Thesis

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

diff --git a/thesis/appendices/tcs_noise.tex b/thesis/appendices/tcs_noise.tex
index 50f3c04..9b6b1e0 100644 (file)
--- a/thesis/appendices/tcs_noise.tex
+++ b/thesis/appendices/tcs_noise.tex
@@ -1,12 +1,12 @@
-\chapter*{Appendix - Effect of Noise on the TCS Curve}
+\section{Effect of Noise on the TCS Curve}
 
-Taking the derivative of discrete data is problematic. Using a centred difference finite derivative approximation:
+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}.
 
-However, if $f_s(x)$ is the result of sampling $f(x)$, with $\Delta f$ the uncertainty in a measurement:
+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} \\
@@ -14,11 +14,18 @@ However, if $f_s(x)$ is the result of sampling $f(x)$, with $\Delta f$ the uncer
 \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.
+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]