### Sufficient Condition for Perfect Reconstruction

Above, we found that, for any integer which divides , a sufficient condition for perfect reconstruction is

 (12.59)

and the output signal is then

 (12.60)

More generally, we allow any nonzero scaling and any additional delay:

 (12.61)

where is any constant and is any nonnegative integer. In this case, the output signal is

 (12.62)

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.

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Hopping Polyphase Filter Bank