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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.,
![$\displaystyle h_0(-n) \eqsp h_0(n)$](img2058.png) |
(12.35) |
in which case the frequency response can be expressed as
![$\displaystyle H_0(\ejo ) \eqsp e^{-j\omega N/2}\left\vert H_0(\ejo )\right\vert.$](img2059.png) |
(12.36) |
Substituting this into the QMF perfect reconstruction constraint
(11.34) gives
![$\displaystyle g\,e^{-j\omega d} \eqsp e^{-j\omega N}\left[ \left\vert H_0(\ejo )\right\vert^2 - (-1)^N\left\vert H_0(e^{j(\pi-\omega)})\right\vert^2\right].$](img2060.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
![$\displaystyle g\,e^{-j\omega d} \eqsp e^{-j\omega N}\left[ \left\vert H_0(\ejo )\right\vert^2 + \left\vert H_0(e^{j(\pi-\omega)}\right\vert^2\right].$](img2061.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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