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 $N$ 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 $W(e^{j\omega T})$ . Then the constant overlap-add property of the $N$ -channel filter bank can be expressed as

$$W \in \hbox{\sc Cola}(2\pi/N)$$ (9.35)

which means

$$S(\omega) = \sum_{k=0}^{N-1} W\left(e^{j(\omega-\omega_k)T}\right) = \hbox{constant}$$ (9.36)

where $\omega_k T\isdef k\cdot 2\pi/N$ as usual. By the dual of the Poisson summation formula, we have

$$\zbox {W \in \hbox{\sc Cola}(2\pi/N) \quad \Leftrightarrow \quad w \in \hbox{\sc Nyquist}(N)} \protect$$ (9.37)

where $w\in \hbox{\sc Nyquist}(N)$ means that $w$ is zero at all nonzero integer multiples of $N$ , i.e.,

$$w(n)=0, \quad n=\pm N,\pm 2N, \pm 3N, \ldots\,.$$ (9.38)

Thus, using the dual of the PSF, we have found that a good $N$ -channel equalizer filter bank can be made using bandpass filters which have zero-crossings at multiples of $N$ 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.