Term 332
EE3010: Signals and Systems Analysis
2. Introduction to Signal and Systems
Dr. Mujahed Al-Dhaifallah
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Dr. Mujahed Al-Dhaifallah
‫ مجاهد آل ضيف هللا‬.‫د‬

Office: Dean Office.
 E-mail: muja2007hed@gmail.com
 Telephone: 7842983
 Office Hours: SMT, 1:30 – 2:30 PM,
or by appointment
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Rules and Regulations
 No



make up quizzes
DN grade == 25% unexcused absences
Homework Assignments are due to the
beginning of the lectures.
Absence is not an excuse for not
submitting the Homework.
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Grading Policy

Exam 1 (10%),
 Exam 2 (15%)
 Final Exam (60%),
 Quizzes (5%)
 HWs (5%)
 Attendance & class participation (5%), penalty for late
attendance
 Note: No absence, late homework submission
allowed without genuine excuse.
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Homework

Send me e-mail

Subject Line: “EE 3010 Student”
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The Course Goal
To introduce the mathematical tools for
analysing signals and systems in the
time and frequency domain and to
provide a basis for applying these
techniques in electrical engineering.
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Course Objectives
1.
2.
3.
4.
5.
6.
Identify the types of signals and their characterization.
Use the Fourier series representation.
Differentiate between the continuous and discretetime Fourier transforms.
Grasp the fundamental concepts of the Laplace and Z
transforms.
Characterize signals and systems in the frequency
domain.
Apply signals and systems concepts in various
engineering applications.
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Course Syllabus
1. Signal and Systems : Introduction,
Continuous and discrete-time signals, Basic
system properties.
2. Linear Time-Invariant (LTI) Systems:
Convolution, LTI systems properties,
Continuous and discrete-time LTI causal
systems.
3. Fourier series Representation of Periodic
Systems: LTI system response to complex
exponentials, Properties of Fourier series,
Applications to filtering, Examples of filters.
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Course Outlines
4. Continuous-Time Fourier Transform:
Fourier transform of aperiodic and periodic
signals, Properties, Convolution and
multiplication properties, Frequency
response of LTI systems.
5. Discrete-Time Fourier Transform:
Overview of Discrete-time equivalents of
topics covered in chapter 4.
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Course Outlines
6. Laplace transform (Laplace transform as
Fourier transform with convergence factor.
Properties of the Laplace transform
7. z transform. Properties of the z transform.
Examples. Difference equations and
differential equations. Digital filters.
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Signals & Systems Concepts

Specific Objectives:
•
•
•
Introduce, using examples, what is a signal
and what is a system
Why mathematical models are appropriate
What are continuous-time and discrete-time
representations and how are they related
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Recommended Reading Material
•
•
Signals and Systems, Oppenheim &
Willsky, Section 1
Signals and Systems, Haykin & Van
Veen, Section 1
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What is a Signal?



Signals are functions that carry information.
Such information is contained in a pattern of
variation of some form.
Examples of signal include:

Electrical signals
–

Acoustic signals
–

Acoustic pressure (sound) over time
Mechanical signals
–

Voltages and currents in a circuit
Velocity of a car over time
Video signals
–
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Intensity level of a pixel (camera, video) over time
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How is a Signal Represented?

Mathematically, signals are represented as a
function of one or more independent variables.
 For instance a black & white video signal intensity
is dependent on x, y coordinates and time t f(x,y,t)
 In this course, we shall be exclusively concerned
with signals that are a function of a single
variable: time f(t)
t
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Example: Signals in an Electrical
Circuit
R
vs

i
C
vc
The signals vc and vs are patterns of variation over time
Step (signal) vs at t=1
RC = 1
First order (exponential)
response for vc
vs, vc

+
-
vs (t )  vc (t )
R
dv (t )
i (t )  C c
dt
dvc (t ) 1
1

vc (t ) 
vs (t )
dt
RC
RC
i (t ) 
Note, we could also have considered the voltage across the resistor
or the current as signals
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Continuous & Discrete-Time Signals

Continuous-Time Signals
x(t)

Most signals in the real world are continuous time,
as the scale is infinitesimally fine.
 Eg voltage, velocity,
 Denote by x(t), where the time interval may be
bounded (finite) or infinite

t
Discrete-Time Signals

Some real world and many digital signals are
discrete time, as they are sampled
 E.g. pixels, daily stock price (anything that a
digital computer processes)
 Denote by x[n], where n is an integer value that
varies discretely

Sampled continuous signal

x[n] =x(nk) – k is sample time
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x[n]
n
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Signal Properties

In this course, we shall be particularly
interested in signals with certain properties:


Periodic signals: a signal is periodic if it repeats
itself after a fixed period T, i.e. x(t) = x(t+T) for all t.
A sin(t) signal is periodic.
Even and odd signals: a signal is even if x(-t) =
x(t) (i.e. it can be reflected in the axis at zero). A
signal is odd if x(-t) = -x(t). Examples are cos(t)
and sin(t) signals, respectively
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Signal Properties
Exponential and sinusoidal signals: a signal is
(real) exponential if it can be represented as x(t) =
Ceat. A signal is (complex) exponential if it can be
represented in the same form but C and a are
complex numbers.
 Step and pulse signals: A pulse signal is one
which is nearly completely zero, apart from a short
spike, d(t). A step signal is zero up to a certain
time, and then a constant value after that time, u(t).
These properties define a large class of tractable,
useful signals and will be further considered in the
coming lectures

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What is a System?
Systems process input signals to
produce output signals
 Examples:

A circuit involving a capacitor can be viewed
as a system that transforms the source
voltage (signal) to the voltage (signal)
across the capacitor
 A CD player takes the signal on the CD and
transforms it into a signal sent to the loud
speaker

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Examples

A communication system is generally composed of
three sub-systems, the transmitter, the channel and
the receiver. The channel typically attenuates and
adds noise to the transmitted signal which must be
processed by the receiver
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How is a System Represented?

A system takes a signal as an input and
transforms it into another signal
Input signal
x(t)

System
Output signal
y(t)
In a very broad sense, a system can be
represented as the ratio of the output signal
over the input signal


That way, when we “multiply” the system by the
input signal, we get the output signal
This concept will be firmed up in the coming weeks
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Continuous & Discrete-Time
Mathematical Models of Systems

Continuous-Time Systems



Most continuous time systems
represent how continuous signals
are transformed via differential
equations.
E.g. circuit, car velocity
Discrete-Time Systems


dvc (t ) 1
1

vc (t ) 
vs (t )
dt
RC
RC
dv(t )
m
 v(t )  f (t )
dt
First order differential equations
y[n]  1.01y[n  1]  x[n]
m

Most discrete time systems
v[n] 
v[n  1] 
f [ n]
represent how discrete signals are
m  
m  
transformed via difference
equations
dv(n) v(n)  v(( n  1))

E.g. bank account, discrete car
dt

velocity system
First order difference equations
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Properties of a System

In this course, we shall be particularly
interested in systems with certain
properties:
•
•
Causal: a system is causal if the output at a
time, only depends on input values up to
that time.
Linear: a system is linear if the output of the
scaled sum of two input signals is the
equivalent scaled sum of outputs
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Properties of a System

Time-invariance: a system is time invariant
if the system’s output signal is the same,
given the same input signal, regardless of
time of application.
These properties define a large class of
tractable, useful systems and will be
further considered in the coming lectures
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How Are Signal & Systems Related (i)?


How to design a system to process a signal in particular
ways?
Design a system to restore or enhance a particular signal
–
–

Assume a signal is represented as


Remove high frequency background communication noise
Enhance noisy images from spacecraft
x(t) = d(t) + n(t)
Design a system to remove the unknown “noise”
component n(t), so that y(t)  d(t)
x(t) = d(t) + n(t)
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System
?
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y(t)  d(t)
25
How Are Signal & Systems Related (ii)?

How to design a system to extract specific
pieces of information from signals
–
–
Estimate the heart rate from an electrocardiogram
Estimate economic indicators (bear, bull) from
stock market values

Assume a signal is represented as
 x(t) = g(d(t))
 Design a system to “invert” the transformation
g(), so that y(t) = d(t)
x(t) = g(d(t))
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System
?
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y(t) = d(t) = g-1(x(t))
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How Are Signal & Systems Related (iii)?

How to design a (dynamic) system to modify or control
the output of another (dynamic) system
–
–

Assume a signal is represented as


Control an aircraft’s altitude, velocity, heading by adjusting throttle,
rudder, ailerons
Control the temperature of a building by adjusting the heating/cooling
energy flow.
x(t) = g(d(t))
Design a system to “invert” the transformation g(), so
that y(t) = d(t)
x(t)
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dynamic
system ?
y(t) = d(t)
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Lecture 2: Exercises

Read SaS OW, Chapter 1. This contains
most of the material in the first three lectures,
a bit of pre-reading will be extremely useful!
 SaS OW:

Q1.1
 Q1.2
 Q1.4
 Q1.5
 Q1.6

In lecture 3, we’ll be looking at signals in
more depth.
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A1. Review of Complex Numbers
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Complex Numbers

Complex numbers: number of the form
z=x+j y
where x and y are real numbers and j   1
 x: real part of z; x = Re {z}
 y: imaginary part of z; y = Im {z}
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Complex Numbers
Two complex numbers z1 and z 2 are equal
if and only if their respective real and
imaginary parts are equal
z1  x1  j y1 ;
z 2  x2  j y 2
z1  z 2
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
 x1  x2

 y1  y2
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Representing Complex numbers
Rectangular representation
Imaginary axis
Imaginary Part
y
s1=x + j y
x
Complex-plane
real axis
Real part
(s-plane)
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Representing Complex numbers
Polar representation
Imaginary axis
s1  x  jy   e j
Imaginary Part
ρ
 : length of s1
θ
  s1
y
x
real axis
 : phase angle
Real part
Complex-plane
(s-plane)
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Conversion between Representations
Example
Imaginary axis
s1  3  j 4  5 e j 0.9273
4
 imaginary part 
  tan 

 real part 
1
θ
3
real axis
magnitude
 s1  32  42  5
phase angle    0.9273 ( radian )
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Euler Formula
j
e  cos(  )  j sin(  )
1 
s1  3  j 4  5 e j 0.9273
4 

  tan
 0.9273
 3
 5 cos(0.9273)  j 5sin (0.9273)
4
θ
3
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Complex Numbers
Addition /Subtraction
z1  x1  j y1 ;
z 2  x2  j y 2
z1  z2  ( x1  x2 )  j ( y1  y2 )
z1  z2  ( x1  x2 )  j ( y1  y2 )
(2  j 3)  (5  j 4)  8  (2  5  8)  j (3  4  0)
 1  j
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Complex Numbers
Multiplication/Division
j1
z1  x1  j y1  r1 e ;
z 2  x2  j y2  r2 e
j 2
z1 z 2  ( x1 x2  y1 y2 )  j ( x1 y2  x2 y1 )
y1 x2  y2 x1
z1 x1 x2  y1 y2
j

2
2
2
2
x2  y 2
x2  y 2
z2
z1 z 2  r1r2 e
j 1  2 
z1 r1 j 1  2 
 e
z 2 r2
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Operations
Examples
s  1  j 3
z  2  j5
s  z  ( 1  2)  j (3  5)  1  j8
s z   1  j 32  j5  17  j
s  1  j 3 13  j11


2  j5
z
29
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More Examples
s  2e
j2
z  3e
j
s z  2e
j2
3e
j
 (2  3)e
j ( 2 1)
 6e
j
j2
s 2e
2 3j
 j  e
z 3e
3
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Conjugate
Imaginary axis
s  x  jy
s : complex conjugate of s
s  x  jy
s  x  jy
real axis
s  x  jy
Complex-plane
(s-plane)
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Conjugate
s  x  jy
s  x  jy
ss  x y
2
2
s s sx y
2
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2
2
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Conjugate
s  1  j 2
s  1  j 2
ss  x y  5
2
2
s s s  x  y 5
2
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2
2
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Conjugate
real/imaginary part
s  x  jy
s  x  jy
(s  s )
Re{ s}  x 
2
(s  s )
Im{ s}  y 
2j
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Operations
Polar coordinate Multiplication/Division
j1
s1  1e ,
s1 s2  1e
s2   2e
j1
j 2
 e     e
j 2
2
1
j 1  2 
2
j1
s1
1e
1 j 1 2 


e
j 2
s2  2e
2
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More Examples
(2  j 2)  (1  j )  (4  0 j )
2(1  j )  (1  j )  2(1  1)  4
2 2 e j / 4
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2 e  j / 4  4 e j 0  4
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Keywords
Conjugate
 Modulus
 Real part
 Imaginary part
 Polar coordinates
 Complex plane
 Imaginary axis
 Pure imaginary

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