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## Mixed MIMO-Pinhole and Rayleigh Channel

We consider a dual-hop multi-antenna cooperative inband 1 transmission [3] where a ns- antennas source is connected to a nd-antennas destination through a nr-antennas relay (ns; nr > 1). The cooperation is assumed half-duplex, spanning two consecutive slots where the source and the relay transmissions are orthogonal in time. The communication between each couple of nodes, 2fs; rg and 02fr; dg, takes place over an independent wireless link 0 experiencing an average propagation loss 0 and a small scale fading eect represented by a channel matrix H0 2 Cn0n , where Hsr is supposed to be a pinhole channel. It is modeled as an outer product of two independent and uncorrelated Rayleigh fading vectors gs 2 Cns1 and gr 2 Cnr1 [52], i.e., Hsr = grgH s : (3.1) Hrd is a full-rank independent standard complex Gaussian matrix. Its entries are conse- quently Rayleigh distributed. Both relay and destination received signals are corrupted by additive white Gaussian noise (AWGN) vectors wr N 0nr1; 2Inr and wd N 0nd1; 2Ind , respectively. The corre- sponding average SNRs per hop are sr = 2 sr=2 and rd = 2 rd=2.

### Project-and-Forward Relaying

Let s 2 Cns1 denote a unitary precoded symbol vector transmitted by the source node. The s r communication model can be accordingly expressed as, yr = srHsrs + wr 2 Cnr1: The key idea of PF relaying is to extract and forward the DoFs of the received signal vector yr via a QR-based orthogonal projection [53]. Given that Hsr is a pinhole channel, a single degree of freedom will be conveyed by the relay to be used in the estimation of the transmit vector s at the destination. Let Hsr = QR denote the QR decomposition of Hsr, where Q 2 Cnrnr is a unitary matrix with q 2 Cnr1 standing for its rst column vector, and R 2 Cnrns is an upper triangular matrix whose (nr 1) bottom rows consist entirely of zeros, i.e.,

#### Asymptotic Behavior

To highlight the eect of channel parameters on both the outage probability and the ergodic capacity, we study their asymptotic behaviors. Invoking [73, Theorem 1.7 and Theorem 1.11], expansions of the Mellin-Barnes integrals involved in the Meijer-G and bivariate Meijer-G functions can be derived by evaluating the residue of the corresponding integrands at the pole closest to the contour; the minimum pole on the right p min for small Meijer-G arguments and the maximum pole on the left p+ max for large ones, as depicted in Fig. 3.3. Moreover, the Inside-Outside theorem [74] states that the obtained result is further multiplied by 1 in case of clockwise-oriented contour (i.e., for small arguments).

**Table of contents :**

Acknowledgements

Abstract

Resume

List of Acronyms

List of Notations

List of Figures

List of Tables

**1 Introduction **

1.1 Context

1.2 Thesis Objective

1.3 Thesis Contributions

1.4 Thesis Outline

**2 Concepts on MIMO-ARQ Relay Communications **

2.1 Introduction

2.2 MIMO

2.3 Hybrid-ARQ

2.4 Cooperative Relaying

2.4.1 Cooperation Protocols

2.4.2 Classical Relaying Schemes

2.4.3 Project-and-Forward Relaying

2.4.4 Complexity Benchmark

2.5 Performance Metrics

2.5.1 Outage Probability

2.5.2 Ergodic Capacity

2.6 Conclusions

**3 Project-and-Forward Relaying over Dual-Hop MIMO Channels: Incentives and Theoretical Performance **

3.1 Introduction

3.2 System Model

3.2.1 Mixed MIMO-Pinhole and Rayleigh Channel

3.2.2 Project-and-Forward Relaying

3.3 Analytical Performance Analysis

3.3.1 Instantaneous SNRs Characterization

3.3.2 Outage Probability

3.3.3 Probability Density Function

3.3.4 Ergodic Capacity

3.4 Asymptotic Behavior

3.4.1 Asymptotic Outage Probability

3.4.2 Asymptotic Ergodic Capacity

3.5 Numerical Results

3.5.1 Settings

3.5.2 Bivariate Meijer-G Implementation

3.6 Conclusions

**4 PF-Based Cooperative Spatial Multiplexing over MIMO Broadband Systems **

4.1 Introduction

4.2 MIMO Relay System Model

4.2.1 Channel Description

4.2.2 Cooperation Protocol

4.3 Broadcast Phase Processing

4.3.1 Signaling Scheme

4.3.2 Broadcast Phase Communication Model

4.4 Relaying Phase Processing

4.4.1 Frequency Domain Transformation

4.4.2 Signal Reduction

4.4.3 Signal-Level Spatial Multiplexing

4.4.4 Relaying Phase Communication Model

4.5 Average Throughput Analysis

4.5.1 Equivalent MIMO Channel

4.5.2 Average Throughput

4.6 Simulation Results

4.6.1 Simulation Settings

4.6.2 Performance Analysis

4.6.2.1 Average throughput versus SNR

4.6.2.2 Average throughput versus distance

4.7 Conclusions

**5 Joint HARQ and PF Relaying for Single Carrier Broadband MIMO Systems **

5.1 Introduction

5.2 Relay ARQ Communication Model

5.2.1 General Framework

5.2.2 Relay-aided ARQ Transmission

5.3 Joint Over Transmissions Project and Forward Relaying

5.3.1 Multi-Transmission Frequency Domain Signal Model

5.3.2 Joint Over Transmissions QRD-based Projection

5.3.3 Signal Normalization

5.3.4 Second Slot Communication Model

5.4 Equivalent MIMO Channel and Theoretical Analysis Tools

5.4.1 Equivalent MIMO Channel

5.4.2 Outage Probability

5.4.3 Average Spectral Eciency

5.5 Simulation Results

5.5.1 Simulation Settings

5.5.2 Performance Analysis

5.6 Conclusions

**6 Conclusions **

6.1 Summary of Contributions

6.2 Future Research Directions

**A Some Generalized Functions Denitions **

A.1 Meijer-G Function

A.2 Bivariate Meijer-G Function

**B Derivation of the Capacity Asymptotic Expressions **

B.1 Case ! +1

B.2 Case ; sr ! +1

**C Bivariate Meijer-G Routine **

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