Circular convolution: Difference between revisions

Revision as of 18:12, 3 February 2024

Revision as of 18:24, 12 April 2024

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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>