Simple Examples of Perfect Reconstruction

If we can arrange to have

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

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

and we have perfect reconstruction.