Variance and correlation of square-law detected allpass channels with bandpass harmonic signals in Gaussian noise

A bandpass harmonic signal consists of a multiple tone complex with uniform spacing of the frequencies. This paper considers the correlation of square-law detected allpass channels with such signals in Gaussian noise. The square-law mixing terms of each allpass filtered channel are low-pass filtered to produce the detector output components, and then are synchronously demodulated at the tone difference frequency to form output components. In this paper, we derive the theoretical variance and cross-correlation between the different channels, before and after synchronous demodulation. Simulations for a three-tone signal demonstrate excellent agreement with the exact and approximate theoretical results. Applications of this work include wide-band system identification, an example of which is detailed in this paper for the estimation of differential phase.