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Necessary and Sufficient Conditions for Perfect Reconstruction

It can be shown [287] that the most general conditions for perfect reconstruction are that

$\displaystyle \zbox {\bold{R}(z)\bold{E}(z) \eqsp c z^{-K} \left[\begin{array}{cc} \bold{0}_{(N-L)\times L} & z^{-1}\bold{I}_{N-L} \\ [2pt] \bold{I}_L & \bold{0}_{L \times (N-L)} \end{array}\right]}$ (12.63)

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

$\displaystyle \left[\begin{array}{cc} \bold{0}_{(N-L)\times L} & z^{-1}\bold{I}_{N-L} \\ [2pt] \bold{I}_L & \bold{0}_{L \times (N-L)} \end{array}\right]^2 \eqsp z^{-1}\bold{I}_N.$ (12.64)

Thus, the general case is the same thing as

$\displaystyle \bold{R}(z)\bold{E}(z) \eqsp c z^{-K} \bold{I}_N.$ (12.65)

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


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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2016-07-18 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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