Practical algorithms for a family of waterfilling solutions

Many engineering problems that can be formulated as constrained optimization problems result in solutions given by a waterfilling structure; the classical example is the capacity-achieving solution for a frequency-selective channel. For simple waterfilling solutions with a single waterlevel and a single constraint (typically, a power constraint), some algorithms have been proposed in the literature to compute the solutions numerically. However, some other optimization problems result in significantly more complicated waterfilling solutions that include multiple waterlevels and multiple constraints. For such cases, it may still be possible to obtain practical algorithms to evaluate the solutions numerically but only after a painstaking inspection of the specific waterfilling structure. In addition, a unified view of the different types of waterfilling solutions and the corresponding practical algorithms is missing. The purpose of this paper is twofold. On the one hand, it overviews the waterfilling results existing in the literature from a unified viewpoint. On the other hand, it bridges the gap between a wide family of waterfilling solutions and their efficient implementation in practice; to be more precise, it provides a practical algorithm to evaluate numerically a general waterfilling solution, which includes the currently existing waterfilling solutions and others that may possibly appear in future problems.