Overlap–add method: Difference between revisions

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{{Short description|Method in signal processing}}

{{hatnoteabout|Thisthe article is about aconvolution method|the of performing convolution, not to be confused with WOLA ("Weight, OverLap, Add)," which is achannelization method of performing channelization. See [[|Discrete-time Fourier transform#Sampling the DTFT|{{!}}Sampling the DTFT]]}}

{{hatnote|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]])}}

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>''':'''

[[Image:Overlap-add algorithm.svg|thumb|500px|Fig 1: A sequence of 5 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 M=16 samples, the length of the segments is L=100 samples and the overlap is 15 samples.]]

{{Equation box 1

:<math>

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

y[n] = x[n] * h[n]

|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}}}}}}

</math>

{{hatnote|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]])}}.

[[Image:Overlap-add algorithm.svg|thumb|right|500px|Fig 1: A sequence of 5five plots depicts one cycle of the Overlapoverlap-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 {{nowrap|''h''[''m''] {{=}} 0}} for ''m'' outside the region {{nowrap|[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\ \begin{cases}

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

0, & \text{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/>

{{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

| chapter=2.25

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

| url-access=registration

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

}}</ref>

:<math>\begin{align}

Line 46 ⟶ 36:

\end{align}</math>

where the linear convolution <math>y_k[n]\ \triangleq\ x_k[n] * h[n]\,</math> is zero outside the region {{nowrap|<math>[1, ''L'' + ''M'' − -1]}}.</math> And for any parameter <math>N \ge L + M - 1,\,</math>{{efn-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 {{nowrap|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

:<math>y_k[n] = \scriptstyle \text{DFT}^{-1} \displaystyle (\ \scriptstyle \text{DFT} \displaystyle (x_k[n])\cdot \scriptstyle \text{DFT} \displaystyle (h[n])\ ),</math>

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

|equation={{NumBlk|:|

:<math>y_k[n]\ =\ \scriptstyle \text{DFT}^{-1IDFT}_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''':'''

* DFT_N and DFTIDFT<supsub>−1N</supsub> 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==

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

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

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

h = FIR_filter

<font color=green>(''Overlap-add algorithm for linear convolution'')</font>

h = FIR_impulse_response

M = length(h)

Nx = length(x)

N = 8 × 2^ceiling( log2(M) ) <span style="color:green;">(see8 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: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

Most of the computation required is involved in complex multiplications. The number performed by an <math>N</math>-length radix-2 FFT is approximately <math>\tfrac{N}{2}\log_2 N.</math> Consequently, the number of complex multiplications of the overlap-add method is given by:

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

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

{{Equation box 1

:<math>C_{OA} = \left\lceil\frac{N_x}{N - M + 1}\right\rceil N\left(\log_2 N + 1\right)\,</math>

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

|equation={{NumBlk|:|

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

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

whichFor includesexample, thewhen FFT,<math>M=201</math> filterand multiplication<math>N=1024,</math> and{{EquationNote|Eq.3}} equals FFT<supmath>-13.67,</math> whereas direct evaluation of {{EquationNote|Eq.1}} would require up to <math>201</supmath> operationscomplex multiplications per output sample, the worst case being when both <math>x</math> and <math>N_xh</math> isare thecomplex-valued. total signalAlso lengthnote 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 <math>C_{OA{EquationNote|Eq.3}}</math> for a range of filter lengths (<math>M</math>).

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:

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

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

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

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 the standard circular convolution method 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 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>N_x\cdot (\log_2(N) + 1)\cdot \frac{N}{N-M+1}.</math>

[[Image:gain oa method.png|frame|none|Fig 3: Gain of the overlap-add method wrt the standard FFT method to evaluate a circular convolution, vs signal length N_x and filter length N_h.]]

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 thea standardsingle, large circular convolution method 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.

== See also ==

*[[Overlap–save method]]

[[Image:gain oa method.png|frame|none|Fig 3: Gain of the overlap-add method wrtcompared theto standarda FFTsingle, methodlarge tocircular evaluateconvolution. a circularThe convolution,axes show values vsof signal length N_x and filter length N_h.]]

== See also ==

* [[Overlap–save method]]

* [[Circular_convolution#Example]]

==Notes==

{{notelist-ua}}

== References ==

{{reflist}}|refs=

<ref name=Rabiner>

{{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

| chapter=2.25

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

| chapter-url-access=registration

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

}}</ref>

}}

== Further reading==

*{{Cite book

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

Line 120 ⟶ 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}}