6.1 Introduction

By adaptive processing, we have in mind a particular, yet very broad class of optimization algorithms which are activated in real time in distance information transmission systems.

The properties of adaptive algorithms are such that, on the one hand, they allow the optimization of a system and its adaptation to its environment without outside intervention, and, on the other hand, this optimization is also assumed in the presence of environmental fluctuation over time.

It is also to be noted that the success of adaptive techniques is such that we no longer meet them only in telecommunications but also in such diverse domains as submarine detection, perimetric detection, shape recognition, aerial networks, seismology, bio-medical instrumentation, speech and image processing, identification of control system, etc.

Amongst the applications cited above, different configurations arise.

Figure 6.1: Prediction

Figure 6.2: Identification

Figure 6.3: Deconvolution

Figure 6.4: Cancellation

In the course of these few pages we will explain the principle of adaptive filtering and establish the first mathematical results.

We will limit ourselves to begin with to WSS processes and to the algorithms called, of the deterministic gradient and the LMS algorithm. We will also give a few examples concerning non-recursive linear adaptive filtering.

Later, we will broaden this concept to non-stationary signals in presenting Kalman filtering in the following chapter.

6.2 Position of problem [ ^{[[WID 85]]}

Starting from observations (or measures) taken at instant K (that we will note y _K: results) of a process X _K issued...