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FilterBank Summation (FBS) Interpretation of the STFT
We can group the terms in the STFT definition differently to obtain
the filterbank interpretation:
As will be explained further below (and illustrated further in
Figures 9.3, 9.4, and 9.5), under the filterbank
interpretation, the spectrum of
is first rotated along the
unit circle in the
plane so as to shift frequency
down
to 0
(via modulation by
in the time domain), thus
forming the heterodyned signal
. Next, the heterodyned signal
is lowpassfiltered to a
narrow band about frequency 0
(via convolving with the timereversed
window
). The STFT is thus interpreted as a
frequencyordered collection of narrowband timedomain
signals, as depicted in Fig.9.2. In other words, the STFT can be
seen as a uniform filter bank in which the input signal
is converted to a set of
timedomain output signals
,
, one for each channel of the
channel filter bank.
Figure 9.2:
Filter Bank Summation (FBS) view of the STFT

Expanding on the previous paragraph, the STFT (9.2) is
computed by the following operations:
The STFT output signal
is regarded as a timedomain
signal (time index
) coming out of the
th channel of an
channel filter bank. The center frequency of the
th channel
filter is
,
. Each channel
output signal is a baseband signal; that is, it is centered
about dc, with the ``carrier term''
taken
out by ``demodulation'' (frequencyshifting). In particular, the
th channel signal is constant whenever the input signal happens to
be a sinusoid tuned to frequency
exactly.
Note that the STFT analysis window
is now interpreted as (the flip
of) a lowpassfilter impulse response. Since the analysis window
in the STFT is typically symmetric, we usually have
.
This filter is effectively frequencyshifted to provide each channel
bandpass filter. If the cutoff frequency of the window transform is
(typically half a mainlobe width), then each channel
signal can be downsampled significantly. This downsampling factor is
the FBS counterpart of the hop size
in the OLA context.
Figure 9.3 illustrates the filterbank interpretation for
(the ``sliding STFT''). The input signal
is frequencyshifted
by a different amount for each channel and lowpass filtered by the
(flipped) window.
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