8.1 Introduction

Advanced statistical methods can result in substantial signal processing gain in wireless systems. Among the most powerful such techniques are the Monte Carlo Bayesian methodologies that have recently emerged in statistics. These methods provide a novel paradigm for the design of low-complexity signal processing techniques with performance approaching theoretical optima, for fast and reliable communication in the severe and highly dynamic wireless environments. Over the past decade or so in the field of statistics, a large body of methods has emerged based on iterative Monte Carlo techniques that is useful, especially in computing the Bayesian solutions to the optimal signal reception problems encountered in wireless communications. When employed in the signal processing engines of the digital receivers in wireless networks, these powerful statistical tools hold the potential of closing the substantial gap between the performance of current state-of-the-art wireless receivers and the ultimate optimal performance predicted by statistical communication theory.

In this chapter we provide an overview of the theories and applications in the emerging field of Monte Carlo signal processing [544]. These methods in general fall into two categories: Markov chain Monte Carlo (MCMC) methods for batch signal processing, and sequential Monte Carlo (SMC) methods for adaptive signal processing. For each category we outline the general theory and provide a signal processing example found in wireless communications to illustrate its application. Specifically, we apply the MCMC technique to the problem of Bayesian multiuser detection in unknown channels; and we apply the SMC technique to the problem of adaptive blind equalization in multiple-input /multiple-output (MIMO) intersymbol interference (ISI) channels. The remainder of this chapter is organized as follows . In Section 8.2 we describe the general Bayesian signal processing framework. In Section 8.3 we introduce the MCMC techniques for Bayesian computation. In Section 8.4 we illustrate the application of MCMC by treating the problem of Bayesian multiuser detection in unknown channels. In Section 8.5 we discuss the SMC paradigm for Bayesian computing. In Section 8.6 we illustrate the application of SMC by treating the problem of blind adaptive equalization in MIMO ISI channels. Finally, Section 8.7 contains some mathematical derivations and proofs.

The following is a list of the algorithms appearing in this chapter.

• Algorithm 8.1 : Metropolis “Hastings algoritm ”Form I

• Algorithm 8.2 : Metropolis “Hastings algorithm ”Form II

• Algorithm 8.3 : Random-scan Gibbs sampler

• Algorithm 8.4 : Systematic-scan Gibbs sampler

• Algorithm 8.5 : Gibbs multiuser detector in Gaussian noise

• Algorithm 8.6 : Gibbs multiuser detector in impulsive noise

• Algorithm 8.7 : Sequential importance sampling (SIS)

• Algorithm 8.8 : Sequential Monte Carlo filter for dynamical systems

• Algorithm 8.9 : Residual resampling

• Algorithm 8.10 : Mixture Kalman filter for conditional dynamical linear models

• Algorithm 8.11 : SMC-based blind adaptive equalizer in MIMO channels