Topic 1: LTI Systems Overview: Introduction to Signals Types of - - PowerPoint PPT Presentation

ELEC361: Signals And Systems

Topic 1: LTI Systems

• Dr. Aishy Amer

Concordia University Electrical and Computer Engineering

Overview:

Introduction to Signals

Types of Signals: CT/DT, analog/digital, periodic/aperiodic Periodic Signals Special signals: Unit Impulse, Unit Step, … Signal Energy and Power Transformations of the Independent Variable Even and Odd Signals Introduction to Systems Basic System Properties Summary

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

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., stock market, human body)

Excitation signals are applied at system inputs Response signals are produced at system outputs

3

What is a signal?

A variable (or multiple variables) that changes in time

Speech or audio signal: A sound 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

Continuous vs. Discrete

The value can vary continuously or take from a discrete set The time and space can also be continuous or discrete

4

Signals: Examples

Example signals

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Outline

• Introduction to Signals
• Types of Signals: CT/DT, analog/digital, periodic/aperiodic
• Periodic Signals
• Signal Energy and Power
• Transformations of the Independent Variable
• Even and Odd Signals
• Special signals: Unit Impulse, Unit Step, …
• Introduction to Systems
• Basic System Properties
• Summary

6

Types of Signals:

Continuous-Time (CT) vs. Discrete-time (DT) Analog vs. digital Periodic vs. non-periodic

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Types of Signals

(Discrete ~ countable)

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Continuous Value and Continuous-Time (CT) Signals

• All continuous signals

are CT but not all CT signals are continuous

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Mathematics and 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,…

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CT and DT Sinusoidal signals

angle Phase : Period l Fundementa : Frequency l Fundementa / 1 2 frequency Angular : at t signal the

• f

value : x(t) instant time : t Signal : x Amplitude Signal : A t) 2 (j e^ A x(t) ) cos( ) ( θ π ω ω π θ ω T T f f f t A t x = = = + =

angle Phase : Period l Fundementa : Frequency l Fundementa : / 1 2 frequency Angular : n at signal the

• f

value : x[n] index time : n Signal : x Amplitude Signal : A ) sin( ] [ θ π ω ω θ ω N N F F n A n x = = + =

Note: a sinusoidal signal has a unique frequency, e.g., 100Hz

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DT Signals: Examples

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Outline

• Introduction to Signals
• Types of Signals: CT/DT, analog/digital, periodic/aperiodic
• Periodic Signals
• Signal Energy and Power
• Transformations of the Independent Variable
• Even and Odd Signals
• Special signals: Unit Impulse, Unit Step, …
• Introduction to Systems
• Basic System Properties
• Summary

13

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

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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

15

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:

• T0=3

is periodic with fundamental period

3 ), 3 cos( ) ( < ≤ = t t t x π

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Periodic CT Signals

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Periodic CT Signals: Examples

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Aperiodic CT Signals

A function that is nor periodic is called

aperiodic

Aperiodic signal Examples:

) ( ) ( nT t x t x + ≠

⎪ ⎩ ⎪ ⎨ ⎧ < ≤ =

• therwise

3 ) 3 cos( ) ( t t t x π

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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

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Periodic CT Signals: Examples

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Periodic DT Signals

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Periodic DT Signals

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

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DT Sinusoids

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DT Sinusoids

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DT Sinusoids

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DT Periodic Sinusoids

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DT Aperiodic Sinusoid

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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

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Random Signals

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Outline

• Introduction to Signals
• Types of Signals: CT/DT, analog/digital, periodic/aperiodic
• Periodic Signals
• Special signals: Unit Impulse, Unit Step, …
• Signal Energy and Power
• Transformations of the Independent Variable
• Even and Odd Signals
• Introduction to Systems
• Basic System Properties
• Summary

31

Singularity Functions

In engineering, we often deal with the idea

• f 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

32

Singularity Functions

Many useful signals are not continuous or

differentiable at every point in time.

For instance, describe the operation of switching on

• r 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:

),... ( ), ( t u t δ

33

Singularity Functions

34

CT Unit Step

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CT Unit Impulse

The CT unit impulse function is represented as The area under the CT unit impulse is equal to 1

36

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

37

Relationship: CT unit step and 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

38

Relationship: CT unit step and unit impulse

The CT unit step is not differentiable at t=0 One can use continuous approximation to the

unit step

Corresponding unit impulse

39

Properties of CT Impulse

Sampling (Shifting) Property: the

value of the function at a point

Scaling Property

40

Properties of CT Impulse

Equivalence Property

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The CT Signum

42

The CT Unit Ramp

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The CT Unit Rectangle

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The CT Unit Triangle

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The CT Unit Sinc Signal

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DT Unit Step

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DT Unit Impulse

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Relationship: DT unit impulse and unit step

It can be shown that

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Outline

• Introduction to Signals
• Types of Signals: CT/DT, analog/digital, periodic/aperiodic
• Periodic Signals
• Special signals: Unit Impulse, Unit Step, …
• Signal Energy and Power
• Transformations of the Independent Variable
• Even and Odd Signals
• Introduction to Systems
• Basic System Properties
• Summary

50

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

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Energy of CT and DT Signals

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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

53

Energy of DT Signal

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Energy of DT Signal

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Energyof CT Signal: Example

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Energy of CT Signal: Example

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Energy of DT Signal: Example

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Signal Power

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Signal Power

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Signal Power: Example

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Outline

• Introduction to Signals
• Types of Signals: CT/DT, analog/digital, periodic/aperiodic
• Periodic Signals
• Special signals: Unit Impulse, Unit Step, …
• Signal Energy and Power
• Transformations of the Independent Variable
• Even and Odd Signals
• Introduction to Systems
• Basic System Properties
• Summary

62

Transformations of CT signals

) ( is what fixed a for

• f

argument the is :

• f

parameter a is :

• f

t variable independen the is :

• f

function a : ) ( τ τ τ τ τ + ⇒ + + t x x t x x t t x t x

63

Transformations of CT signals

Transform x(t): Transform the independent variable

• e.g., x(t/2)

Combine Signals: z(t) = x(t) y(t) z(t) = x(t)/y(t)

64

Transformations of CT signals: Examples

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Transformations of CT signals: Combination of Signals

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Transformation of CT Signals

67

Transformation of CT Signals

68

Transformation of CT Signals

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Transformations of CT signals:Time Shifting

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

70

Transformation of CT Signals

71

Transformations of CT signals: Time Scaling

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

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Multiple Transformation of CT Signals

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Transformations of CT signals: Examples

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Transformations of CT signals: Examples

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Transformation of CT Signals: Examples

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Transformation of CT Signals: Differentiation

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Transformation of CT Signals: Integration

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Transformation of DT Signals

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Transformations of DT signals: Time Shifting

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Transformations of DT signals: Time Scaling

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Transformations of DT signals: Time Scaling

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Transformations of DT signals: Differencing

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Transformations of DT signals: Accumulation

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Outline

• Introduction to Signals
• Types of Signals: CT/DT, analog/digital, periodic/aperiodic
• Periodic Signals
• Special signals: Unit Impulse, Unit Step, …
• Signal Energy and Power
• Transformations of the Independent Variable
• Even and Odd Signals
• Introduction to Systems
• Basic System Properties
• Summary

85

Even and Odd Signals

An even signal is identical to its time reversed Example: An odd signal has the property Example :

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Even and Odd CT Signals

87

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

88

Even and Odd Signals: Example

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Products of Even and Odd CT Functions

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Product of 2 Odd Functions

91

Integrals of Even and Odd CT Functions

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Even and Odd DT Signals

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Outline

• Introduction to Signals
• Types of Signals: CT/DT, analog/digital, periodic/aperiodic
• Periodic Signals
• Special signals: Unit Impulse, Unit Step, …
• Signal Energy and Power
• Transformations of the Independent Variable
• Even and Odd Signals
• Introduction to Systems
• Basic System Properties
• Summary

94

Introduction to 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

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Introduction to Systems: Examples

96

Introduction to Systems: 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

97

Introduction to Systems: Image System to Aid Perception

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Introduction to Systems: Sound Recording

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Introduction to Systems: CT and DT Systems

CT system: CT input signals are applied and result

in CT output signals

The input-output relation of a CT system is DT system: Transforms DT inputs into DT outputs. The input-output relation of a DT system is

100

Introduction to Systems: 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?

101

Introduction to Systems: Interconnections of Systems

Systems can be interconnected in series (cascade),

parallel, feedback, or combination

102

Outline

• Introduction to Signals
• Types of Signals: CT/DT, analog/digital, periodic/aperiodic
• Periodic Signals
• Special signals: Unit Impulse, Unit Step, …
• Signal Energy and Power
• Transformations of the Independent Variable
• Even and Odd Signals
• Introduction to Systems
• Basic System Properties
• Summary

103

Basic System Properties

Linearity Linear system

possesses the property of superposition

Superposition property

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

104

Basic System Properties

105

Basic System Properties: Examples

Linearity:

linear? ) ( ) ( Is

2 t

x t y =

[ ]

) ( ) ( 2 ) ( ) ( )] ( ) ( [ ) ( Then ) ( ) ( ) ( Let

2 1 2 2 2 2 1 2 1 2 2 1 2 2 1 2 1

t y t y x b x abx x a t y b t y a t x b t x a t y t x b t x a t x + + = + + = + ≠ + = + = L

TI? ) ( ) ( Is

2 t

x t y =

Time Invariance:

Let x(t) =

) ( )) ( ( ) ( ) (

2

t t y t t x t w t t x − = − = ⇒ −

106

Basic System Properties

Linear Time-Invariant (LTI) Systems

107

Basic System Properties

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); DT system with memory: a unit delay

] 1 [ ] [ − = n x n y

) ( ) ( t Ri t v =

108

Basic System Properties

Invertibility: System is invertible if distinct inputs lead to

distinct outputs

If a system is invertible

• inverse system exists, when cascaded with the
• riginal system, yields an output equal to the input

to the first system.

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

109

Basic System Properties

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

110

Basic System Properties

Stability:

Stable system: small inputs lead to responses that do

not diverge

BIBO stable system: bounded input results in a

bounded output.

Stable system is always BIBO stable but a BIBO

stable system is not necessarily stable

Is the accumulator system stable? Examples:

111

Summary: Major sub-topics:

Transformation of signals and Properties of systems

112

Summary: a quiz

• 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( ) ( t x t t y =

1 ) 2 / sin( 1 ≤ ≤ − t

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

2 1 2 1 2 1 3 3 2 1 3 2 2 1 1

t by t ay t x t b t x t a t bx t ax t t x t t y t bx t ax t x let t x t t y t x t t y + = + = + = = + = = =

113

Summary: a quiz

• The system is not Memoryless: Output does not

solely depend on current input values (i.e. depends

• n past input values).
• The system is not Time-invariant:

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

1 1 1

t t y t t x t t x t t y t t x t x let t t x t t t t y t x t t y ≠ − = = − = − − = =

114

Summary: a quiz

The system is not Invertible: There

are Different Inputs which lead to the same Outputs

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

2 2 1 2 1 1 1 1

= = = − = = = = = = t x t x t t y t t x t x t x t t y t t x t x t t y π π δ δ