TABLE OF CONTENTS We start by reviewing some basic tools that will be helpful in deriving the optimal coding strategies. A very elegant solution to the joint linear transmit/receiver optimization problem was given by Palomar et al. in [1, 2], where the optimization problems were greatly simplified by using two basic mathematical tools, namely convex optimization and majorization theory. Two basic references for convex optimization and majorization theory are [3] and [4], respectively. Here, we will only review the basic properties that are useful for our optimization problems.
An nth dimensional set A ? ? n is convex if, for any pair of vectors x, y ? A and any real parameter ? ? [0, 1], the vector z = ? x + (1 - ?) y ? A. An example of convex and nonconvex sets, defined on a two-dimensional space, is sketched in Figure 6.1.
A function f( x) of an n-size vector x, defined over a set A, is convex over A if, for all x, y ? A and any real parameter ? ? [0, 1], it satisfies the following inequality
| (6.1) |  |
We generalize now the concept of convexity.
Given a real vector x of size n, we denote with
| (6.2) |  |
the components of x
