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

- (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).

A second condition is that the analysis window be COLA at the hop size used:
![]() |
(9.25) |
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Poisson Summation Formula
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Example of Overlap-Add Convolution