EC204: Networks & Systems
Problem Set 4
1. A signal x(t) can be expressed as the sum of even and odd components as x(t) = x e (t) +
xo (t). (a) If x(t) X ( ), show that for real x(t), xe (t) Re[X ( )]
EC204: Networks & Systems
Problem Set 5
1. x1 (t) = 104 rect(104 t) and x2 (t) = (t) are
applied as inputs to LTI systems with frequency responses H1 ( ) = rect(/40000 ) and
H2 ( ) = rect(/20000 ).
x1
EC204: Networks & Systems
Solutions to Problem Set 4
1. (a) x(t) is real x(t) = x (t) X ( ) = X ( ) X ( ) = X ( )
x (t ) + x ( t )
2
x (t ) x ( t )
x o (t ) =
2
X ( ) + X ( )
xe (t)
2
X ( ) X ( )
xo
EC204: Networks & Systems
Problem Set 6
1. Find the Laplace transform of x(t) and y (t) shown below.
x(t)
y (t)
4
0
1
2
4
t
T /2
0
T
3T /2 2T
5T /2
2. Consider the partial fraction expansions shown be
EC204: Networks & Systems
Problem Set 7
1. Determine i(t) for t 0 in the circuit shown below given that i(0 ) = . Identify the
steady state and transient components of the response.
R
L
t=0
+
i (t )
1
EC204: Networks & Systems
Problem Set 2
1. u(t) is the step function. Find u(t) u(t).
2. Find y (t) = x(t) h(t).
x(t)
h(t)
1
1
0
2
3
-3
t
0
3
t
Figure 1:
3. Find x(t) h(t), where h(t) = (et + 2e2t )u(
EC204: Networks & Systems
Problem Set 3
1. (a) Determine the coecients of the Fourier series (in exponential form) of the periodic
signal x(t) shown below. Sketch the magnitude and phase spectrum.
x(t
EC204: Networks & Systems
Problem Set 1
1. For the signal x(t) illustrated in Fig. 1, sketch (a) x(t 4), (b) x(t/1.5), (c) x(t), (d)
x(2t 4), and (e) x(2 t).
x(t)
4
2
4
0
2
t
Figure 1:
2. Consider the
EC204: Networks & Systems
Problem Set 9
1. Find i(t) for t 0 using (a) Thevenins theorem, and (b) substitution and superposition
theorems.
1H
t=0
1
+
1V
1
i(t)
1F
1
2. The galvanometer current Ig is z
EC204: Networks & Systems
Solutions to Problem Set 2
1.
u (t ) u (t ) =
u( )u(t )d =
0
if t < 0
t
d = t if t 0
0
2.
y (t) = x(t) h(t)
1
-1
0
5
Figure 1: Solution to problem 2
1
6
t
3.
+
x (t ) h (t )
EC204: Networks & Systems
Solutions to Problem Set 8
1. We have
Y (s) = K
(s a)(s b)
(s + 1 j 1)(s + 1 + j 1)
I (s) = V (s)Y (s)
Y (s)|s=0 = 0 =
Kab
2
= a = 0 or b = 0
Without loss of generality, we t
EC204: Networks & Systems
Solutions to Problem Set 9
1. (a) To solve for i(t), t 0 using Thevenins theorem, we rst transform the given
network to the Laplace domain, as in gure (1).
We obtain values f
EC204: Networks & Systems
Solution to Problem Set 7
1. The transformed network is shown below.
I (s)
+
L
Ls
R
+
10
s 2 + 2
I (s) can be determined as follows.
10
+ L
10/L
s2 + 2
I (s) =
+
=
R
2 + 2 )
EC204: Networks & Systems
Problem Set 10
1. Consider the system with the state equation
x1
x2
=
0
1
2 3
x1
x2
+
10
11
u1
u2
and the output equation
10
y1
y2 = 1 1
02
y3
x1
x2
00
+ 1 0
01
u1
u2
.
Fi
EC204: Networks & Systems
Solutions to Problem Set 6
1. (i) x(t) = 2r(t) 4r(t 2) + 2r(t 4)
X (s) =
4
2
2
2
2 e2s + 2 e4s = 2 [1 e2s + e2s ]
2
s
s
s
s
(ii) Dene y1 (t) as
y1(t)
1
t
T
2
0
Figure 1: Pro
EC204: Networks & Systems
Solutions to Problem Set 3
cn ej 2nt . The d.c. value c0 =
1. (a) The period of x(t) is 1. Therefore, x(t) =
n=
A/2. Now, let the rst derivative of x(t) (shown in gure below)
EC204: Networks & Systems
Problem Set 8
1. The admittance function Y (s) has poles at s = 1 j 1 and two zeros.
i(t)
+
LTI
1port
Network
v (t)
Y (s)
The steady state current to a 6V dc input and a sinu