Optimum filtering

The engineer must often consider the common case where, from a raw message or observed signal m (t), containing a useful signal - the desired signal - and noise, we wish to determine the best - optimal - receiver for discriminating the signal from the noise. By optimal receiver or filter, we mean a filter that satisfies a certain optimality criterion under certain assumptions that we will specify.

By filter, we mean a mathematical description of the processing operations that the signal mixed with noise undergoes.

First, we must specify :

1×) that the inputs to these filters will be either random processes, or a combination of deterministic and random signals. In general, we will have a minimum amount of information characterizing these inputs;

2×) that we only consider linear stationary systems. In cases where a hardware realization is required, it will be necessary to consider the filter's realizability.

It is often useful to know the optimal system, even if it is not physically feasible. Knowing it will enable us to measure and assess the performance of systems that are feasible but not optimal.

We deal with three types of filters:

1 - the right filter ;

2 - the Wiener filter ;

3 - Kalman filter.

These different filters correspond respectively to a solution in the cases where :

1×) the desired signal is of known shape. It is mixed with either white or colored noise;

2×) the signal is, like noise, a random process. The filter developed by Norbert Wiener is a non-recursive solution that is difficult to implement on a computer;

3×) signal and noise are random. The Kalman filter is a recursive solution to the filtering problem that generalizes Wiener filtering.

The development of these filters requires a priori information on both the signals and the noise. In particular, we need to know the autocorrelation functions or matrices. In the absence of such information, adaptive filters can be used as an alternative. The latter "learn" the characteristics of signals as they occur. However, it has recently been shown that a family of adaptive filters commonly used in applications can also be considered optimal.