### Conjugate Quadrature Filters (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

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

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

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.)

**Next Section:**

Orthogonal Two-Channel Filter Banks

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Linear Phase Quadrature Mirror Filter Banks