EC2102 Networks and Systems HW 3
August 23, 2012
1F
1. Perform the following convolutions where
indicates convolution.
(a) For u(t) a unit step function, nd
r(t) = u(t) u(t).
(b) Find x(t) h(t), where h(t) =
(et + 2e2t )u(t) and x(t) =
10e3t u(t).
(c) Fin
EC2102 Networks and Systems HW 6
October 3, 2012
1. Show that the Fourier tranform of the
triangular pulse x(t) shown below is
1
X ( ) = 2 (ej + jej 1).
4. Sampling is the process whereby continuoustime signals are converted to discrete-time
signals, whil
HW5-solutions
Q1)
Figure 1: rect
Figure 2: rect
Figure 3: sinc
1
t
5
t
2
t10
8
rect
t
10
(Q2)
Let x (t) =
W
sinc
Wt
, then,
X ( j ) = 1, f or | | W ,
= 0, elsewhere
X ( j ) =
Let =
Wt
,
then
X ( j ) =
Wt
W
sinc
sinc( )e
(1)
e j t dt
j
W
d
Now the fouri
EC2102 Networks and Systems HW 5
September 24, 2012
1. Sketch the following functions:
x(t)
(a) rect(t/2), (b)rect(t 10)/8), and (c)
sinc(t/5) rect(t/10). [sinc(x) = sin(x) ]
x
2. Show that
sinc(x)dx =
1
sinc2 (x)dx = 1.
d
2
3. For each of the following s
HW - 4 solutions (draft)
Q1)
(a) x[n]
(b) h[n]
Figure 1: Signals h[n] and x[n]
a)
If y [n] is the ouput of the system, we get
k=
x[k ]h[n k ]
y [n] =
k=
This gives us the following terms for the output
n
y [n]1
-1
x[0]h[1] = 2
0
x[1]h[1] = 4
1
x[0]h[1] =
HW - 3 solutions (draft)
Q1)
a)
r(t) =
u( )u(t )d
t
=
1 d, t > 0
0
= t, t > 0,
= 0, t < 0
(1)
b)
y (t) = x(t) h(t)
t
(10e3 )(et+ + 2e2t+2 )d
0
t
t
2
2t
t
e d
(e )d + 20e
= 10e
=
0
0
(2)
After simplifying the above integral, we get
y (t) = 5et + 20e2t 15e3
HW - 2 solutions (draft)
Q1)
Given x1 (t) = x1 (t + T1 ) and x2 (t) = x2 (t + T2 ) x(t) = x1 (t) + x2 (t)
Let the period of x(t) be T x(t) = x(t + T )
x(t + T ) = x1 (t + mT1 ) + x2 (t + nT2 )
mT1 = nT2 = T where m, n are least integers.
The period of x(t
EC2102 Networks and Systems HW 2
August 9, 2012
1. Let x1 (t) and x2 (t) be periodic signals
with periods T1 and T2 . Derive the conditions under which the sum x(t) = x1 (t) +
x2 (t) is periodic. What is the fundamental period of x(t) ?
6. Consider an LTI
EC2102 Networks and Systems HW 1
August 2, 2012
y (t)
1. Carefully sketch the following signals.
Mark all the critical points.
(a) g (t) = tu(t 1) u(t 1)
(b) h(t) = etu(t) ,
1
1t1
2. Given a continuous-time signal specied
by
x(t) =
1 |t|,
0,
1 t 1
otherwi