Model-Based Processing

2Random Signals and Systems

2.1 Introduction

We can apply a filter to a random signal, but since its output is still random, we must find a way to eliminate or reduce this randomness in order to employ the powerful techniques available from systems theory. We shall show that techniques from statistics combined with linear systems theory can be applied to extract the desired signal information and reject the disturbance or noise. In this case, the filter is called an estimation filter or simply an estimator that is required to extract the useful information (signal) from noisy or random measurements.

Techniques similar to linear deterministic systems theory hold when the random signal is transformed to its covariance sequence and its Fourier spectrum is transformed to its power spectrum. Once these transformations are accomplished, then the techniques of linear systems theory can be applied to obtain results similar to deterministic signal processing. In fact, we know that the covariance sequence and power spectrum are a discrete Fourier transform (DFT) pair, analogous to a deterministic signal and its corresponding spectrum, that is, we have that

and as in the deterministic case, we can analyze the spectral content of a random signal by investigating its power spectrum.

The power spectral density (PSD) function for a discrete random process is defined as:

(2.1)

where is the ...