Incollection,
Mean Field Theory of Communication
Abstract Book of the XXIII IUPAP International Conference on Statistical Physics, Genova, Italy, (9-13 July 2007)
Abstract
In this talk we will review statistical-mechanics approach
to problems in the field of wireless communication.
Emphasis is placed on describing how the concept of
mean field theory---reduction of a many-body problem
into a one-body problem---will be applied
to the problems of wireless communication.
We consider the generic vector channel
$\boldmath$y$=H\boldmath$x$+\boldmath$n$$,
where $\boldmath$x$$ and $\boldmath$y$$ are
$K$- and $N$-dimensional vectors representing channel input
and output, respectively.
The $N$-dimensional vector $\boldmath$n$$ denotes
channel noise with independent entries.
The $N$-by-$K$ matrix $H$, the channel matrix,
maps the channel input $\boldmath$x$$
to the channel output $\boldmath$y$$
if the channel is noise free.
The generic vector channel can be regarded as a mathematical model
of various wireless communication schemes.
They include the so-called code-division multiple-access (CDMA) channel
and the so-called multiple-input multiple-output (MIMO) channel,
which have been adopted in a wide range of commercial wireless systems,
such as the so-called third-generation (3G) mobile phone standards,
IEEE 802.11 wireless LAN standards, Bluetooth,
the Global Positioning System (GPS), and so on.
The receiver has to estimate the channel input $\boldmath$x$$
from the channel output $\boldmath$y$$,
using knowledge of the channel matrix $H$ and
of statistics of the channel noise $\boldmath$n$$.
In various applications the channel matrix $H$ should be
regarded as random, and consequently,
statistical formulation of the estimation problem
reveals that the posterior distribution of the channel input
$\boldmath$x$$ given the channel output $\boldmath$y$$
induces random correlations among elements of $\boldmath$x$$.
That is, the resulting system has frustration,
so that statistical mechanics comes into play in understanding
the estimation problem as an example of disordered systems.
One can perform analysis of the system with the replica method,
assuming independent entries of the channel matrix $H$
and the ``thermodynamic'' limit.
The result obtained with the assumption of replica symmetry
can be interpreted as follows: The random vector channel
is decomposed into a bundle of $K$ scalar Gaussian channels.
This is nothing other than a mean field theory, but what is
interesting here is that the one-body problem itself, obtained by
reduction from the original many-body problem, now makes sense
as a communication system, bearing the expression of
Gaussian channels.
Guo and Verdú (2005) called this property
the ``decoupling principle,'' and it turns out that
the concept has a significantly wide applicability in wireless
communication.
We will give an example of a MIMO-CDMA channel,
where the one-body problem after reduction is a bundle of CDMA channels,
each of which can further be reduced to a bundle of Gaussian channels.
Tags
- code-division
- communication
- field
- mean
- method
- multiple-access
- multiple-input
- multiple-output
- replica
- statphys23
- theory
- topic-11
- wireless
