## Filter Banks Equivalent to STFTs

We now turn to various practical examples of perfect reconstruction
filter banks, with emphasis on those using FFTs in their
implementation (*i.e.*, various STFT filter banks).

Figure 11.28 illustrates a generic filter bank with
channels,
much like we derived in §9.3.
The analysis filters
,
are bandpass filters
derived from a lowpass prototype
by modulation (*e.g.*,
), as
shown in the right portion of the figure. The channel signals
are given by the convolution of the input signal with
the
th channel impulse response:

From Chapter 9, we recognize this expression as the sliding-window STFT, where is the flip of a sliding window ``centered'' at time , and is the th DFT bin at time . We also know from that discussion that remodulating the DFT channel outputs and summing gives perfect reconstruction of whenever is Nyquist(N) (the defining condition for Portnoff windows [213], §9.7).

Suppose the analysis window
(flip of the baseband-filter impulse
response
) is length
. Then in the context of overlap-add
processors (Chapter 8),
is a Portnoff
window, and implementing the window with a length
FFT requires
that the windowed data frame be *time-aliased* down to length
prior to taking a length
FFT (see §9.7). We can
obtain this same result via polyphase analysis, as elaborated in the
next section.

### 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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MPEG Filter Banks

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Paraunitary Filter Banks