The previous section derived an efficient polyphase implementation of an FIR filter whose output was downsampled by the factor . The derivation was based on commuting the downsampler with the FIR summer. We now derive the polyphase representation of a filter of any length algebraically by splitting the impulse response into polyphase components.
The simplest nontrivial case is channels. Starting with a general linear time-invariant filter
we may separate the even- and odd-indexed terms to get
We define the polyphase component filters as follows:
Now write in terms of its polyphase components:
As a simple example, consider
Then the polyphase component filters are
For the general case of arbitrary , the basic idea is to decompose into its periodically interleaved subsequences, as indicated schematically in Fig.11.9. The polyphase decomposition into channels is given by
where the subphase filters are defined by
The signal can be obtained by passing through an advance of samples, followed by downsampling by the factor . as shown in Fig.11.10.
The polyphase decomposition of into channels in (11.11) may be termed a ``type I'' polyphase decomposition. In the ``type II'', or reverse polyphase decomposition, the powers of progress in the opposite direction:
We will see that we need type I for analysis filter banks and type II for synthesis filter banks in a general ``perfect reconstruction filter bank'' analysis/synthesis system.
leading to the result shown in Fig.11.11.
Next, we commute the : downsampler through the adders and upsampled (stretched) polyphase filters to obtain Fig.11.12. Commuting the downsampler through the subphase filters to obtain is an example of a multirate noble identity.
Figure 11.13 illustrates the so-called noble identities for commuting downsamplers/upsamplers with ``sparse transfer functions'' that can be expressed a function of . Note that downsamplers and upsamplers are linear, time-varying operators. Therefore, operation order is important. Also note that adders and multipliers (any memoryless operators) may be commuted across downsamplers and upsamplers, as shown in Fig.11.14.
Critically Sampled Perfect Reconstruction Filter Banks
Upsampling and Downsampling