### PSF Dual and Graphical Equalizers

Above, we used the Poisson Summation Formula to show that the
constant-overlap-add of a window in the time domain is equivalent to
the condition that the window transform have zero-crossings at all
harmonics of the frame rate. In this section, we look briefly at the
dual case: If the window *transform* is COLA in the frequency
domain, what is the corresponding property of the window in the time
domain? As one should expect, being COLA in the frequency domain
corresponds to having specific uniform zero-crossings in the time
domain.

Bandpass filters that sum to a constant provides an ideal basis for
a *graphic equalizer*. In such a filter bank, when all the
``sliders'' of the equalizer are set to the same level, the filter
bank reduces to no filtering at all, as desired.

Let denote the number of (complex) filters in our filter bank, with pass-bands uniformly distributed around the unit circle. (We will be using an FFT to implement such a filter bank.) Denote the frequency response of the ``dc channel'' by . Then the constant overlap-add property of the -channel filter bank can be expressed as

(9.35) |

which means

(9.36) |

where as usual. By the

*dual*of the Poisson summation formula, we have

where means that is zero at all nonzero integer multiples of ,

*i.e.*,

(9.38) |

Thus, using the dual of the PSF, we have found that a good -channel equalizer filter bank can be made using bandpass filters which have zero-crossings at multiples of samples, because that property guarantees that the filter bank sums to a constant frequency response when all channel gains are equal.

The duality introduced in this section is the basis of the
*Filter-Bank Summation* (FBS) interpretation of the short-time
Fourier transform, and it is precisely the Fourier dual of the
OverLap-Add (OLA) interpretation [9]. The FBS
interpretation of the STFT is the subject of Chapter 9.

**Next Section:**

PSF and Weighted Overlap Add

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Frequency-Domain COLA Constraints