# PF-Based Cooperative Spatial Multiplexing over MIMO Broadband Systems

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## Bivariate Meijer-G Implementation

The bivariate Meijer G-function, presented in Annex C, was developed to enable the numerical evaluation of (3.22) in MATLAB environment. In the generalized Meijer-G and H functions, the complex integration contour is closely (and critically) dependent on the parameters of the function. The implementations of the bivariate Meijer-G function in [68] and even that of the bivariate H-function [75], are settling for a static manual contour de nition which is just not scalable for complicated and dynamic parameters (e.g., inside loops). In contrast, the proposed MATLAB code automates the integration contour de nition, enabling a simple usage of the bivariate Meijer-G function in dynamic codes. A second contribution of our proposed code, is that it is based on the fast MATLAB routine quad2d rather than the slow dblquad. The di erence is notable and appreciable from a running time point of view. Our proposed code is hence an improved (more general and automated) version of the existing implementation.

### Simulation Settings

In this section, average throughput performance of the presented signal-level cooperative spatial multiplexing scheme is evaluated via Monte-Carlo simulations. As a benchmark, we consider the half-duplex orthogonal AF relaying function, that actually, presents the same constraints as our system while being also signal-level oriented. To ensure a fair comparison, nodes of both systems must perceive the same SNRs. Let 0 denote the average SNR per receive antenna over link 0. CM and AF SNR measurements are similar for links s d and s r, i.e., CM AF s 2 0ns ; 0 2 fr; dg, while in link r CM rd2kns rd2nr AF s 0 = s 0 = 2 d we have rd = 2 2 = rd : To balance the relay-destination links, we increase the average transmit power of CM by a factor nr . The outage probability for AF relaying is computed using a 1=2 pre-log factor, i.e., kns 2IAF < S PoutAF (S; ) = Pr : (4.36) 1.
In all simulation scenarios, the source node is equipped with a single antenna (ns = 1) since it is the typical uplink transmission scheme in MIMO broadband systems (e.g., LTE). Links s d, s r, and r d have the same length Lsd = Lsr = Lrd = 3. The path loss exponent is set to = 3, and T = 128 c.u. The average throughput we are computing corresponds to a target spectral e ciency S of ns (bit/s/Hz) that could stem, for example, from a BPSK modulation (uncoded) or a higher order modulation with coding (e.g., QPSK with a 12 -rate encoder). The performance of CM over AF is however valid in both cases, since the two relaying schemes operate at the signal-level. Additional gains after decoding will scale-up respectively.

#### Average throughput versus SNR

It is noteworthy that the relay is located at the midpoint between nodes s and d so that the performance behavior can be relatively decorrelated with node r position. In the case of antennas con guration (ns; nr; nd) = (1; 2; 2), the CM scheme shrinks the relaying phase duration to the half (k = 2) thus leading to a gain of 3 to 4 dB compared to AF over the entire SNR range. Such a di erence becomes more accentuated when (ns; nr; nd) = (1; 3; 3). In Fig. 4.4 and 4.5, a 4 to 5 dB gap is observed between CM and AF since the rst relay performs a spatial multiplexing on its 3 antennas (k = 3). The CM throughput saturates at 9 dB (T = 1), whereas with AF one it only reaches 0:1 bit/s/Hz.

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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
4.3.1 Signaling Scheme
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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