Chapter 9

Linear Estimation, System Modelling and Adaptive Filters

9.1 Introduction

In this chapter, a central problem in signal processing is addressed, namely: the estimation of some signal of interest from a set of received noisy data signals [11, 12, 20, 21]. If the signal is deterministic with known spectrum and this spectrum does not overlap with that of the noise, then the signal can be recovered by the conventional filtering techniques discussed earlier. However, this situation is very rare. Instead, we are often faced with the problem of estimating an unknown random signal in the presence of noise, and this is usually accomplished so as to minimize the error in the estimation according to a certain criterion. This leads to the area of adaptive filtering [21]. A closely related area is that of the modelling or simulation of the behaviour of an unknown system (or process) by a linear system. Initially, the principles of linear estimation and modelling are discussed, then it is shown how these can be implemented using adaptive algorithms. In linear estimation theory, we use techniques that are derived from the classical mean-square approximation method for deterministic functions. Therefore this chapter begins by a discussion of such methods, in preparation for extension to stochastic signals. The exposition in this chapter is intended to serve as an introduction to linear estimation by analog and digital techniques. Emphasis is laid on the Wiener filter and the associated ...

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