Sufficient Condition for Perfect Reconstruction

Above, we found that, for any integer $1\leq R\leq N$ which divides $N$, a sufficient condition for perfect reconstruction is

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

and the output signal is then

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

More generally, we allow any nonzero scaling and any additional delay:

$$\bold{P}(z) \isdef \bold{R}(z)\bold{E}(z) = c z^{-K}\bold{I}_N$$

$$\hbox{(Perfect Reconstruction Constraint)} \protect$$ (11.11)

where $c\neq 0$ is any constant and $K$ is any nonnegative integer. In this case, the output signal is

$${\hat x}(n) = c\frac{N}{R} x(n-N+1-K)$$

Thus, given any polyphase matrix $\bold{E}(z)$, we can attempt to compute $\bold{R}(z) = \bold{E}^{-1}(z)$: If it is stable, we can use it to build a perfect-reconstruction filter bank. However, if $\bold{E}(z)$ is FIR, $\bold{R}(z)$ will typically be IIR. In §10.5 below, we will look at paraunitary filter banks, for which $\bold{R}(z)$ is FIR and paraunitary whenever $\bold{E}(z)$ is.