ee3010_lecture2 al-dhaifallah_term332 1 2. introduction to signal and systems dr. mujahed...
TRANSCRIPT
Al-Dhaifallah_Term332 1EE3010_Lecture2
2. Introduction to Signal and Systems
Dr. Mujahed Al-Dhaifallah
EE3010: Signals and Systems Analysis
Term 332
Al-Dhaifallah_Term332 2EE3010_Lecture2
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
Al-Dhaifallah_Term332 3EE3010_Lecture2
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.
EE3010_Lecture2
Al-Dhaifallah_Term332 5
Homework
Send me e-mailSubject Line: “EE 3010 Student”
EE3010_Lecture2
Al-Dhaifallah_Term332 6EE3010_Lecture2
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.
Al-Dhaifallah_Term332 7EE3010_Lecture2
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.
Al-Dhaifallah_Term332 8EE3010_Lecture2
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.
Al-Dhaifallah_Term332 9EE3010_Lecture2
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.
Al-Dhaifallah_Term332 10EE3010_Lecture2
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.
Al-Dhaifallah_Term332 11
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
EE3010_Lecture2
Al-Dhaifallah_Term332 12
Recommended Reading Material
• Signals and Systems, Oppenheim & Willsky, Section 1
• Signals and Systems, Haykin & Van Veen, Section 1
EE3010_Lecture2
Al-Dhaifallah_Term332 13
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
EE3010_Lecture2
Al-Dhaifallah_Term332 14
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)
Al-Dhaifallah_Term332 15
Example: Signals in an Electrical Circuit
EE3010_Lecture2
+-
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
Al-Dhaifallah_Term332 16
Continuous & Discrete-Time Signals
EE3010_Lecture2
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
Al-Dhaifallah_Term332 17
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
EE3010_Lecture2
Al-Dhaifallah_Term332 18
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
EE3010_Lecture2
Al-Dhaifallah_Term332 19
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
EE3010_Lecture2
Al-Dhaifallah_Term332 20
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
EE3010_Lecture2
Al-Dhaifallah_Term332 21
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)
Al-Dhaifallah_Term332 22
Continuous & Discrete-Time Mathematical Models of Systems
EE3010_Lecture2
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
Al-Dhaifallah_Term332 23
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
EE3010_Lecture2
Al-Dhaifallah_Term332 24
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
EE3010_Lecture2
Al-Dhaifallah_Term332 25
How Are Signal & Systems Related (i)?
EE3010_Lecture2
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)
Al-Dhaifallah_Term332 26
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)
EE3010_Lecture2
System?
x(t) = g(d(t)) y(t) = d(t) = g-1(x(t))
Al-Dhaifallah_Term332 27EE3010_Lecture2
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)
Al-Dhaifallah_Term332 28EE3010_Lecture2
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.
Al-Dhaifallah_Term332 29EE3010_Lecture2
A1. Review of Complex Numbers
Al-Dhaifallah_Term332 30EE3010_Lecture2
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
Al-Dhaifallah_Term332 31EE3010_Lecture2
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
Al-Dhaifallah_Term332 32EE3010_Lecture2
Representing Complex numbersRectangular representation
Complex-plane
(s-plane)
s1=x + j y
Imaginary axis
real axis
Imaginary Part y
xReal part
Al-Dhaifallah_Term332 33EE3010_Lecture2
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
Al-Dhaifallah_Term332 34EE3010_Lecture2
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
Al-Dhaifallah_Term332 35EE3010_Lecture2
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
Al-Dhaifallah_Term332 36EE3010_Lecture2
Complex NumbersAddition /Subtraction
j
jjj
yyjxxzz
yyjxxzz
yjxzyjxz
1
)043()852(8)45()32(
)()(
)()(
;
212121
212121
222111
Al-Dhaifallah_Term332 37EE3010_Lecture2
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
Al-Dhaifallah_Term332 38EE3010_Lecture2
Operations Examples
29
1113
52
31
175231
81)53()21(
52
31
j
j
j
z
s
jjjzs
jjzs
jz
js
Al-Dhaifallah_Term332 39EE3010_Lecture2
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
Al-Dhaifallah_Term332 40EE3010_Lecture2
Conjugate
jyxs
s
jyxs
s of conjugatecomplex :
jyxs
Complex-plane
(s-plane)
Imaginary axis
real axis
jyxs
Al-Dhaifallah_Term332 41EE3010_Lecture2
Conjugate
222
22
yxsss
yxss
jyxs
jyxs
Al-Dhaifallah_Term332 42EE3010_Lecture2
Conjugate
5
5
21
21
222
22
yxsss
yxss
js
js
Al-Dhaifallah_Term332 43EE3010_Lecture2
Conjugate real/imaginary part
j
ssys
ssxs
jyxs
jyxs
2
)(}Im{
2
)(}Re{
Al-Dhaifallah_Term332 44EE3010_Lecture2
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
Al-Dhaifallah_Term332 45EE3010_Lecture2
More Examples
44222
4)11(2)1()1(2
)04()1()22(
04/4/
jjj eee
jj
jjj
Al-Dhaifallah_Term332 46EE3010_Lecture2
Keywords
ConjugateModulusReal partImaginary partPolar coordinatesComplex planeImaginary axisPure imaginary