Noise, oscillators, and algebraic randomness : from noise in communication systems to number theory : lectures of a school held in Chapelle des Bois, France, April 5-10, 1999

Noise Is Ubiquitous In Nature And In Man-made Systems. Noise In Oscillators Perturbs High-technology Devices Such As Time Standards Or Digital Communication Systems. The Understanding Of Its Algebraic Structure Is Thus Of Vital Importance. The Book Addresses Both The Measurement Methods And The Understanding Of Quantum, 1/f And Phase Noise In Systems Such As Electronic Amplifiers, Oscillators And Receivers, Trapped Ions, Cosmic Ray Showers And In Commercial Applications. A Strong Link Between 1/f Noise And Number Theory Is Emphasized. The Twenty Papers In The Book Are Comprehensive Versions Of Talks Presented At A School In Chapelle Des Bois (jura, France) Held From April 6 To 10, 1999 By Engineers, Physisicts And Mathematicians. 1. Introduction -- 2. Classical And Quantum Noise -- 3. Noise In Oscillators, 1/f Noise And Arithmetic -- 4. Algebraic Randomness. Edited By Michel Planat. Chapter 1 Chapter 2 1 Introduction 2 A new look at the exponential 2.1 The pow r of exponentials 2.2 Taylor ’s formula and exponential 2.3 Leibniz ’s formula 2.4 Exponential vs. logarithm 2.5 Infinitesimals and exponentials 2.6 Differential equations 3 Operational calculus 3.1 An algebraic digression: umbral calculus 3.2 Binomial sequences of polynomials 3.3 Transformation of polynomials 3.4 Expansion formulas 3.5 Signal transforms 3.6 The inverse problem 3.7 A probabilistic application 3.8 The Bargmann-Segal transform 3.9 The quantum harmonic oscillator 4 The art of manipulating infinite series 4.1 Some divergent series 4.2 Polynomials of infinite degree and summation of series 4.3 The Euler-Riemann zeta function 4.4 Sums of powers of numbers 4.5 Variation I: Did Euler really fool himself? 4.6 Variation II: Infinite products 5 Conclusion: From Euler to Feynman Acknowledgements References Chapter 3 1 Non-ideal Quantum Measurements 2 Coupling with the Environment 2.1 Dissipation and fluctuations 2.2 Treatment with quantum .elds 2.3 Quantum networks 3 Fluctuations in Amplifiers 4 The Cold Damped Accelerometer References Chapter 4 1 Introduction 2 The Microscopic Model 2.1 Second quantized formalism Holstein –Primakov representation 2.2 Dynamics of the model 2.3 Form factor and conductivity Form factor Macroscopic response function Conductivity 2.4 Conclusion Appendix A Invariance of the commutation relations Appendix B Casimir operator References Chapter 5 1 Introduction 2 Dynamics of Stored Ions 2.1 Ion motion in a pure quadrupole field 2.2 Ion motion in experimental conditions 2.3 Experimental observations 2.4 Simplified motion equation for radial-axial couplings 3 Application to Frequency Standards 3.1 The radiofrequency domain 3.2 The optical domain 3.3 The Ca$^+$ at PIIM, Marseille 4 Quantum Optics with Laser-Cooled Ions 5 Conclusion References Chapter 6 1 Introduction 2 Extensive Air Showers 3 Simulation of EAS Events 4 Analysis of Fluctuations 5 Multifractal Analysis of the 1/f Noise 6 Conclusions Acknowledgements References Chapter 7 1 Introduction 2 Periodic SR in Bistable Dynamic Systems 3 Periodic SR in Static Nonlinear Systems 4 Aperiodic SR in a Nonlinear Information Channel 5 Aperiodic SR in Image Transmission 6 Outlook References Chapter 8 1 Introduction 2 Detrended Fluctuation Analysis Techniques 3 Multiaffine Analysis Techniques 4 Moving Average Techniques 5 Sandpile Model for Rupture and Crashes 6 (m,k )-Zipf Techniques 7 Basics of i-Variability Diagram Techniques Acknowledgements References Chapter 9 1 Introduction 2 Basic Definitions and Principles 2.1 The pendulum as a frequency reference 2.2 Damped and stationary oscillations 2.3 Frequency stability 2.4 Accuracy 3 Electronic Oscillators 4 Frequency-Domain Characterization of Frequency Stability 4.1 The power law model 5 Time-Domain Characterization of Frequency Stability True variance Two-sample variance References Chapter 10 1 Introduction 2 Background 2.1 Double sideband (DSB) representation of noise 2.2 Single sideband (SSB) representation of noise 3 Traditional Methods 3.1 Instrument sensitivity 3.2 Additional instrument limitations 4 Useful Schemes General two port devices Amplifiers High insertion-loss two port devices Equal DUT pair Discriminator and delay line Oscillator pair Frequency multiplier Narrow tuning range oscillators 5 Interferometric Noise Measurement Method 5.1 Design strategies 5.2 Further remarks 6 Correlation Techniques 6.1 Double interferometer 6.2 Noise theory of the double interferometer 6.3 Noise properties of the double interferometer Noise floor Noise measurement below the thermal floor Noise of an attenuator References Chapter 11 1 Introduction 2 Mobility Fluctuation 1/f Noise Induced by Lattice Scattering 3 Phonon Fine Structure in the 1/f Noise of Semiconductor Devices 3 Phonon Fine Structure in the 1/f Noise of Semiconductor Devices 4 Surface and Bulk Phonons in the 1/f Noise of Metals 5 1/f Noise Induced by Surface and Bulk Atomic Motion 6 Physical Significance of the 1/f Noise Parameter 7 Image of Phonon Spectrum in 1/f Noise 7.1 Metals 7.2 Semiconductors 8 Conclusion References Chapter 12 1 Introduction 2 Conventional Quantum 1/f Effect 3 Derivation of the Coherent Quantum 1/f Noise Effect 4 Sufficient Criterion for Fundamental 1/f Noise 5 Application to QED: Quantum 1/f Effect as a Special Case 6 Derivation of the Conventional Quantum 1/f Noise Effect in Second Quantization 7 Physical Derivation of Coherent Quantum 1/f Noise Effect 8 Derivation of Mobility Quantum 1/f Noise in $n^+ - p$ Diodes 9 Quantum 1/f Noise in SQUIDS 10 Quantum 1/f Noise in Bulk Acoustic Wave and SAW Quartz Resonators 11 A Different Approach to 1/f Noise from Frequency Mixing Experiments 12 Discussion References Chapter 13 1 The Communication Receiver 1.1 Theoretical background 1.2 Experiments 2 Arithmetic of Amplitude–Frequency Relationships 2.1 The frequency of beat signals from diophantine approximation 2.2 The amplitude of beat signals and the Franel–Landau shift 2.3 Diophantine signal processing and 1/f frequency fluctuations 2.4 The Riemann zeta function and the Riemann hypothesis and physics References Chapter 14 1 Introduction 2 Experimental and Computed Data 3 Time Series Analysis Methods 3.1 False nearest neighbour percentage 3.2 Correlation dimension 4 Detection of Chaos in Experimental and Computed Data 4.1 Experimental time series 4.2 Computed time series 5 Discussion and Conclusion References Chapter 15 Chapter 16 1 Hamiltonian Chaos and the Standard Map 2 The Critical Constants 3 Complex Analytic Maps 4 Continued Fractions and the Brjuno Function 5 The Brjuno Series and Diophantine conditions 6 The Brjuno Operator 7 Application to Hölder–continuous Functions 8 The Complexification of the Brjuno Function Acknowledgements References Chapter 17 1 A Few “Principles ” 2 Algebraic Randomness 2.1 Block complexity 2.2 Some algebraic “algorithms ” Morphisms and finite automata Cellular automata Some links between these algorithms 2.3 Algebraicity and transcendence The Thue-Morse sequence revisited Real numbers with automatic base $b$ expansion Real numbers with automatic continued fraction expansion 3 Analytic Randomness 3.1 Normality 3.2 Topological entropy 3.3 Fourier analysis The Wiener spectrum A few examples References Chapter 18 1 Introduction 2 Symbolic Dynamics at the Feigenbaum Points 3 Self–Similarity 4 Entropy Analysis and Complexity at a Feigenbaum Point 5 Other Coarse–Gr ined Statistical Properties at the Feigenbaum Points 6 Replacements and Morphisms 7 Digital Approach,Transcendence and Non-Normality 8 On the Feigenbaum Constants $delta$ and $lpha$ 9 The “Standard ”Conjecture of Chaos 10 Conclusion Acknowledgements References Chapter 19 1 Transcendental Values of Böttcher Functions 2 Lehmer’s Problem and the Entropy of Algebraic Dynamical Systems 3 Canonical Heights and Dynamical Systems References Chapter 20 1 Introduction 2 Notation 3 Results 4 An Algorithm 5 An Example References Chapter 21 1 The Modular Function and Elliptic Curves 2 Complex Multiplication 3 Schneider’s Theorem 4 Generalisations References Chapter 22 1 Introduction 2 The Case of the Classical Markoff Theory 3 More General Diophantine Equations 4 Solving the Generalized Equations 5 Application to the Analysis of the Markoff Spectrum 6 A Link with the Representation of Free Groups References Annotation Noise is ubiquitous in nature and in man-made systems. Noise in oscillators perturbs high-technology devices such as time standards or digital communication systems. The understanding of its algebraic structure is thus of vital importance. The book addresses both the measurement methods and the understanding of quantum, 1/f and phase noise in systems such as electronic amplifiers, oscillators and receivers, trapped ions, cosmic ray showers and in commercial applications. A strong link between 1/f noise and number theory is emphasized. The twenty papers in the book are comprehensive versions of talks presented at a school in Chapelle des Bois (Jura, France) held from April 6 to 10, 1999, by engineers, physisicts and mathematicians