In signal processing, overlap–save is the traditional name for an efficient way to evaluate the discrete convolution between a very long signal x[n] and a finite impulse response (FIR) filter h[n]: 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 t (see Convolution). The concept is to compute short segments of y[n] of an arbitrary length L, and concatenate the segments together. That requires longer input seqments that overlap the next input segment. The overlapped data gets "saved" and used a second time. First we describe that process with just conventional convolution for each output segment. Then we describe how to replace that convolution with a more efficient method. 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 j = n-kL, the task is reduced to computing y_k[j] 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 x_k*h, are equivalent in the region. It is therefore sufficient to compute the N-point circular (or cyclic) convolution of x_k[n], with h[n], 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**:** L is customarily chosen such that is an integer power-of-2, and the transforms are implemented with the FFT algorithm, for efficiency.

Pseudocode

(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

Efficiency considerations

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. Each iteration produces N-M+1 output samples, so the number of complex multiplications per output sample is about**:** For example, when M=201 and N=1024, equals 13.67, whereas direct evaluation of would require up to 201 complex multiplications per output sample, the worst case being when both x and h are complex-valued. Also note that for any given M, has a minimum with respect to N. Figure 2 is a graph of the values of N that minimize for a range of filter lengths (M). Instead of, we can also consider applying to a long sequence of length N_x samples. The total number of complex multiplications would be: Comparatively, the number of complex multiplications required by the pseudocode algorithm is: Hence the cost of the overlap–save method scales almost as while the cost of a single, large circular convolution is almost.

Overlap–discard

Overlap–discard and Overlap–scrap are less commonly used labels for the same method described here. However, these labels are actually better (than overlap–save) to distinguish from overlap–add, because both methods "save", but only one discards. "Save" merely refers to the fact that M − 1 input (or output) samples from segment k are needed to process segment k + 1.

Extending overlap–save

The overlap–save algorithm can be extended to include other common operations of a system: