Noise, Oscillators and Algebraic Randomness

✍ Scribed by Michel Planat

• Publisher: Springer
• Year: 2000
• Tongue: English
• Leaves: 409
• Edition: 2000
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

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