text:
\section{Introduction}
The probe of the Epoch of Reionization (EoR) remains one of the outstanding
aims of modern cosmology. In the past decade, intriguing details have emerged
about this epoch from a host of cosmological observables. Gunn-Peterson (GP)
tests on spectra of quasi-stellar objects (QSOs) ((\citet{Fan et al}) in the redshift range $5.7< z <6.3$ suggest that the universe
as making a transition to full reionization during this period. On the
other hand, Cosmic microwave background radiation (CMBR) temperature and polarization anisotropy measurements (\citet{Komatsu et al}, \citet{Planck 2015}) suggest that the universe might have
been fully ionized in a redshift range $8 < z < 10$ (\citet{Planck 2016}). Both these observables
have their strengths and weaknesses. The GP test, based on Lyman-$\alpha$
absorption, is not able to distinguish between a fully neutral medium from
the one ionized to one part  in a thousand. CMBR anisotropies, based on
photon scattering off free electrons, are  sensitive to the integrated
Thompson scattering optical depth and therefore cannot reliably
construct the  tomography of the reionization epoch.

In the recent past, major experimental efforts have been undertaken to
study the EoR  in  redshifted 21-cm line emission from the epoch. In particular,
many radio interferometers in frequency range $80 < \nu < 300 \, \rm MHz$
are currently operational that specifically  aim to detect the EoR, for example, Low Frequency Array (LOFAR, \citet{LOFAR}), 21 Centimeter Array (21CMA, \citet{21cma}), Giant Meterwave Radio Telescope (GMRT, \citet{GMRT}), Donald C. Backer Precision Array for Probing the Epoch of Reionization (PAPER, \citet{PAPER}), and the Murchison Widefield Array (MWA, \citet{MWA, Lonsdale et al, Bowman et al 2009}). Even though
the detection of redshifted HI line from the EoR remains the most direct
and,  possibly the most promising, way to delineate the details of the epoch,
this method is beset with its own set of issues. First, unlike
 CMBR anisotropies, the theoretical modelling of the HI signal from the
 EoR is considerably harder principally  owing to uncertainty in the nature of ionizing sources and the details of their formation and evolution. Second,
the signal is expected to be weak with brightness temperature $T_{\rm B} \simeq 10 \, \rm mK$. Many hundred hours of observation is needed to detect such a
signal with  current interferometers, with the attendant  complication of
maintaining  instrumental stability for  such long periods.  Third, the foreground are expected to be many orders of magnitude larger than the signal (for details on the three issues see e.g.  \citet{Morales Wyithe, Morales Hewitt, Furlanetto et al, Barkana 2001, Zaroubi 2013} and references therein).

The use of  radio interferometers
 to  estimate   the underlying power spectra has been successfully
employed for CMB data analysis (\citet{Hobson CMB}). This method has also
been suggested as a possible probe of the intensity correlations  of the
redshifted HI line, including from the EoR (\citet{Bharadwaj and Sethi, Datta et al, Bharadwaj and Ali}).

Many different approaches have been discussed to detect the HI signal in the
presence of dominant foregrounds \citet{Hazelton foreground, Jelic 2008, Harker 2009, Liu 2011, Morales 2012, Trott 2012, Dillon 2013, Delay 3}. They are all based on the expectation
that  foregrounds are  smooth in frequency space  as they arise from continuum
emission, e.g. Synchrotron radiation,  in both our Galaxy and extra-galactic
sources. On the other hand the HI signal has significant structure in
 the frequency space. It is conceivable that all these sources, both point and diffuse,
can be subtracted from the images, leaving behind the HI signal and Gaussian
noise, and LOFAR partly relies upon this technique (\citet{Chapman 2012, Chapman 2013}).
 Another possible method is based on the isolation of foregrounds
from the HI signal using power spectrum of the observed signal in conjugate space to the observed frequency (\citet{Parsons 2012a, Delay 1, Pober PAPER, Thyagrajan et al}).  Variants of this `delay space' (\citet{Parsons 2012a, Delay 1}) method  are  particularly relevant for interferometers such as MWA that have low angular resolution and have been
used extensively for the analysis of PAPER data. In this approach the observed interferometric data---visibilities for each antenna pair as a function of
frequency---is Fourier transformed along the frequency axis. The Fourier conjugate variable  effectively captures signal delay between  antenna pairs, which  allows one to isolate foregrounds.  In  the context 3d HI power spectrum this variable can  be related to  cosmological distance along line of sight (for details of this  mathematical correspondence see  \citet{Delay 1, Delay 2}). The `delay spectrum' constructed from `delay space' approach can be used to recover the  cosmological 3d  HI power spectrum. In this approach,    one deals  with visibilities directly, which are primary data products of  radio interferometers.

In this paper we propose a new method based on the delay space approach  to
extract the power spectrum of the HI signal in the presence of noise and
foregrounds. Our method  is based on  modelling the HI signal taking
into account the impact of  the $w$-term (arising from  non-coplanarity of the array, \citet{W projection}) and the distortion of  intensity pattern
during a tracking run. The information from  the HI signal
is used as weights  to cross-correlate the measured visibilities.  The proposed
method (`Delay Spectrum with Imaging Arrays (DSIA)')
 is a general method  applicable for tracking  with
  radio interferometers with wide  primary beams and  arbitrary array configuration  (e.g. MWA, LOFAR) and can also be
applied to interferometers with redundant baselines.  We apply the proposed
 method to analyse 3~hours of  MWA data  and compare our
results with noise and foreground simulations.

The outline of this paper is as follows. In the next
section, we describe the method of visibility correlation using
delay space  approach in detail. In particular,
this method is applied to the HI signal. In section~3, we describe  the
 MWA data and the initial analysis of this data based on the publicly-available image processing
software: Common Astronomical Software Applications (CASA). In section~4   the pipeline to extract the HI
power spectrum is discussed. In section~5, the results from MWA data are
 discussed and compared with  simulations of foregrounds
and noise. In the final section, we summarize our results and indicate possible
future directions.

Throughout this paper, we have used the Planck+WP best fit values of
cosmological parameters: $\Omega_{\rm m0}=0.3183$,
$\Omega_{\rm \Lambda0}=0.6817$ , $\Omega_{\rm b0}\,h^2=0.02203$,
$h=0.6704$, $\sigma_8=0.8347$, and $n_{\rm s}=0.9619$
\citet{Planck 2015, Planck 2013}.

\section{HI signal and its correlations}

\label{sec:h1signal}

In this section we study the HI signal using visibility correlations in
delay space. Using
our formulation we derive, in addition to new results,
 many results already known in the literature. The main new results
are: the impact of $w$-term  (subsection~2.1) and
changing intensity pattern in a tracking run  (subsection~2.2) on the
HI correlations.  Our results are valid for any radio interferometer
but our aim here is to underline their applicability  to MWA.

The most important inputs from MWA array configuration for our study are:
(a) the MWA primary beam and (b) the bandwidth. Other properties of the MWA array that have a bearing on our analysis are the non-coplanarity of the array
 and its
baseline distribution.

In delay space, spectrally smooth foregrounds lend themselves to ready
interpretation.  It can be shown that  visibilities computed in delay
space allow isolation of such foregrounds from the regions dominated
by the EoR signal and noise  (e.g. see \citet{wedge 1, wedge 2, Parsons 2009, Parsons 2012a, Delay 1, Delay 2, Liu 2014b, Delay 3, Thyagrajan et al, Delay 4}).
This can be achieved by Fourier transforming the raw visibilities in frequency space.
However, being 3-dimensional and  statistical in nature, the  properties of the HI signal can
only be inferred  by correlating the observed visibilities. Our approach, which is
based on visibility correlations in delay space,   allows us to develop a unified method  to deal with both the HI
signal and the foregrounds, which  are discussed in Appendix~\ref{sec:foregrounds}.

Radio interferometers measure the spatial correlation of the electric fields from the sky, the visibility $V_\nu({\bf u}_\nu)$:
\begin{equation}
 V_\nu({\bf u}_\nu)=\int A_\nu(\vec{\theta}) I_\nu (\vec{\theta})e^{-i2\pi {\bf u}_\nu \cdot{\vec{\theta}}} d^2\theta \label{visibility}
\end{equation}
Here ${\bf u}_\nu=(u_\nu,v_\nu)$ is the distance vector between the antennas of the interferometer (also called baseline vector) in units of wavelength, $\vec{\theta}$ denotes the position vector on the sky plane (which can be be expressed
as a pair of direction cosines $\{l,m\}$)  and $\nu$ is the frequency of observation.

We neglect the impact of $w$-term in this section in writing the relation
between the visibility and specific intensity in Eq.~(\ref{visibility}). The $w$-term arises from
non-coplanarity of the interferometric array and changes as the   interferometer
tracks a region.
In section~\ref{sec:h1sigwterm} we show  that the inclusion of $w$-term causes an effective shrinking of the primary beam.   For our study  we  calculate how  the HI signal is affected owing  to  non-zero $w$  as compared to $w=0$ case.

We explicitly express the frequency dependence of all the quantities. These quantities are: the background
specific intensity $\bar{I_\nu}$, the primary beam $A_\nu(\vec\theta)$ and the
baseline ${\bf u}_\nu$.  In terms of brightness temperature $T_B$, $I_\nu (\vec{\theta}) = 2kT_{B}(\vec{\theta})/\lambda^2$.

 The sky intensity can be decomposed as:
\begin{equation}
 I_\nu(\vec\theta)=\bar{I}_\nu + \Delta I_\nu(\vec\theta)
\end{equation}
where $\bar{I}_\nu$ and $\Delta I_\nu(\vec\theta)$ are the isotropic and fluctuating parts of the intensity distribution. Since the isotropic component does not contribute to interferometric measurement \footnote{However some methods have been discussed in recent literatures to extract monopole signal from interferometric measurements (\citet{Presley 2015, Singh 2015}).}, the visibility recorded at frequency $\nu$ can be written as:
\begin{equation}
 V_\nu({\bf u}_\nu)=\int A_\nu (\vec\theta) \Delta I_\nu(\vec\theta)e^{-i2\pi\vec\theta.{\bf u}_\nu} d^2\theta
 \label{vis_a}
\end{equation}

The measured visibility receives contributions from the redshifted HI line,
the foregrounds, and the noise.

For the HI signal the observed intensity fluctuations can be
related to the HI perturbations in Fourier space, $\delta_{\rm HI}({\bf k})$, as:
\begin{equation}
 \Delta I_\nu(\vec\theta)= \bar{I}_\nu\int\frac{d^3k}{(2\pi)^3}\delta_{\rm HI}({\bf k})
 e^{i{\bf k}\cdot {\bf r}}
 \label{deli_a}
\end{equation}
Here ${\bf r} =\{\vec\theta,r_\nu\}$ specifies the three-dimensional position of
the the HI emission; $r_\nu$ is  the coordinate distance to the point of observation: $r_\nu = \int dz/H(z)$ with the limit of this integral extending from zero to redshift  $z = \nu_e/\nu_0 - 1$.
$\delta_{\rm HI}({\bf k})$  comprises of many physical effects: density fluctuations, ionization inhomogeneity, density-ionization fraction cross-correlation, etc. (\citet{Furlanetto et al, Zaldarriaga 2004}). Together this can be expressed as:
\begin{equation}
 \delta_{\rm HI}({\bf k})=\beta_b\delta_b+\beta_x\delta_x+\beta_\alpha\delta_\alpha+\beta_T\delta_T-\delta_{\partial v}
\end{equation}
Each quantity in the above equation corresponds to the fractional variation of a particular physical quantity:  $\delta_b$ refers to fluctuation in baryonic density,
$\delta_\alpha$ for the Ly$\alpha$ coupling coefficient $x_\alpha$, $\delta_x$ for the neutral fraction, $\delta_T$ for $T_K$, and
$\delta_{\partial v}$ for the line of sight peculiar velocity gradient. $\beta$ factors denote the expansion coefficients of the corresponding quantity (\citet{Furlanetto et al}).

Current experiments such as MWA, LOFAR and PAPER aim
statistical detection of the EoR signal. The quantities of interest here
are the correlation functions of the HI fluctuations. The most important
correlation function one seeks to detect in an EoR experiment
is  the power spectrum, $P_{\rm HI}$, defined as:
\begin{equation}
 \langle\delta_{\rm HI}^*({\bf k})\delta_{\rm HI}({\bf k'})\rangle=(2\pi)^3\delta^3({\bf k}-{\bf k'})P_{\rm HI}(k)
\label{powspec3d}
\end{equation}
 The HI power spectrum can be constructed from the correlation of the
observed visibilities.  Substituting the form of fluctuation $\Delta I_\nu$ from Eq.~(\ref{deli_a}) in the visibility expression (Eq.~(\ref{vis_a})), we get:
\begin{eqnarray}
 V_\nu({\bf u}_\nu)&=&\bar{I}_\nu\int\frac{d^3k}{(2\pi)^3}\delta_{\rm HI}({\vec {k}})e^{ir_\nu k_\parallel}
 \int  A_\nu(\vec\theta) \nonumber \\
 &&\times\exp\left[-2\pi i\left({\bf u}_\nu-\frac{{\bf k_\perp}r_\nu}{2\pi}\right)\ldotp\vec{\theta}\right]d^2\theta
\end{eqnarray}
Here we have decomposed the wave vector ${\bf k}$ as
components on the plane of the sky, ${\bf k}_\perp$ and along the line of
sight, $k_\parallel$. The integral over angles  is  the Fourier transform of the primary beam $A_\nu(\vec{\theta})$, which allows us to re-write this equation as:
\begin{equation}
 V_\nu({\bf u}_\nu)=\bar{I_\nu}\int\frac{d^3k}{(2\pi)^3}\delta_{\rm HI}({\bf k})
 e^{ir_\nu k_\parallel}a\left({\bf u}_\nu-\frac{{\bf k_\perp}r_\nu}{2\pi}\right)
\label{visdef}
\end{equation}
where
\begin{eqnarray}
a\left({\bf u}_\nu- \frac{{\bf k_\perp}r_\nu}{2\pi}\right)&& \equiv \int  A_\nu(\vec\theta) \nonumber \\
&&\times\exp\left[-2\pi i\left({\bf u}_\nu-\frac{{\bf k_\perp}r_\nu}{2\pi}\right)\ldotp\vec{\theta}\right]d^2\theta
\end{eqnarray}

Using Eq.~(\ref{powspec3d}), the visibility correlation function can
be computed:
\begin{eqnarray}
\langle V_\nu({\bf u}_\nu) V_{\nu'}^* &&({\bf u}'_{\nu'}) \rangle = \bar{I_\nu}^2\int\frac{d^3k}{(2\pi)^3}P_{\rm HI}(k) e^{i\Delta r_\nu k_\parallel}\nonumber \\
&&\times a\left({\bf u}_\nu-\frac{{\bf k_\perp}r_\nu}{2\pi}\right) a\left({\bf u'}_\nu-\frac{{\bf k_\perp}r_{\nu'}}{2\pi}\right)
\label{viscorrfreq}
\end{eqnarray}
Here $\Delta r_\nu = |r_{\nu'}-r_\nu|$ which for $|\nu'-\nu| \ll \nu$ can be simplified to: $\Delta r_\nu =r_\nu'|\nu'-\nu|$, where $r_\nu' = |dr_\nu/d\nu|$.
Eq.~(\ref{viscorrfreq}) gives the correlation of the HI signal in three dimensions in which the two coordinates ${\bf u}$ correspond to  Fourier components
of the HI signal while the third $\nu$ refers to  the coordinate of
the fluctuation in position,  $r_\nu$, space  (\citet{Bharadwaj and Sethi}).

To isolate the impact of foregrounds and obtain regions dominated by
the HI signal  and the noise (`EoR window'), we compute the
the visibilities in delay space (\citet{Parsons 2009, Parsons 2012a, Delay 1, PAPER, Delay 2}):
\begin{equation}
V_\tau({\bf u}) =  \int \exp(i2\pi \tau \nu) V_\nu({\bf u}_\nu) d\nu
\label{deftau}
\end{equation}
Here $\tau$, the conjugate variable of $\nu$, defines the relevant variable
in delay space. The delay space approach can be applied to
 data to isolate spectrally smooth foregrounds; we discuss the delay space approach as applied to
such foregrounds in Appendix~A.  In Eq.~(\ref{deftau}), we have suppressed the frequency dependence of the baseline on
the LHS as  the frequency dependence of all the quantities has been integrated.
The baseline vector can be expressed as: ${\bf u}_\nu = {\bf u}_{\nu_0} \nu/\nu_0$,where $\nu_0$ is some fixed frequency that lies within  the bandwidth. On the
LHS of Eq.~(\ref{deftau}), the frequency independent baselines,  ${\bf u} \equiv {\bf u}_{\nu_0}$. Throughout this paper, we  assume: $\nu_0 = 154 \, \rm MHz$,  the central frequency of the bandwidth we use for MWA data analysis.

The autocorrelation of $V_\tau({\bf u})$ can be written as:
\begin{eqnarray}
&&\langle V_\tau({\bf u}) V_\tau ^* ({\bf u'}) \rangle  =  \int  \int d\nu d\nu'  \bar{I_\nu}\bar{I_{\nu'}} \int {d^3k \over (2\pi)^3} P_{\rm HI}(k)\nonumber \\
&\times & \exp\left(i2\pi\tau(\nu'-\nu)+i(k_\parallel(r_{\nu'}-r_\nu) \right) \nonumber \\
& \times &  \int d^2\theta  \exp\left(i2\pi\vec\theta\cdot\left({\bf u}_\nu-\frac{{\bf k_\perp}r_\nu}{2\pi}\right) \right) A_\nu(\vec\theta)\nonumber \\
&\times &\int d^2\theta' \exp\left(- i2\pi\vec\theta'\cdot\left({\bf u'}_{\nu'}-\frac{{\bf k_\perp}r_{\nu'}}{2\pi}\right)\right) A_{\nu'}(\vec\theta')
\label{etacorrela}
\end{eqnarray}
 To make further progress, frequency
dependent quantities are Taylor expanded. For baselines, this is
a straightforward  re-expression of the baseline vector as the vector is
linear in frequency:   ${\bf u'}_{\nu'} = {\bf u'}_\nu +\Delta\nu d{\bf u'}_\nu/d\nu$ where $\Delta\nu = \nu'-\nu$. It should be  noted that $d{\bf u'}_\nu/d\nu$ is the physical baseline length measured in the  units of time.

After the Taylor expansion of relevant quantities, $r_\nu$ and
${\bf u}_\nu$, we obtain:
\begin{eqnarray}
&&\langle V_\tau({\bf u}) V_\tau ^* ({\bf u'}) \rangle  =  \int  \int d\nu d\nu'  \bar{I_\nu}\bar{I_{\nu'}}   \int {d^3k \over (2\pi)^3} P_{\rm HI}(k)\nonumber \\
&\times & \int d^2\theta  \exp\left(i2\pi\vec\theta\cdot\left({\bf u}_\nu-\frac{{\bf k_\perp}r_\nu}{2\pi}\right) \right) A_\nu(\vec\theta) \nonumber \\
& \times & \int d^2\theta' \exp\left(- i2\pi\vec\theta'\cdot\left({\bf u'}_{\nu}-\frac{{\bf k_\perp}r_{\nu}}{2\pi}\right)\right) A_{\nu'}(\vec\theta')\nonumber \\
&\times & \exp\left[i\Delta\nu \left(2\pi\tau+k_\parallel dr_\nu/d\nu + d{\bf u'}_\nu/d\nu\cdot\vec\theta'\right. \right. \nonumber \\
&+ &  \left.\left.   dr_\nu/d\nu  {\bf k}_\perp\cdot\vec\theta'/(2\pi)  \right)\right]
\label{etacorrela12}
\end{eqnarray}
Here $\Delta\nu = \nu'-\nu$;  all the quantities in Eq.~(\ref{etacorrela12})
 have been written
as explicit functions of $\nu$ and $\Delta\nu$. This allows us to simplify the
integral further by making the coordinate transform $y = (\nu+\nu')/2$ and
$x= (\nu'-\nu)/2$; the Jacobian of this transformation is unity. We can make
further simplification by using  $y \simeq \nu$. This is justified for our case
as we assume the bandwidth  to be $\simeq 10 \, \rm MHz$ around a central
frequency of $\simeq 150 MHz$. All the frequency dependent variables change by
less than $10\%$ for this case. This
reduces  Eq.~(\ref{etacorrela12}) to:
\begin{eqnarray}
&&\langle V_\tau({\bf u}) V_\tau ^* ({\bf u'}) \rangle  =  \int  \int d\nu dx  \bar{I_\nu}^2   \int {d^3k \over (2\pi)^3} P_{\rm HI}(k) \nonumber \\
&\times &\int d^2\theta  \exp\left(i2\pi\vec\theta\cdot\left({\bf u}_\nu-\frac{{\bf k_\perp}r_\nu}{2\pi}\right) \right) A_\nu(\vec\theta) \nonumber \\
& \times & \int d^2\theta' \exp\left(- i2\pi\vec\theta'\cdot\left({\bf u'}_{\nu}-\frac{{\bf k_\perp}r_{\nu}}{2\pi}\right)\right) A_\nu(\vec\theta')\nonumber \\ &\times & \exp\left[i2x \left(2\pi\tau+k_\parallel dr_\nu/d\nu + d{\bf u'}_\nu/d\nu\cdot\vec\theta'\right. \right. \nonumber \\
&+ &  \left.\left.  dr_\nu/d\nu  {\bf k}_\perp\cdot\vec\theta'/(2\pi)  \right)\right]
\label{etacorrela1}
\end{eqnarray}
Given the HI power spectrum $P_{\rm HI}(k)$ this integral could be
computed numerically. However, it is possible to determine the correlation scales
in both the transverse and line of sight directions by carefully examining
Eq.~(\ref{etacorrela1}). The integral over angles  shows that the dominant
contribution comes from wavenumbers such that ${\bf k}_\perp \simeq  2\pi {\bf u}_\nu/r_\nu$. This relation allows us to simplify the integrals over $\theta$, $\theta'$ and $x$. In particular, different terms in the
 exponent containing $\tau$ can be estimated. Using  $d{\bf u}_\nu/d\nu = {\bf u}_\nu/\nu$,  the last two terms
in the exponents are  on the order of ${\bf k}_\perp.\vec\theta' r_\nu/(2\pi\nu)$ (the term containing $dr_\nu/d\nu$ is slightly smaller because $\nu/r_\nu dr_\nu/d\nu  \simeq 0.3$). For MWA primary beam,  $\theta_0 \simeq 0.3$, and
for MWA baseline distribution, the term $k_\parallel dr_\nu/\nu$ generally dominates over these terms, especially in the regions dominated by EoR. For all our calculations we use parameters specific to MWA, in particular, the primary beam of MWA. However, the formulation presented here is general enough
to be applicable to other arrays.

By dropping the last  two  terms, which are subdominant,  in the exponent containing $\tau$, we can
separate the integrals over $x$ and angles, this gives us:
\begin{eqnarray}
&&\langle V_\tau({\bf u}) V_\tau ^* ({\bf u'}) \rangle  =  \int   d\nu   \bar{I_\nu}^2   \int {d^3k \over (2\pi)^3} P_{\rm HI}(k)\nonumber \\
&\times &\int d^2\theta  \exp\left(i2\pi\vec\theta\cdot\left({\bf u}_\nu-\frac{{\bf k_\perp}r_\nu}{2\pi}\right) \right) A_\nu(\vec\theta) \nonumber \\
& \times & \int d^2\theta' \exp\left(- i2\pi\vec\theta'\cdot\left({\bf u'}_{\nu}-\frac{{\bf k_\perp}r_{\nu}}{2\pi}\right)\right) A_\nu(\vec\theta')\nonumber \\
 &\times &\int dx \exp\left[i2x \left(2\pi\tau+k_\parallel dr_\nu/d\nu \right) \right ]
\label{etacorrela2}
\end{eqnarray}
The integral over $x$ can readily be carried out now. The dominant
contribution to the integral comes from $\tau \simeq k_\parallel/(2\pi) dr_\nu/d\nu$, which establishes the correlation scale in the direction along the line
of sight. The  variation of  frequency dependence of integrals
over $\theta$ and $\theta'$  is expected to be small for the bandwidth
of MWA and therefore these integrals can be computed at some frequency
that lies within the bandwidth.     In this
paper we assume   these values to be  fixed at the central frequency $\nu = 154 \, \rm MHz$ and use them to compute the relations in Eq~(\ref{ukrelations}).  If the
frequency dependence of the primary beam and the background intensity
are neglected, the integral over $\nu$ in trivial. As noted above, this is a good  approximation for  MWA.  The power spectrum of the HI signal, based on Eq.~(\ref{etacorrela2}), is shown in Figure~\ref{signal_ps}. In this calculation, we use the HI power spectrum $P_{\rm HI}(k)$ given by the simulation of \citet{Furlanetto et al}.

\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{signal_k_pl.png}
\caption{Power spectra of the expected Cosmological HI signal. The power is plotted as $log_{10}P$ where P is in units  $mK^2 (Mpc/h)^3$.}
\label{signal_ps}
\end{figure}

Eqs.~(\ref{etacorrela2}) and~(\ref{viscorrfreq}) can be used to infer
many important properties of the HI signal:
\begin{itemize}
\item[1.] There exists a near one-to-one relation between {\bf u}, $\tau$
and the Fourier components of the HI power spectrum (\citet{Parsons 2012a, Delay 1, Drift paper, Morales Hewitt}):
\begin{equation}
 u_\nu=\frac{k_{\perp 1} r_\nu}{2\pi},\hspace{10 pt} v_\nu=\frac{k_{\perp 2} r_\nu}{2\pi},\hspace{10 pt} \tau=\frac{k_\parallel c(1+z)^2}{2\pi H_0 \nu_{21}E(z)}
\label{ukrelations}
\end{equation}
where $\nu_{21}$ is the rest frame frequency of the 21 cm line, $r_\nu$ is the transverse comoving distance, $E(z)=\sqrt{\Omega_M (1+z)^3+\Omega_k(1+z)^2+\Omega_\Lambda}$ and z is the redshift corresponding to the observed frequency $\nu$.
The relation between $k_\parallel$ and $\tau$ follows from the relation:
$2\pi\tau = k_\parallel dr_\nu/d\nu$ (Eq.~(\ref{etacorrela2})). As noted above,
all the frequency dependent quantities in Eq.~(\ref{ukrelations}) are computed
as a fixed frequency $\nu_0 = 154 \, \rm MHz$.
\item[2.] The correlations in the sky plane and along the line of sight
are nearly separable. This allows us to compute weights in the plane of the
sky owing to  $w$-term  and the distortion of intensity pattern in a tracking run (the next two subsections) without the
additional complication owing to frequency dependence of these quantities.
\item[3.] Eqs.~(\ref{etacorrela2}) and~(\ref{ukrelations})  allow us to simplify the relation between visibility correlation and the HI power spectrum. Eq.~(\ref{etacorrela2}) can be solved in the limit defined by Eq.~(\ref{ukrelations})
to give (e.g. \citet{Delay 4, Pen et al 2009, Morales 2005, McQuinn et al 2006}):
\begin{equation}
\langle V_\tau({\bf u}) V_\tau ^* ({\bf u}) \rangle = {\bar{I_\nu}^2 \Delta\nu \theta_0^2  \over r_\nu^2 dr_\nu/d\nu} P_{\rm HI} (k)
\label{normali}
\end{equation}
Here the MWA primary beam solid angle  $\theta_0^2 = \lambda^2/A_{\rm eff}$.
For MWA $A_{\rm eff} = 21.5 \, \rm m^2$ at 150 MHz (\citet{MWA}). $\Delta\nu = 10.2 \, \rm MHz$ is the
total band width we use in this work.  The mean specific intensity ${\bar{I_\nu}} = 2k_BT_B/\lambda^2$. This allows us to express the HI signal as the square of the  product
of mean brightness temperature and the HI power spectrum $T_B^2 P_{\rm HI}(k) $  in the units $(mK)^2 (Mpc/h)^3$. It should be emphasized that Eq.~(\ref{normali})  provides the suitable normalization only when ${\bf u'} = {\bf u}$, $w=0$,
as has been assumed throughout this section, and the impact of sky intensity distortion while  tracking a region
is not considered. All these effects act to  lower  the RHS of Eq.~(\ref{normali}),  the measured visibility correlation,  for a fixed signal $T_B^2 P_{\rm HI}(k) $. This is accounted for by appropriate weights we discuss in the
next two sections.

\end{itemize}
\subsection{HI signal and w-term}

\label{sec:h1sigwterm}

From Eqs.~(\ref{etacorrela1}) and~(\ref{etacorrela2}), we can gauge the  impact of the $w$-term.
These equations and the discussion following them  shows that the angular
integrals depend only weakly on  the line of sight variables.
 The main effect of the $w$-term is to alter the
integrals over angles which we study here.

 For a given baseline ${\bf b}$:
$w_\nu = {\bf b} \cdot {\bf s}_0 \nu/c$, where ${\bf s}_0$ is the phase center at
any time. As a region is tracked, the $w$-term changes owing to the drift of the phase center. For a tracking run, ${\bf u}_\nu^2 + w_\nu^2$ is left invariant at
any frequency; this result simply follows from the fact the the baseline
length is fixed.

After the inclusion of the $w$-term, the measured visibility for a given intensity distribution is given as:
\begin{eqnarray}
 && V_\nu({\bf u}_\nu,w_\nu)=\int A_\nu (l,m) \Delta I_\nu(l,m)\nonumber \\
 &\times &\exp{\left[-i2\pi\left\{u_\nu l+v_\nu m-w_\nu/2(l^2+m^2)\right\}\right]} dldm
 \label{visw}
\end{eqnarray}
Here we have replaced $\vec{\theta}$ with its components (l,m) and also made the approximation:$\sqrt{1-l^2-m^2}-1\simeq -(l^2+m^2)/2$, which is valid for
MWA primary beam.
After substituting Eq.~(\ref{deli_a}) into Eq.~(\ref{visw}) gives us:
\begin{eqnarray}
 && V_\nu({\bf u}_\nu,w_\nu)=\bar{I}_\nu\int\frac{d^3k}{(2\pi)^3}\delta_{\rm HI}({\vec {k}})e^{ir_\nu k_\parallel}
 \int  A_\nu(l,m) \nonumber \\
 && \times\exp\biggl[-2\pi i \biggl \{\left(u_\nu-\frac{{k_{\perp 1}}r_\nu}{2\pi}\right)l+\left(v_\nu-\frac{{k_{\perp 2}}r_\nu}{2\pi}\right)m\nonumber \\
 && -  {w_\nu\over2}(l^2+m^2) \biggl \}\biggl]dldm
\label{viswithw}
\end{eqnarray}
Each MWA tile being approximately a square aperture, the primary beam $A_\nu(l,m)$ can be written as:
\begin{equation}
 A_\nu(l,m)=\frac{\sin(\pi L_x l)}{\pi L_x l}\frac{\sin(\pi L_y m)}{\pi L_y m} \label{pribeam}
 \end{equation}
Here $L_x$ and $L_y$ are dimensionless. They correspond  to
the ratio of the length of the tile along x- and y-axis  to the wavelength. For central
wavelength of the observation  $L_x = L_y \simeq  2$. Eqs~(\ref{visw}) and~(\ref{pribeam}) show
that integrals over $l$ and $m$ are separable and identical. These integrals
cannot be done analytically but under certain approximations meaningful
analytic expressions can be found. Let us define:
\begin{eqnarray}
Q_\nu(k_{\perp 1};u_\nu,w_\nu)  &=& \int dl \exp\biggl [ -2\pi i \biggl \{\left(u_\nu-\frac{{k_{\perp 1}}r_\nu}{2\pi}\right)l \nonumber \\
&& - {w_\nu\over2}l^2 \biggl \} \biggl ] \frac{\sin(\pi L_x l)}{\pi L_x l}
\label{intprib}
\end{eqnarray}
$Q_\nu(k_{\perp 1};u_\nu,w_\nu)$ is a function of $k_{\perp 1}$ and is parametrized by
$u_\nu$ and $w_\nu$. First we consider, $w=0$. In this case, it can be shown that
if the limits of the integral are  allowed to go from minus infinity to plus infinity, we obtain,
\begin{eqnarray}
Q_\nu(k_{\perp 1};u_\nu,0) & = &  {1 \over L_x} \, \, \>  {\rm if} \, \,  |(u_\nu - r_\nu k_{\perp 1}/(2\pi))|/L_x < 1/2  \nonumber \\
& = & 0 \, \, \> \hbox{otherwise}
\end{eqnarray}
We notice that the approximation used is good  because the function
has a compact support provided by the primary beam. As $L_x \simeq 1/\theta_0$
where $\theta_0$ is the extent of the primary beam, this result means that,
for a given $u_\nu$, the wavenumbers that contribute to the integral are the ones
that are bounded by the extent of the primary beam. This result is already
implied by  Eq.~(\ref{etacorrela1}).

Eq.~(\ref{intprib}) cannot
be analytically approximated so readily for non-zero $w$. We  use the
stationary phase approximation to analytically evaluate the integral. For this assumption
to hold, the phase of the exponent  should be much larger  than the
slow variation of the primary beam. This would be the case if $wl^2$ is large.
In this approximation, we obtain:
\begin{eqnarray}
Q_\nu(k_{\perp 1};u_\nu, && w_\nu) = \sqrt{{2i\over w}}{\sin(\pi L_x (u_\nu-k_{\perp1} r_\nu/(2\pi))/w_\nu) \over \pi L_x (u-k_{\perp 1} r_\nu/(2\pi))/w_\nu} \nonumber \\
&& \times\exp\left [-i(u_\nu-k_{\perp 1} r_\nu/(2\pi))^2/(4\pi w_\nu)\right ]
\label{wtermpb}
\end{eqnarray}
The main impact of the inclusion of the $w$-term can be discerned from this expression. In the
limit of large $w$, the impact  of the $w$-term is to shrink the MWA beam and the primary beam  tends to $1/\sqrt{w}$ (\citet{W projection}). \footnote{The
impact of $w$-term can be more readily computed if the beam is Gaussian (e.g.
Appendix B of \citet{Drift paper}) In this case, the primary beam  approaches
$(1/\theta_0^2+\pi w_\nu)^{-1/2}$ for non-zero $w$.}  However, this also means that the
spread of $k_{\perp 1}$ for which the integral is non-zero also increases, as
seen in the terms involving the  $sin$ function. If the decrease of primary beam
results in a loss of signal-to-noise, an increase in the  correlation length $|{\bf u} -{\bf u'}|$
gains  signal-to-noise. We can write visibility correlation  for pairs of ${\bf u}, w$ and ${\bf u'}_\nu, w_\nu'$ as:
\begin{eqnarray}
&&\langle V_\nu({\bf u}_\nu, w_\nu) V_\nu ^* ({\bf u'}_\nu, w_\nu') \rangle = \int {d^3k \over 2\pi^3} P_{\rm H1}(k)  Q(k_{\perp 1}; u_\nu, w_\nu)\nonumber \\
&&\times Q(k_{\perp 2}; v_\nu, w_\nu) Q(k_{\perp 1}; u_\nu', w_\nu') Q(k_{\perp 2}; u_\nu', w_\nu')
\label{wtermvisvis}
\end{eqnarray}
Eq.~(\ref{wtermvisvis}) can be computed numerically. In Figure~\ref{fig:wterm1}, we show how the HI correlation function is affected in the presence of
$w$-term.
These expressions are also valid for diffuse foregrounds which  have
 a different 2-dimensional power spectrum and frequency dependence, e.g.
 optically thin Synchrotron radiation for which  the angular and frequency dependence is  separable.
As Eq.~(\ref{wtermvisvis}) can be used to compute the impact of $w$-term at
any frequency, it can readily be generalized to study diffuse foregrounds.

\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{viscorr.png}
\caption{The HI signal is plotted as a function $|{\bf u} -{\bf u'}|$ for
different values of w. The impact of w-term is to decrease the overall
signal owing to shrinking of the primary beam and increase the correlation
length $|{\bf u} -{\bf u'}|$.}
\label{fig:wterm1}
\end{figure}

\subsection{Time dependent coordinate system and w-term}
\label{sec:movinggrid}

In a tracking  interferometric observation, a phase
center is tracked and snapshots are taken at regular intervals with
short duration. Each of these snapshots can be imaged and the images
added if the successive   fields of view can be  assumed to be coplanar.
 This approximation breaks down for wide field-of-view instruments such as MWA. One manifestation of the wide field-of-view is the $w$-term whose
impact was studied in the previous sub-section. In this section we generalize the discussion of the last sub-section to take into account the time
dependence of the  non-coplanarity of the tracked region (\citet{1999ASPC..180.....T}).

As the region is tracked,  the relation between the image and
astronomical  coordinates changes which distorts the intensity
pattern with respect to the phase center being tracked. More specifically, this effect arises from the  projection of a non-coplanar array on a plane,
which is necessary to perform the Fourier transform for imaging. It is best
illustrated  with a set of point sources. These sources appear to move with respect to the phase center (e.g. Fig. 19-9 in \citet{1999ASPC..180.....T}).
The distortion of the intensity pattern corresponds to non-uniform
stretching and  it increases  for sources further away
from the phase center. Thus this effect can not be corrected  by a standard shift of coordinate.  The non-uniform stretching makes the situation complex, and the standard grid approach is difficult to implement in this case. For
a set of point sources, the correction for  this effect could   be applied
iteratively in the image plane (Chapter 19, \citet{1999ASPC..180.....T}).

For a small field-of-view, this effect can be neglected
and a unique coordinate system (e.g. time independent direction cosines $\{l,m\}$) can be used to relate
the image coordinates with the sky intensity pattern for a long tracking run.
However, it is not possible to define such a coordinate system when either
the field of view is large or the tracking period is long.

Our aim here is not to correct for this effect but rather to estimate its impact on
the correlation of visibilities at two different times during a tracking run:
suppose we measure visibilities within a  small cell in the $u$--$v$ plane (the size of the cell  will be discussed in a later section)  centered around  a baseline $\{u,v\}$ at
$t=0$. At a later time $t = t'$ another baseline might enter this cell. From the discussion in the previous subsections (e.g. Figure~\ref{fig:wterm1}) the two
 visibilities are expected to correlate strongly with each other  (even if the values of
$w$  differ significantly for these two sets  this statement is generally true). However, visibilities measured at two different times do not
correspond to the same  intensity pattern. Our aim here is to estimate  the level of de-correlation caused
by the distortion of intensity pattern during a tracking run. In this paper, we construct a time-dependent coordinate system  which allows
us to analyse this distortion of intensity pattern. We assess the impact of this
effect when a region   is tracked using the MWA primary beam. In particular,
we consider this effect on the visibilities produced by the EoR HI signal.

We start by recalling the definition of  direction cosines for a point on the sky whose coordinates, declination and hour angle $\{\delta,h\}$, are:
written as (\citet{Chris}):
\begin{eqnarray}
l & = & \sin(h)\cos(\delta) \\
m & = & \cos(\delta) \cos(h) \sin(\phi) -\sin(\delta)\cos(\phi) \\
n & = & \cos(\delta) \cos(h) \cos(\phi) +\sin(\delta)\sin(\phi)
\end{eqnarray}
It can be shown that $n = \sqrt{1-l^2-m^2}$. The phase center is
always defined as $l = 0$, $m = 0$, $n=1$; for the coordinates defined
above it is: $h =0$ and $\delta = \phi$.

As a phase center is tracked  owing to the rotation of the Earth, $\delta$
remains fixed but the hour angle changes. For a wide field of view,
this can result in distortion of the intensity pattern of the sky.
To take into account this effect, we can define a time-dependent coordinate
system:
\begin{equation}
l(\delta h)= \sin(h+\delta h)\cos(\delta) - \sin(h_0+\delta h)\cos(\delta_0)
\end{equation}
Here $h_0$ and $\delta_0$ define the phase center for $\delta h = 0 $; $\delta h$
defines the flow of time. $m$ can be similarly defined and $n$ can be computed from $l$ and $m$. This definition gives  a time dependent
coordinate system where the coordinates are  always defined with respect to the
phase center. It is easy to verify that for small field of view
and for small tracking times,  which corresponds to  cases when higher order terms in $h$, $\delta h$ and $\delta$ can be dropped, $l$ is independent of
time which means that the distance of a point from the phase center is left
invariant under tracking. In such cases, the intensity pattern on the sky
corresponds to the unique  intensity pattern defined by sky coordinates
$\delta$ and $h$ and  remains unchanged  as the phase center is tracked.

However, when this approximation breaks down, $l$ becomes a function of
time and it is impossible to define a unique relation between  direction
cosines and sky coordinates. This means that any quantities defined
with respect sky coordinate (e.g. intensity pattern) become time dependent.
The visibility for the HI signal is given by:
\begin{eqnarray}
&& V_\nu({\bf u}_\nu, w_\nu,\delta h)=\bar{I}_\nu\int\frac{d^3k}{(2\pi)^3}\delta_{\rm HI}({\vec {k}})e^{ir_\nu k_\parallel}
 \int  A_\nu(l,m) \nonumber \\
 && \times\exp\biggl[-2\pi i \biggl \{\left(u_\nu-\frac{{k_{\perp 1}}r_\nu}{2\pi}\right)l+\left(v_\nu-\frac{{k_{\perp 2}}r_\nu}{2\pi}\right)m \nonumber \\
 && -  {w_\nu\over2}(l^2+m^2) \biggl \}\biggl]\sin\delta d\delta dh
\label{viswithw}
\end{eqnarray}
The direction cosines $l$ and $m$  are now  functions of time. The angular
integral is carried out  over $\delta$ and $h$. Unlike the earlier case (fixed
grid) this is not a product of two one-dimensional integrals. The correlation
of the visibilities $V_\nu({\bf u}_\nu, w_\nu,\delta h)$ can be computed using the
same methods as outlined in the previous sections.

In Figure~\ref{wterm2} we show the results when the effect of
the time dependent coordinate system ('moving grid')
is included.  The initial
phase center ($\delta h =0$) is chosen  to be $h=0$ and $\delta = \phi$.
The results are shown for two different values of $\delta h$ and a range of
$w$ values. We only show the auto-correlation function for a given value of
$\delta h$. But the results shown in Figure~\ref{wterm2} can be used
to assess the  cross-correlation of  visibilities measured at two different times.  For our case the  value of this cross-correlation
 lies between the auto-correlations of visibilities  measured at the same time. The moving
grid doesn't introduce another scale in the problem and the results
in this case are not very different from the case for a fixed grid. In both
cases the dominant correlations occur for  ${\bf u} - {\bf u'} < 1.5$.

Figures~\ref{fig:wterm1} and~\ref{wterm2} are based on MWA primary beam. However, it is possible to glean generic information applicable for other primary beams from them.  First,  the decorrelation length $|{\bf u}-{\bf u'}|$ scales as the inverse of the primary beam (e.g. \citet{Drift paper}). So for a smaller beam, the decorrelation seen in the Figures as a function of $|{\bf u}-{\bf u'}|$  would be shallower. The impact of the $w$-term for a different primary beam can be partially gauged from Eq.~(\ref{wtermpb}), which is valid for large values of $w$.
In this limit, the primary beam tends to $1/\sqrt{w}$, irrespective of the primary beam of the telescope. It is difficult to analytically estimate the impact of the $w$-term when this limit does not hold. But it can be shown that the impact of $w$-term diminishes for a smaller primary beam, e.g. a Gaussian beam for
which the primary beam tends to $(1/\theta_0^2+\pi w_\nu)^{-1/2}$ for non-zero $w$
(e.g. \citet{Drift paper}). As noted above, the distortion of intensity pattern during a tracking run is a wide field effect. For a smaller primary beam, the level of decorrelation seen in Figure~\ref{wterm2} would be  smaller but it is difficult to analytically estimate it.

\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{viscorr_movinggrid.png}
\caption{The HI signal is plotted as a function $|{\bf u} -{\bf u'}|$ for
different values of w and $\delta h$. The initial
phase center is chosen  to be $h=0$ and $\delta = \phi$. }
\label{wterm2}
\end{figure}

\subsection{Weights for cross-correlation}

\label{sec:weightscross}
Eq.~(\ref{viswithw}) can be used to compute the counterpart of Eq.~(\ref{etacorrela1})  which takes into account the impact of non-zero $w$-term and the distortion of intensity pattern. We compute this expression for the
visibility correlation in delay space numerically. In this formulation,
the measured visibility is a function of five parameters: $\tau, {\bf u}, w, t$.
Here, as noted above, ${\bf u}$ and $w$ are the values of these variables
at a fixed frequency which we choose to be $\nu_0 = 154 \, \rm MHz$.

We define the weight on a given cross-correlation as:
\begin{equation}
{\cal W}({\bf u},w,t;{\bf u'},w',t') = {\langle V_\tau({\bf u},w,t) V_\tau ^*({\bf u'},w',t') \rangle  \over \langle V_\tau({\bf u},0,t) V_\tau ^*({\bf u'},0,t') \rangle }
\label{weightsdef}
\end{equation}
The weights are defined with respect to the HI cross-correlation
computed  in Eq.~(\ref{etacorrela1}) for ${\bf u} = {\bf u'}$, $w = w' = 0$,
$t = t'$ and $\tau = \tau'$. We only consider the case $\tau' = \tau$ for the
computation of weights.

 Using Eq.~(\ref{weightsdef}) allows one to recover the HI power spectrum
for a fixed wave number $k$ from visibility cross-correlations.

\section{Analysis of MWA data}

The MWA is a low frequency radio interferometer array located in  Western Australia. It consists of 128 antenna tiles with each tile comprising of 16 crossed dipole antennas over a metal ground screen in 4 x 4 grid. MWA bandwidth  is 30.72 MHz, divided into 24 coarse channels of width 1.28 MHz each. The total bandwidth is  divided into 768 fine channels. With the use of an analog beamformer appropriate phase delays are introduced in each individual dipole antenna to track the pointing center of the beam across the sky. For more information on MWA please see \citet{MWA, Lonsdale et al}.

To minimize the effect of Galactic synchrotron emission, the MWA EoR community has chosen three fields on the sky away from the Galactic plane. These fields have been named as EoR0, EoR1 and EoR2 and  are  shown in Figure~\ref{eor_fields}. In this paper we present 3~hours of tracking analysis of the
EoR1 field centered at RA $=$ 4h, Dec $= -27 ^\circ$.

Many  research groups are currently developing pipelines to extract statistical information from radio interferometric data, with an aim to detect the HI
signal from EoR (\citet{Jacobs et al, Hazelton et al, Dillon et al, Trott et al}). These can be divided broadly into two categories: image based and  visibility based pipelines.
For foreground subtraction and  imaging  these pipelines use  the following imaging algorithms: Real Time System (RTS; \citet{Mitchell et al, Ord et al}) and  Fast Holographic Deconvolution (FHD, \citet{Sullivan et al}).

The image based pipelines (\citet{Dillon et al, Hazelton et al}) use source catalog created through the deconvolution of the data which is subtracted to obtain a residual image cube. The Fourier transform of this image cube with some further processing yields the power spectra. On the other hand, the visibility based pipelines (\citet{Trott et al}) use the data in visibility domain
 for power spectra estimation, after the initial processing in the
image domain   for obtaining the foreground model.  The detailed comparison of the outputs from all the methods described above is provided in \citet{Jacobs et al}. \citet{Delay 4, Thyagrajan 2015b} describe the impact of wide field of view in power spectra estimation.

A special variant among the visibility based estimators is `delay spectrum' (\citet{Pober PAPER, Delay 1, PAPER}), which directly Fourier transforms each calibrated visibility along its frequency channels. The radio interferometer PAPER uses this approach extensively; it relies upon redundant baselines to calibrate the interferometer and  East-West and near East-West baselines for power spectrum estimation (\citet{Delay 1, PAPER, Ali et al PAPER}). This particular scheme has been discussed and implemented only for  redundant drift scan observations.  In this paper we explore the possibility of applying this approach for non-redundant imaging arrays and tracking measurements.

In this and the next section, we discuss in detail our method of MWA data analysis and power spectrum estimation from the  data.

 We summarize below the major ingredients of the  method
and then describe each of the stpng in detail in subsequent sections:
\begin{itemize}
\item[1.]  CASA (\citet{CASA}) is used for initial processing of the data to calibrate raw visibility measurements. This
is followed by the creation of a model sky image from clean components. This
model is then subtracted in the visibility domain to obtain residual visibilities. We use both the calibrated and residual  visibilities for computing the
power spectrum.
\item[2.]   Each visibility is then Fourier transformed in frequency space (Eq.~(\ref{visetaspace})). This process is needed for isolation of foregrounds
in the  $k_\perp\hbox{--}k_\parallel$ plane. We note the our method
utilizes both the subtraction of foregrounds and their isolation. But  it does not employ an external point source catalog.
\item[3.] The procedure outlined above yields complex visibilities as a function of five variables: $V_\tau(u,v,w,t)$. For computing the power spectrum we cross-correlate these visibilities for $t'\ne t$ to remove the noise bias. To weigh each cross-correlation we assume that there exist regions in $k_\perp\hbox{--}k_\parallel$ plane which are dominated by only noise and the HI signal.  This allows us to compute
a weight for each cross-correlation based on the expected HI signal. For computing these weights we take into account the impact of $w$-term and the distortion of intensity pattern  in a tracking scan.  The relevant
method  is elaborated in detail in sections~\ref{sec:h1signal}, \ref{sec:h1sigwterm}, \ref{sec:movinggrid}, and~\ref{sec:weightscross}  and summarized in section~\ref{sec:weights}.
\item[4.] In section~\ref{sec:powspecest}, we describe the power spectrum estimator,
taking into account weights given by the expected HI signal, in 3-, 2-
and 1-dimension. We also discuss our method to compute the errors on
the estimated power spectrum.
\end{itemize}

\subsection{CASA processing}

\begin{figure*}
\centering
\includegraphics[width=0.9\textwidth]{eor_fields.png}
\caption{EoR fields are shown on the Galactic map at 408 MHz (\citet{Haslam map}). Three fields, away from the galactic plane, have been identified for the MWA reionization study. These are EoR0, EoR1 \& EoR2 respectively as shown in circles. In this paper we focus on the EoR1 field centered at RA $=$ 4h, Dec $= -27 ^\circ$.}
\label{eor_fields}
\end{figure*}

 MWA data were  collected at 2-minute intervals with a time resolution of 0.5 seconds and frequency resolution of 40 kHz.  The central frequency of these observations is 154.24 MHz.
For preprocessing we have used the Cotter pipeline (\citet{Offringa et al}) to average to 10 seconds of integration;  we have not performed any averaging over the frequency channels. Cotter also uses the in-built AOFlagger to flag and remove radio frequency interference.  The edge channels of each coarse band are flagged with Cotter due to aliasing effects. After this  preprocessing the
 Cotter pipeline  delivers the data in the CASA readable `Measurement set (ms)' format for further processing.

Once the `ms' files are produced for each 2-minute data set, we process each of these 2-minute data in CASA to produce an image. The Hydra A source is used to calculate the bandpass solutions which are applied to the uncalibrated data. We next   construct   a  sky model from these data so that we could subtract it to obtain the residual visibility. After the bandpass calibration the  first round of `clean' is applied on each 2-minute data set. The multi-scale multi-frequency synthesis algorithm (\citet{MSMFS}) has been used for imaging. We have created images of size 3072 x 3072 pixels with 1 arc-minute cell size using the Cotton-Schwab CLEAN (\citet{Schwab}) with uniform weighting scheme. After the first round of clean we have performed self calibration (both phase and amplitude+phase) and apply the  clean loop until the  RMS value of the residual image converges. The threshold limits for the clean stpng were chosen to be 5 sigma. The W-projection algorithm (\citet{W projection}) was also used to correct for the errors arising due to non-coplanarity of baselines.  Once we obtain the best model of the sky for each 2 minute observation, the model visibilities are  then subtracted from the calibrated data using the UVSUB algorithm in CASA to obtain the residual data. This process is followed for both XX \& YY polarizations separately.
 A flow chart of the data pipeline is shown in Figure~\ref{pipeline}. In Figure~\ref{eor1_image}, we present a sample image of  2 minute deconvolution.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{pipeline.pdf}
\caption{A schematic of the  power spectra pipeline is shown.}
\label{pipeline}
\end{figure}

\begin{figure*}
\centering
\subfloat[]{\includegraphics[width=0.9\textwidth]{EOR1.jpg}}\\
\subfloat[]{\includegraphics[width=0.5\textwidth]{EOR1_center.png}}
\subfloat[]{\includegraphics[width=0.5\textwidth]{EOR1_fornax.png}}
\caption{The image of EoR1 field at 154.24 MHz is shown in the top
panel for two minutes of data.  The bottom panels display two regions from the image.
The   right panel displays  the region containing  the Fornax~A.}
\label{eor1_image}
\end{figure*}

As noted above we process the data for only  2 minutes to
ensure the primary beam doesn't substantially change during the run.
For a 2-minute scan  we obtain an RMS of
nearly 40~mJy/beam.

The residual visibility $V_\nu(u_\nu,v_\nu,w_\nu,t)$ is a function of five variables.
We compute the discrete Fourier transform of the residual visibilities
in the frequency space weighted by the Blackman-Nuttall (\citet{BN}) window $B_\nu$ to suppress leakage into the EoR window (\citet{Thyagrajan et al, Thyagrajan 2016}):
\begin{equation}
V_\tau(u,v,w,t) = \sum \Delta \exp(i2\pi\nu\tau) V_\nu(u_\nu,v_\nu,w_\nu,t) B_\nu
\label{visetaspace}
\end{equation}
Notice that in  Eq.~(\ref{visetaspace})  the frequency dependence of
the baseline vector ${\bf b}_\nu =  \{u_\nu,v_\nu,w_\nu\}$ is integrated over.
Therefore, the labels
 $\{u,v,w\}$ on the LHS of Eq.~(\ref{visetaspace}) need further explanation.
As noted above (the discussion following Eq.~(\ref{deftau}))
they    can be chosen to denote a given baseline vector  at a fixed frequency, $\nu_0$. We choose this frequency to be
the central frequency of the band $\nu_0 = 154 \, \rm MHz$. \citet{Parsons 2012a, Delay 1} provide detail implications of the frequency dependence of the baseline vector.
Here $\Delta = 40 \, \rm kHz$ and 256~channels are used for our study, which
correspond to total bandwidth 10.24~MHz in the frequency range 149.09~MHz to 159.34~MHz.

\section{Power spectrum}

\label{sec:weights}

The visibilities (Eq.~\ref{visetaspace}) are cross-correlated  with
weights determined from the HI signal (section~\ref{sec:weightscross})  to estimate the power spectrum. For each
pair of parameters, e.g. $\{u,u'\}$, the weights are generally different.
It is computationally prohibitive to deal with weights for all cross-correlations. We make several simplifying assumptions to make the problem tractable based
on the properties of the HI signal. In  sections~\ref{sec:h1signal}, \ref{sec:h1sigwterm}, and~\ref{sec:movinggrid} we discuss in detail the HI signal and
how it is affected by the inclusion of the $w$-term and the additional complication arising from distortion of the field of view as a region is tracked for
MWA.

We  summarize the main results of these sections as applied to the data:
\begin{itemize}
\item[1.] In section~\ref{sec:h1signal} the HI signal and its correlations
are discussed in detail. Eq.~(\ref{etacorrela2}) shows that correlations in sky
plane are nearly independent of correlations along the line of sight. This allows us to compute weights for correlations in the plane of the sky independent of
the third axis. Eq.~(\ref{etacorrela2}) allows us to derive a relation
between the measured correlation and the inferred HI power spectrum (Eq.~\ref{normali}).  Eq.~(\ref{normali}) defines the scale of cross-correlation weights.
The weight function ${\cal W}(u,v,w,t; u',v',w',t')$ is unity when
${\bf u} = {\bf u'}$, $w = w' = 0$ and $t\simeq t'$. Eq.~(\ref{normali}) refers
to this case.
\item[2.] In section~\ref{sec:h1sigwterm}, the impact of $w$-term on the HI
signal is computed. Eq.~(\ref{wtermpb}) and Figure~\ref{fig:wterm1} capture
the effect of non-zero $w$ on HI correlations. The $w$-term diminishes
the signal by shrinking the effective primary beam and increases the correlation
length scale $|{\bf u} -{\bf u'}|$. We use the analytic
expression based on Eq.~(\ref{wtermpb}) for computing weights for $w > 30$.
\item[3.] In section~\ref{sec:movinggrid}, we attempt to assess the
impact of  time-dependent distortion of intensity pattern in a tracking run
for MWA. Figure~\ref{wterm2} shows the combined effect of moving grid and
$w$-term. The distortion of intensity pattern  generally acts to enhance decorrelation but is  found to
be not significant and  doesn't alter the main features
of the signal.  For our computation, we only  update the weights after every 10~minutes to account for this effect.
\end{itemize}

\subsection{Power spectrum estimator}

\label{sec:powspecest}

As shown above each  correlation receives a different weight depending on the values of $\{u, v,w,t\}$ of the baselines being correlated. As noted above,  we define
the weights ${\cal W}(u,v,w,t;u',v',w',t')$ such that they approach  unity
when $u = u'$, $v=v'$, $w = w' = 0$ and $t\simeq t'$ such that the effect of
the moving grid is not important ($t\ne t'$ for all correlations
to remove noise bias).

 The HI power spectrum is a function of
$k  = \sqrt{{\bf k}_\perp^2 + k_\parallel^2}$; Eq.~(\ref{ukrelations})
gives the relation between the Fourier components of the HI signal  and
$\{{\bf u}, \tau\}$. All cross correlations for which the wave vector lies
in some range $k$ and $k+dk$ can be used to construct the unbiased
HI signal:   $1/N\sum V_\tau(u,v,w,t)V_{\tau'}(u',v',w',t')/{\cal W}$; here $N$ is
the number of   all pairs for which  $k$ lies in the range specified above.   However, this estimator, though unbiased for the HI
signal, could be dominated by small values of weights ${\cal W}$, which
doesn't make it the lowest noise (or  optimal) estimator.

As the observed signal is dominated by noise, we consider  an optimal estimator
for our study:

\begin{equation}
\phat{P}_\tau(u,v)={1\over\sigma}\sum_{u',v',w,w',t,t'}\left( V_\tau V_\tau ^* \times {\cal W} \right)
\label{powspecdef}
\end{equation}
where $\sigma=\sum_{u',v',w,w',t,t'} {\cal W}^2(u,v,w,t,u',v',w',t')$.
To avoid noise bias, $t \ne t'$ for
all cross-correlation. For a given $\{u,v,\tau\}$, Eq.~(\ref{powspecdef}) allows us to compute the
power spectrum by optimally weighing over all the cross correlations. However,
as Figures~\ref{fig:wterm1} and~\ref{wterm2} show the correlations
fall substantially for ${\bf u} - {\bf u'} \ge 1.5$ (see also \citet{Drift paper}
and references therein). This motivates us to pixelize the ${\bf u}$-plane
and consider only those visibility pairs  for which the correlations are significant.  We consider cells of different sizes and present results here
for $\Delta u=\Delta v = 0.5$.  The number of visibility measurements  in a cell vary  depending on the (u,v) values. The shortest baselines  have higher population as expected for  MWA. For 3 hours of analysis and $u, v <  50$, the
 number of visibilities in a cell lie in the range $\sim 1000\hbox{--}3500$ where each visibility has a time resolution of $\Delta t = 10$ sec. All the cross-correlation within a cell
are computed using Eq.~(\ref{powspecdef}).

 For averaging over different
cells,    each cell is assigned an average  weight corresponding to
the RMS of the power spectrum  for a  cell,  $\sigma_p$.  These  weights
are then used for optimally averaging the power spectrum
 (Eq.~(\ref{powspecdef}))  over other cells (For details see Appendix~B).  Note that this procedure
allows us to separate  large  correlations of the HI signal, the
ones for which ${\cal W}$ is close to unity,  from
 the ones which are expected to be  incoherent because ${\cal W}$ is small.

The schematic of the two  processes---the computation of power
spectrum in 3- and 2-dimensions---is displayed  in figure (\ref{schem}):
the top panel  delineates the process of computing cross-correlations
within each cell and the bottom  panel depicts how azimuthal average for
a fixed baseline length $\sqrt{u^2+v^2}$ is computed. For MWA data, $k_\parallel \gg {\bf k}_\perp$, which means the value of $k$ is dominated by the value of
$k_\parallel$. This suggests the following method for  computing the 1-dimensional
power spectrum, which we adopt: all the cells for a given  $\tau$ are  optimally
averaged  using the method described above. This procedure yields a complex
number. In the Figures that display 2- and 1-d power spectra we plot the absolute value of the estimated power spectrum.

\begin{figure}
\centering
\subfloat[]{\includegraphics[width=0.5\textwidth]{schem_1.png}}\\
\subfloat[]{\includegraphics[width=0.5\textwidth]{schem_2.png}}\\
\caption{(a)This shows the population of visibilities within different uv bins for each $\tau$. The small dots denote visibilities due to individual baselines. These are cross-correlated with each other within a given uv bin. (b) The black dots within a bin are the power value at each bin. An
optimal azimuthal average, based on weights of each cell,  is done to collapse the uv axes into a single axis of baseline length.}
\label{schem}
\end{figure}

The error on power spectrum in 1-dimension is computed by first estimating
the RMS for each cell, $\sigma_p$. $\sigma_p$ are then  used as weights for
 optimal averaging over all the cells
for a fixed $\tau$. The resultant RMS after averaging over the cells approaches $\sigma_{\rm fin} \simeq  (1/\sum\sigma_p^2)^{-1/2}$ if the power spectrum
across cells is  uncorrelated. This holds for noise but, as noted above,
is an approximation for the HI signal. We expect this assumption to
be valid in our case as the observed signal is dominated by noise (for detailed
explanation see Appendix~B).

\section{Results}

In Figure~\ref{ps}, the power spectra computed from 3~hours of data are shown in the $k_\perp\hbox{--}k_\parallel$ plane. The power spectra for both  XX and YY polarization are shown for calibrated and residual visibilities.

\begin{figure*}
\centering
\subfloat[dirty XX]{\includegraphics[width=0.5\textwidth]{3hrs_xx_corrected.png}}
\subfloat[dirty YY]{\includegraphics[width=0.5\textwidth]{3hrs_yy_corrected.png}}\\
\subfloat[residual XX]{\includegraphics[width=0.5\textwidth]{3hrs_xx_residual.png}}
\subfloat[residual YY]{\includegraphics[width=0.5\textwidth]{3hrs_yy_residual.png}}\\
\caption{Two-dimensional power spectra for 3~hrs of data on the EoR1 field
 for XX \& YY polarization. The power is plotted as $log_{10}P$ where P is in units  $mK^2 (Mpc/h)^3$.}
\label{ps}
\end{figure*}

\begin{figure*}
\centering
\subfloat[XX]{\includegraphics[width=0.5\textwidth]{1d_power_xx.png}}
\subfloat[YY]{\includegraphics[width=0.5\textwidth]{1d_power_yy.png}}\\
\caption{One dimensional power spectra  ($\Delta^2 = k^3 P(k)/(2\pi^2)$ in
units $(mK)^2$) for the XX (Left panel)  and YY (Right panel) polarization are  shown  along with the errors
for each band. The dot-dashed (red) curve show the expected HI signal (\citet{Furlanetto et al}).}
\label{1d_power}
\end{figure*}

We first discuss discernible features in  the power spectra:
\begin{itemize}
\item[1.] In Fourier space the foreground contributions occupy a wedge shaped region (also called `foreground wedge') owing to the smooth spectral characteristics of foreground sources (\citet{wedge 1, wedge 2, Delay 1, Delay 2, Delay 3, Thyagrajan et al, Delay 4}). The region beyond the foreground wedge is expected to be free from the foreground contamination and dominated by thermal noise and expected HI signal. This foreground isolation approach is particularly useful for the MWA as it has low angular resolution. The strongest HI signals lie in the shortest baselines (low $k_\perp$ values) and it decreases rapidly with increasing $k_\perp$ values.

 The upper  and lower  panels of  Figure~8 display  the dirty (calibrated with no foreground subtraction) and  the residual (clean components subtracted) power spectra, respectively. The Figure~\ref{ps} bear out the assumption that foregrounds
have smooth spectral characteristics as they are seen to form the `foreground wedge', this separation is in good agreement with the expectation  from foreground simulations (Figure~\ref{forenoise}). The first few $k_\parallel$ modes exhibit maximum foreground contributions, the $k_{\parallel}=0$ mode being the strongest. The amplitude at this mode is roughly $10^{14}\hbox{--}10^{15} \, \rm mk^2 \,  (Mpc/h)^3$ which is in good agreement  with the results of  other MWA EoR pipelines
 (\citet{Jacobs et al}). A clear decrement in power in the `foreground wedge' is visible in residual power spectra as compared to the dirty one.

\item[2.]  As described in the previous  section, MWA  has missing channels
on either side of   coarse bands of width 1.28~MHz. This leads to a periodicity of missing data across the frequency axis in visibility, the effect of which is reflected
 in the Fourier-transformed  power spectra as the horizontal bright lines at fixed $k_{\parallel}$.

\end{itemize}

Figure~\ref{1d_power} shows the 1-dimensional  power spectra; the 1-dimensional
power spectrum is obtained from regions that exclude   the foreground
wedge and bright coarse  bands  in  Figure~\ref{ps}. For computing the 1-d power spectrum, the foreground wedge and the bright coarse  horizontal bands of the 2-d power spectrum (Figure~\ref{ps})  are rejected.  More specifically, the channels corresponding to $k_\parallel \le  0.14 h Mpc^{-1}$ and $k_\perp > 0.1 h Mpc^{-1}$ are not considered. For each coarse band, the central  brightest channel along with two channels on either side are excluded.  The remaining contiguous regions are used in estimation of the 1-d power spectrum. For instance, for a given $k_\parallel$ that meets the criterion outlined above, all the cells that correspond to  $k_\perp < 0.1 h Mpc^{-1}$ are used for the computation of 1-d power spectrum.   The error
on the binned  power spectra are computed using a  scheme  outlined in Appendix~\ref{sec:powspecest1}.

 The 2-d power spectra obtained from data (Figure~\ref{ps}) can be compared with Figure~\ref{forenoise} that shows the expected power spectrum based  on
 simulations of foregrounds and noise. In particular, this comparison allows
us to assess the structure of coarse channels and the foreground wedge.
It also indicates the range of scales of the 2-d power spectrum.
 However,
Figure~\ref{forenoise} is based on a single realization of noise and a model of foregrounds
based on random distribution of point sources, and therefore a more detailed
comparison between the data and simulations is not possible. We shall return
to this detailed comparison in future work.

\section{Conclusions}

In this paper,  we propose a new method  to extract  the HI power spectrum from MWA visibility data in delay
space. The proposed method  is applicable when a region is tracked using
 imaging radio interferometers.

 One of the crucial factors in power spectrum estimation is how the $w$-term is dealt with within the pipeline. Our findings are that the $w$-term causes an effective shrinking of primary beam which reduces   the contribution of the
 HI signal. We carefully model the HI signal by taking the $w$-term into account, the weights calculated are then applied to cross-correlate the measured visibilities. Moreover, the cross-correlation approach is particularly useful to minimize various systematics in the system.  We also model and account for the impact of
changing intensity pattern in a tracking run. We find this effect to be
sub-dominant to the $w$-term correction.

We analyse three hours of MWA data from
the EoR1 field, one of the field identified by the MWA community for EoR science. CASA has been used for calibration and to create foreground model using the clean components. Both the dirty (calibrated with no foreground subtraction) \& residual (foreground model subtracted) power spectrum in delay space are presented. Our  results (Figures~\ref{ps} and~\ref{1d_power}) are in good agreement with the    analyses  of other MWA EoR pipelines (\citet{Jacobs et al}).

In the future   we  plan  to apply the method proposed here for more
integration time and, in particular, to  a longer  single tracking  run.
The decorrelation caused by the $w$-term and the changing   intensity pattern would be more  dominant in the latter  case. This will allow us to
 test the efficacy
of our method for more extreme cases and might indicate the best possible
way of detecting the HI signal from the epoch of reionization.

\section{Acknowlegements}
We thank the referee for useful comments which helped us to improve the paper.

This scientific work makes use of the Murchison Radio-astronomy Observatory, operated by CSIRO. We acknowledge the Wajarri Yamatji people as the traditional owners of the Observatory site. Support for the operation of the MWA is provided by the Australian Government Department of Industry and Science and Department of Education (National Collaborative Research Infrastructure Strategy: NCRIS), under a contract to Curtin University administered by Astronomy Australia Limited. We acknowledge the iVEC Petabyte Data Store and the Initiative in Innovative Computing and the CUDA Center for Excellence sponsored by NVIDIA at Harvard University.

redpajama_set_name: RedPajamaArXiv


North County always welcome new members in both playing and non-playing capacities. You can find out more from the membership section of our website here.
We are located just south of Balbriggan town in our beautiful ground of the Inch, please find us here on Google Maps.
North County is one of Ireland's most successful cricket club, located at the Inch, Balrothery, Balbriggan, Co. Dublin.
Five times all Ireland champions, including an unprecedented back to back the County have continued to set new standards for Irish cricket both on and off the pitch. Whether you're an aspiring first team member or a complete beginner drop into the clubhouse for information on how to get started, you'll always be welcome at the Inch.
Among our impressive facilities is a purpose-built Centre of Excellence indoor training centre that is used by National, International and Provincial sides.
North County CC was founded in 1986 when Balrothery CC and Man O War CC amalgamated.


text:
Hovie Lister (1926-2001)
Greg Freeman, Southern Edition,
A southern gospel music pioneer
Hovie Lister and the Statesmen
who paved the way for contemporary Christian music, Hovie Lister lived much of his life in or near Atlanta. The founder of the renowned Statesmen Quartet in 1948, he devoted more than fifty years of his life to gospel music as a pianist, vocalist, emcee, and music publisher.
Born on September 17, 1926, in Greenville, South Carolina, Hovie Franklin Lister was reared in a textile-mill village. At an early age he learned to play piano. Immersed in the music of the traditional gospel singing conventions, Lister's singing family played a decisive role in his budding career.
In 1941 the Reverend Mordecai Ham (who had converted future evangelist Billy Graham a few years earlier) conducted a citywide revival in Greenville. At this event a teenaged Lister was invited to accompany Ham and his music director, noted hymn writer C. Austin Miles, as pianist for the entire event.
With aspirations for a musical career, Lister enrolled the following year in the Stamps-Baxter School of Music in Dallas, Texas, where he sufficiently impressed key gospel-music figures, eventually receiving an invitation to perform temporarily with the Lone Star Quartet of Raleigh, North Carolina. Other early stints included playing piano for Greenville's Lister Brothers and Connor Hall and the Church of God Quartet. Lister soon moved to Atlanta to perform with such acts as the Homeland Harmony Quartet, the LeFevres, the Swanee River Boys, and the Rangers.
Befriending William Barrett Howell (son of Clark Howell Jr., then the editor-in-chief of the Atlanta Constitution) proved beneficial for Lister, who desired to form his own gospel quartet. A meeting with the elder Howell prompted Lister to assemble a group that would be known as the Statesmen with the assurance of employment at WCON, a radio station newly established by the Constitution. The Statesmen, consisting of Hovie Lister (piano) and vocalists Gordon Hill (bass), Bervin Kendrick (baritone), Mosie Lister (lead), and Bobby Strickland (tenor), soon enjoyed immense popularity in the late 1940s with their live morning show. At this time the group also actively pursued the concert circuit with equal success. Jake Hess replaced Mosie Lister, who did not want to travel outside Atlanta, and Claris G. "Cat" Freeman replaced Strickland.
In his autobiography the Christian-music mogul Bill Gaither recounts the influence Hovie Lister and the Statesmen had upon him as a boy in 1949, when he and his parents attended Wally Fowler's All-Night Gospel Singing, a regular event at the Ryman Auditorium in Nashville, Tennessee. Describing Lister's charismatic stage presence, Gaither writes, "He was bouncing all over the piano stool and pulling at his pant legs, revealing his trademark red socks, when his fingers weren't flying up and down the keyboard."
With the onset of the 1950s, another new configuration, featuring James "Big Chief" Wetherington (bass), Doy Ott (baritone), Jake Hess (lead), and Denver Crumpler (tenor), signified the crux of the Statesmen. The group's fancy, innovative vocal arrangements and tight four-part harmonies, combined with Lister's piano work, defied imitation and created the signature sound for which the group would become known. Recording for major labels RCA and Capitol, they toured nationally and hosted Singin' Time in Dixie, a syndicated television show. Popular Statesmen recordings include "Wait Till You See Me in My New Home," "Faith Unlocks the Door," "Oh, What a Time," "The Love of God," and "Goodbye, World, Goodbye."
Other gospel-music endeavors for Lister included managing and playing piano for the star-studded Masters V, a group consisting of J. D. Sumner (bass), James Blackwood (baritone), Jake Hess (lead), and Roland "Rosie" Rozell (tenor). In 1982 the Masters V garnered a Grammy Award for Best Traditional Gospel Performance. Recording such songs as "Everyday" and "Think on Good Things," the Masters V, like the Statesmen decades earlier, capably intertwined traditional roots with fresh material and distinctive song arrangements.
Lister was the quintessential crowd-pleaser onstage. His "sense of style and flair for showmanship gave this music class and chic," according to former U.S. senator and Georgia governor Zell Miller in his book They Heard Georgia Singing. An ordained Southern Baptist preacher, Lister also took seriously his ministerial responsibilities, pastoring the Mount Zion Baptist Church in Cobb County and even taking his music into Georgia prisons.
The Nashville-based Gospel Music Association inducted Lister into the Gospel Music Hall of Fame in 1984; Hovie Lister and the Statesmen were inducted in 1998, the first year groups were made eligible. Lister was inducted into the Georgia Music Hall of Fame in 1986.
Through Bill and Gloria Gaither's Homecoming music series and concert tours, Lister came out of retirement in 1992 and experienced a career resurgence late in life. He even resurrected the Statesmen name by recruiting new personnel to record several projects to the delight of old and new fans alike.
Lister died of lymphoma at his Tucker residence, in DeKalb County, on December 28, 2001. He and his wife, Ethel Abbott, had two children, Lisa and Hovie "Chip" Jr.
Southern Gospel Music
Lee Roy Abernathy (1913-1993)
The LeFevres
More in Religious
Dottie Peoples (b. ca. 1950)
The Sacred Harp
Babbie Mason (b. 1955)
Media Gallery: Hovie Lister (1926-2001)
Bill Gaither, with Ken Abraham, It's More Than the Music: Life Lessons on Friends, Faith, and What Matters Most (New York: Warner Faith, 2003).
Derrick Henry, "Hovie Lister, 75, Famous Gospel Musician, Preacher," Atlanta Journal-Constitution, December 30, 2001.
David L. Taylor, Happy Rhythm: A Biography of Hovie Lister and the Statesman Quartet (Lexington, Ind.: TaylorMade WRITE, 1994).
Freeman, Greg. "Hovie Lister (1926-2001)." New Georgia Encyclopedia. 16 July 2018. Web. 14 January 2021.
Gospel Music Association Hall of Fame: Hovie Lister
Southern Gospel Music Association: Hovie Franklin Lister
Georgia State Capitol
Isaac Hayes (1942-2008)
Kathy Yancey
Fletcher Henderson (1897-1952)
Tobacco Road and God's Little Acre
John Wesley (1703-1791)
George Warren
Robert Conrad Detamore (b. 1947)
Lena Horne (1917-2010)
John D. Young
Howard Swanson (1907-1978)
Robert G. Jackson
Charles B. Cluskey (ca. 1808-1871)
Hank Margeson
Arlan Oftedahl
Ocmulgee Mounds
Peter Tondee (ca. 1723-1775)
Macon State College
Branch Mint at Dahlonega
Sidney J. Marcus (1928-1983)
Lemuel Grant (1817-1893)
Vernacular Gardens
Rural Electrification Act
Georgia Civil War Commission
Public Archaeology Programs
Natasha Trethewey (b. 1966)
Clay Eating
Tull Charitable Foundation
Harold Bright

redpajama_set_name: RedPajamaCommonCrawl


text:
Osoby o tym nazwisku:
 Armido Gasparini – rzymskokatolicki duchowny
 Francesco Gasparini – włoski kompozytor, kapelmistrz, chórmistrz i pedagog
 Michele Gasparini – włoski kierowca wyścigowy
 Mitja Gasparini – słoweński siatkarz
 Rich Gaspari – amerykański kulturysta
 Sandra Gasparini – włoska saneczkarka

redpajama_set_name: RedPajamaWikipedia


text:
Movie Review: 'Fast & Furious 6'
by Nat Almirall, May 24 2013
So Dominic Toretto (Vin Diesel) and his crew are back. Luke Hobbs (Dwayne Johnson) and his crew are back. And some new faces, Hobbs' partner Riley (Gina Carano) and villain Owen Shaw (Luke Evans) dd to the franchise's ever-growing roster.
I thought the vast number of characters would be too much of a distraction in the last installment, and, to an extent, it was. But I liked Fast 5, and I'm happy to say that 6 is even better.
The premise is all but unnecessary. Hobbs recruits Toretto to take down Shaw, a gangster/terrorist/arms dealer whose goal is not particularly clear, but then it doesn't need to be, does it? Shaw's latest plan is to steal a military whatsit that can cause blackouts, and his gang of thugs, the minute you meet them, seem like the evil twins of Toretto's — and it's a credit to the movie's cleverness that they actually address the parallels and get some great character beats in from Roman (Tyrese Gibson) and Tej (Chris Parker), two fellows whose roles in the previous films were to stand around and wait for something to do.
Hobbs' carrot is a photo of Toretto's former lover Letty (Michelle Rodriguez), knowing that Toretto, big softy that he is, won't be able to resist…uh…chatting up old flames? You'd wonder why she never bothered to call him when he figured her dead, but apparently she's under a nasty bout of amnesia and, big reveal, she's working for Shaw's team!
And O'Connor (Paul Walker), Toretto's brother in law has a baby. The rest of them haven't done much since their big score in Rio, save spent their cash and bask in opulence. For those keeping count, Han (Sung Kang) and his gal Gisele (Gal Gadot) round out the rest.
So Toretto 'n' his gang head to London, engage Shaw and his group in a great chase sequence, and discover Letty among the ne'er-do-wells. There's the usual explosions, crashes, weird cars (Shaw's custom job has a modified cow catcher which he uses to blast others either out of the way or straight over top of his own vehicle), and an interesting weapon that, once shot at the hood of a car, can be used to slam the brakes — it's a great idea, but it never shows up again.
Shaw gets away, so Toretto and co. go on recon, leading to a great fight between Riley and Letty and Tej, Roman, and Shaw stooge Jah (The Raid: Redemption's Joe Taslim) in a tube station, and a bit later Toretto and Letty square off at one of London's underground racing events (why one exists is an oddity, especially since the characters often point out how there's a camera monitoring every corner of the city). And then there's…
Basically a movie like this exists for the action, the fast cars, the characters, and the explosions. Fast & Furious 6 delivers. The story barely sticks together even if you try to ignore it (just how damn long is that runway in the last bit?), but its action sequences are less erratic (director Justin Lin has picked up a few things in interim between this and Fast 5), the characters work together so much more — well, they already worked, but we get to spend some more time with them — and the cars, while they're close to being wedged in, are used with just enough cleverness that they justify the scenes even when nothing in the script does. For a series that could and likely is often written off as big, dumb action, few others who share a similar derision continue to improve like the Fast & Furious franchise — it knows where it's going, and at every marker, it presses the gas just a little harder.
And please give Roman and Tej their own spin-off.
Posted in: Action · Movies · Reviews · Universal Pictures
Tagged: Chris Bridges, Clara Paget, Dwayne Johnson, Elsa Pataky, Fast & Furious, Fast and Furious, Fast Furious 6, Gal Gadot, Gina Carano, Joe Taslim, John Ortiz, Jordana Brewster, Kim Kold, Luke Evans, Michelle Rodriguez, Paul Walker, Rita Ora, Shea Whigham, Sung Kang, the fast & the furious, The Fast and the Furious, Tyrese Gibson, Vin Diesel
One Response to "Movie Review: 'Fast & Furious 6'"
Fast and furious 6 hack says:
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Our Model 3LO-882 is a highly reliable latching inertia/impact switch used for various military and commercial aviation applications. The specified g level is preset at the factory. When predetermined g levels are exceeded by the aircraft in any direction, this switch will automatically turn on or off certain systems such as: voice or data recording, emergency lighting, and fire control. The device remains latched until manually reset. A D38999 connector ensures reliable termination in harsh environments. This unit is an excellent choice for helicopter applications.


Mid-Thursday afternoon, the city announced Thursday's public hearing on the plan has been postponed to April 12.
Delnor Hospital in Geneva has plans for a $39.6 million expansion and renovation of its surgical services department.
Representatives from the hospital will come before the Geneva Plan Commission in a public hearing April 12. It was rescheduled from Thursday night.
The hospital, at 300 S. Randall Road, needs to amend its original development plan to allow for a two-story addition.
The work would be the most significant change to the surgical area since the hospital moved to Randall Road 25 years ago.
According to city and state documents, Delnor will remodel its 10 inpatient operating rooms and several procedure rooms. The current operating rooms vary in size, and some are too small to accommodate some specialized medical equipment. The new rooms will all be the same size.
It will also add another 11 pre- and postoperative stations to meet current state-mandated ratios for stations-to-operating rooms. One of the recovery bays will be designated as an isolation area, and another will be designed for pediatric patients.
The renovation does not involve the outpatient surgery center, except that its operating rooms may handle some of the caseload as the other operating rooms are unavailable.
According to its state application, it would like to start construction in April and finish by August 2021.
A state review of the application shows surgical cases in the hospital increased by about 17 percent from 2011 to 2016. More than one-third of the total cases in 2016 were for people 65 and older, and the population of people that age and older in the hospital's service area will likely increase by 20 percent through 2022, according to a state review.
The Illinois Health Facilities and Services Review Board approved the plan in January.


text:
10 of the Craziest Traditional Remedies from Across the Globe
by Jamie Waddell
We like to think that living in the scientifically enlightened twenty-first century means the end of superstitious hokum and crack-pot medical theories. But we'd be wrong. We've compiled a list of the 10 most bizarre traditional remedies that the world has to offer. Do any of them hold scientific merit?
1. Bee Sting Therapy – You Better Believe it
I'm no stranger to imbibing a honey and lemon concoction form to cure my sore throat, but the idea of willingly being stung by a bee doesn't hold quite the same appeal. Firstly, the poor bee leaves his guts attached to the stinger which, unsurprisingly, kills the little blighter. Secondly, venom's bad, right?
The history of using bee venom is a varied one, utilized all across the ancient world. The greatest societies in history: Greece, Egypt and China all swore by the curative powers of bee venom. Are you really going to doubt the ancient Egyptians?
Now, I'm not volunteering or anything, but modern scientific study has shown that bee sting therapy can help those suffering from wasting diseases like multiple sclerosis.
Pat Wagner was diagnosed with MS in 1970, and for 22 years she found little relief from the heaps of medications prescribed to her. Then, in 1992, she decided to take the drastic measure of this obscure therapy. She willingly had, wait for it, 200 bee stings in one sitting! The results, she said, were almost instant: within a couple of hours she no longer felt cold; within a couple of days she had a noticeable increase in energy; and within a couple of weeks she regained the hearing in her right ear.
2. Turtle Power!
Despite living in filthy swamps and carrying salmonella, the terrapin is held in high esteem by the people of Cambodia.
The evidence for the medicinal benefits of terrapins is purely anecdotal. There is nothing to suggest that the touch of a terrapin can cure rheumatism, acne, or any of the ailments that the people of Cambodia claim.
Before you laugh at the customs of another nation, remember that there are still plenty of westerners who believe that toads give you warts. They don't, it's a fairytale.
3. It's the Eye of the Tiger
No lead in your pencil? No wind in your sail? Little soldier refusing to stand to attention? Despite the many lighthearted euphemisms, erectile dysfunction can of course be a seriously damaging condition for a man's self-esteem.
But take a moment to think of the poor tiger, whose eyes and penis bone (baculum) are removed and used to cure impotence and virility problems in China. That can't be good for their self-confidence.
The tiger has long been traditional Chinese medicine's majestic walking pharmacy. The tiger's nose is taken for epilepsy, the tail for skin diseases and powdered tiger bones for arthritis muscle conditions.
China's population more than tripled during the 20th century. Is it down to the miraculous power of tiger penis? Doubtful, and with current worldwide tiger numbers estimated to be around 3,200 it might be time for Chinese men to look for a different means of floating their boat.
4. Just a load of hot air
The technique of fire cupping has been around for thousands of years, with archaeologists having discovered evidence that it was practiced as much as 3000 years ago.
It involves creating a vacuum of fire-heated air between the patient's body and a bell-shaped glass cup. Whilst not quite as insane a number as bee sting therapy, around 12 of these cups can be used in a single treatment.
Popular across Asia, fire cupping is used to treat arthritis, congestion and even depression.
No clinical trial has ever found that fire cupping can provide any form of pain relief or serves any medicinal purpose. Any relief is purely psychosomatic.
5. Tip the Scales
This impressive looking beast is the proud pangolin. Looking like an armadillo on steroids, the pangolin is native to tropical regions of Africa and Asia.
Unfortunately for the pangolin, its scales (a unique adaptation amongst mammals) are believed to cure all sorts of ailments, from promoting lactation, reducing swelling and even resisting cancer.
International trade in pangolins has been banned by CITES (Convention on International Trade in Endangered Species) since 2000, but illegal trade still carries on unabated in China and Vietnam. The illegality of the trade in the scales and meat of the pangolin making them an expensive status symbol.
Most sadly, the pangolin's scales hold no medicinal benefit whatsoever. Made of keratin, the same substance as human hair and fingernails, they are becoming endangered simply for having an ostentatious adaptation.
Verdict: Hokum
6. Dr. Maggot M.D.
Another stomach-turner, this ancient practice involves the application of disinfected fly larvae to treat wounds which are failing to heal. Often referred to as just larvae therapy, the larvae are applied to the wound and are kept there for around three days in a special dressing that prevents them from escaping. During this time the maggots begin eating away at the patient's flesh.
Unlike many of the entries on this list, you've probably already heard of this treatment. That's because it actually works. The maggots not only spped up the rate at which the wound heals, but eat away infected tissue, kill bacteria and thus clean the wound.
7. Med-Fishinal
In Hyderabad, India, the Bathini Goud Brothers offer their famous Bathini Fish Medicine free of charge to anyone suffering from asthma. The family have been doing this for over 160 years since Somalingam Goud was given the secret recipe by a Hindu holy man.
The recipe has been kept top secret since that day, as the family believe that revealing it would cause imitators to begin charging for the treatment and the medicine would lose its magical powers.
The bizarre aspect of this treatment is how it is administered. A small murrel fish is fed the medine and then inserted into the patient's mouth whilst it is still alive. This live fish, wagging its tail and fins, swims down the patient's throat and allegedly clears the phlegm that is causing the patient's distress.
Under pressure from doctors around the world, the family freely gave the medicine for scientific analysis in 2004. Although no medicinal quaIities were observed there were also no harmful ingredients in the concoction.
8. Sand in your pants
People in the Siwa Oasis, an area northwest of Cairo believe that being buried alive in sand in the baking heat of the desert can cure rheumatism and sexual impotency.
It doesn't. It just makes you really hot.
9. Up and Atom!
I was under the impression that being blasted by radiation isn't ideal for one's well-being.
Apparently this is not a worry for the people of the Czech Republic, with 20,000 Czechs traveling to the small town of Jachymov every year to submerge themselves in radioactive water. The water itself is run off from a nearby uranium mine, and they believe that it can cure arthritis. I imagine it would be less effective for impotency.
But this is no folk remedy. No, they've built a lavish spa around the radioactive waters. Crazy
10. On the right track
Another folk remedy without an ancient history, the people of Indonesia have taken to lying on train tracks to cure pretty much every disease in the book. This trend kicked off after a man reportedly recovered from a stroke by regularly laying himself down on the tracks. This happened 2 years ago. People have swarmed to train yards across the islands, believing that minor shocks from the rails are medicinal.
10 of the Craziest Traditional Remedies from Across the Globe was last modified: October 10th, 2019 by CPPN
Category Health Tips | Tags: , Bathini Fish Medicine, Bee Sting Therapy, crazy traditional remedies, curious remedies, folk medicines, folk remedies, funny remedies, larvae therapy, medicinal benefits of terrapins, radiation therapy, tiger organs in Chinese medicine, traditional remedies
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