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
![$\displaystyle g\,e^{-j\omega d} \eqsp e^{-j\omega N}\left[ \left\vert H_0(e^{j\omega})\right\vert^2 - (-1)^N\left\vert H_0(e^{j(\pi-\omega)})\right\vert^2\right].$](http://www.dsprelated.com/josimages_new/sasp2/img2068.png) |
(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
![$\displaystyle g\,z^{-j\omega d} \eqsp e^{-j\omega N}\left[ \left\vert H_0(e^{j\omega})\right\vert^2 + \left\vert H_0(e^{j(\pi-\omega)}\right\vert^2\right].$](http://www.dsprelated.com/josimages_new/sasp2/img2069.png) |
(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)
