\section{Effect of Noise on the TCS Curve}

Taking the derivative of discrete data is problematic. Consider a function $f(x)$. Using a centred difference finite derivative approximation:
\begin{align*}
	\der{f}{x} &= \frac{f(x + h) - f(x - h)}{h} + O(h^2)
\end{align*}
The accuraracy of this approximation increases as $h \to 0$\footnote{Ignoring any effects due to rounding of floating point numbers}.

If $f_s(x)$ is the result of sampling $f(x)$, with $\Delta f$ the uncertainty in a measurement, then we can calculate the final uncertainty when finite differences are used to approximate $\der{f}{x}$:
\begin{align*}
	f_s(x) &= f(x) \pm \Delta f \\
	\der{f_s}{x} &\approx \der{f}{x} \\
	&= \frac{f(x + h) - f(x - h)}{h} + O(h^2) \pm \frac{\Delta f}{h}
\end{align*}
The uncertainty in the sampled derivative has a pole at $h = 0$.


{\emph Note: I now suspect that this is a major reason why Komolov has used Lock-in amplifiers}


Smoothing of the sampled points $f_s(x)$ (by application of a moving average) will reduce the deviation of points the smooth curve which best fits the data; We can think of the points $f_s(x)$ as sampling a \emph{different} function to $f(x)$, but with smaller uncertainties. Smoothing of the original sampled points removes fine structure.

The alternative is to increase $h$.

As shown in Figures \ref{siI.eps} and \ref{siI_tcs.eps}, smoothing of $f_s(x)$ has a far greater effect on the derivative of $f_s$ than on $f_s$ itself.

\emph{TODO: Calculate MSE for both curves}
\emph{TODO: Show curves created with large $h$}


\begin{figure}[H]
	\centering
	\includegraphics[width=0.5\textwidth, angle=270]{figures/tcs/plots/siI.eps}
	\caption{An unprocessed and smoothed I(E) curve for a Si sample.}
	\label{siI.eps}
\end{figure}

\begin{figure}[H]
	\centering
	\includegraphics[width=0.5\textwidth, angle=270]{figures/tcs/plots/siI_tcs.eps}
	\caption{The effect of smoothing the original I(E) curve on its derivative.}
	\label{siI_tcs.eps}
\end{figure}

