On the Capacity of Amplitude Modulated Soliton Communication over Long Haul Fibers

It is predicted that the capacity of data transfer network, mainly consists of optical fibers, will fall behind the data traffic demands in the near future [1]. The prediction implies the need for exploiting current optical fiber infrastructure to their limits before migrating to the next generation of optical fiber systems. However, the fundamental information transmission capacity of the most basic optical fiber link (i.e., standard single-mode fiber) is not fully known in the nonlinear regime. Different approaches have been used to tackle this problem in the literature including the recent application of nonlinear Fourier transform (NFT) to approach the limits of the nonlinear optical fiber [2,3]. Using NFT, the nonlinear dispersive fiber channel, defined by the nonlinear Schr$\ddot{\mathrm{o}}$dinger equation (NLSE), is transformed to linear channels in nonlinear spectral domain, redefining the capacity problem formulation for nonlinear optical fibers.

By applying NFT, the available degrees of freedom in temporal domain are transformed to two types of spectra in the nonlinear spectral domain, namely the discrete and continuous spectra. Therefore, NFT is regarded as a base for development of new techniques of data transmission, and different communication system designs have been proposed using NFT [4,5,6,7,8,9,10,11,12,13]. The performance of such NFT-employed system for long-haul communication is investigated by simulation and experiment [14,15]. However, it has been observed that the noise behavior is not trivial in these systems [16,17,18], and the performance largely depends on the design. Moreover, the application of NFT in estimating the capacity of nonlinear optical fibers is not straightforward since the NFT and inverse NFT (INFT) must be performed numerically and are computationally complex [19,20].

An estimation of the capacity of the nonlinear optical fiber by only signaling on its continuous spectrum defined by NFT is provided in [21,22]. Achievable rates have been predicted, but it has been shown that due to the signal dependency of the noise, the capacity will be saturated at high power. Moreover, several works in the literature have been focused on estimating the achievable information rates (AIR) of the fiber when the discrete spectrum (i.e., soliton transmission) is used as the signal space. In [23], a capacity lower bound for amplitude modulated first-order soliton communication system is estimated using a half-Gaussian input distribution. In [24], an achievable rate is estimated taken into account the Gordon–Haus effect that leads to timing jitter at the receiver. In [18], AIR is estimated for a more complicated system that modulates both the eigenvalue and the norming constant in the discrete spectrum. Assuming a receiver capable of detecting variable pulse duration, in [25], the time-scaled mutual information (MI) is numerically optimized considering the memoryless channel model for soliton communication.

In this paper, we investigate the capacity of the optical fiber channel when only a single discrete spectrum point is encoded and the data is mapped on the imaginary part of the corresponding eigenvalue. This is essentially equivalent to the amplitude modulated soliton communication in [26]. As mentioned above, a number of capacity bounds for such channel has been derived previously [18,23,24], and AIR in bits per second were also discussed in [25]. However, some intrinsic limitations, such as dependence of bandwidth on soliton amplitude and the interaction between neighboring soliton pulses have been ignored. Compared to the state-of-art works in the literature (e.g., [23,25]), we investigate the effect of channel memory induced by solitonic interaction, which is mostly ignored in the literature. In order to incorporate the time-bandwith degrees of freedom into the capacity problem formulation, we study the maximization of time-scaled MI similar to [25] but by assuming a more practical communication system that uses a fixed symbol duration (i.e., soliton pulse width). A general form of capacity-approaching input distributions are proposed through the optimization of an approximated normalized channel model, providing important insights into the optimal design of soliton communication systems. In addition, an analytical estimation of the capacity of amplitude modulated soliton transmission is provided.

This paper is structured as follows: In Section 2, we initially consider a commonly used memoryless non-Gaussian channel model for the imaginary part of the eigenvalue [16]. By applying the variance normalizing transform (VNT) [22,27], the original channel is transformed into an equivalent channel with normalized noise power, which is then approximated by a unit-variance additive white Gaussian noise (AWGN) model in Section 3. Taking into account a peak amplitude constraint imposed by bandwidth limitations, the capacity in bits/normalized time and its corresponding input distribution are estimated using the proposed AWGN model and also an approximate analytical approach. Next, in the Section 4, we consider the effect of channel memory by developing the mismatch capacity bounds based on the split-step simulation of single soliton and soliton sequence transmissions over the NLSE. Based on the mismatch capacity results, the impact of inter-soliton interaction and Gordon–Haus effects on the capacity of soliton communication systems is studied.

At a low launch power, the optical fiber channel can be modeled as a linear dispersive channel impaired by AWGN noise. However, the Kerr nonlinearity becomes significant when the signal power increases to allow transmission over long haul fibers. The propagation of the complex envelope of a narrowband optical field in a standard single-mode fiber can be described by the stochastic nonlinear Schr$\ddot{\mathrm{o}}$dinger equation (NLSE), as discussed in ([28], Chapter 4). Assuming the fiber loss to be perfectly compensated by an ideal distributed Raman amplification, the NLSE is given as where $Q(T,Z)$ denotes the complex envelope of the optical field, $N(T,Z)$ represents the amplifier spontaneous emission (ASE) noise term, T and Z are time and propagation distance, and ${\beta }_2$ and $\gamma$ indicate group velocity dispersion and Kerr nonlinearity respectively. Note that the fiber loss term $\alpha$ here is omitted since ideal distributed Raman amplification is assumed. The ASE noise is modeled by a zero mean white Gaussian noise with autocorrelation $\mathsf{E}\left[N(T,Z)N^{*}(T^{\prime },Z^{\prime })\right]=N_{\mathrm{ASE}}\delta (T-T^{\prime })\delta (Z-Z^{\prime })$. The spectral density of the noise in [W/(km · Hz)] is $N_{\mathrm{ASE}}=\alpha h{\nu }_0{K}_{\mathrm{T}}$ for the ideal distributed Raman amplification assumed in this work, where $h{\nu }_0$ denotes the photon energy and $K_{\mathrm{T}}$ denotes the phonon occupancy factor. The NLSE could be normalized into the form with the corresponding normalized parameters as where dispersion length is defined as $L_{\mathrm{D}}=T_0^2/\left|{\beta }_2\right|$, and normalizing time $T_0$ can be selected independent of other parameters. Consequently, the autocorrelation of the normalized noise is, where ${\sigma }^2=N_{\mathrm{ASE}}\frac{2\gamma L_D^2}{T_0}$ according to the normalization (3).

Using the inverse scattering method, NFT transforms the time domain optical signal into scattering data, consisting of continuous spectrum $\rho (\lambda ,z)$, eigenvalues ${{\lambda }_m\left(z\right)}_{m=1}^M$ and corresponding norming constants ${C_m\left(z\right)}_{m=1}^M$ which evolve linearly along the fiber in nonlinear spectral domain. It can be shown that, in a noise-free and interaction-free scenario, the eigenvalues ${\lambda }_m$ are preserved during the evolution along the fiber [29]. If only one eigenvalue exists at $z=0$ and $\rho (\lambda ,0)=0$, the solution of NLSE is a first-order soliton, which can be described analytically as where the only eigenvalue is ${\lambda }_1=\zeta +i\eta$$(\eta >0)$. Also, $e^{2ϵ}=\frac{C_1}{2\eta }$ and $\psi =\arg C_1\left(z\right)$ where $C_1$ denotes the norming constant corresponding to eigenvalue ${\lambda }_1$.

The Energy of the soliton in (5) is equal to $4\eta$, where the temporal width and bandwidth are proportional to $1/\eta$ and $\eta$ respectively. Note that within this work, only the imaginary part of the eigenvalue is modulated and the real part is set to zero, i.e., $\eta =A$, $\zeta =0$. Thus, at $z=0$, the input pulse can be expressed as

The propagation of the soliton pulse over the fiber is described by NLSE, and at the receiver side, the eigenvalue can be detected by NFT or pulse energy estimation. If the detected eigenvalue is denoted as R, the channel model for this amplitude modulated first-order soliton transmission system can be described by a conditional PDF $P_{R|A}\left(r\right|a)$, which is non-Gaussian with a variance dependent on its mean [16,30]. Ignoring inter-soliton interactions, a memoryless channel model can be defined for the amplitude modulated soliton system based on a noncentral chi-squared distribution (NCX2) with 4 degrees of freedom as [16,23] where $I_1(·)$ denotes the modified first order Bessel function of the first kind. The mean and variance of this distribution for large a are ${\mu }_{\mathrm{NCX}2}\left(a\right)={\sigma }_{\mathrm{N}}^2+a$ and ${\sigma }_{\mathrm{NCX}2}^2\left(a\right)=\frac{1}{2}{\sigma }_{\mathrm{N}}^4+a{\sigma }_{\mathrm{N}}^2$ respectively, where ${\sigma }_{\mathrm{N}}^2=\frac{1}{2}{\sigma }^2\frac{L}{2L_{\mathrm{D}}}$ at distance $Z=L$ and ${\sigma }^2$ is the power spectral density of the normalized ASE noise as defined in Equation (4). It can be seen that the channel model (7) for the imaginary part of the eigenvalue (soliton amplitude, or soliton energy) is non-Gaussian with signal dependent variance. In the next section, we develop different approaches to estimate the capacity of the channel described by (7).

Here, the capacity problem for the channel defined by the conditional PDF (7) is formulated considering a peak amplitude constraint since the bandwidth occupied by soliton pulses is directly related to their amplitudes. That is, the modulating data on higher amplitudes requires larger bandwidth while the maximum signal bandwidth is restricted by physical limitations. Moreover, in practical scenarios, peak power is also constrained due to device limitations. Another important issue that needs to be considered for soliton communications systems is that soliton pulses defined as in (6) are not time-limited, and thus, they should be truncated for practical implementations.

We define the practical width of a soliton pulse (denoted by $t_{\mathrm{s}}$) as the temporal width that contains $1-\delta$ of the soliton energy. Recalling the energy of the normalized soliton (6) is equal to $4A$, this practical width can be obtained by solving the equation below for $t_{\mathrm{s}}$ which is given by where the fixed value $\delta$ should be sufficiently small to make the truncation error negligible compared to noise. For example, assuming that the soliton pulse width is defined based on containing $99.9\%$ of its energy ($\delta =0.001$), we have $t_s=3.8/A$. Noting that the temporal width of soliton pulses is inversely related to their amplitudes, we can also introduce a minimum amplitude constraint to limit the utilization of the temporal resources. Based on the constraints mentioned above, the capacity problem can be formulated as where $C_{\mathrm{bpcu}}$ denotes the capacity in bits per symbol per channel use, $I(A;R)$ represents the MI. Denoting the transmitted and received eigenvalues with random variables A and R respectively, $A_{\mathrm{ub}}$ is the maximum amplitude constraint determined by maximum bandwidth or peak power and $A_{\mathrm{lb}}$ is the minimum amplitude constraint determined by the maximum allowed symbol duration. Note that we also consider the possibility of transmitting no soliton over a symbol duration (i.e., off symbol) with probability $p_0$, which is denoted by $A=0$ here.

Noting that the signal space and the temporal resources are inter-related in the underlying soliton communication system, we will use an alternative capacity formulation that maximizes time-scaled MI [25] to get better insights into AIRs of the system in bits per second. Unlike [25], we assume a fixed symbol duration for all transmitted solitons to facilitate practical implementation. Since the pulse width is inversely related to the amplitude of the soliton, the minimum nonzero soliton amplitude $A_{\min }\ge A_{\mathrm{lb}}$ (i.e., maximum pulse width) in a given input distribution determines the symbol duration. Note that $A_{min}$ is not necessarily equal to the minimum amplitude constraint $A_{\mathrm{lb}}$ and $P(A<A_{min})=p_0$. The time-scaled MI (MI) is thus defined as where MI is divided by the normalized symbol duration, resulting in a unit of [bits/normalized time]. The data rate in [bits/second] can be estimated by dividing the time-scale MI (11) with the normalizing time $T_0$ in (3). The corresponding time-scaled capacity formulation is then given by

Note that the minimum amplitude constraint $A_{\mathrm{lb}}$ can be also relaxed, since it is already inherently imposed by the modified objective function, i.e., the time-scaled MI. This is because the optimal solution would not include the small soliton amplitudes that consume the available temporal resources inefficiently due to their very large pulse width. Hence the capacity problem can be also written as

In Section 3.2, it is shown that a minimum nonzero soliton amplitude $A_{min}$ naturally appears in the optimal distribution of the capacity problem in (13).

So far, we have focused on the capacity estimation of the first-order soliton transmission based on the commonly used memoryless channel model defined by the noncentral chi-squared distribution in (7). In this section, we study the capacity limits of the soliton transmission over a more realistic description of the fibre-optic channel defined by the NLSE. Hence, both the Gordon–Haus effect and the nonlinear interactions between adjacent soliton pulses can be incorporated into the capacity analysis. For this purpose, we use the numerical evaluation of mismatch capacity bounds based on split-step simulation of the NLSE. The mismatch capacity approach is commonly used to provide a lower bound on the capacity of a communication system, by assuming a mismatch distribution for decoding the received signal [32,37]. If the mismatch distribution is denoted by $Q_{Y|X}\left(y\right|x)$ and the real channel statistics is denoted by $P_{Y|X}\left(y\right|x)$, the time-scaled mismatch capacity bound for a discrete input signal is expressed as where $p_X\left(x_j\right)$ denotes the input probability of symbol $x_j$ taken from optimization (26), and ${\mathsf{E}}_{P_{Y|X}\left(y\right|x_k)}[·]$ denotes an expectation operation over the channel model $P_{Y|X}\left(y\right|x_k)$. Recall from Section 3.1 that the unit-variance Gaussian distribution and the NCX distribution are well matched for the interested range of interest. Thus, a unit-variance Gaussian distribution $Q_{Y|X}\left(y\right|x)$ is a reasonable mismatch distribution to be employed in the calculation of the mismatch capacity.

To take into account the impairments introduced by ASE noise, such as Gordon–Haus timing jitter, as well as intersoliton interaction effects, we use the split-step method to simulate the propagation of single soliton or soliton sequence transmission over the fiber. Hence many realizations of the fiber-optic channel can be generated based on the simulation of NLSE to establish the statistics of the realistic channel given the capacity-approaching input distribution obtained in Section 3.2, (i.e., $P\left(y\right|x_k)$). The generated channel statistics can then be used to numerically estimate the mismatch capacity in (36) through a Monte Carlo approach. Noting that the input distribution applied here is not necessarily the optimal distribution for the realistic channel, our results, $C_{\mathrm{Mismatch}}$, provide a lower bound on the mismatch capacity, which in turn gives a lower bound on the capacity of the realistic soliton communication system. The simulation of the channel realization required for the Monte Carlo estimation of mismatch capacity is generated following each function block of the proposed system as in Figure 1. The pulses correspond to the input alphabet will be transmitted into a simulated fiber perturbed by ASE noise via split step Fourier method based on NLSE (1). The output pulse from the simulated fiber will then be put through an NFT detector, which extracts the eigenvalue R from the detected pulse. The received eigenvalue R will then be VNT transformed into the transformed domain for decoding the information. Unless otherwise mentioned, $\delta =0.001$ is assumed to calculate the soliton duration, i.e., $99.9\%$ soliton energy pulse-width.

In this paper, we proposed a number of new approaches for estimating the capacity of the amplitude-modulated soliton communication systems. We provided insights into the AIRs of such systems when effects such as Gordon–Haus and inter-soliton interaction are present. The non-central Chi squared channel model that is commonly used in the literature was initially considered and was then approximated by a unit-variance AWGN channel by applying VNT. Using the approximated channel model and subject to a peak amplitude constraint, optimal input distribution and the corresponding capacity were obtained numerically. The optimized distributions are discrete with a mass point at zero corresponding to no soliton transmission as well as an almost uniform distribution of mass points spread in a range away from zero up to the peak amplitude constraint. Using this general form of the optimal distribution based on the approximate AWGN model and applying some mathematical simplifications, we developed an analytical expression to estimate the capacity of the soliton communication system. Despite the additional approximations, the analytical approach provides a close match to the results obtained numerically based on the AWGN model. The optimal input distribution based on AWGN model were also used to calculate the mismatch capacity of the soliton communication system using the split-step simulation of the realistic channel defined by the NLSE. The results show that the effect of inter-soliton interaction caused by limiting the soliton pulse width is stronger than the Gordon–Haus effect for long haul fibers operating in a range of launch powers up to 10 dBm. They also show the trade-off between extending the pulse width to avoid inter-soliton interaction and compressing the pulse width to improve the temporal efficiency.

In future works, the soliton pulse truncation factor $\delta$ can be included in the capacity problem formulation as an additional variable. This allows for a more comprehensive analysis of the soliton interaction effects. Moreover, the capacity problem based on the assumption of variable pulse width can be considered in the presence of soliton interaction effects. Another interesting problem related to this work is the capacity analysis of higher-order soliton transmissions.