Random Matrix Theory Lecture 3 Free Probability Theory. Symeon Chatzinotas March 4, 2013 Luxembourg
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1 Random Matrix Theory Lecture 3 Free Probability Theory Symeon Chatzinotas March 4, 2013 Luxembourg
2 Outline 1. Free Probability Theory 1. Definitions 2. Asymptotically free matrices 3. R-transform 4. Additive Convolution 5. Sigma-transform 6. Multiplicative Convolution 2. Examples 1. Spectrum Sensing 2. Relay Channel 3. Cochannel interference
3 Introduction Free probability theory A form of independence for non commutative algebras Applications in random Hermitian matrices Expressions that include sums or products of asymptotically free matrices 3
4 Non-commutative Spaces Non commutative probability space (A,φ) A non-commutative unital algebra A A linear function φ:a C with φ(1)=1 Moments Probability space (A N,τ N ) Random Hermitian Matrices A N Real random eigenvalues Functional τ N τ N (I)=1 Moments τ N (Xk) 4
5 Asymptotic Freeness A family of matrices {X N,1,, X N,n } is asymptotically free in (A N,τ N ) if: X N,n has a non random limit distribution For every family of polynomials where It applies 5
6 Asymptotically Free Matrices Any random matrix and the identity matrix Independent Wigner matrices Independent Gaussian matrices Independent Haar matrices Independent Unitarily Invariant (Wishart) matrices Standard Winger and deterministic diagonal Standard Gaussian and deterministic (diagonal) Haar matrices and deterministic matrix Unitarily invariant and deterministic matrix 6
7 R Transform: Definition z belongs to the complex plane Reminder: Stieltjes definition 7
8 R transform: Basic Laws For any positive alpha Semicircle law MP Law 8
9 Additive Free Convolution If matrices A,B are asymptotically free, the R- transform of the matrix sum equals the sum of the R-transforms: 9
10 Sigma transform: Definition For -1<x<0 (eta definition) Reminder: Eta definition 10
11 Sigma transform: Properties & Basic Laws AB non-negative definite MP law 11
12 Multiplicative Free Convolution If matrices A,B are asymptotically free, the Sigma-transform of the matrix product equals the product of the Sigma-transforms: 12
13 Transform Interconnections R Stieltjes (G) Capacity Sigma (S) η MMSE 13
14 Examples Spectrum sensing Addition of Wishart matrix functions Relay channel, Cochannel interference channel Product of Wishart matrix functions 14
15 Spectrum Sensing Matrix I/O model Covariance of received signal 15
16 Spectrum Sensing Pdf trough inversion formula Application: SNR estimation Measure max eigenvalue from received signal Compare to analytic pdf Recover SNR p 16
17 Spectrum Sensing Sum of Standard Wishart matrix Scaled Wishart matrix 17
18 Spectrum Sensing Stieltjes transform Cubic polynomial β dimension ratio p SNR z Stieltjes argument 18
19 Relay Channel Vector I/O model Mutual information 19
20 Relay Channel Asymptotically 20
21 Relay channel Product of Standard Wishart Scaled Wishart plus Identity Auxiliary variables 21
22 Relay Channel Multiplicative Free Convolution Eta transform of M Change of variables in MP law Eta definition 22
23 Relay Channel Quartic polynomial for Stieltjes transform and inversion formula 23
24 Relay Channel 24
25 Cochannel Interference Vector I/O model Mutual information 25
26 Cochannel Interference Product of Standard Wishart Inverse of Scaled Wishart plus Identity Auxiliary variables 26
27 Cochannel Interference Asymptotically 27
28 Cochannel Interference Multiplicative Free Convolution Eta transform of M Change of variables in MP law Eta definition 28
29 Cochannel Interference 29
30 Summary Free probability is a generalization of independence for random matrices First, we have to establish that two matrices are asymptotically free Matrix sums can be tackled through additive free convolution in R-transform domain Matrix products can be tackled through multiplicative free convolution in Sigma-transform domain Applications in: Spectrum Sensing, SNR Estimation Cochannel interference, relay channels 30
31 Questions? Random Matrix Theory Lecture 3 Free Probability Theory 31
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