Joint Angle and Delay Estimation by a single-snapshot: 2D Matrix Pencil Approach

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Detection by MMDL

It has been shown in [41] that the sample eigenvalues ^l1 : : :^lN extracted from the sample covariance matrix ^R are (N;L)-inconsistent estimators of the true eigenvalues of the covariance matrix Rxx, that is, the sample eigenvalues do not converge towards the true ones as (N;L) 􀀀! 1 at the same rate (0 < c = N L < 1). The MDL estimator in (2.21) depends on the sample eigenvalues of ^R , therefore, it seems natural that the performance of the MDL estimator would perform poorly in the asymptotic regime, i.e. (N;L) 􀀀! 1 at the same rate (0 < c = N L < 1). In other words, when insucient number of snapshots L are available with respect to the number of antennas N in such a
way that the ratio c = N L is not negligible, then the MDL estimator would perform poorly. In this section, we present a modied MDL estimator to cope with this aforementioned issue. The modied MDL estimator is based on using improved estimators of eigenvalues of the covariance matrix Rxx, which turn out to be (N;L)-consistent, as shown in [42].

Pursuit-type algorithms

Pursuit-type algorithms are popular algorithms in the eld of compressed sensing. More specically, matching pursuit algorithms deal with an approximate solution of the l0- optimisation problem. For uniqueness of the l0 problem, we refer the reader to [44]. However, basis pursuit relax the l0-optimisation problem to an l1-optimisation one. The l1-optimisation problem is also known as LASSO [45]. For uniqueness of the l1 problem, we refer the reader to [46]. An advantage of this relaxation is that the problem is now convex. It remains to see when the unique solution provided by the l1-optimisation problem coincides with that of the l0 one. The papers in [44, 47] give sucient conditions for ^s to be a unique solution of the l0 and l1-optimisation problems. Moreover, the necessary conditions for that to happen are found in [48, 49]. The pursuit algorithms that are evaluated in the context of AoA estimation in this paper are the following:
• Matching Pursuit (MP) [50]
• Orthogonal MP (OMP) [51]
• Gradient, or directional, Pursuit (GP) [52]
• Basis Pursuit De-Noising (BPDN) [53]

A Newton-type Forward Backward Greedy Method

In this section, we present a new Greedy method, which is inspired from the Adaptive Forward Backward (AdFoBa) [74] Greedy method. The dierence is in the cost function itself, and therefore the forward step would be modied. In addition, we propose a dierent backward scheme, which seems to correct false peaks.

Table of contents :

Abstract
Acknowledgements
List of Figures
Abbreviations
Symbols
1 Introduction 
1.1 Motivation
1.1.1 Brief history
1.1.2 Parameters inferring location
1.2 Parameter Estimation: Problems and Methods
1.3 Contributions of this dissertation
2 Angle-of-Arrival Detection 
2.1 System model
2.1.1 Problem formulation
2.1.2 Assumptions
2.1.3 Problem statement
2.2 Background of main result
2.3 Detection by MMDL
2.3.1 Additional assumptions
2.3.2 The MMDL criterion
2.4 Conclusions and future directions
3 Angle-of-Arrival Estimation by Compressed Sensing Techniques 
3.1 System model
3.1.1 Problem formulation
3.1.2 Problem statement
3.2 Background of existing methods
3.2.1 Pursuit-type algorithms
3.2.2 Thresholding-type algorithms
3.2.3 Bayesian-based algorithms
3.3 Sparse Recovery via Iterative Variational Bayes
3.3.1 The Bayesian perspective
3.3.2 Variational Bayes methodology
3.3.3 The Iterative Variational Bayes method
3.4 A Newton-type Forward Backward Greedy Method
3.4.1 Optimization problem
3.4.2 Forward step
3.4.3 Backward step
3.5 Conclusions and future directions
4 Joint Angle and Delay Estimation 
4.1 System Model
4.1.1 Problem formulation
4.1.2 Assumptions
4.1.3 Problem statement
4.2 Ecient Maximum Likelihood Joint AoA and ToA estimation
4.2.1 Parameterisation of the Noise Subspace
4.2.2 2D Iterative Quadratic ML (2D-IQML)
4.2.3 2D-Denoised IQML (2D-DIQML)
4.3 Joint Angle and Delay Estimation by a single-snapshot: 2D Matrix Pencil Approach
4.3.1 ToA Estimation using 2D Matrix Pencil
4.3.2 AoA Estimation using 2D Matrix Pencil
4.3.3 Proposed Algorithms
4.4 Spatio-Frequential smoothing: A Remedy for coherent sources
4.4.1 The JADE-MUSIC Algorithm: A Recap
4.4.2 The Spatio-Frequential Preprocessing Technique
4.5 Conclusions
5 Joint Angle and Delay Estimation and Detection 
5.1 System model
5.2 JADED-RIP Algorithm Derivation
5.2.1 Data Manipulation
5.2.2 Introducing Orthogonal Projectors
5.2.3 Least-Square Estimator
5.3 Computationally Ecient Single Snapshot JADED-RIP (CESS-JADEDRIP)
5.4 Identiability conditions
5.5 Peak Detection and Resolvability
5.6 Conclusions and future directions
6 Mutual Coupling 
6.1 System model
6.1.1 Problem formulation
6.1.2 Assumptions
6.1.3 Problem statement
6.2 The MUSIC Algorithm
6.2.1 Preliminaries
6.2.2 Mutual Coupling in the sense of MUSIC
6.3 A suboptimal MUSIC-based approach
6.3.1 Algorithm derivation
6.3.2 Discussion
6.4 An optimal MUSIC-based approach
6.4.1 Preliminaries
6.4.2 Algorithm derivation
6.4.3 Properties of the algorithm
6.4.4 MSE Analysis
6.4.5 Comparison with the Cramer-Rao Bound
6.4.6 Rening the AoA estimates by alternating minimisation
6.5 Mutual Coupling Agnostic AoA estimator
6.5.1 The Maximum Likelihood Estimator
6.5.2 Proposed iterative method
6.6 Conclusions and future directions
7 Localizing via Wi-Fi 
7.1 Analytical Modelling
7.1.1 Transmit Signal
7.1.2 Channel Propagation
7.1.3 Receiver
7.1.4 Summary
7.2 Oine calibration approach
7.2.1 Step 1: Detect Frame/Symbol
7.2.2 Step 2: Estimating and Compensating the CFO
7.2.3 Step 3: Compensating Tx/Rx Filter Eects
7.2.4 Step 4: Estimating and Compensating the SFO and antenna phases
7.3 Online method
7.3.1 Why JADED and CSI?
7.3.2 Main blocks
7.3.3 Real data
7.4 Conclusions and future directions
8 Conclusions 
9 Resume en Francais 
9.1 Motivation
9.1.1 Bref historique
9.1.2 Parametres inferring location
9.2 Estimation des parametres: problemes et methodes
9.3 Contributions de cette dissertation
9.4 Conclusions
A Proof of Theorem 4.3
B Proof of Theorem 6.1
C Proof of Theorem 6.3
D Proof of Theorem 6.4
E Proof of Theorem 6.5
F Proof of Theorem 6.6
G Proof of Property 2
H Proof of Property 3
I Proof of Property 5
J Proof of Theorem 6.7
K Proof of Theorem 6.9
L Proof of Theorem 6.10
Bibliography

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