A Covariance Matrix Method for Calculating Accurate Bit Error Rates in a DWDM chirped RZ System
R. Holzlohner, C. Menyuk, V. Grigoryan, Univ. of Maryland Baltimore County, Baltimore, MD; W. Kath, Northwestern University, Chicago, IL.

1 Introduction
Amplifier spontaneous emission (ASE) noise sets the lower limit on the allowed system power in current optical fiber communication system [1]. At the receiver, ASE noise leads to amplitude and timing jitter and deteriorates the bit error rate (BER). The ASE noise spectrum is white over the spectral width of one channel. However, due to the nonlinearity of the optical fibers leading to cross-phase modulation and four-wave mixing, the noise interacts with the signal in a complex way that usually increases the signal degradation. The traditional computation of the probability density function (pdf) of the electrical signal in the receiver, based on Monte-Carlo simulations, only works for a limited range of BERs, beyond which the BER has to be extrapolated [2]. Extrapolation methods to date assume that the receiver voltage is Gaussian distributed. This assumption is not always reliable. Moreover, the Monte Carlo methods are computationally expensive. Thus, they are usually replaced in practice by approximations-the most common of which is to simply neglect noise altogether during propagation and add Gaussian white noise at the entry to the receiver. This and similar approaches are usually not well-validated.
In this contribution, we numerically study a 10Gb/s 5-channel chirped return to zero (CRZ) system with a 50GHz channel spacing, transmitting data over 6,100km. This system resembles a DWDM submarine system [3]. We reduced the number of channels to five in keeping with earlier results indicating that it suffices to simulate a limited number of channels [4]. The covariance matrix method relies on the assumption that the interaction of the optical noise in the fiber with itself is negligible. Noise-noise interactions must be kept in the receiver model. We calculate the linearized evolution of the noise around the noise-free signal without the use of Monte Carlo simulations. The optical noise at the entry to the receiver is multivariate Gaussian distributed after the phase jitter is removed from the noise and is completely described by a covariance matrix [5]. We have previously applied this method to a highly nonlinear dispersion-managed soliton system (DMS) over 24,000km [5], [6] and to a single-channel CRZ system [7], [8] with the same parameters as the central channel in the 5-channel system that we study in this work. The validity of our approach was proven in these references for a wide range of single-channel systems. Hence the focus of this paper is on extending the covariance matrix method to multiple channels.
2 Theory and Results
Our simulated transmission line is identical to [7], [8] and consists of 34 dispersion maps each of length 180km for a total distance of 6,120km. Every 45km, the loss is compensated by an EDFA with a spontaneous emission factor of n_sp=2.0. We transmit the same pseudorandom bit sequence of 2⁵=32 bits in all channels, exhausting all possible bit patterns of length 5. Fig. 1 shows the noise-free optical power spectrum at the receiver.

Fig. 1. Noise-free optical power spectrum at the end of the transmission line. The channel spacing is 50GHz, and the channels are indicated by numbers. The noise analysis is carried out for channel 0.
We compute the covariance matrix as described in [7] by perturbing the noise-free signal by a small amount in each frequency separately. We separate the phase jitter from the covariance matrix at each amplifier. When dealing with strings that contain many overlapping bits, one needs to consider that each pulse in the signal of each channel has a different phase. In a system in which pulses do not overlap, such as a soliton system, the phase jitter can be easily separated from each pulse individually. However, in the present system the maximum FWHM pulse duration is about 210ps, leading to a significant pulse overlap [7]. The phase jitter is separated in the simulation after applying artificial dispersion compensation. This procedure compresses the pulses, and we then separate the phase jitter individually for each pulse. Finally, the opposite dispersion is used to restore the signal shape. We find that timing jitter is small and does not have to be separated from the rest of the noise. In other words, we employ the same algorithm as for the single-channel CRZ system [7], but before we perform the phase jitter separation at each amplifier, we filter out the four surrounding channels since they would otherwise distort the pulse phase of the central channel. As previously, the frequency range of the noise in the covariance matrix is ±22GHz, equaling about half the channel spacing in either direction, which is the relevant range after demultiplexing the central channel. By only keeping the covariance matrix for the central channel, we neglect the linear coupling of the noise modes in the central channel with the noise modes in the other channels. However, we do take into account the interaction of the noise modes in the central channel with the noise-free signal in all channels, because the evolution of the perturbation in each frequency is computed in the presence of all five channels. Using the covariance matrix, we compute the pdf of the electrical voltage. At the receiver, we assume that there is an ideal square law detector followed by a 5-th order Bessel filter with a bandwidth of 8.6GHz.
Fig. 2 shows the smoothed power spectrum of the noise-free signal in the upper curve and the matrix principal diagonal that represents the power spectrum of the noise, in the lower thick curve. The OSNR in channel 0 is 11.2dB, where we define the OSNR as the total signal power divided by the total noise power each computed in the frequency range of ±22 GHz. The noise spectrum is a minimum at zero frequency and has peaks at ±13 GHz that are due to parametric gain; the ratio of the spectral power at the noise peaks to that in the minimum is 1.12. The optical noise entering the receiver is often assumed to be white, corresponding to a covariance matrix with a constant diagonal and zeros elsewhere. This crude approximation can lead to inaccurate BERs, because in reality the diagonal matrix elements vary, and the off-diagonal elements are nonzero.

Fig. 2. Upper thin curve: same spectrum as in Fig. 1, but smoothed, lower thick curve: noise power spectrum at the receiver in channel 0.
We validate our results by comparing the pdfs from the covariance matrix method to extensive Monte Carlo simulations. In Fig. 3, the dots represent a histogram of the receiver voltage from 100,000 Monte Carlo noise realizations. The voltage is normalized to the noise-free peak voltage. The dashed curves show a Gaussian fit to the dots, based on their mean and variance. The solid curves show the accurate pdfs from our method. All pdfs are averaged over the 16 marks and the 16 spaces in the 32-bit signal. The agreement between the covariance matrix method and the Monte Carlo results is excellent. By integrating the pdfs we obtain an optimal BER of 4.7×10⁻¹² from the covariance matrix method and 4.1×10⁻¹¹ from the Gaussian fit of the Monte Carlo data. These values are within a factor of 3 of what we found in a single-channel simulation [7]. The Gaussian BER corresponds to a Q-factor of 6.50. Note that the reason that the Gaussian BER does not differ very strongly from the accurate result in this case is that it overestimates the pdf of the marks and underestimates it in the spaces, and hence the Q-factor method relies on the accidental partial cancellation of two errors. We also computed the covariance matrix and the pdf for channel 1, which yields very similar results.

Fig. 3. Histogram of a traditional Monte Carlo simulation (dots) with Gaussian fits of the data points in the marks and spaces based on their mean and variance (dashed lines) and the pdf of the covariance matrix method (solid line).
3 Conclusions
We extended the covariance matrix method to a 10Gb/s chirped RZ system with a transmission distance of 6,100km with 5 channels, spaced 50GHz apart. We were able to compute the accurate pdfs and the BER in a given channel of a signal with 32 strongly overlapping bits. The simulation time increase, compared to a single-channel system, depends exclusively on the step size reduction of the split-step algorithm due to the additional channels. The computational cost of our method equals that of a Monte Carlo simulation with twice as many noise realizations as the number of complex Fourier modes in the covariance matrix; in this work we used 140 modes. Therefore we propose that traditional Monte Carlo simulations be abandoned in favor of the covariance matrix method, both for calculating accurate BERs and for checking the validity of simplified approaches like using Gaussian white noise at the entry to the receiver.
References
[1] J. P. Gordon and H. A. Haus, "Random walk of coherently amplified solitons in optical fiber transmission," Opt. Lett. 11, 665-667 (1986).
[2] N. S. Bergano, F. W. Kerfoot, and C. R. Davidson, "Margin measurements in optical amplifier systems," IEEE Photon. Technol. Lett. 5, 304-306 (1993).
[3] M. Vaa et al., "Demonstration of a 640Gbit/s×7000 km submarine transmission system technology ready for field deployment," In Proc. OFC'01, paper WF5 (Anaheim, CA, 2001).
[4] T. Yu, W. Reimer, V. S. Grigoryan, and C. R. Menyuk, "A mean field approach for simulating wavelength-division multiplexed systems," IEEE Photon. Technol. Lett. 12, 443-445 (2000).
[5] R. Holzlöhner, V. S. Grigoryan, C. R. Menyuk, and W. L. Kath, "Accurate calculation of eye diagrams and bit error rates in long-haul transmission systems using linearization," J. Lightwave Technol. 20, 389-400 (2002).
[6] R. Holzlöhner, C. R. Menyuk, V. S. Grigoryan, and W. L. Kath, "Direct calculation of the noise evolution in a highly nonlinear transmission system using the covariance matrix," In Proc. CLEO'02, talk CThG5 (Long Beach, CA, 2002).
[7] R. Holzlöhner, C. R. Menyuk, W. L. Kath, and V. S. Grigoryan, "Efficient and accurate calculation of eye diagrams and bit-error rates in a single-channel CRZ system," IEEE Photon. Technol. Lett. 14, 1079-1081 (2002).
[8] R. Holzlöhner, C. R. Menyuk, V. S. Grigoryan, and W. L. Kath, "Efficient and accurate computation of eye diagrams and bit error rates in a single-channel CRZ system," In Proc. CLEO'02, poster CThO44 (Long Beach, CA, 2002)

Abstract:
We present a covariance matrix method to compute accurate error rates in a 10Gb/s DWDM chirped RZ transmission system with channels spaced 50GHz apart at the computational cost of only 280 Monte Carlo noise realizations.