Chaotic Signals and Signal Processing - ArchiMeDes

15 Pages·1999·0.25 MB·English

Oppenheim, A.V. & Cuomo, K.M. “Chaotic Signals and Signal Processing”
Digital Signal Processing Handbook
Ed. Vijay K. Madisetti and Douglas B. Williams
Boca Raton: CRC Press LLC, 1999
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71
Chaotic Signals and Signal
Processing
71.1 Introduction
71.2 ModelingandRepresentationofChaoticSignals
71.3 EstimationandDetection
AlanV.Oppenheim 71.4 UseofChaoticSignalsinCommunications
MassachusettsInstituteofTechnology Self-SynchronizationandAsymptoticStability(cid:15)Robustness
and Signal Recovery in the Lorenz System (cid:15) Circuit Imple-
KevinM.Cuomo mentationandExperiments
71.5 SynthesizingSelf-SynchronizingChaoticSystems
MIT
LincolnLaboratory References
71.1 Introduction
Signalsgeneratedbychaoticsystemsrepresentapotentiallyrichclassofsignalsbothfordetectingand
characterizingphysicalphenomenaandinsynthesizingnewclassesofsignalsforcommunications,
remotesensing,andavarietyofothersignalprocessingapplications.
Inclassicalsignalprocessingarichsetoftoolshasevolvedforprocessingsignalsthataredeter-
ministicandpredictablesuchastransientandperiodicsignals, andforprocessingsignalsthatare
stochastic. Chaoticsignalsassociatedwiththehomogeneousresponseofcertainnonlineardynamical
systemsdonotfallineitheroftheseclasses. Whiletheyaredeterministic,theyarenotpredictablein
anypracticalsenseinthatevenwiththegeneratingdynamicsknown,estimationofpriororfuture
values from a segment of the signal or from the state at a given time is highly ill-conditioned. In
manywaysthesesignalsappeartobenoise-likeandcan,ofcourse,beanalyzedandprocessedusing
classicaltechniquesforstochasticsignals. However, theyclearlyhaveconsiderablymorestructure
thancanbeinferredfromandexploitedbytraditionalstochasticmodelingtechniques.
Thebasicstructureofchaoticsignalsandthemechanismsthroughwhichtheyaregeneratedare
describedinavarietyofintroductorybooks,e.g.,[1,2]andsummarizedin[3].
Chaoticsignalsareofparticularinterestandimportanceinexperimentalphysicsbecauseofthe
wide range of physical processes that apparently give rise to chaotic behavior. From the point of
viewofsignalprocessing,thedetection,analysis,andcharacterizationofsignalsofthistypepresent
asignificantchallenge. Inaddition,chaoticsystemsprovideapotentiallyrichmechanismforsignal
designandgenerationforavarietyofcommunicationsandremotesensingapplications.
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71.2 Modeling and Representation of Chaotic Signals
The state evolution of chaotic dynamical systems is typically described in terms of the nonlinear
state equation xP.t/ D FTx.t/U in continuous time or xTnU D F.xTn−1U/ in discrete time. In a
signalprocessingcontext,weassumethattheobservedchaoticsignalisanonlinearfunctionofthe
stateandwouldtypicallybeascalartimefunction. Indiscrete-time,forexample,theobservation
equationwouldbeyTnU D G.xTnU/. FrequentlytheobservationyTnUisalsodistortedbyadditive
noise,multipatheffects,fading,etc.
Modelingachaoticsignalcanbephrasedintermsofdeterminingfromcleanordistortedobser-
vations,asuitablestatespaceandmappingsF.(cid:1)/andG.(cid:1)/thatcapturetheaspectsofinterestinthe
observedsignaly. Theproblemofdeterminingfromtheobservedsignalasuitablestatespacein
whichtomodelthedynamicsisreferredtoastheembeddingproblem. Whilethereis,ofcourse,no
uniquesetofstatevariablesforasystem,somechoicesmaybebettersuitedthanothers. Themost
commonlyusedmethodforconstructingasuitablestatespaceforthechaoticsignalisthemethod
ofdelaycoordinatesinwhichastatevectorisconstructedfromavectorofsuccessiveobservations.
It is frequently convenient to view the problem of identifying the map associated with a given
chaoticsignalintermsofaninterpolationproblem. Specifically,fromasuitablyembeddedchaotic
signal it is possible to extract a codebook consisting of state vectors and the states to which they
subsequently evolve after one iteration. This codebook then consists of samples of the function
F spaced, in general, non-uniformly throughout state space. A variety of both parametric and
nonparametric methods for interpolating the map between the sample points in state space have
emergedintheliterature,andthetopiccontinuestobeofsignificantresearchinterest. Inthissection
webrieflycommentonseveraloftheapproachescurrentlyused. Theseandothersarediscussedand
comparedinmoredetailin[4].
OneapproachisbasedontheuseoflocallylinearapproximationstoFthroughoutthestatespace[5,
6]. Thisapproachconstitutesageneralizationofautoregressivemodelingandlinearpredictionand
iseasilyextendedtolocallypolynomialapproximationsofhigherorder. Anotherapproachisbased
onfittingaglobalnonlinearfunctiontothesamplesinstatespace[7].
Afundamentallyratherdifferentapproachtotheproblemofmodelingthedynamicsofanem-
bedded signal involves the use of hidden Markov models [8, 9, 10]. With this method, the state
spaceisdiscretizedintoalargenumberofstates,andaprobabilisticmappingisusedtocharacterize
transitionsbetweenstateswitheachiterationofthemap. Furthermore,eachstatetransitionspawns
astate-dependentrandomvariableastheobservationyTnU. Thisframeworkcanbeusedtosimulta-
neouslymodelboththedetailedcharacteristicsofstateevolutioninthesystemandthenoiseinherent
in the observed data. While algorithms based on this framework have proved useful in modeling
chaoticsignals,theycanbeexpensivebothintermsofcomputationandstoragerequirementsdue
tothelargenumberofdiscretestatesrequiredtoadequatelycapturethedynamics.
While many of the above modeling methods exploit the existence of underlying nonlinear dy-
namics,theydonotexplicitlytakeintoaccountsomeofthepropertiespeculiartochaoticnonlinear
dynamicalsystems. Forthisreason,inprinciple,thealgorithmsmaybeusefulinmodelingabroader
classofsignals. Ontheotherhand,whenthesignalsofinterestaretrulychaotic,thespecialprop-
erties of chaotic nonlinear dynamical systems ought to be taken into account, and, in fact, may
oftenbeexploitedtoachieveimprovedperformance. Forinstance,becausetheevolutionofchaotic
systemsisacutelysensitivetoinitialconditions,itisoftenimportantthatthisnumericalinstability
be reflected in the model for the system. One approach to capturing this sensitivity is to require
thatthereconstructeddynamicsexhibitLyapunovexponentsconsistentwithwhatmightbeknown
aboutthetruedynamics. Thesensitivityofstateevolutioncanalsobecapturedusingthehidden
Markov model framework since the structural uncertainty in the dynamics can be represented in
termsoftheprobabilisticstatetransactions. Inanycase,unlesssensitivityofthedynamicsistaken
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intoaccountduringmodeling,detectionandestimationalgorithmsinvolvingchaoticsignalsoften
lackrobustness.
Another aspect of chaotic systems that can be exploited is that the long term evolution of such
systemsliesonanattractorwhosedimensionisnotonlytypicallynon-integral,butoccupiesasmall
fractionoftheentirestatespace. Thishasanumberofimportantimplicationsbothinthemodeling
ofchaoticsignalsandultimatelyinaddressingproblemsofestimationanddetectioninvolvingthese
signals. For example, it implies that the nonlinear dynamics can be recovered in the vicinity of
theattractorusingcomparativelylessdatathanwouldbenecessaryifthedynamicswererequired
everywhereinstatespace.
Identifying the attractor, its fractal dimension, and related invariant measures governing, for
example,theprobabilityofbeingintheneighborhoodofaparticularstateontheattractor,arealso
importantaspectsofthemodelingproblem. Furthermore,wecanoftenexploitvariousergodicity
andmixingpropertiesofchaoticsystems. Thesepropertiesallowustorecoverinformationabout
theattractorusingasinglerealizationofachaoticsignal,andassureusthatdifferenttimeintervals
ofthesignalprovidequalitativelysimilarinformationabouttheattractor.
71.3 Estimation and Detection
Avarietyofproblemsinvolvingtheestimationanddetectionofchaoticsignalsarisesinpotential
application contexts. In some scenarios, the chaotic signal is a form of noise or other unwanted
interferencesignal. Inthiscase,weareofteninterestedindetecting,characterizing,discriminating,
andextractingknownorpartiallyknownsignalsinbackgroundsofchaoticnoise. Inotherscenarios,
it is the chaotic signal that is of direct interest and which is corrupted by other signals. In these
casesweareinterestedindetecting,discriminating,andextractingknownorpartiallyknownchaotic
signalsinbackgroundsofothernoisesorinthepresenceofotherkindsofdistortion.
The channel through which either natural or synthesized signals are received can typically be
expectedtointroduceavarietyofdistortionsincludingadditivenoise,scattering,multipatheffects,
etc. Thereare,ofcourse,classicalapproachestosignalrecoveryandcharacterizationinthepresence
ofsuchdistortionsforbothtransientandstochasticsignals. Whenthedesiredsignalinthechannelis
achaoticsignal,orwhenthedistortioniscausedbyachaoticsignal,manyoftheclassicaltechniques
willnotbeeffectiveanddonotexploittheparticularstructureofchaoticsignals.
Thespecificpropertiesofchaoticsignalsexploitedindetectionandestimationalgorithmsdepend
heavilyonthedegreeofaprioriknowledgeofthesignalsinvolved. Forexample,indistinguishing
chaoticsignalsfromothersignals,thealgorithmsmayexploitthefunctionalformofthemap,the
Lyapunov exponents of the dynamics, and/or characteristics of the chaotic attractor such as its
structure,shape,fractaldimensionand/orinvariantmeasures.
To recover chaotic signals in the presence of additive noise, some of the most effective noise
reductiontechniquesproposedtodatetakeadvantageofthenonlineardependenceofthechaotic
signalbyconstructingaccuratemodelsforthedynamics. Multipathandothertypesofconvolutional
distortioncanbestbedescribedintermsofanaugmentedstatespacesystem. Convolutionorfiltering
ofchaoticsignalscanchangemanyoftheessentialcharacteristicsandparametersofchaoticsignals.
Effectsofconvolutionaldistortionandapproachestocompensatingforitarediscussedin[11].
71.4 Use of Chaotic Signals in Communications
Chaoticsystemsprovidearichmechanismforsignaldesignandgeneration, withpotentialappli-
cationstocommunicationsandsignalprocessing. Becausechaoticsignalsaretypicallybroadband,
noise-like,anddifficulttopredict,theycanbeusedinvariouscontextsincommunications. Apartic-
ularlyusefulclassofchaoticsystemsarethosethatpossessaself-synchronizationproperty[12,13,14].
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Thispropertyallowstwoidenticalchaoticsystemstosynchronizewhenthesecondsystem(receiver)
isdrivenbythefirst(transmitter). Thewell-knownLorenzsystemisusedbelowtofurtherdescribe
andillustratethechaoticself-synchronizationproperty.
The Lorenz equations, first introduced by E. N. Lorenz as a simplified model of fluid convec-
tion[15],aregivenby
xP D (cid:27).y−x/
yP D rx−y−xz (71.1)
zP D xy−bz;
where(cid:27);r;andbarepositiveparameters. Insignalprocessingapplications,itistypicallyofinterest
to adjust the time scale of the chaotic signals. This is accomplished in a straightforward way by
establishingtheconventionthatxP;yP;andzP denotedx=d(cid:28);dy=d(cid:28),anddz=d(cid:28),respectively,where
(cid:28) Dt=T isnormalizedtimeandT isatimescalefactor. Itisalsoconvenienttodefinethenormalized
frequency! D(cid:127)T,where(cid:127)denotestheangularfrequencyinunitsofrad/s. Theparametervalues
T D400(cid:22)sec;(cid:27) D16;r D45:6;andbD4areusedfortheillustrationsinthischapter.
ViewingtheLorenzsystem(71.1)asasetoftransmitterequations, adynamicalreceiversystem
thatwillsynchronizetothetransmitterisgivenby
xP D (cid:27).y −x /
r r r
yPr D rx.t/−yr −x.t/zr (71.2)
zP D x.t/y −bz :
r r r
Inthiscase,thechaoticsignalx.t/fromthetransmitterisusedasthedrivinginputtothereceiver
system. In Section 71.4.1, an identified equivalence between self-synchronization and asymptotic
stabilityisexploitedtoshowthatthesynchronizationofthetransmitterandreceiverisglobal,i.e.,
thereceivercanbeinitializedinanystateandthesynchronizationstilloccurs.
71.4.1 Self-Synchronization and Asymptotic Stability
A close relationship exists between the concepts of self-synchronization and asymptotic stability.
Specifically,self-synchronizationintheLorenzsystemisaconsequenceofgloballystableerrordy-
namics. AssumingthattheLorenztransmitterandreceiverparametersareidentical,asetofequations
thatgoverntheirerrordynamicsisgivenby
eP D (cid:27).e −e /
x y x
ePy D −ey −x.t/ez (71.3)
eP D x.t/e −be :
z y z
where
e .t/ D x.t/−x .t/
x r
e .t/ D y.t/−y .t/
y r
e .t/ D z.t/−z .t/:
z r
Asufficientconditionfortheerrorequationstobegloballyasymptoticallystableattheorigincanbe
determinedbyconsideringaLyapunovfunctionoftheform
1 1
E.e/D 2.(cid:27)ex2Cey2Cez2/:
Since(cid:27) andbintheLorenzequationsarebothassumedtobepositive,EispositivedefiniteandEP
isnegativedefinite. ItthenfollowsfromLyapunov’stheoremthate.t/ ! 0ast ! 1. Therefore,
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synchronizationoccursast ! 1regardlessoftheinitialconditionsimposedonthetransmitter
andreceiversystems.
Forpracticalapplications,itisalsoimportanttoinvestigatethesensitivityofthesynchronizationto
perturbationsofthechaoticdrivesignal. NumericalexperimentsaresummarizedinSection71.4.2,
whichdemonstratestherobustnessandsignalrecoverypropertiesoftheLorenzsystem.
71.4.2 Robustness and Signal Recovery in the Lorenz System
When a message or other perturbation is added to the chaotic drive signal, the receiver does not
regenerateaperfectreplicaofthedrive;thereisalwayssomesynchronizationerror. Bysubtracting
theregenerateddrivesignalfromthereceivedsignal,successfulmessagerecoverywouldresultifthe
synchronization error was small relative to the perturbation itself. An interesting property of the
Lorenzsystemisthatthesynchronizationerrorisnotsmallcomparedtoanarrowbandperturbation;
nevertheless, the message can be recovered because the synchronization error is nearly coherent
withthemessage. Thissectionsummarizesexperimentalevidenceforthiseffect; amoredetailed
explanationhasbeengivenintermsofanapproximateanalyticalmodel[16].
The series of experiments that demonstrate the robustness of synchronization to white noise
perturbationsandtheabilitytorecoverspeechperturbationsfocusonthesynchronizingproperties
ofthetransmitterEqs.(71.1)andthecorrespondingreceiverequations,
xP D (cid:27).y −x /
r r r
yPr D rs.t/−yr −s.t/zr (71.4)
zP D s.t/y −bz :
r r r
Previously,itwasstatedthatwiths.t/equaltothetransmittersignalx.t/,thesignalsxr;yr;andzr
willasymptoticallysynchronizetox;y;andz,respectively. Below,weexaminethesynchronization
errorwhenaperturbationp.t/isaddedtox.t/,i.e.,whens.t/Dx.t/Cp.t/.
First, we consider the case where the perturbation p.t/ is Gaussian white noise. In Fig. 71.1,
we show the perturbation and error spectra for each of the three state variables vs. normalized
frequency!. Notethatatrelativelylowfrequencies,theerrorinreconstructingx.t/slightlyexceeds
the perturbation of the drive but that for normalized frequencies above 20 the situation quickly
reverses. Ananalyticalmodelcloselypredictsandexplainsthisbehavior[16]. Thesefiguressuggest
that the sensitivity of synchronization depends on the spectral characteristics of the perturbation
signal. Forsignalsthatarebandlimitedtothefrequencyrange0 < ! < 10,wewouldexpectthat
thesynchronizationerrorswillbelargerthantheperturbationitself. Thisturnsouttobethecase,
althoughthenextexperimentsuggeststhereareadditionalinterestingcharacteristicsaswell.
Inasecondexperiment,p.t/isalow-levelspeechsignal(forexampleamessagetobetransmitted
andrecovered). Thenormalizingtimeparameteris400(cid:22)secandthespeechsignalisbandlimited
to4kHzorequivalentlytoanormalizedfrequency!of10. Figure71.2showsthepowerspectrum
ofarepresentativespeechsignalandthechaoticsignalx.t/. Theoverallchaos-to-perturbationratio
inthisexperimentisapproximately20dB.
To recover the speech signal, the regenerated drive signal is subtracted at the receiver from the
receivedsignal. Inthiscase, therecoveredmessageispO.t/ D p.t/Cex.t/. Itwouldbeexpected
thatsuccessfulmessagerecoverywouldresultifex.t/wassmallrelativetotheperturbationsignal.
FortheLorenzsystem, however, althoughthesynchronizationerrorisnotsmallcomparedtothe
perturbation,themessagecanberecoveredbecauseex.t/isnearlycoherentwiththemessage. This
coherencehasbeenconfirmedexperimentallyandanexplanationhasbeendevelopedintermsofan
approximateanalyticalmodel[16].
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FIGURE71.1: Powerspectraoftheerrorsignals: (a)Ex.!/. (b)Ey.!/. (c)Ez.!/.
71.4.3 Circuit Implementation and Experiments
In Section 71.4.2, we showed that, theoretically, a low-level speech signal could be added to the
synchronizingdrivesignalandapproximatelyrecoveredatthereceiver. Theseresultswerebasedon
ananalysisoftheexactLorenztransmitterandreceiverequations. Whenimplementingsynchronized
chaoticsystemsinhardware,thelimitationsofavailablecircuitcomponentsresultinapproximations
of the defining equations. The Lorenz transmitter and receiver equations can be implemented
relativelyeasilywithstandardanalogcircuits[17,20,21]. Theresultingsystemperformanceisin
excellent agreement with numerical and theoretical predictions. Some potential implementation
difficultiesareavoidedbyscalingtheLorenzstatevariablesaccordingtou D x=10;v D y=10;and
w Dz=20. Withthisscaling,theLorenzequationsaretransformedto
uP D (cid:27).v−u/
vP D ru−v−20uw (71.5)
wP D 5uv−bw:
Forthissystem,whichwerefertoasthecircuitequations,thestatevariablesallhavesimilardynamic
rangeandcircuitvoltagesremainwellwithintherangeoftypicalpowersupplylimits. Below, we
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FIGURE71.2: Powerspectraofx.t/andp.t/whentheperturbationisaspeechsignal.
discussanddemonstratesomeappliedaspectsoftheLorenzcircuits.
InFig.71.3,weillustrateacommunicationscenariothatisbasedonchaoticsignalmaskingand
recovery[18,19,20,21]. Inthisfigure,achaoticmaskingsignalu.t/isaddedtotheinformation-
bearingsignalp.t/atthetransmitter,andatthereceiverthemaskingisremoved. Bysubtractingthe
regenerateddrivesignalur.t/fromthereceivedsignals.t/atthereceiver,therecoveredmessageis
pO.t/Ds.t/−ur.t/Dp.t/C[u.t/−ur.t/]:
In this context, eu.t/, the error between u.t/ and ur.t/, corresponds directly to the error in the
recoveredmessage.
FIGURE71.3: Chaoticsignalmaskingandrecoverysystem.
Forthisexperiment,p.t/isalow-levelspeechsignal(themessagetobetransmittedandrecovered).
The normalizing time parameter is 400 (cid:22)sec and the speech signal is bandlimited to 4 kHz or,
equivalently,toanormalizedfrequency!of10. InFig.71.4,weshowthepowerspectrumofp.t/
andpO.t/,wherepO.t/isobtainedfrombothasimulationandfromthecircuit. Thetwospectrafor
pO.t/areinexcellentagreement,indicatingthatthecircuitperformsverywell. BecausepO.t/includes
considerableenergybeyondthebandwidthofthespeech,thespeechrecoverycanbeimprovedby
lowpassfilteringpO.t/. WedenotethelowpassfilteredversionofpO.t/bypOf.t/. InFig.71.5(a)and(b),
weshowacomparisonofpOf.t/frombothasimulationandfromthecircuit,respectively. Clearly,the
circuitperformswelland,ininformallisteningtests,therecoveredmessageisofreasonablequality.
AlthoughpOf.t/isofreasonablequalityinthisexperiment,thepresenceofadditivechannelnoise
willproducemessagerecoveryerrorsthatcannotbecompletelyremovedbylowpassfiltering;there
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willalwaysbesomeerrorintherecoveredmessage. Becausethemessageandnoisearedirectlyadded
tothesynchronizingdrivesignal,themessage-to-noiseratioshouldbelargeenoughtoallowafaithful
recoveryoftheoriginalmessage. Thisrequiresacommunicationchannelthatisnearlynoisefree.
FIGURE71.4: Powerspectraofp.t/andpO.t/whentheperturbationisaspeechsignal.
Analternativeapproachtoprivatecommunicationsallowstheinformation-bearingwaveformto
beexactlyrecoveredattheself-synchronizingreceiver(s),evenwhenmoderate-levelchannelnoise
ispresent. Thisapproachisreferredtoaschaoticbinarycommunications[20,21]. Thebasicidea
behindthistechniqueistomodulateatransmitterparameterwiththeinformation-bearingwaveform
andtotransmitthechaoticdrivesignal. Atthereceiver,theparametermodulationwillproducea
synchronizationerrorbetweenthereceiveddrivesignalandthereceiver’sregenerateddrivesignal
withanerrorsignalamplitudethatdependsonthemodulation. Usingthesynchronizationerror,
themodulationcanbedetected.
Thismodulation/detectionprocessisillustratedinFig.71.6. Toillustratetheapproach,weusea
periodicsquare-waveforp.t/asshowninFig.71.7(a). Thesquare-wavehasarepetitionfrequency
of approximately 110 Hz with zero volts representing the zero-bit and one volt representing the
one-bit. The square-wave modulates the transmitter parameter b with the zero-bit and one-bit
parameters given by b.0/ D 4 and b.1/ D 4:4, respectively. The resulting drive signal u.t/ is
transmittedandthenoisyreceivedsignals.t/isusedasthedrivinginputtothesynchronizingreceiver
circuit. InFig.71.7(b),weshowthesynchronizationerrorpowere2.t/. Theparametermodulation
producessignificantsynchronizationerrorduringa“1”transmissionandverylittleerrorduringa
“0”transmission. Itisplausiblethatadetectorbasedontheaveragesynchronizationerrorpower,
followedbyathresholddevice,couldyieldreliableperformance. WeillustrateinFig.71.7(c)that
thesquare-wavemodulationcanbereliablyrecoveredbylowpassfilteringthesynchronizationerror
powerwaveformandapplyingathresholdtest. Thethresholddeviceusedinthisexperimentconsisted
ofasimpleanalogcomparatorcircuit.
Theallowabledatarateofthiscommunicationtechniqueis,ofcourse,dependentonthesynchro-
nizationresponsetimeofthereceiversystem. Althoughwehaveusedalowbitratetodemonstrate
thetechnique,thecircuittimescalecanbeeasilyadjustedtoallowmuchfasterbitrates.
Whiletheresultspresentedaboveappearencouraging,therearemanycommunicationscenarios
where it is undesirable to be restricted to the Lorenz system, or for that matter, any other low-
dimensionalchaoticsystem. Inprivatecommunications,forexample,theabilitytochoosefroma
wide variety of synchronized chaotic systems would be highly advantageous. In the next section,
webrieflydescribeanapproachforsynthesizinganunlimitednumberofhigh-dimensionalchaotic
systems. Thesignificanceofthisworkliesinthefactthattheabilitytosynthesizehigh-dimensional
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FIGURE71.5: (a)Recoveredspeech(simulation). (b)Recoveredspeech(circuit).
chaoticsystemsfurtherenhancestheirapplicabilityforpracticalapplications.
71.5 Synthesizing Self-Synchronizing Chaotic Systems
Aneffectiveapproachtosynthesisisbasedonasystematicfourstepprocess. First,analgebraicmodel
isspecifiedforthetransmitterandreceiversystems. Asshownin[22,23],thechaoticsystemmodels
can be very general; in [22] the model represents a large class of quadratically nonlinear systems,
whilein[23]themodelallowsforanunlimitednumberofLorenzoscillatorstobemutuallycoupled
viaanN-dimensionallinearsystem.
The second step in the synthesis process involves subtracting the receiver equations from the
transmitterequationsandimposingaglobalasymptoticstabilityconstraintontheresultingerror
equations. UsingLyapunov’sdirectmethod,sufficientconditionsfortheerrorsystem’sglobalsta-
FIGURE71.6: Communicatingbinary-valuedbitstreamswithsynchronizedchaoticsystems.
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