Perfect Reconstruction Filter Banks

We now consider filter banks with an arbitrary number of channels, and ask under what conditions do we obtain a perfect reconstruction filter bank? Polyphase analysis will give us the answer readily. Let's begin with the $N$-channel filter bank in Fig.11.21. The downsampling factor is $R\leq N$. For critical sampling, we set $R=N$.

Figure: $N$-channel filter bank.

The next step is to expand each analysis filter $H_k(z)$ into its $N$-channel ``Type 1'' polyphase representation:

$$H_k(z) = \sum_{l=0}^{N-1} z^{-l} E_{kl}(z^N)$$

or

$$\underbrace{\left[\begin{array}{c} H_0(z) \\ [2pt] H_1(z) \\ [2pt... ...} \\ [2pt] \vdots \\ [2pt] \!\!z^{-(N-1)}\!\!\end{array}\right]}_{\bold{e}(z)}$$

which we can write as

$$\bold{h}(z) = \bold{E}(z^N)\bold{e}(z).$$

Similarly, expand the synthesis filters in a Type II polyphase decomposition:

$$F_k(z) = \sum_{l=0}^{N-1} z^{-(N-l-1)}R_{lk}(z^N)$$

or

$$\underbrace{\left[\begin{array}{c} F_0(z) \\ [2pt] F_1(z) \\ [2pt... ...-1,1}(z^N) & \cdots & R_{N-1,N-1}(z^N)\!\! \end{array}\right]}_{\bold{R}(z^N)}$$

which we can write as

$$\bold{f}^T(z) = {\tilde{\bold{e}}}(z)\bold{R}(z^N).$$

The polyphase representation can now be depicted as shown in Fig.11.22. When $R=N$, commuting the up/downsamplers gives the result shown in Fig.11.23. We call $\bold{E}(z)$ the polyphase matrix.

Figure: Polyphase representation of the $N$-channel filter bank.

Figure: Efficient polyphase form of the $N$-channel filter bank.

As we will show below, the above simplification can be carried out more generally whenever $R$ divides $N$ (e.g., $R=N/2, N/3,\ldots, 1$). In these cases $\bold{E}(z)$ becomes $\bold{E}(z^{N/R})$ and $\bold{R}(z)$ becomes $\bold{R}(z^{N/R})$.