Convolution Sum

​
enter data value
0
x
0
enter filter value
0
h
0
n
0
reversed and shifting x
0
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
h
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
0
y
0
0
0
0
0
0
1
2
2
2
2
2
1
0
0
0
0
0
The
th
n
component of the convolution of
h
and
x
is defined by
y
n
=
∞
∑
k=-∞
h
k
x
n-k
. Note that
x
-k
is the sequence
x
k
written in reverse order, and
x
n-k
shifts this sequence
n
units right for positive
n
. Thus one can think of the component
y
n
as an inner product of
h
and a shifted reversed
x
. For purposes of illustration
x
and
h
can have at most six nonzero terms corresponding to
n=0,1,2,3,4,5
. These terms are entered with the controls above the delimiter. In the table the gray-shaded cells mark the position
n=0
. The bold number in the table and larger point on the plot indicate
y
n
.

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
x
is the data or input signal and the kernel
h
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
x
and
h
. Students might want to think about and then experiment with this Demonstration to answer the following questions: (1) what
h
scales
x
by a constant? (2) what
h
would cause
y
to be a delayed version of
x
? and (3) what interpretation would you give to convolving a signal with itself?
Except for padded zeros at the beginning and end of
y
, 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

Bruce Atwood
​
​"Convolution Sum"​
​http://demonstrations.wolfram.com/ConvolutionSum/​
​Wolfram Demonstrations Project​
​Published: June 29, 2009