Perfect Reconstruction Filter Banks

We now consider filter banks with an arbitrary number of channels, and ask under what conditions do we obtain a perfect reconstruction filter bank? Polyphase analysis will give us the answer readily. Let's begin with the -channel filter bank in Fig.11.20. The downsampling factor is . For critical sampling, we set .

The next step is to expand each analysis filter into its -channel ``type I'' polyphase representation:

(12.49) |

or

(12.50) |

which we can write as

(12.51) |

Similarly, expand the synthesis filters in a type II polyphase decomposition:

(12.52) |

or

(12.53) |

which we can write as

(12.54) |

The polyphase representation can now be depicted as shown in
Fig.11.21. When
, commuting the up/downsamplers gives
the result shown in Fig.11.22. We call
the
*polyphase matrix*.

As we will show below, the above simplification can be carried out
more generally whenever
*divides*
(*e.g.*,
). In these cases
becomes
and
becomes
.

- Simple Examples of Perfect Reconstruction
- Sliding Polyphase Filter Bank
- Hopping Polyphase Filter Bank
- Sufficient Condition for Perfect Reconstruction (PR)
- Necessary and Sufficient Conditions for PR
- Polyphase View of the STFT
- Polyphase View of the Overlap-Add STFT
- Polyphase View of the Weighted-Overlap-Add STFT

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