Shift-Variance and Nonstationarity of Linear Periodically Shift-Variant Systems and Applications to Generalized Sampling-Reconstruction Processes

Abstract

We present a systematic analysis on shift-variance of linear $T$ -periodically shift-variant ( $T$ -LPSV) systems with continuous-time input and output. We first determine how far a $T$ -LPSV system $H$ is away from being shift-invariant in terms of two measures, namely, the distance and the angle between $H$ and the subspace of linear shift-invariant systems. We then consider the level and the index of shift-variance (SVI) of $H$ via commutator of $H$ and the shift-operator, and examine the amount of shift-variance of $H$ under particular input. All these quantities are characterized in terms of the so-called shift-variant part and the shift-variance kernel of $H$ . Moreover, we study nonstationarity of wide-sense cyclostationary random processes, possibly induced by $T$ -LPSV systems. We define and calculate a measure of nonstationarity (NSt) of the output as the SVI of the autocorrelation operator. The Nst can be used to characterize performance loss when the output is treated with a WSS approach. The expected shift-variance of $H$ subject to particular WSS random process is also calculated. We apply the above results to generalized sampling-reconstruction processes (GSRPs). In addition, we consider the approximation error of the GSRP and relate it to the shift-variance measures. Three sampling schemes (namely, the orthogonal, the consistent, and the minimax regret sampling) are examined in detail. We show that for both deterministic and WSS random inputs, the minimax regret GSRP always introduces less shift-variance than the consistent counterparts. Numerical results for the GSRPs of B-splines of various orders are provided.