Analysis of M-channel time-varying filter banks

In this paper we present a new theory to analyze M-channel time-varying filter banks in the transform-domain. Through studying the behavior of time-varying building blocks at a fixed time point we have established two models to analyze the time-varying building block. Model 1 gives us the possibility to analyze the spectra changing in the time-varying building blocks. Model 2 shows that the time-varying building block at a fixed time point can be realized by a time-invariant filter bank. Such observation gives us chance to analyze the time-varying filter bank using the theory for time-invariant filter banks. Setting the time-varying building blocks into M-channel time-varying filter banks we find that an M-channel time-varying filter bank at a fixed time point can be implemented with an N-channel time-invariant vector filter bank or a L-channel single filter bank if the filter length is equal to L=NM. Based on the established models, in this paper we analyze the M-channel time-varying filter bank using the classical z-transform in detail. The perfect reconstruction conditions for an M-channel time-varying filter bank are given based on the polyphase-domain analysis. In the modulation-domain analysis we provide the condition for a distortion-free M-channel time-varying filter bank. The relationship between the time- and transform-domain analysis is presented. Finally, a design example is given to show how to use the established theory by designing time-varying filter banks.