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 (FIR) filter

This article uses common abstract notations, such as

in which it is understood that the functions should be thought of in their totality, rather than at specific instants

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.

[1] 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

, the task is reduced to computing

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.[B] If we periodically extend xk[n] with period N  ≥  L + M − 1, according to: the convolutions

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

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: 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.

[E] Each iteration produces N-M+1 output samples, so the number of complex multiplications per output sample is about: For example, when

complex multiplications per output sample, the worst case being when both

Figure 2 is a graph of the values of

that minimize Eq.3 for a range of filter lengths (

Instead of Eq.1, we can also consider applying Eq.2 to a long sequence of length

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[2] and Overlap–scrap[3] 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.

The overlap–save algorithm can be extended to include other common operations of a system:[F][4]

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. [ A ] 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.
Fig 2: A graph of the values of N (an integer power of 2) that minimize the cost function