Blind Identification of FIR Systems and Deconvolution of White Input Sequences U. SOVERINI, P. CASTALDI, R. DIVERSI and R. GUIDORZI Dipartimento di Elettronica, Informatica e Sistemistica Università di Bologna Viale del Risorgimento 2, 40136 Bologna ITALY Abstract: This paper deals with the problem of blind identification and deconvolution of FIR channels driven by a white input sequence with unknown variance, in an unbalanced noise environment. By using the structural properties of the covariance matrix of the input, an estimate of the channel coefficients is obtained. The subsequent deconvolution of the unknown input signal is then performed by means of a minimum variance equalizer. The performance of the proposed algorithm is tested by means of Monte Carlo simulations. Key-Words: Blind identification, blind equalization, multichannel FIR systems, white input sequences. 1 Introduction The problem of identifying transmission channels whose input is unknown has remarkable relevance in the communication field. The knowledge of channel models is, in fact, at the basis of blind deconvolution and equalization procedures that, particularly in mobile applications, overcome the bandwith limitation due to intersymbol interference caused by time varying multipaths 1, 2]. Some of the procedures proposed in the literature to solve this problem are based on higher order statistics since it is well known that the second order stationary statistics of a scalar signal do not contain sufficient information for identifying non minimum phase systems 2]. The associated algorithms, however, provide satisfactory results only if a large number of data is available. More recently, techniques based on second order statistics have been proposed by taking advantage of the cyclostationarity of the signals in an oversampling context 3] 10]. Most of the previous methods are based on the assumption that the unknown input signal is a white process with known variance 3, 5, 7]. This paper proposes an approach to blind identification and deconvolution that can be applied to multichannel environments and assumes a zero mean white input with unknown variance. Moreover, this method allows to deal with output measurements affected by different amounts of noise on the channels; this feature has been introduced in the literature only recently 9, 10]. In this paper only the case of two channels is considered; the extension to an arbitrary number of channels will be described in the future. An advantage of this method consists in its high numerical efficiency, as it is shown by the results obtained in Monte Carlo simulations. The content of the paper is organized as follows. Section 2 introduces the mathematical notation and defines the identification and deconvolution problems. In Section 3 the blind identification problem is solved in the case of two FIR channels while Section 4 treats the subsequent blind deconvolution phase. The results of the performed Monte Carlo simulations are reported in Section 5 and short concluding remarks are finally given in Section 6. 2 Statement of the problem Consider a two channel FIR system whose outputs are linked to the scalar input u(t) by the convolution model L x i (t) = h i (k) u(t k), i = 1, 2 (1) k=0 h 1 (k), h 2 (k) (k = 0,,L) are causal im-
pulse responses with finite support L. The observations of x 1 (t), x 2 (t) are affected by additive noises, so that only the noisy measurements y i (t) = x i (t) + n i (t) i = 1, 2 (2) are available (see Figure 1). It is well known that u(t) h 1 (z) n 1 (t) x 1 (t) y 1 (t) + It is worth pointing out that hypothesis 3) assures the system identifiability 11]. Under these assumptions, the blind identification and equalization problem can be stated as follows. Problem 1 GivenN samples of the noisy outputs y 1 (t), y 2 (t), estimate the coefficients h i (k) (k = 0,...,L)(i = 1, 2) and the input variance σu 2 (blind identification). Determine also the unknown input sequence u( ) (blind deconvolution). Model (1) (2) can be rewritten by using the following matrix notation y(t) = x(t) + n(t) = H L u(t) + n(t) t 0, (4) h 2 (z) x 2 (t) y 2 (t) + n 2 (t) Figure 1: Structure of the considered system model (1) (2) can be adopted to describe either twochannel data acquisition systems or a single oversampled channel. The use of FIR models is very common in practical applications since they can approximate an infinite impulse response (IIR) system by assuming L large enough. If the order L is too large, IIR models would be preferable because fewer parameters are necessary. Nevertheless, FIR models lead to simpler implementations of signal processing algorithms. With reference to model (1) (2) we will introduce the following assumptions: 1) the input u(t) is a zero mean white process with unknown variance σ 2 u ; 2) the additive noises n i (t) (i = 1, 2) are zero mean white noises with unknown variances σ1 2,σ2 2, mutually uncorrelated and uncorrelated with the unknown input u(t); 3) the transfer functions of the channels h i (z) = h i (0) + h i (1)z 1 + +h i (L) z L (i = 1, 2), (3) z denotes the unitary advance operator, do not share common zeros. y(t) = y 1 (t),...,y 1 (t + L), y 2 (t),...,y 2 (t + L) ] T (5) x(t) = x 1 (t),...,x 1 (t + L), x 2 (t),...,x 2 (t + L) ] T (6) u(t) = u(t L),...,u(t + L) ] T (7) n(t) = n 1 (t),...,n 1 (t + L), n 2 (t),...,n 2 (t + L) ] T (8) and H L is the following generalized Sylvester matrix with dimension 2(L + 1) (2L + 1) (1) ] H L H L = H (2), (9) L H (i) L (i = 1, 2) is the (L + 1) (2L + 1) Sylvester matrix of the i-th channel h i (L) h i (0) 0 H (i) L =.......... (10) 0 h i (L) h i (0) 3 Blind identification From the structural properties of H L it is easy to verify that HL H c L = 0, (11) c L = c2 T ] T, ct 1 (12) = h 2 (L),...,h 2 (0), h 1 (L),..., h 1 (0) ] T
and ( ) H denotes the conjugate transpose operator. Consider now the 2(L + 1) 2(L + 1) covariance matrix x of x(t) x x(t) = E x H ] (t) = H L u HH L, (13) u u(t) = E u H ] (t) = σu 2 I 2L+1 (14) is the covariance matrix of the unknown input. When assumption 3) is satisfied, x has rank 2L + 1 and from (11) we obtain x c L = 0. (15) This relation allows to determine c L (up to a scalar factor) if x is known. In the noisy case, it is possible to verify that, because of assumption 2), the covariance matrix y of y(t) is given by y = E y(t)y H ] (t) = x + n, (16) n n(t)n = E H ] (t) = diag σ1 2 I L+1, σ2 2 I ] L+1 (17) is the unknown covariance matrix of the noise vector n(t). Since matrix y is positive definite, the coefficient vector c L can no longer be extracted from a relation of type (15). In this case, the channel coefficients can be obtained as follows. Let us consider the family of non negative definite diagonal matrices n = diag σ1 2 I L+1, σ2 2 I L+1] with unknown σ1 2, σ 2 2 such that x = y n 0. (18) With reference to this family, the following result has been proved 12]. Theorem 1 - The set of all matrices n satisfying relation (18) is defined by the points of a convex curve S( y ), belonging to the first quadrant of the noise plane (σ1 2,σ2 2 ). Every point P = (σ 1 2,σ2 2 ) of the curve defines the FIR models c 1 (P ), c 2 (P ) given by the relation: x (P ) ] c2 (P ) = 0, (19) c 1 (P ) x (P ) = y diag σ 2 1 I L+1, σ 2 2 I L+1 ] 0. (20) By taking into account (13), it is also possible, for every P, to determine, the matrix u (P ) = H L (P ) + x (P ) H L (P ) H +, (21) H L (P ) is constructed with the coefficients obtained from (19) and ( ) + denotes pseudoinversion. Note that, among all admissible solutions, only that corresponding to the point P = (σ1 2,σ2 2 ), is associated with the matrix x and thus with the true coefficient vector c L. This implies that only P defines, through (21), a diagonal matrix u (P ) (equal to u up to a scalar factor). By using these properties it is now possible to introduce the cost function J(P) = u (P ) 2L+1 det ( u (P )) I 2L+1, (22) F F denotes the Frobenius norm and determine univocally the point P from the condition J(P ) = 0. The search for P along S( y ) can be performed by using the following result 13]. Theorem 2 - Let ξ = (ξ 1,ξ 2 ) be a generic point of the first quadrant and r the straight line from the origin through ξ. Its intersection with S( y ) is given by the point P = (σ 2 1,σ2 2 ) with σ 2 i = ξ i λ M, (23) λ M = max eig y 1 diag σ1 2 I L+1,σ2 2 I L+1] ]. (24) On the basis of previous considerations, the following algorithm can be devised for solving Problem 1. Algorithm 1 1) Estimate y by means of an average operation ˆ y = 1 N L 1 y(t)y H (t). (25) N L t=0 2) Start from a generic point P on the curve S( ˆ y ) and compute the model c L (P ) = c T 2 (P ), ct 1 (P )]T by means of (19). 3) Construct the matrix H L (P ) and compute, from (21), the matrix u (P ).
4) Compute the value of the cost function (22). 5) Use a search procedure on the curve S( ˆ y ) in order to obtain the point P = ( σ 1 2, σ 2 2) associated with the minimum of the cost function (22). 6) Compute an estimate σ u 2 relation σ 2 u = trace of σ 2 u ( ) u ( P) 2L + 1 4 Blind deconvolution by means of the (26) When the whole set of available samples is considered in relation (4), the two FIR channel model can be rewritten as y = Hu+ n, (27) y = y 1 (0),...,y 1 (N 1), y 2 (0),...,y 2 (N 1) ] T (28) x = x 1 (0),...,x 1 (N 1), x 2 (0),...,x 2 (N 1) ] T (29) u = u( L),...,u(N 1) ] T (30) n = n 1 (0),...,n 1 (N 1), n 2 (0),...,n 2 (N 1) ] T, (31) and H isa2n (N + L) Sylvester matrix of type (9). The minimun variance estimate of the vector u that can be obtained from the observation vector y is given by the mean value of u conditioned by y u MV = E u y ]. (32) Under the assumption of gaussianity for the random variables u(t), n 1 (t), n 2 (t),wehave u MV = E uy H ] E yy H ] y = σ 2 u HH σu 2 HHH + ] y, (33) σ 2 = 1 I ] N 0 0 σ2 2 I. (34) N When this assumption is not satisfied, expression (33) constitutes, however, the best linear estimate of u that can be obtained from y. Once the noise variances and the channel coefficients have been estimated using Algorithm 1, it is possible to reconstruct the unknown input sequence u( ) from (33) by replacing H,σu 2,σ2 1,σ2 2 with their estimated values: ū = σ u 2 H( P) H σ u 2 H( P)H( P) H ] + ( P) y, (35) ( P) σ 2 ] = 1 I N 0 0 σ 2 2 I. (36) N 5 Experimental results The efficiency of the proposed procedure has been tested by means of Monte Carlo simulations. 100 symbols of a BPSK source have been used as input for the two FIR channels defined by the the transfer functions: h 1 (z) = 1.1836 z 5 + 0.4906 z 4 0.3093 z 3 + 0.4011 z 2 + 0.1269 z 1.8522 h 2 (z) = 0.8221 z 5 + 0.0333 z 4 + 0.2162 z 3 0.0165 z 2 + 0.2531 z + 0.5591. The transfer function h 2 (z) has been scaled in order to set the ratio of the standard deviations of x 1 and x 2 at 7 db so that the same signal to noise ratio (SNR) corresponds to an unbalance of 7 db on the actual amounts of noise on the channels. Note that the considered system is nonminimum phase. The output sequences x 1 ( ) and x 2 ( ) have been corrupted by adding white noises n 1 (t) and n 2 (t) corresponding to SNR ranging from 5 db to 30 db. For each value of the SNR a Monte Carlo simulation of 100 indipendent trials has been performed. The normalized root mean square error (NRMSE) NRMSE = 1 c L 1 R ĉ R L i c L 2 (37) i=1 has been employed as a measure of the performance of the channel estimation. R denotes the number of runs while ĉl i is the estimate, in the the i-th trial, of the coefficient vector c L = c2 T ] T ct 1. Figure 2 reports the obtained NRMSE versus the SNR. As a deconvolution performance index the biterror-rate (BER) has then been considered. The BER is defined as the frequency of error averaged over 100
NRMSE 0.8 0.7 0.6 0.5 0.4 0.3 by means of a minimum variance equalizer. The results of Monte Carlo simulations have confirmed the good performance of the whole procedure also when the data are characterized by poor signal to noise ratios and a small number of samples is available. This performance is remarkably superior to that of existing procedures when the additive noises on the channels are unbalanced. BER 0.2 0.1 0 5 10 15 20 25 30 SNR (db) 0.35 0.3 0.25 0.2 0.15 0.1 0.05 Figure 2: NRMSE versus SNR 0 5 10 15 20 25 30 SNR (db) Figure 3: BER versus SNR Monte Carlo runs. Figure 3 reports the BER versus the SNR. The obtained results show good estimates of the FIRs coefficients even with low signal to noise ratios. 6 Conclusions In this work a solution to the blind identification of multiple FIR channels from noisy output measurements has been presented. The proposed method allows to deal with output measurements affected by different amounts of noise on the channels. The associated procedure relies on the structural properties of the covariance matrix of the input and does not require any knowledge of its variance. The deconvolution of the input signal is then performed References 1] L. Tong and S. Perreau, Multichannel blind identification: from subspace to maximum likelihood methods, Proceedings of the IEEE, Vol. 86, No. 10, 1998, pp. 1951 1968. 2] N. Kalouptsidis, Signal processing systems: theory and design, John Wiley & Sons, 1997. 3] L. Tong, G. Xu and T. Kailath, Blind identification and equalization based on secondorder statistics: a time domain approach, IEEE Transactions on Information Theory, Vol. 40, No. 2, 1994, pp. 340 349. 4] E. Moulines, P. Duhamel, J.F. Cardoso and S. Mayrargue, Subspace methods for the blind identification of multichannel FIR filters, IEEE Transactions on Signal Processing, Vol. 43, No. 2, 1995, pp. 516 525. 5] L. Tong, Blind sequence estimation, IEEE Transactions on Communications, Vol. 43, No. 12, 1995, pp. 2986 2994. 6] G. Xu, H. Liu, L. Tong and T. Kailath, A least squares approach to blind channel identification, IEEE Transactions on signal processing, Vol. 43, No. 12, 1995, pp. 2982 2993. 7] K. Abed-Meraim, E. Moulines and P. Loubaton, Prediction error method for second-order blind identification, IEEE Transactions on signal processing, Vol. 45, No. 3, 1997, pp. 694 705. 8] M.I. Gürelli, C.L. Nikias, EVAM: an eigenvector based algorithm for multichannel blind deconvolution of input colored signals, IEEE Transactions on signal processing, Vol. 43, No. 1, 1995, pp. 134 149.
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