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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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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

Mutual Information:

Ergodic case:

Non-Ergodic case:

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

Gaussian Erasure Channels

Random erasure mechanisms:

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

i.i.d. with uniform

distribution in [0, 1]d,

- Sensor networks
- Multiantenna multiuser communications
- Detection of distributed Targets

an i.i.d sequence

= 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:

Key Tool: Grenander-Szego theorem on the distribution of the eigenvalues of

large Toeplitz matrices

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

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

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.

The - 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.

Theorem:

The Shannon transform and -transforms are related through:

where is defined by the fixed-point equation

- Abe a nonnegative definite random matrix.

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

Theorem:

The -transform of is

where is the solution to:

Theorem:

The Shannon-transform of is

where a and n are the solutions to:

Stationary Gaussian inputs

with power spectral

Flat Fading & Deterministic ISI:

Theorem:

The mutual information is:

with

Stationary Gaussian inputs

with power spectral

Flat Fading & Deterministic ISI:

Theorem:

The mutual information is:

with

Stationary Gaussian inputs

with power spectral

Flat Fading & Deterministic ISI:

Theorem:

The mutual information is:

with

Stationary Gaussian inputs

with power spectral

Special case: Gaussian Erasure Channels

Theorem:

The mutual information is:

with

Stationary Gaussian inputs

with power spectral

with :

Flat Fading & Deterministic ISI:

Let

Theorem:

The mutual information is:

•

─

n = 200

•

n = 1000

─

with so that:

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

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

Corollary:

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.

Theorem:

The -transform of is

where is the solution to:

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

Matrix inversion lemma:

Lemma:

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

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

where

High-SNR dB offset

High-SNR slope

Theorem:

Let ,

and the generalized bandwidth,

- 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

Theorem:

For any output power spectral densityand

Theorem:

For sporadic erasures:

Theorem:

Optimizingover with

with the maximum channel gain

Theorem:

S(f) =1

Theorem:

The mutual information rate is lower bounded by:

Equality

Theorem:

The mutual information rate is upper bounded by:

Diagonal matrix

(either random or deterministic)

with supported compact measure

Diagonal matrix

(either random or deterministic)

with supported compact measure

Theorem:

The -transform of is

The Shannon-Transformr is

Theorem:

The p-moment of is:

- 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