Commit aff2f93d authored by hannahm's avatar hannahm
Browse files

paper tidy up from old files

parent e4c074c4
\documentclass[main-lines.tex]{subfiles}
\begin{document}
The sensitivity plot for LIGO Hanford and LIGO Livingston is shown in figure .....
\begin{figure}
\begin{center}
\includegraphics[width=0.49\textwidth]{images/o2catalog-asd.pdf}
\end{center}
\caption{\label{fig:o2ASD}
Sensitivity plot for LIGO-Hanford (red) and LIGO-Livingston (blue) for a snapshot of O2 (see Ref.~\cite{GWOSC:2019, GWOSC:online}.
The inset shows an enlargement of the region around the $60\,{\rm Hz}$ noise line in each detector. }
\end{figure}
% the script for this plot is here on CIT
%/home/hannah.middleton/repositories/powerlines/powergridlines/plots/asd
The waterfall plot ,,,,,
\end{document}
\documentclass[main-lines.tex]{subfiles}
\begin{document}
Data from gravitational wave (GW) observatories such as Advanced LIGO and Advanced Virgo contain persistent noise spectral lines.
Noise lines pose a particular difficulty to seraches for continuous gravitaitonal waves (CWs); persistent, periodic gravitational wave signals which are expected to be emitted by rotating neutron stars.
The noise lines are often loud and can obscure a CW signal in the affected frequency bins.
Mitigation and removal of these lines is an area of active study.
Much effort has been made to identify the source of lines and, where ever possible, to mitigate them.
For example, in Ref.~\cite{CovasEtAl:2018}, noise lines are identified from blinking LEDs ..... and the effect eliminated by ,......
However it is not always possible to prevent the process causing these lines and in many cases the cause remains unidentified.
Another approach is to remove the effect of the line after data aquisition.
GW detectors can be affected by their environment and therefore have a wealth of physical environmental monitors (PEM) to record any environmental effects on the instrument and assess the quality of the data at any one time.
This information is also useful removing noise from the GW channel.
Recent work ....
In this study we investigate the prospects of adaptive noise cancellation~\cite{WidrowEtal:1975} (ANC) for line removal in GW data.
Similar to ...., we focus on the removal of the $60\,{\rm Hz}$ noise line, which is caused by interference of the United States power grid.
ANC can be applied to situations where a reference signal provides an independent measurement of the noise interference.
The PEM data recorded at each observatory provides an excellent opportunity to test this method in GW data.
In Sec.~\ref{} we describe the method of ANC used in this work.
In Sec.~\ref{} we discuss the behavoir of the power line in the GW data and the avalaible environmental monitors.
In .,..... apply to a toy problem.
In ..... apply to real data. ...
\end{document}
\documentclass[main-lines.tex]{subfiles}
\begin{document}
Adaptive noise cancellation provides a means of estimating a signal which has been corrupted by some interference or noise.
The method can be applied in the situation where there is a primary signal and a reference signal.
The primary signal contains the information of interest, plus some additional unwanted interference.
The reference signal contains a measurement of the interference which is correlated in some unknown way to the noise in the primary signal.
The ANC computes an estimate of the noise as it appears in the primary and subtracts it.
In this section, we describe the implementation of ANC used in this work.
The primary timeseries $d_i$ (evenly sampled at time steps $i = 0,\dots,N$) is known to contain an unwanted interference, which we label the `clutter' $c_i$.
The primary may also contain a signal of interest $h_i$, as well as additional noise $n_i$, so that
\begin{equation}
d_i = h_i + c_i + n_i\,.
\end{equation}
Our objective is to remove $c_i$ with the aid of a reference measurement $r_i$, 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 interferometer maybe extremely complex.
We use $r_i$ to construct an estimate $y_i$ of $c_i$ and remove it in the time domain.
The filtered timeseries is
\begin{eqnarray}
e_i &~=~& d_i - y_i \,, \\
&~\approx~& h_i + n_i \,.\\
\end{eqnarray}
The value of $y_i$ is the scalor product of the tap-input vector $\mathbf{u}$ and the tap-weight vector $\mathbf{w}$ defined as
\begin{eqnarray}
u_k = [r_k, r_{k-1}, \dots, r_{k-M+1}]\,,
w_? = [\dots]\,,
\end{eqnarray}
where $M$ is the order of the filter.
The vector $\mathbf{u}$ comes directly from the reference signal.
We now wish to compute $\mathbf{w}$ at each timestep.
\subsection{Adaptive Recursive Least Squares Method}
In this work we use the adaptive recursive least squares (ARLS) method to compute the estimated signal.
There are two input parameters, the order $M$ and $\lambda$, the meaning of which is discussed below.
Here we describe the algorithm as pseudo code.
\begin{enumerate}
\item Initialise zero value $M$-by-$1$ vectors $\mathbf{u(n=0)}$ and $\mathbf{w(n=0)}$ for the tap-input and tap-weights respectively for timestep $n=0$. .
\item Initialise empty $M$-by-$M$ matrix $P$, which is referred to as the inverse correlation matrix.
\item Read in the values of $d$ and $r$ at the current time $n$.
\begin{enumerate}
\item The tap inputs are updated as
\begin{equation}
u_{k=0} = r(n) \,,
\end{equation}
\item and the estimated signal subtracted,
\begin{equation}
e_n = d_n - \mathbf{w}\mathbf{u}.
\end{equation}
\item An $M$-by-$1$ vector $\mathbf{K}$ (the gain vector) is computed
\begin{equation}
K = { \mathbf{P} \mathbf{u} }{ \lambda + (\mathbf{u} \mathbf{P})\mathbf{u}}
\end{equation}
\item and the value of $\mathbf{P}$ updated
\begin{equation}
\mathbf{P} = \frac{1}{\lambda} ( \mathbf{I} - \mathbf{K}^T \mathbf{u} ) \mathbf{P}\,.
\end{equation}
\item The tap weights are updated
\begin{equation}
\mathbf{w(n+1)} = \mathbf{w(n)} + y(n)*\mathbf{K}
\end{equation}
\item and the elements of the tap inputs are shifted by one so that the next reference value takes the first position (see step 3a).
\end{enumerate}
\item Repeat step 3 until the $n=N$
\end{enumerate}
%A note on the pareamster
More intuitively, $\lambda$ can be thought of as the forgetting factor of the filter.
It is a positive constant, close to but less than $1$.
A value of $\lambda=1$ corresponds to infinite memory and the filter becomes an ordinary least squares method.
$\mathbf{P}$ is
\han{how to make the e}
\end{document}
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