On the Capacity of b-Modulated Nonlinear Frequency Division Multiplexing System

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.