Overlap–add method explained
In signal processing, the overlap–add method is an efficient way to evaluate the discrete convolution of a very long signal
with a
finite impulse response (FIR) filter
:where
for
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
(see Convolution#Notation).
The concept is to divide the problem into multiple convolutions of
with short segments of
:xk[n] \triangleq \begin{cases}
x[n+kL],&n=1,2,\ldots,L\\
0,&otherwise,
\end{cases}
where
is an arbitrary segment length. Then
:
and
can be written as a sum of short convolutions
:\begin{align}
y[n]=\left(\sumkxk[n-kL]\right)*h[n]
&=\sumk\left(xk[n-kL]*h[n]\right)\\
&=\sumkyk[n-kL],
\end{align}
where the linear convolution
yk[n] \triangleq xk[n]*h[n]
is zero outside the region
And for any parameter
it is equivalent to the
-point
circular convolution of
with
in the region
The advantage is that the circular convolution can be computed more efficiently than linear convolution, according to the circular convolution theorem
:where:
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.
Pseudocode
The following is a pseudocode of the algorithm:
(Overlap-add algorithm for linear convolution) h = FIR_filter M = length(h) Nx = length(x) N = 8 × 2^ceiling(log2(M))
(8 times the smallest power of two bigger than filter length M. See next section for a slightly better choice.) step_size = N - (M-1)
(L in the text above) H = DFT(h, N) position = 0 y(1 : Nx + M-1) = 0
while position + step_size ≤ Nx
do y(position+(1:N)) = y(position+(1:N)) + IDFT(DFT(x(position+(1:step_size)), N) × H) position = position + step_size
endEfficiency 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–add method scales almost as
while the cost of a single, large circular convolution is almost
. The two methods are also compared in Figure 3, created by Matlab simulation. The contours are lines of constant ratio of the times it takes to perform both methods. When the overlap-add method is faster, the ratio exceeds 1, and ratios as high as 3 are seen.
See also
Further reading
- Book: Oppenheim, Alan V. . Schafer, Ronald W. . Digital signal processing . 1975 . Prentice-Hall . Englewood Cliffs, N.J. . 0-13-214635-5 .
- Book: Hayes, M. Horace . Digital Signal Processing . Schaum's Outline Series . 1999 . McGraw Hill . New York . 0-07-027389-8 .
- Senobari . Nader Shakibay . Funning . Gareth J. . Keogh . Eamonn . Zhu . Yan . Yeh . Chin-Chia Michael . Zimmerman . Zachary . Mueen . Abdullah . Super-Efficient Cross-Correlation (SEC-C): A Fast Matched Filtering Code Suitable for Desktop Computers . Seismological Research Letters . 90 . 1 . 2019 . 0895-0695 . 10.1785/0220180122 . 322–334 .