Fourier and Laplace
453
c. cos(3w)
d. 2/(jw + 2)
e. 3/(jw + 2)
2
f. −1/w
2
P.4.25 Using the MATLAB symbolic toolbox, repeat P.4.24.
P.4.26 Obtain the following ILTs using Table 4.2, and indicate the Laplace properties used
in the process:
a. (s + 3)/(s
2
+ 4s + 5)
b. δ(s)
c. cos(3s)
d. (2s
2
+ 4s + 4)/(s
2
+ 3s + 2)
e. s + 3/(s + 2)
2
f. −1/s
2
P.4.27 Verify that f(t) and F(s) given as follows constitute a LT pair
f t
e
t
t e
F s
s
e
s
s
t
t
s
( )
sin(
)
( )
(
)
ϭ
ϩ
ϩ
ϭ ϩ
ϩ ϩ
ϩ ϩ
3
4
4
3
3
2
1 6
2
3
2
2 3
3
2
3
Ϫ
Ϫ
Ϫ
Ϫ
↔
P.4.28 Find the partial fraction expansion of each of the following functions:
a. 2s/(s
2
+ 5s + 4)
b. (2s
2
+ 1)/(s
2
+ 5s + 4)
c. 2s/(s
2
+ 6s + 8)
d. 2s/(s
2
+ 1s + 1)
e. s
2
/(s
2
+ 5s + 4)
f. s/(s
2
+ 3s + 4)
P.4.29 Verify the following LT pairs:
a. f t
t
t
F s
s
e
s
s
( )
cos(
)
sin(
)
( ) (
)
ϭ
ϩ
ϭ
ϩ
ϩ
5
3
2 4
3
2
5
12
9
2
2
Ϫ
Ϫ
Ϫ
↔
b. f t
t
F s
s
s
s
( ) sin ( )
( )
(
)(
)(
)
ϭ
ϭ
ϩ
ϩ
ϩ
5
2
2
2
2
2
120
1
3
5
↔
c. f t
t
F s
s s
s
s
( ) sin ( )
( )
(
)(
)(
)
ϭ
ϭ
ϩ
ϩ
ϩ
6
2
2
2
2
2
2
720
2
4
6
↔
d. f t
t
t
F s
s
s
( )
sin ( )
( )
ln
ϭ
ϭ
ϩ
2
2
2
2
1
4
2
↔

 

 
P.4.30 Verify that the current i(t) shown in the circuit diagram of Figure 4.95 is given by
i t
I s
s
s
( )
( )
= £
−
=
{
}
1
2
50
2
5
ϩ ϩ
assuming that all the initial conditions are zero.
CRC_47760_CH004.indd 453
CRC_47760_CH004.indd 453
7/28/2008 12:26:25 PM
7/28/2008 12:26:25 PM