\chapter{Results}\label{chap:results}

\section{Fields Around the Body}

\subsection{Dipole Analysis}

In the dipole simplification, the person is represented by a point of
charge $q$, at a distance of $h$ from the oppositely charged ground
image.  For large distances $d >> h$, the field falls off as $d^3$, as
with any dipole.  In the near field, however, the field of the dipole
is better evaluated as the field of the two distinct charges.  The
receiver is considered to be a distance $d$ from the body, where $d <<
h$.  The two cases of the receiver being placed between the body an
the ground image (Equation \ref{eqn:dipole1}) and perpendicular to the
dipole moment, near the body (Equation \ref{eqn:dipole2} are shown.

\begin{equation}\label{eqn:dipole1}
|\vec{E}| = q(\frac{1}{d^2} - \frac{1}{(h-d)^2})
% \vec{E} = \frac{q(\cos(\theta)\hat{y} + \sin(\theta)\hat{x})}{d^2} + \frac{q(\cos(\theta)\hat{y} - \sin(\theta)\hat{x})}{(s-d\cos(\theta))^2}
\end{equation}

\begin{equation}\label{eqn:dipole2}
|\vec{E}| = q\sqrt{\frac{1}{d^4} + \frac{1}{s^4}  }
\end{equation}

In both cases, the receiver is optimally positioned as closely to the
body charge as possible.

\subsection{Full Simulation}


A PAN receiver essentially measures the modulation of the electric field
in the vicinity of the body.  As the electric field lines are
necessarily perpendicular to the surface of the body (a conductor),
the best geometries place one electrode close to the body and the
other the maximal reasonable distance away from the body perpendicular
to the bodies surface.

\par

The essential tradeoff for signal strength lies in the relative
coupling of the ground electrode to the ground, and to the body.  When
the electrodes are directly between the body and ground, as in the
receiver in configuration {\bf A} in Figure \ref{pan:geom}, the ground
electrode is better coupled to the body than it is to the ground.  In
geometries such as those in configuration {\bf B}, however, the
coupling of the ground electrode to ground ($C_{26}$) gets weaker and
weaker as it is moved up the body, while the ground electrode to body
coupling ($C_{16}$) stays nearly the same.

\begin{figure}\label{pan:simple}
\begin{center}
\mbox{\epsfysize=3in \epsfbox{simple.eps}}
\end{center}
\caption{Simplified View of PAN}
\end{figure}

\begin{table}\label{table:heights}
\begin{center}
\begin{tabular}{||r|r|r|r||}\hline
{\bf Height}& {\bf $V_r (\frac{\rm mV}{\rm V})$}& {\bf $C_{16}$ (pF)}& {\bf $C_{26}$ (pF)}\\
\hline
10 cm& 7.37& 3.66& 6.13\\\hline
20 cm& 7.72& 3.99& 5.45\\\hline
30 cm& 7.87& 4.29& 4.90\\\hline
40 cm& 7.87& 4.58& 4.44\\\hline
50 cm& 7.76& 4.83& 4.04\\\hline
60 cm& 7.58& 5.06& 3.70\\\hline
70 cm& 7.35& 5.26& 3.40\\\hline
80 cm& 7.11& 5.43& 3.13\\\hline
90 cm& 6.84& 5.57& 2.90\\\hline
100 cm& 6.59& 5.66& 2.69\\\hline
110 cm& 6.33& 5.72& 2.50\\\hline
120 cm& 6.10& 5.73& 2.32\\\hline
130 cm& 5.87& 5.69& 2.17\\\hline
140 cm& 5.66& 5.59& 2.02\\\hline
150 cm& 5.55& 5.23& 1.80\\\hline
160 cm& 5.43& 4.84& 1.58\\\hline
170 cm& 5.33& 4.42& 1.38\\\hline
\end{tabular}
\end{center}
\caption{Simulation results at various receiver heights}
\end{table}

In the simplified version of PAN shown in Figure \ref{pan:simple}, the
relationship between changes in height of the receiver and the values
of $C_{16}$ and $C_{26}$, the above mentioned capacitances, and $V_r$,
the coefficient corresponding to the number of millivolts at the
receiver per volt at the transmitter, are fundamental to understanding
the geometry constraints of PAN.

\par


The results of a full simulation with various receiver heights appear
in Table \ref{table:heights}.  The simulation was done with a 170cm
high, 40cm radius cylinder approximating a person.  The transmit
electrodes were beneath the person and the receive were beside them at
various heights.

\par

\begin{figure}\label{graph:sig}
\begin{center}
\mbox{\epsfysize=3in \epsfbox{graph.eps}}
\end{center}

\caption{Signal Strength vs. Receiver Height}
\end{figure}

\par

As seen in Figure \ref{graph:sig}, receiver voltage changes linearly
with height.  The curving of the line at the beginning and end of
graph are due to edge effects near the persons feet and head.
Specifically, near the feet, both receive electrodes become very
highly ground coupled.  This decreses the necassary break in symmetry
at the receiver, and causes a decrease in signal strength.

\par

A foot to foot configuration is far better, with a value of $V_r$ on
the order of $50 \frac{\rm mV}{\rm V}$.

\section{Noise Characterization}

As anticipated, the noise is dominated by amplifier noise, with
possible substantial contributions from external interference.
Thermal noise is sufficiently small it is possibly unmeasurable in
practice.

\par

The amplifier noise depends substantially on the quality of the op-amp
itself.  The noise density from the amplifier varies from 30 to 150
$\frac{{\rm nV}}{\sqrt{Hz}}$ for reasonable op-amps.  External
interference, even in a noisy lab environment typically remains below
50 $\frac{{\rm nV}}{\sqrt{Hz}}$.  Therefore, the total noise at the
receiver is between 30 $\frac{{\rm nV}}{\sqrt{Hz}}$ (low noise op-amp
and negligible interference) and 200 $\frac{{\rm nV}}{\sqrt{Hz}}$
(generic op-amp and electrically noisy environment).

\section{Bandwidth}

The bandwidth of the electric field around the human body is
fundamentally limited by the effective low pass filter caused by the
capacitance between the receiver and ground.  This limit however, is
very high due to the miniscule capacitances involved, and allows for a
bandwidth of many megahertz.

\par

This bandwidth, however, is not realistic for PAN, because as the
frequency increases, the body begins to exhibit antenna-like
properties and radiate, which is in direct conflict with the goal of
PAN.  In practice, the bandwidth is limited by the bandwidth of a filter on the front
end of the receiver which is typically a few kilohertz.

\par

In practice, the bandwidth could likely be usefully pushed up into the
tens or perhaps hundreds of kilohertz.  Above that, pickup of high
frequency interference is likely to become a serious problem as well as
the problem of the body behaving as an antenna.

\section{Channel Capacity}

The channel capacity, as seen in Equation \ref{eqn:channelcap}, and
repeated here, can now be calculated.

\begin{equation}
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}

Using the estimated values for noise, and assuming a transmitter
voltage on the order of 10 V, a channel capacity can be calculated for
a foot-to-foot configuration or as a function of receiver height.  For
the height dependent case, the channel capacity decreases roughly
linearly with height.

\par

Using the most current experimental hardware (not a low noise op-amp)
with a front end low pass filter to limit the bandwidth to 1.2 kHz, and
an interference estimate of white noise at 50 $\frac{{\rm
nV}}{\sqrt{Hz}}$, the channel capacity is roughly 30.8 kilobits per
second (kbps).  A low noise op-amp would increase this to roughly 33.9
kbps.  The thermal noise has almost no impact -- less than 10 bits per
second decrease relative to an ideal system without thermal noise.
Using a wider bandwidth is the most effective for increasing channel
capacity.  Spreading the signal to a 10 kHz band, with a low noise
op-amp yields a channel capacity of 252 kbps.  At 100kHz, the capacity
is in excess of 2000 kbps.

\section{Conclusion}

As expected, foot-to-foot configurations far outperform other
configurations.  Also, amplifier noise and interference are the
primary sources of noise in the system.  Finally, channel capacities
are capable of being substantially boosted by improvements in coding,
but more substantial gains can be made by increasing the bandwidth
over which the PAN signal is spread.


