Overlap–add method

In signal processing, the overlap–add method (OA, OLA) is an efficient way to evaluate the discrete convolution of a very long signal with a finite impulse response (FIR) filter :

where h[m] = 0 for m outside the region [1, M].

The concept is to divide the problem into multiple convolutions of h[n] with short segments of :

where L is an arbitrary segment length. Then:

and y[n] can be written as a sum of short convolutions:

where    is zero outside the region [1, L + M − 1].  And for any parameter    it is equivalent to the -point circular convolution of with   in the region [1, N].

The advantage is that the circular convolution can be computed very efficiently as follows, according to the circular convolution theorem:

where FFT and IFFT refer to the fast Fourier transform and inverse fast Fourier transform, respectively, evaluated over discrete points.

The algorithm

Fig. 1 sketches the idea of the overlap–add method. The signal is first partitioned into non-overlapping sequences, then the discrete Fourier transforms of the sequences are evaluated by multiplying the FFT of with the FFT of . After recovering of by inverse FFT, the resulting output signal is reconstructed by overlapping and adding the as shown in the figure. The overlap arises from the fact that a linear convolution is always longer than the original sequences. In the early days of development of the fast Fourier transform, was often chosen to be a power of 2 for efficiency, but further development has revealed efficient transforms for larger prime factorizations of L, reducing computational sensitivity to this parameter. A pseudocode of the algorithm is the following: