@@ -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

%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.

%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.

%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.