\chapter{Noise Model}

As described in Section \ref{model:noise}, the three sources of noise
for an intrabody signalling system are external interference, thermal
system noise, and amplification noise.  The first of these is highly
dependent on the particular environment in which it is used.  Used on
a body with multiple transmitters, the external interference would be
drastically higher than an isolated receiver far from other radiative
systems.  The thermal and amplification noise are easier to quantitatively
characterize.

\section{Interference}

Estimating interference magnitudes is necessarily an approximate task.
The possibility of an intentionally interfering source yields possible
noise values with essentially no upper limit, so this discussion will
focus on non-intentional sources of interference.

\par

Transmitters other than the one trying to be listened to are an
obvious source of interference.  The magnitude of this noise could be
expected to be roughly equal to the intended received signal strength.

\par

Other electronic devices are a likely source of interference as well.
Typical values measured on an electrode near a body (a typical PAN
configuration) in the lab (admittedly a noisy electrical environment)
in general did not exceed 50 $\frac{{\rm nV}}{\sqrt{Hz}}$.

\section{Thermal Noise}

Each effective capacitance has associated with it an effective noise
current ($i_n$) in parallel with a power of $\frac{1}{2} kT$.  Each of
these noise currents is zero centered and has a root mean square
amplitude dependent on the actual capacitance.  Equation
\ref{eqn:matrix} is then solved for each of the noise sources
independently.  Each of the noise currents across the pair of receiver
electrodes is then RMS summed:

\begin{equation}
i_{n_{tot}} = \sqrt{\sum_j (i_{n_j})^2}
\end{equation}

This final noise current can then be used with the receiver impedance
to get an mean square voltage amplitude.

\par

The resultant thermal noise is confirmed to be as expected, very
small.  A typical value for this mean square voltage is $10^{-18} {\rm
V}^2$.

\section{Amplification Noise}

The source noise generated by a feedback amplifier, from
\cite{horohill}, is a function of some noise parameters of the op-amp,
and the values of the feedback network.  In Equation
\ref{eqn:ampnoise}, $e_A^2$ is the noise voltage squared per hertz,
$e_n$ is the voltage noise from the amplifier, and $i_n$ is the
current noise.  $e_A$, for an inverting amplifier, is calculated as:

\begin{equation}\label{eqn:ampnoise}
\begin{array}{c}
i_A^2 = i_n^2 + \frac{4kT}{R_2}\\
e_A^2 = e_n^2 + R_1^2i_A^2
\end{array}
\end{equation}

Using values for a typical low noise op-amp ($i_n = .01 {\rm
pA}/\sqrt{{\rm Hz}}, e_n = 30 {\rm nV}/\sqrt{{\rm Hz}}$) at an
inverting gain of 10, the total input referred noise is approximately
30 $\frac{{\rm nV}}{\sqrt{{\rm Hz}}}$.  This represents roughly as
much noise power per hertz as there is thermal noise total.  In
practice, a higher noise op-amp was used which had a total input
referred noise voltage of approximately 150 $\frac{{\rm
nV}}{\sqrt{{\rm Hz}}}$.


