The regular waterfilling(WF) policy maximizes the mutual information of parallel channels, when the inputs are Gaussian. However, Gaussian input is ideal, which does not exist in reality. Discrete constellations are usually used instead, such as $ M $-PAM and $ M $-QAM. As a result, the mercury/waterfilling (MWF) policy is introduced, which is a generalization of the regular WF. The MWF applies to inputs with arbitrary distributions, while the regular WF only applies to Gaussian inputs. The MWF-based optimal power allocation (OPA) is presented, for which an algorithm called the internal/external bisection method is introduced. The constellation-constrained capacity is discussed in the thesis, where explicit expressions are presented. The expression contains an integral, which does not have a closed-form solution. However, it can be evaluated via the Monte Carlo method. An approximation of the constellation-constrained capacity based on the sphere packing method is introduced, whose OPA is a convex optimization problem. The CVX was used initially, but it did not generate satisfactory results. Therefore, the bisection method is used instead. Capacities of the MWF and its sphere packing approximation are evaluated for various cases, and compared with each other. It turns out the sphere packing approximation has similar performances to the MWF, which validates the approximation. Unlike the MWF, the sphere packing approximation does not suffer from the loss of precision due to the structure of MMSE functions, which demonstrates its robustness.