Ergodic properties for multirate linear systems

Stochastic analysis of a multirate linear system typically requires the signals in the system to possess certain ergodic properties. Among them, ergodicity in the mean and ergodicity in the correlation are the most commonly used ones. We show that multirate operations and time-variant linear filtering can destroy these ergodic properties. Motivated by this fact, we introduce the notions of strong ergodicity in the mean and strong ergodicity in the correlation. We show that these properties are preserved under a number of operations, namely, downsampling, upsampling, addition, and uniformly stable linear (time-variant) filtering. We also show that white random processes with uniformly bounded second moments are strongly ergodic in the mean and that mutually independent random processes with uniformly bounded fourth moments are jointly strongly ergodic in the correlation. The main implication of these results is that if a multirate linear system is driven by white (independent) random processes with uniformly bounded second (fourth) moments, then every signal in the system is strongly ergodic in the mean (correlation) and therefore ergodic in the mean (correlation). An application of these results is also discussed.