Convolution Sum

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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 .

Contributed by: Bruce Atwood (Beloit College) (June 2009)
Suggested by: Patrick Van Fleet (University of St. Thomas)
Open content licensed under CC BY-NC-SA



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.

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