Shift-Variance Measure of Multichannel Multirate Systems

This study is concerned with quantifying shift variance of linear discrete-time multirate systems. Based on commutators of a system and the shift operators, we introduce a shift-variance measure for the multirate system. The measure is defined as the half of the ratio of the maximal norm of the commutators to the norm of the multirate system. It ranges from zero to unity, corresponding to minimal shift variance (i.e., full shift invariance) and maximal shift variance, respectively. We derive analytical expressions on the shift-variance measure of multirate filterbanks. The relation between shift variance of the individual channels of multichannel filterbanks is then studied. We also analyze the shift invariance of some particular filterbanks, including biorthogonal filterbanks and complex wavelet transforms.