HANDBOOK OF HIGH-ORDER OPTICAL MODULATIONS : signal and spectra for coherent multi-terabit optical... fiber transmission

This book highlights many fundamental aspects of optical fiber transmission engineering while also focusing on current state of the art applications and working examples of digital coherent optical communications. Major engineering themes are reviewed and analyzed in this book, including spectral and time-domain characteristics of multi-level pseudo-random PAM signals, optical QAM and SSB complex modulations and impulse response engineering of linear amplifiers used in next-generation Gbaud transmission systems. This book is balanced between theoretical and numerical simulation approaches, showing numerous working examples developed in Matlab. Presents an in-depth analysis of pseudo-random multi-level signals and high-order complex modulations to support coherent terabit transmission systems; Provides a unified approach to challenging engineering issues encountered in the design of Giga-baud coherent optical transmission systems using high-order complex modulation formats; Reviews engineering themes and provides in-depth analysis, modeling and quantitative examples and solutions of state of the art and future applications. Preface Acknowledgments Contents Chapter 1: Introduction to Multi-terabit Optical Transmission Systems 1 Introduction 2 Challenges in Terabit Optical Transmission 2.1 The Intrinsic Spectral Efficiency 2.2 Optical Modulation Factor 2.3 Intrinsic Spectral Efficiency of the Nyquist Signal 2.4 Subcarriers, Spectral Efficiency, and Symbol Rate 2.5 Optical Modulation Performance Matrix 2.6 Coherent System Design Trade-Off 2.6.1 The Modulation Order 2.6.2 Optical Subcarriers 2.6.3 The System Complexity Factor 2.6.4 Chasing Higher Baud Rate 2.6.5 DSP Drawbacks 3 Direct Detection Transmission Systems 3.1 IM-DD: OOK 3.2 IM-DD: ODB 3.3 PM-DD: DPSK 3.4 PM-DD: DQPSK 3.5 CWDM 100GBASE LR4-ER4 3.6 Needs for Limiting Driver and TIA 4 Overview of Coherent Transmission Systems 4.1 64Gb/s PM-PSK 4.2 128Gb/s PM-QPSK 4.3 128Gb/s PM-QDB 4.4 256Gb/s PM-16QAM Glossary References Chapter 2: Theory of Pseudo-random PAM Sequences and Power Spectra 1 Introduction 1.1 NRZ-to-PAM Encoder 1.2 PAM-to-NRZ Decoder 2 Cyclostationary PAM Signals 2.1 Normalized PAM Levels 2.2 Shifted Cyclostationary Processes 2.3 Autocorrelation and Power Spectral Density 2.3.1 White Noise PAM Sequence 3 Normalized White Noise PAM Sequences 3.1 Mean 3.2 Variance 3.3 Autocorrelation 3.4 Power Spectral Density 3.5 Symbol Rate 3.6 Average Power 3.6.1 Ideal Square Pulse 4 Pseudo-random PAM Sequences 4.1 Level Completeness of the PAM Sequence 4.2 Probability of Levels 4.3 Statistical Assumptions 4.4 Mean 4.5 Variance 4.6 Autocorrelation 4.6.1 Matrix of Joints Occurrences 4.6.2 Products Involving Inner Levels 4.6.3 Products Between the Upper Level and Any Inner Level 4.6.4 Products Between the Lower Level and Any Inner Level 4.6.5 Products Between the Outer Levels 4.6.6 Autocorrelation Lineshape Matlab Function MJO Matlab Function PAM_OSA Matlab Function PRBS 4.7 Temporal Averages 4.7.1 Mean 4.7.2 Autocorrelation Outside the Peak Region: kM - h L = log2n, k Inside the Peak Region: kM - h < L = log2n, k Limit for Indefinitely Long Periods Autocorrelation Algorithm in Matlab 4.7.3 Calculation of the Autocorrelation Inside the Peak Region PAM4 Sequence PAM8 Sequence 4.7.4 A Unified Equation for the Autocorrelation PAM16 Sequence PAM32 Sequence 4.7.5 A Recursive Equation for the Autocorrelation 4.7.6 Matlab Function: PAM_AUTOCORRELATION 4.7.7 Matlab Function: K_FACTOR 4.8 Power Spectrum 4.8.1 Spectral Shaping 4.8.2 PAM2 Sequence 4.8.3 Matlab Script: PAM_autocorrelation_plot 4.9 PAM Sequences with Finite Number of Periods 4.9.1 PAM2 Sequence Matlab Script: PAM2_spectrum_ analytical 4.10 The Master Equation (OSA Procedure) 4.10.1 Spectral Amplitudes Singularities 4.10.2 Sequence of Infinite Length 4.10.3 Sequence of Finite Length 4.10.4 PAM4 Sequence Autocorrelation Spectral Amplitudes Spectral Shaping Power Spectrum: Infinite Length Sequences Power Spectrum: Finite Length Sequences 4.10.5 PAM8 Sequence Autocorrelation Spectral Amplitudes Spectral Shaping Power Spectrum: Infinite Length Sequences Power Spectrum: Finite Length Sequences 4.10.6 PAM16 Sequence Autocorrelation Spectral Amplitudes Spectral Shaping Power Spectrum: Infinite Length Sequences Power Spectrum: Finite Length Sequences 4.11 Spectral Comparison 5 PRBS to PAM Conversion Algorithms 5.1 Incomplete Sequences 5.2 Complete Sequences 5.3 The Block Sequential Algorithm (BSA) 5.3.1 Matlab Function: PAM_BSA 5.4 The One-Step Sequential Algorithm (OSA) 5.4.1 Matlab Function PAM_OSA 5.5 Comparison Between OSA and BSA 5.5.1 Data Patterns 5.5.2 Autocorrelations 5.5.3 Power Spectra 5.5.4 Explaining the BSA Power Spectrum 5.6 The DAC Algorithm (DAC) 5.6.1 PAM Sequences 5.6.2 Autocorrelation 5.7 Level Completeness of the PAM Sequence 5.8 Transition Completeness of the PAM Sequence 5.8.1 The Matlab Color Scale Transition Matrix Representation 5.8.2 Matlab Script: PAM_levels_transitions 5.8.3 Matlab Function: PAM_TRANSITION References Chapter 3: Statistical Modeling of PAM Signals and Power Spectra 1 Statistics Properties of PAM Signals 1.1 Signal Model 1.2 Mean 1.3 Autocorrelation 1.4 Power Spectrum 2 The Harmonic Series Expansion (HSE) 2.1 Method 1 2.2 Method 2 3 Nyquist Pulse and Spectrum 3.1 White Noise PAM Signals 3.1.1 Power Spectrum 3.1.2 Average Power 3.1.3 Power Bandwidths Full-Width Power Bandwidth FWHM Power Bandwidth Bandwidths Comparison 3.1.4 Simulations Nyquist Pulses White Noise PAM2 Nyquist Signal White Noise PAM4 Nyquist Signal White Noise PAM8 Nyquist Signal White Noise PAM16 Nyquist Signal 3.2 Jitter Induced Penalty 3.2.1 Probability Density Function of the Eye Closure 3.2.2 Jitter with Uniform Density 3.2.3 Jitter with Normal Density Mean Variance 3.3 Pseudo-Random PAM Signals 3.3.1 Infinite Length PAM Sequences Pseudo-Random PAM2 Nyquist Signal Pseudo-Random PAM4 Nyquist Signal Pseudo-Random PAM8 Nyquist Signal Pseudo-Random PAM16 Nyquist Signal 3.3.2 Finite Length PAM Sequences 3.3.3 Matlab Script: PAM_power_spectrum_Nyquist 4 ERF Pulse and Spectrum 4.1 Time Domain Representation 4.1.1 FWHM 4.2 Frequency Domain Representation 4.3 Pulse Simulations 4.3.1 Time Domain 4.3.2 Frequency Domain 4.4 The Frequency Scaling Theorem 4.4.1 Role of the Trapezoid Wavefront 4.5 White Noise PAM Signals 4.5.1 Power Spectrum 4.5.2 Average Power 4.5.3 Power Bandwidth Main Lobe Bandwidth FWHM Bandwidth 4.5.4 Simulations White Noise PAM2 ERF Signal White Noise PAM4 ERF Signal White Noise PAM8 ERF Signal White Noise PAM16 ERF Signal 4.6 Pseudo-Random PAM Signals 4.6.1 Infinite Length PAM Sequences Pseudo-Random PAM2 ERF Signal Pseudo-Random PAM4 ERF Signal Pseudo-Random PAM8 ERF Signal Pseudo-Random PAM16 ERF Signal 4.6.2 Finite Length PAM Sequences 5 Delta Impulse and Spectrum 5.1 White Noise PAM Signals 5.1.1 Power Spectrum 5.1.2 Average Power 5.1.3 Simulations White Noise PAM2 Delta Signal White Noise PAM4 Delta Signal White Noise PAM8 Delta Signal White Noise PAM16 Delta Signal 5.2 Pseudo-Random PAM Signals 5.2.1 Infinite Length PAM Sequences 5.2.2 Simulations Pseudo-Random PAM2 Delta Signal (Fig. 3.59 and 3.60) Pseudo-Random PAM4 Delta Signal (Fig. 3.61 and 3.62) Pseudo-Random PAM8 Delta Signal (Fig. 3.63 and 3.64) Pseudo-Random PAM16 Delta Signal (Fig. 3.65 and 3.66) 5.2.3 Finite Length PAM Sequences (OSA Procedure) 6 Comparison of Nyquist and ERF Signals 6.1 Power Bandwidths Comparison 6.1.1 Full-Width and Main Lobe Bandwidths 6.1.2 FWHM Bandwidths Bibliography Chapter 4: Theory and Modeling of Complex Optical Modulations 1 The Electromagnetic Field Model 1.1 Field Phasors 1.2 Wave Impedance 1.3 Slowly Varying Envelope Approximation 1.4 Linear Modulations 2 Complex Optical Modulations 2.1 Spectrum of the Phasor 2.1.1 Amplitude Modulation 2.1.2 Phase Modulation 2.2 Spectrum of the Electric Field 2.2.1 Amplitude Modulation 2.2.2 Phase Modulation 3 Spectral Properties 3.1 In-Phase and Quadrature Representation 3.2 Baseband Equivalents 3.3 Spectrum of the QAM Signal 3.3.1 Spectrum Generated by Two Delayed Identical PAM Signals 4 Detection of QAM Signals 4.1 In-Phase Component 4.2 Quadrature Component 4.3 Spectral Theorems 4.3.1 In-Phase Coherent Detection 4.3.2 Quadrature Coherent Detection 4.4 Coherent Demodulation of the QAM Signal 4.5 Block Diagram of the QAM Modulator 4.6 Block Diagram of the QAM Demodulator 5 Overview of Stochastic Processes 5.1 General Properties of Stationary Random Processes 5.1.1 Stationarity 5.1.2 Autocorrelation 5.1.3 Autocovariance 5.1.4 Power Spectrum 5.1.5 Average Power 5.1.6 Linear Systems 6 Statistical Properties of QAM Signals 6.1 Stationarity 6.2 Autocorrelation 6.3 Power Spectrum 6.4 Average Power 7 Statistical Properties of Scalar Modulations 7.1 Amplitude Modulation 7.1.1 Mean 7.1.2 Autocorrelation 7.1.3 The Shifted Amplitude Modulation Process 7.1.4 The Integrated Amplitude Modulation Process 7.2 Phase Modulation 7.2.1 Auxiliary Harmonic Processes 7.2.2 Mean 7.2.3 Autocorrelation 7.2.4 The Shifted Phase Modulation Process 7.2.5 The Integrated Phase Modulation Process 7.2.6 Phase Increments 7.2.7 Ergodic Phase Increments 7.2.8 Autocorrelation Phasor of the Autocorrelation 7.2.9 Power Spectrum 7.2.10 Sinusoidal Phase Modulation: Field Spectrum 7.2.11 Sinusoidal Phase Modulation: Power Spectrum 8 Simulations of Scalar Modulations 8.1 Amplitude Modulation 8.1.1 Finite Energy Signals (Pulses) ERF Pulse Nyquist pulse 8.1.2 Finite Power (Periodic) Signals AM-PAM2 (2ASK) Signal with ERF Shaping AM-PAM4 (4ASK) Signal with ERF Shaping AM-PAM16 (16ASK) Signal with ERF Shaping AM-PAM2 (2ASK) Signal with Nyquist Shaping (Roll-Off 0.10) AM-PAM2 (2ASK) Signal with Nyquist Shaping (Roll-Off 1.0) AM-PAM4 (4ASK) Signal with Nyquist Shaping (Roll-Off 0.10) AM-PAM4 (4ASK) Signal with Nyquist Shaping (Roll-Off 1.0) AM-PAM16 (16ASK) Signal with Nyquist Shaping (Roll-Off 0.10) AM-PAM16 (16ASK) Signal with Nyquist Shaping (Roll-Off 1.0) 8.1.3 Carrier-Suppressed Amplitude Modulation (CS-AM) CS-AM-PAM2 (CS-2ASK) Signal with ERF Shaping CS-AM-PAM8 (CS-8ASK) Signal with ERF Shaping CS-AM-PAM2 (CS-2ASK) Signal with Nyquist Shaping CS-AM-PAM4 (CS-4ASK) Signal with Nyquist Shaping CS-AM-PAM8 (CS-8ASK) Signal with Nyquist Shaping 8.2 Phase Modulation 8.2.1 Finite Energy Signals (Pulses) Even Pulse Odd Pulse ERF Pulse (Even Pulse) Nyquist Pulse (Even Pulse) Derivative of the ERF Pulse (Odd Pulse) 8.2.2 Finite Power (Periodic) Signals Sinusoidal Phase Signal Series Approximation of the Square Wave Phase Signal 8.2.3 Pseudo-Random Phase Signals Phase Modulated Field Autocorrelation Power Spectrum Field Phasor Phase Constellations, Angular Amplitude, and Phase Noise Laser Phase Noise and Linewidth PM-PAM2 (BPSK) Signal with ERF Shaping PM-PAM4 (4PSK) Signal with ERF Shaping PM-PAM8 (8PSK) Signal with ERF Shaping PM-PAM2 (BPSK) Signal with Nyquist Shaping PM-PAM4 (4PSK) Signal with Nyquist Shaping PM-PAM8 (8PSK) Signal with Nyquist Shaping 9 Simulations of QAM Signals 9.1 4QAM (QPSK) with Heterogeneous Tributaries 9.2 4QAM (QPSK) with Homogeneous Tributaries 9.2.1 Nyquist Shaping 9.2.2 ERF Shaping 9.3 16QAM with Homogeneous Tributaries 9.4 64QAM with Homogeneous Tributaries 10 QAM Engineering 10.1 Complete Square Constellations 10.2 Incomplete Constellations 10.2.1 8QAM 8QAM-Shape1 (Rhombic Constellation) 8QAM-Shape 2 (Circular Constellation) Comments 10.2.2 32QAM 10.2.3 128QAM 10.2.4 QAM Constellations at Glance 10.3 Average Power and Symbol Distance 10.3.1 Complete Square QAM Constellations 10.3.2 Incomplete Square QAM Constellations The Shape Factor 10.3.3 Generic QAM Constellations 8QAM-Shape1 (Rhombic) 8QAM-Shape2 (Circular) 10.4 QAM Gauges 10.4.1 Average Power Gauge (APG) 2QAM (PSK) Constellation Complete Square QAM Constellations Incomplete Square QAM Constellations 32QAM 128QAM 8QAM Constellation (Rhombic) 8QAM Constellation (Circular) APG Equations Summary 10.4.2 Symbol Distance Gauge (SDG) 10.4.3 Maximum Amplitude Gauge (MAG) Appendix A - Phase Noise and Spectral Linewidth The White Noise Frequency Deviations The Integral Phase Process Mean Autocorrelation Power Spectrum The Phase Noise Process Mean Autocorrelation Power Spectrum Lorentzian Power Spectrum Approximation Spectral Linewidth Mean Square Phase Error and Lorentzian Linewidth Summary and Conclusions MATLAB scripts: Phase_noise Bibliography Chapter 5: Advanced Topics in Complex Optical Modulations 1 Hilbert Transforms 1.1 Hilbert Transforms in the Frequency Domain 1.2 Hilbert Transforms in the Time Domain 1.3 The Harmonic Conjugate Function 1.4 A Causal Spectrum: The Modulated Phasor 1.4.1 Narrowband Modulation 1.4.2 Broadband Modulation 1.5 Single-Side-Band Signals 2 Single-Side-Band (SSB) Modulation 2.1 Upper-SSB 2.2 Lower-SSB 2.3 Block Diagrams of the SSB Modulator 3 Simulations of SSB Signals 3.1 Nyquist Pulse with Zero Roll-Off 3.2 Nyquist Pulse with Unit Roll-Off 3.3 Rayleigh Pulse 3.4 The Exponential Causal Pulse 4 Coherent Demodulation of SSB Signals 4.1 Phase Offset of the Local Oscillator 4.2 Block Diagram of the SSB Demodulator 5 Twin-SSB Modulation 5.1 Complementary Spectral Allocations 5.2 Simulations of Twin-SSB Spectra 5.2.1 Nyquist-Rayleigh 5.2.2 Nyquist-Exponential 5.2.3 Rayleigh-Exponential 5.2.4 Nyquist-Nyquist 5.2.5 Rayleigh-Rayleigh 5.2.6 Exponential-Exponential 5.3 Block Diagram of the Twin-SSB Modulator 5.4 Spectrum of the Twin-SSB Modulation 5.5 The Hilbert Transform Process 5.5.1 The Hilbert Filter 5.5.2 Mean 5.5.3 Power Spectra 5.5.4 Correlation Functions 5.5.5 Average Power 5.6 Stationarity Theorem of the Twin-SSB Signal 5.6.1 Mean 5.6.2 Autocorrelation Cross-Correlation Cross-Correlation Cross-Correlation 5.6.3 The Sufficient Condition 5.6.4 The Necessary Condition 5.6.5 Comments 5.7 Power Spectrum of the Twin-SSB Modulation 5.8 Average Power of the Twin-SSB Modulation 5.9 Comparison of QAM and Twin-SSB Modulations 5.9.1 Power Spectra 5.9.2 Average Power Tributary Average Power Gauge Modulation Average Power Gauge 6 Coherent Demodulation of Twin-SSB Signals 6.1 Demodulation of Homogeneous Twin-SSB 6.2 Demodulation of Heterogeneous Twin-SSB 6.3 Chasing the Quadrature Twin-SSB Demodulation 6.4 Demodulation Theorems of Twin-SSB Signals 6.4.1 Frequency Domain Theorem 6.4.2 Time Domain Theorem 6.5 Block Diagram of the Twin-SSB Demodulator 6.5.1 Twin-SSB Demodulation by Frequency Domain Processing 6.5.2 Twin-SSB Demodulation by Time Domain Processing 6.6 Comparison of Demodulated QAM and Twin-SSB 6.6.1 Tributary Average Power Gauge 6.6.2 Modulation Average Power Gauge 7 Simulation of Pulse Twin-SSB Demodulation 7.1 Computation Procedure 7.2 Homogeneous T-SSB with Nyquist-Rayleigh Pulses 7.3 Homogeneous T-SSB with Nyquist-Nyquist Pulses 7.4 Heterogeneous T-SSB with Nyquist-Nyquist Pulses 8 Simulation of PAM Twin-SSB Demodulation 8.1 PAM2 with Nyquist-Nyquist Pulse Loading 8.1.1 Nyquist-Null Pulse Loading 8.1.2 Null-Nyquist Pulse Loading 8.2 PAM2 with ERF-ERF Pulse Loading 8.3 A Theorem on the Demodulated Baseband Spectra 8.4 PAM4 with Nyquist-Nyquist Loading 8.5 PAM4 with Nyquist-ERF Loading 8.6 PAM16 with Nyquist-Nyquist Loading 8.7 PAM16 with ERF-ERF Loading 8.8 Mixed PAM Signals 9 Comparison of 16QAM vs. 4PAM Twin-SSB 9.1 Tributary Average Power Gauge 9.2 Simulation: Modulation Power Gauge Appendix: Class of PAM Sequences Bibliography Chapter 6: Impulse Response Engineering and Statistical Eye Diagram Analysis: An Optimum Design Approach 1 Introduction 2 Phase Response Modeling 2.1 Polynomial Phase Approximation 2.2 Polynomial Group Delay Approximation 2.3 Example of Analytical Phase Response 2.4 MMSE Cubic Phase Interpolation 2.4.1 Example 2.5 MMSE High-Order Polynomial Phase Interpolation 2.6 The MMSE Algorithm 2.6.1 A Relevant Symmetry Between Elements Indices of a Square Matrix 2.6.2 Computation Procedure 2.6.3 MMSE Fitting of Ad Hoc Polynomial Phase Model 2.6.4 Polynomial Interpolation of Measured Phase Responses 2.6.5 The Group Delay Constant 2.6.6 Comment on the RMS Error 2.6.7 Polynomial Interpolation of the Group Delay 3 Linear Systems Dynamic 3.1 Basic Theorems of the Fourier Transform 3.1.1 Existence 3.1.2 Cartesian Representations 3.1.3 Real Time Functions Real and Even Time Functions Real and Odd Time Functions 3.1.4 Imaginary Time Functions Imaginary and Even Time Functions Imaginary and Odd Time Functions 3.1.5 Convolution Theorems Time Domain Convolution Theorem Frequency Domain Convolution Theorem 3.1.6 Symmetry Theorem 3.1.7 Time Shifting Theorem 3.1.8 Frequency Shifting Theorem 3.1.9 Causality 3.2 The Symmetry Theorem of the Impulse Response 3.3 Center of the Pulse 3.3.1 Group Delay Constant 3.3.2 Center of Energy 3.3.3 Center of Gravity 3.3.4 Maximum Amplitude 3.3.5 Threshold Level 3.4 Symmetric Pulses 3.5 Examples of Impulse Response Extraction 3.5.1 Impulse Response of MAOM-003411 Transfer Function and Extracted Impulse Response Comparison of Impulse Responses vs. Polynomial Phase Interpolation Eye Diagrams Comparison 3.5.2 Impulse Response of MAOM-006408 Phase Response Correction Polynomial Interpolation of the Phase Response Transfer Function and Extracted Impulse Response Comparison Versus Polynomial Phase Interpolations Eye Diagrams Comparison 4 Linear Modeling 4.1 Modulus Response Equations 4.1.1 Up-slope Peaking 4.1.2 Supergaussian Roll-off 4.1.3 Sinusoidal Ripple 4.1.4 Cumulative Modulus Response 4.2 Phase Response Equations 4.2.1 Linear Phase Component 4.2.2 Cubic Phase Component 4.2.3 Sinusoidal Phase Ripple 4.2.4 Power Series Expansion of the Sinusoidal Phase Ripple 4.2.5 Sinusoidal Ripple of the Group Delay 4.2.6 Effect of the Group Delay Specifications 4.3 96 Gbaud Linear Driver Model 4.4 128 Gbaud Linear Driver Model 5 Eye Diagram Statistics and Q-Factor Analysis 5.1 Definitions 5.1.1 Vertical Q-Factor 5.1.2 Horizontal Q-Factor 5.1.3 Relative Eye Open 5.1.4 Eye Open Penalty 5.1.5 Relative Eye Width 5.1.6 Eye Width Penalty 5.1.7 Eye Diagram Penalties vs. Q-Factor 5.2 The Eye Diagram Matrix Representation 5.3 Eye Diagram Statistics: 215-1 PAM2 RC-03 Signal 5.3.1 Vertical Statistics 5.3.2 Horizontal Statistic 5.4 Eye Diagram Statistics: 215-1 PAM4 RC-03 Signal 5.5 Eye Diagram Statistics: 215-1 PAM8 RC-03 Signal 5.6 Eye Diagram Statistics: 215-1 PAM16 RC-03 Signal Appendix Appendix 1 S4P to SMM Conversion Equations The Transmission Line Model Uncoupled Transmission Line Pairs References Index