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Equivalence of Linear and Circular Convolutions

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This Demonstration studies the equivalence of linear and circular convolutions. In signal processing, linear convolution (or simply convolution) refers to the convolution between infinitely supported sequences and filters, while circular convolution refers to the convolution between finitely supported and circularly extended sequences and filters (circular extension makes such sequences and filters periodic).

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Given a sequence of length and a filter with an impulse response of length , linear and circular convolutions are equivalent when the period of the circular convolution, , satisfies

In this Demonstration, the first graphic shows the sequence of length , the second graphic shows the filter with impulse response of length , and the third graphic shows the results of linear convolution, (in black), and circular convolution, (in red, repeated with period ). For , linear and circular convolutions are equivalent (black and red stems are identical within one period); for , linear and circular convolutions are not equivalent (black and red stems are not identical within a single period).

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Contributed by: Jelena Kovacevic (July 2012)
Open content licensed under CC BY-NC-SA

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Details

Given a sequence and a filter with an impulse response , linear convolution is defined as

The discrete-time Fourier transform (DTFT) of the linear convolution is the product of the DTFT of the sequence and the DTFT of the filter with impulse response ; in other words, linear convolution in the time domain is equivalent to multiplication in the frequency (DTFT) domain.

Given a length- sequence and a filter with a length- impulse response , circular convolution is defined by

.

The discrete Fourier transform (DFT) of the circular convolution is the product of the DFT of the sequence and the DFT of the filter with impulse response ; in other words, circular convolution in the time domain becomes multiplication in the frequency (DFT) domain.

Reference

[1] M. Vetterli, J. Kovačević, and V. K. Goyal, Foundations of Signal Processing, Cambridge: Cambridge University Press, 2014. www.fourierandwavelets.org.

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