145
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w
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g&
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w
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dw R
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X$9 u p
p
o
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The particular solution obtaine
Solutions to Problems from Chapter 6
Problem 6.1
We rst convolve the signal f2(t) and the signal p2 (t 0 1) and nd f (t) = f2 (t) 3 p2 (t 0 1). To get the required
result f2 (t) 3 p2 (t 0 2) = f2 (t) 3 f1 (t) we will shift the result obtained, f (t), by o
Solutions to Problems from Chapter 3
Problem 3.1
For the periodic signal in FIGURE 3.191, the trigonometric Fourier series coefcients are obtained as follows
t
x(t) = x(t + T ); x(t) = ;
T
an =
=
2
T
ZT
x( ) cos (n!0 )d =
0
2
bn =
2
2 cos
0)
T
=
ZT
2
2 si
332:347 Linear Systems Lab Fall 2016
Lab 3 S. J. Orfanidis
In this lab, besides the theoretical and numerical questions, you will need to produce three graphs
for problem 1, and 18 graphs for problem 2.
1. In this part, you will study the sinusoidal stead
332:345 Linear Systems and Signals Fall 2016
Solved Examples Set 2 S. J. Orfanidis
In this set, we consider the numerical implementation of the examples of Set-1 by converting
their analog transfer functions into digital ones and implementing them in MATL
332:345 Linear Systems and Signals Fall 2016
Convolution Examples Set 3 S. J. Orfanidis
The convolution between two signals h(t) and x(t), denoted by y(t)= h(t)x(t), is defined by
the following integrals,
y(t)=
h(t )x(t t )dt =
h(t t )x(t )dt
(1)
In this
332:345 Linear Systems and Signals Fall 2016
Solved Examples Set 1 S. J. Orfanidis
Problem 1
Consider the following linear system, driven by the input x(t)= 10 e3t u(t), and subject to
(0 )= 5,
the initial conditions at t = 0 , y(0 )= 0, y
(t)+3
(t) ,
y
y
RUTGERS UNIVERSITY
School of Engineering
Department of Electrical & Computer Engineering
332:345/347 Linear Systems and Signals Fall 2016
Course Description:
This course is an introduction to the basic principles and applications of linear systems. It
cov
Solutions to Problems from Chapter 5
Problem 5.1
By the denition of the
Z
transform, we have
Zff [k + k0]u[k]g =
1
X
f [k + k0]u[k]z 0k
k=0
Introducing the change of variables as
1
X
k + k0
=
i,
the above sum is equal to
f [i]u[i 0 k0]z 0(i0k0) = z k0
=
z
~
The corresponding magnitude spectrum is presented in Figure 9.1. Note that the magnitude spectrum is
237
periodic.
o ( Y&#
7 | %
ji y
o 817
m6
o p( o " p(
7
7
j id
i
a ez a j i y d j Cd y
#
e
X
xW
X
PW
U
U
v d#
e#
G
XPW #
q
cfw_
x
gI
&G
U"
i
QS
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x&
257
m
q yW
c|
k B qT
r
W k B qT W
#r x
qT
r
WqT
Ur
W
m
n
m
yruT p UT p m
qyW k B q Wq
s
qT
uA
n
0|T j
Wk
s
g
WqT
UuAD~UT p
Wq
Problem 10.6
Note that the procedure of evaluating the autocorrelation function is similar to the convolution procedure. The
Solutions to Problems from Chapter 4
Problem 4.1
By the denition of the Laplace transform we have
Lff (a(t 0 t0 )u(t 0 t0)g =
1
Z
f (a(t 0 t0)u(t 0 t0 )e0stdt
0
Using the change of variables as
Lff (a(t 0 t0 )u(t 0 t0)g =
at 0 t0 ; a > 0, we obtain
=
1
Z
Solutions to Problems from Chapter 12
Problem 12.1
3
&
1
)653 ) 2 %&$ " $0 )
1 4 '%#" ! (
The sensitivity of the closed-loop system transfer function with respect to parameter
is given by
3C
1
) 3P ) 2 ` ) ) ) 2
IVU6 a1 #53 C P HY3X 1 I#53 C HP !3X 3 C
Solutions to Problems from Chapter 11
Problem 11.1
The frequency domain equivalents for the inductor and the capacitor are presented, respectively, in Figures 11.1a
and 11.1.b.
uw q
$"ts v rp
ghf
i
dce a
b
`
YX5U V
W
8
C
BA9
@
&
'%
#$!"
)
0(
1
S
TR
I
Solutions to Problems from Chapter 8
Problem 8.1
From FIGURE 1.10, we can write
i
e (t) = L
di1 (t)
dt
o
+ R1 i1 (t) + e ( t);
i1 (t) = i 2 (t) + i 3 (t) = C
o
de (t)
dt
+
1
R2
o
e (t)
which implies
di1 (t)
dt
=
0
R1
L
i1 (t)
0
1
L
o
e (t) +
1
L
dt
=
0
R1
1
U0 Y a q' Y H R&' Y V A 9 B A 9 e 2 E B A 9 V A 9 B A 9 e 2 E B A 9 V A 9 B A 9 6
W' '
#
6
6 5Cb`Tk3"CGjb`Rdb`b`&' 34GCdb`3C1#
2EB A 9 a V A 9 B A 9 S # 2 A D B A 9 B A ' # 6
# 8
e "CjC7 b`3C7 W 54GP GcEb`7b`9 7W t W 5n&# a W $"yu
S # d A D B
t W 542
pjo1S
-1-2.
W
k)
1
)(
01
(ci)
(-i)
:2.
I
2-
(I,)
.
2
.t1
(c)
Figure S1.23
i,t,
a-
I -k
,
0
a-
.
e
p
?
I
(Li.)
etb)
4.
I
1
fri.j
.4
11111
111111!
Il.
1 It 1
i_.i OI hII1ITSp
d
:?
n
2f
Figure S1.24
1.24. The even and odd parts are sketch
ed in Figiire S1.2