4.1 Two-branch quadvalue mirror filterbank (qmf)

Quadrature mirror filterbanks

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

• $H_0(z)$ : causal real-coefficient FIR
• $H_1(z)=H_0(-z)$
• $C(z)=2$

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

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 $C(z)=2$ .