Summing STFT Bins

In the Short-Time Fourier Transform, which implements a uniform FIR filter bank (Chapter 9), each FFT bin can be regarded as one sample of the filter-bank output in one channel. It is elementary that summing adjacent filter-bank signals sums the corresponding pass-bands to create a wider pass-band. Summing adjacent FFT bins in the STFT, therefore, synthesizes one sample from a wider pass-band implemented using an FFT. This is essentially how a constant-Q transform is created from an FFT in [30] (using a different frequency-weighting, or ``smoothing kernel''). However, when making a filter bank, as opposed to only a transform used for spectrographic purposes, we must be able to step the FFT through time and compute properly sampled time-domain filter-bank signals.

The wider pass-band created by adjacent-channel summing requires a higher sampling rate in the time domain to avoid aliasing. As a result, the maximum STFT ``hop size'' is limited by the widest pass-band in the filter bank. For audio filter banks, low-frequency channels have narrow bandwidths, while high-frequency channels are wider, thereby forcing a smaller hop size for the STFT. This means that the low-frequency channels are heavily oversampled when the high-frequency channels are merely adequately sampled (in time) [30,88]. In an octave filter-bank, for example, the top octave, occupying the entire upper half of the spectrum, requires a time-domain step-size of no more than two samples, if aliasing of the band is to be avoided. Each octave down is then oversampled (in time) by an additional factor of 2.