ee3010_lecture2 al-dhaifallah_term332 1 2. introduction to signal and systems dr. mujahed...

Al-Dhaifallah_Term332 1EE3010_Lecture2

2. Introduction to Signal and Systems

Dr. Mujahed Al-Dhaifallah

EE3010: Signals and Systems Analysis

Term 332


Dr. Mujahed Al-Dhaifallahالله. ضيف آل مجاهد د

Office: Dean Office. E-mail: [email protected] Telephone: 7842983 Office Hours: SMT, 1:30 – 2:30 PM,

or by appointment


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.

Al-Dhaifallah_Term332 4

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-mailSubject 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.


Course Objectives

1. Identify the types of signals and their characterization.

2. Use the Fourier series representation.

3. Differentiate between the continuous and discrete-time Fourier transforms.

4. Grasp the fundamental concepts of the Laplace and Z transforms.

5. Characterize signals and systems in the frequency domain.

6. Apply signals and systems concepts in various engineering applications.


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.


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.


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.


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– Voltages and currents in a circuit

Acoustic signals– Acoustic pressure (sound) over time

Mechanical signals– Velocity of a car over time

Video signals– 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

EE3010_Lecture2

t

f(t)


Example: Signals in an Electrical Circuit

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+-

i vcvs

R

C

)(1

)(1)(

)()(

)()()(

tvRC

tvRCdt

tdvdt

tdvCti

R

tvtvti

scc

c

cs

The signals vc and vs are patterns of variation over time

Note, we could also have considered the voltage across the resistor or the current as signals

Step (signal) vs at t=1 RC = 1 First order (exponential) response for vc

v s,

v c


Continuous & Discrete-Time Signals

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Continuous-Time Signals 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

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

x(t)

t

x[n]

n


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

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

EE3010_Lecture2

SystemInput signal

x(t)

Output signal

y(t)


Continuous & Discrete-Time Mathematical Models of Systems

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Continuous-Time Systems Most continuous time systems

represent how continuous signals are transformed via differential equations.

E.g. circuit, car velocity

Discrete-Time Systems Most discrete time systems

represent how discrete signals are transformed via difference equations

E.g. bank account, discrete car velocity system

)(1

)(1)(

tvRC

tvRCdt

tdvsc

c

)()()(

tftvdt

tdvm

First order differential equations

][]1[01.1][ nxnyny

][]1[][ nfm

nvm

mnv

First order difference equations

))1(()()( nvnv

dt

ndv


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)?

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How to design a system to process a signal in particular ways?

Design a system to restore or enhance a particular signal– Remove high frequency background communication noise– Enhance noisy images from spacecraft

Assume a signal is represented as x(t) = d(t) + n(t)

Design a system to remove the unknown “noise” component n(t), so that y(t) d(t)

System?

x(t) = d(t) + n(t) y(t) d(t)


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)

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System?

x(t) = g(d(t)) y(t) = d(t) = g-1(x(t))


How Are Signal & Systems Related (iii)?

How to design a (dynamic) system to modify or control the output of another (dynamic) system– 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.

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)

dynamicsystem ?

x(t) y(t) = d(t)


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.


A1. Review of Complex Numbers


Complex Numbers

Complex numbers: number of the form

z = x + j y

where x and y are real numbers and x: real part of z; x = Re {z} y: imaginary part of z; y = Im {z}

1j


Complex Numbers

21

2121

222111

21

;

equal are partsimaginary

and real respective their ifonly and if

equal are z and z numberscomplex Two

yy

xxzz

yjxzyjxz


Representing Complex numbersRectangular representation

Complex-plane

(s-plane)

s1=x + j y

Imaginary axis

real axis

Imaginary Part y

xReal part


Representing Complex numbersPolar representation

anglephase

s

soflength

:

:

1

1

Complex-plane

(s-plane)

Imaginary axis

real axis

Imaginary Part

y

x

Real part

ρ

θ

jejyxs 1


Conversion between Representations Example

)(9273.0

543 221

radiananglephase

smagnitude

0.9273 1 543 jejs

Imaginary axis

real axis

4

3

θ

part real

partimaginary tan 1


Euler Formula

)sin()cos( je j

0.9273)(in5j0.9273)cos(5

543 0.9273 1

s

ejs j

4

3

θ

9273.03

4tan

1


Complex NumbersAddition /Subtraction

j

jjj

yyjxxzz

yyjxxzz

yjxzyjxz

1

)043()852(8)45()32(

)()(

)()(

;

212121

212121

222111


Complex NumbersMultiplication/Division

21

21

21

2

1

2

1

2121

22

22

122122

22

2121

2

1

1221212121

22221111

)()(

;

j

j

jj

er

r

z

z

errzz

yx

xyxyj

yx

yyxx

z

z

yxyxjyyxxzz

eryjxzeryjxz


Operations Examples

29

1113

52

31

175231

81)53()21(

52

31

j

j

j

z

s

jjjzs

jjzs

jz

js


More Examples

jj

j

jjjj

j

j

ee

e

z

s

eeeezs

ez

es

32

)12(2

2

3

2

3

2

6)32(32

3

2


Conjugate

jyxs

s

jyxs

s of conjugatecomplex :

jyxs

Complex-plane

(s-plane)

Imaginary axis

real axis

jyxs


Conjugate

222

22

yxsss

yxss

jyxs

jyxs


Conjugate

5

5

21

21

222

22

yxsss

yxss

js

js


Conjugate real/imaginary part

j

ssys

ssxs

jyxs

jyxs

2

)(}Im{

2

)(}Re{


Operations Polar coordinate Multiplication/Division

21

2

1

2121

21

2

1

2

1

2

1

212121

2211 ,

jj

j

jjj

jj

ee

e

s

s

eeess

eses


More Examples

44222

4)11(2)1()1(2

)04()1()22(

04/4/

jjj eee

jj

jjj


Keywords

ConjugateModulusReal partImaginary partPolar coordinatesComplex planeImaginary axisPure imaginary

ee3010_lecture2 al-dhaifallah_term332 1 2. introduction to signal and systems dr. mujahed...

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