Bifrequency and Bispectrum Maps: a New Look at Multirate Systems with Stochastic

In multirate digital signal processing, we often encounter decimators, interpolators and complicated interconnections of these with LTI lters. We also encounter cyclo-wide-sense stationary (CWSS) processes and linear periodically time-varying (LPTV) systems. It is often necessary to understand the effects of multirate systems on the statistical properties of their input signals. Some of these issues have been addressed earlier. For example it has been shown that a necessary and suucient condition for the output of an L-fold interpolation lter to be wide sense stationary (WSS) for all WSS inputs, is that the lter have an aliasfree(L) support. However several questions of this nature remain unanswered. For example, what is the necessary and suucient condition on a pair (or more generally a bank) of interpolation lters so that their outputs are jointly WSS (JWSS) for all jointly WSS inputs? What is the condition if only the sum of their outputs is required to be WSS? When is the output of an LPTV system (for example a uniform lter-bank) WSS for all WSS inputs? Some of these questions may appear to be simple generalizations of the above-mentioned result for a single interpolation lter. However the frequency domain approaches that proved this result are quite diicult to generalize to answer these questions. The purpose of this paper is to provide these answers using analysis based on bifrequency maps and bispectra. These tools are two-dimensional Fourier transforms that characterize all linear time-varying (LTV) systems and nonstationary random processes respectively. We show that the questions raised above can be addressed elegantly and in a geometrically insightful way using these tools. We also derive a bifrequency characterization of lossless LTV systems. This may potentially lead to an increased understanding of these systems.