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:
![]() |
(12.49) |
![]() |
(12.50) |
![]() |
(12.51) |
Similarly, expand the synthesis filters in a type II polyphase decomposition:
![]() |
(12.52) |
![]() |
(12.53) |
![]() |
(12.54) |
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
.