JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 30, NO. 5, MARCH 1, Houbing Song, Student Member, IEEE, and Maïté Brandt-Pearce, Senior Member, IEEE

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1 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 30, NO. 5, MARCH 1, A 2-D Discrete-Time Model of Physical Impairments in Wavelength-Division Multiplexing Systems Houbing Song, Student Member, IEEE, and Maïté Brandt-Pearce, Senior Member, IEEE Abstract Dense wavelength-division multiplexing (DWDM) is a promising approach to design ultrahigh-capacity fiber-optic communication systems ( Tb/s). However, DWDM gives rise to severe physical impairments that adversely affect system performance. To mitigate various physical impairments in DWDM systems and exploit their system capacity, there is a need to develop a 2-D (time and wavelength) discrete-time input output modelofphysicalimpairmentsthat can become the foundation of signal processing for optical communications. This paper develops such a model based on the Volterra series transfer function (VSTF) method. We overcome the well-known triple integral problem associated with the VSTF method and reduce it to a simple integral. This model takes into account multiple channel effects, fiber losses, frequency chirp, optical filtering, and photodetection, which are ignored in the current literature. The model is in excellent agreement with results obtained by split-step Fourier simulation. Furthermore, with this model, we define coefficients that capture intersymbol interference, interchannel interference, self-phase modulation, intrachannel cross-phase modulation (XPM), intrachannel four-wave mixing (FWM), XPM, and FWM to characterize the impact of these effects individually on the system performance. We also apply this model to analyze the effects of varying system parameters and pulse shapes on the individual physical impairments. Index Terms Chromatic dispersion, fiber nonlinear optics, Kerr effect, optical propagation, wavelength-division multiplexing (WDM). I. INTRODUCTION HIGH-CAPACITY optical backbone networks are needed to support dramatically increasing demand for internet data traffic. A promising solution is dense wavelength-division multiplexing (DWDM) in which data are first time-division multiplexed (TDM) to form a channel centered at a given wavelength and many channels at different wavelengths are then wavelength-division multiplexed (WDM) together for transmitting high-throughput data on a single optical fiber. A total capacity of 69.1 Tb/s with 432 channels and 171 Gb/s Manuscript received July 02, 2011; revised October 03, 2011; accepted November 28, Date of publication December 16, 2011; date of current version February 03, This work was supported by the National Science Foundation under Grant CCF This work was presented in part at the IEEE International Conference on Communications 2011, Kyoto, Japan, and in part at the IEEE Global Communications Conference 2011, Houston, TX. The authors are with the Charles L. Brown Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, VA USA ( song@virginia.edu; mb-p@virginia.edu). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /JLT per channel has been reported [1]. All-optical communications eliminate the bottleneck of optical-to-electrical-to-optical conversion over long-haul DWDM systems. However, with periodic dispersion management and amplification, this inevitably gives rise to severe physical impairments, which in turn adversely affect system performance [2]. These physical impairments include not only linear effects due to dispersion, but also nonlinear effects due to fiber nonlinearities, which further consist of both intrachannel effects and interchannel effects. The performance of long-haul DWDM systems is fundamentally limited by dispersion, fiber nonlinearity, and noise [3] [5]. Sophisticated signal processing techniques are needed to mitigate the physical impairments and fully exploit the system capacity [6]. These techniques cannot be developed without a mathematical model which describes the input output relationship of the long-haul DWDM systems and characterizes the physical impairments experienced. To better explore the digital communications potential of these systems, this model should be a discrete-time model so that various mature digital signal processing (DSP) techniques can be applied. This model should also consider the two dimensions (2-D: time and wavelength) so that both intrachannel and interchannel effects can be simultaneously mitigated. Such a model has the potential to be applied in multichannel signal processing for intersymbol interference (ISI) and interchannel interference (ICI) mitigation, constrained coding for WDM systems, multiuser coding, multichannel detection, and path diversity for all-optical networks. The nonlinear Schrödinger (NLS) equation describes the propagation of optical pulses inside single-mode fibers (SMF). The NLS equation is a nonlinear partial differential equation whose exact analytic solutions generally are difficult to obtain except for some specific cases, such as soliton solutions in which the inverse scattering method can be employed [7] [9]. It is an implicit and continuous time expression, both of which limit its usefulness for signal processing applications. A large number of approximate analytical and numerical methods have been developed to solve the NLS equation. Linearization is a widely used approximate analytical approach. Most linearization methods can be classified into two categories: the Volterra series transfer function (VSTF) method [10] and the regular perturbation (RP) method [11]. The Volterra series is a polynomial expansion that represents the input output relationship of a nonlinear system with memory [12]. The VSTF method expresses the NLS equation as a polynomial expansion in the frequency domain and retains the most significant terms (Volterra kernels) in the resulting /$ IEEE

2 714 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 30, NO. 5, MARCH 1, 2012 transfer function. Since the NLS equation was first formulated as the Volterra series expansion form in 1997 [10], the VSTF method has been applied to investigate system design issues [13], fiber nonlinearities [14] [17], channel capacity [18] [21], dispersion compensation [22], pulse broadening [23], and filtering [24]. The convergence of the VSTF method was also investigated in [25]. The RP method is applicable to differential equations when the nonlinearity is weak [26]. Since 1998, the RP method has been used to solve the NLS equation [11], [27], [28] and investigate the channel capacity [29] and fiber nonlinearities [30] [32]. It has been shown that when the nonlinearities are due to the Kerr effect alone, the -order RP solution coincides with the -order Volterra series solution [28]. Both the third-order VSTF method and the first-order RP method result in the same triple integral. An alternative is to use a purely numerical method. Numerical methods can be classified into two categories: finite-difference methods and pseudospectral methods [33]. The split-step Fourier (SSF) method is one pseudospectral method that has been used extensively to solve the NLS equation. This method runs much faster than most finite-difference methods due in part to the use of the fast-fourier-transform (FFT) algorithm [34]. The SSF method is usually taken as the standard of accuracy for validating other methods in the absence of experimental data due to its well-established ability to accurately simulate the pulse propagation in fibers. However, the SSF method has three important drawbacks: 1) it is unable to display the effects of varying parameters on system output analytically; 2) it is unable to isolate what impairment is causing the most degradation; and 3) it is computationally expensive. Both the VSTF method and the RP method have the same complexity and computational efficiency. However, the triple integral involves massive numerical evaluation of iterated integration. We must reduce the triple integral to a much simpler form; otherwise, there is little computational advantage in using either the VSTF method or the RP method over the SSF method. Many researchers have attempted to solve the triple integral problem. However, most attempts have failed to reach a much simpler form with the exception of a simple integral for a simplified single-channel single-span case in which both fiber losses and pulse chirp are ignored [35], [22], [23]. A common scheme that deals with the triple integral problem is to introduce the concept of the nonlinear transfer function [36] (called the dispersion-managed (DM) kernel in [37]) and approximate it by asymptotic approximations [38] [40]. In this way, the triple integral can be simplified to a double integral but at the cost of accuracy. Three improvements of the RP method have been proposed but they apply to the single-channel single-span case only. The first is the enhanced RP method [28] whose solution contains a simple integral involving a complicated Fourier transform. The second is the multiplicative approximation method [41] whose solution includes a simple integral of a complicated convolution. The third is a recursive method [42], which asymptotically approaches the exact solution of the NLS equation but whose computational complexity remains an issue. The purpose of this paper is to develop a general deterministic analytical model of physical impairments in WDM systems, specifically multichannel multipulse multispan systems with periodic dispersion management and amplification. It is assumed that chirped Gaussian pulses are used at the transmitter and Gaussian optical filters are used at the receiver. We have previously developed, validated, and applied a 1-D discrete-time model of single-channel multipulse multispan systems [43]. In this paper, we extend the VSTF method further to the general multichannel multipulse multispan case and reduce the triple integral to a simple integral, gaining computational efficiency advantage over the SSF method with comparable accuracy. The resulting model is a polynomial model which takes into account fiber losses, dispersion, fiber nonlinearities, multiple channel effects, pulse chirp, and multiple spans. This model can be used to facilitate the suppression of various physical impairments. This paper is organized as follows. In Section II, we introduce the NLS equation, various linear and nonlinear effects, and solution methods. In Section III, we extend the VSTF method from a single-channel single-span case to a multichannel multispan case with periodic dispersion compensation and periodic amplification. We then derive an analytical model for each physical impairment in WDM systems. We extend the model to include photodetection for three popular modulation formats: ON OFF keying (OOK), differential binary phase-shift keying (DBPSK), and differential quadrature phase-shift keying (DQPSK). In Section IV, we validate the accuracy of the model compared to the SSF simulation. In Section V, we apply the model to analyze the impact of two system parameters (symbol rate and channel spacing) and pulse shapes on various physical impairments. Section VI concludes this paper and offers avenues for future work. II. NLS EQUATION The NLS equation models the propagation of optical pulses inside SMF. For pulse widths ps, the NLS equation is given by [33]

3 SONG AND BRANDT-PEARCE: 2-D DISCRETE-TIME MODEL OF PHYSICAL IMPAIRMENTS 715 TABLE I LIST OF SYMBOLS The nonlinear terms on the RHS of (2) can be identified as follows: when and,wehavespm;when or and,itisxpm;when or B. Nonlinear Effects Fiber nonlinearities can be classified into two types: Kerr effect and stimulated scattering [44]. Stimulated scattering leads to intensity-dependent gain or loss, the most detrimental of which is stimulated Raman scattering. The Kerr effect is due to the intensity dependence of the refractive index and causes an intensity-dependent phase shift experienced by an optical field during propagation in optical fibers [33]. Stimulated scattering is relatively small compared with the Kerr effect and is ignored in this paper. In the case of successive transmissions of TDM signals in a WDM system, the Kerr effect leads to the nonlinear interaction among optical pulses on the same channel (intrachannel effects), and among pulses on neighboring channels in a WDM system (interchannel effects). Intrachannel effects can be further divided into three types: self-phase modulation (SPM), intrachannel cross-phase modulation (IXPM), and intrachannel four-wave mixing (IFWM); interchannel effects can also be further separated into two types: cross-phase modulation (XPM), and four-wave mixing (FWM). The field of a multichannel multipulse system can be represented as a double summation of the fields of all individual pulses in all individual channels,, where is the field representing the th of pulses centered at located in the th of channels centered at in our baseband representation, where is the symbol duration and is the channel spacing. By substituting this summation into (1) we obtain (2)

4 716 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 30, NO. 5, MARCH 1, 2012 Fig. 1. Schematic of a typical WDM system with periodic dispersion compensation and amplification. The idea behind the RP method is that when an equation is changed by only a small amount, the solution will often only change by a small amount. This method is applicable to differential equations with a small nonlinear parameter and yields a series of terms of decreasing magnitude that approximate the solution of the original differential equation [26]. With this method, the first-order RP approximation to the output optical field has been obtained with the result being the same triple integral as earlier[28], [29]. Furthermore, it is found that the -order RP solution coincides with the VSTF solution for any integer [28]. 2) Numerical Methods: The SSF method obtains an approximate solution by assuming that in propagating the optical field over a small distance, the dispersive and nonlinear effects can be assumed to act independently. More specifically, propagation from to is carried out in two steps: in the first step, the nonlinearity acts alone; in the second step, the dispersion acts alone. A common improvement in the accuracy of the generic SSF method is a scheme called the symmetrized SSF method [33]. In this paper, we use the SSF simulation to validate the accuracy of our model. III. MODEL DEVELOPMENT This section describes our analytical model of multichannel multipulse multispan systems. Fig. 1 shows a schematic of a typical WDM system with periodic dispersion compensation and amplification. Our method can be applied to other systems than the one studied here; periodic equalization and homogeneous fiber are assumed here for notational simplicity. Consider a -channel WDM system. At the optical transmitter, a bank of laser diodes and a WDM multiplexer convertthedataofthe th channel and th pulse to be transmitted

5 SONG AND BRANDT-PEARCE: 2-D DISCRETE-TIME MODEL OF PHYSICAL IMPAIRMENTS 717 integral, as shown in Appendix A, we obtain the output field of the WDM demultiplexer in the frequency domain, given in (9).

6 718 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 30, NO. 5, MARCH 1, 2012 TABLE II PHOTODETECTOR OUTPUT FOR OOK AND DBPSK including advanced demodulation schemes such as those using heterodyne detection. Although these expressions are obtained under the assumption of chirped Gaussian pulses at the transmitter and the results will change when non-gaussian pulse shapes are used, they can serve as a powerful analytical tool to get an insight into the relative comparison of the intensities among all physical impairments for non-gaussian pulse shapes for which it is difficult to obtain simple integrals similar to (9) and (12). The assumption of Gaussian optical filters at the receiver can be easily relaxed. For any form of optical filters, the output of the fiber (in the frequency domain) can be multiplied by before the square-law operation of the photodetector. In this way, the expressions of the output of the WDM demultiplexer specific to the given optical filters can be obtained using a simple integral similar to (9) plus an integral needed to return to the time domain. To convert the receivedsignal back into electrical form and recover the data transmitted through the system, is passed through a photodetector and sampled at discrete times.thephotodetector output function varies for different modulation schemes: for OOK, the sampled output is simply ; for DBPSK, if a balanced photodetector is used, ; for DQPSK, if a balanced photodetector is used, the in-phase output is the same as the output of DBPSK and the quadrature output is 1. The aforementioned model can be easily extended to a coherent receiver. For a balanced coherent BPSK receiver, if a local oscillator (LO) is given by

7 SONG AND BRANDT-PEARCE: 2-D DISCRETE-TIME MODEL OF PHYSICAL IMPAIRMENTS 719 TABLE III PHOTODETECTOR OUTPUT FOR QUADRATURE CHANNEL OF DQPSK The sampled photodetector outputs for OOK, DBPSK, and DQPSK are summarized in Tables II and III. Appendixes C, D, and E contain the simplifying functions and the definitions of the impairment coefficients used in these tables. One common aspect of these three modulation formats is that the sampled photodetector output consists of seven terms: the contribution of the original transmitted bit, followed by ISI, ICI, the interaction between the targeted bit and the ISI, the interaction between the targeted bit and the ICI, the interaction between the ISI and the ICI, and, finally, the nonlinear effects. The latter can be further separated into five different effects: SPM, IXPM, IFWM, XPM, and FWM. In this way, we establish a mapping from the input to the sampled photodetector output

8 720 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 30, NO. 5, MARCH 1, 2012 Fig. 3. Effect of symbol rate on individual physical impairments for (a) OOK. and (b) DBPSK VSTF method is a nonrecursive method in which the computation is a one-shot process involving in general a triple integral. The numerical evaluation of the double integral with respect to and using a trapezoidal rule dominates the computation, where is the number of with complexity of order of frequency steps needed; the evaluation of the simple integral, where is the step with respect to is proportional to size used in the trapezoidal rule. Therefore, the computational complexity of the third-order VSTF method for a WDM system, where are the number is of other symbols interacting with any given symbol, i.e., the number of terms needed in the triple summations in (15). The and needed so that all significant physical values of impairments are included have been determined in [47]: at 40 ; for IFWM, ; at 50 GHz, Gs/s, for IXPM, ; for FWM,. In this paper, the double for XPM, and given by (17) in Appendix A integral with respect to has been solved and its closed-form solution (11) is a simple fraction function easily evaluated. In this way, the need to evaluate numerically the double integral is eliminated and the triple integral is simplified to a simple integral with respect to. The since and computational complexity is only are constants independent of and. For a multichannel multipulse multispan system, our model is computationally more efficient than both the VSTF method and the SSF method. V. MODEL APPLICATION In this section, we apply our 2-D discrete-time model to analyze the average impact of varying certain system parameters on Fig. 4. Effect of channel spacing on individual physical impairments for. (a) OOK and (b) DBPSK the individual physical impairments in a WDM system. The two most important system parameters for WDM systems are the symbol rate and the channel spacing. For a WDM system with channels and pulses, there are possible input matrices. We calculate the impairment characteristic coefficients using the definitions in Appendix E for OOK or Appendix F for DBPSK, and substitute these coefficients into the 2-D discrete-time model in Table II to obtain the various individual physical impairments for each input matrix. Then, we take the average of these individual impairments. In this section, mw. Other parameters are given in Table I. In addition, we analyze the effect of pulse shape on individual physical impairments. A. Effect of Symbol Rate We choose a three-channel system with seven pulses to investigate how the individual physical impairments imposed on the

9 SONG AND BRANDT-PEARCE: 2-D DISCRETE-TIME MODEL OF PHYSICAL IMPAIRMENTS 721 Fig. 5. Effect of pulse shape on individual physical impairments for OOK ( Gsps, GHz, and is the edge sharpness parameter of the super-gaussian pulses [33]). i.e., IXPM and IFWM, are generally worsened by the decrease in the symbol period. B. Effect of Channel Spacing Similarly, we choose a three-pulse WDM system with seven channels to investigate how the individual physical impairments imposed on the symbol located on the center channel change when the c(the)-362.9(c.44-e) he c.44-e change

10 722 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 30, NO. 5, MARCH 1, 2012 (18) By solving the double integral, the triple integral can be simplifiedtoasimpleintegral. We first separate all terms containing from

11 SONG AND BRANDT-PEARCE: 2-D DISCRETE-TIME MODEL OF PHYSICAL IMPAIRMENTS 723 APPENDIX C SIMPLIFYING FUNCTIONS USED IN TIME DOMAIN OUTPUT This appendix gives expressions for,and used in (14). is defined in Appendix B.

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