National University of Singapore
Department of Electrical & Computer Engineering
EE2023 Signals & Systems Tutorial 5
Section I : Exercises that are straightforward applications of the concepts covered
in class. Please attempt these problems on your own.
1

EE2023 Signals & Systems TUTORIAL 3 (Solutions)
Page 1 of 7
EE2023 TUTORIAL 3 (SOLUTIONS)
Solution to Q.1
2
2cos ( 0.5 t ) ; t 1
x (t ) =
t >1
0;
-1
(a)
0
1
t
Method 1: By applying direct Fourier transform:
X ( f=
)
x ( t ) exp ( j 2 ft ) dt= 1 2cos ( 0

EE2023 Signals & Systems TUTORIAL 4 (Problems)
Page 1 of 3
EE2023 TUTORIAL 4 (PROBLEMS)
Q.1 The signal x ( t ) shown in Fig.Q.1 is sampled at 5-Hz to
form a continuous-time signal x ( t ) . Let X ( f ) be the
spectrum of x ( t ) . Sketch and label x ( t )

National University of Singapore
Department of Electrical & Computer Engineering
EE2023 Signals & Systems Tutorial 6
Section I : Exercises that are straightforward applications of the concepts covered
in class. Please attempt these problems on your own.
1

EE2023 Signals & Systems TUTORIAL 3 (Problems)
Page 1 of 4
EE2023 TUTORIAL 3 (PROBLEMS)
Q.1 A half-cosine pulse x ( t ) and a sine pulse y ( t ) are shown in Fig.Q.1.
x (t )
y (t )
2
1
Fig.Q.1
-1
0
-1
0
-1
t
1
t
1
(a) Derive the spectrum of x ( t ) using

EE2023 Signals & Systems TUTORIAL 2 (Solutions)
Page 1 of 4
EE2023 TUTORIAL 2 (SOLUTIONS)
Solution to Q.1
Description of x ( t ) :
x ( t ) is a REAL & EVEN function of t
x ( t ) has an average (or DC) value of 2
x ( t ) is APERIODIC
Spectrum is REAL and

Lecture 1 (8 March 2016)
Tuesday, 8 March 2016
2:03 PM
Lecturer
A/Prof Tan Woei Wan
Email - wwtan@nus.edu.sg
Phone - 6516 8323
Agenda
Definition of systems
Types of systems
Differential equation models of systems, in particular electrical circuits

Lecture 4 (17 March 2016)
Tuesday, 15 March 2016
11:07 PM
Reminder:
Tutorial 5 will be discussed on 22 March (Tuesday).
Recap:
Transfer function is defined as the ratio of the Laplace Transform of the _ over the Laplace Transform of the
_. Implicit assum

Lecture 2 (10 March 2016)
Thursday, 10 March 2016
12:46 PM
Recap :
This course focuses on _ (LTI) systems.
Input-output relationship of a dynamic LTI system is described by a _.
Output signal of a dynamic LTI system comprises
Transient response :
Ste

EE2023 Signals & Systems TUTORIAL 2 (Problems)
Page 1 of 2
EE2023 TUTORIAL 2 (PROBLEMS)
Q.1 The discrete-frequency spectrum of a signal x ( t ) is shown in Fig.Q.1. Classify x ( t ) based on
inferences drawn from Fig.Q.1 alone. What is the Fourier series

EE2023 Signals & Systems TUTORIAL 1 (Solutions)
Page 1 of 5
EE2023 TUTORIAL 1 (SOLUTIONS)
Solution to Q.1
Write z in polar form:
z =x + jy =z exp ( jz ) .
Since adding integer multiples of 2 to z does not affect the value of z , we may also express z as
=

Lecture 3 (15 March 2016)
Monday, 14 March, 2016
3:46 PM
Announcement:
Tutorial 5 will be discussed on 22 March (Tuesday).
Recap:
Input-output relationship of a dynamic LTI system is governed by a linear ordinary differential equation.
Output signal of

EE2023 Signals & Systems TUTORIAL 1 (Problems)
Page 1 of 3
EE2023 TUTORIAL 1 (PROBLEMS)
Q.1 Let z= x + jy where x and y are real numbers. Provide a formula for computing the N distinct
values of z1 N . Hence, or otherwise, determine 641 6 and ( j81)
14
.

EE2023 Signals & Systems (Chapter 5)
Unit Impulse and Sampling
Theorem
Introduce the concept of sampling using a mathematical
function called the Dirac- function.
This leads to the Nyquist sampling theorem which
stipulates the minimum sampling frequency r

EE2023 Signals & Systems (Chapter 4)
Energy Spectrum, Power Spectrum and
Bandwidth
The energy of aTsignal x(t) , denoted as E, is defined as
2
E lim x (t ) dt if the limit exists and is non-zero.
T
T
The signal, x(t), is an energy signal
0 if E
signal.

Review Sys Properties Sys Model DE Model
Learning Points from first half of course
I
Signals are time-varying mathematical functions that represents the
change in a physical quantity/variable by which information can be
conveyed.
I
Fourier Transform enabl

EE2023 Signals & Systems (Chapter 3)
Fourier
Transform
Fourier Transform (FT) is the equivalent of Fourier Series (FS)
but it applies to continuous time signal which are non-periodic
or aperiodic.
Fundamental difference is that the FT leads to a spectrum

EE2023 Signals & Systems (Chapter 2)
Time and Frequency Domain Representation of
Signals
A signal is a function which represents the time variation of a
physical variable.
A continuous time signal is one that is defined for all time
instants, t, in an int

Motivation Definition Rules Inverse LT DE Soln
What is Laplace Transform ?
Mathematical tool used for solving linear ordinary differential equation.
Solution of
Difficult Problem
Difficult
Problem
Inverse
Laplace
Transform
Laplace
Transform
Easy
Problem
I

EE2023 Signals & Systems (Chapter 1)
Definitions
A signal is a function which represents the time variation of a
physical variable eg pressure, sound, temperature, data, speech,
music, etc. A signal is thus a measured quantity that represents
the physical

Overview Impulse Response Step Response Zero
Common Responses Common Systems
Overview : Output Responses of LTI Systems
Three types of output responses will be studied :
I
Impulse Response is the output signal when the input is an impulse
function.
I
Step

Sine i/p Freq Response Bode Diag Application
Intuition LT Analysis
Response due to sinusoidal input
What is the steady-state output signal of a stable system when the input
signal is a sinusoidal function ?
xo sin(t)
I
G(s)
?
Intutively, the steady-state

EE2023 Signals & Systems (Chapter 8)
Parameters of First-order Systems
D.E. of a linear first-order system is generally written as
dy(t )
T
y(t ) Kx(t )
dt
where: K is the steady-state/static gain,
T is called the time-constant.
Y (s)
K
The transfer fu

EE2023 Signals & Systems (Chapter 10)
Quadratic factor
G( j )
1
j
j
1 2 ( ) ( )2
n
n
Magnitude of a second-order system is
1
j
j 2
1 2 ( n ) ( n )
1
(1
) (2
2 2
2
n
20log10 (1
2
n
)
) (2
2 2
2
n
2
n
)
Bode diagram when 0 < 1 (Magnitude response)

EE2023 Signals & Systems (Chapter 9)
Sinusoidal Response
What is the output when the input is a sinusoidal?
Asin(t)
?
G(s)
Intuitively, the output will also be a sinusoidal signal as the solution
to the system differential equations can be interpreted u

EE2023 Signals & Systems (Chapter 6)
General Physical Systems
Physical systems are interconnections of components, devices or subsystems
Physical systems have many different properties.
We can write mathematical expressions which approximately describes t

EE2023 Signals & Systems (Chapter 5)
Unit Impulse and Sampling Theorem
Introduce the concept of sampling using a mathematical function
called the Dirac- function.
This leads to the Nyquist sampling theorem which stipulates the
minimum sampling frequency r

EE2023 Signals & Systems (Chapter 7)
Use of Laplace Transform in Solving DE
Use Laplace Transform to derive the output, vc (t ), if v (t ) = V (constant
voltage) and the initial condition is vc(0)>0.
We have : V i (t )R v c (t )
C
Using v c (t ) 1
i