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Spectral Audio Signal Processing
    Multirate Polyphase Filter Banks
       Perfect Reconstruction Filter Banks
          Necessary and Sufficient Conditions for Perfect Reconstruction

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Necessary and Sufficient Conditions for Perfect Reconstruction

It can be shown [264] that the most general conditions for perfect reconstruction are that

$\displaystyle \zbox {\bold{R}(z)\bold{E}(z) = c z^{-K} \left[\begin{array}{cc} ... ...d{I}_{N-L} \\ [2pt] \bold{I}_L & \bold{0}_{L \times (N-L)} \end{array}\right]} $

for some constant $ c$ and some integer $ K\geq 0$, where $ L$ is any integer between 0 and $ N-1$.

Note that the more general form of $ \bold{R}(z)\bold{E}(z)$ above can be regarded as a (non-unique) square root of a vector unit delay, since

$\displaystyle \left[\begin{array}{cc} \bold{0}_{(N-L)\times L} & z^{-1}\bold{I}... ...bold{I}_L & \bold{0}_{L \times (N-L)} \end{array}\right]^2 = z^{-1}\bold{I}_N. $

Thus, the general case is the same thing as

$\displaystyle \bold{R}(z)\bold{E}(z) = c z^{-K} \bold{I}_N. $

except for some channel swapping and an extra sample of delay in some channels.

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Next: Polyphase View of the STFT
written by 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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