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

$$\bold{P}(z)\isdefs \bold{R}(z)\bold{E}(z) \eqsp \bold{I}_N$$ (12.59)

and the output signal is then

$${\hat x}(n) \eqsp \frac{N}{R} \, x(n-N+1).$$ (12.60)

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

$$\bold{P}(z) \eqsp \bold{R}(z)\bold{E}(z) \eqsp c\, z^{-K}\, \bold{I}_N \protect$$ (12.61)

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

$${\hat x}(n) \eqsp c\,\frac{N}{R} \, x(n-N+1-K)$$ (12.62)

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 §11.5 below, we will look at paraunitary filter banks, for which $\bold{R}(z)$ is FIR and paraunitary whenever $\bold{E}(z)$ is.