In signal processing, the overlap–add method 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 ${\displaystyle h[m]=0}$ for ${\displaystyle m}$ outside the region ${\displaystyle [1,M].}$ 

This article uses common abstract notations, such as ${\textstyle y(t)=x(t)*h(t),}$ or ${\textstyle y(t)={\mathcal {H}}\{x(t)\},}$ in which it is understood that the functions should be thought of in their totality, rather than at specific instants ${\textstyle t}$ (see Convolution#Notation).

The concept is to divide the problem into multiple convolutions of ${\displaystyle h[n]}$ with short segments of ${\displaystyle x[n]}$:

where ${\displaystyle L}$ is an arbitrary segment length. Then:

and ${\displaystyle y[n]}$ can be written as a sum of short convolutions:

where the linear convolution ${\displaystyle y_{k}[n]\ \triangleq \ x_{k}[n]*h[n]\,}$ is zero outside the region ${\displaystyle [1,L+M-1].}$ 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 ${\displaystyle [1,N].}$  The advantage is that the circular convolution can be computed more efficiently than linear convolution, according to the circular convolution theorem:

where: