## 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.

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