- Overlap-add method
The overlap-add method (OA, OLA) is an efficient way to evaluate the discrete
convolutionbetween a very long signal with a finite impulse response(FIR) filter :
where h [m] =0 for m outside the region [1, M] .
The concept is to divide the problem into multiple convolutions of h [n] with short segmentsof :
where L is an arbitrary segment length. Then:
and y [n] can be written as a sum of short convolutions:
where is zero outside the region [1,L+M-1] . And for any parameter it is equivalent to the -point circular convolution of with in the region [1,N] .
The advantage is that the circular convolution can be computed very efficiently as follows, according to the circular convolution theorem:
where FFT and IFFT refer to the fast Fourier transform and inversefast Fourier transform, respectively, evaluated over discretepoints.
Fig. 1 sketches the idea of the overlap-add method. Thesignal is first partitioned into non-overlapping sequences,then the discrete Fourier transforms of the sequences are evaluated by multiplying the FFT of with the FFT of. After recovering of by inverse FFT, the resultingoutput signal is reconstructed by overlapping and adding the as shown in the figure. The overlap arises from the fact that a linearconvolution is always longer than the original sequences. Note that should be chosen to have a power of 2 which makesthe FFT computation efficient. A
pseudocodeof the algorithm is thefollowing:
Algorithm 1 ("OA for linear convolution") Evaluate the best value of N and L H = FFT(h,N) ("zero-padded FFT") i = 1 while i <= Nx il = min(i+L-1,Nx) yt = IFFT( FFT(x(i:il),N) * H, N) k = min(i+N-1,Nx) y(i:k) = y(i:k) + yt ("add the overlapped output blocks") i = i+L end
Circular convolutionwith the overlap-add method
When sequence x [n] is periodic, and Nx is the period, then y [n] is also periodic, with the same period. To compute one period of y [n] , Algorithm 1 can first be used to convolve h [n] with just one period of x [n] . In the region M ≤ n ≤ Nx, the resultant y [n] sequence is correct. And if the next M-1 values are added to the first M-1 values, then the region 1 ≤ n ≤ Nx will represent the desired convolution. The modified pseudocode is:
Algorithm 2 ("OA for circular convolution") Evaluate Algorithm 1 y(1:M-1) = y(1:M-1) + y(Nx+1:Nx+M-1) y = y(1:Nx) end
Cost of the overlap-add method
The cost of the convolution can be associated to the number of complexmultiplications involved in the operation. The major computationaleffort is due to the FFT operation, which for a radix-2 algorithmapplied to a signal of length roughly calls for complex multiplications. It turns out that the number of complex multiplicationsof the overlap-add method are:
accounts for the FFT+filter multiplication+IFFT operation.
The additional cost of the sections involved in the circularversion of the overlap-add method is usually very small and can beneglected for the sake of simplicity. The best value of can be found by numerical search of the minimum of by spanning the integer in the range .Being a power of two, the FFTs of the overlap-add methodare computed efficiently. Once evaluated the value of itturns out that the optimal partitioning of has .For comparison, the cost of the standard circular convolution of and is:
Hence the cost of the overlap-add method scales almost as while the cost of the standard circular convolution method is almost. However such functions accountsonly for the cost of the complex multiplications, regardless of theother operations involved in the algorithm. A direct measure of thecomputational time required by the algorithms is of much interest.Fig. 2 shows the ratio of the measured time to evaluatea standard circular convolution using EquationNote|Eq.1 withthe time elapsed by the same convolution using the overlap-add methodin the form of Alg 2, vs. the sequence and the filter length. Both algorithms have been implemented under
Matlab. Thebold line represent the boundary of the region where the overlap-addmethod is faster (ratio>1) than the standard circular convolution.Note that the overlap-add method in the tested cases can be threetimes faster than the standard method.
frame|none|Figure 2: Ratio between the time required by ">EquationNote|Eq.1 and the time required by the overlap-add Alg. 2 to evaluatea complex circular convolution, vs the sequence length andthe filter length .
*Weigthed Overlap Add (WOLA), an efficient filterbank method which uses FFT, that can be used to split a continuous signal stream into multiple equally-spaced subbands signal streams.
author=Rabiner, Lawrence R.; Gold, Bernard
title=Theory and application of digital signal processing
location=Englewood Cliffs, N.J.
author=Oppenheim, Alan V.; Schafer, Ronald W.
title=Digital signal processing
location=Englewood Cliffs, N.J.
author=Hayes, M. Horace
title = Digital Signal Processing
series = Schaum's Outline Series
* [http://www.mathworks.com Matlab] for the implementation of the overlap-add method through the function fftfilt.m.
Wikimedia Foundation. 2010.