### 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

the perfect reconstruction requirement reduces to

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

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 [287, 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 .

**Next Section:**

Linear Phase Quadrature Mirror Filter Banks

**Previous Section:**

Polyphase Decomposition of Haar Example