sid sigsys 1- introduction to signals and systems

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College of Engineering and Computing

ENG 343 Signals and SystemsWinter 2014

Introduction

Dr. Sidra A. Shaikh


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Reference Books

Fundamentals of Signals and Systems: Using theWeb and MATLAB

Edward W. Kamen. Ans Bonnie S. Heck.

Pearson Prentice Hall, 2007. ISBN 013168737-9

Signals and Systems

Oppenheim, A. V, Willsky, A.S., and Nawab, S H. 2003.

3rd Edition, Prentice-Hall. ISBN: 0138097313

2Wednesday, 26 February 2014 Dr. Sidra A Shaikh


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Week # Topic/Material Covered in Class/LabReference in the

Textbook/

Course Material1 Introduction: Continuous and discrete-time signals Operations on signals Time Domain Chapter 1, Textbook #1

2 Properties of signals Elementary signals Continuous- and discrete-time systems

Interconnections of systems.

System Properties.

Chapter 1, Textbook #1

3. & 4 Representations for Linear Time Invariant Systems: Convolution Properties of

convolution

Difference and differential equations

Chapter #1, Textbook #1

5 - characterizing solutions

Fourier Representations of Signals: Discrete time periodic signals - the discrete time

Fourier series

Chapter #2, Textbook #1,

6 Continuous time periodic signals Chapter #2, Textbook #1

7 - the Fourier series Discrete time non-periodic signals Chapter #2, Textbook #1,

8 - the discrete time Fourier transform Continuous time non-periodic signals Chapter #2, Textbook #1,

9 - the Fourier transform Properties of Fourier representations Chapter #3, Textbook #1,

10 Applications of Fourier Representations: Frequency response from time-domain system

descriptions


11 Fourier transform representations for periodic signals Convolution and modulation

revisited

Chapter #5, Textbook #1.

12 - mixing periodic and non-periodic signals Chapter 2, Ref book #1.

13 The Fourier transform representation for discrete-time signals Chapter 5 Textbook #1, Lab Manual.

14 Sampling continuous-time signals

Reconstruction of continuous-time signals from samples


15 Revision

Topics to be covered



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Marks Distribution

Assessment Tool/Component

Learning

Outcome to be

Assessed

Weightage

Class/homework, assignments 10%

Class tests 1 & 2 30%

Practical/laboratory/field work 20%

Semester-end final examination 40%



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SIGNALS?

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What is a Signal? A signal is a pattern of variation of some form

Signals are variables that carry information

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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Signals Represent Information

Whether analog or digital, information is

represented by the fundamental quantity in

electrical engineering: the signal .

Stated in mathematical terms, a signal is merely afunction.

Analogue signals are continuous-valued;

Digital signals are discrete-valued.



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Analog signals are usually signals defined overcontinuous independent variable(s).

For Example, Speech is produced by yourvocal cords exciting acoustic resonances in your

vocal tract.

The result is pressure waves propagating inthe air, and the speech signal thus corresponds toa function having independent variables of spaceand time and a value corresponding to airpressure:

s (x, t) (Here we use vector notation x todenote spatial coordinates).

Analog Signals



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Digital Signals

The word digital means discrete-valued and implies

the signal has an integer-valued independent variable.

Digital information includes numbers and symbols

(characters typed on the keyboard, for example). Computers rely on the digital representation of

information to manipulate and transform information.

Symbols do not have a numeric value, and each is

represented by a unique number.



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SIGNAL CLASSIFICATION



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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 onx, ycoordinates and time tf(x,y,t)

On this course, we shall be exclusively concerned with signals

that are a function of a single variable: time

t

f(t)

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Example: Signals in an Electrical Circuit

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

+

-

i vcvs

R

C

)(1

)(1)(

)()(

)()()(

tvRC

tvRCdt

tdv

dt

tdvCti

R

tvtvti

sc

c

c

cs

Step (signal) vsat t=1

RC = 1

First order (exponential) response

for vc

vs,

vc

t



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Continuous & Discrete-Time Signals Continuous-Time Signals Most signals in the real world are

continuous time, as the scale isinfinitesimally fine.

Eg voltage, velocity,

Denote byx(t), where the time intervalmay be bounded (finite) or infinite

Discrete-Time Signals Some real world and many digital

signals are discrete time, as they aresampled

E.g. pixels, daily stock price (anything

that a digital computer processes) Denote byx[n], where nis an integer

value that varies discretely

Sampled continuous signalx[n] =x(nk)kis sample time

x(t)

t

x[n]

n



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Signal Classification

Continuous-time

Discrete-time



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Signal Properties Particularly interested in signals with certain properties:

Periodic signals: a signal is periodic if it repeats itself after a fixed periodT, i.e.x(t) =x(t+T) for all t. A sin(t) signal is periodic.

Even and odd signals: a signal is even ifx(-t) = x(t) (i.e. it can be reflectedin the axis at zero). A signal is odd ifx(-t) = -x(t). Examples are cos(t) andsin(t) signals, respectively.

Exponential and sinusoidal signals: a signal is (real) exponential if it canbe represented asx(t) = Ceat. A signal is (complex) exponential if it can berepresented in the same form but Cand aare complex numbers.

Step and pulse signals: A pulse signal is one which is nearly completelyzero, apart from a short spike, d(t). A step signal is zero up to a certaintime, and then a constant value after that time, u(t).

These properties define a large class of tractable, useful signals and willbe further considered in the coming lectures



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EVEN AND ODD SIGNALS



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Even an Odd properties of Signals



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Even and Odd functions

A real functionxe(t) is said to be an even function of t if

A real functionxo(t) is said to be an odd function of t if



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Even an Odd properties of Signals

Even and odd functions have the following

properties: Even x Odd = Odd

Odd x Odd = Even

Even x Even = Even

Every signal x(t) can be expressed as a

sum of even and odd components

because:



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Even and Odd components

Given the definition, any signal can be

decomposed into a sum of even xe(t) signal

and odd xo(t) signal.



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USEFUL SIGNAL OPERATIONS



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Useful Signal Operations



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Time Reversal



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Time Shifting



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Time Shifting



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Signal may be delayed by time T:

or advanced by time T:

Time Shifting



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Amplitude Scaling


C f i i C i


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Addition of Signals


C ll f E i i d C ti


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Addition of Signals


C ll f E i i d C ti


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Multiplication of Signals


C ll f E i i d C ti


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College of Engineering and Comp ting


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Time Scaling




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Time Scaling




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Time Scaling




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Signal may be compressed in time

(by a factor of 2):

or expanded in time (by a factor of 2):

Time Scaling




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COMBINATION OF SIGNALS




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Combination of Operations Combination of operations on signals

- Easier to determine the final signal in stages

- Create intermediary signals in which one operation is

performed

Note: Both Method 1 and Method 2 give the same answer




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Combination of OperationsEx. Time shifting and Time scaling (1/2)




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Combination of Operations

Ex. Time shifting and Time scaling (2/2)

Note: Both Method 1 and Method 2 give the same answerWednesday, 26 February 2014 41Dr. Sidra A Shaikh



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SYSTEM?




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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 itinto a signal sent to the loud speaker

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 thetransmitted signal which must be processed by thereceiver




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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 berepresented as the ratio of the output signalover the input signal

That way, when we multiply the system by theinput signal, we get the output signal

This concept will be firmed up in the coming weeks

SystemInput signal

x(t)

Output signal

y(t)




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Example: An Electrical Circuit System

Simulink representation of the electrical circuit

+

-

i vcvs

R

C

)(1

)(1)(

)()(

)()()(

tvRC

tvRCdt

tdv

dt

tdvCti

R

tvtvti

sc

c

c

cs

vs(t) vc(t)

first order

system

v

s,

vc

t




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References:

Online Resources

Fundamentals of Signals and Systems: Using the Web and MATLAB

Edward W. Kamen. Ans Bonnie S. Heck.

Pearson Prentice Hall, 2007. ISBN 013168737-9

Signals and SystemsOppenheim, A. V, Willsky, A.S., and Nawab, S H. 2003.

3rd Edition, Prentice-Hall. ISBN: 0138097313

sid sigsys 1- introduction to signals and systems

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