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Spectrumlocalization in the Gaussian information plus noisemodel
The complex Gaussian information plus noise model and notations
Subspace and DoA estimation in large sensor networks
Table of contents :
Rermerciements
Introduction
1 Notations and basic tools
1.1 Notations
1.2 The Stieltjes transformand its properties
1.3 Standard results of matrix analysis
1.3.1 Various inequalities
1.3.2 Resolvent formulas
1.3.3 Eigenvalues perturbation
1.4 Standard probability results
1.4.1 Tools for Gaussian variables
1.4.2 Results forWishart matrices
2 Asymptotic spectral distribution of complex Gaussian inf. plus noisemodels
2.1 The complex Gaussian information plus noise model and notations
2.2 Resolvent of §N§∗N and convergence of the empirical spectral measure
2.3 FunctionmN and density of μN
2.4 Functions wN, φN and support of μN
2.5 The special case of spiked models
2.6 Discussion and numerical examples
2.7 Appendix
2.7.1 Bound on kTN(z)k
2.7.2 Proof of theorem2.2.2
2.7.3 Proof of theorem2.2.3
2.7.4 Proof of theorem2.3.1: 0 does not belong to the support if cN < 1
2.7.5 Proof of property 2.3.1: lower bound formN(z)
2.7.6 Proof of property 2.4.1: function wN
2.7.7 Proof of property 2.4.2: increase of the local extrema of φN
2.7.8 Proof of property 2.4.3: mass of the clusters
2.7.9 Proof theorem2.5.1: support in the spiked model case
3 Spectrumlocalization in the Gaussian information plus noisemodel
3.1 Preliminary results
3.2 Localization of the eigenvalues
3.2.1 Absence of eigenvalues outside the support
3.2.2 Escape probability of the eigenvalues
3.2.3 Separation of the eigenvalues
3.3 Applications to the spiked models
3.4 Discussions and numerical examples
4 Subspace and DoA estimation in large sensor networks
4.1 Statistical model and classical subspace estimation
4.2 Generalized subspace estimation
4.2.1 Preliminary results
4.2.2 The general case
4.2.3 The spiked model case
4.3 DoA estimation
4.3.1 Regularization of the spectrum
4.3.2 Uniformconsistency of the subspace estimate
4.3.3 Consistency of the angles estimates
4.4 Discussion and numerical examples
4.5 Appendix
4.5.1 Proof of lemma 4.2.1: some uniformconvergences
4.5.2 Computations of the residues and formula of the estimator
4.5.3 Proof of lemma 4.2.2: another uniformconvergences
4.5.4 Proof of lemma 4.2.3: Behaviour of ˆωk,N under the spiked model assumption
4.5.5 Proof of lemma 4.3.1: Escape of ˆω1,N,
4.5.6 Proof of lemma 4.3.2 . .
