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.
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 .
[Image:Overlap-save algorithm.svg|thumb|right|500px|Fig 1: A sequence of four plots depicts one cycle of the overlap–save convolution algorithm. The 1st plot is a long sequence of data to be processed with a lowpass FIR filter. The 2nd plot is one segment of the data to be processed in piecewise fashion. The 3rd plot is the filtered segment, with the usable portion colored red. The 4th plot shows the filtered segment appended to the output stream. The FIR filter is a boxcar lowpass with M=16 samples, the length of the segments is L=100 samples and the overlap is 15 samples.]
The concept is to compute short segments of y 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 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. It is therefore sufficient to compute the N-point circular (or cyclic) 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 Discrete Fourier transform#Circular [convolution theorem and cross-correlation theorem|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


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 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 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:
  • 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 can be accomplished by rearranging frequency bins