### 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
:

Let us define the downsampled time index as so that

(10.25) |

*i.e.*, is simply evaluated at every sample, as shown in Fig.9.17.

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