King Fahd University
of Petroleum & Minerals
Electrical Engineering Department
EE 207 Signals and Systems
Second Semester (112)
Exam II
Saturday, 21 April 2012
7:00 pm – 9:00 pm
Name:
ID:
Section:
Instructors:
Dr. Azzedine Zerguine
Dr. Mohamed Deriche
Dr. Adil Balghonaim
Dr. Wajih Abu-Al-Saud
Problem
Score
(Section 1)
(Section 2)
(Sections 3 & 5)
(Section 4)
Out of
1
30
2
35
3
35
Total
100
Good luck!
Problem 1: [30 points]
A signal x (t ) has complex exponential Fourier series expansion
2
6
x (t )  
e j 2 kt
k 2 2  jk
a)
Sketch magnitude and phase line spectra of x (t ) showing all important values on the sketchs.
k
Ck
-2
6
2 j 2
-1
6
2 j
6
2
6
2 j
0
1
6
2 j 2
2
k
Ck
6
 2 
 0  tan 1    0.7854 rad  45
2 j 2
 2 
6
6

 2.1213
2 j 2
22  22
6
6

 2.6833
2 j
22  12
6
 1 
 0  tan 1    0.4636 rad  26.57
2 j
 2 
6
3
2
6
 0  0  0 rad  0
2
6
6

 2.6833
2 j
22  12
6
6

 2.1213
2 j 2
22  22
6
1
 0  tan 1    0.4636 rad  26.57
2 j
2
6
2
 0  tan 1    0.7854 rad  45
2 j 2
2
Magnitude Spectrum
Phase Spectrum
45.0°
3.00
2.68
2.12
2.68
2.12
26.6°
-4
-3
-2
-1
1
2
13
4
k
-4
-3
-2
-1
1
2
-26.6°
-45.0°
b)
find the average power P of the signal x (t ) .
P
2

k 2
Ck
2
2
 C 0  2 C k
2
2
k 1
  3  2  2.6833  2  2.1213
2
 32.4 W
2
2
13
4
k
c)
Find the trigonometric Fourier series coeficients A0, A1, B2, and B3
A k  jB k
2
Ak  C k  C k
Ck 
A k  jB k
2
B k  j C k  C  k
C k 
and
and

A0  C 0  3
A1  C 1  C 1 
6
6

 4.8
2 j 2 j
 6
6 
B 2  j C 2  C 2   j 

  2.4
 2 j 2 2 j 2
B3  0
d)
Sketch the magnitude line spectrum of the siganl y (t ) 
1
x (t )  3 showing all important values
2
on the sketchs.
Magnitude Spectrum
4.50
1.06
-4
-3
-2
1.34
1.34
-1
1
1.06
2
13
4
k
Problem 2: [35 points]
a)
The signal x t  has a Fourier transform
X   
1
2    j 3
2
Find the Fourier tranform of the
i)
x  2t 
x (2t )
ii)
F .T .





x t   cos  3t 
x (t )  cos  3t 
iii)

1   1
1
X   
2

j 3
2  2  2

 2

4
2
F .T .
1
 X   3  X   3  
2

1
1
1
 


2  2    32  j 3   3 2    32  j 3   3 
x  5t  2 
x (5t  2)
F .T .
1     j  ( 2) /( 5)
X   e
5  5 

1
1
 
2

j 3
5

 2
25
5


  j2/5
 e



b)
The signal x t  has a Fourier transform X   given by
1, 1    1
X    
0, otherwise
The signal y t  is related to x t  by the relation y t  

 y t 
2
dt of the signal y t  .

Since y t  
d 2 x (t )
, the F.T. of y t  can be written as
dt 2
Y     j   X     2 X  
2
So,
1, 1    1
Y    
0, otherwise
 2 , 1    1

otherwise
 0,
Total Energy of y t  is

E y (t ) 

y t 
2


1
2
1
 
1
dt 
2
2 2
d
1
1
1

 4d 

2 1
1
1
 5 
1
10
1

1  (1)
10
1

5


 Y   d 

2
d 2 x (t )
. Find the total energy
dt 2
Problem 3: [35 points]
a)
A simple circuit used as a filter is shown below
L
R
x(t)
y(t)
Determine the frequency response of the system above, H   , assuming R/L = 3,
i)
H ( ) 
R
R  j L
R 
 
L
  
R 
   j
L 
3

3 j
Plot the magnitude and phase of H   over the frequency range –16 to 16 rad.
i)
H ( ) 
3
3
 
  tan 1  
3 j
3
 ( ) 
32   2
Magnitude
Phase
/2
1
-16
-12
-8
-4
4
8
12
16

-16
-12
-8
-4
4
/2
8
12
16

i)
Explain what type of filters the circuit represents? Why?
The filter is a lowpass filter (LPF) because it passes low frequencies around 0 rad/s unattenuated and stops high frequencies.
ii)
Determine the exact value of the 3-dB (half-power) cut-off frequency.
H ( ) 
3
32   2
1
2

9
1

2
3 
2
2
2
3    18
2
2  9
So, the 3-dB cutoff frequency (Bandwidth of the LPF) is
b)
 3
A linear time-invariant (LTI) system has the following impulse response
h t   e 2t u t 
i)
Find the systems transfer function H ( )
H   
i)
1
2  j
Without using convolution, find the output of the system y (t ) if input to the system is
x t   t 3e 2t u t 
X   
6
2  j 
4
Y    H    X  


6
2  j 

5

6
2  j 
1
2  j
4
1
y t   t 4e 2t u t 
4
1
24
4  2  j  5
ii)
Without using convolution, find the output of the system y (t ) if input to the system is
x t   cos  2t 
Write the answer in the simplest form.
X        2      2  
Y    H    X  
     2      2   
1
2  j

1
1 
     2 
    2 
2  j
2  j  


1
1 
     2 
    2 
2 j 2
2  j 2 



j
j

4
4
e
e
     2 
    2 

8
8







j
j 
 
4

    2  e 4 
   2  e
8
h t  
1



j  2t  
4

1
e

e
2 8
2 8
1



cos  2t  
4
8



 j  2t  
4
