### 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 :

(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