Circular convolution: Difference between revisions

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Revision as of 15:43, 16 June 2020

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Let ''x'' be a function with a well-defined periodic summation, ''x''_''T'', where:

:<math>x_T(t) \ \triangleq \ \sum_{k=-\infty}^\infty x(t - kT) = \sum_{k=-\infty}^\infty x(t + kT).</math>{{efn-la<ref name=McGillem/>

|[[#refMcGillem|McGillem and Cooper]], p 183 (4-51)

}}

If ''h'' is any other function for which the convolution ''x''_''T'' ∗ ''h'' exists, then the convolution ''x''_''T'' ∗ ''h'' is periodic and identical to''':'''

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}}

where ''t''_o is an arbitrary parameter and ''h''_''T'' is a [[periodic summation]] of ''h''.&nbsp; The second integral is called the '''periodic convolution'''<ref name=Jeruchim/><ref name=Udayashankara/> of functions ''x''_''T'' and ''h''_''T''.&nbsp; When ''x''_''T'' is expressed as the [[periodic summation]] of another function, ''x'', the same operation may also be referred to as a '''circular convolution'''<ref name=Udayashankara/><ref name=Priemer/> of functions ''h'' and ''x''.{{efn-ua

|This terminology is not consistent across all authors. Some authors constrain both <math>h</math> and <math>x</math> to the interval <math>[0,T]</math> and call &nbsp;<math>\int_{0}^{T} h(\tau)\cdot x(\mathrm{mod}_T(t - \tau))\ d\tau</math>&nbsp; a ''circular convolution''.}}

The second integral is called the '''periodic convolution'''<ref name=Oppenheim/><ref name=Jeruchim/><ref name=Udayashankara/> of functions ''x''_''T'' and ''h''_''T''.&nbsp; When ''x''_''T'' is expressed as the [[periodic summation]] of another function, ''x'', the same operation may also be referred to as a '''circular convolution'''<ref name=Oppenheim/><ref name=Udayashankara/><ref name=Priemer/> of functions ''h'' and ''x''.{{efn-ua

|This terminology is not consistent across all authors. Some authors consider them synonymous.{{efn-la

|[[#refMcGillem|McGillem and Cooper]], p 172 (4-5 & 4-6)}}

}}

== Discrete sequences ==

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}}</ref>

<ref name=Oppenheim>

{{cite book

|ref=refOppenheim

|last=Oppenheim

|first=Alan V.

|authorlink=Alan V. Oppenheim

|last2=Schafer

|first2=Ronald W.

|author2-link=Ronald W. Schafer

|last3=Buck

|first3=John R.

|title=Discrete-time signal processing

|pages=[https://archive.org/details/discretetimesign00alan/page/548 548],571

|year=1999

|publisher=Prentice Hall

|___location=Upper Saddle River, N.J.

|isbn=0-13-754920-2

|edition=2nd

|url-access=registration

|url=https://archive.org/details/discretetimesign00alan

}}&nbsp; Also available at https://d1.amobbs.com/bbs_upload782111/files_24/ourdev_523225.pdf

</ref>

<ref name=Priemer>

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| isbn =978-8-12-034049-7

}}</ref>

<ref name=OppenheimMcGillem>

}}

{{cite book

{{refbegin}}

#<li value="6">{{cite book

| ref=refMcGillem

| last1 =McGillem

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| last2 =Cooper

| first2 =George R.

| page =183 (4-51)

| title =Continuous and Discrete Signal and System Analysis

| publisher =Holt, Rinehart and Winston

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| date =1984

| isbn =0-03-061703-0

}}</ref>

}}

{{refbegin}}

#<li value="6">{{cite book

|ref=refOppenheim

|last=Oppenheim

|first=Alan V.

|authorlink=Alan V. Oppenheim

|last2=Schafer

|first2=Ronald W.

|author2-link=Ronald W. Schafer

|last3=Buck

|first3=John R.

|title=Discrete-time signal processing

|pages=[https://archive.org/details/discretetimesign00alan/page/548 548],571

|year=1999

|publisher=Prentice Hall

|___location=Upper Saddle River, N.J.

|isbn=0-13-754920-2

|edition=2nd

|url-access=registration

|url=https://archive.org/details/discretetimesign00alan

}}&nbsp; Also available at https://d1.amobbs.com/bbs_upload782111/files_24/ourdev_523225.pdf

</li>