Bifrequency and bispectrum maps: a new look at multirate systems with stochastic inputs

In multirate digital signal processing, we often encounter decimators, interpolators, and complicated interconnections of these with LTI filters. We also encounter cyclo-wide-sense stationary (CWSS) processes and linear periodically time-varying (PTV) 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 sufficient addition for the output of an L-fold interpolation filter to be wide sense stationary (WSS) for all WSS inputs is that the filter was an alias-free (L) support. However, several questions of this nature remain unanswered. For example, what is the necessary and sufficient condition on a pair (or more generally a bank) of interpolation filters 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 filter-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 filter. However, the frequency domain approaches that proved this result are quite difficult 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 (2-D) Fourier transforms that characterize linear time-varying (LTV) systems and nonstationary random processes, respectively. We show that the questions raised above are addressed elegantly and in a geometrically insightful way using these tools. We also derive a bifrequency characterization lossless LTV systems. This may potentially lead to an increased understanding of these systems.