On-line Blind Equalization via On-line Blind ... - Semantic Scholar
PUBLISHED IN SIGNAL PROCESSING 68(1998),PP.271-281
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On-line Blind Equalization via On-line Blind Separation
Howard Hua YANG Department of Computer Science and Engineering Oregon Graduate Institute PO Box 91000, Portland, OR 97291-1000, USA
November 16, 1998
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PUBLISHED IN SIGNAL PROCESSING 68(1998),PP.271-281
Abstract A blind equalization method based on the theory of independent component analysis is presented. The blind equalization of an unknown FIR channel with possibly non-minimum phase is formulated as a blind separation problem. The inputs to the equalizer are formed by stacking the fractional samples of the channel outputs. New on-line blind equalization algorithms are developed from on-line blind separation algorithms, thus they inherit the equivariant property from the equivariant blind separation algorithms. Due to this property, the performance of the new algorithms is independent of the channel parameters. Therefore, they are useful for equalizing some ill-conditioned channels. It is shown by simulations that the proposed algorithm works well when the input sequence has some degree of correlation.
blind equalization, blind separation, on-line algorithm, fractional sample, equivariant property, condition number Running head: Blind Equalization via Blind Separation
Keywords:
I. Introduction
The constant modulus (CMA) and Bussgang algorithms are typical blind equalization algorithms for SISO FIR channels[12]. The Bussgang-type blind clustering algorithms [8], [15] are also for SISO channels. Other blind deconvolution algorithms based on high-order statistics (HOS) are reviewed in [6]. These algorithms can be easily implemented by on-line algorithms to equalize SISO FIR channels. However, these algorithms are only applicable for equalizing invertible SISO FIR channels. They are not applicable to the deconvolution of an SIMO linear time-invariant system in which each FIR sub-channel may not be invertible. For the SIMO system or an FIR channel with fractionally sampled channel outputs, several blind identi cation methods such as those in [13], [17], [18], [19] have been proposed based on second-order statistics. In contrast to the HOS methods, these second-order methods assume less priori information about transmitted signals and generally need shorter observations to identify the system. The source signals can be recovered via the Moore-Penrose inverse after the system is identi ed. But these methods are batch type algorithms and require matrix decompositions in the implementations. The system identi cation and equalization processes are two pipeline operations. The equalization can only be started after the channel is identi ed. For some ill-conditioned systems, the pseudo-inverse fails due to numerical errors even if the system identi cation is perfect. Compared to the algorithms in [18], [19], the algorithms in [11], [16] also use only the second order statistics but they are designed for the blind deconvolution of MIMO FIR channels which is a more challenging problem than the deconvolution of SIMO FIR channels. The approaches in [11], [16] assume that the number of sensors is greater than the number of sources. Under this assumption, an MIMO FIR channel can be identi ed up to an instantaneous mixing matrix. After the channel is identi ed, blind separation algorithms can be used to extract sources. Based on the general theory of independent component analysis (ICA) [9], several blind separation algorithms have been proposed and analyzed in [2], [5], [7], [21] for example. The blind separation algorithms in [2], [7] have the equivariant property [7]. The performance of these algorithms is equivariant DRAFT
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to the unknown mixing matrix. The robust performance of these algorithms is needed in the approaches in [11], [16] to equalize ill-conditioned channels. The equivariant blind separation algorithms were rst derived in [7] based on the relative gradient approach. These algorithms were also derived in [2] based on the natural gradient approach. Similar to the approaches in [18], [19], the approaches in [11], [16] also have two pipeline operations. The performance of the algorithms based on these approaches is not equivariant to the unknown channel parameters. One equivariant algorithm for blind equalization is proposed in [23] for equalizing SIMO FIR channels. Recently, an equivariant algorithm for equalizing MIMO FIR channels is derived in [3], [4] using an approach di erent from that in [23]. In this paper, the blind equalization of an SIMO FIR system is reformulated as a blind separation problem so that the blind separation algorithms can be applied for blind equalization [22], [23]. Di erent from the approaches in [11], [16], [18], [19], the approach in this paper gives the on-line blind equalization algorithms which do not need the channel identi cation stage. This approach is more e ective for equalizing ill-conditioned SIMO systems because of the equivariant property inherited from the blind separation algorithms. The rest of the paper is organized as follows. The continuous-time and discrete-time channel models are de ned in section II. The blind equalization algorithms based on blind separation and least-squares approaches are formulated in section III-A and section III-B respectively. Simulations and some remarks on the conditional numbers of the channels are given in section IV. Finally, conclusions are made in section V. II. Channel Models
Consider the following time invariant channel de ned in [18]:
x(t) =
X kT t
s(k)h(t ? kT ) + n(t)
(1)
where fs(k)g is an input sequence, T the symbol interval, n() the additive noise, and h() the channel impulse response function. Assume h(t) = 0 if t 2= [0; LT ]. The time interval [0; LT ] is called the duration of the continuous FIR. The system (1) is an SISO channel with a discrete-time input and a continuoustime output. The problem of the blind equalization is to recover the input sequence from the history of the channel output x(t). When the sampling rate is M times faster than the baud rate, the model (1) becomes an SIMO system:
xm (k) =
L X l=0
hm (l)s(k ? l) + nm (k); k = 1; ; N;
(2)
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where xm (k) is the output of the m-th channel, s(k) the common input to M channels, N the data length, nm (k) a zero mean Gaussian noise, and for each m the parameters
m T ); l = 0; ; L; hm(l) = h(lT + M
are the impulse responses of the m-th channel. Now the problem is to recover the input from the outputs of the SIMO system (2). III. Algorithms
A. Blind separation for blind equalization (BSBE)
Assume a linear mixture model:
(t) = As(t)
x
where A is an unknown n n non-singular mixing matrix, s(t) = [s1 (t); ; sn (t)]T the vector of unknown independent sources and x(t) = [x1 (t); ; xn (t)]T the vector of mixtures. In order to recover the original sources from the mixtures, we use a linear transform y
(t) = W x(t):
The following learning algorithm (LA) is used to update the weight matrix W :
W k+1 = (1 + )W k ? (yk )yTk W k (3) (y) = (f (y1 ); ; f (yn ))T . This algorithm has been derived in [2] by f
where is a learning rate and f minimizing the mutual information of the outputs based on the natural gradient descent method. The algorithm of this type can also be derived by other approaches such as info-max[5] and maximum likelihood. Di erent approaches may give di erent function forms for f () such as instantaneous functions in [1] and adaptive ones in [20]. To apply the algorithm (3) to the blind equalization problem, we write the system (2) in a vector form:
xk = H 1sk + nk
(4)
where H 1 = [ hL hL?1 h0 ], hl = (h1 (l); ; hM (l))T , l = 0; 1; ; L, xk = (x1 (k); ; xM (k))T ; sk = (s(k ? L); ; s(k))T ; nk = (n1(k); ; nM (k))T :
De ne a p (p + L) block matrix H p as: 3 2 hL hL?1 h0 0 0 7 66 66 0 hL hL?1 h0 0 777 Hp = 6 64 775 hL?1 h0 0 0 hL
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where each block in H p is an M 1 vector. In this paper, the parameter p is called as a stacking level. Fixing p and stacking p observation vectors x1; ; xp , from (4) we form a joint observation vector 2 2 3 3 2 3 x s (1 ? L ) n 1 1 6 6 7 7 6 7 u1 = 664 ... 775 = H p 664 ... 775 + 664 ... 775 : (5) xp s(p) np The next joint observation vector u2 is formed by removing x1 in u1 and appending xp+1 to u1 . Generally, stacking p observation vectors xk ; ; xp+k?1 , we obtain
uk = H p sk + nk
(6)
where uk = (xTk ; ; xTp+k?1 )T ; sk = (s(k ? L); ; s(p + k ? 1))T ; nk = (nTk ; ; nTp+k?1 )T : Note H p is of pM (p + L). It is a square matrix when p = L and M = 2. In this paper, we assume p ML?1 , H p has a full column rank, and the input is a non-Gaussian iid sequence. These are conditions for using the BSBE. The block diagram of the BSBE is shown in Fig. 1 in which W k and V k are of (p + L) (p + L) and (p + L) pM , respectively. When p > ML?1 , the output dimension pM of the system (6) is greater than its input dimension p + L. The matrix V k is used to whiten the output and at the same time to reduce the dimensionality of the joint observation vector from pM to p + L. V k is updated by the following whitening algorithm (WA): V k+1 = (1 + )V k ? vk vTk V k (7) where is a learning rate and vk = V k uk . Considering the following system
vk = V k H p sk + V k nk
(8)
as a mixture model with p + L independent sources, we apply the equivariant blind separation algorithms such as the algorithm (3) to update the weight matrix W k and recover the source vector by a linear transform sfk = W k vk = W k V k uk : Since both sk and H p are unknown, we cannot obtain the exact inverse of H p . However, we can use any blind separation algorithm to obtain
W 1 = DP (V 1H p )?1 November 16, 1998
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where D is a diagonal matrix with non-zero diagonal elements and P is a permutation matrix. Asymptotically, we have
sfk = W 1V 1 uk = DP (V 1 H p)?1 V 1 H psk + W 1V 1nk = DPsk + W 1V 1nk : Therefore, we obtain a possibly delayed and scaled input in each dimension of the output vector sfk . Since the permutation matrix P and the scaling matrix D cannot be determined by the blind separation algorithm, we cannot use sfk as an estimate for sk . Nevertheless, each dimension of sfk is a possibly delayed and scaled input. We can x any dimension of sfk as the output of the BSBE equalizer. Since the BSBE inherited the equivariant property from the equivariant blind separation algorithms, the performance of the BSBE is independent of the parameters in H p . For this reason, the BSBE is very useful for equalizing ill-conditioned channels. B. Least-squares approach for blind equalization (LSBE)
Applying the least-squares approach in [19], we estimate the channel impulse responses. Since H p is uniquely determined up to a constant factor by the least-squares approach, we can use the Moore-Penrose inverse to estimate the input dp )+ uk : sck = (H (9) Here ()+ denotes the Moore-Penrose inverse. Since sck may di er from sk only by an unknown constant factor, the whole vector sk is recovered up to this constant factor. This makes the LSBE equalizer more attractive than the BSBE equalizer. However, the LSBE is not equivariant to the parameters in H p . The sensitivity of the solution (9) is measured by the condition number [10] of H p , denoted by (H p ), which is de ned as the ratio of the largest and the smallest singular values of H p . When (H p ) is large, the solution to the linear equation (6) is not reliable. Therefore, even if the identi cation of the channel parameters in H p is perfect, the LSBE may fail to equalize some ill-conditioned channels. On the contrary, when (H p ) is large the BSBE still works as long as H p has a full column rank. For this reason, the BSBE performs better than the LSBE for equalizing some ill-conditioned channels. C. Stacking level p
In practice, we need to select the stacking level p for both the BSBE and the LSBE. It is shown by simulation that when p increases the condition number function c(p) = (H p ) rst decreases and then becomes saturated around some level. Since the BSBE is not sensitive to the condition of H p , for the BSBE we choose the smallest admissible stacking level
p = d M L? 1 e
where dxe denotes the smallest integer which is not small than x. DRAFT
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For the LSBE, we choose
p = maxfp : p L=(M ? 1); (H p ) ? 1 ? " > 0g where
1 = maxf(H p ) : p L=(M ? 1)g
and " > 0 is a tolerance number. Note (H p ) 1 . For some ill-conditioned channels, the LSBE cannot be improved by choosing p because 1 's for these channels are too large. IV. Simulation
A. Case 1: A two-channel model
Let us consider the blind equalization of a single-input and two-output system with a vector impulse responses 2 3 h ( l ) 0 5 ; l = 0; ; L; h(l) = 4 (10) h1 (l) where L = 11, fh0 (l)g =[?0:0166 0:0189 0:0226 ? 0:0354 ? 0:0231 0:1799 0:4339 0:4596 0:2185 ? 0:0248 ? 0:0639 0:0150], fh1 (l)g =[?0:0008 0:0300 ? 0:0036 ? 0:0497 0:0562 0:3193 0:4868 0:3599 0:0779 ? 0:0708 ? 0:0260 0:0378]. In this case, M = 2 and L = 11. The distributions of the zeros of the two channels are shown in Fig. 2. Both channels are non-minimum phase systems with several zeros almost on the unit circle. When p = L, the condition number (H p ) = 2:88 104. We test the BSBE algorithm by choosing the simplest function f (y) = y3 among all other options for f (y). The noise in the observation is a Gaussian noise with zero mean and variance 2 . When = 0:0005, after using 2000 symbols to equalize the two-channel model, we obtain the outputs of the two algorithms and their ISI errors illustrated in Fig. 3. After the channel is equalized by the BSBE and the LSBE respectively, 104 symbols are transmitted to count bit-error-rate (BER). We obtain BER=0.03% for the BSBE and BER=23% for the LSBE. It is clearly shown in Fig. 3 that the BSBE performs better than the LSBE for the two-channel model with the vector impulse responses given by (10). The LSBE fails to equalize the two-channel model. The stacking level p is a control parameter. However, the simulation shows that we cannot improve the performances of the BSBE and the LSBE by increasing p. To justify the BSBE, we have assumed that the input sequence is non-Gaussian and iid. But the iid assumption may not hold in practice. A correlated binary input sequence was constructed in [14] to test a source correlation compensation algorithm for blind channel identi cation. We use this sequence to check November 16, 1998
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whether the BSBE may tolerate the correlated input to some degree. The sequence is constructed from an iid sequence su (k) uniformly distributed on [?0:5; 0:5]. De ne 8 < 1; if su (k) + su (k ? 1) > 0 s(k) = : ?1; otherwise. where j j 1 is a constant. It is easy to see that p(s(k) = 1) = p(su (k) + su (k ? 1) > 0) = 0:5 for any . So the sequence s(k) has zero mean. It can be shown that the sequence s(k) has the following autocorrelation sequence: 8 > > for = 0 > 1; < rs ( ) = > =2 ? 3 =6; for = 1 > > : 0; otherwise. The one step correlation of the input sequence s(k) is de ned as = =2 ? 3 =6. For example, = 0:3321 and 0:3333 when = 0:95 and 1, respectively. Taking s(k) as the common input to the two channels and applying the BSBE, we obtain the plots in Figure 4 which shows that the BSBE works well when = 0:3321 but breaks down when = 0:3333. This means that the iid assumption for the input may be relaxed to some degree in practice. B. Case 2:A four-channel model
In this case, we consider a single-input and four-output system. Fig. 5 shows the distributions of the zeros of the four channels. All the sub-channels are non-minimum phase systems of order L = 5. When p = L, the condition number (H p ) = 77:79 is much lower than that in Case 1. The outputs of the BSBE and the LSBE and their ISI errors are plotted in Fig. 6 which shows that both algorithms are successful. In this case, the LSBE has a better performance with much lower ISI than the BSBE. C. Remarks
The condition number (H p ) is an objective quantity to measure the condition of SIMO systems. For a linear system Ax = b, the condition number (A) is related to the perturbation bound[10]. Let x and y be the solution of the linear equation Ax = b and the solution of the perturbated equation (A + A)y = b + b respectively with kAk kAk, kbk kbk, > 0 and (A) < 1. Then the Theorem 2.7.2 in [10] gives the perturbation bound for the linear system Ax = b:
kx ? yk 2(A) : kxk 1 ? (A)
In Fig. 7, the condition numbers (H p ) in Case 1 are compared with those in Case 2 for di erent stacking levels. The legitimate range for the stacking levels in Case 1 is di erent from that in Case 2. But it is still meaningful to compare the condition numbers in the two cases because the perturbation bound does not depend on the dimensionality of the linear system. DRAFT
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The BSBE is successful in both cases. The performances of the BSBE in the two cases are illustrated by Fig. 3(b) and Fig. 6(b). Due to the equivariant property, the BSBE performs roughly the same in the two cases. Because the noise added at the channel output becomes the noise in the mixture model (8), the equivariant property does not hold completely. So the BSBE has slightly di erent performances in the two cases. On the other hand, it is illustrated by Fig. 3(c) and (d) that the LSBE fails in Case 1 because the condition numbers shown by solid line in Fig. 7 are too large. Nevertheless, the LSBE works very well in Case 2 and achieves a much smaller ISI error than the BSBE. V. Conclusion
The on-line blind equalization algorithm is proposed for equalizing an SIMO system and an SISO system with fractionally sampled channel outputs. Based on the on-line blind separation algorithms, the BSBE performs the equalization directly without the channel identi cation stage. For ill-conditioned channels, it achieves the equalization with better quality than the batch type least-squares algorithm LSBE does. Since the BSBE inherits the equivariant property from the equivariant blind separation algorithms, it is very robust in equalizing the ill-conditioned channels. But when the channel condition is moderately good, the performance of the batch algorithm LSBE is better than that of the on-line algorithm BSBE. To justify the BSBE, we have assumed that the input sequence is non-Gaussian and iid. But, the iid assumption is not crucial for the BSBE in practice. It is shown by the simulations that the BSBE allows certain degree of correlation in the input sequence. References [1] S. Amari, A. Cichocki, and H. H. Yang. Recurrent neural networks for blind separation of sources. In Proceedings of NOLTA 1995, volume I, pages 37{42, December 1995. [2] S. Amari, A. Cichocki, and H. H. Yang. A new learning algorithm for blind signal separation. In Advances in Neural [3] [4] [5] [6] [7] [8] [9] [10]
Information Processing Systems, 8, eds. David S. Touretzky, Michael C. Mozer and Michael E. Hasselmo, MIT Press: Cambridge, MA., pages 757{763, 1996. S. Amari, S. Douglas, A. Cichocki, and H. H. Yang. Multichannel blind deconvolution and equalization using the natural gradient. In Proc. of IEEE SPAWC'97, pages 101{104, Paris, France, April 1997. S. Amari, S. Douglas, A. Cichocki, and H. H. Yang. Novel on-line adaptive learning algorithms for blind deconvolution using the natural gradient approach. In Proc. of SYSID'97 , 11th IFAC Symposium on System Identi cation, Fukuoka, Japan, July 1997. A. J. Bell and T. J. Sejnowski. An information-maximisation approach to blind separation and blind deconvolution. Neural Computation, 7:1129{1159, 1995. J. A. Cadzow. Blind deconvolution via cumulant extrema. IEEE SP MAGAZINE, 13(3):24{42, May 1996. J.-F. Cardoso and B. Laheld. Equivariant adaptive source separation. IEEE Trans. on Signal Processing, 44(12):3017{ 3030, December 1996. S. Chen, S. McLaughlin, P. M. Grant, and B. Mulgrew. Multi-stage blind clustering equaliser. IEEE Trans. on Communications, 43(2/3/4):701{705, February/March/April 1995. P. Comon. Independent component analysis, a new concept? Signal Processing, 36:287{314, 1994. G. H. Golub and C. F. Van Loan. Matrix Computation . The Johns Hopkins University Press, 1989.
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[11] A. Gorokhov, P. Loubaton, and E. Moulines. Second order blind equalization in multiple input multiple output r systems: a weighted least squares approach. In ICASSP-96, pages 2415{2418. IEEE, May 1996. [12] S. Haykin. Blind Deconvolution. Prentice-Hall, Inc., 1994. [13] Y. Hua. Fast maximum likelihood for blind identi cation of multiple r channels. IEEE Trans. on Signal Processing, 44(3):661{672, March 1996. [14] Y. Hua, H. Yang, and W. Qiu. Source Correlation Compensation for Blind Channel Identi cation Based on SecondOrder Statistics . IEEE Signal Processing Letters, 1(8):119{120, August 1994. [15] J. Karaoguz and S. H. Ardalan. A soft decision-directed blind equalization algorithm applied to equalization of mobile communication channels. In Proc. ICC'92, Vol.3 , pages 343.4.1{343.4.5, Chicago, 1992. [16] A. Mansour, C. Jutten, and P. Loubaton. Subspace method for blind separation of sources in convolutive mixture. In G. Ramponi et al, editor, Signal Processing VIII, Vol.III, pages 2081{2084. Trieste, Italy, 1996. [17] E. Moulines, P. Duhamel, J.-F. Cardoso, and S. Mayrargue. Subspace Methods for the Blind Identi cation of Multichannel FIR Filters . IEEE Trans. on Signal Processing, 43(2):516{525, February 1995. [18] L. Tong, G. Xu, and T. Kailath. Blind Identi cation and Equalization Based on Second-Order Statistics: A Time Domain Approach . IEEE Trans. on Information Theory, 40(2):340{349, March 1994. [19] G. Xu, H. Liu, L. Tong, and T. Kailath. A least-squares approach to blind channel identi cation. IEEE Trans. on Signal Processing, 43(12):2982{2993, December 1995. [20] H. H. Yang and S. Amari. Two Gradient Descent Algorithms for Blind Signal Separation. In Proceedings of ICANN96, The Lecture Notes in Computer Science Vol.1112, pp.287-292. Springer-Verlag, 1996. [21] H. H. Yang and S. Amari. Adaptive on-line learning algorithms for blind separation: Maximum entropy and minimum mutual information. Neural Computation, 9(7):1457{1482, 1997. [22] H. H. Yang and S. Amari. Blind equalization of switching channels based on ICA and learning of learning rate. In Proc. of IEEE ICASSP'97, Munich, Germany, pages 1849{1852, April 1997. [23] H. H. Yang and E.-S. Chng. An on-line learning algorithm for blind equalization. In Progress in Neural Information Processing: Proceedings of ICONIP*96, pages 317{321. Springer, September 1996.
List of gure captions: Figure 1: Figure 2: Figure 3: Figure 4: Figure 5: Figure 6: Figure 7: DRAFT
The block diagram of the BSBE. The zeros of the two channels. Comparing the BSBE and the LSBE in Case 1 with an iid input sequence. The performances of the BSBE in Case 1 with a correlated input sequence. The zeros of the four channels. Comparing the BSBE and the LSBE in Case 2 with an iid input sequence. Comparing the condition numbers in Case 1 with those in Case 2. November 16, 1998
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noise
? ? ? sk - H p - uk- V k vk- W k - sfk ? ?
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YANG:ON-LINE BLIND EQUALIZATION VIA ON-LINE BLIND SEPARATION
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On-line Blind Equalization via On-line Blind Separation
Howard Hua YANG Department of Computer Science and Engineering Oregon Graduate Institute PO Box 91000, Portland, OR 97291-1000, USA
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Abstract A blind equalization method based on the theory of independent component analysis is presented. The blind equalization of an unknown FIR channel with possibly non-minimum phase is formulated as a blind separation problem. The inputs to the equalizer are formed by stacking the fractional samples of the channel outputs. New on-line blind equalization algorithms are developed from on-line blind separation algorithms, thus they inherit the equivariant property from the equivariant blind separation algorithms. Due to this property, the performance of the new algorithms is independent of the channel parameters. Therefore, they are useful for equalizing some ill-conditioned channels. It is shown by simulations that the proposed algorithm works well when the input sequence has some degree of correlation.
blind equalization, blind separation, on-line algorithm, fractional sample, equivariant property, condition number Running head: Blind Equalization via Blind Separation
Keywords:
I. Introduction
The constant modulus (CMA) and Bussgang algorithms are typical blind equalization algorithms for SISO FIR channels[12]. The Bussgang-type blind clustering algorithms [8], [15] are also for SISO channels. Other blind deconvolution algorithms based on high-order statistics (HOS) are reviewed in [6]. These algorithms can be easily implemented by on-line algorithms to equalize SISO FIR channels. However, these algorithms are only applicable for equalizing invertible SISO FIR channels. They are not applicable to the deconvolution of an SIMO linear time-invariant system in which each FIR sub-channel may not be invertible. For the SIMO system or an FIR channel with fractionally sampled channel outputs, several blind identi cation methods such as those in [13], [17], [18], [19] have been proposed based on second-order statistics. In contrast to the HOS methods, these second-order methods assume less priori information about transmitted signals and generally need shorter observations to identify the system. The source signals can be recovered via the Moore-Penrose inverse after the system is identi ed. But these methods are batch type algorithms and require matrix decompositions in the implementations. The system identi cation and equalization processes are two pipeline operations. The equalization can only be started after the channel is identi ed. For some ill-conditioned systems, the pseudo-inverse fails due to numerical errors even if the system identi cation is perfect. Compared to the algorithms in [18], [19], the algorithms in [11], [16] also use only the second order statistics but they are designed for the blind deconvolution of MIMO FIR channels which is a more challenging problem than the deconvolution of SIMO FIR channels. The approaches in [11], [16] assume that the number of sensors is greater than the number of sources. Under this assumption, an MIMO FIR channel can be identi ed up to an instantaneous mixing matrix. After the channel is identi ed, blind separation algorithms can be used to extract sources. Based on the general theory of independent component analysis (ICA) [9], several blind separation algorithms have been proposed and analyzed in [2], [5], [7], [21] for example. The blind separation algorithms in [2], [7] have the equivariant property [7]. The performance of these algorithms is equivariant DRAFT
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to the unknown mixing matrix. The robust performance of these algorithms is needed in the approaches in [11], [16] to equalize ill-conditioned channels. The equivariant blind separation algorithms were rst derived in [7] based on the relative gradient approach. These algorithms were also derived in [2] based on the natural gradient approach. Similar to the approaches in [18], [19], the approaches in [11], [16] also have two pipeline operations. The performance of the algorithms based on these approaches is not equivariant to the unknown channel parameters. One equivariant algorithm for blind equalization is proposed in [23] for equalizing SIMO FIR channels. Recently, an equivariant algorithm for equalizing MIMO FIR channels is derived in [3], [4] using an approach di erent from that in [23]. In this paper, the blind equalization of an SIMO FIR system is reformulated as a blind separation problem so that the blind separation algorithms can be applied for blind equalization [22], [23]. Di erent from the approaches in [11], [16], [18], [19], the approach in this paper gives the on-line blind equalization algorithms which do not need the channel identi cation stage. This approach is more e ective for equalizing ill-conditioned SIMO systems because of the equivariant property inherited from the blind separation algorithms. The rest of the paper is organized as follows. The continuous-time and discrete-time channel models are de ned in section II. The blind equalization algorithms based on blind separation and least-squares approaches are formulated in section III-A and section III-B respectively. Simulations and some remarks on the conditional numbers of the channels are given in section IV. Finally, conclusions are made in section V. II. Channel Models
Consider the following time invariant channel de ned in [18]:
x(t) =
X kT t
s(k)h(t ? kT ) + n(t)
(1)
where fs(k)g is an input sequence, T the symbol interval, n() the additive noise, and h() the channel impulse response function. Assume h(t) = 0 if t 2= [0; LT ]. The time interval [0; LT ] is called the duration of the continuous FIR. The system (1) is an SISO channel with a discrete-time input and a continuoustime output. The problem of the blind equalization is to recover the input sequence from the history of the channel output x(t). When the sampling rate is M times faster than the baud rate, the model (1) becomes an SIMO system:
xm (k) =
L X l=0
hm (l)s(k ? l) + nm (k); k = 1; ; N;
(2)
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where xm (k) is the output of the m-th channel, s(k) the common input to M channels, N the data length, nm (k) a zero mean Gaussian noise, and for each m the parameters
m T ); l = 0; ; L; hm(l) = h(lT + M
are the impulse responses of the m-th channel. Now the problem is to recover the input from the outputs of the SIMO system (2). III. Algorithms
A. Blind separation for blind equalization (BSBE)
Assume a linear mixture model:
(t) = As(t)
x
where A is an unknown n n non-singular mixing matrix, s(t) = [s1 (t); ; sn (t)]T the vector of unknown independent sources and x(t) = [x1 (t); ; xn (t)]T the vector of mixtures. In order to recover the original sources from the mixtures, we use a linear transform y
(t) = W x(t):
The following learning algorithm (LA) is used to update the weight matrix W :
W k+1 = (1 + )W k ? (yk )yTk W k (3) (y) = (f (y1 ); ; f (yn ))T . This algorithm has been derived in [2] by f
where is a learning rate and f minimizing the mutual information of the outputs based on the natural gradient descent method. The algorithm of this type can also be derived by other approaches such as info-max[5] and maximum likelihood. Di erent approaches may give di erent function forms for f () such as instantaneous functions in [1] and adaptive ones in [20]. To apply the algorithm (3) to the blind equalization problem, we write the system (2) in a vector form:
xk = H 1sk + nk
(4)
where H 1 = [ hL hL?1 h0 ], hl = (h1 (l); ; hM (l))T , l = 0; 1; ; L, xk = (x1 (k); ; xM (k))T ; sk = (s(k ? L); ; s(k))T ; nk = (n1(k); ; nM (k))T :
De ne a p (p + L) block matrix H p as: 3 2 hL hL?1 h0 0 0 7 66 66 0 hL hL?1 h0 0 777 Hp = 6 64 775 hL?1 h0 0 0 hL
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where each block in H p is an M 1 vector. In this paper, the parameter p is called as a stacking level. Fixing p and stacking p observation vectors x1; ; xp , from (4) we form a joint observation vector 2 2 3 3 2 3 x s (1 ? L ) n 1 1 6 6 7 7 6 7 u1 = 664 ... 775 = H p 664 ... 775 + 664 ... 775 : (5) xp s(p) np The next joint observation vector u2 is formed by removing x1 in u1 and appending xp+1 to u1 . Generally, stacking p observation vectors xk ; ; xp+k?1 , we obtain
uk = H p sk + nk
(6)
where uk = (xTk ; ; xTp+k?1 )T ; sk = (s(k ? L); ; s(p + k ? 1))T ; nk = (nTk ; ; nTp+k?1 )T : Note H p is of pM (p + L). It is a square matrix when p = L and M = 2. In this paper, we assume p ML?1 , H p has a full column rank, and the input is a non-Gaussian iid sequence. These are conditions for using the BSBE. The block diagram of the BSBE is shown in Fig. 1 in which W k and V k are of (p + L) (p + L) and (p + L) pM , respectively. When p > ML?1 , the output dimension pM of the system (6) is greater than its input dimension p + L. The matrix V k is used to whiten the output and at the same time to reduce the dimensionality of the joint observation vector from pM to p + L. V k is updated by the following whitening algorithm (WA): V k+1 = (1 + )V k ? vk vTk V k (7) where is a learning rate and vk = V k uk . Considering the following system
vk = V k H p sk + V k nk
(8)
as a mixture model with p + L independent sources, we apply the equivariant blind separation algorithms such as the algorithm (3) to update the weight matrix W k and recover the source vector by a linear transform sfk = W k vk = W k V k uk : Since both sk and H p are unknown, we cannot obtain the exact inverse of H p . However, we can use any blind separation algorithm to obtain
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where D is a diagonal matrix with non-zero diagonal elements and P is a permutation matrix. Asymptotically, we have
sfk = W 1V 1 uk = DP (V 1 H p)?1 V 1 H psk + W 1V 1nk = DPsk + W 1V 1nk : Therefore, we obtain a possibly delayed and scaled input in each dimension of the output vector sfk . Since the permutation matrix P and the scaling matrix D cannot be determined by the blind separation algorithm, we cannot use sfk as an estimate for sk . Nevertheless, each dimension of sfk is a possibly delayed and scaled input. We can x any dimension of sfk as the output of the BSBE equalizer. Since the BSBE inherited the equivariant property from the equivariant blind separation algorithms, the performance of the BSBE is independent of the parameters in H p . For this reason, the BSBE is very useful for equalizing ill-conditioned channels. B. Least-squares approach for blind equalization (LSBE)
Applying the least-squares approach in [19], we estimate the channel impulse responses. Since H p is uniquely determined up to a constant factor by the least-squares approach, we can use the Moore-Penrose inverse to estimate the input dp )+ uk : sck = (H (9) Here ()+ denotes the Moore-Penrose inverse. Since sck may di er from sk only by an unknown constant factor, the whole vector sk is recovered up to this constant factor. This makes the LSBE equalizer more attractive than the BSBE equalizer. However, the LSBE is not equivariant to the parameters in H p . The sensitivity of the solution (9) is measured by the condition number [10] of H p , denoted by (H p ), which is de ned as the ratio of the largest and the smallest singular values of H p . When (H p ) is large, the solution to the linear equation (6) is not reliable. Therefore, even if the identi cation of the channel parameters in H p is perfect, the LSBE may fail to equalize some ill-conditioned channels. On the contrary, when (H p ) is large the BSBE still works as long as H p has a full column rank. For this reason, the BSBE performs better than the LSBE for equalizing some ill-conditioned channels. C. Stacking level p
In practice, we need to select the stacking level p for both the BSBE and the LSBE. It is shown by simulation that when p increases the condition number function c(p) = (H p ) rst decreases and then becomes saturated around some level. Since the BSBE is not sensitive to the condition of H p , for the BSBE we choose the smallest admissible stacking level
p = d M L? 1 e
where dxe denotes the smallest integer which is not small than x. DRAFT
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For the LSBE, we choose
p = maxfp : p L=(M ? 1); (H p ) ? 1 ? " > 0g where
1 = maxf(H p ) : p L=(M ? 1)g
and " > 0 is a tolerance number. Note (H p ) 1 . For some ill-conditioned channels, the LSBE cannot be improved by choosing p because 1 's for these channels are too large. IV. Simulation
A. Case 1: A two-channel model
Let us consider the blind equalization of a single-input and two-output system with a vector impulse responses 2 3 h ( l ) 0 5 ; l = 0; ; L; h(l) = 4 (10) h1 (l) where L = 11, fh0 (l)g =[?0:0166 0:0189 0:0226 ? 0:0354 ? 0:0231 0:1799 0:4339 0:4596 0:2185 ? 0:0248 ? 0:0639 0:0150], fh1 (l)g =[?0:0008 0:0300 ? 0:0036 ? 0:0497 0:0562 0:3193 0:4868 0:3599 0:0779 ? 0:0708 ? 0:0260 0:0378]. In this case, M = 2 and L = 11. The distributions of the zeros of the two channels are shown in Fig. 2. Both channels are non-minimum phase systems with several zeros almost on the unit circle. When p = L, the condition number (H p ) = 2:88 104. We test the BSBE algorithm by choosing the simplest function f (y) = y3 among all other options for f (y). The noise in the observation is a Gaussian noise with zero mean and variance 2 . When = 0:0005, after using 2000 symbols to equalize the two-channel model, we obtain the outputs of the two algorithms and their ISI errors illustrated in Fig. 3. After the channel is equalized by the BSBE and the LSBE respectively, 104 symbols are transmitted to count bit-error-rate (BER). We obtain BER=0.03% for the BSBE and BER=23% for the LSBE. It is clearly shown in Fig. 3 that the BSBE performs better than the LSBE for the two-channel model with the vector impulse responses given by (10). The LSBE fails to equalize the two-channel model. The stacking level p is a control parameter. However, the simulation shows that we cannot improve the performances of the BSBE and the LSBE by increasing p. To justify the BSBE, we have assumed that the input sequence is non-Gaussian and iid. But the iid assumption may not hold in practice. A correlated binary input sequence was constructed in [14] to test a source correlation compensation algorithm for blind channel identi cation. We use this sequence to check November 16, 1998
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whether the BSBE may tolerate the correlated input to some degree. The sequence is constructed from an iid sequence su (k) uniformly distributed on [?0:5; 0:5]. De ne 8 < 1; if su (k) + su (k ? 1) > 0 s(k) = : ?1; otherwise. where j j 1 is a constant. It is easy to see that p(s(k) = 1) = p(su (k) + su (k ? 1) > 0) = 0:5 for any . So the sequence s(k) has zero mean. It can be shown that the sequence s(k) has the following autocorrelation sequence: 8 > > for = 0 > 1; < rs ( ) = > =2 ? 3 =6; for = 1 > > : 0; otherwise. The one step correlation of the input sequence s(k) is de ned as = =2 ? 3 =6. For example, = 0:3321 and 0:3333 when = 0:95 and 1, respectively. Taking s(k) as the common input to the two channels and applying the BSBE, we obtain the plots in Figure 4 which shows that the BSBE works well when = 0:3321 but breaks down when = 0:3333. This means that the iid assumption for the input may be relaxed to some degree in practice. B. Case 2:A four-channel model
In this case, we consider a single-input and four-output system. Fig. 5 shows the distributions of the zeros of the four channels. All the sub-channels are non-minimum phase systems of order L = 5. When p = L, the condition number (H p ) = 77:79 is much lower than that in Case 1. The outputs of the BSBE and the LSBE and their ISI errors are plotted in Fig. 6 which shows that both algorithms are successful. In this case, the LSBE has a better performance with much lower ISI than the BSBE. C. Remarks
The condition number (H p ) is an objective quantity to measure the condition of SIMO systems. For a linear system Ax = b, the condition number (A) is related to the perturbation bound[10]. Let x and y be the solution of the linear equation Ax = b and the solution of the perturbated equation (A + A)y = b + b respectively with kAk kAk, kbk kbk, > 0 and (A) < 1. Then the Theorem 2.7.2 in [10] gives the perturbation bound for the linear system Ax = b:
kx ? yk 2(A) : kxk 1 ? (A)
In Fig. 7, the condition numbers (H p ) in Case 1 are compared with those in Case 2 for di erent stacking levels. The legitimate range for the stacking levels in Case 1 is di erent from that in Case 2. But it is still meaningful to compare the condition numbers in the two cases because the perturbation bound does not depend on the dimensionality of the linear system. DRAFT
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The BSBE is successful in both cases. The performances of the BSBE in the two cases are illustrated by Fig. 3(b) and Fig. 6(b). Due to the equivariant property, the BSBE performs roughly the same in the two cases. Because the noise added at the channel output becomes the noise in the mixture model (8), the equivariant property does not hold completely. So the BSBE has slightly di erent performances in the two cases. On the other hand, it is illustrated by Fig. 3(c) and (d) that the LSBE fails in Case 1 because the condition numbers shown by solid line in Fig. 7 are too large. Nevertheless, the LSBE works very well in Case 2 and achieves a much smaller ISI error than the BSBE. V. Conclusion
The on-line blind equalization algorithm is proposed for equalizing an SIMO system and an SISO system with fractionally sampled channel outputs. Based on the on-line blind separation algorithms, the BSBE performs the equalization directly without the channel identi cation stage. For ill-conditioned channels, it achieves the equalization with better quality than the batch type least-squares algorithm LSBE does. Since the BSBE inherits the equivariant property from the equivariant blind separation algorithms, it is very robust in equalizing the ill-conditioned channels. But when the channel condition is moderately good, the performance of the batch algorithm LSBE is better than that of the on-line algorithm BSBE. To justify the BSBE, we have assumed that the input sequence is non-Gaussian and iid. But, the iid assumption is not crucial for the BSBE in practice. It is shown by the simulations that the BSBE allows certain degree of correlation in the input sequence. References [1] S. Amari, A. Cichocki, and H. H. Yang. Recurrent neural networks for blind separation of sources. In Proceedings of NOLTA 1995, volume I, pages 37{42, December 1995. [2] S. Amari, A. Cichocki, and H. H. Yang. A new learning algorithm for blind signal separation. In Advances in Neural [3] [4] [5] [6] [7] [8] [9] [10]
Information Processing Systems, 8, eds. David S. Touretzky, Michael C. Mozer and Michael E. Hasselmo, MIT Press: Cambridge, MA., pages 757{763, 1996. S. Amari, S. Douglas, A. Cichocki, and H. H. Yang. Multichannel blind deconvolution and equalization using the natural gradient. In Proc. of IEEE SPAWC'97, pages 101{104, Paris, France, April 1997. S. Amari, S. Douglas, A. Cichocki, and H. H. Yang. Novel on-line adaptive learning algorithms for blind deconvolution using the natural gradient approach. In Proc. of SYSID'97 , 11th IFAC Symposium on System Identi cation, Fukuoka, Japan, July 1997. A. J. Bell and T. J. Sejnowski. An information-maximisation approach to blind separation and blind deconvolution. Neural Computation, 7:1129{1159, 1995. J. A. Cadzow. Blind deconvolution via cumulant extrema. IEEE SP MAGAZINE, 13(3):24{42, May 1996. J.-F. Cardoso and B. Laheld. Equivariant adaptive source separation. IEEE Trans. on Signal Processing, 44(12):3017{ 3030, December 1996. S. Chen, S. McLaughlin, P. M. Grant, and B. Mulgrew. Multi-stage blind clustering equaliser. IEEE Trans. on Communications, 43(2/3/4):701{705, February/March/April 1995. P. Comon. Independent component analysis, a new concept? Signal Processing, 36:287{314, 1994. G. H. Golub and C. F. Van Loan. Matrix Computation . The Johns Hopkins University Press, 1989.
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[11] A. Gorokhov, P. Loubaton, and E. Moulines. Second order blind equalization in multiple input multiple output r systems: a weighted least squares approach. In ICASSP-96, pages 2415{2418. IEEE, May 1996. [12] S. Haykin. Blind Deconvolution. Prentice-Hall, Inc., 1994. [13] Y. Hua. Fast maximum likelihood for blind identi cation of multiple r channels. IEEE Trans. on Signal Processing, 44(3):661{672, March 1996. [14] Y. Hua, H. Yang, and W. Qiu. Source Correlation Compensation for Blind Channel Identi cation Based on SecondOrder Statistics . IEEE Signal Processing Letters, 1(8):119{120, August 1994. [15] J. Karaoguz and S. H. Ardalan. A soft decision-directed blind equalization algorithm applied to equalization of mobile communication channels. In Proc. ICC'92, Vol.3 , pages 343.4.1{343.4.5, Chicago, 1992. [16] A. Mansour, C. Jutten, and P. Loubaton. Subspace method for blind separation of sources in convolutive mixture. In G. Ramponi et al, editor, Signal Processing VIII, Vol.III, pages 2081{2084. Trieste, Italy, 1996. [17] E. Moulines, P. Duhamel, J.-F. Cardoso, and S. Mayrargue. Subspace Methods for the Blind Identi cation of Multichannel FIR Filters . IEEE Trans. on Signal Processing, 43(2):516{525, February 1995. [18] L. Tong, G. Xu, and T. Kailath. Blind Identi cation and Equalization Based on Second-Order Statistics: A Time Domain Approach . IEEE Trans. on Information Theory, 40(2):340{349, March 1994. [19] G. Xu, H. Liu, L. Tong, and T. Kailath. A least-squares approach to blind channel identi cation. IEEE Trans. on Signal Processing, 43(12):2982{2993, December 1995. [20] H. H. Yang and S. Amari. Two Gradient Descent Algorithms for Blind Signal Separation. In Proceedings of ICANN96, The Lecture Notes in Computer Science Vol.1112, pp.287-292. Springer-Verlag, 1996. [21] H. H. Yang and S. Amari. Adaptive on-line learning algorithms for blind separation: Maximum entropy and minimum mutual information. Neural Computation, 9(7):1457{1482, 1997. [22] H. H. Yang and S. Amari. Blind equalization of switching channels based on ICA and learning of learning rate. In Proc. of IEEE ICASSP'97, Munich, Germany, pages 1849{1852, April 1997. [23] H. H. Yang and E.-S. Chng. An on-line learning algorithm for blind equalization. In Progress in Neural Information Processing: Proceedings of ICONIP*96, pages 317{321. Springer, September 1996.
List of gure captions: Figure 1: Figure 2: Figure 3: Figure 4: Figure 5: Figure 6: Figure 7: DRAFT
The block diagram of the BSBE. The zeros of the two channels. Comparing the BSBE and the LSBE in Case 1 with an iid input sequence. The performances of the BSBE in Case 1 with a correlated input sequence. The zeros of the four channels. Comparing the BSBE and the LSBE in Case 2 with an iid input sequence. Comparing the condition numbers in Case 1 with those in Case 2. November 16, 1998
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? ? ? sk - H p - uk- V k vk- W k - sfk ? ?
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