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POLITECNICO DI TORINO
Master’s thesis
Flex-PAM Modulation
Formats for flexible optical networks
Author: Supervisors: Paula Cortada Prof. Andrea Carena May, 2015 Prof. Vittorio Curri
Abstract
Flex-PAM modulation format is an innovative proposal that offers network flexibility using the very well know PAM modulation. We propose four strategies for Flex-PAM Tx operation, having three of them very similar behavior. Back-to-back simulations verify the theoretical predictions without extra penalties. Uncompensated links are the state of the art nowadays in coherent optical communications systems. Analysis of nonlinear propagation is done using GN-model as a reference prediction, which experiences a limited extra penalty for unbalanced power, inherent in this format. A proper power ratio tuning enables better performance reaching the GN-model bound.
Contents Introduction .......................................................................................................................... 1
1. Optical Fibers ............................................................................................................... 3
1.1 Electromagnetic waves .............................................................................................. 3
1.2 Geometric description ................................................................................................ 3
1.3 Propagation modes ..................................................................................................... 5
1.4 Fiber effects: .............................................................................................................. 6
2. Coherent Systems ....................................................................................................... 10
2.1 Evolution from IIMD to coherent systems .............................................................. 10
2.2 Modulator technologies ........................................................................................... 11
2.3 I-Q modulator ........................................................................................................... 12
2.4 Coherent detection ................................................................................................... 13
2.5 Coherent receiver scheme ........................................................................................ 14
2.6 PM-QPSK example .................................................................................................. 16
3. Flexible-PAM Modulation Formats ........................................................................... 18
3.1 Pulse-Amplitude Modulation (PAM) ...................................................................... 18
3.2 Flexible-PAM Modulation Formats ......................................................................... 21
3.3 Strategies for Transmitter Operation ....................................................................... 24
3.4 Best Combinations ................................................................................................... 25
4. The Gaussian Noise Model ......................................................................................... 28
4.1 Uncompensated Transmission (UT) ........................................................................ 28
4.2 FWM-like effects ..................................................................................................... 29
4.3 The GN-model reference formula: ........................................................................... 31
4.4 WDM-channel spectrum: ......................................................................................... 32
4.5 Observations ............................................................................................................ 34
5. Back-to-back Simulation Performance ....................................................................... 35
5.1 Theoretical analysis of power strategies .................................................................. 35
5.2 Simulation Setup Parameters and Procedure ........................................................... 39
5.3 Back-to-back simulation results ............................................................................... 41
6. Non-linear Propagation Simulation ............................................................................ 47
6.1 System Setup ............................................................................................................ 47
6.2 Maximizing the number of spans ............................................................................. 49
6.3 Comparison between Model and Simulation Results for “Same-BER” strategy .... 50
6.4 Pre-distortion technique ........................................................................................... 54
6.5 Conclusions of Pre-distortion in Flex-PAM modulation systems ........................... 56
7. Conclusions and Future work ..................................................................................... 57
7.1 Tuning PR Test for BpS=6 (Rb=150 Gbit/s) bps=[1212]........................................ 57
Bibliography ...................................................................................................................... 59
Introduction Since optical networks goes towards the direction of maximum capacity and flexibility, it is very important for transceivers to be able to maximize spectral efficiency (SE)
SE = BpS ⋅ Rs ∆f by adapting to the actual conditions of the network and data rate for each given traffic demand [9]. To simplify transceiver implementation it is convenient to keep both the channel spectral
∆f and symbol rate
Rs as constants [6], so we can only vary BpS by changing the modulation format. Standard multilevel PM-mQAM modulation formats do not satisfy flexibility demand, being BpS limited to a few values. One possible solution is Time-Division Hybrid Modulation Formats (TDHMF) [6]. They operate mixing different PM-mQAM formats in time domain in order to achieve flexibility, which requires less hardware cost than the code-rate-variable FEC [19]. Using TDHMF, any SE that falls between the SE of the two regular QAMs can be realized easily by appropriately designing the frame length and its cardinality, allowing huge granularity [18]. Gaining flexibility requires an increase of the transceiver complexity, particularly in de-multiplexing the two formats. We propose a new solution: Flexible-PAM modulation format (Flex-PAM), which uses four different M-PAM modulation formats, one for each quadrature of the optical field [in-phase and quadrature components for both polarizations: X and Y]. Although, Flex-PAM can only work at integer
BpS , with a reduction of flexibility with respect to TDHMF, optical networks based on it will benefit for two main reasons:
1. From a networking point of view, each dimension can be assigned to a specific tributary and select the M-PAM level based on the traffic request, which leads to an independency between dimensions
2. It is simpler in the Tx/Rx structure and DSP compared to TDHMF because does
not vary the format in time. The transmitter structure of Flex-PAM modulation is the same as for standard PM-mQAM formats, which is based on two classic I-Q modulators.
Firs purpose of this work is to define the possible strategies for the operation of Flex-PAM. We ran then an optimization process setting down the best combination and power strategies. The back-to-back bit error rate (BER) performance vs. signal-to-noise ratio (SNR) is simulated. We chose the most adequate strategy for real applications and we tested its propagation along the fiber.
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In order to predict the maximum transmission distance, we based on the Gaussian-Noise model for nonlinear interference, considering uncompensated uniform links. This work is organized as follows: Chapter 1 is a brief review of the basic concepts of the optical fiber. Chapter 2 compares the traditional detection scheme vs. the coherent technology as well as an explanation of the revolutionary PM-QPSK modulation format. Chapter 3 explain what is the Flex-PAM modulation and defines four power strategies. Chapter 4 introduces the basis of one of the most used model predictions of maximum reach, as well as in this work, i.e., GN-model. Chapter 5 is the back-to-back simulation performance and in chapter 6 the propagation simulation results are analyzed. Finally, in Chapter 7 conclusions on achieved results and a new proposal for future works is reported.
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1. Optical Fibers
1.1 Electromagnetic waves Light is an electromagnetic field whose evolution can be written as:
E(t,z) = Ao ⋅ cos(2πfot − koz −φo)
Where
Ao is the amplitude,
fo the frequency,
φo the initial phase and
ko =2πλo
is the
propagation constant.
Figure 1.1: Propagation of an electromagnetic wave
The polarization is the evolution of the electric field in the xy, the transversal plane with respect to the propagation direction.
E (t,z) = Ax ⋅ cos(2πfot − koz −φx ) ˆ x + Ay ⋅ cos(2πfot − koz −φy ) ˆ y The shape and orientation defines the polarization state, which can be circular, linear and elliptic.
Figure 1.2: Plot of the evolution in the direction of propagation of the Ex and Ey
components
1.2 Geometric description Definition: An optical fiber is a very thin cylindrical glass (SiO2) wavelength consisting of two differential parts: the core material and the cladding material.
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The core and cladding are designed so as to keep the light signals guided inside the fiber, allowing the light signal to be transmitted for reasonably long distances before the signal degrades in quality. Both the core and the cladding are made primarily of silica, which has a refractive index of approximately 1.45. The refractive index of a material is the ratio of the speed of light in a vacuum to the speed of light in that material. The refractive index is slightly higher in the core than in the cladding [20].
Figure 1.3: Geometrical form of fiber
Types: STEP INDEX AND GRADED INDEX We can difference between two kinds of optical fiber, “step-index fiber” in which there is an abrupt index change at the core-cladding interference and the “graded-index fiber” in which the refractive index decreases gradually inside the fiber core. In figure 1.4 the index profile and the core section for the two kinds of fiber are represented.
Figure 1.4: Cross section and refractive index profile for step-index fiber and grades-
index We will focus on the step-index fiber because it is the most used in optical communications systems [21]. STEP-INDEX FIBERS: the most common optical fibers type. They are characterized by an abrupt change in the refractive index between core and the cladding
The relative refractive index difference:
∆ =n1 − n2
n1
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A light beam incised at the core center making an angle
φi with the fiber axis. Because of refraction at the fiber-air interference, the beam bends toward the normal. The angle of refracted beam is
φr :
no ⋅sin(φi) = n1 ⋅sin(φr ) (Snell law) Where
n1 and
no are the refractive index of the fiber core and air, respectively. The refracted beam hits the core-cladding interface and is refracted again. However, refraction is possible only for an angle of incidence
φi such that
sin(φi) < n2 n1 (where
n2 is the cladding index). The critical angle is defined as the angle that produces the total internal reflection at the core-cladding interface of the beam:
sin(φc ) = n2 n1 .
Figure 1.5: Light beams propagation inside the fiber
1.3 Propagation modes The electromagnetic field propagates in an optical fiber according to “modes”. To each mode corresponds a specific field in the cross section. A simpler interpretation: light consists of a number of rays. A mode then is defined as a ray that propagates along the fiber with a specific angle. Under particular conditions, only a single mode propagates along the fiber, parallel to the fiber axes: SINGLE MODE FIBERS (the most used in telecommunications networks).
Figure 1.6: Multi-mode fiber and single-mode fiber Typical multimode fiber has a core much larger than a wavelength of light. Multimode fiber carries hundreds of modes, which can be thought of as independency propagating paths of the optical signal. The signal consists in multiple light rays, each one with a different path through the fiber: MULTI-MODE FIBER. Each path corresponds to a propagation mode. Each mode therefore travels with a slightly different speed compared to the other modes, resulting in intermodal dispersion. This leads to the overlap of pulses representing adjacent bits, distorting the signal. This phenomenon is called Inter-Symbol Interference (ISI).
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Single-mode fiber has a core on the same scale as a wavelength that restricts itself to a single “fundamental” spatial core. SMF is used for the highest bandwidth and longest distance transmission. The cutoff condition:
V =2πλ
a n12 − n2
2 < 2.405
1.4 Fiber effects: Fiber losses: The loss incurred by propagating down a fiber can be modeled easily as follows: the output power Pout at the end of a fiber of a length L is related to the input power Pin by
Pout = Pine−αL , where the parameter alpha represents the fiber attenuation. Usually the loss is expressed in units of dB/km. The two main loss mechanisms in an optical fiber are material absorption and Rayleigh scattering. Fiber losses represent a limiting factor because they reduce the signal power reaching the receiver. As optical receivers need a certain minimum amount of power for recovering the signal accurately, the transmitted distance is inherently limited by fiber loss. Fiber losses depends on the wavelength of transmitted signals: There are three windows:
1st: 800 nm, alpha=2.5 db/km 2nd: 1330 nm, alpha=0.4 db/km 3rd: 1150 nm, alpha=0.22 db/km
Figure 1.7: Attenuation loss in silica as a function of wavelength
• Material absorption includes absorption by silica as well as the impurities in the
fiber. The material absorption of pure silica is negligible in the entire 800-1600 nm band that is used for optical communication systems. The reduction of the loss due to material absorption by the impurities in silica has been very important in making optical fiber the remarkable communication medium that it is today. The loss has now been reduced to negligible levels at the wavelengths of interest for optical communication (so much so that the loss due to Rayleigh scattering is the dominant component in today’s fibers in all three wavelength bands used for optical communication:
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Chromatic Dispersion: The general concept of dispersion lies in any effect wherein different components of the transmitted signal travel at different velocities in the fiber, arriving at different times at the receiver, and consequently distorting the signal. We will study chromatic dispersion in SMF, and it depends on two main effects.
1. MATERIAL DISPERSION: The refractive index of silica is frequency dependent. Thus different frequency components travel at different speeds. 2. WAVGUIDE DISPERSION: It relies on two different phenomenon. First, the light energy of a mode propagates partially in the core and partially in the cladding. Second, the effective index of a mode lies between the refractive index of the core and of the cladding. Consequently, the actual value of the effective index between these two limits depends on the proportion of power contained in both of them. So, the most power contained in one of them, the most effective index is closer to that index of refraction. As also power depends on the wavelength it produces that if the wavelength changes, the power distribution change and the effective index change.
The shape of the pulses propagating in optical fiber is not preserved. The parameter
governing the evolution of pulse shape is
β2 =d 2βd 2w
: GROUP VELOCITY DISPERSION
PARAMETER. If
β2 > 0 CD is said to be normal, instead, if
β2 < 0 is said to be anomalous.
The CD parameter is related to the GVD parameter as:
D = −2πcλ2
β2 in
psnm ⋅ km
and
expresses the temporal spread per unit propagation distance, per unit pulse spectrum width. Polarization Mode Dispersion: The cause of polarization mode dispersion lies behind the Birefringence concept. In real fibers there is a variation in the shape of the cylindrical core along the fiber length. They may also experience non-uniform stress. Degeneracy between the orthogonally polarized fiber modes is removed because of these factors, and the fiber acquires birefringence. The modal birefringence is defined by
Bm = nx − ny , where
nx and
ny are the mode indices for the orthogonally polarized fiber modes. Birefringence leads to a periodic power exchange between the two polarization
components. The period is called beat length and defined by
LB =λ
Bm.
Its state of polarization changes along the fiber length from linear to elliptical and then back to linear, as
LB . Figure 1.9 shows such period change in the state of polarization for a fiber of constant birefringence B.
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Figure 1.8: Propagation of an optical pulse in a fiber with constant birefringence
In conventional SMF, birefringence is not constant along the fiber but changes randomly, both in magnitude and direction, because of variations in the core shape and the anisotropic stress acting on the core. As a result, light launched into the fiber with linear polarization quickly reaches a state of arbitrary polarization. Moreover, different frequency components acquire different polarization states, resulting in pulse broadening. This effect is called PMD.
Figure 1.9: Propagation of a pulse in a fiber with random birefringence
Nonlinear effects: The nonlinear effect arises due to the dependence of the refractive index on the intensity of the applied electric field, through the Kerr effect. Fiber nonlinearities are generated by the Kerr effect, which produce a change in the refractive index of the fiber due to the variation of the signal power. The following expression represents that effect as:
n(z, t) = nL + n2P(z, t)Aeff
Where
nL is the refractive index,
n2 is the nonlinear index coefficient, which depends on the material,
P(z, t) is the optical power, varying with distance according to the attenuation of the fiber and with the time according to the modulation. The effective area
Aeff can be approximate by the area of the fiber core as
Aeff ≅ π ⋅ r 2, where
r is the fiber core radius. The most important nonlinear effects in this category are self-phase modulation (SPM) and four-wave mixing (FWM).
• Self-Phase Modulation (SPM):
SPM arises because the refractive index of the fiber has an intensity-dependent component. This nonlinear refractive index causes an induced phase shift that is proportional to the intensity of the pulse. Thus different parts of the pulse undergo different phase shifts, which gives rise to chirping of the pulses. Pulse chirping in turn enhances the pulse-broadening effects of chromatic dispersion. This chirping effect is proportional to the transmitted signal power so that SPM effects are more pronounced in systems using high power transmitted.
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The SPM-induced chirp affects the pulse broadening effects of chromatic dispersion and thus is important to consider for high-bit-rate systems that already have significant chromatic dispersion limitations.
• Four-Wave Mixing (FWM): In WDM systems using the angular frequencies ω1,...,ωn, the intensity dependence of the refractive index not only induces phase shifts within a channel but also gives rise to signals at new frequencies such as 2ωi − ωj and ωi + ωj − ωk. This is the Four-Wave mixing effect and it is independent of the bit rate but is critically dependent on the channel spacing and fiber chromatic dispersion. Decreasing the channel spacing increases the four-wave mixing effect, and so does decreasing the chromatic dispersion. Thus the effects of FWM must be considered even for moderate-bit-rate systems when the channels are closely spaced and/or dispersion-shifted fibers are used.
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2. Coherent Systems
2.1 Evolution from IIMD to coherent systems Since 1970s, IIMD has been commonly employed in current optical communication systems. Such systems used intensity modulation of semiconductor lasers, and the transmitted optical signal intensity was detected by a photodiode. In 1980s, coherent optical receivers were deeply investigated, allowing multilevel modulation formats. But the technical difficulties inherent in coherent receivers and the invention of EDFA made an interruption of coherent research for 20 years. The invention of EDFA made the shot noise limited receiver sensitivity of the coherent receiver less significant. The EDFA-based system started to take benefit from WDM techniques to increase the transmission capacity of a single mode fiber. For this reason, coherent technologies have restarted to attract interest over the last years. Nowadays however, coherent systems come again into the focus of interest, due to the recent development of high-speed digital integrated circuits, offering the possibility of treating the electrical signal in digital signal processing (DSP). The digital coherent receiver allows the detection of in-phase and quadrature components of both polarizations, and therefore doubles the capacity. It detects the entire optical field in the digital domain, i.e. is both amplitude and phase, which allows the signal to be processed by DSP algorithms. The most important advantage of coherent detection is the ability to detect higher-order modulation formats (PM-mQAM family), which use IQ modulators to access the in-phase and quadrature components in both polarizations [2]. The preservation of the temporal phase enables new methods for adaptive electronic compensation of chromatic dispersion. When concerning WDM systems, coherent receivers offer tunability and allow channel separation via steep electrical filtering. Furthermore, only the use of coherent detection permits to converge to the ultimate limits of spectral efficiency. To reach higher spectral efficiencies the use of multi-level modulation is required. Concerning this matter coherent systems are also beneficial, because all the information of the optical field is available in the electrical domain [13].
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2.2 Modulator technologies
Three basic modulator technologies are widely in use today: directly modulated lasers (DML) and Mach-Zehnder Modulators (MZM). 1. DMLs:
Direct modulation of lasers is the easiest way to imprint data on an optical carrier. Here, the transmit data is modulated onto the laser drive current, which then switches on and off the light emerging from the laser. The resulting modulation format is binary intensity modulation (OOK). The power of the optical signal is used to encode digital information. Today, DMLs are widely available up to modulation speeds of 2.5 Gbit/s, with some limited availability at 10 Gb/s and research demonstrations up to 40 Gb/s. The main drawback of DMLs for high-bitrate transmission beyond short-reach access applications is their inherent, highly component-specific chirp, i.e., a residual phase modulation accompanying the desired intensity modulation; laser chirp broadens the optical spectrum, which impedes dense WDM channel packing and can lead to increased signal distortions caused by the interaction with fiber CD [17].
2. MZMs:
Are based on the same principle of MZ filters: Constructive or Destructive interference between two different optical paths. The realization is based on Lithium niobate properties. Electro-optic effect: refractive index depends on the applied electrical field. In the push-pull configuration, the output signal is amplitude modulated without any spurious phase/frequency modulation: chirp free.
Figure 2.1: MZM principle of operation.
MZM is formed by cascading two couplers and delay interferometers in between. Performs the sum of the input signal with a phase modulated copy of itself. Phase and intensity are modulated as response to a voltage signal. Asymmetric MZ interferometer can be viewed as a filter that converts phase modulation into amplitude modulation, which is then detected by the photodiodes. In the push-pull operation 𝐸𝐸𝑜𝑜𝑜𝑜𝑜𝑜 = 𝐸𝐸𝑖𝑖𝑖𝑖 · cos (𝑘𝑘𝑘𝑘) and 𝑃𝑃𝑜𝑜𝑜𝑜𝑜𝑜 = 𝑃𝑃𝑖𝑖𝑖𝑖 · cos (𝑘𝑘𝑘𝑘)2 ,the output signal is amplitude modulated without any spurious phase/frequency modulation: it is chirp free. MZMs are the most widely used modulators in high performance transmission systems [3].
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2.3 I-Q modulator
Using a nested MZM structure together with a Phase Modulator, a general I-Q modulator can be developed and used for the implementation of multilevel modulation formats in optical communications.
Light is an electric field that can be expressed
E(t) = A(t)e jϕ(t ) ⋅e jwot Being 𝐴𝐴(𝑡𝑡) the amplitude, 𝜑𝜑(𝑡𝑡) the phase and 𝑤𝑤𝑜𝑜 the optical carrier. Neglecting the carrier:
E(t) = A(t)e jϕ(t ) = ER (t) + jEI (t) .
Figure 2.2: Optical IQ modulator
The modulator follows the principles of an external modulator, using a CW laser as an optical source. Also contains two Mach-Zehnder modulators (MZMs) integrated in two different arms. In one arm is generated the in-phase component EIx(t) and in the other is carried the quadrature part EQx(t). Both components are separated by 90 degrees using an optical phase modulator. Finally both signal components are sum by an optical combiner obtaining the QPSK signal.
Using Polarization Multiplexing we can use both polarizations to carry independent multilevel modulation formats. It doubles the capacity: BpS’=2·BpS The fiber carries two independent electrical fields: they travel onto orthogonal
polarizations:
E(t) = [ERX(t) + jEI X
(t)] x∧+ [ERY
(t) + jEIY(t)] y
∧
PM-mQAM modulator is composed of two IQ modulators. Both mQAM polarizations generated are added by a Polarization Beam Combiner (PBC), but first one of the polarizations needs to be rotated through a Polarization Rotator to obtain orthogonally between both electromagnetic fields.
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Figure 2.3: PM-mQAM modulator
2.4 Coherent detection
Single-photodiode converts the input optical power into an output current. All phase and polarization information is lost: i(t) is proportional to P(t). IIMD: For all the modulation formats based on the intensity modulation, the optimal receiver (ASE limited system) is the one based on the optical matched filter. The simple direct detection receivers are limited by thermal noise and do not achieve the shot noise limited sensitivities of ideal receivers. Sensitivity could be improved significantly by using an optical pre-amplifier. Another way to improve the receiver sensitivity is using coherent detection [11]. The basic idea behind coherent detection consists of combining the optical signal coherently with a continuous wave (CW) optical field before it falls on the photo-detector. The CW field is generated locally at the receiver using a laser: Local Oscillator (LO) [1]. The signal and the local oscillator beams must be coherent in order to recover the information contained in the phase of the signal beam. The dominant noise at the receiver becomes the shot noise due to the LO, allowing the receiver to achieve the shot noise limited sensitivity.
Figure 2.4: Local oscillator
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2.5 Coherent receiver scheme The transmitted signal interferes inside an optical hybrid with a LO signal, converting both quadratures of X and Y polarization into the electrical domain. Balanced photo-detection is often employed, a hybrid is used and the pair of output is then differentially amplified to eliminate direct-detection components in the signal. The signal is digitized by the use of analog-to-digital converters (ADCs) [2]. Parts:
• Local oscillator (LO): a signal aligned with that component is added. • Two 90 degree Hybrid: creates two copies of the LO, then it adds them to the
signal. This guarantees that we are detecting two orthogonal components • Receivers need an active polarization control (pol. Adjust) • Four balanced photo-detector8 photodiodes
Figure 2.5: Coherent receiver for PM-mQAM modulation formats
Digital Signal Processing (DSP): There is no guarantee that the phase and polarization axes of such 4 components coincide with those transmitted, because of the fluctuations of light state of polarization (SOP). The performance of DSP equipment has improved over last years, and fast ADC makes possible to re-align the reference axes to the transmitted ones, and decode the signal correctly. After the electro-optical front-end, the signal has become in four electrical currents, which are digitally converted with a high-speed analog-to-digital converter (ADC). The mitigation of transmission impairments and equalization is carried out on a real time digital signal processing (DSP). After the DSP, the original electrical wave is obtained.
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Figure 2.6: Coherent receiver structure
The aim of DSP for optical communications is to process the transmitted digital data (in the electrical domain) so as to correctly detect it, ideally compensating or at least mitigating all impairments. DSP main goals are Digital filtering of transmission impairments: Phase and polarization reference recovery. Also the optical channel is affected by several amounts of the following linear effects:
• Polarization mode dispersion (PMD) • Chromatic dispersion (CD)
Figure 2.7: Digital signal processing concept form [10]
The progress in the increased performance, speed and reliability with reduction in size and cost of integrated circuits now makes DSP an attractive approach to recover the information from the base-band signal. The DSP circuit must operate the sequence operations of the figure 2.7 to retrieve the information from the modulated signal. Equalization: Multiple-input-multiple output (MIMO) systems are widely used to combat multi-path propagation effects such as PMD. This is sometimes referred to as butterfly structure. It can be implemented by four FIR filters, the filters have adaptive taps, due to the time-dependent nature of PMD. Generally, an adaptive equalizer tries to estimate the Jones matrix of the channel and apply the inverse of it to the signal. To adapt the taps to the changing channel conditions we used the LMS algorithm. The equalizer tries to minimize the amplitudes of the error signals on both polarizations [11].
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2.6 PM-QPSK example
• QPSK modulation: The quaternary-phase-shift keying (QPSK) is a form of angular digital modulation at constant amplitude. Is the M=4 case for MPSK schemes and include 4 possible phases having a constant frequency carrier. Due to 4 different output phases, QPSK need 4 different inputs to codify. 2 bits generate a symbol group of cardinality 4 that are: 00, 01, 10 and 11. QPSK uses a constellation of four points in a circumference and thanks to the use of gray coding the amplitude of the signal never goes to zero. The constellation points are described by the equation: E(t) = A · I(t) + A · Q(t), where I(t) and Q(t) are called in-phase and quadrature components respectively: I(t)=cosθi ·cosωct and Q(t) = sin θi · sin ωct, where θi is the phase for each symbol.
Figure 2.8: QPSK constellation
• PM-QPSK: Taking the advantage of the Polarization Multiplexing, we can use two QPSKs and double the spectral efficiency.
Figure 2.9: PM-QPSK constellation
The solution must be search in multilevel modulation format, looking to transmit more bits in each pulse. The problem now translates to the OSNR requirement and implementation complexity of each multilevel modulation format. High values of OSNR implies higher power values and consequently it will collapse our transmission performance due to non-linear effects. On the other hand it is important to study if such multilevel modulation format will be possible to implement without waste its performance due to implementation impairments. The study of the first key aspect it can be done through an inspection of the Shannon bound for each modulation format.
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Figure 2.10: Spectral efficiency vs. OSNR over 0.5 nm
In blue is shown several kinds of modulation formats, and in red it can be seen polarization multiplexing (PolMux). PolMux formats have the advantage of double the spectral efficiency. The modulation formats near to his corresponding Shannon bound has the best spectral efficiency. And the modulation formats with lower OSNR has the feature of longer efficient transmission distances. Then we are looking for a trade-off between both characteristics.
Conclusions: Finally PM-QPSK modulation format highlights as the better trade-off between OSNR demand and spectral efficiency. A DSP-based implementation allows getting close to such limits, thanks to mitigation through post-processing. This format is very near to his corresponding Shannon bound, allowing it to carry 4 bits per pulse, producing a considerable reduction on symbol rate. Also his OSNR requirement is very low, to obtain a Bit-Error-Rate (BER) of 10−3 only 6.5dB over 0.5nm allowing to increase transmission distances over 3000 km.
PM-QPSK uses efficiently all four available degrees of freedom and this is why it performs so well on the Shannon “drawing board”. All pulses have the same powerthe nonlinearities that depend on abrupt power changes are mitigate. Its very good sensitivity allows transmission at reduced launched power. PM-QPSK is a top-performer from the viewpoint of Shannon’s drawing board. PMD and CD can be compensated electronically 100%. PM-QPSK is resilient to non-linear effects and 100% EDC is competitive with optical dispersion managements.
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3. Flexible-PAM Modulation Formats
3.1 Pulse-Amplitude Modulation (PAM) PAM is a base-band digital modulation format, where the information is encoded in the
amplitude of the pulse as:
s(t) = a[n]p(t − nTs)n =−∞
∞
∑ .
Where
a n[ ] is the symbol, and the pulse
p(t) is a Nyquist filter, particularly has been employed a square-root raised cosine filter, in order to satisfy the Nyquist criteria (no ISI).
Figure 3.1: Block diagram of a digital transmission based on PAM modulation system
with AWGN
Every bit time
Tb it is transmitted 1 bit of a sequence
b n[ ]. This sequence is formed by ‘0’s and ‘1’s with same probability of being transmitted, with a bit rate
rb = 1 T b . The cardinality of the modulation format is defined as
M = 2BpS , where
BpS = log2 (M ) is the number of bits per symbol. For instance, if the number of bits per symbol is
BpS = 3, the correspond number of symbols is
M = 8. The symbol is transmitted every symbol period
Ts, expressed in seconds, with a symbol rate
rs = rb BpS = 1 Ts . The alphabet defines all the possible signals of a certain modulation:
sm (t) ∈ {s1(t),...,sM (t)} assuming
Pr{sm (t)} = Pr{sm } = 1 M , i.e., the transmitted symbols are equally likely. The average
symbol energy follows this expression
Es = Pr{sm } ⋅ Em =1M
m =1
M
∑ sm2 (t)dt
−∞
∞
∫m =1
M
∑ and the average
bit energy is
Eb =Es
BpS.
In figure 1 it is considered that power received is exactly the power transmitted
Prx = Ptx . The signals of the alphabet can be represented as vectors, through the space of the signal. This is to find a set of functions, which form an orthonormal basis generator of signals to be transmitted. In that way, it is convert a temporal signal
sm (t) into a vector signal
sm .you can use such representation. For the particular case of transmitting an M-PAM modulation the parameters are:
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• Pulse symbols:
sm =2m − M −1
2d
• Average Energy symbol:
Es =1M
(sm )2
m =1
M
∑ =M 2 −1
12d 2
• Average Energy bit:
Eb =M 2 −112BpS
d 2
Where
d represents the minimum distance between symbols in their geometric representation, and it is related with the average bit energy and it is used a as a parameter of reference.
Example of 4-PAM case
Figure 3.2: 4-PAM Constellation
The coordinates of the space signal for a 4-PAM modulation expressed in terms of distance are:
s4 = −s1 =3d2
,
s3 = −s2 =d2
.
The average symbol energy is
Es =14
2 ⋅d2
2
+ 2 ⋅3d2
= 5 d
2
2
, and with the relation
Eb = Es BpS , the average bit energy is
Eb =52
d2
2
.
The coordinates can be then represented as
s4 = −s1 =185
Eb ,
s3 = −s2 =25
Eb .
Assumption: Bit Error Rate (BER) is analysed considering an ideal channel with Additive-White Gaussian Noise (AWGN) and MAP criteria is used in the detection, assuming equally likely symbols.
• Symbol Error Probability:
Pe =2M − 2
MQ( d
2σ) =
2M − 2M
Q( 6BpSM 2 −1
EbN o
) , analysis of the
error probability in terms of the average bit energy over the noise PSD (
Eb N o ) o Variance noise:
σ 2 =N o2
o Noise Power:
PN =N o2
⋅ Rs
o Average bit energy of an M-PAM modulation
Eb =M 2 −112BpS
d 2
o Q-function related to the complementary error function is
Q(x) =12
erfc( x2
)
• Symbol Error Probability:
Pe =(M −1)
Merfc( 3BpS
M 2 −1EbN o
)
19
• Signal to Noise Ratio:
SNR =PsPN
=Ps
N o2
Rs
=2 ⋅ EsN o
=2EbN o
⋅ BpS
Where the relation between power and energy is
Ps = Es ⋅ Rs and the relation between energy bit and energy symbol is
Es = Eb ⋅ BpS , had been used. Considering:
• Gray coding: the difference of two adjacent symbols is just in one bit.
• The quotient is supposed to be
EbN o
>> 1, so when an error occurs, it happens just
between two adjacent symbols.
This leads to the conclusion that one symbol error generates one bit error:
BER =Pe
BpS
• Bit Error Rate M-PAM:
BER =(M −1)M ⋅ BpS
erfc( 3M 2 −1
SNR2
) , analysis of the error
probability in terms of the signal-to-noise ratio. The BER for M=2,4,8 is plotted in figure 3.
Figure 3.3: BER over SNR of an M-PAM modulation
Figure 3 shows the dependence on the M level on the BER curve. For a particular target BER (BERt), the sensitivity required (SNRt) is higher for the highest M level. As instance, for BERt=-3 dB, for M=2 the SNRt=9.8 dB, for M=4 is 16.55 dB and for M=8 is 22.5 dB, which is approximately a difference of 6 dB between Mi levels.
20
3.2 Flexible-PAM Modulation Formats
We propose to use an advanced modulation format called Flex-PAM. It is based on the use of four independent and different M-PAM formats transmitted at the same time. Channel spacing and symbol rate are considered constants for all dimensions. With the advantage of coherent detection systems, we can detect information on all quadratures of the optical field, but usually it is used QAM on each polarization. Our proposal is to use one M-PAM for each of the dimensions of the optical field (in-phase and quadrature on both polarizations). The transmitter used to generate the Flex-PAM follows the same structure as the standard PM-mQAM modulation formats (see chapter 2), using two I-Q modulators, one for each polarization, based on nested Mach-Zehnder. The addition of flexibility while keeping the TX structure simple, makes this new modulation format very useful. Some constellation examples are showed in Figure 4,5 and 6.
BpS=4, bps=[1 1 1 1]
Figure 3.4: Flex-PAM BpS=4 Constellation
BpS=5, bps=[1 1 1 2]
Figure 3.5: Flex-PAM BpS=5 Constellation
BpS=6, bps=[1 2 2 1]
Figure 3.6: Flex-PAM BpS=6 Constellation
21
In order to prove the quality of the system, the analytical expression of the overall bit-error rate
BERF −PAM over the total signal-to-noise ratio
SNRF −PAM , for a Flex-PAM symbol is
BERF −PAM = Φ(M i ,Pi ,SNRF −PAM ) , is required. We can express the parameters related to the system in vectors of four components:
•
M = M1M 2M 3M 4 ][ :
M i defines the cardinality •
bps = bps1bps2bps3bps4 ][ :
bpsi is the number of bit-per-symbol •
P = P1P2P3P4 ][ :
Pi defines the individual average power
•
BER = BERM iBERM 2
BERM 3BERM 4 ][ :
BERM iis the partial bit-error rate
•
SNR = [SNR1SNR2SNR3SNR4 ] :
SNRi is the partial signal-to-noise ratio
Figure 3.7: Block diagram of a digital transmission of a Flex-PAM modulation system with
AWGN Where the power received is equal as the power transmitted
Prx = Pi and the noise power is split in the four dimensions
PNi=
N o2
Rs .
Analysis for one dimension
• Symbol Error Probability Mi-PAM:
PeM i=
(M i −1)M i
erfc( 3bpsi
M i2 −1
EbiN o
)
Where
M i is the number of symbols, the number of bits-per-symbol is
bpsi = log2 (M i) and
the quotient
EbiN o
express the relation between the average bit energy over the spectral
density of noise. • The signal-to-noise ratio is
SNRi =Pi
PNi
=Pi
N o2
Rs
[1]
Where
Pi is the signal power and
PNi is the noise power of one dimension, assuming the
relation
Rs = 1/Ts and
Esi = Pi ⋅Ts , we obtain
SNRi = 2 EsiN o
in terms of energy.
22
The following relation let us express the
BERM i in terms of
SNRi :
EbiN o
=EsiN o
1bpsi
=SNRi2bpsi
through the relation between the average bit energy and the average symbol energy.
• Bit Error Rate Mi-PAM:
BERM i=
PeM i
bpsi=
(M i −1)(M i ⋅bpsi)
erfc( 3M i
2 −1SNRi
2) [2]
Analysis for all dimensions
• Number of Bits per Symbol Mi-Flex-PAM:
BpS = log2 (M i)i=1
4
∑
• Total Signal to Noise Ratio:
SNRFlex −PAM =
Pii=1
4
∑
PNi
i=1
4
∑=
Pii=1
4
∑N o2
Rsi=1
4
∑=
Pii=1
4
∑2N oRs
[3]
o
Ptot = Pii=1
4
∑
SNRFlex −PAM =Ptot
2N oRs [4]
Where
Ptot is the total power of the symbol, i.e., the sum of the power over the four quadratures.
• Relation between
SNRFlex −PAM and
SNRi :
o Substituting
N oRs from [4] on [1] it is obtained
SNRi = 4 pi × SNRFlex −PAM [5].
Where the fraction power
pi =Pi
Ptot is the fraction of power transmitted per
dimension respect the total power of the system.
o Substituting
PiN oRs
from [1] to [3] it is obtained another relation as
SNRFlex −PAM =12
SNRi2
i=1
4
∑ =14
SNRii=1
4
∑ [6]
• Bit Error Rate Mi-Flex PAM:
BERFlex −PAM =
BerMi ⋅bpsii=1
4
∑BpS
=
1BpS
(Mi −1)(Mi)
erfc( 6Mi 2 −1
pi ⋅ SNRFlex −PAMi=1
4
∑ ) [7]
o Substituting [5] on the
BERMi [2] expression we obtained the relation with the
BERFlex −PAM over
SNRFlex −PAM .
Demonstration of the
BERFlex −PAM expression:
23
BERFlex −PAM =# errors
# bits=
eii=1
4
∑
Nbitii=1
4
∑=
eiNbiti
=i=1
4
∑ eibpsi ⋅ Nsimi
=ei
Nsimibpsi
=i=1
4
∑i=1
4
∑Pe Mi
i=1
4
∑BpS
=
BERMi ⋅bpsii=1
4
∑BpS
Where
N bit is the number of bits per symbol on the dimension-i and is expressed as
Nbiti = Nsimi ⋅bpsi , and
Pe Mi =# errors
# symbols.
Comment: There is a relation between optical signal-to-noise ratio and signal-to-noise-ratio as
OSNR =Pamp
PN=
P2N oBN
OSNR =P
2N oBN⋅
RsRs
= SNR ⋅RsBN
. In this work, we consider the
optical signal-to-noise ratio OSNR as referred to the noise bandwidth
BN = Rs, reason why we express the
BER in terms of
SNR .
3.3 Strategies for Transmitter Operation Flex-PAM has a degree of freedom in transceiver design with respect to standard modulation formats, i.e. the need of settle down the individual power level Pi for each dimension. Once the Mi levels are chosen, the
BpS is consequently fixed and so it is the overall
SE . The power launched in each dimension must be chosen which will depend or not on the Mi level, and consequently determine the m-PAM signal-to-noise ratio
SNRi and the overall bit error rate
BERFlex −PAM performance. The way the total power is distributed affects in the sensitivity, that’s why it is important to choose a good strategy. Four different strategies are explained, two of them have a similar behaviour, one is clearly not useful and the fourth one is the optimal solution.
a) Same-
Pi: The same quantity of power is delivered to all the M-PAM formats: pi=1/4 independently of their value. Pi does not depend on the target BER. It is a waste of resources, because higher Mi levels needs more power to achieve a given target BER.
b) Same-
di : The minimum Euclidean distance
di is kept equal for all M-PAM modulation
formats. The relation between distance and power is
Pi = (Mi 2 −13
) di2
⋅ Rs.
As the symbol rate Rs is constant and in this case di is constant, Pi just depends on the Mi level. Therefore, the dimensions with higher m-pam need more power. c) Same-
BERMi: All four M-PAM modulation formats are forced to work at same-
BERi. This strategy is similar with “Same-
di “, but in this case, Pi slightly depends on the target BER.
24
d) Min-
SNRFlex −PAM : The combination of the power ratios is obtained by minimizing the overall SNR at a given target BER for each of the BpS values. Pi varies with the target BER.
3.4 Best Combinations
The m-PAM order is chosen to vary from 2 to 8, i.e. bit-per-symbol possible values
bpsi = 1,2,3. Therefore, BpS granularity is any integer in between 1 and four times )max( ibps , which is 12.
Since, for each possible BpS, there are several different Mi-PAM combinations of modulation formats, suboptimal combinations are discard by comparing the required SNR at target BER. Consequently, we ran an optimization process setting down the best bpsi combination and the best PRi for every possible BpS and for all four operation strategies described before, choosing the minimum SNR required. Taking into account that for a given BpS, there are several combinations of the same M levels, but in different order, we can say that the order does not affect the symbol performance. But, somehow the order is settled down by imposing the definition of the power ratio
referred to the first dimension:
PRi =PiP1
, and consequently, the bps vector goes from the
minim m-PAM level to the maxim. Observation: When all dimensions have the same
bpsi value, all different power strategies become the “Same-Pi” and consequently those cases are the standard modulation formats based on the polarization-multiplexing m-QAM family (PM-mQAM). With this new modulation, the standard modulation formats based on the polarization-multiplexing in the m-QAM family, can also be implemented.
BpS bps PM-mQAM family PR “same-Pi” [dB] 1 [1000] SP-BPSK [0 - - -] 2 [1010] PM-BPSK [0 - 0 -] 4 [1111] PM-QPSK [0 0 0 0] 8 [2222] PM-16QAM [0 0 0 0]
12 [3333] PM-64QAM [0 0 0 0] Table 3.1: PM-mQAM family same power strategy
NOTE: in the table, when there is no PAM symbol transmitted, there is no power launched on that dimension, so in the PR vector appears ‘-‘.
25
Figure 3.8: BER over SNR for PM-mQAM standard formats for BpS=2,4,8 and 12
Best Results for Flex-PAM symbols:
BpS bps PR “same-di” [dB] PR “same-BER” [dB] PR “min-SNR” [dB]
5 [1112] [0 0 0 6.74] [0 0 0 6.74] [0 0 0 6.02] 6 [1122] [0 0 6.74 6.74] [0 0 6.74 6.74] [0 0 6.02 6.02] 7 [1222] [0 6.74 6.74 6.74] [0 6.74 6.74 6.74] [0 6.02 6.02 6.02] 9 [2223] [0 0 0 6.23] [0 0 0 6.01] [0 0 0 4.77]
10 [2233] [0 0 6.23 6.23] [0 0 6.01 6.01] [0 0 5.44 5.44 ] 11 [2333] [0 6.23 6.23 6.23] [0 6.01 6.01 6.01] [0 5.44 5.44 5.44]
Table 3.2: PR vector for Flex-PAM symbols at target BER=1·10^-3 for “same-di”, “same-ber” and “min-snr” strategy
There is a special Flex-PAM case, the BpS=3 with combination bps=[1 0 1 1] and PR[dB]=[0 - 0 0] where all strategies are also the same, but it does not form part of the PM-mQAM family. Note 2: For all Flex-PAM symbols, “same-Pi” strategy has a constant power ratio vector PR [dB]=[0 0 0 0]
Figure 3.9 and figure 3.10 shows BER vs. SNR plot for all the strategies, in two cases (BpS=5 and BpS=6). Except for the “same power” strategy, the other three present very similar performances, with Min-SNR being the optimal choice, as expected.
26
Figure 3.9: BER over SNR of Flex-PAM modulation, Bps=5, bps=[1 1 1 2] for all the strategies
Figure 3.10: BER over SNR of Flex-PAM modulation, Bps=6, bps=[1 1 2 2] for all the
strategies
27
4. The Gaussian Noise Model
4.1 Uncompensated Transmission (UT)
The development of digital signal processing (DSP) in last year allows the introduction of coherent detection. Using Coherent Receivers, chromatic dispersion can be completely compensated at the receiver (electronically), in the DSP part, avoiding the in-line DCU (dispersion management). Such systems are called Uncompensated Transmissions (UT) and had been deeply investigated last years. UT turns to be more economic and provides better signal performance than traditional in-line dispersion compensation for multilevel modulation formats with coherent receivers. A simulation example that validates this affirmation is done in [8], which consists on a 9 Nyquist Wavelength Division Multiplexing (NyWDM) channels with a PM-QPSK modulation at 111 Gbit/s, over different kind of fibers with 20 spans of 90 km with EDFA. In order to predict the system maximum reach, different types of model can be used. GN-model is one of the several models valid for UT, and several experiments have tested its good prediction. Key aspects of UT:
1. It arises from the simulations that the four components of the transmitted optical signal, due to the large amount of dispersion accumulated, appears to quickly take on identical, statistically independent, zero mean Gaussian distribution, as they propagate among the link. The signal itself becomes Gaussian noise.
Figure 4.1: Electrical field distribution of one of the components with PM-QPSK
modulation at 32 GBaud after 500 km of SMF [4].
28
2. It has been proved that even if there is no ASE noise in the link, the statistical received constellation points appears to be Gaussian, with independent components. Nonlinearity becomes approximately Gaussian and additive and it is called: Nonlinear interference (NLI).
Figure 4.2: back-to-back received constellation in the absence of ASE noise [1].
This assumption allows redefining the expression
SNR =PCH
PASE + PNLI , because NLI is uncorrelated with the signal and the ASE noise. The formula is valid considering perturbative approach, i.e., nonlinearity is relatively small versus the useful signal.
Where: •
Pch is the launched power per channel •
PASE is the power of the ASE noise generated by the EDFA. ASE noise is defined as
PASE = N s ⋅ hf ⋅ F (G −1)BN , where
N s is the number of the frequency of the optical signal,
F is the EDFA noise figure, i.e. a measure of the amount of noise introduced by the amplifier,
G is the gain of EDFA and
BN is the noise bandwidth. •
PNLI is the non-linear interference (NLI) noise.
3. It was simulative found that the NLI power follows the following expression:
PNLI = ηNLI ⋅ N s ⋅ Pch3 , which suggests that the NLI is generated similarly to the FWM
effects.
4.2 FWM-like effects The FWM-like models are based on ideally slicing up the signal spectrum into spectral components, whose non-linear beating during propagation is then analytically expressed in a fashion similar to the classical formulas of FWM [12].
29
Figure 4.3: WDM signal spectrum of the FWM-like model
In case of propagation of multilevel modulation formats on UT, the analysis of non-linearity studied in Chapter 1 cannot be applied as UT changes the properties of the signal propagation. New classification of non-linear noise: In UT the NLI can be distinguished between three non-linear contributions. We are focusing on the central frequency of the center channel (f=0), because it is subjected to the highest number of contributions from other frequencies. (f1,f2,f3): generating signal components and (f=0): generated NLI contributions.
• Self Channel Interference (SCI): interference on a channel generated by the beating of different spectral lines of the channel. (f1,f2,f3) belong to the center channel and whose beating adds up at the center f of the same channel.
Figure 4.4: SCI effect
• Cross Channel Interference (XCI): disturbance on a channel induced by the
beating of spectral lines of that channel with the ones of other channels. (f1,f2,f3) belong to the center channel and one other channel
Figure 4.5: XCI effect
• Multi Channel Interference (MCI): interference coming from the beating of
spectral lines of different channels falling on another one. (f1,f2,f3) involves at least two channels apart from the center channel.
30
Figure 4.6: MCI effect
All these effects have the same qualitative impact on the signal, adding Gaussian noise, whereas in systems using in-line DCU, nonlinear contribution is different depending on the nonlinear effect.
4.3 The GN-model reference formula: The GN-model supplies the analytical formula for NLI Power Signal Density (PSD) in order to calculate the NLI power
PNLI . GN-model premises for the GNRF formula:
• Dual polarization • All identical spans • EDFA lumped amplification. • Loss exactly compensated for by amplification • Odd number of channels
N ch in the WDM comb and
f = 0 represents the center frequency of the center channel
GN-model parameters:
•
α : fiber loss coefficient
[Km−1], describes the attenuation of the signal among the fiber span as
e−2αLs . •
β2 : dispersion coefficient
[ ps2Km−1] •
γ : fiber non-linear coefficient
[W −1Km−1] •
Ls: span length
[Km−1] •
N s: number of spans in the optical link.
•
Leff : effective length
[Km] , defined as
1− e−2αLs
2α The GN-model reference formula is the power spectral density of the NLI noise at the end of the link for a multi-span system.
GNLI ( f ) =1627
γ 2L2eff ⋅ GWDM ( f1)GWDM ( f2)GWDM ( f1 + f2 − f ) ⋅ ρ( f1, f2, f ) ⋅ χ ( f1, f2, f )df2df1
−∞
∞∫
−∞
∞∫
The GNRF can be physically interpreted as describing the beating of each thin spectral slice of the WDM signal with all others through a FWM process.
31
Where:
•
ρ( f1, f2, f ) models the efficiency of the NLI generated at the frequency
f created by the three spectral components
f1,
f2 and
f3 = f1 + f2 − f . It decreases if the channel spacing increases and if increases the dispersion.
ρ( f1, f2, f ) =1− e−2αLs e j4π 2β 2Ls ( f1 − f )( f2 − f )
2α − j4π 2β2 ( f1 − f )( f2 − f )
2
⋅ L−2eff
• The integrand factor
GWDM ( f1)GWDM ( f2)GWDM ( f1 + f2 − f ) represents the power spectral density of the three spectral lines involved in the generation of the NLI.
• The Phased-array factor:
χ ( f1, f2, f ) =sin2 (2N sπ
2 ( f1 − f )( f2 − f )β2Ls)sin2 (2π 2 ( f1 − f )( f2 − f )β2Ls)
this term is
related with the noise accumulation at frequency
f along the link. In a single span system there are just one number of spans Ns=1 it, consequently the the phased-array factor disappears from the formula:
GNLI(1span ) ( f ) =
1627
γ 2L2eff ⋅ GWDM ( f1)GWDM ( f2)GWDM ( f1 + f2 − f ) ⋅ ρ( f1, f2, f )df2df1
−∞
∞∫
−∞
∞∫
4.4 WDM-channel spectrum: There are two differenent types of WDM-channel spectrum:
• Nyquist Limit (NyWDM): all channels have a rectangular spectrum with channel spacing (
∆f ) equal to the bandwidth (
Bch). Considering a perfect rectangular shape, the bandwidth is equal to the symbol rate.
Figure 4.7: NyWDM channel spectrum with a rectangular shape
• Non-Nyquist Limit: channel spacing is any value and the bandwidth is equal to
the symbol rate, if has a rectangular perfect shape.
32
Figure 4.8: Non-NyWDM channel spectrum with a rectangular shape NyWDM channel approximation:
a) Square root raise cosine: Ideally we want a perfectly squared spectrum with
Bch = Rs, and consequently the SE=BpS. Practically it cannot be obtained a perfect rectangular pulse, using a PSD of a root raised cosine we loss a little of SE but keeping the absence of ISI and
Bch = Rs ⋅ (1+ β) , where beta is the roll-off factor and SE depends on that value. Using a
Nyquist spectrum:
H ( f ) =r cos(β)
Sinc(π fRs
) is proportional to the square-root of a raised
cosine because we want an ISI free eye diagram at the receiver after the equalizer.
b) Local white shape: In the center of the central channel (f=0), NLI PSD is locally flat (white) and the noise power can be calculate as
PNLI = GWDM (0) ⋅ BWDM c) NLI noise accumulation:
NLI noise contributions from each span propagate linearly through the rest of the link and in the receiver the contributions are summed in two different ways that follows this expression:
GNLI ( f ) = GNLI(1span ) ( f ) ⋅ N s
1+ε
o Incoherent law (
ε = 0): when the NLI noise generated in any given span is summed in power.
o Coherent law (
0 < ε ≤ 1): the sum cannot be carried out in power, because it matters the phase relationships accumulate in propagation.
Considering a Ny-WDM channel comb and supposing local white shape in the central channel (f=0), and incoherent noise accumulation, the PSD of the NLI noise can be expressed:
GNLI(1span ) (0) ≈ γ 2Leff
2 (23
)3GWDM3
a sinh(12
π 2 β2 Leff ,aBWDM2 )
π β2 Leff ,a
GNLI (0) = N s ⋅GNLI(1span ) (0)
Where, all parameters have been defined in previous sections, but the asymptotic
effective length
Leff ,a =1
2α
33
4.5 Observations
• It has been validated for standard modulation formats: The GN-model quantitative predictions have been proved in [Andrea’s paper] in which four different coherent modulation formats, with three different fiber types among different channel spacing, at 32 GBaud tested successfully the GN-model bounds.
• Theoretically, the GN-model is independent from the modulation formats, at
least for standard modulation formats it has been proved. We will test if is still a good prediction for Flex-PAM modulation.
• In practice, the signal behaves as a Gaussian noise process, which shows a power
spectral density equal to the one of the transmitted signal itself. This phenomenon is verified if the dispersion and the symbol rate are high enough, for instance, in SSMF with 30 GBaud the signal takes about 200 Km to become Gaussian noise. In GN-model the signal is assumed Gaussian from the beginning, although this assumption can lead to non-imperceptible errors in the prediction.
• In our simulation system we have considered incoherent accumulation because
the simulations hinted at a better accuracy of that model, although the difference between them is not strong, and it is easier to calculate.
34
5. Back-to-back Simulation
Performance
5.1 Theoretical analysis of power strategies The purpose of this chapter is to verify the Flex-PAM modulation, showing the back-to-back bit error rate (BER) performance vs. signal-to-noise ratio (SNR). A given target BER is defined in all strategies and for each of them we calculate the correspondent target SNR, which will be named Sensitivity. In the end we will choose the strategies that require less sensitivity and evaluate which are the best ones to real applications. The back-to-back transmission consists in launching the signal and immediately received it, without being propagated through the optical fiber. This procedure generates a signal that is just distorted for noise impact (ASE noise).
Figure 5.1: Sensitivity curve
The bit error rate expression
BERFlex −PAM = Φ(PRi ,bpsi ,SNRFlex −PAM ) depends on the combination of the four different M-PAM modulation formats and on the power strategy applied. Forcing to work at a certain target BER will also change the target SNR value. The procedure of choosing the bps vector combinations has been explained in chapter 3 and de corresponding power ratio vector is shown in table 3.1 and 3.2. Having a target 𝐵𝐵𝐸𝐸𝐵𝐵 = 1 · 10−3, we calculate all target SNR for different BpS cases.
35
Example of “Same-BER” strategy for BpS=6
Figure 5.2: Plot of the different sensitivity curves for different combinations at a target
BER = 1 ⋅10−3. Comment: As we can see in figure 5.2, in the "same BER strategy, the less difference of M-PAM level there are between dimensions, the lower is the sensitivity required. For instance, the combination bps=[1 0 2 3], which contains three different PAM modulation, has a SNR=18.5 dB, on the contrary, the bps=[1 1 2 2], has the SNR=14.9 dB. Not transmitting in one of the dimensions may not be efficient, because in bps=[2 2 2 0], although there is just one M-PAM level, the overall contribution is not good. In conclusion, it seems that the more balanced is the bpsi, the better results it gets. The Sensitivity for the four strategies presented in Chapter 3 is compared here, and we will choose the best strategy for real applications. The theoretical results show that three of them have similar behaviour, as one is clearly not effective. “Same-Pi” strategy for PM-mQAM symbols:
BpS bps SNR “same-Pi” [dB]
1 [1000] 3.779 2 [1010] 6.789 4 [1111] 9.8 8 [2222] 16.542
12 [3333] 22.548 Table 5.1: Target SNR for standard modulation formats at target BER=1·10^-3
36
Best Results for Flex-PAM symbols:
BpS bps SNR “same-di” [dB] SNR “same-BER” [dB] SNR “min-SNR” [dB] 5 [1112] 12.721 12.657 12.57 6 [1122] 14.416 14.366 14.305 7 [1222] 15.615 15.589 15.561 9 [2223] 19.027 18.964 18.861
10 [2233] 20.565 20.51 20.431 11 [2333] 21.678 21.648 21.608 Table 5.2: Target SNR for Flex-PAM symbols at target BER=1·10^-3 for three power
strategies Note: SPECIAL CASE: Flex-PAM symbol with BpS=3, bps=[1011], PR=[0 - 0 0] “Same Pi” strategy and SNR=8.55 dB.
BpS bps SNR “same-Pi” [dB]
5 [1112] 15.092 6 [1122] 15.293 7 [1222] 16.405 9 [2223] 21.299
10 [2233] 22.051 11 [2333] 22.359
Table 5.3: Target SNR for Flex-PAM symbols at target BER=1·10^-3 for “same-power” strategy
Figure 5.3: SNR required vs. BpS for all the strategies
37
Observations:
1. Same power strategy does not work well for those BpS that are not the PM-mQAM modulation formats, and has been discarded for real applications.
2. Results between the same ber and min snr are quite similar.
Figure 5.4: SNR required for both “Same-BER” and “Min-SNR” strategies over BpS
As it is shown in figure 5.4, the difference between them is almost negligible. Taking a look on figure 5.5, where the sensitivity difference between Same-BER and Min-SNR strategy is plotted over BpS, we can see precisely the difference between them, and the maxim sensitivity difference is around 0.1 dB at 9 bit-per-symbol, which is nearly imperceptible.
Figure 5.5: SNR difference between Same-BER and Min-SNR theory
38
Same-BER strategy has the simplest implementation, having selected same FEC for all quadratures, they will require the same pre-FEC BER. Min-SNR strategy is the optimal one, but it implies some more complexity because each format delivers a different pre-FEC BER and should work with a specific FEC. Min snr shows the better performance, but it implies more complexity. From a practical implementation point of view the same ber is simpler.
5.2 Simulation Setup Parameters and Procedure
A theoretical analysis of Flex-PAM has been done in chapter 2 and we validate our results using a time-domain simulator called OptSim. OptSim is a software tool for the design and simulation of optical communications systems at the physical level. The whole procedure has three differential parts, and it is valid also for propagation simulation (chapter 6).
1. Pre-processing:
The pre-processing part is done in Matlab, in which the Flex-PAM signal is shaped with a Nyquist filter (sqrt-raised cosine spectrum with roll-off 0.2).
2. TX and RX:
Then the signal is modulated and transmitted for being received with the standard coherent receiver. The Flex-PAM modulation is based on the use of an IQ modulator for each of the polarization, formed with nested MZMs (see chapter 2). Using an optical coupler, we join the signal with the AWGN generator and send them to the coherent receiver, where it is mixed with the LO in two 90º hybrid. The signal is electrically filtered using a 5th order Bessel Low-Pass filter (LPF) after the four-balanced photo-detectors (BPD). 3. Post-processing: After detecting the signal, it is sampled by an ADC at 2 samples per symbol. Dispersion is electrically compensated by the DSP and there, the four signal components are processed to estimate the channel by a multiple-input multiple-output (MIMO) equalizer, which consists in four 51-tap FIR filters. The coefficients are estimated using LMS algorithm. The signals out of the equalizer were used for decision and BERs were evaluated using direct error counting.
Setup Parameters for back-to-back performance:
• The net symbol rate is Rsnet=25 Gbauds, and taking into account the 26% of protocol overhead and 2% of FEC the total symbol rate is Rs=32 Gbauds.
39
• A variable bit rate
Rb = BpS ⋅ RsNET
BpS 1 2 3 4 5 6 7 8 9 10 11 12
Rb [Gbit/s] 25 50 75 100 125 150 175 200 225 250 275 300
• PR vector depending on the power strategy (see chapter 2) • Target
BERt = 1 ⋅10−3 • Number of channels
N ch = 1 • Power transmitted
Ptx = 0dBm • Four Components: px,qx,py,qy
Figure 5.7: OptSim back-to-back simulation blocks
40
5.3 Back-to-back simulation results
- PM-mQAM standard modulation formats:
Figure 5.8: Sensitivity curves for standard modulation formats. Dashed lines correspond to target
BER = 1 ⋅10−3
41
- Min-SNR strategy:
42
Figure 5.9: Sensitivity curves for the Min-SNR strategy. Dashed lines correspond to target
BER = 1 ⋅10−3
43
- Same-BER strategy:
44
Figure 5.10: Sensitivity curves for the “Same-BER” strategy. Dashed lines correspond to
target
BER = 1 ⋅10−3
- Conclusions: Both strategies showed very limited penalties in all Flex-PAM combinations, but for the reasons explained before, “Same-BER” strategy is the best candidate for applications.
Figure 5.11: Comparison of SNR required between Simulation and Theory over BpS As we can see in Figure 5.11: the expectations have accomplished for all BpS, there is no any extra penalty, although at BpS=10 (Rb=250 Gbit/s) and BpS=11 (Rb=275 Gbit/s), and for both strategies the difference is quite big respect other BpS.
45
Figure 5.12: SNR difference between simulation and theory for both strategies
As we can see in figure 5.12, for almost all BpS cases the results are satisfactory, except for two cases, i.e., BpS=10 (Rb=250 Gbit/s) and BpS=11 (Rb=275 Gbit/s). Same BER seems to have more penalty than Min SNR, being the maxim difference of 0.4 dB. The simple “Same BER” strategy shows very limited penalties in all Flex-PAM combinations and it is the best candidate for applications.
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6. Non-linear Propagation
Simulation
6.1 System Setup The following analysis is based on the evaluation of non-linear propagation performance for Flex-PAM modulation format at target
BER = 1 ⋅10−3, considering only the “Same-BER” strategy. The motives why we have chosen “Same-BER” against the others are:
1. The maximum difference between “Min-SNR” and “Same-BER” in terms of SNR
required is of 0.1 dB, as we can see in figure 5.5 in the chapter 5. 2. It is easier in DSP to use “Same-BER”, because it is used the same FEC code for
each dimension. Where we assume that the FEC code does not have any impact on the m-PAM level (modulation format).
Simulation test set-up: We employed the commercial optical system simulator OptSim to carry out the simulations. We set the TX according to the “Same-BER” strategy. For all modulation formats we used standard I-Q transmitters based on nested Mach-Zehnder modulators. We used 13-NyWDM channels comb with 50 GHz of channel spacing and shaped with a roll-off equal to 0.2, at a symbol rate of 32 Gbauds.
Figure 6.1: Transmitted signal comb 6]
The optical link was composed of Ns identical uncompensated spans. Each span was composed of Ls=100 km of transmission fiber followed by an EFDA with noise-figure F=5 dB that completely recovered the span loss Aspan. The fiber type is a standard SMF with fiber parameters:
• Dispersion Coefficient: D=16.7 (ps·km)/nm • Loss Coefficient: alpha=0.22 dB/km • Nonlinear Coefficient: gamma=1.3 1/W/km
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Figure 6.2: Optical link scheme [6]
We use a standard coherent receiver, and it is exactly the same as we used for the back-to-back simulation in chapter 5. Linear Propagation: if NLI noise is absent, the signal-to-noise ratio is expressed as:
SNR =Pch
PASE, where
PASE = N s ⋅ PASE(1span ) ≈ N s ⋅ As ⋅ hfo ⋅ BN
Non-linear Propagation: taking into account that the NLI noise is uncorrelated with the
signal and the ASE noise we can define the new
SNRNLI =Pch
PASE + PNLI, where
PNLI = N s ⋅ηNLI
⋅ Pch3
• Where
Pch is the power channel,
PASE is the ASE noise introduced by the EDFA and
PNLI is the Gaussian noise introduced by the nonlinearities of the fiber. •
N s is the number of spans,
As is the span loss,
h is the Planck constant,
fo is the central frequency and
BN is the noise bandwidth. •
ηNLI is the efficiency, which depends on the
α ,
β2 ,
γ ,
leff and
∆f and varies with the chosen dispersion map.
Figure 6.3: Equivalent optical link [6]
The simulation aim is to find the maximum reach, when different powers are transmitted, and compare the results with the incoherent GN-model, explained in chapter 4, which is used as a reference for our predictions. The analysis performs tests for several transmission power levels, varying from -0.5 dBm to 1.5 dBm in steps of 0.5 dBm, in order to analyze the increasing amount of distortion introduced by nonlinear effects, at target BER=1·10^-3.
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6.2 Maximizing the number of spans The procedure of evaluate the maximum reach, given the span loss and target BER, is running several simulations at a fixed power for an increasing number of spans, looking for the maxim that obtains
BER ≤ 1 ⋅10−3. A graphical example of this procedure is shown in the following figure:
Figure 6.4: BER vs. Nspan of bps=[1122] at Ptx=0 dBm.
Repeating the procedure for each
Ptx , we can find the maximum reach and the optimal power. Theoretically for lower power, the span number match the theoretical bound, but for higher power, the increase of non-linear effects, produce a decreasing of
N span value.
Figure 6.5: theoretical curve of Number of spans vs. power transmitted [7]
Imposing BER=BERt is equivalent as SNR=SNRt. The nonlinear signal-to-noise ratio can be expressed:
SNRNLI =Pch
N s (PASE(1span ) + ηNLI Pch
3 )
Being
PASE(1span ) = hf ⋅ AsFBN = FAsPbase the number of spans expression is:
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N s =1
SNRNLI
Pch
(FAsPbase + ηNLI Pch3 )
And deriving we find the maxim power launched (Popt) and the maxim number of spans (NsMAX):
N smax =23
1SNRt
12ηNLI (FAsPbase )23
Pch,opt =FAsPbase
2ηNLI3
When the transmitted power is the optical launched, the ASE power is 2 times the NLI power. In the Nyquist Limit, the optical power is the same for all formats, and is independent of the link length. The capacity is influenced by the total WDM bandwidth.
6.3 Comparison between Model and Simulation Results for “Same-BER” strategy Modulation formats:
In this section we compare simulative results with model predictions. For each of the Flex-PAM modulation formats at “Same-BER” strategy, for a fixed frequency spacing and fiber type, we calculate the number of spans
N span as a function of the launched power per channel
Ptx , using as a reference the incoherent GN-model. We have not considered testing the standard modulation formats, as a broad simulative validation effort has done in [] and the results show the GN-model to yield very accurate prediction. As an example, we show a set of such curves for PM-QPSK over SMF. Each curve refers to a different spacing. The model (solid lines) matches rather closely the simulative results (markers).
Figure 6.6: PM-QPSK over SMF, 9 channels at 32 Gbaud, BER=1·10^-3 from [1]
These 6 cases have been selected to obtain the propagation performance of the Flex-PAM modulation format using “Same-BER” strategy. The corresponding PR vector is showed, for every bps combination.
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Rb [Gbit/s] bps PR [Db] 125 [1 1 1 2] [0 0 0 6.74] 150 [1 1 2 2] [0 0 6.74 6.74] 175 [1 2 2 2] [0 6.74 6.74 6.74] 225 [2 2 2 3] [0 0 0 6] 250 [2 2 3 3] [0 0 6 6] 275 [2 3 3 3] [0 6 6 6]
Table 6.1: bps and PR for “Same-BER” strategy
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Figure 6.7: System reach Nspan vs. Ptx for Rb=125,150,175,225,250 and 275 [Gbit/s], using the “Same-BER” strategy.
It can be observed that with the increasing of M the nonlinear impairments grow, consequently reducing the reach.
In TDHMF, due to the power unbalancing in time between generating formats, it was observed a performance penalty when propagating in a nonlinear fiber [5]. Being Flex-PAM based on same principle to balance the performance of different M-PAM, we carried out a simulative analysis to verify if a similar effects impacts on this new format. Flex-PAM shows a penalty compared to the GN-model prediction that could be ascribed to the power unbalancing between formats used to build the Flex-PAM. This is the main difference with respect to a standard modulation format, where GN-model predicts very accurately the reach. Observation about the Bit Error Rate Some BER analyse in the “Same-BER” strategy has been done in order to examine the penalties that suffer each dimension.
Rb [Gbit/s] bps PR [dB] 125 [1 1 1 2] [0 0 0 6.74] 150 [1 1 2 2] [0 0 6.74 6.74] 150 [1 2 1 2] [0 6.74 0 6.74]
Table 6.2: bps and PR
If we take a look at Rb=125 Gbit/s: • The BER on Y polarization, “BERy” is larger than “BERx”. • “BERpx” and “BERqx” have a similar value, but “BERpy” is larger than “BERqy”.
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• Although py there is just one bit transmitted, in qy there are two bits transmitted.
This lead us to guess that the dimension with more power have a BER effect on the other dimension of the same polarization. Rb=225 has the same results than this one.
Another interesting example is the Rb=150 Gbit/s with bps=[1 1 2 2]:
• “BERy” is larger than “BERx”. But in terms of power, the polarizations are balanced, because their dimensions transmit the same amount of power.
• BERpx and BERqx are almost equal, and BERpy and BERqy too.
Changing the bps combination of the Rb=150 Gbit/s, i.e., bps=[1 2 1 2]: • BERy is almost the same as BERx. • But BERpx is larger than BERqx, and the same for the y polarization. The same
happens with Rb=250 Gbit/s, with bps=[2 2 3 3] and bps=[2 3 2 3].
Figure 6.8: BER vs. Nspan propagation curve for each dimension and the overall
expression In conclusion, in one polarization, the dimension with less power has a larger BER. When compare the two polarizations, the polarization with more power has a larger BER. From this observation we thought it would be a good idea changing the combination bps to reach this “balanced power” between the two polarizations. And then we did the simulation of Rb=150 and 250 balancing the bps combination and the result was the following:
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Figure 6.9: Propagation performance in Rb=250 [Gbit/s] with “Same-BER” strategy, for
bps classic combination and bps with power balanced
Figure 6.10: Propagation performance in Rb=150 [Gbit/s] with “Same-BER” strategy, for
bps classic combination and bps with power balanced
6.4 Pre-distortion technique
Difference between in-line Compensation and Pre-distortion:
1. In IIMD systems, dispersion has big effect, so often it was necessary to used Compensation Units before the receiver, which basically consists in adding dispersion on the opposite sign (negative) in order to compensate it.
54
2. Since coherent detection has become available, with the use of DSP, uncompensated systems have been used lately because it has been proved the increment in distance achieved, and in the receiver it is used electronic compensation (EDC).
In order to counteract the impact of nonlinearities, the introduction of pre-distortion at the transmitter is considered. The simulations were performed introducing an increasingly higher amount of chromatic dispersion Dpre. Its effect is the mitigation of the nonlinearities at high values of power launched and the consequent increase of maximum reach. Let’s analyze pre-distortion for a few cases and see how the signal behaves:
Figure 6.11: Propagation simulation with pre-distortion at D=5000 and 10000 for Rb=125for Rb=150 at bps=[1212], and D=10000 and 30000 for Rb=175
Figure 6.12: Propagation simulation with pre-distortion at D=10000 and 30000 and D=10000 and 30000 for Rb=150 at bps=[1122]
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6.5 Conclusions of Pre-distortion in Flex-PAM modulation systems
We evaluated NspanMAX for a range of Dpre from 200 to 20000 at Rb=150 Gbit/s and bps=[1122]:
Figure 6.13: Maxim reach vs. pre-distortion
The aim of the simulation was to find the maximum reach achievable by the Flex-PAM under investigation when different average powers are transmitted, and to evaluate possible discrepancies with respect to the reach predicted when the nonlinearities impact is evaluated with the GN-model described in chapter 4. What we were expecting was to increase linearly the reach with the amount of pre-distortion, but it maximum capacity goes down while Dpre increases. The idea of pre-distortion is based on the GN-model, that considers that the signal is Gaussian enough from the beginning, so we if we add dispersion we should obtain a better performance. In TDHMF the used of pre-distortion makes more close the simulation to the prediction of GN-model. The idea is that in TDHMF power in time is not balanced, and using pre-distortion the power becomes balanced. But In flex-PAM the power is already constant in time, while it is not between dimensions. In conclusion, pre-distortion has an opposite effect in Flex-PAM modulation formats, because it is already balanced the power in time, and what pre-distortion does is exactly this.
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7. Conclusions and Future work
In this work, a new modulation format has been proposed as a solution for the demanding of flexibility in long-haul optical systems. The degree of freedom of this new format, i.e. the election of the power strategy, was the most important aspect to deal with, because of the limiting penalties of the fiber with the change of the transmitted power. On the other hand, an exhausted research on the best combination of the four m-PAMs levels has been done, and properly setting the power. The second part was focused to prove the correct performance of the simulations comparing the theoretical and the practical results, for both the back-to-back and the propagation simulation. We show that the back-to-back has no any extra penalty. On the contrary, in the propagation simulation, seems to be some impairments caused by the power unbalanced between dimensions, and consequently this influences on the maxim distance achievable. We showed that the incoherent GN-model is not good enough accurate for predicting the performance of Flex-PAM modulation formats. After testing the efficiency of the pre-distortion technique and realize it’s not functionality for Flex-PAM modulation, we focused on a new line of investigation for improving the maxim reach, called Tuning PR. The goal is to reduce the impact of nonlinear propagation impairments, testing the effectiveness of reducing power unbalance between formats. The estimation of the number of spans over power launched has been done using the theoretical formula:
N span =Ptx
SNRt ⋅ (PASE(1span ) + ηPtx
3)
Fixing the combination bps and the power ratio PR in the back-to-back simulation we obtain a target SNR to accomplished the target BER at 1·10^-3. This value is used in order to estimate the Nspan. Changing the eta value, and using the formulas from chapter 5, of the Nspanmax and Popt, we chose the maxim performance for each of the PR. The propagation simulation has done as in previous work, adding the estimation of eta using the LMS algorithm.
7.1 Tuning PR Test for BpS=6 (Rb=150 Gbit/s) bps=[1212] Figure 7.1 shows the SNRT at target BER in back-to-back conditions a function of the power ratio between formats. The optimal power ratio using “Same-BER” strategy is 6.7 dB, as expected, is the minimum SNR required. Moving away from this optimal value has the following effect:
57
back-to-back sensitivity is worse while nonlinear penalty due to the power unbalance is reduced allowing an increase of the optimal power. Figure 2: the optimum power launched Ptx,opt vs. PR It can be clearly observed that a reduction in power ratio improves the optimal power because of a reduction of nonlinear impact. Figure 3: nspanmax vs. PR, obtained by mixing figure 1 and 2. It is showed the maximum reach improvement achieved with this proposed technique that is the combination of the two effects described above. The improvement due to a reduction of the non-linear effects dominates the poorer back-to-back performance: at 150 Gbps, we observe an increase in maximum reach from 13 to 14 spans.
Figure 1: SNRt vs. PR Figure 2: Ptx,opt vs. PR
Figure 3: NspanMAX vs. PR
In conclusion, nonlinear propagation on uncompensated uniform links impacts Flex-PAM due to the power unbalance between generating formats. Tuning PR is a promising solution for achieving propagation penalty reduction, tuning the adequate power ratio: it allows increase the maximum reach to equalling GN- prediction based on back-to-back performance. The investigation line is now focused on this new scenario. In conclusion, Flex-PAM adds flexibility for future optical networks and keeping a limited complexity while maintaing propagation performance. Acknowledgments: Thanks to Prof. Andrea Carena
4 4.5 5 5.5 6 6.5 70.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
PR (dB)
Opt
imum
Pow
er(d
Bm
)
4 4.5 5 5.5 6 6.5 712
12.5
13
13.5
14
14.5
PR (dB)
Nsp
an
3 4 5 6 7 8 914
14.5
15
15.5
16
16.5
17
PR (dB)
SN
R (d
B)
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