inde-pendent/random onesfrom a purely statistical point of view.

The need for specialized theorems regarding chaotic-FMCEW signals derives from the fact that chaotic sequencescannot be seamlessly substituted for random ones at the inputof the FM modulator.1 It is very easy to find evidence that theusage of chaotic sequences changes the CEW signal spectralproperties. For instance, consider a modulating sequence builtusing the so calledtent map

(17)

The corresponding chaotic-FM spectrum (for MHz,MHz a modulation index set to 0.425) is shown in

Fig. 4 and confronted to the spectrum that would have been ob-tained by a random-FM modulation characterized by an iden-tical . As can be seen, the two spectra are very different fromeach other, with the chaotic one showing noticeable peaks.

To the best of our knowledge, these effects associated withthe FM modulation of chaotic sequences where never reportedbefore in the Literature. A notable fact is that peaking occurswith many chaotic maps and that the peaks tend to correspondto their unstable equilibrium points, as Fig. 5 depicts. Further-more, it has been empirically observed that the peaks are moreevident for equilibrium points occurring where is smallin modulus or is negative.

Our work to analyze the PDS of chaotic-FM CEW signal isstill in progress, but we are already able to partial explore thetwo bottom-most regions in Fig. 2, by means of three theorems.

1Unless appropriate quantizations are applied [21].


Fig. 5. An example of how peaks in the spectrum correspond to unstableequilibrium points in the chaotic system.

Theorem 4 (Chaotic-FM Random-FM for ):Assumptions: is a CEW signal obtained by an FM

modulator with frequency deviation , driven by a chaoticsequence having a pulse period and generated by meansof a chaotic system , whose rate of mixing is

and whose invariant density is .Statement:

xK2 x f

x K3 x f

(18)

where the kernels , , and are defined as in (10),Theorem 1.

Theorem 5 (Slow Modulation Chaotic-FM Spectrum):Assumptions:same as in Theorem 4.Statement:for all those so that

(19)

Theorem 6 (Chaotic-FM Spectrum Symmetry):Assumptions:same as in Theorem 4.Statement:

(20)

Commentary:Theorem 4 does formally state what was intu-itively introduced a few paragraphs above: chaotic systems char-acterized by very good (low) rate of mixing produce sequenceswhich, in the context of the generation of FM-CEW signals, arecompletely assimilable to random ones. In fact, the limit PDSshown in (18) is the same as the random-FM PDS in (9). Notethat by no means does one need to set exactlyto zero, or soclose to zero to hinder the possibility of practically implementa suitable chaotic source. As a rule of thumb, if is smooth,it is generally sufficient to let be in defect of 0.1– 0.2, inorder to have

xK2 x f

x K(21)

Fig. 6. Convergence to the random-FM spectrum for diminishing values ofr (f = 100 MHz;�f = 8:5 MHz;m = 0:425). Left column: fromtop to bottom, chaotic maps withr = 0:5;0:25;0:125; (�̂ remaining thesame). Right column: Corresponding chaotic-FM spectrum. The plot on the verybottom reports the spectrum of a random-FM signal with identical�̂.

Furthermore, from an implementation point of view, it is veryeasy and inexpensive to design chaotic sources characterizedby extremely low values, thanks to askippingtechnique.Suppose that one has a chaotic circuit characterized by a certain

value which delivers a sequence. By introducingsuitable re-sampling, it is very easy to make this circuit delivera new sequence , where is the skipping factor. Itis then easy to see that is characterized by an valuewhich is . For a full picture about skipping andother implementation-oriented techniques, see [22].

In order to exemplify, Fig. 6 shows three chaotic mapscharacterized by an identical(the same as in the example ofFig. 3) and by very similar spectral properties (left column).Only the values are changed, decreasing from top tobottom. The right column shows the corresponding chaotic-FMsignal spectra converging to the random-FM one.

The significance of Theorem 4 is thus obvious. In gen-eral terms, we do not have an analysis tool for the PDS ofchaotic-FM CEW signals. However, whenever is verylow, the results that are known to be valid for the random-FMcase can be borrowed. Alternatively, if we have an FM CEWsignal generator that is known to work with random sequences,Theorem 4 states that chaotic sequences can be substituted for


Fig. 7. Spectrum comparison for CEW signals generated driving the FMmodulator(m = 0:5) with chaotic systems characterized by identical��,identicalr , and different chaotic maps. Plot A: Tent map. Plot B: Bernoullishift.

them, with no loss in performance,provided that the rate ofmixing of the chaotic source is sufficiently low. This is veryconvenient since chaotic sources are generally regarded to havea much lower implementation cost than random sources [10].

Please notice that in any case Theorem 4 describes alimitbehavior. It would be incorrect to assume that different sys-tems characterized by identical and behave similarly.On the contrary, if is large, very different spectra can beobtained, as highlighted by the example in Fig. 7 which com-pares a tent map and a Bernoulli shift modbased system. In both cases, (uniform PDF) and

.Theorems 5 and 6 are the equivalent of Theorems 2 and 3 for

the chaotic-FM case. Particularly, the enunciation of Theorem5 is identical to that of Theorem 2, apart the convergence nowbeingpunctualand validalmosteverywhere, with the exceptionof a set of points having null measure but infinite cardinality.Note that, considered together, Theorems 2 and 5 tell us that achaotic-FM CEW source behaves identically to a random-FMCEW source not only when its is low, but also when theFM modulation index is large, independently from .

In general terms, the convergence process for slow modula-tion in the chaotic-FM case is a much more complicated issuethan in the random-FM case. This is illustrated in the plot se-quence in Fig. 8, where the same chaotic sequence is fed toan FM modulator at decreasing pulse rates: evidently the se-quence converges, but the way in which this happens is wild,with spectrum peaks appearing and possibly disappearing asis increased.

Notwithstanding the illustrated complications, it is inter-esting to try to understand the mechanics of the convergenceprocess. As a first-order approximation, assume that the PDSof a chaotic-FM signal is assimilable to that of a random-FMsignal (with identical ) with the exception of some intervals,where the power density over shots or under shots (spectrumpeaks). When the modulating sequence pulse period is in-creased, such excess (or defect) power is eventually reduced.However, it is not that the peaks disappear: peaks do alwaysexist in correspondence to the map equilibrium points or limitcycles (spectrum singularities). Merely, the peak power reducesbecause the peaks shrink in width. Figs. 8 and 9 illustrate thisbehavior very well. In Fig. 9, it is shown how peaks shrinkin power, but not inheightas is increased. Fig. 8 confirmsthat the peaks correspond to the stable equilibrium points ofthe chaotic map. In fact, stable equilibrium is at1 and 1

Fig. 8. Convergence to a spectrum shaped as��(x) by means of an increasingmodulation index. Plot A:m = 0:425. B:m = 0:5. C:m = 0:85. D:m = 1.Peaks do intermittently appear and disappear.

Fig. 9. Effects of changes in the modulation index.

for the Bernoulli shift used in this simulation and the peaksfall accordingly at and , apart from echosoutside the interval. The peaks here are very sharpcompared to those in Fig. 5, due to the much higher modulationindexes which are being adopted.

Finally, Theorem 6 is the equivalent of Theorem 3 for thechaotic-FM case. It is interesting to observe that for random-FMCEW signals, the even-symmetry of the PDF of the modulatingsequence is a sufficient condition for achieving an even-sym-metric PDS. For chaotic-FM CEW signals, this is not the caseand an additional requirement need to be fulfilled: the chaoticmap must be odd symmetric. Incidentally, note that the odd sym-metry of the map implies the even symmetry of. It is inter-esting to remark that while all the other chaos related theoremsassume limit conditions, this latter one requires no approxima-tion in any application condition.


Fig. 10. Application example for Theorem 2.

IV. SYNTHESIS TOOLS

The next few paragraphs are devoted to show how the analysistools presented in Section III can drive a design process.

It is trivial to see that whenever a slow FM modulation can beselected, the synthesis of a CEW signal conforming to particularspectral requirements can be accomplished relatively easily, ei-ther in the random-FM or in the chaotic-FM case. As a matterof fact, whenever is large enough, we can write

(22)

which is an invertible relation, and hence obtain from. An example will help clarify.

Suppose that for some reason, one needs a CEW signalwhose PDS is very much attenuated within a certain fre-quency range, as shown in the mask at the left-top of Fig. 10.Then, it is sufficient to chose an appropriate carrier-frequency,frequency-deviation couple (in this case, MHz,

MHz) and to transform the spectrum mask into anappropriate PDF mask following (11) (top right of the figure).At this point, a PDF can be selected (synthesis operation). Theadoption of a large modulation index (for example, 10) andthe consequent determination of the input sequence samplerate k sample/s completes the design. The bottom graphin the figure shows the output spectrum as obtained from asimulated experiment.

An obvious question at this point is when a modulation indexcan be assumed to be large enough to allow the usage of theapproximated expressions (11) and (19). What has been empir-ically observed is thatfastmodulations result in a phenomenonresembling power leakage to neighboring frequencies: theymake it impossible to have sharp transitions in the PDS of therandom-FM signal (Fig. 11). Consequently, as a rule of thumb,if one needs spectra with sharp transitions, really large valuesmay need to be selected for (10, 20 or even more). In othercases, a few units may suit.

Fig. 11. Effect of variations in the modulation index. Plot A: Input PDF. PlotsB, C, D: output spectrum form = 1; 4; 10, respectively.

Fig. 12. Probability density increase versus power density increase inrandom-FM signals, for the fast modulation case. Spectrum in plot Ccorresponds to the PDF in plot A. Spectrum in plot D to the PDF in plot B.

For all those cases where large modulation indexes cannotbe selected, it is impossible to analytically determine the PDFwhich is required to get a pre-assigned PDS. However, it is in-teresting to notice that a “monotonic” relationship does anywayappear to exist between the PDF of the modulating sequenceand the PDS of the random/chaotic-FM signal. In other words,it can be empirically verified that even in fast modulation cases,an increase in the value of at some corresponds to an in-crease in the power localized at the frequency ,as shown in Fig. 12. In most cases, thisinformation may beenough to drive a “trial-and-adjust” iterative synthesis proce-dure for producing a random-FM signal whose PDS satisfiescertain constraints. The basic algorithm is given in Fig. 13. Forfurther information see [23].

Finally, in any case, whenever the only constraint about thePDS is to be “sufficiently wide and symmetric” as it may oftenbe the case, Theorems 3 and 6 are by themselves sufficient to


TABLE ISUMMARY OF CURRENT KNOWLEDGE ABOUT FM BASED CEW SIGNALS. NOVEL RESULTS AREEVIDENCED IN GRAY

Fig. 13. Main lines of an algorithm to iteratively compute the PDF required todrive an FM modulator, in order to obtain a CEW signal conforming to a givenPDS whenT is low.

impose sufficient conditions on the modulating PDF to achievethis result.

V. CONCLUSION

In this paper, we have presented a menagerie of results aboutthe spectral characterization of CEW signals generated bymeans of FM modulators driven by noise-like sequences. Twocases have been analyzed. The case in which the modulatingsequence can be assumed to be built of samples independentof each other (random case) and the case in which the mod-ulating sequence is chaotic and produced by a 1-D nonlineardynamical system. The second case is much harder to consider,but also worth approaching since chaotic sources have lowerimplementation costs than other noise generators. To deal withthe two cases in a coherent fashion, a framework of simplifyingassumptions has been presented and some mathematical toolshave been developed within it.

Table I summarizes what we know at the end of the discussion.Results presented in this work for the first time are highlighted.The table also highlights how some of the proposed resultsare useful both as analysis and synthesis tools. In fact, tosome extent, they allow the design of CEW signals generatorsbased on either random or chaotic sequence sources, capableof delivering spectral properties decided in advance.

APPENDIX

PROOFS OF THEPROPOSEDTHEOREMS

The purpose of this Appendix is to provide formal proofsfor the theorems presented in this paper. Due to the fact thatthe chaotic-FM case is by far more complicated than therandom-FM one, and thanks to the possibility of consideringthe latter as a specialization of the former, it is convenient toprovide the proofsout of orderwith regard to the theoremsnumbering.

The theorem proofs that will be shortly presented require afew prerequisites. In particular, we shall re-express (6) by eval-uating its integrand as

l

(23)

By splitting the two sums into the cases , and


��fTk 1

j k 1xj

(24)

Exploiting the stationarity of the modulating sequence,canthus be expressed as

(25)

where only one of the two latter sums can be nonnull, dependingon the sign of . Furthermore, as autocorrelations demand,

meaning that “one side” of the autocorre-lation function is sufficient to carry all the information. Withthis, setting , can be expressed as

(26)

where is simplified into

(27)

by defining

(28)

Proof of Theorem 4:Define

(29)

(30)

With this, the term in (27) can be recast as

(31)

Now, consider a generic correlation term built as

(32)

From the very definition of [20] and (16), sothat for any and the following inequality holds:

(33)

Now, define

(34)

so that (33) can be rewritten as

(35)Exploiting the inequality , this becomes

(36)By means of the recursive definition (30), the term can beexpressed as

(37)Hence, we have the recursive inequality

(38)Since is obviously zero, for the case one can write

(39)

which vanishes for . Since the product in the definitionof (34) is a product of expectations, this means that

(40)

In order to exploit this factorization, the summands in (27) mustbe considered. First of all, it is possible to swap the integral andthe sum, thanks to the fact that for any givenand the sum isactually made of a finite number of terms. The general summandis thus

(41)where and stands for convolution. Thelast equality is obtained by substituting for and ex-ploiting the vanishing of when its argument lays outside .Note that for any given the limit holds uniformly in . Nowconsider the expression of the PDS when (27) is substi-tuted into (26). Since it is not obvious whether it is legitimate tocompute by moving the limit inside the infi-nite sum, we shall split the PDS into the following terms:

T

T

H T e2 f

d


(42)

so that for any finite. The first term can always be rewritten as

Te 2 �fx f T d x dx

sinc (43)

The second term, for any givenand for is

T

n

H n 1 T e 2 fT n 1

(44)

Finally, the third term vanishes for for any . Thisis due to the fact that is nothing but the PDS itself,which must obviously converge for any. Therefore, for any

given . The latterlimit can be recast considering that

i (45)

and that

T e 2 1

(46)

Note that (46) is valid thanks to the fact that . Nowcompare , and in the theorem statement with (43),(45), and (46) to see that the thesis has been proven.

Proof of Theorem 5:For proving this theorem, it is conve-nient to recast the averaged autocorrelation by breakingthe evaluation of (25) into cases depending on the value of.Obviously, it suffices to consider only the positive values of,as mentioned at the beginning of this Appendix. If we define

to be the largest integer not greater than, we easily obtain

(47)

where

if

if

(48)

and

(49)The representation of by means of the functions

and allows us to analytically capturesome noteworthy property of the PDS of chaotic-FM signals.

Preliminarily, note that we may write the base-band PDS as

k 1 T

kT

f

x0 Ak T x0 f Bk T x0 f (50)

where we have implicitly defined

(51)

and

(52)

Since we are interested in investigating the behavior ofand for

once is given and , we integrate (48) in toobtain

if

w Tx xk

f x0 xkw if

(53)

From now on, we shall focus only on and provideformulas with are valid under such an assumption. By takingthe Fourier integral of (53) we get

w

kf

f x0 xk x0f

�fxk

f

�f


(54)

The squared modulus of the quantity above evaluates to

(55)

which, defining , andcan be rewritten as

(56)

This last formulation allows to highlight the singularitiesof which occur whenever anyone of, , or

zeros. Note that, if we exclude those valuesthat makea periodic point of , these conditions cannot happen

simultaneously. Consequently, under our hypothesis, we canassume that the singularities are approached separately when

spans . With this we can independently compute thethree limits , and .

The derivation of the above-mentioned limit expressions iscumbersome. We shall thus report only the case . Pleasenote that the other cases can be dealt with in a totally similarmanner, thanks to the symmetry of with respectto , , .

For dealing with it is convenient to write

(57)

Now note that the term has a localmaximum at which evaluates to while theterm behaves approx-imately as and thus is bounded by

in a neighborhood of .Hence, for large , is dominated by

in a finite neighborhood of . Analogousconsiderations lead to discover that the same modulus is domi-nated by in a finite neighborhood of andby in a finite neighborhood of .

Since for any finite the number of singularities ofis finite, the value of the function in a neighbor-

hood of those singularities is uniformly bounded by forsome constant .

Furthermore, it can be easily seen that in general

(58)

which compels to vanish for at anynonsingular point of . Putting together the twolatest results, we may conclude that

(59)For , the right-hand side of the above inequality

tends to zero at any excluding a finite set of pointswhere it is nevertheless bounded. Hence, when ,the quantity vanishes causing also

x0 Ak T x0 f to annihilate.

Consequently, Ak T x0 f can bemade smaller than any prescribed for any finite ,simply by choosing large enough.

To complement this result, note also that, from theexistence of the PDS, we already know that the series

Ak T x0 f must converge and thus thatAk T x0 f can be made smaller than any

prescribed by choosing large enough (a similarderivation has been used also for the proof of Theorem 4).

Hence, as all the terms (from the most significant tothe less significant) of the series accounting for the contributionof the for tend to vanish, eventually makingsuch contribution arbitrarily close to zero.

As far as is concerned, analogous cal-culations starting from (49) lead to

ww T

k x0 xk 1 1

f xk 1 x0

(60)

so that

wk

x

f x0 xk 1 x0f

�fxk 1

f

�f

(61)

to which we may apply the same procedure as for ,to conclude that the terms of the series accounting for the contri-bution of for tend to vanish when .

Consequently, we have that, for all the frequencies not corre-sponding to periodic points of and , only playsa role in determining the chaotic-FM signal PDS. In order to


obtain from , the same derivation process that hasalready been used for the function in the proof of Theorem4 can be adopted, as in (43). We obtain

A T f

sinc (62)

Therefore, it is sufficient to take the limit for ofand, to this aim, it is convenient to rewrite

(63)

Since

(64)

the conditions hold for writing

(65)

and, thus, for proving the theorem.Proof of Theorem 6:First of all, note that

, i.e., if the map is odd symmetric,then the invariant density is even symmetric [24]. Also, consider2 sequences and . If isodd symmetric , , i.e.,a change in the sign of the initial value causes a change in thesign of all the following values.

Now, consider the definition of in (28) and note thatit can be recast as

(66)

and, thus, if the map is odd symmetric, is the sum of2 complex conjugated quantities and is real. If is real,also the autocorrelation function (27) is so. Since the Fouriertransform of real functions is even, we have the thesis.

Proof of Theorem 1:The proof of Theorem 1 is a special-ization of the proof of Theorem 4. All the required steps areformally identical, with the exception of (40) which, thanks tothe sample independence, can be written without the limit, as

(67)

The thesis follows.

Proof of Theorem 2:The required proof can be obtainedtrivially by taking the limit of (9). Increasing , the kernelannihilates, so that only remains. In turn, the limitof the latter is . The derivation required for thislimit is the same as at the end of the proof of Theorem 5, from(63) on.

Alternatively, note that the proof of Theorem 5 remains validfor the random case, without the need of excluding anyvalue.

Proof of Theorem 3:This proof can be easily obtained byconsidering the very expression of in (9). By substituting

for , we have

(68)

because c c . Furthermore

x K2 x f

x K3 x fx K2 x f

x K3 x f

x K2 x f

x K3x f

x K2 x f

x K3 x f

x K2 x f

x K3 x f(69)

By adding up the two terms we have the proof.

REFERENCES

[1] R. A. Scholtz, “The origin of spread spectrum,”IEEE Trans. Commun.,vol. 30, pp. 822–854, May 1982.

[2] , “The spread-spectrum concept,”IEEE Trans. Commun., vol. 25,pp. 748–755, Aug. 1977.

[3] J. G. Proakis,Digital Communications, 4th ed. New York: McGrawHill, 2000.

[4] R. Rovatti, G. Setti, and S. Graffi, “Chaos based FM of clock signals forEMI reduction,” inProc. ECCTD’99, vol. 1, Stresa, Italy, Sept. 1999,pp. 373–376.

[5] G. Setti, M. Balestra, and R. Rovatti, “Experimental verification ofenhanced electromagnetic compatibility in chaotic-FM clock signals,”in Proc. IEEE ISCAS’00, vol. 3, Lausanne, Switzerland, 2000, pp.229–232.

[6] M. P. Kennedy, G. Kolumban, and Z. Jákó, “Chaotic modulationschemes,” in Chaotic Electronics in Telecommunications. BocaRaton, FL: CRC, 2000, ch. 6.

[7] K. B. Hardin, J. H. Fessler, D. R. Bush, and J. J. Booth, “Spread Spec-trum Clock Generator and Associated Method,” U.S. Patent 5 488 627,1996.

[8] J. B. Anderson, Y. Aulin, and C. F. Sundberg,Digital Phase Modula-tion. New York: Plenum, 1986.

[9] G. Mazzini, R. Rovatti, and G. Setti, “Spectrum of randomly frequencymodulated signals,” inProc. NDES’00, Catania, Italy, May 2000, pp.146–150.

[10] M. Delgado-Restituto and A. Rodríguez-Vásquez, “Chaos-Based NoiseGeneration in Silicon,” inChaotic Electronics in Telecommunica-tions. Boca Raton, FL: CRC, 2000, ch. 6.

[11] S. Callegari, R. Rovatti, and G. Setti, “A comparison of chaos-basedand random-bascd modulations in EMI reduction tasks,” inProc. EMC2002, Wroclaw, Poland.

[12] G. Kolumbán, P. K. Kennedy, and G. Kis, “Performance Evaluationof FMDCSK,” in Chaotic Electronics in Telecommunications. BocaRaton, FL: CRC, 2000, ch. 7.

[13] S. Callegari, R. Rovatti, and G. Setti, “Robustness of chaos in analog im-plementations,” inChaotic Electronics in Telecommunications. BocaRaton, FL: CRC , 2000, ch. 12.

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[17] M. Dellnitz, G. Froyland, and S. Sertl, “On the isolated spectrum of theperron-frobenius operator,”Nonlinearity, vol. 13, no. 4, pp. 1171–1188,2000.

[18] C. Livcrani, “Decay of correlations,”Ann. Math., vol. 142, pp. 239–301,1995.

[19] G. Froyland, “Extracting dynamical behavior via Markov models,” inNonlinear Dynamics and Statistics: Proceedings Newton Institute, Cam-bridge, 1998. Berlin, Germany, 2000, pp. 283–324.

[20] R. Rovatti, G. Setti, and G. Mazzini, “Chaotic complex spreading se-quences for asynchronous DSCDMA. II Some theoretical performancebounds,”IEEE Trans. Circuits Syst. I, vol. 45, pp. 496–506, Apr. 1998.

[21] T. Kohda and A. Tsundeda, “Information sources using chaotic dy-namics,” inChaotic Electronics in Telecommunications. Boca Raton,FL: CRC , 2000, ch. 4.

[22] S. Callegari, R. Rovatti, and G. Setti, Generation of constant-enve-lope spread-spectrum signals via chaos-based FM: applications andimplementation issues, in Internal Report of CEG-DEIS, University ofBologna, Bologna, Italy, 2001.

[23] >S. Callegari, “Generation of band-pass constant-envelope signals witha pre-assigned spectrum: a synthesis procedure,”Int. J. Circuit TheoryApp., vol. 90, no. 5, pp. 481–486, 2002, to be published.

[24] W. Schwarz, M. Götz, K. Kelber, A. Abel, T. Falk, and F. Dachselt, “Sta-tistical analysis and design of chaotic systems,” inChaotic Electronicsin Telecommunications. Boca Raton, FL: CRC, 2000, ch. 9.

[25] M. P. Kennedy, R. Rovatti, and G. Setti, Eds.,Chaotic Electronics inTelecommunications. Boca Raton, FL: CRC International Press, 2000.

Sergio Callegari(M’01) was born in Bologna, Italy,on May 26, 1970. He received the Dr.Eng. degree(with honors) in electronic engineering and the Ph.Ddegree in electronic engineering and computer sci-ence from the University of Bologna, Bologna, Italy,in 1996 and 2000, respectively.

In 1996, he was a Visiting Student at King’sCollege London, London, U.K. He is currently aResearcher at the second Faculty of Engineering,University of Bologna, where he has also recentlyjoined the Advanced Center for Electronic Systems.

His current research interests include nonlinear signal processing, internallynonlinear, externally linear networks, and chaotic maps. He has authored orcoauthored more than 25 papers in international conferences and journals andalso a book chapter.

Riccardo Rovatti (M’99–SM’02) was born on Jan-uary 14, 1969. He received the Dr.Eng. degree in elec-tronic engineering (with honors) and the Ph.D. degreein electronic engineering and computer science fromthe University of Bologna, Bologna, Italy, in 1992,and 1996, respectively.

He is currently an Associate Professor of AnalogElectronics at the University of Bologna, where hewas a Lecturer of Digital Electronics from 1997 to2000, and an Assistant Professor from 2000 to 2001.His research interests include fuzzy theory founda-

tions, learning, and CAD algorithms for fuzzy and neural systems, statisticalpattern recognition, function approximation, nonlinear system theory, and iden-tification as well as theory and applications of chaotic systems. He has authoredor coauthored more than 130 international scientific publications. He is Co-Ed-itor of the book,Chaotic Electronics in Telecommunications(Boca Raton: FL,CRC, 2000), and one of the Guest-Editors of the May 2002 special issue of thePROCEEDINGS OF THEIEEE onApplications of Nonlinear Dynamics to Elec-tronics and Information Engineering.

Gianluca Setti (S’89–M’91–SM’02) received theDr.Eng. degree in electronic engineering (withhonors) and the Ph.D. degree in electronic engi-neering and computer science from the University ofBologna, Bologna, Italy, in 1992, and 1997, respec-tively, working on the study of neural networks andchaotic systems.

From May 1994 to July 1995, he was a VisitingResearch Assistant with the Circuits and SystemsGroup, Swiss Federal Institute of Technology,Lausanne, Switzerland. He is currently an Associate

Professor of Circuit Theory and Analog Electronics at the University of Ferrara,Ferrara Italy, where he was a Lecturer from 1997 to 1998, and an AssistantProfessor of Analog Electronics from 1998 to 2001. His research interestsinclude nonlinear circuit theory, recurrent neural networks, and design andimplementation of chaotic circuits and systems, as well as their applications toelectronics and signal processing. He is also Co-Editor of the book,ChaoticElectronics in Telecommunications(Boca Raton: FL, CRC, 2000).

Dr. Setti received the 1998 Caianiello prize for the best Italian Ph.D. disser-tation on neural networks. In 2002, he was Chair of the IEEE Technical Com-mittee on Nonlinear Circuits and Systems. He is as an Associate Editor of theIEEE TRANSACTIONS ONCIRCUITS AND SYSTEMS—I, from 1999 to 2001, forthe area of nonlinear circuits and systems, and since 2002, for the area of chaosand bifurcation. He has also been one of the Guest-Editors of the May 2002special issue of the PROCEEDINGS OF THEIEEE onApplications of NonlinearDynamics to Electronics and Information Engineering.

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