Shift Variance Measures for Multirate LPSV Filter Banks With Random Input Signals

Abstract

The linear periodically shift variant (LPSV) properties within critically sampled multirate FIR filter banks are generally analysed using deterministic signals. Periodic shift variance is, however, closely related to cyclostationarities introduced by the LPSV system into originally wide sense stationary (WSS) random signals passing through the system. We provide first a unified framework to measure both shift variance of the LPSV system and the amount of cyclostationarity it generates. In this respect, the key concept is the covariance operator associated to a random variable. Cyclostationarity of the variable translates to LPSV properties of the operators, and vice versa. We study several related concepts for the quantification of shift variance in operators and their interpretation in the stochastic setting. We then introduce a new concept called expected shift variance. Fourier-analytic expressions for the various measures are given, and subsequently used to derive explicit formulae for the case of critically sampled two-channel filter banks, as well as sharp upper bounds for unitary two-channel filter banks. Numerical evaluations of the measures show that they provide largely consistent rankings between various critically sampled two-channel filter banks.