Circular convolution: Difference between revisions

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

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

:<math>x_T(t) \ \triangleq \ \sum_int_{k=-\inftyt_o}^{t_o+T} h_T(\inftytau)\cdot xx_T(t - kT\tau) = \sum_{k=-,d\infty}^\inftytau,</math> x(t&nbsp; +<ref kT).<name=Jeruchim/math><ref name=McGillemUdayashankara/>

where ''t''_o is an arbitrary parameter.&nbsp; <math>h_T(t)</math> and <math>x_T(t)</math> can also be expressed in terms of some aperiodic components <math>h</math> and <math>x</math>, i.e.:

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''':'''

:<math>h_T(t) \ \triangleq \ \sum_{k=-\infty}^\infty h(t - kT) = \sum_{k=-\infty}^\infty h(t + kT),</math>

:<math>

\begin{align}

which is called a [[periodic summation]]. Then the periodic convolution can be expressed in the notation of normal ''linear'' convolution:

(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,

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

\end{align}

</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>

}}

The term ''circular convolution'' 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''<ref name=McGillem/>{{efn-la

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

}}:

:<math>x_T(t) = x(t_{\mathrm{mod}\ T}), \quad -\infty < t < \infty.</math>

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

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><ref name=McGillem/>

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''':'''

:<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

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

Line 128 ⟶ 152:

| ___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 ⟶ 175:

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

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

<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>

</li>