Downsampled STFT Filter Bank
So far we have considered only
(the ``sliding'' DFT) in our
filter-bank interpretation of the STFT. For
we obtain a
downsampled version of
:
![\begin{eqnarray*}
X_{mR}(\omega_k) &=& \sum_{n=-\infty}^\infty [x(n)e^{-j\omega_kn}]\tilde{w}(mR-n)
\hspace{1.2cm} (\tilde{w} \mathrel{\stackrel{\Delta}{=}}\hbox{\sc Flip}(w)) \\
&=& (x_k \ast {\tilde w})(mR)
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/sasp2/img1659.png)
Let us define the downsampled time index as
so
that
![]() |
(10.25) |
i.e.,



Note that this can be considered an implementation of a phase vocoder filter bank [212]. (See §G.5 for an introduction to the vocoder.)
Filter Bank Reconstruction
Since the channel signals are downsampled, we generally need
interpolation in the reconstruction. Figure 9.18
indicates how we might pursue this. From studying the overlap-add
framework, we know that the inverse STFT is exact when the
window
is
, that is, when
is constant.
In only these cases can the STFT be considered a perfect
reconstruction filter bank. From the Poisson Summation Formula in
§8.3.1, we know that a condition
equivalent to the COLA condition is that the window
transform
have notches at all harmonics
of the frame rate, i.e.,
for
. In the
present context (filter-bank point of view), perfect reconstruction
appears impossible for
, because for ideal reconstruction
after downsampling, the channel anti-aliasing filter (
) and
interpolation filter (
) have to be ideal lowpass filters.
This is a true conclusion in any single channel, but not for the
filter bank as a whole. We know, for example, from the overlap-add
interpretation of the STFT that perfect reconstruction occurs for
hop-sizes greater than 1 as long as the COLA condition is met. This
is an interesting paradox to which we will return shortly.
What we would expect in the filter-bank context is that the
reconstruction can be made arbitrarily accurate given better and
better lowpass filters
and
which cut off at
(the folding frequency associated with down-sampling by
). This is
the right way to think about the STFT when spectral
modifications are involved.
In Chapter 11 we will develop the general topic of perfect reconstruction filter banks, and derive various STFT processors as special cases.
Next Section:
Downsampling with Anti-Aliasing
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Uniform Running-Sum Filter Banks