Simple Examples of Perfect Reconstruction

If we can arrange to have

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

then the filter bank will reduce to the simple system below:

When $R=N$ , we have a simple parallelizer/serializer (or de-multiplexor/multiplexor, or de-interleaver/re-interleaver), which is perfect-reconstruction by inspection:

• 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'' (delay line) 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 parellelizer/serializer is therefore

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

and we have perfect reconstruction.