12.10 • CONVOLUTION 573Figure 12.28 The region of integration in the convolution theorem.
Table 12 .4 lists the properties of convolution.Commutative
Distributive
Associative
Zerofg=gf
f * (g + h) = f * g + f * h
U*u) *h = t * (9* h)
f *0=0Table 12 .4 Properties of Convolution
•EXAMPLE 12.29 Show that .c-^1 (
28
(s^2 + 1)^2 ) = tsint.Solution lfweletF(s)=- 2
1
,G(s)=- 2
28,/(t)=sint,g(t)=2cost,
s +l s +l
respectively, and apply the convolution theorem, we get
c-^1 (~-#--) = c-^1 (F (s) G (s)) = f
1
2sin (t - r) COST dr
s + 1 s + 1 } 0= 1t [2sintcos^2 r - 2costsinrcosr] dr
= sin t (r +sin T cos r) -cost sin^2 rl~~~
= tsint+sin^2 tcost-costsin^2 t = tsint.- EXAMPLE 12 .30 Use the convolution theorem to solve the integral equa-
tion
f(t)=2cost-1
1