Synchronization in Digital Communications

Transcription

1 Synchronization in Digital Communications Volume 1 Phase-, Frequency-Locked Loops, and Amplitude Control Heinrich Meyr Aachen University of Technology (RWTH) Gerd Ascheid CADIS GmbH, Aachen WILEY A Wiley-lnterscience Publication JOHN WILEY & SONS New York Chicester Brisbane Toronto Singapore

2 CONTENTS Preface Acknowledgements XIII xvii PART 1 1 Chapter 1 Introduction 1.1. Topical Outline Possible Approaches Implementation of Synchronizers 1.4. Outline References to Volume PART 2 19 Chapter 2 Phase-Locked Loop Fundamentals Automatic Phase Control The Phase-Locked Loop The Linear Approximation Basic Transfer Functions Steady-State Phase Error Design of Feedback Systems Using the Bode Diagram Second-Order Phase-Locked Loop Transfer Functions Stability Considerations State Variable Description Transient Loop Response Under Linear Conditions 48 VII

3 viii CONTENTS 2.5. Third-Order, Type-3 Phase-Locked Loop Transfer Functions Stability Considerations State Variable Description Transient Loop Response Under Linear Conditions Comparison of the Transient Response of a Second-Order and a Third-Order Loop Phase Detectors Multiplier Type Phase Detectors Sequential Logic Phase Detectors Charge Pump Phase-Locked Loops Principles of Charge Pump Phase-Locked Loops Quasi-Continuous Analysis of the Charge Pump Phase-Locked Loop Phase Accuracy of a Practical Second-Order Charge Pump Phase-Locked Loop Exact Analysis of a Second-Order Charge Pump Phase-Locked Loop Random Phase and Frequency Modulation 95 Chapter 3 Phase-Locked Loop Tracking Performance in the Presence of Noise Narrowband Gaussian Noise Process Phase Detector Operation in the Presence of Additive Noise Sinusoidal Phase Detector Characteristics Nonsinusoidal Phase Detector Characteristics Additive Noise in Linear Model State Variable Equations in the Presence of Additive Noise Time and Frequency Stability of Signal Generators Characterization of Time Properties Standard Parameters Characterizing Random Fluctuations of Oscillators Time Domain to Frequency Domain Interconnections Frequency Domain Model of Oscillator Phase Noise Effect of Oscillator Phase Noise on the Phase-Locked Loop Tracking Performance 147

4 CONTENTS JX 3.7. Optimization of the Tracking Performance in the Presence of Noise 150 Appendix 3.2A. The Closed-Loop Phase-Locked Loop with Multiplier-Type Phase Detector and the Exact Noise Model 155 Appendix 3.2B. Complex Envelope Representation of Signals 157 Chapter 4 Unaided Acquisition Introduction First-Order Loop Phase Acquisition in the Absence of Noise Phase Acquisition in the Presence of Additive Noise Second-Order Loop Frequency Acquisition Generalized Study of Frequency Acquisition Failure 188 Appendix 4.2A. Phase Acquisition Probability of a First- Order Phase-Locked Loop with Sinusoidal Phase Detector 193 Chapter 5 Aided Acquisition Phase Acquisition Frequency Acquisition Sweeping Frequency Discriminator Aided Acquisition Acquisition Aid Using a Nonlinearity 218 Chapter 6 Loop Threshold Introduction Understanding Cycle Slips Loops with Small Damping Factors Overdamped Loops ( >1) Frequency Detuning Cycle Slip Statistics for a Wideband Noise Disturbance Measurement of Cycle Slips: Experimental Configuration Experimental Results Theoretical Results Loop Parameters for Maximum Meantime between Cycle Slips 255

5 X CONTENTS PART3 261 Chapter 7 Amplitude Control Limiters Bandpass Limiters Limiter Followed by a Phase Detector , Automatic Gain Control Circuits Gain Controlled Amplifiers Detectors The Automatic Gain Control Loop Steady-State Analysis Linear Approximations Exact Dynamics Acquisition of Coherent Automatic Gain Control and Phase-Locked Loop Miscellaneous Modifications of Automatic Gain Control 295 Appendix 7. 1A. Series Representation of the Hard Appendix 7.1С. Limiter Output Signal 297 Expected Values E[g 0 (4> + 0J], Е[8 2 0(Ф + в )} 298 The Modified Bessel Functions / (*) 302 Appendix 7. IB. Chapter 8 Automatic Frequency Control Introduction Structures of Frequency Detectors Optimal Frequency Estimator Suboptimal Frequency Estimation Methods Performance in the Presence of Additive Noise 316 Appendix 8.2A. Maximum Likelihood (ML) Parameter Estimation 326 Appendix 8.2B. Optimal Phase Estimator 329 Appendix 8.ЗА. Evaluation of Gaussian Moments 330 PART Chapter 9 Brief Review of Some Mathematical Fundamentals 9.1. Introduction Stochastic Differential Equations Fokker-Planck Equation Derivation of the Fokker-Planck Equation Intensity Coefficients

6 CONTENTS XI Physical Interpretation of the Fokker-Planck Equation N-Dimensional Fokker-Planck Equation Formal Derivation of the Intensity Coefficients for a Vector Process Initial and Boundary Conditions Disturbance of Systems by Impulsive Noise 359 Chapter 10 Relaxation Times, Meantime Between Cycle Slips, Transition Rates, and Eigenvalues of Fokker-Planck Operators Introduction Modulo 2тг Phase Error Process Renewal Process Bistable and Multistable Cyclic Models "Coarse-Grained" Model 371 Chapter 11 Renewal Process Approach First-Order Systems with Periodic Phase Detector Characteristic Modeling the Phase Error as a Renewal Process Probability Laws of the Single Process Basic Recurrence Relations for the Renewal Process Modified Fokker-Planck Equation of the Renewal Process Equations for the Steady State Stationary Phase Error Distribution, Meantime between Cycle Slips and Mean Cycle Slip Rate Time-Dependent Solution of the Fokker- Planck Equation of the Single Process Distribution of Renewal Epochs t and Associated Jumps rj n Time-Dependent Probability Density Function of the Renewal Process Numerical Example: First-Order Phase- Locked Loop Higher Order Systems with Periodic Phase Detector Characteristic Modeling of the Phase Error as a Vector Renewal Process 412

7 xii CONTENTS Probability Laws of the Single Process Basic Recurrence Relations of the Vector Renewal Process Modified Fokker-Planck Equation of the Renewal Process Equations for the Steady State Meantime between Cycle Slips Approximative Use of Renewal Theory in the Strict Sense Stability, Persistance, and Steady-State Distribution Systems with Aperiodic Phase Detector Characteristic Mathematical Model of the Delay-Locked Loop Fokker-Planck Equation Modeling the Operation of the Delay- Locked Loop as a Renewal Process Fokker-Planck Equation for a Process with Distributed Sinks Numerical Example: First-Order Delay- Locked Loop 445 Appendix 11.1 A. Normalized Stochastic Differential Equation of First-Order Systems with State-Dependent Noise Intensity 452 Chapter 12 The Matrix Eigenvalue Approach Eigenfunctions of the Operator L Moderate Noise and Coarse-Graining to a Markovian Jump Process M-Attractor Cyclic Models Numerical Computation of the Eigenvalues The Matrix A for First-Order Systems (M = 2 Attractors) Decomposition of the Matrix A and Geometric Interpretations (M = 2 Attractors) The Matrix A for the Mh Order System Decomposition of the Matrix A for an M-Attractor Model of an Nth Order System Numerical Examples 488 Appendix 12.A. A Brief Account on Weak Noise Theories 498 Epilogue: Unexplored Topics 501 Index 505