Adaptive Filters

CHAPTER 8 Steepest–Descent Technique

The earlier chapters discussed in some detail the theory of least-mean-squares estimation and highlighted several applications in the context of channel equalization, channel estimation, and antenna beamforming. While the chapters in Part I (Optimal Estimation) studied optimal estimators, which are generally nonlinear functions of the observations, the chapters in Part II (Linear Estimation) focused on linear estimators with and without constraints. In all cases, the estimators were optimal in the sense that they minimized the mean-square error.

Now there are situations where a designer may be interested in other performance criteria, other than the mean-square error criterion. Several examples to this effect are provided in the problems at the end of this part (e.g., Probs. III. 12–III. 18). In most of these cases it is generally not possible to describe the optimal solution in closed-form in terms of the moments of the underlying variables, and it often becomes necessary to approximate the optimal solution iteratively. The iterative procedure would start from an initial guess for the solution and then improve upon it from one iteration to another. The purpose of this chapter is to describe one class of iterative schemes known as steepest-descent methods, which is at the core of most adaptive filtering techniques.

The steepest-descent methods ...