# Optimization of MIMO Systems Capacity Using Large Random Matrix Methods

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## Abstract

**:**

## 1. Introduction

## 2. General Principles

#### 2.1. Optimization of the Average Mutual Information

#### 2.2. Large System Approximation of $I(\mathbf{Q})$

#### 2.2.1. The Case $\mathbf{Q}=\mathbf{I}$

**Background on the behaviour of the eigenvalue distribution of**$\mathbf{H}{\mathbf{H}}^{H}$ Let ${\mathbf{H}}_{r,t}$ be a $r\times t$ complex Gaussian random matrix representing the channel matrix of a MIMO system (we indicate that $\mathbf{H}$ is a $r\times t$ matrix by denoting $\mathbf{H}$ by ${\mathbf{H}}_{r,t}$), and consider its associated Gram matrix defined by ${\mathbf{H}}_{r,t}{\mathbf{H}}_{r,t}^{H}$. Under certain assumptions on the probability distribution of ${\mathbf{H}}_{r,t}$ (see the examples below), it appears that the empirical eigenvalue distribution ${\widehat{\mu}}_{r,t}$ of ${\mathbf{H}}_{r,t}{\mathbf{H}}_{r,t}^{H}$ defined as $\frac{1}{r}{\sum}_{i=1}^{r}\delta \left(\lambda -{\lambda}_{i}({\mathbf{H}}_{r,t}{\mathbf{H}}_{r,t}^{H})\right)$ tends to have a deterministic behaviour when the dimensions r and t converge to $+\infty $ in such a way that $\frac{t}{r}\to c$, where $0<c<+\infty $. In order to shorten the notations, $t\to +\infty $ stands in the following for r and t converge to $+\infty $ in such a way that $\frac{t}{r}\to c$. This is equivalent to saying that it exists deterministic probability measures ${\mu}_{r,t}$ (which depend on the dimensions r and t) for which

**Large system approximation of**I We finally consider the problem of approximating $I=\mathbb{E}\left(log\mathrm{det}({\mathbf{I}}_{r}+\frac{1}{{\sigma}^{2}}\mathbf{H}{\mathbf{H}}^{H})\right)$. We first note that

**Examples**

#### 2.2.2. The General Case $\mathbf{Q}\ne \mathbf{I}$

- for model (15), matrix $\tilde{\mathbf{C}}$ by ${\mathbf{Q}}^{1/2}\tilde{\mathbf{C}}{\mathbf{Q}}^{1/2}$,
- for model (27), matrices ${({\tilde{\mathbf{C}}}_{l})}_{l=1,\dots ,L}$ by ${({\mathbf{Q}}^{1/2}{\tilde{\mathbf{C}}}_{l}{\mathbf{Q}}^{1/2})}_{l=1,\dots ,L}$,
- for model (36), matrices $\mathbf{A}$ and $\tilde{\mathbf{C}}$ by $\mathbf{A}{\mathbf{Q}}^{1/2}$ and ${\mathbf{Q}}^{1/2}\tilde{\mathbf{C}}{\mathbf{Q}}^{1/2}$ respectively,
- for model (51), matrices ${({\tilde{\mathbf{C}}}_{l})}_{l=1,\dots ,L}$ by ${({\mathbf{Q}}_{l}^{1/2}{\tilde{\mathbf{C}}}_{l}{\mathbf{Q}}_{l}^{1/2})}_{l=1,\dots ,L}$ (where $\mathbf{Q}=\mathrm{Diag}({\mathbf{Q}}_{1},\dots ,{\mathbf{Q}}_{L})$)

## 3. Concavity of the Large System Approximation

## 4. Properties of $\overline{I}({\overline{\mathbf{Q}}}_{*})$

#### 4.1. Comparison between $I({\mathbf{Q}}_{*})$ and $I({\overline{\mathbf{Q}}}_{*})$

**Theorem**

**1**

- (1)
- ${sup}_{t}\parallel {\mathbf{Q}}_{*}\parallel <+\infty $
- (2)
- ${sup}_{t}\parallel {\overline{\mathbf{Q}}}_{*}\parallel <+\infty $
- (3)
- $I({\mathbf{Q}}_{*})-I({\overline{\mathbf{Q}}}_{*})=\mathcal{O}\left(\frac{1}{t}\right)$

#### 4.2. Structure of $I({\overline{\mathbf{Q}}}_{*})$

**Theorem**

**2**

**Proof**

**of**

**Theorem 2.**

**Definition**

**1**

**Proposition**

**1**

- φ is Gâteaux differentiable at $\mathbf{Q}$ in the direction $\mathbf{P}-\mathbf{Q}$ for each $\mathbf{P},\mathbf{Q}\in {\mathcal{C}}_{1}$,
- ${\mathbf{Q}}_{opt}$ is the unique maximizer of φ on ${\mathcal{C}}_{1}$ if and only if it verifies:$$\forall \mathbf{Q}\in {\mathcal{C}}_{1},\phantom{\rule{0.166667em}{0ex}}\langle {\varphi}^{\prime}({\mathbf{Q}}_{opt}),\mathbf{Q}-{\mathbf{Q}}_{opt}\rangle \le 0$$

## 5. Optimization Algorithms of $\overline{I}(\mathbf{Q})$

- Initialization: choose ${\delta}^{(0)}$ and ${\tilde{\delta}}^{(0)}$ as vectors with strictly positive components
- Evaluation of $({\delta}^{(k+1)},{\tilde{\delta}}^{(k+1)})$ from $({\delta}^{(k)},{\tilde{\delta}}^{(k)})$:$$\begin{array}{c}\hfill {\delta}^{(k+1)}=f(\mathbf{Q},{\delta}^{(k)},{\tilde{\delta}}^{(k)})\end{array}$$$$\begin{array}{c}\hfill {\tilde{\delta}}^{(k+1)}=\tilde{f}(\mathbf{Q},{\delta}^{(k)},{\tilde{\delta}}^{(k)})\end{array}$$

- Initialization: ${\mathbf{Q}}^{(0)}=\mathbf{I}$
- Evaluation of ${\mathbf{Q}}^{(m+1)}$ from ${\mathbf{Q}}^{(m)}$: ${\delta}^{(m+1)}$ and ${\tilde{\delta}}^{(m+1)}$ are defined as the solutions of the canonical Equation (73, 74) when $\mathbf{Q}={\mathbf{Q}}^{(m)}$, and ${\mathbf{Q}}^{(m+1)}$ coincides with the argument of the maximum w.r.t. $\mathbf{Q}$ of $log\mathrm{det}\left(\mathbf{I}+\mathbf{Q}\mathbf{G}({\delta}^{(m+1)},{\tilde{\delta}}^{(m+1)})\right)$.

- Initialization: choose ${\delta}^{(0)}$ and ${\tilde{\delta}}^{(0)}$ as vectors with strictly positive components
- Evaluation of ${\delta}^{(n+1)}$ and ${\tilde{\delta}}^{(n+1)}$ from ${\delta}^{(n)}$ and ${\tilde{\delta}}^{(n)}$: ${\mathbf{Q}}^{(n)}$ coincides with the argument of the maximum w.r.t. $\mathbf{Q}$ of $log\mathrm{det}\left(\mathbf{I}+\mathbf{Q}\mathbf{G}({\delta}^{(n)},{\tilde{\delta}}^{(n)})\right)$ and ${\delta}^{(n+1)}$ and ${\tilde{\delta}}^{(n+1)}$ are defined by:$$\begin{array}{c}\hfill {\delta}^{(n+1)}=f({\mathbf{Q}}^{(n)},{\delta}^{(n)},{\tilde{\delta}}^{(n)})\end{array}$$$$\begin{array}{c}\hfill {\tilde{\delta}}^{(n+1)}=\tilde{f}({\mathbf{Q}}^{(n)},{\delta}^{(n)},{\tilde{\delta}}^{(n)})\end{array}$$

$l=1$ | $l=2$ | $l=3$ | $l=4$ | $l=5$ | |
---|---|---|---|---|---|

mean departure angle | 4.44 | 0.20 | 1.74 | 0.29 | 0.61 |

departure angle spread | 0.09 | 0.08 | 0.04 | 0.11 | 0.01 |

mean arrival angle | 4.12 | 0.22 | 5.34 | 5.87 | 4.26 |

arrival angle spread | 0.09 | 0.08 | 0.05 | 0.08 | 0.03 |

$r=t=4$ | $r=t=8$ | |
---|---|---|

Vu–Paulraj | $503s$ | $1877s$ |

Frozen Water-filling | $17.3ms$ | $20.1ms$ |

Water-filling of [8] | $5.10ms$ | $7.13ms$ |

## 6. Concluding Remarks

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Dupuy, F.; Loubaton, P. Optimization of MIMO Systems Capacity Using Large Random Matrix Methods. *Entropy* **2012**, *14*, 2122-2142.
https://doi.org/10.3390/e14112122

**AMA Style**

Dupuy F, Loubaton P. Optimization of MIMO Systems Capacity Using Large Random Matrix Methods. *Entropy*. 2012; 14(11):2122-2142.
https://doi.org/10.3390/e14112122

**Chicago/Turabian Style**

Dupuy, Florian, and Philippe Loubaton. 2012. "Optimization of MIMO Systems Capacity Using Large Random Matrix Methods" *Entropy* 14, no. 11: 2122-2142.
https://doi.org/10.3390/e14112122