Design of a chaos-based spread-spectrum communication system using dual Unscented Kalman Filters. S. Azou 1 and G. Burel 2

Transcription

1 Design of a chaos-base srea-sectrum communication system using ual Unscente Kalman Filters S. Azou 1 an G. Burel 2 1 L.E.S.T. - UMR CNRS 6165, Pôle Universitaire Per Jaez Helias, Creac'h Gwen, 29 QUIMPER, France 2 L.E.S.T. - UMR CNRS 6165, Université e Bretagne Occientale, BP 89, BREST ceex, France {Stehane.Azou, Gilles.Burel}@univ-brest.fr Abstract - It has been emonstrate recently than use of chaotic sreaing coes can significantly increase transmission rivacy for irect-sequence srea sectrum systems. In this note, we consier the roblem of receiver synchronization as a ual estimation of the clean state an the unerlying moel arameters from the observe noisy chaotic signal. An efficient imlementation of the emoulator is investigate owing to the Unscente Kalman Filter, a recent alternative to the Extene Kalman Filter, roviing suerior erformance at an equivalent comutational comlexity. Numerical simulations show the erformance of this novel emoulator against noise for a single user on Gaussian channels. I. Introuction In the last few years, a great research effort has been evote towars the eveloment of efficient chaos-base moulation techniques [1][2]. The motivation to use chaos for transmission is mainly ue to its wieban nature an its noise-lie aearance. Hence, as for common srea-sectrum techniques, chaotic signals rovie robustness against frequency selective faing in multiath channels an narrowban interference. Owing to the intricate ynamic of chaos, the rivacy of communications can be significantly increase in comarison with stanar seuo-noise coes use in srea-sectrum [3]. Without nowlege of the tye of nonlinearity on which the transmission is base, it will be extremely ifficult for the unauthorize user aware of the transmission to access the information. Many other otential benefits have to be notice, among others the reuce comlexity of transmission evices an a sharing of channel resources via Coe Division Multile Access (CDMA). Chaos can be use in multile ways in a igital communication system [4]. As in conventional communication systems, the transmitte symbols can be recognize at the receiver using either coherent or noncoherent emoulation techniques. Whereas a noncoherent receiver relies only on some statistics of the receive signal, a coherent receiver requires a comlete nowlege about the transmitter to recover the original chaotic sreaing coe, by a synchronization rocess. Although the latter aroach suffers from sensitivity to arameter mismatches between the transmitter an the receiver, an also from signal istortion inuce by the channel, the robability of intercetion is reuce in this way. This aer eals with emoulator esign in a Chaotic Direct-Sequence Srea Sectrum system [5]. As in revious aers, the logistic ma is recommene here as the sreaing coe generator, ue to its favorable correlation roerties. In orer to get reliable communication for realistic roagation conitions, a robust synchronization roceure has to be eveloe at the receiver sie. A recursive nonlinear estimation scheme is referre here to the stanar master-slave configuration, initially roose by Pecora an Caroll [6]. The iea of using state-sace aative filtering to erform chaotic synchronization is not new ; Cuomo et al. [7] have reviously mentione the interest of the Extene Kalman Filter in such a context, for a Lorenz system. More recently, a metho of synchronizing two chaotic Duffing systems is escribe in [8] by imlementing an EKF for continuous-time systems. In this

2 wor, communication is achieve by moulating a arameter within the transmitter an augmenting the moel orer within the EKF to estimate that arameter. In [9][1], similar results are reorte using a iscrete-time formalism. Given a stochastic linear iscrete-time moel in state-sace, the Kalman filtering roblem [11] is to rouce an estimate xˆ of the true state x, given a sequence of noisy observations u to time. In the linear case, the Kalman filter yiels the otimal estimate in the minimum mean-square error (MMSE) sense. For alication to nonlinear moels, the so-calle Extene Kalman Filter is often use in ractice. In the EKF, the state istribution is aroximate by a Gaussian ranom variable, which is roagate analytically through a first-orer linearization of the nonlinear system. As a consequence of these successive aroximations, large errors in the true osterior mean an covariance of the transforme ranom variable can occur, which may lea to sub-otimal erformance an sometimes ivergence of the filter. Motivate by the eficiencies of the EKF, we roose to use the more robust Unscente Kalman Filter (UKF) to erform the synchronization in a CD3S receiver. The UKF, recently roose by Julier et al. in the context of nonlinear control [12], aresses the aroximation issues of the EKF. The state istribution is again reresente by a Gaussian ranom variable, but is now secifie using a minimal set of carefully chosen samle oints. At each ste of the recursion, these samle oints are roagate through the true nonlinear functions of the moel. Hence, osterior mean an covariance are cature accurately to the secon orer (Taylor series exansion), whatever the nonlinearity is [13]. Sreaing the sectrum of the information signal by irect-sequence can be consiere as a secial case of switching the arameter of a ynamic moel for communicating with chaos. It follows from this observation that CD3S signals can be emoulate through a ual estimation scheme. This rocess will be imlemente in an original manner here owing to the UKF. The aer is organize as follows. In section II we resent rinciles of a CD3S transmitter an the re-rocessing that have to be one at the receiver before emoulation. The ual estimation scheme for receiver synchronization an imlementation etails using UKF are treate in section III. The unscente transform, laying a central role in the UKF, is overviewe in section IV. Then, in section V, the metho is illustrate an some erformances for Gaussian channels are given. Finally, in section VI we raw some concluing remars. II. Chaotic Direct-Sequence Srea Sectrum (CD3S) Signals Figure 1 illustrates the rinciles of a CD3S moulator. At the moment, ata have been moulate through BPSK. A ifferential encoing may be erforme to eliminate the hase ambiguity at the recetion. The sreaing oeration is one by multilication of the ata symbols with the chaotic signal, evolving at a rate Fc>>F, F being the ata rate. The rocessing gain W = Fc / F must be an integer. Its value eens on the banwith available for the roagation channel, notably. In orer to facilitate the receiver synchronization, the information to transmit is structure in frames ; In this way, chaotic marers, whose length is ientical to the rocessing gain, are regularly inserte after the sreaing oeration. This means that the receiver can reconstruct the marers in an autonomous way. A basic solution is to reeat the same marer for each new frame an to store the marer signal at the receiver sie. Then, an usamling rocess by zeros inserting is accomlishe an a square-root raise cosine shaing filter is alie, with a rolloff factor α of.5, before a carrier moulation at

3 central frequency F. To avoi aliasing, the signal has to be samle at a minimum value of 2F + (1 + α)f c. Fig. 1 Bloc iagram of a CD3S transmitter. As in [14], the logistic ma is suggeste as the sreaing sequence generator, ue to its favorable correlation roerties. Hence, the wieban signal resulting from the sreaing oeration (before marers insertion) is given by 2 x (1 2x ) = 1 where x enotes the state of the ynamical system an where the arameter symbol (e.g. ± 1 for a BPSK encoing). is the ata III. A ual estimation scheme for CD3S emoulation In this section, we focus on the emoulator esign for CD3S signals. A simultaneous estimation of the state of the noisy receive chaotic signal an the ata symbol is roose to recover the information. As illustrate by figure 2, this ual estimation roblem is solve owing to Unscente Kalman Filters, ue to their moerate comutational cost an their robustness in the resence of strongly nonlinear signals. Fig. 2 Dual estimation scheme for CD3S emoulation. The receive signal, samle at frequency F s has first to be brought bac to baseban an lowass filtere (square root raise cosine filter) before any rocessing. Then, the frames are synchronize, by correlations, thans to the set of chaotic marers that has been inserte by the transmitter. Also, the carrier hase an the signal ower fluctuations have to be ajuste before the emoulation rocess. In what follows, a binary ata moulation (e.g. BPSK) is assume ; In this case, the information is emboie in the real art of the signal only at the outut of the re-rocessing bloc. The receive ownsamle signal is assume to evolve accoring to the following system moel : 2 x f x ) + v = (1 2x v = ( 1 1 ) + 1

4 where x an are the clean chaotic signal state an true symbol resectively, an where v is a zero mean, White Gaussian Noise (WGN) sequence with variance Q, ineenent of the ast an current state. This moel uncertainty is necessary to tae into account the istortions resulting from transmitter/receiver filters, the analog-igital conversions an the channel multiath. The available observations are moelle as z = h( x ) + n = x + n where x is the true state an where n is a WGN whose variance R eens uon signal-tonoise ratio (SNR). As the information symbol to evolve as is constant over W = F c / F consecutive samles, it is assume ( ) = f 1 + w 1 = 1 + w 1 The aitional term w is a WGN. Its variance Q will influence the aatability of the symbol filter ; A low value will result in slow changes whereas a larger value will result in rai variations of the symbol estimates. Finally, the true symbol is observe accoring to the moel z = h ( ) + n = (1 2 2 x ) 1 + n The imlementation of the ual estimation roceure is outline at figure 3. Fig. 3 Imlementation of the emoulator using Unscente Kalman Filters. Fig. 4 True state vs. Estimate state for a ual UKF synchronization of a CD3S receiver. IV. Overview of the Unscente Kalman Filter (UKF) The UKF, eveloe by Julier et al. [12], aresses the aroximation issues of the EKF. The reaer is referre to [11] for a etaile exlanation of the EKF. The state istribution is reresente by a Gaussian ranom variable x, but is secifie using a minimal set of carefully samle oints. These oints comletely cature the mean an covariance of the ranom variable x. Then, owing to the Unscente Transformation (UT), it is ossible to cature recisely the statistics of the ranom variable y resulting from a nonlinear transformation. The general roblem of aroximating nonlinear transformations of robability n istributions can be state as follows : given a r.v. x R with mean x an covariance P xx, m we woul lie to reict the mean y an covariance Pyy of a r.v. y R, where y is relate to x by the nonlinear transformation y = f(x). To solve this roblem, Julier ha an intuition that with a fixe number of arameters it shoul be easier to aroximate a Gaussian

5 istribution than it is to aroximate an arbitrary non linear transformation. The roceure can be summarize as follows : is chosen to aroximate the r.v. x : 1. A set of weighte oints { i} i= 1,...,2n+ 1 where κ R ; = x W = x + = κ /( n + κ ) ( ( n + κ ) Pxx ) Wl = 1/ 2( n + κ ) l ( ( n + κ ) P ) W = 1/ 2( n + ), l = 1,..., n l l+ n = x xx l l+ n κ 2. Each oint is then transforme as = f ( ) Y ; i 3. The mean y is given by the weighte average of the transforme oints : y = WiY ; i 4. The reicte covariance yy i P is comute as : P { Y y}{ Y } T 2n = yy Wi i i y i= In comarison with Monte Carlo techniques, the funamental ifference is that the samles are not rawn at ranom but accoring to a eterministic algorithm. A very small number of samles is then require to aroximate the robability istribution. This metho has many others avantages : owing to the arameter κ the scaling of the fourth an higher orer moments can be influence; the function f, which may be imlicit, is not aroximate through a truncation of its series exansion an the imlementation is very easy since no evaluation of Jacobians is neee. It is shown that the mean an covariance of y are cature accurately to the secon orer (Taylor series exansion) for any nonlinearity. The Unscente Kalman Filter is an extension of the UT to the recursive estimation scheme of a Kalman filter. More etails about this subject can be foun in [13]. V. Illustrations an Performance evaluation for Gaussian channels Numerical simulations have been conucte to evaluate erformances of the roose CD3S receiver. In this section, results for a single user on a Gaussian channel are escribe only. Performances in a realistic context of transmission unerwater are reorte in [15]. Figures 4 & 5 illustrate the receiver behaviour for an aitive white Gaussian noise with SNR equal to 8 B an a rocessing gain of 63. The UKF configuration was chosen as : Q =. 1, R =. 5 an Q =.2. As shown by figure 4, the chaotic synchronization erforms well at this noise level. Figure 5 shows few transmitte symbols an their estimates ; A goo aatativity of the UKF is notice on this figure. Monte Carlo simulations have been conucte to evaluate the Bit Errror Rate for larger noise levels. The results are given in figure 6. It shoul be notice that the variance R was ajuste for each SNR, in orer to get the uer boun of the erformances. Nevertheless, it has been observe that a rough estimation of the noise level is sufficient to aroach this boun. The roose UKF emoulator has a goo allowance concerning its configuration (choice of Q, Q, R ). VI. Conclusion A novel emoulator has been investigate for Chaotic Direct-Sequence Srea Sectrum (CD3S) signals. This emoulator relies on a ual estimation of the state of the receive chaotic signal an the associate ata symbol (arameter of the system moel). An imlementation base on Unscente Kalman Filters is roose. In this way, a goo robustness against noise is achieve at a limite comutational cost. The BER erformance has been evaluate via Monte Carlo simulations, for a single user on a Gaussian channel.. 2n i=

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