John L. Stensby

Professor, Electrical and Computer Engineering

Ph.D., Texas A&M University, 1981

(205) 890-6258

stensby@ebs330.eb.uah.edu

Research Areas Communication Systems and Signal Processing

Dr. Stensby's research in phase coherent systems has produced fundamental contributions on the problem of false lock in phase locked loops (PLLs). He has developed a new lock detector which significantly out performs the classical phase quadrature PLL lock detector (which is a first-order approximation of the new detector).

Dr. Stensby is also active in the area of satellite communication systems. He has constructed a ground station capable of effecting computer file transfer with existing low earth orbit store-and-forward packet radio satellites.

Class Notes on the Web! Click on the following links to download some really great class notes.

EE420/500/603 Class Notes on Probability and Random Processes

EE448/528/628 Class Notes on Numerical Linear Algebra

EE426/506 Class Notes on Communication Systems

Selected Reviewed Publications

"Phase-Locked Loops, Theory and Applications", CRC Press, 1997 (description follows below).

"The Half-plane Pull-in Range of a Second-order Phase-locked Loop," Bassam Harb and John Stensby, J. Franklin Inst., 333(B), No. 2, pp. 191-199 (1996).

"Computing the Half-Plane Pull-In Range of Second-Order PLLS," J. Stensby and Bassam Harb, Electronics Letters, 31, No. 11 (1995).

"Modulated High Sensitivity Infrared Polarimeter," US Patent #5,210,417, J. Grisham, F. Clark, C. Christensen, J. Stensby (1993).

"Saddle Node Bifurcation at a Nonhyperbolic Limit Cycle in a Phase Locked Loop," J. Stensby, J. Franklin Institute, 330, No. 5, pp. 775-786 (1993).

"Lock Detection in Phase Locked Loops," J. Stensby, SIAM J. on Appl. Mathematics, 52, No. 5 (1992).

"A Parametrically Driven PLL Lock Detector," J. Stensby, J. Franklin Institute, 329, No. 2 (1992).

"A General Second-Order Lock Detector for Type I PLLs,"Southeastcon '92, Birmingham, AL (1992).

"Multiple Fourier Transform Generation for Coherent Optical Correlators," J. Upatnieks, J. Abshier, C. Christensen and J. Stensby, Appl. Optics, 29, No. 11 (1990).

"False Lock and Bifurcation in Costas Loops," J. Stensby, SIAM J. on Appl. Mathematics, 49, No. 2 (1989).

"Optimum 8087 Stack Use in Solving Equations," J. Stensby, Access - J. Micro-computer Applications, 8, No. 1, Issue 43 (1989).

"A Discrete Extended Kalman Filter for Radar Pointing Error Reduction," M. Greene, J. Walls and J. Stensby, IEEE Trans. on Aerospace and Electronic Systems, AES-24, No. 1 (1988).

"Further Results on Radar Painting Error Reduction Using Discrete Extended Kalman Filtering," M. Greene, J. Walls, J. Stensby, IEEE Transactions on Aerospace and Electronic Systems, AES-24, No. 6 (1988).

"False Lock and Bifurcation in the Phase Locked Loop," J. Stensby, SIAM J. on Appl. Mathematics, 47, No. 6 (1987).

"Radar Target Pointing Error Reduction Using Extended Kalman Filtering," J. Stensby and M. Greene, IEEE Trans. on Aerospace and Electronic Systems, AES-23, No. 3 (1987).

"On the PLL Spectral Purity Problem," J. Stensby, IEEE Transactions on Circuits and Systems, CAS-30, No. 4 (April 1983).

"Floquet Theory of the Linear Synchronous Machine," J.L. Stensby, R.K Cavin, C.W. Brice, J. Franklin Institute, 316, No. 1 (July 1983).

J.L. Stensby, Phase-Locked Loops, Theory and Application, CRC Press, Bocca Raton, 1997.

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Contents

Index of Important Symbols

Preface

Part I - Elementary Theory and Applications

Chapter 1 Introduction

1.1 The Phase and Frequency of a Signal Relative to a Reference

1.2 A Generic Problem

1.3 The Phase-Locked Loop

1.4 Basic Applications

1.4.1 Coherent Demodulation of Amplitude Modulated Signals

1.4.2 Frequency Synthesis

1.4.3 Demodulation of BPSK Signals

1.4.4 Phase-Locked Receivers

1.5 Phase-Locked Loop Literature

1.6 Topical Outline of the Text

Chapter 2 Modeling the Phase-Locked Loop

2.1 Modeling PLL Components

2.1.1 Modeling the Analog Phase Detector

2.1.2 Modeling the Loop Filter

2.1.3 Modeling the Voltage Controlled Oscillator

2.2 Modeling the Nonlinear PLL

2.2.1 PLL Description Based on an Ordinary Differential Equation

2.2.2 PLL Description Based on a First-Order Nonlinear System

2.2.3 PLL Description Based on an Integral Differential Equation

2.3 Modeling the Linear PLL

2.4 Modeling a PLL With an Angle Modulated Reference Source

2.5 Modeling a PLL With a Noisy Reference Source

2.5.1 Reference Signal Corrupted By Additive Noise

2.5.2 Output of the Sinusoidal Phase Detector

2.5.3 Nonlinear Model of the PLL With a Noisy Reference

2.5.4 Linear Model of the PLL With a Noisy Reference

2.6 Modeling the Limiter Phase Detector

2.6.1 Hard Limiting the Input and VCO Signals

2.6.2 The Output of the Limiter Phase Detector

2.6.3 Splitting the Detector's Output into Control and Noise Components

2.6.4 Practical Application of the Limiter Phase Detector Model

2.6.5 Example: Limiter Phase Detector with Sinusoidal g(f )

2.6.6 Example: Limiter Phase Detector with Triangular g(f )

2.7 Modeling the Long Loop

2.7.1 Baseband Model of the Long Loop

Appendix 2.5.1 Narrowband Signals and Systems

2.5.1.1 Modeling Bandpass Signals and Systems

2.5.1.2 Lowpass Equivalent Signals and Systems

2.5.1.3 Symmetrical Bandpass Filter

2.5.1.4 Phase and Group Delays of a Bandpass System

2.5.1.5 Bandpass Input/Output

Appendix 2.5.2 Narrowband Noise

2.5.2.1 Relationship Between R_h , Autocorrelations of h_c and h_s , and the Cross Correlation of h_c and h_s

2.5.2.2 Symmetrical Bandpass Processes

Appendix 2.6.1 Evaluation of E[cosq_n] and E[sinq_n] for the Gaussian Noise Case

Chapter 3 Linear Analysis of Common First and Second Order PLLs

3.1 The First-Order PLL

3.1.1 Closed Loop Transfer Function

3.1.2 Transient and Steady-State Tracking Errors

3.1.3 Noise Equivalent Bandwidth

3.1.4 Summary of the First-Order PLL Linear Model

3.2 The Second-Order PLL with a Perfect Integrator

3.2.1 Transfer Functions

3.2.2 Loop Stability

3.2.3 Transient and Steady-State Tracking Errors

3.2.4 Noise Equivalent Bandwidth

3.2.5 Summary for the Second-Order PLL Containing a Perfect Integrator Loop Filter

3.3 The Second-Order PLL with Imperfect Integrator

3.3.1 Transfer Functions

3.3.2 Loop Stability

3.3.3 Transient and Steady-State Tracking Errors

3.3.4 Noise-Equivalent Bandwidth

3.3.5 Summary for the Second-Order PLL Based on the Imperfect Integrator Loop Filter

Chapter 4 Phase-Locked Loop Components and Technologies

4.1 Phase Detectors - Analog and Digital

4.1.1 Integrated Circuit Four Quadrant Analog Multipliers

4.1.2 Diode-Ring Mixer

4.1.3 Exclusive OR Gate

4.1.4 RS Flip Flop

4.1.5 Sequential Phase/Frequency Detectors

4.2 Loop Filters

4.3 Voltage Controlled Oscillators

4.3.1 Voltage Controlled Crystal Oscillators

4.3.2 RC Multivibrators

4.4 Lock Detection

Part II - Nonlinear PLL Analysis

Chapter 5 Nonlinear PLL Behavior in the Absence of Noise

5.1 First-Order PLL With Constant Frequency Reference

5.1.1 Phase Plane Analysis of a First-Order PLL

5.1.2 Phase Acquisition in a First-Order PLL

5.2 A Second-Order PLL Using a Perfect Integrator

5.2.1 Stable and Unstable Equilibrium Points

5.2.2 A Phase Plane Analysis of the Perfect Integrator Case

5.2.3 Pull-In Properties of a Second-Order Type II PLL

5.3 A Second-Order PLL Containing an Imperfect Integrator Loop Filter

5.3.1 Stable and Unstable Equilibrium Points

5.3.2 Phase Plane Structure Dependent on w_D - The Values W_p, W₂, and W_h

5.3.3 The High-Gain Case

5.3.4 The Low-Gain Case

5.3.5 General Phase Plane Characteristics for the Low-Gain Case

5.3.6 Computing the Separatrix Cycle and Determining its Stability

5.4 Effects of IF Filtering on the Long Loop

Appendix 5.2.1 Pull-In Time For a Second-Order PLL

Appendix 5.3.1 Pull-In Range of a Second-Order PLL

Appendix 5.3.2 Computation of Separatrices For a Second-Order PLL

Appendix 5.3.3 The Separatrix Cycle of a Second-Order PLL

5.3.3.1 Computing the Separatrix Cycle

5.3.3.2 Stability of the Separatrix Cycle

5.3.3.3 Bifurcation of a Periodic Limit Cycle from the Separatrix Cycle

Chapter 6 Stochastic Methods For the Nonlinear PLL Model

6.1 The Random Walk - A Simple Markov Process

6.1.1 The Wiener Process As a Limit of the Random Walk

6.1.2 The Diffusion Equation For the Transition Density Function

6.1.3 An Absorbing Boundary On the Random Walk

6.1.4 An Absorbing Boundary On the Wiener Process

6.1.5 Gaussian White Noise as the Formal Derivative of the Wiener Process

6.2 The First-Order Markov Process

6.2.1 An Important Application of Markov Processes

6.2.2 The Chapman-Kolmogorov Equation

6.2.3 The One-Dimensional Kramers-Moyal Expansion

6.2.4 The One-Dimensional Fokker-Planck Equation

6.2.5 Transition Density Function

6.2.6 Natural, Periodic and Absorbing Boundary Conditions

6.2.7 Steady-State Solution to the Fokker-Planck Equation

6.2.8 The One-Dimensional First Passage Time Problem

6.2.9 The Distribution and Density of the First Passage Time Random Variable

6.2.10 The Expected Value of the First Passage Time Random Variable

6.2.11 Ratio of Boundary Absorption Rates

6.3 The Vector Markov Process

6.3.1 The n-Dimensional Kramers-Moyal Expansion

6.3.2 The n-Dimensional Fokker-Planck Equation

6.3.3 A Simple Example

Chapter 7 Noise In the Nonlinear PLL Model

7.1 Qualitative Nature of and Models for the Phase Error

7.1.1 Modulo-2p Phase Error Model

7.1.2 Bistable Cyclic Model

7.1.3 Generalizations: The Multistable/m-Attractor Cyclic Model

7.1.4 Qualitative Properties of the Multistable Model for the Moderate Noise Case

7.1.5 Phase Error Model Containing Absorbing Boundaries

7.2 Noise in the First-Order PLL

7.2.1 Modeling the First-Order PLL with a Noisy Reference

7.2.2 Fokker-Planck Equation for the Time-Normalized, First-Order PLL Model

7.2.3 Steady-State Density for the Modulo-2p Phase Error

7.2.4 Steady-State Distribution for the Modulo-2p Phase Error

7.2.5 Mean of the Steady-State, Modulo-2p Phase Error

7.2.6 Variance of the Steady-State, Modulo-2p Phase Error

7.2.7 Noise-Induced Cycle Slips

7.3 Noise in Second - Order PLLs

7.3.1 Nonlinear Model for a Second-Order PLL with a Noisy Reference

7.3.2 Fokker-Planck Equation for the Second-Order PLL

7.3.3 A Simple Example: A PLL With an RC Low-Pass Loop Filter

7.3.4 Equation of Flow in the f₁ Direction

7.3.5 Conditional Expectation Method for Approximating p₁(f₁)

7.3.6 Approximating the Conditional Expectation

7.3.7 Approximating p₁(f₁) for the Perfect Integrator Case

7.3.8 Noise Induced Cycle Slips in Second-Order PLLs

7.3.9 Average Cycle Slip Rate in the Forward Direction

7.3.10 Ratio of Cycle Slip Rates for the Second-Order PLL

Chapter 8 Numerical Methods for Noise Analysis in the Nonlinear PLL Model

8.1 Computing an Approximation to Steady-State p₁(f₁,Y)

8.1.1 Expansion of the Joint Density in a Complete Orthonormal Set

8.1.2 A Coupled System of Differential Equations for the p_k

8.1.3 Computing a 2p-Periodic Solution of the Coupled System

8.1.4 The Case n ³ 2

8.2 Approximating p₁(f₁) for a PLL Containing an RC Low-Pass Loop Filter

8.2.1 Transformation of the Fokker-Planck Operator

8.2.2 Form of Series Expansion

8.2.3 Truncation of the Infinite Series

8.2.4 Computing the Periodic Solution and Approximating p₁(f₁)

8.2.5 Practical Limitations of the Algorithm

8.3 Approximating p₁(f₁) for a PLL Containing a Perfect Integrator Loop Filter

8.3.1 Transformation of the Fokker-Planck Operator

8.3.2 Development of the Coupled System of Differential Equations

8.3.3 Computing an Approximation to the Marginal Density p₁(f₁)

8.4 Modeling Cycle Slips as State Transitions of a Markov Jump Process

8.4.1 Modeling the Cycle Slip Problem as a Finite State Markov Jump Process

8.4.2 The Transition Rate Matrix

8.4.3 Physical Interpretation and Properties of the Transition Rates

8.4.4 The Dynamics of State Occupancy

8.4.5 Eigenvalues and Eigenfunctions of the Fokker-Planck Operator

8.4.6 Eigenvalues and Eigenfunctions Under Moderate Noise Conditions

8.4.7 Calculation of the Transition Rates a_ij

8.4.8 An m-Attractor Markov Jump Model with a Cyclic Structure

8.4.9 Relationship of the Cyclic Jump Model to an Unrestricted Jump Model

8.4.10 Computing the Slip Rates in the "Coarse-Grained", m-Attractor, Cyclic State Model

8.5 The First-Order PLL as a Simple Example

8.5.1 The Eigenvalue Problem Formulated in Terms of a First-Order System of Differential Equations

8.5.2 Decomposition of the Vector Space of 2pm-Periodic Functions

8.5.3 An Algorithm for Computing the l₀^(k)

8.5.4 Numerical Results

8.6 Eigenvalue Method Applied to a Second-Order System

8.6.1 Development of the System Model and Fokker-Planck Equation

8.6.2 Development of the Coupled System of Differential Equations

8.6.3 Algorithm for Computing the p_n, h_n and l_k

8.6.4 Numerical Results

Appendix 8.1.1 Hermite Polynomials

8.1.1.1 Definitions and Elementary Properties

8.1.1.2 Weighted Hermite Polynomials as Eigenfunctions of a Differential Operator

Appendix 8.4.1 Least Positive Residue Function

Appendix 8.5.1 Computing Eigenvalues of the Fokker-Planck Operator

References

Index

Extramural Support

US Army Missile Command, DEPSCOR, AFOSR

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