Modulation formats and digital signal processing for fiber-optic communications with coherent detection

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Université libre de Bruxelles Ecole Polytechnique de Bruxelles

Service OPERA

Modulation formats and digital signal processing for fiber-optic communications with coherent

detection

A thesis submitted for the degree of Docteur en Sciences de l’Ingénieur

by

Jessica Fickers

Jury:

Dr. Gabriel Charlet

Prof. Philippe Emplit (Co-promoteur) Assoc. Prof. Yann Frignac

Prof. Simon-Pierre Gorza (Président du jury)

Assoc. Prof. François Horlin (Promoteur)

Prof. Jérôme Louveaux (Secrétaire du jury)

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First of all, I would like to thank my supervisors, the Professors François Horlin and Philippe Emplit as the initiators of my thesis project. Thank you for trusting me to bridge your respective areas of expertise and for the assistance you provided at all levels of the research project.

Second, this project would have been impossible without the help of our partners at the Alcatel-Lucent Bell Labs in France. I would like express my gratitude to Dr. Jean-Pierre Hamaide who agreed to receive me at Alcatel-Lucent and Dr. Gabriel Charlet who really integrated me in his research group during my stays. I never felt a stranger at Villarceaux and I am very grateful for that. It was a privilege for me to be able to work with expert researchers who were willing to share their knowledge with me: in no particular order of importance, I express my gratitude to Dr. Oriol Bertran-Pardo, Dr. Annalisa Morea, Dr. Jeremie Renaudier, Haïk Mardoyan, Patrick Brindel and Rafael Rios-Müller. My special thanks go to Dr. Amirhossein Ghazisaidi, Dr. Massimiliano Salsi and - again - Dr. Gabriel Charlet. You never hesitated to take time to help me in my research and in the laboratory, thank you for your trust, help and interest.

This thesis was funded by the Belgian Fonds National de la Recherche Scientifique (FNRS) to whom I address my sincere gratitude. Appreciation also goes to my alma mater the Université libre de Bruxelles and especially the OPERA department. Thanks go out to the Professors, colleagues and friends at the wireless communications group:

Professor Philippe De Doncker, Professor Jean-Michel Dricot, Thibault Deleu, Ourouk Jawad, Theodoros Mavridis, Luca Petrillo, Jonathan Verlant-Chenet, Marc Bauduin, Sul- livan Derenne and Jonathan Bodard. I know I am not the most outgoing of collegues, but rest assured I appreciate the welcoming atmosphere at the group. My special thanks go to Natascha Vander Heyden and Ibtissame Malouli for all the instances in which their assistance helped me along the way.

Last but not least kommt die Familie. Da mich ab jetzt keiner von den Nasen da oben mehr verstehen kann, bin ich mal ehrlich: Ihr wisst ja hoffentlich dass Ihr viel wichtiger seid als all die anderen zusammen? Mama und Papa, danke dass Ihr die allerbesten Eltern seid die man sich vorstellen kann. Hallo Jerome und Vicky! Un tout grand merci va aussi à ma deuxième famille Christian, Christine, Yves et Maud. Xavier, que dire?

Nous avons traversé cette aventure ensemble. Merci und setzen wir die Segel für das nächste Abenteuer!

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List of acronyms

ADC Analog to digital converter ASE Amplified spontaneous emission

BER Bit error rate

BPSK Binary phase shift keying

CD Chromatic dispersion

CFE Carrier frequency estimation CMA Constant modulus algorithm CPE Carrier phase estimation DAC Digital to analog converter

DD Decision directed

DD Direct detection

DFE Decision feedback equalizer DSP Digital signal processing

DU DIspersion-unmanaged

DU Dispersion unmanaged

EVM Error vector magnitude

FB Feedback

FD Frequency domain

FEC Forward error code

FF Feedforward

FFT Fast Fourier transform FIR Finite impulse response ICI Inter-carrier interference IFFT Inverse fast Fourier transform ISI Intersymbol interference

ITU-T International telecommunications union - telecommunication sector

LO Local oscillator

LUT Look-up table

MAP Maximum a posteriori

MC Multicarrier

ML Maximum likelihood

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MMSE Minimum mean square error

MZM Mach Zehnder modulator

N-WDM Nyquist-wavelength division multiplexing NLSE Non linear Schrödinger equation

NLT Nonlinear threshold

OFDM Orthogonal frequency division multiplexing

OOK On-off keying

OQAM Offset quadrature amplitude modulation OSNR Optical signal to noise ratio

PD Photodiode

PDM Polarization division multiplexing PMD Polarization mode dispersion

PME Polarization divisison multiplexing emulation PRBS Pseudo-random binary sequence

PS Polarization-switched

QAM Quadrature amplitude modulation QPSK Quaternary phase shift keying

ROADM Reconfigurable optical add-drop multiplexer

RRC Root-raised-cosine

SBS Symbol by symbol

SC Single carrier

SE Spectral efficiency

SMF Single mode fiber

SSMF Standard single mode fiber

TD Time domain

V+V Viterbi and Viterbi algorithm WDM Wavelength division multiplexing WSS Wavelength selective switch

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Chapter 1 Introduction

Optical fiber telecommunications form the backbone of the global data network. As such, they have enabled the exponential growth of the data transfer capacity over the last decades. Thanks to major technological breakthroughs such as single mode fibers and the erbium doped fiber amplifier, the transmission distances were progressively increased and today, long-haul optical links span oceans and continents.

Nowadays, with the speed of electronics progressively catching up with the data rates of optical signals, a new revolution is taking place: the advent of digital signal processing for optical communications. Impairments that were to be accounted for by analogical means - or not at all - can now be corrected after the signal is translated to the digital domain at the receiver. Moreover, the very form in which the information is transmitted over the fiber can be re-defined by software. As we will see, these techniques can be used to increase the data rate on the communication link.

The next-generation optical communication networks will respond to higher require- ments in both spectral efficiency and flexibility than the current systems. Moreover, with an increasing awareness of the need to decrease the power consumption of optical networks, the computational complexity of digital signal processing becomes a research focus.

The context of this PhD thesis is the potential of digital signal processing to achieve these goals.

In the second chapter of this thesis, we review the fiber-optic transmission channel and fiber-optic communication architecture for long-haul distances. First, the optical fiber is introduced as a transmission medium. We study the main effects that distort the optical signal as it is transmitted over the fiber. We see how digital data can be imprinted on laser light and injected into the link as a first example of the use of digital signal processing.

Wavelength and polarization division multiplexing are introduced as means to enhance the communication capacity of a single optical fiber. Next, the global architecture of a point-to-point optical communication link is introduced. We will see how the same optical signal can be transmitted over transatlantic distances thanks to optical amplification. We introduce coherent detection thanks to which both amplitude and phase of the signal can be detected.

In the third chapter, we present the state of the art in digital signal processing for coherent optical communications. We present a diagram of the most commonly accepted digital signal processing architecture for coherent detection. Each block in the diagram is targeted towards one type of impairment: chromatic dispersion, polarization effects and

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2 Chapter 1. Introduction

phase offsets. In the sections of the chapter, one or several algorithms are presented that are typically used to mitigate each of the impairments.

The remaining chapters each present a study, experimental or by simulation, of a digital signal processing solution to mitigate impairments, enhance the spectral efficiency or the flexibility of a long-haul optical communication link. They progressively evolve from being very similar to the state of the art to more novel techniques. It should be mentioned that while the simulations and original digital signal processing algorithms presented here were designed and implemented by the author of this work, all experimental results presented in this thesis were carried out at the Alcatel-Lucent Bell Labs in Villarceaux, France with much help from experienced Bell Labs scientists and technicians for both laboratory work and state of the art digital signal processing. Luckily, I was also able to help out with a few digital signal processing solutions (see n. 9 in Publications for example) that are not included in this thesis because my contributions were minor.

In chapter 4, transmitter digital signal processing is used to optimize the pulse shaping in order to reduce the optical channel spacing. We use numerical simulations to show that a trade-off exists between spectral efficiency and the mitigation of hardware impairments.

In chapter 5 we enhance the equalization at the receiver so as to mitigate heavy in-line filtering. In this study, we first show by numerical simulation that an equalizer that uses the symbol decisions in a feed-back loop can mitigate the performance penalties arising from bandwidth reduction at an acceptable computational cost. The proposed equalizer is then validated in an experimental study in the context of in-line filtering stemming from reconfigurable optical add-drop multiplexers, components that route the optical channels at the network nodes. In this study, the experimental waveforms were measured by A.

Ghazisaeidi and M. Salsi at Bell Labs.

In chapter 6 we modify the mode of data transmission in order to increase the flexibility of the transmission scheme. We digitally divide the electrical bandwidth of one channel into subcarriers. Each subcarrier is modulated independently using Offset-QAM.

In order to successfully detect the multicarrer Offset-QAM, the state of the art digital signal processing can no longer be used. We propose an equalizer based on the minimum mean square error criterion and trial phases. We experimentally show the feasibility of multicarrier Offset-QAM in long-haul link. For this study, measurements were carried out by the author of this work at the transmissions laboratory of Bell Labs with much help from A. Ghazisaeidi and P. Brindel.

In chapter 7, we extend the study of chapter 6 to a higher density of subcarriers in order to significantly reduce the computational complexity of the receiver digital signal processing. In order to successfully decode the symbols in the presence of the channel impairments, we propose a hybrid modulation architecture using a single carrier pilot combined with low-complexity frequency domain equalization and prove its feasibility by simulation. The different contributions of this thesis are summarized in Fig. 1.1 according to their main goal(s).

Most of the results presented in this manuscript have already been published and presented in different peer-reviewed journal articles and conference contributions which I have authored or co-authored. These articles are listed in the last chapter of the manuscript named Publications.

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3

Spectral efficiency

Flexibility

Reduction of computational

complexity Chapter 4

Chapter 5 Chapter 6

Chapter 7

Figure 1.1: Summary of the contributions presented in this thesis according to their main goals.

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Chapter 2 Fiber-optic communications

2.1 Introduction

In today’s telecommunications, silica fibers are the preferred medium to transmit large amounts of data over long distances. First proposed in 1966 for the transfer of information [1], optical fibers are waveguides based on the principle of total internal reflection.

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6 Chapter 2. Fiber-optic communications

Compared to free-space transmission, they offer very low attenuation of the optical signal in a broad range of frequencies in the near-infrared range. The data to be transmitted is imprinted on the laser light which is then injected into the fiber. At the other end of the link, photodiodes are used to translate the received light into electrical signals so that the messages can be decoded. Through this basic principle, several Tbit/s can be transmitted on the same fiber over distances ranging up to tens of thousands of kilometers. This chapter provides a basis for the understanding of the fiber-optical communication link.

First, we define the long-haul optical network in section 2.2. In section 2.3, we describe the silica fiber as a transmission medium. In section 2.4, we detail the transmission impairments in optical fibers. In particular, we describe the effects of attenuation, chromatic dispersion (CD) polarization effects and nonlinear distortions originating in the nonlinear optical Kerr effect. In section 2.5, the general architecture of point-to-point optical transmission links is introduced. We discuss laser sources and optical modulation by which the numerical data is imprinted on the carrier wave. We introduce digital modulation formats. Those can be understood as alphabets used to transmit information. Next, we will see how optical amplifiers compensate the attenuation losses in order to transmit the same optical signal over thousands of kilometers. Finally, we introduce coherent detection, a receiver architecture that allows us to detect both phase and amplitude of the optical signal.

2.2 Fiber-optic networks

All optical communication systems propagate data encoded on light waves over a series of optical fibers. They can be categorized according to the distances covered, the envi- ronment in which the fiber is deployed and the network architecture.

• In terrestrial long-haul networks, the distances involved are in the order of several hundreds to several thousands of kilometers. Terrestrial long-haul networks are conceived to cover the surface of a continent, with large cities at their nodes. In these systems, each fiber section also called span measures about 80 to 100 kilometers.

Singlemode fibers (see section 2.3) are typically used. Between fiber spans, amplifiers compensate for the attenuation of the signal power during transmission. At the nodes of the system, electrical and/or optical routing devices such as wavelength selective switches (WSS) and reconfigurable add-drop multiplexers (ROADM) insert or retrieve individual part of the transmitted signal called channels. Each channel occupies a specific frequency range defined by the laser source providing its carrier wave. The switching devices are often located at the same places as the amplifiers and inserted between amplifying stages.

• Submarine long-haul networks are conceived to transmit data between continents.

As a consequence, their transmission distances range from 6000 (transatlantic) to 13000 (longest transpacific) kilometers. They also use singlemode fiber, but the individual spans are shorter than in terrestrial systems (45-60 km). Amplification is typically done in a single step. Submarine networks have very few ramifications.

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2.3. The optical fiber as a transmission medium 7

• Short-distance networks. On the scale of cities, metropolitan networks are designed to link the long-distance networks to access networks.

The techniques developed in this work and the experimental setups used are representative of terrestrial long-distance networks.

2.3 The optical fiber as a transmission medium

125 µm 250

µm

8µm core

cladding

coating x

y z

Refractive index

Fiber radius core

cladding cladding

Figure 2.1: Silica fiber

Optical fibers are cylindrical dielectric waveguides made of silica glass. As can be seen in Fig. 2.1, they are composed of fiber core, cladding and coating. The dielectric properties of the cladding differ from those of the core. The refractive index is higher in the core than in the cladding. As a consequence, light propagating approximately in the direction of the main axis of the fiber z is confined to the core by total internal reflection [2].

The modes of wave propagation inside the optical fiber can be determined by solving Maxwell’s equations for the electric field and magnetic fields, taking into account the boundary conditions at the limits of core and cladding and the cylindrical symmetry. This yields a discrete number of solutions of guided waves inside the fiber. In singlemode fibers, the distribution of the refractive index is designed so that only one mode of propagation is permitted. Singlemode fibers are the most widely used fibers in telecommunications because the presence of several modes of propagation leads to intermodal interference of the transmitted signal. The transverse energy distribution of the guided mode in a singlemode optical fiber is illustrated in Fig. 2.2. It is uniform along the azimuthal direction and has an approximately Gaussian distribution of energy with respect to the fiber radius.

After the transverse distribution, we now examine the axial distribution, in other words the propagation of the guided mode. In order to understand the propagation of

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core diameter

Figure 2.2: Energy distribution of the fundamental mode of propagation in singlemode fibers. Illustration from [3].

optical signals through the fiber, we introduce the concept of wave-packets. A wave-packet is a superposition of plane waves of different frequencies around a central frequency. In optical telecommunications, the central frequency is the frequency of the continuous laser source that is injected into the fiber. The components of the wave-packet that occupy the frequencies around the central frequency carry the information that is to be transmitter over the link. The laser source is also called the "carrier" for this information.

The wave-packetA(z, t)at timetand axial positionz is expressed as the superposition of its spectral components:

A(z, t)∝ Z +∞

ω=−∞

A(ω) exp (j(β(ω)z˜ −ωt))dω (2.1) In this equation,β is the propagation constant, andωdenotes the pulsation. To calculate the group velocity of the wave-packet, i.e. the speed at which the information propagates, we must make the assumption that the wave-packet is quasi-monochromatic around the carrier pulsation ω0:

β(ω)≈β₀+β₁.(ω−ω₀) (2.2)

In this equation, β₀ .

= β(ω₀) and β₁ .

= ^∂β_∂ω

ω=ω0. In this case, it is easily shown [2] that the wave-packet travels at the group-velocity v_g = 1/β₁ without distortion.

In modern optical transmission links, optical signals emitted by different laser sources at different wavelengths are independently modulated and simultaneously propagated over the same optical fiber [4]. This technique is called wavelength division multiplexing (WDM) and exploits the wide frequency range in which optical fibers show very small attenuation. The telecommunication standardization sector of the international telecommunication union (ITU-T) defined six bands for transmission using singlemode fiber. Fig. 2.3 illustrates these bands as well as the attenuation as a function of wavelength/frequency.

The main communication band is the C band (see Fig. 2.3). The bandwidth can however be enlarged by using the L-band. Combining these two bands, approximately 160 wavelength channels can be transmitted on a 50-GHz grid.

The polarization of light refers to the direction of pulsation of the transverse electrical field as a function of time. In the general case, the electrical field traces an ellipse in the

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2.4. Transmission impairments in optical fibers 9

Figure 2.3: Attenuation of singlemode fiber as a function of wavelength and associated transmission bands. Illustration from [5].

(x, y)-plane. If the electrical field pulsates along a line, it is said to be linearly polarized along this direction [6]. Any state of polarization of a wave-packet propagating through the fiber can be expressed as the combination of two fields that are linearly polarized along orthogonal axes. Seen this way, the "unique" mode of propagation in singlemode fiber is in fact the superposition of two degenerate modes polarized along two orthogonal axes in the transverse plane of the fiber. In today’s optical telecommunications, the two linearly polarized modes of propagation are used to transmit independent streams of information.

First demonstrated in 1992 [7], this technique is called polarization division multiplexing (PDM).

2.4 Transmission impairments in optical fibers

The undisturbed transmission of the wave-packet in Eq. (2.1) is only a first approximation.

In optical fibers, the main physical effects leading to the distortion of the wave-packet as it propagates are attenuation, chromatic dispersion, polarization mode dispersion and the non-linear Kerr effect. The evolution of the optical field along its propagation axis is described by a partial differential equation called the coupled non-linear Schrödinger equation [8]:

∂A

∂z =−α 2A

| {z }

(1)

−j∆β₀(z)

2 J(z)A− ∆β₁(z)

2 J(z)∂A

∂τ

| {z }

(3)

+jβ₂ 2

∂²A

∂τ² +β₃ 6

∂³A

∂τ³

| {z }

(2)

−jγ

|A|²A− 1

3(A^∗σ₃A)σ₃A

| {z }

(4)

(2.3)

In this equation, the boldAdesignates the propagated wave-packet as Jones vector. In Jones formalism, the electrical field is expressed using the linearly polarized components along orthogonal directions. τ =t−z/v_g is the time frame moving along the wave-packet referential. J(z) is a 2×2 unitary Jones matrix. σ₃ is the third Pauli matrix. α is the attenuation coefficient, γ is called the nonlinear coefficient, β_i .

= _∂ω^∂ⁱ^βi

ω=ω0

where ω₀ is

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the central pulsation and β(ω) is the propagation constant. In Eq. (2.3), the different right-hand terms refer to (1) attenuation, (2) chromatic dispersion (limited to the third order) (3) polarization mode dispersion and (4) the non-linear Kerr effect. These fiber impairments will be detailed in the subsequent paragraphs.

2.4.1 Attenuation

Ignoring all right-hand terms in Eq. (2.3) but attenuation, we have:

∂A

∂z =− α

2A (2.4)

It is easily seen that the power of the electric field A will decrease exponentially with transmission distance. When an electromagnetic wave propagates through an optical fiber, its amplitude is attenuated by absorption and scattering. While attenuation does not result in signal distortion in itself, the need for amplification units along the transmission line results in signal corruption by noise as we will see in section 2.5.4.

2.4.2 Chromatic dispersion

In an optical fiber, the quasi-monochromatic assumption Eq. (2.2) does only approximately hold. Around 1550 nm (transmission in the C band, see Fig. 2.3) and for signal bandwidths of several tens of GHz, it is necessary to take into account at least one more term in the development of the dispersion relation around the central pulsation. This effect is called group-velocity dispersion and results in distortion of the transmitted signal because the spectral components of the pulses travel at different speeds. The scalar NLSE ignoring all effects except CD becomes:

∂A

∂z =jβ₂ 2

∂²A

∂τ² +β₃ 6

∂³A

∂τ³ (2.5)

We define the Fourier transform of the wave-packet:

A(z, τ) = Z ∞

−∞

A(z, ω)e˜ ^jωτdω (2.6) In the frequency domain, Eq (2.5) becomes:

∂A˜

∂z =−j

β1ω+ β₂

2 ω²+ β₃ 6 ω³

A˜ (2.7)

We see that the equation is easily solved in the frequency domain:

A(z, ω) = ˜˜ A(0, ω)e^−j(^β²²^ω²⁺^β⁶³^ω³^)z (2.8) As a conclusion, chromatic dispersion manifests as the accumulation of a quadratic phase for the second order dispersion and a cubic phase for third order dispersion in the frequency domain and can be inverted by multiplication with the exponential of the opposite phase in frequency domain. In the fiber-optic community, CD is generally quantified as the

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2.4. Transmission impairments in optical fibers 11

dispersion factor D [ps/nm.km] which is linked to β₂ and the central wavelength of the signal λ in the following way:

D=−2πc

λ² β₂ (2.9)

As an example, the dispersion factor is given as a function of wavelength for standard singlemode fiber (SSMF) [9]:

D(λ)≈S_z(λ− λ⁴_z

λ³) (2.10)

In this equation, λ_z = 1320 nm is the zero-dispersion wavelength and S_z = 0.092 ps/(nm²km)is called the zero dispersion slope.

2.4.3 Polarization mode dispersion

Birefringence is the dependency of the refractive index of the fiber on the signal polarization. This phenomenon is caused by fabrication imperfections, stress and temperature variations. In the presence of birefringence, the propagation constant depends on the polarization of the light. At each axial position z, the fiber is characterized by an eigenmode corresponding to the field polarization with the slowest propagation constant [10].

At the same time, the orthogonal eigenmode is the mode with the highest propagation constant. The birefringent fiber model is illustrated in Fig. 2.4. The distribution of the eigenmodes as a function of distance is stochastic in nature and changes slowly with time.

For singlemode fibers, the correlation length, i.e. the distance after which the eigenmodes have completely independent orientation, is in the order of 100 m [11].

slow axis

fast axis

Figure 2.4: Fiber model of randomly birefringent sections

Looking at Eq. (2.3),δβ₀(z)is the difference in propagation constant between the slow and fast polarization axes at the central pulsation. δβ₁(z) is the slope of the propagation constant difference as a function of pulsation. The Jones matrix J(z), which is defined on the eigenmodes at position z, acts as projector on the eigenmodes at position z. As a consequence of PMD, the pulses transmitted on the two polarization modes are both mixed and broadened. The strength of PMD in a fiber is generally expressed by the PMD coefficient, in [ps/√

km]. The PMD coefficient, multiplied by the square root of the fiber length, gives the mean differential group delay between orthogonal polarization states. Modern optical fibers have a PMD coefficient in the order of 0.05 [ps/√

km]. The distribution of the eigenmodes along the fiber also varies with time on a ms scale [12, 13]

which means that PMD parameters must be tracked at the receiver.

2.4.4 Nonlinear interactions

The optical Kerr effect is the main source of nonlinear distortion in optical communication systems [14]. It originates in the nonlinear response of the electronic polarization of the

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medium. As a consequence, the refractive index depends on the power of the electric field propagating through the fiber. Other nonlinear effects arise from the interaction of the optical field with the silica molecules: stimulated Raman scattering and stimulated Brillouin scattering [8]. These nonlinear effects are due to the inelastic scattering of photons towards lower energy photons. The energy difference is absorbed by molecular vibrations or phonons in the silica. However, nonlinear Raman or Brillouin scattering is only efficient above power thresholds that are not obtained in the framework of this thesis and are therefore ignored.

In Eq. (2.3) the right-hand terms under-braced with (4) define the Kerr effect. The transmission impairments suffered by a WDM complex propagation through an optical fiber and stemming from the optical Kerr effect can be categorized:

• Self Phase Modulation. The electromagnetic field of one optical channel modifies the refractive index according to its own instantanenous power. As a consequence, the field is phase shifted according to Eq. (2.3).

• Cross Phase Modulation. Similarly to Self Phase Modulation, Cross Phase Modu- lation originates from the dependency of the refractive index on the instantaneous optical power. Here, all co-propagating WDM channels induce a phase modulation.

• Four Wave Mixing. Two photons of different energies may be annihilated and two photons of different energy appear if the net energy and momentum of the transfor- mation are conserved [8]. Phase matching between the different spectral components is required for the Four Wave Mixing interaction to be effective. On the macroscopic scale, Four Wave Mixing manifests as a non-linear interaction between up to four WDM channels.

In addition to this general classification, Eq. (2.3) shows that nonlinear interaction also occurs between polarization components both in intra-channel and inter-channel frame- works, an effect called Cross Polarization Modulation.

The existence of inter-channel effects such as Cross Phase Modulation and Four Wave Mixing make the Kerr effect very difficult to compensate in the digital domain because the information sent over the neighboring WDM channels is generally unknown at the receiver. To this date, the Kerr effect is not completely compensated by signal processing in coherent optical communication systems [15].

2.5 The fiber-optic communication system

Fig. 2.5 summarizes the typical architecture of an optical point-to-point long-haul communications link. The information to be transmitted takes the form of a binary sequence.

This sequence undergoes digital signal processing before being translated into an analog waveform. A continuous laser wave is modulated by this waveform and the resulting signal is fed into the optical fiber. Typical transmission distances range from 1000 to 5000 km for terrestrial transmission links. At the receiver, the optical signal is transformed back into the electrical domain by photodiodes. The resulting analog waveform is sampled. The digital sequences undergo signal processing again before they are converted

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2.5. The fiber-optic communication system 13

DAC

! N

Demodulator

Laser

ADC

Fiber

...1010011... ...1010011...

Laser

Modulator Ampli

Transmitter Transmission link Receiver

Photodiode Decision

Figure 2.5: Architecture of the fiber-optical communication link

back into a binary sequence. In this section, we introduce the relevant aspects of the physical long-haul communication system.

2.5.1 Laser sources

Laser sources for telecommunications must operate continuously for long times at room temperature. They are of small diameter because the light must be fed into the core of the fiber. The most commonly used sources are light-emitting diodes and semiconductor lasers. In a semiconductor laser, the performance is mostly limited by spontaneous emission. This results in both amplitude and phase noise and thus broadening of the laser linewidth. When coherent detection is used (see section 2.5.5), the phase noise beating of transmitter and receiver lasers imprints itself on the modulated signal. As a consequence, the received signal shows a phase drift in accordance with the laser linewidths. The laser frequency noise of telecommunication lasers is commonly assumed a Brownian movement which results in a lorentzian line shape [16]. The typical linewidth of optical communication lasers is in the MHz range [17]. The frequency noise ∆f(t) is easily computed as white noise with a spectral density equal to γ/π, where γ is the full width at half maximum linewidth. The optical field of the laser is then given in complex notation:

E(t) = E₀e^j(2πf⁰^t+∆φ(t)) (2.11)

∆φ(t) = Z t

−∞

∆f(t⁰)dt⁰ (2.12)

In this equation f0 is frequency at the maximum of the lorentzian line.

Although the laser frequency can be controlled directly based on a feedback loop, this type of design is usually complicated and inefficient. Free-running laser sources also exhibit slow fluctuations of the emitting frequency over the day [18]. The accuracy of temperature-stabilized lasers is approximately 1 GHz [19].

2.5.2 Optical modulation

The function of the modulator is to convert the electrical data into the optical domain. In most long-haul high-speed transmission systems, distributed feedback lasers are externally modulated using a Mach-Zehnder modulator (MZM). The MZM is an interferometer composed of two 3-dB couplers and two waveguides with equal lengths as shown in Fig. 2.6.

The incoming optical energy is equally divided between the two arms. Each arm comprises

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an electro-optical cell which induces a phase-shift between the optical signals in each arm depending on the applied voltage called drive voltage. These phase-shifts translate into amplitude fluctuations once the optical signals on the two arms are recombined. As a consequence, the electrical drive voltage can be translated into optical modulation of the continuous wave. The input-output characteristic of the MZM is given by a cosine function of the difference of the drive voltagesV1 andV2. The drive voltage difference leading from maximum output power to zero is calledV_π. When the variations of V₁−V₂ are centered aroundV_π/2and small with respect toV_π, the characteristic can be considered linear and the distortion between the drive voltage input and the optical output minimized.

Two MZM can be combined in order to imprint complex information on the optical wave. Again, two 3-dB couplers and two waveguides are used to equally divide the optical energy. The real("I") and imaginary ("Q") parts of the information streamI +iQare fed into their respective MZMs. After phase shifting the imaginary arm by π/2, the optical fields are recombined. This setup is called an IQ-modulator. In polarization-multiplexed transmission systems (see section 2.2), two IQ-modulators are used. They are followed by polarizers which impose a linearly polarized state of polarization on two orthogonal transverse axes. The two signals are then combined in a polarization combiner to compose the polarization-multiplexed optical signal.

V

₁

V

2

waveguide

3-dB coupler 3-dB coupler

electrode

Light in Light out

Figure 2.6: Mach-Zehnder modulator

2.5.3 Digital modulation formats

In the previous paragraph, we described how different degrees of freedom - amplitude, phase, polarization - can be used to transmit information on optical waves. This paragraph deals with the modulation formats, i.e. the way binary sequences can be digitally translated into these entities before the optical modulation. We review the main system- relevant modulation formats. They are illustrated in Fig. 2.7.

• On-Off Keying with direct detection (OOK-DD). This modulation format is used in most legacy networks operating at 10 Gbit/s without detecting the phase of the signal ("direct detection"). While historically important, it is no longer competitive since neither polarization modes nor optical phase are exploited, which results in a limited spectral efficiency.

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R I

OOK BPSK

QPSK 16-QAM

Figure 2.7: Modulation formats for optical communications

• Polarization Division Multiplexed Binary Phase Shift Keying (PDM-BPSK). With the advent of coherent detection and digital signal processing, PDM has become a standard feature permitting to double the data rate at the same symbol rate as OOK [20]. BPSK encodes one bit per symbol per polarization. Comparing to higher-order modulation formats, PDM-BPSK is therefore the most suitable choice for long-distance links with moderate data rate such as 40 Gbit/s [5].

• Polarization Division Multiplexed Quaternary Phase Shift Keying (PDM-QPSK).

Encoding two bits per symbol per polarization, PDM-QPSK is the key modulation format in order to achieve a 100 Gbit/s data rates while keeping the symbol rate as low as 28 Gbd/s on each optical channel [21]. QPSK and BPSK are part of the M-phase shift keying modulation format family, characterized by a constant modulus and equally spaced phase values. This geometry is particularly interesting for receiver DSP polarization demultiplexing (see section 3.4) and phase equalization (see section 3.6).

• Polarization Division Multiplexed 16-Quadrature Amplitude Modulation (PDM- 16QAM). To further increase the data rate per optical channel beyond 100 Gbit/s, high-order modulation formats are currently researched [22, 23]. PDM-16QAM is illustrated in Fig. 2.7 as an example. The main challenges of these high-order modulation formats are reduced transmission distances due to increased noise and phase sensitivity with respect to PDM-QPSK [24].

In this listing, we only consider modulation formats in which the two polarization components are independent (PDM). In fact, the noise sensitivity of PDM-QPSK can be

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further optimized over the three degrees of freedom phase, amplitude, polarization using a polarization coding scheme called polarization switched quaternary phase shift keying (PS-QPSK) [25, 26], which is beyond the scope of this work.

2.5.4 Loss compensation

Nowadays, long optical transmission distances are achieved by periodically incorporating optical amplification in order to compensate fiber loss and to amplify the whole WDM multiplex at once. The transmission link consists of consecutive fiber spans separated by optical amplifiers. Optical amplifiers enhance the weak input signal from the previous span and launch it into the next span at a high power. Their principle of operation is stimulated emission. In the optical amplifier medium, electrons are pumped to a high energy level.

This results in population inversion, i.e. a predominance of high-energy electrons between two energy levels. In these conditions, stimulated emission can take place: the incoming photons are duplicated during their transmission through the amplifier. At the same time, spontaneous emission corrupts the signal. Spontaneous photons are decorrelated from the signal and have arbitrary frequency and polarization. Moreover, the spontaneous photons are also amplified along the link (ASE for amplified spontaneous emission). The span length varies depending on system configuration but most of the terrestrial systems use spans between 80 and 100 kilometers. The most widely used type of amplifier is the erbium-doped fiber amplifier [27]. It consists of a singlemode fiber of about 10 meters doped with Erbium ions which is pumped using a laser to achieve the population inversion necessary to amplify the signal through stimulated emission. The amount of corruption caused by optical amplification on the transmitted signal is quantified by the optical signal to noise ratio (OSNR):

OSNR = P_signal

P_noise (2.13)

Where P_signal is the signal power and P_noise the noise power. The OSNR is usually normalized to the reference bandwidth of 0.1nm. Through this work, the signal to noise ratio is noted as OSNR_(0.1nm).

2.5.5 Coherent detection

Coherent detection relies on detecting a signal through its beating with a reference carrier supplied by a local oscillator (LO) [28]. As a consequence, both phase and amplitude of the optical field can be detected. It follows that the information to be transmitted can be encoded on both amplitude and phase of the optical signal as in the digital modulation formats introduced in section 2.5.3. While legacy networks still use a simple photodiode (direct detection) to detect channels at 10 Gbits/s, newly installed transmission links make use of coherent detection to enhance the throughput of the link and achieve channel rates of 100 Gbit/s and more. In this work, only coherent detection will be considered.

The architecture and operation of a coherent mixer is explained in Fig. 2.8.

Be E_{CohM ix}(t) the optical field at the entry of the coherent mixer. Be E_LO(t) the unmodulated light of the LO. Both signals first pass polarizers (45^◦ in Fig. 2.8). At the

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45°

λ/4

1 2

3 4

LO

Signal

Figure 2.8: The coherent mixer

output of the polarizer, polarized light at45^◦between the horizontal (→) and vertical () axes is obtained. Thanks to polarization beam splitters, the PD1 and PD3 photodiodes receive the → polarization whereas the PD2 and PD4 photodiodes receive . As an example, we calculate the signal received by PD1:

Useful signal The signal is reflected and accumulates a π/2 phase shift

E_signal,out(t) =E_{CohM ix}(t)e^jπ/2e^jω^c^t (2.14) Local oscillator The signal of the LO goes through the quarter wavelength plate and is

not reflected:

Eol,out(t) =ALO(t)e^jπ/2e^jω^ol^t (2.15) The output currentI_{P D1} of the photodiode is proportional to the incident power of light.

We define ECohM ix(t) = ACohM ix(t)e^jφ(t)

I_{P D1}(t) ∝ (E_s,out+E_ol,out)(E_s,out+E_ol,out)^∗ (2.16)

= |A_{CohM ix}|²+|A_ol|²+ 2|A_{CohM ix}||A_ol|cos((ω_c−ω_ol)t+φ(t)) (2.17) Doing the same forI_{P D2}, I_{P D3} and I_{P D4} yields:

I1(t) =IP D1(t)−IP D3(t) = 4|ACohM ix||Aol|cos((ωc−ωol)t+φ(t)) (2.18) I₂(t) =I_{P D2}(t)−I_{P D4}(t) = 4|A_{CohM ix}||A_ol|sin((ω_c−ω_ol)t+φ(t)) (2.19) The currentsI₁(t)and I₂(t)represent the in-phase and quadrature components of the incoming signal [29]. The frequency offset between the LO and the signal ω_c −ω_ol is corrected via digital signal processing (see section 3.5). In a practical coherent mixer, the coherent architecture is duplicated as shown in Fig. 2.9. The dual-polarization signal is separated into two polarization components fed into two coherent mixers. In any realistic system, the transmitter and receiver polarization components are not aligned. All polarization effects can however be corrected via digital signal processing (see section 3.4).

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ADC

⊕

⊗

_j

�

MIXER 1

�

ADC

⊕

⊗

_j

�

MIXER 2

�

ADC

Ix,1(t)

Ix,2(t)

Ix,3(t)

Ix,4(t)

Sx,RX[m]

Sy,RX[m]

PBS LO

Signal

ADC

Figure 2.9: Coherent detection for polarization-multiplexed systems

2.5.6 Trade-off between noise and nonlinear distortions

As we will see in the subsequent chapters, many linear transmission impairments in optical fibers can be mitigated using digital signal processing (hereafter abbreviated DSP).

The system performance is ultimately limited only by ASE noise and the nonlinear Kerr effect. When the input power per optical channel is low, the system is limited by noise.

However, a high input power will result in unacceptable performance penalties stemming from nonlinear distortions. As a consequence, a trade-off must be found between the noise-limited and the nonlinear limited regimes as illustrated in Fig. 2.10. The nonlinear threshold is defined as the power resulting in optimum performance. Its value depends on the system architecture but also on DSP (modulation format, presence of a mitigation algorithm for nonlinear effects, etc.).

Limited by

ASE noise

Limited by non linear ef

fects Link input power per channel [dBm]

Q²-factor [dB] NLT

Figure 2.10: Performance as a function of the link input power

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2.6. Mathematical description of the fiber-optic communication system 19

2.6 Mathematical description of the fiber-optic com- munication system

2.6.1 Nonlinear fiber simulation and linear transfer function

In order to calculate the optical field at the output of an optical fiber given an input field, the nonlinear Schrödinger equation Eq. (2.3) must be solved. The most commonly used technique to propagate the optical field is called the split-step Fourier method or SSFM [8]. Its principle is to propagate the field over successive fiber segments or "steps".

In each step, one assumes that the linear terms of Eq. (2.3) (contributions (1), (2) and (3)) can be applied separately from the nonlinear terms (contribution (4)). The SSFM then consists in applying in each step the two operators separately (splitting). FFT and IFFT are used to compute the linear part of the step in frequency domain. Split-step methods are easily accommodated for taking into account polarization effects. The steps are then used to emulate the fiber model of randomly birefringent sections as described in section 2.4.3. In practice, a random Jones matrix J(z) and birefringence parameters

∆β₀(z) and ∆β₁(z) (see Eq. (2.3)) must be drawn for each fiber segment. More details on this computation can be found in [20]. The step size of the SSFM triggers its accuracy and is often optimized empirically, important parameters include the coherence length of the PMD, i.e. the distance over which the principal states of polarization of the fiber stay constant and the nonlinear coefficient γ. In simulations for fiber-optic communications with SSMF, the step size is in the order of 100-200 meters.

In fiber-optic communication systems, the nonlinear effects usually have a significant effect on the transmitted signal because of the ASE noise/Kerr effect trade-off pointed out in section 2.5.6. However, only linear effects are generally mitigated using digital signal processing. It is therefore interesting to write out the solution of Eq. 2.3 ignoring the nonlinear term (4). In this derivation, we will further ignore attenuation (1) and write out the Jones components as A = [A₁ A₂]^T. Over the l-th split-step of length h, where the birefringence parameters can be considered constant, Eq. 2.3 writes out in the frequency domain:

A˜₁(z+h) A˜₂(z+h)

=







H₁₁(z) H₁₂(z) H₂₁(z) H₂₂(z)

| {z }

H[l]





 .

A˜₁(z) A˜₂(z)

(2.20)

Where

H_ii(z) .

=e^j(−^δβ⁰⁽²^z)^+ω^δβ¹⁽²^z)^)Jⁱⁱ^(z)−ω²^β²²^−ω³^β⁶³^)h (2.21) And

H_ik(z) .

=e^j(−^δβ⁰⁽²^z)^+ω^δβ¹⁽²^z)^)J^ik^(z))h i6=k (2.22) The input-output relation can then be obtained by concatenation:

A˜₁(L) A˜₂(L)

= Y

l=1:L/h

H[l].

A˜₁(0) A˜₂(0)

(2.23)

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The fiber impulse response is defined as the inverse Fourier transform of Eq. 2.20.

2.6.2 Mathematical models of the optical communication chain

The object of this section is a mathematical description of the optical transmission link shown in Fig. 2.6.2 where polarization multiplexing is ignored for the sake of simplicity.

If the effects of transmitter, channel and receiver can be assumed linear, their transfer functions can be described as impulse responses [29]. The fiber impulse response has been defined in the preceding section. In a real transmission line, noise is added progressively at each amplification stage and is propagated and amplified along the transmission line.

Here, we use noise loading at the end of the transmission line. This assumption is valid in a linear model where the noise is assumed to be additive, white and Gaussian. Different studies [30, 31] proved this assumption to be verified, so that we systematically adopt it in this work. In Fig. 2.6.2 (b), the noise is represented by n(t), a continuum of white Gaussian noise. Modulator and demodulator impulse responses are called g(t) and p(t) respectively. The analog system model between DAC and ADC operations can now be written out:

z(t) =g(t)⊗(h(t)⊗p(t)⊗s(t) +w(t)) (2.24) Where ⊗ is the continuous convolution operator.

! N

Demodulator

Laser

ADC

Fibre

Laser

Modulator Ampli

DAC

DAC g(t) h(t) p(t) ADC

w(t)

s(t)

⊕

z(t)

g[n'] h[n'] p[n']

w[n']

⊕

↑2 ↓2

s[n] z[n^�]

(a)

(b)

(c)

Figure 2.11: (a) Architecture of the communication chain (b) Continuous input-output model (c) Digital input-output model

We can define the equivalent digital channel model [29] in Fig. 2.6.2 where the operations of pulse shaping, channel and receiver filtering are identified as convolution operations with digital FIR filters g[n⁰], h[n⁰] and p[n⁰] which are sampled version of g(t),

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2.7. Summary 21

h(t)and p(t)respectively at twice the symbol rate. In order to avoid aliasing, the symbol sequences are up (↑ 2)- and down-sampled (↓ 2) by two. w[n⁰] are the digitized noise samples. We can define the input-output model of the communication chain:

z[n⁰] =

∞

X

m=−∞

s[m]q[n⁰−m] +w⁰[n⁰] (2.25) q[n⁰] = (g⊗h⊗p) [n⁰] (2.26) w⁰[n] = (w⊗p) [n⁰] (2.27) (2.28) The digital input-output model is easily generalized to 2×2multiple-input multiple- output for taking into account PDM:

z_i[n⁰] = X

k=1,2

∞

X

m=−∞

s_i[m]q_ik[n⁰−m] +w_i⁰[n⁰] (2.29) qik[n⁰] = (g⊗hik⊗p) [n⁰] (2.30) w⁰_i[n] = (w_i⊗p) [n⁰] (2.31) (2.32) In this equation, z_i[n]is the received signal on polarization i,h_ik[n]is the inverse Fourier transform of H_ik(t) (see preceding section), sampled at twice the symbol rate. w_i[n⁰] is the sampled noise on polarization i. In optical communications, the signal to noise ratio is often identified by theOSNR_(0.1nm) (see section 2.5.4) which is particularly suitable for measurements. In numerical simulations, E_b/N₀, the energy per bit divided by the noise spectral density, is very useful because it takes into account the modulation format. It can be calculated from the relative variances of received signal and noise in Eq. 2.29 and the number of bits per symbol K. By definition, the noise power can be decomposed into (noise spectral density: N₀) × (spectral width of the receiver filter ∆f_s). Similarly, we decompose the signal power into (Energy per symbol: E_s) × (spectral spectral width of the signal ∆f_s). Here, we supposed that the receiver filter and signal have the same bandwidth.

<|z[n]|² >−<|w[n]|² >

K <|w[n]|² = Es∆fs

KN₀∆f_s = Es

KN₀ = Eb

N₀ (2.33)

2.7 Summary

In this chapter, we review the principles of optical telecommunications. We describe the optical fiber as a transmission medium. We detail the main impairments corrupting the signal in the fiber channel: CD, PMD, ASE noise and the nonlinear Kerr effect. We out- line the architecture of a point-to-point long-haul optical telecommunications link using coherent detection. We detail the principle of optical modulation and introduce digital modulation formats. We introduce a mathematical model of the fiber-optic communication link.

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Chapter 3 Digital signal processing in coherent optical communication systems

3.1 Introduction . . . 23 3.2 Time/frequency acquisition . . . 24 3.3 Chromatic dispersion compensation . . . 25 3.4 Polarization demultiplexing . . . 26 3.5 Carrier frequency estimation . . . 27 3.6 Carrier phase estimation . . . 28 3.6.1 Viterbi-Viterbi carrier recovery algorithm . . . 28 3.6.2 Maximum-Likelihood Phase Estimator . . . 28 3.7 Symbol identification and FEC . . . 29 3.8 Summary . . . 30

3.1 Introduction

In optical telecommunications, the development of DSP is closely related to technological advances. Its use in association with coherent detection permitted a tremendous increase in spectral efficiency since the 1990ies [32]. Thanks to coherent detection, both phase and amplitude of the optical signal can be detected (see chapter 2). As a consequence, complex and high-order modulation formats can be used [33]. DSP is needed to correct the transmission impairments for the complex modulation formats. In addition, in order to benefit from polarization multiplexing, cross-polarization DSP is compulsory [34]. Finally, before the advent of DSP in optical communications, dispersion compensating fiber were used to compensate for in-line CD [35]. Thanks to the digital compensation of CD, dispersion-unmanaged (DU) links have become standard use for newly installed links [36].

Today, the performance of DSP benefits from continuously increasing electrical processing capacities.

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24

Chapter 3. Digital signal processing in coherent optical communication systems

g[n']

⊕

p[n']

↑2 z1[n^�] ↓2

g[n']

⊕

p[n']

↑2 ↓2

s2[n]

s1[n]

z2[n^�] w1[n^�]

w2[n^�]

hik[n^�] eik[n^�]

r1[n]

r2[n]

Figure 3.1: Digital transmission model and equalization

Fig. 3.1 presents the digital channel model defined in the preceding chapter for a polarization-multiplexed link. The role of the cross-polarization equalizere_ik is to recover the data symbols s_i[n]. If the channel impulse responses are known, we can compute the equalizer guaranteeing the minimum mean square error betweens_i[n]andr_i[n]. It is most conveniently expressed in matrix form. Be Hik(z) the z-transform of hik[n⁰] and Eik(z) the z-transform of the equalizer. We define MIMO matricesH(z) and E(z):

H(z) =

H11(z) H12(z) H₂₁(z) H₂₂(z)

(3.1)

E(z) =

E₁₁(z) E₁₂(z) E₂₁(z) E₂₂(z)

(3.2) The linear minimum mean square error (MMSE) equalizer is:

E(z) = (H^H(z)H(z) + σ_n²

σ²_sH(z))⁻¹H^H(z) (3.3) In this equation, σ²_n is the noise variance and σ²_s the signal variance.

In principle, Eq. 3.3 solves the question of DSP in optical transmission systems. How- ever, the parameters of several impairments present in these links cannot be known a priori at the receiver. Phase effects vary very rapidly. CD is known and constant but requires very long equalizer filters. This variance in impairment results in a DSP architecture much more complex than shown in Fig. 3.1.

The common configuration of optical coherent receiver associated with DSP algorithms is shown in Fig. 3.2 [37, 38]. Due to the specificity of the linear impairments in optical communications, a block architecture has been widely adopted. Each of the blocks ad- dresses only one linear impairment. The architecture is described in the sections of this chapter.

3.2 Time/frequency acquisition

After passing through the dual-polarization coherent mixer (see section 2.5.5), the electrical signal is first sampled by analog-to-digital converters (ADC). In order to avoid

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3.3. Chromatic dispersion compensation 25

ADC

⊕

⊗_j

�

MIXER 1

�

ADC

⊕

⊗_j

�

MIXER 2

�

PBS ADC LO

Signal

ADC

CD comp

Pol Demux

CFE / CPE

Decision

z1

z₂

r1

r₂ e₁₁ e₁₂ e₂₁ e₂₂

Figure 3.2: Main DSP blocks in the state-of-the-art coherent receiver

aliasing, the sampling frequency must exceed the bandwidth occupancy of the signal [29].

The signal is usually down-sampled to one sample per symbol after equalization of the main channel perturbations. If the sampling rate is a multiple of the symbol rate, no in- terpolation is needed to achieve the conversion. As a consequence, the sampling frequency is often chosen as twice the symbol rate.

3.3 Chromatic dispersion compensation

Chromatic dispersion, as introduced in section 2.4.2, is a static impairment with a well- known expression in the frequency domain (Eq. (2.8)). As can be observed in the nonlinear Schrödinger equation (2.3), it is a polarization-independent phenomenon. Moreover, being directly determined by the length of the link and its type of fiber, the strength of CD is roughly known at the receiver [39]. Moreover, CD compensation represents the most complex algorithm in DSP for coherent optical communications [40,41]. These arguments justify the implementation of an exclusive CD compensation block ("CD comp") as shown in Fig. 3.2. It is the only static block of the DSP suite.

Using Eq. (2.8) and the expression of the dispersion parameter Dfrom Eq. (2.10), the frequency-domain filter to compensate for CD is easily designed [42]:

G(ω) = exp(−jλ²z

4πcω²) (3.4)

In this equation, λ is the central wavelength of the optical signal, c is the speed of light z is the fiber length, and j = √

−1. In practice, the filter Eq. (3.4) is truncated to the effective bandwidth of the signal and a static finite impulse response (FIR) filter is designed based on the truncated response. CD compensation can be implemented in the time domain (TD) or the frequency domain (FD). In a TD implementation, the signal is convolved with a finite impulse response (FIR) filter implementing a truncated inverse Fourier transform of Eq. (3.4) [43]. In a FD implementation the incoming data is transformed with a fast fourier transform (FFT). The result is multiplied with the filter frequency response Eq. (3.4) and the signal is transformed back into TD by inverse fast fourier transform (IFFT). The number of taps of the TD equalizer or the size of the FFT in the FD equalizer is directly proportional to the amount of residual CD [44]. For

Modulation formats and digital signal processing for fiber-optic communications with coherent detection