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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 $ \omega$ whenever its impulse response $ h_0(n)$ is symmetric, i.e.,

$\displaystyle h_0(-n) \eqsp h_0(n)$ (12.35)

in which case the frequency response can be expressed as

$\displaystyle H_0(\ejo ) \eqsp e^{-j\omega N/2}\left\vert H_0(\ejo )\right\vert.$ (12.36)

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

$\displaystyle g\,e^{-j\omega d} \eqsp e^{-j\omega N}\left[ \left\vert H_0(\ejo )\right\vert^2 - (-1)^N\left\vert H_0(e^{j(\pi-\omega)})\right\vert^2\right].$ (12.37)

When $ N$ is even, the right hand side of the above equation is forced to zero at $ \omega=\pi/2$ . Therefore, we will only consider odd $ N$ , for which the perfect reconstruction constraint reduces to

$\displaystyle g\,e^{-j\omega d} \eqsp e^{-j\omega N}\left[ \left\vert H_0(\ejo )\right\vert^2 + \left\vert H_0(e^{j(\pi-\omega)}\right\vert^2\right].$ (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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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University