  This paper proposes a joint design of probabilistic constellation shaping
(PCS) and precoding to enhance the sum-rate performance of multi-user visible
light communications (VLC) broadcast channels subject to signal amplitude
constraint. In the proposed design, the transmission probabilities of bipolar
$M$-pulse amplitude modulation ($M$-PAM) symbols for each user and the transmit
precoding matrix are jointly optimized to improve the sum-rate performance. The
joint design problem is shown to be a complex non-convex problem due to the
non-convexity of the objective function. To tackle the problem, the firefly
algorithm (FA), a nature-inspired heuristic optimization approach, is employed
to solve a local optima to the original non-convex optimization problem. The
FA-based approach, however, suffers from high computational complexity.
Therefore, we propose a low-complexity design based on zero-forcing (ZF)
precoding, which is solved using an alternating optimization (AO) approach.
Simulation results reveal that the proposed joint design with PCS significantly
improves the sum-rate performance compared to the conventional design with
uniform signaling. Some insights into the optimal symbol distributions of the
two joint design approaches are also provided.
