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 |  موضوع: كتاب Noise and Signal Interference in Optical Fiber Transmission Systems الجمعة 20 أبريل 2018, 11:25 pm | |
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Noise and Signal Interference in Optical Fiber Transmission Systems
An Optimum Design Approach
Stefano Bottacchi
Photonics AG, Germany
ويتناول الموضوعات الأتية :
Contents
Note to the Reader xvii
Acknowledgements xix
Introduction xxi
1 Introductory Concepts and Noise Fundamentals 1
1.1 Introduction: The Noise Concept 1
1.2 Functional Classification of Noise 2
1.2.1 Constant Term: Thermal Noise 3
1.2.2 Linear Term: Shot Noise 4
1.2.3 Quadratic Term: Beat Noise 5
1.3 Total Noise 7
1.4 Bit Error Rate Performance 10
1.5 Timing Jitter 18
1.6 Partition of Noise Sources 22
1.7 Conclusions 30
References 30
2 Noise Principles in Optical Fiber Communication 31
2.1 Introduction 31
2.2 Receiver Thermal Noise 32
2.3 Dark Shot Noise 34
2.4 Signal Shot Noise 35
2.4.1 Optimized Multilevel Quantum Detection 35
2.4.1.1 Level Optimization 38
2.5 Multiplication Shot Noise 41
2.6 Optical Amplification and Beat Noises 47
2.6.1 ASE Spectrum 47
2.6.2 Population Inversion and the Spontaneous Emission Factor 48
2.6.3 Amplifier Gain 49
2.6.4 Optical Bandwidth 52
2.6.5 Photocurrent Equivalent 54
2.6.6 Signal-Spontaneous Beat Noise 55
2.6.7 Spontaneous-Spontaneous Beat Noise 58
2.6.8 Optically Amplified Receivers 592.7 Optical Noise and Coherence 65
2.7.1 Instantaneous Frequency Deviation 65
2.7.2 Integral Phase Process 67
2.7.3 Phase Noise 71
2.7.3.1 Mean 71
2.7.3.2 Autocorrelation 73
2.7.3.3 Power Spectrum 74
2.7.4 Coherence Time and Length 75
2.7.5 Frequency Chirping and Laser Linewidth 78
2.7.5.1 Linewidth Enhancement Factor 79
2.7.5.2 Chirped Source Spectrum 80
2.8 Relative Intensity Noise 83
2.9 Mode Partition Noise 85
2.10 Modal Noise 89
2.11 Reflection Noise 94
2.11.1 Fabry–Perot Interferometer 95
2.11.2 Interferometric Intensity Noise 98
2.11.2.1 Autocorrelation Function 99
2.11.2.2 Power Spectrum 101
2.12 Polarization Noise in Multimode Fibers 103
2.12.1 Theoretical Concepts 105
2.12.1.1 Mode Group Power 106
2.12.1.2 Modal Power Coupling 107
2.12.1.3 Source Polarization and Axis Rotation 109
2.12.2 Heuristic Approach 110
2.12.3 Azimuth Scanning Compliant Test (ASCOT) 111
2.12.4 Experimental Survey 111
2.12.5 Comments 115
References 115
3 Theory of Stochastic Processes 117
3.1 Introduction 117
3.2 Fundamentals of Random Processes 120
3.2.1 Definition 121
3.2.2 Probability Density Functions 122
3.2.3 Expectation Operator 123
3.2.4 Mean 124
3.2.5 Variance 125
3.3 Autocovariance Function 125
3.3.1 Statistical Uncorrelation and Independence 126
3.3.2 Definition and Basic Properties 127
3.3.2.1 Relation with the Variance 128
3.3.2.2 Conjugate Symmetric Property 128
3.3.3 Cross-Covariance Function 128
3.4 Degree of Coherence 130
3.4.1 Uncorrelated (Incoherent) Processes 131
3.4.2 Correlated (Coherent) Processes 131
3.5 Autocorrelation Function 134
3.5.1 Uncorrelated (Incoherent) Process 135
3.5.2 Ensemble Average Power 135
viii Contents3.5.3 Symmetric Conjugate Property 137
3.5.4 Example: Harmonic Field with Random Amplitude 137
3.6 Linear Combination of Random Processes 139
3.6.1 Statistical Homogeneity 139
3.6.2 Mutual Statistical Independence 140
3.6.3 Convolution Theorem of Probability Densities 140
3.6.3.1 Example: Gaussian Densities 141
3.6.4 Theorem of Composite Probability Density 143
3.6.4.1 Example: Quadratic Function 145
3.6.4.2 Example: Harmonic Function 148
3.6.5 Scaling Theorem of Probability Density 151
3.6.5.1 Example: Gaussian Density 152
3.6.6 Central Limit Theorem 153
3.6.6.1 Example: Gaussian Process Estimator 153
3.7 Characteristic Function 155
3.7.1 Conjugate Symmetry 156
3.7.2 Upper Bound 156
3.7.3 Shifting Theorem 158
3.7.4 Expectation Form 158
3.7.5 Inversion Formula 159
3.7.6 Scaling Theorem of Characteristic Function 159
3.7.7 Examples of Probability Densities 160
3.7.7.1 Uniform Density 160
3.7.7.2 Triangular Density 161
3.7.7.3 Gaussian Density 162
3.7.7.4 Symmetric Exponential Density 162
3.7.8 Linear Combination of Stochastic Processes 164
3.7.9 Process Estimator for Uniform Density 165
3.8 Stationary Processes 174
3.8.1 Stationary Processes of the Nth Order 174
3.8.2 Strict-Sense Stationary (SSS) Processes 175
3.8.3 Wide-Sense Stationary (WSS) Processes 177
3.8.4 Characterization of WSS Random Processes 177
3.8.4.1 Mean 178
3.8.4.2 Variance 178
3.8.4.3 Autocovariance 178
3.8.4.4 Degree of Coherence 179
3.8.4.5 Autocorrelation 179
3.8.4.6 Ensemble Average Power 180
3.9 Ergodic Processes 180
3.9.1 Temporal Averages of Stochastic Processes 182
3.9.1.1 Mean Ergodic Processes 182
3.9.1.2 Functional Ergodic Processes 185
3.9.1.3 Power Ergodic Processes 186
3.9.1.4 Autocovariance Ergodic Processes 187
3.9.1.5 Autocorrelation Ergodic Processes 188
3.9.2 Wide-Sense Ergodic (WSE) Processes 189
3.10 Spectral Representation 190
3.10.1 Mean 191
3.10.2 Autocorrelation 191
Contents ix3.10.3 Finite Energy Processes: An Integral Theorem 192
3.10.3.1 Example: Smoothed Random-Amplitude Harmonic Field 195
3.10.4 Stationary Processes: The Wiener–Khintchin Theorem 197
3.10.4.1 Example: Harmonic Field with Random Amplitude 203
3.11 Normal Processes 204
3.11.1 Jointly Normal Random Variables 204
3.11.2 Definition of Normal Processes 207
3.11.3 Properties of Normal Processes 210
3.12 White Noise Modeling 212
3.12.1 Stationary White Noise 213
3.12.2 Ergodic White Noise 217
3.12.3 Gaussian White Noise 217
3.13 Conclusions and Remarks 218
References 219
4 Linear Systems and Noise 221
4.1 Introduction 221
4.2 Linear Systems with Stochastic Inputs 225
4.2.1 Definitions 226
4.2.2 Output Mean 227
4.2.2.1 Example: Filtered Additive Noise 228
4.2.3 Output Autocorrelation 228
4.2.3.1 Stationary Processes 229
4.3 Noise Bandwidth 231
4.4 Basic Electrical Filters 235
4.4.1 Ideal Low-Pass (NYQUIST) Filter 236
4.4.1.1 Noise Bandwidth 238
4.4.2 Single-Pole Filter 238
4.4.2.1 Noise Bandwidth 239
4.4.3 Gaussian Filter 241
4.4.3.1 Noise Bandwidth 242
4.4.3.2 Relative Frequency Attenuation Ratio 243
4.4.4 Conjugate Gaussian Relationships 245
4.4.4.1 Impulse Response 246
4.4.4.2 Fourier Transform Pairs 246
4.4.4.3 Normalization Properties 246
4.4.4.4 Uncertainty Relationship 246
4.4.4.5 Conjugate FWHM 246
4.4.4.6 Unilateral HWHM Bandwidth 247
4.4.4.7 Cut-Off Bandwidth 247
4.4.4.8 Transient Times 248
4.4.4.9 Gaussian Integrals 250
4.4.4.10 Noise Bandwidth 251
4.4.4.11 Applications 251
4.4.4.12 Example: 10 ps RMS Pulse 253
4.4.4.13 Example: 10 GbE Optical Link 254
4.5 Raised-Cosine Filter 259
4.5.1 Frequency Response 259
4.5.2 Impulse Response 260
x Contents4.5.2.1 First Set of Zeros 261
4.5.2.2 Second Set of Zeros 264
4.5.3 Shaping Factor and Windowing Function 264
4.5.4 Noise Bandwidth 268
4.5.4.1 A Noteworthy Integral 270
4.6 Bessel–Thompson Filter 271
4.6.1 Transfer Function 272
4.6.2 Cut-off Frequency 278
4.6.3 Group Delay 280
4.6.4 Half-Bit Delay Line 281
4.6.5 Computer Simulations 282
4.6.5.1 Frequency Responses 282
4.6.5.2 MATLAB1 Source Code 287
4.6.6 Impulse Response 288
4.6.7 Center of Gravity and Temporal Average 289
4.6.8 Real-Time Representations 292
4.6.8.1 MATLAB1 Source Code 293
4.6.9 Fourth-Order Bessel–Thompson Filter 298
4.6.10 Noise Bandwidth 305
4.6.10.1 MATLAB1 Code: Noise Bandwidth 307
4.7 Noise Bandwidth Comparison 311
4.8 Conclusions 312
References 313
5 Statistical Theory of Intersymbol Interference 315
5.1 Introduction 315
5.2 Theory of Signal Interference 317
5.2.1 Definitions 318
5.2.2 Space and Time Representations 319
5.2.3 Definition of the Interfering Terms 321
5.2.4 Signal Sample 323
5.2.5 Analysis of the Interfering Terms 323
5.2.5.1 Isolated Symbol Sample 324
5.2.5.2 Postcursor Interference 324
5.2.5.3 Precursor Interference 325
5.2.6 Random Binary Sequence 326
5.2.7 Postcursor Interfering Term 329
5.2.7.1 Mean 329
5.2.7.2 A Theorem for the Variance 329
5.2.7.3 Variance 331
5.2.8 Precursor Interfering Term 332
5.2.9 Probability Density Function 333
5.2.9.1 Postcursor Interference 333
5.2.9.2 Precursor Interference 335
5.2.10 Stationarity of the Random Sequences 337
5.2.11 Cyclostationary Binary Sequence 338
5.2.11.1 Pseudorandom Binary Sequence 340
5.2.11.2 Example 341
5.2.11.3 Random Sequences of Finite Length 341
Contents xi5.2.11.4 Tail Resolution 342
5.2.11.5 Postcursor Population 343
5.2.11.6 Precursor Population 344
5.3 Matrix Representation 345
5.4 Variable-Width Raised-Cosine Pulse 348
5.4.1 Frequency Response 348
5.4.1.1 Frequency Normalization 349
5.4.2 Impulse Response 351
5.4.3 Interfering Terms 351
5.4.4 Mean and Variance 352
5.4.5 Probability Density Function (Histogram) 353
5.4.6 Simulations 355
5.4.7 Comments on the Density Profile of the VWRC 359
5.4.8 Fine Structure of the Subhistograms 363
5.4.9 MATLAB1 Code: VWRC_HISTO 374
5.4.10 MATLAB1 Code: LOCAL_MEAN 377
5.4.11 Conclusions 377
5.5 Concluding Remarks 378
References 378
6 Modeling and Simulation of Intersymbol Interference 379
6.1 Introduction 379
6.2 Single-Pole NRZ Pulse (SP-NRZ) 381
6.2.1 Normalized Frequency Response 381
6.2.2 Normalized Impulse Response 382
6.2.3 Synthesis of the SP-NRZ Pulse 382
6.2.4 Pulse Properties and FWHM 383
6.2.5 Interfering Terms 385
6.2.6 Mean of the Interference 388
6.2.6.1 Asymptotic Linear Behavior 388
6.2.7 Variance of the Interference 390
6.2.7.1 Approximations and Limiting Behaviors for Large 392
6.2.7.2 Variance Series S() 393
6.2.7.3 Comments on Calculation of the Variance 406
6.2.8 Probability Density Functions 406
6.2.8.1 Time Constant ¼ 0:6 $ c ffi 0:265 408
6.2.8.2 Time Constant ¼ 1:0 $ c ffi 0:159 409
6.2.8.3 Time Constant ¼ 2:0 $ c ffi 0:080 411
6.2.8.4 Time Constant ¼ 3:0 $ c ffi 0:053 411
6.2.8.5 MATLAB1 Code: SPNRZ_HISTO 412
6.2.8.6 Comments on the Simulation Results 414
6.2.9 Summary of the SP-NRZ Pulse Modeling 415
6.2.10 Conclusions 415
6.3 Gaussian NRZ Pulse (GS-NRZ) 415
6.3.1 Gaussian Frequency Response 417
6.3.2 Gaussian Impulse Response 417
6.3.3 Synthesis of the GS-NRZ Pulse 418
6.3.4 Properties and Pulse Profiles 420
6.3.4.1 MATLAB1 Code: GSNRZ_Pulse 425
6.3.5 Finite Sequences and Amplitude Fluctuations 425
xii Contents6.3.5.1 Analysis of the Interfering Terms 428
6.3.5.2 Amplitude Fluctuations 428
6.3.5.3 MATLAB1 Code: GSNRZ_Multi_Sequences 431
6.3.6 Full-Width-at-Half-Maximum 433
6.3.6.1 MATLAB1 Code: GSNRZ_FWHM 435
6.3.7 Transition Time 436
6.3.7.1 Rise Time and Fall Time 438
6.3.7.2 Relation between the Rise Time and the FWHM 440
6.3.8 Interfering Terms 441
6.3.9 Mean of the Interference 445
6.3.9.1 Theorem I: Power Series of Error Functions 445
6.3.10 Variance of the Interference 447
6.3.11 Two Mathematical Theorems 448
6.3.11.1 Theorem I: Power Series of the Error Function 448
6.3.11.2 Application of Theorem I to the Calculus of the Variance 449
6.3.11.3 Theorem II: Series Solution by Parts 452
6.3.11.4 Applications of Theorem II to the Calculus
of the Variance 454
6.3.12 Limiting Behavior of the Variance 462
6.3.12.1 Large Value of the Standard Deviation: 1 462
6.3.12.2 Theorem III – Series of Gaussian Samples: 1 464
6.3.12.3 Small Value of the Standard Deviation: 1=?2
ffiffiffi
p2? 467
6.3.12.4 MATLAB1 Code: GSNRZ_Variance 471
6.3.13 On the Solution of the Series of Gaussians 472
6.3.14 Probability Density Functions (Histograms) 473
6.3.14.1 Standard Deviation: ¼ 0:5 477
6.3.14.2 Standard Deviation: ¼ 3:0 478
6.3.14.3 Standard Deviation: ¼ 5:0 479
6.3.14.4 Comments 479
6.3.14.5 MATLAB1 Code: GSNRZ_HISTO 479
6.3.15 Summary of the GS-NRZ Pulse Modeling 482
6.3.16 Conclusions 484
6.4 Solution of the Variance Series 485
6.5 Bessel–Thompson NRZ Pulse 488
6.5.1 Frequency Response 490
6.5.2 Impulse Response 492
6.5.3 Synthesis of the BT-NRZ Pulse 494
6.5.3.1 BT-NRZ Pulse Alignment Procedure 494
6.5.3.2 Center-of-Gravity Theorem 498
6.5.3.3 Group Delay Theorem 501
6.5.3.4 Application to the BT-NRZ Pulse 503
6.5.4 Self-Aligned BT-NRZ Pulse 506
6.5.4.1 MATLAB1 Code: BTNRZ_CAP_Pulse 507
6.5.5 Pulse Profiles and Characteristic Parameters 509
6.5.5.1 Normalized Cut-off c ¼ 0:25 509
6.5.5.2 Normalized Cut-off c ¼ 0:75 509
6.5.5.3 Normalized Cut-off c ¼ 2:0 510
6.5.6 Full-Width-at-Half-Amplitude (FWHA) 512
6.5.6.1 MATLAB1 Code: BTNRZ_FWHA 515
6.5.7 Interfering Terms 517
Contents xiii6.5.7.1 Postcursor Sequence 517
6.5.7.2 Precursor Sequence 517
6.5.8 Mean of the Interference 519
6.5.8.1 Mean Value of the Postcursor Intersymbol Interference 521
6.5.8.2 Mean Value of the Precursor Intersymbol Interference 521
6.5.9 Variance of the Interference 522
6.5.9.1 Variance of the Postcursor Intersymbol Interference 522
6.5.9.2 MATLAB1 Code: BTNRZ_Variance_PST 525
6.5.9.3 Variance of the Precursor Intersymbol Interference 527
6.5.10 Probability Density Functions (Histograms) 528
6.5.10.1 Low Cut-off Frequency: c ¼ 0:05 530
6.5.10.2 Mid Cut-off Frequency: c ¼ 0:20 533
6.5.10.3 High Cut-off Frequency: c ¼ 1:0 534
6.5.10.4 MATLAB1 Code: BTNRZ_HISTO 536
6.5.11 Conclusions 540
References 542
7 Frequency Representation of Intersymbol Interference 543
7.1 Introduction 543
7.2 ISI in the Frequency Domain 544
7.2.1 Fourier Series Kernel 544
7.2.1.1 Solutions and Properties of the Partial Sums 545
7.2.1.2 Solution and Properties of the Symmetric Sum 548
7.2.1.3 Limiting Behavior of the Symmetric Sum for N ! 1 551
7.2.1.4 Limiting Behavior of the Partial Sums for N ! 1 553
7.2.2 Sum of the Interfering Terms 560
7.2.2.1 Sum of Precursors 561
7.2.2.2 Sum of Postcursors 563
7.2.2.3 Discussion: The Spectral Overlapping Mechanism 564
7.3 Total ISI Theorem 572
7.3.1 Coincidence of the Sums of Partial Intersymbols 573
7.3.1.1 Application to the Even-Symmetric Pulse 573
7.3.2 Sum of All Intersymbol Interferences 573
7.3.3 Concept and Definition of the Total ISI 574
7.3.4 Corollary I: NRZ Synthesized Pulse 575
7.3.5 Proof of the Total ISI Theorem 576
7.4 Applications and Examples 579
7.4.1 Fourth-Order Bessel–Thompson NRZ Pulse 579
7.4.1.1 Reference Amplitude 0 < 1 579
7.4.1.2 Reference Amplitude 0 ¼ 1 582
7.4.1.3 Reference Amplitude 0 > 1 584
7.4.1.4 MATLAB1 Code: BTNRZ_Total_ISI_Theorem 586
7.4.2 Gaussian NRZ Pulse (Error Function) 587
7.4.2.1 Comments on the Spectrum of the NRZ Synthesized Pulses 588
7.4.2.2 MATLAB1 Code: GSNRZ_Total_ISI_Theorem 589
7.4.3 Variable-Width Raised-Cosine Pulse 590
7.4.3.1 Narrowband VWRC Pulse 593
7.4.3.2 Broadband VWRC Pulse 594
7.4.3.3 Simulation Results 594
7.4.3.4 MATLAB1 Code: VWRC_Total_ISI 597
xiv Contents7.5 Statistical Theorems 598
7.5.1 Mean Value Theorem 601
7.5.2 Comments and Applications 603
7.5.2.1 SP-NRZ Pulse 603
7.5.2.2 GS-NRZ Pulse 603
7.5.2.3 BT-NRZ Pulse 604
7.5.2.4 Narrowband VWRC Pulse 604
7.5.2.5 MATLAB1 Code: Total_ISI_Mean_Theorem 611
7.5.3 Variance Theorem 616
7.5.3.1 Solution for the Series S1 618
7.5.3.2 Series S3 668
References 670
8 DBRV Method for Calculation of the ISI Statistic 671
8.1 Introduction 671
8.2 Matrix Method for the Total ISI 672
8.2.1 Characteristic Function 674
8.2.2 Mean of the Intersymbol Interference 675
8.2.3 Variance of the Intersymbol Interference 675
8.2.4 Correction Term for Finite Binary Sequences 679
8.2.4.1 Calculation of the Mean 679
8.2.4.2 Calculation of the Variance 681
8.3 Simulations of ISI PDF 682
8.3.1 BT-NRZ Pulse 684
8.3.1.1 Low Cut-off Frequencies 684
8.3.1.2 High Cut-off Frequency 686
8.3.2 VWRC Pulse 688
8.3.2.1 Synchronized Pulse ¼ 1:0 689
8.3.2.2 Lower Detuned Rate Parameter ¼ 0:95 689
8.3.2.3 Higher Detuned Rate Parameter ¼ 1:05 690
8.3.2.4 Low Rate Parameter ¼ 0:35 692
8.3.3 SP-NRZ Pulse 694
8.3.4 MATLAB1 Code: ISI_Histo_TOT 696
8.4 Concepts for the Gaussian Statistic 706
8.4.1 Central Limit Theorem 706
8.4.1.1 A Theorem on the Characteristic Function 707
8.4.1.2 Heuristic Approach to the Central Limit Theorem 709
8.4.1.3 Series of Independent and Uniform Random Variables 711
8.4.1.4 Normalized Sum of Independent Random Variables 713
8.4.2 Statistical Estimator of a Random Sequence 713
8.4.2.1 Mean and Variance 714
8.4.2.2 Characteristic Function 715
8.4.2.3 Probability Density Function 716
8.4.2.4 Summary 716
8.4.3 Gaussian Estimator of a Random Sequence 717
8.4.4 Repeated Trials 718
8.4.4.1 Characteristic Function 719
8.4.4.2 Probability Density Function 720
8.4.4.3 Summary 720
8.4.5 Examples and Simulations 721
Contents xv8.4.5.1 Rayleigh Probability Density 721
8.4.5.2 Exponential Probability Density 730
8.4.5.3 Uniform Probability Density 734
8.4.5.4 Calculation of the Gaussian Estimator 737
8.4.5.5 Mating the Three Probability Densities 743
8.4.5.6 MATLAB1 Code: Gaussian_Estimator_Theory 745
8.5 DBRV Method for the Total ISI 753
8.5.1 Discrete Binary Random Variable 753
8.5.1.1 Two Independent Discrete Binary Random Variables 755
8.5.2 ISI Comb-Like Statistic 756
8.5.2.1 Probability Density Function 756
8.5.2.2 Fine Structure of the Probability Density 758
8.5.2.3 Mean and Variance 761
8.5.2.4 Characteristic Function 762
8.5.2.5 Summary of the DBRV Method 765
8.5.3 Heuristic Approach to the Gaussian ISI 765
8.5.4 Some ISI Generating Sequences 767
8.5.4.1 Square-root Decaying Sequence 767
8.5.4.2 Sinc Decaying Sequence 771
8.5.4.3 Power-law Decaying Sequence 775
8.5.4.4 MATLAB1 Code: Binary_Char_Func 783
8.5.5 ISI Generation of the VWRC Pulse 786
8.5.5.1 Averages 788
8.5.5.2 Comparison between the Matrix and DBRV Methods 789
8.5.5.3 MATLAB1 Code: VWRC_ISI_STAT_Comparison 796
8.5.6 General Recommendations for the DBRV Method 801
8.5.6.1 Application to the Causal WVRC Pulse 803
8.5.6.2 MATLAB1 Code: VWRC_ISI_DBRV_Method 804
8.5.6.3 Flow Chart of the DBRV Method 808
8.5.7 Gaussian Estimator of the VWRC ISI 808
8.5.7.1 Theory 809
8.5.7.2 Calculations 810
8.5.7.3 A Remark on the Scaled Characteristic Function 815
8.5.7.4 MATLAB1 Code: VWRC_Gaussian_Estimator 817
8.6 Conclusions 822
References 823
Index
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