Digital Signal Processing, 10th Edition

13Signal Analysis and Modeling

The modeling of systems is one of the most important areas of signal processing. Furthermore, modeling is an alternative approach to signal analysis, with properties differing from those of the Fourier transform and those of filters defined in the frequency domain. Linear prediction, in particular, is a simple and efficient tool to characterize some signal types and then compress them. The processing is specified in the time domain, using statistical parameters and, particularly, correlation.

13.1 Autocorrelation and Intercorrelation

The degree of similarity between two signals can be described using a correlation coefficient, which should logically take the values +1 for two identical signals, zero for two signals which have no relationship to each other, and −1 for signals in opposition to each other. When time-dependent signals are compared, the correlation coefficient becomes a function of time. It is called the intercorrelation function if the signals are different and the autocorrelation (AC) function if they are the same.

Some definitions and properties will now be restated in order to recapitulate and supplement the discussions in Sections 1.8 and 4.4.

As shown in Section 1.8, the AC function of a random discrete signal x(n) is the set r_xx(n) such that:

(13.1)

where E[x] denotes the expectation value of x.

With the assumption of ergodicity, ...