Dominating broadcasting is a domination-type structure that models a transmission antenna network. In this paper, we study a limited version of this structure, that was proposed as a common framework for both broadcast and classical domination. In this limited version, the broadcast function is upper bounded by an integer k and the minimum cost of such function is the dominating k-broadcast number. Our main result is a unified upper bound on this parameter for any value of k in general graphs, in terms of both k and the order of the graph. We also study the computational complexity of the associated decision problem