Transcript Diapositiva 1 - University of Colorado Boulder

Workshop on Random Matrix Theory
and Wireless Communications
Bridging the Gaps:
Free Probability
and
Channel Capacity
Antonia Tulino
Università degli Studi di Napoli
Chautauqua Park, Boulder, Colorado, July 17, 2008
Linear Vector Channel
noise=AWGN+interference
N-dimensional
output
K-dimensional
input
(NK) channel matrix
Variety of communication problems by simply
reinterpreting K, N, and H
 Fading
 Wideband
 Multiuser
 Multiantenna
Role of the Singular Values
Mutual Information:
Ergodic case:case:
Non-Ergodic
Role of the Singular Values
Minumum Mean-Square Error (MMSE)
H-Model

entries

Independent and Identically distributed
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 ISI:
Flat
Fading
& Deterministic
Random erasure mechanisms:




link congestion/failure (networks)
cellular system with unreliable wired infrastructure
impulse noise (DSL)
faulty transducers (sensor networks)
d-Fold Vandermonde Matrix
Sensor networks
 Multiantenna multiuser co
 Detection of distributed T

i.i.d. with uniform
distribution in [0, 1]d,
Flat Fading & Deterministic ISI:
an i.i.d sequence
Formulation
= asymptotically circulant matrix (stationary input
(with PSD
)
= asymptotically circulant
Grenandermatrix
Szego
theorem
Eigenvalues of S
Deterministic ISI & |Ai|=1
where
is the waterfilling input power spectral
density given by:
the water level
is chosen so that:
Key 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 largedimensional random matrix:
 S = asymptotically circulant
matrix
 A = random diagonal fading
matrix
Ai =ei ={0,1}
Key Question: The distribution of the eigenvalues of a largedimensional random matrix:
 S = asymptotically circulant
matrix
 E = random 0-1 diagonal matrix
RANDOM MATRIX THEORY: - & Shannon-Transform
The - and Shannon-transform of an
nonnegative
definite random matrix , with asymptotic ESD
with X a nonnegative random variable whose distribution
is
while g is a nonnegative real number.
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.
RANDOM MATRIX THEORY: Shannon-Transform

A be a
nonnegative definite random matrix.
Theorem:
The Shannon transform and -transforms are related
through:
where
is defined by the fixed-point equation
Property of



is monotonically increasing with g
which is the solution to the equation
is monotonically decreasing with y
Theorem: -Transform
Theorem:
The -transform of
where
is
is the solution to:
Theorem: Shannon-Transform
Theorem:
The Shannon-transform of
is
where a and n are the solutions to:
Flat Fading & Deterministic ISI:
Stationary Gaussian
inputs
with power spectral
Theorem:
The mutual information is:
with
Flat Fading & Deterministic ISI:
Stationary Gaussian
inputs
with power spectral
Theorem:
The mutual information is:
with
Flat Fading & Deterministic ISI:
Stationary Gaussian
inputs
with power spectral
Theorem:
The mutual information is:
with
Special Case: No Fading


Special Case: Memoryless Channels


Special case: Gaussian Erasure Channels
Stationary Gaussian
inputs
with power spectral
Theorem:
The mutual information is:
with
Flat Fading & Deterministic ISI:
Stationary Gaussian
inputs
with power spectral
Theorem:
Let
The mutual information is:
with
:
Example n=200
•
n = 200
─
f xi g
H (f )
f ui g
Example n=1000
•
n = 1000
─
f xi g
H (f )
f ui g
Input Optimization
Theorem:
 be an

with
so that:
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
Input Optimization
Theorem:
The capacity-achieving input power spectral density is:
where
and
is chosen so that
Input Optimization
Corollary:
Effect of fading on the capacity-achieving input power
spectral density = SNR penalty
with
the waterfilling solution for g
the fading-free water level for g
k< 1 regulates amount of water
admitted
on each frequency tailoring the
waterfilling
for no-fading to fading channels.
H20
Theorem: -Transform
Theorem:
The -transform of
where
is
is the solution to:
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:
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
slope
High-SNR dB
offset
Asymptotics: High-SNR
Theorem:
Let
and
,
the generalized bandwidth,
Asymptotics


Sporadic Erasure (e !0)
Sporadic Non-Erasure (e !1)
Asymptotics: Sporadic Erasures (e0)
Theorem:
For any output power spectral density
Theorem:
For sporadic erasures:
and
Memoryless noisy erasure channel
Low SNR
High
SNR
where
is the water level of the PSD that achieves
Asymptotics: Sporadic Non-Erasures (e1)
Theorem:
Theorem:
Optimizing over
with
with
the maximum
channel gain
Bounds:
Theorem:
The mutual information rate is lower bounded by:
Equality
S(f) =1
Bounds:
Theorem:
The mutual information rate is upper bounded by:
d-Fold Vandermonde Matrix
Diagonal matrix
(either random or deterministic)
with supported compact measure
Diagonal matrix
(either random or deterministic)
with supported compact measure
d-Fold Vandermonde Matrix
Theorem:
The -transform of
The Shannon-Transformr is
is
d-Fold Vandermonde Matrix
Theorem:
The p-moment of
is:
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)