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In signal processing, overlap–save is the traditional name for an efficient way to evaluate the discrete convolution between a very long signal x {\displaystyle x} and a finite impulse response filter h {\displaystyle h} :
where h = 0 for m outside the region.This article uses common abstract notations, such as y = x ∗ h , {\textstyle y=x*h,} or y = H { x } , {\textstyle y={\mathcal {H}}\{x\},} in which it is understood that the functions should be thought of in their totality, rather than at specific instants t {\textstyle t} .
The concept is to compute short segments of y of an arbitrary length L, and concatenate the segments together. Consider a segment that begins at n = kL + M, for any integer k, and define:
Then, for k L + M + 1 ≤ n ≤ k L + L + M {\displaystyle kL+M+1\leq n\leq kL+L+M} , and equivalently M + 1 ≤ n − k L ≤ L + M {\displaystyle M+1\leq n-kL\leq L+M} , we can write: