Overlap–add method: Difference between revisions

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{{Short description|Method in signal processing}}

{{hatnote|This article is about a method of performing convolution, not to be confused with WOLA (Weight, OverLap, Add), which is a method of performing channelization. See [[Discrete-time_Fourier_transform#Sampling_the_DTFT|Sampling the DTFT]]}}

{{about|the convolution method|the "Weight, OverLap, Add" channelization method|Discrete-time Fourier transform#Sampling the DTFT{{!}}Sampling the DTFT}}

In [[signal processing]], the '''overlap–add method''' is an efficient way to evaluate the discrete [[convolution]] of a very long signal <math>x[n]</math> with a [[finite impulse response]] (FIR) filter <math>h[n]</math>''':'''

{{hatnote|This article uses common abstract notations, such as &nbsp;<math>\scriptstyle y(t) = x(t) * h(t),</math>&nbsp; or &nbsp;<math>\scriptstyle y(t) = \mathcal{H}\{x(t)\},</math>&nbsp; in which it is understood that the functions should be thought of in their totality, rather than at specific instants <math>t.</math> (see [[Convolution#Notation]])}}

{{Equation box 1

In [[signal processing]], the '''overlap–add method (OA, OLA)''' is an efficient way to evaluate the discrete [[convolution]] of a very long signal <math>x[n]</math> with a [[finite impulse response]] (FIR) filter <math>h[n]</math>''':'''

|indent= |cellpadding= 0 |border= 0 |background colour=white

|equation={{NumBlk|:|

<math>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],</math> &nbsp; &nbsp;

|{{EquationRef|Eq.1}}}}}}

where <math>h[m] = 0</math> for <math>m</math> outside the region <math>[1,M].</math>&nbsp; This article uses common abstract notations, such as <math display="inline">y(t) = x(t) * h(t),</math> or <math display="inline">y(t) = \mathcal{H}\{x(t)\},</math> in which it is understood that the functions should be thought of in their totality, rather than at specific instants <math display="inline">t</math> (see [[Convolution#Notation]]).

:<math>

\begin{align}

y[n] = x[n] * h[n] \ \stackrel{\mathrm{def}}{=} \ \sum_{m=-\infty}^{\infty} h[m] \cdot x[n-m]

= \sum_{m=1}^{M} h[m] \cdot x[n-m],

\end{align}</math>

[[Image:Overlap-add algorithm.svg|thumb|right|500px|Fig 1: A sequence of five plots depicts one cycle of the overlap-add convolution algorithm. The first 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, including the filter rise and fall transients. The 4th plot indicates where the new data will be added with the result of previous segments. The 5th plot is the updated output stream. The FIR filter is a boxcar lowpass with <math>M=16</math> samples, the length of the segments is <math>L=100</math> samples and the overlap is 15 samples.]]

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

The concept is to divide the problem into multiple convolutions of ''<math>h''[''n'']</math> with short segments of <math>x[n]</math>''':'''

:<math>x_k[n] \ \triangleq\ \stackrelbegin{\mathrm{defcases}}{=}

x[n + kL], & n = 1, 2, \ldots, L\\

\begin{cases}

0, & \text{otherwise},

x[n+kL] & n=1,2,\ldots,L\\

0 & \textrm{otherwise},

\end{cases}

</math>

where ''<math>L''</math> is an arbitrary segment length. Then''':'''

:<math>x[n] = \sum_{k} x_k[n - kL],\,</math>

and ''<math>y''[''n'']</math> can be written as a sum of short convolutions''':'''<ref name=Rabiner/>

:<math>\begin{align}

y[n] = \left(\sum_{k} x_k[n - kL]\right) * h[n]

\begin{align}

y[n] = \left(\sum_{k} x_k[n-kL]\right) * h[n] &= \sum_{k} \left(x_k[n - kL] * h[n]\right)\\

&= \sum_{k} y_k[n - kL],

\end{align}</math>

</math>

where &nbsp;the linear convolution <math>y_k[n] \ \stackrel{\mathrm{def}}{=} triangleq\ x_k[n] * h[n]\,</math>&nbsp; is zero outside the region <math>[1,&nbsp;''L''&nbsp;+&nbsp;''M''&nbsp;&minus;&nbsp;-1].</math> &nbsp;And for any parameter &nbsp;<math>N \ge L + M - 1,\,</math>&nbsp; it is equivalent to the <math>N\,</math>{{efn-point [[circular convolution]] of <math>x_k[n]\,</math> with <math>h[n]\,</math>&nbsp; in the region&nbsp;[1,&nbsp;''N''].ua

|This condition implies that the <math>x_k</math> segment has at least <math>M-1</math> appended zeros, which prevents circular overlap of the output rise and fall transients.

}} it is equivalent to the <math>N</math>-point [[circular convolution]] of <math>x_k[n]\,</math> with <math>h[n]\,</math> in the region <math>[1,N].</math>&nbsp; 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]]''':'''

{{Equation box 1

The advantage is that the [[circular convolution]] can be computed very efficiently as follows, according to the [[Discrete Fourier transform#Circular convolution theorem and cross-correlation theorem|circular convolution theorem]]''':'''

|indent= |cellpadding= 0 |border= 0 |background colour=white

|equation={{NumBlk|:|

<math>y_k[n]\ =\ \scriptstyle \text{IDFT}_N \displaystyle (\ \scriptstyle \text{DFT}_N \displaystyle (x_k[n])\cdot\ \scriptstyle \text{DFT}_N \displaystyle (h[n])\ ),</math> &nbsp; &nbsp;

|{{EquationRef|Eq.2}}}}}}

where''':'''

{{NumBlk|:|<math>y_k[n] = \textrm{IFFT}\left(\textrm{FFT}\left(x_k[n]\right)\cdot\textrm{FFT}\left(h[n]\right)\right)</math>|{{EquationRef|Eq.1}}}}

* DFT_N and IDFT_N refer to the [[Discrete Fourier transform]] and its inverse, evaluated over <math>N</math> discrete points, and

* <math>L</math> is customarily chosen such that <math>N=L+M-1</math> is an integer power-of-2, and the transforms are implemented with the [[Fast Fourier transform|FFT]] algorithm, for efficiency.

==Pseudocode==

where FFT and IFFT refer to the [[fast Fourier transform]] and inverse

The following is a [[pseudocode]] of the algorithm''':'''

fast Fourier transform, respectively, evaluated over <math>N</math> discrete

points.

<span style="color:green;">(''Overlap-add algorithm for linear convolution'')</span>

== The algorithm ==

h = FIR_filter

M = length(h)

Nx = length(x)

N = 8 × 2^ceiling( log2(M) ) <span style="color:green;">(8 times the smallest power of two bigger than filter length M. See next section for a slightly better choice.)</span>

step_size = N - (M-1) <span style="color:green;">(L in the text above)</span>

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

'''end'''

==Efficiency considerations==

[[Image:Depiction of overlap-add algorithm.png|frame|none|Figure 1: the overlap–add method]]

[[Image:FFT_size_vs_filter_length_for_Overlap-add_convolution.svg|thumb|400px|Fig 2: A graph of the values of N (an integer power of 2) that minimize the cost function <math>\tfrac{N\left(\log_2 N + 1\right)}{N - M + 1}</math>]]

When the DFT and IDFT are implemented by the FFT algorithm, the pseudocode above requires about {{nowrap|'''N (log₂(N) + 1)'''}} complex multiplications for the FFT, product of arrays, and IFFT.{{efn-ua

Fig. 1 sketches the idea of the overlap–add method. The

|1=Cooley–Tukey FFT algorithm for N=2^k needs (N/2) log₂(N) – see [[Fast Fourier transform#Definition|FFT – Definition and speed]]

signal <math>x[n]</math> is first partitioned into non-overlapping sequences,

}} Each iteration produces {{nowrap|'''N-M+1'''}} output samples, so the number of complex multiplications per output sample is about''':'''

then the [[discrete Fourier transform]]s of the sequences <math>y_k[n]</math>

are evaluated by multiplying the FFT of <math>x_k[n]</math> with the FFT of

<math>h[n]</math>. After recovering of <math>y_k[n]</math> by inverse FFT, the resulting

output signal is reconstructed by overlapping and adding the <math>y_k[n]</math>

as shown in the figure. The overlap arises from the fact that a linear

convolution is always longer than the original sequences. In the early days of development of the fast Fourier transform, <math>L</math> was often chosen to be a power of 2 for efficiency, but further development has revealed efficient transforms for larger prime factorizations of L, reducing computational sensitivity to this parameter. A [[pseudocode]] of the algorithm is the

following:

{{Equation box 1

'''Algorithm 1''' (''OA for linear convolution'')

|indent= |cellpadding= 0 |border= 0 |background colour=white

Evaluate the best value of N and L (L>0, N = M+L-1 nearest to power of 2).

|equation={{NumBlk|:|

Nx = length(x);

<math>\frac{N (\log_2(N) + 1)}{N-M+1}.\,</math> &nbsp; &nbsp;

H = FFT(h,N) <span style="color:green;">(''zero-padded FFT'')</span>

|{{EquationRef|Eq.3}}}}}}

i = 1

y = zeros(1, M+Nx-1)

'''while''' i <= Nx <span style="color:green;">(''Nx: the last index of x[n]'')</span>

il = min(i+L-1,Nx)

yt = IFFT( FFT(x(i:il),N) * H, N)

k = min(i+N-1,M+Nx-1)

y(i:k) = y(i:k) + yt(1:k-i+1) <span style="color:green;">(''add the overlapped output blocks'')</span>

i = i+L

'''end'''

This algorithm is based on the linearity of the DFT

For example, when <math>M=201</math> and <math>N=1024,</math> {{EquationNote|Eq.3}} equals <math>13.67,</math> whereas direct evaluation of {{EquationNote|Eq.1}} would require up to <math>201</math> complex multiplications per output sample, the worst case being when both <math>x</math> and <math>h</math> are complex-valued. Also note that for any given <math>M,</math> {{EquationNote|Eq.3}} has a minimum with respect to <math>N.</math> Figure 2 is a graph of the values of <math>N</math> that minimize {{EquationNote|Eq.3}} for a range of filter lengths (<math>M</math>).

== Circular convolution with the overlap–add method ==

Instead of {{EquationNote|Eq.1}}, we can also consider applying {{EquationNote|Eq.2}} to a long sequence of length <math>N_x</math> samples. The total number of complex multiplications would be:

When sequence ''x''[''n''] is periodic, and ''N''_''x'' is the period, then ''y''[''n''] is also periodic, with the same period. &nbsp;To compute one period of y[n], Algorithm 1 can first be used to convolve ''h''[''n''] with just one period of ''x''[''n'']. &nbsp;In the region ''M'' ≤ ''n'' ≤ ''N''_''x'', &nbsp;the resultant ''y''[''n''] sequence is correct. &nbsp;And if the next ''M''&nbsp;&minus;&nbsp;1 values are added to the first ''M''&nbsp;&minus;&nbsp;1 values, then the region 1 ≤ ''n'' ≤ ''N''_''x'' will represent the desired convolution. The modified pseudocode is''':'''

:<math>N_x\cdot (\log_2(N_x) + 1).</math>

'''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'''

Comparatively, the number of complex multiplications required by the pseudocode algorithm is:

== Cost of the overlap-add method ==

:<math>N_x\cdot (\log_2(N) + 1)\cdot \frac{N}{N-M+1}.</math>

The cost of the convolution can be associated to the number of complex

multiplications involved in the operation. The major computational

effort is due to the FFT operation, which for a radix-2 algorithm

applied to a signal of length <math>N</math> roughly calls for <math>C=\frac{N}{2}\log_2 N</math>

complex multiplications. It turns out that the number of complex multiplications

of the overlap-add method are:

Hence the ''cost'' of the overlap–add method scales almost as <math>O\left(N_x\log_2 N\right)</math> while the cost of a single, large circular convolution is almost <math>O\left(N_x\log_2 N_x \right)</math>. The two methods are also compared in Figure 3, created by [[MATLAB|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.

:<math>C_{OA}=\left\lceil \frac{N_x}{N-M+1}\right\rceil

N\left(\log_2 N+1\right)\,</math>

[[Image:gain oa method.png|frame|none|Fig 3: Gain of the overlap-add method compared to a single, large circular convolution. The axes show values of signal length N_x and filter length N_h.]]

<math>C_{OA}</math> accounts for the FFT+filter multiplication+IFFT operation.

==See also==

The additional cost of the <math>M_L</math> sections involved in the circular

* [[Overlap–save method]]

version of the overlap–add method is usually very small and can be

* [[Circular_convolution#Example]]

neglected for the sake of simplicity. The best value of <math>N</math>

can be found by numerical search of the minimum of <math>C_{OA}\left(N\right)=C_{OA}\left(2^m \right)</math>

by spanning the integer <math>m</math> in the range <math>\log_2\left(M\right)\le m\le\log_2 \left(N_x\right)</math>.

Being <math>N</math> a power of two, the FFTs of the overlap–add method

are computed efficiently. Once evaluated the value of <math>N</math> it

turns out that the optimal partitioning of <math>x[n]</math> has <math>L=N-M+1</math>.

For comparison, the cost of the standard circular convolution of <math>x[n]</math>

and <math>h[n]</math> is:

==Notes==

:<math>C_S=N_x\left(\log_2 N_x+1\right)\,</math>

{{notelist-ua}}

==References==

Hence the cost of the overlap–add method scales almost as <math>O\left(N_x\log_2 N\right)</math>

{{reflist|refs=

while the cost of the standard circular convolution method is almost

<ref name=Rabiner>

<math>O\left(N_x\log_2 N_x \right)</math>. However such functions accounts

{{Cite book

only for the cost of the complex multiplications, regardless of the

| author1=Rabiner, Lawrence R.

other operations involved in the algorithm. A direct measure of the

| author2=Gold, Bernard

computational time required by the algorithms is of much interest.

| title=Theory and application of digital signal processing

Fig. 2 shows the ratio of the measured time to evaluate

| year=1975

a standard circular convolution using &nbsp;{{EquationNote|Eq.1}} with

| publisher=Prentice-Hall

the time elapsed by the same convolution using the overlap–add method

| ___location=Englewood Cliffs, N.J.

in the form of Alg 2, vs. the sequence and the filter length. Both algorithms have been implemented under [[Matlab]]. The

| isbn=0-13-914101-4

bold line represent the boundary of the region where the overlap–add

| chapter=2.25

method is faster (ratio>1) than the standard circular convolution.

| pages=[https://archive.org/details/theoryapplicatio00rabi/page/63 63–65]

Note that the overlap–add method in the tested cases can be three

| chapter-url-access=registration

times faster than the standard method.

| chapter-url=https://archive.org/details/theoryapplicatio00rabi/page/63

}}</ref>

}}

==Further reading==

[[Image:gain oa method.png|frame|none|Figure 2: Ratio between the time required by &nbsp;{{EquationNote|Eq.1}} and the time required by the overlap–add Alg. 2 to evaluate

a complex circular convolution, vs the sequence length <math>N_x</math> and

the filter length <math>M</math>.]]

== See also ==

*[[Overlap–save method]]

== References ==

*{{Cite book

|author1=Rabiner, Lawrence R. |author2=Gold, Bernard | title=Theory and application of digital signal processing

| year=1975

| publisher=Prentice-Hall

| ___location=Englewood Cliffs, N.J.

| isbn=0-13-914101-4

| pages=63–67

}}

*{{Cite book

|author1=Oppenheim, Alan V. |author2=Schafer, Ronald W. | title=Digital signal processing

Line 162 ⟶ 139:

| pages=

}}

* {{cite journal | last1=Senobari | first1=Nader Shakibay | last2=Funning | first2=Gareth J. | last3=Keogh | first3=Eamonn | last4=Zhu | first4=Yan | last5=Yeh | first5=Chin-Chia Michael | last6=Zimmerman | first6=Zachary | last7=Mueen | first7=Abdullah | title=Super-Efficient Cross-Correlation (SEC-C): A Fast Matched Filtering Code Suitable for Desktop Computers | journal=Seismological Research Letters | volume=90 | issue=1 | date=2019 | issn=0895-0695 | doi=10.1785/0220180122 | pages=322–334 | url=https://www.cs.ucr.edu/~eamonn/SuperEfficientCrossCorrelation.pdf }}

== External links ==

{{DEFAULTSORT:Overlap-Add Method}}