Elsevier

Signal Processing

Volume 83, Issue 2, February 2003, Pages 325-338
Signal Processing

A neuro-evolutionary framework for Bayesian blind equalization in digital communications

https://doi.org/10.1016/S0165-1684(02)00418-8Get rights and content

Abstract

The application of the Bayesian formulation to the joint data and channel estimation in digital communication is not feasible in practice because the computational complexity and memory requirements of the estimation process grow exponentially with time. However, the evolution with time of the channel conditional density model suggests the application of pruning, selection, crossover and other concepts from evolutionary computation and neural networks, which drastically reduce the complexity of the Bayesian equalizer without severe performance degradation. Although some problems of convergence to wrong channel estimates may arise, Bayesian equalizers can detect those situations by estimating, during operation, the overall symbol error probability. If suboptimal convergence is detected, the estimation process is automatically re-started.

Introduction

The development of efficient equalization techniques to compensate the effects of noise and inter-symbol interference is an important issue in modern digital communications. Recently, there has been much interest in blind (also called non-data-aided (NDA)) methods [17], [22], [28] where the equalizer parameters have to be adjusted without using any training sequence [15], [22]. They are specially important in wireless mobile communications [21], where the channel response may change along time, as well as in multi-point communications, where the re-transmission of a training signal every time severe fading interrupts a link with a user is not feasible.

There are two main approaches to data equalization: the first is to estimate the channel response and use a maximum a posteriori (MAP) or a maximum likelihood (ML) detector to recover the data. The second approach is to sidestep the channel estimation and invert (equalize) the channel to estimate the data directly. This paper deals precisely with the first method.

The joint data and channel estimation is an ill-posed problem. Most techniques are based on the use of qualitative information: for instance, the linearity of the channel response, the Gaussian distribution of the noise, or the finite alphabet used by the transmitter (which is known by the receiver). Many efforts have been made to extend the statistical approach used for data detection to the estimation of the channel response, i.e., to apply a Bayesian formulation [7], [6], [16], [23] or the ML criteria [13], [18], [28], [27] to both channel estimation and data detection. Besides the problems of local convergence, one of the main difficulties of this approach is the computational complexity: simplification techniques are required to avoid the memory and computational requirements growing exponentially with time.

This paper analyzes the application of evolutionary computation (EC) strategies to limit the complexity of Bayesian blind equalizers. We show that, under the Bayesian formulation, the problem of estimating the channel response can be interpreted as the selection of an efficient estimate of the channel response among a growing population of estimates. Selection, crossover, mutation and other concepts from EC can be used to efficiently estimate the channel response while the computational complexity remains constant along time.

In contrast to ML detectors, Bayesian equalizers can estimate the overall bit error rate (BER) of the detection process during operation. In this paper, we show how this feature can be applied to monitor the estimation process, identify problems of convergence to unsatisfactory channel estimates and, if necessary, re-start the estimation algorithm.

The application of EC ideas to equalization is not new: in [3] a genetic algorithm is used to expand the search space in order to escape from the usual local minima problems arising in gradient-based methods. The same purpose motivates in [5] the combination of genetic algorithms and ML detection. In the context of blind Bayesian estimation, the simplifying mechanisms discussed in [7], [6], [16] can be re-formulated under an evolutionary perspective, as will be shown later. Our work expands some preliminary results in [23], [24]. To the best of our knowledge, no other attempts to make explicit use of evolutionary computation strategies to reduce the computational complexity of Bayesian blind equalizers have been published.

This article is structured as follows: Section 2 states and discusses, separately, the problems of Bayesian detection and channel estimation. Section 3 uses the Bayesian formulation to solve (theoretically) the joint problem. The expressions obtained in this section constitute the core of the paper, since they are the starting point for all EC strategies proposed and discussed in Section 4. The capability of estimating the BER during operation is also analyzed. Section 5 analyzes the performance of the methods proposed by means of simulations. Finally, some conclusions are outlined in Section 6.

Section snippets

Digital channel model

Consider the channel model represented in Fig. 1. We assume that a digital sequence {xk} of symbols from alphabet A={a0,…,aM−1} is passed through a stationary linear dispersive channel with a finite impulse response given by coefficient vector h=(h0,…,hm). Defining vector x(k)={xk,xk−1,…,xk−m}, the sample received at time k can be expressed asrk=hTx(k)+nk,where {nk} is a sequence of zero-mean additive, white and Gaussian noise (AWGN) samples.

A usual time-variant model for non-stationary channel

Bayesian formulation

In the analysis carried out in the previous Section, h is an unknown parameter and xk is a fixed known symbol sequence. When the channel response h is unknown and no training sequence is available, a joint detection and estimation problem must be solved. Analyzing the evolution of the conditional pdf p(h|rk) as new samples arrive, the Bayesian approach can be extended to this case.

Since xk is unknown, the conditional pdf of h must be expanded over the set of hypotheses about the transmitted

Individuals

Vector ci(k) in Eq. (19) is the MAP channel estimate under the hypothesis that xk=aik. Therefore, it represents a “channel candidate”. All the information about the channel model pdf at time k can be encoded in set Pk={(aik,ci(k),Si(k)),i=0,…Lk−1} and probabilities λi(k). In the following sections, Pk is called the population at time k and every element is an individual. Probability λi is the fitness value of the ith individual.

Note that the genetic information of an individual encodes: (1) a

Performance analysis and numerical results

This Section analyzes the performance of several equalizer configurations based on the Bayesian-EC formulation. The numerical comparisons are focused on: (i) the selection strategy (DSS vs. MSS), (ii) the channel updating method (RLS vs. LMS), (iii) the effect of CC, (iv) the complexity of the algorithm (number of individuals per generation).

Conclusions

This article gathers under a common framework several quasi-Bayesian algorithms for blind equalization in the light of the complexity inherent to the pure Bayesian method. We have developed a new Bayesian equalizing algorithm with the capability of in-service BER monitoring. Special emphasis on the evolutionary interpretation and flexibility of the pruning step has been taken. Monitoring blind BER estimates is useful to auto-detect the situations of convergence to non-optimal solutions (channel

Acknowledgements

The authors would like to thank Dr. A. Bravo-Santos for his useful comments on the manuscript, and Ms. M. Eugenia Llamas for her help with the english version.

References (28)

  • J Cid-Sueiro et al.

    Recurrent RBF networks for optimal symbol-by-symbol equalization

    Signal Processing

    (November 1994)
  • J Cid-Sueiro et al.

    Soft decision Bayesian equalizers for in-service error rate monitoring

    Signal Processing

    (April 2000)
  • K Abend et al.

    Statistical detection for communication channels with intersymbol interference

    Proc. IEEE

    (May 1970)
  • L. Chambers (Ed.), The Practical Handbook of Genetic Algorithms Applications, 2nd Edition, Chapman & Hall/CRC,...
  • S Chen

    Genetic algorithm optimization for blind channel identification with higher order cumulant fitting

    IEEE Transactions Evolut. Comput.

    (November 1997)
  • R Chen et al.

    Mixture Kalman filters

    J. Roy. Statist. Soc. B

    (2000)
  • S Chen et al.

    Maximum likelihood joint channel and data estimation using genetic algorithms

    IEEE Trans. Signal Process.

    (May 1998)
  • J. Cid-Sueiro, A.R. Figueiras-Vidal, Recurrent radial basis function networks for optimal blind equalization,...
  • A.P Clark

    Equalizers for Digital Modems

    (1985)
  • D Crisan et al.

    Discrete filtering using branching and interacting particle systems

    Markov Proc. Rel. Fields

    (1999)
  • A Doucet et al.

    On sequential Monte Carlo sampling methods for Bayesian filtering

    Statist. Comput.

    (2000)
  • G.D Forney

    The Viterbi algorithm

    Proc. IEEE

    (May 1973)
  • M Ghosh et al.

    Maximum-likelihood blind equalization

    Opt. Eng.

    (June 1992)
  • J.F Hayes et al.

    Optimal sequence detection and optimal symbol-by-symbol detectionsimilar algorithms

    IEEE Trans. Comm.

    (January 1982)

    • Cited by (6)

    View full text
    Copyright © 2002 Elsevier Science B.V. All rights reserved.