GAUSSIAN models are often used in optical communication

Transcription

1 4650 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 21, NOVEMBER 1, 2009 A Gaussian Polar Model for Error Rates of Differential Phase Detection Impaired by Linear, Nonlinear, and Laser Phase Noises Yuval Atzmon and Moshe Nazarathy, Senior Member, IEEE, Member, OSA Abstract We develop a simple analytic gaussian model, predicting BER performance of optical DPSK receivers with high accuracy in the wake of all three phase noise sources impairing differential phase detection: linear phase noise (ASE-induced), nonlinear phase noise (the Gordon Mollenauer effect), and laser phase noise Brownian fluctuations. We validate our analytical Q-factor based formulas using known analytical cases and importance-sampling based Monte-Carlo simulations. Index Terms Bit error rate (BER), differential phase shift keying (DPSK), gaussian noise, laser linewidth, laser noise, optical fiber communication, optical filtering, phase noise, Q-factor, self-phase modulation. I. INTRODUCTION GAUSSIAN models are often used in optical communication in order to simplify the BER performance evaluation of optical receivers. Such simple descriptions, based on Q-factor representations for the presumed gaussian decision variable, are known to work reasonably well for Intensity Modulated (IM) systems, despite the fact that the noise in the intensity domain is not actually gaussian. To meet the ever-increasing demand for high data rates, a new generation of transmission systems has emerged, based on differential phase detection: differential phase shift keying (DPSK) systems [1] are currently deployed, while differentially coherent advanced modulation formats such as multisymbol DSPK with decision-feedback are being contemplated [2] [4]. The precise analytical or quasi-analytical analysis of differential detection systems is quite involved (even in the simplest binary-dpsk case), and is generally not amenable to yielding accurate predictions based on the gaussian assumption, due to the non-gaussian statistics of the phase noise mechanisms [5]. The most natural description of differential phase transmission systems is in the polar (angular) domain, although simplified cartesian (I&Q components based) analyses are often used to obtain first-order insight, e.g., by neglecting noise noise terms it is simple to derive a gaussian approximation in the cartesian domain for the DPSK decision variable in the presence of additive white gaussian noise (AWGN). Unfortunately, such an approximation is quite inaccurate, e.g., at, bit error rate Manuscript received January 19, 2008; revised May 10, First published June 30, 2009; current version published September 10, The authors are with the Department of Electrical Engineering, The Technion Israel Institute of Technology, Haifa 32000, Israel ( nazarat@ee. technion.ac.il; naza@ieee.org; yuvval@gmail.com). Digital Object Identifier /JLT (BER), the so-called differential detection penalty (signal to noise ratio (SNR) between differential (DPSK) and coherent phase shift keying (PSK) detection) is 1 db for optical binary-dpsk (BDPSK), and 2.4 db for quaternary-dpsk (QDPSK), rather than the 3 db predicted by the gaussian Cartesian approximation. Differential detection performance is essentially determined by the phase noise fluctuations of the angle of the DPSK decision variable. There are three relevant sources of phase noise impacting DPSK receivers. (i) Linear (LN) phase noise, mainly due to the fluctuations in the quadrature component of the amplified spontaneous emission (ASE). (ii) Nonlinear (NL) phase noise due to the conversion of the ASE power fluctuations into phase-shifts via the mechanisms of self-phase-modulation (SPM), and cross-phase-modulation (XPM). This is the so-called Gordon-Mollenauer effect [6]. (iii) The random fluctuations of the phase of the optical source: laser phase noise (LPN). Notice that with the advent of high-speed optical transmission, as bitrates gradually increased, the LPN impairment in DPSK transmission had become insignificant (as the shorter bit duration allows less time for the phase to wander off). However, with the recent advent of advanced differential phase modulation methods processing information over many symbol intervals, LPN has made a comeback, reassuming its role as a limiting impairment [3], [7]. This paper aims at DPSK designers interested in simple and intuitive yet accurate analytical modeling of their systems under certain simplifying assumptions. We develop an analytic gaussian model of BER generation, with very high accuracy ( 0.1 db LN and LPN) accounting for the combined effect of all three phase noise sources (LN, NL, and laser) impairing DPSK detection. The model applies to arbitrary transmit pulses and optical filter (OF) responses, subject to the following limitations: (a) the transmitted pulse shapes are assumed to be limited to the symbol duration [e.g., return to zero (RZ) and nonreturn to zero (NRZ) pulses], and so is the duration of the optical filter impulse response, i.e., the optical bandwidth is relatively large such that intersymbol interference is neglected. (b) the effect of the electrical postdetection filter is not included. (c) The DPSK receiver front-end is assumed ideal. Imbalances and other receiver impairments are not considered here, but were treated in [7] based on the applying methods of the detailed theory first described in this paper. The formulation of such an all-gaussian analytic model for all sources of phase noise is far from trivial, e.g., the statistics of nonlinear phase noise [5] has been found both theoretically /$ IEEE

2 ATZMON AND NAZARATHY: GAUSSIAN POLAR MODEL FOR ERROR RATES OF DIFFERENTIAL PHASE DETECTION 4651 [8] and experimentally [9] not to be gaussian distributed. For non-gaussian noise neither the variance nor the conventional Q-factor are sufficient to characterize system performance, rather the full probability density function (PDF) or cumulative density function (CDF) are, in principle, necessary. The usage of Q-factor for describing DPSK performance is not accurate [10]. A correction to the Q-factor was proposed in [11]. We should note that these papers the usage of Q-factor based models (implying gaussian statistics) is relegated to numerical application of estimating the BER from (Cartesian domain) simulated eye-diagrams, whereas in our approach we develop analytic formulas with Q-factor evaluated from the variances of the various phase noise mechanisms in the polar domain. Although the DPSK complex-decision variable is not gaussian in the wake of nonlinear phase noise [i.e., the cartesian I&Q samples of the optical field are not jointly gaussian random variables (RV)], nevertheless, upon converting from a cartesian to a polar description, the nonlinear phase noise component (the noise in the angle of the decision statistic) was identified to be nearly gaussian [12] [14]. Considering next laser phase noise, we recently developed a much simplified, yet very accurate, LPN analysis [15], based on the phase-noise exponent commutation (PNEC) premise that the receive filtering may be effectively applied in the polar (angular) domain, modeling for the first time arbitrary transmit pulses (e.g., RZ) and optical filter responses. The PNEC concept directly implies that the angle of the DPSK decision statistic is gaussian distributed, which assumption was verified numerically as very accurate, by means of importance-sampling based Monte-Carlo simulations of the phase-noise BER floor. Thus, two of our three phase noise sources are nearly gaussian. It remains to address the linear (ASE-induced) phase noise, and the interplay among all three noise sources. An analytic formula for the applicable distribution of the phase noise of a fixed phasor perturbed by circular gaussian noise has been long known [16]. It is only for very small angular decision regions that the phase distribution tends to gaussian. However, we managed to show that the non-gaussian LN phase noise distribution may be very closely fit (over a wide range of SNRs of interest and angular decision regions) by that of a gaussian RV in its tails (which is the only region which matters for BER evaluation), albeit with an effective SNR lower by as much as 34% for BDPSK and 7% for QDPSK, relative to the SNR suggested by the naïve gaussian phase model. With this caveat, all three (independent) phase noise sources are nearly gaussian (in their tails); hence, their sum is even more closely gaussian [as expected of the repeated convolution of PDFs, in the spirit of the Central Limit Theorem (CLT)]. The paper is structured as follows. In Section II, we develop approximations to linear phase noise in the polar and cartesian domains. In Section III, we evaluate the accuracy of these approximations and refine the polar gaussian approximation of Section II, conceiving a highly accurate analytic gaussian model for linear phase noise, further justified in the Appendix. In Section IV, we consider SPM-induced nonlinear phase noise, and its interplay with linear phase noise. In Section V, we incorporate laser phase noise into the model, obtaining the full jointly gaussian formulation for all three phase noise sources, and providing accurate analytical evaluation of the BER based on a polar-domain Q-factor. Section VI concludes the paper indicating future extension directions. II. LINEAR PHASE NOISE GAUSSIAN POLAR AND CARTESIAN APPROXIMATIONS We start by modeling linear phase noise in an optically amplified fiber transmission system employing differential phase detection. The transmitted optical signal complex-envelope is PAM modulated, using a time-limited transmit pulse-shape (with an indicator function assuming the value 1 over, 0 outside it) The receiver front-end is assumed to include an OF after the last optical amplifier (OA). The OF impulse response is also approximated as time-limited to the interval, as its bandwidth is larger than the symbol rate The assumed time-limitations of, imply that there is no intersymbol interference (ISI). In this study we do not model the effect of the postdetection electrical filter; refer to [17] [19] for generalized approaches treating ISI and postdetection electrical filtering while accounting for ASE-induced phase noise. The sampling is assumed to occur directly at the output of the delay interferometer(s) (DI) optical front-end, which generates the well-known decision variable for DPSK detection,, where is the th sample of the received optical field at the OF output. In the absence of additive noise and phase noise impairments, the received sample then acquires the simple form where with denoting convolution. Introducing now additive noise, first assume that the fiber is linear, such that the amplified spontaneous emission (ASE) contributed by all the OFs along the link, additively superposes at the OF input to form an AWGN process, with two-sided spectral density. The noise samples at the OF output,, are added up to the noiseless signal samples (3), yielding the noisy samples where is a normalized version of the optically filtered AWGN samples, with complex circular gaussian (CCG) statistics, and variance given by (1) (2) (3) (4) (5) (6)

3 4652 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 21, NOVEMBER 1, 2009 where, and we used the fact that the transmitted constellation symbols are normalized to unity magnitude for differential phase signaling (the amplitude of the optical carrier is absorbed in ). The real and imaginary parts,, of are independent identically distributed gaussian RVs:. The received noisy samples (5) may then be expressed in terms of the I&Q components, as Since for reasonable optical SNR (OSNR) we have, (7) is further approximated as Using the first-order Taylor series expansion, (8) is finally approximated in a form separating amplitude and phase noises The amplitude noise is seen to be independent of the linear phase noise, which is given by the Phase Noise Exponent (PNE),, further expressed in a form highlighting the useful property that the filtering of ASE effectively occurs in the polar (angular) domain (7) (8) (9) where (11) was used, is the differentially phase modulated information, and describes the phase noise (fluctuating component) of the DPSK angular decision variable. Being given by the difference of two independent gaussian RVs, is itself gaussian distributed,, with variance (6), double that of (12), as the both successive observations contribute to the phase noise (14) This model for the DPSK angular decision variable is henceforth referred to as the gaussian polar approximation (GPA). 1 Symbol-by-symbol detection of -ary DPSK in AWGN is optimally based on quantizing the gaussian angle in the polar domain, yielding the pairwise error probability (PWEP) (15) with the Gaussian-Q function. For a matched filter (MF) OF,, the linear phase noise variance (6) reduces to the inverse of he SNR 2 (here, is the received symbol energy) (16) Consider detection of Binary DPSK (BDPSK),, with an ideally matched OF. The resulting bit-error-rate (BER) is (17) The angle of the observed noisy sample, expressed (defining ) (10) (9), is then simply The inaccuracy incurred in this polar domain gaussian approximation is evaluated in the next section. We end this section by reviewing an alternative gaussian approximation for (differential) phase transmission, called here the cartesian gaussian approximation (CGA) This well-known model provides a first-order approach to deriving DPSK performance, by evaluating Cartesian (I&Q) gaussian noise components of the decision variable,, which is expressed, using (8), as (11) It is apparent that represents the fluctuating (linear phase noise) component of the total observed angle,, which provides a sufficient statistic for coherent PSK detection, directly quantized to obtain the PSK decision. The variance of the angle of the observation is half the variance of (6) Expanding the product, and discarding the (retaining just the gaussian terms), yields (18) terms (12) A sufficient statistic for DPSK detection is the angle of the decision variable, expressed as the difference of the phase angles of successive observations,, (13) i.e., the noiseless signal gaussian noise with variance (19) is perturbed by circular in each of its I&Q independent 1 GPA may be alternatively interpreted as Gaussian Phase Approximation. 2 E =N is the signal to noise ratio of matched filtering, and in our case, it is the inverse of the variance of the normalized AWGN, since the signal reduces to unity.

4 ATZMON AND NAZARATHY: GAUSSIAN POLAR MODEL FOR ERROR RATES OF DIFFERENTIAL PHASE DETECTION 4653 Fig. 2. Horizontal displacement (along the log SNR axis) between the exact and GPA binary DPSK BER curves in Fig. 1. Fig. 1. Various gaussian phase noise approximations versus exact BER for B-DPSK. linear scale correction factor of the GPA model to the SNR components. The CGA predicts that compared with BPSK, the BER of B/Q-DPSK exhibits 3 db degradation (as its variance is doubled relative to the PSK case) (20) We show in the next section that between these two gaussian approximations (GPA versus CGA), the CGA is the less accurate one. A novel approach refining the GPA model, making it very accurate, is further introduced. III. REFINED GAUSSIAN POLAR MODEL In the previous section, some approximations led us to a simplified GPA model, whereby the phase noise of the observation,, and that of the DPSK decision variable, are approximated as gaussian distributed. To assess the accuracy of the GPA, we compare its BER prediction (17) with the exact analytic expression available for an ideal BDPSK receiver with matched optical filtering and no electrical filter [5] (21) The comparison is illustrated in Fig. 1. A key graphical observation (somewhat masked by optical illusion) is that the horizontal shift between the two curves is nearly constant. To better visualize this, the horizontal displacement (along the log SNR axis) between the two curves is plotted in Fig. 2, indicating an average shift of 1.28 db, with 0.1 db variation over a wide scale of BER levels of interest. The observed constant shift suggests that the actual BER may be far more accurately described by a refined gaussian polar approximation (RGPA), whereby a horizontal shift is applied to the gaussian-q function predicted by the GPA, displacing the GPA BER curve to the right by 1.28 db. This displacement along the horizontal log SNR scale amounts to analytically obtaining the actual BER curve of BDPSK by multiplicatively applying a (22) As demonstrated in Fig. 1 the RGPA model (22) (circles) yields remarkably accurate results, almost perfectly coinciding with the true analytic formula (21) (solid line). Fig. 1 also plots the CGA-predicted BER, revealing that this approximation is the least precise, incurring an inaccuracy of 2.2 db for BDPSK. We also note that the GPA and CGA curves bracket the analytic curve (which coincides with the RGPA) providing lower and upper bounds for the exact BER. Strictly speaking, the angular noise of the decision variable,, is not precisely gaussian distributed, as predicted by the RGPA model, but is very nearly so, albeit with an effective variance larger by 34% than the GPA-predicted variance,. Fig. 3 illustrates that although the PDF of the angle of the received signal is not exactly Gaussian, its logarithm maintains a quadratic shape over its mid range, fitting a gaussian with the variance, starting to deviate from the gaussian shape down in the tails. Our procedure amounts to fitting a complementary cumulative gaussian distribution of modified variance to better approximate the tails (which are the only part of the PDF curve impacting the BER evaluation). Further mathematical insight justifying the accuracy of the RGPA method is presented in the Appendix. We next consider the modeling of QDPSK BER for which a precise analytic formula exists, e.g., [20, Eq. 35]. As shown in Figs. 1 and 4 the solid exact analytic BER curve almost perfectly coincides with the RGPA prediction (circles), bracketed from below and above by the GPA and CGA (with the CGA again yielding the least accurate approximation). Figs. 1 and 4 indicate that the (uncorrected) GPA incurs the greatest inaccuracy for BPDSK, whereas for QDPSK it incurs a small inaccuracy of 0.3 db, relative to the exact and RGPA results, i.e., the RGPA correction is not that significant for QDPSK. Extending the RGPA model to -ary DPSK constellations, we must apply constellation-dependent correction factors

5 4654 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 21, NOVEMBER 1, 2009 Fig. 4. Various gaussian phase noise approximations versus exact BER for QDPSK. Fig. 3. Exact PDF versus a gaussian approximation of the phase of the received field in the presence of linear phase noise, for SNR of 7 and 20 db., with the angular distance from the mean angular signal to the decision boundary in the polar domain. We summarize these observations by tabulating the RGPA correction factor for db db db (24) This completes our treatment of linear phase noise, whereby we have shown that gaussian approximations for the DPSK decision statistic are more accurate in the polar rather than the Cartesian domain, further refining the gaussian approximation in the polar domain, making it almost perfect (up to 0.1 db) by applying a multiplicative correction to the effective variance (amounting to a shift of the BER curve along the horizontal logarithmic or db scale). The key result is a very accurate analytic formula (23) for -ary DPSK detection in the linear transmission regime, expressed as a gaussian complementary CDF (CCDF). As another example, we consider coherent QPSK transmission, analytically deriving the RGPA correction factor by reformulating the BER in the gaussian polar domain. The exact analytic BER for match-filtered QPSK detection is known to be (25) where (16) was used in the last equality. Since the variance of the coherent QPSK decision statistic is given by (12), using, the RGPA model BER for QPSK is expressed in the polar-domain form (23) The general trend is that as the order of the -ary constellation is increased, quickly converges to unity, i.e., the GPA gaussian approximation for the phase noise becomes more and more accurate. In fact, for, the correction is minor, and for constellations such as 8-DPSK or higher, the inaccuracy incurred by the GPA is negligible i.e., we may set (26) The correction factor is then analytically extracted by equating the arguments (Q-factors) in (25) and in (26), yielding (27)

6 ATZMON AND NAZARATHY: GAUSSIAN POLAR MODEL FOR ERROR RATES OF DIFFERENTIAL PHASE DETECTION 4655 This analytic result (27) is consistent with the high-snr numerical results of Fig. 7, as further discussed in the Appendix. Finally, we remark that the linear phase noise analysis carried out here appears to be just a first-order approximation. However, since it is carried out in the polar rather than the angular domain, and as it involves a modified variance better fitting the distribution tails, its accuracy is substantially higher than that of a conventional Cartesian domain first-order approximation, as further verified in the Appendix. In the following sections we extend our phase noise model by incorporating additional phase noise mechanisms induced by the fiber nonlinearity and by the laser source. IV. SPM INDUCED NONLINEAR PHASE NOISE Over a realistic fiber optic link with Kerr nonlinearity, the ASE noise interacts with the nonlinear SPM mechanism to generate nonlinear (NL) phase noise this is the so-called Gordon-Mollenauer (GM) effect, in effect converting amplitude or power noise (fluctuations in the radial component) to phase noise (fluctuations in the angular component). In the absence of dispersion, the contributions from multiple spans of an optically amplified link compound to yield the total NL phase noise at the receiver as follows: (28) where is the effective length, is the NL coefficient, is the transmitted optical field and is the th sample of the ASE noise optical field due to the th OA. The th term in the summation may be simplified by a formalism similar to that used in deriving (9) the nonlinear phase noise component; hence, it may be accurate to neglect its impact on the linear phase noise. As shown in [12] [14], the NL phase noise has a nearly Gaussian distribution in the polar (angular) domain. This may be simply seen by neglecting the noise noise terms in (29). The remaining signal noise fluctuating term is gaussian. The fact that the NL phase noise is nearly gaussian is reminiscent of the nearly gaussian nature of the linear phase noise, as established in the previous sections. The total (linear + NL) phase noise of the received field sample, being the sum of two independent gaussian processes, is itself gaussian, with variance (30) where is the linear phase noise variance (6) (for both I&Q components summed up) and denotes the variance of the nonlinear phase noise (with the factor of half introduced for consistency with the LN case). Notice that may be evaluated by plugging (29) into (28), and then evaluating the variance of the resulting expression by conventional statistical techniques [5]. Moreover, the statistical dependence between different NL phase noise samples is negligible. In fact both the NL and linear phase noise sample may be well approximated as white discrete-time processes (their samples at different times are independent). It follows that the linear+nl phase noise in has double the variance of the linear+nl phase noise (30) of the observation (31) The correction factor applied to the linear phase noise will hardly matter when the NL phase noise is dominant; however, when the NL phase noise is not too large (relative to the linear term), the correction may become significant. The resulting BER for -ary DPSK in the wake of linear + nonlinear phase noise is, therefore, expressed using the RGPA model as follows: (29) The nonlinear phase noise is seen to be dependent on the in-phase (re) ASE noise components injected by the OAs, whereas, as seen in Section II, the linear phase noise is essentially driven by the quadrature (im) components of the ASE. As the ASE is circular gaussian, its I&Q components are independent of each other. It then follows that the NL and linear phase noises are statistically independent. Previous results [11, Fig. 3] indicate that such model, neglecting the in-phase radial real-valued component, provides an accurate description for the combination of linear and nonlinear phase noise (without laser phase noise, unlike our treatment here). In fact the real-valued ASE noise component contributes considerably to the nonlinear phase noise component via the self-phase modulation effect, translating intensity into phase variations. Therefore, the contribution of the real-valued component to the linear phase noise component is negligible relative to its dominant contribution to (32) V. JOINTLY GAUSSIAN LINEAR, NONLINEAR AND LASER PHASE NOISE TREATMENT It remains to incorporate the treatment of laser phase noise (LPN). This impairment usually impacts high-speed -ary DPSK systems using distributed feedback (DFB) laser sources only for high values of (e.g., 8-DPSK or higher-order constellations). However, LPN may be significant for advanced modulation formats using differential phase detection over longer windows than two adjacent bits [2] [4]. In a recent paper [15], we have shown that for DPSK detection in the presence of LPN, the angle of the DPSK decision statistic is nearly gaussian distributed. Our results in [15] generalized LPN modeling to arbitrary pulse shapes and OF impulse responses,, subject to the same time-limitation to the symbol duration, as assumed in Section II. The following integral expression (readily evaluated numerically or in certain

7 4656 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 21, NOVEMBER 1, 2009 cases analytically) was provided for the variance of the DPSK decision statistic (33) with and defined as (34) Fig. 5. BER versus SNR performance of an NRZ BDPSK system using a matched I&D optical filter, accounting for linear and laser phase noise sources. The five continuous curves plot the analytic gaussian-based RGPA model (36), parameterized by various normalized laser linewidths. Good agreement is apparent with the data points of the IS Monte Carlo simulations. The five thin gray curves plot the corresponding analytic formulas of Einarsson [21]. In the special case of NRZ pulses and matched integrate and dump (I&D) optical filtering, the general expression (33) reduces to. Here is the laser linewidth normalized through the symbol rate. Laser phase noise generation is evidently independent of the two other mechanism of ASE induced phase noise (LN and NL), which were in turn seen to be nearly independent of each other. So, all three phase noises (LN, NL, LPN) are well approximated as independent gaussian RVs. We shall refer to them as jointly gaussian, in the sense of a convenient figure of speech (namely that all three are gaussian), rather than making a statistical statement. 3 Since all three noise sources are gaussian and independent, we conclude that the decision statistic is normally distributed with variance (35) Hence, the BER in the wake of all three phase noise sources is very well approximated by a gaussian CCDF with (36) (37) This formulation (36), (37) provides a simple yet accurate tool to evaluate the combined effects of all three phase noise in -ary DPSK detection. A simple Q-factor,, (with given by (37)) may be used to equivalently represent the system performance, with the usual relation between Q-factor and BER applying. We reiterate that our formulation is valid for arbitrary pulse and optical filter response shapes (time-limited to ). 3 It is also formally correct, though trivial, that these three RVs are jointly gaussian in the statistical sense, as each RV is normal and the three RVs are mutually independent. Figs. 5 and 6 numerically validate the accuracy of our jointly gaussian formulation (36) for linear + laser phase noise (no NL noise included) for Binary and Quaternary NRZ-DPSK modulation, respectively, assuming an I&D matched optical filter. The figures compare the plots of the simple analytical formulas of the RGPA model, with numerical data points obtained by Importance Sampling (IS) based Monte-Carlo (MC) numerical simulations. The high accuracy of the gaussian formulation is apparent. Fig. 5 further compares our BDPSK BER results with those obtained by Einarsson [21] (the gray dotted lines). The approach used in [21] is analytically complex yet not very accurate: a firstorder Taylor expansion is used to derive the moment generating function (MGF) of the DPSK decision statistic, then a saddle point MGF inversion algorithm is applied to derive the resulting BER. Although we use a simplified gaussian formulation for the BER, Fig. 5 indicates that our results (as validated by the numerical IS MC data points) are far more accurate than those in [21]. Notice that we performed our comparison in Fig. 5 for I&D matched filtering, which to the best of our knowledge is the only filter shape supported by the approach [21]. However, as already indicated, our model covers more general transmission pulse shapes and optical filter impulse responses (subject to the time-limitation to the symbol duration). We further note that a comparison to the results of Kaiser [22] is not directly possible, as the baseline laser-phase-noisefree curve (i.e, just AWGN, turning the LPN off) [22] disagrees with the exact analytical expression (21). However, our results display trends similar to those of Kaiser s. Finally, we remark that our correction factor (24), verified by our own MC simulations, nearly coincides with the correction factor derived by Hiew and Abbou [11] by means of estimating the DPSK BER with linear and nonlinear phase noises from simulated eye-diagrams (the correction factor in [11] is 0.87, apparently biased to provide higher accuracy for lower SNR, according to Fig. 7, whereas our factor covers a wider range of SNRs). In contrast, we developed our Q-factor analytic expression directly in terms of the variances of the various phase

8 ATZMON AND NAZARATHY: GAUSSIAN POLAR MODEL FOR ERROR RATES OF DIFFERENTIAL PHASE DETECTION 4657 APPENDIX FROM THE EXACT PDF OF THE LINEAR PHASE NOISE TO THE REFINED GAUSSIAN POLAR APPROXIMATION For the AWGN (ASE) only case, the signal at the optical filter output is additive Gaussian in the Cartesian domain. The distribution of its phase is known to be [16] (38) Fig. 6. Analytic gaussian-based RGPA predictions (36) accounting for linear and laser phase noise sources versus IS Monte Carlo simulations for NRZ- QDPSK modulation with matched I&D optical filtering. where. Inspecting the numerical properties of we make the following observations: 1) For moderate phase values (up to (38) is approximately Gaussian distributed noise mechanisms in the polar domain, and thoroughly investigated the applicability of a fixed correction factor in the Appendix. Remarkably, we managed to derive the factor without even resorting to simulations for the nonlinear phase noise. The reason is that the correction is just associated with the LN phase noise the NL phase noise automatically comes out as Gaussian in the polar domain as already shown in [12] [14]; hence, when deriving the correction factor, it is not necessary to consider the NL noise at all. Moreover, unlike [11] our model further extends the treatment by incorporating laser phase noise. VI. SUMMARY AND DISCUSSION The topic of evaluating the BER of DPSK systems based on simplified gaussian models or equivalent Q-factor representations has been actively pursued [10] [12] yet heretofore not fully clarified. This paper contributed a simple gaussian analytic model of DPSK detection, accounting for all three phase noise sources, while providing remarkable accuracy. In particular, the effect of laser phase noise was demonstrated to be readily and accurately assessed in conjunction with linear phase noise, while avoiding complex MGF analytic evaluation and inversion [21], which nevertheless provides less accuracy. Our gaussian model formulates a simple Q-factor based on the analytically (or possibly numerically) estimated variances of the three phase noise mechanisms (including the correction factor applied to the linear phase noise variance). This is distinct from approaches deriving a Q-factor for purposes of numerical evaluation of BER based on empirical Q-factors extracted out of simulated eye-diagrams [11], [12]. Following the principles outlined in this paper, our analytic formulas will be adapted in future work to evaluating the nominal performance and degradations of advanced modulation formats based on multisymbol DPSK with decision feedback. Additional study will be devoted to lifting the limitations of our method, namely incorporating the effect of electrical postdetection filtering and accounting for intersymbol interference. This will be accomplished by using Volterra-series techniques (up to second-order) to model propagation through the detector quadratic nonlinearity. 2) For SNR exceeding 5 db, the ratio of the logarithms of (38) and of the Gaussian pdf, (39) for any given phase value, is approximately independent of the (square-root) SNR value. This is apparent by noting that the contour lines in the contour plot in Fig. 7 remain vertical, not bending excessively. Moreover, the horizontal and vertical gradients of the contour plot in Fig. 7 are very mild (the vertical gradient is almost zero even on the logarithmic scale). It follows from (39) that the phase PDF is given by and its CCDF (for a specified value )is (40) (41) As observed above, varies significantly more slowly than either or. We may then approximate (41) by replacing with its value at the beginning of the integration interval (42) where we neglected the CCDF tail beyond, and further discarded the factor preceding the gaussian-q function, since the quadratically-exponential decay of the integrand dominates.

9 4658 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 21, NOVEMBER 1, 2009 Fig. 7. (a) Contour plot of the ratio k(; s) between the logarithms of the exact and Gaussian approximated distribution functions of the phase of linear phase noise. (b) Cross section of (a) for two different SNR values. As mentioned above, since the vertical gradient is almost zero over the useful domain, we define an SNR averaged ratio with the overbar denoting averaging over the given range. Replacing in (42) by its average value (43), yields (43) (44) This justifies the high accuracy of a Gaussian model for the phase noise of the observed sample at the output of the optical filter, (e.g., as used in coherent PSK detection), albeit with a modified variance relative to that predicted by the first-order Cartesian approximation. A coherent PSK detection system would then display an error probability approximately following the Gaussian CCDF, albeit with a modified scale parameter. In particular, for a coherent QPSK system, we take and in (44), and further reformulate the applicable correction factor there as (45) yielding an expression coinciding with (26). We now verify the consistency of this approach with the value (27) previously obtained for, namely, which implies using (45). This value may indeed be read off in Fig. 7 as the ordinate at abscissa for high-snr. Consider next differential detection involving the subtraction of a preceding symbol phase from the current symbol phase. As both phase terms are statistically independent, the resulting PDF of the angle of the DPSK decision variable is the autocorrelation of (38) (autocorrelation rather than self-convolution because of the minus rather than plus sign). Although we have not obtained an analytic expression for the exact PDF, we intuitively argue that the resulting PDF may be even better fit in its tails by a Gaussian, than (38) may, and that the logarithmic ratio property is maintained (the ratio (39) slowly varies with, now assumed to stand for the angle of the decision variable, where ). Therefore, the Gaussian CCDF form (44) for the angle of the decision variable is applicable to this case as well. A supporting argument, in the spirit of the Central Limit Theorem, is that since, may be fit in their tails with gaussian variables, the autocorrelation of their PDFs, describing the PDF of their difference, is even better fit by a Gaussian, with an appropriately modified variance. Although we recognize that the current argument is not entirely rigorous, the curves of Figs. 1 and 4, empirically support its validity, as the GPA curve (17) displays an almost constant shift along the logarithmic horizontal scale (SNR [db]) indicating that it is possible to treat the phase of the observation or the DPSK decision statistic as Gaussian, once a correction factor, as graphically extracted from (24), is applied. REFERENCES [1] A. H. Gnauck and P. J. Winzer, Optical phase-shift-keyed transmission, J. Lightw. Technol., vol. 23, p. 115, [2] X. Liu, S. Chandrasekhar, and A. Leven, Digital self-coherent detection, Opt. Exp., vol. 16, pp , [3] M. Nazarathy, X. Liu, L. Christen, Y. Lize, and A. Willner, Self-coherent decision-feedback-directed 40 Gbps DQPSK receiver, Photon. Technol. Lett., vol. 19, pp , [4] M. Nazarathy, X. Liu, L. Christen, Y. Lize, and A. Wilner, Self-coherent optical detection of multi-symbol differential phase-shift-keyed transmission, J. Lightw. Technol., vol. 26, no. 13, pp , Jul [5] K.-P. Ho, Phase Modulated Optical Communication Systems. New York: Springer, [6] J. P. Gordon and L. F. Mollenauer, Phase noise in photonic communications systems using linear amplifiers, Opt. Lett., vol. 15, no. 23, pp , Dec [7] Y. K. Lize, L. Christen, M. Nazarathy, Y. Atzmon, S. Nuccio, P. Saghari, R. Gomma, J.-Y. Yang, R. Kashyap, A. E. Willner, and L. Paraschis, Tolerances and receiver sensitivity penalties of multibit delay differential-phase shift-keying demodulation, IEEE Photon. Technol. Lett., vol. 19, no. 23, pp , Dec. 1, [8] A. Mecozzi, Limits to long-haul coherent transmission set by the kerr nonlinearity and noise of the in-line amplifiers, J. Lightw. Technol, vol. 12, no. 11, pp , Nov [9] H. Kim and A. Gnauck, Experimental investigation of the performance limitation of DPSK systems due to nonlinear phase noise, IEEE Photon. Technol. Lett., vol. 15, no. 2, pp , Feb [10] Bosco and Poggiolini, On the Q-factor inaccuracy in the performance analysis of optical direct-detection DPSK systems, IEEE Photon. Technol. Lett., vol. 16, no. 2, pp , Feb

10 ATZMON AND NAZARATHY: GAUSSIAN POLAR MODEL FOR ERROR RATES OF DIFFERENTIAL PHASE DETECTION 4659 [11] Q. C. C. Hiew, F. M. Abbou, H. T. Chuah, S. P. Majumder, and A. A. R. Hairul, BER estimation of optical WDM RZ-DPSK systems. Through the differential phase, IEEE Photon. Technol. Lett., vol. 16, no. 12, pp , Dec [12] X. Wei, X. Liu, and C. Xu, Numerical simulation of the SPM penalty in a. 10-Gb/s RZ-DPSK system, IEEE Photon. Technol. Lett., vol. 15, 11, Nov [13] Y. Yadin, M. Shtaif, and M. Orenstein, Bit-error rate of optical DPSK in fiber systems by multicanonical Monte Carlo simulations, IEEE Photon. Technol. Lett., vol. 17, no. 6, pp , Jun [14] Y. Yadin, M. Shtaif, and M. Orenstein, Nonlinear phase noise in phase-modulated WDM fiber-optic communications, IEEE Photon. Technol. Lett., vol. 16, no. 5, pp , May [15] Y. Atzmon and M. Nazarathy, Laser phase noise in coherent and differential optical transmission revisited in the polar domain, J. Lightw. Technol., vol. 27, no. 1, pp , Jan [16] J. P. Thomas, An Introduction to Statistical Communication Theory. New York: McGraw-Hill, [17] J. Wang and J. M. Kahn, Accurate bit-error-ratio computation in nonlinear CRZ-OOK and CRZ-DPSK systems, IEEE Photon. Technol. Lett., vol. 16, no. 9, p. 2165, Sep [18] P. Serena, A. Orlandini, and A. Bononi, Parametric-gain approach to the analysis of single-channel DPSK/DQPSK systems with nonlinear phase noise, IEEE J. Lightw. Technol., vol. 24, no. 5, pp , May [19] M. Nazarathy, B. Livshitz, Y. Atzmon, M. Secondini, and E. Forestieri, Optically amplified direct detection with pre- and post- filtering: A Volterra series approach, IEEE J. Lightw. Technol., vol. 26, no. 22, pp , [20] F. Edbauer, Bit error rate of binary and quaternary DPSK signals with multiple differential feedback detection, IEEE Trans. Commun., vol. 40, no. 3, pp , Mar [21] G. Einarsson, J. Strandberg, and I. T. Monroy, Error probability evaluation of optical systems disturbed by phase noise and additive noise, J. Lightw. Technol., vol. 13, no. 9, pp , Sep [22] C. P. Kaiser, M. Shafi, and P. J. Smith, Analysis methods for optical heterodyne DPSK receivers corrupted by laser phase noise, J. Lightw. Technol., vol. 11, no. 11, pp , Nov Yuval Atzmon received the B.Sc. degree (cum laude) in computer engineering and the M.Sc. degree in electrical engineering from The Technion Israel Institute of Technology, Haifa, in 2003 and 2008, respectively. During , he was with the R&D Directorate, Israeli Ministry of Defense, conducting research in the field of radio-frequency microelectronics and monolithic microwave integrated circuits. Moshe Nazarathy (SM 05) received the B.Sc. degree (cum laude) and a Doctor of Science EE degree from The Technion Israel Institute of Technology, Haifa. He is a tenured Associate Professor with the Electrical Engineering Department, The Technion. He was a visiting Associate Professor with the same department during From 1982 to 1984, he held a postdoctoral position at Stanford University s Information Systems Laboratory, Stanford, CA. From 1984 to 1988, he was with Hewlett Packard s Photonics and Instruments Laboratory, attaining the rank of Principal Engineer. He co-founded Harmonic, Inc., served as Senior VP R&D, and corporate CTO and General Manager of the company s Israeli subsidiary and was a member of Harmonic s board of directors from 1988 to He also serves as a Technology Venture Partner with Giza Ventures, a leading VC firm in Israel, and on the advisory board of four start-up companies.