Le Nguyen BINH
Centre for Telecommunications and Information Engineering, Department of Electrical.
电信与信息工程中心，电气系。
DOI: 10.4236/ijcns.2009.22012   PDF    HTML     38,577 Downloads   86,660 Views   Citations
DOI： 10.4236/ijcns.2009.22012 PDFHTML 38,577下载 86,660次浏览引用

Abstract 抽象

High speed and ultra-high capacity optical communications have emerged as the essential techniques for backbone global information transmission networks. As the bit rate of the transmission system gets higher and higher 40 Gb/s to 100 Gb/s the modeling of proposed modulation techniques is very important so as to avoid costly practical demonstration. The search for a universal modeling platform for such systems is urgent. Matlab Simulink has become the universal mathematical and modeling tools in most universities and re-search laboratories around the world. This paper thus describes the modeling techniques for advanced photonic transmission systems and Simulink is proven to be very effective platform for development of photonic communications systems due its comprehensive blocksets. The simulation is based mainly on the physical phenomena and understanding of its concepts of communications and photonics. Simulink models are given as examples of various sub-systems of the photonic transmission systems. Some simulated trans-mission performances are demonstrated as examples of final results obtained from Simulink models of the transmission systems.
高速、超高容量光通信已成为骨干全球信息传输网络的基本技术。随着传输系统的比特率越来越高，从 40 Gb/s 到 100 Gb/s，所提出的调制技术的建模非常重要，以避免昂贵的实际演示。为此类系统寻找通用建模平台是当务之急。Matlab Simulink 已成为世界各地大多数大学和重新搜索实验室的通用数学和建模工具。因此，本文描述了高级光子传输系统的建模技术，Simulink 由于其全面的模块集而被证明是开发光子通信系统的非常有效的平台。仿真主要基于物理现象以及对其通信和光子学概念的理解。Simulink 模型作为光子传输系统的各种子系统的示例给出。一些模拟的传输性能作为从传动系统的 Simulink 模型获得的最终结果的示例进行演示。

Keywords 关键字

Communications Systems, MATLAB Simulink Simulation, Optical Communications
通信系统， MATLAB Simulink 仿真， 光通信

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High speed and ultra-high capacity optical communications have emerged as the essential techniques for backbone global information transmission networks. As the bit rate of the transmission system gets higher and higher 40 Gb/s to 100 Gb/s the modeling of proposed modulation techniques is very important so as to avoid costly practical demonstration. The search for a universal modeling platform for such systems is urgent. Matlab Simulink has become the universal mathematical and modeling tools in most universities and research laboratories around the world. This paper thus describes the modeling techniques for advanced photonic transmission systems and Simulink is proven to be very effective platform for development of photonic communications systems due its comprehensive blocksets. The simulation is based mainly on the physical phenomena and understanding of its concepts of communications and photonics. Simulink models are given as examples of various sub-systems of the photonic transmission systems. Some simulated transmission performances are demonstrated as examples of final results obtained from Simulink models of the transmission systems.
高速、超高容量光通信已成为骨干全球信息传输网络的基本技术。随着传输系统的比特率越来越高，从 40 Gb/s 到 100 Gb/s，所提出的调制技术的建模非常重要，以避免昂贵的实际演示。为此类系统寻找通用建模平台是当务之急。Matlab Simulink 已成为全球大多数大学和研究实验室的通用数学和建模工具。因此，本文描述了高级光子传输系统的建模技术，Simulink 由于其全面的模块集而被证明是开发光子通信系统的非常有效的平台。仿真主要基于物理现象以及对其通信和光子学概念的理解。Simulink 模型作为光子传输系统的各种子系统的示例给出。一些仿真的传输性能作为从传输系统的 Simulink 模型获得的最终结果的示例进行演示。

1.  Introduction 1. 引言

1.1.  Overview of a Digital Photonic System
1.1. 数字光子系统概述

Any study on digital photonic transmission systems requires in-depth understanding of operational principles of system components which involve: 1) modulation/ demodulation or generation/detection of the optical signals modulated by proposed formats and the detection here implies the incoherent direct detection; 2) impairments in either electronic or photonic domains, especially the dynamics of optical fiber and the noise sources contributed by optical amplifiers and receiver electronic noise; 3) effects of optical and electrical filters. The schematic diagram of a DWDM digital photonic system is illustrated in Figure 1.
任何关于数字光子传输系统的研究都需要深入了解系统组件的工作原理，其中包括：1） 由建议格式调制的光信号的调制/解调或生成/检测，这里的检测意味着不相干的直接检测;2） 电子或光子域的损伤，尤其是光纤的动力学以及光放大器和接收器电子噪声贡献的噪声源;3） 光学和电滤波器的影响。DWDM 数字光子系统的示意图如图 1 所示。

The transmission medium may consist a variety of fiber types such as the standard SMF ITU-G.652 or nonzero dispersion shifted fibers (NZ-DSF) ITU-G.655 or the new type of fiber: Corning Vascade fiber. The dispersion and distortion of the lightwave signals are usually compensated by dispersion compensating fibers (DCF). The DCFs are normally accompanied by two discrete optical amplifiers, the Erbium-doped optical amplifiers (EDFA), one is for pre-amplification to compensate the attenuation of the transmission span, and the other is a booster amplifier for boosting the optical power of the channels to an acceptable, below the nonlinear limit level. It is assumed in this work that the amplifiers are operating in the saturation region.
传输介质可以由多种光纤类型组成，例如标准 SMF ITU-G.652 或非零色散偏移光纤 （NZ-DSF） ITU-G.655 或新型光纤：康宁 Vascade 光纤。光波信号的色散和失真通常由色散补偿光纤 （DCF） 补偿。DCF 通常配有两个分立式光放大器，掺铒光放大器 （EDFA），一个用于预放大以补偿传输跨度的衰减，另一个是升压放大器，用于将通道的光功率提升到可接受的非线性极限水平以下。在这项工作中，假设放大器在饱和区域工作。

The receiving sub-system would take on: 1) single detector direct detection optical receiver 2) the balanced detector receiving structure. The first type of the receiver is widely used for detection of ASK modulated optical signals. For the later case, the structure acts as an optical phase comparator employing a delay interferometer. Detailed description of these direct detection receivers for novel modulation formats are presented. In addition, especially for contemporary systems with high capacity, high bit rate and requiring high performance, electronic equalizers can be employed as part of the receiver. Section 5 gives insight and performance of one of the most effective electronic equalizers which is maximum likelihood sequence estimation (MLSE) with Viterbi algorithm.
接收子系统将承担：1） 单探测器直接检测光接收器 2） 平衡探测器接收结构。第一种接收器广泛用于检测 ASK 调制光信号。对于后一种情况，该结构充当采用延迟干涉仪的光学相位比较器。详细介绍了这些用于新型调制格式的直接检测接收机。此外，特别是对于具有高容量、高比特率和需要高性能的现代系统，电子均衡器可以用作接收器的一部分。第 5 节提供了最有效的电子均衡器之一的见解和性能，它是使用 Viterbi 算法的最大似然序列估计 （MLSE）。

1.2. Matlab Simulink® Modeling Platform

High speed and high capacity modern digital photonic systems require careful investigations on the theoretical performance against various impairments caused by either electronics or fiber dynamics before they are deployed in practice. Thus, the demand for a comprehensive modeling platform of photonic systems is criticalespecially a modeling platform that can structure truly the photonic sub-systems. A simulation test-bed is necessary for detailed design, investigation and verification on the benefits and shortcomings of these advanced modulation formats on the fiber-optic transmission systems Furthermore the modeling platform should take advantage of any user-friendly software platform that are popular and easy for use and further development for all the operators without requiring very much expertise in the physics of the photonic systems. Besides, this platform would offer research community of optical communication engineering a basis for extension and enhance the linkages between research groups.
高速和高容量的现代数字光子系统在实际部署之前，需要仔细研究针对电子或光纤动力学引起的各种损伤的理论性能。因此，对光子系统综合建模平台的需求至关重要，尤其是能够真正构建光子子系统的建模平台。仿真测试台对于详细设计、调查和验证光纤传输系统上这些高级调制格式的优缺点是必要的。此外，建模平台应利用任何流行的用户友好型软件平台，这些软件平台易于使用，并为所有操作员进一步开发，而无需光子系统物理学方面的专业知识。此外，该平台将为光通信工程研究团体提供扩展基础，并加强研究小组之间的联系。

Thus, it is the principal incentive for the development of a simulation package based on Matlab Simulink® platform¹ To the best of my knowledge, this is the first Matlab© Simulink-platform photonic transmission testbed for modeling advanced high capacity and long-haul digital optical fiber transmission systems. The simulator is used mainly for investigation of performance of advanced modulation formats, especially the amplitude and/or phase shift keying modulation with or without the continuity at the phase transition. Here, a single channel optical system is of main interest for implementation of the modeling in this paper.
因此，它是开发基于 Matlab Simulink® 平台¹ 的仿真包的主要动力。据我所知，这是第一个用于模拟高级高容量和长距离数字光纤传输系统的 Matlab© Simulink 平台光子传输测试平台。该仿真器主要用于研究高级调制格式的性能，尤其是幅度和/或相移键控调制，无论是否具有相变的连续性。在这里，单通道光学系统是实现本文建模的主要兴趣点。

Several noticeable advantages of the developed Matlab Simulink&regmodeling platform are listed as follows:
开发的 Matlab Simulink®modeling 平台的几个显着优势如下：

•The simulator provides toolboxes and blocksets adequately for setting up any complicated system configurations under test. The initialization process at the start of any simulation for all parameters of system components can be automatically conducted. The initialization file is written in a separate Matlab file so that the simulation parameter can be modified easily.
•仿真器提供了足够的工具箱和模块集，用于设置任何复杂的被测系统配置。系统组件的所有参数在任何模拟开始时的初始化过程都可以自动进行。初始化文件被写入一个单独的 Matlab 文件中，因此可以很容易地修改仿真参数。

•Signal monitoring is especially easy to be carried out. Signals can be easily monitored at any point along the propagation path in a simulation with simple plug-and-see monitoring scopes provided by Simulink®.
•信号监控特别容易进行。使用 Simulink® 提供的简单即插即用监控示波器，可以在仿真中沿传播路径的任何点轻松监控信号。

•Numerical data including any simulation parameters and the numerical results can be easily stored for later processing using Matlab toolboxes. This offers a complete package from generating the numerical data to processing these data for the achievement of final results.
•数值数据，包括任何仿真参数和数值结果，都可以轻松存储，以便以后使用 Matlab 工具箱进行处理。这提供了一个完整的软件包，从生成数值数据到处理这些数据以获得最终结果。

•A novel modified fiber propagation algorithm has been developed and optimized to minimize the simulation processing time and enhance its accuracy.
•开发并优化了一种新的改进光纤传播算法，以最大限度地减少模拟处理时间并提高其准确性。

•The transmission performance of the optical transmission systems can be automatically and accurately evaluated with various evaluation methods. These methods, especially proposal of novel statistical evaluation techniques are to be presented in Section 6.
•可以使用各种评估方法自动准确地评估光传输系统的传输性能。这些方法，特别是新颖的统计评估技术的建议，将在第 6 节中介绍。

Several Matlab Simulink® modeling frameworks are demonstrated in the Appendix of this paper. A Simulink model of a photonic transmission system can be shown in Figure 1(b).
本文附录中演示了几种 Matlab Simulink® 建模框架。光子传输系统的 Simulink 模型如图 1（b） 所示。

2.Optical Transmitters 2.光发射机

The transmitters would consist of a narrow linewidth laser source to generate lightwaves of wavelength conformed to the ITU grid. These lightwaves are combined and then modulated. This form is for laboratory experiments only. In practice each laser source would be modulated by an external modulation sub-system. The MZIM can bee a single or dual drive type. The schematic of the modulator is shown in Figure 2(a) and the Simulink model is in Figure 2(a) for generation of photonic signals by multi-level amplitude and phase shift keying modulation formats.
发射器将由一个窄线宽激光源组成，以产生符合 ITU 网格波长的光波。这些光波被组合然后调制。此表格仅用于实验室实验。在实践中，每个激光源都将由外部调制子系统调制。MZIM 可以采用单驱动或双驱动类型。调制器的原理图如图 2（a） 所示，Simulink 模型如图 2（a） 所示，用于通过多级幅度和相移键控调制格式生成光子信号。

In 1980s and 1990s, direct modulation of semiconductor lasers was the choice for low capacity coherent optical systems over short transmission distance. However, direct modulation induces chirping which results in severe dispersion penalties. In addition, laser phase noise and induced from non-zero laser linewidth also limit the advance of direct modulation to higher capacity and higher bit rate transmission.
在 1980 年代和 1990 年代，半导体激光器的直接调制是短传输距离内低容量相干光学系统的选择。然而，直接调制会引起啁啾声，从而导致严重的色散损失。此外，激光相位噪声和非零激光线宽引起的噪声也限制了直接调制向更高容量和更高比特率传输的进展。

Overcoming the mentioned issues, external modulation techniques have been the preferred option for digital photonic systems for over the last decade. External modulation can be implemented using either electro-absorption modulator or electro-optic modulators (EOM). The EOM whose operation is based on the principles of electro-optic effect (i.e. change of refractive index in solid state or polymeric or semiconductor material is proportional to the applied electric field) has been the preferred choice of technology due to better performance in terms of chirp, extinction ratio and modulation speed. Over the years, the waveguides of the electrooptic modulators are mainly integrated on the material platform of lithium niobate (LiNbO3) which has been the choice due to their prominent properties of low loss, ease of fabrication and high efficiency [4].
克服上述问题，在过去十年中，外部调制技术一直是数字光子系统的首选。可以使用电吸收调制器或电光调制器 （EOM） 实现外部调制。基于电光效应原理（即固态或聚合物或半导体材料中的折射率变化与施加的电场成正比）操作的 EOM 由于在啁啾、消光比和调制速度方面具有更好的性能，因此一直是首选的技术选择。多年来，电光调制器的波导主要集成在铌酸锂 （LiNbO3） 的材料平台上，由于其低损耗、易于制造和高效率的突出特性而成为人们的选择 [4]。

These LiNbO3 modulators have been developed in the early 1980s, but not popular until the advent of the Erbium-doped optical fiber amplifier (EDFA) in the late 1980s. Prior to the current employment of LiNbO3 modulators for advanced modulation formats, they were employed in coherent optical communications to mitigate the effects of broad linewidth due to direct modulation of the laser source. These knowledges have recently been applied to the in-coherent advanced modulation formats for optically amplified transmission systems.
这些 LiNbO3 调制器是在 1980 年代初期开发的，但直到 1980 年代末掺铒光纤放大器 （EDFA） 出现才流行起来。在目前将 LiNbO3 调制器用于高级调制格式之前，它们被用于相干光通信，以减轻由于激光源的直接调制而导致的宽线宽的影响。这些知识最近已应用于光放大传输系统的相干高级调制格式。

EOMs are utilized for modulation of either the phase or the intensity of the lightwave carrier. The later type is a combination of two electro-optic phase modulators (EOPMs) forming an interferometric configuration.
EOM 用于调制光波载流子的相位或强度。后一种类型是两个电光相位调制器 （EOPM） 的组合，形成干涉配置。

2.1. Optical Phase Modulator
2.1. 光相位调制器

Electro-optic phase modulator employs a single electrode as shown in Figure 3. When a RF driving voltage is applied onto the electrode, the refractive index changes accordingly inducing variation amount of delays of the propagating lightwave. Since the delays correspond to the phase changes, EOPM is used to carry out the phase modulation of the optical carrier.
电光相位调制器采用单个电极，如图 3 所示。当 RF 驱动电压施加到电极上时，折射率会相应地发生变化，从而引起传播光波的延迟变化量。由于延迟对应于相位变化，因此 EOPM 用于执行光载波的相位调制。

The induced phase variation is governed by the following equation:
感应相位变化由以下公式控制：

(1)

where V_π is the RF driving voltage required to create a π phase shift of the lightwave carrier and typically has a value within a range of 3V to 6V. The optical field at the output of an EOPM is generated given in following equation:
其中 V_π 是产生光波载波π相移所需的射频驱动电压，值通常在 3V 至 6V 范围内。EOPM 输出端的光场由以下公式给出：

(2)

where E_o is the transmitted optical field at the output the MZIM and noted in the low pass equivalent representation i.e the carrier is removed from the expression; V(t) is the time-varying signal voltage, Vbias is the DC bias voltage applied to the phase modulator.
其中 E_o 是输出端 MZIM 处的透射光场，并以低通等效表示形式表示，即从表达式中去除载波;V（t） 是时变信号电压，Vbias 是施加到相位调制器的直流偏置电压。

Recently, EOPMs operating at high frequency using resonant-type electrodes have been studied and proposed in [2,3]. Together with the advent of high-speed electronics which has evolved with the ASIC technology using 0.1µm GaAs P-HEMT or InP HEMTs [4], the contemporary EOPMs can now exceed 40Gb/s operating rate without much difficulty.
最近，[2,3] 研究并提出了使用谐振型电极在高频下工作的 EOPM。随着使用 0.1μm GaAs P-HEMT 或 InP HEMT [4] 的 ASIC 技术发展而来的高速电子设备的出现，现代 EOPM 现在可以毫不费力地超过 40Gb/s 的运行速率。

Such phase modulation can be implemented in MATLAB Simulink as shown in Figure 2(b) using a phase shift block of the Common Blockset. The phase bias is in one phase shift block and then the signal modulation or time dependent is fed into another phase shift block. The signals of the two parallel phase shift/modulation blocks are then combined to represent the interferometric construction and destruction, thus an intensity modulation can be achieved as described in the next sub-section.
这种相位调制可以在 MATLAB Simulink 中使用 Common Blockset 的相移模块实现，如图 2（b） 所示。相位偏置位于一个相移模块中，然后信号调制或瞬态输入到另一个相移模块中。然后将两个并联相移/调制模块的信号组合在一起，以表示干涉构造和破坏，从而可以实现下一小节中描述的强度调制。

2.2.  Optical Intensity Modulator
2.2. 光强度调制器

Optical intensity modulation is operating based on the principle of interference of the optical field of the two lightwave components. A LiNbO3 optical intensity modulator thus employs the interferometric structure as
光强度调制是根据两个光波分量的光场干涉原理运行的。因此，LiNbO3 光强度调制器采用干涉结构为

shown in Figure 4 and is most popularly well-known as the Mach-Zehnder interferometer (MZIM). The operational principles are briefly explained in the following paragraph. For the rest of the chapters in this paper, unless specifically indicated, the term of optical modulator is referred to the external LiNbO3 MZIM modulator.
如图 4 所示，最广为人知的名称是马赫-曾德尔干涉仪 （MZIM）。下一段将简要说明操作原理。对于本文的其余部分，除非特别说明，否则光调制器的术语是指外部 LiNbO3 MZIM 调制器。

The lightwave is split into two arms when entering the modulator. The power slitter is normally a 3-dB type i.e equally splitting the power of the optical signals. Each arm of the LiNbO3 modulator employs an electro-optic phase modulator in order to manipulate the phase of the optical carrier if required. At the output of the MZIM, the lightwaves of the two arm phase modulators are coupled and interfered with each other. The transfer curve of an MZIM is shown in Figure 4(c). A LiNbO3 MZIM modulator can be a single or dual drive type.
当光波进入调制器时，光波被分成两个臂。功率分路器通常是 3 dB 型，即平均分路光信号的功率。LiNbO3 调制器的每个臂都采用电光相位调制器，以便在需要时操纵光载波的相位。在 MZIM 的输出端，两个臂相位调制器的光波相互耦合并相互干扰。MZIM 的传输曲线如图 4（c） 所示。LiNbO3 MZIM 调制器可以是单驱动或双驱动类型。

In the case of single-drive MZIM, there is only a single RF voltage driving one arm of the MZIM. For instance, there is no RF driving voltage on arm 1, hence V₁(t) = 1 and the RF voltage V₂(t) applied on arm 2 is noted as V(t). The transmitted optical field E(t) at the output a single-drive MZIM as a function of the driving voltage V(t) and a bias DC voltages V_bias can be written as
在单驱动 MZIM 的情况下，只有一个 RF 电压驱动 MZIM 的一个臂。例如，臂 1 上没有射频驱动电压，因此 V₁（t） = 1，施加在臂 2 上的射频电压 V₂（t） 记为 V（t）。输出端的传输光场 E（t） 单驱动 MZIM 作为驱动电压 V（t） 和偏置直流电压 V_bias 的函数，可以写为

(3)

where V_π. is the required driving voltage to obtain a π phase shift in the lightwave carrier.
其中 V_π。是在光波载波载波中获得π相移所需的驱动电压。

It can be seen that the phase term in Equation (1) implies the existence of the modulation of the optical carrier phase and commonly known as the chirping effect. Thus, by using a single-drive MZIM, generated optical signals is not chirp-free. Furthermore, it is reported that a z-cut LiNbO3 MZIM can provide a modest amount of chirping due to its asymmetrical structure of the electrical field distributions whereas its counterpart x-cut
可以看出，方程（1）中的相位项暗示了光载波相位调制的存在，通常称为啁啾效应。因此，通过使用单驱动器 MZIM，生成的光信号并非没有啁啾。此外，据报道，由于其电场分布的不对称结构，z 切割 LiNbO3 MZIM 可以提供适量的啁啾，而其对应的 x 切割

MZIM is a chirp-free modulator thanks to the symmetrical or push-pull configuration of the electrical fields. Furthermore, also having a push-pull arrangement, complete elimination of chirping effect in modulation of the lightwave can be implemented with use of a dual-drive MZIM. The transmitted optical field E(t) at the output a MZIM as a function of the driving and bias voltages can be written as
MZIM 是一种无啁啾调制器，这要归功于电场的对称或推挽配置。此外，还具有推挽式布置，通过使用双驱动 MZIM，可以完全消除光波调制中的啁啾效应。输出 a MZIM 处的透射光场 E（t） 作为驱动电压和偏置电压的函数，可以写为

(4)

In a dual-drive MZIM, the RF driving voltage V₁(t) and V₂(t) are inverse with each other i.e V₂(t)=-V₁(t). Equation (4) indicates that there is no longer phase modulation component, hence the chirping effect is totally eliminated.
在双驱动 MZIM 中，射频驱动电压 V₁（t） 和 V₂（t） 彼此相反，即 V₂（t）=-V₁（t）。公式 （4） 表明不再有相位调制分量，因此完全消除了啁啾效应。

3.  Fiber Transmission Dynamics
3. 光纤传输动力学

3.1.  Chromatic Dispersion (CD)
3.1. 色散 （CD）

This section briefly presents the key theoretical concepts describing the properties of chromatic dispersion in a single-mode fiber. Another aim of this section is to introduce the key parameters which will be commonly mentioned in the rest of the paper.
本节简要介绍了描述单模光纤中色散特性的关键理论概念。本节的另一个目的是介绍本文其余部分通常提到的关键参数。

The initial point when mentioning to the chromatic dispersion is the expansion of the mode propagation constant or “wave number” parameter, β, using the Taylor series:
提到色散时，首先要说的是用泰勒级数扩展模式传播常数或“波数”参数，β：

(5)

where ω is the angular optical frequency, n(ω) is the frequency-dependent refractive index of the fiber. The parameters
其中 ω 是角光学频率，n（ω） 是光纤的频率相关折射率。参数 have different physical meanings as 1) β_o is involved in the phase velocity of the optical carrier which is defined as
具有不同的物理含义，因为 1） β_O 参与光载流子的相速度，定义为; 2) β₁determines the group velocity ν_g which is related to the mode propagation constant β of the guided mode by [5,6]
;2） β₁确定群速度 ν_g，它与 [5,6] 的引导模态的模式传播常数 β 有关

(6)

And 3) β2 is the derivative of group velocity with respect to frequency. Hence, it clearly shows the frequency-dependence of the group velocity. This means that different frequency components of an optical pulse travel at different velocities, hence leading to the spreading of the pulse or known as the dispersion. β₂ is therefore is known as the famous group velocity dispersion (GVD). The fiber is said to exhibit normal dispersion for β₂>0 or anomalous dispersion if β₂<0.
3） β2 是群速度相对于频率的导数。因此，它清楚地显示了群速度的频率依赖性。这意味着光脉冲的不同频率分量以不同的速度传播，从而导致脉冲的扩散或称为色散。因此，β₂ 被称为著名的群速度色散 （GVD）。据说该纤维在 β₂>0 时表现出正常色散，如果β₂<0，则表现出异常色散。

A pulse having the spectral width of is broadened by. In practice, a more commonly used factor to represent the chromatic dispersion of a single mode optical fiber is known as D (ps/nm.km). The dispersion factor is closely related to the GVD β2 and given by:
光谱宽度为 的 脉冲被拓宽 。在实践中，表示单模光纤色散的更常用的因子称为 D （ps/nm.km）。色散因子与 GVD β2 密切相关，由下式给出：at the operating wavelength λ; where β₃ defined as
在工作波长 λ;其中β₃ 定义为 contributes to the calculations of the dispersion slope, , which is an essential dispersion factor for high-speed DWDM transmission. can be obtained from the higher order derivatives of the propagation constant as
有助于计算色散斜率 ，色散斜率是高速 DWDM 传输的基本色散因子。 可以从传播常数的高阶导数中获得，如

(7)

A well-known parameter to govern the effects of chromatic dispersion imposing on the transmission length of an optical system is known as the dispersion length LD. Conventionally, the dispersion length LD corresponds to the distance after which a pulse has broadened by one bit interval. For high capacity long-haul transmission employing external modulation, the dispersion limit can be estimated in the following Equation [8].
控制色散对光学系统传输长度影响的一个众所周知的参数称为色散长度 LD。通常，色散长度 LD 对应于脉冲扩大一个比特间隔的距离。对于采用外部调制的高容量长距离传输，色散极限可以用下面的公式 [8] 来估计。

(8)

where B is the bit rate (Gb/s), D is the dispersion factor (ps/nm km) and L_D is in km.
其中 B 是比特率 （Gb/s），D 是色散因子 （ps/nm km），L_D 以 km 为单位。

Equation (8) provides a reasonable approximation even though the accurate computation of this limit that depends the modulation format, the pulse shaping and the optical receiver design. It can be seen clearly from (8) that the severity of the effects caused by the fiber chromatic dispersion on externally modulated optical signals is inversely proportional to the square of the bit rate. Thus, for 10 Gb/s OC-192 optical transmission on a standard single mode fiber (SSMF) medium which has a dispersion of about ±17 ps/nm.km, the dispersion length L_D has a value of approximately 60 km i.e corresponding to a residual dispersion of about ±1000 ps/nm and less than 4 km or equivalently to about ± 60 ps/nm in the case of 40Gb/s OC-768 optical systems. These lengths are a great deal smaller than the length limited by ASE noise accumulation. The chromatic dispersion therefore, becomes the one of the most critical constraints for the modern high-capacity and ultra long-haul transmission optical systems.
等式 （8） 提供了一个合理的近似值，即使该限值的精确计算取决于调制格式、脉冲整形和光接收器设计。从（8）中可以清楚地看出，光纤色散对外部调制光信号的影响的严重性与比特率的平方成反比。因此，对于在色散约为 ±17 ps/nm.km 的标准单模光纤 （SSMF） 介质上的 10 Gb/s OC-192 光传输，色散长度 L_D 的值约为 60 km，即对应于约 ±1000 ps/nm 的残余色散，小于 4 km 或相当于约 ± 60 ps/nm（在 40Gb/s OC-768 光学系统的情况下）。这些长度比 ASE 噪声累积限制的长度小得多。因此，色散成为现代高容量和超长距离传输光学系统最关键的限制因素之一。

3.2.  Polarization Mode Dispersion (PMD)
3.2. 偏振模色散 （PMD）

Polarization mode dispersion (PMD) represents another type of the pulse spreading. The PMD is caused by the
偏振模色散 （PMD） 代表了另一种类型的脉冲扩散。PMD 是由

differential group delay (DGD) between two principle orthogonal states of polarization (PSP) of the propagating optical field.
传播光场的两个主要正交偏振态 （PSP） 之间的差分群延迟 （DGD）。

One of the intrinsic causes of PMD is due to the asymmetry of the fiber core. The other causes are derived from the deformation of the fiber including stress applied on the fiber, the aging of the fiber, the variation of temperature over time or effects from a vibration source. These processes are random resulting in the dynamic of PMD. The imperfection of the core or deformation of the fiber may be inherent from the manufacturing process or as a result of mechanical stress on the deployed fiber resulting in a dynamic aspect of PMD.
PMD 的内在原因之一是由于纤芯的不对称性。其他原因来自纤维的变形，包括施加在纤维上的应力、纤维的老化、温度随时间的变化或振动源的影响。这些过程是随机的，导致了 PMD 的动态。纤芯的缺陷或纤维的变形可能是制造过程所固有的，也可能是由于展开的纤维上的机械应力导致 PMD 的动态方面。

The delay between these two PSP is normally negligibly small in 10Gb/s optical transmission systems. However, at high transmission bit rate for long-haul and ultra long-haul optical systems, the PMD effect becomes much more severe and degrades the system performance  [9-12]. The DGD value varies along the fiber following a stochastic process. It is proven that these DGD values complies with a Maxwellian distribution as shown in Figure 7 [10,13,14].
在 10Gb/s 光传输系统中，这两个 PSP 之间的延迟通常可以忽略不计。然而，在长距离和超长距离光学系统的高传输比特率下，PMD 效应变得更加严重，并降低了系统性能 [9-12]。DGD 值在随机过程之后沿纤维变化。事实证明，这些DGD值符合图7所示的麦克斯韦分布[10,13,14]。

(9)

where is differential group delay over a segment of the optical fiber. The mean DGD value is commonly termed as the “fiber PMD” and normally given by the fiber manufacturer.
其中 是光纤 一段上的差分群延迟。平均 DGD 值 通常称为“纤维 PMD”，通常由纤维制造商给出。

An estimate of the transmission limit due to PMD effect is given as:
由于 PMD 效应引起的传输限制的估计值如下：

(10)

where R is the transmission bit rate. Therefore, =1 ps/km (older fiber vintages); Bit rate = 40 Gbit/s; Lmax=12.5 Km; Bit rate =10 Gbit/s; Lmax=200 Km; =0.1 ps/km (contemporary fiber for modern optical systems); Bit rate = 40 Gbit/s; Lmax=1250 Km ; thence for Bit rate = 10 Gbit/s ; Lmax=20.000 Km.
其中 R 是传输比特率。因此， =1 ps/km（较旧的纤维年份）;比特率 = 40 Gbit/s;Lmax=12.5 公里;比特率 = 10 Gbit/s;Lmax=200 公里; =0.1 ps/km （用于现代光学系统的现代光纤）;比特率 = 40 Gbit/s;Lmax=1250 公里;因此，比特率 = 10 Gbit/s ;Lmax=20.000 公里。

Thus PMD is an important impairment of ultra long distance transmission system even at 10 Gb/s optical transmission. Upgrading to higher bit rate and higher capacity, PMD together with CD become the most two critical impairments imposing on the limitation of the optical systems.
因此，即使在 10 Gb/s 光传输时，PMD 也是超长距离传输系统的重要损伤。升级到更高的比特率和更高的容量，PMD 和 CD 成为对光学系统限制施加的两大关键损伤。

3.3.  Fiber Nonlinearity 3.3. 纤维非线性

The fiber refractive index is not only dependent of wavelength but also of intensity of the lightwave. This well-known phenomenon which is named as the Kerr effect is normally referred as the fiber nonlinearity. The power dependence of the refractive index n_r is shown in the following expression
光纤折射率不仅取决于波长，还取决于光波的强度。这种众所周知的现象被称为克尔效应，通常被称为纤维非线性。折射率 n_r 的功率依赖性显示在下面的表达式中

(11)

P is the average optical intensity inside the fiber, is the nonlinear-index coefficient and A_eff is the effective area of the fiber.
P 是光纤内部的平均光强度， 是非线性指数系数，A_eff 是光纤的有效面积。

There are several non-linearity phenomena induced from the Kerr effects including intra-channel self-phase modulation (SPM), cross phase modulation between inter-channels (XPM). four wave mixing (FWM), stimulated Raman scattering (SRS) and stimulated Brillouin scattering (SBS). SRS and SBS are not main degrading factors compared to the others. FWM effect degrades performance of an optical system severely if the local phase of the propagating channels are matched with the introduction of the ghost pulse. However, with high local dispersion parameter such as in SSMF or even in NZDSF, effect of the FWM becomes negligible. XPM is strongly dependent on the channel spacing between the channels and also on local dispersion factor of the optical fiber [refs]. [ref] also report about the negligible effects of XPM on the optical signal compared the SPM effect. Furthermore, XPM can be considered to be negligible in a DWDM system in the following scenarios: 1) highly locally dispersive system e.g SSMF and DCF deployed systems; 2) large channel spacing and 3) high spectral efficiency [15-19]. However, the XPM should be taken in to account for the systems deploying Non-zero dispersion shifted fiber (NZ-DSF) where the local dispersion factor is low. The values of the NZ-DSF dispersion factors can be obtained from Figure 5. Among nonlinearity impairments, SPM is considered to be the major shortfalls in the system.
Kerr 效应会引起几种非线性现象，包括通道内自相位调制 （SPM）、通道间交叉相位调制 （XPM）。四波混合 （FWM）、受激拉曼散射 （SRS） 和受激布里渊散射 （SBS）。与其他因素相比，SRS 和 SBS 不是主要的降解因素。如果传播通道的局部相位与虚影脉冲的引入相匹配，则 FWM 效应会严重降低光学系统的性能。然而，对于高局部色散参数，例如 SSMF 甚至 NZDSF，FWM 的影响变得可以忽略不计。XPM 在很大程度上取决于通道之间的通道间隔以及光纤的局部色散因数 [refs]。[ref] 还报告了与 SPM 效应相比，XPM 对光信号的影响可以忽略不计。此外，在以下情况下，XPM 在 DWDM 系统中可以被认为可以忽略不计：1） 高度局部分散的系统，例如 SSMF 和 DCF 部署系统;2） 大通道间隔和 3） 高光谱效率 [15-19]。但是，应考虑 XPM 部署局部色散因子较低的非零色散偏移光纤 （NZ-DSF） 的系统。NZ-DSF 色散因子的值可以从图 5 中获得。在非线性损伤中，SPM 被认为是系统的主要缺陷。

In this paper, only the SPM non-linearity is generally considered. This is the main degradation factor for high bit rate transmission system where the signal spectrum is broadened. The effect of SPM is normally coupled with the nonlinear phase shift which is defined as
在本文中，通常只考虑 SPM 非线性。这是信号频谱变宽的高比特率传输系统的主要劣化因素。SPM 的效应通常与非线性相移耦合，其定义为

(12)

where w_c is the lightwave carrier, L_eff is the effective transmission length and α is the attenuation factor of a SSMF which normally has a value of 0.17-0.2 dB/km for the currently operating wavelengths within the 1550nm window. The temporal variation of the non-linear phase f_NL while the optical pulses propagating along the fiber results in the generation of new spectral components far apart from the lightwave carrier w_c implying the broadening of the signal spectrum. The spectral broadening dw which is well-known as frequency chirping can be explained based on the time dependence of the nonlinear phase shift and given by the expression:
其中 w_c 是光波载波，L_eff 是有效传输长度，α 是 SSMF 的衰减系数，对于 1550nm 窗口内的当前工作波长，其值通常为 0.17-0.2 dB/km。当光脉冲沿光纤传播时，非线性相位 f_NL 的时间变化导致产生远离光波载波 w_c 的新光谱分量，这意味着信号光谱的拓宽。频谱展宽 dw，即众所周知的频率啁啾，可以根据非线性相移的时间依赖性来解释，表达式如下：

           (13)

From (13), the amount of dw is proportional to the time derivative of the signal power P. Correspondingly, the generation of new spectral components may mainly occur the rising and falling edges of the optical pulse shapes, i.e. the amount of generated chirp is larger for an increased steepness of the pulse edges.
从（13）中，dw 的量与信号功率 P 的时间导数成正比。相应地，新光谱分量的产生可能主要发生在光脉冲形状的上升沿和下降沿，即产生的啁啾量更大，脉冲边缘的陡峭度增加。

4.  Modeling of Fiber Propagation
4. 纤维传播建模

4.1 Non-linear Schrodinger Equation (NLSE)
4.1 非线性薛定谔方程 （NLSE）

Evolution of the slow varying complex envelope A(z,t) of the optical pulses along a single mode optical fiber is governed by the well-known nonlinear Schroedinger equation (NLSE):
光脉冲沿单模光纤的慢速变化复包络 A（z，t） 的演变由众所周知的非线性薛定谔方程 （NLSE） 控制：

(14)

where z is the spatial longitudinal coordinate, α accounts for fiber attenuation, indicates the differential group delay (DGD), and represent 2^nd and 3^rd order factors of the group velocity dispersion (GVD) and is the nonlinear coefficient. Equation (14) involves the following effects in a single-channel transmission fiber: 1) the attenuation, 2) chromatic dispersion, 3) 3^rd order dispersion factor i.e the dispersion slope, and 4) self phase modulation nonlinearity. Other critical degradation factors such as the non-linear phase noise due to the fluctuation of the optical intensity caused by ASE noise via Gordon-Mollenauer effect [20] is mutually included in the equation.
其中 z 是空间纵坐标，α考虑光纤衰减， 表示差分群延迟 （DGD）， 表示群速度色散 （GVD） 的 2^阶和 3^阶因子， 是非线性系数。公式 （14） 涉及单通道传输光纤中的以下影响：1） 衰减，2） 色散，3） 3^阶色散因数，即色散斜率，以及 4） 自相位调制非线性。其他关键的劣化因素，例如由 ASE 噪声通过 Gordon-Mollenauer 效应 [20] 引起的光强度波动引起的非线性相位噪声，也包含在方程中。

4.2.  Symmetrical Split Step Fourier Method
4.2. 对称分步傅里叶法

In this Paper, solutions of the NLSE and hence the model of pulse propagation in a single mode optical fiber is numerically solved by using the popular approach of the split step Fourier method (SSFM) [5] in which the fiber length is divided into a large number of segments of small step size.
在本文中，NLSE 的解以及单模光纤中脉冲传播的模型通过使用流行的分步傅里叶法 （SSFM） [5] 方法进行数值求解，其中光纤长度被分成大量小步长 的段。

In practice, dispersion and nonlinearity are mutually interactive while the optical pulses propagate through the fiber. However, the SSFM assumes that over a small length, the effects of dispersion and the nonlinearity on the propagating optical field are independent. Thus, in SSFM, the linear operator representing the effects of fiber dispersion and attenuation and the nonlinearity operator taking into account fiber nonlinearities are defined separately as
在实践中，当光脉冲通过光纤传播时，色散和非线性是相互交互的。然而，SSFM 假设在小长度 上，色散和非线性对传播光场的影响是独立的。因此，在 SSFM 中，表示光纤色散和衰减影响的线性算子和考虑光纤非线性的非线性算子分别定义为

(15)

where A replace sfor simpler notation and T=t-z/v_g is the reference time frame moving at the group velocity. The NLSE Equation (14) can be rewritten as
其中 A 替换 s 表示更简单的符号，T=t-z/v_g 是以群速度移动的参考时间框架。NLSE 方程 （14） 可以改写为

(16)

and the complex amplitudes of optical pulses propagating from z to z+ is calculated using the approximation as given:
从 z 传播到 z+ 的光脉冲的复振幅使用给定的近似值计算：

(17)

Equation (14) is accurate to second order in the step size. The accuracy of SSFM can be improved by including the effect of the nonlinearity in the middle of the segment rather than at the segment boundary as illustrated in Equation (17) can now modified as
方程 （14） 精确到步长 的二阶。通过将非线性的影响包含在段的中间而不是在段边界处，可以提高 SSFM 的精度，如方程 （17） 所示，现在可以修改为

(18)

This method is accurate to third order in the step size. The optical pulse is propagated down segment from segment in two stages at each step. First, the optical pulse propagates through the first linear operator (step of /2) with dispersion effects taken into account only. The nonlinearity is calculated in the middle of the segment. It is noted that the nonlinearity effects is considered as over the whole segment. Then at z+/2, the pulse propagates through the remaining /2 distance of the linear operator. The process continues repetitively in executive segments until the end of the fiber. This method requires the careful selection of step sizes to reserve the required accuracy.
此方法精确到步长 的三阶。光脉冲在每一步分两阶段逐段向下传播。首先，光脉冲通过第一个线性算子（步长 /2）传播，仅考虑色散效应。非线性在段的中间计算。值得注意的是，非线性效应被认为是在整个 Segment 上。然后在 z+ /2 处，脉冲通过线性算子的剩余 /2 距离传播。该过程在执行部门 中重复继续，直到纤维结束。这种方法需要仔细选择步长 ，以保留所需的精度。

The Simulink model of the lightwave signals propagation through optical fiber is shown in Figure 8(b). All parameters required for the propagation model are fed as the inputs into the block. The propagation algorithm split-steps and FFT are written in .m files in order to simplify the model. This demonstrates the effectiveness of the linkage between MATLAB and Simulink. A Matlab program is used for modeling of the propagation of the guided lightwave signals over very long distance is given in the Appendix.
光波信号通过光纤传播的 Simulink 模型如图 8（b） 所示。propagation model 所需的所有参数都作为 input 馈送到 block 中。传播算法 split-steps 和 FFT 被写入 .m 文件中，以简化模型。这证明了 MATLAB 和 Simulink 之间链接的有效性。附录中给出了一个 Matlab 程序，用于模拟引导光波信号在很长距离上的传播。

4.3.  Modeling of Polarization Mode Dispersion (PMD)
4.3. 偏振模色散 （PMD） 建模

The first order PMD effect can be implemented by splitting the optical field into two distinct paths representing two states of polarizations with different propagating delays, then implementing SSFM over the segment before superimposing the outputs of these two paths for the output optical field.
一阶 PMD 效应可以通过将光场分成两条不同的路径来实现，这些路径代表具有不同传播延迟 的两种偏振状态，然后在将这两条路径的输出叠加到输出光场之前在段 上实施 SSFM。

The transfer function for first-order PMD is given by [21].
一阶 PMD 的传递函数由 [21] 给出。

(19)

  and 和

with is the splitting ratio. The usual assumption is =1/2. Finite impulse response filter blocks of the digital signal processing blocksets of Simulink can be applied here without much difficulty to represent the PMD effects with appropriate delay difference.
with 是分光比。通常的假设是 =1/2。Simulink 的数字信号处理模块集的有限脉冲响应滤波器模块可以在这里应用，以适当的延迟差表示 PMD 效应，没有太大困难。

4.4.  Fiber Propagation in Linear Domain
4.4. 线性域中的光纤传播

Here, the low pass equivalent frequency response of the optical fiber, noted as H(f) has a parabolic phase profile and can be modeled by the following equation, [22]
在这里，光纤的低通等效频率响应，记为 H（f），具有抛物线相位曲线，可以用以下公式建模 [22]

(20)

where, represents the Group Velocity Distortion (GVD) parameter of the fiber and L is the length of the fiber. The parabolic phase profile is the result of the chromatic dispersion of the optical fiber [23]. The 3^rd order dispersion factor b₃ is not considered in this transfer function of the fiber due to negligible effects on 40Gb/s transmission systems. However, if the transmission bit rate is higher than 40Gb/s, the b₃ should be taken into account.
其中， 表示纤维的群速度失真 （GVD） 参数，L 是纤维的长度。抛物线相位分布是光纤色散的结果 [23]。由于对 40Gb/s 传输系统的影响可以忽略不计，因此在光纤的此传递函数中不考虑 3^阶色散因子 b₃。但是，如果传输比特率高于 40Gb/s，则应考虑 b₃。

In the model of the optical fiber, it is assumed that the signal is propagating in the linear domain, i.e. the fiber nonlinearities are not included in the model. These nonlinear effects are investigated numerically. It is also assumed that the optical carrier has a line spectrum. This is a valid assumption considering the stateof-the-art laser sources nowadays with very narrow linewidth and the use of external modulators in signal transmission.
在光纤的模型中，假设信号在线性域中传播，即光纤非线性不包括在模型中。这些非线性效应是通过数值研究的。还假设光载波具有线谱。考虑到当今最先进的激光源具有非常窄的线宽，并且在信号传输中使用外部调制器，这是一个有效的假设。

A pure sinusoidal signal of frequency f, propagating through the optical fiber, experiences a delay of |2πf_Db₂L|. The standard fibers used in optical communications have a negative and thus, in low pass equivalent representation, sinusoids with positive frequencies (i.e. frequencies higher than the carrier) have negative delays, i.e. arrive early compared to the carrier and the ones with negative frequencies (i.e. frequencies lower than the carrier) have positive delays and arrive delayed. The dispersion compensating fibers have positive and so have reverse effects. The low pass equivalent channel impulse response of the optical fiber, has also followed a parabolic phase profile and is given as,
频率为 f 的纯正弦信号，通过光纤传播，会经历 |2πf_Db₂L|的延迟。光通信中使用的标准光纤具有负 光纤，因此，在低通等效表示中，具有正频率（即频率高于载波）的正弦波具有负延迟，即与载波相比，到达时间较早，而具有负频率（即频率低于载波）的正延迟和延迟到达。分散补偿纤维具有积极 作用，因此具有相反的效果。光纤 的低通等效通道脉冲响应也遵循抛物线相位曲线，其公式为：

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5.  Optical Amplifier 5. 光放大器

5.1.  ASE Noise of Optical Amplifier
5.1. 光放大器的 ASE 噪声

The following formulation accounts for all noise terms that can be treated as Gaussian noise
以下公式说明了所有可以被视为高斯噪声的噪声项

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G =amplifier gain; nsp = spontaneous emission factor; m =number of polarization modes (1 or 2); PN =mean noise in bandwidth; OSNR at the output of EDFA.
G = 放大器增益;nsp = 自发排放因子;m = 偏振模式的数量（1 或 2）;PN = 带宽中的平均噪声;EDFA 输出的 OSNR。

5.2.  Optical Amplifier Noise Figure
5.2. 光放大器噪声系数

Amplifier Noise Figure (NF) is defined at the output of the optical amplifier as the ratio between the output OSNR on the OSNR at the input of the EDFA.
放大器噪声系数 （NF） 在光放大器的输出端定义为输出 OSNR 与 EDFA 输入端的 OSNR 之比。

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A Simulink model of the optical amplifier is shown in that represents all the system operational parameters of such amplifier. Only blocks of the Common Blockset of Simulink are used.
显示了光放大器的 Simulink 模型，该模型表示此类放大器的所有系统操作参数。仅使用 Simulink 的 Common Blockset 的模块。

6.  Optical Filter 6. 滤光片

In this paper, optical filtering of the noise-corrupted optical signals is conducted with a Gaussian-type filter whose 3dB bandwidth is governed by
在本文中，使用高斯型滤波器对噪声损坏的光信号进行光滤波，其 3dB 带宽由

(24)

where  哪里 in which B is the Gaussian filter’s 3-dB bandwidth and T is the bit rate. The BT product parameter is B times the input signal’s bit period.
其中 B 是高斯滤波器的 3 dB 带宽，T 是比特率。BT 乘积参数是 B 乘以输入信号的位周期。

The modeling of an electrical filter can also use a Gaussian filter with similar impulse response as defined in (24) or a conventional analog 5^th order Bessel filter which can be easily designed using filter design toolbox in Matlab. The Matlab pseudo-codes for designing an analog 5^th order Bessel filter are shown as follows:
电滤波器的建模也可以使用 （24） 中定义的具有类似脉冲响应的高斯滤波器或传统的模拟 5^阶贝塞尔滤波器，可以使用 Matlab 中的滤波器设计工具箱轻松设计。用于设计模拟 5^阶贝塞尔滤波器的 Matlab 伪代码如下所示：

[b,a] = besself(5^thorder,2*pi*BT_b/os_fac); %Analog filter
[b，a] = besself（5^阶，2*pi*BT_b/os_fac）;%模拟滤波器

[bz,az] = impinvar(b,a,1); %Digital filter
[bz，az] = impinvar（b，a，1）;%数字滤波器

[hf t1] = impz(bz,az,2*delay*os_fac+1,os_fac);
[hf t1] = impz（bz，az，2*延迟*os_fac+1，os_fac）;

In the above pseudo-codes, the BT product parameter is defined similarly to that in the case of a Gaussian filter. Alternatively, the transfer function of an analog 5^th order Bessel filter can be referred from [24].
在上面的伪代码中，BT 乘积参数的定义类似于高斯滤波器中的定义。或者，模拟 5^阶贝塞尔滤波器的传递函数可以从 [24] 中引用。

7.  Optical Receiver 7. 光接收器

The demodulation of the original message is carried out in electrical domain, thus the conversion of lightwaves to electrical signals is required. In digital optical communication, this process has been widely implemented with a PIN photodiode in a coherent or incoherent detection. The first type requires a local oscillator to coherently down-convert the modulated lightwave from optical frequency to IF frequency. The second type which has been the preferred choice for currently deployed systems is the incoherent detection which is based on square-law envelop detection of the optical signals. For incoherent detection, the recovery of clock timing is critical. In the rest of this Paper and in the simulations, ideal clock timing is assumed.
原始消息的解调是在电域中进行的，因此需要将光波转换为电信号。在数字光通信中，这一过程已通过 PIN 光电二极管在相干或非相干检测中得到广泛实施。第一种类型需要一个本振器，将调制光波从光频率相干地下转换到 IF 频率。第二种类型是当前部署系统的首选非相干检测，它基于光信号的平方律包络检测。对于非 coendent detection， clock timing 的恢复至关重要。在本文的其余部分和 simulations 中，假设了理想的 clock timing 。

After detection, the electrical current is normally amplified with a trans-impedance amplifier before passing through an electrical filter which is normally of Bessel type. The bandwidth of the electrical filter generally varies between 0.6 and 0.8 R. At this point, electrical eye diagrams are normally observed for the assessment of signal quality. Sampling of electrically filtered received signals is next carried out. Without use of electronic equalizers, hard decision which compares the received signal level to a pre-set threshold for making the decision is implemented.
检测后，电流通常用跨阻放大器放大，然后通过通常为贝塞尔型的电气滤波器。电滤波器的带宽通常在 0.6 到 0.8 R 之间变化。此时，通常会观察电眼图来评估信号质量。接下来对电滤波的接收信号进行采样。在不使用电子均衡器的情况下，将接收到的信号电平与预设的阈值进行比较以进行决策的硬决策。

For advanced phase modulation formats such as DPSK, CPFSK or MSK, a MZDI-based balanced receiver with two photodiodes connected back-to-back is required. Excluding the distortions of waveform due to fiber dynamics and from the analytical point of view, the received electrical signals are corrupted with noise from several sources including 1) shot noise (), 2) electronic noise of trans-impedance amplifier, 3) dark current noise and 4) interactions between signals and ASE noise () and between ASE noise itself as
对于 DPSK、CPFSK 或 MSK 等高级相位调制格式，需要一个基于 MZDI 的平衡接收器，其中两个光电二极管背靠背连接。从分析的角度来看，不包括由于光纤动力学引起的波形失真，接收到的电信号会被来自多个来源的噪声破坏，包括 1） 散粒噪声 （ ），2） 跨阻放大器的电子噪声 ，3） 暗电流噪声 和 4） 信号与 ASE 噪声之间的相互作用 （ ） 以及 ASE 噪声本身 之间的相互作用

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These noise sources are usually modeled with normal distributions whose variances representing the noise power are defined as
这些噪声源通常使用正态分布进行建模，其方差表示噪声功率定义为

1) Shot noise is caused by the intrinsic electro-optic phenomenon of the semiconductor photodiode in which a random number of electron-hole pairs is generated with the receipt of photons causing the randomness of the induced photo-current. The shot noise is given in the following formula:
1） 散粒噪声是由半导体光电二极管的固有电光现象引起的，其中接收光子时会产生随机数量的电子-空穴对，从而导致感应光电流的随机性。散粒噪声由以下公式给出：

(26)

where Be is the 3dB bandwidth of the electrical filter, < i_s> is the average signal-only photo-current after the photodiodes.
其中 Be 是电滤波器的 3dB 带宽，< i_s> 是光电二极管之后的平均仅信号光电流。

2) The electronic noise source is injected from the trans-impedance amplifier. It is modeled with an equivalent noise current density iNeq over the bandwidth of the electrical filter. The unit of iNeq is A/ and the value of is obtained as (iNeq) 2Be.
2） 电子噪声源 从跨阻放大器注入。它是在电滤波器的带宽上以等效的噪声电流密度 iNeq 建模的。iNeq 的单位是 A/ ，的值 为 （iNeq） 2Be。

3) Value of dark current idark is normally specified with a particular photo-diode and has the unit of A/Hz. Hence, the noise power is calculated as idark Be.
3） 暗电流 idark 的值通常由特定的光电二极管指定，单位为 A/Hz。因此，噪声功率 计算为 idark Be。

4) The variances of amplitude fluctuations due to the beating of signal and ASE noise and between ASE noise itself are governed by the following expressions:
4） 由于信号和 ASE 噪声的跳动以及 ASE 噪声本身之间的幅度波动的变化由以下表达式控制：

(27)

(28)

where B_opt is the 3dB bandwidth of the optical filter and i_N is the noise-induced photo-current. In practice, the value of is normally negligible compared to the value of and can be ignored without affecting the performance of the receiver.
其中 B_opt 是滤光片的 3dB 带宽，i_N 是噪声感应的光电流。在实践中，与 value 相比，该值 of 通常可以忽略不计 ，并且可以忽略，而不会影响接收器的性能。

It is worth noting that in an optically pre-amplified receiver, i.e. the optical signal is amplified at a stage before the photo-detector, is the dominant factor compared to other noise sources.
值得注意的是，与其他噪声源相比，在光学预放大接收器中，即光信号在光电探测器之前的阶段被放大， 是主导因素。

8.  Performance Evaluation
8. 性能评估

Performance evaluation of an optical transmission system via the quality of the electrically detected signals is an essential aspect in both simulation and experiment scenarios. The key metrics reflecting the signal quality include optical signal to noise ratio (OSNR) and OSNR penalty, eye opening (EO) and eye opening penalty (EOP) where as bit error rate (BER) is the ultimate indicator for the performance of a system.
通过电检测信号的质量来评估光传输系统的性能是仿真和实验场景中的一个重要方面。反映信号质量的关键指标包括光信噪比 （OSNR） 和 OSNR 损失、眼图张开度 （EO） 和眼图张开度损失 （EOP），其中误码率 （BER） 是系统性能的最终指标。

In an experimental set-up and practical optical systems, BER and the quality factor Q-factor can be obtained directly from the modern BERT test-sets and data can be exported to a portable memory for post-processing. However, it is noted that these experimental systems need to be run within at least a few hours so that the results are stable and accurate.
在实验装置和实际光学系统中，BER 和品质因数 Q 因子可以直接从现代 BERT 测试装置中获得，数据可以导出到便携式存储器进行后处理。但是，需要注意的是，这些实验系统至少需要在几个小时内运行，以便结果稳定和准确。

For the case of investigation of performance of an optical transmission system by simulation, several methods have been developed such as
对于通过仿真研究光传输系统性能的情况，已经开发了几种方法，例如

1) Monte Carlo numerical method
1） 蒙特卡洛数值法

2) Conventional method to calculate Q-factor, Q dB and hence BER based on assumption of Gaussian distribution of noise.
2） 基于噪声高斯分布假设计算 Q 因子、Q dB 和 BER 的传统方法。

3) Methods based on statistical processes taking into account the distortion from the dynamic effects of the optical fibers including the ISI induced by CD, PMD and tight optical filtering.
3） 基于统计过程的方法，考虑了光纤动态效应的失真，包括 CD、PMD 和紧密滤波诱导的 ISI。

•The first statistical technique implements the Expected Maximization theory in which the pdf of the obtained electrical detected signal is approximated as a mixture of multiple Gaussian distributions.
•第一种统计技术实现了预期最大化理论，其中获得的电检测信号的 pdf 近似为多个高斯分布的混合。

•The second technique is based on the Generalized Extreme Values theorem. Although this theorem is wellknown in other fields such as financial forecasting, meteorology, material engineering, etc to predict the probability of occurrence of extreme values, it has not much studied to be applied in optical communications.
•第二种技术基于广义极值定理。虽然这个定理在金融预测、气象学、材料工程等其他领域广为人知，可以预测极值出现的概率，但它在光通信中的应用并没有得到太多的研究。

8.1. Monte Carlo Method 8.1. 蒙特卡洛法

Similar to the bit error rate test (BERT) equipment commonly used in experimental transmission, the BER in a simulation of a particular system configuration can be counted. The BER is the ratio of the occurrence of errors (N_error) to the total number of transmitted bits N_total and given as:
与实验传输中常用的误码率测试 （BERT） 设备类似，可以计算特定系统配置模拟中的 BER。BER 是差错发生次数（N_个错误）与传输比特总数 N_个总数的比率，公式为：

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Monte Carlo method offers a precise picture via the BER metric for all modulation formats and receiver types. The optical system configuration under a simula-
蒙特卡洛方法通过 BER 度量为所有调制格式和接收机类型提供精确的图像。模拟 - 下的光学系统配置

tion test needs to include all the sources of impairments imposing to signal waveforms including the fiber impairments and ASE (optical)/electronic noise.
测试需要包括对信号波形施加的所有损伤来源，包括光纤损伤和 ASE（光学）/电子噪声。

It can be seen that a sufficient number of transmitted bits for a certain BER is required and leading to exhaustive computational time. In addition, time-consuming algorithms such as FFT especially carried out in symmetrical SSFM really contribute to the long computational time. A BER of 1e-9 which is considered as ’error free’ in most scientific publications requires a number of at least 1e10 bits transmitted.
可以看出，对于某个 BER 需要足够数量的传输比特，这会导致计算时间穷举。此外，耗时的算法（如 FFT），尤其是在对称 SSFM 中执行，确实会导致计算时间较长。在大多数科学出版物中，1e-9 的 BER 被认为是“无差错”的，它需要传输至少 1e10 比特的数量。

However, 1e-6 even 1e-7 is feasible in Monte Carlo simulation. Furthermore, with use of forward error coding (FEC) schemes in contemporary optical systems, the reference for BERs to be obtained in simulation can be as low as 1e-3 provided no sign of error floor is shown. This is normally known as the FEC limit. The BERs obtained from the Monte Carlo method is a good benchmarking for other BER values estimated in other techniques. The time required for completion of the simulation may take several hours to reach BER of 1e-9. Thus statistical methods can be developed to determine the BER of transmission systems to save time. This is addressed in the next section.
但是，1e-6 甚至 1e-7 在 Monte Carlo 仿真中是可行的。此外，通过在现代光学系统中使用前向误差编码 （FEC） 方案，只要没有显示本底误差的迹象，则在仿真中获得的 BER 参考可以低至 1e-3。这通常称为 FEC 限制。从 Monte Carlo 方法获得的 BER 是其他技术中估计的其他 BER 值的良好基准。完成模拟所需的时间可能需要几个小时才能达到 1e-9 的 BER。因此，可以开发统计方法来确定传输系统的 BER 以节省时间。这将在下一节中讨论。

8.2. BER and Q-Factor from Probability Distribution Functions (PDF)
8.2. 概率分布函数的 BER 和 Q 因子 （PDF）

This method implements a statistical process before calculating values of BER and quality Q-factor to determine the normalized probability distribution functions (PDF) of received electrical signals (for both “1” and “0” and at a particular sampling instance). The electrical signal is normally in voltage since the detected current after a photo-diode is usually amplified by a transimpedance electrical amplifier. The PDFs can be determined statistically by using the histogram approach.
该方法在计算 BER 值和质量 Q 因子之前实施一个统计过程，以确定接收电信号的归一化概率分布函数 （PDF）（对于“1”和“0”以及在特定采样实例下）。电信号通常为电压，因为光电二极管后检测到的电流通常由跨阻电放大器放大。可以使用直方图方法对 PDF 进行统计确定。

A particular voltage value as a reference for the distinction between “1” and “0” is known as the threshold voltage (V_th). The BER in case of transmitting bit “1” (receiving as “0” instead) is calculated from the wellknown principle [25], i.e. the integral of the overlap of normalized PDF of “1”exceeding the threshold. Similar calculation for bit “0” is applied. The actual shape of the PDF is thus very critical to obtain an accurate BER. If the exact shape of the PDF is known, the BER can be calculated precisely as:
作为区分“1”和“0”的参考的特定电压值称为阈值电压 （V_th）。发送位 “1” （接收为 “0”） 时的 BER 是根据众所周知的原理 [25] 计算的，即 “1” 的归一化 PDF 重叠超过阈值的积分。对位 “0” 应用类似的计算。因此，PDF 的实际形状对于获得准确的 BER 非常关键。如果知道 PDF 的确切形状，则 BER 可以精确计算为：

(30)

where: is the probability that a “1” is sent; is the probability of error due to receiving “0” where actually an “1” is sent; is the probability that a “0” is sent; is the probability of error due to receiving “1” where actually a “0” is sent; As commonly used, the probability of transmitting a “1” and “0’ is equal i.e.
其中： 是发送 “1” 的概率; 是由于收到 “0” 而导致出错的概率，而实际上发送的是 “1”; 是发送 “0” 的概率; 是由于接收 “1” 而实际发送 “0” 而出错的概率;通常，传输 “1” 和 “0” 的概率相等，即 。

A popular approach in both simulation and commercial BERT test-sets is the assumption of PDF of “1” and “0” following Gaussian/normal distributions, i.e noise sources are approximated by Gaussian distributions. If the assumption is valid, high accuracy is achieved. This method enables a fast estimation of the BER by using the complementary error functions [25]:
仿真和商业 BERT 测试集中的一种流行方法是假设 PDF 为“1”和“0”，遵循高斯/正态分布，即噪声源由高斯分布近似。如果假设有效，则获得高准确度。该方法通过使用互补误差函数 [25] 来快速估计 BER：

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where and are the mean values for PDF of “1” and “0” respectively whereas and are the variance of the PDFs. The quality factor - Q-factor which can be either in linear scale or in logarithmic scale can be calculated from the obtained BER through the expression:
其中 和 分别是 PDF 的平均值 “1” 和 “0”，而 是 PDF 的方差。品质因数 - Q 因子可以是线性刻度或对数刻度，可以通过以下表达式从获得的 BER 计算出来：

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8.2.1.  Improving Accuracy of Histogram
8.2.1. 提高直方图的准确性

The common objective is to search for the proper values for number of bins and bin-width to be used in the approximation of the histogram so that the bias and the variance of the estimator can be negligible. According to [26], with a sufficiently large number of transmitted bits (N₀), a good estimate for the width (W_bin) of each equally spaced histogram bin is given by:.
共同目标是搜索 bins 数量和 bin-width 的适当值，以用于直方图的近似，以便估计器的偏差和方差可以忽略不计。根据 [26]，在传输比特数量足够多的情况下 （N₀），每个等距直方图 bin 的宽度 （W_bin） 的良好估计由下式给出： 。

8.3.  Optical Signal-to-Noise Ratio (OSNR)
8.3. 光信噪比 （OSNR）

The optical signal-to-noise ratio (OSNR) is a popular benchmark indicator for assessment of the performance of optical transmission systems, especially those limited by the ASE noise from the optical amplifiers – EDFAs. The OSNR is defined as the ratio of optical signal power to optical noise power. For a single EDFA with output power, Pout, the OSNR is given by:
光信噪比 （OSNR） 是评估光传输系统性能的常用基准指标，尤其是那些受光放大器 （EDFA） 的 ASE 噪声限制的系统。OSNR 定义为光信号功率与光噪声功率的比率。对于具有输出功率 Pout 的单个 EDFA，OSNR 由下式给出：

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where NF is the amplifier noise figure, G is the amplifier gain, hf is the photon energy, is the optical measurement bandwidth.
其中 NF 是放大器噪声系数，G 是放大器增益，hf 是光子能量， 是光学测量带宽。

However, OSNR does not provide good estimation to the system performance when the main degrading sources involve the dynamic propagation effects such as dispersion (including both CD and PMD) and Kerr nonlinearity effects (eg. SPM). In these cases, the degradation of the performance is mainly due to waveform distortions rather than the corruption of the ASE or electronic noise. When addressing a value of an OSNR, it is important to define the optical measurement bandwidth over which the OSNR is calculated. The signal power and noise power is obtained by integrating all the frequency components across the bandwidth leading to the value of OSNR. In practice, signal and noise power values are usually measured directly from the optical spectrum analyzer (OSA), which does the mathematics for the users and displays the resultant OSNR versus wavelength or frequency over a fixed resolution bandwidth. A value of = 0.1 nm or =12.5GHz is widely used as the typical value for calculation of the OSNR.
然而，当主要劣化源涉及动态传播效应，如色散（包括 CD 和 PMD）和 Kerr 非线性效应（例如。SPM） 的在这些情况下，性能下降主要是由于波形失真，而不是 ASE 或电子噪声的损坏。在对 OSNR 的值进行寻址时，定义计算 OSNR 的光测量带宽非常重要。信号功率和噪声功率是通过对带宽上的所有频率分量进行积分而获得的，因此 OSNR 的值。在实践中，信号和噪声功率值通常直接从光谱分析仪 （OSA） 测量，该分析仪为用户进行数学运算，并在固定分辨率带宽上显示合成的 OSNR 与波长或频率的关系。 = 0.1 nm 或 =12.5GHz 的值被广泛用作计算 OSNR 的典型值。

OSNR penalty is determined at a particular BER when varying value of a system parameter under test. For example, OSNR penalty at BER=1e-4 for a particular optical phase modulation format when varying length of an optical link in a long-haul transmission system configuration.
当被测系统参数的值发生变化时，OSNR 损失是在特定的 BER 下确定的。例如，在长距离传输系统配置中改变光链路的长度时，特定光相位调制格式的 BER=1e-4 处的 OSNR 损失。

8.4.  Eye Opening Penalty (EOP)
8.4. 眼部张开惩罚 （EOP）

The OSNR is a time-averaged indicator for the system performance where the ratio of average power of optical carriers to noise is considered. When optical lightwaves
OSNR 是系统性能的时间平均指标，其中考虑了光载波的平均功率与噪声的比率。当光学光波

propagate through a dispersive and nonlinear optical fiber channel, the fiber impairments including ISI induced from CD, PMD and the spectral effects induced from nonlinearities cause the distortion of the waveforms. Another dynamic cause of the waveform distortion comes from the ISI effects as the results of optical or electrical filtering. In a conventional OOK system, bandwidth of an optical filter is normally larger than the spectral width of the signal by several times.
通过色散和非线性光纤通道传播，包括 CD、PMD 引起的 ISI 和非线性引起的频谱效应在内的光纤损伤会导致波形失真。波形失真的另一个动态原因来自光或电滤波结果的 ISI 效应。在传统的 OOK 系统中，滤光片的带宽通常比信号的光谱宽度大几倍。

The eye-opening penalty (EOP) is a performance measure defined as the penalty of the “eye” caused by the distortion of the electrically detected waveforms to a reference eye-opening (EO). EO is the difference between the amplitudes of the lowest mark and the highest space.
眼图张开度损失 （EOP） 是一种性能衡量标准，定义为由电气检测波形失真到参考眼图张开度 （EO） 而导致的“眼图”损失。EO 是最低标记和最高空间的振幅之间的差值。

The benchmark eye opening is usually obtained from a back-to-back measurement when the waveform is not distorted at all by any above impairments. The eye opening penalty at a particular sampling instance is normally calculated in log scale (dB unit) and given by:
基准眼图张开度通常是从背靠背测量中获得的，此时波形完全没有被上述任何损伤所扭曲。特定采样实例的眼图张开度损失通常以对数刻度（dB 单位）计算，并由下式给出：

(34)

The EOP is useful for noise-free system evaluations as a good estimate of deterministic pulse distortion effects. The accuracy of EOP indicator depends on the sampling instance in a bit slot. Usually, the detected pulses are sampled at the instance giving the maximum eye opening. If noise is present, the calculation of the EOP become less precise because of the ambiguity of the signal levels which are corrupted by noise.
EOP 可用于无噪声系统评估，作为确定性脉冲失真效应的良好估计。EOP 指示符的精度取决于位槽中的采样实例。通常，在眼图张开度最大的实例处对检测到的脉冲进行采样。如果存在噪声，则 EOP 的计算会变得不那么精确，因为信号电平会受到噪声的破坏。

9.  MATLAB Statistical Evaluation Techniques
9. MATLAB 统计评估技术

The method using a Gaussian-based single distribution involves only the effects of noise corruption on the detected signals and ignores the dynamic distortion effects such as ISI and non-linearity. These dynamic distortions result in a multi-peak pdf as demonstrated in Figure 11, which is clearly overlooked by the conventional single distribution technique. As the result, the pdf of the electrical signal can not be approximated accurately. The addressed issues are resolved with the proposal of two new statistical methods.
该方法使用基于高斯的单一分布，仅涉及噪声损坏对检测到的信号的影响，而忽略了 ISI 和非线性等动态失真效应。这些动态失真导致多峰 pdf，如图 11 所示，传统的单一分布技术显然忽略了这一点。因此，无法准确估计电信号的 pdf。通过提出两种新的统计方法解决了所解决的问题。

Two new techniques proposed to accurately obtain the pdf of the detected electrical signal in optical communications include the mixture of multi-Gaussian distributions (MGD) by implementing the expectation maximization theory (EM) and the generalized Pareto distribution (GPD) of the generalized extreme values (GEV) theorem. These two techniques are well-known in fields of statistics, banking, finance, meteorology, etc. The implementation of required algorithms is carried out with MATLAB functions. Thus, these novel statistical methods offer a great deal of flexibility, convenience, fast-processing while maintaining the errors in obtaining the BER within small and acceptable limits.
为准确获取光通信中检测到的电信号的 pdf 而提出的两种新技术包括通过实施期望最大化理论 （EM） 的多高斯分布 （MGD） 和广义极值 （GEV） 定理的广义帕累托分布 （GPD） 的混合。这两种技术在统计、银行、金融、气象等领域广为人知。所需算法的实现是通过 MATLAB 函数进行的。因此，这些新颖的统计方法提供了极大的灵活性、便利性、快速处理，同时将获得 BER 的误差保持在较小且可接受的范围内。

9.1.  Multi-Gaussian Distributions (MGD) via Expectation Maximization (EM)Theorem
9.1. 通过期望最大化 （EM） 定理的多高斯分布 （MGD）

The mixture density parameter estimation problem is probably one of the most widely used applications of the expectation maximization (EM) algorithm. It comes from the fact that most of deterministic distributions can be seen as the result of superposition of different multi distributions. Given a probability distribution function for a set of received data, can be expressed as the mixture of M different distributions:
混合密度参数估计问题可能是期望最大化 （EM） 算法应用最广泛的应用之一。它来自这样一个事实，即大多数确定性分布可以被视为不同多分布叠加的结果。给定一组接收数据的概率分布函数 ， 可以表示为 M 个不同分布的混合：

(35)

where the parameter are such that and each is a PDF by and each pdf having a weight, i.e probability of that PDF.
其中参数是 这样的 和 每个 都是 PDF， 并且每个 PDF 都有一个权重 ，即该 PDF 的概率。

As a particular case adopted for optical communications, the EM algorithm is implemented with a mixture of multi Gaussian distributions (MGD). This method offers great potential solutions for evaluation of performance of an optical transmission system with following reasons: 1) In a linear optical system (low input power into fiber), the conventional single Gaussian distribution fails to take into account the waveform distortion caused by either the ISI due to fiber CD and PMD dispersion, the patterning effects. Hence, the obtained BER is no longer accurate. These issues however are overcome by using the MGD method. 2) Computational time for implementing MGD is fast via the EM algorithm which has become quite popular.
作为光通信采用的一种特殊情况，EM 算法是通过多高斯分布 （MGD） 的混合实现的。该方法为评估光传输系统的性能提供了巨大的潜在解决方案，原因如下：1） 在线性光学系统（光纤输入功率低）中，传统的单高斯分布未能考虑光纤 CD 和 PMD 色散引起的 ISI 引起的波形失真，以及图案效应。因此，获得的 BER 不再准确。然而，这些问题可以通过使用 MGD 方法来解决。2） 通过已经非常流行的 EM 算法实现 MGD 的计算时间很快。

The selection of Number of Gaussian distributions for MGD Fitting can be conducted as follows. The critical step affecting the accuracy of the BER calculation is the process of estimate of the number of Gaussian distributions applied in the EM algorithm for fitting the received signal pdf. This number is determined by the estimated number of peaks or valleys in the curves of 1^st and 2^nd derivative of the original data set. Explanation of this procedure is carried out via the well-known “Hemming Lake Pike” example as reported in [27,28]. In this problem, the data of five age-groups give the lengths of 523 pike (Esox lucius), sampled in 1965 from Hemming Lake, Manitoba, Canada. The components are heavily overlapped and the resultant pdf is obtained with a mixture of these 5 Gaussian distributions as shown in Figure 12(a). The figures are extracted from [29] for demonstration of the procedure.
MGD 拟合的高斯分布数的选择可以按如下方式进行。影响 BER 计算精度的关键步骤是估计 EM 算法中用于拟合接收信号的高斯分布数量的过程 pdf。该数字由原始数据集的 1^阶和 2^阶导数曲线中的估计峰值或谷值确定。通过[27\u201228]中报道的著名的 “Hemming Lake Pike” 例子来解释这一过程。在这个问题中，五个年龄组的数据给出了 523 条梭子鱼 （Esox lucius） 的长度，1965 年从加拿大曼尼托巴省的海明湖取样。这些组件严重重叠，结果的 pdf 是用这 5 个高斯分布的混合物获得的，如图 12（a） 所示。这些数字是从 [29] 中提取的，用于演示该过程。

Estimation of number of Gaussian distributions in the mixed pdf based on 1^st and 2^nd derivatives of the original data set (courtesy from [29]). As seen from Figure 12, the 1^st derivative of the resultant pdf shows clearly 4 pairs of peaks (red) and valleys (blue), suggesting that there should be at least 4 component Gaussian distributions contributing to the original pdf. However, by taking the 2^nd derivative, it is realized that there is actually up to 5 contributed Gaussian distributions as shown in Figure 12.
根据原始数据集的 1^阶和 2^阶导数估计混合 pdf 中高斯分布的数量（由 [29] 提供）。从图 12 中可以看出，所得 pdf 的 1^阶导数清楚地显示了 4 对峰（红色）和谷值（蓝色），这表明至少应该有 4 个分量高斯分布有助于原始 pdf。然而，通过取 2^阶导数，可以意识到实际上有多达 5 个贡献的高斯分布，如图 12 所示。

In summary, the steps for implementing the MGD technique to obtain the BER value is described in short as follows: 1) Obtaining the pdf from the normalized histogram of the received electrical levels; 2) Estimating the number of Gaussian distributions (N_Gaus) to be used for fitting the pdf of the original data set; 3) Applying EM algorithm with the mixture of N_Gaus Gaussian distributions and obtaining the values of mean, variance and weight for each distribution; 4) Calculating the BER value based on the integrals of the overlaps of the Gaussian distributions when the tails of these distributions cross the threshold.
总之，实施 MGD 技术获得 BER 值的步骤简要描述如下：1） 从接收到的电电平的归一化直方图中获取 pdf;2） 估计用于拟合原始数据集 pdf 的高斯分布 （N_Gaus） 的数量;3） 将 EM 算法与 N_高斯高斯分布混合，并得到每个分布的均值、方差和权重值;4） 当高斯分布的尾部超过阈值时，根据这些分布的重叠积分计算 BER 值。

9.2.  Generalized Pareto Distribution (GPD)
9.2. 广义帕累托分布 （GPD）

The GEV theorem is used to estimate the distribution of a set of data of a function in which the possibility of extreme data lengthen the tail of the distribution. Due to the mechanism of estimation for the pdf of the extreme data set, GEV distributions can be classified into two classes consisting of the GEV distribution and the generalized Pareto distribution (GPD).
GEV 定理用于估计函数的一组数据的分布，其中极端数据的可能性会拉长分布的尾部。由于极端数据集的 pdf 估计机制，GEV 分布可分为两类，分别是 GEV 分布和广义帕累托分布 （GPD）。

There has recently been only a countable number of research studies on the application of this theorem into optical communications. However, these studies only reports on the GEV distributions which only involves the effects of noise and neglect the effects of dynamic distortion factors.
最近关于该定理在光通信中的应用的研究数量不计其数。然而，这些研究仅报告了 GEV 分布，仅涉及噪声的影响，而忽略了动态失真因素的影响。

Unlike the Gaussian-based techniques but rather similar to the exponential distribution, the generalized Pareto distribution is used to model the tails of distribution. This section provides an overview of the generalized Pareto distribution (GPD). The probability density function for the generalized Pareto distribution is defined as follows:
与基于高斯的技术不同，但与指数分布非常相似，广义帕累托分布用于对分布的尾部进行建模。本节概述了广义帕累托分布 （GPD）。广义 Pareto 分布的概率密度函数定义如下：

(36)

 

where k is shape parameter k ≠ 0, σ is scale parameter and the threshold parameter θ.
其中 k 是形状参数 k ≠ 0，σ 是比例参数和阈值参数 θ。

Equation (36) has significant constraints given as
方程 （36） 具有重要的约束，给定为

•When k>0: i.e there is no upper bound for x
•当 k>0 时： 即 x 没有上限

• When k<0:  • 当 k<0 时：and zero probability for the case
并且这种情况的概率为零

•When k = 0, i.e Equation turning to:
•当 k = 0 时，即方程变为：

•If k = 0 and θ = 0, the generalized Pareto distribution is equivalent to the exponential distribution.
•如果 k = 0 且 θ = 0，则广义帕累托分布等价于指数分布。

•If k > 0 and θ = σ, the generalized Pareto distribution is equivalent to the Pareto distribution.
•如果 k > 0 且 θ = σ，则广义 Pareto 分布等同于 Pareto 分布。

The GPD has three basic forms reflecting different class of underlying distributions.
GPD 有三种基本形式，反映不同类别的基础分布。

•Distributions whose tails decrease exponentially, such as the normal distribution, lead to a generalized Pareto shape parameter of zero.
•尾部呈指数级减小的分布（如正态分布）会导致广义 Pareto 形状参数为零。

•Distributions with tails decreasing as a polynomial, such as Student’s t lead to a positive shape parameter.
•尾部作为多项式递减的分布（如 Student 的 t）会导致正形状参数。

• Distributions having finite tails, such as the beta, lead to a negative shape parameter.
• 具有有限尾部的分布 （如 beta） 会导致负形状参数。

GPD is widely used in fields of finance, meteorology, material engineering, etc… to for the prediction of extreme or rare events which are normally known as the exceedances. However, GPD has not yet been applied in optical communications to obtain the BER. The following reasons suggest that GPD may become a potential and a quick method for evaluation of an optical system, especially when non-linearity is the dominant degrading factor to the system performance.
GPD 广泛应用于金融、气象、材料工程等领域。to 用于预测通常称为超标的极端或罕见事件。然而，GPD 尚未应用于光通信以获得 BER。以下原因表明，GPD 可能成为评估光学系统的一种潜在且快速的方法，尤其是当非线性是系统性能的主要降级因素时。

1) The normal distribution has a fast roll-off, i.e. short tail. Thus, it is not a good fit to a set of data involving exeedances, i.e. rarely happening data located in the tails of the distribution. With a certain threshold value, the generalized Pareto distribution can be used to provide a good fit to extremes of this complicated data set.
1） 正态分布具有快速滚降，即短尾。因此，它不适合一组涉及 exeedances 的数据，即位于分布尾部的很少出现的数据。具有一定的阈值，广义 Pareto 分布可用于很好地拟合此复杂数据集的极值。

2) When nonlinearity is the dominating impairment degrading the performance of an optical system, the sampled received signals usually introduce a long tail distribution. For example, in case of DPSK optical system, the distribution of nonlinearity phase noise differs from the Gaussian counterpart due to its slow roll-off of the tail. As the result the conventional BER obtained from assumption of Gaussian-based noise is no longer valid and it often underestimates the BER.
2） 当非线性是降低光学系统性能的主要损伤时，采样的接收信号通常会引入长尾分布。例如，在 DPSK 光学系统的情况下，非线性相位噪声的分布与高斯相位噪声的分布不同，因为它的尾部滚降很慢。因此，从基于高斯的噪声假设中获得的常规 BER 不再有效，并且经常低估 BER。

3) A wide range of analytical techniques have recently been studied and suggested such as importance sampling, multi-canonical method, etc. Although these techniques provide solutions to obtain a precise BER, they are usually far complicated. Whereas, calculation of GPD has become a standard and available in the recent Matlab version (since Matlab 7.1). GPD therefore may provide a very quick and convenient solution for monitoring and evaluating the system performance. Necessary preliminary steps which are fast in implementation need to be carried out the find the proper threshold.
3） 最近研究和提出了广泛的分析技术，例如重要性抽样、多规范方法等。尽管这些技术提供了获得精确 BER 的解决方案，但它们通常要复杂得多。然而，GPD 的计算已成为标准，并在最近的 Matlab 版本（自 Matlab 7.1 起）中可用。因此，GPD 可以为监控和评估系统性能提供非常快速和方便的解决方案。需要执行必要的初步步骤，这些步骤在实施中要快，以找到合适的阈值。

4) Evaluation of contemporary optical systems requires BER as low as 1e-15. Therefore, GPD can be seen quite suitable for optical communications.
4） 对现代光学系统的评估需要低至 1e-15 的 BER。因此，可以看出 GPD 非常适合光通信。

9.2.1.  Selection of Threshold for GPD Fitting
9.2.1. 选择 GPD 拟合的阈值

Using this statistical method, the accuracy of the obtained BER strongly depend on the threshold value (V_thres) used in the GPD fitting algorithm, i.e. the decision where the tail of the GPD curve starts.
使用这种统计方法，获得的 BER 的准确性在很大程度上取决于 GPD 拟合算法中使用的阈值 （V_thres），即 GPD 曲线尾部开始的决定。

There have been several suggested techniques as the guidelines aiding the decision of the threshold value for the GPD fitting. However, they are not absolute techniques and are quite complicated. In this paper, a simple technique to determine the threshold value is proposed. The technique is based on the observation that the GPD tail with exceedances normally obeying a slow exponential distribution compared to the faster decaying slope of the distribution close to the peak values. The inflection region between these two slopes gives a good estimation of the threshold value for GPD fitting. This is demonstrated in Figure 13.
有几种推荐的技术作为指南，有助于确定 GPD 拟合的阈值。但是，它们不是绝对的技术，并且相当复杂。在本文中，提出了一种确定阈值的简单技术。该技术基于以下观察结果：与接近峰值的分布的快速衰减斜率相比，超标的 GPD 尾部通常服从缓慢的指数分布。这两个斜率之间的拐点区域可以很好地估计 GPD 拟合的阈值。如图 13 所示。

Whether the selection of the V_thres value leads to an adequately accurate BER or not is evaluated by using the cumulative density function (cdf-Figure 14) and the quantile-quantile plot (QQ plot Figure 15). If there is a high correlation between the pdf of the tail of the original data set (with a particular V_thres) and pdf of the GPD, there would be a good fit between empirical cdf of the data set with the GPD-estimated cdf with focus at the most right region of the two curves. In the case of the QQ-plot, a linear trend would be observed. These guidelines are illustrated in Figure 14. In this particular case, the value of 0.163 is selected to be V_thres.
通过使用累积密度函数（cdf-图 1、4）和分位数-分位数图（QQ 图、图 1、5）来评估 V_thres 值的选择是否导致足够准确的 BER。如果原始数据集尾部的 pdf（具有特定的 V_阈值）与 GPD 的 pdf 之间存在高度相关性，则数据集的经验 cdf 与 GPD 估计的 cdf 之间将有很好的拟合，焦点在两条曲线的最右侧区域。在 QQ 图的情况下，将观察到线性趋势。这些准则如图 14 所示。在此特定情况下，选择 0.163 的值为 V_thres。

Furthermore, as a demonstration of improper selection of V_thres, the value of 0.2 is selected. Figure 16 and Figure 17 show the non-compliance of the fitted curve with the GPD which is reflected via the discrepancy in the two cdfs and the nonlinear trend of the QQ-plot.
此外，为了演示 V_thres 选择不当，选择了 0.2 的值。图 16 和图 17 显示了拟合曲线与 GPD 的不一致性，这通过两个 cdf 的差异和 QQ 图的非线性趋势反映出来。

9.3.  Validation of the Statistical Methods
9.3. 统计方法的验证

A simulation test-bed of an optical DPSK transmission system over 880 km SSMF dispersion managed optical link (8 spans) is set up. Each span consists of 100 km SSMF and 10 km of DCF whose dispersion values are +17 ps/nm.km and -170 ps/nm.km at 1550 nm wavelength respectively and fully compensated i.e zero residual dispersion. The average optical input power into each span is set to be higher than the nonlinear threshold of the optical fiber. The degradation of the system performance hence is dominated by the nonlinear effects which are of much interest since it is a random process creating indeterminate errors in the long tail region of the pdf of the received electrical signals.
建立了超过 880 km SSMF 色散管理光链路（8 个跨度）的光 DPSK 传输系统的仿真测试台。每个跨度由 100 km SSMF 和 10 km DCF 组成，在 1550 nm 波长下的色散值分别为 +17 ps/nm.km 和 -170 ps/nm.km，并且是完全补偿的，即零残余色散。每个跨度的平均光输入功率设置为高于光纤的非线性阈值。因此，系统性能的退化主要由非线性效应主导，这非常有趣，因为它是一个随机过程，在接收到的电信号的 pdf 的长尾区域产生不确定的误差。

The BER results obtained from the novel statistical methods are compared to that from the Monte-Carlo simulation as well as from the semi-analytical method. Here, the well-known analytical expression to obtain the BER of the optical DPSK format is used, given as [30].
将从新颖的统计方法获得的 BER 结果与蒙特卡洛模拟和半分析方法获得的 BER 结果进行比较。这里使用了众所周知的解析表达式来获得光学 DPSK 格式的 BER，如 [30]。

(37)

where is the obtained OSNR and is the variance of nonlinear phase noise.
其中 是获得的 OSNR， 是非线性相位噪声的方差。

In this case, in order to calculate the BER of a optical DPSK system involving the effect of nonlinear phase noise, the required parameters including the OSNR and the variance of nonlinear phase noise etc are obtained from the simulation numerical data which is stored and processed in Matlab. The fitting curves implemented with the MGD method for the pdf of bit 0 and bit 1 (input power of 10 dBm) as shown in Figure 18 and illustrated in Figure 19 for bit 0 and bit 1 respectively.
在这种情况下，为了计算涉及非线性相位噪声影响的光学 DPSK 系统的 BER，所需的参数包括 OSNR 和非线性相位噪声的方差等，是从 Matlab 中存储和处理的仿真数值数据中获得的。使用 MGD 方法对位 0 和位 1（输入功率为 10 dBm）的 pdf 实现的拟合曲线，如图 18 所示，位 0 和位 1 分别如图 19 所示。

The selection of optimal threshold for GPD fitting follows the guideline as addressed in detail in the previous section. The BER from various evaluation methods are shown in Table 1. The input powers are controlled to be 10 dBm and 11 dBm.
GPD 拟合的最佳阈值的选择遵循上一节中详细讨论的准则。表 1 显示了各种评估方法的 BER。输入功率控制为 10 dBm 和 11 dBm。

Table 1 validates the adequate accuracy of the proposed novel statistical methods with the discrepancies compared to the Monte-Carlo and semi-analytical BER to be within one decade. In short, these methods offer a great deal of fast processing while maintaining the accuracy of the obtained BER within the acceptable limits.
表 1 验证了所提出的新颖统计方法的足够准确性，与蒙特卡洛和半分析 BER 相比，差异在 10 年以内。简而言之，这些方法提供了大量的快速处理，同时将获得的 BER 的准确性保持在可接受的范围内。

10.  Conclusions 10. 结论

We have demonstrated the Simulink modeling of amplitude and phase modulation formats at 40 Gb/s optical fiber transmission. A novel modified fiber propagation algorithm has been used to minimize the simulation processing time and optimize its accuracy. The principles of amplitude and phase modulation, encoding and photonic-opto-electronic balanced detection and receiving modules have been demonstrated via Simulink modules and can be corroborated with experimental receiver sensitivities.
我们已经演示了 40 Gb/s 光纤传输时幅度和相位调制格式的 Simulink 建模。使用了一种新颖的改进光纤传播算法来最大限度地减少仿真处理时间并优化其精度。幅度和相位调制、编码以及光子光电平衡检测和接收模块的原理已通过 Simulink 模块进行了演示，并且可以与实验接收器灵敏度相印证。

The XPM and other fiber nonlinearity such as the Raman scattering, four wave mixing are not integrated in the Matlab Simulink models. A switching scheme between the linear only and the linear and nonlinear models is developed to enhance the computing aspects of the transmission model.
XPM 和其他纤维非线性（如拉曼散射、四波混频）未集成到 Matlab Simulink 模型中。开发了线性模型与线性和非线性模型之间的切换方案，以增强传输模型的计算方面。

Other modulations formats such as multi-level MDPSK, M-ASK that offer narrower effective bandwidth, simple optical receiver structures and no chirping effects would also be integrated. These systems will be reported in future works. The effects of the optical filtering components in DWDM transmission systems to demonstrate the effectiveness of the DPSK and DQPSK formats, have been measured in this paper and will be verified with simulation results in future publications. Finally, further development stages of the simulator together with simulation results will be reported in future works.
其他调制格式，如多电平 MDPSK、M-ASK等，提供更窄的有效带宽、简单的光接收器结构和无啁啾效应。这些系统将在未来的工作中报告。本文已经测量了 DWDM 传输系统中光滤波组件对证明 DPSK 和 DQPSK 格式有效性的影响，并将在未来的出版物中通过仿真结果进行验证。最后，模拟器的进一步开发阶段以及仿真结果将在未来的工作中报告。

We have illustrated the modeling of various schemes of advanced modulation formats for optical transmission systems. Transmitter modules integrating lightwaves sources, electrical pre-coder and external modulators can be modeled without difficulty under MATLAB Simulink. As the popularity of MATLAB becoming a standard computing language for academic research institutions throughout the world, the models reported here would contribute to the wealth of computing tools for modeling optical fiber transmission systems and teaching undergraduates at senior level and postgraduate research scholars. The models can integrate photonic filters or other photonic components using blocksets available in Simulink. Furthermore we have used the developed models to assess the effectiveness of the models by evaluating the simulated results and experimental transmission performance of long haul advanced modulation format transmission systems.
我们已经说明了光传输系统高级调制格式的各种方案的建模。集成光波源、电预编码器和外部调制器的发射器模块可以在 MATLAB Simulink 下轻松建模。随着 MATLAB 成为全球学术研究机构的标准计算语言的普及，这里报告的模型将有助于为光纤传输系统建模和教授高级本科生和研究生研究学者提供丰富的计算工具。这些模型可以使用 Simulink 中提供的模块集集成光子滤波器或其他光子组件。此外，我们通过使用开发的模型通过评估长距离高级调制格式传输系统的模拟结果和实验传输性能来评估模型的有效性。

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Appendix: A Matlab program of the split-step propagation of the guided lightwave signals
附录：引导光波信号的分步传播的 Matlab 程序

function output = ssprop_matlabfunction_raman(input)
函数输出 = ssprop_matlabfunction_raman（输入）

nt = input(1); nt = 输入 （1）;

u0 = input(2:nt+1); u0 = 输入（2：nt+1）;

dt = input(nt+2); dt = 输入（nt+2）;

dz = input(nt+3); dz = 输入（nt+3）;

nz = input(nt+4); nz = 输入（nt+4）;

alpha_indB = input(nt+5);
alpha_indB = 输入（nt+5）;

betap = input(nt+6:nt+9);
betap = 输入（nt+6：nt+9）;

gamma = input(nt+10); 伽玛 = 输入（nt+10）;

P_non_thres = input(nt+11);
P_non_thres = 输入（nt+11）;

maxiter = input(nt+12); maxiter = 输入 （nt+12）;

tol = input(nt+13); tol = 输入（nt+13）;

%Ld = input(nt+14); %Ld = 输入 （nt+14）;

%Aeff = input(nt+15); %Aeff = 输入 （nt+15）;

%Leff = input(nt+16); %Leff = 输入 （nt+16）;

tic; 抽搐;

%tmp = cputime; %tmp = cpu时间;

%----------------------------------------------------------

%----------------------------------------------------------

% This function ssolves the nonlinear Schrodinger equation for % pulse propagation in an optical fiber using the split-step % Fourier method % % The following effects are included in the model: group velocity % dispersion (GVD), higher order dispersion, loss, and self-phase % modulation (gamma).
% 此函数使用分步 % 傅里叶法 % % 求解光纤中脉冲传播的非线性薛定谔方程 % 模型中包括以下效应：群速度 % 色散 （GVD）、高阶色散、损耗和自相位 % 调制 （gamma）。

% % USAGE % % u1 = ssprop(u0,dt,dz,nz,alpha,betap,gamma);
% % 使用情况 % % u1 = ssprop（u0，dt，dz，nz，alpha，betap，gamma）;

% u1 = ssprop(u0,dt,dz,nz,alpha,betap,gamma,maxiter);
% u1 = ssprop（u0，dt，dz，nz，alpha，betap，gamma，maxiter）;

% u1 = ssprop(u0,dt,dz,nz,alpha,betap,gamma,maxiter,tol);
% u1 = ssprop（u0，dt，dz，nz，alpha，betap，gamma，maxiter，tol）;

% % INPUT % % u0 - starting field amplitude (vector)
% % INPUT % % u0 - 起始磁场振幅（矢量）

% dt - time step - [in ps]
% dt - 时间步长 - [以 PS 为单位]

% dz - propagation stepsize - [in km]
% dz - 传播步长 - [以 km 为单位]

% nz - number of steps to take, ie, ztotal = dz*nz % alpha - power loss coefficient [in dB/km], need to convert to linear to have P=P0*exp(-alpha*z)
% nz - 要采取的步数，即 ztotal = dz*nz % alpha - 功率损耗系数 [以 dB/km 为单位]，需要转换为线性以获得 P=P0*exp（-alpha*z）

% betap - dispersion polynomial coefs, [beta_0 ... beta_m] [in ps^(m-1)/km]
% BETAP - 离散多项式系数， [beta_0 ...beta_m] [单位为 ps^（m-1）/km]

% gamma - nonlinearity coefficient [in (km^-1.W^-1)]
% gamma - 非线性系数 [in （km^-1.W^-1）]

% maxiter - max number of iterations (default = 4)
% maxiter - 最大迭代次数（默认值 = 4）

% tol - convergence tolerance (default = 1e-5)
% TOL - 收敛容差（默认值 = 1E-5）

% % OUTPUT % % u1 - field at the output
% % OUTPUT % % u1 - 输出处的字段

%--------------

% Convert alpha_indB to alpha in linear domain
线性域中alpha_indB转换为 Alpha 的百分比

%-------------

-alpha = 1e-3*log(10)*alpha_indB/10;         
-alpha = 1e-3*log（10）*alpha_indB/10;

% alpha (1/km) - see Agrawal p57
% alpha （1/km） - 参见 Agrawal p57

%--------------%

P_non_thres = 0.0000005;

ntt = length(u0); ntt = 长度（u0）;

w = 2*pi*[(0:ntt/2-1),(-ntt/2:-1)]'/(dt*nt);
w = 2*pi*[（0：ntt/2-1），（-ntt/2：-1）]'/（dt*nt）;

%t = ((1:nt)'-(nt+1)/2)*dt;
%t = （（（1：nt）'-（nt+1）/2）*dt;

gain = numerical_gain_hybrid(dz,nz);
增益 = numerical_gain_hybrid（dz，nz）;

for array_counter = 2:nz+1
array_counter = 2：nz+1

    grad_gain(1) = gain(1)/dz;
grad_gain（1） = 增益（1）/dz;

    grad_gain(array_counter) = (gain(array_counter)-gain(array_counter-1))/dz;
grad_gain（array_counter） = （增益（array_counter）-增益（array_counter-1））/dz;

end gain_lin = log(10)*grad_gain/(10*2);
结束gain_lin = log（10）*grad_gain/（10*2）;

clear halfstep 清除半步

  halfstep = -alpha/2; 半步 = -alpha/2;

    for ii = 0:length(betap)-1;
对于 ii = 0：长度（betap）-1;

        halfstep = halfstep - j*betap(ii+1)*(w.^ii)/factorial(ii);
halfstep = 半步 - j*betap（ii+1）*（w.^ii）/factorial（ii）;

    end 结束

     square_mat = repmat(halfstep, 1, nz+1);
square_mat = repmat（halfstep， 1， nz+1）;

     square_mat2 = repmat(gain_lin, ntt, 1);
square_mat2 = repmat（gain_lin， ntt， 1）;

     size(square_mat); 大小 （square_mat）;

     size(square_mat2); size（square_mat2）;

     total = square_mat + square_mat2;
总计 = square_mat + square_mat2;

    

 clear LinearOperator 清除 LinearOperator

    % Linear Operator in Split Step method
“拆分步骤”方法中的 % 线性运算符

    LinearOperator = halfstep;
LinearOperator = 半步;

    halfstep = exp(total*dz/2);
半步 = exp（总计*dz/2）;

 u1 = u0;

ufft = fft(u0); UFFT = fft（u0）;

% Nonlinear operator will be added if the peak power is greater than the % Nonlinear threshold iz = 0;
如果峰值功率大于 % 非线性阈值 iz = 0，则将添加 % 非线性运算符;

while (iz < nz) && (max((gamma*abs(u1).^2 + gamma*abs(u0).^2)) > P_non_thres)
while （iz < nz） && （max（（gamma*abs（u1）.^2 + gamma*abs（u0）.^2）） > P_non_thres）

  iz = iz+1;

  uhalf = ifft(halfstep(:,iz).*ufft);
uhalf = ifft（halfstep（：，iz）.*uft）;

   

  for ii = 1:maxiter    uv = uhalf .* exp((-j*(gamma)*abs(u1).^2 + (gamma)*abs(u0).^2)*dz/2);
对于 II = 1：最大 uv = uhalf .* exp（（-j*（gamma）*abs（u1）.^2 + （gamma）*abs（u0）.^2）*dz/2）;

    ufft = halfstep(:,iz).*fft(uv);
ufft = halfstep（：，iz）.*fft（uv）;

    uv = ifft(ufft); uv = ifft（ufft）;

   

    if (max(uv-u1)/max(u1) < tol)
如果 （max（uv-u1）/max（u1） < tol）

      u1 = uv; u1 = 紫外线;

      break; 破;

    else 还

      u1 = uv; u1 = 紫外线;

    end 结束

   

  end 结束

 % fprintf('You are using SSFM\n');
% fprintf（'您正在使用 SSFM\n'）;

  if (ii == maxiter) if （ii == maxiter）

     

   fprintf('Failed to converge to %f in %d iterations',tol,maxiter);
fprintf（'在 %d 次迭代中无法收敛到 %f'，tol，maxiter）;

  end 结束

 

  u0 = u1;

 end if (iz < nz) && (max((gamma*abs(u1).^2 + gamma*abs(u0).^2)) < P_non_thres)
结束 if （iz < nz） && （max（（gamma*abs（u1）.^2 + gamma*abs（u0）.^2）） < P_non_thres）

  

%  u1 = u1.*rectwin(ntt);  % u1 = u1.*rectwin（ntt）;

   ufft = fft(u1); 乌夫特 = fft（u1）;

   ufft = ufft.*exp(LinearOperator*(nz-iz)*dz);
uft = ufft.*exp（线性运算符*（nz-iz）*dz）;

   u1 = ifft(ufft); u1 = ifft（ufft）;

 

   %fprintf('Implementing Linear Transfer Function of the Fibre Propagation');
%fprintf（'实现光纤传播的线性传递函数'）;

end 结束

NOTES 笔记

Conflicts of Interest 利益冲突

The authors declare no conflicts of interest.
作者声明没有利益冲突。

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