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Two-Branch Quadvalue Mirror Filterbank (QMF)

Module by: Phil Schniter

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.

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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