Overlap–save method

In signal processing, overlap–save is the traditional name for an efficient way to evaluate the discrete convolution between a very long signal and a finite impulse response (FIR) filter : where h[m] = 0 for m outside the region [1, M]. This article uses common abstract notations, such as or in which it is understood that the functions should be thought of in their totality, rather than at specific instants (see Convolution#Notation). The concept is to compute short segments of y[n] 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 , and equivalently , we can write: With the substitution , the task is reduced to computing for . These steps are illustrated in the first 3 traces of Figure 1, except that the desired portion of the output (third trace) corresponds to 1 ≤ j ≤ L. If we periodically extend xk[n] with period N ≥ L + M − 1, according to: the convolutions and are equivalent in the region . It is therefore sufficient to compute the N-point circular (or cyclic) convolution of with in the region [1, N]. The subregion [M + 1, L + M] is appended to the output stream, and the other values are discarded. The advantage is that the circular convolution can be computed more efficiently than linear convolution, according to the circular convolution theorem: where: DFTN and IDFTN refer to the Discrete Fourier transform and its inverse, evaluated over N discrete points, and L is customarily chosen such that N = L+M-1 is an integer power-of-2, and the transforms are implemented with the FFT algorithm, for efficiency. The leading and trailing edge-effects of circular convolution are overlapped and added, and subsequently discarded. (Overlap-save algorithm for linear convolution) h = FIR_impulse_response M = length(h) overlap = M − 1 N = 8 × overlap (see next section for a better choice) step_size = N − overlap H = DFT(h, N) position = 0 while position + N ≤ length(x) yt = IDFT(DFT(x(position+(1:N))) × H) y(position+(1:step_size)) = yt(M : N) (discard M−1 y-values) position = position + step_size end When the DFT and IDFT are implemented by the FFT algorithm, the pseudocode above requires about N (log2(N) + 1) complex multiplications for the FFT, product of arrays, and IFFT.