Overlap–save method

Overlap–save is the traditional name for an efficient way to evaluate the discrete convolution between a very long signal $x[n]$ and a finite impulse response (FIR) filter $h[n]$:

$y[n]=x[n]*h[n]\ \triangleq \ \sum _{m=-\infty }^{\infty }h[m]\cdot x[n-m]=\sum _{m=1}^{M}h[m]\cdot x[n-m],\,$ Eq.1

where h[m]=0 for m outside the region [1, M].

A sequence of 4 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[n] 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:

$x_{k}[n]\ \triangleq {\begin{cases}x[n+kL]&1\leq n\leq L+M-1\\0&{\textrm {otherwise}}.\end{cases}}$

$y_{k}[n]\ \triangleq \ x_{k}[n]*h[n]=\sum _{m=1}^{M}h[m]\cdot x_{k}[n-m].$

Then, for kL + M  ≤  n  ≤  kL + L + M − 1, and equivalently M  ≤  n − kL  ≤  L + M − 1, we can write:

$y[n]=\sum _{m=1}^{M}h[m]\cdot x_{k}[n-kL-m]\ \ \triangleq \ \ y_{k}[n-kL].$

With the substitution  j ≜ n-kL,  the task is reduced to computing y_k(j), for M  ≤  j  ≤  L + M − 1. The process described above is illustrated in the accompanying figure.

Also note that if we periodically extend x_k[n] with period N  ≥  L + M − 1, according to:

$x_{k,N}[n]\ \triangleq \ \sum _{\ell =-\infty }^{\infty }x_{k}[n-\ell N],$

the convolutions  $(x_{k,N})*h\,$  and  $x_{k}*h\,$  are equivalent in the region M  ≤  n  ≤  L + M − 1. It is therefore sufficient to compute the N-point circular (or cyclic) convolution of $x_{k}[n]\,$ with $h[n]\,$  in the region [1, N].  The subregion [M, L + M − 1] 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:

$y_{k}[n]=\scriptstyle {\text{DFT}}^{-1}\displaystyle (\ \scriptstyle {\text{DFT}}\displaystyle (x_{k}[n])\cdot \scriptstyle {\text{DFT}}\displaystyle (h[n])\ ),$

where:

• DFT and DFT⁻¹ refer to the Discrete Fourier transform and its inverse, evaluated over N discrete points, and
• L is customarily chosen such that N = L+M-1 is an integer power-of-2, and the transforms are implemented with the FFT algorithm, for efficiency.
  • Optimal N is in the range [4M, 8M].
  • As shown in the figure (3rd trace), the leading and trailing edge-effects of convolution are overlapped and added (and subsequently discarded). In other words, the first output value is a weighted average of the last M-1 samples of the input segment (and the first sample of the segment). The next M-2 outputs are weighted averages of both the beginning and the end of the segment. The M^th output value is the first one that combines only samples from the beginning of the segment.

Pseudocode

(Overlap-save algorithm for linear convolution)
h = FIR_impulse_response
M = length(h)
overlap = M − 1
N = 4 × overlap    (or a nearby power-of-2)
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

When the DFT and its inverse is implemented by the FFT algorithm, the pseudocode above requires about N log₂(N) + N complex multiplications for the FFT, product of arrays, and IFFT. Each iteration produces N-M+1 output samples, so the number of complex multiplications per output sample is about:

${\frac {N\log _{2}(N)+N}{N-M+1}}.\,$ Eq.2

For example, when M=201 and N=1024, Eq.2 equals 13.67, whereas direct evaluation of Eq.1 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, Eq.2 has a minimum with respect to N. It diverges for both small and large block sizes.

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 may be extended to include other common operations of a system:

• additional 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

• Overlap–add method

References

4. Rabiner, Lawrence R.; Gold, Bernard (1975). "2.25". Theory and application of digital signal processing. Englewood Cliffs, N.J.: Prentice-Hall. pp. 63–67. ISBN 0-13-914101-4.
5. US patent 6898235, Carlin, Joe; Collins, Terry & Hays, Peter et al., "Wideband communication intercept and direction finding device using hyperchannelization", published 1999, issued 2005