### Quadrature Mirror Filters (QMF)

The well studied subject of Quadrature Mirror Filters (QMF) is entered by imposing the following symmetry constraint on the analysis filters:That is, the filter for channel 1 is constrained to be a -rotation of filter 0 along the unit circle. This of course makes perfect sense for a two-channel band-splitting filter bank, and can form the basis of a

*dyadic tree*band splitting, as we'll look at in §11.9.1 below. 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 [51], 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 [214]. 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) [287]. Combining the QMF symmetry constraint with the aliasing-cancellation constraints, given by

Now, all four filters are determined by . It is easy to show using the polyphase representation of (see [287]) that the only causal FIR QMF analysis filters yielding exact perfect reconstruction are

*two-tap FIR filters*of the form

*Haar filters*, which we saw gave perfect reconstruction in the amplitude-complementary case, are also examples of a QMF filter bank:

**Next Section:**

Linear Phase Quadrature Mirror Filter Banks

**Previous Section:**

Polyphase Decomposition of Haar Example