On Bounds of Shift Variance in Two-Channel Multirate Filter Banks

Critically sampled multirate FIR filter banks exhibit periodically shift variant behavior caused by nonideal antialiasing filtering in the decimation stage. We assess their shift variance quantitatively by analysing changes in the output signal when the filter bank operator and shift operator are interchanged. We express these changes by a so-called commutator. We then derive a sharp upper bound for shift variance via the operator norm of the commutator, which is independent of the input signal. Its core is an eigensystem analysis carried out within a frequency domain formulation of the commutator, leading to a matrix norm which depends on frequency. This bound can be regarded as a worst case instance holding for all input signals. For two channel FIR filter banks with perfect reconstruction (PR), we show that the bound is predominantly determined by the structure of the filter bank rather than by the type of filters used. Moreover, the framework allows to identify the signals for which the upper bound is almost reached as so-called near maximizers of the frequency-dependent matrix norm. For unitary PR filter banks, these near maximizers are shown to be narrow-band signals. To complement this worst-case bound, we derive an additional bound on shift variance for input signals with given amplitude spectra, where we use wide-band model spectra instead of narrow-band signals. Like the operator norm, this additional bound is based on the above frequency-dependent matrix norm. We provide results for various critically sampled two-channel filter banks, such as quadrature mirror filters, PR conjugated quadrature filters, wavelets, and biorthogonal filters banks.