THESIS

[matches/honours.git] / thesis / appendices / tcs_noise.tex

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+\chapter*{Appendix - Effect of Noise on the TCS Curve}
+
+Taking the derivative of discrete data is problematic. 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}.
+
+However, if $f_s(x)$ is the result of sampling $f(x)$, with $\Delta f$ the uncertainty in a measurement:
+\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}
+
+The problem may be fixed [dodged?] by increasing $h$ (in which case the resolution of the derivative is decreased dramatically), or application of smoothing averages (which also decrease the resolution, but not as much). \emph{Needs rephrasing}
+
+Smoothing of the sampled curve $f_s(x)$ (by application of a moving average) will reduce the deviation of points the smooth curve which best fits the data. 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.
+
+
+\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}
+