Statistical Approach to Signal Analysis

Abstract

Discrete-time signals often derive from periodically repeated measurements of some quantity over a finite time span, and we are interested in the characteristics of the quantity and of the process that generates it, rather than in those of the particular sequence we measured. The measured record, affected by random errors, is interpreted as a segment of a persistent discrete-time random power signal, conceptually resulting from sampling a continuous-time random signal (process) possibly varying with time. The value assumed by a random signal is not exactly specified at a given past or current instant, and future values are not predictable with certainty on the basis of past behavior. This chapter provides a brief introduction to the basic theory of discrete-time random processes, which can be described using theoretical average quantities. The latter could be calculated if the probabilistic laws associated with the random process were known, but usually they are unknown; the problem is then simplified by assuming stationarity, i.e., no dependence on time, and ergodicity, a property that allows for substituting theoretical averages with time averages performed on a single finite-length data record. This path leads to a spectral representation for the discrete-time random power signal, i.e., to the power spectrum. Also the representation of the common spectral content of two random signals is introduced.