Circular convolution: Difference between revisions

Revision as of 14:04, 1 July 2020

Revision as of 17:08, 1 July 2020

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Both forms can be called ''periodic convolution''.{{efn-la

|[[#refMcGillemMcGillem|McGillem and Cooper]], p 172 (4-6)

}} The term ''circular convolution''<ref name=Udayashankara/><ref name=Priemer/> arises from the important special case of constraining the non-zero portions of both <math>h</math> and <math>x</math> to the interval <math>[0,T].</math> Then the periodic summation becomes a ''periodic extension''{{efn-la

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

}}, which can also be expressed as a ''circular function''''':'''

:<math>x_T(t) = x(t_{\mathrm{mod}\ T}), \quad t\in\mathbb{R}\,</math> ([[Number#Real_numbers|any real number]]){{efn-la

|[[#refOppenheimOppenheim|Oppenheim and Shafer]], p 559 (8.59)

}}

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:<math>(h *x_T)(t) = \int_{0}^{T} h(\tau)\cdot x((t - \tau)_{\mathrm{mod}\ T})\ d\tau.</math>{{efn-la

|[[#refOppenheimOppenheim|Oppenheim and Shafer]], p 571 (8.114), shown in digital form

}}{{efn-la

|[[#refMcGillemMcGillem|McGillem and Cooper]], p 171 (4-22), shown in digital form

}}

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<ref name=Rabiner>

{{cite book

| author1=Rabiner, Lawrence R.

| ref=refRabiner

| author2=Gold, Bernard

| author1=Rabiner, Lawrence R.

| title=Theory and application of digital signal processing

| author2=Gold, Bernard

| pages=63–67

| title=Theory and application of digital signal processing

| year=1975

| pages=63–67

| publisher =Prentice-Hall

| year=1975

| ___location=Englewood Cliffs, N.J.

| publisher=Prentice-Hall

| isbn=0-13-914101-4

| ___location=Englewood Cliffs, N.J.

| url-access=registration

| isbn=0-13-914101-4

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

| url-access=registration

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

}}</ref>

<ref name=Priemer>

{{cite book

| last =Priemer

| ref=refPriemer

| first =Roland

| last =Priemer

| title =Introductory Signal Processing

| first =Roland

| pages=286–289

| title =Introductory Signal Processing

| publisher =World Scientific Pub Co Inc.

| pages=286–289

| series =Advanced Series in Electrical and Computer Engineering

| publisher =World Scientific Pub Co Inc.

| volume =6

| series =Advanced Series in Electrical and Computer Engineering

| date =July 1991

| volume =6

| ___location =Teaneck, N.J.

| date =July 1991

| url =https://books.google.com/?id=QBT7nP7zTLgC&printsec=frontcover&dq=Priemer,+Roland

| ___location =Teaneck, N.J.

| isbn =9971-50-919-9

| url =https://books.google.com/?id=QBT7nP7zTLgC&printsec=frontcover&dq=Priemer,+Roland

| isbn =9971-50-919-9

}}</ref>

<ref name=Jeruchim>

{{cite book

| last =Jeruchim

| ref=refJeruchim

| first =Michel C.

| last =Jeruchim

| last2 =Balaban

| first =Michel C.

| first2 =Philip

| last2 =Balaban

| last3 =Shanmugan

| first2 =Philip

| first3 =K. Sam

| last3 =Shanmugan

| title =Simulation of Communication Systems: Modeling, Methodology and Techniques

| first3 =K. Sam

| pages=73–74

| title =Simulation of Communication Systems: Modeling, Methodology and Techniques

| publisher =Kluwer Academic Publishers

| pages=73–74

| edition =2nd

| publisher =Kluwer Academic Publishers

| date =October 2000

| edition =2nd

| ___location =New York

| date =October 2000

| isbn =0-30-646267-2

| ___location =New York

| isbn =0-30-646267-2

}}</ref>

<ref name=Udayashankara>

{{cite book

| last =Udayashankara

| ref=refUdayashankara

| first =V.

| last =Udayashankara

| title =Real Time Digital Signal Processing

| first =V.

| page=189

| title =Real Time Digital Signal Processing

| publisher=Prentice-Hall

| page=189

| date =June 2010

| publisher =Prentice-Hall

| ___location =India

| date =June 2010

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

| ___location =India

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

}}</ref>

}}

{{refbegin}}

#<li value="5">{{cite book

|ref=refOppenheimOppenheim

|last=Oppenheim

|first=Alan V.

Line 157 ⟶ 153:

|year=1999

|publisher=Prentice Hall

|___location=Upper Saddle River, N.J.

|isbn=0-13-754920-2

|edition=2nd

Line 164 ⟶ 160:

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

#{{cite book

| ref=refMcGillemMcGillem

| last1 =McGillem

| first1 =Clare D.

| last2 =Cooper

| first2 =George R.

| title =Continuous and Discrete Signal and System Analysis

| publisher =Holt, Rinehart and Winston

| edition =2

| date =1984

| isbn =0-03-061703-0

}}

</li>

== Further reading ==

*{{cite book |author1=Oppenheim, Alan V. |author2=Willsky, with S. Hamid | title=Signals and Systems | publisher=Pearson Education | year=1998 | isbn=0-13-814757-4}}

{{refend}}