Polyphase Resampling with a Rational Factor

Recall that resampling by rational rate LM LM can be accomplished through the following three-stage process(see Figure 1).

Figure 1

If we implemented the upsampler/LPF pair with a polyphase filterbank, we would still waste computations due to eventual downsampling by M M. Alternatively, if we implemented the LPF/downsampler pair with a polyphase filterbank, we would waste computations by feeding it the (mostly-zeros) upsampler output. Thus, we need to examine this problem in more detail.

Assume for the moment that we implemented the upsampler/LPF pair with a polyphase filterbank, giving the architecture in Figure 2.

Figure 2

Keeping the "parallel-to-serial" interpretation of the upsampler/delay ladder in mind, the input sequence to the decimator ql ql has the form as in Figure 3

Figure 3

leading to the observation that ql= v l L ⌊lL⌋ q l v l L l L

Thus, to calculate the resampled output at output index mm, we should calculate only the output of branch number mMmodL m M L at input index ⌊mML⌋m ML. No other branch outputs are calculated, so that no computations are wasted. The resulting structure is depicted in Figure 4.

Figure 4:

An equally-efficient structure could be obtained if we implemented the LPF/downsampler using the MM-branch polyphase decimator which was fed with the proper sequence of input samples. However, this structure is not as elegant: rather than computing the output of one particular polyphase branch per output sample, we would need to add all branch outputs, but where each branch output was calculated using a particular subset of polyphase taps.