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Wikipedia / Overlap–save method

In signal processing, overlap–save is the traditional name for an efficient way to evaluate the discrete convolution and a finite impulse response:

where for m outside the region (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:

, we can write:

. 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 xk[n] with period N ≥ L+M−1, according to:

the convolutions and . It is therefore sufficient to compute the N-point circular (or cyclic) convolution 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

  • is customarily chosen such that 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.

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 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:

whereas direct evaluation of that minimize ).

Instead of , we can also consider applying samples. The total number of complex multiplications would be:

Comparatively, the number of complex multiplications required by the pseudocode algorithm is:

Hence the cost.

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:

  • additional IFFT channels can be processed more cheaply than the first by reusing the forward FFT

  • sampling rates can be changed by using different sized forward and inverse FFTs

  • frequency translation (mixing) can be accomplished by rearranging frequency bins

See also

Notes

References

  1. , also available at https://patentimages.storage.googleapis.com/4d/39/2a/cec2ae6f33c1e7/US6898235.pdf