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Embedded SystemsFPGAElectronics

Spectral Audio Signal Processing
    Multirate Filter Banks
       Critically Sampled Perfect Reconstruction Filter Banks
          Linear Phase Quadrature Mirror Filter Banks

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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) = h_0(n) $

in which case the frequency response can be expressed as

$\displaystyle H_0(e^{j\omega}) = e^{-j\omega N/2}\left\vert H_0(e^{j\omega})\right\vert $

Substituting this into the QMF perfect reconstruction constraint (10.7) gives

$\displaystyle \hbox{constant} = 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]. $

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 \hbox{constant} = 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] $

We see that perfect reconstruction is obtained in the linear-phase case whenever the analysis filters are power complementary. See [266] for further details.

Previous: Quadrature Mirror Filters (QMF)
Next: Conjugate Quadrature Filters (CQF)

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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