### Linear Phase Quadrature Mirror Filter Banks

Linear phase filters delay all frequencies by equal amounts, and this
is often a desirable property in audio and other applications. A
filter phase response is linear in
whenever its impulse
response
is *symmetric*, *i.e.*,

(12.35) |

in which case the frequency response can be expressed as

(12.36) |

Substituting this into the QMF perfect reconstruction constraint (11.34) gives

(12.37) |

When is even, the right hand side of the above equation is forced to zero at . Therefore, we will only consider odd , for which the perfect reconstruction constraint reduces to

(12.38) |

We see that perfect reconstruction is obtained in the linear-phase case whenever the analysis filters are

*power complementary*. See [287] for further details.

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Conjugate Quadrature Filters (CQF)

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Quadrature Mirror Filters (QMF)