Generalising the Polyphase Identity

In this paper, we generalise the well known polyphase identity in multirate signal processing. A cascade of expander, linear system and a decimator is considered for simplification. The nature of the kernel of the linear system resulting in the cascade is studied. The linear system is chosen to be (p, q) shift invariant for further simplification. The different subcases of the problem which are equally important are also highlighted. The properties of cascade of shift invariant systems are derived, simultaneously stressing the important subcases. Interestingly two other problems, decimating a block Toeplitz matrix and a 2-D periodic signal were also addressed in the same framework. As an extension for expanding the kernel, we proposed a new multirate structure.