... ... @@ -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.