In signal processing, overlapâÂÂsave 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 for m outside the region . 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. That requires longer input segments 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 , 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 ⤠⤠L.
If we periodically extend x<sub>k</sub>[n] with period N âÂÂ¥ L + M â 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 <u>discarded</u>. The advantage is that the circular convolution can be computed more efficiently than linear convolution, according to the circular convolution theorem:
where:
<span style="color:green;">(Overlap-save algorithm for linear convolution)</span> h = FIR_impulse_response M = length(h) overlap = M â 1 N = 8 àoverlap <span style="color:green;">(see next section for a better choice)</span> step_size = N â overlap H = DFT(h, N) position = 0
while position + N ⤠length(x) yt = IDFT(DFT(x(position+(1:N))) àH) y(position+(1:step_size)) = yt(M : N) <span style="color:green;">(discard MâÂÂ1 y-values)</span> position = position + step_size end
When the DFT and IDFT are implemented by the FFT algorithm, the pseudocode above requires about complex multiplications for the FFT, product of arrays, and IFFT. Each iteration produces output samples, so the number of complex multiplications per output sample is about:
For example, when and equals whereas direct evaluation of would require up to complex multiplications per output sample, the worst case being when both and are complex-valued. Also note that for any given has a minimum with respect to Figure 2 is a graph of the values of that minimize for a range of filter lengths ().
Instead of , we can also consider applying to a long sequence of length 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 overlapâÂÂsave method scales almost as while the cost of a single, large circular convolution is almost .
OverlapâÂÂdiscard and OverlapâÂÂscrap are less commonly used labels for the same method described here. However, these labels are actually better (than overlapâÂÂsave) to distinguish from overlapâÂÂadd, because <u>both</u> methods "save", but only one discards. "Save" merely refers to the fact that M â 1 input (or output) samples from segment k are needed to process segment k + 1.
The overlapâÂÂsave algorithm can be extended to include other common operations of a system: