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Course by: Phil Schniter. E-mail the author

# Perfect Reconstruction QMF

Module by: Phil Schniter. E-mail the author

## Perfect Reconstruction QMF

The system transfer function for a QMF bank is

Tz= H 0 2z H 1 2z=4z-1 P 0 z2 P 1 z2 T z H 0 z 2 H 1 z 2 4 z -1 P 0 z 2 P 1 z 2
(1)
For perfect reconstruction, we need Tz=zl T z z l for some lN l , which implies the equivalent conditions 4z-1 P 0 z2 P 1 z2=zl 4 z -1 P 0 z 2 P 1 z 2 z l P 0 z2 P 1 z2=14z(l1) P 0 z 2 P 1 z 2 1 4 z l 1 P 0 z P 1 z=14zl12 P 0 z P 1 z 1 4 z l 1 2 For FIR polyphase filters, this can only be satisfied by P 0 z= β 0 z n 0 P 0 z β 0 z n 0 P 1 z= β 1 z n 1 P 1 z β 1 z n 1 where we have n 0 + n 1 =l12 n 0 n 1 l 1 2 and β 0 β 1 =14 β 0 β 1 1 4 .

In other words, the polyphase filters are trivial, so that the prototype filter H 0 z H 0 z has a two-tap response. With only two taps, H 0 z H 0 z cannot be a very good lowpass filter, meaning that the sub-band signals will not be spectrally well-separated. From this we conclude that two-channel 1 perfect reconstruction QMF banks exist but are not very useful.

## Footnotes

1. It turns out that MM-channel perfect reconstruction QMF banks have more useful responses for larger values of MM.

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