Random Beamforming over Quasi-Static and Fading Channels: A Deterministic Equivalent Approach

In this work, we study the performance of random isometric precoders over quasi-static and correlated fading channels. We derive deterministic approximations of the mutual information and the signal-to-interference-plus-noise ratio (SINR) at the output of the minimum-mean-square-error (MMSE) receiver and provide simple provably converging fixed-point algorithms for their computation. Although these approximations are only proven exact in the asymptotic regime with infinitely many antennas at the transmitters and receivers, simulations suggest that they closely match the performance of small-dimensional systems. We exemplarily apply our results to the performance analysis of multi-cellular communication systems, multiple-input multiple-output multiple-access channels (MIMO-MAC), and MIMO interference channels. The mathematical analysis is based on the Stieltjes transform method. This enables the derivation of deterministic equivalents of functionals of large-dimensional random matrices. In contrast to previous works, our analysis does not rely on arguments from free probability theory which enables the consideration of random matrix models for which asymptotic freeness does not hold. Thus, the results of this work are also a novel contribution to the field of random matrix theory and applicable to a wide spectrum of practical systems.Comment: to appear in IEEE Transactions on Information Theory, 201

Random Beamforming over Correlated Fading Channels

We study a multiple-input multiple-output (MIMO) multiple access channel (MAC) from several multi-antenna transmitters to a multi-antenna receiver. The fading channels between the transmitters and the receiver are modeled by random matrices, composed of independent column vectors with zero mean and different covariance matrices. Each transmitter is assumed to send multiple data streams with a random precoding matrix extracted from a Haar-distributed matrix. For this general channel model, we derive deterministic approximations of the normalized mutual information, the normalized sum-rate with minimum-mean-square-error (MMSE) detection and the signal-to-interference-plus-noise-ratio (SINR) of the MMSE decoder, which become arbitrarily tight as all system parameters grow infinitely large at the same speed. In addition, we derive the asymptotically optimal power allocation under individual or sum-power constraints. Our results allow us to tackle the problem of optimal stream control in interference channels which would be intractable in any finite setting. Numerical results corroborate our analysis and verify its accuracy for realistic system dimensions. Moreover, the techniques applied in this paper constitute a novel contribution to the field of large random matrix theory and could be used to study even more involved channel models.Comment: 35 pages, 5 figure

Iterative Deterministic Equivalents for the Performance Analysis of Communication Systems

In this article, we introduce iterative deterministic equivalents as a novel technique for the performance analysis of communication systems whose channels are modeled by complex combinations of independent random matrices. This technique extends the deterministic equivalent approach for the study of functionals of large random matrices to a broader class of random matrix models which naturally arise as channel models in wireless communications. We present two specific applications: First, we consider a multi-hop amplify-and-forward (AF) MIMO relay channel with noise at each stage and derive deterministic approximations of the mutual information after the Kth hop. Second, we study a MIMO multiple access channel (MAC) where the channel between each transmitter and the receiver is represented by the double-scattering channel model. We provide deterministic approximations of the mutual information, the signal-to-interference-plus-noise ratio (SINR) and sum-rate with minimum-mean-square-error (MMSE) detection and derive the asymptotically optimal precoding matrices. In both scenarios, the approximations can be computed by simple and provably converging fixed-point algorithms and are shown to be almost surely tight in the limit when the number of antennas at each node grows infinitely large. Simulations suggest that the approximations are accurate for realistic system dimensions. The technique of iterative deterministic equivalents can be easily extended to other channel models of interest and is, therefore, also a new contribution to the field of random matrix theory.Comment: submitted to the IEEE Transactions on Information Theory, 43 pages, 4 figure

A Deterministic Equivalent for the Analysis of Non-Gaussian Correlated MIMO Multiple Access Channels

Large dimensional random matrix theory (RMT) has provided an efficient analytical tool to understand multiple-input multiple-output (MIMO) channels and to aid the design of MIMO wireless communication systems. However, previous studies based on large dimensional RMT rely on the assumption that the transmit correlation matrix is diagonal or the propagation channel matrix is Gaussian. There is an increasing interest in the channels where the transmit correlation matrices are generally nonnegative definite and the channel entries are non-Gaussian. This class of channel models appears in several applications in MIMO multiple access systems, such as small cell networks (SCNs). To address these problems, we use the generalized Lindeberg principle to show that the Stieltjes transforms of this class of random matrices with Gaussian or non-Gaussian independent entries coincide in the large dimensional regime. This result permits to derive the deterministic equivalents (e.g., the Stieltjes transform and the ergodic mutual information) for non-Gaussian MIMO channels from the known results developed for Gaussian MIMO channels, and is of great importance in characterizing the spectral efficiency of SCNs.Comment: This paper is the revision of the original manuscript titled "A Deterministic Equivalent for the Analysis of Small Cell Networks". We have revised the original manuscript and reworked on the organization to improve the presentation as well as readabilit

A Deterministic Equivalent for the Analysis of Correlated MIMO Multiple Access Channels

In this article, novel deterministic equivalents for the Stieltjes transform and the Shannon transform of a class of large dimensional random matrices are provided. These results are used to characterise the ergodic rate region of multiple antenna multiple access channels, when each point-to-point propagation channel is modelled according to the Kronecker model. Specifically, an approximation of all rates achieved within the ergodic rate region is derived and an approximation of the linear precoders that achieve the boundary of the rate region as well as an iterative water-filling algorithm to obtain these precoders are provided. An original feature of this work is that the proposed deterministic equivalents are proved valid even for strong correlation patterns at both communication sides. The above results are validated by Monte Carlo simulations.Comment: to appear in IEEE Transactions on Information Theor

Free probability theory: deterministic equivalents and combinatorics

The topic of this thesis work is free probability theory. The main goal is to understand asymptotic eigenvalue distributions of large classes of random matrices. For the models discussed in [SpVa12], we obtain a quite general algorithm to plot their distributions. We also apply the tools from [BSTV14] to give a general numerical algorithm to compute the asymptotic distribution of some other types of matrix models, such as the block-linearly modified models which have been considered in [Au12,BaNe12,BaNe12b]. Classical, free and non-commutative probability can be jointly understood through the combinatorics of multiplicative functions with respect to different lattices of set partitions. The second goal of this thesis is to survey on the basic topics on the combinatorics of free probability. Our basic reference is [NiSp06]. We present new results which allow to compute cumulants of products of free and Boolean independent random variables in terms of the posets of k-divisible set partitions [ArVa12]. We also find formulas relating the different types of cumulants in non-commutative probability [AHLV14]. In connection to random matrix theory, we make particular use of the combinatorial approach to operator-valued free probability ([NSS02]) to compute Cauchy-Stieljes transforms of the asymptotic eigenvalue distributions of the matrix ensembles introduced in [SpVa12]. We do this to show that our definition of a free deterministic equivalent as a concrete operator, introduced in [SpVa12], agrees with the more widespread notion of deterministic equivalents which are being used, for example, to describe recent matrix models for wireless communications [CoDe11]. Voiculescu introduced free probability in 1985 in the context of operator algebras. In 1991, he found realizations of his free circular, semi-circular and Haar-unitary operators through limits of eigenvalue distributions of quite remarkable random matrix models, such as independent (self-adjoint and non-self-adjoint) Wigner and Haar-unitary random matrices. This allowed to understand Wigner's semicircle law as a special, single-variable case of a very general phenomenon on joint non-commutative distributions of large random matrices. In 1995 he introduced operator-valued free probability, where the limiting behaviors of much more general random matrix models can be realized. A rich class of random matrix models arises from considering a polynomial P(x_1,...,x_n) in non-commutative indeterminates x_1,...,x_n,x_1^*,...,x_n^* and evaluating it on random and deterministic matrices. In this work we are specially concerned about these kind of models. We refer to them as polynomial models''. If the inputs are (self-adjoint or non-self-adjoint) Wigner matrices, Wishart matrices, and deterministic matrices, we may consider a deterministic operator Q by evaluating P on certain operators (y_1,...,y_n) in the context of Voiculescu's free probability theory. Provided that the size of the matrices is large (but not necessarily too large), the spectral measure of the simplified model Q becomes a good approximation of the averaged eigenvalue distribution of P. The dimensions of the matrices can also be different. The free deterministic equivalent Q of P was defined in [SpVa12], based on the generalizations of [Vo91] to rectangular spaces [BG09,BG09b], and using mostly combinatorial tools from [NSS02]. The method of deterministic equivalents (DE) was introduced by Girko at the level of Cauchy-Stieltjes transforms of the considered matrix models. In contrast to DE, the simplification from P to Q can be explained very easily and does not require the polynomial to have a specific form. It will turn out that our definitions from [SpVa12] can be very effectively combined with all the elements of method described in [BMS13] for the distributions of polynomials on self-adjoint, square, asymptotically free random matrices. Throughout this work, we comment on how the different assumptions on the distributions that we input to the random matrix models affect the quality and the type of convergence of the model to its FDE. In particular, we discuss this at the combinatorial level: The different assumptions on the model determine the classes of partitions (or cumulants) that show up on the matrix sums, and hence the moments and the nature of the fixed point equations that we will get for its FDE. To be numerically efficient one needs to understand how freeness restricts and extends to different operator-valued levels. For this, the combinatorial methods from [NSS02] are quite important.Gegenstand dieser Arbeit ist die freie Wahrscheinlichkeitstheorie. Ihr Hauptziel ist es, die asymptotische Eigenwertverteilung einer großen Klasse von Zufallsmatrizen zu verstehen. Für die in [SpVa12] diskutierten Modelle erhalten wir einen sehr allgemeinen Algorithmus zur graphischen Darstellung ihrer Verteilungen. Wir wenden auch Methoden aus [BSTV14] an, um einen allgemeinen numerischen Algorithmus zur Berechnung der asymptotischen Verteilungen anderer Typen von Matrizenmodellen formulieren zu können, wie etwa für die Block-linear modifizierten Verteilungen, die in [Au12, BaNe12, BaNe12b] betrachtet wurden. Klassische, freie und nicht-kommutative Wahrscheinlichkeitstheorie können einheitlich über die Kombinatorik multiplikativer Funktionen bezüglich verschiedener Verbände von Partitionen von Mengen verstanden werden. Das zweite Ziel dieser Arbeit ist es, eine Übersicht über einige der grundlegenden kombinatorischen Strukturen in der freien Wahrscheinlichkeitstheorie zu geben. Unsere wesentliche Referenz hierfür ist [NiSp06]. Wir stellen neue Resultate vor, die die Berechnung der Kumulanten von Produkten freier und Boolesch unabhängiger Variablen mittels Posets k-teilbarer Partitionen ermöglichen [ArVa12]. Darüber hinaus geben wir Formeln an, die verschiedene Typen von Kumulanten zueinander in Verbindung setzen [AHLV14]. In Verbindung mit der Zufallsmatrizentheorie nutzen wir speziell den kombinatorischen Zugang zur operatorwertigen freien Wahrscheinlichkeitstheorie ([NSS02]), um die Cauchy-Stieltjes-Transformierten der asymptotischen Eigenwertverteilungen der in [SpVa12] eingeführten Matrizenensembles zu berechnen. Wir tun dies, um zu zeigen, dass unsere Definition eines freien deterministischen Äquivalents als konkreter Operator, wie er in [SpVa12] eingeführt wurde, mit dem weitverbreiteten Begriff des deterministischen Äquivalents übereinstimmt, wie er beispielsweise zur Beschreibung neuerer Matrizenmodelle in der drahtlosen Kommunikation verwendet wird [CoDe11]. Voiculescu begründete die freie Wahrscheinlichkeitstheorie im Jahr 1985 im Kontext von Operatoralgebren. 1991 fand er Realisierungen seiner freien Kreis-, Halbkreis- und Haar-unitären Operatoren als Grenzwerte von Eigenwertverteilungen bemerkenswerter unabhängigen Zufallsmatrizenmodelle. So ermöglichte Voiculescu das Verständnis des Wignerschen Halbkreisgesetzes als den einvariabligen Sonderfall eines wesentlich allgemeineren Phänomens bei gemeinsamen nicht-kommutativen Verteilungen großer Zufallsmatrizen. 1995 führte er die operatorwertige freie Wahrscheinlichkeitstheorie ein. Damit wurden verschiedene Zufallsmatrixmodelle auch durch Mittel der freien Wahrscheinlichkeitstheorie beschreibbar. Dasselbe gilt für Produkte von Block-Halbkreis Matrizen [BSTV14] und Block-modifizierte Zufallsmatrizen [ANV], welche in dieser Arbeit kurz betrachtet werden. Viele hermiteschen Zufallsmatrixmodelle P ergeben sich durch Auswertung eines selbst-adjungierten Polynoms P(x_1,...,x_n) nicht-kommutierender Variablen x_1,...,x_n,x_1^*,...,x_n^* in zufälligen und deterministischen Matrizen. Falls die Auswertung in unabhängigen Wigner Matrizen, Wishart Matrizen, zufälligen Haar Unitären und deterministischen Matrizen erfolgt, kann man eine deterministische, operator-algebraische Vereinfachung Q von P im Rahmen der freien Wahrscheinlichkeitstheorie betrachten, um damit eine Approximation der Eigenwertverteilung von P zu erhalten. Die Dimensionen der Matrizen x_1,...,x_n dürfen dabei unterschiedlich sein. Das freie deterministische Äquivalent Q von P wurde in [SpVa12] eingeführt, basierend auf der in [BG09, BG09b] beschriebenen Verallgemeinerung von [Voi91] auf rechteckige Räume und unter hauptsächlicher Verwendung der kombinatorischen Werkzeuge aus [NSS02]. Die Methode deterministischer Äquivalente (DE) wurde von Girko auf der Ebene der Cauchy-Stieltjes-Transformierten der betrachteten Matrizenmodelle eingeführt. Im Gegensatz zu seinen deterministischen Äquivalenten kann unsere Vereinfachung Q von P sehr leicht beschrieben werden und setzt darüber hinaus auch keine spezielle Gestalt des betrachteten Polynoms voraus. Es wird sich zeigen, dass unsere Definition aus [SpVa12] sehr effektiv mit allen Elementen der in [BMS13] beschriebenen Methode zur Berechnung der Verteilung selbst-adjungierter Polynome in quadratischen, asymptotisch freien Zufallsmatrizen kombiniert werden kann. Im Verlauf dieser Arbeit werden wir anmerken, wie verschiedene Annahmen über die Verteilungen, die wir in die Zufallsmatrizenmodelle einsetzen, die Qualität und die Art der Konvergenz des Modells zu seinem freien deterministischen Äquivalent beeinflussen. Dies diskutieren wir insbesondere auf kombinatorischer Ebene. Um auch numerisch effizient zu sein, muss man verstehen, wie sich Freeness einschränkt und fortsetzt zwischen verschiedenen operator-wertigen Ebenen. Zu diesem Zweck sind die Methoden von [NSS02] besonders wichtig

Free probability theory: deterministic equivalents and combinatorics

The topic of this thesis work is free probability theory. The main goal is to understand asymptotic eigenvalue distributions of large classes of random matrices. For the models discussed in [SpVa12], we obtain a quite general algorithm to plot their distributions. We also apply the tools from [BSTV14] to give a general numerical algorithm to compute the asymptotic distribution of some other types of matrix models, such as the block-linearly modified models which have been considered in [Au12,BaNe12,BaNe12b]. Classical, free and non-commutative probability can be jointly understood through the combinatorics of multiplicative functions with respect to different lattices of set partitions. The second goal of this thesis is to survey on the basic topics on the combinatorics of free probability. Our basic reference is [NiSp06]. We present new results which allow to compute cumulants of products of free and Boolean independent random variables in terms of the posets of k-divisible set partitions [ArVa12]. We also find formulas relating the different types of cumulants in non-commutative probability [AHLV14]. In connection to random matrix theory, we make particular use of the combinatorial approach to operator-valued free probability ([NSS02]) to compute Cauchy-Stieljes transforms of the asymptotic eigenvalue distributions of the matrix ensembles introduced in [SpVa12]. We do this to show that our definition of a free deterministic equivalent as a concrete operator, introduced in [SpVa12], agrees with the more widespread notion of deterministic equivalents which are being used, for example, to describe recent matrix models for wireless communications [CoDe11]. Voiculescu introduced free probability in 1985 in the context of operator algebras. In 1991, he found realizations of his free circular, semi-circular and Haar-unitary operators through limits of eigenvalue distributions of quite remarkable random matrix models, such as independent (self-adjoint and non-self-adjoint) Wigner and Haar-unitary random matrices. This allowed to understand Wigner's semicircle law as a special, single-variable case of a very general phenomenon on joint non-commutative distributions of large random matrices. In 1995 he introduced operator-valued free probability, where the limiting behaviors of much more general random matrix models can be realized. A rich class of random matrix models arises from considering a polynomial P(x_1,...,x_n) in non-commutative indeterminates x_1,...,x_n,x_1^*,...,x_n^* and evaluating it on random and deterministic matrices. In this work we are specially concerned about these kind of models. We refer to them as polynomial models''. If the inputs are (self-adjoint or non-self-adjoint) Wigner matrices, Wishart matrices, and deterministic matrices, we may consider a deterministic operator Q by evaluating P on certain operators (y_1,...,y_n) in the context of Voiculescu's free probability theory. Provided that the size of the matrices is large (but not necessarily too large), the spectral measure of the simplified model Q becomes a good approximation of the averaged eigenvalue distribution of P. The dimensions of the matrices can also be different. The free deterministic equivalent Q of P was defined in [SpVa12], based on the generalizations of [Vo91] to rectangular spaces [BG09,BG09b], and using mostly combinatorial tools from [NSS02]. The method of deterministic equivalents (DE) was introduced by Girko at the level of Cauchy-Stieltjes transforms of the considered matrix models. In contrast to DE, the simplification from P to Q can be explained very easily and does not require the polynomial to have a specific form. It will turn out that our definitions from [SpVa12] can be very effectively combined with all the elements of method described in [BMS13] for the distributions of polynomials on self-adjoint, square, asymptotically free random matrices. Throughout this work, we comment on how the different assumptions on the distributions that we input to the random matrix models affect the quality and the type of convergence of the model to its FDE. In particular, we discuss this at the combinatorial level: The different assumptions on the model determine the classes of partitions (or cumulants) that show up on the matrix sums, and hence the moments and the nature of the fixed point equations that we will get for its FDE. To be numerically efficient one needs to understand how freeness restricts and extends to different operator-valued levels. For this, the combinatorial methods from [NSS02] are quite important.Gegenstand dieser Arbeit ist die freie Wahrscheinlichkeitstheorie. Ihr Hauptziel ist es, die asymptotische Eigenwertverteilung einer großen Klasse von Zufallsmatrizen zu verstehen. Für die in [SpVa12] diskutierten Modelle erhalten wir einen sehr allgemeinen Algorithmus zur graphischen Darstellung ihrer Verteilungen. Wir wenden auch Methoden aus [BSTV14] an, um einen allgemeinen numerischen Algorithmus zur Berechnung der asymptotischen Verteilungen anderer Typen von Matrizenmodellen formulieren zu können, wie etwa für die Block-linear modifizierten Verteilungen, die in [Au12, BaNe12, BaNe12b] betrachtet wurden. Klassische, freie und nicht-kommutative Wahrscheinlichkeitstheorie können einheitlich über die Kombinatorik multiplikativer Funktionen bezüglich verschiedener Verbände von Partitionen von Mengen verstanden werden. Das zweite Ziel dieser Arbeit ist es, eine Übersicht über einige der grundlegenden kombinatorischen Strukturen in der freien Wahrscheinlichkeitstheorie zu geben. Unsere wesentliche Referenz hierfür ist [NiSp06]. Wir stellen neue Resultate vor, die die Berechnung der Kumulanten von Produkten freier und Boolesch unabhängiger Variablen mittels Posets k-teilbarer Partitionen ermöglichen [ArVa12]. Darüber hinaus geben wir Formeln an, die verschiedene Typen von Kumulanten zueinander in Verbindung setzen [AHLV14]. In Verbindung mit der Zufallsmatrizentheorie nutzen wir speziell den kombinatorischen Zugang zur operatorwertigen freien Wahrscheinlichkeitstheorie ([NSS02]), um die Cauchy-Stieltjes-Transformierten der asymptotischen Eigenwertverteilungen der in [SpVa12] eingeführten Matrizenensembles zu berechnen. Wir tun dies, um zu zeigen, dass unsere Definition eines freien deterministischen Äquivalents als konkreter Operator, wie er in [SpVa12] eingeführt wurde, mit dem weitverbreiteten Begriff des deterministischen Äquivalents übereinstimmt, wie er beispielsweise zur Beschreibung neuerer Matrizenmodelle in der drahtlosen Kommunikation verwendet wird [CoDe11]. Voiculescu begründete die freie Wahrscheinlichkeitstheorie im Jahr 1985 im Kontext von Operatoralgebren. 1991 fand er Realisierungen seiner freien Kreis-, Halbkreis- und Haar-unitären Operatoren als Grenzwerte von Eigenwertverteilungen bemerkenswerter unabhängigen Zufallsmatrizenmodelle. So ermöglichte Voiculescu das Verständnis des Wignerschen Halbkreisgesetzes als den einvariabligen Sonderfall eines wesentlich allgemeineren Phänomens bei gemeinsamen nicht-kommutativen Verteilungen großer Zufallsmatrizen. 1995 führte er die operatorwertige freie Wahrscheinlichkeitstheorie ein. Damit wurden verschiedene Zufallsmatrixmodelle auch durch Mittel der freien Wahrscheinlichkeitstheorie beschreibbar. Dasselbe gilt für Produkte von Block-Halbkreis Matrizen [BSTV14] und Block-modifizierte Zufallsmatrizen [ANV], welche in dieser Arbeit kurz betrachtet werden. Viele hermiteschen Zufallsmatrixmodelle P ergeben sich durch Auswertung eines selbst-adjungierten Polynoms P(x_1,...,x_n) nicht-kommutierender Variablen x_1,...,x_n,x_1^*,...,x_n^* in zufälligen und deterministischen Matrizen. Falls die Auswertung in unabhängigen Wigner Matrizen, Wishart Matrizen, zufälligen Haar Unitären und deterministischen Matrizen erfolgt, kann man eine deterministische, operator-algebraische Vereinfachung Q von P im Rahmen der freien Wahrscheinlichkeitstheorie betrachten, um damit eine Approximation der Eigenwertverteilung von P zu erhalten. Die Dimensionen der Matrizen x_1,...,x_n dürfen dabei unterschiedlich sein. Das freie deterministische Äquivalent Q von P wurde in [SpVa12] eingeführt, basierend auf der in [BG09, BG09b] beschriebenen Verallgemeinerung von [Voi91] auf rechteckige Räume und unter hauptsächlicher Verwendung der kombinatorischen Werkzeuge aus [NSS02]. Die Methode deterministischer Äquivalente (DE) wurde von Girko auf der Ebene der Cauchy-Stieltjes-Transformierten der betrachteten Matrizenmodelle eingeführt. Im Gegensatz zu seinen deterministischen Äquivalenten kann unsere Vereinfachung Q von P sehr leicht beschrieben werden und setzt darüber hinaus auch keine spezielle Gestalt des betrachteten Polynoms voraus. Es wird sich zeigen, dass unsere Definition aus [SpVa12] sehr effektiv mit allen Elementen der in [BMS13] beschriebenen Methode zur Berechnung der Verteilung selbst-adjungierter Polynome in quadratischen, asymptotisch freien Zufallsmatrizen kombiniert werden kann. Im Verlauf dieser Arbeit werden wir anmerken, wie verschiedene Annahmen über die Verteilungen, die wir in die Zufallsmatrizenmodelle einsetzen, die Qualität und die Art der Konvergenz des Modells zu seinem freien deterministischen Äquivalent beeinflussen. Dies diskutieren wir insbesondere auf kombinatorischer Ebene. Um auch numerisch effizient zu sein, muss man verstehen, wie sich Freeness einschränkt und fortsetzt zwischen verschiedenen operator-wertigen Ebenen. Zu diesem Zweck sind die Methoden von [NSS02] besonders wichtig

On Linear Transmission Systems

This thesis is divided into two parts. Part I analyzes the information rate of single antenna, single carrier linear modulation systems. The information rate of a system is the maximum number of bits that can be transmitted during a channel usage, and is achieved by Gaussian symbols. It depends on the underlying pulse shape in a linear modulated signal and also the signaling rate, the rate at which the Gaussian symbols are transmitted. The object in Part I is to study the impact of both the signaling rate and the pulse shape on the information rate. Part II of the thesis is devoted to multiple antenna systems (MIMO), and more specifically to linear precoders for MIMO channels. Linear precoding is a practical scheme for improving the performance of a MIMO system, and has been studied intensively during the last four decades. In practical applications, the symbols to be transmitted are taken from a discrete alphabet, such as quadrature amplitude modulation (QAM), and it is of interest to find the optimal linear precoder for a certain performance measure of the MIMO channel. The design problem depends on the particular performance measure and the receiver structure. The main difficulty in finding the optimal precoders is the discrete nature of the problem, and mostly suboptimal solutions are proposed. The problem has been well investigated when linear receivers are employed, for which optimal precoders were found for many different performance measures. However, in the case of the optimal maximum likelihood (ML) receiver, only suboptimal constructions have been possible so far. Part II starts by proposing new novel, low complexity, suboptimal precoders, which provide a low bit error rate (BER) at the receiver. Later, an iterative optimization method is developed, which produces precoders improving upon the best known ones in the literature. The resulting precoders turn out to exhibit a certain structure, which is then analyzed and proved to be optimal for large alphabets