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.

[How to cite this work] [Order a printed hardcopy] [Comment on this page via email]

Copyright ©

Center for Computer Research in Music and Acoustics (CCRMA), Stanford University