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Two-Branch Quadvalue Mirror Filterbank (QMF)

Module by: Phil Schniter. E-mail the author

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

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Quadrature Mirror Filterbanks

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

  • H 0 z H 0 z : causal real-coefficient FIR
  • H 1 z= H 0 -z H 1 z H 0 z
  • Cz=2 C z 2
Combining the various design rules, it is easy to see that all filters will be causal, real-coefficient, and FIR. The QMF choices yield the system transfer function
Tz= H 0 2z− H 1 2z= H 0 2z− H 0 2-z T z H 0 z 2 H 1 z 2 H 0 z 2 H 0 z 2 (1)
The name "QMF" is appropriate for the following reason. Note that | H 1 ⅇⅈω|=| H 0 ⅇ-ⅈω|=| H 0 ⅇⅈω−π|=| H 0 ⅇⅈπ−ω| 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 magnitude responses | H 0 ⅇⅈω| H 0 ω and | H 1 ⅇⅈω| H 1 ω from a mirror-image pair, symmetric around ω=π2=2π4 ω 2 2 4 (the "quadrature frequency"), as illustrated in Figure 1.

Figure 1
Figure 1 (qmf_f1.png)

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

H 0 z= P 0 z2+z-1 P 1 z2 H 0 z P 0 z 2 z -1 P 1 z 2 (2)
so that H 1 z= H 0 -z= P 0 z2−z-1 P 1 z2 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 z2+2z-1 P 1 z2 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 z2+2z-1 P 1 z2 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 Figure 1:

Figure 2
Figure 2 (qmf_f2.png)

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

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