Correlation Our study of signal processing systems has been dominated by the concept of ‘convolution’, and we have somewhat neglected its close relative the ‘correlation’. While formally similar (in fact convolution by a symmetric FIR filter can be considered a correlation as well), the way one should think about the two is different. Convolution is usually between a signal and a filter; we think of it as a system with a single input and stored coefficients. | Digital Signal Processing A Computer Science Perspective Jonathan Y. Stein Copyright 2000 John Wiley Sons Inc. Print ISBN 0-471-29546-9 Online ISBN 0-471-20059-X 9 Correlation Our study of signal processing systems has been dominated by the concept of convolution and we have somewhat neglected its close relative the correlation . While formally similar in fact convolution by a symmetric FIR filter can be considered a correlation as well the way one should think about the two is different. Convolution is usually between a signal and a filter we think of it as a system with a single input and stored coefficients. Crosscorrelation is usually between two signals we think of a system with two inputs and no stored coefficients. The difference may be only in our minds but nonetheless this mind-set influences the way the two are most often used. Although somewhat neglected we weren t able to get this far without mentioning correlations at all. We have already learned that crosscorrelation is a measure of similarity between two signals while autocorrelation is a measure of how similar a signal is to itself. In Section we met the autocorrelation for stochastic signals which are often quite unlike themselves and in Section we used the crosscorrelation between input and output signals to help identify an unknown system. Correlations are the main theme that links together the present chapter. We first motivate the concept of correlation by considering how to compare an input signal to a reference signal. We find that the best signal detector is the correlator. After formally defining both crosscorrelation and autocorrelation and calculating some examples we prove the important Wiener-Khintchine theorem which relates the autocorrelation to the power spectral density PSD . Next we compare correlation with convolution and discover that the optimal signal detector can be implemented as a matched filter. The matched filter was invented for radar and a digression into this .
