Our main purpose in this chapter is to determine the optimal filter needed to extract the signal from the noise. This optimal filter can be defined as follows: it is a mathematical description of the signal processing operations that have to be conducted on the noisy signal. This description should respect the criteria of optimality that will be described in this chapter.
As a prelude, the following points should be noted:
– the inputs of these filters are either random signals, or combinations of random and deterministic signals;
– we will only cover stationary linear systems in this chapter. When the final aim is to obtain a physical implementation, we will also consider the realizability issues of the filter.
We will consider two main types of filters: the matched filter and the Wiener filter. These two classes are, respectively, the solution to the following cases:
– detecting the desired signal, whose shape is already known, when it is disturbed by a white or colored noise;
– extracting the signal when both the signal and the noise are random processes.
When designing these filters, the autocorrelation functions and matrices are assumed to be known.
The chapter is organized as follows. In the first section, we will look at matched filters and treat the two successive cases where the signal is disturbed by a white additive noise and by a colored one. In the second section, we recall the traditional presentation of the Wiener ...