DSP-Based Coherent Optical Systems: Receiver Sensitivity and Coding Aspects

DSP-based Coherent Optical Systems: Receiver Sensitivity and Coding Aspects

MIU YOONG LEONG

Licentiate Thesis in Physics Stockholm, Sweden 2015 KTH School of Information and Communication Technology TRITA-ICT 2015:03 SE-164 40 Stockholm ISBN 978-91-7595-551-3 SWEDEN

Akademisk avhandling som med tillstånd av Kungl Tekniska Högskolan framläg- ges till oﬀentlig granskning för avläggande av teknologie licentiatexamen i fysik onsdagen den 10 juni 2015 klockan 10.00 i Sal C, Electrum 229, Kungl Tekniska Högskolan, Isafjordsgatan 22, Stockholm.

Tryck: Universitetsservice US AB iii

Abstract

User demand for faster access to more data is at a historic high and rising. One of the enabling technologies that makes the information age possible is fiber-optic communications, where light is used to carry information from one place to another over optical fiber. Since the technology was first shown to be feasible in the 1970s, it has been constantly evolving with each new generation of fiber-optic systems achieving higher data rates than its predecessor. Today, the most promising approach for further increasing data rates is digital signal processing (DSP)-based coherent optical transmission with multi-level modulation. As multi-level modulation formats are very susceptible to noise and distortions, forward error correction (FEC) is typically used in such systems. However, FEC has traditionally been designed for additive white Gaussian noise (AWGN) channels, whereas fiber-optic systems also have other impairments. For example, there is relatively high phase noise (PN) from the transmitter and local oscillator (LO) lasers. The contributions of this thesis are in two areas. First, we use a unified approach to analyze theoretical performance limits of coherent optical receivers and microwave receivers, in terms of signal-to-noise ratio (SNR) and bit error rate (BER). By using our general framework, we directly compare the performance of ten coherent optical receiver architectures and five microwave receiver architectures. In addition, we put previous publications into context, and identify areas of agreement and disagreement between them. Second, we propose straightforward methods to select codes for systems with PN. We focus on Bose-Chaudhuri-Hocquenghem (BCH) codes with simple − implementations, which correct pre-FEC BERs around 10 3. Our methods are semi-analytical, and need only short pre-FEC simulations to estimate error statistics. We propose statistical models that can be parameterized based on those estimates. Codes can be selected analytically based on our models.

Acknowledgments

I would like to thank my main supervisor Assoc. Prof. Sergei Popov. His op- timism and can-do attitude were both encouraging and infectious. He brought a unique perspective, which improved both my understanding and the clarity of my explanations. I would also like to thank my co-supervisor Prof. Gunnar Jacobsen, another optimist. His drive to keep things moving ensured that we made steady progress. He was always accessible, providing me with invaluable opportunities to tap into his wealth of knowledge and experience. Much of this research was conducted in collaboration with the Technical Uni- versity of Denmark (DTU). I would especially like to thank Prof. Knud J. Larsen, whose expertise and well-considered input were vital to the success of this work. I would also like to thank Assoc. Prof. Darko Zibar for his constructive comments. My thanks also to the good people at Aston University. For the opportunity to conduct part of my research there, I would like to thank Dr. Sergey Sergeyev. I would also like to thank Tatiana Kilina for the engaging conversations. A nod to my colleagues at Acreo Swedish ICT and KTH Royal Institute of Tech- nology, especially those at the Networking and Transmission Laboratory (Netlab) at Acreo, for the useful discussions in a pleasant atmosphere. Additionally, I would like to thank my friends, especially Carrie for being sup- portive and dependable through the years. Lastly, I would like to thank my family, especially my father for his quiet love and unrelenting determination to put me through school.

Contents

Acknowledgments v

List of Publications ix

Contributions and Structure of the Thesis xi

List of Abbreviations xiii

List of Figures xvii

List of Tables xix

1 Introduction 1 1.1 HistoricalBackground ...... 1 1.1.1 The Rise, Demise, and Rebirth of Coherent Optical Systems 1 1.1.2 Forward Error Correction in Coherent Systems ...... 3 1.2 State-of-the-art and Challenges ...... 3

2 Coherent Optical Systems 7 2.1 Overview ...... 7 2.2 Transmitter...... 7 2.3 Channel...... 10 2.3.1 Fiber Losses and Ampliﬁcation ...... 10 2.3.2 LaserPhaseNoise ...... 11 2.3.3 ChromaticDispersion ...... 12 2.3.4 Polarization-Mode Dispersion ...... 12 2.3.5 Nonlinear Eﬀects ...... 13 2.4 Receiver...... 14 2.4.1 Optical and Electrical Front-end ...... 14 2.4.2 Analog-to-Digital Converter ...... 23 2.4.3 Digital Signal Processing ...... 24 2.5 Electrical Front-end in Microwave Systems ...... 25 2.6 Theoretical Performance Limits ...... 28

vii viii CONTENTS

3 Forward Error Correction 31 3.1 General Principles ...... 31 3.2 LinearBlockCodes...... 34 3.3 CyclicCodes ...... 35 3.4 BCHCodes...... 35 3.5 Serially-concatenated Codes ...... 36 3.6 TurboCodes ...... 37 3.7 LDPCCodes ...... 37 3.8 CodedModulation ...... 37 3.9 Interleaving ...... 38

4 BCH Codes for Coherent Optical Systems 41 4.1 System Model and General Approach ...... 41 4.2 BinaryBCHcodes ...... 42 4.3 RScodes ...... 48

5 Conclusion and Future Research 51

Bibliography 55 List of Publications

Publications included in the thesis I. M. Y. Leong, G. Jacobsen, S. Popov, and S. Sergeyev, “Receiver sensitivity in optical and microwave, heterodyne and homodyne systems,” Journal of Optical Communications, vol. 35, no. 3, pp. 221–229, Mar. 2014.

II. M. Y. Leong, K. J. Larsen, G. Jacobsen, S. Popov, D. Zibar, and S. Sergeyev, “Dimensioning BCH codes for coherent DQPSK systems with laser phase noise and cycle slips,” J. Lightw. Technol., vol. 32, no. 21, pp. 4048–4052, Nov. 2014.

III. M. Y. Leong, K. J. Larsen, G. Jacobsen, S. Popov, D. Zibar, and S. Sergeyev, “Interleavers and BCH Codes for Coherent DQPSK Systems with Laser Phase Noise,” IEEE Photon. Technol. Lett., vol. 27, no. 7, pp. 685–688, Apr. 2015.

IV. M. Y. Leong, K. J. Larsen, G. Jacobsen, S. Popov, D. Zibar, and S. Sergeyev, “Dimensioning RS codes for mitigation of phase noise induced cycle slips in DQPSK systems,” in Proc. Asia Communications and Photonics Conference (ACP), Nov. 2014, ATh4D.4.

Related publications not included in the thesis V. M. Y. Leong, K. J. Larsen, G. Jacobsen, S. Popov, D. Zibar, and S. Sergeyev, “Novel BCH code design for mitigation of phase noise induced cycle slips in DQPSK systems,” in Proc. Conf. Lasers Electro-Opt. (CLEO), Jun. 2014, STu3J.6.

VI. M. Y. Leong, S. Popov, G. Jacobsen, and S. Sergeyev, “SNR comparison of coherent optical receivers,” in Proc. Progress in Electromagnetics Research Symposium (PIERS), Aug. 2014, p. 966.

VII. M. Y. Leong, K. J. Larsen, G. Jacobsen, S. Popov, D. Zibar, and S. Sergeyev, “Interleaving to Reduce Code Overhead in DQPSK Systems,” in Proc. Progress in Electromagnetics Research Symposium (PIERS), Jul. 2015, in print.

Contributions and Structure of the Thesis

The contributions of this thesis are in two areas: 1. theoretical performance limits for uncoded systems, and 2. simple methods to select simple Bose-Chaudhuri-Hocquenghem (BCH) codes for coherent optical systems with laser phase noise (PN). The thesis is structured as follows. A brief introduction to the history and state-of-the-art is in Chap. 1. An overview of coherent optical systems is in Chap. 2. This covers the transmitter, channel, and receiver. In addition, coherent optical receivers are compared with microwave receivers. A general introduction to coding is in Chap. 3. In Chap. 4, we describe methods for selecting BCH codes for coherent systems with PN. Finally, in Chap. 5, we conclude and provide suggestions for future research. The research presented in this thesis is supported by results which we have published in the following papers.

Journal Papers I. M. Y. Leong, G. Jacobsen, S. Popov, and S. Sergeyev, “Receiver sensitivity in optical and microwave, heterodyne and homodyne systems,” Journal of Optical Communications, vol. 35, no. 3, pp. 221–229, Mar. 2014.

In Paper I, we derive theoretical limits of signal-to-noise ratio (SNR) and bit error rate (BER) for various coherent optical and microwave receiver architectures using a uniﬁed approach. We consider Gaussian noise sources and uncoded systems in our analysis. Based on our framework, we put previous publications in context, and identify areas of agreement/disagreement between them. The author of this thesis was responsible for the mathematical derivations, the comparisons with previous publications, and writing the manuscript.

II. M. Y. Leong, K. J. Larsen, G. Jacobsen, S. Popov, D. Zibar, and S. Sergeyev, “Dimensioning BCH codes for coherent DQPSK systems with laser phase noise

xi xii CONTRIBUTIONS AND STRUCTURE OF THE THESIS

and cycle slips,” J. Lightw. Technol., vol. 32, no. 21, pp. 4048–4052, Nov. 2014.

In Paper II, we propose a statistical model to describe the bit errors in a diﬀerential quadrature phase-shift keying (DQPSK) system with laser PN. We use the model to select binary BCH codes that meet a target post-forward error correction (FEC) BER. The author of this thesis was responsible for the choice of statistical model, developing the method for code selection, verifying the method using simulations, and writing the manuscript.

In Paper III, we propose a method to identify combinations of binary BCH codes and interleavers that achieve a target post-FEC BER. We obtain general results for theoretical uniform interleaving for DQPSK systems with laser PN. We also establish that a trade-oﬀ exists between code overhead and interleaver delay. The author of this thesis was responsible for developing the method for code selection, verifying the method using simulations, and writing the manuscript.

Conference Papers IV. M. Y. Leong, K. J. Larsen, G. Jacobsen, S. Popov, D. Zibar, and S. Sergeyev, “Dimensioning RS codes for mitigation of phase noise induced cycle slips in DQPSK systems,” in Proc. Asia Communications and Photonics Conference (ACP), Nov. 2014, ATh4D.4.

In Paper IV, we propose a method for selecting Reed-Solomon (RS) codes for coherent DQPSK systems with laser PN. The author of this thesis was responsible for modifying the method for code selection (from Paper V, which is not included in the thesis), verifying the method using simulations, and writing the manuscript. List of Abbreviations

ADC analog-to-digital converter

AGC automatic gain control

ASE ampliﬁed spontaneous emission

AWGN additive white Gaussian noise

BCH Bose-Chaudhuri-Hocquenghem

BER bit error rate

BPF bandpass ﬁlter

BPSK binary phase-shift keying

DQPSK diﬀerential quadrature phase-shift keying

DSP digital signal processing

DTU Technical University of Denmark

EDFA erbium-doped ﬁber ampliﬁer

FEC forward error correction

FFT fast Fourier transform

FWHM full width half maximum

FWM four-wave mixing

GVD group-velocity dispersion

I in-phase

IF intermediate frequency

IFFT inverse fast Fourier transform

xiii xiv LIST OF ABBREVIATIONS i.i.d. independent identically distributed LDPC low-density parity-check LFSR linear feedback shift register LO local oscillator LNA low-noise amplifier LPF low-pass filter MSB most-significant bit MZM Mach-Zehnder modulator NCG net coding gain OBPF optical bandpass filter OFDM orthogonal frequency-division multiplexing OOK on-off keying OPLL optical phase-locked loop PAPR peak-to-average power ratio PBC polarization beam combiner PBS polarization beam splitter PC polarization controller PDF probability density function PE phase estimate PLL phase-locked loop PMD polarization-mode dispersion PN phase noise PSD power spectral density PSK phase-shift keying Q quadrature QPP quadratic permutation polynomial QPSK quadrature phase-shift keying xv

QAM quadrature amplitude modulation

RF radio frequency

RS Reed-Solomon

SBS stimulated Brillouin scattering SC-FDMA single-carrier frequency-division multiple access

SNR signal-to-noise ratio

SPM self phase modulation

SRS stimulated Raman scattering

VV Viterbi-Viterbi WDM wavelength-division multiplexing

XPM cross phase modulation

List of Figures

2.1 Coherentsystem ...... 8 2.2 Electro-optical transmitter ...... 8 2.3 Single-carrier modulation formats ...... 9 2.4 QPSKvs.DQPSK...... 9 2.5 QPSK constellation with AWGN ...... 11 2.6 QPSK constellation with laser PN ...... 12 2.7 Chromaticdispersion...... 12 2.8 Polarization-mode dispersion ...... 13 2.9 Single-polarization heterodyne without image rejection ...... 15 2.10 Single-polarization heterodyne with image-reject OBPF ...... 15 2.11 Single-polarization heterodyne with optical image-reject receiver . . . . 16 2.12 Single-polarization dual-quadrature homodyne ...... 17 2.13 Single-polarization single-quadrature homodyne ...... 17 2.14 Dual-polarization heterodyne without image rejection ...... 18 2.15 Dual-polarization heterodyne with image-reject OBPF ...... 19 2.16 Dual-polarization heterodyne with optical image-reject receiver . . . . . 20 2.17 Dual-polarization dual-quadrature homodyne ...... 21 2.18 Dual-polarization single-quadrature homodyne ...... 22 2.19 Optical image-reject mixer using a high-birefringence ﬁber ...... 23 2.20DSPsubsystems ...... 24 2.21 Theoretical BER for QPSK/DQPSK with AWGN ...... 25 2.22 Microwave heterodyne without image rejection ...... 26 2.23 Microwave heterodyne with image reject BPF ...... 26 2.24 Microwave heterodyne with image-reject receiver ...... 26 2.25 Microwave dual-quadrature homodyne ...... 26 2.26 Microwave single-quadrature homodyne ...... 27 2.27 Hartley image-reject receiver ...... 27

3.1 Signal space and code space ...... 32 3.2 Grosscodinggain ...... 32 3.3 Hard-decision decoding vs. soft-decision decoding ...... 33 3.4 Systematic linear block code ...... 34

xvii xviii List of Figures

3.5 BinaryBCHcode...... 36 3.6 RScode...... 36 3.7 Serially-concatenated code ...... 36 3.8 Parallel-concatenated code ...... 37 3.9 Cosetpartitioning ...... 38 3.10Codedmodulation ...... 38 3.11 Interleaver using random permutation ...... 39 3.12Blockinterleaver ...... 39

4.1 Systemmodel...... 42 4.2 Pre-FECBER ...... 43 4.3 Correlated bivariate binomial PDF ...... 45 4.4 BER performance of binary BCH codes ...... 46 4.5 Trade-oﬀ between code overhead and interleaver length ...... 47 4.6 BER performance of RS codes ...... 49 List of Tables

2.1 Single-/dual-polarization optical receiver SNRs ...... 28 2.2 Microwave receiver SNRs ...... 29 2.3 BERasafunctionofSNR...... 29 2.4 Comparison of coherent optical results with other publications . . . . . 30 2.5 Comparison of microwave results with other publications ...... 30

4.1 Binary BCH codes and uniform interleavers ...... 47 4.2 RScodes ...... 49

xix

Chapter 1

Introduction

The first digital communication system was the electric telegraph, invented by Samuel Morse in 1837. By the late 19th century, efforts were well underway to develop radio frequency (RF) telecommunication systems. In 1909, the Nobel Prize in Physics was awarded to Guglielmo Marconi and Karl Ferdinand Braun “in recogni- tion of their contributions to the development of wireless telegraphy”. By contrast, interest in fiber-optic communication systems did not take off until the 1970s. Ex- actly one century after Marconi and Braun, in 2009, Charles K. Kao was awarded the Nobel Prize in Physics “for groundbreaking achievements concerning the transmission of light in fibers for optical communication”. In this chapter, we trace the development of fiber-optic systems from their birth in 1970 to the present day.

1.1 Historical Background

Early ﬁber-optic systems were uncoded. Only in later systems was forward error correction introduced to improve system performance in terms of bit error rate (BER). As the design of uncoded systems is a specialization in itself, and often distinct from the design of codes, we present the two historical tracks separately.

1.1.1 The Rise, Demise, and Rebirth of Coherent Optical Systems In 1970, two key breakthroughs were made in the field of fiber-optic communications. The first was the development of optical fiber with losses below 20 dB/km [1], thereby finally achieving that which previous studies had indicated was possible [2]. The second was the development of compact GaAs semiconductor lasers [3]. To- gether, these achievements spurred the research community to develop practical fiber-optic communication systems [4–10]. First generation systems operated at a wavelength of around 0.8 µm using GaAs semiconductor lasers. In 1980, the first commercial system had a data rate of 45

1 2 CHAPTER 1. INTRODUCTION

Mbps and repeater spacing of 7 km over multi-mode fiber [11]. Second generation systems operated around 1.3 µm over single-mode fibers and used InGaAsP semiconductor lasers. At this wavelength, fibers had lower losses (typically 0.5 dB/km) and minimal chromatic dispersion. Additionally, the switch to single-mode fibers eliminated modal dispersion. As a result, commercial systems in 1987 had a maximum data rate of 1.7 Gbps and a repeater spacing of 50 km [4]. Third generation systems operated at 1.55 µm. At this wavelength, fiber losses could be reduced to 0.2 dB/km [12], but fiber chromatic dispersion was high. The solution to the dispersion problem came in the form of dispersion-shifted fibers and limiting the laser spectrum to a single longitudinal mode. In 1990, commercial systems had data rates of 2.5 Gbps [4]. The rise of coherent systems occurred as scientists searched for ways to increase the repeater spacing of third generation systems, which were typically 60–70 km. At this point, the main motivation for using coherent systems was their improved sensitivity compared to direct-detection systems [13–15]. Coherent optical systems are similar to microwave systems, in the sense that the signal is mixed with a local oscillator (LO) at the receiver. By increasing the power of the LO laser in coherent optical receivers, receiver sensitivity approaches the shot noise limit [15, 16]. However, the main drawback of coherent systems is receiver complexity. For example, polarization-matching is needed between the received signal and LO laser [4]. Additionally, coherent receivers that use homodyne optical front-ends— where the optical signal is down-mixed directly to baseband—needed accurate phase tracking between the LO and signal carrier [4, 16]. Early homodyne receivers used optical phase-locked loops (OPLLs), which were difficult to implement. The invention of erbium-doped fiber amplifiers (EDFAs) led to the demise of coherent systems [4], as EDFAs provided a cheaper and simpler way to increase repeater spacing. Additionally, EDFA systems are limited by amplified spontaneous emission (ASE) instead of shot noise, so shot-noise limited coherent receivers became less interesting [16]. Instead, fourth generation systems used optical amplification such as EDFAs, and wavelength-division multiplexing (WDM) [4]. In a 1991 experiment, bit rates of 5 Gbps over 14 300 km were achieved in a recirculating loop [17]. Since commercial systems became available in 1996, data rates have continued to increase. By 2000, commercial systems had bit rates of 1.6 Tbps [4]. The rebirth of coherent systems occurred around the turn of the century. Unlike previously, interest this time was mainly driven by the quest for higher data rates. A 2002 experiment showed a doubling in spectral efficiency by using differential quadrature phase-shift keying (DQPSK) instead of on-off keying (OOK) [18]. By encoding two bits per symbol, DQPSK has double the bit rate for the same spectral bandwidth. Higher-order modulation formats have even higher spectral efficiencies. Another difference compared to the earlier coherent systems is the use of digital signal processing (DSP) [19]. These had improved over the years and could now operate at speeds high enough to process the baseband signal. Problems such as polarization- and carrier phase-compensation, which were difficult to solve previ- 1.2. STATE-OF-THE-ART AND CHALLENGES 3 ously, could now be solved digitally [16,20]. So far, experiments have achieved data rates of over 100 Tbps in a single-core fiber [21, 22], and over 1 Pbps in a 12-core fiber [23].

1.1.2 Forward Error Correction in Coherent Systems One of the earliest experiments involving forward error correction (FEC) in fiber- optic systems was published in 1988 [24, 25]. First generation FEC used hard- decision decoding, where the demodulator made firm decisions on the received constellation points. The detected bit sequence of 1’s and 0’s was then passed onto the decoder. Such systems were initially deployed in submarine fiber-optic systems in the 1990s [26]. Among first-generation codes, the Reed-Solomon (RS) code RS(255, 239) recommended by ITU-T G.975 became widely adopted in industry. It had a net coding gain (NCG)—coding gain adjusted for the increase in bit rate 15 due to redundancy—of 6.2 dB at a post-FEC BER of 10− [27]. Second generation FEC used serially concatenated codes with hard-decision decoding [24]. First, an inner code corrected most of the errors. Then, an outer code corrected any remaining errors. Performance could be further improved using techniques such as interleaving and iterative decoding. The FEC that was widely used in 10 and 40 Gbps WDM systems had an NCG of 8.5 dB at a post-FEC BER 15 of 10− [24]. Third generation FEC used soft-decision and iterative decoding to achieve NCGs above 10 dB [24]. When soft-decisions are used, the demodulator output is quan- tized to Q bits, where Q is an integer greater than 1. The most-significant bit (MSB) is the hard decision, whereas the other Q 1 bits indicate the level of confidence in the decision. The first experiment involving− soft-decision based FEC in optical systems was conducted in 1999 [24]. In that system, an inner soft-decision convolutional code with rate r = 1/2 and constraint length k = 7 was paired with an outer RS(255, 239) code [28]. This gave an NCG of 10 dB. Currently, 100 Gbps systems use either soft-decision block-turbo codes or soft-decision low-density parity-check (LDPC) codes, with the latter being more widespread [29]. Recently, it has been shown that spatially-coupled LDPC codes can achieve capacity [30,31]. 15 So far, an NCG of 12 dB at post-FEC BER of 10− has been shown in numerical simulations [32]. That system used a soft-decision based FEC with concatenated LDPC and Bose-Chaudhuri-Hocquenghem (BCH) codes.

1.2 State-of-the-art and Challenges

Since fiber-optic systems were first shown to be feasible in the 1970s, the technology has been constantly evolving with each new generation achieving higher data rates than its predecessor. Currently, the most promising technology for further increasing data rates is DSP-based coherent systems with multi-level modulation. By switching from OOK to quadrature phase-shift keying (QPSK), 16-quadrature 4 CHAPTER 1. INTRODUCTION amplitude modulation (QAM), and higher-order modulation formats, data rates can be increased without increasing spectral bandwidth. Channel impairments in fiber-optic systems include laser phase noise, nonlinear phase noise, chromatic dispersion, and polarization-mode dispersion (PMD). Various DSP algorithms have been proposed to estimate and compensate these impairments [33–35]. Additionally, there is additive white Gaussian noise (AWGN) from ASE in fiber amplifiers, shot noise in photodetectors, and thermal noise in electrical circuits. In our research, we analyze receiver sensitivity for various architectures. Increasing the order of the constellation without reducing noise and distortions gives worse BER. However, noise and distortions are often limited by practical considerations. For example, long-haul systems are typically ASE-limited, making it difficult to further reduce AWGN. By contrast, FEC can be easily implemented in DSP-based systems, and is a straightforward way to improve BER. However, much of coding theory assumes AWGN channels, and therefore ignores the relatively high phase noise (PN) in coherent optical systems. Various algorithms have been proposed for estimating and compensating PN [20, 36]. These can be divided into feedback and feedforward schemes, depending on how modulation is removed. Feedback schemes are decision-directed, whereas feedforward schemes exploit other properties to remove modulation. Due to practical issues such as feedback delay [37], and difficulties in implementing, verifying, and troubleshooting feedback loops, feedforward schemes are often preferred. A popular feedforward scheme for M-ary phase-shift keying (PSK) is the Viterbi-Viterbi (VV) algorithm [38], which removes modulation by raising to the M-th power, followed by a smoothing filter. Several variants of VV have been proposed [34], including the use of Wiener filtering [36] and the low-complexity Barycenter approximation [39]. Another feedforward approach is blind phase search [37], which tests a number of candidate phase shifts and selects the best one. Naturally, phase estimation can also be done using pilot symbols, at the cost of extra overhead. Blind phase search and pilot-based estimation can be directly applied to higher-order modulation formats beyond QPSK. However, to use VV on 16-QAM, the constellation is first partitioned into QPSK subsets [40]. This approach becomes unwieldy for square constellations beyond 16-QAM. Instead, it is more straightforward to use VV with circular QAM [41]. As blind phase search has good performance but high complexity whereas VV-based methods have lower complexity but poorer performance, two-stage algorithms have been proposed [20, 42, 43]. These use coarse- and fine-estimation stages, with the aim of lowering complexity and/or improving performance. Phase estimation also resolves the phase ambiguity in a constellation. For example, square constellations have a 4-fold rotational symmetry – the constellation looks identical when rotated by integer multiples of 90◦. When resolved incor- rectly, this is known as a cycle slip. Typically, differential encoding is used in systems where cycle slips occur frequently, and not used if they occur infrequently. This gives rise to different types of errors, and hence affects the choice of FEC. 1.2. STATE-OF-THE-ART AND CHALLENGES 5

When differential encoding is not used, e.g. QPSK, cycle slips result in catas- trophic bit errors after demodulation. In [44], the authors propose an approach to detect and correct cycle slips. In [45], pilots are inserted periodically into the data sequence in order to limit the length of an error burst, and a burst error correcting code is used. In [46], a stronger code is used on some of the data and a weaker code is used on the rest. The strongly-coded data is then used as extra pilots. In [47], the authors propose a coded modulation scheme using LDPC codes. When differential encoding is used, e.g. DQPSK, differential decoding in the receiver corrects the subsequent errors, leaving a single symbol error at the position of each cycle slip. However, there is a BER penalty for AWGN errors. For DQPSK, a nearest-neighbor error of 90◦ gives two bit errors. The BER penalty decreases as the order of the constellation± increases, so very high order modulation formats have no penalty [48]. Several soft-decision-based schemes using LDPC codes have been proposed for coherent systems with PN [49–51]. Ultimately, the choice of FEC scheme depends on many factors, including the classic trade-off between performance and cost/complexity. This design space has not yet been fully explored, especially when it comes to systems with PN. In our research, we focus on such systems, and in particular on simple methods to design simple codes with simple implementations.

Chapter 2

Coherent Optical Systems

Fiber-optic systems use light to transmit information from one place to another over optical fiber. As with all communication systems, the physical layer of fiber-optic systems can be modeled in three parts: the transmitter, channel, and receiver. However, the details in each block are unique to the particular system. In this chapter, we describe these blocks as they apply to coherent fiber-optic systems. Additionally, we draw parallels between coherent optical receivers and microwave receivers, as both have much in common.

2.1 Overview

The blocks in a coherent optical system are shown in Fig. 2.1. The distinguishing feature of a coherent system is the presence of a local oscillator (LO) laser at the receiver. By ﬁrst combining the received optical signal with an LO, the output of the photodetectors contains information on the amplitude and phase of the electric ﬁeld. These are modulated in the transmitter, in order to send information to the receiver. In digital signal processing (DSP)-based coherent optical systems, digital signal processing is used in the receiver to compensate impairments and demodulate the data.

2.2 Transmitter

In the transmitter, data bits modulate the amplitude and phase of a transmitter laser using a Mach-Zehnder modulator (MZM). Either one or both polarizations of light may be modulated, as shown in Fig. 2.2. Either single- or multi-carrier modulation formats may be used (on each polarization). A few examples of single-carrier formats are shown in Fig. 2.3. More formats can be found in [52,53]. Formats optimized for phase noise are considered in [41,54–56]. For each of these formats, data bits can be mapped onto constellation points using either non-differential or differential encoding. When non-differential

7 8 CHAPTER 2. COHERENT OPTICAL SYSTEMS

Optical Electrical Modulator ADC DSP 1 1 0 1 0 front-end front-end 1 1 0 1 0 Photo- detectors

Transmitter LO laser laser

Figure 2.1: Coherent system. In the transmitter, data bits modulate a transmitter laser. The signal is sent through an optical ﬁber. In the receiver, the signal is combined with an LO laser in the optical front-end. Balanced photodetectors convert the optical signal to an electrical signal. This is followed by an electrical front-end. The analog electrical signal is then converted to a digital signal by analog-to-digital converters (ADCs). Finally, the signal is processed and demodulated using digital signal processing.

MZM (a) 0 0 1 0 1 Transmitter laser

MZM 1 0 1 0 0 (b) Transmitter PBS PBC laser MZM 1 1 0 1 0

Figure 2.2: Electro-optical transmitter. (a) Single-polarization. Data bits modulate a transmitter laser via a Mach-Zehnder modulator (MZM). (b) Dual-polarization. The transmitter laser is split into two orthogonal polarizations using a polarization beam splitter (PBS). Each polarization is modulated with data bits in MZMs. The two polarizations are combined with a polarization beam combiner (PBC). 2.2. TRANSMITTER 9

Q Q Q

I I I

(a) QPSK (b) 8-PSK (c) 16-QAM

Figure 2.3: Example single-carrier modulation formats: (a) quadrature phase-shift keying (QPSK), (b) 8-phase-shift keying (PSK), and (c) 16-quadrature amplitude modulation (QAM). Constellations are shown on the in-phase (I)-quadrature (Q) plane.

Q Q 00 01 00 01

I I 11 10 11 10

(a) QPSK (b) DQPSK

Figure 2.4: Difference between quadrature phase-shift keying (QPSK) and differential quadrature phase-shift keying (DQPSK). In (a) QPSK, data bits are mapped to absolute phases. In (b) DQPSK, data bits are mapped to phase transitions, relative to the previous symbol (shown in red). 10 CHAPTER 2. COHERENT OPTICAL SYSTEMS encoding is used, data bits are directly mapped to constellation points. When differential encoding is used, data bits are mapped to transitions between points, and at least one reference symbol must be transmitted at the beginning of the data sequence. An example of non-differential encoding (QPSK) and differential encoding (DQPSK) is shown in Fig. 2.4. Differential encoding for other single-carrier formats can be found in [48]. In multi-carrier modulation, several electrical subcarriers are used. A popular scheme is orthogonal frequency-division multiplexing (OFDM) [57], in which orthogonal subcarriers corresponding to the frequency components of a fast Fourier transform (FFT) are modulated. The time-domain signal is obtained by inverse fast Fourier transform (IFFT). This is then transmitted through the channel. A disadvantage of OFDM is its high peak-to-average power ratio (PAPR), which is problematic in the presence of nonlinearities in e.g. the fiber and electrical front- end. In systems where lower PAPR is required, hybrid modulation formats such as single-carrier frequency-division multiple access (SC-FDMA) may be used [58]. In addition, pulse shaping is used to limit the spectral bandwidth of the signal.

2.3 Channel

The main channel effects in fiber-optic systems are additive white Gaussian noise, laser phase noise, chromatic dispersion, polarization-mode dispersion, and fiber nonlinearities. In this section, we limit our discussion to single-mode fibers.

2.3.1 Fiber Losses and Amplification As the signal propagates through the fiber, the average optical power is attenuated. After traveling a length of fiber L, the output power is [4]

P = P exp ( αL) (2.1) out in − where Pin is the input power, and α is the attenuation coeﬃcient (linear, not dB). The attenuation coeﬃcient is, however, often expressed in dB/km

10 P α = log out 4.343α (2.2) [dB] − L 10 P ≈ in Fiber loss is wavelength-dependent. The two most important contributors to fiber loss are material absorption and Rayleigh scattering [4]. Material absorption can be subdivided into intrinsic absorption which is the loss in fused silica, and extrinsic absorption which is caused by impurities in the silica. Rayleigh scattering occurs because fluctuations in density in the silica result in fluctuations in the refractive index over distances much smaller than the optical carrier wavelength. To counter fiber loss, in-line fiber amplifiers can be used [4]. A popular type of fiber amplifier is erbium-doped fiber amplifier (EDFA), in which a gain medium is formed by doping the fiber core with Erbium. Another type of fiber amplifier is 2.3. CHANNEL 11 the Raman amplifier, which uses stimulated Raman scattering. Due to amplified spontaneous emission (ASE) from fiber amplifiers, the signal is impaired with independent identically distributed (i.i.d.) additive white Gaussian noise (AWGN) (Fig. 2.5).

Figure 2.5: QPSK constellation with additive white Gaussian noise (AWGN).

2.3.2 Laser Phase Noise In semiconductor lasers, laser frequency noise is dominated by carrier noise [59]. Carrier density fluctuations change the refractive index in the active region of the laser. This changes the optical length of the cavity, thus causing the laser frequency to fluctuate. In communication systems, laser frequency noise is typically assumed to be flat up to a bandwidth B. As B , the power spectral density (PSD) becomes → ∞ Lorentzian. To model this, an ideal laser tone at frequency fo can be written in complex form as exp (j2πfokTs), where 1/Ts is the sampling rate. Sampling occurs at time instants t = kTs, where k 0, 1, 2,..., and j = √ 1. With laser phase ∈ − noise (PN), this becomes exp (j2πfokTs + jθp[k]), where θp[k] is a Wiener process (random walk) of the form [60,61]

θp[k] = θp[k 1] + ∆θp[k]. (2.3) −

The phase changes ∆θp[k] are i.i.d. Gaussian random variables with zero mean and variance 2πTs∆ν. The full width half maximum (FWHM) laser linewidth of the Lorentzian power spectrum is ∆ν. Initial phase θp[0] is uniformly distributed [ π, π). The eﬀect of laser PN on the signal is shown in Fig. 2.6. − Unlike AWGN (Sec. 2.3.1), laser PN θp[k] is not i.i.d.. The coherence time of a laser is [59] 1 t = (2.4) c π∆ν The Lorentzian spectrum does not model the relaxation oscillation peak that occurs at the resonant frequency of the laser, or the eﬀects of 1/f noise. 12 CHAPTER 2. COHERENT OPTICAL SYSTEMS

Figure 2.6: QPSK constellation with laser phase noise (PN).

2.3.3 Chromatic Dispersion Chromatic dispersion is the phenomenon where different wavelengths of light traverse a fiber at different speeds, as illustrated in Fig. 2.7. In a communication system, the main concern is the temporal spreading of an information pulse. This is known as group-velocity dispersion (GVD) [4]. The dispersion parameter has units ps/(km-nm). It has two contributing factors: material dispersion and waveguide dispersion. Material dispersion is a property of the silica that is used to make fibers, specifically that the refractive index of silica is wavelength-dependent. Waveguide dispersion depends on the structure of the waveguide, e.g. core radius, index difference, etc. Basically, different wavelengths see slightly different waveguides.

Fiber

time

Figure 2.7: Illustration of chromatic dispersion. Different wavelengths traverse the fiber at different speeds. Wavelengths that are time-aligned at the input of the fiber are no longer aligned at the output.

2.3.4 Polarization-Mode Dispersion Polarization-mode dispersion (PMD) refers to the phenomenon where the two polarizations of light propagate through the fiber at different speeds, as illustrated in Fig. 2.8. This is related to fiber birefringence, where the refractive index of the fiber is polarization-dependent due to variations in (asymmetric) core shape and stress [4]. Birefringence is not constant through the fiber but changes randomly e.g. due to temperature drift, leading to random variations in the polarization state of light as it propagates through the fiber. The PMD parameter has units ps/√km. 2.3. CHANNEL 13

Electric field polarization 1

Fiber

Electric field polarization 2

time

Figure 2.8: Illustration of polarization-mode dispersion (PMD). The two orthogonal polarizations of light traverse the fiber at different speeds. Signals on each polarization that are time-aligned at the input of the fiber are no longer aligned at the output.

2.3.5 Nonlinear Eﬀects Fiber nonlinearities can be divided into nonlinear refraction and stimulated inelastic scattering. Nonlinear refraction occurs because polarization P is a nonlinear function of the electric ﬁeld E [62]

. (1) (2) (3). P = εo χ E + χ : EE + χ .EEE + (2.5) · · · · (i) where εo is the vacuum permitivity, and χ is the i-th order susceptibility. Linear susceptibility χ(1) is dominant and included in the refractive index of the fiber (2) n(ωo), which depends on optical angular frequency ωo. As χ is negligibly small for symmetric molecules like silica, second-order effects are typically ignored in optical fibers. The third-order susceptibility χ(3) is the main term responsible for nonlinear refraction, where the refractive index depends on the intensity of light. With it included, the total refractive index becomes

2 2 n˜ ωo, E = n(ωo) + n2 E (2.6) | | | | 2 where E is optical intensity, and n2 is the nonlinear-index coefficient that is related| to| χ(3) [62]. Nonlinear refraction gives rise to self phase modulation (SPM) and cross phase modulation (XPM). Changes in intensity affect the refractive index, which affects the phase of the signal. When the changes in intensity are from the signal itself, this is known as SPM. When the changes in intensity are from other signals in the fiber with different wavelength, polarization, or direction of propagation, this is known as XPM. Nonlinear refraction also gives rise to the phenomenon of four-wave mixing (FWM), where waves at three frequencies generate a wave at a fourth frequency ωo, = ωo, + ωo, ωo, . In order for FWM to occur, the waves ωo, , ωo, , and 4 1 2 − 3 1 2 ωo,3 must be phase-matched. This only occurs when chromatic dispersion is near 14 CHAPTER 2. COHERENT OPTICAL SYSTEMS zero e.g. dispersion-shifted fibers. The phenomenon of FWM in optical fibers is analogous to third-order intermodulation in electrical circuits (Sec. 2.4.1). Stimulated inelastic scattering includes stimulated Raman scattering (SRS) and stimulated Brillouin scattering (SBS) [62]. In both cases, a pump photon is ab- sorbed, creating a photon at lower frequency and a phonon. Optical phonons are involved in SRS, whereas acoustic photons are involved in SBS. Another difference is that SRS can occur in both forward and backward directions, whereas SBS only occurs in the backward direction.

2.4 Receiver

A coherent optical receiver consists of an optical and electrical front-end, analog- to-digital converters, and digital signal processing block.

2.4.1 Optical and Electrical Front-end The optical front-end can be constructed in diﬀerent ways. Possible architectures include [63]

1. heterodyne without image rejection, 2. heterodyne with image-reject optical bandpass ﬁlter (OBPF),

3. heterodyne with optical image-reject receiver,

4. dual-quadrature homodyne, and

5. single-quadrature homodyne.

These are shown for single-polarization in Figs. 2.9–2.13, and for dual-polarization in Figs. 2.14–2.18. Heterodyne means that, after optical down-mixing at the photodetectors, the signal is at an intermediate frequency (IF). In this case, down-mixing is also needed in the electrical front-end in order to obtain the electrical baseband signal (Figs. 2.9–2.11 and Figs. 2.14–2.16). The degrees in the 180◦ and 90◦ hybrids refer to the relative phases at the output ports of the hybrids [64]. 2.4. RECEIVER 15

Figure 2.9: Single-polarization heterodyne without image rejection. The (white) polarization controller (PC), balanced photodetectors, and low-pass ﬁlter (LPF) are present in ◦ all architectures. Heterodyne without image rejection (green) additionally includes a 180 hybrid and electrical mixer.

Figure 2.10: Single-polarization heterodyne with image-reject OBPF. The (white) PC, balanced photodetectors, and LPF are present in all architectures. Heterodyne with ◦ image-reject OBPF (yellow) additionally includes an OBPF, 180 hybrid, and electrical mixer. 16 CHAPTER 2. COHERENT OPTICAL SYSTEMS

Figure 2.11: Single-polarization heterodyne with optical image-reject receiver. The (white) PC, balanced photodetectors, and LPF are present in all architectures. Heterodyne with optical image-reject receiver (pink) additionally includes a λ/4 plate on the LO path, optical image-reject mixer (Fig. 2.19), and electrical mixer.

When down-mixing to an IF, frequencies above and below the LO that are equidistant to the LO frequency will mix down to the same IF. On one side of the LO, we have the wanted signal. The unwanted noise on the other side of the LO is called the image. To prevent the image from mixing into the signal band and degrading signal-to-noise ratio (SNR), an OBPF can be used to ﬁlter out the image before the 180◦ hybrid (Fig. 2.10 and Fig. 2.15). Alternatively, an optical image-reject receiver can be used (Fig. 2.11, Fig. 2.16, and Fig. 2.19). Homodyne means that the signal after the photodetectors is at baseband (zero IF), and no down-mixing is needed in the electrical front-end (Figs. 2.12–2.13 and Figs. 2.17–2.18). When the signal has both in-phase and quadrature components, a 90◦ hybrid must be used (Fig. 2.12 and Fig. 2.17). However, when the signal has only in-phase component but no quadrature component, a 180◦ hybrid can be used (Fig. 2.13 and Fig. 2.18). An example modulation format with only in-phase component is binary phase-shift keying (BPSK). 2.4. RECEIVER 17

Figure 2.12: Single-polarization dual-quadrature homodyne. The (white) PC, balanced photodetectors, and LPF are present in all architectures. Dual-quadrature homodyne ◦ (blue) additionally includes a 90 hybrid.

Figure 2.13: Single-polarization single-quadrature homodyne. The (white) PC, balanced photodetectors, and LPF are present in all architectures. Single-quadrature homodyne ◦ (purple) additionally includes a 180 hybrid and does not use Ir,q(t). 18 CHAPTER 2. COHERENT OPTICAL SYSTEMS

Figure 2.14: Dual-polarization heterodyne without image rejection. The (white) balanced photodetectors, LPFs, and LO polarization beam splitter (PBS) are present in all archi- ◦ tectures. Heterodyne without image rejection (green) additionally includes 180 hybrids and electrical mixers. 2.4. RECEIVER 19

Figure 2.15: Dual-polarization heterodyne with image-reject OBPF. The (white) balanced photodetectors, LPFs, and LO PBS are present in all architectures. Heterodyne with ◦ image-reject OBPF (yellow) additionally includes an OBPF, 180 hybrids, and electrical mixers. 20 CHAPTER 2. COHERENT OPTICAL SYSTEMS

Figure 2.16: Dual-polarization heterodyne with optical image-reject receiver. The (white) balanced photodetectors, LPFs, and LO PBS are present in all architectures. Heterodyne with optical image-reject receiver (pink) additionally includes a PBS on the signal path, λ/4 plates on the LO path, optical image-reject mixers (Fig. 2.19), and electrical mixers. 2.4. RECEIVER 21

Figure 2.17: Dual-polarization dual-quadrature homodyne. The (white) balanced photodetectors, LPFs, and LO PBS are present in all architectures. Dual-quadrature homo- ◦ dyne (blue) additionally includes 90 hybrids. 22 CHAPTER 2. COHERENT OPTICAL SYSTEMS

Figure 2.18: Dual-polarization single-quadrature homodyne. The (white) balanced photodetectors, LPFs, and LO PBS are present in all architectures. Single-quadrature homo- ◦ dyne (purple) additionally includes 180 hybrids and does not use the quadrature outputs. 2.4. RECEIVER 23

Figure 2.19: Optical image-reject mixer using a high-birefringence ﬁber [63]. The ﬁber changes the polarizations of the signal band, LO, and image band. Solid polarizations are for single- (Fig. 2.11) and top branch of the dual-polarization receiver (Fig. 2.16). Dotted polarizations are for the bottom branch of the dual-polarization receiver (Fig. 2.16). At the output, the signal band and LO are polarization-aligned. They are also orthogonal to the image band. Polarizations for the single-polarization receiver are from [65].

Impairments in the optical front-end include AWGN from LO-ASE beat noise and photodetector shot noise. Optical path lengths may be mismatched. The optical signal and LO laser may drift in frequency or be spatially misaligned. Addition- ally, IQ imbalance occurs when e.g. 90◦ hybrids are not precisely 90◦. Impairments in the electrical front-end are described in Sec. 2.5.

2.4.2 Analog-to-Digital Converter

According to the sampling theorem, a baseband signal with single-sided bandwidth B (i.e. from B to +B Hz) can be reconstructed when sampled at the Nyquist − rate of 2B samples/s [52]. For single-carrier modulation with symbol rate 1/Ts symbols/s, the theoretical minimum sampling rate is 1/Ts Hz. In practice, the sampling rate is determined by two opposing requirements. On the one hand, a higher sampling rate is preferable for pulse-shaping, equalization, and asynchronous timing [34,52,66]. On the other hand, it is diﬃcult to design high-speed analog-to- digital converters (ADCs) for the data rates used in ﬁber-optic systems. Currently, a sampling rate of 2/Ts Hz is typically used [34]. An ADC adds quantization noise to the signal, and clips large signals. In a well- designed system, quantization noise should be much lower than front-end noise. The amount of clipping that can be tolerated by the demodulator depends on the modulation format and pulse shape used. To ensure that the signal into the ADC is kept at the right level—neither too high (clipping) nor too low (quantization noise)—automatic gain control (AGC) is used. 24 CHAPTER 2. COHERENT OPTICAL SYSTEMS

2.4.3 Digital Signal Processing Typical blocks for digital signal processing are shown in Fig. 2.20 [34]. Front- end compensation corrects some of the impairments from the optical and electrical front-ends. For example, optical path length mismatch and IQ imbalance can be compensated, but AWGN cannot be compensated. Static channel equalization compensates fiber chromatic dispersion and nonlinearities [34,35,66,67]. Dynamic channel equalization compensates time-varying fiber effects such as PMD and state of polarization [34]. Symbol timing synchronization determines the optimum sampling points, and resamples the signal to one sample/symbol. Carrier frequency and phase are estimated and compensated [20, 34]. Combinations of impairments and algorithms can give rise to additional effects. For example, the combination of PN, chromatic dispersion, and their compensation results in equalization-enhanced phase noise [68–71].

Static Dynamic Carrier Front-end Symbol timing Frequency channel channel phase Demodulation compensation synchronization estimation equalization equalization estimation

Figure 2.20: Typical DSP subsystems. Imperfections in the optical and electrical front- ends are compensated. Static and dynamic channel eﬀects are equalized. Symbol timing, carrier frequency, and carrier phase are estimated. Finally, the signal is demodulated.

Finally, the signal is demodulated. Non-differentially encoded QPSK is coherently demodulated. This means that carrier phase estimation is done, and bit decisions are made based on the modulator mapping (e.g. Fig. 2.4(a)). Coherent demodulation is not to be confused with coherent optical, which refers to systems with an LO laser at the receiver (Sec. 2.1). When DQPSK modulation is used, demodulation can be either coherent or differential. In the case of coherently-demodulated DQPSK, carrier phase estimation and demodulation is first done as with QPSK. Then, differential decoding translates the bit mapping from QPSK (Fig. 2.4(a)) to DQPSK (Fig. 2.4(b)). In the case of differentially-demodulated DQPSK, carrier phase estimation is not done. Instead, phase differences between successive symbols are calculated, and bit decisions are made directly using the mapping in Fig. 2.4(b). The theoretical bit error rate (BER) performance for these demodulators with AWGN only is shown in Fig. 2.21. The BER of coherently-demodulated DQPSK is 2 that of coherently-demodulated QPSK. This is the differential encoding penalty× for QPSK, which decreases for higher-order quadrature amplitude modulation (QAM) [48]. For the same BER, differentially-demodulated DQPSK needs higher SNR than coherently-demodulated QPSK. Intuitively, we expect this SNR penalty to be around 3 dB because differentially demodulating DQPSK involves calculating phase differences between successive symbols, both of which are equally noisy. The actual penalty for DQPSK turns out to be around 2.3 dB at high SNR [52]. Demodulators for other modulation formats and their performance can be found in [52,72]. 2.5. ELECTRICAL FRONT-END IN MICROWAVE SYSTEMS 25

100 coherently−demodulated QPSK coherently−demodulated DQPSK 10−1 differentially−demodulated DQPSK

10−2

10−3 BER

10−4

10−5

10−6 9 10 11 12 13 14 15 16 SNR (dB)

Figure 2.21: Theoretical BER performance for QPSK and DQPSK with AWGN only.

2.5 Electrical Front-end in Microwave Systems

Coherent optical receivers are similar to microwave receivers. For each of the optical front-end architectures in Sec. 2.4, we show equivalent microwave architectures in Figs. 2.22–2.26 [63, 73]. There are two key differences between coherent optical front-ends and microwave front-ends. First, at sensitivity, the dominant AWGN in microwave receivers is thermal noise from the antenna and low-noise amplifier (LNA), instead of the LO-ASE beat noise and shot noise in optical receivers. Second, unlike optical systems which can be either single- or dual-polarization, single-antenna microwave systems are always single-polarization. Otherwise, there are many parallels between coherent optical and microwave receivers. As in optical front-ends, heterodyne in microwave front-ends means that the signal is first down-mixed to IF, before down-mixing a second time to baseband (Figs. 2.22–2.24). To prevent noise in the image band from mixing into the signal band, a bandpass filter (BPF) can be used to filter out the image before the first mixer (Fig. 2.23). Alternatively, an electrical image-reject receiver can be used (Fig. 2.24 and Fig. 2.27). Homodyne means that the radio frequency (RF) signal is down-mixed only once, 26 CHAPTER 2. COHERENT OPTICAL SYSTEMS

Figure 2.22: Microwave heterodyne without image rejection. The (white) LPF is present in all architectures. Heterodyne without image rejection (green) additionally includes a ﬁrst mixer LO1 and second mixer LO2.

Figure 2.23: Microwave heterodyne with image reject BPF. The (white) LPF is present in all architectures. Heterodyne with image-reject BPF (yellow) additionally includes an image-reject BPF, LO1 mixer, and LO2 mixer.

Figure 2.24: Microwave heterodyne with image-reject receiver. The (white) LPF is present in all architectures. Heterodyne with image-reject receiver (pink) additionally includes an LO1 mixer, LPF, image cancellation, and LO2 mixer. An example structure for the LO1 mixer, LPF, and image cancellation is shown in Fig. 2.27.

Figure 2.25: Microwave dual-quadrature homodyne. The (white) LPF is present in all architectures. Dual-quadrature homodyne (blue) additionally includes an LO2 mixer. 2.5. ELECTRICAL FRONT-END IN MICROWAVE SYSTEMS 27

Figure 2.26: Microwave single-quadrature homodyne. The (white) LPF is present in all architectures. Single-quadrature homodyne (purple) additionally includes an LO1 mixer and does not use Vr,q(t).

Figure 2.27: Hartley image-reject receiver [63,73]. At the RF input, both signal and image are present. In the upper branch, they are multiplied by sin(ωLO1t), filtered with an LPF, ◦ and shifted by 90 . In the lower branch, they are multiplied by cos(ωLO1t), and filtered with an LPF. The signal components at the output of both the upper and lower branches ◦ are in phase, whereas the image components are 180 out of phase. Summing the upper and lower branches therefore yields the signal at IF without the image. directly to baseband, i.e. zero-IF (Figs. 2.25–2.26). In dual-quadrature homodyne (Fig. 2.25), two mixers are used to multiply with sin(ωct) and cos(ωct), where ωc is the angular RF carrier frequency. This recovers both the in-phase and quadrature components of the signal. In single-quadrature homodyne (Fig. 2.26), only one mixer is used to multiply with cos(ωct), assuming that that is phase-locked to the RF signal. Like its coherent optical counterpart, single-quadrature homodyne can be used when the signal has only in-phase component but no quadrature component. Another architecture that is used in microwave receivers is known as low-IF [73]. In this case, the IF is only slightly higher than the single-sided bandwidth of the baseband signal, i.e. the LO is placed near the edge of the RF signal. As this architecture is typically used when channels are narrowband, there is no equivalent in optical systems, and we will not discuss it further here. Besides AWGN from thermal noise, other impairments in the electrical front- end include nonlinear effects such as harmonic distortion, gain compression, cross modulation, and intermodulation [73]. Third-order intermodulation is analogous to 28 CHAPTER 2. COHERENT OPTICAL SYSTEMS

FWM in the fiber (Sec. 2.3.5), with the difference that the phase-matching criterion is always fulfilled in electrical circuits. Imperfect mixers and mismatches between in-phase (I) and quadrature (Q) branches result in IQ imbalance. Phenomena such as LO leakage and second-order nonlinearities result in DC offsets. Phase noise also occurs in electrical phase-locked loops (PLLs), although this is less severe than that from lasers.

2.6 Theoretical Performance Limits

In coherent optical receivers, the theoretical limit of receiver sensitivity is set by AWGN in the optical front-end. For the receiver architectures in Sec. 2.4.1, theoretical SNRs at the demodulator are shown in Table 2.1 [63]. Optical signal power and ASE PSD are at position (i) in Figs. 2.9–2.18. When LO-ASE beat noise is dominant (ﬁrst term in the denominator in Table 2.1), the SNR for heterodyne without image rejection is half that of the other architectures. This is because ASE from the image band mixes into the signal band. When shot noise is dominant (second term in the denominator in Table 2.1), the SNR for single-quadrature homodyne is double that of the other architectures. Thermal noise can be neglected because it can be made much smaller than shot noise by increasing the power of the optical LO.

SNRIQ Heterodyne no image reject Ps/ [(2ξ + q/R) B] Heterodyne with image reject Ps/ [(ξ + q/R) B] Dual-quadrature homodyne Ps/ [(ξ + q/R) B] Single-quadrature homodyne Ps/ [(ξ + q/ [2R]) B]

Table 2.1: Single-/dual-polarization optical receiver SNRs [63]. Noise from both in-phase and quadrature branches are included in SNRIQ. Optical signal power per polarization is Ps, passband signal bandwidth (single-sided deﬁnition) is B, and ASE two-sided PSD per polarization is ξ/2. Photodetector responsivity is R and electron charge is q. The ﬁrst (ξ) term in the denominator is from LO-ASE beat noise. The second (q/R) term is from shot noise.

In microwave receivers, the theoretical limit of receiver sensitivity is set by AWGN from thermal noise in the antenna and LNA (Sec. 2.5). For the receiver architectures in Sec. 2.5, theoretical SNRs at the demodulator are shown in Table 2.2 [63]. Thermal noise is at position (i) in Figs. 2.22–2.26. Thermal noise in microwave receivers (Table 2.2) is analogous to LO-ASE beat noise in coherent optical receivers (Table 2.1). Unlike coherent optical receivers, thermal noise in microwave receivers cannot be made negligible by increasing LO power. In Table 2.3, BER is given as a function of SNR for diﬀerent modulation formats [63]. For example, from Table 2.1 and Table 2.3, a dual-quadrature homodyne coherent optical receiver for QPSK modulation has BER (1 [1 ≈ − − 2.6. THEORETICAL PERFORMANCE LIMITS 29

SNRIQ Heterodyne no image reject Ps/(2ξB) Heterodyne with image reject Ps/(ξB) Dual-quadrature homodyne Ps/(ξB) Single-quadrature homodyne Ps/(ξB)

Table 2.2: Microwave receiver SNRs [63]. Noise from both in-phase and quadrature branches are included in SNRIQ. The RF signal power is Ps, passband signal bandwidth (single-sided deﬁnition) is B, and two-sided thermal noise PSD is ξ/2.

Modulation BER BPSK Q( 2SNRIQ) 2 QPSK (1 [1 Q( SNRIQ)] )/ log 4 ≈ − − p 2 M-PSK [2Q( 2SNRIQ sin[π/M])]/ log2 M ≈ p 2 M-QAM 1 1 2 1 1 Q 3 SNR log M p√M M 1 IQ 2 ≈ − − − − h q i . Table 2.3: BER as a function of SNRIQ in Tables 2.1–2.2 [63]. The function Q(x) = 0.5 erfc(x/√2), where erfc is the complementary error function.

2 Q( Ps/ [(ξ + q/R) B])] )/2. From Table 2.2 and Table 2.3, a dual-quadrature homodyne microwave receiver for QPSK has BER (1 [1 Q( P /(ξB))]2)/2. p s This is the same as the coherent optical receiver when≈ LO-ASE− − beat noise is domi- p nant. For the same BER, a receiver for 16-QAM requires higher SNR compared to QPSK. Using our unified framework [63], we compare our results in Tables 2.1–2.3 with previous publications [4, 65, 74–78]. This is summarized in Tables 2.4–2.5. In some publications, when modulation formats with only in-phase component are considered (e.g. BPSK), only noise on the in-phase branch is included in SNR. We denote this as SNRI , which is 2 the SNRIQ in Tables 2.1–2.3. × Comparing our results with [74], in which SNRI was derived for optical receivers, we find that we are in agreement. For microwave receivers, it is mentioned in [74] that heterodyning and homodyning have the same SNR, but this was not derived. According to our results, this is true if the microwave heterodyne uses image rejection, which is commonly the case. In [75], a mixture of SNRIQ and SNRI was used. The former was used in microwave heterodyne, optical heterodyne, and optical dual-quadrature homodyne. The latter was used in microwave single-quadrature homodyne, optical heterodyne, and optical single-quadrature homodyne. Our passband signal bandwidth B matches [75] when SNRIQ was used. However, our B is half that in [75] when SNRI was used. After accounting for these differences, our results agree with [75]. In [65], SNRs at IF for ASE were derived. This can be compared to our SNRIQ. Our results agree with [65] for ASE. Also in [65], the SNR for shot noise was stated 30 CHAPTER 2. COHERENT OPTICAL SYSTEMS

[74] [75] [65] [76] [77] [78] [4] Heterodyne no image reject: Shot noise A A D A A LO-ASE beat noise A Heterodyne with image reject: Shot noise D A LO-ASE beat noise A D Dual-quadrature homodyne: Shot noise A A A LO-ASE beat noise D A Single-quadrature homodyne: Shot noise A A A A A LO-ASE beat noise D

Table 2.4: Comparison between the coherent optical results in [63] and other publications. A: agree, D: disagree.

[74] [75] [65] [76] [77] [78] [4] Heterodyne no image reject Heterodyne with image reject A A Dual-quadrature homodyne Single-quadrature homodyne A A

Table 2.5: Comparison between the microwave results in [63] and other publications. A: agree, D: disagree. without derivation. Our results do not agree with [65] for shot noise. In [76], single-polarization receivers were analyzed. Our derivations agree with [76] on shot noise but disagree on LO-ASE beat noise. Also in [76], the authors state that performance of dual-polarization is the same as single-polarization. However, no derivation was shown and only ASE was mentioned. According to our results, this is true if optical signal power per polarization is the same in single- and dual- polarization. Lastly, we agree with [76] on how to calculate BER from SNR. In [77], SNRIQ and BER for BPSK were derived. The receiver was an optical phase-diversity homodyne. This is comparable to our single-polarization dual- quadrature homodyne, and our results are in agreement. Our results agree with [78] in which SNRIQ and BER for BPSK were derived, and with [4] in which SNRI and BER for BPSK were derived. Additionally, we note that the BERs in [78] and [4] also agree with each other, as the factor of 2 diﬀerence in SNR was accounted for when relating SNR to BER. Chapter 3

Forward Error Correction

In general, higher-order modulation formats require higher signal-to-noise ratios (SNRs) for the same bit error rate (BER), However, it is often impractical to increase SNR. For example, long-haul systems are typically limited by amplified spontaneous emission (ASE), making it difficult to further reduce additive white Gaussian noise (AWGN). Instead, a more feasible solution is to combine higher- order formats with forward error correction (FEC). For the same SNR, higher-order formats have higher uncoded BERs, but this is improved by FEC so the BER of the total coded system is low. In this chapter, we introduce several widely-used codes. However, we do not cover convolutional codes as they are rarely used in fiber optic systems [24].

3.1 General Principles

The basic idea in coding is to map points from a k-dimensional signal space (with 2k points in the space) to an n-dimensional code space (with 2n points), where n > k. By number of dimensions, we mean the number of bits needed to represent a point in the space. For example, the signal space in Fig. 3.1(a) has eight points, i.e. k = 3. The code space in Fig. 3.1(b) has 64 points, i.e. n = 6. However, the transmitter only uses eight codewords, depicted as black points in Fig. 3.1(b). The unused white points form a “buffer zone”. In a well-designed code, errors most likely result in a white point that is closer to the transmitted codeword than any other codeword. In this way, errors can be detected and corrected. In the transmitter, the process of mapping from signal space to code space is called encoding. In the receiver, the reverse process of mapping from code space back to signal space is called decoding. There are two figures of merit for evaluating codes. The first is net coding gain (NCG), which is the gross coding gain in dB minus the bit rate increase due to code overhead in dB [24]. Gross coding gain is the difference in SNR between the coded and uncoded system at a particular BER, as shown in Fig. 3.2. Code

31 32 CHAPTER 3. FORWARD ERROR CORRECTION

S6 C6

S5 C5 S2 C2

S1 C1

S7 C7 S3 C3 S4 C4

S8 C8 (a) (b)

Figure 3.1: Points S1–S8 in signal space (a) are mapped to codewords C1–C8 in code space (b). Code space is a higher-dimensional space than signal space. The extra degrees of freedom are used to choose a mapping such that a codeword with errors is usually a white point that is closer to the error-free codeword than any other codeword. This allows errors to be detected and corrected.

0 10 Uncoded Coded −5 10

−10 10 BER gain −15 10

−20 10 5 10 15 20 SNR (dB)

Figure 3.2: Gross coding gain at a particular BER is the difference in SNRs between the coded and uncoded systems. 3.1. GENERAL PRINCIPLES 33 overhead1 is (n k)/k. The second figure of merit is latency, which is the additional processing delay− introduced by encoding and decoding. A good code has high NCG and low latency. Decoding can be done on either hard or soft decisions. The distinction is in the number of bits per symbol at the output of the demodulator as shown in Fig. 3.3. When soft decisions are used, more information is passed from the demodulator to the decoder. For example, the most-significant bits (MSBs) in Fig. 3.3(b) are the same as the hard decisions in Fig. 3.3(a), while the extra bits indicate degree of confidence in those decisions. This extra information is used by the decoder to improve coding gain, at the cost of higher complexity. 1 0 7 6 5 4 3 2 1 0

1 0 7 6 5 4 3 2 1 0 (a) (b)

Figure 3.3: (a) Hard-decision decoding. The output of the demodulator (Fig. 2.20) has the minimum number of bits required to represent the signal. Shown here is quadrature phase-shift keying (QPSK) with AWGN. Decision boundaries are marked with dashed red lines. The demodulator output is two bits per symbol, i.e. one bit for in-phase and one bit for quadrature. (b) Soft-decision decoding. The output of the demodulator has more than the minimum number of bits required to represent the signal. Shown here is six bits per symbol, i.e. three bits for in-phase and three bits for quadrature.

In order to design complex systems efficiently, engineers partition them into smaller subsystems that are designed independently. To enable such partitioning between the uncoded system in Chap. 2 and FEC in this chapter, FEC limits are used. The FEC limit of a code is the pre-FEC BER level at which that code achieves a given target post-FEC BER. As long as the uncoded system can achieve a pre- FEC BER lower than the FEC limit, the coded system can achieve a post-FEC BER less than or equal to the target. The FEC limit is determined by the choice of code and target post-FEC BER. For example, product codes aiming for a post-FEC 15 3 2 target of 10− have FEC limits around 4.3 10− for 7% overhead and 1.3 10− × × 1In coherent optical systems, the metric for redundancy is code overhead. In other fields, such as coding theory and wireless communications, the metric for redundancy is code rate k/n. 34 CHAPTER 3. FORWARD ERROR CORRECTION for 20% overhead [79, 80]. In general, codes either aim to correct pre-FEC BERs 3 2 around 10− or around 10− , with the latter requiring more complicated codes. It is important to note that the use of FEC limits assumes that bit errors are independent identically distributed (i.i.d.)—which models non-differential modulation with AWGN (Chap. 2)—and may not be applicable when this is not the case.

3.2 Linear Block Codes

A code is linear when the set of used codewords forms a subspace that is closed under modulo addition. We consider binary linear block codes, where data and codewords are sequences of bits 0 and 1. An encoder segments the input bit sequence into blocks of length k bits. Each block is a message vector u. The corresponding codeword v of length n bits is obtained by multiplying u with generator matrix G [81] v = uG (3.1) The code is systematic if G has the form

G = I P (3.2) where I is the identity matrix, and P is a matrix that determines the redundant bits. An illustration of systematic encoding is shown in Fig. 3.4.

Input: ......

Output: ......

Figure 3.4: An encoder for a systematic linear block code divides the input bit sequence into blocks. For each block, the encoder adds (blue) redundant bits that are a function of the (white) data bits.

The received vector r is the sum of codeword v and error vector e

r = v + e (3.3)

Determining the error vector e involves two steps. The ﬁrst step is syndrome calculation, in which the syndrome s is obtained by multiplying the received vector with parity check matrix H = PT I

s = rH T = (r + e)HT = eHT (3.4) where T denotes transpose. The second step is to use the syndrome to determine the error vector e. There are 2k error patterns that give the same syndrome s [81]. Among these, the decoder guesses that the error pattern with the fewest bit errors is the correct one, as this is the error pattern that is most likely to occur. When this is indeed the case, then the received vector is correctly decoded. 3.3. CYCLIC CODES 35

3.3 Cyclic Codes

Cyclic codes are a subclass of linear block codes, with the property that a cyclic shift of a codeword v results in another codeword. When analyzing cyclic codes, instead of working with blocks as vectors u = u0 u1 ... uk−1 , it is convenient to work with polynomials [81,82] 2 k 1 u(X) = u + u X + u X + + uk− X − . (3.5) 0 1 2 · · · 1 The generator polynomial has the form 2 n k 1 n k g(X) = 1 + g X + g X + + gn−k− X − − + X − (3.6) 1 2 · · · 1 and the code polynomial is 2 n 1 v(X) = u(X)g(X) = v + v X + v X + + vn− X − . (3.7) 0 1 2 · · · 1 The received polynomial is r(X) = v(X) + e(X) (3.8) where e(X) is the error polynomial. The syndrome polynomial s(X) is the re- mainder when r(X) is divided by g(X). Encoding and syndrome calculation of cyclic codes can be done with linear feedback shift registers (LFSRs), resulting in low-complexity implementations.

3.4 BCH Codes

Bose-Chaudhuri-Hocquenghem (BCH) codes are an important subclass of cyclic linear block codes. There are binary and non-binary BCH codes. The most important class of non-binary BCH codes are the Reed-Solomon (RS) codes. µ A full-length binary BCH code with block length nB = 2 1 and µτ redundant bits corrects up to at least τ bit errors (Fig. 3.5). Its generator− polynomial g(X) is the lowest-degree polynomial that has α, α2, α3,...,α2τ as its roots, i.e. g(αi) = 0 for 1 i 2τ [81,83,84]. For≤ the≤ received polynomial (3.8), the i-th component of the syndrome is i i Si = r(α ) = e(α ) (3.9) for 1 i 2τ. There are 2kB error patterns that result in the same syndrome. How- ever,≤ when≤ the true error pattern has τ or fewer bit errors, the error pattern with the smallest number of errors is the correct one. A full-length code BCH(nB , kB ) may be shortened to BCH(n , k ), where n = n l , k = k l , and B,S B,S B,S B − B B,S B − B nB,S kB,S µτ. While− binary≤ BCH codes use Galois ﬁeld GF(2), RS codes use GF(2m) [81,85]. m A full-length RS code with m-bit symbols, block length nR = 2 1 symbols, and 2t redundant symbols corrects up to t symbol errors (Fig. 3.6). A− full-length code

RS(nR, kR) may be shortened to RS(nR,S , kR,S ), where nR,S = nR lR, kR,S = kR lR, and n k = 2t. − − R,S − R,S 36 CHAPTER 3. FORWARD ERROR CORRECTION

coded bits

k data bits B redundant bits

Figure 3.5: A full-length binary BCH code BCH(nB , kB ) with µτ redundant bits corrects up to at least τ bit errors.

n m R = 2 - 1 coded symbols

m bits/symbol

k R data symbols 2t redundant symbols

Figure 3.6: A full-length RS code RS(nR, kR) with m bits/symbol and 2t redundant symbols corrects up to t symbol errors.

3.5 Serially-concatenated Codes

Serially-concatenated codes consist of an inner code and an outer code, as shown in Fig. 3.7 [82]. The inner code corrects most of the errors, and the outer code corrects any remaining errors. Interleaving may be used between inner and outer codes (Sec. 3.9).

Outer Inner Interleaver encoder encoder 1 1 0 1 0

Figure 3.7: Encoder for a serially-concatenated code. Data bits are ﬁrst encoded by an outer code, followed by an inner code. Interleaving may also be done between outer and inner encoders. 3.6. TURBO CODES 37

3.6 Turbo Codes

Turbo codes are parallel-concatenated codes as shown in Fig. 3.8 [82,86]. The two encoders are separated by an interleaver (Sec. 3.9).

1 1 0 1 0 Encoder 1

Interleaver

Encoder 2

Figure 3.8: Parallel-concatenated encoder for a Turbo code, consisting of two encoders and an interleaver. The output of the Turbo encoder is the data bits, followed by the output from Encoder 1 and output from Encoder 2.

In the receiver, a turbo decoder consists of two decoders, corresponding to each of the parallel encoders. Decoder 1 uses the received bits that correspond to the uncoded message and output from Encoder 1 to determine the transmitted message and generate reliability information. It shares the reliability information with Decoder 2. Decoder 2 uses the uncoded message and output from Encoder 2 to determine the transmitted message and its own reliability information, sharing the latter with Decoder 1. This process continues iteratively. Ideally, both decoders eventually agree on the transmitted message.

3.7 LDPC Codes

Low-density parity-check (LDPC) codes were invented by Gallager in the 1960s [87]. However, they were largely ignored until rediscovered in the 1990s [88, 89]. They are linear block codes where the parity check matrix H has the property that each column of the matrix has a small number dc 3 of 1’s, and each row has a small ≥ number dr > dc of 1’s. The matrix is sparse, hence the term “low-density”. Like turbo codes, LDPC codes use iterative decoding. However, LDPC codes tend to have lower decoding complexity than turbo codes, but higher encoding complexity [82].

3.8 Coded Modulation

Instead of treating coding and modulation separately, they can be done jointly. In coded modulation [52, 82, 90–93], a modulation format is partitioned into cosets in such a way that the minimum Euclidean distance between points in the cosets increases with each level of partitioning. An example of 16-quadrature amplitude 38 CHAPTER 3. FORWARD ERROR CORRECTION modulation (QAM) partitioning is shown in Fig. 3.9. In this case, the minimum Euclidean distance increases by √2 at each level. An encoder is shown in Fig. 3.10. It encodes k bits into k+r bits. The latter are used to select a coset. The remaining n k uncoded bits are used to select a point in that coset. This is the signal point to− be transmitted. Although Fig. 3.9 shows partitioning to the limit, where each coset consists of a single point, it is not necessary to go all the way. For example, we could stop at level “B”, using two cosets of eight points each.

B0 B1

C0 C2 C1 C3

D0 D4 D2 D6 D1 D5 D3 D7

Figure 3.9: Partitioning 16-QAM into cosets with increasing Euclidean distance between points. Red points are in the coset; white points are not in the coset.

k + r coded bits Select k uncoded bits Encoder coset

Select signal n - k uncoded bits signal point point

Figure 3.10: Combined encoder and modulator. Some data bits are encoded and used to select a coset. The remaining uncoded bits select a signal point in that coset.

3.9 Interleaving

The purpose of interleaving is to break up burst errors. In the transmitter, an interleaver re-orders bits using a known permutation. In the receiver, a deinterleaver applies the opposite permutation, thereby returning the bits to their original order. In doing so, the deinterleaver re-distributes the errors from the channel. 3.9. INTERLEAVING 39

There are several ways to construct interleavers, including uniform interleaving, pseudorandom interleaving, S-random interleaving, quadratic permutation polynomial (QPP) interleaving, and block interleaving. Uniform interleaving is a theoretical construct. It can be approximated in simulations by generating the interleaver permutation randomly for each interleaver frame. This approximates the performance over the ensemble of interleavers, but is too complex for practical implementation. An interleaver that is suitable for practical implementation can be made using a single known random permutation, where the input-output mapping is generated by a pseudorandom sequence (Fig. 3.11).

Interleaver:

Deinterleaver:

Figure 3.11: Interleaving by random permutation. In the transmitter, the interleaver re- orders data bits using a known random permutation. In the receiver, the deinterleaver returns the bits to their original order.

An S-random interleaver uses a single random permutation, with the constraint that input bits that are within a distance of S from each other at the input, will be at a distance greater than S from each other at the output [94]. In order to generate the interleaver mapping, S must be less than Ni/2, where Ni is the number of bits in an interleaver frame. p A QPP interleaver has the form [95,96]

2 P (x) = (a0 + a1x + a2x ) mod Ni (3.10)

where a0, a1, and a2 are non-negative integers. Naturally, for this to be an interleaver, the mapping from x 0, 1,...,Ni 1 to P (0),P (1),...,P (Ni 1) must be a permutation. ∈ { − } { − } In a block interleaver, data bits are read in by columns and read out by rows, as shown in Fig. 3.12.

read in by columns

read out by rows

Figure 3.12: Block interleaver. Data bits are read in by columns, and read out by rows. 40 CHAPTER 3. FORWARD ERROR CORRECTION

For coded modulation, interleaving may either be done on the bits before encoding, or on the symbols after encoding [82]. The former is known as bit-interleaved coded modulation, and the latter as symbol-interleaved coded modulation. Chapter 4

BCH Codes for Coherent Optical Systems

Traditionally, much of coding theory focuses on additive white Gaussian noise (AWGN) channels. However, coherent optical systems have relatively high phase noise (PN) from the transmitter and local oscillator (LO) lasers, in addition to AWGN. This changes the error statistics which aﬀect the performance of forward error correction (FEC) codes. In this chapter, we consider codes speciﬁcally for such systems. More precisely, we describe a family of straightforward methods for selecting Bose-Chaudhuri-Hocquenghem (BCH) codes with simple implementations.

4.1 System Model and General Approach

The system model is shown in Fig. 4.1 [97–99]. The modulation format is Gray- coded diﬀerential quadrature phase-shift keying (DQPSK) (Fig. 2.4(b)). Channel impairments are laser PN (Sec. 2.3.2) and AWGN (Fig. 2.5). Several carrier phase estimation techniques have been proposed [20,34,36]. We adopt the commonly-used Viterbi-Viterbi algorithm [38], where the phase estimate is

k+N θˆ[k] = unwrap arg r4[i] 4. (4.1) " i=k N !#! X− . Unwrapping keeps phase jumps within [ π, π) by adding integer multiples of 2π. The received DQPSK signal is coherently-demodulated− (Sec. 2.4.3). To select BCH codes for this system, the general approach that we use consists of three steps:

1. identify the worst-case operating point from pre-FEC simulations,

2. extract error statistics from pre-FEC simulations, and

41 42 CHAPTER 4. BCH CODES FOR COHERENT OPTICAL SYSTEMS

BCH Differential QPSK Interleaver encoder encoder modulator

BCH Differential QPSK Deinterleaver Viterbi-Viterbi decoder decoder demodulator

Figure 4.1: System model. Bits are BCH encoded, diﬀerentially encoded, and quadrature phase-shift keying (QPSK) modulated. This yields signal s[k]. Channel impairments are transmitter laser PN θt[k], AWGN n[k], and LO laser PN θr[k]. Phase estimation on the received signal r[k] is by the Viterbi-Viterbi algorithm. The signal is QPSK demodulated, diﬀerentially decoded, and BCH decoded. Additionally, interleaving and deinterleaving may be done after the BCH encoder and before the BCH decoder respectively.

3. use the error statistics to parameterize a statistical model, based on which codes are selected analytically.

By parameterizing a statistical model based on pre-FEC simulations, the methods in this chapter require less simulation eﬀort than methods that rely on post-FEC simulations to select codes. For Step 1, for a given linewidth-symbol time product, ∆νTs, there is an optimum N in (4.1) that results in performance closest to “ideal phase estimate (PE) pre-FEC” in Fig. 4.2. However, after optimization, any increase in ∆ν degrades performance. Based on the linewidth variations that we want to accommodate, we simulate a worst-case “poor PE pre-FEC” curve. We select a point on this curve as the worst-case operating point, and select codes for that point. Steps 2 and 3 are described for binary BCH codes in Sec. 4.2, and for Reed- Solomon (RS) codes in Sec. 4.3.

4.2 Binary BCH codes

In this section, we describe Steps 2 and 3 (Sec. 4.1) for binary BCH codes [97,98]. An AWGN error of 90◦ gives bit error patterns 0101, 1010, 0110, 1001 with ± { } probability 1/4 each, and a cycle slip of 90◦ gives one bit error. ± For Step 2, we extract the following error statistics:

1. probability of AWGN error pg,

2. probability of cycle slips pc, and

3. correlation coeﬃcient ρ. 4.2. BINARY BCH CODES 43

−2 10

−3 Ideal PE, pre−FEC 10 Poor PE, pre−FEC Pre−FEC BER

−4 10 9 10 11 12 13 SNR (dB)

Figure 4.2: “Ideal phase estimate (PE) pre-FEC” means PN is ideally removed. This is the theoretical bit error rate (BER) for coherently-demodulated DQPSK with AWGN only (Fig. 2.21). There are AWGN errors but no cycle slips. “Poor PE pre-FEC” is an example when PN is not ideally removed. Errors are a combination of AWGN errors and cycle slips.

By segmenting bit errors into blocks of length nB,S bits (Sec. 3.4), we record the probability of AWGN errors in the κ-th block pg[κ], and the probability of cycle slip errors in the κ-th block pc[κ]. The mean probability of AWGN error is κ 1 pg = κ pg[κ], (4.2) κ=1 X where κ is the number of blocks used in pre-FEC bit error rate (BER) simulations. The mean probability of cycle slips is κ 1 pc = κ pc[κ]. (4.3) κ=1 X This gives pre-FEC BER ppre = 2pg + pc. The sample correlation coeﬃcient is κ (pg[κ] pg) (pc[κ] pc) ρ = κ=1 − − (4.4) κ 2 κ 2 P (pg[κ] pg) (pc[κ] pc) κ=1 − κ=1 − q For Step 3, we use pg, Ppc, and ρ to parameterizeP a correlated bivariate binomial probability density function (PDF). Let Yg be a random variable representing the number of AWGN errors in a code block, and Yc be a random variable representing the number of cycle slips in a block. Their joint PDF is

nB,S yg nB,S yg Pr(Yg = yg,Yc = yc) = p (1 pg) − f(yc yg) (4.5) y g − | g 44 CHAPTER 4. BCH CODES FOR COHERENT OPTICAL SYSTEMS where

yg nB,S yg f(yc yg) = − | γg γc (γg ,γc) Γ X∈ γg pc + β(pc pg) + β × { − } yg γg 1 pc β(pc pg) − × { − − − } (4.6) γc pc + β(pc pg) × { − } nB,S 1 pc β(pc pg) + β − − − × 1 + β yg γc 1 pc β(pc pg) + β − − × { − − − }

Γ is the set of (γg, γc) such that

Γ = (γg, γc) : { γg + γc = yc; (4.7) γg = 0, 1,...,yg;

γc = 0, 1,...,nB,S yg − } and 1 − 1 pg(1 pg) β = − 1 . (4.8) ρ pc(1 pc) − " s − # An example Pr(Yg = yg,Yc = yc) is shown in Fig. 4.3. We collapse the two-dimensional joint PDF Pr(Yg = yg,Yc = yc) into a one- dimensional PDF Pr(Ya = ya), where Ya is a random variable representing the total number of bit errors in a code block.

Pr(Ya = ya) = Pr(Yg = yg,Yc = yc). (4.9) (yg ,yc):2Xyg +yc=ya

A factor of 2 multiplies yg because AWGN error patterns have two bit errors.

A uniform interleaver (Sec. 3.9) of length D code blocks re-orders the nB,S D bits within those code blocks. When D = 1, this is the case of no interleaving.·

Let YD be a random variable representing the total number of bit errors in D code blocks. The PDF of YD is the D-fold convolution of Pr(Ya = ya) with itself

⋆D Pr(YD = yD ) = Pr(Ya = ya) (4.10) = Pr(Ya = ya,1) Pr(Ya = ya,2) Pr(Ya = ya,D ) × × · · · × (ya, ,ya, ,...,ya,D ) Ψ 1 2X ∈ where Ψ is the set of (y ,y ,...,y ) such that y + y + + y = y , a,1 a,2 a,D a,1 a,2 · · · a,D D and D 1, 2, 3,... . For D = 1, Pr(YD = ζ) = Pr(Ya = ζ), i.e. in this case ∈ { } 4.2. BINARY BCH CODES 45

y ≥ 0, y ≥ 0 g c 0.1

) 2y +y ≥ τ+1 c g c =y c ,Y

g 0.05 =y g

Pr(Y 0 0

10 0 10 20 y 20 y g c

Figure 4.3: Correlated bivariate binomial PDF Pr(Yg = yg, Yc = yc) for pre-FEC point “B” in Fig. 4.4.

yD = ya = ζ. For D = 2, Pr(YD = yD ) = Pr(Ya = ya) ⋆ Pr(Ya = ya). Convolution is denoted by ⋆.

Deinterleaving distributes the yD bit errors over D code blocks. Let Zℓ be a random variable representing the number of bit errors in the ℓ-th code block after deinterleaving.

∞ Pr(Z = z ) = Pr(Z = z Y = y ) Pr(Y = y ) (4.11) ℓ ℓ ℓ ℓ| D D D D y =0 XD where Pr(Z = z Y = y ) ℓ ℓ| D D yD 1 zℓ 1 yD zℓ 1 − , if zℓ yD , (4.12) = zℓ D − D ≤ (0, otherwise. If the average number of bit errors in a code block is much less than the code error correcting capability τ (Sec. 3.4), post-FEC BER can be approximated as [98] τ + 1 Pb,post Pr(τ + 1 Zℓ τ + 3). (4.13) ≈ nB,S ≤ ≤

For a (possibly shortened) code of block length nB,S bits, we calculate τ (4.13) and

D ((4.10), (4.12)) to meet a post-FEC BER target (4.13). The combination of nB,S and τ speciﬁes the BCH code (Sec. 3.4), and D speciﬁes the uniform interleaver (Sec. 3.9). 46 CHAPTER 4. BCH CODES FOR COHERENT OPTICAL SYSTEMS

As an example, we assume the system was optimized for symbol rate 1/Ts = 28 Gbaud and combined transmitter-and-LO laser linewidths ∆ν < 100 kHz. For those system speciﬁcations, Viterbi-Viterbi works well with N = 20 (4.1). We further assume that the worst-case pre-FEC performance occurs with a linewidth of ∆ν = 19.6 MHz (Fig. 4.4). In this example, we consider only one worst- case pre-FEC curve, as this is suﬃcient to illustrate the principle. In practice, both optimization and worst-case conditions must be carefully considered. Using the method in this section to select codes for pre-FEC points “B” and “C” in 5 Fig. 4.4, we aim for a post-FEC target of 10− , and obtain the interleaver-code combinations in Table 4.1. Error statistics and pre-FEC simulations are based on 106 bits. Post-FEC simulations are based on 107 post-FEC bits, except at the target signal-to-noise ratio (SNR) (10 or 12 dB) where 108 bits are used for better accuracy. Simulations are modeled in VPI [100].

−2 10 C Poor PE, pre−FEC −3 B B1, post−FEC 10 B2, post−FEC C1, post−FEC BER −4 C2, post−FEC 10 Bivariate B, post−FEC Bivariate C, post−FEC −5 10 9 10 11 12 13 SNR (dB)

Figure 4.4: BER performance. The interleaver-code combinations used for post-FEC simulations are in Table 4.1.

There is a trade-oﬀ between code overhead and interleaver length D, as shown in Fig. 4.5. With no interleaver (D = 1), bit errors are correlated because ρ is positive and AWGN error patterns have 2 bit errors (100% correlation). So code blocks with errors have many errors. To correct these errors, codes with higher overhead are needed. Interleaving re-distributes the errors more evenly between blocks. As interleaver length increases, bit errors become more uniformly-distributed, resulting in lower overheads. However, as bit errors become more uniformly-distributed, the incremental improvement from further increasing interleaver length diminishes. Therefore, the curves in Fig. 4.5 ﬂatten out at high D. The discrepancy in code overhead between Table 4.1 and Fig. 4.5 for some codes is because of the inequality nB,S kB,S µτ in Sec. 3.4 (which is < for some codes). The most accurate way of− calculating≤ overhead is that used in Table 4.2. BINARY BCH CODES 47

Post-FEC curve Code τ Overhead Interleaver in Fig. 4.4 (%) length D (code blocks) B1 BCH(8190,7969) 17 2.8 2 B2 BCH(8190,7995) 15 2.4 8 C1 BCH(8190,7384) 62 10.9 2 C2 BCH(8190,7423) 59 10.3 6 Bivariate B BCH(8190,7943) 19 3.1 1 Bivariate C BCH(8190,7345) 66 11.5 1

Table 4.1: Interleaver-code combinations used for post-FEC simulations in Fig. 4.4. Code overhead is calculated as (nB,S kB,S )/kB,S . −

B C 3.2 12 Bivariate B Bivariate C 3 11.5

B1 2.8 11 C1

Overhead (%) 2.6 Overhead (%) 10.5 C2 B2

2.4 10 0 2 4 6 8 0 2 4 6 8 10 12 Interleaver length, D (code blocks) Interleaver length, D (code blocks) (a) (b)

Figure 4.5: Trade-oﬀ between code overhead and interleaver length D for points (a)

B and (b) C in Fig. 4.4. Code overhead is calculated as µτ/(nB,S µτ), where − µ = log2 (nB,S + 1) , and x is the smallest integer greater than or equal to x. ⌈ ⌉ ⌈ ⌉ −5 Interleaver-code combinations that give post-FEC BER 10 are marked with a dot ( ). • The combinations B1, B2, Bivariate B, C1, C2, and Bivariate C are listed in Table 4.1. 48 CHAPTER 4. BCH CODES FOR COHERENT OPTICAL SYSTEMS

4.1. However, this can only be done after speciﬁc codes BCH(nB,S , kB,S ) have been identiﬁed. In the early stages of code selection, it is typical to work with nB,S and

τ instead. Then, overhead is estimated assuming equality nB,S kB,S = µτ as in Fig. 4.5, although this can be slightly inaccurate for some codes.−

4.3 RS codes

In this section, we describe Steps 2 and 3 (Sec. 4.1) for RS codes [99]. An AWGN error of 90◦ gives bit error patterns 11, 101, 1001 with probability 1/4 1/2 1/4 , and a± cycle slip of 90 gives one{ bit error. } ◦ For Step 2, we extract± the error statistics

1. probability of AWGN error pg, and

2. probability of cycle slips pc by classifying errors based on their error patterns, as in Sec. 4.2.

A (possibly shortened) RS code RS(nR,S , kR,S ) with m bits/RS-symbol corrects up to t symbol errors (Sec. 3.4). For Step 3, we use the following model. A cycle slip occurs in a symbol with probability

pe,c mpc (4.14) ≈ and AWGN patterns {11, 101, 1001} occur in a symbol with probability

pe,g mpg/4 mpg/2 mpg/4 . (4.15) ≈ These patterns span one RS symbol with probability

p1 (m 1)/m (m 2)/m (m 3)/m (4.16) ≈ − − − and span two symbols with probability

(1 p1) 1/m 2/m 3/m (4.17) − ≈ where 1 is the vector 1 1 1 . The probability of an RS symbol error is

T T pr,s p1pe,g + 2(1 p1)pe,g + pe,c = (m + 2)pg + mpc. (4.18) ≈ − If the average number of symbol errors in a code block is much less than the code error correcting capability t, post-FEC BER can be approximated as [99]

2[t + 1] nR,S t+1 nR,S t 1 Pr,post pr,s (1 pr,s) − − . (4.19) ≈ m nR,S t + 1 − ·

For an RS code with m bits/symbol and (possibly shortened) block length nR,S , we calculate t to meet a post-FEC BER target (4.19). The combination of m, nR,S , and t speciﬁes the RS code (Sec. 3.4). 4.3. RS CODES 49

We use the same example here as in Sec. 4.2. Error statistics and pre-FEC BER in Fig. 4.6 are calculated using 106 bits. Using the method in this section and 5 aiming for a post-FEC target of 10− , we obtain the codes in Table 4.2. Post-FEC BERs are calculated using 107 post-FEC bits (Fig. 4.6). Simulations are modeled in VPI [100].

−2 10 C

−3 B 10 Poor PE, pre−FEC BR, post−FEC BER −4 CR, post−FEC 10

−5 10 9 10 11 12 13 SNR (dB)

Figure 4.6: BER performance. The codes used for post-FEC simulations are listed in Table 4.2.

Post-FEC curve Code m t Overhead (%) in Fig. 4.6 BR RS(819,797) 10 11 2.8 CR RS(819,741) 10 39 10.5

Table 4.2: Codes used for post-FEC simulations in Fig. 4.6. Overhead is (nR,S − kR,S )/kR,S .

Comparing Fig. 4.4 and Fig. 4.6, the selected binary BCH and RS codes have similar post-FEC performance. In other words, the methods for selecting binary BCH codes (Sec. 4.2) and RS codes (Sec. 4.3) work equally well for the case considered. The overheads for the RS codes in Table 4.2 are lower than those for the binary BCH codes with no interleaving (D = 1) in Table 4.1. This is due to the correlation coeﬃcient ρ being positive (Sec. 4.2), and AWGN error patterns having two bit errors (100% correlation). Correlation means that bit errors tend to bunch together, thus favoring RS codes which encode symbols. The situation changes when uniform interleaving is used. Uniform interleaving re-distributes the bit errors, spreading them out more evenly. The longer the interleaver length D, the more evenly distributed the bit errors. This eventually favors binary BCH codes, 50 CHAPTER 4. BCH CODES FOR COHERENT OPTICAL SYSTEMS which encode bits. The overheads for the binary BCH codes B2/C2 in Table 4.1 are lower than those for the RS codes in Table 4.2. In general, RS codes are better when bit errors occur in short bursts, whereas binary BCH codes are better when bit errors occur individually. Unlike binary BCH codes (Sec. 3.4, Sec. 4.2), equality n k = 2t always R,S − R,S holds for RS codes (Sec. 3.4). So overhead can be calculated using either nR,S and kR,S , or nR,S and t. Lastly, as RS codes work well when bit errors occur in short bursts, uniform interleaving (Sec. 3.9, Sec. 4.2) should not be used with RS codes. Chapter 5

Conclusion and Future Research

Coherent optical transmission using digital signal processing (DSP) and multi-level modulation is the most promising candidate for future high-data-rate fiber optic systems. However, multi-level modulation formats are very susceptible to noise and distortions. In this thesis, we address a number of questions on system performance and impairment mitigation in such systems. On the topic of system performance, we derive theoretical performance limits of coherent optical receivers in the presence of additive white Gaussian noise (AWGN). These limits are set by local oscillator (LO)-amplified spontaneous emission (ASE) beat noise and shot noise in the optical front-end. We analyze five receiver architectures: 1. heterodyne without image rejection, 2. heterodyne with image-reject optical bandpass filter (OBPF), 3. heterodyne with optical image-reject receiver, 4. dual-quadrature homodyne, and 5. single-quadrature homodyne. For each architecture, we consider both single- and dual-polarization structures. Using a unified approach, we derive theoretical signal-to-noise ratios (SNRs) and bit error rates (BERs) for all configurations. When LO-ASE beat noise is dominant, the SNR for heterodyne without image rejection is half that of the other architectures. When shot noise is dominant, the SNR for single-quadrature homodyne is double that of the other architectures. As coherent optical receivers are similar to microwave receivers, we also analyze equivalent microwave architectures. Unlike optical systems which can be either single- or dual-polarization, single-antenna microwave systems are always single- polarization. In microwave receivers, the theoretical limit of receiver sensitivity is set by AWGN from thermal noise in the antenna and low-noise amplifier (LNA). This is analogous to LO-ASE beat noise in coherent optical receivers, so those results can be directly applied to microwave receivers. Furthermore, we compare our results with those of previous publications. By putting previous publications into the context of our unified framework, we identify areas of agreement and disagreement between them. On the topic of impairment mitigation, we propose methods for selecting Bose-

51 52 CHAPTER 5. CONCLUSION AND FUTURE RESEARCH

Chaudhuri-Hocquenghem (BCH) codes for mitigating cycle slips induced by laser phase noise (PN). Specifically, we propose a family of semi-analytical methods that only require short pre-forward error correction (FEC) simulations for code selection. The pre-FEC simulations are used to estimate error statistics, which are in turn used to parameterize statistical models, based on which codes can be selected analytically. Our methods are straightforward, and require significantly less simulation effort than methods which rely on post-FEC simulations to select codes. They are applicable to coherently-demodulated differential quadrature phase-shift 3 keying (DQPSK) systems, and pre-FEC BERs around 10− . Our methods cover both binary BCH codes and Reed-Solomon (RS) codes. For binary BCH codes, we consider both systems without interleaving and systems with uniform interleaving. However, as RS codes encode symbols, they work best when errors occur in short bursts, so we consider RS codes for systems without interleaving. In order to select codes with little simulation effort, our methods rely on statistical models of the errors in the system. For binary BCH codes, we propose a novel correlated bivariate binomial probability density function (PDF) for this purpose. For RS codes, we model the occurrence of RS symbol errors as a sequence of Bernoulli trials (biased coin tosses). As an example, we evaluate our methods for a 28 Gbaud DQPSK system with linewidths ranging from < 100 kHz to 19.6 MHz. For a target post-FEC BER of 5 10− , codes selected with our methods give BERs around 3 target, and achieve the target with around 0.2 dB extra SNR. In other words, our× methods accurately identify codes with minimum overhead that give performance close to target. Due to the presence of correlation in the statistical models, RS codes have lower overheads than binary BCH codes in systems without interleaving. However, with uniform interleaving, the deinterleaver spreads bit errors out more evenly. The longer the interleaver, the more evenly distributed the bit errors. With sufficiently long interleaving, bit errors tend to occur individually instead of in short bursts. This favors binary BCH codes, as they encode bits. While the contributions of this thesis add to our knowledge on the topics above, there are still many open questions in this research area. For example, the methods for selecting BCH codes in this thesis have been developed theoretically, and verified using Monte-Carlo simulations. A logical next step would be to verify the methods experimentally. For binary BCH codes with interleaving, uniform interleaving is considered. This is a theoretical concept which is useful for gaining general insight into the problem, but not suitable for practical implementation. For real systems, other interleaver structures must be considered. Our methods for code selection are for coherently-demodulated DQPSK systems. However, to further increase data rates, practical systems are turning to higher-order modulation formats, such as 16-quadrature amplitude modulation (QAM) and beyond. In light of this, our work can be viewed as a base from which to develop methods for higher-order formats. 53

So far, we have considered laser PN and AWGN. The impact of other channel impairments have yet to be investigated.Likewise, our methods assume that the popular Viterbi-Viterbi phase estimation algorithm is used. The impact of using other carrier phase estimation techniques can be considered.Additionally, the inter- play between channel impairments and DSP algorithms can give rise to additional eﬀects, for example equalization-enhanced phase noise. Due to simulation limitations, we simulate at most 108 post-FEC bits. Although the accuracy of our results indicate that our proposed statistical models capture the main behavior of the system,the models have not been veriﬁed down to practical 15 post-FEC BERs of 10− , and it is possible that rare events that are not included in our models could occur at very low post-FEC BERs. Thus, model accuracy at very low post-FEC BERs is an open question for investigation. Lastly, we use our statistical models to select BCH codes. Future work could involve using the models to select other classes of codes or code ensembles, as well as to design new coding schemes.

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