ELEC264: Signals And Systems Topic 1: LTI Systems · ELEC264: Signals And Systems Topic 1: LTI...

ELEC264: Signals And Systems

Topic 1: LTI Systems

Aishy Amer

Concordia University

Electrical and Computer Engineering

Figures and examples in these course slides are taken from the following sources:

• A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997

• M.J. Roberts, Signals and Systems, McGraw Hill, 2004

• J. McClellan, R. Schafer, M. Yoder, Signal Processing First, Prentice Hall, 2003

Introduction

Types of signals

CT Signals

Sinusoidal and exponential signals

Periodic and aperiodic signals

Signal energy and power

Transformation of the independent variable

Even and odd signals

Special signals

CT Systems & basic properties

DT Signals






Special signals

DT systems & basic properties

2

Signals and Systems

A signal is any physical phenomenon which conveys information (e.g., human voice)

Systems respond to signals and produce new signals (e.g., Aircraft, human body)

Excitation signals are applied at system inputs

Response signals are produced at system outputs

3

Signals & Systems

Representation, transformation and

manipulation of signals and

information they contain

Modifying and analyzing information

with computers

4

What is a signal ?

A flow of information

Mathematically, x(t)

a function of independent variables

such as time (e.g. speech signal),

position (e.g. image)

t : a common convention is to refer to

the independent variable as time,

although may in fact not

5

Example signals

Speech: 1-Dimension signal as a function of time s(t)

Grey-scale image: 2-Dimension signal as a function of

space I(x,y)

Video: 3 x 3-Dimension signal as a function of space and

time {R(x,y,t), G(x,y,t), B(x,y,t)}

6

Signals: Examples

Example signals

7

Continuous to discrete?

Computers and other digital devices are restricted to discrete time

Take samples of the continuous signal => discrete

Analog to digital conversion (ADC)

]5.42.34.25.0[][

)( 4

nx

etx

t

8

Continuous to discrete?

From http://www.ece.rochester.edu/courses/ECE446

9

Key History of Signals &Systems

Prior to 1950’s: analog signal processing using

electronic circuits or mechanical devices

1950’s: computer simulation before analog

implementation, thus cheap to try out

1965: Fast Fourier Transforms (FFTs) by

Cooley and Tukey – make real time digital

signal processing (DSP) possible

1980’s: IC technology boosting digital signal

processing (DSP)

10

Typical system components

Physical signals Analog signals Digital signals

Transducers

e.g. microphones

Analog-to-digital

converters

Digital-to-Analog

converters

Output devices

11

Applications of Signals &Systems

Speech processing

Enhancement – noise filtering

Coding, synthesis and recognition

Image processing

Enhancement, coding, pattern recognition (e.g. OCR)

Multimedia processing

Media transmission, digital TV, video conferencing

Communications

Biomedical engineering

Navigation, radar, GPS

Control, robotics, machine vision

12

Pros and cons of DT Signals

&System

Pros Easy to duplicate

Stable and robust: not varying with temperature,

storage without deterioration

Flexibility and upgrade: use a general computer or

microprocessor

Cons Limitations of ADC and DAC

High power consumption and complexity of a DSP

implementation: unsuitable for simple, low-power

applications

Limited to signals with relatively low bandwidths

13

Outline Introduction

Types of signals

CT Signals






Special signals


DT Signals






Special signals


14

Types of signals

A variable (or multiple variables) that changes in time

Speech or audio signal: Amplitude that varies in time

Temperature readings at different hours of a day

Stock price changes over days

…

More generally, a signal may vary in time, 1D, and/or in space, 2-D

A picture: the color varies in a 2-D space

A video sequence: the color varies in 2-D space and in time

15

Types of signals

The independent variable may be either continuous or discrete

Continuous-time signals: The time varies continuously

Discrete-time signals are defined at discrete times

• Represented as sequences of numbers

• Discrete ~ Countable

The signal amplitude may be either continuous or discrete

Analog signals: both time and amplitude are continuous

Digital signals: both are discrete

Signal processing systems classification follows the same lines

16

Types of signals

Signals are functions x(t) = et/4

to manipulate them we apply

calculus on Continuous-Time (CT) signals

algebra on Discrete-Time (DT) signals

CT Signal x(t) is a continuous-value function

DT signal x[n] is a sequence of real or complex numbers

x[n] = [0.5 2.4 3.2 4.5]• x[0] =0.5, x[1]=2.4,…

17

Basic Signals

Sinusoidal

Exponential

Unit Impulse

Unit Step

An arbitrary signal can be expressed as a sum of manysinusoidal signals with different frequencies, amplitudesand phases

00

01)(

t

ttu

1)( dtt

)cos()( 0ttx

tjetx 0)(

00 2, f

00

01)(

t

tt

18

Periodic CT Signals

A CT signal is periodic if there is a positive value

for which

The fundamental period of is the smallest positive

value of for which the equation above holds

Examples:

is periodic with fundamental period

30),3

cos()( tttx

19

Sinusoidal Signals

Sinusoidal signals: important because they can be used to synthesize any signal Phase shift: how much the sinusoidal signal is shifted away from t=0

Music notes are essentially sinusoids at different frequencies

f0 = 1000Hz

f0 = 2000Hz

20

Constant Signal

Case:

•Let the fundamental frequency be zero,

i.e.,constant signal (d.c) has zero

rate of oscillation

•x(t) is periodic with period T for any positive

value of T

Thus fundamental period is undefined

0o

fT

10

f=0

Illustration of Frequency

21

22

Frequency content in signals

A constant : only zero frequency (DC) component

A sinusoid : Contain only a single frequency component

Slowly varying : contain low frequency only

Fast varying : contain very high frequency

Sharp transition : contain from low to high frequency

Music: :

contain both slowly varying and fast varying components

23

What is frequency of an

arbitrary signal?

Sinusoidal signals have a distinct (unique) frequency

An arbitrary signal x(t) does not have a unique frequency

x(t) can be decomposed into many sinusoidal signals

with different frequencies, each with different magnitude

and phase

)32

3cos(7)

4cos()( tttx

24

Deterministic vs. random

signals

A deterministic signal is a signal in which each value of the signal is fixed and can be determined by a mathematical expression, rule, or table

Future values of the signal can be calculated from past values with complete confidence

A random signal cannot be described by a mathematical formula

has a lot of uncertainty about its behavior

Future values of a random signal cannot be accurately predicted

Future values can usually only be guessed based on the averages of sets of signals

atetx )(

25

Random Signals

26

Types of Signals:Continuous-Time (CT) vs. Discrete-time (DT)

Analog vs. digital

Periodic vs. non-periodic

27

Types of Signals

(Discrete ~ countable)

28


Types of signals

CT Signals






Special signals


DT Signals






Special signals


29

Sinusoidal & Exponential

•Play a central role in signals &

systems

•Serve as building block for many

other signals

30

Periodic Complex Exponential

& Sinusoidal Signal

•Sinusoidal signal

A CT sinusoidal signal

1. has one unique frequency

Two signals with different

frequencies are never

identical

2. is always periodic for any

tjte oo

tj o sincos

angle Phase :

Period lFundementa :

2 Frequency lFundementa /1

2 frequency Angular :

at t signal theof value: x(t)

instant time: t

Signal : x

Amplitude Signal :A

)cos()(

0

000

000

0

T

TTf

f

tAtx

• Complex exponential signal:

They can be written in terms of sinusoidal

signals

0

Src: Wikipedia

31

Complex numbers

sin2

cos2

sincos

:relation '

. , |z|r

z, of phaseor angle theis and

z, of magnitude theis 0r where

-:formPolar

numbers, real are b and a and 1

,

:forr rectangulaor

jee

ee

je

sEuler

z

rez

jwhere

jbaz

Cartesian

jj

jj

j

j

32

tjjtjj

ooo ee

Aee

AtA

22)cos(

:lexponentiacomplex

periodic of in termswritten

becan period, lfundamenta

with signal, sinusoidal The


& Sinusoidal Signal

33

ondcycles/secor Hertzin frequency lfundamenta theof

condradians/sein freq.angular lfundamenta of2 o

period lfundamenta the

with

})(

{.)sin(

})(

Re{.)cos(

:followsIt

oT

tojeIMAtoAor

tojeAtoA

Periodic Complex Exponential & Sinusoidal Signal

34


A necessary condition for a complex

exponential tje to be periodic with period To is 1oTj

e

,...2,1,0,2T i.e. .2 of

multiple a is that implieswhich

o kk

To

00 .)( TjtjTtj

eee

35

Harmonically Complex

Exponential Signals

0o

k

,.,frequency positive single

a of multiples arethat sfrequencie

lfundamenta with lsexponentia

periodic ofset :)( 0

kei

Aetx

k

tjk

k

angei

To

oo

o

k .. of multipleinteger an bemust then

,2

define weIf

K

TT 0

36


Exponential Signals

||

periods lfundamenta and |k| sfrequencie

lfundamenta with periodic is )( 0, k

constant. a is )( 0,kFor

2,...1,0,k ,)(

o

k

TT

tx

tx

etx

o

k

k

tjk

ko

37


Exponential Signals

.Tlength of interval any time

during periods lfundamenta its of |k|exactly

throughgoesit as well,as T periodwith

periodic still is )( harmonickth The

o

o

txk

38

General Complex Exponential

Signals

.|||C|

)(Then

.ja and |C|C

:formcartesian in a

and formpolar in expressed is C If

numbers.

complex are a'' and C''both where,)(

)()(

o

tjttjj

at

j

at

oo eeCee

Cetx

e

Cetx

39

General Complex Exponential

Signals

l.exponentia decayingby multipled signals sinusoidal is x(t)

0, If

l.exponentia growingby multipled signals sinusoidal is x(t)

0, If

.sinusoidal are partsimaginary & real the

0, If

)sin(||)cos(||)(

relation, sEuler' Using

teCjteCCetx o

t

o

tat

40

σ>0

σ<0

Growing & Decaying

Sinusoidal Signals

41

pi=3.142;

t=-10:.1:10;

f=2000;

w=2*pi*f;

sigma=0.1;

x=zeros(size(t));

x=exp((sigma+w*i)*t);

theta=pi/4;

c=1*exp(i*theta);

y=c*x;

subplot(2,1,1);

plot(t,y);

grid;

sigma=-0.1;

x=zeros(size(t));

x=exp((sigma+w*i)*t);

theta=pi/4;

c=1*exp(i*theta);

y=c*x;

subplot(2,1,2);

plot(t,y);

grid;

end;

Matlab Program for

Growing & Decaying Sinusoids

42


Types of signals

CT Signals






Special signals


DT Signals






Special signals


43

Periodic Signals: Examples

Planet and satellite orbital positions

Phases of the moon

Firing pattern of spark plugs in a car traveling at a constant speed

Blinker lights in automobiles

Angular position of a pendulum in antique clocks

Migration pattern of birds

44

Periodic Signals:

Sinusoidal Signals

Sinusoidal signal:

Unique frequency e.g., 10Hz

An arbitrary x(t):

No unique frequency

x(t) = summation of sine or cosine

functions at different frequencies

45

Periodic CT Signals

,Example:

46

Periodic CT Signals:

Examples

47

Sum of CT periodic signals

The period of the sum of CT periodic functions is the

least common multiple of the periods of the individual

functions summed

If the least common multiple is infinite, the sum is aperiodic

48

Periodic CT Signals:

Examples

49

Aperiodic CT Signals

A function that is not periodic is called

aperiodic

Aperiodic signal

Examples:

)()( nTtxtx

otherwise 0

30)3

cos()(

tttx

50


Types of signals

CT Signals






Special signals


DT Signals






Special signals


51

Signal Power and Energy

Signal: A function of a time-varying amplitude

Signal: Many different physical entities

No unit for energy/power

Often, a signal is a function of varying amplitude over time

A good measurement of the strength of a signal would be the area under the function

But this area may have a negative part which does not have less strength than a positive signal of the same size

This suggests either squaring the signal or taking its absolute value, then finding the area under that curve

Energy/power: strength of the signal

52

Energy of CT Signals

53

Energy of

CT Signal

54

Signal Energy and Power

A signal with finite signal

energy is called an energy

signal

If the signal does not

decay infinite energy

A signal with infinite signal

energy and finite average

signal power is called a

power signal

55

Energyof

CT Signal: Example

56

Energy of

CT Signal: Example

57

Signal Power

58

Signal Power

59

Signal Power: Example

60


Types of signals

CT Signals






Special signals


DT Signals






Special signals


61

Transformations of CT

signals

Modifying a signal x(t) through

1. Transformation of the independent

variable t e.g., x(t/2)

2. Combination of signals:

y(t) = x(t) w(t)

y(t) = x(t)/w(t)

y(t)=sin(t)x(t)

y(t)=dx/dt

62

Transformations of CT signals:

Combination of Signals

63

Transformation of CT Signals:

Differentiationt

txttx

dtd x

)()(

64

Transformation of CT

Signals: Integration

65

Transformation of Independent

Variable or Modification of

independent variable t

Modifying signals through elementary

transformations

Examples of elementary transformation

time shift, x(t-t0)

time reversal, x(-t)

time scaling, x(0.5t)

66

Shifting right or lagging

signal x(t)

X(t)

t

X(t-t0)

0

0 t

t0is a positive value

67

Shifting left or leading signal

x(t)

X(t)

tX(t+t1)

0

0 t

-t1

68

Folded or Flipped x(t)

=x(-t), time reversal

69

Time scaling of continuous

signal

x(t)

x(2t)

x(t/2)

t

t

t

Compression a>1

Linearly stretching a<1

70


Variable: Applications

Play music at faster or slower speed

Aircraft control system:

Input correspond to pilot action

Actions are transformed by electrical &

mechanical system of the aircraft to

changes to aircraft trust or position control

surfaces such as the rudder & ailerons

Finally these changes affect the dynamics

& kinematics such as the aircraft velocity

and heading

71


Time Shifting Applications

Time-shifting occurs in many real physical systems:

Listening to someone talking 2m away

Received signal will be delayed, but the delay won’t be noticeable

Satellite communication systems (delay can be noticeable if ground stations are not directly below the satellite)

Radar systems:• Transmitted signal Ax(t)

• Received signal Bx(t-to), with B<A, due to attenuation

72


Time Scaling Applications

Examples:

Playing an audio tape at a faster or slower speed

Doppler effect: standing by the side of a road

while a fire truck approaches and then passes by

73

Modification of independent

variable (time axes)

)( is what fixed afor


ofargument theis :

of parameters are :,

of t variableindependen theis :

offunction a :

)()(

tx

tx

xt

x

xt

tx

txty

• Recommended approach:

o Sketch y(t) for a selected set of t until y(t) becomes clear

o Steps:

1. Rewrite: y(t) = x(α(t+β/ α))

2. Scale by |α|: x(|α|t)

3. Invert x(|α|t) if α<0

4. Shift to the LEFT by |β/ α| if β/ α>0

5. Shift to the RIGHT by |β/ α| if β/ α<0

74



75

Transformation of CT Signals

76

Transformation of CT Signals

77


Signals

78


Signals

79


Signals: Examples x(t)

x(t+1), x(t) shifted left by 1sect

t

1

1

0

0

1 2

1 2-1

80

Tables of x(t) & x(t+1) & x(-t+1)

t x(t) x(t+1) x(-t+1)

-2 0 0 0

-1 0 1 0

0 1 1 1

1 1 0 1

2 0 0 0

3 0 0 0

81

Example

x(t+1) is x(t) shifted left by 1

x(-t+1) is x(t+1) flipped about t=0t

t

1

1

0

0

1 2

1 2-1

-1

82

Example: Alternative 1

x(t-1) is x(t) shifted right by 1sec

x(-t+1)=x(-1(t-1))

Flip about axis t=1

t

t

1

1

0

0

1 2

1 2-1

83

Example , Method 2

x(-t), flip about axis t=0

x(-t+1), shift right (because -t) by 1t

t

1

1

0

0

1 2

1 2-1

-1

84

Example

x(t)

x(t+1), x(t) shifted left by 1sect

t

1

1

0

0

1 2

1 2-1

85

Example

x(3t/2), x(t) compressed by 2/3

t

1

0 1 2-1 2/3 4/3

x((3/2)*(t+2/3)),

x(t) compressed by 2/3

& shifted left by 2/3

t

1

0 1-1 2/3-2/3

86


signals: Examples

87


signals: Examples

88


Signals: Examples

89


Types of signals

CT Signals






Special signals


DT Signals






Special signals


90

Even and Odd Signals

An even signal is identical to its time reversed

Example:

An odd signal has the property

91

Even and Odd CT Signals

92

Even and Odd Parts of CT

Signals

The even part of a CT function is

The odd part of a CT function is

A function whose even part is zero is odd and a function whose odd part is zero is even

The derivative of an even CT function is odd and the derivative of an odd CT function is even

The integral of an even CT function is an odd CT function, plus a constant, and the integral of an odd CT function is even

93

Even and Odd Signals:

Example

94

Products of Even and Odd CT

Functions

95

Product of 2 Odd Functions

96

Integrals of Even and Odd CT

Functions

97


Types of signals

CT Signals






Special signals


DT Signals






Special signals


98

Singularity Functions

In engineering, we often deal with the idea of an action occurring at a point

Whether it be a force at a point in space or a signal at a point in time, it becomes worth while to develop some way of quantitatively defining this

This leads us to the idea of a unit impulse

Unit impulse & complex exponential functions are the two most important functions in systems and signals courses

99


Many useful signals are not continuous or

differentiable at every point in time.

For instance, describe the operation of switching on

or off a signal at some specified time

Singularity functions are a set of functions that

are related to one another via integrals/derivatives

and can be used to mathematically describe signals

with discontinuities.

Example:

),...(),( tut

100


101

CT Unit Impulse

The CT unit impulse function is represented as

The area under the CT unit impulse is equal to 1

00

01)(

t

tt

1)( dtt

102

Properties of CT Impulse

Sampling (Shifting) Property:

the value of the function at a point

Scaling Property

)/()(||

1 abtbata

103

Properties of CT Impulse

Equivalence Property

104

CT Unit Step

105

Relationship: CT unit step and

unit impulse

The CT unit impulse is the first derivative of the

continuous-time unit step

The area under the CT unit impulse is equal to 1

The CT unit impulse function is represented as

106


unit impulse

Example: consider a mass with zero velocity. Assume

that a force is applied to it to change its velocity from

zero to 1 on a surface with no friction. The acceleration

of the mass will be a unit impulse

It can be easily verified that

107


unit impulse

The CT unit step is not differentiable at t=0

One can use continuous approximation to the unit

step

Corresponding unit impulse

108

The CT Signum

109

The CT Unit Ramp

110

The CT Unit Rectangle

111

The CT Unit Triangle

112

The CT Unit Sinc Signal

113


Types of signals

CT Signals






Special signals


DT Signals






Special signals


114

Introduction to Systems

To get the output y(t) Apply the system S{} on input x(t)

y(t) is the response of S{} to x(t)

A system: An integrated whole composed of diverse, interacting, specialized parts

System performs a function not possible with any of the individual parts

Any system has objectives

Systems respond to particular signals by producing other signals or some desired behavior

115

Introduction to Systems:

Examples

116


a Communication System

A communication system has an information

signal plus noise signals

This is an example of a system that consists of

an interconnection of smaller systems

Cellphones are based on such systems

117


Image System to Aid

Perception

118


Sound Recording

119


Interconnections of Systems

Systems can be interconnected in series (cascade),

parallel, feedback, or combination

120


Response of Systems

Systems respond to signals and produce new

signals

Real signals are applied at system inputs and

response signals are produced at system outputs

Example: What is the response of a system to a

unit impulse?

121

Response of systems

The response of a system to an impulse is

called "impulse response" h(t)

The impulse response h(t) completely

characterize a Linear Time-Invariant (LTI)

system

122

Basic System Properties

Linearity or superposition property

Linear system

possesses the property of superposition

any constant values a and b, the following equation is satisfied

It can be easily verified that for linear systems:

an input which is zero for all time,

results in an output which is zero for all time

123


124

Basic System Properties:

Examples

Linearity:

linear? )()( Is 2 txty

)()(

2

)()()]()([)(Then

)()()(Let

21

2

2

2

21

2

1

2

21

2

21

21

tyty

xbxabxxa

tybtyatxbtxaty

txbtxatx

TI? )()( Is 2 txty Time Invariance:

x(t) ~ (x(t))2

x(t-t0) ~ (x(t-t0))2 = y(t-t0)

125


Memory:

System is memoryless if its output for each value of the independent variable at a given time is dependent only on the input at that same time

Memoryless CT system: the input-output relationship of a resistor

Examples: y(t) = x(t-1); y(t) = x(t/2);

)()( tRitv

126


Invertibility:

System is invertible if distinct inputs lead to

distinct outputs

If a system is invertible

• inverse system exists, when cascaded with the

original system, yields an output equal to the input

to the first system.

Example

x(t) MP3 y(t) MP3 x(t) ?

INV

SSx(t) y(t) x(t)

127


Causality: A system is causal (or non-anticipative) if the output

at any time depends only on values of the input at the present time and in the past

All memoryless systems are causal

Examples of causal systems

Examples of non-causal systems

Causal signals are zero for all negative t

Anti-causal signals are zero for all positive t

Non-causal signals have non-zero values in both positive and negative t

128


Stability: Stable system: small inputs lead to responses that do

not diverge

BIBO stable system: bounded input results in a

bounded output. If |x(t)|<∞ |y(t)|<∞

Stable system is always BIBO stable but a BIBO

stable system is not necessarily stable

Is the accumulator system stable?

Examples:

129

CT Systems: Example

A system is defined by the following relationship: Is this system: BIBO Stable; Casual; Linear; Memoryless; Time-Invariant;

Invertible?

All answers must be justified (i.e. a simple “Yes” or “No” is not sufficient).

The system is Stable:

So, for any bounded input, the output is bounded.

The system is Casual: Output at time t depends on input at time t/2 - which is the past. The system is Casual.

The system is Linear: Consider:

)2/()2/sin()( txtty

1)2/sin(1 t

)()()2/()2/sin()2/()2/sin(

))2/()2/()(2/sin()2/()2/sin()(

)()()(

)2/()2/sin()(

)2/()2/sin()(

2121

2133

213

22

11

tbytaytxtbtxta

tbxtaxttxtty

tbxtaxtxlet

txtty

txtty

130

CT Systems: Example

The system is not Memoryless: Output does not solely depend on current input values (i.e. depends on past input values).

The system is not Time-invariant:

)_(

)2/()2/sin()2/()2/sin()(

)()(

)2/)(()2/)sin(()_(

)2/()2/sin()(

0

011

01

000

tty

ttxttxtty

ttxtxlet

ttxtttty

txtty

131

CT Systems: Example

The system is not Invertible: There

are Different Inputs which lead to the

same Outputs

0)2/()sin()2/()2/sin()(

)2()(

0)2/()0sin()2/()2/sin()(

)()(

)2/()2/sin()(

221

2

111

1

txtxtty

ttx

txtxtty

ttx

txtty

132


CT Signals






Special signals


DT Signals






Special signals


133

Discrete-time signals

Sequences of numbers

Periodic sampling of an

analog signal

integeran is where

]},[{

n

nnxx

period. sampling thecalled is where

),(][

T

nnTxnx a

x[1]

x[2]

x[n]x[-1]

x[0]

ADC

134

DT Signals: Examples

135

Periodic DT Signals

136

Basic DT Signals

Sinusoidal

Exponential

Unit Impulse

Unit Step00

01][

n

nnu

)cos(][ 0nnx

njenx 0][

00 2, f

00

01][

n

nn

137

DT Sinusoidal signals

angle Phase :

Period lFundementa :

Frequency lFundementa : /1

2 frequency Angular :

nat signal theof value:x[n]

index time:n

Signal :x

Amplitude Signal :A

)sin(][

0

00

000

0

N

NF

F

nAnxA DT sinusoidal signal

1. is NOT always periodic

** Periodic only if its frequency

is a rational number

2. Two signals with different

frequencies maybe identical

njnj oo ee)2(

138

Sinusoidal sequences

nnAnx allfor ),cos(][ 0

139

Discrete-time Sinusoidal

Signals

)12/2cos(][ nnx

)31/8cos(][ nnx

)6/cos(][ nnx

140

DT Sinusoidal Signals

DT signals possess:

1) Infinite total energy

2) Finite average power

141

Decaying or

Damped Sinusoids

Examples:

oResponse of RLC circuits

oMechanical systems having both

damping & restoring forces e.g.

automotive suspension system

x[n] = eσn.cos(ω0n)σ>0 : grows

σ<0 : decays

142

Growing Real Exponential Signalnnxge 1.1*2][..1 where*][ nAnx

143

Decaying Real Exponential Signal10 where*][ nAnx

nnx 9.0*2][

144

Real Exponential Signal

Real-valued discrete exponentials

are used to describe:

1) Population growth as function of

generation

2) Total return on investment as a

function of day, month or

quarter

145

Real Exponential Signal01- where*][ nAnx

nnx )9.0(*2][

146

Real Exponential Signal1 where*][ nAnx

nnx )1.1(*2][

147

Real Exponential Signal1 where*][ nAnx nnx )1(*2][

148

Complex exponential signals

njjnjj

o

oo

nj

o

o

nj

o

n

oo

o

o

eeA

eeA

nA

njne

nAnx

enx

Aenx

SignallExponentia

22)cos(

sincos:relation sEuler' From

radians of units have and

both then ess,dimensionl asn Taking

)cos(][

:signal sinusoidal torelatedclosely is signal This

][

imaginary)purely ( j & 1Alet ,][

149

General Complex

Exponential Sequence

)sin(||||)cos(||||

||||

||||][

,|| and || If

numbers.complex generalin are andA where

,][

00

)( 0

0

0

nAjnA

eA

eeAAnx

eAAe

Anx

nn

njn

njnjn

jj

n

150

General complex exponential

signals

151


sequence

By analogy with the continuous-time case,

the quantity is called the frequency of

the complex sinusoid or complex

exponential and is the phase

n is always an integer

differences between discrete-time and

continuous-time

0

)sin(||)cos(||||][

,1||When

00

)( 0 nAjnAeAnxnj

152


signals

1

1

153

Periodicity Properties of DT

Complex Exponentials

There are differences in each of the

above properties for the discrete-time

case of nj oe

o

j

o

j

of any valuefor periodic is e 2)

noscillatio of rate theishigher the, islarger The)1

et counterpar CT of properties Two

o

o

t

t

154



signals DTfor 2 of intervalfrequency

consider only to need we,2 ofy periodicit thisof Because

of aluesdistinct v allfor distinct all are signals the

wherebycase CT thefromdifferent very is This

on so and,4,2 sfrequencieat Similarly

at that as same theis 2 frequency at lexponentia the

:2 frequency with lexponentiacomplex DT heConsider t

o

oo

oo

2)2(

o

njnjnjnj ooo eeee

155



2at signal d.c.or sequenceconstant a i.e. 0 todecrease will

n oscillatio r the thereafte, until increasesn oscillatio the

n)oscillatio no sequence,constant (d.c., 0 from Increasing

magnitudein increased is as

noscillatio of rate increasingy continuall a havenot does signal the

signal, DT ofy periodicit implied thisof Because

o

o

o

o

nj oe

156



in timepoint each at sign changing rapidly, oscillates signal the

,)1()(e ,of multiple odd,for :Note

of multiple odd,3,at are sfrequencieHigh 2.

of multipleeven

,2 ,0at occurs sfrequencie low 1.

Therefore,

nj

o

o

o

nnje

or 20

:2 of rangefrequency lFundamenta

157

524.069269

)4.0cos(69cos :Examples

) same thegive(or same theare

- ;2 sfrequencie All

or 20

:2 of rangefrequency lfundamenta a has

integeran is ;)2(

ππ.kπn.

πnπn.(

e

k

e

kee

o

nj

ook

nj

njnkj

k

oo



158



).....3

as same theappears )2

2 as same theappears 0)1

or 20


00

00

qq

qq

159



qq 2 as same theappears 0


00

160



1)0cos(][ nnx )8/cos(][ nnx )4/cos(][ nnx

)2/cos(][ nnx)2/3cos(][ nnx

)4/7cos(][ nnx )8/15cos(][ nnx )2cos(][ nnx

)cos(][ nnx

161



otherwise periodicnot is andnumber rational a is

2/ if periodic is e signal that themeans This

,2

y equvalentlor ,2 i.e.

2 of multiple a bemust

.1ely equivalentor ,ee

0,N period with periodic be toefor order In

o

j

oo

o

jj)(j

j

o

ooo

o

n

NnNn

n

N

mmN

N

This is also true for the DT sinusoids

162

Fundamental Period & Frequency of

DT complex exponential

)2

m(N

as written is period lfundamenta The

,N

2 isfrequency lfundamenta its

N, period lfundamenta with periodic ise x[n]If

o

j o

m

o

n

163

DT Sinusoidal Signals

12

1

2

,12/2 because periodic

)12/2cos(][

o

o

nnx

31

4

2 ,31/8 because periodic

)31/8cos(][

oo

nnx

number rational2

,6/1 because periodicnot

)6/cos(][

o

o

nnx

164

Periodic & Aperiodic

DT sinusoidal signals

165

DT Sinusoids

166

DT Sinusoids

167

DT Sinusoids

168

DT Periodic Sinusoids

169

nt oo jje and e signal theof Comparison

o

jj

of values

distinctfor signalsDistinct

e e oo nt

o of choiceany for Periodic

ofrequency lFundamenta

o

undefined

2:0

:0

period lFundamenta

o

o

2 of multiplesby separated

of for values signals Identical o

m and 0N integers somefor

,2

ifonly Periodic oN

m

mfrequency lFundamenta 0

)2

(:0

:0

period lFundamenta

o

o

o

m

undefined

170

CT vs. DT : Frequency

Consider a frequency

More generally being an integer,

Same for sinusoidal sequences

So, only consider frequencies in an interval of

such as

njnjnjnjAeeAeAenx 000 2)2(

][

)2( 0

20or 00

rr ,)2( 0

)cos(])2cos[(][ 00 nAnrAnx

2

njrnjnjnrjAeeAeAenx 000 2)2(

][

171

CT vs. DT: Frequency

For a CT sinusoidal signal

For the DT sinusoidal signal

rapidly more and more oscillates )( increases, as

),cos()(

0

0

tx

tAtx

slower become nsoscillatio the,2 towards from increases as

rapidly more and more oscillates ][ , towards0 from increases as

),cos(][

0

0

0

nx

nAnx

172

CT vs. DT: periodicity

CT case: a sinusoidal signal and a complex

exponential signal are both periodic

DT case: a periodic sequence is defined as

where the period N is necessarily an integer.

For sinusoid,

integeran is where

/2or 2 that requireswhich

)cos()cos(

00

000

k

kNkN

NnAnA

nNnxnx allfor ,][][

173

CT vs. DT: periodicity

For complex exponential sequence

Complex exponential and sinusoidal sequences

are not necessarily periodic in n with period

depending on the value of , may not be periodic at

all

Consider

kN

eenjNnj

2for only trueiswhich

,

0

)( 00

)/2( 0

0

16 of period ath wi),8/3cos(][

8 of period ath wi),4/cos(][

2

1

Nnnx

Nnnx

Increasing frequency increasing period!

174


Types of signals

CT Signals






Special signals


DT Signals






Special signals


175

Periodic DT Signals

A periodic DT function is one which is invariant to the

transformation , nn+mN , where N is a period of the

function and m is any integer

A DT signal is periodic with period where is a

positive integer if

The fundamental period of is the smallest positive

value of for which the equation holds

Example:

Is periodic with fundamental period

)3

8()4

3

2m()

2m(N

o

m

number rational a is

2/ if periodic is sinusoid DT o

176


Types of signals

CT Signals






Special signals


DT Signals






Special signals


177

Energy of DT Signals

178

Energy of

DT Signal

179

Energy of

DT Signal: Example

180

DT Signal Power

181

DT Signal Power

182


Types of signals

CT Signals






Special signals


DT Signals






Special signals


183

Transformations of DT

signals

Modifying a signal x[n] through

1. Transformation of the independent

variable t e.g., y[n]=x[2n+1]

2. Combination of signals:

y[n] = cos[wn](x[n]

y[n]=x[n]-x[n-1]

184

Combination of signals

The product & sum of two sequences x[n] and y[n]:

z[n] = x[n]+y[n]

sample-by-sample production and sum

Multiplication of a sequence x[n] by a number

:multiplication of each sample value by

185

Combination of signals:: Differencing

186

Combination of signals:: Accumulation

187

Summation Operation of x[n]

188


Variable or Modification of

independent variable n

Modifying signals through elementary

transformations

Examples of elementary transformation

time shift, x[n-n0)

time reversal, x[-n]

time scaling, x[n/2]

189





ofargument theis :

of parameters are :,

of t variableindependen theis :

offunction a :

)()(

integeran bemust :

nx

nx

xn

x

xn

nx

nxny

n

• Recommended approach:

o Sketch y[n] for a selected set of n until y[n] becomes clear

o Steps:

1. Rewrite: y(n) = x(α(n+β/ α))

2. Scale by |α|: x(|α|n)

3. Invert x(|α|n) if α<0

4. Shift to the LEFT by |β/ α| if β/ α>0

5. Shift to the RIGHT by |β/ α| if β/ α<0

190

Signal Flip about y- axes

x[-n], time reversal

191

Signal delay

x[n]

x[n-5]

192

Example of Delayed Signal

193

Signal advance

x[n]

x[n+5]

194

Signal shift and reversal

x[n]

x[-n+5]

195

Transformation of DT

Signals

196


signals: Time Shifting

197


signals: Time Scaling

198


signals: Time Scaling

199


Types of signals

CT Signals






Special signals


DT Signals






Special signals


200

Even and Odd DT Signals

An even signal is identical to its time reversed

An odd signal has the property

Example :

201

Even and Odd DT Signals

202


Types of signals

CT Signals






Special signals


DT Signals






Special signals


203

DT Unit Impulse

204

DT Unit Impulse

Unit sample sequence

(discrete-time impulse, impulse)

Any sequence can be represented as a sum of

scaled, delayed impulses

More generally

]5[...]2[]3[][ 523 nanananx

k

knkxnx ][][][

,0,1

,0,0][

n

nn

205

DT Unit Step

206

Defined as

Related to the impulse by

Conversely,

Relationship: DT unit impulse

and unit step

,0,0

,0,1][

n

nnu

0

][][][][

or

...]2[]1[][][

kk

knknkunu

nnnnu

]1[][][ nunun

207

Relationship: DT unit

impulse and unit step

It can be shown that

208

Exponential

Extremely important in representing and analyzing LTI systems

Defined as

If A and are real numbers, the sequence is real

If and A is positive, the sequence values are positive and decrease with increasing n

If , the sequence values alternate in sign, but again decrease in magnitude with increasing n

If , the sequence values increase with increasing n

10

1||

nAnx ][

01

n

n

n

nx

nx

nx

22][

)5.0(2][

)5.0(2][

209

Combining basic sequences

An exponential sequence that is zero for n<0

0 ,0

,0,][

n

nAnx

n

][][ nuAnx n

210

Outline

Introduction

Types of signals

CT Signals






Special signals


DT Signals






Special signals


211

DT Systems

To get the output y[n] Apply the system S{} on input x(n)

y[n] is the response of S{} to x[n]

A system: An integrated whole composed of diverse, interacting, specialized parts

System performs a function not possible with any of the individual parts

Any system has objectives

Systems respond to particular signals by producing other signals or some desired behavior

212

DT systems

A transformation or operator that maps input into

output

Examples:

The ideal delay system

A memoryless system

]}[{][ nxTny

T{.}x[n] y[n]

nnnxny d ],[][

nnxny ,])[(][ 2

213


Linearity:

A Linear system possesses the property of superposition

any constant values a and b, the following equation is satisfied

It can be easily verified that for linear systems:

an input which is zero for all time,

results in an output which is zero for all time

214


A system is linear if and only if

Combined into superposition

constantarbitrary an is where

][]}[{]}[{

and

][][]}[{]}[{]}[][{ 212121

a

naynxaTnaxT

nynynxTnxTnxnxT

additivity property

scaling property

][][]}[{]}[{]}[][{ 212121 naynaynxaTnxaTnbxnaxT

215

Examples

Accumulator system – a linear system

A nonlinear system

)10log()1log()101log(][3

21

10

10][ and 1][Consider

|)][(|log][

ny

nxnx

nxny

][][])[][(][

][][,][][

][][

21213

2211

nbynaykbxkaxny

kxnykxny

kxny

n

k

n

k

n

k

n

k

216


o A TI system is a system for which a time shift or delay of the input

sequence causes a corresponding shift in the output sequence

217

Basic System Properties:

Example

Accumulator system

][][][][ 0101 nnynynnxnx

][][

][][][

][][

01

011

0

0

1

0

nnykx

nkxkxny

kxnny

nn

k

n

k

n

k

nn

k

218

Basic System Properties:Causality

The output sequence value at the index

n=n0 depends only on the input sequence

values for n<=n0

Example

Causal for nd>=0

Noncausal for nd<0

nnnxny d ],[][

219

Basic System Properties: Stability

A system is stable in the BIBO sense if

and only if every bounded input sequence

produces a bounded output sequence

Example

stable

nnxny ,])[(][ 2

220


Types of signals

CT Signals






Special signals


DT Signals






Special signals


221

Summary: Major sub-topics:

Transformation of signals and

Properties of systems

ELEC264: Signals And Systems Topic 1: LTI Systems · ELEC264: Signals And Systems Topic 1: LTI...

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Transcript of ELEC264: Signals And Systems Topic 1: LTI Systems · ELEC264: Signals And Systems Topic 1: LTI...