Performance of MLSE-Based Receivers in Lightwave Systems with Nonlinear Dispersion and Amplified Spontaneous Emission Noise

Transcription

1 Performance of MLSE-Based Receivers in Lightwave Systems with Nonlinear Dispersion and Amplified Spontaneous Emission Noise Mario R. Hueda Diego E. Crivelli Hugo S. Carrer Digital Communications Research Laboratory - National University of Cordoba - Argentina Av. Vélez Sarsfield 6 - Córdoba X56GCA - Argentina - mhueda@com.uncor.edu Abstract Maximum likelihood sequence estimation MLSE has been proposed in earlier literature to combat the effects of nonlinear dispersion in intensity modulation/direct detection IM/DD optical channels. In this paper, we develop a theory of the bit error rate BER of MLSE-based IM/DD receivers operating in the presence of nonlinear intersymbol interference ISI and amplified spontaneous emission ASE noise. Numerical results confirm the predictions of the theory developed in this work. Based on the new analysis, we also investigate the loglikelihood ratio LLR of the received signal yielded by a softinput/soft-output SISO front-end decoder. Our study shows that traditional channel codes designed for transmissions over AWGN channels, in combination with SISO front-end detectors, achieve asymptotically optimal performance in transmissions over IM/DD optical channels. I. INTRODUCTION Long haul high-speed optical fiber transmission systems suffer from impairments such as chromatic dispersion CD, polarization mode dispersion PMD, and the amplified spontaneous emission ASE noise introduced by optical amplifiers []. In intensity modulation/direct detection IM/DD schemes, fiber dispersion combined with the square-law response of the photodetector gives rise to nonlinear intersymbol interference ISI. Additionally, after the photodetector, ASE noise becomes nongaussian and signal-dependent. Recently, there has been a great deal of interest in using digital equalization to compensate the dispersion of optical channels e.g., []-[5]. The feasibility of a very large scale integrated VLSI implementation of digital signal processing based equalization for optical channels has been demonstrated by Agazzi et al. in [6]. It has been shown in [] that feed forward equalization FFE and decision feedback equalization DFE are severely degraded in the presence of nonlinearity, whereas MLSE equalization is not. MLSE-based receivers for optical channels have already been reported in [4]-[7]. Unlike in traditional dispersive linear channels with additive white Gaussian noise AWGN, in IM/DD optical channels ISI is nonlinear and noise is nongaussian and signal-dependent [4]. Therefore, new studies are needed to characterize the performance of MLSE-based IM/DD receivers. Weiss [5] reported computer simulations of MLSE receivers in the presence of CD, PMD, and ASE noise. A semi-analytical method to evaluate the bit error rate BER of MLSE based IM/DD receivers operating in the presence of nonlinearities and generic nongaussian signal-dependent noise, is introduced in [4]. The accuracy of this technique is shown to be excellent in all the cases considered. However, closed-form analytical expressions for the bit error probability similar to those available for AWGN channels [8] have not been reported so far. Analytical expressions would be useful not only to predict system performance, but also to provide information for the design of other aspects of the transmission system such as channel codes. In this paper, we introduce a closed-form analytical expression for the bit error probability of MLSE-based optical receivers in the presence of nonlinear dispersion and ASE noise. Numerical results show a close agreement with the predictions of the theory. Building on the new analysis, we also investigate the log-likelihood ratio LLR of the received signal provided by a soft-input/soft-output SISO front-end decoder. Our study shows that traditional channel codes designed for transmission over AWGN channels could be used with SISO front-end detectors in IM/DD optical channels achieving asymptotically optimal performance. The rest of this paper is organized as follows. In Section II we present the channel model and a brief description of MLSE receivers. Performance analysis is introduced in Section III. The predictions of the theory are compared with simulation results in Section IV, while concluding remarks are given in Section V. II. CHANNEL MODEL AND MLSE-BASED RECEIVERS This work focuses on long-haul or metro links spanning several hundred kilometers of single-mode fibers with optical amplifiers. Fig. shows a simplified model of the system under consideration. The transmitter modulates the intensity of the transmitted signal using a binary alphabet e.g., On-Off eying OO modulation. Let {a n } = a,a,..., a N and N denote, respectively, the bit sequence to be sent on the optical channel a n A= {, } and the total number of transmitted symbols. The optical power ratio between the pulses representing a logical and a logical, r, is called the extinction ratio. We assume that the intensity level for a logical a n = is different from zero, which is usual in practical transmitters [] e.g., r = r. [9]. The optical fiber introduces chromatic and polarization mode dispersion, as well Globecom /4/$. 4 IEEE Authorized licensed use limited to: UNIVERSIDAD CORDOBA. Downloaded on August 7, 9 at 4:4 from IEEE Xplore. Restrictions apply.

2 Laser Optical Filter Fig.. Modulator T T LPF dt Optical Amplifiers A/D Optical channel model. MLSE Detected Bits as attenuation. Optical amplifiers are deployed periodically along the fiber to compensate the attenuation, also introducing ASE noise in the signal. ASE noise is modeled as AWGN in the optical domain []. At the receiver, the optical signal is filtered and then converted to a current with a PIN diode or avalanche photodetector. The resulting photocurrent is filtered by an integrate-and-dump electrical filter []. The output of the filter is sampled at the symbol rate and applied to the MLSE-based detector. The samples of the received signal after the optical-toelectrical conversion can be expressed as y n = I n + z n = fa n +z n, where I n = fa n represents the noise-free electrical received signal, which is in general a nonlinear function of a group of δ consecutive transmitted bits a n = a n,a n,..., a n δ+ note that I n I = {I,I,..., I δ } with I = f,,...,, I = f,,...,,..., I δ = f,,..., ; z n are samples of the nongaussian signaldependent noise originated by the direct detection process of the optical signal and the ASE noise. Thermal noise, shot noise, and noise contributions from any other source are ignored since it is assumed that ASE noise is dominant []. The probability density function pdf of y n is noncentral chi-square with M degrees of freedom []: f y a y n a n = M e +In I N n J M, I n y n >, where is related to the variance of the noise in the electromagnetic field domain, M is the ratio of the optical B o to electrical B e bandwidth of the front-end i.e., M Bo B e, and J m is the m-th modified Bessel function of the first kind. Assuming that is sufficiently small and r > [9], in the Appendix we show that is well approximated by f y a y n a n e n, 3 π n n = I n + I sp 3, n >, 4 The maximum likelihood sequence detector chooses, among all possible transmitted bit sequences, the sequence {â n } that minimizes the metric [8] M = N n= ln f y a y n â n, 5 with â n = â n, â n,...,â n δ+. The minimization can be efficiently implemented using the Viterbi algorithm VA. Using 3, it is simple to show that minimizing 5 is equivalent to minimize the metric N ˆM = ˆ n + ln ˆ n, 6 n= where ˆ n = În + I sp 3 and În = fâ n. III. PERFORMANCE ANALYSIS In this section, we analyze the performance of MLSE in the presence of dispersion and ASE noise in IM/DD optical channels. In our case each detection error causes exactly one bit error, therefore the probability of bit error of the Viterbi decoder [] is upper bounded by P b Ψ =ˆΨ W H Ψ, ˆΨ Pr{ ˆΨ Ψ} Pr{Ψ}, 7 where Ψ={a n } represents the transmitted sequence, ˆΨ = {â n } is an erroneous sequence, Pr{ ˆΨ Ψ} is the probability of the error event Ψ ˆΨ i.e., the error event that occurs when the Viterbi decoder chooses sequence ˆΨ instead of Ψ, and W H Ψ, ˆΨ is the Hamming weight of Ψ ˆΨ or, in other words, the number of bit errors in the error event is the exclusive OR operator. Pr{Ψ} is the probability that the transmitter sent sequence Ψ. Assuming that the incorrect path through the trellis diverges from the correct path at time k and remerges with the correct path at time k + l l δ, from 6 note that the MLSE-based receiver will choose the incorrect sequence Ψ ˆΨ if ˆ n +ĝ n < n +gn, 8 where g n = N ln n and ĝ n = N ln ˆ n. After binomial expansion and some manipulation, we obtain n u n ˆ n < n ˆ n + G n, 9 where u n = y n and G n = g n ĝ n. The probability density function of u n can be obtained [] replacing 3 in f u a u n a n =u n f y a u n a n u n >, where I sp M. Note that n = {,,..., δ } with s = I s +I sp 3,s=,,..., δ. yielding u n n f u a u n a n = e un. πn Globecom /4/$. 4 IEEE Authorized licensed use limited to: UNIVERSIDAD CORDOBA. Downloaded on August 7, 9 at 4:4 from IEEE Xplore. Restrictions apply.

3 In practical situations i.e., r. [9], when the power of optical noise is sufficiently low, that is, I s, s =,,..., δ, it has been shown in [3] that is well approximated by f u a u n a n e un n. 3 πn From 3 note that u n can be modeled as a Gaussian random variable with mean and variance respectively given by η u n and σu. 4 Let r n be the random variable defined by r n = Then, from 9 notice that { { } Pr ˆΨ Ψ = Pr r n < n u n ˆ n. 5 n ˆ n + G n }. 6 Taking into account the independence of the samples u n given the transmitted sequence, from 4 and 5 we conclude that r n is also a Gaussian random variable with mean and variance respectively given by η r = σ r = n n ˆn, 7 n ˆ n. 8 Moreover, since the power of the optical noise is assumed sufficiently small, from 4 we verify that G n, n I n, and ˆn În. 9 Thus, based on 7-9 and the fact that r n is Gaussian, it is possible to show that 6 reduces to { } Pr ˆΨ Ψ Q In Î n, where Qx = π x e t dt. Let I and I be the current generated by each constellation symbol in a nondispersive optical channel. Then, defining Extinction Ratio: r = r = I, I Optical Signal-to-Noise Ratio: OSNR= I, I sp Normalized OSNR []: SNRT M OSNR, 3 d = I I In Î n 4 Λy n Λy n 5 a OSNR = db Exact Our Analysis y n b OSNR = 5 db Exact 3 Our Analysis Fig y n LLR versus y n. a: OSNR =db. b: OSNR =5dB. expression can be rewritten as { } Pr ˆΨ Ψ Q SNRT β d, 5 where β = r. Parameters β and d take into account the reduction of SNRT owing to the extinction ratio and channel dispersion. In particular, for nondispersive optical channels δ = l =, note that I n În = I I I n În, therefore from 4 we obtain d =. Based on 7 and 5 it is possible to derived an upper bound for P b []. In the following we provide a tight lower bound for P b. Defining d min as the minimum distance of an error event, and taking into account that W H Ψ, ˆΨ, the bit error probability can be lower bounded by P b JQ SNRT β d min, 6 where J = Ψ D Pr{Ψ} and D is the set of transmitted sequences that have a minimum distance error event []. The lower bound 6 with J =will be used in Section IV to evaluate the performance of MLSE-based receivers []. Analysis of the Log-Likelihood Ratio LLR Expressions 3 and 3 can also be used to provide useful information for the design of other aspects of the transmission system such as channel decoders. Consider a nondispersive optical channel i.e., a n = a n. Then, the LLR of the received signal provided by a SISO front-end decoder is given by Λy n =ln f y ay n a n = f y a y n a n =. 7 From 3, note that the LLR 7 can be approximated by Λy n y n + + ln, 8 Globecom /4/$. 4 IEEE Authorized licensed use limited to: UNIVERSIDAD CORDOBA. Downloaded on August 7, 9 at 4:4 from IEEE Xplore. Restrictions apply.

4 log BER r =db Lower Bound 6 Simulation Exact Simulation Approximate r =5dB OSNR [db] Fig. 3. BER versus OSNR for an optical channel with no dispersion. M =3. where s = I s + I sp 3, s =,. Fig.shows Λy n versus y n for OSNR= and 5 db with M =3and r =db. We present the exact function 7 derived from and values obtained from 8. In both cases, we verify the excellent accuracy of approximation 8. The pdf of Λy n can be easily derived from and 8. However, if condition is satisfied, from 3 we verify that the random variable y n i.e., u n is Gaussian with parameters defined by 4, therefore Λy n is also a Gaussian random variable with mean and variance respectively given by N η Λ + ln a n = 9 + ln a n = σ Λ, a n. 3 As an application of this analysis, we derive simple closedform analytical expressions for the bit error probability. Based on 9, 9, 3, and neglecting term ln in 9 i.e., high OSNR, it is simple to verify that P b Q SNRTβ, 3 which agrees with 6 when J = and d min = i.e., nondispersive optical channel. Since Qx πx e x [], from 3 we get P b πsnrtβ e SNRTβ. 3 Note that 3 agrees with [4, Eq. 56] when i it is evaluated at high OSNR and ii ASE noise is dominant. IV. SIMULATION RESULTS AND DISCUSSION In this section, we confirm the theoretical analysis introduced previously by using computer simulations. Our analysis is valid for both linear and nonlinear channel dispersion including, for example, nonlinear err effect and it is not limited to the photodetector nonlinearity. However the pdf of log BER r =db Lower Bound 6 Simulation Exact Simulation Approximate r =5dB OSNR [db] Fig. 4. BER versus OSNR for 5 km fiber span. M =3. the noise is assumed to be a chi-square. The photodetector response is the only source of nonlinearity considered in this section. We present results for an optical channel with no dispersion and a typical 5 km single-mode fiber as specified by the ITU G.65 Recommendation [5]. The latter is used in the third telecommunications window 55 nm, which leads to a dispersion parameter D =7ps/km-nm [6]. We consider a data rate of Gb/s, M =3, OO RZ modulation, r =5and db. The pulse shaping function used in the simulations is an unchirped Gaussian with T FWHM =6ps [6]. MLSE is implemented with an eight-state VA where a perfect knowledge of the channel response is assumed. Figs. 3 and 4 show the bit error rate versus the optical signal-to-noise ratio for an optical channel with no dispersion and a typical 5 km of fiber span, respectively. We present theoretical values obtained from 6, and simulation results of the entire system using both i the optimum metrics based on the exact pdf of the received signal, and ii the approximate metrics defined by 6. Parameter d min in 6 is found from the Viterbi algorithm []. In all cases, comparisons between the values derived from the theory and simulation confirm the excellent accuracy of the lower bound 6. Furthermore, from Figs. 3 and 4 it can be noted the high accuracy of the simulation results based on the approximate pdf derived in the Appendix. Fig. 5 presents pdfs of the signal at the output of a SISO/MLSE equalizer i.e., the LLR, obtained from both computer simulation and the Gaussian approximation. The latter is evaluated using mean and variance values obtained from the simulated samples. We consider no dispersion and two values of OSNR, and db, with r =db and M =3. In all cases, the excellent agreement between values obtained from the Gaussian approximation and the simulations confirms the validity of the theoretical study presented in the previous section. Also note that the accuracy of the Gaussian approximation for Λy n improves as OSNR increases, in agreement with our analysis. In particular, the accuracy of the Gaussian approximation is slightly better for a n =than for a n =. This is because the ratio I is smaller than Globecom /4/$. 4 IEEE Authorized licensed use limited to: UNIVERSIDAD CORDOBA. Downloaded on August 7, 9 at 4:4 from IEEE Xplore. Restrictions apply.

5 PDF PDF I Fig. 5. a OSNR = db Simulation Gaussian Approx Λy n b OSNR = db Simulation Gaussian Approx Λy n PDFs of Λy n in optical channels with no dispersion., therefore the accuracy of assumption decreases for a n =. From Fig. 5 we infer that channel coding schemes designed for transmission over AWGN channels could be used with SISO/MLSE equalizers in IM/DD optical channels with negligible performance degradation [3]. This result is of great interest for turbo decoding techniques, which have been considered for optical channel transmission in earlier literature e.g., see [3] and references thereof. V. CONCLUSIONS In this paper, we have developed a new theoretical performance analysis of lightwave systems in the presence of nonlinear dispersion and ASE noise. Unlike previous contributions, we have derived a closed-form analytical expression for the bit error probability. Its accuracy has been confirmed by comparison with values derived from computer simulations. As a second important contribution of this paper, we have analyzed the LLR of the received signal yielded by a SISO front-end decoder. Our study has concluded that traditional channel codes designed for transmission over AWGN channels in combination with SISO/MLSE-based front-end detectors, could achieve asymptotically optimal performance in IM/DD optical channels. Finally, although the analysis presented here considers ideal optical/electrical filters e.g., integrate-anddump electrical filters, simulation results not presented in this work have shown that the accuracy of the Gaussian approximation for the LLR is also satisfactory for more realistic filters such as, for example, five-pole Bessel electrical filters [7]. APPENDIX In this Appendix we derive approximation 3. Since the m-th modified Bessel function of the first kind can be approximated by J m x πx e x for x m [8], we obtain I n N J M π e In, I n >, 33 y n I n with y n >. In practical situations, approximation 33 is verified when r > and I sp = M I s s. Using 33 in and rearranging terms, it is simple to obtain M f y a y n a n I n π e In. y n I n 34 Since r > and I sp I s s, we can verify that Pr{ y n I n <ξ} with ξ>and arbitrary small. Then, using a similar analysis to that reported in the Appendix of [3], it can be shown that 34 reduces to f y a y n a n e In. 35 πi n The mean value obtained from 35 I n + 3 is different from the exact value I n +I sp []. This difference negligible at high OSNR, can be overcome by replacing I n with I n + I sp 3 in 35, yielding thus expression 3. The accuracy of 3 has been found to be highly satisfactory in most cases of practical interest e.g., r. [9], OSNR>3 db. REFERENCES [] R. Ramaswami and. Sivarajan, Optical Networks: a Practical Perspective. Morgan aufmann,. [] O. E. Agazzi and V. Gopinathan, The impact of nonlinearity on electronic dispersion compensation of optical channels, in Proc. of OFC, Feb. 4. [3] W. 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Agrawal, Fiber-Optic Communication Systems. Wiley- Interscience, 997. [7] E. Forestieri, Evaluating the error probability in lightwave systems with chromatic dispersion, arbitrary pulse shape and pre-and postdetection filtering, J. Lightwave Technol., vol. 8, pp , Nov.. [8] W. H. Press et al., Numerical Recipies in C: The Art of Scientific Computing, nd ed. Cambridge, 99. Globecom /4/$. 4 IEEE Authorized licensed use limited to: UNIVERSIDAD CORDOBA. Downloaded on August 7, 9 at 4:4 from IEEE Xplore. Restrictions apply.