Simple Examples of Perfect Reconstruction

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Index: Spectral Audio Signal Processing

Spectral Audio Signal Processing

Perfect Reconstruction Filter Banks

Sliding Polyphase Filter Bank

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A delay line is used to delay a signal in time. Delay lines for digital signals are typically implemented in software using a circular buffer. — Click for https://ccrma.stanford.edu/~jos/pasp/Delay_Lines.html

A set of filters that decompose a signal into a set of components — Click for http://en.wikipedia.org/wiki/Filter_bank

A filter in the audio signal processing context is any operation that accepts a signal as an input and produces a signal as an output. Most practical audio filters are linear and time invariant, in which case they can be characterized by their impulse response or their frequency response. — Click for https://ccrma.stanford.edu/~jos/filters/What_Filter.html

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Index: Spectral Audio Signal Processing

Spectral Audio Signal Processing

Perfect Reconstruction Filter Banks

Sliding Polyphase Filter Bank

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 inverts .
$\includegraphics{eps/polyNchanI}$

Thus, when 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 as ``filling'' a length delay line over sample clocks. At time 0 , the downsamplers and upsamplers ``fire'', transferring (and zeros) from the delay line to the output delay chain, summing with zeros. Over the next clocks, 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 . At time , makes it to the output. At time , the downsamplers ``fire'' again, transferring a length ``buffer'' [ : ] to the upsamplers. On the same clock pulse, the upsamplers also ``fire'', transferring samples to the output delay chain. The bottom-most sample [ ] goes out immediately at time . Over the next sample clocks, the length output buffer will be ``drained'' and refilled by zeros. Simultaneously, the input buffer will be replaced by new samples of . At time , the downsamplers and upsamplers ``fire'', and the process goes on, repeating with period . The output of the -way parallelizer/serializer is therefore