1.12 Polyphase resampling with a rational factor

Polyphase resampling with a rational factor

Recall that resampling by rational rate $\frac{L}{M}$ can be accomplished through the following three-stage process(see ).

If we implemented the upsampler/LPF pair with a polyphase filterbank, we would still waste computations due to eventual downsampling by $M$ . 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 .

Keeping the "parallel-to-serial" interpretation of the upsampler/delay ladder in mind, the input sequence to the decimator $q(l)$ has the form as in

leading to the observation that $$q(l)=v_{{〈l〉}_L}(\lfloor \frac{l}{L}\rfloor )$$

Thus, to calculate the resampled output at output index $m$ , we should calculate only the output of branch number $mM\mod L$ at input index $\lfloor \frac{mM}{L}\rfloor$ . No other branch outputs are calculated, so that no computations are wasted. The resulting structure is depicted in .

An equally-efficient structure could be obtained if we implemented the LPF/downsampler using the $M$ -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.