A chaotic spreading code and its application to blindchannel estimation

A. Müller and J.M.H. ElmirghaniDepartment of Electrical and Electronic Engineering,

University of Wales Swansea, Singleton Park, Swansea, SA2 8PP, UK

AbstractThe proposed system introduces a new approach tochaotic-based signal transmission and blind channelestimation. It is now well known that channel estimatorsbased on FIR-digital adaptive-filters (FIR-DAF)approach the upper bound to their performance whendriven by samples of chaotic signals. A fixed chaotic-system parameter is assigned to every m-valuedinformation symbol and thus after symbol detection thechaotic sequence can be reconstructed. Assuming propersymbol synchronisation, the complete transmittedsequence (which is in general deteriorated by noise) canbe reconstructed and employed as drive signal for theblind channel estimation algorithm. The performance ofthe proposed system is evaluated for a channel with lowpass characteristics and a 10 MHz bandwidth. Channelestimation is carried out using the finite correlation leastsquares (FCSL) algorithm recently introduced by theauthors.

I. IntroductionChaotic signals have recently received muchattention in the context of communication andsignal processing. In particular, the observation ofsynchronization in chaotic systems has fuelledmuch of the interest witnessed in the last decade[1]. The work has since developed and a number ofways in which chaotic signals can usefully beexploited in communication systems wereproposed. Chaotic-based spread spectrumapplications were investigated [2] and transmitterand receiver design were also evaluated based on avariety of methods including adaptive filterapproaches and neural network methods [3-7].Apart from using chaotic signals as informationcarriers or as signature codes in spread spectrum,other applications in channel estimation have alsoevolved [8-10]. The latter applications were mainlymotivated by the white power spectral density(PSD) and the noise-like properties of certainchaotic signals. These properties make chaoticsignals ideal as drive or excitation signals forchannel estimation approaches based on the leastmean square (LMS) algorithm [11]. In classical channel estimation approaches, a finiteimpulse response digital adaptive filter (FIR-DAF)is adapted by an LMS algorithm where the bestperformance is obtained when the estimator isdriven by a white noise (WN) process. The meanvalue of the ith tap weight coefficient of the FIR-DAF converges proportional to the i th eigenvalue ofthe drive signal autocorrelation matrix and thus theadaptation performance is determined by theeigenvalue spread of the autocorrelation matrix.

Therefore the tap weights adaptation depends onthe drive signal PSD. Furthermore, the drive signalfourth order statistics determine the fluctuation(variance) of the coefficient vector from theoptimum. As a result, the drive signal PSD (asecond order statistic) is important in determiningadaptation and the rate of convergence, while thedrive signal fourth order statistics have importantimplications on stability [11]. As such the drivesignal statistics are crucial. Chaotic signals areweakly correlated and can further have a white PSDand thus represent ideal drive signals. Some chaoticsystems have a f1 PSD, however, we areinterested in chaotic systems that possess a whitePSD such as systems based on the Logistic map [3,8]. While a closed form expression for the PSD ofthe Logistic map can be found for fixed systemparameters, a similar analysis is not possible forgeneral chaotic systems. Here we employ theLogistic map and the discrete chaotic circuitproposed by Yamakawa [6]. A statisticalevaluation, however, has shown that samples of theYamakawa system are weakly correlated and thusthe resulting PSD is suited for channel estimationand the circuit is available as a chip [6].

II. Proposed SystemShift keying schemes (e.g. ASK, PSK, FSK) assigna fixed waveform to each possible value of a digitalsignal and in analogy, our chaotic transmissionsystem assigns a chaotic code taken from a set ofquasi-orthogonal chaotic codes to each possibleinformation symbol. Discrete chaotic systems aredescribed by a set of nonlinear difference equations.A simple example is the Logistic map described by

( )x x x xk k k k= − ≤ ≤− −λ 1 11 0 1, (1)

where λ∈[3.7,4] is a parameter, the bifurcationparameter, which together with the initialconditions of the system determines the systemdynamics. Moreover, small changes in λ yieldabsolutely different time evolutions. The idea ofchaotic coding is to relate the value of λ to thevalue of an information signal. This wassuccessfully introduced for speech coding intelephony networks [3, 8]. In case of digitalinformation signals, λ can assume only finitediscrete values. While for continuos informationsignals several demodulation algorithms wereproposed, digital information can be decoded in adifferent manner as will be shown in this work.

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0-7803-7206-9/01/$17.00 © 2001 IEEE

Figure 1. Digital communication scheme with chaotic spreading codes

Consider a set of m chaotic circuits and take the LM(1) as an example. Let λi, i=1,...,m be the parameterassociated with the i th system such that each systemshows different dynamics. A feature of sampledchaotic signals is their weak correlation. Theautocorrelation of the LM sequence {xk} with λ=4is known to be [9, 12]

E x xk i k i[ , ] ,=1

8δ (2)

so that samples of this sequence are uncorrelated.Empirical statistical evaluations of the LM withλ≠4 have shown similar effects. Thus samples ofdifferent LM systems will be mutually uncorrelatedexcept if the underlying systems have the sameparameters and initial conditions. This effect can beemployed to classify systems according to theparameter employed. Every parameter in turn isassigned to one of the m possible values of a digitalm-ary code. For a chaotic m-ary coder, m sequencesof a chaotic system are generated and thetransmitted sequence is taken from this setaccording to the digital value of the actual codeword to be transmitted. A possible choice for the λi

set is

λi

i

mi m= + =3 7 0 3 1. . , , ,� (3).

Each chaotic system has a different but fixedparameter λ. Due to the parameter sensitivity ofchaotic systems, the time evolutions of thosesystems diverge from each other. If such sequencesare transmitted over a communication channel thisallows the receiver side to detect which one of thepossible systems a received sequence belongs to.Sequences belonging to different digital values willyield a much lower empirical correlation sum

compared to those which stem from identicalsystems. At the receiver side, the same set ofsequences are generated using chaotic circuits withthe same parameter constellation. An empiricalcorrelation sum can be used to determine thecorrelation of the received and the candidatesequences. Provided that the receiver andtransmitter are synchronized (symbolsynchronization), it is possible to classify themembership of the received sequence to one of thepossible chaotic sequences which are generated atthe receiver and thus to decode the original digitalsequence that was encoded. The scheme is shownin Figure 1.Chaotic signals have an almost constant PSD due tothe weak autocorrelation. The large number ofsamples of the chaotic sequence necessary to obtaina likelihood estimation of the correlation introducea spreading effect. That is, the bandwidth of theoriginal digital signal is spread over a wide band.The resultant wide band spectrum associated withthe chaotic codes is however ideal as a drive signalfor channel estimation [9]. Moreover, using theproposed scheme, the transmitted signal can bereconstructed at the receiver based on the decoder’sdecision. This reproduced signal can then be usedto drive a channel estimation algorithm while thereceived sequence serves as channel output. Theoverall structure is depicted in Figure 1, where sk isthe m-valued digital signal. This set-up amounts toblind channel estimation.

III. Chaotic spreading codeHere we employ the LM (1) and the Yamakawacircuit described by [6]

187

( )11

11

−−

−−−=

−=

kkk

kkk

yxy

yxfx

βα

(4)

where

( )( )

( )

≤+−<≤

<−=

22223

212

111

ExEkExk

ExExk

xEExk

xf (5)

is a piecewise linear function and k1, k2, k3, E1,E2,α, β are system parameters.Figure 2 shows the autocorrelation of samples,{xk},of the Yamakawa system with k1=4.0, k2=-1.5,k3=2.5, E1=-1.3, E2=0.8, α=0.11, β=0.196.

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

0 200 400 600 800 1000

C(k

)

k

Figure 2. Autocorrelation of samples {xk} of theYamakawa-system

Due to the high parameter sensitivity, the timeevolution of a chaotic system is completelychanged after a small perturbation of theseparameters. The theoretically infinite number ofpossible parameter changes is the basis for chaoticmodulation and coding. Having a finite number ofcode words to be encoded, the parameter range isdiscretized such that to every code word there is afixed parameter value.In spread coding, to every channel is assigned achip, e.g. taken from the set of Walsh codes. Whenusing chaotic spreading codes, to every chip isassigned a different parameter of the chaoticsystem. For the Yamakawa-system (4) a possiblecoding scheme is

iiii sask += 01 (6)where si and ai are the m-valued information signaland a scaling factor to ensure that the parametervalue remains in the allowed range and does notoverlap with those of the other channels. Theparameter level is determined by s0

i. Because of thehigh sensitivity to changes in system parameters,the trajectories of each system assigned to each chipwill diverge very soon even with the same initialconditions. Moreover the samples of each systemare uncorrelated. This is the counterpart of theorthogonality of usual spreading basis codes.

IV. Blind channel estimation with chaoticspreading codes

Orthogonal spreading codes like Walsh codes arewidely used, e.g. in CDMA. These codes allowspread spectrum communication together withpower control of the transmitted sequence. As for ageneral channel, it is important in mobileapplications to continuously estimate thetransmission channel in order to meet thetransmission/reception specifications. This has to bedone in real time and so an estimation algorithmhas to be simple in implementation but efficient inperformance. The simplest update algorithm is theLMS algorithm. It is well known that the LMSachieves its maximum performance when driven byuncorrelated samples obtained for example bysampling a white noise signal or certain chaoticsignals [8, 9, 11]. As pointed out in Section II,samples of the chaotic spreading code are weaklycorrelated and hence constitute ideal LMS drivesignals. Further it is possible to combine the robustfinite correlation LMS (FCLS) [8] with the newspreading scheme. This is appropriate as FCLS canbe made to default to the LMS at low projectionswhile yielding better adaptation performance athigher projections. With an FIR model of order nhaving a filter coefficient vector g=(g0,...,gn-1), theoutput is given by

kT

ky xg= (7)

with the sample vector(8)

The FCLS algorithm is given by

( )g g X X X y X gk k kT

k kT

k k ktr+ = + −1

µΦ

(9)where

( )( ) ( )

y

X x x X X X X

k k k n

k k k m kT

k kT

k

y y=

= =− +

− +

, . . . ,

, . . . , ,

1

1 Φ(10)

The approach adopted consists of decoding thereceived spread spectrum signal thus determiningthe digital codeword transmitted. This in turnenables the receiver to construct the broadbandsignal at the channel’s input. The accuracy of thereconstruction depends on the decoding errorsincurred. The reconstructed signal and the channeloutput then drive the channel estimation algorithmwithout explicit knowledge of the original signalpassing through the channel.Figure 1. shows the schematic methodology forchannel estimation with chaotic spreading codes ina multilevel coding scenario. The transmitterswitches between m chaotic carriers to encode them-level code to be transmitted. At the receiver thesame set of chaotic circuits with equal parametersare iterated. Due to the uncorrelated nature of the

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different sequences, the integrator output are smallexcept when the test sequence matches the one usedat the transmitter side. The number of necessarysamples can be large depending on the actual SNRof the channel. Thus proper detection is at theexpense of an increase in bandwidth. This,however, is not uncommon since an increase ofaround 100 times is expected in CDMAapplications.Additionally, the m-ary chaotic coded signal can beused as a form of thin overlay spread spectrumadditional channel that spreads its spectrum thinlyover that of many existing channels. This mayresult in an acceptable deterioration in theperformance of these channels provided that thechannels have a good SNR. In which case, theproposed scheme gives the bonus of an addedinformation channel and a channel estimator thatserves the existing channels as well.

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0 5 10 15 20 25 30 35 40 45 50

BE

R

SNR [dB]

30 samples40 samples50 samples55 samples60 samples

Figure 3. Bit error rate for different numbers ofsamples of the YS used for symbol detection.

Having decoded the m-level code, the original drivesignal is known and can be fed into the channelestimation algorithm. Obviously the performance ofthe estimator depends on the accuracy of thedecoding procedure and thus on the segment lengthused to construct the correlation.

V. Simulation resultsTwo systems were considered, the LM andYamakawa system. The first because of itssimplicity and excellent PSD and the latter becauseit is accessible as a silicon chip [6]. Our testscenario consists of a 4-level coding regime suchthat si∈{1,2,3,4}. The system parameters for theYamakawa system were set according to

k s k k

E E

i1 2 3

1 2

15

41 5 2 5

0 1 1 0 1 9 6

1 3 0 8

= + = − =

= == − =

, . , .

. , .

. , .

α β (11)

and the parameters for the LM were in accordanceto (3). The x-state of (4) was taken as chaoticsequence. As pointed out earlier the accuracy of thecoding procedure using chaotic spreading codes

thoroughly depends on the number of samplesemployed to estimate the correlation of the receivedsequence with the set of candidate samples assignedto each symbol. To assess the coding strategy interms of error probability, Figures 3 and 4 werecompiled and show the BER for the LM and theYamakawa system, respectively, for several SNRvalues and several numbers of samples used toconstruct the correlation sums for classification.The number of transmitted samples was 106 toensure objectivity. Comparing Figures 3 and 4, it isclear that the Logistic map based system offersbetter performance compared to the Yamakawasystem. For example, at 50 chaotic samples per bitand at a channel SNR of 5 dB, the Yamakawa

system offers a probability of error of 3105.1 −× ,whereas the system based on the Logistic map

offers a probability of error of 4107.1 −× . Theperformance improvement can be attributed to theweak correlation properties of the Logistic map.

0

0.0002

0.0004

0.0006

0.0008

0.001

0 5 10 15 20 25 30 35 40

BE

R

SNR [dB]

40 samples50 samples60 samples80 samples

120 samples

Figure 4. Bit error rate for different numbers ofsamples of the LM used for symbol detection.

-20

-15

-10

-5

0

0 1000 2000 3000 4000 5000 6000

20 samples50 samples30 samples

Figure 5. Model misadjustment for a channel SNRof about 15 dB for the YS

To assess the channel estimation capability, a 10MHz low pass filter was used to emulate theunknown channel to be estimated. 4 MHz sampledsequences of the both systems were then used toencode a random 4-valued digital code sequence.The channel estimator was a 128 tap FIR DAFdriven by the FCLS algorithm. Since it turned out

189

that the probability of error of the Yamakawasystem is high for a small number of samples, themodel misadjustment improvement was consideredfor several numbers of samples per information bitin the case of the Yamakawa system. When usingthe LM the number was fixed to 50. The modelmisadjustment is defined as the Euclidean distancebetween the tap weight vector to be estimated andthe tap weight vector produced by the estimator. Itis a useful measure and minimizing its valueensures accurate channel estimates.

-100

-80

-60

-40

-20

0

20

0 500 1000 1500 2000 2500 3000

Mis

adju

stm

ent [

dB]

k

noise free-10 dB SNR

0 dB SNR10 dB SNR30 dB SNR60 dB SNR80 dB SNR

Figure 6. Model misadjustment for the LM

Figure 5 shows the model misadjustment for a SNRvalue of 15 dB when the Yamakawa system is usedas the chaotic signal generator. It is clear that whenthe number of samples incorporated in the symboldetection reaches a lower bound the estimation iscorrupted. Good performance is obtained when 50samples per bit were used in the encoding process.In particular a model misadjustment of about –15dB is achieved after 1000 iterations (250 µs). Theestimation accuracy, however, increases to only –17dB after 4000 iterations (1 ms). In Figure 6, themodel misadjustment for the LM-based system isshown versus the number of iterations with thechannel SNR as a parameter. The performanceachieved is greatly enhanced (compared to Figure5) and a model misadjustment of –40 dB isachieved after 1000 iterations (250 µs). Thisperformance is significant and is comparable to thatobtained for a white noise driven 128 tap FIR DAF[8]. This is further confirmed when variouschannels SNRs are considered in Figure 6. Theresults show that the model misadjustment saturatesat an SNR that is approximately the negative of thechannel SNR as observed for a white noise drivingprocess. Moreover, it should be observed that blindchannel estimations is achieved.

VI. ConclusionA novel chaotic-based multi-level signal transmission andblind channel strategy has been proposed and analyzed.The system transmits a number of chaotic samples in lieuof the digital information bits. At the receiving end, acorrelative receiver decides on the transmitted word bycomparing the output of a bank of correlators driven by

the received signal and local chaotic generators. Thedecision is used to drive a channel estimator that employsthe FCLS algorithm. Probability of error results haveshown that the weakly correlated sequences produced bythe Logistic map yield a performance better than thatoffered by a system based on the Yamakawa system(correlated, see Figure 2). At a channel SNR of 5 dB and

using 50 samples per bit, error rates of 3105.1 −× and4107.1 −× were achieved for the Yamakawa and LM

based systems respectively. Blind channel estimation wasalso achieved where the Yamakawa and the LM basedsystems offered model misadjustment values of –15 dBand –40 dB respectively after 1000 iterations (250 µs)for a 128 tap FIR DAF channel estimator model. Aproposal for a thin overlay spread spectrum systembased on this approach was made and will bepursued.

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