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A SIGNAL THEORETICINTRODUCTION TORANDOM PROCESSES


ROY M HOWARDDepartment of Electrical Engineering amp ComputingCurtin University of TechnologyPerth Australia

Copyright copy 2016 by John Wiley amp Sons Inc All rights reserved

Published by John Wiley amp Sons Inc Hoboken New JerseyPublished simultaneously in Canada

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Library of Congress Cataloging-in-Publication Data

Howard Roy MA signal theoretic introduction to random processes Roy M Howard

pages cmIncludes bibliographical references and indexISBN 978-1-119-04677-6 (cloth)

1 Signal processing 2 Signal theory (Telecommunication) 3 Stochastic processes4 Random noise theory I TitleTK51029H695 2015003 54ndashdc23

2014049440

Set in 1012pt Times by SPi Global Pondicherry India

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

1 2016

ABOUT THE AUTHOR

Clarity follows from rigourAll research is preliminary

Roy M Howard has been awarded a BE and a PhD in Electrical Engineering as wellas a BA (Mathematics and Philosophy) by the University ofWestern Australia PerthAustralia To date his academic career has been at the University ofWestern Australiaand then in the School of Electrical and Computer Engineering at Curtin UniversityPerth Australia Over recent years he has been a regular visitor to the TechnicalUniversity of Darmstadt Darmstadt Germany He is the author of Principles ofRandom Signal Analysis and Low Noise Design The Power Spectral Density andits Applications Wiley 2002 He has expertise in low-noise amplifier design signaltheory and modeling and characterization of random phenomena

CONTENTS

Preface xiii

1 A Signal Theoretic Introduction to Random Processes 111 Introduction 112 Motivation 213 Book Overview 8

2 Background Mathematics 1121 Introduction 1122 Set Theory 1123 Function Theory 1324 Measure Theory 1825 Measurable Functions 2426 Lebesgue Integration 2827 Convergence 3728 LebesguendashStieltjes Measure 3929 LebesguendashStieltjes Integration 50210 Miscellaneous Results 61211 Problems 62

3 Background Signal Theory 7131 Introduction 7132 Signal Orthogonality 7133 Theory for Dirichlet Points 75

34 Dirac Delta 7835 Fourier Theory 7936 Signal Power 8237 The Power Spectral Density 8438 The Autocorrelation Function 9139 Power Spectral DensityndashAutocorrelation Function 95310 Results for the Infinite Interval 96311 Convergence of Fourier Coefficients 103312 Cramerrsquos Representation and Transform 106313 Problems 125

4 Background Probability and Random Variable Theory 15341 Introduction 15342 Basic Concepts Experiments-Probability Theory 15343 The Random Variable 16044 Discrete and Continuous Random Variables 16245 Standard Random Variables 16546 Functions of a Random Variable 16547 Expectation 16648 Generation of Data Consistent with Defined PDF 17249 Vector Random Variables 173410 Pairs of Random Variables 175411 Covariance and Correlation 186412 Sums of Random Variables 191413 Jointly Gaussian Random Variables 193414 Stirlingrsquos Formula and Approximations to Binomial 194415 Problems 199

5 Introduction to Random Processes 21951 Random Processes 21952 Definition of a Random Process 21953 Examples of Random Processes 22154 Experiments and Experimental Outcomes 22555 Prototypical Experiments 22856 Random Variables Defined by a Random Process 23257 Classification of Random Processes 23358 Classification One-Dimensional RPs 23659 Sums of Random Processes 239510 Problems 239

6 Prototypical Random Processes 24361 Introduction 24362 Bernoulli Random Processes 24363 Poisson Random Processes 246

viii CONTENTS

64 Clustered Random Processes 25565 Signalling Random Processes 25766 Jitter 26267 White Noise 26568 1f Noise 27269 BirthndashDeath Random Processes 275610 Orthogonal Increment Random Processes 278611 Linear Filtering of Random Processes 282612 Summary of Random Processes 283613 Problems 285

7 Characterizing Random Processes 28971 Introduction 28972 Time Evolution of PMF or PDF 29173 First- Second- and Higher-Order Characterization 29274 Autocorrelation and Power Spectral Density 29775 Correlation 30876 Notes on Average Power and Average Energy 31077 Classification Stationarity vs Non-Stationarity 31678 Cramerrsquos Representation 32379 State Space Characterization of Random Processes 335710 Time Series Characterization 347711 Problems 347

8 PMF and PDF Evolution 36981 Introduction 36982 Probability MassDensity Function Estimation 37083 NonSemi-parametric PDF Estimation 37284 PMFPDF Evolution Signal Plus Noise 37885 PMF Evolution of a Random Walk 38186 PDF Evolution Brownian Motion 38487 PDF Evolution Signalling Random Process 38888 PDF Evolution Generalized Shot Noise 39089 PDF Evolution Switching in a CMOS Inverter 396810 PDF Evolution General Case 400811 Problems 405

9 The Autocorrelation Function 41791 Introduction 41792 Notation and Definitions 41793 Basic Results and Independence Information 41994 Sinusoid with Random Amplitude and Phase 42195 Random Telegraph Signal 42396 Generalized Shot Noise 424

ixCONTENTS

97 Signalling Random Process-Fixed Pulse Case 43498 Generalized Signalling Random Process 44199 Autocorrelation Jittered Random Processes 453910 Random Walk 456911 Problems 457

10 Power Spectral Density Theory 481101 Introduction 481102 Power Spectral Density Theory 481103 Power Spectral Density of a Periodic Pulse Train 485104 PSD of a Signalling Random Process 487105 Digital to Analogue Conversion 501106 PSD of Shot Noise Random Processes 505107 White Noise 509108 1f Noise 510109 PSD of a Jittered Binary Random Process 5131010 PSD of a Jittered Pulse Train 5171011 Problems 525

11 Order Statistics 553111 Introduction 553112 Ordered Random Variable Theory 557113 Identical RVs With Uniform Distribution 574114 Uniform Distribution and Infinite Interval 584115 Problems 590

12 Poisson Point Random Processes 621121 Introduction 621122 Characterizing Poisson Random Processes 623123 PMF Number of Points in a Subset of an Interval 625124 Results From Order Statistics 630125 Alternative Characterization for Infinite Interval 634126 Modelling with Unordered or Ordered Times 636127 Zero Crossing Times of Random Telegraph Signal 638128 Point Processes The General Case 639129 Problems 639

13 BirthndashDeath Random Processes 649131 Introduction 649132 Defining and Characterizing BirthndashDeath Processes 649133 Constant Birth Rate Zero Death Rate Process 656134 State Dependent Birth Rate - Zero Death Rate 662135 Constant Death Rate Zero Birth Rate Process 665

x CONTENTS

136 Constant Birth and Constant Death Rate Process 667137 Problems 669

14 The First Passage Time 677141 Introduction 677142 First Passage Time 677143 Approaches Establishing the First Passage Time 681144 Maximum Level and the First Passage Time 685145 Solutions for the First Passage Time PDF 690146 Problems 695

Reference Material 709

References 717

Index 721

xiCONTENTS

PREFACE

This book has arisen from research over a 30 year period related to modeling andcharacterizing random phenomena From this research I have formed the view whichunderpins the book that an introduction to random processes should be groundedin signal theory as well as probability theory results for random processes shouldbe established on the finite interval and mathematical rigor is of fundamentalimportance

Writing of material for the book commenced around 2005 with an initial goal ofextending my book Principles of Random Signal Analysis and LowNoise Design ThePower Spectral Density and its Applications Wiley 2002 Around 2006 it becameclear that a new book was required and when time permitted research and writingfor this book was undertaken Significant impetus for the book was given with a Sab-batical taken between August 2011 and January 2012 which was spent at The Tech-nical University of Darmstadt Darmstadt Germany

The book assumes a prior introduction to probability theory and random variabletheory and is suited to final year Electrical Electronic and Communications Engi-neering students postgraduate students and researchers with an interest in character-izing randomness

The following people have contributed to the book in various ways FirstProf A Zoubir Head of the Signal Processing Group The Technical Universityof Darmstadt Darmstadt Germany has generously supported my career and visitsto The Technical University of Darmstadt where a significant amount of writingand research for the book was undertaken Second Prof K Fynn Head of SchoolElectrical Engineering and Computing Curtin University Perth Australia for sup-port for my research and travel that has facilitated the writing of the book Thirdas Head of the Electrical and Computer Engineering Department Curtin University

Prof S Nordholm has provided good collegial support for the book Fourth the staffat John Wiley (Susanne Steitz-Filler Senior Editor Sari Friedman Senior EditorialAssistant F Pascal Raj Assistant Account Manager) have supported and managedthe publication of the book I also wish to note the contribution of the staff at a count-able number of cafes (countable but large) for their indirect support whilst a signif-icant level of writing and editing was in progress

Finally one can only lament the era of unparalleled wealth and knowledgewhere important scholarly endeavors increasingly are not supported and where manyacademic values which underpin important aspects of a society that one can valuehave been seriously eroded For those who have supported in this context myendeavors I am thankful

ROY M HOWARD

August 2014

xiv PREFACE

1A SIGNAL THEORETICINTRODUCTION TO RANDOMPROCESSES

11 INTRODUCTION

Consistent with the nature of the physical universe the technical world and the socialworld and at the usual levels of observation randomness is ubiquitous Consistentwith this randomness has received significant attention over human history and pre-history The appearance of gods and associated offerings to such deities in early civi-lizations was in part an attempt to understand the randomness inherent in nature andto have control over this variability As our understanding of the nature of the physicaluniverse has expanded interest in random phenomena and its characterizationhas increased significantly Today the accumulated knowledge on characterizingrandomness is vast Cohen (2005) provides a good overview of part of the more recentscientific endeavor to understand and characterize random phenomena Meyer (2009)provides a historical perspective on the mathematical development of stochasticprocess theory from the 1940s

Random phenomena occur widely in the physical social and technical worlds andwell-known examples include the variation with time in measures of the weather(wind velocity temperature humidity etc) economic activity (inflation rate stockindexes currency exchange rates etc) an individualrsquos physical state (heart rateblood pressure feeling of well-being etc) and technical entities (the informationflow to a mobile communication device the fuel economy of a car etc) Other tech-nical examples include the lifetime of a set product manufactured by a manufacturervariations in the queue length of data packets waiting to be forwarded at a given

A Signal Theoretic Introduction to Random Processes First Edition Roy M Howardcopy 2016 John Wiley amp Sons Inc Published 2016 by John Wiley amp Sons Inc

network node the effective information rate on a given communication channel thevoltage variation between the terminals of a conductive medium due to the randommovement of electrons unwanted signal variations at the receiver of a communicationsystem or at the output of a sensor etc Random phenomena generally inhibit theperformance of a system and for example limit the transmission distance forelectronicphotonic communication systems the sensitivity of sensors for monitoringphenomenon in the natural world the timing accuracy of all timing reference sourcesetc Randomness is not always detrimental to system operation and the introductionof a random signal component to a system can in certain circumstances enhancesystem performance and stochastic resonance is a common term associated with suchan outcome

12 MOTIVATION

The transformation of the world to the information age has been underpinned byadvances in electrical electronic and photonic technology Such technology is basedon controlling the random movement of electrons and photons As electrons have amass of 9 11 times 10minus31 kg they potentially can move with very high velocities (of theorder of 105 ms) between collisions when subject to an electric field andor due toambient thermal energy (Such velocities should not be confused with the averagedrift velocity of electrons in a conducting mediamdashof the order of 10minus4m s in a copperconductor) One extraordinary achievement of the modern era has been the develop-ment of sophisticated devices and systems with robust well-defined performancebased on controlling electrons Figure 11 provides a perspective

Solid foundation

Sure

FIGURE 11 On the foundation of modern electrical and electronic technology

2 A SIGNAL THEORETIC INTRODUCTION TO RANDOM PROCESSES

The following examples illustrate in small part the reality of engineering in thecontext of the nature of our physical world and the diverse nature of the manifestationof randomness

First the voltage at a node in an electricalelectronic system will exhibit random-ness consistent for example with white Gaussian noise or 1f noise The nature ofsuch randomness is illustrated in Figure 12 Note the regular nature of a whiteGaussian noise signal in comparison to a 1f noise signal The irregularity of 1f noiseis consistent with the nature of randomness in complex systems with multiple sourcesof noise and such noise has been found in signals from a diverse range of systemsincluding graphene electronic devices the human heart and brain the humanresponse to stimuli phenomena in the natural world and economic activity The spec-trum of many forms of music is consistent with a 1f noise spectrum and 1f noise isoften a signature of complexity The introduction and cited references in Grigoliniet al (2009) provides a good overview of the diverse sources of such noise

Second as an example of the importance of randomness in electrical engineeringconsider the random movement of electrons that leads to a signal at the output of anelectronic amplifier illustrated in Figure 13 having the form illustrated in Figure 14In a communication context the noise floor (the output noise signal in the absence ofan input signal) of an amplifier limits the distance information can be transmitted and

0 200 400 600 800ndash10

ndash5

0

5

10

1000 t

FIGURE 12 Upper white Gaussian noise signalmdashoffset by 5 Lower 1f noise signalmdashoffset by minus5

VS +Vo

100+

ndash

RS

FIGURE 13 Schematic diagram of a signal source connected to an amplifier

3MOTIVATION

recovered In a sensor context the amplifier noise limits the sensitivity of the sensorThe effect of the noise on a signal depends on the nature of the noise the bandwidth ofthe noise the noise level the signal form and the information required from the sig-nal Modelling and characterization of noise is fundamental to designing amplifiers tominimize the noise generated and for subsequent processing that potentially can limitthe effect of the introduced noise

Third the randomness of the movement of electrons and the randomness inherentin the natural environment (eg temperature variations) results in the drift ofall electronic-based timing references This necessitates for example circuitry oralgorithms to maintain synchronization and hence reliable communication betweenthe nodes in a communication network The first-order model of an electronicoscillator is

x t =Ao sin 2πfot +ϕ t 1 1

where fo is the oscillator frequency and ϕ(t) is the phase variation due to the influenceof random electrical and thermal variations As a first-order model the phase ϕ(t)variation will exhibit a random walk behavior as illustrated in Figure 15

Noise in a system often leads to the delay or advance of the timing of a signal whencompared with the zero noise case This leads to jitter of the time a signal crosses a setthreshold as illustrated in Figure 16 When the jitter is associated with a clock signalwhich serves as a timing reference for a system constraints on the switching rate forreliable system operation result In a communication context jitter in the timingreference at a receiver leads to an increase in the probability of error for the receivedinformation

Fourth the time a signal first reaches a set threshold level is called the first passagetime and the first passage time is of interest in many areas including the spread ofdiseases (when does a measure of the disease reach a set level for the first time)

0 000001 000002 000003 000004 t

ndash00005

00000

00005

00010

00015

Voltage

Signal

Signal plus noise

FIGURE 14 Example of a signal corrupted by the noise inherent in an amplifier at the

amplifier output for the case where the maximum signal level is 10 times the rms noise level


finance (when does a set measure of economic activity reach a set level for the firsttime) neuronal firing (a neuron fires when the collective inputs reach a set level) etcThe first passage time for a random walk is illustrated in Figure 15

Fifth in many contexts our intuition and experience are consistent with eventsoccurring at random and at a regular rate Examples include the arrival of peopleat a supermarket queue the timing of incoming phone callsemails during a set periodof the day etc In a technical context examples include the arrival of photons on aphotodetector the crossing times of electrons in a PN junction the arrival of datapackets at a network node etc The simplest model for such timing is the Poisson pointprocess (random timing of points but at a set rate) and examples of times defined bysuch a process are shown in Figure 17 Such point processes underpin for examplethe examples noted above

Sixth the efficient and inexpensive conveyance of information underpins in sig-nificant measure the latest transformation of our world and is based on agreed pro-tocols with respect to encoding of information in signals Efficient communicationrequires the use of signals that are spectrally efficient with a unit of information taking

0 2 4 6 8 10 t

ndash10

ndash05

00

05

10

15

20

25

Threshold

tFP

ɸ(t)

FIGURE 15 Three trajectories consistent with the random-walk behavior of the phase of anoscillator (arbitrary units) The time tFP is the time the upper random walk first reaches thethreshold level of unity

0 1 2 3 4 500

02

04

06

08

10

t

Threshold

FIGURE 16 Three jittered binary signals

5MOTIVATION

the smallest possible amount of the available capacity of a communication channelThree different signals each with different spectral efficiency but encoding the sameinformation are shown in Figure 18 Further the nature of the signals used has impli-cations for the ability of the receiver to recover the sent information in the context ofnoise and interference Signal theory random process theory and methods of character-izing the spectral content of communication signals underpin modern communication

121 Usefulness of Randomness

Randomness is not always detrimental to the outcomes of a system One example isillustrated in Figure 19 where the irregularity of the main traffic flow facilitates themovement of traffic from the right

Another application where noise is useful is illustrated in Figures 110 and 111 andarises when an input level is to be estimated from samples provided by a device for

0 5 10 15 2000

05

10

15

20

25

30

t

FIGURE 17 Four examples of the times defined by a Poisson point random process with arate of one point per unit time

0 2 4 6 8 100

1

2

3

4

5

6

7

t

FIGURE 18 Three signalling waveforms for encoding the data 1 0 1 1 1 1 0 0 1 0 at arate of 1 bit per second Bottom signalling with rectangular pulsesmdashreference level of zeroMiddle signalling with return to zero pulses and polar codingmdashreference level of threeTop signalling with raised cosine pulses using bipolar codingmdashreference level of six


example an analogue to digital converter that produces quantized levels In theabsence of noise the output level will differ from the input level by the quantizationerror When noise with a variation greater than the quantization resolution is presentit is possible to recover the input level accurately by averaging output values taken attimes greater than the correlation time of the noise The resolution can be improvedaccording to 1 N where N is the number of data values averaged

FIGURE 19 Regular and irregular traffic flow

InputOutput

Quantization error

Threshold

Quantization level

FIGURE 110 Illustration of the quantization error arising from sampling a signal with finiteamplitude resolution

0 1 2 3 4 5 6

Signal level

t

Sample times

Threshold

Quantizationlevel

Quantizationlevel

Output levelSignal + noise

00

05

10

15

20

25

FIGURE 111 Illustration of the output values arising from sampling a constant signal levelplus additive noise The sampling rate is unity

7MOTIVATION

Additive noise or the noise inherent in a system can aid subthreshold detection ofa periodic signal as illustrated in Figure 112 Such detection occurs for example inneuronal networks The phrase stochastic resonance is widely used when noise aidsthe detection of a signal

122 Engineering

The first requirement for engineering in the context of random phenomena is themodelling of the random phenomena The second requirement is the characterizationof the random phenomena such that measures of system performance can be ascer-tained and the effects of the randomness minimized or utilized as appropriate Thegoal of this book is to provide a theoretical basis that leads to the facilitation of boththe modelling and the characterization of the prototypical random phenomenaencountered in electrical engineering

13 BOOK OVERVIEW

The theory for modelling and characterizing random phenomena is vast and a singlebook can at best provide a modest introduction in a specific area This book and theapproach taken for introducing and characterizing random phenomena has arisen outof long-term research in the electronics and communications field The theory andexamples included are consistent with the broad electricalelectroniccommunicationsengineering discipline and include the prototypical random phenomena of these

0 1 2 3 4 5 6 t

Signal

Output Signal + noise

Threshold

ndash1

0

1

2

3

FIGURE 112 Illustration of the beneficial effect of additive noise on the detection of a pulsetrain signal that is below a threshold level The output level (offset) is a pulse when the signalplus noise is above the threshold


disciplines the random walk Brownian motion the random telegraph signal thePoisson point process the Poisson counting process shot noise white noise 1f noisesignalling random processes (which underpin most forms of communication) jitterrandom clustering and birthndashdeath random processes

The rationale for the book is threefold first random process theory should begrounded in signal theory as well as probability theory Second results for randomprocesses should be established on the finite interval and results for the infiniteinterval obtained by taking an appropriate limit Third attention to mathematical rigorprovides clarity and facilitates understanding and such rigor is well suited to themodelling and characterization of random phenomena Detailed proofs of results havebeen included to provide a comprehensive treatment of material at a moderatemathematical level The book assumes a prior introduction to probability theoryand random variable theory

The approach taken with a strong mathematical and signal theory basis provides afoundation for random process theory that facilitates the continued development ofrandom process theory in the context of many unsolved problems and the increasingimportance of modelling and characterizing random phenomena The modelling ofrandom processes on the finite interval allows transient as well as steady-state resultsto be obtained Importantly it allows the use of finite-dimensional functions for char-acterizing random phenomena and this facilitates the development of random processtheory The use of a signal theory basis provides a general framework for definingfunctions used for characterizing random phenomena including the autocorrelationfunction and the power spectral density This has pedagogical value as the definitionshave a signal basis rather than a random process basis and is less problematic forstudents The use of a signal basis set approach for defining the power spectral densitywhich is the most widely used measure for characterizing random phenomenaprovides a simple and natural interpretation of this function for the general caseand for the usual case where a sinusoidal basis set is assumed

Chapters 2ndash5 provide the necessary mathematical theory background signaltheory random variable theory and random process theory for subsequent discussionof random processes Chapter 6 details the prototypical random processes that are fun-damental to electrical electronic and communication engineering Chapters 7ndash10provide a basis for the characterization of random phenomena Chapter 7 providesa general overview Chapter 8 details probability mass functionprobability densityfunction evolution Chapter 9 details the autocorrelation function and Chapter 10details the power spectral density Chapter 11 provides an introduction to orderstatistics and this provides the background for a discussion of the Poisson pointrandom process in Chapter 12 Chapter 13 provides an introduction to birthndashdeathrandom processes while Chapter 14 provides an introduction to first passage timetheory

9BOOK OVERVIEW

2BACKGROUND MATHEMATICS

21 INTRODUCTION

This chapter details relevant mathematical theory andmathematical results to providethe basis for analysis in subsequent chapters and to ensure a coherent and systematicdevelopment of material A summary of mathematical notation is detailed in thenotation table in the reference section at the end of the book

A basic introduction to set theory and function theory is provided and this isfollowed by an introduction to measure theory Two types of measure are consideredThe first Lebesgue measure underpins Lebesgue integration and this is fundamentalto signal theorywhich is discussed inChapter 3 The secondLebesguendashStieltjesmeas-ure provides the basis for LebesguendashStieltjes integration LebesguendashStieltjes integra-tion underpins Cramerrsquos representation of a signal which is discussed in Chapter 3The discussion in general complements that of Howard (2002 ch 2)

22 SET THEORY

221 Basic Definitions

Definition Set A set is a collection of distinct entitiesThe notation α1 α2hellip αN is used for the set of distinct entities α1 α2hellip αN Thenotation x f(x) is used for the set of elements x for which the property f(x) is trueThe notation x S means that the entity denoted x is an element of the set S


The empty set is denoted by The complement of a set S denoted SC isdefined as SC = x x S where S is usually a subset of a larger setmdashoften theuniversal set The union and intersection of two sets are defined as follows

A B = x x A or x BA B = x x A and x B

2 1

2211 Numbering and Notation Number theory (eg Feferman 1964 Kranz2005 ch 2) provides one of the foundations of modern analysis and the followingnotation is widely used for the set of natural numbers integer numbers and rationalnumbers

N = 123hellipZ = hellip minus3 minus2 minus10123hellipQ = p q pq Zq 0 gcd pq = 1

2 2

where gcd is the greatest common divisor functionThe set of positive integers Z + is defined as being equal to N and depending on

the context may include the number 0The set of rational numbers however is not complete in the sense that it does not

includeusefulnumberssuchas the lengthof thehypotenuseofa right trianglewhosesideshave unity length or the area of a circle of unit radius By augmenting the set of rationalnumbers with irrational numbers the set of real numbers denoted R can be defined

The set of complex numbers denoted C is the set of possible ordered pairs that canbe generated from real numbers that is

C= αβ α β R 2 3

When representing a complex number in the plane the notation xy = x + jy isused where j= 01 The algebra of complex numbers is governed by the rules ofvector addition and scalar multiplication that is

x1y1 + x2y2 = x1 + x2y1 + y2a x1y1 = ax1ay1 a Rx1y1 x2y2 = x1x2minusy1y2x1y2 + y1x2

2 4

From these definitions the familiar result of j2 = minus1 or j= minus1 follows Theconjugate of a complex number (x y) by definition is x minusy

2212 Countability

Definition Countable and Uncountable Sets A set is a countable set if eachelement of the set can be associated uniquely with a natural number that is anelement of N (Sprecher 1970 p 29) If such an association is not possible thenthe set is an uncountable set

The sets N Z and Q are countable sets The sets R and C are uncountable sets

12 BACKGROUND MATHEMATICS

222 Infinity

The term infinity written infin does not denote a number but is notation for that which isarbitrarily large Consider a sequence of numbers x1 x2hellipwhere xi gt ximinus1 and whichhas no upper bound The notation lim

i infinxi = infin has the meaning

Io Z+ xi S such that xi gt Io S = x1x2hellip 2 5

2221 Infinite Set Operations The following notation is widely used for the

union and intersection of an infinite number of setsinfin

i=1Ai and

infin

i=1Ai The following

definitions apply

infin

i=1Ai = lim

N infin

N

i=1Ai

infin

i=1Ai = lim

N infin

N

i=1Ai 2 6

223 Supremum and Infimum

Definition Supremum and Infimum The supremum of a set A of real numbersdenoted sup(A) is the least upper bound of that set and is such that (Marsdenand Hoffman 1993 p 45)

sup A ge x x A

εgt 0 x A such that sup A minusxlt ε2 7

The infimum of a set A of real numbers denoted inf(A) is the greatest lower bound ofthat set and is such that

inf A le x x A

ε gt 0 x A such that xminus inf A lt ε2 8

23 FUNCTION THEORY

Random phenomena define signals and accordingly signal and function theory areof fundamental importance The following provides a definition for a function definescommon functions and defines some useful properties of functions

231 Function Definition

Definition Function A function f is a mapping as illustrated in Figure 21 from aset SD called the domain to a set SR called the range such that only one element inthe range is associated with each element in the domain Such a function is writtenf SD SR If x SD and y SR with x mapping to y under f then the notationy = f x is used

13FUNCTION THEORY

232 Common Functions

The following are widely used functions

Definition Kronecker Delta The Kronecker delta function δK is defined accord-ing to

δK R 01 δK x =0 x 01 x = 0

2 9

Definition Sign or Signum Function The sign or signum function sgn isdefined according to

sgn R minus101 sgn x =minus1 x lt 00 x = 01 x gt 0

2 10

Definition Unit Step Function The unit step function u is defined according to

u R 01 u x =0 xlt 01 x ge 0

2 11

Definition Indicator Function The indicator function IA for a set A R is definedaccording to

IA R 01 IA x =1 x A0 x A

2 12

One example of an indicator function is illustrated in Figure 22 Note that indicatorfunctions are measurable functions (to be defined later)

Definition Sinc Function The sinc function is defined according to

sinc R R sinc x =sin πxπx

2 13

f

x y

SD SR

FIGURE 21 Illustration of the mapping of a function















2014049440



10 9 8 7 6 5 4 3 2 1

1 2016

ABOUT THE AUTHOR



CONTENTS

Preface xiii








viii CONTENTS





ixCONTENTS






x CONTENTS




References 717

Index 721

xiCONTENTS

PREFACE







ROY M HOWARD

August 2014

xiv PREFACE


11 INTRODUCTION





12 MOTIVATION


Solid foundation

Sure






0 200 400 600 800ndash10

ndash5

0

5

10

1000 t


VS +Vo

100+

ndash

RS


3MOTIVATION







0 000001 000002 000003 000004 t

ndash00005

00000

00005

00010

00015

Voltage

Signal

Signal plus noise







0 2 4 6 8 10 t

ndash10

ndash05

00

05

10

15

20

25

Threshold

tFP

ɸ(t)


0 1 2 3 4 500

02

04

06

08

10

t

Threshold


5MOTIVATION





0 5 10 15 2000

05

10

15

20

25

30

t


0 2 4 6 8 100

1

2

3

4

5

6

7

t





InputOutput

Quantization error

Threshold

Quantization level


0 1 2 3 4 5 6

Signal level

t

Sample times

Threshold

Quantizationlevel

Quantizationlevel


00

05

10

15

20

25


7MOTIVATION


122 Engineering


13 BOOK OVERVIEW


0 1 2 3 4 5 6 t

Signal


Threshold

ndash1

0

1

2

3







9BOOK OVERVIEW


21 INTRODUCTION



22 SET THEORY






2 1



2 2





C= αβ α β R 2 3



2 4


2212 Countability




222 Infinity






i=1Ai and

infin

i=1Ai The following

definitions apply

infin

i=1Ai = lim

N infin

N

i=1Ai

infin

i=1Ai = lim

N infin

N

i=1Ai 2 6



sup A ge x x A



inf A le x x A


23 FUNCTION THEORY




13FUNCTION THEORY




δK R 01 δK x =0 x 01 x = 0

2 9



2 10


u R 01 u x =0 xlt 01 x ge 0

2 11



2 12




2 13

f

x y

SD SR



ABOUT THE AUTHOR



CONTENTS

Preface xiii








viii CONTENTS





ixCONTENTS






x CONTENTS




References 717

Index 721

xiCONTENTS

PREFACE







ROY M HOWARD

August 2014

xiv PREFACE


11 INTRODUCTION





12 MOTIVATION


Solid foundation

Sure






0 200 400 600 800ndash10

ndash5

0

5

10

1000 t


VS +Vo

100+

ndash

RS


3MOTIVATION







0 000001 000002 000003 000004 t

ndash00005

00000

00005

00010

00015

Voltage

Signal

Signal plus noise







0 2 4 6 8 10 t

ndash10

ndash05

00

05

10

15

20

25

Threshold

tFP

ɸ(t)


0 1 2 3 4 500

02

04

06

08

10

t

Threshold


5MOTIVATION





0 5 10 15 2000

05

10

15

20

25

30

t


0 2 4 6 8 100

1

2

3

4

5

6

7

t





InputOutput

Quantization error

Threshold

Quantization level


0 1 2 3 4 5 6

Signal level

t

Sample times

Threshold

Quantizationlevel

Quantizationlevel


00

05

10

15

20

25


7MOTIVATION


122 Engineering


13 BOOK OVERVIEW


0 1 2 3 4 5 6 t

Signal


Threshold

ndash1

0

1

2

3







9BOOK OVERVIEW


21 INTRODUCTION



22 SET THEORY






2 1



2 2





C= αβ α β R 2 3



2 4


2212 Countability




222 Infinity






i=1Ai and

infin

i=1Ai The following

definitions apply

infin

i=1Ai = lim

N infin

N

i=1Ai

infin

i=1Ai = lim

N infin

N

i=1Ai 2 6



sup A ge x x A



inf A le x x A


23 FUNCTION THEORY




13FUNCTION THEORY




δK R 01 δK x =0 x 01 x = 0

2 9



2 10


u R 01 u x =0 xlt 01 x ge 0

2 11



2 12




2 13

f

x y

SD SR



CONTENTS

Preface xiii








viii CONTENTS





ixCONTENTS






x CONTENTS




References 717

Index 721

xiCONTENTS

PREFACE







ROY M HOWARD

August 2014

xiv PREFACE


11 INTRODUCTION





12 MOTIVATION


Solid foundation

Sure






0 200 400 600 800ndash10

ndash5

0

5

10

1000 t


VS +Vo

100+

ndash

RS


3MOTIVATION







0 000001 000002 000003 000004 t

ndash00005

00000

00005

00010

00015

Voltage

Signal

Signal plus noise







0 2 4 6 8 10 t

ndash10

ndash05

00

05

10

15

20

25

Threshold

tFP

ɸ(t)


0 1 2 3 4 500

02

04

06

08

10

t

Threshold


5MOTIVATION





0 5 10 15 2000

05

10

15

20

25

30

t


0 2 4 6 8 100

1

2

3

4

5

6

7

t





InputOutput

Quantization error

Threshold

Quantization level


0 1 2 3 4 5 6

Signal level

t

Sample times

Threshold

Quantizationlevel

Quantizationlevel


00

05

10

15

20

25


7MOTIVATION


122 Engineering


13 BOOK OVERVIEW


0 1 2 3 4 5 6 t

Signal


Threshold

ndash1

0

1

2

3







9BOOK OVERVIEW


21 INTRODUCTION



22 SET THEORY






2 1



2 2





C= αβ α β R 2 3



2 4


2212 Countability




222 Infinity






i=1Ai and

infin

i=1Ai The following

definitions apply

infin

i=1Ai = lim

N infin

N

i=1Ai

infin

i=1Ai = lim

N infin

N

i=1Ai 2 6



sup A ge x x A



inf A le x x A


23 FUNCTION THEORY




13FUNCTION THEORY




δK R 01 δK x =0 x 01 x = 0

2 9



2 10


u R 01 u x =0 xlt 01 x ge 0

2 11



2 12




2 13

f

x y

SD SR







viii CONTENTS





ixCONTENTS






x CONTENTS




References 717

Index 721

xiCONTENTS

PREFACE







ROY M HOWARD

August 2014

xiv PREFACE


11 INTRODUCTION





12 MOTIVATION


Solid foundation

Sure






0 200 400 600 800ndash10

ndash5

0

5

10

1000 t


VS +Vo

100+

ndash

RS


3MOTIVATION







0 000001 000002 000003 000004 t

ndash00005

00000

00005

00010

00015

Voltage

Signal

Signal plus noise







0 2 4 6 8 10 t

ndash10

ndash05

00

05

10

15

20

25

Threshold

tFP

ɸ(t)


0 1 2 3 4 500

02

04

06

08

10

t

Threshold


5MOTIVATION





0 5 10 15 2000

05

10

15

20

25

30

t


0 2 4 6 8 100

1

2

3

4

5

6

7

t





InputOutput

Quantization error

Threshold

Quantization level


0 1 2 3 4 5 6

Signal level

t

Sample times

Threshold

Quantizationlevel

Quantizationlevel


00

05

10

15

20

25


7MOTIVATION


122 Engineering


13 BOOK OVERVIEW


0 1 2 3 4 5 6 t

Signal


Threshold

ndash1

0

1

2

3







9BOOK OVERVIEW


21 INTRODUCTION



22 SET THEORY






2 1



2 2





C= αβ α β R 2 3



2 4


2212 Countability




222 Infinity






i=1Ai and

infin

i=1Ai The following

definitions apply

infin

i=1Ai = lim

N infin

N

i=1Ai

infin

i=1Ai = lim

N infin

N

i=1Ai 2 6



sup A ge x x A



inf A le x x A


23 FUNCTION THEORY




13FUNCTION THEORY




δK R 01 δK x =0 x 01 x = 0

2 9



2 10


u R 01 u x =0 xlt 01 x ge 0

2 11



2 12




2 13

f

x y

SD SR







ixCONTENTS






x CONTENTS




References 717

Index 721

xiCONTENTS

PREFACE







ROY M HOWARD

August 2014

xiv PREFACE


11 INTRODUCTION





12 MOTIVATION


Solid foundation

Sure






0 200 400 600 800ndash10

ndash5

0

5

10

1000 t


VS +Vo

100+

ndash

RS


3MOTIVATION







0 000001 000002 000003 000004 t

ndash00005

00000

00005

00010

00015

Voltage

Signal

Signal plus noise







0 2 4 6 8 10 t

ndash10

ndash05

00

05

10

15

20

25

Threshold

tFP

ɸ(t)


0 1 2 3 4 500

02

04

06

08

10

t

Threshold


5MOTIVATION





0 5 10 15 2000

05

10

15

20

25

30

t


0 2 4 6 8 100

1

2

3

4

5

6

7

t





InputOutput

Quantization error

Threshold

Quantization level


0 1 2 3 4 5 6

Signal level

t

Sample times

Threshold

Quantizationlevel

Quantizationlevel


00

05

10

15

20

25


7MOTIVATION


122 Engineering


13 BOOK OVERVIEW


0 1 2 3 4 5 6 t

Signal


Threshold

ndash1

0

1

2

3







9BOOK OVERVIEW


21 INTRODUCTION



22 SET THEORY






2 1



2 2





C= αβ α β R 2 3



2 4


2212 Countability




222 Infinity






i=1Ai and

infin

i=1Ai The following

definitions apply

infin

i=1Ai = lim

N infin

N

i=1Ai

infin

i=1Ai = lim

N infin

N

i=1Ai 2 6



sup A ge x x A



inf A le x x A


23 FUNCTION THEORY




13FUNCTION THEORY




δK R 01 δK x =0 x 01 x = 0

2 9



2 10


u R 01 u x =0 xlt 01 x ge 0

2 11



2 12




2 13

f

x y

SD SR








x CONTENTS




References 717

Index 721

xiCONTENTS

PREFACE







ROY M HOWARD

August 2014

xiv PREFACE


11 INTRODUCTION





12 MOTIVATION


Solid foundation

Sure






0 200 400 600 800ndash10

ndash5

0

5

10

1000 t


VS +Vo

100+

ndash

RS


3MOTIVATION







0 000001 000002 000003 000004 t

ndash00005

00000

00005

00010

00015

Voltage

Signal

Signal plus noise







0 2 4 6 8 10 t

ndash10

ndash05

00

05

10

15

20

25

Threshold

tFP

ɸ(t)


0 1 2 3 4 500

02

04

06

08

10

t

Threshold


5MOTIVATION





0 5 10 15 2000

05

10

15

20

25

30

t


0 2 4 6 8 100

1

2

3

4

5

6

7

t





InputOutput

Quantization error

Threshold

Quantization level


0 1 2 3 4 5 6

Signal level

t

Sample times

Threshold

Quantizationlevel

Quantizationlevel


00

05

10

15

20

25


7MOTIVATION


122 Engineering


13 BOOK OVERVIEW


0 1 2 3 4 5 6 t

Signal


Threshold

ndash1

0

1

2

3







9BOOK OVERVIEW


21 INTRODUCTION



22 SET THEORY






2 1



2 2





C= αβ α β R 2 3



2 4


2212 Countability




222 Infinity






i=1Ai and

infin

i=1Ai The following

definitions apply

infin

i=1Ai = lim

N infin

N

i=1Ai

infin

i=1Ai = lim

N infin

N

i=1Ai 2 6



sup A ge x x A



inf A le x x A


23 FUNCTION THEORY




13FUNCTION THEORY




δK R 01 δK x =0 x 01 x = 0

2 9



2 10


u R 01 u x =0 xlt 01 x ge 0

2 11



2 12




2 13

f

x y

SD SR






References 717

Index 721

xiCONTENTS

PREFACE







ROY M HOWARD

August 2014

xiv PREFACE


11 INTRODUCTION





12 MOTIVATION


Solid foundation

Sure






0 200 400 600 800ndash10

ndash5

0

5

10

1000 t


VS +Vo

100+

ndash

RS


3MOTIVATION







0 000001 000002 000003 000004 t

ndash00005

00000

00005

00010

00015

Voltage

Signal

Signal plus noise







0 2 4 6 8 10 t

ndash10

ndash05

00

05

10

15

20

25

Threshold

tFP

ɸ(t)


0 1 2 3 4 500

02

04

06

08

10

t

Threshold


5MOTIVATION





0 5 10 15 2000

05

10

15

20

25

30

t


0 2 4 6 8 100

1

2

3

4

5

6

7

t





InputOutput

Quantization error

Threshold

Quantization level


0 1 2 3 4 5 6

Signal level

t

Sample times

Threshold

Quantizationlevel

Quantizationlevel


00

05

10

15

20

25


7MOTIVATION


122 Engineering


13 BOOK OVERVIEW


0 1 2 3 4 5 6 t

Signal


Threshold

ndash1

0

1

2

3







9BOOK OVERVIEW


21 INTRODUCTION



22 SET THEORY






2 1



2 2





C= αβ α β R 2 3



2 4


2212 Countability




222 Infinity






i=1Ai and

infin

i=1Ai The following

definitions apply

infin

i=1Ai = lim

N infin

N

i=1Ai

infin

i=1Ai = lim

N infin

N

i=1Ai 2 6



sup A ge x x A



inf A le x x A


23 FUNCTION THEORY




13FUNCTION THEORY




δK R 01 δK x =0 x 01 x = 0

2 9



2 10


u R 01 u x =0 xlt 01 x ge 0

2 11



2 12




2 13

f

x y

SD SR



PREFACE







ROY M HOWARD

August 2014

xiv PREFACE


11 INTRODUCTION





12 MOTIVATION


Solid foundation

Sure






0 200 400 600 800ndash10

ndash5

0

5

10

1000 t


VS +Vo

100+

ndash

RS


3MOTIVATION







0 000001 000002 000003 000004 t

ndash00005

00000

00005

00010

00015

Voltage

Signal

Signal plus noise







0 2 4 6 8 10 t

ndash10

ndash05

00

05

10

15

20

25

Threshold

tFP

ɸ(t)


0 1 2 3 4 500

02

04

06

08

10

t

Threshold


5MOTIVATION





0 5 10 15 2000

05

10

15

20

25

30

t


0 2 4 6 8 100

1

2

3

4

5

6

7

t





InputOutput

Quantization error

Threshold

Quantization level


0 1 2 3 4 5 6

Signal level

t

Sample times

Threshold

Quantizationlevel

Quantizationlevel


00

05

10

15

20

25


7MOTIVATION


122 Engineering


13 BOOK OVERVIEW


0 1 2 3 4 5 6 t

Signal


Threshold

ndash1

0

1

2

3







9BOOK OVERVIEW


21 INTRODUCTION



22 SET THEORY






2 1



2 2





C= αβ α β R 2 3



2 4


2212 Countability




222 Infinity






i=1Ai and

infin

i=1Ai The following

definitions apply

infin

i=1Ai = lim

N infin

N

i=1Ai

infin

i=1Ai = lim

N infin

N

i=1Ai 2 6



sup A ge x x A



inf A le x x A


23 FUNCTION THEORY




13FUNCTION THEORY




δK R 01 δK x =0 x 01 x = 0

2 9



2 10


u R 01 u x =0 xlt 01 x ge 0

2 11



2 12




2 13

f

x y

SD SR





ROY M HOWARD

August 2014

xiv PREFACE


11 INTRODUCTION





12 MOTIVATION


Solid foundation

Sure






0 200 400 600 800ndash10

ndash5

0

5

10

1000 t


VS +Vo

100+

ndash

RS


3MOTIVATION







0 000001 000002 000003 000004 t

ndash00005

00000

00005

00010

00015

Voltage

Signal

Signal plus noise







0 2 4 6 8 10 t

ndash10

ndash05

00

05

10

15

20

25

Threshold

tFP

ɸ(t)


0 1 2 3 4 500

02

04

06

08

10

t

Threshold


5MOTIVATION





0 5 10 15 2000

05

10

15

20

25

30

t


0 2 4 6 8 100

1

2

3

4

5

6

7

t





InputOutput

Quantization error

Threshold

Quantization level


0 1 2 3 4 5 6

Signal level

t

Sample times

Threshold

Quantizationlevel

Quantizationlevel


00

05

10

15

20

25


7MOTIVATION


122 Engineering


13 BOOK OVERVIEW


0 1 2 3 4 5 6 t

Signal


Threshold

ndash1

0

1

2

3







9BOOK OVERVIEW


21 INTRODUCTION



22 SET THEORY






2 1



2 2





C= αβ α β R 2 3



2 4


2212 Countability




222 Infinity






i=1Ai and

infin

i=1Ai The following

definitions apply

infin

i=1Ai = lim

N infin

N

i=1Ai

infin

i=1Ai = lim

N infin

N

i=1Ai 2 6



sup A ge x x A



inf A le x x A


23 FUNCTION THEORY




13FUNCTION THEORY




δK R 01 δK x =0 x 01 x = 0

2 9



2 10


u R 01 u x =0 xlt 01 x ge 0

2 11



2 12




2 13

f

x y

SD SR




11 INTRODUCTION





12 MOTIVATION


Solid foundation

Sure






0 200 400 600 800ndash10

ndash5

0

5

10

1000 t


VS +Vo

100+

ndash

RS


3MOTIVATION







0 000001 000002 000003 000004 t

ndash00005

00000

00005

00010

00015

Voltage

Signal

Signal plus noise







0 2 4 6 8 10 t

ndash10

ndash05

00

05

10

15

20

25

Threshold

tFP

ɸ(t)


0 1 2 3 4 500

02

04

06

08

10

t

Threshold


5MOTIVATION





0 5 10 15 2000

05

10

15

20

25

30

t


0 2 4 6 8 100

1

2

3

4

5

6

7

t





InputOutput

Quantization error

Threshold

Quantization level


0 1 2 3 4 5 6

Signal level

t

Sample times

Threshold

Quantizationlevel

Quantizationlevel


00

05

10

15

20

25


7MOTIVATION


122 Engineering


13 BOOK OVERVIEW


0 1 2 3 4 5 6 t

Signal


Threshold

ndash1

0

1

2

3







9BOOK OVERVIEW


21 INTRODUCTION



22 SET THEORY






2 1



2 2





C= αβ α β R 2 3



2 4


2212 Countability




222 Infinity






i=1Ai and

infin

i=1Ai The following

definitions apply

infin

i=1Ai = lim

N infin

N

i=1Ai

infin

i=1Ai = lim

N infin

N

i=1Ai 2 6



sup A ge x x A



inf A le x x A


23 FUNCTION THEORY




13FUNCTION THEORY




δK R 01 δK x =0 x 01 x = 0

2 9



2 10


u R 01 u x =0 xlt 01 x ge 0

2 11



2 12




2 13

f

x y

SD SR




12 MOTIVATION


Solid foundation

Sure






0 200 400 600 800ndash10

ndash5

0

5

10

1000 t


VS +Vo

100+

ndash

RS


3MOTIVATION







0 000001 000002 000003 000004 t

ndash00005

00000

00005

00010

00015

Voltage

Signal

Signal plus noise







0 2 4 6 8 10 t

ndash10

ndash05

00

05

10

15

20

25

Threshold

tFP

ɸ(t)


0 1 2 3 4 500

02

04

06

08

10

t

Threshold


5MOTIVATION





0 5 10 15 2000

05

10

15

20

25

30

t


0 2 4 6 8 100

1

2

3

4

5

6

7

t





InputOutput

Quantization error

Threshold

Quantization level


0 1 2 3 4 5 6

Signal level

t

Sample times

Threshold

Quantizationlevel

Quantizationlevel


00

05

10

15

20

25


7MOTIVATION


122 Engineering


13 BOOK OVERVIEW


0 1 2 3 4 5 6 t

Signal


Threshold

ndash1

0

1

2

3







9BOOK OVERVIEW


21 INTRODUCTION



22 SET THEORY






2 1



2 2





C= αβ α β R 2 3



2 4


2212 Countability




222 Infinity






i=1Ai and

infin

i=1Ai The following

definitions apply

infin

i=1Ai = lim

N infin

N

i=1Ai

infin

i=1Ai = lim

N infin

N

i=1Ai 2 6



sup A ge x x A



inf A le x x A


23 FUNCTION THEORY




13FUNCTION THEORY




δK R 01 δK x =0 x 01 x = 0

2 9



2 10


u R 01 u x =0 xlt 01 x ge 0

2 11



2 12




2 13

f

x y

SD SR






0 200 400 600 800ndash10

ndash5

0

5

10

1000 t


VS +Vo

100+

ndash

RS


3MOTIVATION







0 000001 000002 000003 000004 t

ndash00005

00000

00005

00010

00015

Voltage

Signal

Signal plus noise







0 2 4 6 8 10 t

ndash10

ndash05

00

05

10

15

20

25

Threshold

tFP

ɸ(t)


0 1 2 3 4 500

02

04

06

08

10

t

Threshold


5MOTIVATION





0 5 10 15 2000

05

10

15

20

25

30

t


0 2 4 6 8 100

1

2

3

4

5

6

7

t





InputOutput

Quantization error

Threshold

Quantization level


0 1 2 3 4 5 6

Signal level

t

Sample times

Threshold

Quantizationlevel

Quantizationlevel


00

05

10

15

20

25


7MOTIVATION


122 Engineering


13 BOOK OVERVIEW


0 1 2 3 4 5 6 t

Signal


Threshold

ndash1

0

1

2

3







9BOOK OVERVIEW


21 INTRODUCTION



22 SET THEORY






2 1



2 2





C= αβ α β R 2 3



2 4


2212 Countability




222 Infinity






i=1Ai and

infin

i=1Ai The following

definitions apply

infin

i=1Ai = lim

N infin

N

i=1Ai

infin

i=1Ai = lim

N infin

N

i=1Ai 2 6



sup A ge x x A



inf A le x x A


23 FUNCTION THEORY




13FUNCTION THEORY




δK R 01 δK x =0 x 01 x = 0

2 9



2 10


u R 01 u x =0 xlt 01 x ge 0

2 11



2 12




2 13

f

x y

SD SR









0 000001 000002 000003 000004 t

ndash00005

00000

00005

00010

00015

Voltage

Signal

Signal plus noise







0 2 4 6 8 10 t

ndash10

ndash05

00

05

10

15

20

25

Threshold

tFP

ɸ(t)


0 1 2 3 4 500

02

04

06

08

10

t

Threshold


5MOTIVATION





0 5 10 15 2000

05

10

15

20

25

30

t


0 2 4 6 8 100

1

2

3

4

5

6

7

t





InputOutput

Quantization error

Threshold

Quantization level


0 1 2 3 4 5 6

Signal level

t

Sample times

Threshold

Quantizationlevel

Quantizationlevel


00

05

10

15

20

25


7MOTIVATION


122 Engineering


13 BOOK OVERVIEW


0 1 2 3 4 5 6 t

Signal


Threshold

ndash1

0

1

2

3







9BOOK OVERVIEW


21 INTRODUCTION



22 SET THEORY






2 1



2 2





C= αβ α β R 2 3



2 4


2212 Countability




222 Infinity






i=1Ai and

infin

i=1Ai The following

definitions apply

infin

i=1Ai = lim

N infin

N

i=1Ai

infin

i=1Ai = lim

N infin

N

i=1Ai 2 6



sup A ge x x A



inf A le x x A


23 FUNCTION THEORY




13FUNCTION THEORY




δK R 01 δK x =0 x 01 x = 0

2 9



2 10


u R 01 u x =0 xlt 01 x ge 0

2 11



2 12




2 13

f

x y

SD SR






0 2 4 6 8 10 t

ndash10

ndash05

00

05

10

15

20

25

Threshold

tFP

ɸ(t)


0 1 2 3 4 500

02

04

06

08

10

t

Threshold


5MOTIVATION





0 5 10 15 2000

05

10

15

20

25

30

t


0 2 4 6 8 100

1

2

3

4

5

6

7

t





InputOutput

Quantization error

Threshold

Quantization level


0 1 2 3 4 5 6

Signal level

t

Sample times

Threshold

Quantizationlevel

Quantizationlevel


00

05

10

15

20

25


7MOTIVATION


122 Engineering


13 BOOK OVERVIEW


0 1 2 3 4 5 6 t

Signal


Threshold

ndash1

0

1

2

3







9BOOK OVERVIEW


21 INTRODUCTION



22 SET THEORY






2 1



2 2





C= αβ α β R 2 3



2 4


2212 Countability




222 Infinity






i=1Ai and

infin

i=1Ai The following

definitions apply

infin

i=1Ai = lim

N infin

N

i=1Ai

infin

i=1Ai = lim

N infin

N

i=1Ai 2 6



sup A ge x x A



inf A le x x A


23 FUNCTION THEORY




13FUNCTION THEORY




δK R 01 δK x =0 x 01 x = 0

2 9



2 10


u R 01 u x =0 xlt 01 x ge 0

2 11



2 12




2 13

f

x y

SD SR







0 5 10 15 2000

05

10

15

20

25

30

t


0 2 4 6 8 100

1

2

3

4

5

6

7

t





InputOutput

Quantization error

Threshold

Quantization level


0 1 2 3 4 5 6

Signal level

t

Sample times

Threshold

Quantizationlevel

Quantizationlevel


00

05

10

15

20

25


7MOTIVATION


122 Engineering


13 BOOK OVERVIEW


0 1 2 3 4 5 6 t

Signal


Threshold

ndash1

0

1

2

3







9BOOK OVERVIEW


21 INTRODUCTION



22 SET THEORY






2 1



2 2





C= αβ α β R 2 3



2 4


2212 Countability




222 Infinity






i=1Ai and

infin

i=1Ai The following

definitions apply

infin

i=1Ai = lim

N infin

N

i=1Ai

infin

i=1Ai = lim

N infin

N

i=1Ai 2 6



sup A ge x x A



inf A le x x A


23 FUNCTION THEORY




13FUNCTION THEORY




δK R 01 δK x =0 x 01 x = 0

2 9



2 10


u R 01 u x =0 xlt 01 x ge 0

2 11



2 12




2 13

f

x y

SD SR





InputOutput

Quantization error

Threshold

Quantization level


0 1 2 3 4 5 6

Signal level

t

Sample times

Threshold

Quantizationlevel

Quantizationlevel


00

05

10

15

20

25


7MOTIVATION


122 Engineering


13 BOOK OVERVIEW


0 1 2 3 4 5 6 t

Signal


Threshold

ndash1

0

1

2

3







9BOOK OVERVIEW


21 INTRODUCTION



22 SET THEORY






2 1



2 2





C= αβ α β R 2 3



2 4


2212 Countability




222 Infinity






i=1Ai and

infin

i=1Ai The following

definitions apply

infin

i=1Ai = lim

N infin

N

i=1Ai

infin

i=1Ai = lim

N infin

N

i=1Ai 2 6



sup A ge x x A



inf A le x x A


23 FUNCTION THEORY




13FUNCTION THEORY




δK R 01 δK x =0 x 01 x = 0

2 9



2 10


u R 01 u x =0 xlt 01 x ge 0

2 11



2 12




2 13

f

x y

SD SR




122 Engineering


13 BOOK OVERVIEW


0 1 2 3 4 5 6 t

Signal


Threshold

ndash1

0

1

2

3







9BOOK OVERVIEW


21 INTRODUCTION



22 SET THEORY






2 1



2 2





C= αβ α β R 2 3



2 4


2212 Countability




222 Infinity






i=1Ai and

infin

i=1Ai The following

definitions apply

infin

i=1Ai = lim

N infin

N

i=1Ai

infin

i=1Ai = lim

N infin

N

i=1Ai 2 6



sup A ge x x A



inf A le x x A


23 FUNCTION THEORY




13FUNCTION THEORY




δK R 01 δK x =0 x 01 x = 0

2 9



2 10


u R 01 u x =0 xlt 01 x ge 0

2 11



2 12




2 13

f

x y

SD SR







9BOOK OVERVIEW


21 INTRODUCTION



22 SET THEORY






2 1



2 2





C= αβ α β R 2 3



2 4


2212 Countability




222 Infinity






i=1Ai and

infin

i=1Ai The following

definitions apply

infin

i=1Ai = lim

N infin

N

i=1Ai

infin

i=1Ai = lim

N infin

N

i=1Ai 2 6



sup A ge x x A



inf A le x x A


23 FUNCTION THEORY




13FUNCTION THEORY




δK R 01 δK x =0 x 01 x = 0

2 9



2 10


u R 01 u x =0 xlt 01 x ge 0

2 11



2 12




2 13

f

x y

SD SR




21 INTRODUCTION



22 SET THEORY






2 1



2 2





C= αβ α β R 2 3



2 4


2212 Countability




222 Infinity






i=1Ai and

infin

i=1Ai The following

definitions apply

infin

i=1Ai = lim

N infin

N

i=1Ai

infin

i=1Ai = lim

N infin

N

i=1Ai 2 6



sup A ge x x A



inf A le x x A


23 FUNCTION THEORY




13FUNCTION THEORY




δK R 01 δK x =0 x 01 x = 0

2 9



2 10


u R 01 u x =0 xlt 01 x ge 0

2 11



2 12




2 13

f

x y

SD SR





2 1



2 2





C= αβ α β R 2 3



2 4


2212 Countability




222 Infinity






i=1Ai and

infin

i=1Ai The following

definitions apply

infin

i=1Ai = lim

N infin

N

i=1Ai

infin

i=1Ai = lim

N infin

N

i=1Ai 2 6



sup A ge x x A



inf A le x x A


23 FUNCTION THEORY




13FUNCTION THEORY




δK R 01 δK x =0 x 01 x = 0

2 9



2 10


u R 01 u x =0 xlt 01 x ge 0

2 11



2 12




2 13

f

x y

SD SR



222 Infinity






i=1Ai and

infin

i=1Ai The following

definitions apply

infin

i=1Ai = lim

N infin

N

i=1Ai

infin

i=1Ai = lim

N infin

N

i=1Ai 2 6



sup A ge x x A



inf A le x x A


23 FUNCTION THEORY




13FUNCTION THEORY




δK R 01 δK x =0 x 01 x = 0

2 9



2 10


u R 01 u x =0 xlt 01 x ge 0

2 11



2 12




2 13

f

x y

SD SR






δK R 01 δK x =0 x 01 x = 0

2 9



2 10


u R 01 u x =0 xlt 01 x ge 0

2 11



2 12




2 13

f

x y

SD SR



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