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


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


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


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?


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?


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


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 ?


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


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?


22

EE312/02A Examination no. 1

Spring 2011

(50 minutes)

Kuwait University


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


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


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?

ee312 old exams

Documents