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
![$ x$](http://www.dsprelated.com/josimages_new/sasp2/img38.png)
![$ z$](http://www.dsprelated.com/josimages_new/sasp2/img3.png)
![$ \omega_k$](http://www.dsprelated.com/josimages_new/sasp2/img100.png)
![$ e^{-j\omega_k n}$](http://www.dsprelated.com/josimages_new/sasp2/img1540.png)
![$ x_k(n)\isdeftext x(n)\exp(-j\omega_k
n)$](http://www.dsprelated.com/josimages_new/sasp2/img1541.png)
![$ x_k(n)$](http://www.dsprelated.com/josimages_new/sasp2/img1542.png)
![$ \hbox{\sc Flip}(w)$](http://www.dsprelated.com/josimages_new/sasp2/img1543.png)
![$ x(n)$](http://www.dsprelated.com/josimages_new/sasp2/img43.png)
![$ N$](http://www.dsprelated.com/josimages_new/sasp2/img61.png)
![$ X_n(\omega_k)$](http://www.dsprelated.com/josimages_new/sasp2/img1544.png)
![$ k=0,1,\ldots,N-1$](http://www.dsprelated.com/josimages_new/sasp2/img260.png)
![$ N$](http://www.dsprelated.com/josimages_new/sasp2/img61.png)
Expanding on the previous paragraph, the STFT (9.2) is computed by the following operations:
- Frequency-shift
by
to get
.
- Convolve
with
to get
:
(10.3)
![$ X_m(\omega_k)$](http://www.dsprelated.com/josimages_new/sasp2/img1286.png)
![$ m$](http://www.dsprelated.com/josimages_new/sasp2/img1168.png)
![$ k$](http://www.dsprelated.com/josimages_new/sasp2/img49.png)
![$ N$](http://www.dsprelated.com/josimages_new/sasp2/img61.png)
![$ k$](http://www.dsprelated.com/josimages_new/sasp2/img49.png)
![$ \omega_k =
2\pi k/N$](http://www.dsprelated.com/josimages_new/sasp2/img102.png)
![$ k=0,1,\ldots,N-1$](http://www.dsprelated.com/josimages_new/sasp2/img260.png)
![$ e^{j\omega_k m}$](http://www.dsprelated.com/josimages_new/sasp2/img1550.png)
![$ k$](http://www.dsprelated.com/josimages_new/sasp2/img49.png)
![$ \omega_k$](http://www.dsprelated.com/josimages_new/sasp2/img100.png)
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
Previous Section:
Overlap-Add (OLA) Interpretation of the STFT