Reg. No:…………………………….
KIGALI INSTITUTE OF SCIENCE AND TECHNOLOGY
INSTITUT DES SCIENCES ET TECHNOLOGIE
Avenue de l'Armée, B.P. 3900 Kigali, Rwanda
INSTITUTE EXAMINATIONS – ACADEMIC YEAR 2012
END OF SEMESTER EXAMINATION: MAIN EXAM
FACULTY OF ENGINEERING
COMPUTER ENGINEERING & INFORMATION TECHNOLOGY
FIRST YEAR SEMESTER I
CIT 3213 SIGNALS & SYSTEMS
DATE:
/ /2012
TIME: 2 HOURS
MAXIMUM MARKS = 60
INSTRUCTIONS
1. This paper contains Four (4) questions.
2. Answer One compulsory question section A and any two (2) questions in
section B
3. The compulsory question mark is 30 Marks and each question of section B is
15 Marks
4. No written materials allowed.
5. Do not forget to write your Registration Number.
6. Do not write any answers on this question paper.
SECTION A
1. a) Answer the following questions:
i.
ii.
iii.
iv.
What is a shift invariant system?
[1 Mark]
What is meant by region of convergence in Z-transform? [1 Mark]
Find the Laplace transform of the unit step function(show how you get it) and its
ROC
[2 Marks]
State Parseval’s relation in Fourier transform.
[3 Marks]
b) Determine whether or not the following signal is periodic. If the signal is periodic,
determine its fundamental period.
[3 Marks]
c) Find the Laplace transform X(S) of the following function, determine the zeros
and poles then sketch the pole-zero plot with the ROC for the following signal:
.
[6 Marks]
d) Using partial-fraction expansion, find the inverse Z-transform .
X ( z) 
1 
3  56 z 1
1
4

z 1 1  13 z 1

,
e) Find the Fourier transform of the signal
and represent x (t) and its Fourier transform.
1
1
 z .
4
3
[7 Marks]
[7 Marks]
SECTION B
2. a) The following are the impulse responses of discrete-time LTI systems. Determine
whether each system is causal. Justify your answer.
1
2
i) h[n]  ( ) u[n]
ii)
n
[4 Marks]
b) Find the Laplace transform X(s)), find the zeros and poles then sketch the pole-zero
plot with the ROC for the following signal:
[6 Marks]
c) The impulse response h[n] of a discrete-time LTI system is shown in the figure below.
Determine and sketch the output y[n] of this system to the input x[n] shown in the
figure below without using the convolution technique.
[5 marks]
3. a) Determine whether the following signal is energy signal, power signal or neither.
Justify your answer.
[5 marks]
b) Determine the discrete-time Fourier transforms (DTFTs) for
[5 marks]
c) State and proof the time-shifting property in continuous time Fourier
transform.
[5 marks]
4. a) Express the following discrete-time sequence as a sum of minimum number of scaled and
shifted unit steps:
[4 Marks]
[6 Marks]
b) Find the Z-transform of the following signal and its ROC.
c) Consider the periodic square wave x (t) shown in Figure below.
Determine the complex exponential Fourier series of x (t)
[5 Marks]
SECTION A
1. a) Answer the following questions:
i. What is a shift invariant system?
A system is called time-invariant if a time shift (delay or advance) in the input
signal causes the same time shift in the output signal.
[1 Mark]
ii.What is meant by region of convergence in Z-transform?
The ROC of X(z) is the sets of values of z for which X(z) attains a finite value.
[1 Mark]
iii. Find the Laplace transform of the unit step function(show how you get it) and its
ROC
[2 Marks]
iv. State Parseval’s relation in Fourier transform.



1
x(t ) dt 
2
2



X ( j ) d
2
Parseval’s relation says that this total energy may be determined either by computing energy
2
x (t )
per unit time (
and integrating over all time or by computing the energy per unit
X ( j ) / 2
2
frequency (
) and integrating over all frequencies.
[3 Marks]
b) Determine whether or not the following signal is periodic. If the signal is periodic,
determine its fundamental period.
c) Find the Laplace transform X(S) of the following function, determine the zeros and poles then
sketch the pole-zero plot with the ROC for the following signal:
.
We see that X(S) has no zeros and two poles at s = 2 and s = -3 and the ROC is -3 < Re(S) < 2, as
sketched.
[4 marks]
(2 marks)
d). Using partial-fraction expansion, find the inverse Z-transform .
X ( z) 
1 
3  56 z 1
1
4

z 1 1  13 z 1

,
1
1
 z .
4
3
Performing the partial-fraction expansion
X ( z) 
1
1  14 z 1

2
1  13 z 1
n
1
1
ZT
  u[n]  1 1 ,
1 4 z
4
n
[5Marks]
z  14
2
1
ZT
 2  u[n  1] 
,
1 1
n
n
 3  1 
 1  1 3 z
z  13
x[n]    u[n]  2  u[n  1]
4
 3  [2 Marks]
e)Find the Fourier transform of the signal
its Fourier transform.
Signal x (t) can be rewritten as
and represent
[7 Marks]
SECTION B
2. a)The following are the impulse responses of discrete-time LTI systems. Determine
whether each system is causal. Justify your answer.
1
h[n]  ( ) n u[n]
2
(i)
The system is causal since h[n]  0 for
n0
(ii)
The system is non-causal because of the term
(0.5)n u[n]
[4 Marks]
c)The impulse response h[n] of a discrete-time LTI system is shown below
Determine and sketch the output y[n] of this system to the input x[n] shown in the figure below
without using the convolution technique.
From the figure we can express:
Since the system is linear and time-invariant and by the definition of the impulse response, we
see that the output y[n] is given by
[5 Marks]
3. a) Determine whether the following signal is energy signal, power signal or neither.
Justify your answer.
For a discrete-time signal
as:
, the normalized energy content
of
is defined
The normalized average power
of
is defined as:
1.
is said to be an energy signal if and only if
2.
is said to be a power signal if and only if
, and so
, and so
.
.
Given a signal
Energy of
Power of
is given by:
is given by:
The energy is finite
, and
, therefore
is an energy signal.
[5 Marks]
b) Determine the discrete-time Fourier transforms (DTFTs) for
the DTFT exists if
As a result, the DTFT exists and has the form of
[5 Marks]
3. State and proof the time-shifting property in Continuous Time Fourier transform.
If
FT
x(t ) 
X ( j )
Then
FT
x(t  t 0 ) 
e  jt0 X ( j )
Verification of the time-shifting property
FT
x(t  t 0 ) 
e  jt X ( j )
0
By the change of variable
Hence,
, we obtain
FT
x(t  t 0 ) 
e  jt0 X ( j )
[5 Marks]
4. a) Express the following discrete-time sequence as a sum of minimum number of scaled
and shifted unit steps:
x [n] = 2 δ [n +2] – 2 δ [n + 1] + 2 δ [n] + 3 δ [n -2]
As
 [n]  u[n]  u[n  1]
x[n] = 2{u[n +2]- u [n +1]} – 2 {u [n +1] – u [n]} + 2 {u [n] – u [n -1]}
+ 3{u [n – 2] – u [n – 1]}
x [n] = 2 u [n +2]- 2 u [n +1] – 2 u [n +1 ] + 2 u [n ] + 2 u [n ] – 2 u [n – 1]
+ 3 u [n – 2] – 3 u [n – 1]
x [n ] = 2 u [n +2] – 4 u [n +1] + 4 u [n ]- 5 u [n -1] + 3 u [n – 2]
[4 Marks]
b) Find the Z-transform of the following signal and its ROC.
Or
[6 Marks]
c) Consider the periodic square wave x(t) shown in Figure below. Determine the complex
exponential Fourier series of x(t)
[5 Marks]
SUPPLEMENTARY EXAM
Section A
Question 1
a) A system has the input-output relation given by
. Determine whether
the system is:
i. Memoryless
ii. Causal
iii. Linear
iv. Time-invariant
[4 Marks]
b) . A continuous-time signal x (t) is shown below. Sketch and label each of the following signals.
( a ) x(t - 2); ( b ) x(2t); ( c ) x(t/2); (d) x ( - t )
[6 Marks]
c) Find the Laplace transform X(S) of the following function and sketch the pole-zero plot with
the ROC for the following signal:
.
[5 marks]
d) A finite sequence x [n] is defined as
Find X (z) and its ROC.
[5 marks]
e) State and prove the time shift property in Discrete Time Fourier transform. [5 marks]
f) Determine the impulse response h (t) using the Fourier transform for the system below:
[5 marks]
Question 2
a) Consider a LTI system with unit sample response h[n] and input x[n], as illustrated in Figure
(a). Calculate the convolution sum of these two sequences graphically.
[5 marks]
b) Find the z-transform of each of the following sequences: In all cases assume that n ≥ 0 unless
otherwise stated.
[5 marks]
c) Find the Fourier transform of the rectangular pulse signal x (t) defined by the following function and represent its
Fourier transform
[5 marks]
Question 3
a) Find the inverse Laplace transform of
b) State and briefly explain Parseval’s relation in Fourier transform.
[5 marks]
[5 marks]
c) Find the z-transform X (z) and sketch the pole-zero plot with the ROC for each of the
following sequences:
[5 marks]
Question 4
a) Find the Fourier transform of the signal
and represent x (t) and its Fourier transform.
[5 Marks]
b) Consider the periodic square wave x (t) shown in Figure below. Determine the trigonometric
Fourier series of x (t) knowing that
[5 marks]
c) Using tabular method, determine the convolution sum of two sequences:
[5 marks]
SUPPLEMENTARY MARKING SCHEME
SECTION A
Question 1
a)
[4 Marks]
b)
[6 Marks]
c)
[3 marks]
We see that X(S) has no zeros and two poles at s = 2 and s = -3 and the ROC is -3 < Re(S) < 2, as
sketched.
[2 Marks]
d)
For z not equal to zero or infinity, each term in X (z) will be finite and consequently X (z)
will converge. Note that X (z) includes both positive powers of z and negative powers of z.
Thus, we conclude that the ROC of X ( z ) is .
[5 Marks]
e) The time shift property in Discrete Time Fourier transform.
Verification of Time Shifting property
Let k = n – no then n = k + no
[5 Marks]
f) Taking the Fourier transforms of the above equation, we get
[5 Marks]
Question 2
a) x [n] = {1,1,1} and h [n] = {0.5, 1, 0.5, 1, 0.5}
x[n] starts at n1 = 0 and h[n] starts at n2 = - 2. Therefore, y[n] will start at n = n1+n2 = - 2.
The length of x[n] is N1 = 3 and that of h[n] is N2 = 5. Hence, the length of y[n] is N = N1+N2-1 = 7.
By definition,
If n <-2
If n = -2,

k=0
 x[k ]h[n  k ]  0
k  
If n = - 1, k = 0, 1
If n =0 , k = 0, 1, 2
If n =1 , k = 0, 1, 2
If n = 2, k = 0, 1, 2
If n = 3, k = 0, 1, 2
If n = 4, k = 2
If n > 4

 x[k ]h[n  k ]  0
k  
y[n] = {0.5, 1.5, 2, 2.5, 2, 1.5, 0.5}
[5 Marks]
b)
[5 Marks]
c)
Represent also its Fourier transform.
[5 Marks]
Question 3
a)
[5 Marks]
b)
Parseval’s relation in Fourier transform establishes the conservation of energy.



x(t ) dt 
2
1
2



X ( j ) d
2
Parseval’s relation says that this total energy may be determined either by computing energy
2
x (t )
per unit time (
and integrating over all time or by computing the energy per unit
X ( j ) / 2
2
frequency (
) and integrating over all frequencies.
[5 Marks]
c)
Question 4
a)
Signal x (t) can be rewritten as
[5 Marks]
b)
[5 Marks]
c)
[5 Marks]