Symulation of Optical Fiber Communication System With Polarization Mode Dispersion

8/3/2019 Symulation of Optical Fiber Communication System With Polarization Mode Dispersion

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This thesis is about the simulation of optical fiber communication systems withthe effect of PMD (Polarization Mode D ispersion) and the application of adaptivechannel equalizers,CPI (Cross Polarization Interference) cancellers and echo cancellerswhich are used to mitigate PMD.

The basic structure of this system is that two QAM signals are transmitted througha long optical fiber with PMD nd they are received by a coherent receiver which consistsof adaptive electrical channel equalizers and adaptive electrical CPI cancellers. The echoin a bidirectional transmission system is studied and adaptive echo cancellers are used tocancel it.

The objective of this thesis is to create the system structure, to simulate the opticalfiber chamel with PMD,o simulate the adaptive channel equalizers, CPI cancellers andecho cancellers and to evaluate the systern performance.

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ACKNOWLEDGEMENTS

I would iike to expressmy profound gratitude andhigh regards to professor BrentR Petersen. His encouragement, fiiendshipand suggestionsduring the course of thisresearchhave played a vital role.

1also would like to express the deepest feeling of gratitudeto my parents, YongWei Lu and Zhao Yun Ding for their support and love.


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...Abstract ................................................................................................................ 111.................................................................................................cknowledgements iv

............................................................................................................ist of Contents v.List ofTables ............................................................................................................... VII...List of Figures .......................................................................................................... vm

List of Symbols .............................................................................................................List of Abbreviations and Acronyrns ............................................................... xviChapter 1. ntroduction ........................................................................................i1.1 Introduction to the Research .......................................................................... 11.2 Thesis Contributions ..................................................................................... 31.3 he sis Outline ................................................................................................4Chapter 2. System Simulation Model ................................................................. 72.1 System Mode1 .................................................................................................2.2 Baseband Simulation and Coherent Demodulation ....................................... 13

2.3 Baseband System Mode1................................................................................ 152.3 System Development Tools: U L\ TL AB % ~ SIMULINK! ......................... 18Chapter 3. Polarization Mode Dispersion ...................................................... 19

..............................................................................................1 What is PMD? 193.1.1 Birefringence and PMD ............................................................. 193.1.2 Mode Coupling in Long Fiber ................................................21....................................................................2 Principal States of Polarization 22

.........................................................................................3 Waveplate Mode1 26....................-4 Cornputer Simulationof PMD ........................................... 283.5 Channel S imulation using MA TL AB ~ nd SIMULINK? ........................... 31

Chapter 4. Adaptive Channel Equalizers and CPI Cancellers ............... 33.............................................1 Adaptive Channel Equalization Technique 3 3

4.1.1 Fundarnentals of Equalization ............................................. 33.............................1.2 Adaptive Equalization ................... ...............-... 35


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4.1.3 Adaptive Equalization Categorization and Algorithm ................... 364.2 Channel Equalization and CPI Cancellation ................................................. 394.3 Simulation of Channel Equalizea and CPI Cancellers .................................46.....3.1 Simulation Structure of Channel Equalizers and CPI Cancellers 46..................3.2 Performance of Channel Equalizm and CPI Cancellea 48

Chapter 5.System Performance ............................................................... 50...................................................................................................... Eye Pattern 505.2 Bit Error Rate Evaluation .............................................................................. 5 1Chapter6. Bidirectional Transmission System ........................................... 596.1 Bidirec tional Transmission System .............................................................. 59...................................................................................1.1 Fiber Coupler 60.......................................................................1.2 Rayleigh Backscatter 616.2 Echo Cancellation ......................................................................................... 636.3 Power Budget .............................................................................................. 64

...................3.1 Pcwer Budget of the System without Echo Cancellers 65........................3.2 Power Budget of the System with Echo Cancellen 666.4 Simulation of Echo Cancellen Using SIMULMK" .................................... 67..........................................................................................hapter 7 Conclusions 69.....................................................................................................1 Conclusions 69................................................................................................2 Future Work 70

...................................................................................................................eferences 7 1Appendix A. SIMULINK"Simulation Modeis ............................................. 75Appendix B. M A T L A B % O U ~ C ~ode ............................................................. 89


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LIST OF TABLESTable 5.1Table 5.2Table 5.3Table 6.1Table A.1Table A.2Table A.3Table A.4Table A.5Table A.6Table A.7Table A.8Table A.9Table A. 0Table A . 1Table A. 2

Jm.. (dB) when DGD is 10ps .................................................................. 53Jm,,(dB)when DGD is 100ps ............................................................... 54J,, (dB) when DG D is 500 ps ................................................................. 56WD 515 senes: 153 11558nrnBidirectional Coupler .............................61Parameters of System Model ............................................................... 7 5

...............................................................aram eten of Transrnitter-A/I3 76Parameters of Transmitter........................................................................ 77Parameters of Channel ............................................................................ 78.............................................................arameters of Impulse Response 80Parameters of Gaussian Noise ................................................................. 81Parameters of Baseband Demodulation................................................. 82Param eten of Adaptive Channel Equalizers. CPI Cancellers and EchoCanceliers ............................................................................................... 84

Parameters of Normalized LMS Adaptive Filter...................................... 85. . . . .................................................................arameters of Decision Circuit 86Parameters of Performance Meter........................................................... 87.........................................................arameters of Square Error Trend 8 8

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LIST OF FIGURESFigure 2.1Figure 2.2Figure 2.3Figure 3.1Figure 3.2Figure 3.3Figure 3.4Figure 3.5Figure 3.6Figure 3.7Figure 3.8Figure 3.9Figure 3.10Figure 4.1Figure 4.2Figure 4.3Figure 4.4Figure 4.5Figure 4.6Figure 4.7Figure J.8aFigure 4.8bFigure 4 . 8 ~Figure 4.9aFigure 4.9bFigure 5 .Figure 5.2Figure 5.3Figure 5.4Figure 6.1Figure 6.2Figure 6.3Figure 6.1Figure 6.5Figure 6.6Figure 6.7Figure 6.8Figure 6.9Figure A .Figure A.2Figure A.3

...........................................................................assband System Model 8Impulse Response of RC Filter................................................................. 12..........................................................................aseband System Mode1 16PMD and DGD ........................................................................................ 20................................................................................................DF of A t 25Waveplate Mode1 ..................................................................................... 27........................................................agnitudes of Frequency Response 29Phases of Frequency Response ............................................................. 29............................................................agnitudesof Impulse Response 30Phases of Impulse Response ................................................................... 30Interferometric Measurement of the 10-ps PMD Emulator ..................... 31Optical Fiber Channel Simulation ............................................................ 32Simulation of Baseband Impulse Responses .......................................... 32Equalization ............................................................................................ 34Categorization of Adaptive Equalization ................................................. 36Linear LMS Transversal Equalizer ........................................................ 37Channel Equalizers and CPI Cancellers ................................................ 39Vertical Channel Eq ud ize n and CPI Cancellers ..................................... 41LM S Adaptive Filter .............................................................................. 47Adaptive Channel Equalizers and CPI Cancellers ................................... 48J (dB) hen ~ 0 . 2.................................................................................. 49J (dB) hen ~ 0 . 4.................................................................................. 49J (dB) when ~ 0 . 6................................................................................. 19J (dB) when Filter Length= 16 ................................................................. 49J (dB) when Filter Length=8 .................................................................. 49...........................................................................ully Open Eye Pattern 50.............................................................................ye Pattern (SN R=30) 51..............................................................................ye Pattern ( S M = 15) 51BER Plot ............................................................................................ 5 8Bidirectional Transmission System ...................................................... 9...........................................................................................iber Coupler 60......................................................................Possible Backscatter Plot 62Echo Cancellation ...................................................63Structure of Echo Cancellers ................................................................... 64Power Budget ........................................................................................... 64Simulation of Echo Cancellers ...................................... 67Eye Pattern without Echo Cancellen ..................................................... 68Eye Pattern with Echo Cancellers ........................................................ 68System Mode1 ................................................................................... 71Transmitter-A .................................... . 72................................................................................................................ransmitter 73

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Figure A.4 Channel .................................................................................................... 74Figure AS Opticd Fiber Channel with PMD .......................................................... 75Figure A.6 Impulse response ................................................................................... 76Figure A.7 Gaussian Noise .......................................................................................77Figure A.8 Baseband Demodulation .................................................................-....... 78Figure A.9a Receiver with Echo Cancellers ................................................................ 79Figure A.9b Receiver without Echo Cancellers ........................................................... 79

..................................igure A. Oa Adaptive Channel Equaliters and CPI Cancellers 80Figure A. Ob Echo Cancellers ..................................................................................... 80Figure A.11 Normalized LMSAdaptive Filter ............................................................81.......................................................................................igure A. 2 Decision Circuit 82Figure A. 3 Performance Meter ................................................................................... 83FigureA. 4 Square Error Trend ..................................................................................84


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LIST OF SYMBOLSreal part of the tap weight c(n). See Eqs. (4.20,4.21).input PAM signal in the horizontal channel.input PAM signal in the vertical channel.imaginary part of tap weight c(n). SeeEqs. (4.20,4.21).bit rate.bandwidth of the message signal.light speed. See Eq. (3.1).filter coefficients.estimated of the equalizer tap weights at step n. See Eq. (4.13).impulse response of the equalizer. See Fig. (4.2).channel equalizers . See Fig. (4.4).CPI cancellen. See Fig. (4.4).estimated tap-weight of the equalizerc,,.(n). See Eq. (1.44).estimated tap-weight of the CPI canceiler c,.,,(n).See Eq. (4.45).complex coefficien ts of PSPs. See Eq. (3.9).Fourier transfomi of c(r).'desiredn signal. See Eq. (4.8).differential of *.phase retarder. See Eq. (3.1 7) and Fig. (3 3).estimated error. See Eq. (4.8).estimated error vector. See Eq. (4.18).input signal power. See Eq. (5.1).mathematical expectation ofinput optical field vector. See Eqs. (3.6.3.7).amplitude of the input PSPs n frequency domain. See Eq. (3.7).amplitude of the output PSPs n frequency domain. See Eq. (3.8).complex unit vector sp ec ifj hg the PSPs. See Eq. (3.7).complex unit vector specifj4ng the PSPs. See Eq. (3.8).amplitude of the time-varying input field. See Eq. (3.9).amplitude of the time-varying output field. See Eq. (3.9).output optical field in frequency domain. See Eqs. (3.6,3.8).unit vectors specifying the two output PSPs. See Eq. (3.9).carrier Frequency. See Eq. (2.3).sampling frequency.receiver filter. See Eqs. (2.10,2. 11) and Figs. (2.1.2.2).overall complex baseband impulse response of the trammitter, channeland matched filter. See Fig. (4.1).Fourier transform o f& (I). See Fig. (4.1).transmitter filter. See Eqs. (2.2.2.11) and Figs. (2.1.2.2).combined impulse response of the transmitter, channel, matched fter and


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equalizer. See Eq. (4.3) and Fig. (4.1).Planck's constant, 6 . 6 3 ~ JlHz. See Eq. (6.8).ma& of the baseband impulse responses of the optical channel. SeeEq. (4.11).complex baseband impulse response. See Eq. (3.2 1).matrix o f he passband channel impulse responses. See Eq. (2.4).baseband horizontal to horizontal impulse response. See Eq. (2.5).baseband horizontal to vertical impulse response. See Eq. (2.5).baseband vertical to horizontal impulse response. See Eq. (2.5).baseband vertical to vertical impulse response. See Eq. (2.5).matrix of the baseband horizontal to horizontal channel impulse responses.See Eq. (2.24).matrix of the baseband horizontal to vertical channel impulse responses.See Eq. (2.23).in-phase of passband horizontal to horizontal impulse response. SeeEq. (2.5).in-phase of passband horizontal to vertical impulse response. SeeEq. (2.5).in-phase of passband vertical to horizontal impulse response. SeeEq. (2.5).in-phase of passband vertical to vertical impulse response. See Eq. (2.5).passband horizontal to horizontal impulse response. See Eq. (2.4) andFig. (2.1).passband horizontal to vertical impulse response. See Eq. (2.4) andFig. (2.1).passband vertical to horizontal impulse response. See Eq. (2.4) andFig. (2.1 ).passband vertical to vertical impulse response. See Eq. (2.4) andFig. (2.1 ).quadrature of passband horizontalEq. (2.5).quadrature O f passband horizontalEq. (2.5).quadrature of passband vertical taEq. (2.5).quadrature of passband vertical toEq. (2.5).

to horizontal impulse response. Seeto vertical impulse response. Seehorizontal impulse response. Seevertical impulse response. See

matrix of thebaseband vertical to horizontal channel impulse responses.See Eq. (2.22).matrix of the baseband vertical to vertical channel impulse responses. SeeEq. (2.21).transfer matrx of the fiber channel in the waveplate model. SeeEq. (3.20).imaginary number, satisQing j = -1.


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minimum MSE. See Eq. (5.10).mean-squared error vector. See Eq. (4.19).coupling length.fiber length.DGD enerator. See Eqs. (3.16,3.17) and Fig. (3.3).system loss.effective index of refiaction for fast mode. See Eq. (3.1).effective index of refraction for slow mode. See Eq. (3.1 ).randorn Gaussian noises. See Eq. (5.2).number of the transmitter or receiver filter taps. See Eq. (2.2).transfer maaix of the segment k n a single mode fiber. See Eq. (3.17).receiver sensitivity (average nurnber of photos per bit).probability density function of DGD. See Eq. (3.12).bit error rate.power in the starting mode. See Eq.(3.3).average power in the orthogonal polarization mode. See Eq(3.3).crosstalk. See Eq. (6.3).directivity. See Eq. (6.4).excess loss. See Eq. (6.1).optical power launched into the fiber. See Eq. (6.6).insertion loss. See Eq. (6.2).Rayleigh backscattered opticai power. See Eq. (6.6).received sensitivity (dBm). See Eq. (6.8).Rayleigh scatter loss.symbol error probability for Clevel QAM signal. SeeEq. (5.4).split ratio. See Eq. (6.5).total power in the fiber. See Eq.(3.3)power of Trammitter-B. See Fig. (6.6).power of Transmitter-A. See Fig. (6.6).signal power after Coupler-A. See Fig. (6.6).signal power received by Coupler-B. See Fig. (6.6).signal power received by Receiver-B. See Fig. (6.6).echo signal power due to coupler crosstalk and the Rayleigh scatter. SeeFig. (6.6).cross-correlation vector between the input vector to the equalizer uJn) andthe desired response x,(n). See Eq. (4.29).instantaneous estimate of P,,,(n). See Eq. (4.40).cross-correlation vector between the input vector to the equalizer uJn) inthe vertical polarjzation channel and the input vector in the verticalpolarization channel u,(n). See Eq. (4.3 1).instantaneous estimate of Q&). See Eq. (4.42).Q-function of 0 .filter rolloff factor. See Eq. (2.1 1) and Fig. (2.2).

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phase rotator. See Eq. (3.7).auto-correlation matnx of the input vector to the equalizer uv(n). SeeEq. (4.30).instantaneous estimate of F&,,(n). See Eq. (4.41).baseband signal vector in horizontal channel. See Eq. (2.19) andFig. (2.3).baseband signal vector in vertical channel. See Eq. (2.19) and Fig. (2.3).modulated signal after the modulator in the horizontal channel. SeeEq. (2.3) and Fig. (2.1).in-phase signal after the transmitter filter in the horizontal channei. SeeEq. (2.2) and Figs. (2.1,2.3).in-phase signal after the trammitter filter in the vertical channel. SeeEq. (2.2) and Figs. (2.1,2.3).quadurate signal d e r he transm itter filter in the horizontal channel. SeeEq. (2.2) and Figs. (2.1, 2.3).quadurate signal af ter the transmitter filter in the vertical channel. SeeEq. (2.2) and Figs. (2.1, 2.3).modulated signal after the modulator in the vertical channel. See Eq. (2.3)and Fig. (2.1).fraction of captured op tical power. SeeEq. (6.6).complex baseband waveform of the modulated signal in the verticalchannel. See Eq. (2.13).three-component unit Stokes vector. See Eq. (3.10).transmitter filter sampling time. See Eq. (2.2).optical fiber complex transfer matnx. See Eq. (3.4).discre te baseband input signal to the equalizer. See Fig. (4.3).K dimensional input discrete signal vector at step n. See Eq. (4.9).output signal vector of the fiber channel. See Eqs. (2.6,J. 12).received baseband signal vector in the horizontal channei. See Eq. (2.25)and Fig. (2.3).received baseband in-phase signal in the horizontal channel. SeeEq. (2.25) and Fig. (2.3).received baseband in-phase signal in the vertical channel. See Eq. (2.25)and Fig. (2.3).received baseband quadrature signal in the horizontal channel. SeeEq. (2.25) and Fig. (2.3).received baseband quadrature signal in the vertical channel. SeeEq. (2.25) and Fig. (2.3).received baseband signal vector in vertical channel. See Eq. (2.25) andFig. (2.3).output of the horizontal channel. See Eqs. (1.6,4.11).output of the vertical channel. See Eqs. (2.6,4.11).input signal vectors to the equalizer in the vertical and horizontal channels.

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See Eq. (4.15) and Fig. (4.4).u~ ( o ) ,~ ~ ( 0 )ornplex quantities of the transfermatrix, T(o).~ ( 0 Gaussian noise signal. See Eq. (2.8) and Fig. (2.1).'2 different group velocity. See Eq. (3.2).VA{) Gaussian noise signai in the horizontal channel. See Eq. (2.8) andFig. (2.1).Gaussian noise signal in the vertical channel. See Eq. (2.8) and Fig. (2.1).angular carrier frequency. See Eq. (3.21) .input optical pulse width. See Fig. (6.6).'desiredm signal vector. See Eq. (4.17).input QAM signal vector for the horizontal channel. SeeEq. (2.1) andFig. (2.1).in-phase signal in the horizontal polarization channel. See Eq. (2.1) andFig. (2.1).in-phase signal in the vertical polarization channel. See Eq. (2.1) andFig. (2.1).quadurate signal in the horizontal polarization channel. See Eq. (2.1) andFig. (2.1).quadurate signal in the vertical polarization channel. See Eq. (2.1) andFig. (2.1 ).input QAM signal vector for the veniczl channel. See Eq. (2.1) andFig. (2.1).output of the adaptive equalizer. See Eq. (1.7).output signal vector. See Eq. (4.10) and Fig. (4.4).output of the equalizer. See Fig. (4.1).output of receiver in the horizontal channel. See Eq. (2.9) and Fig. (2.1).demodulated in-phase signal in the horizontal channel. See Eq. (2.10) and

Fig. (2.1).demodulated in-phase signal in the vertical channel. See Eq. (2.10) andFig. (2.1).demodulated quadurate signal in the horizontal channel. See Eq. (2.10)and Fig. (2.1 ).demodulated quadurate signal in the vertical channel. See Eq. (2.10) andFig. (2.1).output of the receiver in the vertical channel. See Eq. (1.9) and Fig. (2.1).received PAM signals in the horizontal channel. SeeFig. (2.1).received PAM signals in thevertical channel. See Fig. (2.1).average differential go u p delay time. See Eq. (3.12).fiber loss.propagation constant for the fast mode. See Eq. (3.1) and Fig. (3.1).propagation constant for the slow mode. See Eq. (3.1) and Fig. (3.1 ).received SNR. See Eq. (5.5).Rayleigh scattering coefficient. See Eq. (6.6).system SNR. See Eq. (5.12).

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atenuation coefficient. See Eq. 6.6).bitefngence. See Eq. (3.1).differential index of fraction. See Eq. (3.1).DGD. See Eq. (3.2) and Fig. (3.1).nn s value of the differential group delay dr. See Eq. 3.14).gradient of J,(n) o c,,(n). See Eq. (4.23).gradient of JJn) o c,,,(n). See Eq. (4.22).independent unifonly distributed random variables, distnbuted frorn -n or. See Eqs. (3.18,3.19).wavelength.step-size, an adaptation constant. See Eq. (4.8).variance of the channel Gaussian noise.variance of n,.variance of n,.Mauwe;elliandistributed DGD. See Eq. (5.16).arriv d times in the two polarization States. See Eq. (3.9).average DG D in each segment of a fiber. See Eq. 3.15).initial phases in the horizontal channel. See Eq. (2.18).initial phases in the vertical channel. See Eq. (2.17).polarization-independentphase. See Eq. (3.4).Ludolphian number, 3.1415926 --.angular frequency.central angular fiequency.phase of the optical field. See Eqs. (3.7.3.8).dispersion vectors, indicates rate and direction of rotation. See Eq. (3.10).Hermitian transpose of *.matrix transpose of a.complex conjugate of *.convolution of and +.multiplication of and +average of 0 .absolute value of a.


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BERCPIDFPDG DDSPEDFAFIRIFIM-DDISILM SLTEMSEOTDRPAMPD FPbIDPNPSPsQAMRLSRMSSNRZF

LIST OF ABBREVIATION AND ACRONYMSBit Error RateCross Polarization Interferenceinverse Discrete Fourier TransformationDifferential Group DelayDigital Signal ProcessingErbium-Doped Fiber AmplifierFinite Impulse ResponseIntem ediate FrequencyIntensity Modulation with Direct DetectionInter-Symbol InterferenceLeast Mean SquareLinear Transversal EqualizerMean Squared ErrorOptical Time Domain R eflectom ewPulse Amplitude ModulationProbability Density FunctionPolarkation Mode DispersionPseudonoisePrincipal Sta tes of PolarizationQuadrature A mplitude M odulationRecursive Least SquaresRoot Mean SquareSignal Noise RatioZero Forcing


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Chapter 1Introduction

1.1 Introduction to the ResearchOptical fiber communication system s have attracted more attention in the recent

years, because of the outstanding advantage of optical fiben. The most significant men tof an optical fiber is its enormous bandwidth. An optical fiber communication systemuses a very high carrier frequency. around 200 T M 2]. which yields a far greaterpotential bandwidth than a cable system; coaxial cables have a bandwidth up toapproximately 500 MHz l]. In optical systems, this carrier frequency is usuallyexpressed as a wavelength, 1.55 Pm. This enormous bandwidth provides the potential totransmit signals at a very high speed.

At present. however, this potential bandwidth cannot be fully utilized. Asignificant reason is the fiber dispersion. Usually, the fiber dispersion includes inter-mode d ispersion, intra-mode dispersion and PMD 2]. In a single mode fiber, inter-modedispersion is absent. In the systems used today, intra-mode dispersion is not a significantproblem. PMD s the main limitation that confines the optical fiber transm ission system sfiom utilizing the bandwidth effectively.

An EDFA (Erbium-Doped Fiber Amplifier) can regenerate weak signa ls, but itmakes the dispersion worse since the dispersion effects accumulate over the muitipleamplifier stages. Therefore, the current task of optical fiber communication system


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design is to solve the fiberdispersion problem, especially PMD. here aremanytechniques developed for PMD ompensation in both the optical domain and theelectrical domain [2]. Early strategies to reduce PMDwere focused on reducing theintrinsic PMD f the fiber by altering the manufacturing process [60,6 1. This led to thelow and stable values of PMD n the new generation of single-mode fibea beingmanufactured [62]. More recent efforts have been examining ways in which lightwavesystems cm be designed to better accomm odate othenvise unacceptable levels of PMDand thus make full use of the existing embedded fiber base. Such strategies ind ud ereduction of ISI (Inter-Symbol Interference) by electronic equalization in the receiver[40,41.431, and optical equalization using an automatic polarization controller at thereceiver or transmitter [42,59].

Winters proposed a strategy for PMD ompensation using optical equalization[42]. He used a polarkation controller to adjust the polarization into a fiber to one of thePSPs (Principal States of Polarization) of the fiber. The receiver detects only one of thePSPs o elim inate first-orderPMD. A method to track the changing PSPs using agradient search algorithm was also presented in his paper. The advantage of opticaltechniques is that they can be bit rate independent, while for electrical techniques, thecomplexity and difficulty will increase dramatically as the bit rate increases. With thedevelopment of high speed DSP (Digital Signal Processing) and Si/SiGe epitaxialtransistors (64,651, an electrical equalizer operating at 10 GHzmay be realized [j 1. Ahi& speed e lectnca l equaiizer becornes an attractive solution for PMDmitigation due toits compact and cost effective implementation.


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Based on previous research, the intent of this thesis is to constnict a simulationmodel of an optical fiber comm unication system, including simulaiing the optical fiberchannel with PMD s well as the electrical channel equalizen and CPI cancellers tocompensate PMD.

1.2 ThesisContributionsThis hesis proposes a novei optical fiber transmission system where two 4-QAM

(Quadrature Amplitude Modulation) signals are transmitted separately over vertical andhorizontal polarization channels. This strategy hasa clear advantage, more systemcapacity and higher bit rate. However, it aiso causes CPI between the two signals in thetwo polarization channels. Actually, CPI is the result of PMD ue to the power coupling.PMD nd CPI c m be compensated using electrical adaptive filters.

The simulationof PMD n an optical fiber channel is an mportant part of thisthesis. By applying the theory of PSPs [45,52] and the waveplate model [42,57,59], thefrequency responses and baseband impulse responses are obtained. The complexbaseband impulse responses are the backbone to construct the simulation of the opticalfiber channel. Gaussian noise signals are also added to represent the thermal noise in thechanne1 and the fiont end of the receiver.

This thesis proposes the combination of channel equalizers and CPI cancellerswhich are applied to compensate the PMD nd CPI in an optical fiber communicationsystem. It shows the potential application of electrical adaptive equalizen and CPIcancellers inanoptical fiber comm unication system and demonstratesa bright funue for


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ihis strategy. The structure of the channel equaiizea and cancellen is described andanalyzed. The contribution that the equalizen and CPI cancellers offer is demonstratedby the performance improvement they create.

The system performance such as theeye pattern and bit error rate are presented.The bit error rate is an important parameter to measure the performance of an optical fibercommunication system. It shows the PMD, PI and the noise effects on the optical fibercommunication system as well as and the contribution of adaptive channel equalizers andCPI cancellers in the receiver. Also, the bit error rate reveals the relation between PMD

and the received SNR (Signal Noise Ratio). and helps us determine the requirement forPMD nd the noise tolerance.

As a funher study, bidirectional transmission through a single t'be r is alsopresented. The crosstalk in the coupler and the backscatter [1 ] in the fiber causes aretumed echo in this system. To eliminate this echo, echo cancellers in the receiver mightbe used. The power penalties when there are echo cancellers and no echo cancelles[19,28] are compared to show the contribution of the echo cancellers.

1.3 ThesisOutlineThe outline of the thesis is as follows:Chapter 2: The system simulation model is consmicted. Signal format.,

trammitter, channel and receiver structuresare defined. Therequirements of the baseband s imulation model are proposed andexplained. Finally, the simulation tools, MATL.AB' and

-4-


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SIMULINKD,are introduced.Chapter 3 PMD oncepts and the PSPsarediscussed first. Based on the

theory of PSPs, a waveplate mode1 is presented, which is used tosimulate the frequency responses and complex baseband impulseresponses. The computer simulation of PMD s achieved by usingthe complex baseband impulse responses. n i e method ofsimulation using MATLAB~ nd sIMULMK' is also shown inthis c hapter.

Chapter 4: At first, the adaptive channel equalizer technique is introduced,which includes hd am enta l equalizer theory and the type ofadaptive equalizers. Then. th e method to determine the adaptivetap weights of the equalizer and CPI canceller using the normalizedLMS Least Mean Square) algorithm is derived. The adaptivechannel equalizers and CPI cancellen are constnicted based onthese results. The performance of the adaptive eq ua liz e~ndcancellen is also analyzed.

Chapter 5: The eye patterns of the received signals are s h o w in this chapter.The system bit error rate curve is d s o shown. The discussion ofthe system performance is presented.

Chapter 6: Bidirectional transmission in a single fiber is proposed. Thecoupler crosstalk and Rayleigh backscatter as well as their effect onthe echo are introduced. Then, the power penalty is calculated. and


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the implementation with or without an echo cancellers is discussed.Chapter 7: The conclusion of this research and the future work are presented.


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Chapter 2System Simulation Model2.1 System Model

In this thesis, Csta te QAM ignais are launched onto a fiber. The benefit of thismodulation scheme is good spectral efficiency. A dually polarized digital optical fibertransmission system with two independent QAM input signals is constructed [21-26].One QAMsignal is transmitted over a horizontal channel and the other is transmittedover a vertical chame l. This system is shown in Fig. (2.1).

The input signals A,(n) and A,,(n)are 4 level PAM (Pulse AmplitudeModulation) signals. They are uniformly distributed random integer signals ranging from1 to 4. These P A ! signals are then converted to QAM signals x,.(n) and x,(n). bymapping,

where x,,(n), xh(n) are QAM ignal vecton in the vertical and horizontal channels.x,,(n) is the in-phase signal in the vertical polarization channel,x,(n) is the quadrature signal in the vertical polarization channel,.r,(n) is the in-phase signal in the horizontal polarization channel, andx,,(n) is the quadurate signal in the horizontal polarization channel.


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x1Jn), , (n) , x,,(n)and x,,(n) are the values of 1 + , -1+ , I - and - - . And j is theimaginary number, satisfying j = -1.

Figure 2.1 Passband System Model

The signals after transmitter filter, g(r ) , are s,,,(t). s&), sJr) and s&),

where sI&)is the in-phase signal after transmitter filter in the vertical channel,sJt) is the quadrature signaiafier transmitter filter in the vertical channel,


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s,(t) is the in-phase signal afier transmitter filter in the horizontal channel,s,,(t) is the quadraturesignal afier transmitter Blter in the horizontal channel,g(t) is the transm itter filter, which can be selected by thedesigner to satisfy the

limitations on transmitted power and bandwidth,T is the transmitter filter signaling rate or symbol period, andN is the number of transm itter filter taps.

The above signals are then passed to modulators. The rnodulated signals are s,.(t) nddo*

where sJ t ) is the modulated signal afier the modulator in the vertical channel.sh(t) s the modulated signal after the modulator in the horizontal chamel. and1;. is the carrier frequency.The optical fiber channel is characterized by a matrix h,(r) ui th four passband

impulse responses,

where hm&) is the passband vertical to vertical impulse response,h,,,(t) is the passband horizontal to horizontal impulse response,h,,(r) is the passband vertical to horizontal impulse response, andhph(t) s the passband horizontal to vertical impulse response.


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InEq. (2.4), the passband channel impulse responses are related to the baseband channelimpulse responses by,

where Re[.] denotes the real part of m. The subscripts of h(t) in Eq. (2.5) are of the foms , s2 ,. The first subscnp t, s, is one of =pn,bn, "in or "qn , denoting the type of impulseresponse. 'passband". 'cornplex baseband", "in-phase baseband" or "quadraturebaseband", respectively. The second and the third subscripts, s2and s,, will be either "vnor "hn. denoting "horizontal" and "verticaln respectively. The subscript s, denotes theinput to the channel and the subscript- enotes the output of the channel.

Having known the modulated signals and channel responses, the output of thechannel. u(t ) ,can be obtained [25],

where denotes convolution,

uv(r) s the output of the verticai channel,uh(f) s the output of the horizontal channel, andif)s the Gaussian noise signal vector with the components in both vertical and


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horizontal channels, v,(t)and vh(t),

The noise signais vv( t ) nd v,,(t) are zero mean uncorrelated with variances of o f .In the receiver, u( t ) is first demodulated and then passed through receiver filters.

The outputs of the receiver are y,(t) and yh(t),

and we have,

where y , , ( t ) s the demodulated in-phase signal in the vertical channel.y&) is the dem odulated quadrature signal in the vertical channel,y&) is the demodulated in-phase signal in the horizontal channel.y,,(!) is the demodulated quadrature signal in the horizontal channel,#.and A are the phase offsets between the trammitter and he receiver in the

vertical and horizontal channels.A t ) is the receiver filter that is used to cut off the fiequency components near 2f,

and preserve the desired com ponent near zero.The PAM signals &(no and Z,(n) for the vertical and horizontal channels can be obtained


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by demapping the received QAMsignals y,(t) and yh(t).&(t) a d Zh(t)are similar instructure to A.(!) and Ah([) .

In digital transmission, signals are ohen represented by ideal steps, and they haveinfinite bandwidth. ' o restrict the spectnim o f the source signal, an RC (Raised Cosine)filter is commonly used [I l] . In this system, the RC ilter is divided into two parts, one isthe transmitter filterg( t ) and the other is the receiver filterf(t). Each part is the squareroot of the magnitude response of the RC filter in the frequency domain. nie impulseresponse of g(t ) and f ( t ) is [Il] ,

g ( t )= /(O

where r is the rolloff factor, O 5 r s 1.The impulse response of RC filter is s h o m in Fig. (2.2) [ I l ] .

Figure 2 2 Impulse Response ofRC Filter


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2.2 Baseband Simulation and CoherentDemodulationThe previous section described the passband model where the carrier frequencyf,

was introduced in the model. According to Nyquist's baseband sarnpling theorem, itrequires that a simulation sampling frequencyf; must be at least twice the m aximumfiequency being modeled. Since the optical carrier fiequency is usually very high, around200 THz,modeling passband comm unication systems involves high com putational loads.TOalleviate this problem, baseband simulation techniques are used.

Baseband simulation, also known as the low-pass equivalent method, uses thecomplex envelope of a passband signal. As an example, consider the vertical channel inFig. (2.1). The equation for modulating the in-phase signal s,,(t) and the quadrature signalS , ~ , . ( I ) y the carrier frequency is [j, 11,

S . ( t ) = JS[S, , . ( t)cos(2e . r )- s ( r ) sin(2;rj))] .which is equivalent to,

where Sb&) is called the complex baseband waveform,

The high frequency componentf;. disappeared in this expression. Baseband simulationmodels he com plex baseband waveform &(t) only. Let B, be the bandwidth of themessage signal. The baseband simulation requires the simulation sampling rate to be


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larger than 2B,. In the baseband simulation, the signal bandwidth is always assumed tobe much smaller than the carrier fiequency, or B,


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where ubth(t)s the received baseband in-phase signal in the horizontal channel, andu4,(t) is the received baseband quadrature signal in the horizontal channel.

2.3 Baseband System Mode1Because of the benefit of baseband simulation, the passband model which is

shown in Fig. (2.1)can be uansfonned to a baseband model. A baseband model isconstnicted which includes a trammitter, a receiver and a chamel? hey are shown inFig. (2.3).

In the baseband model, the complex signals can be expressed as the mauix forms.

I Figure 2.3 Baseband System Modei


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Thus, in the trammitter, the baseband modulated signals s,,(t) and s,(r) are,

where s,,.(t) is the baseband signal vector in the vertical channel, ands&) is the baseband signal vector in the horizontal channel.Corresponding to the passband channel impulse responses in Eqs. (2.4,2.5), the

baseband impulse responses are expressed in a matrx f o m ,

where h(t) is the overall baseband cornplex impulse response matrix,h,,.(t) is die baseband impulse response mauix, from the vertical to the vertical

channel.

h,.,,(r) is the baseband impulse response matrix, fiom the vertical to the horizontalchannel,

hdt ) is the baseband impulse response mauix. fiom the horizontal to the verticalchannel,


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h,(r) is the baseband impulse response matrix, fiom the horizontal to thehorizontal channel,

In the receiver, the received baseband signai vectors uh(t) and uhh ( t )are,

From Eqs. 2.17,2.18), we have,Y (0 cos$,. - sin 4,. tibfv tm=[&) ]= [ sin$,. cos#,. ][ lw ( t ) ] and

Thus, in the baseband model, there is a phase rotation between u&) and y , . ( [ ) , u,,(r) andyh( t )which can be expressedas a matriu. Therefore, the basebandsystem rnodel isconstnicted as that s h o w in Fig. (2.3) which is used as a foundationof the cvholesimulation system.


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2.3 System DevelopmentTools:MATLABDand SIMULINKBMATLAB' is a high-performance language for technical computing [13]. It

integrates computation, visualization, and programrning in an easy-to-use environmentwhere problems and solutions are expressed in familiar mathematical notation. TheSignal Processing Toolbox is a collection of tools built on MATLAB" [14]. This toolboxsupports a wide range of signal processing operations, from waveform generation to filterdesign and irnplernentation,parametric modeling,and spectral ana lysis [1 5,161.

SIMULINK% a s o h a r e package in MATLAB' for modeling, simulating, andanalyzing dynamical systems [IO]. It supports linear and nonlinear systerns, modeled incontinuous time, sarnpled time, or a hybrid of the two. sIMuLMK' is a powerful toolfor real-time simulation. The communicationsToolbox and DSP BIockset are the mostus eh l tools for simulation of telecommunication systems. The Communications Toolboxis a collection of computation functions and simulation blocks for research, development.system design. analysis, and simulation in the communicationsarea [ I I l . The DSPBlockset is a collection of block libraries which have been designed specifically for DSPapplications. and include operations such as classical, multirate, and adaptive filtering,compler and matrk arithmetic, tmnscendental and statiaical operations, convolution, andFourier transforms [121.


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Chapter 3Polarization Mode Dispersion3.1What i s PMD?

3.1.1 Birefringence and PMDIn single-m ode fibers there are two different polarization m odes, the fast mode

and the slow mode, due to birefnngence which is caused by intemal or externa l stresses

or by non-perfect circularity of the fiber core. Each polarization mode has its ownpropagation constantPT nd 4,which are shown in Fig. (3.1). Usually the state ofpoiarization of an arbiirary optical field can be represented by the vector sum of the fieldcomponents aligned with the two polarkation modes.

The difference in propagation constants is called birefnngence [3].

where A p is birefnngence.p, is the propagation constant of the slow mode,P, is the propagation constant of the fast mode,Anefl s the differential index of fiaction'n, s the effective index of refiaction of the slow mode,n, is the effective index of re hc tio n of the fast mode,c is the speed of light, 3x l O' m/s,

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O is the angular frequency, and/ 1 denotes the absolute value of 0 .

The birefnngence, AB is related to the different group velocity v ,

where d(m)denotes the differential of 0 .The different group velocitiesof the modes limit the bandwidth of the optical

fibercommunication system, because they cause pulse broadening. Thus the differencein the group delays of the fast and slow modes is,

where d is also called the differential group delay (DGD). nd L is the fiber length.

1 DGD differentidp u p delayI Figure 3.1 PMD and DCD 1701

Because of the existenceof birefringence, if the input pulse excites bothpolarization cornponents, it becomes broader at the fiberoutput since the two componentsdisperse along the fiber because of theDGD. see Fig. (3.1). This phenornenon is called


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Polarization Mode Dispersion (PMD). And the delay between the two components iscalled the DGD.

3.1.2 Mode coupag in long fiberIn Eq. (3.2), for a specific Anef,DGD s proportional to the fiber length, L .

However, this phenornenon occurs only in short fiber. In long-distance optical fibertransmission systems, there is a polarization mode coupling due to power of onepolarization mode leaking into the other mode. The mode coupling is a random process,but the average power will grow with the distance.

A couplin g length, 1, is defined to be the length at which the average power in theorthogonal polarization mode, Ph(( . ) , s within l/e2 of the power in the starting mode. Pu

where P,o,,i is the total power in the fiber. and(a ) stands for the ensemble average of e.The coupling length. [,cm be used to distinguish between a long fiber and a shon

fiber. When a fiber length is less than l, , it can be considered to be a short-length fiber.In a short-length fiber, the polarization effects are deterministic and PMD grows linearlywith the fiber length. When a fiber length is much larger than [,we can regard it as along-length fiber. In a long-length fiber, the fiber shows a statistical variation in thepolarization due to both PMD and mode coupling. Aiso, the DGD ecomes proportionalto the square root of the fiber length [57].


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3.2 Principal States of PolarizationPoole proposed a well known model, "Principal States of Polarization", in 1986 to

describe PMD SZ]. It revealed that 'in any linear optical transmission medium that hasno polarization-dependent loss there exist orthogonal input states of polarization forwhich the corresponding output states of polarization are orthogonal and show nodependence on wavelength to first order*. Such states are called principal states ofpolarization (PSPs).

Under the assumptions of this model, the linear medium can be described by acomplex transfer matrixT(o) 67],

where a, s the fiber loss,HO, ) is the polarization-independent phase,1 1 , ( 0 ) and u,(o) are complex quantities, and they satisfy the relation,

1% (dl2[ri,(dl21, and0 ' denotes complex conjugate of 0 .Suppose the input optical field vector is Eu(@), nd the output opticai field is

Edo). The relationship between these optical fields is,Edo) T(o) E X 4 , ( 3-6 )

and the optical field Eu(@) nd E,(o) can be expressed in the form [2],E, (O)= E, ( o ) eJPU (O), ( 3.7 )


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EbO) = Ea #)elRh m),where Eu(#) is the amplitudeof the input PSPs n frequency domain,

E,(o) is the amplitude of the output PSPs in frequency domain,a i s he phase of the input PSPs in fiequency domain,p,,is the phase of the output PSPs in frequency domain, (o)s the complex unit vector specifj6ng the input PSPs, andEh(O ) s the complex unit vector specifying the output PSPs.The Eq. (3.6) is a frequency model used to describe the PSPs. The important

contribution of this model is that it points out the property of polarization invariance withfrequency for the fi n t order, that an optical pulse aligned with a principal state at theinput of a fiber will emerge at the output with its spectral components al1 having the samestate of polarkation [52]. This means that the only distortion in the pulse is the phasedistortion and it will not change the shape of the pulse.

Because of the existence of PSPs, any input signal can be expressed as the sum oftwo states. In either of these polarization states there is no pulse shape distonion. but thetwo states have d ifferent time delays. Thus we also describe the model in the timedomain as [54],

where E,(t) is the output PSP s in the tirne domain.Eu@)are the amplitudes of the time-varying input PSPsin the time domain,c . , -are complex coefficients, which are the fraction of the power of the input


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signal into each of the PSPs, ,-are unit vectors sjxcifying the two output PSPs, ndr , q are the &val times in the two polarization states.

The diffetence of the arriva1 times is DGD, A r = r. - z-, which shows two different PSPscause the pulse broadening at the output of a fiber. Eq. (3.9) describes the fiat-ordereffect of PMD in time dornain [45], which is shown to induce a dual splitting of the initialpulse. In short fiben, the principal states correspond to the polarization modes of thefiber. In the long fiber spans, these states are deterrnined by the cumulative effects of thebirefringence over the entire span [2].

In frequency domain, the polarization mode dispersion is manifested as afrequency dependent state of polarization at the output of a fiber [54,55]. Over a narrowfrequency range, it takes the fonn of a rotation of the polarization on the Poincar sphere,

where S is the three-component unit Stokes vector. representing the states of polarizationof the output on the Poincar sphere,

0 ndicates the rate and direction of rotation, and is usually referred to asdispersion vector, and

x+ denotes the multiplication of and + .The relationship between the fiequency domain and the time domain behaviors is

that the rate of rotation, 101,s equal to the differential group delay time experienced byspectrally narrow pulses having polarzation aligned at the input with two orthogonalPSPs S I ,


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Using the theory of PSPs, the statistical properties of DGD were studied. ThePDF Probability D ensity Function) of DGD, p(Ar), has k e n proven to be a Mavwelliandistribution [58],

10 otherwise,where a s the average differential group delay time,

Ar,, is the mis value of the DGD,

where 4, and Q, are dispersion vec toa which have independent Gaussian distributionswith zero mean and identical variance [56]. Therefore, to generate a M aw ell iandistributed variable, first we should pn era te three independent Gaussian randomvariables, then take the square root of sum of three squared Gaussian random variables.

Figure 3.2 PDF ofAr-25-


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Fig. (3.2) shows a probability distribution of the random Maxwellian distributedDGDs in every segment of a fiber whose average DGD through the whole fiber is 10 0 ps,and the number of segments,K, s 100. Thus, the average DGD in each segment, ( r ) is,

3.3 Waveplate Mode1The waveplate mode1 is a practical and effective approach to measure and

simulate the first-order PMD n a long single mode fiber and was discussed in papers[57,59]. According to the theocy of PSPs. a fiber which is polarization loss independent,excited by one of its PSPsdefined at the optical angular frequency o. ehaves as a simpledelay. Thus if we suppose the input field is an input reference frarne which consists oftwo orthogonal input PSPs, nd the output field is anoutput reference frarnewhichconsists of two orthogonal output PSPs, then the phase shifi of a fiber can be described bya Jones matrix, iCl,(o)which we cal1 the "DGD enerator",

where r, is the differential time delay of one segment, andK s the number of fiber segments.

Considering the existence of mode coupling which causes random otations, weshould add a phase rotator and phase retarder. Thus each segment of a single mode fibercmbe simulated as a combination of a DGD generator iCh(o),a phase rotator R, andphase retarder4 which is s h o w in Fig (3.9,

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where N,(o ) is the transfer matrix of the segment k in a single mode fiber, and0, and pk re independent unifomly distributed random variables. distributed

from -75 to IC.

1Figure 3.3Waveplate Mode1

n i e overall transfer matrix, H(w) ,is the concatenation of ail the segments s h o win Fig. (3.3),

where K m ) is determineci by the polarization-independentphase shift and the fiber loss.


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3.4 CornputerSimulation of PMDBased on the waveplate model, a computer simulation is used to get both

frequency responsesand impulse responses.The waveplate model provides the Frequency transfer response H ( o ) with central

frequency,/, = ciAc,where the central wavelength, JO is 1.55 Pm. n the digitaisimulation performed in this thesis, to balance the computing speed and the accuracy, weused 128 frequency points in the frequency domain, which means the sampling frequency,f;, is128xB,where B is the signal bit rate. Thus a passband Frequency is in the range from/; - j 7 2 t0f ; .~1; /2 .

In this simulation, the bit rate, B. is 2.5 GHz, he DGD is 100 ps, and K s 100segments. Fig. (3.1) shows the magnitudes of the frequency responses. H(o ) ,andFig. (3.5) shows the phases of these frequency responses.

Our objective is to get the baseband impulse responses of the optical fiber channelwith PMD. o achieve this objective, the passband frequency responses should be shifiedto baseband. Then the baseband impulse responses cm be obtained from,

h ~ t )D FT - I [ H ( ~ -,)],where h,(t) is the complex baseband impulse response,

DFT' denotes inverse discrete Fourier transformation, andq, s the central angular frequency.Thuswe can get four complex baseband impulse responses, hbbv(t), b,(t), h,,(t)

and hbM(t).Fig. (3 .6)shows the magnitudes of impulse responses, and Fig. (3.7) showsphases of impulse responses.


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Frequency Response: Hw

14Frequency Response: ~hv

Frequency Response: Hvh1 i

14Frequency Response: ~ilh#

Figure 3.4 Magnitudes of Frequency Responses

Frequency Response: Hw4

14Frequency Response: fihv

Frequency Response: Hvh4 , t

14Frequency Response: ~ilttH)

Figure 3.5 Phases of Frequency Responses


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Impulse Response: Hw Impulse Response: H\rti0.25

I

Impulse Response: H ~ J"O lm pulse Response: ~ k h-"0.25 0.250.2 O .2aJ au O3 0.15 3 0.15.- --

c c0 .1 Rf 0 .15 I0.05 0.05

O O-2 O 2 -2 O 2

Figure 3.6 Magnitudes of ImpulseResponses

lmpulse Response: Hw4nmpulse Response: HvhImpulse Rerponss: H R J ~ - ' O - f OImpulse Response: nhhl0

Figure 3.7 Phases of Impulse Responses


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Fig. (3.8) shows the interferometric measurement of PMD 1= 1.3 pm) f a 10psemulator in a IM-DD Intensity Modulationwith Direct Detection) transmission system[68].Although it is a different type oftransmission system , wavelength, and value ofPMD (1 O ps or 100 ps), there is a qualitative -20 O 20Relative delaybs]agreement between the measurement and the C Figure 3.8 InterferometrieMeasurement of the IO-ps PMDsimulation. Emulator (68)

3.5 Channel Simulation usingMATLAB' and SIMULINKCBased on the waveplate model and the baseband impulse responses. the optical

fiber channel simulation is constructed in Fig. (3.9). It is just the implementation of thefiber model in Fig. (2.3).

In Fig. (3.9). V,, nd H,,epresent the inputs to the vertical and horizontalchannels. and V , , and Ho,,epresent the outputs of the vertical and horizontal channels.The impulse responses are complex. and the implementation of these impulse responsesis shown in Fig. (3.10). A complex impulse response is comprised of four real impulseresponses. Convolution of the input signals in Fig. (3.10) with these real impulseresponses is realized by calling the MATLAB~unction,fdter(), as in

y( t 1= fi[te&(t)? x(t)),where y(() is the output of fiber chamel?

h,(t) is the real o r imaginary part of baseband channel impulse response,which-3 1-


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hvv(t)vout

Impu t s a Responsehvh(t)

I L.Impu l se Raseonsehhh(t ). -outIFigure3.9Optical Fiber Channel Simulation

In Fig. (;.IO), S-Function. a SIMULMK~lock. is used to cal1a MATLAB'function. The purpose of the "Buffern nd 'Unbuffer" block is to convert scalars andvectors.

4-8-i I

0 lR.In1SplH

Figure 3.10 Simulation of Baseband Impulse Responses

Join


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Chapter 4Adaptive Channel Equalizersand CPI Cancellers4.1 Adaptive Channel Equalization Technique

4.1.1 Fundamentals of EqualizationThe transmission of digital data through a linear communication c h a ~ e isuaily

suffers deterioration due to two major limits, K I and additive thermal noise. 1st is causedby multipath in a bandlimited time dispersion channel which distons the tm sm itt edsignai. causing bit erron at the receiver. In an optical fiber. the factors that can cause ISIare inter-mode dispersion, chromatic dispersion, and PMD.

Equalization is a technique to overcom e ISI. Since the fading channel is usuallyrandom and time varying, equalizers must track the time varying charactenstics of thefading channel. and this kind of equalizer is called adaptive equalizer which providesprecise control over th e time response of the channel.

Fig. (4.1) shows a communication system with an adaptive equalizer c ( t ) [j].~ ( t )is the data input, andf,(t) is the overall complex baseband impulse response of thetransmitter, channel and matched filter. The signal received by the equalizer is,

y ( [ )=w*,W + v u ) ,where v ( t ) is the baseband Gaussian noise signal.

If the impulse response of equalizer is c(t) , then the output of the equalizer is


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~ q ( 0 9Y $ ) = y(O * d t )

= x(t)* ( 1)* c(t ) + v ( t)*c(t ) ( 4.2 )= x w * g. (0+ v(O* 4 0 ,

where g,,(t) is the combined impulse response o f the transmitter, channel, matched filter

x,W. - Hl) y - 1Channel Mstcbed + Equalizer -*() Decision40 M t) film - A d0 - ircuit 2 - > m

fd(') x2( t )

I Figure 4.1 Equalizatioaand equalizer,

g.O) = f.O* ~ ( 0 . ( 4.3 )The complex impulse response of equalizer c(t) can be expressed as a transversal

FIR filter with coefficientsc,,,

where N is the number of filter taps.Ideally, the desued output of equalizer,y,(t), should be equal to the original input

x(t). If v(t) is equal to zero, then to forcey,(t) equal to x(r), Eq. (4.3) is used to get

In frequency domain, Eq. (4.5) can be expressed as,


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F , ( f )C ( f ) = 9where F,V) is the Fourier transform o f ' t ), and

CU) is the Fourier transfomi of c(t).The result we get in Eq. (4.6) shows that when the baseband noise v ( t ) is ignored,

the equalizer filter CV) ctually is an inverse filter of F,CT)which is the fiequencyresponse of the combination of fiber channel, transmitter and m atched filter.

41.2 Adoptive EqualizatioaThe charactenstic of the time varying opticd fiber channel requires the use of an

adaptive equalizer. An adaptive equalizer needs the knowledge of the desired input so asto compare it with the output of the equalizer to get the error signal which is used in theadaptive algorithm. However, because of the different locations of trammitter andreceiver, it is difficult to get the original data input to the transmitter in the receiver.There are two methods which c m generate a "desired" signal as that in the transmitter [7].

1. The first method is called training. In this method, before data transmission. asequence of PN (pseudonoise) signals are sent fiom transmitter as test signals. In thereceiver, this PN ignal is pre-stored and it is the sarne as that in the transrnitter. The PNsignal is called a training sequence. and the time to transmit the training sequence iscalled the training period. In he training perod, the adaptive equalizer c m adjust itsfilter coefficients and reaches a steady state where the smallest en or is achieved. Merthe ainingperiod, the filter coefficients are fixed. Sometimes. the training sequence willbe resent perodically to allow the adaptive equalizer to adapt to the tirne-varying channel.In Fig. (4.1), when the switch is switched to "1". it means the receiver is using the

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training sequence, x , t).2. h e second method is called decision-directed. Usually this method is used

along with the training method. In the training method, the equalizer is adapted using atraining sequence. When the training penod is over, the equalizer will be adapted usingthe decision-directed method at the receiver. Afier the training period, the error and ISIare very srnail. Therefore, the decisions made by the receiver are correct enough to beused as a replica of the desired signal. This means that the adaptive equalizer is able toimprove the tap weight settings by virtue of the correlation procedure built into its

feedback control loop. In Fig. (4.1) , afier the training period, the switch is then switchedto "2" rom "1". nd x,(t) is used to estimate the error e( t) .which will be used to updatethe tap weights.

4.13 Adaptive Equalization Categorization and AlgorithmThe general categorization of adaptive equalization techniques accord ing to the

types. structures and algorithms is s h o w in Fig. (4.2) [j].

Zcm Forcing GradientRLS LMS Griui icarW LMSLM S RLS RLSRLS Algorithm

Figure i 2 -~ n t e ~ o r i z n t i o af ~ d k t i v c q u i b t i o n [SI-36-


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Equalizers can be divided into two general categories, linear and nonlinearequalizers. In Fig. (4.1), there is no feedback fiom the output of receiver, z(t), thus it is alinear equalizer. On the other hand, if z(t) is fed back in order to control the equalizer, theequa lizer is nonlinear.

In this thesis, a discrete L E Linear Transversal Equalizer) structure is usedwhich is s h o w in Fig. (4.3). A discrete LTE s made up of K apped delay lines whichhave a symbol period TV,nd the number of taps is K. In Fig. (4.3), the output of the LTE,y(n) is,

where n is time index,u(n) is the baseband input signal, andc , (n ) are the cornplex coefficients.

I Figure 4 3 LinearLMS TransversalEqualizer


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Classic adaptive equalizer algorithms includeZF Zero Forcing),LMS LeastMean Squares)and RLS Recursive Least Squares). In this thesis, four linear LMStransversal adaptive filters are used as cbannel equalizers and CPI cancellen. Thecomplex normalized LMS algorithm is descnbed mathematically as [7,12],

where an denotes Hermitian transpose of e.u(n) is the K dimensional input signal vector at step n.

u(n) = [ tr(n) cr(n- 1) ... ic(n- K + [ ) I r . ( 4.9c (n ) is the K dimensional vector of estimated of the equalizer tap weights at step n.

d(n) is the "desiredw ignal,y(n) is the output of adaptive equalizer,e (n) is the estimated error,p is the adaptation constant, som etimes it is cailed step-size, anda is a small positive constant ( e - ' 9 used to overcome the potential numerical

ins~abilityn the tap weight update.


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4.2 ChannelEqualht ion and CPI CancellationThe existence of PMD n an optical fiber introduces CPI between the dually

polarized signals and ISI. This chapter contributes to realize CPI cancellation as well asISI equalization.

Fig. (4.4) shows the discrete baseband structure of the channel equdizers and CPIcancellen, where cw(n) nd cVl(n) re the channel equalizers and we cal1c d n ) and c,(n)are the CPI cancellers.

1 Figure 4.4 Channel Equalizers and CPI Cancellers

The complex inputs to the vertical and horizontal polarization channels arexv(n)and x,(n), and the outputs of channels are u l n ) and u,(n). Because this is a basebandmodel, the IF (Intem ediate Frequency) stage and the rnodulator/demodulator are notshown. nius the channel is followed by channel equalizen and CPI cancellers whoseoutputs areyJn) and yh(n). We can express the relationship between these signals inmatrix form,


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where y(n) is the output signal vector,c(n) s the matrix of adaptive channel eqdizers and CPI cancellers,

u(n) is the fiber channel output at step n,

u(n) is the input signal vector to the equalizer,

u,(n) and u,(n) are the input signal vectors in vertical and horizontal channels tothe equalizer,

eT stands for the matrix transpose of *,


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r ( n ) is the desired signal vector,

h(n) is the matrix of the complex baseband channel impulse responses, seeEq. (2.21).

The error, e(n), is the difference between the desired response r(n) and output ofthe cancellers and equalizers, y(n),

Next is defined the MSE (mean-squared error), J(n),J (n ) = ~ ( e ( n ) e ' ( n ) ) ~ ( l e ( n f )

where E(.) denotes mathematical expectation of 0.

I Figure 4.5 Vertical Channel Equalizorand CPI Caaceller


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To simpliQ the analysis, consider only the vertical polarizationchannel equalizerc,(n) and the CPI canceller c,.(n) fiom the horizontal to the vertical channel, as s h o w inFig. (4.5).

Suppose the complex filter tap weight vector for channel equalizer c,(n), is,

where a(n) s the real part and b(n) is the imaginary part of tap weight c(n) . The --dientof J,.(n)with respect to c,(n) and c,,,,(n)s [7],

where v , ~ ( J , ( ~ ) )s the gradient of J,(n)to c,Jn), andv ,+, (J , (~) )s the gradient ofJ,,(n)to c,.,,(n).

Substituting the JJn)of E q . (4.19) in Eq. (4.22),

According to Eqs. 4.12,4.18,4.20) [7],


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Therefore, the final result will be.

To study the channel equalizer c,,.(n),expand Eq. (4.22). giving,

where P,,.(n) is the cross-correlation vcctor between the input vector to the equalizer u,(n)and the desired responsexV(n),

RJn) s the auto-correlationmatnv of the input vector to the equalizer u,.(n)R,,.(n)= ~ [ ~ , - ( n ) u % o ] . ( 4.30 )


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Q,.,,(n)is the cross-correlation vector between the input vector to the equalizeruJn) in the vertical polarization channel and the input vector in thevertical polarization ~ h m n e l ,,(n),

Sim ilar to above, Vvh ~ ~ ( n ) )m be expanded to get,

According to the method of steepest descent, the updated value of the tap-weightvectors for equaiizer cJ n ) and canceller c d n ) are,

and

where p is called step size, an adaption constant which controls the size of theincremental correction applied to the tap-weight vector [q. Subnituting in V , ~ ( J , ( ~ ) )nd


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In Eqs. (4.38,4.39), the adaptive tap-weights for channel equalizer c,(n) and CPIcanceller c A n )are established. In reality, however, this method will not work becausethe exact measurement of the gradient vec or V, (~,(n))and ( ~ ~ ( n ) )s impossiblesince it requires the correlation matrix R(n) and cross-correlation matrixes P(n) and Q(n).Therefore, O, ( ~ , . ( n ) )ndO,.,,~ ~ ( n ) )ust be estimated from the available data. This ideais realized in the LMS lgorithm.

In he LMS lgorithm. the instantaneous estimates are used instead of thecorrelation matrix R(n) and cross-conelation matrices P(n) nd Q(n), and for channelequalizer cJn) . they are defined by, respectively.

P,,. n)= u,w x : (4k ( n ) u,(n)u. (n) ,Q ,&O = u v ( , ) 4 ' ( n )

The corresponding instantaneous estimate of the gradient vector is then,_ ( ~ ~ . ( n ) )-2u, . (n)x:(n)+ u , ( n ) u ~ ( n ) , . Zuv(n)u:(n)e,, ( 4.43 )

where S,(n) is the estimated tap-weight of channel equalizer cJn) , and ,(n) is theestimated tap-weight of CPI cancellercvh(n).


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Substituthg the estimated gradient vector +,&,,(n)) in Eq. (4.43) for theV &,An)) in Eq. (4.36),gives get the adaptive tap-weight in the LMS lgorith,

k ( n + 1) = , ( n ) + p u , ( n ) [ x : ( n )- u : ( n ) , ( n ) - u : ( n ) ~ , ( n ) ] , ( 4.44 )and similar to Eq. (4.43),

t , ( n + 1) = ,(n) + pu , (n) [ . r i (n )- u * ( n ) c , ( n ) - u ~ ( n ) , , . ( n ) ] . ( 4.47 )When the input u(n) s large, the LMS algorithm experiences a gradient noise

amplification problem [7]. Therefore. the normalized algorithm is used to overcome thisproblem. The normalized algorithmsof Eqs. (4.41 -4.4 4) are,

6,Jn + 1 ) = ,(n) + PU.^ [xi( n )- u (n )Qhhn )- u: (n)t,, n ) ] ( 4.50a + u : ( n ) u , ( n )

where p is the step size, O cp


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4.3 Simulation ofChannel Equalizers and CPI Cancellers4.3.1 Simulation StructureofChannel Equalizers and CPI CanceUersBased on the Eqs. 4.48-4.5 l) , theNomalized LMS hannel equalizers or CPI

cancellen can be constnicted using SIMULMK~s shown in Fig. (4.6).Fig. (4.6) shows the model of the nonnalized LM S filter. In this model, the first

input is u(n), the second input is error e(n)and the output isy@). The buffer is used togenerate input signal vector u(n).

Figure 4.6 LMS Adaptive FilterThe overall adaptive equalization and cancellation system is s h o w in Fig. (4.7).

Complex delays, 2" and ?,are used to synchronize the output of the equalizer and thedesired input. The factors that cause the delay include the mansrnitter filter. receiver filterand channel delay. So the desired input h m he information generator has to be delayedby the same time. The input signals are sarnpled at the rate 1;=B which is 128 timeslower than the channel sampling rate before they go into the equalizer. T w witches areused to se lect the training or decision-directed rnethod. At the beginning of thissimulation, the nvitches are set to the training method. In every bit period, the tap-weights are adjusted to minimize the enoa.After the MSE eaches the steady state,


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switches are then set to decision-directed method where ou tput signals of the decisioncircuits are used as the desired input.I

av I

C

Figure 4.7 Adaptive Channel Equalizersand CPI Cancellers4.3.2 Performance of Channel Equalizers and CPI CancelIersFrom Eqs. (4.48-4.5), we can see that the value of step-size p is very important.

Fig. (4.8) shows hree leaning c w e s wth erron (dB) on the ordinates, when p is 0.2.0.4and 0.6 respectively. By comparing these three cases, it is clear we have to tradeoff thetraining time versus the residuai error. When p is 0.2, the error is the smallest. bu t ittakes longer time to train. When p is 0.6, the equalizer can reach the steady state quickly,but the error is the largest. Thus we choose p equal to 0.4, which is the best compromise.

Also, the number of taps in the equalizer and canceller is the factor that affects theperfotmance and accuracy of the receiver. In the simulation system, ive choose 8 iaps for


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equalizers and cancelles rather than 16 taps, because when we use 8 taps, the filter takesless time to reach steady state and has smaller variance. The errors (dB) of the equalizerand canceller when numberof filter number of taps is 8 and 16 are s h o w inFigs. (4.9~4.9b).

J (dB) when p 0 . 2Figure 4.8 (a) J(dB) when ~ 0 . 4Figure J d (b)

wben Filter Leagth=l6Figure 1.9 (a)

...........~............... ............;.. .............;.......... .............

.......... .& ......... . . . . . . ..... .................a . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .I.p.. .............;-................-...............

. . . . . . . . . .,, . . . . . . . . . . : . ;.. . . .- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .II'Y. , ! - . '. : .: a4 , RI' ,aJ dB) when PO. 6Figure 4.8 (c)

when FiFi@ lter Lengthtsire 4 9 (b)


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Chapter 5Sysiem Performance5.1 Eye Pattern

The eye pattem is a convenient way to see the distortion on a channel, which isusually obtained by displaying the data pulse stream on an oscilloscope. A two levelsignal after an RC filterwith filter rolloff factor of 0.8 in the transmitter is s h o w inFig. (5.1). In this eye pattern, one symbol interval is displayed, and the central verticalline indicates the decision point where one canget the lowest error and best BER. Thussampling should occur at this point where eye isopen the widest. If al1 of the lines in the traces goihrough the desired pulse amplitude at thedecision point. then the eye is said to be fullyopen, such as the one in Fig (5.1).

Figure 5.1 Fully Open EyePatternThrough the eye pattem, Ive can roughly mesure the performance of the received

signai in the receiver, which suffers fiom the PMD nd noise. When the input SNR is30 dB, the eye pattem in the receiver isobtained as s h o w in F ig (5.2). By companngthese two figures, it is clear that the width of eye pattern d2 in Fig. (5.2) is smaller thand l , which indicates the latter traces deviate fiom the completely open eye pattern.Fie. (5.3) is the eye patternwhen input SNR s 15 dB. Thewidth of eye pattern d3 is


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much smaller than d2, so the en or probability is larger, because there is a smaller space todistinguish between the two levels.

Figure 5.2 Eye Pattern Figure 5 3 Eye Pattern(SNR=30) (SNR=15)

5.2 Bit Error Rate EvaluationThe eye pattern is only a rough method to measure the enor. To analyze

accurately the performance of a telecommunication system, the bit error rate (BER) m u t

The simplestway to measure BER s error counting. or counting the number oferror bits in a penod. This is simple, but it may not be accurate. Also because the BERin a optical fiber communication system is very low, usually lower than it requires arather long tirne to get a satisfactory re sd t by simulation.

An alternative way to measure BER is by calculation. Consider the case in whichthe tmsmit ted signal over the vertical or horizontalchannel is s(t) ,


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where E is the input signai power. The correspondhg received signal in receiver, r ( t ) , s

where n, and+ are the error signais with zero mean and the same variance,d,

where~ and 0;are the variances of n, nd -, respectively. This distortion in thedetected bit is due to ISI, CPI and noise. To simpliQ the calculations. the distortion istreated as Gaussiannoise. For high signal-to-noise ratios, this assumption will producean acceptable estimate of the BER

Thus the symbol error probability for Clevel QAM signal is [8], P,,

wherea*)tands for Q-function, which is defined as,

Suppose, the received SNR, y, is,

then the symbol error probability in Eq. (5.4) is,

The relationship between bit error rate, Pb, and P, s,b ( 1 - pb)Z = c.


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Thus, we have,

The relationship between the teceived SNR, y, and minimum MSE, J, is [8],

Thus, BER can be obtained by measuring the J,

Therefore, it is possible to measure the minimum eceived MSE ,, when the channelequalizen and CPI cancellers reach the steady state, and to get the BER according toEq. (5.1 1).

In this simulation system, the channel and receiver thermal noises are treated asGaussian noises, the input signal power is O dB. The signal bit rate is 2.5 GHz. Thesystem SNR, , is from 4 to 32 dB , where.-

ty , =- and2 qis the variance of channel Gaussian noise. The DGD s 10ps, 100ps and 500 ps,separately. The number of the adaptive filter taps is 8, and step-size, p is 0.4. Thetraining time is longer than 3 x 1O4 sec for 1O-ps PMD, x IO4 sec for 100-psPMD nd10 104 sec fo r 500-ps PMD. We measure the J,, four times (No. 1 to No. 4) for everyy,. Each t h e , the J,, in vertical in-phase signal, vertical quadrature signal, horizontal in-phase and horizontal quadrature si& are obtained. The experimental results of J,,, areIisted in Tables (5.1,5.2.5.3).


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Table 5.1 J , (dB)whenDGD is 10p3Avg.1 Jmtn(dB)-10.2

-11.6

-13.5

-15.3

-18.5

-23.6

t

123412341234

1234123412

VerticalIn-phase-6.78-12.3-13.1-1 1.8-13.9-10.2-1 1.3- 13.2-12.8-12.3-14.0-14.1-14.0-16.9-13.5-18.8-1 7.0-17.6-17.3-16.9-24.1-23.1

HorizontalIn-phase-13.2-8.50-9.45-7.9 1-9.76-10.8- 10.7-13.0-1 1.3-14.9-14.3-15.8-1 1.8-17.0-13.0-16.4-19.2-22.6-19.7-18.3-27.7-36.5

VerticalQuadurate-7.84-12.0-1 1.9-1 1.1-13.1- 10.5-1 1.6-13.1-12.8-12.5-15.1-13.6-13.5-16.4-14.0-17.1-17.3-1 8.9-17.0-17.7-23-6-25.5

HorizontalQuadurate-12.7-8.5 1-7.22-8.94-12.7-9.50-12.3-10.8-12.9-14.7- 12.8-12.2-14.7-16.2-17.3-13.9- 17.3-18.0-16.0-21.9-21.7-30.1

Avg.

-10.1-10.3- 10.4 .-9.94-12.4-10.2

I

-1 1.4-12.5-12.4-13.6-14.0

1

-13.9-13.5-16.6-14.4-16.6-17.7-19.2

-17.55-- 18.7-24.3-26.3


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1

YS(dB)

4

1

8

12

16

NO.

123

1

41.7-3412341

I

2

Avg.~m(dB)

-7.75

-9.13

-10.8

-12.5

Ji. dB)Avg.

-8.89-6.32

1

-7.83-7.67-10.4- 10.2-8.13-7.84-11.5

1

-1 1.2-IO.-10.2-11.6-12.5

HorizontalQuadurate-9.47-5.5 1-7.94-7.43-8.9 1-9.66-8.77-7.20-1 1.8-1 1.9-10.6-10.1-12.5-1 1.5

(DGD = 100 ps)HorizontalIn-phase

-8.22-5.54-7.5 1-7.77-1 1.5-10.4-8.05-7.70-1 1.0-10.2-9.57-10.2-10.5-13.2

rVerticalIn-phase

-9.57-7.46-8.27-8.27-12.0- 1 1.2-8.41-8.89- t2.6-12.3-1 1.0-10.8-12.7-14.2

VerticalQuadurate-8.3 1-6.77-7.58-7.23-9.05-9.42-7.34-7.58-10.7-10.3-9.3 1-9.89-10.5-1 1.0


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Table 5.3 J , (dB) when DGD s 500 ps


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From the above tables and Eq. (5.1 1), the BER plot is s h o w in Fig. (5.4). Thisfigure shows the BER when DGD is 10,100 and 500 ps. From the BER plot, to make theBER maller than 1c9,he system SNR, y,, should be larger than 17 dB when DGD is 10ps, larger than 21 dB when DGD is 100 ps, and larger than 27 dB when DGD s 500 ps.Also the BER plot shows the trends of'BER for these three cases. With the increase of y,,the plot of BER when DGD s 10 and 100 ps decrease steeply, however. when DGD s500 ps, it decreases slowly.

Figure 5.4 BER Plot


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Chapter 6Bidirectional Transmission System6.1 Bidirectional Transmission System

Until now, what was discussed is an unidirectional data transmission systemthrough a fiber. Ordinarily, to reaiize a duplex transmission using only one waveiength.two fibers are required, one for the f o m d ransmission, the other for the backwardtransmission.

In this thesis, a different strategy, a bidirectional transmission system is presented.In this str at ea . a transceiver is used, which consists a transmitter. a receiver and acoupler. This strategy is s h o w in Fig. (6.1).

Reccivcr Coupler

Ttranscciver- . -.

Coupler - Recriva

Ttransceivcr -Figure6.1 Bidirectional Transmission System

Clearly, the bidirectional transmission ystem can improve the synern capacity,and Save fiber resources. However, there aresomeproblems which need to be solved inthe bidirectional transm ission system. Themost important limit is the echo ahich is

-59-


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caused by the use of a fiber coupler and the existence of Rayleigh backscatter in fiber.6.1.1 Fiber CouplerFig. (6.2) is a basic structure of a four-port fiber coupler [l]. The o

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Symulation of Optical Fiber Communication System With Polarization Mode Dispersion