The well studied subject of Quadrature Mirror Filters (QMF) is entered by imposing the following symmetry constraint on the analysis filters:
In the time domain, the QMF constraint (11.33) becomes , i.e., all odd-index coefficients are negated. If is a lowpass filter cutting off near (as is typical), then is a complementary highpass filter. The exact cut-off frequency can be adjusted along with the roll-off rate to provide a maximally constant frequency-response sum.
Two-channel QMFs have been around since at least 1976 [#!CroisiereEtAl76!#], and appear to be the first critically sampled perfect reconstruction filter banks. Moreover, the Princen-Bradley filter bank, the initial foundation of MPEG audio as we now know it, was conceived as the Fourier dual of QMFs [#!PrincenBradley86!#]. Historically, the term QMF applied only to two-channel filter banks having the QMF symmetry constraint (11.33). Today, the term ``QMF filter bank'' may refer to more general PR filter banks with any number of channels and not obeying (11.33) [#!Vaidyanathan93!#].
Combining the QMF symmetry constraint with the aliasing-cancellation constraints, given by
the perfect reconstruction requirement reduces to
It is easy to show using the polyphase representation of (see [#!Vaidyanathan93!#]) that the only causal FIR QMF analysis filters yielding exact perfect reconstruction are two-tap FIR filters of the form
where and are constants, and and are integers. Therefore, only weak channel filters are available in the QMF case [ ], as we saw in the amplitude-complementary case above. On the other hand, very high quality IIR solutions are possible. See [#!Vaidyanathan93!#, pp. 201-204] for details. In practice, approximate ``pseudo QMF'' filters are more practical, which only give approximate perfect reconstruction. We'll return to this topic in §11.7.1.
The scaled Haar filters, which we saw gave perfect reconstruction in the amplitude-complementary case, are also examples of a QMF filter bank:
In this example, , and .