### Summary of Overlap-Add FFT Processing

Overlap-add FFT processors provide efficient implementations for FIR filters longer than 100 or so taps on single CPUs. Specifically, we ended up with: (9.24)

where is acyclic in this context. Stated as a procedure, we have the following steps in an overlap-add FFT processor:
(1)
Extract the th length frame of data at time .
(2)
Shift it to the base time interval (or ).
(3)
Optionally apply a length analysis window (causal or zero phase, as preferred). For simple LTI filtering, the rectangular window is fine.
(4)
Zero-pad the windowed data out to the FFT size (a power of 2), such that , where is the FIR filter length.
(5)
Take the -point FFT.
(6)
Apply the filter frequency-response as a windowing operation in the frequency domain.
(7)
Take the -point inverse FFT.
(8)
Shift the origin of the -point result out to sample where it belongs.
(9)
Sum into the output buffer containing the results from prior frames (OLA step).
The condition is necessary to avoid time aliasing, i.e., to implement acyclic convolution using an FFT; this condition is equivalent to a minimum sampling-rate requirement in the frequency domain.

A second condition is that the analysis window be COLA at the hop size used: (9.25)

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