Logical convolution and discrete Walsh and Fourier power spectra

The Walsh power spectrum of a sequence of random samples is defined as the Walsh transform of the logical autocorrelation function of the random sequence. The "logical" autocorrelation function is defined in a similar form as the "arithmetic" autocorrelation function. The Fourier power spectrum, which is defined as the Fourier transform of the arithmetic autocorrelation function, can be obtained from the Walsh power spectrum by a linear transformation. The recursive relations between the logical and arithmetic auto-correlation functions are derived in this paper. For a given process with computed or modeled autocorrelation function the Fourier and Walsh power spectra are computed by using the fast Fourier and Walsh transforms, respectively. Examples are given from the speech and imagery data.