1. Introduction

Space division multiplexing (SDM) is an attractive technique to increase the capacity of long-haul coherent optical systems [1]. A wide variety of experiments and demonstrations using different types of fibers have been proposed for SDM: coupled-core multicore [2]; few-mode; combination of multi-core and few-mode [3]; multi-mode fibers (MMF) [4], among others. SDM systems using coupled channels requires compensation of inter-symbol interference (ISI) and crosstalk, generated by the modes or MIMO streams mixing, modal dispersion (MD) and chromatic dispersion (CD) [5]. Multiple-input multiple-output (MIMO) linear receivers based on the minimization of the mean squared error criteria, also known as MIMO minimum mean square error (MMSE) linear receivers, are commonly selected due to its limited complexity and relatively low power consumption [6]. Moreover, this type of receiver is optimal, in the sense of minimum error probability and, hence, equivalent to a matched filter, for optical channels with non-significant mode-dependent loss (MDL) or mode-dependent gain (MDG) [5].

The combined effect of MDG and MDL has been identified as one of the main limitations of long-haul SDM systems performance [7,8]. Although the origin of MDG/MDL are all kinds of inline components that may imply MDG and/or MDL (mode multiplexers/demultiplexers, switches, couplers, multimode or multicore erbium doped fiber amplifiers) [9,10], in this paper we focus on the amplification imbalance between modes and, for the sake of simplicity, we will hereafter refer to the combined effect of MDL and MDG simply as MDL.

The presence of MDL in the optical channel not only reduces the average channel capacity, in terms of achievable bit rate [11,12], but also introduces randomness in the power of the different modes. This fact conditions the communication system reliability, because it may generate outages [8,13]. The MDL random effects in the channel are usually modelled by considering the channel as a concatenation of spans that includes a random gain according to a certain probability density function (PDF).

In the presence of MDL, a linear MIMO MMSE receiver is no longer equivalent to a matched filter and, therefore, residual ISI and crosstalk, in addition to the noise enhancement inherent in MMSE equalizers, is unavoidable [5].

Several solutions have been proposed in the literature to alleviate the effect of the MDL on the system performance, from non-linear MIMO receivers based on maximum likelihood criteria [11,14,15] to spatial modulations to mitigate the MDL effect [16,17], all of them with a significant increase in the system complexity [11]. Orthogonal-frequency-division-multiplexing (OFDM) for SDM systems has been also proposed to deal with the residual ISI and crosstalk in optical channels with non-negligible MDL, but new challenges arise such as the high peak-to-average power ratio management [18,19] and the losses associated to the cyclic prefix insertion [20].

Some methods have been proposed to measure the MDL-induced signal degradation in OFDM SDM systems, based on error vector magnitude [9] and for MIMO maximum likelihood sphere decoding receivers [21]. In [22], a method for the evaluation of the channel MDL level by using the obtained MMSE equalization coefficients is proposed. One of the objectives of estimating the channel overall MDL is to obtain a lower bound on the per-span MDL standard deviation (that we denote as $\sigma _g$) from closed expressions that are valid for low-to-medium levels of MDL, and from which to derive a bound for the outage probability [7].

However, the relationship between the per-span MDL standard deviation $\sigma _g$, the SNR in terms of energy per bit to noise power spectral density ratio at the receiver input ($\gamma _{in}=E_b/N_0$), and the PDF of the communication system bit error rate (BER) for MMSE MIMO receivers has not been addressed in the literature to the best authors’ knowledge. There are works giving upper bounds for the BER by approximating the overall MDL effect of the channel by means of a diagonal matrix whose elements follow a uniform PDF [16,21,23], or by taking the average BER of the distribution to obtain an SNR penalty due to the overall channel MDL [17]. However, a theoretical approach of the statistical distribution of the BER or the SNR after the MMSE MIMO receiver for MDL channels, as the one described for other random channels, like the ones with Rayleigh fading [24], is still missing.

The main contribution of this paper is to present a novel statistical analysis of the BER for a long-haul SDM optical system using a MIMO MMSE linear receiver in a channel with MDL. A detailed knowledge of the statistical properties of BER after the MIMO MMSE receiver is key for the system design. As a first element of the analysis, the residual ISI and crosstalk at the receiver output are calculated, and then its modeling as a random Gaussian term that is independent and additive to the noise is assessed by means of numerical simulations. In addition, the paper contributes with expressions for the BER computation and approximations by means of semi-analytical methods based on the Gauss quadrature rule (GQR). Also, the BER PDF is estimated by numerical simulations of single carrier 2-PAM systems with pulse-shaping and using the fractionally-spaced equalizer (FSE) MIMO implementation of the ideal MIMO linear receiver in discrete-time. This simulations are carried out for different levels of per-span MDL standard deviation $\sigma _g$ and SNR $\gamma _{in}$. Moreover, a closed-form expression for the BER statistical distribution is provided, which is based on a polynomial fit of the generalized extreme value (GEV) distribution parameters for a wide range of per-span MDL and $\gamma _{in}$ values. The last contribution is the proposal of maximum BER contour maps, according to the targeted outage probability, which are a useful representation of the expected system performance depending on the physical channel parameters of per-span MDL and SNR.

The structure of the document is as follows. Section 2 describes the SDM communication model based on a generalized PAM transmitter and a MIMO MMSE linear receiver used to cope with an optical channel model that includes CD, MD and MDL impairments. The discrete-time equivalent channel is given as a result of this section. In section 3, the analysis of the system BER and its PDF is addressed. The residual ISI and crosstalk after an ideal MIMO MMSE linear receiver is presented, and the GQR method extended to MIMO systems is used to approximate the BER computation. In section 4, we describe the numerical simulations of a system implementation with a FSE MIMO linear receiver carried out to estimate BER PDF curves. Section 5 presents a fit of the estimated BER PDF with an analytical function, the GEV distribution, whose parameters are matched to polynomials as a function of $\sigma _g$ and $\gamma _{in}$. In section 6, we show BER contour maps obtained for different levels of system outage and for a range of $\sigma _g$ and $\gamma _{in}$. Finally, some conclusions are summarized in section 7.

2. SDM communication system model

The whole system model considered in this work is shown in Fig. 1 [25]. In long-haul optical communication systems, the end-to-end channel state information (CSI) is expected to change faster than the time needed for the system to collect such information at receiver, send it backwards, and process it at transmitter. Therefore, the proposed SDM system model does not consider the use of CSI at transmission for exploitation of MIMO multiplexing and diversity features [26].

 

Fig. 1. Spatial division multiplexing (SDM) communication system model generic PAM transmitter and linear multiple-input multiple-output (MIMO) receiver.

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2.1 System model description

The binary data symbols, denoted as $\textbf {s}[n]=\left [s_{1}[n], s_{2}[n], \dots, s_{D}[n]\right ]^T$, are PAM modulated in parallel for each of the $i \in \{1,\dots,D\}$ modes by using the same transmitter pulse $P(\omega )$ that is a square-root raised cosine with roll-off factor equal to $\alpha$. The blocks with $P(\omega )$ in Fig. 1 represent $D$ parallel PAM modulators working at the symbol rate $1/T$, being $T$ the symbol period, and they, by nature, already include the discrete-time to continuous-time conversion. The PAM modulated signals are denoted by the column vector $\textbf {x}(t)=\left [x_{1}(t), x_{2}(t), \dots, x_{D}(t)\right ]^T$, and they are subject to ISI and crosstalk introduced by the MIMO channel, modeled on the channel response matrix, once they reach the receiver noted as $\textbf {y}(t)=\left [y_{1}(t), y_{2}(t), \dots, y_{D}(t)\right ]^T$ [5].

The channel frequency response matrix is

The approach to model the MDL effect is with a different amplification for each $i$-th mode and $k$-th span, which is known as uncoupled modal gains. We define a vector $\textbf {g}^{(k)}=[g_1^{(k)},g_2^{(k)},\dots,g_D^{(k)}]$, where $g_i^{(k)}$ for $i \in \{0,\dots,D\}$ is expressed as natural logarithm of power gain and considered as a random variable with Gaussian distribution, zero mean, and standard deviation, $\sigma _{g}$. We use later the notation $\sigma _{g}(dB)$ when referring to the value of $\sigma _{g}$ adequately converted to decibels. The sum of all factors $\sum _{i=1}^D{g_i^{(k)}}$ is set to 0 for normalization purposes. Therefore, the diagonal matrix $\boldsymbol {\Lambda }^{(k)}(\omega )$ includes the MDL effects and the MD of each mode w.r.t. the mode-averaged value [26], and can be expressed as

Therefore, this channel model considers that the MDL in each mode and span is statistically independent. Our approach is different from the one used in [21,23], where MDL is defined as the ratio between the maximum and minimum mode gain values taken from the overall channel matrix. The overall standard deviation of the MDL for the end-to-end optical channel, $\sigma _{mdl}$, can be obtained from numerical simulations or using a deterministic relationship between $\sigma _g$ and $\sigma _{mdl}$ assuming a strong mode coupling regime and small MDL [7,28]. However, in this paper we only use $\sigma _g$ as the parameter that models the MDL effects in the channel.

We also consider that the noise at different MIMO ports is uncorrelated and represent it as a vector $\textbf {n}(t)=\left [n_{1}(t), n_{2}(t), \dots, n_{D}(t)\right ]^T$ of additive white Gaussian noise (AWGN) with variance equal to $\frac {N_0}{2}$ [26]. Neither phase noise or non-linear effects have been included in the simulations.

The MIMO receiver provides the estimation of the transmitted symbols $\textbf {s}[n]$, denoted as $\boldsymbol {\hat {s}}[n]=\left [\hat {s}_{1}[n], \hat {s}_{2}[n], \dots, \hat {s}_{D}[n]\right ]^T$, being $n$ the discrete-time symbol index, from the continuous-time received signal vector $\textbf {y}(t)$. This symbol estimation is obtained with a generic linear filter of response $\textbf {W}(\omega )$, working at continuous time and followed by a sampler at the symbol rate, whose coefficients are computed subject to a certain criterion, usually the MMSE.

2.2 Discrete-time equivalent channel

The normalized received signal in each mode $i$, $y_i(t)$, can be written from the respective transmitted symbols $s_{j}[n]$ as

Using the system model in Fig. 1 and Eq. (4), we can express the estimated symbol at the discrete-time $n$ for the received mode $i$, $\hat {s}_i[n]$, by assuming perfect synchronization, as

Hence, $o_{ij}[n]$ represents the equivalent discrete-time channel of each MIMO stream. Note that $w_{ij}(t)$ is the $(i,j)$ element of the continuous-time domain matrix $\textbf {W}(t)$ that is the impulse response of the linear MIMO receiver, represented in Fig. 1 with its Fourier transform $\textbf {W}(\omega )$. Similarly, $v_i[n]=v_i (n T)$ is the discrete-time noise obtained after sampling the filtered noise

This output noise, $v_i(t)$, incorporates the noise enhancement associated with the MMSE receiver and it is taken into account in the subsequent BER calculation.

Summarizing, Eq. (6) describes the discrete-time equivalent channel and is structured in four factors: the first is the transmitted symbol multiplied by a constant $o_{ii}[0]$; the second is the residual ISI due to past and future symbols $s_i[n-m]$ transmitted in the same mode $i$; the third is the residual crosstalk generated by other modes $j \neq i$ into mode $i$ with contributions of symbols $s_j[n-m]$ transmitted at any time; and the fourth is the output noise filtered by the MIMO receiver.

3. BER probability density function calculation

Once identified the components of the received signal, we focus on the analysis of the residual ISI and crosstalk terms and how they influence the BER of the system. In principle, these terms cannot be assumed directly as being normally distributed and independent random variables. In the next subsections, we analyze in detail if this assumption can be done for SDM communication systems using $D$ modes.

From Eq. (6), it is observed how the ISI and crosstalk terms depend on the symbols transmitted before and after the $n$-th symbol and in other modes $j\neq i$. Considering that the channel is invariant for a particular realization of $\textbf {H}_{tot}(t)$ (which in fact is a random process due to MDL and MD randomness), the overall $\textbf {O}[n]$ matrix is also invariant, and the random nature of the ISI and crosstalk terms comes from the random values of $s_j[m]$ for $j \neq i$ and $m \neq n$. We can extend the single-input single-output (SISO) case described in [30] to MIMO systems and define a random signal $X_i[n]$ that includes both residual ISI and crosstalk for mode $i$ at sample $n$ as

Assuming that the received symbols $s_{i}[n]$ are stationary, the statistical characteristics of $X_i[n]$ are the same independently of the sample index $n$, and therefore we can drop this index from now on. For a 2-PAM antipodal modulation with amplitudes $\{-A,A\}$, the slicer in the receiver make their decisions to estimate the received symbols $\hat {s}_i[n]$ by using a zero threshold. The error probability $P_i(e)$, for mode $i$ and a particular realization of the channel, can be then calculated as [31]

An alternative strategy for the BER computation is the assumption that $X_i$ can be considered as a Gaussian random variable with zero mean and standard deviation $\sigma _{X_i}$ that is superimposed to the AWGN. In such a case, the Eq. (10) can be approximated as [30]

In Subsection 4.1, we compare the BER PDF obtained using both approximations in Eq. (12) and Eq. (13) from numerical simulations of the system implementation described in the following section.

4. Numerical simulations

This section describes the numerical simulations carried out for the modeled system aiming at estimating the BER PDF.

4.1 System implementation by using a fractionally-spaced MIMO equalizer

To simulate the system model, we employ an FSE-based receiver, which is the most common implementation of the optimal MMSE linear receiver in Fig. 2. For this purpose, the oversampling factor $r_{ov}$ is set to 2, the symbol rate $1/T$ is 64 GBauds, and the roll-off factor $\alpha$ of the raised-cosine-spectrum pulse shaping function $P(\omega )$ is fixed to 0.9, according to the analysis of its impact on MDL systems in [5]. The number of taps of the FSE-MMSE linear receiver is set to 800, since we have tested that beyond this number of taps the performance does not improve any further. Hence, this can be considered a good reference of the theoretical performance of a MIMO MMSE linear receiver.

 

Fig. 2. Fractionally-spaced equalizer (FSE) implementation of a minimum mean square error (MMSE) MIMO receiver.

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The channel model implementation is composed by $K_{amp}=100$ spans of length $\ell _{span}=50$ km, given a total length of $\ell _{tot}=5000$ km, with $D=6$ spatial modes. At each span, as mentioned in subsection 2.1, the MDL is modeled with a Gaussian random variable with standard deviation $\sigma _g$. The remaining optical channel model parameters required to obtain $\sigma _\tau$ and $H_{CD}(\omega )$ in Eq. (1) are taken from [20,33], and are summarized in Table 1.

Table 1. Optical channel model parameters

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The noise power spectral density $\frac {N_0}{2}$ of $\textbf {n}(t)$ is adjusted to have a given $\gamma _{in}=E_b/N_0$, the averaged value for all the modes, at the MMSE receiver for each channel realization $\textbf {H}_{tot}(\omega )$. Hence, the optical system implementation for the numerical simulation presented here, contains $2D$ random variables per each of the $K_{amp}$ spans. The first set of $D$ random variables per span $\{\tau ^{(k)}_i\}_{i=1}^D$ follows a Gaussian distribution with standard deviation equal to $\sigma _\tau$ that models the MD. The second set of $D$ random variables per span $\{g^{(k)}_i\}_{i=1}^D$ models the MDL and also follows a Gaussian distribution with standard deviation equal to $\sigma _g$. As described in [7,16,21], the performance metrics of the optical channel become random variables that depend on the MDL level.

Under this conditions, we compute numerically the BER PDF from a total of $R$=10000 different realizations of the random channel $\textbf {H}_{tot}(\omega )$, incorporated to the proposed FSE-MMSE receiver. These BER results will be used in the following sections for different analyses.

4.2 Residual ISI and crosstalk normality test

The probability of error $P_i(e)$ has been obtained numerically by Montecarlo simulations for each of the $D=6$ modes and each of the $R$ realizations of the random channel $\textbf {H}_{tot}(\omega )$ given certain per-span MDL of $\sigma _g$ and $\gamma _{in}$, but has also been computed by using the GQR approximation of Eq. (12), and the Gaussian approximation of Eq. (13). In particular, we calculate the mean squared error in Eq. (14), to be plugged in Eq. (13), by using the semi-analytical expression available for the MIMO FSE-MMSE receiver in [34] with the number of taps specified in previous subsection 4.1. The resulting $R\cdot D$ values of $P_i(e)$ for each pair of optical system parameters $\{\sigma _g, \gamma _{in}\}$ conforms the obtained set of BER PDF.

We show in Fig. 3 both the BER PDF computed for the GQR approximation of Eq. (12) (in black) and computed for the Gaussian approximation of Eq. (13) (in red) for $\gamma _{in}$ = 12 dB and different values of $\sigma _g$. For the sake of clarity, numerical simulations results obtained by using the Montecarlo method are not shown, since the plots are almost identical to the ones provided by the GQR and Gaussian approximation. Later on, in Fig. 4, we show results of the BER PDF extimated by means of the simulations.

 

Fig. 3. Probability density function (PDF) of $BER$ for $D=6$, $\alpha =0.9$, $\gamma _{in}=12$ and ideal equalizer. BER calculated using Gauss quadrature rule (GQR) estimation (Eq. (12)) versus BER calculated considering residual ISI and crosstalk as an additional Gaussian noise (Eq. (13)).

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Fig. 4. GEV optimal PDF fit for $\gamma _{in}$ = 6 and 16 dB.

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A first observation is the widening of the PDF when increasing $\sigma _g$ that highlights the natural dependency of BER dispersion on the MDL level.

A second observation is the non-symmetrical shape of the PDF, which represents and interesting and innovative result.

The fact that there are some points in the estimated PDF of the BER for $\sigma _g\neq 0$ with lower BER values than for the ideal $\sigma _g=0$ case is because we represent the distribution of the BER for a large pull of $D$ modes and $R$ random channels. If we had plotted only the mean BER among the $D$ modes of each of the $R$ channels, none would have outperformed the ideal case. As it can be also observed, the mean of the BER PDFs are always worse than the case with $\sigma _g=0$.

The last conclusion from Fig. 3 is that the results of both methods show a strong resemblance. Such equivalence is also observed for other values of $\gamma _{in}$ not shown in Fig. 3. Therefore, we can conclude that both approximations are equivalent to estimate the BER PDF and, from now, we assume that the residual ISI and crosstalk are an extra additive Gaussian term and use the expression in Eq. (13), particularized for the MIMO FSE-MMSE receiver in Fig. 2.

5. Closed-form approximation to the BER probability density function

The purpose of this section is to try to approximate the BER PDF presented in Fig. 3 to a close-form expression, to achieve a better understanding, for instance, of the relation between the optical channel parameters and the resulting BER for SDM communication systems. In the process to select a correct PDF distribution, we have discarded after several trials some well-known distributions such as Gamma, inverse Gaussian, and Birnbaum-Saunders. Although these distributions offer a reasonable fit, evaluated in terms of negative log-likelihood [35] and Akaike information criterion (AIC) [36] for values of $\sigma _g>0$, some of their parameters are not well defined or present numerical problems in the vicinity of $\sigma _g=0$, which represents the negligible MDL case that corresponds to a 2PAM system with AWGN and is the performance reference.

5.1 Generalized extreme value distribution

We have chosen the generalized extreme value (GEV) family of distributions, also known as Fisher-Tippet distribution [37], for its goodness of fit for the optical system parameters, $\gamma _{in}$ and $\sigma _g$, range of interest and, in particular, for small $\sigma _g$ values and its smooth behavior of the distribution parameters over this entire range.

This family of distributions has three parameters ($\xi$, $\sigma$ and $\mu$), and allows three different types of behavior in the tails of the distribution, depending on the value of the parameters. First type allows distributions whose tails decays exponentially. Second type includes a tail decrease as a polynomial, and a third type includes distributions with finite tails, such as the Beta distribution. The PDF of a GEV distribution $GEV(\mu,\sigma,\xi )$ is

The support of the random variable $supp(x)$ for this GEV family of distributions is given by

And the cumulative density function (CDF) expression is

5.2 GEV parameters approximation

Next we explore a closed-form fit for the GEV PDF expression that depends only on $\sigma _g$ and $\gamma _{in}$. The first step is to compute the three parameters $\xi$, $\sigma$, and $\mu$ in Eq. (15) that optimize, in the sense of minimizing the negative log-likelihood goodness-of-fit for each of the numerically simulated cases resulting from a pair of $\sigma _g$, $\gamma _{in}$ values. We have considered ranges for $\sigma _g \in \{0,0.1,0.2,0.3,0.4,0.5,0.6\},$ and $\gamma _{in} \in \{6, 8, 10, 12, 14, 16, 18\}$ dB, and we have calculated the optimal values of $\xi$, $\sigma$, and $\mu$ for each combination of the simulated cases.

In Fig. 4, we present the matching of the GEV PDF with the optimal fit, and the BER PDF estimated by simulation for $\gamma _{in} \in \{6, 16\}$ dB that we select for illustrative purposes. As mentioned, the fit is rather good. Note that the PDF of the BER is different depending on the value of $\gamma _{in}$ and $\sigma _g$; therefore, a potential close-form fit must depend on those two system variables.

Then, we propose to look for a close-form approximation based on a polynomial fit for each of the GEV parameters $\xi$, $\sigma$, and $\mu$ as a function of $\sigma _g$ and $\gamma _{in}$, denoted as $\xi _{app}$, $\sigma _{app}$, and $\mu _{app}$, aiming at a more useful GEV formulation of the BER that is detached from our optical system simulations. We achieve this by using mathematical techniques for two variables interpolation as described in [38] (Chapter 6, Section 10). After this process, the polynomial fit that we propose for each GEV parameter is denoted in a compact matricial form as

Figure 5 shows the values of $\textbf {P}$ matrices and the number of coefficients for each of the polynomial fits for each of the GEV parameters, and Table 2 summarizes the complexity of the polynomials proposed. Finally, Fig. 6 presents by means of a 3-D representation, the optimal values of the GEV parameters $\xi$, $\sigma$ and $\mu$ for each of the simulated optical system pair {$\gamma _{in},\sigma _g$} (which are the rounded dots) superimposed to the resulting surface that represents the two-dimensional function with the polynomial approximation according to Eq. (19) and $\textbf {P}$ matrices in Fig. 5 (which is the colored surface). It is observed a reasonable matching between the dots and the interpolating surface that support, with this graphical test, our proposal.

 

Fig. 5. Matrix $\textbf {P}$ for the polynomial fit of the GEV parameters $\xi _{app}$, $\sigma _{app}$, and $\mu _{app}$ according to Eq. (19).

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Fig. 6. GEV parameters $\xi$, $\sigma$, and $\mu$ obtained from the optimal fit from the system simulations (rounded dots) and GEV parameters two-variables polynomial interpolation according to Eq. (19) and Fig. 5 (colored surface).

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Table 2. Complexity of the polynomial approach for GEV parameters

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6. Maximum BER contour maps for different outage probabilities

After the proposal of the polynomial approach of the GEV parameters distribution defined in Eq. (19) and Fig. 5, we can extract the CDF required value from the analytical expression in Eq. (18) that can also be seen as the system availability probability, i.e. the probability of reaching the abscissa value. We are going to use the BER in logarithm scale as

For a given value of $CDF(x)$, or a certain target of system availability probability (SAP), the following relation holds

Therefore, we can express the BER value in terms of the target $SAP$ and the GEV PDF parameters $\xi$, $\sigma$, and $\mu$ as

The outage probability (OP) is easily related to SAP by the expression $OP = 1 - SAP$.

We use this result, together with the polynomial fit of the BER PDF according to Eq. (19), to plot a BER contour map that consists in a representation as shown in Fig. 7. In these maps, we present the maximum system BER, as iso-level curves, for the range of $\gamma _{in}$ and $\sigma _g$ of interest, given a certain OP. Four BER contour maps can be observed for OP values {0.2, 0.15, 0.1, 0.05}.

 

Fig. 7. Maximum system BER contour map for different outage probability (OP). GEV parameter polynomial approximation defined in Table 2.

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It is worth mentioning that the degradation that an increase of the MDL in each span $k$ given by $\sigma _g$ causes on the BER is different depending on the $\gamma _{in}$ selected for the system, and the relations among these factors are far from being linear. For instance, let us consider a system that requires a maximum outage probability of 0.05 and an output BER of $10^{-3}$ or less, and assume that the MDL level is given by $\sigma _g = 0.1$; then, under this conditions, a $\gamma _{in}$ of approximately 7 dB is needed. In case that the MDL is increased up to $\sigma _g = 0.2$, an increment of only 0.8 dB in $\gamma _{in}$ is required to maintain the system performance. However, to support higher MDL, like, $\sigma _g = 0.4$, a much higher increase on $\gamma _{in}$ is needed, in the order of 4 dB.

Moreover, the desired value for the BER in the SDM communication system, without coding, determines to a great extent the system sensitivity to the MDL, e.g. if a receiver output BER of $10^{-3}$ is chosen, the system exhibits a worse degradation due to a MDL increase than another one designed to have an output BER of $10^{-12}$.

These two observations stem from the fact that the combined effect of filtered noise, residual ISI and crosstalk at the MIMO MMSE output in the presence of MDL depends on the desired BER and maximum outage probability, the MDL level represented by $\sigma _g$, and the SNR $\gamma _{in}$. Since the relationship is not straightforward, we propose the contour plots in Fig. 7 as a graphical tool to better understand these interactions.

7. Conclusions

In this paper, we have analyzed the performance of SDM systems using PAM with pulse shaping over MDL impaired channels. We have computed the remaining distortion at the output of linear MIMO MMSE receivers, in terms of residual ISI and crosstalk that are caused by the optical channel impairments: CD, MD and MDL. For this purpose, we have employed the simulation model of the long-haul optical SDM system that was developed in [5]. We have analyzed the system BER probability distribution by both analytical expressions and approximations, and numerical simulations, in which we employ an implementation of the MMSE MIMO receiver based on a FSE. It has been demonstrated that residual ISI and crosstalk at the output of the MIMO MMSE receiver can be assumed as an independent Gaussian term additive to the noise, enhanced by the MMSE equalizer, at least for systems with a number of modes $D$=6. An important contribution is the proposal of a BER PDF closed-form modeling by using a polynomial approximation of the GEV distribution parameters depending on the optical system metrics for the SNR and MDL level {$\gamma _{in}, \sigma _g$}. The results show a complex relationship between MDL, SNR, BER, and outage probability; therefore BER contour maps have been proposed as a graphical tool which provides an interesting insight for long-haul SDM systems design. For instance, the SNR increase needed to guarantee a given BER and system outage probability when MDL is larger, is not linear; moreover, it depends on the particular BER and system outage probability chosen for the system. Furthermore, the choice of the target BER may condition the need to compensate for the MDL, with adequate receiver implementation, or to reduce it, e.g. by including better amplification stages at each span or by limiting the span distance.