Commit 1912b31f authored by Hannah Middleton's avatar Hannah Middleton
Browse files

adding notest on simulated data

parent 3331b8eb
......@@ -12,6 +12,7 @@ The primary timeseries $d_i$ (evenly sampled at time steps $i = 0,\dots,N$) is k
The primary also contains other noise $n_i$, and may contain a signal of interest $h_i$ (the GW), so that
\begin{equation}
d_i = h_i + c_i + n_i\,.
\label{eqn:primary}
\end{equation}
The objective is to remove $c_i$ with the aid of a reference measurement $r_i$ whilst leaving $h_i$ intact.
It is likely that the relationship between $r_i$ and $c_i$ is unknown; the coupling of the mains interference into the detector may be extremely complex.
......
The ingredients for producing simulated data for this work are a reference signal and a primary signal.
The primary contains three components as described by Eqn.~\ref{eqn:primary}: noise, a signal of interest, and clutter which is related to the reference in some unknown way.
In Section~\ref{sec:simData} we describ how we produce simulated data and we it with the ANC in Section~\ref{sec:simResults}.
\subsection{Generating simulated data}
\label{sec:simData}
The simulated reference signal is produced by summing $N$ weighted sinusoids.
\begin{eqnarray}
\label{eqn:simRef}
r_{\rm s}(t) &=& \sum_{m=0}^{m=N} \omega_m * \exp \left[ 2 i \pi \left(\fline + a_m + \phi_m \right) t \right]\,, \\
r_{\rm s}(t) &=& \Re r_c(t)\,,
\end{eqnarray}
%\han{from Andrew: Would they all be phased locked? Should there also be $\phi_m$ in each too? How is $r_i$ related to $c_i$ here? -> Equation}
where $\fline$ is the central frequency of the line (e.g., $60\,\Hz$ power line), $a_m$ is a small offset to $\fline$, $\phi_m$ is the phase, and $\omega_m$ is the amplitude of the summed sinusoid.
Equation~\ref{eqn:simRef} represents the auxillery channel data.
The simulated injected GW signal is
\begin{equation}
\label{eqn:simGW}
g_{rm s}(t) = h \sin \left(2 \pi \fgw t + \phi \right)\,,
\end{equation}
where $h$ is the GW strain amplitude as seen by the detector, $\fgw$ is the GW frequency and $\phi_{\rm gw}$ is the phase.
The next step is to construct the primary signal.
The reference signal given by Eqn~\ref{eqn:simRef} is not neccessarily the same as the clutter that appears in the primary data.
We therefore shift the phase by some value $\psi$ and scale up the amplitude by a factor of $A_{\rm c}$ so that
\begin{equation}
c_{\rm s}(t) = A_{\rm c} \Re \left\{ r_c(t)\exp(i\psi) \right\} \,.
\end{equation}
The resulting simulated primary signal is the altered reference signal, plus the injected GW signal (Eqn.\ref{eqn:simGW}) and Gaussian noise $n(t)$.
\begin{equation}
d_{\rm s}(t) = g(t) + c(t) + n(t)\,.
\end{equation}
\subsection{Simulated data results}
\label{sec:simResults}
%Some preliminary results are show in figure~\ref{fig:toyprobresults1}.
%We have an injected signal of interest at $4\Hz$ and a wandering unwanted signal centred around $5.2\Hz$.
%The full set of parameters are listed in Table~\ref{tab:toyprobparams}.
%\begin{table}
%\begin{tabular}{lll}
%\hline
%\hline
%data parameters & value & notes \\
%\hline
%$\fgw/\Hz$ & $4.0$ & \\
%$\fline/\Hz$ & $5.2$ & \\
%$h$ & $0.05$ & \\
%$N$ & $5$ & \\
%$a_n$ & $(0.03, 0.10, -0.16, 0.08, -0.07)$ & $G[0,0.1]$ \\
%$\omega_n$ & $(0.41, 0.04, 0.22, 0.57, 0.35)$ & $U[0,1]$ \\
%$\phi$ & $0.34$ & \\
%$\psi$ & $0.235$ & \\
%random seed & $81$ & \\
%range$/\secs$ & $0$--$80$ & \\
%timestep$/\secs$& $0.0001$ & \\
% & & \\
%\hline
%\hline
%ANC parameters & & \\
%\hline
%$\mu$ & $0.0001$ & \\
%$p$ (order) & $100$ & \\
%\hline
%\hline
%\end{tabular}
%\caption{ \label{tab:toyprobparams}
%Showing the parameters used for Figure~\ref{fig:toyprobresults1}.
%The top set of parameters correspond to the data (the injected signal of interest and unwanted %interference) and the bottom set are used for the adaptive noise cancelling filter.
%}
%\end{table}
%\begin{figure}
%\begin{center}
%\includegraphics[width=0.48\textwidth]{images/fft_order100_mu0p0001_freqGW4p0_freqLine5p2.png}
%\end{center}
%\caption{\label{fig:toyprobresults1}
%Top: the time series data containing a wandering line around $\fline=5.2\Hz$ and a signal of interest at $\fgw=4.0\Hz$ showing the measured primary signal (blue), the reference signal (red), the injected smaller signal (black) and the result of the ANC filter (yellow).
%Bottom: Fourier transform of the primary (blue), reference (red) and filtered (yellow) signals.
%The signal at $4.0\Hz$ is relatively untouched by the filtering, however the unwanted lines are reduced.}
%\end{figure}
%In Figure~\ref{fig:toyprobresults1}, the signal of interest can clearly be seen at $4\Hz$ and is quite separated in frequency from the unwanted wandering line at $5.2\Hz$.
%After filtering, the unwanted lines are reduced.
%The signal of interest is reduced by only a small amount.
%\begin{figure}
%\begin{center}
%\includegraphics[width=0.48\textwidth]{images/fft_order100_mu0p0001_freqGW5p9_freqLine5p2.png}
%\end{center}
%\caption{\label{fig:toyprobresults2}
%Identical to Figure~\ref{fig:toyprobresults1}, but here with $\fgw=5.9\Hz$.}
%\end{figure}
%\begin{figure}
%\begin{center}
%\includegraphics[width=0.48\textwidth]{images/fft_order100_mu0p0001_freqGW5p4_freqLine5p2.png}
%\end{center}
%\caption{\label{fig:toyprobresults3}
%Identical to Figure~\ref{fig:toyprobresults1}, but here with $\fgw=5.4\Hz$.}
%\end{figure}
%\begin{figure}
%\begin{center}
%\includegraphics[width=0.48\textwidth]{images/fft_order100_mu0p0001_freqGW5p3_freqLine5p2.png}
%\end{center}
%\caption{\label{fig:toyprobresults4}
%Identical to Figure~\ref{fig:toyprobresults1}, but here with $\fgw=5.3\Hz$.}
%\end{figure}
%Figures~\ref{fig:toyprobresults2},~\ref{fig:toyprobresults3} and ~\ref{fig:toyprobresults4} shows the same signal placed at progressively closer frequencies to the unwanted lines.
%When $\fgw=5.4\Hz$ (in Figure~\ref{fig:toyprobresults3}), the signal can still be seen in the filtered result.
%However when $\fgw=5.3\Hz$, overlapping with the unwanted signal it becomes difficult to judge whether the signal is still there in the filtered data.
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