Table Of ContentNoise and Signal Interference in
Optical Fiber Transmission Systems
Noise and Signal Interference in Optical Fiber Transmission Systems: An Optimum Design Approach Stefano Bottacchi
© 2008 John Wiley & Sons, Ltd. ISBN: 978-0-470-06061-2
Noise and Signal Interference in
Optical Fiber Transmission Systems
An Optimum Design Approach
Stefano Bottacchi
u2t Photonics AG, Germany
Thiseditionfirstpublished2008.
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LibraryofCongressCataloging-in-PublicationData
Bottacchi,Stefano.
Noiseandsignalinterferenceinopticalfibertransmissionsystems:anoptimumdesignapproach/Stefano
Bottacchi.
p.cm.
Includesbibliographicalreferencesandindex.
ISBN978-0-470-06061-2(cloth)
1. Opticalcommunications.2. Fiberoptics.3. Noise(Electronics) I.
Title.
TK5103.59.B692008
621.382075–dc22
2008024772
AcataloguerecordforthisbookisavailablefromtheBritishLibrary.
ISBN978-0-470-06061-2(HB)
Typesetin10/12ptTimesbyThomsonDigitalNoida,India.
PrintedinSingaporebyMarkonoPrintMediaPteLtd.
To my wife Laura
and my charming daughters
Francesca and Alessandra
Contents
NotetotheReader xvii
Acknowledgements xix
Introduction xxi
1 IntroductoryConceptsandNoiseFundamentals 1
1.1 Introduction:TheNoiseConcept 1
1.2 FunctionalClassificationofNoise 2
1.2.1 ConstantTerm:Thermal Noise 3
1.2.2 LinearTerm:ShotNoise 4
1.2.3 QuadraticTerm: BeatNoise 5
1.3 TotalNoise 7
1.4 BitErrorRatePerformance 10
1.5 TimingJitter 18
1.6 PartitionofNoise Sources 22
1.7 Conclusions 30
References 30
2 NoisePrinciples inOpticalFiberCommunication 31
2.1 Introduction 31
2.2 ReceiverThermalNoise 32
2.3 DarkShot Noise 34
2.4 SignalShotNoise 35
2.4.1 OptimizedMultilevelQuantumDetection 35
2.4.1.1 LevelOptimization 38
2.5 MultiplicationShot Noise 41
2.6 OpticalAmplification andBeatNoises 47
2.6.1 ASESpectrum 47
2.6.2 PopulationInversionandtheSpontaneousEmissionFactor 48
2.6.3 AmplifierGain 49
2.6.4 OpticalBandwidth 52
2.6.5 PhotocurrentEquivalent 54
2.6.6 Signal-SpontaneousBeatNoise 55
2.6.7 Spontaneous-SpontaneousBeatNoise 58
2.6.8 Optically AmplifiedReceivers 59
viii Contents
2.7 OpticalNoiseandCoherence 65
2.7.1 InstantaneousFrequency Deviation 65
2.7.2 IntegralPhase Process 67
2.7.3 PhaseNoise 71
2.7.3.1 Mean 71
2.7.3.2 Autocorrelation 73
2.7.3.3 Power Spectrum 74
2.7.4 CoherenceTimeandLength 75
2.7.5 Frequency ChirpingandLaserLinewidth 78
2.7.5.1 LinewidthEnhancementFactor 79
2.7.5.2 ChirpedSourceSpectrum 80
2.8 RelativeIntensityNoise 83
2.9 ModePartitionNoise 85
2.10 ModalNoise 89
2.11 ReflectionNoise 94
2.11.1 Fabry–PerotInterferometer 95
2.11.2 InterferometricIntensityNoise 98
2.11.2.1 AutocorrelationFunction 99
2.11.2.2 Power Spectrum 101
2.12 Polarization NoiseinMultimodeFibers 103
2.12.1 Theoretical Concepts 105
2.12.1.1 ModeGroupPower 106
2.12.1.2 ModalPowerCoupling 107
2.12.1.3 SourcePolarization andAxisRotation 109
2.12.2 HeuristicApproach 110
2.12.3 AzimuthScanningCompliantTest(ASCOT) 111
2.12.4 ExperimentalSurvey 111
2.12.5 Comments 115
References 115
3 TheoryofStochasticProcesses 117
3.1 Introduction 117
3.2 Fundamentals ofRandomProcesses 120
3.2.1 Definition 121
3.2.2 ProbabilityDensityFunctions 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 StatisticalUncorrelationandIndependence 126
3.3.2 Definition andBasicProperties 127
3.3.2.1 RelationwiththeVariance 128
3.3.2.2 Conjugate SymmetricProperty 128
3.3.3 Cross-Covariance Function 128
3.4 DegreeofCoherence 130
3.4.1 Uncorrelated(Incoherent)Processes 131
3.4.2 Correlated(Coherent) Processes 131
3.5 AutocorrelationFunction 134
3.5.1 Uncorrelated(Incoherent)Process 135
3.5.2 EnsembleAveragePower 135
Contents ix
3.5.3 Symmetric ConjugateProperty 137
3.5.4 Example:HarmonicFieldwithRandomAmplitude 137
3.6 LinearCombinationofRandomProcesses 139
3.6.1 StatisticalHomogeneity 139
3.6.2 MutualStatisticalIndependence 140
3.6.3 ConvolutionTheoremofProbabilityDensities 140
3.6.3.1 Example: GaussianDensities 141
3.6.4 TheoremofCompositeProbability Density 143
3.6.4.1 Example: QuadraticFunction 145
3.6.4.2 Example: HarmonicFunction 148
3.6.5 ScalingTheoremofProbability Density 151
3.6.5.1 Example: GaussianDensity 152
3.6.6 CentralLimitTheorem 153
3.6.6.1 Example: GaussianProcessEstimator 153
3.7 Characteristic Function 155
3.7.1 ConjugateSymmetry 156
3.7.2 UpperBound 156
3.7.3 ShiftingTheorem 158
3.7.4 Expectation Form 158
3.7.5 InversionFormula 159
3.7.6 ScalingTheoremofCharacteristicFunction 159
3.7.7 ExamplesofProbability Densities 160
3.7.7.1 UniformDensity 160
3.7.7.2 Triangular Density 161
3.7.7.3 GaussianDensity 162
3.7.7.4 SymmetricExponentialDensity 162
3.7.8 LinearCombinationofStochastic Processes 164
3.7.9 ProcessEstimator forUniform Density 165
3.8 StationaryProcesses 174
3.8.1 StationaryProcesses oftheNthOrder 174
3.8.2 Strict-Sense Stationary(SSS)Processes 175
3.8.3 Wide-SenseStationary(WSS)Processes 177
3.8.4 Characterization ofWSS RandomProcesses 177
3.8.4.1 Mean 178
3.8.4.2 Variance 178
3.8.4.3 Autocovariance 178
3.8.4.4 DegreeofCoherence 179
3.8.4.5 Autocorrelation 179
3.8.4.6 EnsembleAveragePower 180
3.9 ErgodicProcesses 180
3.9.1 TemporalAveragesofStochasticProcesses 182
3.9.1.1 MeanErgodicProcesses 182
3.9.1.2 FunctionalErgodicProcesses 185
3.9.1.3 PowerErgodicProcesses 186
3.9.1.4 Autocovariance ErgodicProcesses 187
3.9.1.5 AutocorrelationErgodicProcesses 188
3.9.2 Wide-SenseErgodic(WSE)Processes 189
3.10 Spectral Representation 190
3.10.1 Mean 191
3.10.2 Autocorrelation 191
x Contents
3.10.3 FiniteEnergyProcesses:AnIntegralTheorem 192
3.10.3.1 Example: SmoothedRandom-Amplitude HarmonicField 195
3.10.4 StationaryProcesses: TheWiener–KhintchinTheorem 197
3.10.4.1 Example: HarmonicFieldwithRandomAmplitude 203
3.11 NormalProcesses 204
3.11.1 JointlyNormalRandomVariables 204
3.11.2 Definition ofNormalProcesses 207
3.11.3 PropertiesofNormalProcesses 210
3.12 WhiteNoiseModeling 212
3.12.1 StationaryWhiteNoise 213
3.12.2 ErgodicWhiteNoise 217
3.12.3 GaussianWhiteNoise 217
3.13 Conclusions andRemarks 218
References 219
4 LinearSystemsandNoise 221
4.1 Introduction 221
4.2 LinearSystems withStochastic Inputs 225
4.2.1 Definitions 226
4.2.2 OutputMean 227
4.2.2.1 Example: FilteredAdditiveNoise 228
4.2.3 OutputAutocorrelation 228
4.2.3.1 Stationary Processes 229
4.3 NoiseBandwidth 231
4.4 Basic ElectricalFilters 235
4.4.1 IdealLow-Pass (NYQUIST)Filter 236
4.4.1.1 NoiseBandwidth 238
4.4.2 Single-PoleFilter 238
4.4.2.1 NoiseBandwidth 239
4.4.3 GaussianFilter 241
4.4.3.1 NoiseBandwidth 242
4.4.3.2 RelativeFrequencyAttenuation Ratio 243
4.4.4 ConjugateGaussianRelationships 245
4.4.4.1 ImpulseResponse 246
4.4.4.2 FourierTransformPairs 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 UnilateralHWHMBandwidth 247
4.4.4.7 Cut-OffBandwidth 247
4.4.4.8 Transient Times 248
4.4.4.9 GaussianIntegrals 250
4.4.4.10 NoiseBandwidth 251
4.4.4.11 Applications 251
4.4.4.12 Example: 10psRMSPulse 253
4.4.4.13 Example: 10GbEOpticalLink 254
4.5 Raised-CosineFilter 259
4.5.1 Frequency Response 259
4.5.2 ImpulseResponse 260
Contents xi
4.5.2.1 FirstSetofZeros 261
4.5.2.2 SecondSetofZeros 264
4.5.3 ShapingFactorandWindowingFunction 264
4.5.4 NoiseBandwidth 268
4.5.4.1 ANoteworthyIntegral 270
4.6 Bessel–Thompson Filter 271
4.6.1 Transfer Function 272
4.6.2 Cut-offFrequency 278
4.6.3 GroupDelay 280
4.6.4 Half-BitDelayLine 281
4.6.5 ComputerSimulations 282
4.6.5.1 FrequencyResponses 282
4.6.5.2 MATLAB1SourceCode 287
4.6.6 ImpulseResponse 288
4.6.7 CenterofGravityandTemporalAverage 289
4.6.8 Real-TimeRepresentations 292
4.6.8.1 MATLAB1SourceCode 293
4.6.9 Fourth-OrderBessel–ThompsonFilter 298
4.6.10 NoiseBandwidth 305
4.6.10.1 MATLAB1Code:NoiseBandwidth 307
4.7 NoiseBandwidthComparison 311
4.8 Conclusions 312
References 313
5 StatisticalTheoryofIntersymbolInterference 315
5.1 Introduction 315
5.2 TheoryofSignalInterference 317
5.2.1 Definitions 318
5.2.2 SpaceandTime Representations 319
5.2.3 Definition oftheInterferingTerms 321
5.2.4 SignalSample 323
5.2.5 AnalysisoftheInterfering Terms 323
5.2.5.1 Isolated SymbolSample 324
5.2.5.2 PostcursorInterference 324
5.2.5.3 PrecursorInterference 325
5.2.6 RandomBinarySequence 326
5.2.7 PostcursorInterferingTerm 329
5.2.7.1 Mean 329
5.2.7.2 ATheorem fortheVariance 329
5.2.7.3 Variance 331
5.2.8 PrecursorInterferingTerm 332
5.2.9 ProbabilityDensityFunction 333
5.2.9.1 PostcursorInterference 333
5.2.9.2 PrecursorInterference 335
5.2.10 StationarityoftheRandomSequences 337
5.2.11 CyclostationaryBinarySequence 338
5.2.11.1 PseudorandomBinarySequence 340
5.2.11.2 Example 341
5.2.11.3 RandomSequencesofFiniteLength 341
xii Contents
5.2.11.4 TailResolution 342
5.2.11.5 PostcursorPopulation 343
5.2.11.6 PrecursorPopulation 344
5.3 MatrixRepresentation 345
5.4 Variable-WidthRaised-Cosine Pulse 348
5.4.1 Frequency Response 348
5.4.1.1 FrequencyNormalization 349
5.4.2 ImpulseResponse 351
5.4.3 InterferingTerms 351
5.4.4 MeanandVariance 352
5.4.5 ProbabilityDensityFunction (Histogram) 353
5.4.6 Simulations 355
5.4.7 CommentsontheDensity ProfileoftheVWRC 359
5.4.8 FineStructureoftheSubhistograms 363
5.4.9 MATLAB1Code:VWRC_HISTO 374
5.4.10 MATLAB1Code:LOCAL_MEAN 377
5.4.11 Conclusions 377
5.5 ConcludingRemarks 378
References 378
6 ModelingandSimulation ofIntersymbolInterference 379
6.1 Introduction 379
6.2 Single-PoleNRZPulse(SP-NRZ) 381
6.2.1 NormalizedFrequency Response 381
6.2.2 NormalizedImpulseResponse 382
6.2.3 SynthesisoftheSP-NRZPulse 382
6.2.4 PulsePropertiesandFWHM 383
6.2.5 InterferingTerms 385
6.2.6 MeanoftheInterference 388
6.2.6.1 AsymptoticLinearBehavior 388
6.2.7 VarianceoftheInterference 390
6.2.7.1 ApproximationsandLimitingBehaviorsforLarge(cid:1) 392
6.2.7.2 Variance SeriesS((cid:1)) 393
6.2.7.3 CommentsonCalculation oftheVariance 406
6.2.8 ProbabilityDensityFunctions 406
6.2.8.1 TimeConstant (cid:1)¼0:6$(cid:2)cffi0:265 408
6.2.8.2 TimeConstant (cid:1)¼1:0$(cid:2)cffi0:159 409
6.2.8.3 TimeConstant (cid:1)¼2:0$(cid:2)cffi0:080 411
6.2.8.4 TimeConstant (cid:1)¼3:0$(cid:2)cffi0:053 411
6.2.8.5 MATLAB1Code:SPNRZ_HISTO 412
6.2.8.6 CommentsontheSimulationResults 414
6.2.9 SummaryoftheSP-NRZPulseModeling 415
6.2.10 Conclusions 415
6.3 GaussianNRZPulse(GS-NRZ) 415
6.3.1 GaussianFrequency Response 417
6.3.2 GaussianImpulseResponse 417
6.3.3 SynthesisoftheGS-NRZ Pulse 418
6.3.4 PropertiesandPulseProfiles 420
6.3.4.1 MATLAB1Code:GSNRZ_Pulse 425
6.3.5 FiniteSequences andAmplitude Fluctuations 425
