# Oversampled polyphase filter banks

i would a confirm of this: with polyphase structure is possible only design a filter bank with a INTEGER oversampling ratio? For non-integer i've seen the weighted overlap-add metod, is right?

thanks S.

If you have a 1D uniform $M$-channel multirate filter bank (FB), and $N$ is the decimation ($M \ge N$), rational oversampling ratios ($M/N$) are indeed possible (see Fig. 2), or from Figure 1 in Optimization of Synthesis Oversampled Complex Filter Banks, J. Gautier et al., IEEE Trans. Signal Processing, 2009 (doi), which provides optimized synthesis FB with any ratio (like $7/4$).

More precisely: for an analysis filter bank, you can choose any decimation you want. But you may loose information, and the invertibility of the analysis/synthesis system. In a lot of practical applications (for instance in speech processing), integer oversampling ratios are quite common, by choosing $N = M/k$, for instance $k=M/2$ or $k=M/4$. Such choices make the computations and some calculations much simpler: for instance, an inverse synthesis filter bank may take a closed form, related to the analysis window for DFT FBs.

However with redundancy, there are an infinity of inverses, and polyphase formulations help finding synthesis FBs.

A set of useful books (to me):

• Thanks! It's awesome paper! I don't Know why the book of Crochiere "Multirate Digital Signal Processing" says only integer oversampling ratio.. – Stefano Jan 5 '16 at 23:09
• 1st edition, page 325 where talks about comparision between polyphase and wola. – Stefano Jan 6 '16 at 10:22
• I understand the chapter as "In this section [...] the polyphase structure is limited primarily to the critically sampled case where $M = K$ or to cases when $MI = K$, where $I$ is an integer". Sooner, they say that "This realization is most easily seen for the case of critically sampled filter banks where $M = K$. Designs for other choices of $M$ (or $K$) are not as straightforward as the critically sampled case". Primarily is important here. It does not mean it is impossible – Laurent Duval Jan 6 '16 at 19:35
• Ok i didn't understand well. Thanks. I have another doubt .(, having M channels, N decimation factor and polyphase structure. In all documents, the analysis polyphase matrix is MxN. But Why? If the prototype filter H0 has length L, the matrix should have M x L/M, or not? Namely, for me each row of this matrix represents one polyphase component of length L/M. Where do i wrong? – Stefano Jan 6 '16 at 21:04
• Polyphase behave like a subsampling of filter coefficients by $N$. So you have the subsequences $0, N, 2N...$, $1, N+1, 2N+1...$ up to $N-1,2N-1, ...$, hence $(N-1)-0+1$ polyphase component per filter. The polyphase size does not depend on the filter size, which is one of its main advantages. – Laurent Duval Jan 6 '16 at 21:30