Spectral Analysis with Sinusoids and Walsh Functions

The Walsh spectrum of a sinusoid of frequency f = p/2q with p odd consists entirely of lines at orders s that are odd multiples of 2/2q, and the Fourier spectrum of a Walsh function of order 2p/2q consists entirely of lines at frequencies that are odd multiples of 1/2q. For all other frequencies or orders, the spectrum contains no lines, but the power spectral density takes all values in the range [0,oo] in every interval, however short, while being almost everywhere zero. For detecting the presence of a sinusoid by means of Walsh analysis, the time scale should therefore be chosen so that the order of the Walsh function is a power of 2 and the Walsh function is a Rademacher function, i.e., a hard-limited sinusoid of the frequency being sought.