\chapter{Modeling}

The primary goal of this thesis is to provide a detailed model for the
channel capacity of a PAN channel.  This analysis will provide no
indication as to what encoding scheme will best acheive the predicted
maximum capacity.  The experimental work described breifly in section
\ref{intro:history} addresses some of this, but it seems likely that
there are encoding schemes which will be able to more effectively
approach the theoretical limit.


In order to evaluate the channel capacity of the body (which is taken
to be a gaussian channel), it is necassary to evaluate:
\begin{equation}\label{eqn:channelcap}
C = {\Delta}f\log_2\left(1+\frac{S}{N}\right)\\
= {\Delta}f\log_2\left(1+\frac{S}{N_{0}{\Delta}f}\right)\frac{{\rm bits}}{{\rm second}}
\end{equation}

In this equation, $C$ is the channel capacity, ${\Delta}f$ is the bandwidth (and
the sampling period is $\frac{1}{2{\Delta}f}$, $S$ is the signal
power, $N$ is the noise power, and $N_0$ is the noise power spectral
density.

\par

In order to calculate $S$, it is necassary to calculate the signal
strength at the receiver.  This will be highly dependent on geometry
and transmission strength.  In order to calculate $N$, a noise
estimate must be made, which will be dependent on geometry but is
expected to be dominated by amplification noise in the receiver.

\par

The signal calculation will be accomplished by an electrostatic model
of the body or bodies with the transmitter, receiver and ground, and a
circuit model of the interconnected capacitances. The noise
calculation will be done from first principles for the noise in the
body and empirical noise estimates for the receiver amplification.

\par

Finally, given these values it will be possible to obtain a reasonable
estimate for the channel capacity of the body given practical
amplification, as well as an estimate for the absolute upper bound,
given ideal amplification.

\section{Electrostatic Physics Model}\label{model:epm}


In the PAN system, the body is effectively modelled as an
equipotential, which has an induced charge caused by the potential on
the transmitter electrode.  This charge on the body is measurable by
the receiver electrodes.

\par

To effectively model this for arbitrary geometry, it is necassary to
numerically solve Poisson's equation:

\begin{equation}\label{eqn:poisson}
{\nabla}^2\phi = \frac{\rho}{\epsilon}
\end{equation}

This is accomplished by successive over-relaxation (SOR) of the
three-dimensional finite difference approximation of the diffusion
equation, which Poisson's equation is a steady-state solution of.
Doing this calculation gives the full values of the potential in the
vicinity of the objects in question.  Given $\phi$ and the definition
of electric field:

\begin{equation}
\vec{E} = -\nabla\phi
\end{equation}

it is possible to calculate the charge on each of the objects

\begin{equation}\label{eqn:charge}
\int_s\vec{E} \cdot d\vec{A} = \frac{q}{\epsilon}
\end{equation}

and from that, and the known potential between each pair of objects
($V = \phi_1 - \phi_2$), the capacitance can be calculated:

\begin{equation}
C = \frac{q}{V}
\end{equation}

\par

More details of the electrostatic model used
can be found in Chapter
\ref{chap:epm}.

\subsection{Dipole/Multipole Approximation}

It is also possible to model the electrostatics as a dipole.  That is,
assuming strong coupling between the body transmit electrode and the
body, and strong coupling between the ground transmit electrode and
the ground, allows the system to be modeled as a charged body with a
reflected, and inversely charged, body image in the ground.

\par

Given this dipole, potentials at various positions in the vicinity of
the body can be calculated.  See Chapter \ref{chap:results} for some
comparisons of the dipole approximation results with full SOR
calculations.

\par

A full multipole expansion can also be done to better approximate the
fields in the vicinity of the body.  This calculation is not done in this thesis.

\section{Circuit Model}

Given a capacitavely fully interconnected network of six nodes, the
circuit is best solved by a Guass-Jordan matrix solver.  The current
node equations are used.

\par

As a linear network, all six equations are unnecassary, as any five
equations fully specify the sixth.  One node must be arbitrarily set
to be the reference voltage, 0 for convenience.  Additionally, an
arbitrary current is run across the transmitter.  The linearity of the
system can be exploited to solve the system given any transmitter
voltage.

\section{Noise Model}\label{model:noise}

The entry of noise into the intrabody signalling system comes from
three sources:

\begin{itemize}

\item{Interference.}  Environmental noise, both background, other 
unintentionally interfering devices, and other intrabody signalling devices
all produce a highly variable amount of noise.

\item{Thermal, in body} noise, as contributed by the actual transmission of
the data through the body.  Each equivalent capacitor contributes
$\frac{1}{2}kT$ noise power.

\item{Amplification} noise.  Whatever amplification system is used in the 
receiver affects the total noise in the channel.

\end{itemize}

Amplification noise is expected to dominate, though clearly external
interference has potential to be significant, depending substantially
on the encoding scheme selected.
