RECURSIVE BLIND IDENTIFICATION AND EQUALIZATION OF FIR CHANNELS FOR CHAOTIC COMMUNICATION SYSTEMS

Transcription

1 6th European Signal Processing Conference (EUSIPCO 008), Lausanne, Sitzerland, August 5-9, 008, copyright by EURASIP RECURSIVE BLIND IDENIFICAION AND EQUALIZAION OF FIR CHANNELS FOR CHAOIC COMMUNICAION SYSEMS Cabir VURAL, Göçen ÇEİNEL Electrical- Electronics Engineering Department, Saarya University 5487 Esentepe, Saarya, urey phone: + (90) , fax: + (90) , cvural@saarya.edu.tr, gcetinel@saarya.edu.tr.saarya.edu.tr ABSRAC In this study, an adaptive autoregressive (AR) algorithm is proposed for the blind identification and equalization of finite impulse response (FIR) channels for chaotic communication systems. he AR method is derived by minimizing a nonlinear prediction error function that is calculated by exploiting the short time predictability of a chaotic signal. he adaptive AR algorithm requires a recursion ithin a recursion. A simplification is provided to remove the costly inner recursion. When convergence taes place, the AR filter coefficients give an approximation of the FIR channel coefficients and the AR filter output is an estimate of the transmitted chaotic signal. Simulation results sho that the method gives promising results.. INRODUCION Broad-band, orthogonality and complexity aspects of chaotic signals motivated researches in the area of communication and signal processing to investigate if chaos based communication offers advantages over classical communication systems in the last to decades. Potential applications of chaos resulting directly from these three aspects are spread- spectrum, multi-user communication, and cryptography [, ]. In chaotic communications, a chaotic sequence is transmitted through the propagation channel. If the channel is not ideal, hich is often the case in practice, the transmitted signal is corrupted before it reaches to the receiver. Hence, channel equalization is required to mae the bit error rate of the receiver as small as possible. In many practical cases, channel parameters are unnon. Hence, channel equalization must be performed from the corrupted signal alone, and this is called the blind channel equalization. In classical communication systems, most of the channel equalization algorithms are based on the statistical properties of the transmitted signal. Hoever, since a chaotic sequence is a deterministic signal, the statistics-based equalization techniques ill not achieve optimum estimation accuracy for chaotic communication systems because they do not tae into account the inherent properties specific to a chaotic signal. Various chaotic blind identification and equalization techniques based on different properties of the transmitted chaotic signal have been developed recently. Leung et al. introduced a ne complexity measure called the Phase Space Volume (PSV) to describe the finite dimensional property of a chaotic signal. hey shoed that the parameter estimates could be obtained by minimizing the PSV. Although the Minimum Phase Space Volume (MPSV) technique can offer very high estimation accuracy, the computation time spent for the PSV is very high [3]. hen, Zhu and Leung proposed an identification approach called the Minimum Nonlinear Prediction Error (MNPE). he MNPE approach is derived by considering the short-term predictability of chaotic signal. Hoever, they performed the identification by representing an FIR channel as an infinite-order AR model. he estimation accuracy is limited by the truncation error in approximating the infinite order AR model by a finite order AR model [4]. In addition to above mentioned studies, Zhu and Leung presented an Extended Kalman Filter (EKF) based adaptive equalization algorithm. Compared ith the MPSV and MNPE, its computational complexity is very lo, but its estimation accuracy is not ell [5]. In this study, an adaptive algorithm is proposed for the blind identification and equalization of FIR channels for chaotic communication systems. Equalization of an FIR channel can be performed by using either an FIR filter or an AR filter. When an AR filter is used, e do not need to find the optimum filter support because it is same as the support of the channel. Furthermore, there is no distortion introduced by the finite support of the FIR filter. Hence, in our study e used AR filter. he AR method is derived by minimizing a nonlinear prediction error function that is calculated by exploiting the short time predictability of a chaotic signal. he algorithm requires a recursion ithin a recursion, hich is computationally complex. We propose a simplification that removes the inner recursion in a manner similar to that described in [6, 7]. When convergence taes place, the AR filter coefficients give an approximation of the FIR channel coefficients and the AR filter output is an estimate of the transmitted chaotic signal. Simulation results sho that the method gives promising results. he study is organized as follos. In Section II, e formulate the blind equalization of FIR channels for chaotic communication systems. he proposed adaptive algorithm is derived in Section III. In Section IV, simulation results are presented to evaluate the effectiveness of the proposed algorithm. Concluding remars are given in Section V.

2 6th European Signal Processing Conference (EUSIPCO 008), Lausanne, Sitzerland, August 5-9, 008, copyright by EURASIP Figure - A model that shos the relationship beteen the received and the transmitted chaotic signals.. BLIND CHANNEL EQUALIZAION Consider a chaotic communication system shon in Figure, in hich a chaotic information signal x[n] is transmitted through a propagation channel c[n] and contaminated further by an additive noise v[n] before it reaches to the receiver. Even though other maps are also possible, ithout loss of generality, the transmitted chaotic signal x[n] is assumed to be generated by the logistic map given by x[n] = f (x[n ]) =λx[n ]( x[n ]). () λ is a constant called the bifurcation parameter. In practice, the ideal channel does not exist. It as found that a typical channel could be modelled by an FIR filter [5]. hus, the received signal y[n] ould be y [n] = x[n]*c[n] + v[n] N = c[]x[n ] + v[n]. () = 0 here c[] s are the channel coefficients, and v[n] is the additive hite Gaussian noise ith zero mean and variance σ, and N is the length of the channel. he obective of channel equalization is to recover the transmitted signal x[n] from the received signal y[n] alone. 3. HE PROPOSED MEHOD For the case of non-chaotic transmitted signals, in most blind channel equalization algorithms a priori information about the statistics of the input signal is exploited. In case of chaotic communication, e need to base the equalization strategy to the nonlinearity of the chaotic signal instead of its statistical properties since a chaotic signal is a deterministic signal. Figure illustrates the proposed recursive blind channel equalization approach here the received signal is applied to an adaptive AR filter hose purpose is to estimate the transmitted chaotic signal. A desired signal is required in all adaptive algorithms in order to update the filter coefficients. One possible ay of obtaining the desired signal is to use a training sequence. his method is usually not preferred since it is not bandidth efficient. he other method is to artificially generate the desired signal from the adaptive filter output. In this study e ill adopt the second approach. he adaptive filter output should satisfy Eq. () assuming that it v Figure - he proposed recursive blind channel equalization approach is a reliable estimate of the transmitted chaotic signal. Hence, a plausible cost function J is the squared difference beteen the adaptive filter output and its prediction obtained by using Eq.(), i.e., J ( [n] f ( [n ])) = (3) here f (.) is the nonlinear map used to generate the transmitted chaotic signal. When the logistic map is used, the cost function can be ritten as J ( = [n] λ [n ]( [n ])). (4) he Gradient Descent (GD) algorithm ill be used to update the filter coefficients [8]. Hence, the derivative of J ith respect to the adaptive filter coefficients must be determined. Note that the equalizer output at time n at th iteration is given by N [n] = y[n] [i] [n i]. (5) here y[n], [n], [i] are the received signal, the output of the AR filter and the adaptive filter coefficients at th iteration, respectively. he general form of the GD algorithm for minimizing the proposed cost function is [] = [] + µ dj dj d[n] = [] µ (6) d[n] here =,,,N-. he first derivative in Equation (6) is dj = ( [n] λ [n ]( [n ])). (7) d [n] It is not possible to rite a closed- form expression for the second derivative in Equation (6), but it can be calculated iteratively using regressor filtering. o derive this term, note that [n] can be ritten as

3 6th European Signal Processing Conference (EUSIPCO 008), Lausanne, Sitzerland, August 5-9, 008, copyright by EURASIP N [n] = y[n] [i] [n i] N i i = = y[n] [] [n ] [i] [n i]. (8) aing the derivative of both sides of Equation (8) ith respect to [] gives N d [n] d [n i] [n ] [i], = (9) i i = Let us define the regressor [n, ] as, d [n] [n] : =. hen, Equation (9) can be ritten in terms of [n, ] as N i,[ n] = [n ] [i],[n i]. (0) Substituting Equation (0) and Equation (7) in Equation (6) yields ] = [ ] + µ (ˆ x [ n] [ n ]( [ n ])) [ ] () + [ λ, n Equation () gives the update equations for one of the adaptive filter coefficients. It is possible to rite update equations for all coefficients in vector form as + = + µ [n] λ [n ]( [n ])) [n] () ( here and [n] are the adaptive filter coefficients vector and the regressor filter vector at time n given by :=[ [], [],..., [N ]] [n]:= [, [n],,[n],...,,n [n]] he presence of regressor filter in Equation () maes implementation of the recursive algorithm costly. A simplified algorithm that bypasses the regressor filtering ould be preferred. An approximate gradient uses the currently available data vector in place of the regressor filtered version, that is [n] [ [n ], [n ],..., [n N ]]. (3) + Equations () and (3) constitute the proposed method. he output of the adaptive AR filter is an estimate of transmitted chaotic signal x[n], and the coefficients [n] provide an estimate of the channel coefficients c[n] at convergence. Conditions under hich the approximation made in Equation (3) need to be determined. In classical communication systems, a sufficient condition as obtained according to hich the channel must be Strictly Positive Real (SPR) [5,6]. A channel is said to be SPR if the real part of the discrete Fourier transform of its impulse response is positive. In our simulations e used SPR and non-spr channels to see hether this condition is valid in chaotic communication systems as ell. We ill not attempt to derive the necessary condition here since the space is limited. We plan to provide the proof (if it exists) in a forthcoming paper. 4. SIMULAION RESULS In this section, e use three computer simulations to evaluate the efficiency of the proposed algorithm. In all simulations the transmitted chaotic signal as generated using logistic map given in Equation (), the bifurcation parameter λ as taen to be 3.8, and the initial value of x [n] as chosen as x[0]=0.78. In the first experiment, performance of the proposed algorithm is discussed for SPR and non SPR channels. Efficiency of the adaptive algorithm for increasing channel length is investigated in the second experiment. Finally, effect of the adaptive AR filter order on the accuracy of the estimation of the channel coefficients is investigated in the third experiment. Experiment : In this case, the channel coefficients are estimated by using the proposed algorithm for the third-order SPR and non SPR channels, he simulation results are shon in able I and Figure 3. As shon in able I, the estimated channel coefficients are very close to the true channel coefficients for the SPR channel. Hoever, the estimated channel coefficients for the non SPR channel are not good approximations of the true channel coefficients. For these to channels the mean square error (MSE) beteen the true channel coefficients and the estimated channel coefficients are shon in Figure 3 as function of the signal-tonoise ratio (SNR). MSE decreases for the SPR channel hen SNR increases, but for the non SPR channel, MSE does not change much ith respect to changes in SNR. In addition, MSE values in the SPR case are much better than those for the non SPR case. Improvement might be obtained if the simplification that bypasses the regressor filtering is not ignored. Experiment : In this experiment, it is shon that the proposed algorithm gives good results for different length channels as ell. Simulations are performed for the fourth (channel ) and the fifth order (channel ) channels. Results are demonstrated in Figure 4. Note that even for SNR values as lo as 0 db the adaptive AR algorithm provides good results for both cases. Hoever, for lo SNR values the

4 6th European Signal Processing Conference (EUSIPCO 008), Lausanne, Sitzerland, August 5-9, 008, copyright by EURASIP able I- Identification results for SPR and non-spr channels of lenght 3. he true channel coefficients for SPR channel are c[0] =, c[] =0.75, c[] =0.356 and for those non-spr channel are c[0] =, c[] = 0.85, c[] =-0.5. SNR(dB) SPR channel Non-SPR channel ĉ [] ĉ [] ĉ [] ĉ [] able- II- Effect of adaptive AR filter order on the accuracy of the estimation of the channel coefficients. he true channel coefficients are c[0] =, c[] =0.75, c[] = AR filter order ĉ [] ĉ [] ĉ [3] ĉ [4] ĉ [5] MSE e e e e e 0.995e e e e e- method gives comparable results for the to channels hile as the SNR increases the performance difference beteen the fourth order and the fifth order channel cases increases. Experiment 3: Effect of adaptive AR filter order on the accuracy of the estimation of the channel coefficients is investigated in this experiment. For this purpose, a third order channel is used. By assuming that the order of the channel is not non a priori, the order of the adaptive AR filter is varied to see its effect on the identification. he estimated channel coefficients are shon in able-ii for different orders for the AR filter. It is clear from able-ii that the first to AR filter coefficients are close to the true channel coefficients and all extra coefficients tae small values that can be ignored. In other ords, the proposed method can also be used to estimate the order of the channel if it is not non. Before concluding this section it is important to mention that in all simulations MSE as computed by MSE= [(ĉi[] ci[]) + (ĉ [] c []) here is the number of iterations used in the adaptive algorithm until convergence taes place. Note that this definition of MSE applies to the length to channels, though it is i i ] easy to extend the definition to orders bigger than to. MSE is the most frequently used measure in order to compare the performance of different channel equalization algorithms hen the channel is non. When the channel is not non MSE does not mae sense, other performance measures must be considered. 5. CONCLUSIONS In this study, an adaptive AR algorithm as developed for blind equalization of FIR channels in chaotic communications. his method exploits the prior noledge about nonlinear dynamics of the chaotic signal to estimate the transmitted chaotic signal. One limitation of the adaptive AR filter is that it requires a recursion ithin a recursion, hich maes its implementation costly. A simplified algorithm to remove the costly inner recursion as proposed, though a rigorous proof on hen this can be done as omitted because of space limitation. he simplified algorithm gives better results for SPR channels than non-spr channels and it may not even or non-spr channels. Hoever, it is easy to implement and provides dramatic improvement on MSE compared to the existing methods.

5 6th European Signal Processing Conference (EUSIPCO 008), Lausanne, Sitzerland, August 5-9, 008, copyright by EURASIP Figure 3- MSE versus SNR for the SPR and the non-spr channels. Figure 4- MSE versus SNR for to channels ith different lengths. REFERENCES [] Abel, A. and Scharz, W. Chaos communicationsprinciples, schemes and system analysis Proceedings of IEEE, Vol. 90, No.5, pp.69-70, May 00 [] P. Stavroulais, Chaos Applications in elecommunications, aylor & Francis, 006 [3] Leung, H. System identification using chaos ith application to equalization of a chaotic modulation system, IEEE rans. on Circuits and Systems, Vol. 45 No.3, March 998. [4] Zhu, Z. and Leung, H. Identification of linear systems driven by chaotic signals using nonlinear prediction, IEEE rans. on Circuits and Systems, Vol. 49, No., pp.70-80, Feb.00 [5] Zhu, Z. and Leung, H. Adaptive blind equalization for chaotic communications systems using Extended Kalman filter, IEEE rans on Circuits and Systems, Vol.48, No.8, pp , Aug. 00 [6] Johnson, Jr. CR. A convergence proof for a hyper stable adaptive recursive filter, IEEE rans. on Information heory I-5, no.6,pp , Nov [7] reichler, JR, Johnson Jr. CR, Laimore MG, Anderson BDO. SHARF convergence properties IEEE rans. on Circuits and System, Vol.8, No.6, pp , June 98. [8] S. Hayin, Adaptive Filter heory, Fourth Edition, Prentice Hall, 00 [9] Wang, B. Y. and Zheng, W.X. Blind adaptive channel identification/equalization in chaotic communications by using nonlinear prediction technique, ICSP 04 Proceedings, pp ,004