Above, we found that, for any integer which divides , a sufficient condition for perfect reconstruction is
and the output signal is then
More generally, we allow any nonzero scaling and any additional delay:
where is any constant and is any nonnegative integer. In this case, the output signal is
Thus, given any polyphase matrix , we can attempt to compute : If it is stable, we can use it to build a perfect-reconstruction filter bank. However, if is FIR, will typically be IIR. In §11.5 below, we will look at paraunitary filter banks, for which is FIR and paraunitary whenever is.
Necessary and Sufficient Conditions for Perfect Reconstruction
Hopping Polyphase Filter Bank