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 ${\displaystyle x[n]}$ with a finite impulse response (FIR) filter ${\displaystyle h[n]}$:

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 ${\displaystyle x[n]}$:

where L is an arbitrary segment length. Then:

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

where  ${\displaystyle y_{k}[n]\ {\stackrel {\mathrm {def} }{=}}\ x_{k}[n]*h[n]\,}$  is zero outside the region [1, L + M − 1].  And for any parameter  ${\displaystyle N\geq L+M-1,\,}$  it is equivalent to the ${\displaystyle N\,}$-point circular convolution of ${\displaystyle x_{k}[n]\,}$ with ${\displaystyle h[n]\,}$  in the region [1, N].

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