The computation of correlation and spectral functions by orthogonal filtering

The increased consideration of random time functions in engineering and science has led to the development of various digital and analog techniques for the computation of autocorrelation and cross-correlation functions or their frequency domain equivalents, spectral density and cross-spectral density. Analog techniques, which are the subject of this paper, have been limited primarily to either the time domain1??3 (employing the operations of time delay, multiplication, and averaging) or the frequency domain4,5 (employing the operations of narrow-band filtering, multiplication, and averaging). A more general but little-known procedure, which was originally introduced by Lampard,6 is to represent the autocorrelation or cross-correlation function by a finite series of predetermined orthogonal approximating functions and to determine coefficients in the series by the operations of orthogonal filtering, multiplication, and averaging. Advantages of the series method include analytic representations for both the correlation and spectral functions and the possible elimination of delay lines or narrow-band filters.