Code-Aided Expectation-Maximization and Probabilistic Constellation Shaping for Fiber-Optic Communication Systems. Chunpo Pan

Transcription

1 Code-Aided Expectation-Maximization and Probabilistic Constellation Shaping for Fiber-Optic Communication Systems by Chunpo Pan A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Electrical and Computer Engineering University of Toronto c Copyright 2016 by Chunpo Pan

2 Abstract Code-Aided Expectation-Maximization and Probabilistic Constellation Shaping for Fiber-Optic Communication Systems Chunpo Pan Doctor of Philosophy Graduate Department of Electrical and Computer Engineering University of Toronto 2016 This thesis provides signal-processing and constellation shaping algorithms to combat equalizer-enhanced phase-noise and nonlinearity-induced interference in long-haul fiberoptic communication systems. In the first part of this work, an iterative phase-estimation algorithm is designed that combines expectation-maximization (EM) with a soft-input soft-output error control decoder. At 280 Gbit/s, this system doubles the optical system reach, and generally enhances phase-noise tolerance. Among three different 16-point constellations, the ring constellation was found to have the best performance. The laser linewidth tolerance gain is improved further by reducing the code rate. In the second part of this work, this code-aided (CA) EM approach is applied to long-haul systems dominated by nonlinearity-induced interference. Based on laboratory measurements, a second-order autoregressive phase-noise model is proposed, and used to modify the EM regularizer term. Simulation and experimental results show that in a dual-polarization wavelength-division-multiplexed 16-QAM system, launch-power tolerance can be increased by 1.5 db, and the optical signal-to-noise ratio requirement can be relaxed by 0.3 db to achieve the same error ratio. The complexity of the CAEM algorithm was found to be about 1/4 of the complexity of digital back-propagation. In the third part of this work, a low complexity probabilistic constellation shaping scheme is proposed to increase the system tolerance to nonlinearity impairments. A theoretical analysis predicts that, with 16- and 64-point constellations, reach imii

3 provements of 7% and 10% respectively can be achieved at a mutual information of 3.2 bit/symbol/polarization. Such probabilistic constellation shaping requires minimal added complexity. iii

4 iv To my parents and my husband.

5 Acknowledgements First, I would like to express my sincerest gratitude to my supervisor Prof. Frank R. Kschischang, for giving me the opportunity to carry this research work, and for his support and guidance throughout the years. He is caring and patient, while he never ceases surprising us by the broadness and depth of his knowledge. Frank s enthusiasm, insightful comments, and sense of humour have made every single one of our meetings enlightening and forever memorable. It has been a true privilege to spend the past years in the FRK group, and to work and study with someone like Frank from undergrad to PhD. I would like to thank the following people for their direct help on my research work: Dr. Vadimir Pechenkin for developing majority of the optical channel simulation code, Lei Zhang for designing the LDPC codes used in Chapter 4 of my thesis, Wilfred Idler for conducting the experiments in Chapter 5, Dr. Laurent Schmalen for designing the LDPC code used in Chapter 5, and Dr. Georg Böcherer for discussions on the practical implementation of probabilistic shaping. A special thank you to Dr. Henning Bülow for supervising me while I was an intern at Bell Labs Stuttgart. His knowledge, patience, and attention to detai a great teacher and supervisor. Without Dr. Bülow s guidance and support, my experience at Bell Labs could not have been so fruitful. I would also like to thank Prof. Alan P. T. Lau for hosting me at Hong Kong Polytechnic University for one semester, where I was exposed to the experimental setup of fiber-optic communication links for the first time. This experience has helped me better understand the physics behind the channel model and practical implementation constraints in the research. I would also like to extend my gratitude to my committee members: Prof. Octavia Dobre, Prof. Stark Draper, Prof. Brendan Frey, Prof. Ashish Khisti, Prof. Raviraj Adve, Prof. Li Qian, and Prof. Prasanth Nair for serving on my PhD defense committee, taking the time to reach my thesis, and giving so many insightful comments. A special thank you to Dr. Kangping Zhong, Zhenhua Dong, Nan Guo, and Dr. Yuliang Gao for their help at Hong Kong Polytechnic University, Jerry Wang and Xiaofeng Lu for their help in Stuttgart, Germany. My experience abroad could not have been so smooth if it was not for their warm help. I had a lot of fun while working and hanging out with them. The same also goes to my friends at University of Toronto: Christopher Blake, Vladimiro Cirillo, Binbin Dai, Prof. Chen Feng, Alice Gao, Siddarth Hari, Dr. Siyu Liu, Dr. Huiyuan Xiong, Lei Zhang, and Dr. Yuhan Zhou. Their help and accompany throughout my PhD study has made it much more colorful and fun-filled. Last but not least, I would like to express my heartfelt appreciation to my parents v

6 for their unconditioned love and support, and to my loving husband for his patience and understanding. Without their encouragement this thesis could not have been possible. vi

7 Contents 1 Introduction Motivation Fiber-Optic Communication Thesis Outline Equalizer-Enhanced Phase-Noise Nonlinearity-Induced Interference Probabilistic Shaping Outline Fiber-Optic Communication Systems A Fiber-Optic Link Transmitter Fiber Channel Receiver Split-Step Fourier Method Digital Back-Propagation Random Symbol Transmission Code-Aided Expectation-Maximization Expectation Maximization Algorithm Code-Aided Expectation-Maximization Equalizer Enhanced Phase Noise Compensation Motivation System Model System Components Source of Equalization-Enhanced Phase Noise Code Aided Expectation Maximization Algorithm vii

8 4.4 Simulations Constellation Design Effect of Code Rate Complexity Summary Nonlinear Phase Noise Compensation Motivation System Model and Parameter Estimation Code-Aided Expectation Maximization Simulations and Experimental Results Complexity Optimization Summary Probabilistic Constellation Shaping Motivation Nonlinear Noise and Optimal Launch Power Optimal Input Distributions An Example Constellation Labelling and Code Design Simulation Results Complexity Extension to Higher Order Modulation Summary Summary and Conclusions Contributions Future Directions List of Publications Appendices 92 A Computing the Gradient of the Objective Function 93 B Computing the Variance of the Nonlinearity-Induced Noise 96 B.1 i 0, l, p, q p B.2 i 0, l, p, q = p B.3 i = 0, l, p l, q l, q p viii

9 B.4 i = 0, l, p l, q = p B.5 i = 0, l, p = l, q l B.6 i = 0, l, p l, q = l B.7 i = 0, l = p = q Bibliography 103 ix

10 List of Tables 4.1 System parameters used in simulation Iteration parameters used in simulation System parameters used in experiment and simulation. See also Table LDPC code properties Experimental Q 2 -factor as a function of the number iterations between LDPC decoder and EM algorithm (columns) vs. the number of iterations between E- and M-step (rows) on validation data set Experimental Q 2 -factor as a function of the number iterations between E- and M-step (columns) vs. the number of gradient-ascent steps (rows) on validation data set Experimental Q 2 -factor as a function of the number iterations between LDPC decoder and EM algorithm (columns) vs. the number of gradientascent steps (rows) on validation data set Minimum number of iterations required in each loop to achieve nearoptimal performance System parameters used in simulation The optimal input distributions for different transmission distances LDPC code properties x

11 List of Figures 1.1 A fiber-optic link with DCF. D means the dispersion coefficient is negative, and D + means the dispersion coefficient is positive Layout of the encoding and decoding modules in a digital communication system Layout of an optical link An example of a fiber-optic link The effect of laser linewidth on an optical system An example power gain/loss profile f(x) for an optical link with EDFAs Block diagram of the transmission system Graphical illustration of the EM algorithm A block diagram of the CAEM algorithm Modules in an optical communication system Physical components in an optical link Channel model of a coherent fiber-optic communication channel The factor graph version of CAEM Different 16-point constellations used in the simulation Comparison of fiber-optic systems System reach comparison. Both the transmitter and receiver lasers had linewidth of 5 MHz. Each span had length of 65 km. LDPC code was identical in all systems, and had a rate of 15/ Effects of code rate. One LDPC code had a rate of 15/16 with 5 pilot symbols in front of every 100 information symbols, the other had a rate of 9/10 with 5 pilot symbols in front of every 400 information symbols. All systems had a rate of and a transmission distance of 975 km xi

12 4.9 Comparison between high complexity version and low complexity version. Two different complexity settings are summarized in Table 4.2. Transmission distance was 975 MHz, and the ring constellation was used Amplitude and phase noise in received signal y[k] Transmission system and CAEM algorithm Autocorrelation function of experimental phase noise Partial autocorrelation function of experimental phase noise Autocorrelation function of simulated phase noise Partial autocorrelation function of simulated phase noise Experimental Q 2 -factor as a function of the regularizer weight α Different iterative loops in the CAEM algorithm: L1 is the iteration between SISO decoder and the EM algorithm; L2 is the loop between E-step and M-step; L3 is the loop for gradient ascent algorithm; L4 is the loop for sum-product algorithm Experimental differential phase noise variance with and without CAEM Input and output BER curve of the LDPC code on a 16-QAM channel Simulation results for 2640 km SSMF transmission Experimental results for 2640 km SSMF transmission Experimental Q 2 -factor as a function of the number sum-product algorithm iterations in LDPC decoder on validation data set Achievable Q 2 -factor as a function of complexity Physical components in an optical link Mutual information I(X; Y ) as a function of the transmission distance for 16-QAM. The upper two curves show results for distances shorter than 40 spans (top x-axis); the lower two curves show results for distances greater than 40 spans (bottom x-axis) The optimal launch power per channel as a function of the number of spans The optimal scaling factor A opt as a function of the number of spans Binary sequence to constellation point mapping Histogram of LLRs for shaped 16-QAM Histogram of LLRs for uniform 16-QAM Input and output BER characteristics of LDPC 1. The test is performed on AWGN channel with the constellation mapping scheme described in Fig xii

13 6.9 Input and output BER characteristics of LDPC 2. The test is performed on AWGN channel with uniform 16-QAM constellation Reach comparison between uniform 16-QAM and shaped 16-QAM Mutual information I(X; Y ) as a function of the transmission distance for 64-QAM. The upper two curves show results for distances shorter than 40 spans (top x-axis); the lower two curves show results for distances greater than 40 spans (bottom x-axis) Contributions of the thesis xiii

14 Acronyms ACF ADC AMIN AR ASE AWGN BCH BCJR BER BICM CA CAEM CD CMA CNLSE CPE CR CWDM DAC DBP DCF DSP DWDM E-step EDFA EEPN EM EXIT FE FEC FFT FIR FWHM FWM autocorrelation function analog-to-digital converter amplitude modulation induced noise auto-regressive amplified spontaneous emission additive white Gaussian noise Bose, Ray-Chaudhuri, Hocquenghem Bahl, Cocke, Jelinek, Raviv bit-error rate bit-interleaved coded-modulation code-aided code-aided expectation-maximization chromatic dispersion constant modulus algorithm coupled nonlinear Schrödinger equation carrier phase estimation carrier phase recovery coarse wavelength-division multiplexing digital-to-analog converter digital back-propagation dispersion compensating fiber digital signal processing dense wavelength-division multiplexing expectation step erbium doped fiber amplifier equalizer enhanced phase noise expectation maximization extrinsic information transfer frequency estimation forward error-correcting code fast Fourier transform finite impulse response full-width at half-maximum four-wave mixing xiv

15 GVD group velocity dispersion IFWM intra-channel four-wave mixing i.i.d. independent and identically distributed ISI inter-symbol-interference KL Kullback-Leibler (divergence) LDPC low-density parity-check LLR log likelihood ratio LO local oscillator M-step maximization step ML maximum likelihood NLIN nonlinearity induced noise NLMS normalized least mean square NLSE nonlinear Schrödinger equation OSNR optical-signal to noise ratio PACF partial autocorrelation function PBC polarization beam combiner PBS polarization beam splitter PMD polarization mode dispersion PolDM polarization-division multiplexing PSP principal states of polarization QAM quadrature amplitude modulation QPSK quadrature phase-shift keying ROADM reconfigurable optical add-drop multiplexer RS Reed-Solomon RX receiver SER symbol error rate SISO soft-input-soft-output SNR signal-to-noise ratio SPM self phase modulation SSMF standard single-mode fiber TX transmitter WDM wavelength-division multiplexing XPM cross phase modulation xv

16 Chapter 1 Introduction 1.1 Motivation Fiber-optic links are widely deployed nowadays both in long-haul communication systems (such as transcontinental networks) and metropolitan networks. In contrast to co-axial links, they have extremely high reliability, and high data throughput. With the help of wavelength-division-multiplexing (WDM) and polarization-division multiplexing (PolDM), a single fiber-optic link is able to carry data rate as high as multi Terabits per second [45]. In recent years, there have been booming applications and technologies (e.g., video conferencing, high-definition video streaming, cloud computing, and real-time video gaming) that heavily rely on optical networks. The global IP traffic will grow from 60 Exabyte per month in 2014 to 170 Exabyte per month in 2019, mostly thanks to the tremendous amount of high-definition videos watched online [1]. Researchers have been striving to push the capacity of fiber-optic links to ensure that cost-effective network traffic scaling can continue to accommodate such a rapid traffic growth. With the development of reconfigurable optical add-drop multiplexers (ROADMs), a single optical fiber is able to collect many short-reach client optical signals, and convert them to long-reach signals [75], thus requiring a high throughput. There are different ways to increase the throughput of fiber links. One may increase the signaling rate. The per wavelength baud rate of optical channel has increased from below 10 Gbaud in the 1990s to over 40 GBaud today [5, 13, 46]. The increased baud rate leads to lower tolerance to phase noise and nonlinearity-induced-interference [9,72]. Any further increase in the baud rate also becomes difficult due to its requirement on the bandwidth of the modulator, and the sampling rate needed for the digital to analog converter (DAC), and the analog to digital converter (ADC). It the signals being sent in parallel so that higher information 1

17 Chapter 1. Introduction 2 rate can be transmitted during each symbol duration. Chapter One way 2. Background to parallelize optical signals is to deploy WDM. Modern WDM systems are 27 able to carry over 100 optical signals simultaneously. Another way to parallelize optical the signals end of is to each deploy span, PolDM. EDFAs not PolDM only systems amplify the cansignal, transmit but in also two introduce orthogonal noise. polarization states In standard independently, single mode thus fiber doubling (SSMF), the CD number broadens of possible the pulse, channels and introduces in a fiber. inter-symbol A third way is to deploy coherent modulation, which utilizes both the quadrature and in-phase interference dimensions (ISI). [36, 47, In68]. our system, Anotherwe great consider advantage the case ofwhere coherent CDdetection is compensated is that using it enables dispersion us tocompensating use powerful fiber digital (DCF) signal (Fig. processing 2.9). DCFs(DSP) are fibers chips withtoβ 1 perform whose magnitude equalization is much and dispersion compensation digitally [35, 42, 64, 65]. greater In traditional than that of fiber-optic SSMF, and communication whose sign is opposite systems, to light that wave of SSMF. propagates In this in case, standard only a relatively single mode short fiber section (SSMF), of DCF andisdispersion required tois compensated for CD using dispersion each span. compensating fiber (DCF) (Fig. 1.1) [2]. Such systems are often referred to as dispersion managed N spans Transmitter D - D + D - D + Receiver DCF EDFA SSMF DCF EDFA SSMF EDFA Figure 1.1: A fiber-optic link with DCF. D means the dispersion coefficient is negative, and D + means the dispersion coefficient is positive. Figure 2.9: The optical transmission system assumed in the thesis. EDFAs are used to amplify the signal, and DCF sections are used to compensate for chromatic dispersion. D means dβ 1 systems. In each span, the signal goes through a spool of DCF at the end to reverse dλ is the negative, effect of andcd D + (Fig. means1.1). dβ 1 is The positive. length of DCF does not contribute to the physical dλ distance the optical signal travels, and usually DCF imposes greater nonlinear distortion than SSMF. This method inevitably adds to the nonlinear distortion accumulated in the link. In dispersion unmanaged systems, the CD is accumulated in the entire link length. In long-haul systems, each symbol could get broadened into hundreds of adjacent Transmitter symbols causing severe inter-symbol-interference (ISI). At the receiver, the dispersion is compensated digitally to reduce the accumulated nonlinearity [42, 64, 65, 78]. In this A transmitter of PolDM systems can be illustrated in Fig A laser source generates thesis, we focus on systems without DCF, where CD is fully compensated using a DSP single equalizer wavelength at the receiver. pulses that are split into two orthogonal polarization components with equal energy When using the a polarization baud rate isbeam-splitter. high (32 to 40 These GBaud), two components and WDMare andmodulated PolDM are independently employed, and higher order modulation formats (such as 16-QAM) are used, the optical signal according becomes more to each susceptible user data sequence. to noise and Then interference. the two polarization Impairments components such as are laser combined phase together noise and using inter-channel a polarization nonlinearity beam-combiner. can no longer be neglected, unlike in old genera- The laser source can generate different pulse shapes. The most commonly used one in optical systems is non-return-to-zero (NRZ). If OOK is used, NRZ accounts for the fact that

18 Chapter 1. Introduction 3 tion systems where the data rate was on the order of 10 Gb/s. Many new phenomena and problems arise from the high rate systems. This thesis addresses three of them: how to compensate for equalizer enhanced phase noise (EEPN); how to compensate for nonlinear-induced phase noise in WDM optical systems; and how to design a probabilistic constellation shaping scheme for WDM optical systems that has high nonlinear noise tolerance. 1.2 Fiber-Optic Communication Source Outer Encoder Interleaver Inner Encoder Modulator Channel Demodulator Inner Decoder De interleaver Outer Decoder Sink Figure 1.2: Layout of the encoding and decoding modules in a digital communication system. Fig. 1.2 shows a typical digital communication channel. There are usually two pairs of encoder/decoder in such a link. The outer encoder/decoder represents a powerful error control code, such as Reed-Solomon code [10, 16]. The target output bit error rate (BER) for the outer decoder is or lower. To achieve this output BER, the outer decoder requires the input sequence to have a raw BER below a certain threshold, BER th. We call this threshold the forward-error-correction (FEC) threshold. The inner encoder/decoder pair represents another error control code, whose target output BER is consequently BER th. For conventional fiber-optic links, BER th is typically on the order of 10 3 [10, 16]. Fig. 1.3 shows the digital communication link between the inner encoder and the inner decoder. The inner encoder adds more redundancy to the binary sequence, and the modulator maps the sequence to an optical signal, given by A(t, 0) = k a[k]g(t kt, 0), (1.1)

19 Chapter 1. Introduction 4 where 1/T is the symbol rate, a[k] is a point from the constellation set, and g(t, 0) is the pulse shaping function. The signal then goes through N spans of fiber, and at the end of each span, the signal attenuation is compensated by an erbium doped fiber amplifier (EDFA). An EDFA can be used to directly amplify multiple WDM channels of the optical signal without going back to the electrical domain (unlike decode and forward mechanism). The amplification is achieved by a stimulated emission process, while the amplifier may also undergo a spontaneous emission process, which produces amplified spontaneous emission (ASE) noise. Another laser is used as a local oscillator at the receiver, which is not phase-locked with the transmitter. The local oscillator signal is used as a reference signal to measure the channel output to achieve coherent detection. Then the signal goes through an equalizer module and is passed into the inner decoder. Inner encoder Channel Data ENC SSMF N spans EDFA DSP Inner decoder Modulator Coupler Equalizer DEC Tx laser Rx laser (local oscillator) Figure 1.3: Layout of an optical link. Fiber-optics systems suffer from both linear impairments and nonlinear impairments [2]. The former includes dispersion and attenuation. The latter affects the system in the form of nonlinearity-induced interference. Let A(t, z) denote the light wave, where t is the time measured in s, and z is the distance measured in m, which goes from 0 to the span length L. In the absence of noise, these effects can be modelled by the nonlinear Schrödinger equation [2] (NLSE): A(t, z) z + jβ A(t, z) t 2 = α 2 A(t, z) + jγ A(t, z) 2 A(t, z), (1.2) where β 2 is the group velocity dispersion (GVD) coefficient in s 2 /m, α is the attenuation coefficient measured in 1/m, and γ is the nonlinear parameter measured in 1/(W m). The attenuation may be compensated by an EDFA at the end of each span, leading to the amplified signal A (t, L) = G A(t, L) [2]. Usually, we have the gain of the EDFA

20 Chapter 1. Introduction 5 G = e α 2 L. A non-zero β 2 means light components with different wavelength (or color) travel at different group velocity in the fiber. The optical pulse gets broadened in the fiber causing inter-symbol-interference (ISI). This phenomenon is called chromatic dispersion (CD) [2]. A non-zero γ means that the refractive index of the fiber is a function of the light power, which leads to nonlinear distortion [2]. A more detailed explanation of these effects will be given in Chapter 2. There is no general analytical solution to NLSE, and NLSE is generally difficult to understand. But if we separate the linear and nonlinear effects, it becomes easier to understand. If we look at the effect of dispersion and nonlinearity separately, we have the following equations. When α = 0, γ = 0, the complex envelope at distance z is A(t, z) = 1 ( ) j Ã(ω, 0) exp 2π 2 β 2zω 2 + jωt dω, (1.3) where Ã(ω, 0) is the Fourier transform of A(t, 0). As we can see, the effect of dispersion is a linear time-invariant process, with a transfer function ( ) j H disp (ω, z) = exp 2 β 2zω 2. (1.4) In the absence of noise, dispersion can be compensated using a linear filter H 1 disp (ω, z) = exp ( j 2 β 2zω 2 ). (1.5) When the system is under the influence of nonlinearity only (when α = 0, β 2 = 0), we have A(t, z) = A(t, 0) exp [jφ NL (t, z)], (1.6) where φ NL (t, z) is the nonlinearity phase noise given by φ NL (t, z) = γ z 0 A(t, u) 2 du. (1.7) In reality, power attenuation, dispersion, nonlinearity, and noise interact with each other, and there is no simple way to reverse their combined effect.

21 Chapter 1. Introduction Thesis Outline Equalizer-Enhanced Phase-Noise Conventional digital CD equalizers often overlook one important factor: receiver laser phase noise [42,64,65]. Laser phase noise comes from the transmitter laser as well as the receiver laser. Their effect is not as significant in systems without digital CD equalizer. However the laser phase noise may get amplified in CD equalizers. In systems, where CD is digitally equalized, transmitter laser phase noise goes through both the CD and the CD equalizer, and the phase noise dispersion interaction is canceled out. However, the local oscillator only goes through the CD equalizer, hence the noise amplification arises. Such amplification gets greater as the link length grows, and eventually the system tolerance to laser phase noise becomes the limiting factor in the system reach. In Chapter 4, we investigate a new equalizer structure for coherent optical communication system without DCF. Both transmit laser and local oscillator phase noises are incorporated in the model. The equalizer takes advantage of the fact that the phase noise is correlated over time. A turbo equalization scheme is set up to mitigate the equalizer enhanced phase noise. A soft-input-soft-output (SISO) FEC is used as the inner code. An expectation-maximization (EM) algorithm is used to combine the output of the FEC with the channel information. This code-aided expectation-maximization (CAEM) algorithm is shown to be able to increase the system laser phase noise tolerance by 40%, or double the system reach. It is also found a particular 16-point constellation design outperforms some others in terms of laser linewidth tolerance Nonlinearity-Induced Interference When the laser phase noise is relatively small, nonlinearity becomes the dominating impairment in an optical system. Because nonlinear impairments interact with dispersion and power attenuation of the link, there is no simple algorithm that can reverse the nonlinear effects. In a single wavelength system, nonlinearity can be partially compensated using a finite element analysis algorithm: digital back-propagation (DBP). Such an algorithm back-propagates the received waveform digitally in small steps. In each step it can be assumed that nonlinearity operates independently of the dispersion and attenuation. However, when wavelength-division-multiplexed (WDM) is employed, interchannel nonlinear effects also play an important role. To fully compensate both intraand inter-channel nonlinearity using DBP requires high sampling rate and fine step size. Given the high baud rate of optical links, the complexity of such algorithms is beyond

22 Chapter 1. Introduction 7 what can be implemented in practice using today s semiconductor technology. In Chapter 5, we propose two methods to compensate for nonlinear impairments in a WDM optical link. The first method is CAEM based method that utilizes an FEC and EM algorithm at the equalizer to compensate for nonlinearity. The dominating nonlinear effects in a WDM optical system is self phase modulation (SPM) and cross phase modulation (XPM). In SPM, the signal of interest gets interfered by the signal in adjacent symbols in the same channel, and all together they change the instantaneous refractive index of the fiber, causing phase distortion in the signal. Similarly, in XPM, the signal of interest interferes with the signal in adjacent channels to cause change in the refractive index. The signal dependent nature of nonlinearity makes the nonlinearinduced phase noise correlated over time. However, it is not clear which model this correlation follows and at what level the correlation is. This thesis studies this correlation under both experimental and simulation setups. An auto-regressive model is used to fit the correlation in the nonlinear-induced phase noise. A modified version of CAEM is also proposed to compensate for it Probabilistic Shaping Another way to increase the system tolerance to nonlinearity-induced-interference is to perform probabilistic constellation shaping at the transmitter. In Chapter 6, we investigate a probabilistic shaping scheme for WDM optical channels, so that the system becomes more robust to nonlinearity-induced-interference. Probabilistic shaping is a wellknown method in additive white Gaussian noise (AWGN) channels, that can be used to increase the system noise tolerance by increasing the Euclidean distance among constellation points for a fixed power. The advantage of probabilistic shaping over geometric shaping is the former does not change the modulator structure, and hence is easier to implement for practical purposes. This is especially important for optical systems, where optical modulators operate at very high frequency, and it is difficult to generate nonuniform constellation points on a complex plane. Results from AWGN channels do not directly apply to optical systems. This is because in an optical link, the nonlinear interference depends on the power and distribution of the input signal. Such dependency imposes a cap on the maximum achievable signal-to-noise ratio (SNR) in the channel. We are no longer interested in a system where the noise power and signal power are fixed. Instead, we are interested in the optical constellation distribution and the launch power that achieves the highest mutual information for a fixed transmission distance. An example of how probabilistic shaping can be implemented in practice is also described.

23 Chapter 1. Introduction Outline The remainder of the thesis is organized as follows. Chapter 2 is devoted to some background knowledge used in this thesis. This chapter explains the components of a fiber-optic link, and the major impairments in it. It also outlines how the wave propagation in the fiber can be modeled using a finite element analysis method (split-step Fourier method), and how DBP can be used to compensate nonlinearity-induced-interference in the system. Lastly, it explains how a transmitted random bit sequence can be used as an equivalent to a specific codeword when we have no control over the transmitted bit sequence in a lab setup. Chapter 3 is devoted to the background on the CAEM algorithm. This chapter explains the EM algorithm and its application. Then it explains the setup of the conventional CAEM algorithm used in wireless communication systems. The CAEM algorithm is later modified by adding a regularizer, which is designed specific to the correlation of the noise, to be suitable for EEPN compensation, and nonlinearity-induced phase-noise compensation in Chapter 4 and Chapter 5. Chapter 4 is one of the three major contributions of this thesis. This chapter focuses on the design of a CAEM algorithm for systems that suffer from EEPN. The reach improvement and linewidth tolerance improvement are tested for three different 16 point constellation designs. The effect of different FEC s is also investigated. Chapter 5 is the second major contribution of this thesis. This chapter focuses on 16-QAM WDM systems that suffer more from nonlinear-induced phase noise than from EEPN. The statistical properties of nonlinear-induced phase noise are studied using both experimental and simulated data. A modified CAEM algorithm is proposed to utilize the correlation in the phase noise to mitigate the effect of nonlinearity. The performance of the algorithm is evaluated both on experimental and simulation data, in terms of launch power tolerance gain and SNR gain respectively. The complexity of the proposed algorithm is also analyzed and compared with the DBP. Chapter 6 is the third major contribution of this thesis. This chapter also addresses the nonlinear impairments in long-haul WDM systems. A low complexity probabilistic constellation shaping scheme is proposed to increase the system tolerance to nonlinearity and noise. Theoretical analysis for 16-QAM and 64-QAM systems is conducted to find the optimal constellation distribution and launch power that maximize the mutual information between the input and the output. The achievable mutual information gain is reported for each transmission distance. An example is also presented to show how the probabilistic shaping scheme can be implemented in practice.

24 Chapter 1. Introduction 9 Chapter 7 summarizes the contribution of this thesis, and briefly explains what possible future work could be done.

25 Chapter 2 Fiber-Optic Communication Systems This chapter is devoted to some background information on fiber-optic communication systems. Section 1 outlines the components in a fiber-optic link, the important physical properties of the fiber, and the role of each module in the receiver. Section 2 explains how waveforms propagate within optical fiber, and how we can simulation this process using the split-step Fourier method. Section 3 describes digital back-propagation, which is a technique that can be used to compensate Channel for nonlinear impairments in the link. Section 4 explains Data how ENCa random sequence N spans of symbols can be used to closely simulate a coded sequence of symbols. This is important in Pol. Chapter & 5, where wedsp Modulator SSMF EDFA have no control phase over the transmitted Tx bit sequence. CD Pol. PBS PBC diversity CR DEC laser Eq Demux 90 Modulator hybrid Data ENC 2.1 A Fiber-Optic Link Rx laser (LO) Data Tx laser ENC Modulator PBS PBC Modulator SSMF Channel N spans EDFA Pol. & phase diversity 90 hybrid CD Eq Pol. Demux FE DSP CR DEC Data ENC Rx laser (LO) Figure 2.1: An example of a fiber-optic link. 10

26 Chapter 2. Fiber-Optic Communication Systems 11 Fig. 2.1 shows an example of a fiber-optic communication link. divided into transmitter (TX), fiber-optic channel, and receiver (RX) parts. The link can be Transmitter User data is passed into the optical link as a random bit stream. In a PolDM system, the bit stream is split into two streams to be transmitted in two orthogonal polarizations. For each polarization, the bit stream gets encoded by some forward error-correcting codes (FECs), to protect it from noise and distortion. The target output bit-error-rate (BER) of an optical channel is usually below 10 15, in contrast to 10 3 to 10 7 in typical radio and satellite channels [50]. The first generation FEC for optical system was mostly studied in late 80 s to late 90 s. It consists of a single hard decision linear block code, such as a shortened Hamming code, a Reed-Solomon (RS) code, or a BCH code [50]. In early 2000 s the second generation FEC, concatenated hard-decision codes (e.g., RS+RS and RS+BCH), have been proposed [50]. With the advancement of digital signal processing (DSP) hardware, the major interest has shifted to the more powerful soft-decision FEC s such as low-density parity-check (LDPC) codes and turbo codes, in the past decade [50]. In this thesis, we focus on the structure where the outer code is a hard-decision linear block code, and the inner code is a soft-decision FEC. We assume a FEC threshold on the order of An unpolarized TX laser beam gets split into two orthogonal polarizations by a polarization beam splitter (PBS), and is passed into two modulators. In each polarization, the polarized beam is further split into two beams to achieve quadrature amplitude modulation (QAM). A Mach-Zehnder modulator driven by the electrical encoded data stream is used to modulate the optical beam, and to generate the optical symbols. Then the modulated signal gets combined by a polarization beam combiner (PBC), and is coupled into the optical fiber [2]. The wavelength of the laser is usually around 1550 nm, and the frequency is approximately 194 THz. All oscillators are characterized by a parameter called the linewidth. It is difficult to make sure the laser only outputs a single wavelength. Instead, there is a fluctuation in the frequency, and the range of fluctuation is characterized by the laser linewidth ν (Fig. 2.2). The laser power spectral density follows a Lorentzian distribution, and the linewidth is defined as the full-width at half-maximum (FWHM) of the power spectral density [63]. A direct impact of the linewidth on the optical signal is the phase noise. The greater the linewidth, the greater the phase noise is. The phase noise follows a Wiener process in the time domain, where the phase noise measured at time t 1

27 Chapter 2. Fiber-Optic Communication Systems 12 and t 2 follows φ(t 1 ) φ(t 2 ) is zero-mean Gaussian with variance σp 2 = 2π ν t 1 t 2 [2]. A typical low-cost distributed feedback laser has a linewidth of a few MHz, while more sophisticated and more expensive lasers, e.g., external-cavity lasers, can have a linewidth of several KHz. Constellation Q Q Phase noise I I Laser power spectral density Δυ Δυ ω ω Figure 2.2: The effect of laser linewidth on an optical system. Multiple channels can be integrated together to co-propagate in a single fiber. One way to achieve this is to employ WDM. There are two types of WDM systems: coarse WDM (CWDM) and dense WDM (DWDM) systems. In a CWDM system, typically up to eight channels are distributed in the wavelength range of 1470 nm to 1610 nm, and the frequency spacing is around 2 THz [37]. In a DWDM system, the channel are typically distributed in the wavelength range of 1530 nm to 1565 nm (the C-band), and 1570 nm to 1600 nm (the L-band) [37]. Each band can support 40 channels at 100 GHz spacing, 80 channels at 50 GHz spacing, or 160 channels at 25 GHz spacing [37]. DWDM is much more bandwidth efficient than CWDM, thus DWDM is often used in systems where the transmission distance is greater than a few hundred kilometers, or more than 16 channels are required to co-propagate in a single fiber. However, since the channel spacing is much narrower in the DWDM system, for a given data baud-rate or channel bandwidth, the inter-channel-interference is more severe in DWDM systems. Hence a lower FEC code rate or a more sophisticated equalizer is required.

28 Chapter 2. Fiber-Optic Communication Systems Fiber Channel Light propagation along a fiber is affected by fiber attenuation, group velocity of the signal and nonlinearity of the fiber. Fiber Loss When light travels inside the fiber, it experiences power attenuation, which is mainly caused by material absorption and Rayleigh scattering [2]. The former is affected by the level of impurity in the fiber. The latter is caused by local fluctuation of the refractive index. Such a fluctuation causes light being reflected into all directions, and a portion of the signal gets lost because it no longer experiences total internal reflection. We use α to denote the attenuation coefficient of the fiber, measured in 1/m or db/km. In practice, modern fibers have minimum loss around λ = 1550 nm, and it is the reason why the WDM channels are distributed around this wavelength [2, 37]. Dispersion The group velocity of light inside a particular fiber depends on both geometry of the core and refractive index of the core. Let ω 0 denote the optical carrier frequency in rad/s, let λ denote the wavelength, and let c denote the speed of light in vacuum. Let n(ω) be the refractive index of the fiber. We define the propagation constant β as β(ω) = n(ω) ω c, (2.1) measured in 1/m. The intensity of an optical signal as a function of the time, t, and the distance, z, can be described as E(t, z) = A(t, z)e j(ω 0t β(ω 0 )z), (2.2) where A(t, z) C is the complex envelope of the signal. It is in general hard to measure β(ω) explicitly as a function of ω. We often use its Taylor expansion about frequency ω 0, β(ω) = β(ω 0 ) + β 1 (ω ω 0 ) + β 2 (ω ω 0 ) , (2.3) where β m = dm β dω m, m {0, 1, 2, 3,...}. (2.4) ω=ω0 The quantity β(ω) characterizes how fast the optical signal components with different

29 Chapter 2. Fiber-Optic Communication Systems 14 frequencies travel along the fiber. We have β 1 = dβ dω ω=ω0 = d ( n(ω) ω ) dω c ω=ω0 ( = 1 n(ω 0 ) + ω dn(ω) c dω ω=ω0 ), (2.5) where β 1 is measured in s/m. The group velocity v g and the parameter β(ω) are related by v g = 1/β 1. A non-zero β 2 indicates that light at different frequencies travel at different group velocities in a fiber, this phenomenon is called chromatic dispersion (CD). Thus β 2 is called the group velocity dispersion (GVD) parameter measured in s 2 /m. Another term, dispersion coefficient D, which is a function of the wavelength λ, is also often used by engineers. These two quantities have the relationship D = dβ 1 dλ = dβ ( ) 1 dω dω dλ = β d 2πc 2 = ω dλ λ λ β 2. (2.6) In this thesis, we assume that the effect of higher order derivatives of β is negligible. Nonlinearity The third factor that governs the light propagation is the nonlinearity of a fiber. It is dominated by Kerr effect, where the refractive index n is also a function of the light power A(t, z) 2. In WDM systems, both the signal in the channel of interest, and the signal in neighbouring channels play an important role. The former is referred to as intra-channel nonlinearity, and the latter is referred to as inter-channel nonlinearity. Examples of nonlinear interaction include self-phase modulation (SPM) (intra-channel), cross-phase modulation (XPM) (inter-channel), and four-wave mixing (FWM) (inter-channel) [3]. We use γ, measured in 1/(W m) to denote the nonlinearity parameter of the fiber. Nonlinearity is significant when γ is large, the launch power is high, the transmission distance is long, or WDM channels are densely packed.

30 Chapter 2. Fiber-Optic Communication Systems 15 Nonlinear Schrödinger Equation The unpolarized optical signal propagation in the fiber is governed by the scalar version nonlinear Schrödinger equation (NLSE) [3] E(t, z) z E(t, z) +jβ 1 + jβ 2 2 E(t, z) + jβ 3 t 2 t E(t, z) α t 3 2 E(t, z) = jγ E(t, z) 2 E(t, z). (2.7) When the frame of reference we choose is moving at the group velocity 1/β 1, we E(t,z) can eliminate the jβ 1 term, and if we also assume that higher order dispersion t coefficients (β 3, β 4,...) are 0, then (2.7) becomes A(t, z) z + jβ 2 2 where A(t, z) is the complex envelope of E(t, z). 2 A(t, z) t 2 + α 2 A(t, z) = jγ A(t, z) 2 A(t, z), (2.8) In a WDM system, both the signal of interest and signals in adjacent channels interfere with the channel of interest to cause intra- and inter-channel nonlinearity-inducedinterference (caused by the term on the right hand side of (2.8)). We may analyze the effect of such interference by using a first order analysis [17]. Without loss of generality, let the channel of interest be the 0-th channel, and the interfering channel be the i-th channel. The frequency spacing between the center frequencies of the two channels is denoted by Ω i. Assuming A(t, 0) = i a i [k]g(t kt, 0)e jωit, (2.9) k then the zeroth order or linear solution (denoted by the superscript (0)) of the total optical signal at time t, distance z, is A (0) (t, z) = k a 0 [k]g (0) (t kt, z) + i 0 k a i [k]e jω it+j β 2 Ω 2 i 2 z g (0) (t kt β 2 Ω i z, z), (2.10) where g(t, 0) is the pulse shaping function, g (0) (t kt, z) is defined as g (0) (t, z) = U(z)g(t, 0), and where U(z) = exp(j 1 2 β 2z 2 t 2 ) is the operator representing the effect of CD. The first order solution for the field is denoted by A (1) (t, z). It is computed by solving (2.8), when the right hand side of (2.8) is replaced with the zeroth order approximation. We have A (1) (t, z) z = j 2 β 2 2 A (1) (t, z) t 2 + jγf(z) A (0) (t, z) 2 A (0) (t, z), (2.11)

31 Chapter 2. Fiber-Optic Communication Systems 16 where f(z) is the power gain/loss profile. An example of f(z) is given in Fig. 2.3 for links with EDFA amplification, and f(z) = 1 for perfectly distributed power compensation. 1 f(z) L 2L 3L 4L 5L Distance z Figure 2.3: An example power gain/loss profile f(x) for an optical link with EDFAs. At z = NL, we have A (1) (t, NL) = A (0) (t, NL) + jγ NL 0 U(NL z)f(z) A (0) (t, z) 2 A (0) (t, z)dz. (2.12) The first order approximation assumes A(t, NL) A (1) (t, NL). After the signal A(t, NL) goes through matched filtering and sampling, it can be shown [17] the resulting signal is y 0 [k] = a 0 [k] + a[k], (2.13) where a[k] = a 0 [k l]a i [k p]a i [k q]χ i,l,p,q. (2.14) i,l,p,q The parameter χ i,l,p,q is a coefficient that depends on the channel parameters, such as γ, β, the link distance, the nonlinear coefficient, the center frequency of the i-th channel, and the link power loss/gain profile [18]. The parameter χ i,l,p,q can be computed as [17] χ i,l,p,q = ρ jγ NL 0 ( f(z) g (0) (t, z) ) g (0) (t lt, z) g (0) (t pt β 2 Ω i z, z)g (0) (t qt β 2 Ω i z, z)dtdz, (2.15) where when i = 0, ρ = 1 and Ω i = 0, and when i 0, ρ = 2.

32 Chapter 2. Fiber-Optic Communication Systems 17 Polarization Polarization is a characteristic of the electromagnetic wave that defines the relationship between the direction of the electrical field and the direction of propagation. The electrical field at time t and distance z can be measured as a vector in a two dimensional space, which is the 2-D plane perpendicular to the direction of propagation. The direction of this vector is called the polarization state. When a wave is linearly polarized, its polarization state does not change over z, and when a wave is circularly or elliptically polarized, its polarization state rotate around the direction of propagation, as z changes. In PolDM systems, data is modulated in two orthogonal polarizations u, and v. Due to the non-circular cross-sectional geometry of the fiber, and non-isotropic stress on the fiber, light in different polarizations travel at different group velocity, i.e., β 1 is polarization dependent. This effect is called polarization mode dispersion (PMD). The PMD can be characterized by the principal states of polarization (PSP), x and y, and light component in the x or y polarization, travels with a group velocity of 1/β 1x or 1/β 1y. When α x α y, it indicates that there is polarization dependent loss. The PSP of the fiber depends on its geometry, the temperature, and the stress/vibration it is under [2]. Thus, the PSP is a slowly time-varying property, with a coherence time of a few ms [2]. In long-haul systems, it is usually assumed that the u-v polarizations are not aligned with the x-y polarization. We may extend (2.7) to its vector version, the coupled nonlinear Schrödinger equation (CNLSE) [3] A x (t, z) z + jβ 1x A x (t, z) t + jβ A x (t, z) + α x t 2 2 A x(t, z) = jγ ( A x (t, z) 2 + W A y (t, z) 2) A x (t, z), (2.16) A y (t, z) z + jβ 1y A y (t, z) t + jβ A y (t, z) + α y t 2 2 A y(t, z) = jγ ( A y (t, z) 2 + W A x (t, z) 2) A y (t, z), (2.17) where W is a coupling coefficient that depends on the angle difference between the u-v polarization states and the PSP. Amplifier The loss of the fiber is compensated by an erbium doped fiber amplifier (EDFA) at the end of each span. An EDFA consists of a piece of fiber, whose core is doped with

33 Chapter 2. Fiber-Optic Communication Systems 18 erbium ions. Such ions are simple two-level systems that have only two energy states: a fundamental state and an excited state. A strong laser light at a proper wavelength (usually 980 nm or 1480 nm) propagates in the core of the fiber to excite the erbium ions. When photons in the optical signal in the channel hit the excited ions, they have a certain probability (depending on the wavelength of the input photon) of releasing photons identical to the hitting one. Such phenomenon is called stimulated emission, which brings the excited ion to its fundamental state. The ratio between the number of the photons at the output and the number of hitting photons is denoted as the gain G. With some probability, sometimes the ions may fall back to the fundamental state without being hit by a photon, during which a photon with a random wavelength is emitted. Such a process is called spontaneous emission. These photons may also get amplified in the EDFA to cause amplified spontaneous emission (ASE) noise [2]. The ASE noise in general follows a circularly symmetric complex Gaussian distribution. Samples of ASE can be considered to be independent and identically distributed (i.i.d.). The SNR degradation caused by ASE is quantified by the amplifier noise figure, F n, defined as F n = SNR in SNR out, (2.18) where SNR in and SNR out are the input and output signal-to-noise ratio respectively. Most practical amplifiers have F n greater than 3 db, and can be as large as 6 to 8 db [2]. Another parameter, the spontaneous-emission factor, n sp is often used to quantify the ASE level too. Its relationship with F n is F n = 2n sp (G 1)/G 2n sp, (2.19) where G is the gain of the EDFA, defined as the ratio between the output power and the input power, which is usually a large number [2]. In practice, the ASE noise, the laser noise, the CD, the PMD, and the nonlinearity all interact with each other to act on the optical signal. The noises also gets dispersed and also causes nonlinearity-induced-interference. In some of the analysis in this thesis, high SNR is assumed, and hence the noise-signal nonlinear interference is ignored. However, in the simulations, all such interactions are properly taken into consideration Receiver At the receiver, the local oscillator (LO) is multiplied with the optical signal to produce one image at the high frequency, and one image at the low frequency. The LO and

34 Chapter 2. Fiber-Optic Communication Systems 19 the transmitter laser should have frequencies that closely match with each other (but in general, they are not strictly the same). Hence the low frequency image is roughly centered around the zero frequency. The high frequency images get filtered out by a low pass filter, which leaves us with the base band signal. The received base band signal is projected into two orthogonal polarization states, which are not necessarily aligned with the u-v polarization states. In each polarization, the optical in-phase and quadrature components are not necessarily aligned with the transmitter either. The optical signal gets converted to electrical signal by photo detectors, and then sampled and quantized to get the digital received signal. The digital signal then goes through a sequence of digital signal processing (DSP) modules: CD equalizer, polarization de-multiplexer, frequency estimation (FE), and carrier phase recovery (CR) module. The CD equalizer, which reverses the ISI can be implemented either in frequency domain, or implemented in time domain as an FIR filter [40]. The LO phase noise, despite that it does not go through CD, also gets filtered by this process. This causes amplification in the phase noise, and it is often referred to as equalizer enhanced phase noise (EEPN) [41]. The in-phase and quadrature component, as well as the two orthogonal polarization states can be realigned with the transmitter using a blind butterfly structure polarization demultiplexer [62]. The resulting signal may have π/2 phase offset with the transmitted signal, and it cannot distinguish the two orthogonal polarizations. Some pilot symbols can be used to correct such misalignment. The RX laser is usually not phase locked or frequency locked with the TX laser, hence frequency offset and additional laser phase noise are added to the signal at the receiver. The signal goes through the FE module that compensates for the frequency offset between TX and RX lasers. The CR module corrects part of the phase noise caused by phase misalignment between the transmitter and LO, the laser linewidth, and the nonlinear effects. The FE module can be implemented independently of CR [85], or they can be combined together in one module [73]. The FE module is sometimes absent in the later chapters, because in simulation, only the envelope of the waveform is propagated in the fiber, and no frequency offset can be observed in the result. Lastly, the signal gets passed into the decoder. 2.2 Split-Step Fourier Method The nonlinear Schrödinger equation (2.8) does not have a known analytical solution. Hence we have to use numerical methods to solve it, if we want to model the light

35 Chapter 2. Fiber-Optic Communication Systems 20 propagation in the fiber. One well-known such numerical method is called the splitstep Fourier method [3]. This is a finite element analysis method, that is based on the assumption that within a small step in the distance z, the linear and nonlinear part of the partial differential equation can operate independent of each other. Note that (2.8) contains a part that represents the dispersive effects, and a part that represents the nonlinear effects. We can separate these two parts and rewrite (2.8) in the following form where A(t, z) z = ( ˆD + ˆN)A(t, z), (2.20) ˆD = α 2 jβ t 2, (2.21) ˆN = jγ A(t, z) 2. (2.22) Over a long fiber, the linear operator ˆD and the nonlinear operator ˆN act together on the signal. However, over a short distance h, the dispersive and nonlinear effects can be assumed to act independently. In this case, we have A(t, z + h) exp(h ˆD) exp(h ˆN)A(t, z). (2.23) The nonlinear step takes the form A(t, z) z = jγ A(t, z) 2 A(t, z). (2.24) Therefore B(t, z) = exp(h ˆN)A(t, z) can be immediately calculated in the time domain as B(t, z) = exp(h ˆN)A(t, z) = A(t, z) exp(jγ A(t, z) 2 h). (2.25) We can calculate the effect of the dispersion operator, exp(h ˆD), in the frequency domain exp(h ˆD)B(t, z) = F 1 [exp(h ˆD ω )F[B(t, z)]], (2.26) where F and F 1 are the Fourier transform and inverse Fourier transform operators, ˆD ω is calculated from (2.32) by replacing t with jω i. The local error of this procedure is bounded by Ch 2 for some constant C, denoted as O(h 2 ) [3]. Such an accuracy is considered to be low in our simulations. The accuracy can be improved using symmetrized

36 Chapter 2. Fiber-Optic Communication Systems 21 split-step Fourier method. Instead of (2.23), we use ( ) h A(t, z + h) exp 2 ˆD ( exp h ˆN ) ( ) h exp 2 ˆD A(t, z). (2.27) We divide the dispersive operator into two symmetric parts each with distance advancement of h/2. A half of the dispersive effect is computed before the nonlinear effect, the other half is computed after the nonlinear effect. The local error of this method is O(h 3 ), which is acceptable in most applications [3]. In the case where PolDM is employed, our goal is to solve (2.17) instead of (2.8). The split-step Fourier method can be modified to accommodate the PMD by defining A x (t, z) z A y (t, z) z = ( ˆD x + ˆN x )A x (t, z), (2.28) = ( ˆD y + ˆN y )A y (t, z), (2.29) where ˆD x = α x 2 jβ 1x t jβ t, (2.30) 2 ˆN x = jγ ( A x (t, z) 2 + W A y (t, z) 2). (2.31) ˆD y = α y 2 jβ 1y t jβ t, (2.32) 2 ˆN y = jγ ( A y (t, z) 2 + W A x (t, z) 2) (2.33) (recall that W is the polarization coupling coefficient). The wave propagation can be simulated as ( h A x (t, z + h) exp A y (t, z + h) exp 2 ˆD x ( h 2 ˆD y ) exp (h ˆN ) ( h x exp ) exp (h ˆN y ) exp ) 2 ˆD x A x (t, z), (2.34) ) A y (t, z). (2.35) ( h 2 ˆD y 2.3 Digital Back-Propagation Since nonlinearity interacts with dispersion in the fiber, there is no simple way to compensate for nonlinear effects completely without involving dispersion. It has been shown in [36, 54, 59] that we can use split-step Fourier method at the receiver to reverse both the dispersive and nonlinear effects in the fiber, often referred to as digital backpropagation (DBP). Effectively, we can take negative steps h from the receiver, and

37 Chapter 2. Fiber-Optic Communication Systems 22 back-propagate the signal until the transmitter is reached. Because none of the PSP, β 1x, β 1y, α x, or α y is available at the receiver, even for PolDM systems, DBP can only back-propagate the scalar version NLSE (2.8). Following (2.27), the DBP algorithm can be described as ( A(t, z h) exp h ) 2 ˆD ( exp h ˆN ) ( exp h ) 2 ˆD A(t, z). (2.36) To achieve high accuracy in DBP the waveform is usually sampled at minimum twice the symbol rate [81], and there should be at least one step per fiber span. Thousands of symbols are usually back-propagated together to minimize the edge effect. Note that the linear step is computed in Fourier domain, while the nonlinear step is computed in time domain. This means for each step we have to compute the Fourier transform and inverse Fourier transform of thousands of samples. This makes the complexity of DBP much higher than other DSP modules in an optical system. For WDM systems, the signal in adjacent channels also influence the refractive index of the fiber. Hence the waveform from all channels need to be co-back-propagated together to fully account for both the intra- and inter-channel nonlinear effects. In a network where there exist add/drop functionalities, it is impossible to obtain the information from all co-propagating channels at each receiver. Even when such information is available, a much higher sampling frequency is required to capture the waveform of the interfering channels. This makes the complexity of full DBP even higher, and more difficult to be realized. 2.4 Random Symbol Transmission In the experimental part of Chapter 5, the optical signal is generated based on a precomputed uniform i.i.d. pseudo random bit sequence, which is mapped to a uniform i.i.d. random sequence of constellation points. It is technically difficult to overwrite this precomputed bit sequence and replace it with a coded bit sequence. Hence, an alternative way is used to use this random symbol transmission to simulation a system, where the bit sequence is encoded. This section explains how this can be done with the help of a dither, and why it is a close approximation to the true performance where we have the freedom to choose the transmitted bit sequence. A bit-interleaved coded-modulation (BICM) system is assumed to be used in a real system, where a large interleaver is used to interleave hundreds of thousands of encoded bits to decorrelate the the bit sequence. This interleaved bit sequence is modulated into

38 Chapter 2. Fiber-Optic Communication Systems 23 a sequence of constellation points. For inner codes like LDPC code, the block length is typically several or tens of thousands of bits. After the interleaver, it can be assumed that a particular bit has negligible correlation with the adjacent bits. Fig. 2.4 shows a block diagram of how a dither can help convert a random bit sequence to a coded bit sequence. More importantly, this dither preserves the magnitude of the log-likelihood-ratio (LLR) of each bit, and whenever the dither is 0, the sign of the LLR is also preserved, otherwise, the sign is negated. Let Q be the set of constellation points, and let n = log 2 Q. The label for each constellation point is a binary n-tuple. Let {a[i]} be a random binary n-tuple sequence with a[i] F2 n that are independent and identically distributed (i.i.d.). We assume that a[i] follows a uniform distribution over all 2 n n-tuples, and can be mapped to a constellation point through a one-to-one mapping M : F2 n Q, M(a[i]) = x[i], where M is fixed for all i. The sequence {x[i]} is transmitted through a channel and a complex sequence y[i] C is received. Then we have the following properties: 1. The symbols x[i] are i.i.d., and follow a uniform distribution over Q. 2. For another fixed binary n-tuple sequence {b[i]}, we can define a dither sequence d[i] = a[i] b[i]. 3. Define the mapping ˆM[i] : F n 2 Q, and ˆM[i](b[i]) = M(b[i] d[i]). Theorem 1. Let a[i, j] be the j-th bit of the i-th n-tuple a[i], and y[i] be the i-th received symbol. Let the LLR of a[i, j] be defined as LLR a[i,j] = log p(a[i, j] = 0 y[i]) p(a[i, j] = 1 y[i]). (2.37) Then, LLR b[i,j] = ( 1) b[i,j] LLR a[i,j]. (2.38) Proof. The LLR of b[i, j] can be written as p(b[i, j] = 0 y[i]) LLR b[i,j] = log p(b[i, j] = 1 y[i]) = p(a[i, j] d[i, j] = 0 y[i]) log p(a[i, j] d[i, j] = 1 y[i]) = p(a[i, j] = d[i, j] y[i]) log p(a[i, j] = 1 d[i, j] y[i]). (2.39) Hence, when d[i, j] = 0, LLR b[i,j] = LLR a[i,j], and when d[i, j] = 1, LLR b[i,j] = LLR a[i,j].

39 Chapter 2. Fiber-Optic Communication Systems 24 During the experiment, we know a[i] and b[i], hence the dither d[i] = a[i] b[i] can hence be computed. While a[i] is i.i.d., b[i] is determined from a set of codewords. Using Theorem 1, we may also compute LLR b[i,j] based on d[i, j] and LLR a[i,j]. When an LDPC code is used, it is as if we can transmit an encoded bit sequence. Let Λ denote the LLR distribution of bit b, and let f Λ b (λ b). We assume that f Λ b (λ 0) = f Λ b ( λ 1), (2.40) i.e., the channel is symmetric, hence fixing b[i] = for all i (for the simplicity of encoding and decoding) is equivalent to using any other codeword. Encoded bit sequence b[i] F 2 n Binary dither sequence n d[i] F 2 Random binary sequence n a[i] F 2 + Bit sequence to symbol mapper M Random symbol sequence x[i] Q Channel P(b[i] = b[i]) y[i]) P(a[i] = a[i]) y[i]) + M 1 ( x[i]) P(x[i] = x[i]) y[i]) j = 1, 2, 2 n Received symbols y[i] C d[i] Figure 2.4: Block diagram of the transmission system.