### Polyphase Analysis of Portnoff STFT

Consider the th filter-bank channel filter

 (12.96)

The impulse-response can be any length sequence. Denote the -channel polyphase components of by , . Then by the polyphase decomposition (§11.2.2), we have

Consequently,

If is a good th-band lowpass, the subband signals are bandlimited to a region of width . As a result, there is negligible aliasing when we downsample each of the subbands by . Commuting the downsamplers to get an efficient implementation gives Fig.11.29.

First note that if for all , the system of Fig.11.29 reduces to a rectangularly windowed STFT in which the window length equals the DFT length . The downsamplers hold off'' the DFT until the length 3 delay line fills with new input samples, then it fires'' to produce a spectral frame. A new spectral frame is produced after every third sample of input data is received.

In the more general case in which are nontrivial filters, such as , for example, they can be seen to compute the equivalent of a time aliased windowed input frame, such as . This follows because the filters operate on the downsampled input stream, so that the filter coefficients operate on signal samples separated by samples. The linear combination of these samples by the filter implements the time-aliased windowed data frame in a Portnoff-windowed overlap-add STFT. Taken together, the polyphase filters compute the appropriately time-aliased data frame windowed by .

In the overlap-add interpretation of Fig.11.29, the window is hopped by samples. While this was the entire window length in the rectangular window case ( ), it is only a portion of the effective frame length when the analysis filters have order 1 or greater.

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