1 \chapter*{Appendix - Effect of Noise on the TCS Curve}
3 Taking the derivative of discrete data is problematic. Using a centred difference finite derivative approximation:
5 \der{f}{x} &= \frac{f(x + h) - f(x - h)}{h} + O(h^2)
7 The accuraracy of this approximation increases as $h \to 0$\footnote{Ignoring any effects due to rounding of floating point numbers}.
9 However, if $f_s(x)$ is the result of sampling $f(x)$, with $\Delta f$ the uncertainty in a measurement:
11 f_s(x) &= f(x) \pm \Delta f \\
12 \der{f_s}{x} &\approx \der{f}{x} \\
13 &= \frac{f(x + h) - f(x - h)}{h} + O(h^2) \pm \frac{\Delta f}{h}
15 The uncertainty in the sampled derivative has a pole at $h = 0$.
17 {\emph Note: I now suspect that this is a major reason why Komolov has used Lock-in amplifiers}
19 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}
21 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.
26 \includegraphics[width=0.5\textwidth, angle=270]{figures/tcs/plots/siI.eps}
27 \caption{An unprocessed and smoothed I(E) curve for a Si sample.}
33 \includegraphics[width=0.5\textwidth, angle=270]{figures/tcs/plots/siI_tcs.eps}
34 \caption{The effect of smoothing the original I(E) curve on its derivative.}