### Filter-Bank Summation (FBS) Interpretation of the STFT

We can group the terms in the STFT definition differently to obtain
the *filter-bank interpretation*:

As will be explained further below (and illustrated further in Figures 9.3, 9.4, and 9.5), under the filter-bank 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 lowpass-filtered to a narrow band about frequency

**0**(via convolving with the time-reversed window ). The STFT is thus interpreted as a

*frequency-ordered collection of narrow-band time-domain 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 time-domain output signals , , one for each channel of the -channel filter bank.

Expanding on the previous paragraph, the STFT (9.2) is computed by the following operations:

- Frequency-shift by to get .
- Convolve
with
to get
:
(10.3)

*baseband signal*; that is, it is centered about dc, with the ``carrier term'' taken out by ``demodulation'' (frequency-shifting). 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 lowpass-filter impulse response. Since the analysis window
in the STFT is typically symmetric, we usually have
.
This filter is effectively frequency-shifted to provide each channel
bandpass filter. If the cut-off frequency of the window transform is
(typically half a main-lobe 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 filter-bank interpretation for (the ``sliding STFT''). The input signal is frequency-shifted by a different amount for each channel and lowpass filtered by the (flipped) window.

**Next Section:**

FBS and Perfect Reconstruction

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Overlap-Add (OLA) Interpretation of the STFT