# Advanced Signal Processing and Coding Techniques for Optical Communications

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## From on-off keying to multi-level modulations

In its most general form, an optical fiber transmission system is composed of a transmitter, an optical channel containing fiber spans and other optical components, and a receiver. A basic transmitter consists of electrical drivers and optical modulators. Each electrical driver transforms a data bit sequence, that may be coded using a forward error correction (FEC) module, to an electrical signal converted to the optical domain by an electro-optical modulator. The optical signal is transmitted in the fiber link where optical amplifiers are regularly inserted to compensate the fiber attenuation and carry the signal over thousands of kilometers. At the optical receiver, the signal is detected by one or more photodiodes and is converted to the electrical domain. After, analog-to-digital conversion is performed and digital signal processing is applied to compensate the transmission impairments. Finally, the information bits are estimated using an appropriate decoder. A schematic architecture of an optical transmission system is given in Fig. 1.3.

### Optical transmitters

A bit sequence at the input of an optical transmitter is converted to an electrical signal that modulates an optical carrier emitted by a laser. The amplitude and phase of the optical signal, at the output, are determined by a certain number of information bits. The electrical field of the optical
signal at the output of a perfect optical modulator is given by: E0(t) = A0(t) exp(w0t +f0t) (1.1) where A0 and f0 are the amplitude and phase of the output field, w0 =2p f0 is the carrier frequency, in rad/s, linked to the more commonly used wavelength l0 = 2pc/w0 with c = 3.108 m/s being the velocity of light in vacuum and f0 the frequency in Hz. Depending on the used modulation format, we define an amplitude, phase or amplitude and phase modulation.
Real optical modulators are characterized by an electro-optical bandwidth defining the maximum frequency at which they keep responding to the electrical driving signal, an extinction ratio between the maximum and the minimum output power and an insertion loss. Various technologies of optical modulation exist [14] including a direct modulation of the laser or the use of external components that modulate the continuous wave emitted by the laser. These components include electro-absorption modulators, phase modulators and Mach-Zehnder modulators (MZMs). While direct intensity modulation of the laser is the easiest method, it causes an unwanted variation of the lasing frequency, called frequency chirp that induces pulse spreading and hence limits the reach of the transmission system. Electro-absorption modulators, based on the modulation of the absorption coefficient of a semiconductor material, have a reduced frequency chirping effect and can be monolithically integrated with the laser providing a compact and low-cost component called the electro-absorption modulated laser (EML). EMLs are well suited for access networks with typical transmission distances of 80 km and 10 to 40 Gb/s rates. The MZM is another external modulation
component that exhibits very low frequency chirping and is a popular device for high-speed longhaul optical transmissions because it offers a large electro-optical bandwidth (up to 40 GHz), small insertion losses (≤ 4 dB) and a high extinction ratio (≥ 20 dB). Moreover, MZMs are known to be weakly wavelength dependent. Their higher cost and power consumption made them unsuitable
for access networks.

#### Mach-Zehnder modulator

A MZM is an interferometer composed of two 3 dB couplers and two arms of Lithium Niobate crystal as shown in Fig. 1.4. The modulation occurs by applying a driving voltage to the arms that modifies their refractive index, controlling hence the phase of the light propagating through the arms and creating constructive and destructive interference at the output of the MZM. This interference translates into amplitude fluctuations of the optical signal. The input-output characteristic of an MZM is given by: where Ein and Eout are the fields at the input and output of the modulator respectively, V1 and V2 are the voltages applied to the arms, and Vp is the drive voltage corresponding to a phase shift from a constructive (maximum) to a destructive (minimum) interference (transmittance). With adequate electrical voltages, MZMs can be used for digital modulations (on-off keying, binary phase shift keying, QAM…) or analog modulations such as OFDM, as shall be seen in the next sections.

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Table of contents :

Glossary
Notations & Symbols
List of Figures
List of Tables
Introduction
1 Evolution of Optical Fiber Transmission Systems
1.1 From on-off keying to multi-level modulations
1.1.1 Optical transmitters
1.1.2 Propagation in optical fiber transmission systems
1.1.3 Optical receivers
1.2 Wavelength Division Multiplexing (WDM)
1.3 Polarization Division Multiplexing (PDM)
1.3.1 Principle of coherent PDM transmissions
1.3.2 Propagation equations
1.3.3 Polarization Mode Dispersion (PMD)
1.3.4 Polarization Dependent Loss (PDL)
1.3.5 Non-linear effects
1.4 Space Division Multiplexing (SDM)
1.4.1 Multi-core fibers
1.4.2 Multi-mode fibers
1.4.3 Linear modal crosstalk
1.4.4 Differential Modal Group Dispersion (DMGD)
1.4.5 Mode Dependent Loss (MDL)
1.4.6 Non-linear effects
1.5 Beyond 400G: WDM-SDM-PDM super-channels
2 Advanced Signal Processing and Coding Techniques for Optical Communications
2.1 Forward Error Correcting (FEC) codes
2.1.1 Basics of FEC
2.1.2 FEC in optical transmission systems
2.2 Basic digital equalization in 100 Gb/s single-carrier PDM-QPSK systems
2.2.1 Dispersion compensation & polarization de-multiplexing
2.2.2 Frequency offset compensation & carrier phase estimation
2.2.3 Transmitter-side DSP
2.3 OFDM: more than a modulation format
2.3.1 Principle
2.3.2 OFDM for optical communications
2.3.3 Capacity of linear MIMO-OFDM channels
2.3.4 Optical OFDM or single-carrier for MIMO schemes?
2.4 MIMO coding techniques
2.4.1 Wireless MIMO channels
2.4.2 Design criteria of Space-Time (ST) codes for Rayleigh fading channels
2.4.3 Space-Time Block Codes
2.4.4 Decoders for MIMO schemes
3 Polarization-Time Coding for PDL Mitigation in Polarization Multiplexed Systems
3.1 Polarization-Time coding: preliminary results
3.2 Numerical study of a Polarization-Time coded optical OFDM transmission
3.2.1 Performance of Silver, Golden and Alamouti coded schemes
3.2.2 Tolerance of PT-coded OFDM for non-linear effects
3.3 Theoretical analysis of the PT coding performance
3.3.1 Computing an upper bound of the pairwise error probability (PEP)
3.3.2 Concatenation of a FEC code and a PT code
3.4 Experimental validation of PT coding gains
3.4.1 Experimental setup
3.4.2 Experimental results
4 ST coding for MDL Mitigation in Mode Division Multiplexed (MDM) Systems
4.1 Description of LP modes in parabolic-index FMFs
4.2 MDL in MDM transmission systems
4.2.1 Best-case scenario: strong-coupling model
4.2.2 Case study 1: fiber misalignments and micro-bends
4.2.3 Case study 2: few mode amplifiers with mode dependent gain (MDG)
4.3 ST coding for various MDL-impaired MDM channel models
4.3.1 Case study 1: fiber misalignments and micro-bends
4.3.2 Case study 2: few mode amplifiers with mode dependent gain (MDG)
4.4 Scalability and complexity issues of ST coding
5 Low-Complexity Schemes for Large Optical MIMO Systems
5.1 Low-complexity sub-optimal decoding of ST codes
5.1.1 Performance of ZF decoding
5.1.2 Performance of ZF-DFE decoding
5.1.3 Performance of MMSE decoding
5.1.4 Performance of sub-optimal decoders in PDL-impaired PDM systems
5.2 Multi-block coding of SDM schemes
5.2.1 Description of multi-block schemes
5.2.2 Performance of ML-decoded multi-block schemes
5.3 Mode selection for SDM transmission systems
5.3.1 Motivation and principle
5.3.2 Application to a 10×10 MDM scheme
A OSNR Sensitivity
B Polarization of light
C Matrix Operations
D The Sphere Decoder
E Condensed French Version
E.1 Introduction
E.1.1 Objectifs et contributions de la thèse
E.2 Codage Polarisation-Temps pour les systèmes de transmission optique PDM-OFDM191
E.2.1 Modèle du système PDM-OFDM codé en polarisation et en temps
E.2.2 Etude numérique de la compensation des effets de PDL
E.2.3 Validation Expérimentale de la compensation des effets de PDL
E.2.4 Conclusion
E.3 Codage Espace-Temps pour les systèmes de transmission optique SDM-OFDM
E.3.1 Systèmes de transmission optique multiplexés en modes spatiaux
E.3.2 Codage Espace-Temps et décodage en sphères
E.3.3 Performance des systèmes MDM codés en espace et en temps
E.3.4 Complexité et extensibilité des systèmes MDM codés
E.3.5 Conclusion
E.4 Perspectives
Bibliography

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