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Simple Examples of Perfect Reconstruction

If we can arrange to have

$\displaystyle \zbox {\bold{R}(z)\bold{E}(z) = \bold{I}_N}$ (12.55)

then the filter bank will reduce to the simple system shown in Fig.11.23.

Figure: Simplified filter bank when $ R(z)$ inverts $ E(z)$ .

Thus, when $ R=N$ and $ \bold{R}(z)\bold{E}(z)=\bold{I}_N$ , we have a simple parallelizer/serializer, which is perfect-reconstruction by inspection: Referring to Fig.11.23, think of the input samples $ x(n)$ as ``filling'' a length $ N-1$ delay line over $ N-1$ sample clocks. At time 0 , the downsamplers and upsamplers ``fire'', transferring $ x(0)$ (and $ N-1$ zeros) from the delay line to the output delay chain, summing with zeros. Over the next $ N-1$ clocks, $ x(0)$ makes its way toward the output, and zeros fill in behind it in the output delay chain. Simultaneously, the input buffer is being filled with samples of $ x(n)$ . At time $ N-1$ , $ x(0)$ makes it to the output. At time $ N$ , the downsamplers ``fire'' again, transferring a length $ N$ ``buffer'' [$ x(1$ :$ N)$ ] to the upsamplers. On the same clock pulse, the upsamplers also ``fire'', transferring $ N$ samples to the output delay chain. The bottom-most sample [ $ x(n-N+1)=x(1)$ ] goes out immediately at time $ N$ . Over the next $ N-1$ sample clocks, the length $ N-1$ output buffer will be ``drained'' and refilled by zeros. Simultaneously, the input buffer will be replaced by new samples of $ x(n)$ . At time $ 2N$ , the downsamplers and upsamplers ``fire'', and the process goes on, repeating with period $ N$ . The output of the $ N$ -way parallelizer/serializer is therefore

$\displaystyle {\hat x}(n) \eqsp x(n-N+1)$ (12.56)

and we have perfect reconstruction.

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2020-07-26 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University