Equivalence of Linear and Circular Convolutions

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).
Given a sequence of length
M
and a filter with an impulse response of length
L
, linear and circular convolutions are equivalent when the period of the circular convolution,
N
, satisfies
N≥M+L-1.
In this Demonstration, the first graphic shows the sequence
x
n
of length
M=6
, the second graphic shows the filter with impulse response
h
n
of length
L=3
, and the third graphic shows the results of linear convolution,
l
n
(in black), and circular convolution,
c
n
(in red, repeated with period
N
). For
N≥8
, linear and circular convolutions are equivalent (black and red stems are identical within one period); for
N<8
, linear and circular convolutions are not equivalent (black and red stems are not identical within a single period).

Details

Given a sequence
x
n
and a filter with an impulse response
h
n
, linear convolution is defined as
l
n
=
∞
∑
k=-∞
x
k
h
n-k.
The discrete-time Fourier transform (DTFT) of the linear convolution is the product of the DTFT of the sequence
x
n
and the DTFT of the filter with impulse response
h
n
; in other words, linear convolution in the time domain is equivalent to multiplication in the frequency (DTFT) domain.
Given a length-
N
sequence
x
n
and a filter with a length-
N
impulse response
h
n
, circular convolution is defined by
c
n
=
N-1
∑
k=0
x
k
h
(n-k)modN
.
The discrete Fourier transform (DFT) of the circular convolution is the product of the DFT of the sequence
x
n
and the DFT of the filter with impulse response
h
n
; in other words, circular convolution in the time domain becomes multiplication in the frequency (DFT) domain.

References

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

External Links

Convolution Sum
Discrete-Time Convolution
Moving-Average Discrete Filters
Linear-Phase Discrete Filters
Ideal Nth-Band Discrete Filters

Permanent Citation

Jelena Kovacevic
​
​"Equivalence of Linear and Circular Convolutions"​
​http://demonstrations.wolfram.com/EquivalenceOfLinearAndCircularConvolutions/​
​Wolfram Demonstrations Project​
​Published: July 6, 2012