Classroom Tips and Techniques: A First Look at Convolution Robert J. Lopez Emeritus Professor of Mathematics and Maple Fellow Maplesoft This document summarizes a presentation of the concept of convolution in the Laplace transform portion of an introductory course in differential equations. The document mode in which this summary is written hides most of the input devices used for creating the learning objects found here. The output can be removed (Edit/Remove Output/From Worksheet) and restored calculation-by-calculation, but the (math) cursor needs to be positioned by hand at the end of each input command. Piecewise functions have been predefined, and graphs have been pre-computed in the auto-execute region below. This device hides all the extraneous information not pertinent to the flow of the presentation. Of course, all this information can be exposed by expanding the various document blocks containing the hidden materials. The conceptual content of the presentation centers on the definition of the convolution integral. An animation is used to provide an intuitive understanding of what the convolution integral does, and two computational devices are presented for evaluating the convolution. K t
For elementary functions such as e and sin t , the convolution integral is easily evaluated in Maple. After all, it's just a definite integral. For piecewise-defined functions, the convolution integral contains entangled conditional statements that easily become too complex for even Maple to unravel unaided. In this case, we show how to interpret the integration process in the framework of integration. Alternatively, we show that use of the convolution theorem for Laplace transforms leads to an immediate evaluation of the convolution for functions amenable to Maple's laplace and invlaplace commands. Maple has been initialized
We begin by defining two piecewise functions and displaying their graphs.
1
f t
g t t ! 0
0 K t
e
t ! 1
0
t ! 2
(1)
0
t ! 0
sin t
t ! π
0
otherwise
(2)
P
1 0.8 0.6 0.4 0.2 0
0
1
2
3
The following table contains a definition of the convolution integral, followed by graphical tools for interpreting this definition.
The Convolution Integral t
f * g
t = f t K x g x dx 0
P1
P2
2
t = 0.
1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2
2
1
0
1 x
2
0 0
1
2
3 x
The figure on the left shows (in blue) the graph of f x , and (in red), the graph of f K x , a reflection across the y-axis of the graph of f x . Thus, f x K t is a translation (to the right) of the graph of f x , and f t K x is its reflection across the y-axis. On the right is an animation. (Click on the figure to expose the animation tool-bar at the top of the Maple window.) This animation shows (in red), f t K x for a progression of t -values. The product of f t K x with g x , itself shown in black, is given by the thick blue curve. This is the integrand of the convolution integral. The area under this integrand, shaded in green, is the value of the convolution integral for each fixed value of t . This particular animation shows that the amount of green starts out as zero, increases, then decreases back to zero. Consequently, this approach to the convolution integral gives the intuitive result that the convolution of f t and g t will be a unimodal function with a single maximum. For more insight into this particular convolution, we need to determine the exact value of the convolution integral. We begin with the direct approach shown below.
Evaluating the Convolution Integral t
0
t
f t K x
g x
K t C x
d x
e
0
0
t K x ! 0
0
x !0
t K x ! 1
sin x
x !π
t K x ! 2
0
otherwise
d x
0
Maple is unable to evaluate the convolution integral because the implied inequalities defining the domain of integ 3
inequalities, we note that the integrand is nonzero exactly when the inequalities t K x R 0 t K x ! 1 x R 0 x ! π
are satisfied. In the xt -plane, a graphical representation of the feasible set is shaded in green in the following figur P3
5
4
3 t
2
1
0 0
1
2
3
4
x For fixed t , the convolution integral defines the following piecewise function.
4
t
e
K
t K x
sin x d x
t % 1
1 Kt 1 e K cos t 2 2
0 t
e FG :=
K
t K x
K
t K x
sin x d x
t % π
t K 1
1 K1 e cos t K 1 2 1 2
π
e
sin x d x t % 1 C π
t K 1
e
K
C
1 K1 e sin t K 1 2
cos t K 1
t K 1 Ke
t K 1
1 sin t 2 K
1 cos t 2
sin t K 1
1 sin t 2
C
π
Ce
K t
e
0 0
t R 1 C π
A graph of the convolution so computed is shown below. plot FG, t = 0 ..2Pi
0.6
0.5
0.4
0.3
0.2
0.1
0 0
1
2
3
4
5
6
t Graph of the convolution f t * g t As we have just seen, it can be a tedious task to obtain a convolution by directly evaluating its defining 5
definite integral. Many convolutions met in the introductory differential equations courses can be evaluated more easily via the following device.
The Convolution Theorem F s G s = L f t * g t
The convolution theorem states that the Laplace transform of a convolution is the product of the transforms of the theorem is written as K1
f t * g t = L
F s G s
we can obtain the convolution of f t and g t by inverting the product of the transforms of the factors f t and g author in his differential equations courses at the Rose-Hulman Institute of Technology. It always surprised him t settled for the graphical approach captured in the animation given earlier. fg := invlaplace laplace f t , t , s simplify convert fg, piecewise
laplace g t , t , s , s, t :
1 Kt 1 e K cos t 2 2 1 K1 e cos t K 1 2
K
1 K1 e sin t K 1 2
1 π K t 1 K1 e C e cos t K 1 2 2
1 sin t 2
C
K
K
1 cos t 2
t % C
1 sin t 2
1 K1 e sin t K 1 2
t % t %π
0
πC
That the two calculations of the convolution are equivalent is established by the following calculation, which sho simplify FG K fg
0
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