10217

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 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).

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.

PERMANENT CITATION

 Share: Embed Interactive Demonstration New! Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details » Download Demonstration as CDF » Download Author Code »(preview ») Files require Wolfram CDF Player or Mathematica.

Related Topics

 RELATED RESOURCES
 The #1 tool for creating Demonstrations and anything technical. Explore anything with the first computational knowledge engine. The web's most extensive mathematics resource. An app for every course—right in the palm of your hand. Read our views on math,science, and technology. The format that makes Demonstrations (and any information) easy to share and interact with. Programs & resources for educators, schools & students. Join the initiative for modernizing math education. Walk through homework problems one step at a time, with hints to help along the way. Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet. Knowledge-based programming for everyone.