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 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.

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 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:

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 .