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 8.3, 8.4, and 8.5), 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.8.2. 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 (8.2) is computed by the following operations:
- Frequency-shift by to get .
- Convolve with
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 8.3 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.
FBS and Perfect Reconstruction
Overlap-Add (OLA) Interpretation of the STFT