Overlap–save method: Difference between revisions

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In [[signal processing]], '''Overlap–save''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>''':'''

{{Equation box 1

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

|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'']}}.

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 4four 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.]]

|Rabiner and Gold, Fig 2.35, fourth trace

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 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. Consider a segment that begins at ''n'' = ''kL''&nbsp;+&nbsp;''M'', for any integer ''k'', and define''':'''

Consider a segment that begins at ''n'' = ''kL''&nbsp;+&nbsp;''M'', for any integer ''k'', and define''':'''

:<math>x_k[n] \ \triangleq

\begin{cases}

x[n+kL], & 1 \le n \le L+M-1\\

0, & \textrm{otherwise}.

\end{cases}

</math>

:<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 <math>j = n-kL</math>, the task is reduced to computing <math>y_k[j]</math> for <math>M+1 \le j \le L+M</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

The task is thereby reduced to computing ''y''_''k''[''n''], for '''''M''''' &nbsp;≤&nbsp; ''n'' &nbsp;≤&nbsp; '''''L''&nbsp;+''&nbsp;M''&nbsp;−&nbsp;1'''. The process described above is illustrated in the accompanying figure.

|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.}}'''

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

:<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>. So itIt 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

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{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 Discrete Fourier transform, respectively, evaluated over ''N'' discrete points, and

*''N''{{math|L}} is customarily chosen tosuch bethat {{math|N {{=}} L+M-1}} is an integer power-of-2, which optimizesand the efficiencytransforms ofare implemented with the [[Fast Fourier transform|FFT]] algorithm, for efficiency.

*The leading and trailing edge-effects of circular convolution are overlapped and added,{{efn-ua

*Optimal N is in the range [4M, 8M].

|Not to be confused with the [[Overlap-add method]], which preserves separate leading and trailing edge-effects.

*Unlike the third graph in the figure above, depicting separate leading and trailing edge-effects, this method causes them to be overlapped and added. So they are discarded together. In other words, with circular convolution, the first output value is a weighted average of the <u>last</u> M-1 samples of the input segment (and the first sample of the segment). The next M-2 outputs are weighted averages of both the beginning and the end of the segment. The M^th output value is the first one that combines only samples from the beginning of the segment.

}} and subsequently discarded.{{efn-ua

|1=The edge effects can be moved from the front to the back of the IDFT output by replacing <math>\scriptstyle \text{DFT}_N \displaystyle (h[n])</math> with <math>\scriptstyle \text{DFT}_N \displaystyle (h[n+M-1]) =\ \scriptstyle \text{DFT}_N \displaystyle (h[n+M-1-N]),</math> meaning that the N-length buffer is ''circularly-shifted'' (rotated) by M-1 samples. Thus the h(M) element is at n=1. The h(M-1) element is at n=N. h(M-2) is at n=N-1. Etc.}}

==Pseudocode==

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

h = FIR_impulse_response

M = length(h)

overlap = M − 1

N = 48 × overlap <span style="color:green;">(orsee next section for a nearbybetter power-of-2choice)</span>

step_size = N − overlap

H = DFT(h, N)

Line 57 ⟶ 69:

'''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 its inverse is implemented by the FFT algorithm, the pseudocode above requires about '''N log₂(N) + N''' complex multiplications for the FFT, product of arrays, and IFFT.{{efn-ua

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

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

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

{{NumBlk|:|<math>\frac{N \log_2(N) + N}{N-M+1}.\,</math>|{{EquationRef|Eq.2}}}}

|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

For example, when '''M'''=201 and '''N'''=1024, {{EquationNote|Eq.2}} equals 13.67, whereas direct evaluation of {{EquationNote|Eq.1}} would require up to 201 complex multiplications per output sample, the worst case being when both '''x''' and '''h''' are complex-valued. Also note that for any given '''M''', {{EquationNote|Eq.2}} has a minimum with respect to '''N'''. It diverges for both small and large block sizes.

|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}}}}}}

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:

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

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

:<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==

''Overlap–discard''<ref>Harris, Fname=f.J. (1987). "Time ___domain signal processing with the DFT". ''Handbook of Digital Signal Processing'', D.F.Elliot, ed., San Diego: Academic Press. pp 633–699. {{ISBN|0122370759}}.<harris/ref> and ''Overlap–scrap''<ref> name=Frerking, Marvin (1994). ''Digital Signal Processing in Communication Systems''. New York: Van Nostrand Reinhold. {{ISBN|0442016166}}.</ref> are less commonly used labels for the same method described here. However, these labels are actually better (than ''overlap–save'') to distinguish from [[Overlap–add method|overlap–add]], because <u>both</u> methods "save", but only one discards. "Save" merely refers to the fact that ''M''&nbsp;−&nbsp;1 input (or output) samples from segment ''k'' are needed to process segment ''k'' + 1.

===Extending overlap–save===

The overlap–save algorithm maycan be extended to include other common operations of a system:{{efn-ua

|[[#refCarlin|Carlin et al. 1999]], p 31, col 20.

}}<ref name=Borgerding/>

}}<ref>{{Citation |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 |url=http://www.3db-labs.com/01598092_MultibandFilterbank.pdf}}</ref>

* additional IFFT channels can be processed more cheaply than the first by reusing the forward FFT

* sampling rates can be changed by using different sized forward and inverse FFTs

* frequency translation (mixing) can be accomplished by rearranging frequency bins

== See also ==

* [[Overlap–add method]]

* [[Circular convolution#Example]]

==Notes==

Line 85 ⟶ 113:

==References==

{{reflist}}|refs=

<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>

<ref name=Frerking>

{{cite book |author=Frerking, Marvin |year=1994 |title=Digital Signal Processing in Communication Systems |___location=New York |publisher=Van Nostrand Reinhold |isbn=0442016166

}}</ref>

<ref name=Borgerding>

{{cite journal

| 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>

}}

{{refbegin}}

#<li value="4">{{Cite book

| ref=refRabiner

| author1=Rabiner, Lawrence R.

| author2=Gold, Bernard

Line 97 ⟶ 144:

| 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

|invent1=Carlin, Joe

| inventor-first =Joe

| inventor2-last invent2=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

|pubdate=1999-12-10

| inventor3-first =Peter

|fdate=1999-12-10

| inventor4-last =Hemmerdinger

|gdate=2005-05-24

| inventor4-first =Barry

|country=US

| inventor5-last =Kellogg

|status=patent

| inventor5-first =Robert

|number=6898235

| inventor6-last =Kettig

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

| 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

| issue-date = 2005

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

| country-code = US

| description = patent

| patent-number = 6898235

}}</li>

==Further reading==

*Rabiner, Lawrence R.; Gold, Bernard (1975). ''Theory and application of digital signal processing''. Englewood Cliffs, N.J.: Prentice-Hall. pp 65–67. {{ISBN|0139141014}}.

{{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}}