On the Capacity and Optimal Signal Constellations for SISO-VLC Systems

In this paper, we investigate the capacity of the SISO-VLC system. The capacity of SISO-VLC system without considering the average intensity constraint is firstly analyzed. It can be written as a functional optimization problem in a normed linear space and the objective function is proved to be concave with convex feasible region. Thus, a unique optimal probability density function (PDF) of the input signal exists and its corresponding necessary and sufficient condition is deduced according to the optimization theorem. It is further proved that the capacity-reaching PDF of the input signal is combined of a finite number of scaled impluse function. Therefore, the capacity-reaching constellation optimization problem is formulated and an algorithm is proposed to solve it. Thereafter, the capacity of the SISO-VLC system with average intensity constraint is considered as a functional optimization problem with an equality constraint and is transformed into a new unconstrained functional optimization problem by method of Lagrange multipliers. The existence and uniqueness of the capacity-reaching PDF still hold. The necessary and sufficient condition of the capacity-reaching PDF is obtained in a similar way. The capacity-reaching PDF is proved to be finite and discrete, too. In addition, the algorithm to solve the capacity-reaching constellation optimization problem is proposed as well.