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 filter :
where for m outside the region.
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 kL + M ≤ n ≤ kL + L + M − 1, and equivalently M ≤ n − kL ≤ L + M − 1, we can write:
With the substitution , the task is reduced to computing, for M ≤ ≤ L + M − 1. These steps are illustrated in the first 3 traces of Figure 1, except that the desired portion of the output corresponds to 1 ≤ ≤ L.
If we periodically extend xk with period N ≥ L + M − 1, according to:
the convolutions and are equivalent in the region M ≤ n ≤ L + M − 1. It is therefore sufficient to compute the N-point circular convolution of with in the region . The subregion 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:'

h = FIR_impulse_response
M = length
overlap = M − 1
N = 8 × overlap
step_size = N − overlap
H = DFT
position = 0
while position + N ≤ length
yt = IDFT
y = yt
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:
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.
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 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 to distinguish from overlap–add, because both methods "save", but only one discards. "Save" merely refers to the fact that M − 1 input 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: