Joint density functions for digital spectra

An analysis is presented for the joint probability density functions of power spectra of random processes computed under various conditions of data/frequency smoothing. Spectral correlation functions are determined which illustrate the effect of smoothing on adjacent spectral ordinates. The technique of segment averaging of spectra is explored and bivariate probability density functions established for correlated spectral ordinates. These distributions are shown to be generalizations of the well known Rician probability density functions. Finally, the case of spectra obtained as moving average values is considered and expressions are derived for the spectral correlation functions in terms of the parameter of the smoothing sequences and the averaging length. It is shown that spectra obtained in this manner introduce a distortion of the spectral correlation function. Several computed results are included to illustrate the analysis.