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Chautauqua Park, Boulder, Colorado,July 17, 2008

Workshop on Random Matrix Theory and Wireless Communications

Bridging the Gaps:Free Probability and Channel Capacity

- Antonia Tulino
- Università degli Studi di Napoli

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Linear Vector Channel

noise=AWGN+interference

N-dimensional

output

K-dimensional

input

(NK)channel matrix

- Variety of communication problems by simply reinterpreting K, N, and H
- Fading
- Wideband
- Multiuser
- Multiantenna

Role of the Singular Values

Minumum Mean-Square Error (MMSE) :

- Independent and Identically distributed entries
- Separable Correlation Model
- UIU-Model with independent arbitrary distrbuted entries
- with which is uniformly distributed over the manifold of complex matrices such that

Flat Fading & Deterministic ISI:

Random erasure mechanisms:

- link congestion/failure (networks)
- cellular system with unreliable wired infrastructure
- impulse noise (DSL)
- faulty transducers (sensor networks)

distribution in [0, 1]d,

d-Fold Vandermonde Matrix- Sensor networks
- Multiantenna multiuser communications
- Detection of distributed Targets

Flat Fading & Deterministic ISI:

Formulation

= asymptotically circulant matrix (stationary input (with PSD )

= asymptotically circulant matrix

Grenander-Szego

theorem

Eigenvalues ofS

the water levelis chosen so that:

where is the waterfilling input power spectral density given by:

Deterministic ISI & |Ai|=1Key Tool: Grenander-Szego theorem on the distribution of the eigenvalues of

large Toeplitz matrices

Deterministic ISI & Flat Fading

Key Question: The distribution of the eigenvalues of a large-dimensional random matrix:

- S = asymptotically circulant matrix
- A = random diagonal fading matrix

Ai =ei ={0,1}

Key Question: The distribution of the eigenvalues of a large-dimensional random matrix:

- S = asymptotically circulant matrix
- E = random 0-1 diagonal matrix

with X a nonnegative random variable whose distribution is while g is a nonnegative real number.

RANDOM MATRIX THEORY:- & Shannon-TransformThe - and Shannon-transform of an nonnegative definite random matrix , with asymptotic ESD

A. M. Tulino and S. Verdú“Random Matrices and Wireless Communications,”Foundations and Trends in Communications and Information Theory, vol. 1, no. 1, June 2004.

The Shannon transform and -transforms are related through:

where is defined by the fixed-point equation

RANDOM MATRIX THEORY:Shannon-Transform- Abe a nonnegative definite random matrix.

- is monotonically increasing with g
- which is the solution to the equation
- is monotonically decreasing with y

with power spectral

Flat Fading & Deterministic ISI:

Theorem:

The mutual information is:

with

with power spectral

Flat Fading & Deterministic ISI:

Theorem:

The mutual information is:

with

with power spectral

Flat Fading & Deterministic ISI:

Theorem:

The mutual information is:

with

with power spectral

Special case: Gaussian Erasure Channels

Theorem:

The mutual information is:

with

with power spectral

with :

Flat Fading & Deterministic ISI:

Let

Theorem:

The mutual information is:

Input Optimization

Theorem:

- be an random matrix
- such that

i-th column of

A. M. Tulino, A. Lozano and S. Verdú“Capacity-Achieving Input Covariance for Single-user Multi-Antenna Channels”,IEEE Trans. on Wireless Communications 2006

Input Optimization

Theorem:

The capacity-achieving input power spectral density is:

where

and is chosen so that

the waterfilling solutionfor g

the fading-free water level for g

Input OptimizationCorollary:

Effect of fading on the capacity-achieving input power spectral density = SNR penalty

with

k< 1regulates amount of water admitted

on each frequency tailoring the waterfilling

for no-fading to fading channels.

Proof: Key Ingredient

- We can replace S by it circulant asymptotic equivalent counterpart, =FLF†
- Let Q = EF, denote by qi the ith column of Q, and let

Proof:

Matrix inversion lemma:

Proof:

Lemma:

Asymptotics

- Low-power ( )
- High-power ( )

Asymptotics: High-SNR

At large SNR we can closely approximate it linearly need and S0

where

High-SNR dB offset

High-SNR slope

Asymptotics

- Sporadic Erasure (e!0)
- Sporadic Non-Erasure (e!1)

Memoryless noisy erasure channel

High SNR

where is the water level of the PSD that achieves

Low SNR

Asymptotics: Sporadic Erasures (e0)Theorem:

For any output power spectral densityand

Theorem:

For sporadic erasures:

Optimizingover with

with the maximum channel gain

Asymptotics: Sporadic Non-Erasures (e1)Theorem:

d-Fold Vandermonde Matrix

Diagonal matrix

(either random or deterministic)

with supported compact measure

Diagonal matrix

(either random or deterministic)

with supported compact measure

Summary

- Asymptotic distribution of A S A ---new result at the intersection of the asymptotic eigenvalue distribution of Toeplitz matrices and of random matrices ---
- The mutual information of a Channel with ISI and Fading.
- Optimality of waterfilling in the presence of fading known at the receiver.
- Easily computable asymptotic expressions in various regimes (low and high SNR)
- New result for d-fold Vandermond matrices and on their product with diagonal matrices