10981

# Convolution Sum

The component of the convolution of and is defined by . Note that is the sequence written in reverse order, and shifts this sequence units right for positive . Thus one can think of the component as an inner product of and a shifted reversed . For purposes of illustration and can have at most six nonzero terms corresponding to . These terms are entered with the controls above the delimiter. In the table the gray-shaded cells mark the position . The bold number in the table and larger point on the plot indicate .

### DETAILS

Convolution is a topic that appears in many areas of mathematics: algebra (finding the coefficients of the product of two polynomials), probability, Fourier analysis, differential equations, number theory, and so on. One important application is processing a signal by a filter. For more information see P. J. Van Fleet, Discrete Wavelet Transformations, Hoboken, New Jersey: John Wiley & Sons, Inc., 2008.
In signal processing the list is the data or input signal and the kernel is a filter or the response to a unit impulse for a linear time-invariant system. There are several examples in the bookmarks to look at and explore by modifying the terms of and . Students might want to think about and then experiment with this Demonstration to answer the following questions: (1) what scales by a constant? (2) what would cause to be a delayed version of ? and (3) what interpretation would you give to convolving a signal with itself?
Except for padded zeros at the beginning and end of , this Demonstration replicates the output of the Mathematica command ListConvolve[h, x, {1, -1}, 0]. Additional interesting applications can be found in the Mathematica help for ListConvolve, at this link.

### 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.