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 -channel filter bank in Fig.11.20. The downsampling factor is . For critical sampling, we set .
The next step is to expand each analysis filter into its -channel ``type I'' polyphase representation:
Similarly, expand the synthesis filters in a type II polyphase decomposition:
The polyphase representation can now be depicted as shown in Fig.11.21. When , commuting the up/downsamplers gives the result shown in Fig.11.22. We call the polyphase matrix.
As we will show below, the above simplification can be carried out more generally whenever divides (e.g., ). In these cases becomes and becomes .