1 Introduction.

This short tutorial presents two mathematical techniques namely Majorization Theory and Matrix-Monotone Functions which are applied to solve communication and information theoretic problems in wireless communications.

1.1 Majorization Theory

Inequalities have been always a major mathematical research area beginning with Gauss Cauchy, and others. Pure and applied mathematical analysis needs inequalities, e.g., absolute inequalities, triangle inequalities, integral or differential inequalities, and so on. The building blocks of Majorization are contained in the book [48]. The complete theory including many applications is presented in [92]. The theory is about the question how to order vectors with nonnegative real components and its order-preserving functions, i.e., functions f which satisfy that for x [greater than or equal to] y it follows f (x) [greater than or equal to] f (y). The characterization of this class of functions is important to exploit the properties of this monotony.

In the wireless communication context, those functions arise naturally in resource allocation for multiple user systems or multiple antenna systems, e.g., sum rate of the multiple access channel (MAC) with K users and channels [[alpha.sub.l], ..., [[alpha].sub.K] as a function of the power allocation [p.sub.1], ..., [p.sub.K] with inverse noise power [rho]

C(p) = log (1 + [rho][K.summation over (k=1)] [p.sub.k][[alpha].sub.k]).

Assume that the sum power is constraint to K, i.e., [[summation].sup.K.sub.(k=1)] [p.sub.k] = K. Order the components [[alpha].sub.l] [greater than or equal to] [[alpha].sub.2] [greater than or equal to] ... [greater than or equal to] [[alpha].sub.K] [greater than or equal to] 0 and [p.sub.i] [greater than or equal to] [p.sub.2] [greater than or equal to] ... [greater than or equal to] [p.sub.K] [greater than or equal to] 0. The function C turns out to be Schur-convex with respect to p, i.e., monotonic decreasing with respect to the Majorization order. If p [greater than or equal to] q then C(p) [greater than or equal to] C(q). Therefore, the maximum value is attained for a power allocation vector with elements, i.e., C([K, 0, ..., 0]) [greater than or equal to] C (p) [greater than or equal to] C(1).

This monotony behavior is illustrated for K = 2 with power allocation p = [2 - p, p] in Figure 1.1. This result implies that TDMA is optimal, because the complete transmit power is optimally allocated to one user [80].

[FIGURE 1.1 OMITTED]

Most of the basic definitions and basic properties can be found in the text books [8, 48, 50, 51, 92]. Majorization theory is a valuable tool and it is successfully applied in many research areas, e.g., in optimization [39, 168], signal processing and mobile communications [59, 105], and quantum information theory [101].

1.2 Matrix-Monotone Functions

More than 70 years have passed since Lowner [88] proposed the notion of matrix-monotone functions. A real, continuous function f : Z [right arrow] R defined on a nontrivial interval I is said to be matrix monotone of order n if

X [greater than or equal to]Y [??] f (X) [greater than or equal to]f(Y)

for any pair of self-adjoint n x n matrices X and Y with eigenvalues in I. Lowner characterized the set of matrix-monotone functions of order n in terms of the positivity of certain determinants (the so-called Lowner determinants and the related Pick determinants), and proved that a function is matrix monotone if and only if it allows an analytic continuation to a Pick function; that is, an analytic function defined in the complex upper half-plane, with nonnegative imaginary part. A function is called matrix monotone if it is matrix monotone for all orders n.

A representation theorem was proven for the class of matrix-monotone functions [34, 83, 88, 156]. Every matrix-monotone function f can be expressed as

f(t) = a + bt + [[integral].sup.[infinity].sub.0] st/s+t d[mu](s) (1.1)

with a positive measure [mu] [member of] [0, [infinity]) and real constants a, b [greater than or equal to] 0.

Representatives of the class of matrix-monotone functions arise naturally in the context of multiple antenna systems in the single- as well as in the multiuser context. The two most important examples are the mutual information and the minimum mean square error (MMSE) in multiple-input multiple-output (MIMO) systems. Consider the mutual information (1) for the vector model y = Hx + n between x and y for independently complex zero-mean Gaussian distributed x and n with covariances Q and I

f(HQ[H.sup.H]) = I(x; y) = log det (I + HQ[H.sup.H]).

The mutual information denoted as the function f (HQ[H.sup.H]) = tr log (I + HQ[H.sup.H]) can be represented by the matrix-monotone function f (t) = log(1 + t) which has the integral representation

f(t) = [[integral].sup.[infinity].sub.1] t/s+t 1/s ds.

Hence, all results that hold for matrix-monotone function also hold for the mutual information and (as we will show later) for the MMSE. This approach allows to unify many recent results and it is possible to extract the main principles and concepts.

Finally, matrix-monotone functions are applied in many other areas, e.g., in optimization [25] and signal processing for communications [71].

1.3 Classification and Organization

1.3.1 Classifcation and Differences to Related Literature

The well-established book [92] contains more results on Majorization than this short tutorial. The main difference is that this tutorial focusses on a subset of topics from [92], especially results regarding averages and distributions of weighted random variables, as well as averages of trace functions. These topics are treated in more detail, new results are added (from subsection 2.2.3 until subsection 2.2.7), and the connection to the application in communication theory is always kept in mind. Furthermore, the fist two tutorial chapters are rigorous in the sense that they contain all necessary definitions and results but additionally contain also many remarks and examples which help the reader to understand the concepts.

There exist approaches in the literature that propose a unifed framework for analysis and optimization of MIMO systems. First, the PhD thesis [104] provides a framework for optimization of linear MIMO systems also by using Majorization theory. The tutorial [107] extends these results to nonlinear decision feedback MIMO systems. Interestingly, the application of Majorization in the other tutorial [107] is not for analysis of impact of fading parameters on system performance but for the optimization of single-user transmit strategies under various objective criteria. Another difference to the tutorial [107] is that the article at hand offers two own full chapters with a tutorial of the mathematical techniques used. Therefore, both tutorial complement one another well.

Another related tutorial is [122] which studies the active field of interference function calculus. An interesting overview presentation is given in the plenary lecture at the workshop on signal processing advances in wireless communications in June 2007 [12].

Furthermore, a unified analytical description of MIMO systems was studied in the PhD thesis [79]. The main focus in [79] is to derive a framework for analytically computing closed-form expressions of MIMO transceiver performances which are then used for optimization. Finally, the connection between the capacity and mean-square-error (MSE) from an estimation and information theoretic point of view was analyzed in the PhD thesis [42]. The thesis contains one part that clearly shows the connection between the capacity and MMSE for various channel models, e.g., discrete, continuous, scalar, and vector channels and different input signals. In subsection 5.1.2 three different relationships between the mutual information and the MMSE are described.

1.3.2 Organization

The fist two chapters present the definitions, properties, and many examples to explain the foundations and concepts of the two techniques. The three main topics discussed are

(a) the partial order on vectors and matrices,

(b) the characterization of order preserving functions,

(c) the optimization of Schur-convex and matrix-monotone functions.

The main goal of these two chapters is to make the reader familiar with the basic concepts and to enable her to apply these techniques to problems in his or her respective research area. The various examples illustrate the theoretical concepts and reconnect to practical problem statements. In "Majorization Theory," we present novel results with respect to Schur-convexity and Schur-concavity for the most general classes of functions and constraints. Later in "Application of Majorization in Wireless Communications," these functions obtain their operational meaning in the context of communication theory. In 'Matrix-Monotone Functions," we present novel results in terms of bounds for matrix-monotone functions, optimization of matrix-monotone functions, and discuss the connection to matrix norms as well as to connections and means.

In "Application of Majorization in Wireless Communications" and "Application of Matrix-Monotone Functions in Wireless Communications," we apply the learned techniques to concrete problem statements from wireless communications. The four main application areas are

(a) spatial correlation in multiple antenna systems,

(b) user distributions in cellular systems,

(c) development of a unified performance measure,

(d) optimization of MIMO system performance.