Modified Volterra series transfer function method

In this letter, we offer a modified version of the Volterra series transfer function (VSTF) method that has been previously proposed as an analytical solution to the nonlinear Schrodinger equation for single-mode fibers. The modified VSTF provides a simple closed-form expression of the output of a mildly nonlinear fiber. It gives orders more accurate result than the original VSTF method and can successfully solve the energy divergence problem experienced by the original method. The result is a stronger analytical tool for modeling signal propagation in optical communication systems     

Volterra series approach for optimizing fiber-optic communicationssystem designs

A new methodology for designing long-haul fiber-optic communication systems is presented. We derive the overall Volterra series transfer function of the system including linear dispersion, fiber nonlinearities, amplified spontaneous emission (ASE) noise from the fiber amplifiers, and the square-law nature of the direct detection (DD) system. Since analytical expressions for the probability of error are difficult to derive for the complex systems being used, we derive analytical expressions for an upper bound on probability of error for integrate-and-threshold detection at the receiver. Using this bound as a performance criterion, we determine the optimal dispersion parameters of each fiber segment required to minimize the effects of linear dispersion, fiber nonlinearities and ASE noise from the amplifiers. We study the dependence of optimal dispersion parameters on the average power levels in the fiber by varying the peak input power levels and the amplifier gains. Analytical expressions give us the freedom to choose system parameters in a practical manner, while providing optimum system performance. Using a simple system as an example, we demonstrate the power of the Volterra series approach to design optimal optical communication systems. The analysis and the design procedure presented in this work can be extended to the design of more complex wavelength division multiplexed (WDM) systems     

The RP method: a new tool for the iterative solution of the nonlinear Schrodinger equation

An original approach to the solution of the nonlinear Schrodinger equation (NLSE) is pursued in this paper, following the regular perturbation (RP) method. Such an iterative method provides a closed-form approximation of the received field and is thus appealing for devising nonlinear equalization/compensation techniques for optical transmission systems operating in the nonlinear regime. It is shown that, when the nonlinearity is due to the Kerr effect alone, the order n RP solution coincides with the order 2n + 1 Volterra series solution proposed by Brandt-Pearce and co-workers. The RP method thus provides a computationally efficient way of evaluating the Volterra kernels, with a complexity comparable to that of the split-step Fourier method (SSFM). Numerical results on 10 Gb/s single-channel terrestrial transmission systems employing common dispersion maps show that the simplest third-order Volterra series solution is applicable only in the weakly nonlinear propagation regime, for peak transmitted power well below 5 dBm. However, the insight in the nonlinear propagation phenomenon provided by the RP method suggests an enhanced regular perturbation (ERP) method, which allows the first order ERP solution to be fairly accurate for terrestrial dispersion mapped systems up to launched peak powers of 10 dBm.     

Volterra kernel estimation for nonlinear communication channels using deterministic sequences

We present a new exact method for the identification of communication channels with nonlinearities. The channel is modelled as a third-order discrete Volterra filter and the Volterra kernels are measured using deterministic input sequences and the corresponding channel outputs. The solutions are in closed form, exact and kernels are estimated in a nonrecursive manner, thus eliminating error propagation. Complex inputs and complex kernels are allowed, which permits the use of PSK or QAM modulated signals for the identification of the bandpass Volterra channel. Simulation examples and comparison with other methods in the literature are provided to verify the method.     

DWDM光纤通信系统中受激喇曼散射的统计特性

随着通信容量的增大,密集波分复用(DWDM)光纤通信系统中的受激喇曼散射(SRS)非线性效应成为影响系统质量的重要因素.采用时域分步法模型,对色散下的SRS统计特性进行了深入研究,得到了在整码元色散走离时,DWDM系统各信道传输功率的统计特性,推导了各信道传输功率的数学期望、标准差,证实了各信道传输功率符合高斯分布,计算了概率密度函数,并且与仿真数据取得了较好的一致.

Abstract：

With the enlargement of communication capacity, the performance of dense wavelength division multiplexed (DWDM) fiber communication systems might be degraded by nonlinear effect of stimulated Raman scattering (SRS) in the optical fiber. To resolve these problems, the time-domain split-step method is applied to investigate the statistical characteristics of SRS. The mean and the standard deviation of transmission powers of all channels are derived analytically in a closed-form formula for DWDM systems with SRS and integer bits dispersion walk-off. The probability density function (PDF) of output powers of all channels is confirmed to be Gaussian distribution. The numerical results show a good correspondence with analytical formulae.     

A Scattering Variable Approach to the Volterra Analysis of Nonlinear Systems

A new mathematical model is developed which extends Volterra series analysis of nonlinear systems with memory to high-frequency systems, including those containing linear distributed component devices. A generalized set of nonlinear scattering parameters is defined which can be used to describe power transfer and distortion in nonlinear multiports, and which reduce to the classical scattering parameters for linear networks. The methodology is based on Volterra functional series, and is most useful for the small-signal case where the response can be approximated by a finite number of terms of the series. Nonlinear scattering kernels, derived by extending the Volterra analysis, are simply related to previously developed nonlinear voltage and current Volterra kernels. For sinusoidal inputs nonlinear scattering parameters are defined which are shown to be particularly helpful when power relationships are studied. The principal applications are for microwave networks terminated in real-valued site reference impedances. To evaluate the average power dissipated in a load at some intermodulation frequency, the concept of nonlinear transducer gain is defined and shown to be proportional to the squared magnitude of a nonlinear scattering parameter. Examples are presented illustrating the analysis procedure for a tunnel diode reflection amplifier and for a linear lossless transmission line terminated by a non-linear network.     

Carrier-induced phase noise in angle-modulated optical-fiber systems

Phase noise in angle-modulated optical-fiber communication systems arising from optical power fluctuations is analyzed. The nonlinear refractive index of silica is the physical mechanism which converts power fluctuations into phase fluctuations. The effects of self-phase modulation (an optical wave acting on itself) and cross-phase modulation in wavelength-division multiplexed (WDM) systems (one optical wave modulating a channel at a different wavelength) have been calculated. The phase noise generated in single-channel systems is negligible for laser fluctuations less than 1-mW rms. In WDM systems containing as few as four channels the phase noise exceeds tolerable levels (0.15 rad) for power fluctuation of 1 mW in each channel.     

Compensating for Nonlinear Effects in Coherent Detection Optical Transmission Systems

Fiber nonlinearity is one of the most important limiters of capacity in coherent optical communications. In this paper, we review two nonlinear compensation methods: digital backward propagation (BP) and nonlinear electrical equalizer (NLEE) based on the time-domain Volterra series. These compensation algorithms are implemented in a single-channel 50 Gb/s coherent optical single-carrier frequency-division multiplexed (CO-SCFDM) system transmitting over 10 × 80 km of standard single-mode fiber (SSMF).     

非线性光孤子传输方程的研究

本文在从描述光孤子在单模光纤中的非线性传输方程（ＮＬＳ，即非线性薛定谔方程）出发，利用Ｍａｘｗｅｌｌ方程和慢变包络近似性，导出ｐｓ量级光孤子高阶修正的非线性薛定谔方程（ＨＭＮＬＳ）可进一步研究色散，损耗等高价效应对孤子传输造成的影响。     

非线性光孤子传输方程

本文基于麦克斯韦方程和利用太勒级数展开严格地建立了描述光孤子传输的非线性薜定谔方程,可以用来分析损耗、高阶线性色散和高阶非线性色散效应对光孤子传输的影响。     

Nonlinear System Identification Using Deterministic Multilevel Sequences

A new exact method of measuring the Volterra kernels of finite-order discrete nonlinear systems is presented. The kernels are rearranged in terms of multivariate crossproducts in vector form. The one-, two-, . . . , and ℓ-dimensional kernel vectors are determined using a deterministic multilevel sequence with ℓ distinct levels at the input of the system. It is shown that the defined multilevel sequence with ℓ distinct levels is persistently exciting for a truncated Volterra filter with nonlinearities of polynomial degree ℓ. Examples demonstrating the rearrangement of the Volterra kernels and a novel method for estimation of the kernels are presented. Simulation results are given to illustrate the effectiveness of the proposed method.     

光学孤子在单模光纤中的非线性传输方程

人们熟知,由非线性薛定谔(NLS)方程描述的光学孤子可以在单模光纤中稳定地传输。本文在准单色近似和慢变包络近似条件下,利拥泰勒级数展开推出了这一非线性传输方程。由于推出的方程包含着高阶线性色散和光纤损耗项,因而可以用来研究短光脉冲(皮秒量级)在单模光纤传输过程中相关的非线性光学孤子效应,比如光学孤子分裂和光学孤子损耗等。     

Accurate performance modeling of subcarrier multiplexed fiber/radiosystems: implications of laser nonlinear distortion and wide dynamicrange

In this paper, we present an analytical approach to accurately assessing the performance of subcarrier multiplexed optical systems for the delivery of future mobile radio broadband services. The existence of a wide dynamic range of power levels in the return link, together with laser nonlinear distortion, represents a major problem and makes performance evaluation difficult. By modeling laser nonlinear distortion using Volterra series analysis, we show that the performance assessment of these systems becomes not only tractable, but also that they can be can be optimized based on the worst case condition. This analysis is performed here for the case of continuous phase frequency shift keying (CPFSK) modulation, for which we calculate the power spectral density (PSD) of the intermodulation products. The overall system performance is that investigated, and its dependence on laser bias and RIN is examined. As an illustrative example we consider a specific 50 channel CPFSK system occupying the bandwidth of 2 to 2.1 GHz     

光纤孤子的“双陷”型非线性管理

为了研究光纤系统中由色散参量和 “双陷”型非线性参量可能导致的孤子现象,通过分步傅里叶数值算法求解了变系数（1+1）维非线性薛定谔方程。结果发现,光纤孤子在两种“双陷”型非线性光学系统中具有一系列有趣的性质,如单孤子脉冲在可变参量背景下由“双陷”型非线性结构诱发的周期性振荡;双孤子脉冲在常参量背景下由“双陷”型非线性结构诱导的走离效应;三个孤子脉冲在两种“双陷”型非线性光学系统中体现的特殊相互作用、融合效应等。     

Volterra series modeling of power conversion systems

The nonlinear control-to-output response of pulse width modulated (PWM) conversion systems is modeled via the Volterra functional series. A brief overview of the series is presented. It is seen that the Volterra series is a power series with memory. Each term in the series represents a convolution integral. The nonlinear response of the system, for any input, can thus be determined from a knowledge of the multidimensional Volterra kernels or impulse responses. The determination of the Volterra kernels in the transform domain is performed on a simplified state-space model of the converter. The dominant component of various harmonic and intermodulation distortion frequency products in the output spectrum is derived and is expressed in terms of these kernels. Experimental results are presented confirming the modeling procedure     

Nonlinear channel modeling and identification using baseband Volterra-Parafac models

Baseband Volterra models are very useful for representing nonlinear communication channels. These models present the specificity to include only odd-order nonlinear terms, with kernels characterized by a double symmetry. The main drawback is their parametric complexity. In this paper, we develop a new class of Volterra models, called baseband Volterra-Parafac models, with a reduced parametric complexity, by using a doubly symmetric Parafac decomposition of high order Volterra kernels viewed as tensors. Three adaptive algorithms are then proposed for estimating the parameters of these models. Some Monte Carlo simulation results are presented to compare the performance of the proposed estimation algorithms, in the case of third-order baseband Volterra systems excited by PSK and QAM inputs.     

孤子间互作用相对间距与相位关系的解析分析

基于光孤子脉冲在光纤中传输所遵从的非线性薛定谔(NLS)方程,在忽略三阶以上色散效应、自陡和自频移效应并通过对方程中相关参量的无量纲化处理之后,得到了总光元场满足的微扰NLS方程,分析了孤子相对间距与初始相位对其传输的影响。     

Linear multichannel blind equalizers of nonlinear FIR Volterrachannels

Truncated Volterra expansions model nonlinear systems encountered with satellite communications, magnetic recording channels, and physiological processes. A general approach for blind deconvolution of single-input multiple-output Volterra finite impulse response (FIR) systems is presented. It is shown that such nonlinear systems can be blindly equalized using only linear FIR filters. The approach requires that the Volterra kernels satisfy a certain coprimeness condition and that the input possesses a minimal persistence-of-excitation order. No other special conditions are imposed on the kernel transfer functions or on the input signal, which may be deterministic or random with unknown statistics. The proposed algorithms are corroborated with simulation examples     

多信道IM—DD密集波分复用光纤传输系统中的交叉相位调制效应

本文研究了交叉相位调制（XPM）对DWDM数字基带主干传输网性的影响,建立了表征XPM效应对任意调制信号在色散信道中传输的指标损伤的等效系统函数模型,并据此推导出了不同系统参数条件下（基带信号速率,波长信道间隔,复用信道数）计算XPM导致和系统光功率代价的计算公式,通过数值模拟计算证明了在基带信号速率高（＞2.5Gbit/s),波长信道间隔窄（－1nm),复用信道数多（＞8）的条件下XPM效应将成为限制系统性能的主要非线性因素。     

光强控制非线性布喇格光纤光栅慢光特性研究

为了在光纤通信网络中控制光的传播速度以实现全光路由及全光缓存,构建了耦合模传输模型,采用含时推移变量的传输矩阵方法,数值研究了光强控制下非线性布喇格光纤光栅(NLBG)的慢光特性。结果表明,NLBG禁带位置具有强烈的光强依赖性,入射光强的变化能够有效改变光的传输群速度,对于输出脉冲未展宽的情况,普遍可以得到100ps以上的输出脉冲时延,且当入射光强一定时,NLBG的长度变化对光的传播群速度也有显著的影响。