### Conjugate QuadratureFilters (CQF)

A class of causal, FIR, two-channel, critically sampled, exact perfect-reconstruction filter-banks is the set of so-called Conjugate Quadrature Filters (CQF). In the z-domain, the CQF relationships are

 (12.39)

In the time domain, the analysis and synthesis filters are given by

That is, for the lowpass channel, and each highpass channel filter is a modulation of its lowpass counterpart by . Again, all four analysis and synthesis filters are determined by the lowpass analysis filter . It can be shown that this is an orthogonal filter bank. The analysis filters and are power complementary, i.e.,

 (12.40)

or

 (12.41)

where denotes the paraconjugate of (for real filters ). The paraconjugate is the analytic continuation of from the unit circle to the plane. Moreover, the analysis filters are power symmetric, e.g.,

 (12.42)

The power symmetric case was introduced by Smith and Barnwell in 1984 [272]. With the CQF constraints, (11.18) reduces to

 (12.43)

Let , such that is a spectral factor of the half-band filter (i.e., is a nonnegative power response which is lowpass, cutting off near ). Then, (11.43) reduces to

 (12.44)

The problem of PR filter design has thus been reduced to designing one half-band filter . It can be shown that any half-band filter can be written in the form . That is, all non-zero even-indexed values of are set to zero. A simple design of an FIR half-band filter would be to window a sinc function:

 (12.45)

where is any suitable window, such as the Kaiser window. Note that as a result of (11.43), the CQF filters are power complementary. That is, they satisfy

 (12.46)

Also note that the filters and are not linear phase. It can be shown that there are no two-channel perfect reconstruction filter banks that have all three of the following characteristics (except for the Haar filters):
• FIR
• orthogonal
• linear phase
In this design procedure, we have chosen to satisfy the first two and give up the third. By relaxing orthogonality'' to biorthogonality'', it becomes possible to obtain FIR linear phase filters in a critically sampled, perfect reconstruction filter bank. (See §11.9.)
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