FFT Filter-Bank Summary and Fourier Duality with OLA

We have seen in this section how to make general FIR filter banks implemented using the STFT. This approach can be seen as the Fourier dual of time-domain overlap-add (OLA) STFT processing (Chapter 8).

To summarize in terms of OLA duality, we perform any desired overlap-add decomposition of the spectrum of an STFT frame in the frequency domain. Each STFT spectrum is multiplied, in the frequency domain, by a time-limited spectral window (i.e., an FIR-filter frequency-response). To obtain FFT efficiency for arbitrary spectral bands, we extend the width each spectral band (either zero-padding or ``stop-band-padding'') to the next power of 2; this is reminiscent of the OLA step of zero-padding a time-domain frame to the next power of 2. An inverse FFT is performed on the extended band, either in its natural location, or translated to baseband. For the channel signals to have a common sampling rate, each band is embedded in the same spectral domain from dc to half the common sampling rate (or the full sampling rate in the complex case), and the same-length IFFT is used for each spectral band. On the other hand, when the channel signals are downsampled to their natural bandwidths (rounded up to the next power of 2 in FFT bins), the aliasing distortion is fully controlled by the stop-band rejection of the band-filter used.