Static and Dynamic Sparse Bayesian Learning usingMean Field Variational Bayes 

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Background on Beamforming in MaMIMO

In this thesis, we use beamforming or precoding to represent the same concept. In short, they represent the usage of an antenna array to transmit one or more spatially directive signals. BF matrix is designed as a function of the estimated channel such that a directive signal (or a beam) is formed towards each user canceling the interference from other user’s signals. First, we would like to mention that in this thesis we are specifically focused towards the weighted sum rate (WSR) maximization problem for the BF design. In fact, in the literature, we can find several op-timization criteria such as WSR, SINR balancing (maximize the minimum SINR), weighted sum energy efficiency (energy efficiency of any user is defined as the ratio of rate and the power con-sumed per user), etc. In our case, we are specifically interested in maximizing the sum through-put across the entire network, with the weights chosen to assign certain priorities to users. Even though we are not interested in optimizing the weights in most of the work proposed here, it is considered to make it more general. By a broadcast channel (BC), we refer to a communi-cation system where a single transmitter (BS) sends independent information through a shared medium to uncoordinated receivers (UEs). An interfering broadcast channel (IBC) refers to a multi-cell network where each BS serves multiple UEs in its network and the UEs in a particular cell are impacted by the inter-cell interference also. For a MaMISO IBC, the received signal at any UE k (assuming the channel between BS and UE k is represented as hk,bk and the BF for user is gk , where bk represents the BS to which user k is associated) can be written as (1.1) yk ˘ pk hkH,bk gk xk ¯ X pi hkH,bi gi xi ¯ vk , vk » N (0,¾2).

HBF Design using WSMSE for Multi-User MIMO

Consider a Multi-User MIMO system with Ntc transmit antennas in cell c and K multi-antenna users. In this section, we shall consider a per stream approach (which in the perfect CSI case would be equivalent to per user). In an IBC formulation, one stream per user can be expected to be the usual scenario. In the development below, in the case of more than one stream per user, we shall treat each stream as an individual user. So, consider an IBC with C cells with a total of K users. We shall consider a system-wide numbering of the users. User k is served by BS bk . User is equipped with Nk antennas. The Nk £1 received signal at user k in cell bk can be written as (2.1) bk bk c X X yk ˘ Hk,bk V gk sk ¯Hk,bk biXbk ¯ Hk,c V c gi si ¯vk V i 6˘k gi si c bk i :bi 6˘ ˘ signal ˘ | {z } | {z } intercell interf. | {z bk intracell interf. }
where sk is the intended (white, unit variance) scalar signal stream, Hk,bk is the Nk £Nt channel from BS bk to user k. Hk,bi represents the Nk £ Ntbi channel from BS bi to user k. BS c serves Uc ˘ i :bi ˘c 1 users. We considered a noise whitened signal representation so that we get for the noise vk » C N (0, INk ). The Ntbk £1 spatial Tx filter or beamformer (BF) is gk . The analog beamformer for base station c, V c is of dimension Ntc £ Mc . Mc is the number of RF chains at BS c. Treating interference as noise, user k will apply a linear Rx filter fk (of dimension Nk £ 1) to maximize the signal power (diversity) while reducing any residual interference that would not have been (sufficiently) suppressed by the BS Tx. The Rx filter output is s fH y , hence sk ˘ fkH Hk,bk Vbk gk sk ¯ K bk ˘ k k (2.2) X fkH Hk,bi Vbi gi si ¯fkH vk . b ˘1,6˘k.

Hybrid Beamforming for Globally Converging Phasor Design

As we saw in Section 2.1, the main issue with WSR/WSMSE optimization for an HBF hybrid de-sign is the high non-convexity of the cost function. This implies that even if it is possible to show convergence to a local optimum [10], convergence to the global optimum cannot be guaran-teed. To avoid the convergence to a local optimum, [11] proposed Deterministic Annealing (DA) for digital BF design in the MIMO interference channel.
In this section, we go one step further and consider a multi-stream approach with dk streams for user k. So, consider an Interfering BroadCast (IBC) (i.e. multi-cell MU downlink) system of cells with a total of K users and Ntc transmit antennas in cell c. User k is equipped with Nk antennas. Hk,c represents the Nk £ Ntc MIMO channel between user k and BS c and we define i E HkH,c Hk,c ˘ £kc . User k receives bk X bi (2.20) yk ˘ Hk,bk V Gk sk ¯ Hk,bi V Gi si ¯vk , i 6˘k where sk , of size dk £1, is the intended signal stream vector (all entries are white, unit variance). X BS c serves Uc ˘ 1 users. We are considering a noise whitened signal representation so i :bi ˘c that we get for the noise vk » C N (0,INk ). The analog beamformer Vc for base station c is of dimension Ntc £Mc where Mc is the number of RF chains at BS c. The Mc £dk digital beamformer i is Gk , where Gk ˘ g(1)k … g(kdk ) and g(ks) represents the beamformer for stream s of user k. The transmit power constraint at base station c can be written as tr{Vc H Vc P K H } • Pc .

Hybrid Beamformer Capabilities

In this section, we analyze to what extent a hybrid BF can achieve the same performance as a fully digital BF. In particular, we shall see that this is possible for a sufficient number of RF chains and with the antenna array responses being phasors. Consider a specular or pathwise channel model with say L multi-paths per link. For notational simplicity, we shall consider a uniform L and Nk ˘ Nr ,8k. Let the antenna array response for BS c be hct (`) for Angle of Departure (AoD)`. We assume that all entries of hct (`) have the same magnitude. This assumption is necessary for the following theorem to be valid and it is necessitated by the unit magnitude constraints on the analog BF. Then the collective Nt £ L multipath Tx array response Ht ,k for the downlink channel of user k is (2.35) Hct,k ˘ £hct ¡`k,1¢ hct ¡`k,2¢ … hct ¡`k,L ¢⁄⁄ , and the concatenated antenna array response matrix to all users can be written as, (2.36) Hct ˘hHct,1 Hct,2 … Hct,K i, of dimension Nt £Np , where we denote the total number of paths Np ˘ LK . Similarly, we define Hcr and Ac for the concatenated Rx antenna array responses and complex path amplitudes. Ac is an Np £ Np block diagonal matrix with blocks of size L £L and Hcr is a K Nr £ Np block diagonal matrix with blocks of size Nr £L. Finally, we can write the K Nr £Nt MIMO channel from BS c to all a users as (2.37) Hc H ˘ Hct Ac H Hcr H .

BF initializations for the algorithm

In this thesis, for the simulations, corresponding to fully or hybrid beamforming schemes, the BFs are initialized as follows. We use the concept of deterministic annealing proposed for a fully digital solution in [32]. At low SNR, an optimal BF solution corresponds to matched filtering. Starting with this solution for the fully digital BF at low SNR, we optimize the BFs using the alter-nating minorization concept proposed here. Further, at any SNR, the converged values from the previous iterations are used as the initialization point. This process is followed in the simulations of all other chapters (where BF techniques are discussed) in this thesis.

Simulations for WSMSE based HBF

In all the figures, the simulations are done for SN R ˘ 20d B, L ˘ 6. Since single cell, C ˘ 1 and the number of users is denoted by U .
The sum spectral efficiency (sum of the rates of the U users) is plotted versus the number of users and compared with the sub-optimal algorithms proposed in [23] and [25] (« ‘MUBS Precod-ing »’, MUBS refers to a multi-user beam steering scheme). In [23], V is computed using covari-ance CSIT and the g are updated with instantaneous CSIT. The interesting part about our work is in showing that the analog beamformer need not be adapted at the fast fading rate. For the com-parison to be fair, the proposed update of the analog beamformer with outdated CSIT and digital beamformer with instantaneous CSIT is compared with prior works which use covariance CSIT for V and instantaneous CSIT for g . It is evident from all the figures that our approach based on Mixed Time Scale WSMSE Adaptation outperforms those of [23] and [25]. In Figure 1, Nt ˘ 32 and the number of users is equal to the number of RF chains (U ˘ M). Simulations are done for the two variants of CSIT. In the first case, V and g are iteratively updated w.r.t the instanta-neous channel. In the second case, we consider two realizations of the channels (hk (1),hk (2)). For hk (1), V and g are updated as per Algorithm 1. For hk (2), the result obtained with hk (1) is used for V and is not updated, whereas the g are updated. So this is the case of outdated CSIT for V . The results are averaged over 40 such pairs of channel realizations, each time with the same channel covariances. From Figure 2.3 we can see that in the second case with outdated CSIT for V , performance is slightly degraded but still much better than the suboptimal algorithms of [23] and [25].

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Table of contents :

List of Figures
List of Tables
I Motivation and Background 
1 Introduction 
1.1 Motivation and State of the Art
1.2 Organization of the thesis
1.3 Background Information
1.3.1 Background on Beamforming inMaMIMO
1.3.2 AlternatingMinorization
1.3.3 Background on Compressed Sensing
II Beamforming Techniques forMassive MIMO 
2 Hybrid Beamforming 
2.0.1 Summary of the Chapter
2.0.2 Phase Shifter Architecture
2.1 HBF Design using WSMSE forMulti-User MIMO
2.1.1 WSR Optimization in terms ofWSMSE
2.1.2 Design of the Analog Beamformer with Perfect CSIT
2.1.3 Mixed Time Scale Adaptation
2.2 Hybrid Beamforming for Globally Converging Phasor Design
2.2.1 AlternatingMinorization Approach
2.2.2 Digital BF Design
2.2.3 Design of Unconstrained Analog BF
2.2.4 Design of Phase Shifter Constrained Analog Beamformer
2.2.5 Simulation results
2.3 Hybrid Beamforming under Realistic Power Constraints
2.3.1 Digital BF Design
2.3.2 Optimization of Power Variables
2.3.3 Design of Unconstrained Analog BF
2.3.4 Hybrid Beamforming Design with Per-Antenna Power Constraints
2.3.5 AlgorithmConvergence
2.3.6 Conclusion
2.4 Hybrid Beamforming Design forMulti-User MIMO-OFDM Systems
2.4.1 MIMO OFDMChannelModel
2.4.2 WSRMaximization viaMinorization and Alternating Optimization
2.4.3 Digital BF Design
2.4.4 Design of Unconstrained Analog BF
2.4.5 AlgorithmConvergence
2.4.6 Analysis on the number of RF Chains and HBF Performance
2.4.7 Simulation Results
2.4.8 Conclusions and Perspectives
3 Hybrid Beamforming for Full-Duplex Systems
3.1 Introduction
3.1.1 Summary of the Chapter
3.2 Full-Duplex BidirectionalMIMO SystemModel
3.2.1 ChannelModel
3.3 WSR maximization through WSMSE
3.3.1 Two-stage transmit BF design
3.3.2 Hybrid Combiner/Two-Stage BF Capabilities for SI Power Reduction
3.3.3 Simulation Results
3.3.4 Conclusion
3.4 Robust Beamforming Design under Partial CSIT
3.4.1 EWSR maximization through alternating minorization
3.4.2 Two-stage transmit BF design
3.4.3 Optimization of streampowers
3.5 Simulation Results
3.6 Conclusion
4 NoncoherentMulti-UserMIMO Communications using Covariance CSIT 
4.1 Introduction
4.2 Streamwise IBC SignalModel
4.3 Max WSR with Perfect CSIT
4.3.1 FromMax WSR toMin WSMSE
4.3.2 Minorization (DC Programming)
4.3.3 PathwiseWirelessMIMO ChannelModel
4.4 MIMO Interference Alignment (IA)
4.5 Expected WSR (EWSR)
4.5.1 Massive EWSR with pwCSIT
4.5.2 Interference management by Tx/Rx
4.5.3 Comparison of instantaneous CSIT and pathwise CSIT WSR at low SNR .
4.5.4 Comparison of instantaneous CSIT and pathwise CSIT WSR at high SNR .
4.6 Simulation Results
4.6.1 Conclusions and Perspectives
5 Rate Splitting for Pilot Contamination 
5.1 Introduction
5.1.1 Summary of this Chapter
5.2 System model
5.2.1 Assumptions on the user channel
5.2.2 Channel estimation
5.2.3 Rate Splitting in Downlink transmissions
5.2.4 Spectral efficiency
5.3 Power optimization and precoding design
5.3.1 Power optimization
5.3.2 Precoding design for common message
5.4 Simulation Results
5.5 Concluding Remarks
III Stochastic Geometry based Large System Analysis 
6 Asymptotic Analysis of Reduced Order Zero Forcing Beamforming 
6.1 Introduction
6.1.1 Summary of this Chapter
6.2 Multi-UserMIMO SystemModel
6.3 Large System Analysis of Optimal BF-WSMSE
6.4 Large System Analysis of Optimal DPC
6.5 Reduced Order ZF
6.6 Large System Analysis for RO-ZF, Full Order ZF and ZF-DPC
6.6.1 Optimization of user powers pk
6.7 Optimization of the ZF Order
6.8 Simulation Results
6.9 Extension of RO-ZF BF to IBC under Partial CSIT
6.9.1 Channel and CSITModel
6.9.2 Partial CSIT BF based on Different Channel Estimates
6.9.3 BF with Partial CSIT
6.9.4 Max EWSR ZF BF in theMaMISO limit (ESEI-WSR)
6.9.5 Reduced Order ZF with Partial CSIT
6.9.6 Large System Analysis for RO-ZF and Full Order ZF
6.9.7 Optimization of the ZF Order
6.9.8 Simulation Results
6.9.9 Conclusions
7 Stochastic Geometry based Large SystemAnalysis 
7.0.1 Summary of this Chapter
7.1 Massive MISO Stochastic Geometry based Large System Analysis
7.1.1 MISO IBC SignalModel
7.1.2 Channel and CSITModel
7.1.3 Various Channel Estimates for Partial CSIT
7.1.4 Beamforming with Partial CSIT
7.1.5 Further Considerations on EWSR Bounds
7.1.6 Asymptotic Analysis: Stochastic GeometryMaMISO Regime
7.1.7 Computation of eigenvalues ofWk,bi
7.1.8 EWSMSE BF in theMaMISO Stochastic Geometry Regime
7.1.9 Deterministic Equivalent of Auxiliary Quantities
7.1.10 Simplified SumRate Expressions with Different BF and Channel Estimators
7.1.11 Simulation Results
7.1.12 Channel Estimation Error/1/P
7.1.13 Constant Channel Estimation Error
7.1.14 Conclusion
IV Approximate Bayesian Inference for Sparse Bayesian Learning 
8 Static and Dynamic Sparse Bayesian Learning usingMean Field Variational Bayes 
8.1 Introduction
8.1.1 Summary of the Chapter
8.2 SignalModel-SBL
8.3 SBL using Type-IIML
8.3.1 Variational Interpretation of SBL
8.3.2 Overview of Fast SBL Algorithms
8.3.3 Variational Bayes
8.4 SAVE Sparse Bayesian Learning
8.4.1 Computational Complexity
8.4.2 Convergence Analysis of SAVE orMean Field Approximation
8.4.3 Sparsity Analysis with SAVE
8.4.4 Simulation results
8.4.5 Conclusion
8.4.6 Open Issues: Reduced Complexity Linear Tx/Rx Computation
8.5 Dynamic SBL-SystemModel
8.5.1 Gaussian PosteriorMinimizing the KL Divergence
8.6 SAVE SBL and Kalman Filtering
8.6.1 Diagonal AR(1) ( DAR(1) ) Prediction Stage
8.6.2 Measurement or Update Stage
8.6.3 Fixed Lag Smoothing
8.6.4 Estimation of Hyperparameters
8.7 VB-KF for Diagonal AR(1) (DAR(1))
8.7.1 DAR(1) Prediction Stage
8.7.2 Measurement or Update Stage
8.7.3 Fixed Lag Smoothing
8.7.4 Simulation Results
9 Sparse Bayesian Learning usingMessage Passing Algorithms 
9.0.1 Summary of this Chapter
9.1 Approximate Inference Cost Functions: An Overview
9.1.1 Region Based Free Energy
9.1.2 Combined BP/MF Approximation
9.2 Dynamic SBL SystemModel
9.2.1 BP-MF based Static SBL
9.2.2 Dynamic BP-MF-EP based SBL
9.3 Optimal Partitioning of BP and MF nodes
9.3.1 Optimal Partitioning for Static SBL:
9.3.2 Optimal Partitioning for DAR-SBL:
9.4 Simulation Results
9.4.1 Conclusions
9.5 Posterior Variance Prediction: Large System Analysis for SBL using BP
9.5.1 Iterations inMatrix Form
9.5.2 Convergence Analysis of BP
9.5.3 Scalar Iterations
9.5.4 Original AMP Iterations and SBL-AMP
9.6 Bayesian SAGE (BSAGE)
9.7 Concluding Remarks on Combined BP-MF-EP DAR-SBL
9.8 Towards a Convergent AMP-SBL Solution
9.8.1 Fixed Points of Bethe Free Energy and GSwAMP-SBL
9.9 GSwAMP-SBL based Dynamic AR-SBL
9.10 GSwAMP-SBL for Nonlinear Kalman Filtering
9.10.1 Diagonal AR(1) ( DAR(1) ) Prediction Stage
9.10.2 Measurement Update (Filtering) Stage
9.10.3 Lag-1 Smoothing Stage
9.11 Simulation Results
9.11.1 ill-conditioned A case:
9.11.2 Non-zero mean A case:
9.11.3 Rank Deficient A case (Figure 9.8):
9.12 Conclusions
9.12.1 Conclusions and Perspectives
10 Sparse Bayesian Learning for Tensor Signal Processing 
10.1 Summary of this Chapter
10.1.1 Tensor Notations
10.2 Hierarchical ProbabilisticModel
10.2.1 Application-MultipathWireless Channel Estimation
10.3 Variational Bayesian Inference for JointDictionary Learning and Sparse Signal Recovery
10.4 Kronecker Structured Dictionary Learning
10.4.1 SAVED-KS Sparse Bayesian Learning
10.4.2 Joint VB for KS Dictionary Learning
10.5 Identifiability of KS Dictionary Learning
10.5.1 Identifiability for mix of parametric and non-parametric KS factors
10.5.2 Simulation Results
10.5.3 Conclusions and Perspectives
10.6 Joint Dictionary Learning and Dynamic Sparse State Vector Estimation
10.6.1 Dynamic BP-MF-EP based SBL
10.6.2 Suboptimality of SAVED-KS DL and Joint VB
10.7 Optimal Partitioning of theMeasurement Stage and KS DL
10.8 Simulation Results
10.9 Conclusions and Perspectives


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