Circular convolution: Difference between revisions

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The '''circular convolution''', also known as '''cyclic convolution''', of two aperiodic functions (i.e. [[Schwartz functions]]) occurs when one of them is [[convolution|convolved in the normal way]] with a [[periodic summation]] of the other function. That situation arises in the context of the [[Discrete Fourier transform#Circular convolution theorem and cross-correlation theorem|circular convolution theorem]]. The identical operation can also be expressed in terms of the periodic summations of ''both'' functions, if the infinite integration interval is reduced to just one period. That situation arises in the context of the [[discrete-time Fourier transform]] (DTFT) and is also called '''periodic convolution'''. In particular, the DTFT of the product of two discrete sequences is the periodic convolution of the DTFTs of the individual sequences.{{efn And each DTFT is a periodic summation of a continuous Fourier transform function. (See [[Discrete-uatime Fourier transform#Definition]].)

|If a sequence, ''x''[''n''], represents samples of a continuous function, ''x''(''t''), with Fourier transform ''X''(ƒ), its DTFT is a periodic summation of ''X''(ƒ). (See [[Discrete-time Fourier transform#Definition]].)

}}

==Definitions==

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

The ''periodic convolution'' of two T-periodic functions, <math>h_T(t)</math> and <math>x_T(t)</math> can be defined as''':'''

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

:<math>\int_{t_o}^{t_o+T} h_T(\tau)\cdot x_T(t - \tau)\,d\tau,</math> &nbsp; <ref name=Jeruchim/><ref name=Udayashankara/>

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''':'''

where ''t''_o is an arbitrary parameter.&nbsp; An alternative definition, in terms of the notation of normal ''linear'' convolution, follows from expressing <math>h_T(t)</math> and <math>x_T(t)</math> as [[periodic summation|periodic summations]] of aperiodic components <math>h</math> and <math>x</math>, i.e.''':'''

:<math>

\begin{align}

(x_T * h)(t)\quad &\triangleq \ \int_{-\infty}^\infty h(\tau)\cdot x_T(t - \tau)\,d\tau \\

&\equiv \int_{t_o}^{t_o+T} h_T(\tau)\cdot x_T(t - \tau)\,d\tau,

\end{align}

</math>{{efn-ua

|Proof:

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

Then''':'''

:<math>\int_{t_o}^{t_o+T} h_T(\tau)\cdot x_T(t - \tau)\,d\tau = \int_{-\infty}^\infty h(\tau)\cdot x_T(t - \tau)\,d\tau\ \triangleq\ (h *x_T)(t) = (x * h_T)(t).</math>{{efn-ua

|Proof:

:<math>\begin{align}

&\int_{-\infty}^\infty h(\tau)\cdot x_T(t - \tau)\,d\tau \\

Line 23 ⟶ 21:

&\sum_{k=-\infty}^\infty \left[\int_{t_o}^{t_o+T} h(\tau + kT)\cdot x_T(t - \tau -kT)\ d\tau\right] \\

={} &\int_{t_o}^{t_o+T} \left[\sum_{k=-\infty}^\infty h(\tau + kT)\cdot \underbrace{x_T(t - \tau-kT)}_{X_T(t - \tau), \text{ by periodicity}}\right]\ d\tau\\

={} &\int_{t_o}^{t_o+T} \underbrace{\left[\sum_{k=-\infty}^\infty h(\tau + kT)\right]}_{\triangleq \ h_T(\tau)}\cdot x_T(t - \tau)\ d\tau \quad \quad \scriptstyle{(QED)}

\end{align}</math>

}}

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

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

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

|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 write &nbsp;<math>\int_{0}^{T} h(\tau)\cdot x((t - \tau)\scriptstyle \mathrm{mod}\ T\displaystyle)\ d\tau,</math>&nbsp; which gives rise to the interpretation of a ''circular convolution''.}}

}} 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

|[[#refMcGillem|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 -\infty < t < \infty.</math>{{efn-la

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

}}

And the limits of integration reduce to the length of function <math>h</math>''':'''

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

|[[#refOppenheim|Oppenheim and Shafer]], p 571 (8.114)

}}{{efn-la

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

}}

== Discrete sequences ==

Line 128 ⟶ 141:

| ___location =India

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

}}</ref>

<ref name=McGillem>

{{cite book

| ref=refMcGillem

| last1 =McGillem

| first1 =Clare D.

| last2 =Cooper

| first2 =George R.

| page =183 (4-51)

| title =Continuous and Discrete Signal and System Analysis

| publisher =Holt, Rinehart and Winston

| edition =2

| date =1984

| isbn =0-03-061703-0

}}</ref>

}}

{{refbegin}}

#<li value="65">{{cite book

|ref=refOppenheim

|last=Oppenheim

Line 165 ⟶ 164:

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

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

#{{cite book

| ref=refMcGillem

<ref | last1 name=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>