Overlap–save method: Difference between revisions

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{{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]], '''''overlap–save''''' is the traditional name for an efficient way to evaluate the [[Convolution#Discrete convolution|discrete convolution]] between a very long signal <math>x[n]</math> and a [[finite impulse response]] (FIR) filter <math>h[n]</math>''':'''

[[Image:Overlap-save algorithm.svg|thumb|500px|Fig 1: A sequence of four 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.{{efn-ua

|[[#refRabiner|Rabiner and Gold]], Fig 2.35, fourth trace.

}} 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

{{NumBlk|:|<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 {{nowrap|''h''[''m''] {{=}} 0}} for ''m'' outside the region {{nowrap|[1, ''M'']}}.

{{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]])}}.

[[Image:Overlap-save algorithm.svg|thumb|right|500px|Fig 1: A sequence of four 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.{{efn-ua|[[#refRabiner|Rabiner and Gold]], Fig 2.35, fourth trace.}} 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. That requires longer input segments that overlap the next input segment. The overlapped data gets "saved" and used a second time.<ref name=OLA/> First we describe that process with just conventional convolution for each output segment. Then we describe how to replace that convolution with a more efficient method.

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''&nbsp;+&nbsp;''M'', for any integer ''k'', and define''':'''

:<math>x_k[n] \ \triangleq

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:<math>y_k[n] \ \triangleq \ x_k[n]*h[n] = \sum_{m=1}^{M} h[m] \cdot x_k[n-m].</math>

Then, for <math>kL+M+1 \le n \le kL+L+M</math>, and equivalently <math>M+1 \le n-kL \le L+M</math>, we can write:

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

:<math>y[n] = \sum_{m=1}^{M} h[m] \cdot x_k[n-kL-m] \ \ \triangleq \ \ y_k[n-kL].</math>

With the substitution &nbsp;{{<math|>j ≜= n-kL}}</math>,&nbsp; the task is reduced to computing {{<math|y{{sub|k}}(>y_k[j)}},]</math> for '''''<math>M'''''+1 &nbsp;≤&nbsp;\le {{mvar|j}} \le &nbsp;≤&nbsp; '''''L''&nbsp;+''&nbsp;M''&nbsp;−&nbsp;1'''</math>. These steps are illustrated in the first 3 traces of Figure 1, except that the desired portion of the output (third trace) corresponds to '''''1''''' &nbsp;≤&nbsp; {{mvar|j}} &nbsp;≤&nbsp; '''''L''.{{efn-ua

|Shifting the undesirable edge effects to the last M-1 outputs is a potential run-time convenience, because the IDFT can be computed in the <math>y[n]</math> buffer, instead of being computed and copied. Then the edge effects can be overwritten by the next IDFT.&nbsp; A subsequent footnote explains how the shift is done, by a time-shift of the impulse response.}}'''

If we periodically extend ''x''_''k''[''n''] with period ''N'' &nbsp;≥&nbsp; ''L''&nbsp;+&nbsp;''M''&nbsp;−&nbsp;1, according to''':'''

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:<math>x_{k,N}[n] \ \triangleq \ \sum_{\ell=-\infty}^{\infty} x_k[n - \ell N],</math>

the convolutions &nbsp;<math>(x_{k,N})*h\,</math>&nbsp; and &nbsp;<math>x_k*h\,</math>&nbsp; are equivalent in the region ''<math> M''+1 &nbsp;≤&nbsp;\le ''n'' &nbsp;≤&nbsp;\le ''L''&nbsp;+&nbsp;''M''&nbsp;−&nbsp;1 </math>.&nbsp; It is therefore sufficient to compute the '''N'''-point [[circular convolution|circular (or cyclic) convolution]] of <math>x_k[n]\,</math> with <math>h[n]\,</math>&nbsp; in the region [1,&nbsp;''N'']. &nbsp;The subregion [''M''&nbsp;+&nbsp;1,&nbsp;''L''&nbsp;+&nbsp;''M''&nbsp;−&nbsp;1] is appended to the output stream, and the other values are <u>discarded</u>.&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

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

|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>|{{EquationRef|Eq.2}}}} &nbsp; &nbsp;

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

where''':'''

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

{{Equation box 1

{{NumBlk|:|<math>\frac{N (\log_2(N) + 1)}{N-M+1}.\,</math>|{{EquationRef|Eq.3}}}}

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

|equation={{NumBlk|:|<math>\frac{N (\log_2(N) + 1)}{N-M+1}.\,</math>|{{EquationRef|Eq.3}}}} &nbsp; &nbsp;

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

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>).

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:

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

Hence the ''cost'' of the overlap–save 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>.

==Overlap–discard==

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== See also ==

* [[Overlap–add method]]

* [[Circular convolution#Example]]

==Notes==

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<ref name=f.harris>

{{cite book |author=Harris, F.J. |year=1987 |title=Handbook of Digital Signal Processing |editor=D.F.Elliot |___location=San Diego |publisher=Academic Press |pages=633–699 |isbn=0122370759

}}</ref>

<ref name=OLA>

{{cite web|url=https://www.dsprelated.com/freebooks/sasp/Overlap_Add_OLA_STFT_Processing.html|title=Overlap-Add (OLA) STFT Processing {{!}} Spectral Audio Signal Processing |website=www.dsprelated.com |access-date=2024-03-02 |quote=The name overlap-save comes from the fact that L-1 samples of the previous frame [here: M-1 samples of the current frame] are saved for computing the next frame.

}}</ref>

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| last =Borgerding |first=Mark |title=Turning Overlap–Save into a Multiband Mixing, Downsampling Filter Bank

| journal =IEEE Signal Processing Magazine |issue= March 2006 |pages=158–161 |year=2006

|volume=23 |doi=10.1109/MSP.2006.1598092 |bibcode=2006ISPM...23..158B }}</ref>

| url =http://www.3db-labs.com/01598092_MultibandFilterbank.pdf

}}</ref>

}}

{{refbegin}}

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| chapter=2.25

| pages=[https://archive.org/details/theoryapplicatio00rabi/page/63 63–67]

| chapter-url-access=registration

| chapter-url=https://archive.org/details/theoryapplicatio00rabi/page/67

}}

#{{cite patent

|ref=refCarlin

|title=Wideband communication intercept and direction finding device using hyperchannelization

|inventor-last=Carlin

|inventor-firstinvent1=Carlin, Joe

|inventor2-lastinvent2=Collins, Terry

|invent3=Hays, Peter

|inventor2-first=Terry

|invent4=Hemmerdinger, Barry E. Kellogg, Robert L. Kettig, Robert L. Lemmon, Bradley K. Murdock, Thomas E. Tamaru, Robert S. Ware, Stuart M.

|inventor3-last=Hays

|publication-datepubdate=1999-12-10

|inventor3-first=Peter

|inventor4-last=Hemmerdinger

|inventor4-first=Barry

|inventor5-last=Kellogg

|inventor5-first=Robert

|inventor6-last=Kettig

|inventor6-first=Robert

|inventor7-last=Lemmon

|inventor7-first=Bradley

|inventor8-last=Murdock

|inventor8-first=Thomas

|inventor9-last=Tamaru

|inventor9-first=Robert

|inventor10-last=Ware

|inventor10-first=Stuart

|publication-date=1999-12-10

|fdate=1999-12-10

|gdate=2005-05-24

|country-code=US

|title=Wideband communication intercept and direction finding device using hyperchannelization

|country-code=US

|description=patent

|status=patent

|patent-number=6898235

|url=}}, <!--template creates link to worldwide.espacenet.com-->also available at https://patentimages.storage.googleapis.com/4d/39/2a/cec2ae6f33c1e7/US6898235.pdf

}}, url2=https://worldwide.espacenet.com/patent/search/family/034590049/publication/US6898235B1?q=pn%3DUS6898235

</li>

{{refend}}

== External links ==

* Dr. Deepa Kundur, [https://www.comm.utoronto.ca/~dkundur/course_info/real-time-DSP/notes/8_Kundur_Overlap_Save_Add.pdf Overlap Add and Overlap Save], University of Toronto

{{DEFAULTSORT:Overlap-save method}}