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This module covers Quadrature Mirror Filterbanks (QMF) and looks at the new design choices they implement and how they are used in perfect reconstruction.

Quadrature mirror filterbanks

The quadrature mirror filterbank (QMF) is an aliasing-cancellation filterbank with the additional designchoices:

  • H 0 z : causal real-coefficient FIR
  • H 1 z H 0 z
  • C z 2
Combining the various design rules, it is easy to see that all filters will be causal, real-coefficient, and FIR. The QMFchoices yield the system transfer function
T z H 0 z 2 H 1 z 2 H 0 z 2 H 0 z 2
The name "QMF" is appropriate for the following reason. Notethat H 1 ω H 0 ω H 0 ω H 0 ω where the last step follows from the DTFT conjugate-symmetry of real-coefficient filters. This implies that the magnituderesponses H 0 ω and H 1 ω from a mirror-image pair, symmetric around ω 2 2 4 (the "quadrature frequency"), as illustrated in .

The QMF design rules imply that all filters in the bank are directly related to the "prototype" filter H 0 z , and thus we might suspect a polyphase implementation. In fact, one exists. Using the standardpolyphase decomposition of H 0 z , we have

H 0 z P 0 z 2 z -1 P 1 z 2
so that H 1 z H 0 z P 0 z 2 z -1 P 1 z 2 G 0 z 2 H 1 z 2 P 0 z 2 2 z -1 P 1 z 2 G 1 z -2 H 0 z -2 P 0 z 2 2 z -1 P 1 z 2 Application of the Noble identity results in the polyphase structure in :

The QMF choice C z 2 implies that the synthesis filters have twice the DC gain ofthe corresponding analysis filters. Recalling that decimation by 2 involves anti-alias lowpass filtering with DC gain equalto one, while interpolation by 2 involves anti-image lowpass filtering with DC gain equal to 2, suggests an explanation for the choice C z 2 .

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Source:  OpenStax, Digital signal processing (ohio state ee700). OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10144/1.8
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