Circular convolution: Difference between revisions

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{{Short description|Mathematical operation}}

'''Circular convolution''', also known as '''cyclic convolution''', is a special case of '''periodic convolution''', which is the [[convolution]] of two periodic functions that have the same period. Periodic convolution arises, for example, in the context of the [[discrete-time Fourier transform]] (DTFT). In particular, the DTFT of the product of two discrete sequences is the periodic convolution of the DTFTs of the individual sequences. And each DTFT is a [[periodic summation]] of a continuous Fourier transform function (see {{slink|DTFT#DefinitionDiscrete-time Fourier transform|Relation to Fourier_Transform}}). Although DTFTs are usually continuous functions of frequency, the concepts of periodic and circular convolution are also directly applicable to discrete sequences of data. In that context, circular convolution plays an important role in maximizing the efficiency of a certain kind of common filtering operation.

==Definitions==

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:<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/>

where ''t''<submath>ot_o</submath> is an arbitrary parameter.&nbsp; An alternative definition, in terms of the notation of normal ''linear'' or ''aperiodic'' 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>h_{_T}(t) \ \triangleq \ \sum_{k=-\infty}^\infty h(t - kT) = \sum_{k=-\infty}^\infty h(t + kT).</math>

Then''':'''

{{NumBlk|:|<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>|{{EquationRef|Eq.1}}}}

{{Equation box 1

{{math proof|title=Derivation of Eq.1|proof=

|indent=:|cellpadding=0|border=0|background colour=white

<math display="block">\begin{align}

|equation={{NumBlk||

<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>|{{EquationRef|Eq.1}}}} &nbsp; &nbsp;

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

{{mathCollapse prooftop|title=Derivation of Eq.1|proof=}}

:<math display="block">\begin{align}

\int_{-\infty}^\infty h(\tau)\cdot x_{_T}(t - \tau)\,d\tau

&=\sum_{k=-\infty}^\infty \left[\int_{t_o+kT}^{t_o+(k+1)T} h(\tau)\cdot x_{_T}(t - \tau)\ d\tau\right] \quad t_0 \text{ is an arbitrary parameter}\\

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&=\int_{t_o}^{t_o+T} \underbrace{\left[\sum_{k=-\infty}^\infty h(u + kT)\right]}_{\triangleq \ h_{_T}(u)}\cdot x_{_T}(t - u)\ du\\

&=\int_{t_o}^{t_o+T} h_{_T}(\tau)\cdot x_{_T}(t - \tau)\ d\tau \quad \text{substituting } \tau \triangleq u

\end{align}</math>}}

{{Collapse bottom}}<br>

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

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== See also ==

*[[Convolution_theoremConvolution theorem#Functions_of_discrete_variable_sequencesFunctions of discrete variable sequences|Convolution theorem]]

*[[Circulant matrix]]

*[[Hilbert transform#Discrete Hilbert transform|Discrete Hilbert transform]]

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|year=1975

|publisher=Prentice-Hall

|___location=Englewood Cliffs, N.J.

|isbn=0-13-914101-4

|url-access=registration

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|volume=6

|date=July 1991

|___location=Teaneck, N.J.

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

|isbn=9971-50-919-9

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|year=1999

|publisher=Prentice Hall

|___location=Upper Saddle River, N.J.

|isbn=0-13-754920-2

|edition=2nd