Sufficient Condition for Perfect Reconstruction

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

$\displaystyle \bold{P}(z)\mathrel{\stackrel{\mathrm{\Delta}}{=}}\bold{R}(z)\bold{E}(z) = \bold{I}_N$

and the output signal is then

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

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

$\displaystyle \bold{P}(z)$	$\displaystyle \mathrel{\stackrel{\mathrm{\Delta}}{=}}$	$\displaystyle \bold{R}(z)\bold{E}(z) = c z^{-K}\bold{I}_N$
		$\displaystyle \hbox{(Perfect Reconstruction Constraint)} \protect$	(10)

where $c\neq 0$ is any constant and

is any nonnegative integer. In this case, the output signal is

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

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