Random Matrix Theory Lecture 3 Free Probability Theory

Symeon Chatzinotas March 4, 2013 Luxembourg

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

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

Non-commutative Spaces

• Non commutative probability space (A,φ)

– A non-commutative unital algebra A

– A linear function φ:AC with φ(1)=1

– Moments

• Probability space (AN,τN)

– Random Hermitian Matrices AN

– Real random eigenvalues

– Functional τN

– τN(I)=1

– Moments τN(Xk)

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Asymptotic Freeness

• A family of matrices {XN,1,…, XN,n} is asymptotically free in (AN,τN) if:

– XN,n has a non random limit distribution

– For every family of polynomials where

It applies

5

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

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R Transform: Definition

• z belongs to the complex plane

• Reminder: Stieltjes definition

7

R transform: Basic Laws

• For any positive alpha

• Semicircle law

• MP Law

8

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

Sigma transform: Definition

• For -1<x<0 (eta definition)

• Reminder: Eta definition

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Sigma transform: Properties & Basic Laws

• AB non-negative definite

• MP law

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Multiplicative Free Convolution

• If matrices A,B are asymptotically free, the Sigma-transform of the matrix product equals the product of the Sigma-transforms:

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Transform Interconnections

13

R Stieltjes

(G) Capacity

MMSE η Sigma (S)

Examples

• Spectrum sensing

– Addition of Wishart matrix functions

• Relay channel, Cochannel interference channel

– Product of Wishart matrix functions

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Spectrum Sensing

• Matrix I/O model

• Covariance of received signal

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Spectrum Sensing

• Pdf trough inversion formula

• Application: SNR estimation

– Measure max eigenvalue from received signal

– Compare to analytic pdf

– Recover SNR p

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Spectrum Sensing

• Sum of

– Standard Wishart matrix

– Scaled Wishart matrix

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Spectrum Sensing

• Stieltjes transform

• Cubic polynomial

– β dimension ratio

– p SNR

– z Stieltjes argument

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Relay Channel

• Vector I/O model

• Mutual information

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Relay Channel

• Asymptotically

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Relay channel

• Product of

– Standard Wishart

– Scaled Wishart plus Identity

• Auxiliary variables

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Relay Channel

• Multiplicative Free Convolution

• Eta transform of M

– Change of variables in MP law

– Eta definition

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Relay Channel

• Quartic polynomial for Stieltjes transform and inversion formula

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Relay Channel

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Cochannel Interference

• Vector I/O model

• Mutual information

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Cochannel Interference

• Product of

– Standard Wishart

– Inverse of Scaled Wishart plus Identity

• Auxiliary variables

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Cochannel Interference

• Asymptotically

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Cochannel Interference

• Multiplicative Free Convolution

• Eta transform of M

– Change of variables in MP law

– Eta definition

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Cochannel Interference

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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

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• Questions?

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Random Matrix Theory Lecture 3 Free Probability Theory