Elsevier

Signal Processing

Volume 68, Issue 3, 24 August 1998, Pages 271-281
Signal Processing

On-line blind equalization via on-line blind separation

https://doi.org/10.1016/S0165-1684(98)00077-2Get rights and content

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.

Zusammenfassung

Es wird eine Methode zur blinden Entzerrung vorgestellt, die auf der Theorie der Analyse unabhängiger Komponenten aufbaut. Die blinde Entzerrung eines unbekannten FIR-Kanals mit einer möglichen nicht-minimalen Phase wird als ein blindes Separationsproblem formuliert. Die Eingänge des Entzerrers werden durch Stapelung der überabgetasteten Werte des Kanalausgangssignals gebildet. Es werden neue Echtzeitalgorithmen zur blinden Entzerrung aus Echtzeit-algorithmen für die blinde Separierung entwickelt. Deshalb haben sie die äquivalenten Eigenschaften der blinden Separationsalgorithmen. Dieser Eigenschaft entsprechend ist die Leistungsfähigkeit des neuen Algorithmus unabhängig von den Kanalparametern. Deshalb sind sie für die Entzerrung einiger schlecht konditionierter Kanäle geeignet. Es wird durch Simulationen gezeigt, daß der vorgeschlagene Algorithmus gut arbeitet, falls die Eingangsfolge eine gewisse Korrelation besitzt.

Résumé

Nous présentons une méthode d’égalisation aveugle basée sur la théorie de l’analyse en composantes indépendantes. L’égalisation aveugle d’un canal FIR inconnu possiblement à phase non minimale est formulée en termes de séparation aveugle. Les entrées de l’égalisateur sont formées en empilant les échantillons fractionnels des sorties du canal. De nouveaux algorithmes d’égalisation aveugle en ligne sont développés à partir d’algorithmes de séparation aveugle en ligne, et de ce fait héritent de la propriété d’équivariance des algorithmes de séparation aveugle. A cause de cette propriété, les performances des algorithmes nouveaux sont indépendantes de paramètres du canal. En conséquence, ils sont utiles pour égaliser certains canaux mal conditionnés. Il est montré par des simulations que l’algorithme proposé fonctionne bien lorsque la séquence d’entrée présente un certain degré de corrélation.

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 identification 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 identified. But these methods are batch type algorithms and require matrix decompositions in the implementations. The system identification and equalization processes are two pipeline operations. The equalization can only be started after the channel is identified. For some ill-conditioned systems, the pseudo-inverse fails due to numerical errors even if the system identification 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 identified up to an instantaneous mixing matrix. After the channel is identified, 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 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 first 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 different 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. Different 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 identification stage. This approach is more effective 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 defined in Section 2. The blind equalization algorithms based on blind separation and least-squares approaches are formulated in Section 3.1and Section 3.2, respectively. Simulations and some remarks on the conditional numbers of the channels are given in Section 4. Finally, conclusions are made in Section 5.

Section snippets

Channel models

Consider the following time invariant channel defined in [18]:x(t)=kT⩽ts(k)h(t−kT)+n(t),where {s(k)} 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[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 continuous-time output. The problem of the blind equalization is to recover the input sequence from the history of

Blind separation for blind equalization (BSBE)

Assume a linear mixture model:x(t)=As(t),where A is an unknown n×n nonsingular 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 transformy(t)=Wx(t).The following learning algorithm (LA) is used to update the weight matrix W:Wk+1=(1+η)Wk−ηf(yk)ykTWk,where η is a learning rate and f(y)=(f(y1),…,f(yn))T. This algorithm has been derived in [2]

Case 1. A two-channel model

Let us consider the blind equalization of a single-input and two-output system with a vector impulse responsesh(l)=h0(l)h1(l),l=0,…,L,where L=11,{h0(l)}=[−0.01660.01890.0226−0.0354−0.02310.17990.43390.45960.2185−0.0248−0.06390.0150],{h1(l)}=[−0.00080.0300−0.0036−0.04970.05620.31930.48680.35990.0779−0.0708−0.02600.0378].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

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 identification 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

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