Page 117 - Advanced engineering mathematics

3.4 Convolution 97

In general the transform of a product of functions does not equal the product of their trans-
forms. However, the transform of a convolution is the product of the transforms of the individual
functions. This fact is called the convolution theorem, and is the rationale for the definition.

THEOREM 3.5 The Convolution Theorem

L[ f ∗ g]= L[ f ]L[g].
Equivalently,

L[ f ∗ g](s) = F(s)G(s).
A proof is outlined in Problem 26.

The inverse transform version of the convolution theorem is
−1
L [FG]= f ∗ g. (3.8)

This states that the inverse transform of a product of two functions F(s) and G(s) is the convolu-
tion f ∗ g of the inverse transforms of the functions. This fact is sometimes useful in computing
an inverse transform.

EXAMPLE 3.14
Compute

1
−1
L .
s(s − 4) 2
Certainly, we can do this by a partial fractions decomposition. To illustrate the use of the
convolution, however, write
1 1
F(s) = and G(s) =
s (s − 4) 2
so we are computing the inverse transform of a product. By the convolution theorem,
1

−1
L = f ∗ g,
s(s − 4) 2
where
1

−1
f (t) = L = 1
s
and
1

−1 4t
g(t) = L = te .
(s − 4) 2
Then
1

−1
L = f (t) ∗ g(t)
s(s − 4) 2
t
4t 4τ
= 1 ∗ te = τe dτ
0
1 1 1
4t
4t
= te − e + .
4 16 16
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October 14, 2010 14:14 THM/NEIL Page-97 27410_03_ch03_p77-120