# Direct and Transpose Polyphase Multirate Processing

Polyphase implementations of upsampling/ interpolation and downsampling/ decimation, after having invoked the Noble identities, are presented as follows (taken from Proakis):

(Three-Channel Polyphase Decimation) (Three-Channel Polyphase Interpolation)

The decimation always seems to use the direct form (delay before filtering), while the interpolation always seems to use the transposed form (delay after filtering). I understand that these two forms are equivalent (or believe that's the case). But I'm unable to find an explicit discussion on why this is canonical; why, for instance, doesn't the interpolation use the direct form? I've seen a couple of suggestions that it has something to do with the critical path, but I'm not clear on precisely what this means.

Could anyone clear this up? I suspect I'm being a bit obtuse...

## 1 Answer

The antialiasing filters here are FIR so there's no difference in the order of operations. If they were IIR there would be implications, see NUMERICAL ROBUSTNESS OF TDF-II. I think the classical representation of polyphase decimators and interpolators here is an elegant way to say that filtering has to occur at the lowest rate (before interpolation or after decimation).