The '''circularCircular 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 [[periodicspecial summation]]case of the'''periodic otherconvolution''', function.which That situation arises in the context ofis the [[Discrete Fourier transform#Circular convolution theorem and cross-correlation theorem|circular convolution theorem]].Theof identicaltwo operationperiodic canfunctions alsothat be expressed in terms ofhave the periodicsame summationsperiod. of''both''Periodic functions.convolution Thatarises, situationfor arisesexample, 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. And each DTFT is a [[periodic summation]] of a continuous Fourier transform function. (See [[Discrete-time Fourier transform#Definition]].) 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.
where ''t''<sub>o</sub> is an arbitrary parameter. 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.''':'''
