Author name used in this publication: Francis C. M. LauAuthor name used in this publication: Chi K. TseRefereed conference paper2003-2004 > Academic research: refereed > Refereed conference paperVersion of RecordPublishe

Kis, G

Kolumbán, G

Lau, FCM

Tse, CKM

PolyU Institutional Repository

OPTIMUM NONCOHERENT FM-DCSK DETECTOR: APPLICATION OF CHAOTIC GMLDECISION RULEGéza Kolumbán & Gábor Kis ∗Department of Measurement andInformation Systems,Budapest University of Technology andEconomics, Budapest, H-1521, Hungary.{kolumban,kisg}@mit.bme.huFrancis C.M. Lau & Chi K. Tse†Department of Electronic andInformation Engineering,The Hong Kong Polytechnic University,Kowloon, Hong Kong, China.{encmlau,encktse}@polyu.edu.hkABSTRACTA new detector configuration is proposed for the detectionof FM-DCSK signals with improved noise performance.Since the new detector configuration is developed directlyfrom the chaotic GML decision rule, it offers the best noiseperformance in AWGN channel in the category of nonco-herent detection if the reception of a single isolated symboland memoryless modulation scheme is considered. The pa-per shows how the a priori information on chaotic signalsmay be exploited in waveform communication to maximizethe noise performance.1. INTRODUCTIONA lot of chaos-based communication schemes have beenproposed recently [1]. Among them the frequencymodulated-differential chaos shift keying (FM-DCSK)modulation scheme assures the best potential noise perfor-mance if the reception of a single isolated symbol and mem-oryless modulation schemes are considered, because theFM-DCSK is the only chaotic modulation scheme whichhas orthonormal basis functions [2].The noise performance of a modulation scheme dependson two factors: on the separation of elements of signalset and the detector configuration. The separation of ele-ments of signal set gives the performance bound on attain-able noise performance. The better the separation, the betterthe attainable noise performance.The detector observes the received signals in the obser-vation space. A well designed detector exploits all the avail-able a priori information on the elements of signal set toseparate the signal to be detected from channel noise, thatis, to suppress the noise. As a rule of thumb we may say,∗ This research was financed by the Hungarian Scientific Foundationof Hungary (OTKA) under Grant number T-038083† This work is supported by a research grant provided by Hong KongPolytechnic University.the more a priori information is exploited, the better noiseperformance is achieved. In this respect the key issue is thedecision rule: the application of maximum likelihood (ML)decision rule assumes that all elements of signal set are ex-actly known at the receiver, while in case of the generalizedmaximum likelihood (GML) decision rule [3] only certainparameters of elements of signal set must be known.In chaotic waveform communication, the elements ofsignal set are not fixed waveforms. The chaotic basis func-tions are different for each transmitted symbol even if thesame symbol is transmitted repeatedly. The continuouslyvarying waveforms make it very hard to find a mathemati-cal model for the detection problem that is simple enough.This problem has been solved in [4], where the Fourieranalyzer concept was introduced to define the observationspace of the detector. Section 2 surveys the Fourier analyzerconcept and introduces the chaotic GML decision rule. Theimportance of chaotic GML decision rule is that it showswhat is the main theoretical difference between chaotic andconventional waveform communication and how the max-imum amount of a priori information may be exploited inchaotic communication by the detector.The new detector configuration is developed in Sec. 3from the chaotic GML decision rule. Since the applicationof GML decision rule results in optimum noise performancein additive white Gaussian noise (AWGN) channel, this de-tector configuration gives the theoretical limit on bit errorrate (BER) that may be achieved with chaotic modulationschemes if chaotic basis functions may not be recoveredat the receiver from the modulated, distorted and noisy re-ceived signal.Section 4 shows the noise performance of the optimumnoncoherent FM-DCSK detector that has been determinedby computer simulation. The results are compared with thenoise performance of classical FM-DCSK and conventionalFSK detectors, conclusions are drawn and the limit on at-tainable performance improvement is discussed.???????????????????????????????? ????????? ??????????Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on December 15, 2008 at 02:14 from IEEE Xplore.  Restrictions apply.2. GENERALIZED MODEL OF WAVEFORMCOMMUNICATIONSEach digital receiver must contain at least a channel filter,whose bandwidth is equal to the bandwidth of modulatedsignal, and a decision circuit. Consequently, the minimumamount of a priori information that is always available at areceiver is the RF bandwidth 2B of channel filter and sym-bol duration T .Let rm(t) = sm(t) + n(t) denote the received signal,where s(t) denotes the transmitted element of signal set andn(t) represents the channel noise. This signal is fed intothe channel filter, the output of channel filter is denoted byr˜m(t). This signal is observed by the detector.2.1. The Fourier Analyzer ConceptEach element sm(t) of signal set is expressed as a linearcombination of N orthonormal basis functions gj(t), j =1, 2, . . . , N [2]. The Fourier analyzer concept [4] has beenderived from the recognition that a detector observes the re-ceived signal only on the interval 0 ≤ t ≤ T to performthe detection. Consequently, the signal may have any valueoutside this interval. Let us exploit this property and let aperiodic basis function be defined asgT,j(t) ={gj(t), for 0 ≤ t ≤ Tgj(t− CT ), otherwisewhere C is an arbitrarily nonzero integer. Then the basisfunctions may be represented over the interval 0 ≤ t ≤ Tby a Fourier seriesgT,j(t) =K2∑k=K1[αj,k cos(k2πTt) + βj,k sin(k2πTt)](1)where αj,k and βj,k denote the coefficients of Fourier seriesexpansion. The constants K1 and K2 are determined by thefrequency band covered by the spectrum of basis functionsgj(t), for a band-pass system K2 −K1 + 1= 2BT .Equation (1) shows that the basis functions have a dis-crete spectra, the fundamental frequency is determined bythe bit duration, and that the number of harmonically relatedfrequencies to be considered is determined by 2BT .The Fourier series expansion defines the observationspace which is a Hilbert space of harmonically relatedcos(·) and sin(·) functions with dimension Dˆ = 4BT . Notethat the dimension of observation space is determined bythose parameters that are always known at the receiver.2.2. Chaotic GML Decision RuleTwo orthonormal basis functions are used in FM-DCSKand each bit is mapped into one basis function. This pa-per assumes that identical chaotic synchronization may notbe used due to the bad propagation conditions. If so thenthe GML decision rule [3] has to be used. The applicationof GML decision rule is summarized in the following.Consider the observation space constructed by theFourier analyzer concept. Each basis function defines a sub-space in the observation space. The received signal is pro-jected into the subspaces of each basis function and the en-ergies measured in the different subspaces are determined.The decision is made in favor of the subspace and, conse-quently, in favor of the bit, that receives the greatest energy.But the GML decision rule known from the theory ofconventional communication systems may not be applieddirectly to chaotic communication, since in the latter thebasis functions vary from symbol to symbol. Consequently,only the mean values of the Fourier coefficients may be de-termined in (1).Let a weight be defined for each harmonic frequency in(1) asWj,k =E[√α2j,k + β2j,k]E[√α2j,k0 + β2j,k0]where k0 = (K1 + K2)/2 assigns the center frequency ofFM-DCSK signal and E denotes the expectation operator.Then the weighted energy received in the subspace of gj(t)is obtained asEj,m =2TK2∑k=K1([Wj,k∫ T0r˜m(t) cos(k2πTt)dt]2+[Wj,k∫ T0r˜m(t) sin(k2πTt)dt]2).(2)Due to channel noise, Ej,m is a random variable. Sincethe spectra of basis functions are not uniform, the signal-to-noise-ratio (SNR) differs for each terms in (2). The dutyof the weights Wj,k is to maximize the SNR for each ba-sis function represented by Fourier series in the observationspace. As a result, the overall SNR is maximized and thebest noise performance is achieved. This decision proce-dure is called chaotic GML decision rule.The weights 0 ≤ Wj,k ≤ 1 are determined from theaveraged spectrum of the jth basis function.3. DEVELOPMENT OF NEW FM-DCSKDETECTOR CONFIGURATIONThis section develops the block diagram of an optimumnoncoherent FM-DCSK detector from (2). To do that, firstthe weights Wj,k have to be determined.3.1. Determination of WeightsEquation (1) gives the Fourier series expansion of basisfunctions. Recall that bits “1” and “0” are mapped into the????????Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on December 15, 2008 at 02:14 from IEEE Xplore.  Restrictions apply.first and second basis functions, respectively. It has beenshown in [5] that the spectra of first and second basis func-tions have only even and odd, respectively, harmonics. Toillustrate this, Fig. 1 shows this separation of basis functionsin the frequency domain. In the figure, the power spectrumof a pure bit “1” sequence is shown for T=2 µs and centerfrequency of 2.4 GHz. Note, only even harmonics appear.2.38 2.385 2.39 2.395 2.4 2.405 2.41 2.415 2.42x 109?40?35?30?25?20?15?10?505Frequency [ Hz ] [ dB ] Fig. 1. Power spectrum of an FM-DCSK signal for a purebit “1” sequence.The first conclusion is that for bit “1” the weights Wj,kin (2) are equal to zero for each odd harmonics since thereis no signal energy at those frequencies. The zero weightssuppresses the noise components at odd harmonics. Sim-ilarly, for bit “0” the weights Wj,k are zero for each evenharmonics.To get the exact values of weights, the voltage spec-trum of an FM-DCSK signal was determined for a very longFM-DCSK signal carrying an equiprobable random bit se-quence. Since an FM-DCSK receiver with the parametersof 2B=30.5 MHz and T=2 µs is considered in our simu-lations, only 61 weights may differ from zero as shown inFig. 2. As examples, the first four values of Wj,k are givenin Table 1.2.38 2.385 2.39 2.395 2.4 2.405 2.41 2.415 2.42x 10900.20.40.60.81Frequency  [ Hz ][ V ]Fig. 2. Voltage spectrum of FM-DCSK signal carrying arandom bit sequence.Table 1. The first four values of the weights Wj,k.Bit “1” Bit “0”Frequencies [MHz] W1,k W2,k2,400.0 1.00 0.002,399.5 2,400.5 0.00 0.99832,399.0 2,401.0 0.9896 0.002,398.5 2,401.5 0.00 0.97483.2. Block Diagram of the New FM-DCSK DetectorConsider the reception of bit “1” first. The subspaceof g1(t) contains only the even harmonics of the clockfrequency 1/T . Substituting j=1 into (2) we obtain forg1(t)TCE1,m =K2∑k=K1([W1,k∫ T0r˜m(t) cos(k2πTt)dt]2+[W1,k∫ T0r˜m(t) sin(k2πTt)dt]2) (3)where TC = T/2, k is an even number and the weightsW1,k may be determined from Fig. 2 and Table 1.In a similar fashion, substituting j=2 into (2) we obtainthe product of TCE2,m for bit “0”. In the case of g2(t), k isan odd number and the values of W2,k may be determinedfrom Fig. 2 and Table 1.The decision is done in favor of bit “1” ifTCE1,m − TCE2,m > 0 . (4)From (3) and (4), the new FM-DCSK detector may beconstructed as shown in Fig. 3. Like the conventional co-herent receivers, theoretically there is no need for channelfilter in this receiver since the bandlimiting of input signalis performed by the appropriate choice of weights.4. LIMIT ON BER IMPROVEMENTThe noise performance of optimum noncoherent FM-DCSKdetector is shown by dotted curve in Fig. 4. For comparison,the noise performance of frequency domain FM-DCSK [5](dash-dot curve), differentially coherent FM-DCSK (solidcurve) and noncoherent FSK detectors (dashed curve) arealso shown.The frequency domain FM-DCSK and differentially co-herent FM-DCSK detectors have the same noise perfor-mance. Recall that in the frequency domain FM-DCSKdetector the weights in (2) have only two values, 0 and 1,while in the optimum noncoherent FM-DCSK detector thethe weights are matched to the spectrum of FM-DCSK sig-nal.From Fig. 4, the following conclusions are drawn. Theapplication of chaotic GML decision rule improves the????????Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on December 15, 2008 at 02:14 from IEEE Xplore.  Restrictions apply.                       		          ﬀ ﬁ   ﬀ ﬁ                              ! " # $ % & '  " ( ( (' #   )   & '  "*  $   +  ,-    .  /  0ﬀ                  % 12 3 4 5 6  7 8 92 3 4 5 6  7 8 6 9 8 7 : : 3 ; 6 4% < ﬀ =% > % ?=/ @%@? ?A B BFig. 3. Block diagram of optimum noncoherent FM-DCSK detector.0 2 4 6 8 10 12 14 1610?410?310?210?1100Eb / No [ dB ]Bit Error RateFig. 4. Noise performance of optimum noncoherent FM-DCSK (dotted curve), frequency domain FM-DCSK (dash-dot curve) and differentially coherent FM-DCSK (solidcurve) detector. BER of noncoherent FSK, upper bound onnoise performance improvement, is shown by dashed curve.noise performance since it minimizes the variance of ob-servation signal by matching the weights Wj,k to the spec-trum of FM-DCSK signal. The noise performance may beimproved by reducing the bandwidth of FM-DCSK signal,the lower the bandwidth, the better the noise performance.In the limit, optimum noncoherent FM-DCSK achieves thenoise performance of noncoherent FSK.However, if the signal bandwidth is reduced then FM-DCSK signal becomes a narrow-band signal and the excel-lent multipath performance of FM-DCSK is lost. The signalbandwidth is one of the most important design parameter,the small bandwidth gives a better noise but worse multi-path performance, while increasing the bandwidth results inworse noise but improved multipath performance.5. REFERENCES[1] M. Hasler, G. Mazzini, M. Ogorzalek, R. Rovatti, and G. Setti,“Special issue on applications of nonlinear dynamics to elec-tronic and information engineering,” Proceedings of the IEEE,vol. 90, no. 5, May 2002.[2] G. Kolumbán, M. P. Kennedy, Z. Jákó, and G. Kis, “Chaoticcommunications with correlator receiver: Theory and perfor-mance limit,” Proceedings of the IEEE, vol. 90, no. 5, pp.711–732, May 2002.[3] L. M. Fink, Discrete Message Transmission Theory, Soviet-skaya Radio, Moscow, 2nd edition, 1970, In Russian.[4] G. Kolumbán, F. C. M. Lau, and M. Small, “A new descriptionof chaotic waveform communications: The Fourier analyzerapproach,” in Proc. ECCTD’03, Cracow, Poland, Septem-ber 1-4 2003, vol. III, pp. 241–244.[5] G. Kolumbán, “A new frequency-domain FM-DCSK detec-tor,” in Proc. ECCTD’03, Cracow, Poland, September 1-42003, vol. III, pp. 253–256.????????Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on December 15, 2008 at 02:14 from IEEE Xplore.  Restrictions apply.

Optimum noncoherent FM-DCSK detector : application of chaotic GML decision rule

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