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## Preliminary Mathematical Tools

In this section, we introduce mathematical tools for analyzing the favorable propa-gation conditions in CF massive MIMO systems. Communication systems modeled by random channel matrices can be e ciently studied via their covariance eigenvalue spectrum [80, 81]. In the subsequent section, we derive a tight approximation of the eigenvalue moments of the channel covariance matrix that can be e ciently applied to the analysis of favorable propagation properties in CF massive MIMO systems and the design and analysis of multi-stage linear detectors. In the following subsection, we recall some theoretical concepts necessary to derive the results in this chapter.

### Free Probability Theory

Free probability is a mathematical theory that studies non-commutative random vari-ables while classical probability theory is concerned with commutative random vari-ables. Free probability theory was initiated by Dan Voiculescu in the 1980’s in order to attack the free group factors isomorphism problem, an important unsolved problem in the theory of operator algebras [82{84]. A few years later, in 1991, Voiculescu dis-covered the relation between random matrices and free probability [85]. An interesting aspect and active research direction of free probability lies in its applications to ran-dom matrix theory (RMT) [86] [87]. Free probability theory provides a very e cient framework to study limiting distributions of some models of large dimensional random matrices. This theory introduces freeness between non-commutative random variables, which is analogous to the independence between classical commutative random vari-ables. Free probability has developed many powerful tools from classical probability to provide new ideas to study random matrices.

#### Eigenvalue Moments for Antennas in LoS

In this subsection, we derive the eigenvalue moments for DASs with transmitters and receivers in LoS and channel attenuation given by path loss. The matrix G (G1D) for transmitters and receivers in LoS admits a decomposition similar to G (G1D), i.e., G= RT T (G1D= R;1D T1D T;1D). As in [32][93], the b b H H eigenvalue moments of the channel covariance matrix are obtained by approximating the random matrices R ( R;1D) and T ( T;1D) by the independent matrices R ( R;1D) and T ( T;1D), respectively, consisting of i.i.d. zero mean complex Gaussian elements with variance

2 ( 1 for 1D systems) to obtain matrix Ge = RT HT (Ge1D = R;1D T1D HT;1D). This approximation enables the application of classical techniques from RMT and free probability [86][97]. The derivation of the eigenvalue moments follows the techniques proposed in [77][98]. The results are summarized in the following proposition. Proposition 2.1 Let g(ri; tj) be the function of channel coe cients in LoS, T (f1; f2) with (f1; f2) 2 [ 1=2; +1=2]2 be the 2D Fourier series of the sequence obtained by g ; regular grid # ; and m(2‘) = +1=2 +1=2 j T (f ; f ) 2‘df df : sampling (ri tj) over the H A1 T H R 1=2 R21=2 1 2 j 1 2 Consider the matrix C = G G with G = RT T . For ; NR; NT ! +1 with NT = 2 ! T 2 ; C (‘) ; k ‘ m(‘); and NR=e ! eR e kk thee -th diagonal element of matrix C and C ‘ of the matrix C converge to a deterministic value the eigenvalue moment of ordere e e given .

**Linear Multi-Stage Detectors**

Systems with favorable propagation can e ciently utilize low complexity matched lters at the central processing unit since this lter achieves almost optimal perfor-mance in such environments. However, when conditions (2.26) are not satis ed, even linear multi-user detectors are expected to provide substantial gains compared to the matched lter. In the following, we consider low complexity multi-stage detectors in-cluding both polynomial expansion detectors, e.g., [78], and multi-stage Wiener lters [79] and we analyze their performance in terms of their signal to interference and noise ratio (SINR) by applying the uni ed framework proposed in [77][100]. In [77], it is shown that both design and analysis of multi-stage detectors with M stages can be described by a matrix S(X) de ned as 2 X(2) + 2X(1) X(M+1) + 2X(M) .

**Eigenvalue Moments for Antennas in LoS**

In this subsection, we derive the eigenvalue moments for DASs with transmitters and receivers in LoS and channel attenuation given by path loss. The matrix G (G1D) for transmitters and receivers in LoS admits a decomposition similar to bG (bG1D), i.e., G = RT HT (G1D = R;1D T1D HT ;1D). As in [32][93], the eigenvalue moments of the channel covariance matrix are obtained by approximating the random matrices R ( R;1D) and T ( T;1D) by the independent matrices R (R;1D) and T (T;1D), respectively, consisting of i.i.d. zero mean complex Gaussian elements with variance 2 (1 for 1D systems) to obtain matrix eG = RTHT (eG1D = R;1D T1D HT ;1D).

This approximation enables the application of classical techniques from RMT and free probability [86][97]. The derivation of the eigenvalue moments follows the techniques proposed in [77][98]. The results are summarized in the following proposition.

Proposition 2.1 Let g(ri; tj) be the function of channel coecients in LoS, T(f1; f2) with (f1; f2) 2 [1=2; +1=2]2 be the 2D Fourier series of the sequence obtained by sampling g(ri; tj) over the regular grid A# 1; and m(2`) T = R +1=2 1=2 R +1=2 1=2 jT(f1; f2)j2`df1df2: Consider the matrix eC = eGHeG with eG = RTHT . For 2;NR;NT ! +1 with NT =2 ! T and NR=2 ! R; e C(`) kk ; the k-th diagonal element of matrix eC` and m(`) eC ; the eigenvalue moment of order ` of the matrix eC converge to a deterministic value.

**Table of contents :**

Abstract

Abrege [Francais]

Acknowledgements

List of Figures

Acronyms

Notations

**1 Introduction **

1.1 Favorable Propagation

1.2 Pilot Contamination

1.3 Expectation Propagation

1.4 Thesis Outline and Main Contributions

**2 Favorable Propagation Analysis and Multi-Stage Linear Detection **

2.1 Introduction

2.2 System Model

2.3 Preliminary Mathematical Tools

2.3.1 Free Probability Theory

2.4 Channel Eigenvalue Moments

2.4.1 Eigenvalue Moments for Antennas in LoS

2.4.2 Eigenvalue Moments for Rayleigh Fading Channels

2.5 Favorable Propagation

2.6 Linear Multi-Stage Detectors

2.7 Simulation Results

2.8 Conclusion

**3 Semi-Blind Pilot Decontamination **

3.1 Introduction

3.2 System Model

3.3 Cramer-Rao Bound Analysis

3.4 Identiability

3.4.1 Message Passing Algorithm

3.5 Bayesian Semi-Blind Iterative Algorithm

3.6 Semi-Blind Approach with Gaussian Inputs

3.6.1 Joint Channel MAP for All Users

3.7 Pilot Based Bayesian Performance Bounds

3.8 Gaussian Inputs Bayesian Semi-Blind CRB

3.9 Gaussian-Gaussian Extrinsic Information Lower Bound

3.10 Simulation Results

3.11 Conclusion

**4 Expectation Propagation Based Bayesian Semi-Blind Approach **

4.1 Introduction

4.2 System Model

4.3 Expectation Propagation Algorithm

4.4 Variable Level Expectation Propagation(VL-EP)

4.5 VL-EP for Gaussian-Gaussian Semi-Blind

4.5.1 Channel VL-EP for GG-SB with Eliminated Inputs

4.6 Simulation Results

4.7 Conclusion

**5 Conclusions and Future work **

**Appendices **

**A Appendices of Chapter 2 **

A.1 Proof of Algorithm 1

A.2 Proof of Rayleigh Fading Eigenvalue Moments

**B Appendices of Chapter 3 **

B.1 Derivation of Deterministic CRB

B.2 Derivation of Algorithm 2