Representations of linear periodically time-varying and multirate systems

The linear switched time-varying (LSTV), polyphase (blocked), and alias-component representations of linear periodically time-varying (LPTV) systems are studied. In particular, alias-components are related to the time-shifted versions of the system. It is shown that in general, a filterbank is equivalent to the cascade connection of two LPTV systems. By generalizing some of the results of the LSTV representation of LPTV systems, it is shown that for relatively coprime integers p and m, a p-channel filterbank that has LPTV filters with period m is equivalent to an mp-channel filterbank with LTI filters. The representation of multirate systems that have LPTV kernels is discussed next. Due to the presence of the upsampler and downsampler, there are some degrees of freedom in the choice of the kernel. This redundancy is dealt with by choosing from various subclasses so that there is a one-to-one relationship between a multirate system and its kernel. Then, we find the LPTV kernel that has the least period.