In this paper, we investigate the capacity of the continuous spectrum of nonlinear frequency division multiplexing (NFDM) systems when data is encoded using b-modulation. We prove that the capacity-achieving distribution is unique, discrete in amplitude and uniform in phase, forming a support of finitely concentric shells. An algorithm is proposed to numerically calculate the optimal number of shells, their amplitudes and their probabilities such that the mutual information is maximized. An analytical lower bound on the capacity of the b-modulation channel is presented. Moreover, the mismatch capacity lower bound is calculated based on a set of realistic channel realizations, generated by the transmission of pulses over a long-haul fiber modeled by nonlinear Schrodinger equation (NLSE) and simulated by split step Fourier method. We also calculate the mutual information for discrete-points constellations. The numerical results show that the gap of mutual information is very small between these two approaches which proves that discrete-points distribution can be used as a tight lower bound instead of shell-based optimal distribution since it is a practical constellation. Finally, practical achievable rates are calculated based on some standard amplitude and phase shift keying (APSK) modulation schemes to show the achievable performance gains.

In this paper, we investigate the capacity of the$b- \mathbf{modulated}$nonlinear frequency division multiplexing (NFDM) systems. Recently, a tractable channel model was proposed for such optical fiber communication systems, describing the received$b-\mathbf{Modulated}$signal by an input-dependent complex Gaussian distributed noise. Considering this channel model, we prove that the capacity achieving input distribution has discrete amplitude, uniform independent phase (DAUIP). Noting that this distribution is supported on a finite number of concentric shells, we find a lower bound for the capacity by assuming a single shell support. The corresponding output distribution and mutual information expressions are derived in closed-form.

An approximated channel model is proposed for direct signaling on the continuous spectrum of a nonlinear frequency division multiplexed (NFDM) communication system, describing the effect of noise and nonlinearity at the receiver. The optimal input distribution that maximizes the mutual information of the proposed approximated channel under the peak amplitude constraint is then studied. We present that, considering the input-dependency of the noise, the conventional amplitude-constrained constellation designs can be geometrically shaped to provide significant mutual information gains. However, it is observed that further probabilistic shaping and constellation size optimization can provide only limited additional gains beyond the best geometrically shaped benchmark scheme, i.e., 64 Amplitude Phase Shift Keying. Then, an approximated channel model that neglects the correlation between subcarriers is proposed for the matched filtered signaling system, based on which the input constellation is geometrically shaped. We demonstrate that although the inter-subcarrier interference in the filtered case is neglected in the channel model, the shaping of the matched filtered case can provide promising gains in mismatch capacity over the unfiltered scenario.

In this paper, we investigate the capacity of the continuous spectrum of nonlinear frequency division multiplexing (NFDM) systems when data is encoded using b-modulation. Recently, a tractable channel model is proposed for such optical fiber communication systems, describing the received b-modulated signal with an input-dependent complex Gaussian distributed noise. Considering this channel model, we prove that the capacity-achieving distribution is unique and discrete. A search algorithm is proposed to determine the optimal discrete input distribution that maximizes the mutual information. The numerical results show that the optimal input distributions form ring-based constellations. We compare these optimal designs with some conventional ring-based amplitude and phase shift keying (APSK) modulation schemes including 16 APSK, 32 APSK and 64 APSK, to show the achievable performance gains.

An approximated channel model is proposed for signalling on the continuous spectrum of a nonlinear frequency division multiplexing communication system, describing the effect of noise and nonlinearity at the receiver. The optimal input distribution that maximizes the mutual information of the proposed approximated channel under peak amplitude constraint is then studied. We show that, considering the input-dependency of noise, the conventional amplitude-constrained constellation designs can be shaped geometrically to show significant mutual information gains. However, it is observed that further probabilistic shaping and input distribution size optimization can only provide limited additional gains beyond the best geometrically shaped benchmark scheme, i.e., 64 Amplitude Phase Shift Keying.

In this paper, a free-space quantum key distribution (QKD) system based on BB84 protocol is studied assuming that the transmitted quantum bits are multiplexed spatially based on orthogonal orbital angular momentum (OAM) carrying beams. The performance of such a system is estimated in terms of quantum bit error rate, outage capacity, and effective data rate taking into account the impact of error correction and privacy amplification. Based on the simulation of OAM-carrying beam propagating through atmospheric channels impaired by turbulence, the crosstalk among the multiplexed channels are investigated and its effect on the performance of the QKD system is evaluated. The presented results provide insight into the efficient design of such multiplexed QKD systems particularly in terms of the choice of the optimal average photon number per pulse and the multiplexing mode set.