## Abstract

We propose a new method for estimating the bit error rate (BER) of optical channels suffering from crosstalk and waveform distortion. We demonstrate experimentally that the new method can provide a good BER estimation with an error range less than one order of magnitude.

©2004 Optical Society of America

## 1. Introduction

In future high-capacity long haul wavelength division multiplexed (WDM) optical networks, a challenging task is to monitor the quality of optical channels, since the signal quality may be affected by linear and non-linear optical effects. The bit-error-rate (BER) is a fundamental performance parameter in determining the signal quality. BER monitoring based on the evaluation of signal’s amplitude histograms has emerged as an attractive scheme and many methods have been demonstrated [1–5]. In these methods, digital signals are sampled synchronously or asynchronously so as to obtain their amplitude histograms [1–5]. Assuming that the signal’s waveform distortion is sufficiently small and the ASE and other noise that are added to the signal can be approximated as Gaussian noise, then the BER is estimated as

where *µ*
_{1} and *µ*
_{0} are the mean values and *σ*
_{1} and *σ*
_{0} the standard deviations of the histogram at the marks (data “1”) and the spaces (data “0”), respectively. However, if the signal suffers from crosstalk or waveform distortion and the resultant noise can not be approximated as a Gaussian noise, the BER estimated by Eq. (1) is very inaccurate [6–8]. Reference [7] reported an improved histogram-based technique where the histograms of marks and spaces are each approximated by multiple Gaussian functions instead of a single one. However, this method requires an optical spectrum analyzer to measure the noise power spectral density so as to obtain the standard deviation of each Gaussian function. In addition, a few other device parameters such as the receiver’s responsivity, receiver’s electrical and optical bandwidth must be measured in advance. In this paper, we propose a new histogram-based BER estimating technique. In this technique, multiple Gaussian functions are used to fit the curves of the signal’s histograms so that the noise power spectral density and the device’s parameters need not be considered and measured, leading to a significant simplification of the BER estimation. The proposed method is experimentally demonstrated and verified. The experimental results show that the new technique can give a very good BER estimation with an error range less than one order of magnitude for various levels of signal distortion due to linear or non-linear optical effects such as crosstalk and dispersion.

## 2. New BER estimation technique

When an optical signal is affected by optical linear or nonlinear effects such as crosstalk and dispersion and its histogram is distorted, the probability density function of the transmitted marks and spaces is no longer a Gaussian distribution. Then Eq. (1) can not be used to estimate the signal’s BER. However, the statistical properties of the signal can be regarded as the superposition of two sets of Gaussian random processes [7]. For this model, the BER is given by the sum of the probabilities that these Gaussian processes are above or below the decision threshold respectively.

Base on the above principle, we fit the normalized histograms of the marks and the spaces with the sum of a set of Gaussian functions, respectively, as shown in Fig. 1. The normalized histogram for the data marks is fitted with the sum of m Gaussian functions as:

where *A*
_{1,m}, *µ*
_{1,m}, and *σ*
_{1,m} are, respectively, the relative weighting, the mean value and the standard deviation of each Gaussian function. Similarly, the normalized histogram for the data spaces is fitted with the sum of n Gaussian functions as:

where *A*
_{0,n}, *µ*
_{0,n}, and *σ*
_{0,n} are, respectively, the relative weighting, the mean value and the standard deviation of each Gaussian function. *A*
_{1,m}, *µ*
_{1,m}, *σ*
_{1,m}, *A*
_{0,n}, *µ*
_{0,n}, and *σ*
_{0,n} can all be directly obtained from the process of curve fitting. Assuming transmitted marks and spaces are of equal probability, *A*
_{1,m} and *A*
_{0,n} can be derived from the histogram and normalized as:

Thus the BER can be estimated by

where *D* is the decision threshold. In order to find the minimum BER, *D* has to be optimized.

## 3. Experiment and results

The experimental setup for the proposed BER monitoring system is shown in Fig. 2. A DFB laser emitting at 1550nm is modulated with a 2^{15}-1 pseudorandom NRZ code generated by a pulse pattern generator (PPG) at 2.48832 Gbit/s. The optical signal is split into two parts by a 10:90 coupler. One part passing through the coupler’s 10% port acts the intraband crosstalk. The other part passing through the coupler’s 90% port, a 40km SMF, an erbium-doped fiber amplifier (EDFA) and a tunable attenuator, acts as the signal. The tunable attenuator adjusts the signal power so that various signal-to-crosstalk ratios can be achieved. The two parts are then combined by a 50:50 coupler. Note that the EDFA in the signal path adds amplified spontaneous emission (ASE) noise to the signal, and the insertion of the 40km SMF is intended to bit decorrelate the signal in the path of the 90% part with the intraband crosstalk in the path of the 10% part and also to introduce certain chromatic dispersion to the signal. Another tunable attenuator is used to adjust the whole optical power to be received by an APD photo-receiver. The signal detected by the photo-receiver is then sampled with a sub-harmonically synchronized sampling technique [9]. The sub-harmonically synchronized sampling technique is to sample the signal at a sub-harmonic rate which is 1/N (N is an integer) of the data rate instead of sampling bit by bit. In our experiment, since the sampling module is triggered at 1/8 clock of the signal, the sampling speed is 311.04 mega samples per second. The amplitudes of all sampled points are then transferred to a Pentium IV PC, where the signal’s histograms of the marks and the spaces are formed. Note that the sub-harmonically synchronized sampling technique significantly reduces the processing speed of electronic circuits, and hence is potentially cost-effective.

Rapid estimation of BER is very important for real networks. However, due to the limitation of the sampling module and other hardware used in the experiment, fast processing of sampled data and rapid estimation of BER are not possible at the present. In the experiment, the sampling module is connected to the PC through a GPIB port. Every time it samples the signal, it will then transfer the data to the PC before it makes the next sampling. Thus the acquisition time for obtaining sufficient number of samples is quite long. We expect that the acquisition time will be shortened greatly if the sampling module can store the data in a RAM memory, and then subsequently transfer the sampled data in a big block to the PC.

The BER estimation is conducted in two steps by using the Matlab-based program. Firstly, we fit the histograms with multiple Gaussian functions. The Gaussian functions’ relative weightings (*A*
_{1,m} and *A*
_{0,n}), mean values (*µ*
_{1,m} and *µ*
_{0,n}) and standard deviations (*σ*
_{1,m} and *σ*
_{0,n}) are continuously adjusted so as to ensure the goodness of the fit statistics is better than 98% [10,11]. Secondly, the BER is calculated by Eq. (5). During the calculation, iterations are performed by changing the decision threshold *D* until the BER is minimized.

Figure 3 shows the estimated BER vs. the received power for different crosstalk levels of -20dB, -23dB, -25dB and -30dB. At each crosstalk level, the Matlab-based program fit the histograms of the marks and the spaces with multiple Gaussian functions and calculated the BER value automatically and respectively. The number of the Gaussian functions was set by the program. For the purpose of comparison, the BER measured by a BER Tester (BERT) and the BER estimated with a single Gaussian function (i.e, the conventional Q-factor method [2–5]) are also shown in Fig. 4. Note that, though the sampled data were not the same as those measured by the BERT, the experiment was carried out under the same condition (the same PPG, the same sampling module, and the same photo-receiver, etc.) with a constant temperature in the laboratory, stable humidity, and very small or no vibration. Therefore, the sampled data should be able to reflect the same statistics of the data measured by the BERT.

As demonstrated in Fig.3, the BER estimated with our new method is closer to the BER measured by the BERT than the BER estimated with a single Gaussian function. Although the BER estimated with a single Gaussian function can also predict the trend of the BER change against the received power, it gives a higher discrepancy from the BER measured by the BERT, as compared with the new method.

Figure 4 shows the difference between the BER estimated with our new method and the BER measured by the BERT (i.e. log(BER estimated)–log(BER measured)). As can be seen in Fig. 4, the estimated BER by our method is in good agreement with that measured by the BERT, and the difference between the estimated BER and the measured BER is within one order of magnitude.

## 4. Summary

We have proposed a new method for estimating BER based on histograms and curve fitting. Experimental investigation has demonstrated that the new method can provide a good estimation of BER with an error range less than one order of magnitude. Furthermore, since sub-harmonically synchronized sampling is utilized, the method is potentially cost effective for monitoring optical signals of various data rates.

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