Solution to ECE Test #5 S05
1.
L
Given that Ae− at sin ( 3t ) u ( t ) ←
→
12
, what are the numerical values of A, a
s + 6s + b
2
and b?
A = 4 , a = 3 , b = 18
L
Ae− at sin ( 3t ) u ( t ) ←
→
2.
3A
( s + a)
2
+9
=
12
⇒ A = 4 , a = 3 and b = 18
s + 6s + b
2
L
→
Let the function x ( t ) be defined by x ( t ) ←
s ( s + 5)
. x ( t ) can be written as
s 2 + 16
the sum of three functions, two of which are sinusoids.
(a)
What is the third function?
An impulse
(b)
What is the radian frequency ω of the sinusoids?
ω =4
The fraction is improper in s. Synthetically dividing,
s ( s + 5)
s
4
5 s − 16
= 1+ 2
= 1+ 5 2
−4 2
2
s + 16
s + 16
s + 16
s + 16
L
δ ( t ) + 5 cos ( 4 t ) − 4 sin ( 4 t ) ←
→1 + 5
s
4
−4 2
s + 16
s + 16
2
Solution to ECE Test #5 S05
1.
L
Let the function x ( t ) be defined by x ( t ) ←
→
s ( s + 2)
. x ( t ) can be written as
s 2 + 64
the sum of three functions, two of which are sinusoids.
(a)
What is the third function?
An impulse
(b)
What is the radian frequency ω of the sinusoids?
ω =8
The fraction is improper in s. Synthetically dividing,
s ( s + 2)
s
2 s − 64
8
= 1+ 2
= 1+ 2 2
−8 2
2
s + 64
s + 64
s + 64
s + 64
L
δ ( t ) + 2 cos (16t ) − 4 sin (16t ) ←
→1 + 2
2.
L
Given that Ae− at sin ( 5t ) u ( t ) ←
→
s
8
−8 2
s + 64
s + 64
2
30
, what are the numerical values of A,
s + 12 s + b
2
a and b?
A = 6 , a = 6 , b = 61
L
Ae− at sin ( 5t ) u ( t ) ←
→
5A
( s + a)
2
+ 25
=
30
⇒ A = 6 , a = 6 and b = 61
s + 12 s + b
2