# Random Matrix Theory for large system analysis and Massive MIMO design

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## Deterministic annealing (DA) and WSMSE-SR (Simple Receiver)

From [23], we recall that the maximization problem in (1.6) is highly non concave. At low SNR (high noise variance), any interference is negligible compared to the noise. Hence, all links can be considered decoupled, and, like in single-user MIMO, rate maximization becomes SNR maximization for a single stream to which all transmit power is devoted. The optimal Tx and Rx filters are the left and right singular vectors corresponding to the largest singular value of the channel between the Tx and Rx. This implies that, for SNR = 0, a convergence to the global optimum is guaranteed. Meanwhile, as soon as the SNR increases, many further local optima getintroduced due to the appearance of the additional streams. Then, as the SNR increases further, more streams and local optima appear. The idea of DA (Figure 1.3) is to initialize the WSMSE (KG) with the solution of the WSMSE (KG) algorithm at lower SNR, starting from very low SNR, which guarantees a convergence to a global optimum. This process goes on until a stream distribution is reached, at some higher SNR, corresponding to a maximal stream distribution for which interference alignment is feasible. Indeed, at very high SNR, the Tx and Rx filters converge to the global solution. In order to prove the efficiency of the DA approach, we plot the sum rate versus SNR corresponding to an IBC system where the signals are precoded by the WSMSE (KG) precoder and by the WSMSE (KG) precoder with DA. This latter means that precoders at lower SNR will serve as initializations for the precoding algorithm at higher SNR. The classical WSMSE precoder is initialized by the right eigenvectors of user’s channel. The classical KG is nitialized by zero matrices. The overall number of iterations will remain the same. In other words, with DA the number of iterations will be the sum of the number of iterations needed to converge at each of the SNR used before achieving our goal SNR. However, using the classical algorithms we do not need to iterate over lower SNRs, but we run our algorithm immediately at the goal SNR; a number of iterations, which equals the total number of iterations in the case of DA is herein needed. We can observe in Figure 1.4 that DA enhances a lot the performance of the WSMSE (or LUO) and KG algorithms.

### Large system analysis

In this section, performance analysis is conducted for the precoder of the previous section. The large-system limit is considered, where the number of transmit antennas M and the numbers of users served per BS K go to infinity while keeping the ratio K/M finite such that limsupMK/M < ∞ and liminfMK/M > 0.
The results should be understood in the way that, for each set of system dimension parameters M and K we provide an approximate expression for the SINR and the achieved sum rate, and the expression is tight as M and K grow large.
Before we continue with our performance analysis of the above precoder, a deterministic equivalent of the SINR of the MF precoder is required. All vectors and matrices should be understood as sequences of vectors and matrices of growing dimensions.

#### Decentralized approach for large dimensions system

The idea is to try to identify the quantities that require global knowledge of the channel vectors, the intercell interference inter,c,k and D in our case, and exchange them (or the quantities related to them) between the different BSs in such a way that the maximum WSR problem will decompose into parallel sub-problems (one per BS). However, it is necessary to limit as much as possible this exchange in order to be backhaul friendly (efficient). The solution in the last section can be reformulated as the following: ac,k = gHc khc,c,k(σ2 +intra,c,k +inter,c,k) This solution can be initialized by a random precoder, e.g., a matched filter (MF) precoder. In general, it needs a central processing node to be implemented because of (3.6) which depends on global channels knowledge as shown in (3.7). In the case of absence of this central node, (3.6) can be detected by each receiver and then fed back using an over-the-air link as in [22]. However, this approach is spectral inefficient. Another way to decentralize consists in that each BS m calculates the quantities in (3.7), inter,m,c,k considered as the interference leakage from BS m to user k of cell c 6= m for all the users and sends them to the BS c using a backhaul link. This procedure is a bit heavy, so it is beneficial to limit as much as possible the number of iterations. However, at high SNR, the solution above requires a lot of iterations to converge, hence, requires an extensive exchange of information using the backhaul link which burdens this latter and makes it practically infeasible. For a limited number of iterations, the solution becomes very sub-optimal.
Thus, in the following we present a new initialization method which accelerates the convergence, hence, few iterations are no more sub-optimal and the backhaul-based decentralization becomes realistic. In this following, performance analysis is conducted for the proposed precoder. The large-system limit is considered, where M and K go to infinity while keeping the ratio K/M finite such that limsupMK/M < ∞ and liminfMK/M > 0.

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Abstract
Acknowledgements
List of Figures
Abbreviations
1 Motivation and Models
1.1 Introduction
1.2 Summary of Contributions
1.3 Notation
1.4 System Model
1.4.1 The WSMSE algorithm
1.4.2 The KG precoding algorithm
1.5 Deterministic annealing (DA) and WSMSE-SR (Simple Receiver)
1.5.1 DA
1.5.2 The WSMSE-SR
1.6 Channel estimation
1.7 Conclusion
I Random Matrix Theory for large system analysis and Massive MIMO design
2 The WSMSE algorithm: A Large System Analysis
2.1 Introduction
2.2 System model: The MISO IBC case
2.3 Large system analysis
2.3.1 Deterministic Equivalent of the SINR for the MF
2.3.2 Deterministic equivalent of the SINR of proposed precoder for correlated channels
2.3.3 Numerical results
2.4 The MIMO single stream case
2.4.1 Applications of the deterministic equivalent of the SINR
2.5 Conclusion
3 Using the Complex Large System Analysis to Simplify Beam- forming
3.1 Decentralized approach for large dimensions system
3.2 Signaling
3.3 Numerical results
3.4 Analytic solution
3.5 Conclusion
II Further Random Matrix Theory exploitation with par- tial CSIT
4 Robust Beamformers for Partial CSIT
4.1 The naive approach : ENAIVEKG
4.2 The EWSMSE approach
4.3 The ESEI-WSR approach
4.3.1 Alternative expression of ˘Ac,k
4.4 Numerical results and interpretation
4.5 Practical decentralized solution
4.6 The MISO case for large system analysis
4.6.1 Max EWSR BF (ESEI-WSR) in the MaMISO limit
4.6.2 EWSMSE and the naive approach
4.7 Large System Approximation of the EWSR
4.7.1 Large system analysis
4.7.2 Numerical results for MISO large system analysis
4.8 Alternative (sub-optimal) approach
4.9 Conclusion
5 Non-Linear Precoding Schemes
5.1 Introduction
5.2 The IBC signal model
5.3 The LA operation
5.4 Solving the EWSR problem
5.4.1 Max WSR with Perfect CSIT : DC approach
5.4.2 Solution with imperfect CSIT
5.5 Simulation results
5.6 Conclusion
III Relaying with Random Matrix Theory
6 Beamformers Design with AF Relays
6.1 Introduction
6.2 The IBRC signal model
6.3 The WSMSE precoder for IBRC
6.4 The KG precoder for IBRC
6.5 Numerical results and short discussion
6.6 Joint ZF-IN feasibility conditions
6.7 Conclusion
7 Conclusions and Future Works
7.0.1 Summary and conclusions
7.0.2 Future work
8 R´esum´e en Fran¸cais
8.1 Introduction
8.2 Probl`eme `a r´esoudre
8.2.1 L’algorithme WSMSE
8.2.2 L’algorithme KG
8.3 Partie I: Random Matrix Theory for large system analysis and Massive MIMO design
8.4 Partie II: Further random matrix theory exploitation with partial CSIT
8.5 Partie III: Relaying with random matrix theory
8.6 Conclusions
A Random Matrix Theory
B Proof of an equivalent deterministic expression: Perfect CSIT
C Proof of an equivalent deterministic expression: Imperfect CSIT
Bibliography

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