Energy-efficient joint source and relay precoding design for cooperative MIMO-AF systems
In this paper, we aim to study in an unified fashion several quadratic vector and matrix equations with nonnegativity hypotheses. Specific cases of such problems have been studied extensively in the past by several authors. For references to the single equations and results, we refer the reader to the following sections, in particular section 3. Many of the results appearing here have already been proved for one or more of the single instances of the problems, resorting to specific characteristics of the problem. In some cases the proofs we present here are mere rewritings of the original proofs with a little change of notation to adapt them to our framework, but in some cases we are effectively able to remove some hypotheses and generalize the results by abstracting the specific aspects of each problem. It is worth noting that Ortega and Rheinboldt [19, Chapter 13], in a 1970 book, treat a similar problem in a far more general setting, assuming only the monotonicity and operator convexity of the involved operator. Since their hypotheses are far more general than the ones of our problem, the obtained results are less precise than the one we are reporting here. Moreover, all of their proofs have to be adapted to our case, since the operator F (x) we are dealing with is operator concave instead of convex.