### 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

*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

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Orthogonal Two-Channel Filter Banks

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Linear Phase Quadrature Mirror Filter Banks