EE312 Old Exams
Transcript of EE312 Old Exams
1
EE312/52 Examination no. 1 Spring 2005
Kuwait University
Electrical Engineering Department
Name :………………………………………
Student I.D. :…………………………………….…
Problem no. Grade
1 20
2 25
3 25
4 20
5 20
Extra
20
Total
130
Remark: for each problem, write your final answer inside the given box.
2
Problem 1:
a) (10 points): Consider the following function ( )f t
plot 1
( 1)2
f t ?
b) (10 points): Consider a linear time invariant system. Its zero state response 1( )y t
due to the input 1( )u t is shown in the following figure:
find the input ( )u t when the output ( )y t is:
-3 -2 -1 0 1 2 3
t
1
t
1
0 1 2 t
2
0 1
Final answer
3
Problem 2:
a) (10 points): Use the Laplace transform to find the zero state response ( )y t of a
system described by the differential equation:
2( ) ( ) ( ) 0y t a y t u t a
and excited by the unit step input ( ) 1u t for 0t .
t
1
0 1 2 3 4
-2
Final answer
4
b) (15 points): Write a differential equation to describe the following block diagram:
where a and b are real numbers.
integrator b
a
+
-
+
-
+
+
Final answer
Final answer
5
Problem 3:
a) (10 points): Find the transfer function from the input ( )U s to the output ( )Y s of the
following system:
where ( )iH s is the ith transfer function of the system i.
b) (15 points): Find the inverse Laplace transform of:
2( )
( 2 )
seF s
s s s b
where b is a real number greater than 1.
+ -
+
+ +
-
Final answer
6
Problem 4: (20 points)
a) Use Laplace transfer to find a differential equation of a system described by the
impulse response:
( ) , 0th t te t
b) Use convolution integral to find a differential equation of a system described by
the impulse response:
( ) , 0th t te t
Final answer
Final answer
7
Problem 5: (20 points)
Consider a linear time invariant causal system with impulse response:
[ ] [ ] , 0, 1, 2,kh k k e k
where [ ]k is the impulse sequence. Find a difference equation to describe the
system?
Extra Problem: (20 points)
Consider a system described by the following block diagram:
-
+
integrator derivative a
b
Final answer
Final answer
8
where ( )u t is the unit step input and 6( ) 2 ty t e , 0t , is the output of the
system. Find the values of a and b?
Final answer
9
EE312/52 Examination no. 2 Spring 2005
Kuwait University
Electrical Engineering Department
Name :………………………………………
Student I.D. :…………………………………….…
Problem no. Grade
1 30
2 20
3 20
4 30
Extra
20
Total
120
10
Problem 1 (30 points) : Consider an LTIL system described by the difference equation:
[ 2] 2cos(3) [ 1] [ ] [ ] cos(3) [ 1]y k y k y k u k u k
With initial conditions [ 2] 2cos(3)sin(3)y , and [ 1] sin(3)y . The system is excited
by the impulse sequence:
1 0
[ ]0 0
for kk
for k
Find
a) (10 points): The zero input response?
b) (10 points): The zero state response?
c) (5 points): The total response?
d) (5 points): The transfer function?
Problem 2 (20 points):
a) (15 points): Find the inverse z-transform of 2
4( )
( 1)( 2)F z
z z z
?
b) (5 points): Find the initial value of [ ]f k , that is [ 0]f k ?
Problem 3 (20 points): Find the transfer function of a system described by the following
block diagram:
- ( )U z + + + ( )Y z
- +
11
Problem 4 (30 points):
a) (15 points): Determine if the polynomial : 4 3 2( ) 2 0.5 0.1D s s s s s is
Hurwitz? Use Routh test method.
b) (15 points): Consider a discrete time system with transfer function:
2
1( )
( 0.5)( 0.5)H z
z z az
, for what values of “a” the system is BIBO stable?
Use Jury test method.
Extra Problem: (20 points): Given the transfer function 1
( )1 2 s
H se
of a
system. Is the system BIBO stable?
12
EE312/52 Final Spring 2005
Kuwait University
Electrical Engineering Department
Name :………………………………………
Student I.D. :…………………………………….…
Problem no. Grade
1 30
2 20
3 20
4 30
Extra
20
Total
120
13
Problem 1 (30 points) : Consider an LTIL system described by the differential equation:
( ) ( ) 2 ( ) 3 ( )y t y t y t u t
where ( )y t is the system output and ( )u t is the system input. The system is excited by
the initial conditions (0 ) 1y , (0 ) 2y , and the unit step input:
1 0
( )0 0
for tu t
for t
Find
e) (10 points): The zero state response?
f) (10 points): The zero input response?
g) (2 points): The total response?
h) (2 points): The transfer function?
i) (2 points): The impulse response?
j) (2 points): The final value of ( )y t that is ( )y t ?
k) (2 points): Is the system BIBO stable?
Problem 2 (20 points): Given the transfer function 3 2
2( )
( 0.5)( 1)
z z zH z
z z
of an LTIL
discrete-time system.
c) (15 points): Find the inverse z-transform of ( )H z ?
d) (5 points): Is the system BIBO stable?
Problem 3 (20 points):
c) (10 points): Determine if the polynomial 4 3( ) 2 0.5D z z z z is Schur? Use
Jury test method.
d) (10 points): Find the state variable equations of the differential equation
2 ( ) ( ) ( ) ( )y t y t y t u t where ( )y t is the system output and ( )u t is the system
input.
14
Problem 4 (30 points): Consider the continuous-time state variable equations:
1 1
2 2
1
2
( ) ( )1 2 0( ) ( )
( ) ( )0 1 1
( )( ) 1 0
( )
x t x tx t u t
x t x t
x ty t
x t
where ( )y t is the system output and ( )u t is the system input. The system is excited by
the initial conditions 0
(0)0
x
and the unit step input:
1 0
( )0 0
for tu t
for t
Find
a) (10 points): The output ( )y t using Laplace transform method?
b) (10 points): The output ( )y t using time domain method?
c) (5 points): The transfer function?
d) (5 points): Is the system BIBO stable?
Extra Problem: (20 points): Consider the state variable equations of a continuous-time
system:
1 1
2 2
3 3
4 4
1
2
3
4
( ) ( )1 1 0 0 1 1
( ) ( )0 0 1 1( ) ( ) ( )
( ) ( )0 0 1 0 1 1
( ) ( )0 0 0 1 1 1
( )
( )( ) 1 0 0 0
( )
( )
x t x t
x t x ta bx t u t u t
x t x t
x t x t
x t
x ty t
x t
x t
Where ( )y t is the system output, ( )u t is the system input, and “a” and “b” are real
numbers. Find the ranges of “a” and “b” such that the system is BIBO stable?
15
EE 312/01 Examination No. 1
Spring 2007
Kuwait University
Collage of Engineering and Petroleum
Name : ………………………………………
Student I. D. : …………………………………….…
Signature : …………………………………….…
Problem No. Grade
1 20
2 20
3 20
4 20
5 20
Total
100
16
Problem 1 (20 points): Given the signal
Plot )3()1( 2 tftf ?
Problem 2 (20 points): Given the following outputs of a system. Is the system linear?
Explain your answer.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 27
8
9
10
11
12
13
14
t
y(t
)
The output y(t) when:
initial condition x(0)=10input u(t)=2q(t)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 212
14
16
18
20
22
24
26
28
t
y(t
)
The output y(t) when:initial condition x(0)=20input u(t)=4q(t)
f(t)
0
3
2
17
Problem 3 (20 points): Given the impulse response
00
01)(
tfor
tforeth
t
for an LTI system. Find the differential
equation that describes the system?
Problem 4 (20 points): Given the state space equations
][
][11][
][0
1
][
][
11
01
]1[
]1[x
2
1
2
1
2
1
nx
nxny
nunx
nx
nx
n
find the impulse response?
Problem 5 (20 points): Given the state transition matrix
t
t
ttt
At
e
e
eee
e
00
00
03
2
3
2
4
4
find the matrix A? ( Hint: use the properties of Ate )
18
EE312/01 Examination No. 2
Spring 2007
Kuwait University
Electrical Engineering Department
Name :………………………………………
Student I.D. :…………………………………….…
Signature :……………………………………….
Problem No. Grade
1 30
2 20
3 25
4 25
Total
100
19
Problem 1 (30 points) : Use the properties of Laplace transform to answer the following
questions:
l) (10 points): why the Laplace transform of ttt eee 52 42 is not
10132
323 sss
?
m) (10 points): Find the Laplace transform of
0
)()()( daytxtf in terms of
)()( sYandsX ?
n) (10 points): Given dt
tdytg
)()( with
164
620)(
23
sss
ssY , find )0( tg ?
Problem 2 (20 points): Find )(sG in terms of )()( sHandsV so that )()( sUsY ?
Problem 3 (25 points):
e) (20 points): Find the inverse z-transform of )5.0)(5.1(
4)(
zzzF ?
f) (5 points): Find the final value of ][nf , that is ][ nf ?
Problem 4 (25 points): Consider a continuous time system with transfer function
)210)((
1)(
23
bsssassH , find the ranges of a and b so that the system is BIBO
stable? Use Routh-Hurwitz test.
+
+
20
EE312/01 Final Spring 2007
Kuwait University
Electrical Engineering Department
Name :………………………………………
Student I.D. :…………………………………….…
Signature :………………………………………
Problem No. Grade
1 25
2 15
3 30
4 30
Total
100
21
Problem 1 (25 points): Consider a system described by the differential equation:
)(3)()()()( tutytatyty ; where ( )y t is the system output, ( )u t is the system input,
and )(ta is a function of time t.
o) (10 points): Is the system linear? Why?
p) (15 points): If 2)( ta , find the state space equations?
Problem 2 (15 points): Given the transfer function 3.02.0
1)(
2
zz
zzH of an LTI
discrete-time system
g) (10 points): Find the difference equation?
h) (5 points): Is the system BIBO stable? Why?
Problem 3 (30 points): Given the state space matrix
600
013
001
A
e) (10 points): is the system asymptotically stable? Why?
f) (20 points): Find the state transition matrix Ate using Laplace transform?
Problem 4 (30 points): Consider the continuous-time state variable equations:
)(
)(01)(
)(1
1
)(
)(
21
01
)(
)(
2
1
2
1
2
1
tx
txty
tutx
tx
tx
tx
The system is excited by the initial conditions
1
0)0(x and the unit step input )(tq :
a) (10 points): Find the zero input response using Laplace transform method?
b) (10 points): Find the zero state response using Laplace transform method?
c) (5 points): Find the impulse response?
d) (5 points): Is the system BIBO stable?
22
EE312/02A Examination no. 1
Spring 2011
(50 minutes)
Kuwait University
Electrical Engineering Department
Name :………………………………………
Student I.D. :…………………………………….…
Signature :………………………………………
Problem no. Grade
1
/25
2
/25
3
/25
4
/25
Total
/100
23
Problem 1: Given the system )()()( tutyty where )(ty is the output and )(tu is
the input.
c) (15 points) Is the system linear? Why
d) (10 points) Is the system time-invariant? Why
Problem 2 (25 points): The input )(tu is applied to the system with the impulse
response )(th
Sketch the convolution
0
)()()( duthty ? Show the values of
)2(),1(),5.0(),5.0( yyyy
Problem 3: (25 points): Find the state space equations of the following difference
equation:
][]1[][3]2[ nununyny ?
Problem 4: Given the following state space equations ( is any real number)
)(01)(
)(1
1)(
0
00)(
txty
tutxtx
C
BA
with
0
0)0(x and )()( tqtu
``step input”
a) (10 points) Find Ate ?
b) (15 points) Find )(ty ?
24
EE312/02A Examination no. 2
Spring 2011
(50 minutes)
Kuwait University
Electrical Engineering Department
Name :………………………………………
Student I.D. :…………………………………….…
Signature :………………………………………
Problem no. Grade
1
/25
2
/25
3
/25
4
/25
Total
/100
25
Problem 1 (25 points): Given the state space equations
)(
)(01)(
)(0
1
)(
)(
65
10
)(
)(
2
1
2
1
2
1
tx
txty
tutx
tx
tx
tx
with the initial state
0
2
)0(
)0(
2
1
x
x and the unit step input )()( tqtu . Find the output
)(ty using the Laplace transform?
Problem 2 (25 points): Find the transfer function of the following block diagram?
Problem 3: (25 points): Use the Routh-Hurwitz test to find the condition on so that
the system described by the transfer function 82
3)(
23
ssssH
is BIBO
stable?
Problem 4: Given the transfer function
s
sH1
)( ( is any positive number)
Answer the following questions in term of :
a) (10 points) Find the magnitude response and the phase response?
b) (10 points) Find the bandwidth?
c) (5 points) Find the steady state output if the input )sin()( ttu ?
Unit-time
delay
Unit-time
delay
26
EE312/02A Final Examination
Spring 2011
Kuwait University
Electrical Engineering Department
Name :………………………………………
Student I.D. :…………………………………….…
Signature :………………………………………
Problem no. Grade
1
/20
2
/20
3
/20
4
/20
5
/20
Total
/100
27
Problem 1: The output )(ty of a linear time-invariant system excited by an impulse
input )()( ttu
is given bytt eety 42 33)( .
e) (10 points) Find the differential equation of the system using the Laplace
transform?
f) (10 points) Find the differential equation of the system using the continuous
convolution?
Problem 2 (20 points): Find the Laplace transform of )()( 2 tfetx t where )(tf is
given by:
Problem 3: Given the state space equations
)(01)(
)(0
1)(
41
02)(
tty
tutt
with the
initial state 0)0( . Find the impulse response?
a) (10 points) Using the Laplace transform?
b) (10 points) Using the time-domain?
Problem 4 (20 points): Use the Jury’s test to tell if the system described by the state
space equation: ][
113.0
1.005.0
100
]1[ nn
is asymptotically stable or not?
Problem 5: Given the following transfer function of the FIR digital filter 431 221)( zzzzH
A: (10 points) is the filter linear phase or generalized linear phase? Why
B: (10 points) find the direct realization of the filter?