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

CONTENTS

Preface

Acknowledgements

XIII

xvii

PART 1

Chapter 1 Introduction

1

3 1.1. Topical Outline 3 1.2. Possible Approaches 8 1.3. Implementation of Synchronizers 1.4. Outline 14 1.5. References to Volume 1 16

11

PART 2 19

Chapter 2 Phase-Locked Loop Fundamentals 21 2.1. Automatic Phase Control 21 2.2. The Phase-Locked Loop 23 2.3. The Linear Approximation 26

2.3.1. Basic Transfer Functions 27 2.3.2. Steady-State Phase Error 28 2.3.3. Design of Feedback Systems Using the Bode

Diagram 30 2.4. Second-Order Phase-Locked Loop 35

2.4.1. Transfer Functions 35 2.4.2. Stability Considerations 42 2.4.3. State Variable Description 44 2.4.4. Transient Loop Response Under Linear

Conditions 48

VII

viii CONTENTS

2.5. Third-Order, Type-3 Phase-Locked Loop 54 2.5.1. Transfer Functions 54 2.5.2. Stability Considerations 55 2.5.3. State Variable Description 57 2.5.4. Transient Loop Response Under Linear

Conditions 59 2.5.5. Comparison of the Transient Response of a

Second-Order and a Third-Order Loop 60 2.6. Phase Detectors 66

2.6.1. Multiplier Type Phase Detectors 67 2.6.2. Sequential Logic Phase Detectors 69

2.7. Charge Pump Phase-Locked Loops 79 2.7.1. Principles of Charge Pump Phase-Locked

Loops 79 2.7.2. Quasi-Continuous Analysis of the Charge

Pump Phase-Locked Loop 82 2.7.3. Phase Accuracy of a Practical Second-Order

Charge Pump Phase-Locked Loop 89 2.7.4. Exact Analysis of a Second-Order Charge

Pump Phase-Locked Loop 91 2.8. Random Phase and Frequency Modulation 95

Chapter 3 Phase-Locked Loop Tracking Performance in the Presence of Noise 99 3.1. Narrowband Gaussian Noise Process 99 3.2. Phase Detector Operation in the Presence of Additive

Noise 106 3.2.1. Sinusoidal Phase Detector Characteristics 106 3.2.2. Nonsinusoidal Phase Detector

Characteristics 110 3.3. Additive Noise in Linear Model 123 3.4. State Variable Equations in the Presence of Additive

Noise 130 3.5. Time and Frequency Stability of Signal

Generators 133 3.5.1. Characterization of Time Properties 134 3.5.2. Standard Parameters Characterizing Random

Fluctuations of Oscillators 140 3.5.3. Time Domain to Frequency Domain

Interconnections 141 3.5.4. Frequency Domain Model of Oscillator Phase

Noise 144 3.6. Effect of Oscillator Phase Noise on the Phase-Locked

Loop Tracking Performance 147

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 n'(t, @(t)) 155

Appendix 3.2B. Complex Envelope Representation of Signals 157

Chapter 4 Unaided Acquisition 163 4.1. Introduction 163 4.2. First-Order Loop 163

4.2.1. Phase Acquisition in the Absence of Noise 163

4.2.2. Phase Acquisition in the Presence of Additive Noise 171

4.3. Second-Order Loop 176 4.3.1. Frequency Acquisition 176

4.4. 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 5.1. Phase Acquisition 195 5.2. Frequency Acquisition 199

5.2.1. Sweeping 199 5.2.2. Frequency Discriminator Aided

Acquisition 211 5.2.3. Acquisition Aid Using a Nonlinearity 218

195

Chapter 6 Loop Threshold 227 6.1. 6.2.

6.3.

231

Introduction 227 Understanding Cycle Slips 229 6.2.1. Loops with Small Damping Factors 6.2.2. Overdamped Loops ( £ > 1 ) 239 6.2.3. Frequency Detuning 242 Cycle Slip Statistics for a Wideband Noise Disturbance 246 6.3.1. Measurement of Cycle Slips: Experimental

Configuration 246 Experimental Results 248 Theoretical Results 251 Loop Parameters for Maximum Meantime between Cycle Slips 255

6.3.2. 6.3.3. 6.3.4.

X CONTENTS

PART3 261

Chapter 7 Amplitude Control 263 7.1.

7.2,

7.2.3. 7.2.4. 7.2.5. 7.2.6. 7.2.7.

7.2.8.

Appendix 7.

Appendix 7.

Appendix 7.1С.

Limiters 263 7.1.1. Bandpass Limiters 263 7.1.2. Limiter Followed by a Phase Detector Automatic Gain Control Circuits 273 7.2.1. Gain Controlled Amplifiers 274 7.2.2. Detectors 276

The Automatic Gain Control Loop 281 Steady-State Analysis 283 Linear Approximations 286 Exact Dynamics 290 Acquisition of Coherent Automatic Gain Control and Phase-Locked Loop 294 Miscellaneous Modifications of Automatic Gain Control 295

1A. Series Representation of the Hard Limiter Output Signal 297

IB. Expected Values E[g0(4> + 0 J ] , Е[82

0(Ф + в„)} 298 The Modified Bessel Functions /„(*) 302

267

Chapter 8 Automatic Frequency Control 305 8.1. Introduction 305 8.2. Structures of Frequency Detectors 305

8.2.1. Optimal Frequency Estimator 305 8.2.2. Suboptimal Frequency Estimation

Methods 310 Performance in the Presence of Additive Noise 316 8.3.

Appendix 8.2A.

Appendix 8.2B. Appendix 8.ЗА.

Maximum Likelihood (ML) Parameter Estimation 326 Optimal Phase Estimator 329 Evaluation of Gaussian Moments 330

PART 4 333

Chapter 9 Brief Review of Some Mathematical Fundamentals 9.1. Introduction 335 9.2. Stochastic Differential Equations 336 9.3. Fokker-Planck Equation 338

9.3.1. Derivation of the Fokker-Planck Equation 338

9.3.2. Intensity Coefficients 341

335

CONTENTS XI

9.3.3. Physical Interpretation of the Fokker-Planck Equation 350

9.3.4. N-Dimensional Fokker-Planck Equation 351 9.3.5. Formal Derivation of the Intensity Coefficients

for a Vector Process 353 9.3.6. Initial and Boundary Conditions 356 9.3.7. Disturbance of Systems by Impulsive

Noise 359

Chapter 10 Relaxation Times, Meantime Between Cycle Slips, Transition Rates, and Eigenvalues of Fokker-Planck Operators 363 10.1. Introduction 363 10.2. Modulo 2тг Phase Error Process 365 10.3. Renewal Process 366 10.4. Bistable and Multistable Cyclic Models 369 10.5. "Coarse-Grained" Model 371

Chapter 11 Renewal Process Approach 373 11.1. First-Order Systems with Periodic Phase Detector

Characteristic 373 11.1.1. Modeling the Phase Error as a Renewal

Process 373 11.1.2. Probability Laws of the Single

Process 376 11.1.3. Basic Recurrence Relations for the

Renewal Process 380 11.1.4. Modified Fokker-Planck Equation of the

Renewal Process 383 11.1.5. Equations for the Steady State 385 11.1.6. Stationary Phase Error Distribution,

Meantime between Cycle Slips and Mean Cycle Slip Rate 388

11.1.7. Time-Dependent Solution of the Fokker-Planck Equation of the Single Process 398

11.1.8. Distribution of Renewal Epochs t„ and Associated Jumps rjn 401

11.1.9. Time-Dependent Probability Density Function of the Renewal Process 406

11.1.10. Numerical Example: First-Order Phase-Locked Loop 407

11.2. Higher Order Systems with Periodic Phase Detector Characteristic 412 11.2.1. Modeling of the Phase Error as a Vector

Renewal Process 412

xii CONTENTS

11.2.2. Probability Laws of the Single Process 413 11.2.3. Basic Recurrence Relations of the Vector

Renewal Process 416 11.2.4. Modified Fokker-Planck Equation of the

Renewal Process 420 11.2.5. Equations for the Steady State 423 11.2.6. Meantime between Cycle Slips 425 11.2.7. Approximative Use of Renewal Theory in

the Strict Sense 427 11.2.8. Stability, Persistance, and Steady-State

Distribution 429 11.3. Systems with Aperiodic Phase Detector

Characteristic 432 11.3.1. Mathematical Model of the Delay-Locked

Loop 432 11.3.2. Fokker-Planck Equation 437 11.3.3. Modeling the Operation of the Delay-

Locked Loop as a Renewal Process 439 11.3.4. Fokker-Planck Equation for a Process with

Distributed Sinks 441 11.3.5. 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 455 12.1. Eigenfunctions of the Operator L 456 12.2. Moderate Noise and Coarse-Graining to a

Markovian Jump Process 457 12.3. M-Attractor Cyclic Models 461 12.4. Numerical Computation of the Eigenvalues 466 12.5. The Matrix A for First-Order Systems (M = 2

Attractors) 467 12.6. Decomposition of the Matrix A and Geometric

Interpretations (M = 2 Attractors) 475 12.7. The Matrix A for the Mh Order System 480 12.8. Decomposition of the Matrix A for an M-Attractor

Model of an Nth Order System 485 12.9. Numerical Examples 488 Appendix 12.A. A Brief Account on Weak Noise

Theories 498

Epilogue: Unexplored Topics 501

Index 505