### 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:

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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Necessary and Sufficient Conditions for Perfect Reconstruction

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Hopping Polyphase Filter Bank