Supelec

Random Matrix Theory for Wireless Communications ´ Merouane Debbah http://www.supelec.fr [email protected] February, 2008

Program

Course 1: Overview and Historical development. Course 2: Probability and convergence measures review. Course 3: Basic Results on Random Matrix Theory Course 4: What about deterministic matrices? Course 5: Stieltjes Transform Method. Course 6: Results on Unitary Random Matrix Theory Course 7: The role of the Cauchy-Stieltjes transform in communications Course 8: Free probability theory and random matrices Course 9: Free deconvolution for signal processing applications Course 10 MIMO Channel Modelling and random matrices Course 11: Asymptotic analysis of (MC)-CDMA systems Course 12: Asymptotic Analysis of MIMO systems Course 13: Asymptotic design of receivers Course 14: Decoding order in receivers Course 15: Game theory and Random matrix theory

1

Presentation

Overview and Historical development

2

Random Matrix Theory

3

Random Matrix Theory

Cited as one of the ”modern tools” in mathematics and used in the proof of an important result in prime number theory

4

Applications of Random Matrix Theory

• Wigner (55) , Dyson (67) : Random matrix theory and the statistical theory of energy levels of nuclei. • Potters (00), Bouchaud (00) : Random matrix theory and financial correlations. • Voiculescu (91) , Biane (00), Hiai, Petz (00): Random matrix theory and Free probability Theory. • Silverstein (89), Pastur (72), Girko (90), Edelman (89): Random matrix theory and Cauchy-Stieltjes transform. • Speicher (92): Random matrix theory and Combinatorics. • Tanaka (01), Moustakas (03), Sengupta (03) : Random matrix Theory and statistical mechanics approach.

5

Random Matrices: Some Dates in Wireless Communications

• Tse & Hanly (99), Evans & Tse (00) : Asymptotic Performance of Linear Receivers for certain CDMA systems • Foschini & Gans (96), Telatar (99) : Shannon Capacity for MIMO systems. • Verdu´ & Shamai (99, 00), Tulino & Verdu´ (04) : Capacity of CDMA and (MC)-CDMA systems. • Tse & Zeitouni (00) : CLT for the output SINR of CDMA receivers. • Guo et. al. (99), Muller & Verdu´ (01), Honig & Xiao (01), ... : Design of reduced-rank ¨ multi-user detectors. • Chuah et.al. (02), Muller (02), Mestre et. al. (03) : Capacity for more realistic MIMO ¨ systems. • Zaidel et.al. (01), Debbah et. al.(04): Design of multi-cell systems. ˆ • O. Leveque et.al. (03): Scaling laws for ad-hoc networks. • V. Poor et al. (05): Game theory with a large number of players.

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Typical Random Matrix Questions

• • • • • • •

Distribution of λ(H). Distribution of λ(HH H). Distribution of λmax(H). Joint distribution of λ1(H), ..., λN (H). Distribution of the spacings between adjacent eigenvalues. Distribution of HH H. Distribution of the matrix of eigenvectors of HH H.

7

The birth of Random Matrix Theory

J. Wishart, ”The generalized product moment distribution in samples from a normal multivariate population”, Biometrika, vol. 20A, pp. 32-52, 1928. Probability density function of:

v1v1

H

+ .... + vnvn

H

where vi are i.i.d Gaussian vectors.

8

Wishart Matrices

Definition., The m × m random matrix A = HHH is a central real/complex Wishart Matrix with n degrees of freedom and covariance matrix Σ (A ∼ Wm(n, Σ)) if the columns of the m × n matrix H are zero mean independent real/complex Gaussian vectors with covariance matrix Σ. The pdf of a complex Wishart matrix A ∼ Wm(n, Σ) for n ≥ m is:

π m(m−1)/2 [−Trace(Σ−1 B)] n−m fA(B) = e detB Q m detΣn i=1(n − i)! Note that if the entries of H have non-zero mean, HHH is a non-central Wishart matrix.

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Eigenvalues of Wishart Matrices

R. A. Fisher, ”The sampling distribution of some statistics obtained from non-linear equations,”, the annals of Eugenics, Vol.9, pp.238-249, 1939. M. A. Girshick, ”On the sampling theory of roots of determinantal equations”, The Annals of Math. Statistics, vol. 10, pp. 203-204, 1939. P.L. Hsu, ”On the distribution of roots of certain determinantal equations,” The Annals of Eugenics, vol. 9, pp.250-258, 1939. S.N. Roy, ”p-statistics or some generalizations in the analysis of variance appropriate to multi-variate problems”, Sankhya, vol. 4, pp. 381-396, 1939 The joint p.d.f. of the ordered strictly positive eigenvalues of the Wishart matrix HHH , where the entries of H are i.i.d complex Gaussian with zero mean and unit variance.

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Eigenvalues of Wishart Matrices

Sir Ronald Fisher, 1890-1962

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Eigenvalues of Wishart Matrices

Theorem Let the entries of H be i.i.d zero mean Gaussian and unit variance. The joint p.d.f of the ordered strictly positive eigenvalues of the Wishart matrix HHH , λ1, ..., λm, equals m m n−m Pm Y Y λ − i=1 λi 2 i e (λi − λj ) (n − i)!(m − i)! i