# Wide field Beamformed Observation with MeerKAT

Weiwei Chen1 Ewan Barr1 Ramesh Karuppusamy1 Michael Kramer1,2 Benjamin
Stappers2 1 Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121
Bonn, Germany
2 Jodrell Bank Centre for Astrophysics, University of Manchester, M13 9PL, UK

###### Abstract

Large scale beamforming with radio interferometers has the potential to
revolutionize the science done with pulsars and fast radio bursts by improving
the survey efficiency for these sources. We describe a wide field beamformer
for the MeerKAT radio telescope and outline strategies to optimally design
such surveys. We show initial results using the beamformer by observing a
globular cluster to track several pulsars simultaneously and demonstrate the
source localization capability of this observation.

###### keywords:

Instrumentation; Pulsar; Interferometry; Transient

##  1 Introduction

Observing the Universe with ever higher time resolution has enabled many
scientific discoveries over the last half century. For example, the 100 -ms
resolution observations that lead to the serendipitous discovery of the first
radio pulsar PSR B1919+21 (Hewish et al., 1968) could aptly be said to have
opened a new window to the Universe. The discovery of pulsars was momentous,
showing for the first time that such short duration astronomical phenomena
could be observed. The discovery further provided empirical evidence for the
existence of neutron stars, thereby opening up a new field in astronomy.
Subsequent sub-ms resolution radio observations uncovered the first
millisecond pulsar (MSP, Backer et al., 1982), establishing a new class of
fast spinning neutron stars. These paved the way to a clearer understanding of
stellar evolution and the physics of dense matter. Owing to their formation
mechanism, a significant fraction of MSPs are found in tight orbits with
massive companions and serve as test beds for theories of gravity (Kramer,
2004). More recently, the discovery of Fast Radio Bursts (FRBs), aided by high
resolution observations, has provided a new tool in the study of the Universe
on cosmic scales (Lorimer et al., 2007). While over 100 FRBs have now been
discovered (Petroff et al., 2016), the nature of the progenitors is unclear.
Recently, FRBs have been used to measure the cosmic baryonic density (Macquart
et al., 2020), thereby making them true cosmological probes. The foregoing
discussion clearly argues for an increase in the number of FRBs and pulsars or
even new unknown phenomenon and thus motivates the importance and need for
further high time resolution observations.

While large single dish telescopes provide the raw sensitivity required, their
small fields of view (FoV) are a major drawback in large pulsar and transient
surveys. To compensate for this, multi-beam receivers are often used on these
telescopes to increase the FoV by a factor of about an order of magnitude,
e.g. the 13-beam receiver on the Parkes radio telescope (Staveley-Smith et
al., 1996). In fact, multi-beam receivers were key to the discovery of FRBs,
as it allowed Lorimer et al. (2007) to distinguish the first discovered FRB
from terrestrially generated radio frequency interference (RFI). The
discrimination of astronomical signals from RFI and the increased FoV combined
with high time resolution has led to a number of successful pulsar surveys;
e.g. the PMPS survey (Manchester et al., 2001) and HTRU survey (Keith et al.,
2010; Barr et al., 2013) have discovered more than 1000 new pulsars using
multi-beam receivers on the Parkes and Effelsberg radio telescopes; the GBNCC
survey (Stovall et al., 2014) with the Green Bank telescope has so far
discovered 173 new pulsars including 22 millisecond pulsars. The PALFA survey
(Cordes et al., 2006) carried out using a 7-beam receiver on the Arecibo radio
telescope has discovered 192 pulsars and the first repeating FRB. The GPPS
survey (Han et al., 2021, submitted) with FAST using a 19-beam receiver has so
far discovered 193 pulsars.

Despite sensitivity and FoV improvements, the discovery of any point-like
sources can greatly benefit from better localisation and a further increase in
FoV. Among other parameters, a larger FoV directly translates to a greater
survey speed, thereby improving the chance of new discoveries. These aspects
are crucial for FRB discoveries because of the largely isotropic sky
distribution of FRBs and their low occurrence rates (Bhandari et al., 2018).
Similarly, a significant number of radio pulsars are found along the Galactic
plane where a large FoV is of value. Equally important in these endeavours is
the need to rapidly localise the sources to arcsecond accuracy. This permits
the unambiguous association of the FRBs to their host galaxy (see e.g.
Macquart et al., 2020). Rapid, precise localisation can also greatly reduce
the time required to converge on an accurate timing model for newly discovered
pulsars. Single dish telescopes have risen to the challenge with the phased
array feed (PAF) – a technological evolution of multi-pixel receivers seeking
to address precisely these requirements, namely larger FoV and better
localisation (Deng et al., 2017). However, the small improvement in
localisation with a PAF is far from the sub–arcsecond accuracy needed to
explore the local environment of FRBs (Marcote et al., 2020).

Radio interferometers comprised of a large number of relatively small
parabolic dishes with sufficiently long baselines can potentially address the
conflicting requirements of a large FoV and the need to simultaneously
localise point sources. Equipped with the right technology, these “small-D,
large-N” interferometers are optimally positioned to become FRB and pulsar
discovery facilities. If $\lambda$ is the observing wavelength, the primary
FoV, $\sim\lambda/D$ from small dishes of diameter $D$, results in a large
FoV. Longer baselines between the dishes modify the interferometric response
resulting in correspondingly narrower synthesised beams with which the primary
FoV can be filled. This increase in FoV and spatial resolution for radio
transients is made possible through the use of digital beamforming. While the
technique is widely used in applications such as radar, sonar and mobile
communications, it has recently become feasible for radio astronomy owing to
affordable signal processing solutions on general purpose graphics processing
units (GPGPUs). We note that further increases in FoV can be achieved if the
dishes in such interferometers are equipped with PAFs (see e.g. ASKAP).
However, in this case the cost of signal processing can increase by an order
of magnitude or more.

##  2 Recent Interferometric FRB/Pulsar Searches

The ability to measure spatial brightness distribution with radio
interferometers and the resulting continuum maps has been indispensable in
targeted pulsar searches. Polarised compact radio sources displaying a steep
spectrum identified from these maps were observed with single dish telescopes
to detect the periodicity, as seen in several interesting pulsars (e.g. Backer
et al., 1982; Staelin and Reifenstein, 1968; Navarro et al., 1995). Similarly,
Roy et al. (2012) report targeted searches of sources identified as potential
pulsars from the FERMI gamma ray observations by forming up to 16 beams on the
sky. Unlike the preceding targeted searches, limited pulsar and transient
surveys were carried out with the ingenious use of regular radio arrays as
outlined in Janssen et al. (2009) and Rubio-Herrera et al. (2013).

Recent advances in technology have made large-scale surveys for pulsars and
transients possible. The LOFAR telescope (Stappers et al., 2011) was conceived
with transient search capability since its inception. Using the facility,
Coenen et al. (2014) reported two surveys for pulsars, one of which covered
$\sim$ 600 deg2 using 19-beams on the sky. Sanidas et al. (2019) reported a
much larger survey using LOFAR’s 222 beams covering the Northern sky. A
similar low frequency survey albeit with a single pixel comprising the
incoherent beam, the GMRT southern sky survey (Bhattacharyya et al., 2016) has
so far discovered 18 pulsars. A unique feature of this survey was the rapid
localisation capability allowing rapid convergence in pulsar timing
characterisation. Using the fast-sample visibility mode of the Karl G. Jansky
Very Large Array (VLA), Wharton (2017) has conducted a deep pulsar search with
300 beams in the Galactic center. The upgraded Molonglo Observatory Synthesis
Telescope (UTMOST, Bailes et al., 2017) generates 352 “fan beams” in the sky
to perform real-time single pulse searches and has made the first FRB
detection with interferometer as well as accruing a growing list of FRB
discoveries. The APERTIF survey (Adams and van Leeuwen, 2019) with the
Westerbork Synthesis Radio Telescope (WSRT) and the Australian Square
Kilometre Array Pathfinder (ASKAP, Johnston et al., 2008) are the first two
interferometers to be equipped with the PAFs to increase their FoV. This has
lead to further FRB discoveries (Bannister et al., 2017; Connor et al., 2020).
The Canadian Hydrogen Intensity Mapping Experiment (CHIME) has led the field
in the hunt for FRBs. It consists of four cylindrical reflecting surfaces
without moving parts which provides an instantaneous field of view of $\sim$
200 deg2. More than 1000 new FRBs have been detected with its hybrid
beamforming pipeline (Ng et al., 2017).

Multi-beam observations have become a requirement for the best FRB searches,
as this allows for robust discrimination against RFI. The latter is usually
detected in multiple beams owing to their near-field, terrestrial origins.
However, much of the interesting cosmological applications of FRBs are
possible only if the FRBs can be associated with a host galaxy often requiring
arcsecond spatial resolutions. Just as in pulsar searching, large-N, small-D
interferometers equipped with transient voltage buffers are ideal tools for
FRB searches and subsequent localisation.

MeerKAT (Camilo, 2018), an interferometer with an array of 64 dishes in the
Northern Cape of South Africa, is a new member of this group. It has baselines
up to 8 km and has a total collecting area of 9161 m2, making it the most
sensitive radio observatory in the Southern Hemisphere. MeerKAT itself can
currently only form one synthesized beam. However, through the use of an
externally developed total power beamformer (Barr, 2018), it can improve the
field of view of beamformed observations by generating up to 1000 beams on the
sky with high time and spatial resolution while achieving sensitivity on par
with large single dishes.

In this paper, we discuss the properties of a general purpose multi-beam
beamformer for a radio interferometer. Observations using such a beamformer
must be carefully planned to account for the time, frequency and array
configuration dependence of the synthesised beam. We present a software tool
able to simulate the synthesised beam for an arbitrary array and generate
optimal tilings of beams for point source surveys. We discuss how this tool
has been integrated into the above-mentioned beamformer developed by the
Max–Planck–Institute for Radio Astronomy (MPIfR).

##  3 Digital Beamforming

Beamforming is a signal processing technique that modifies the spatial
response of an array of sensors such as hydrophones, microphones, radar and
radio astronomical receptors, resulting in an enhanced sensitivity in a
particular direction. This is done by the phase coherent summation of the
signals (see e.g. Van Veen and Buckley, 1988) from the individual receptors.
For a monochromatic signal, this can be expressed as

$F(t_{ref})=\sum_{n=1}^{N}f(t_{ref}-\tau_{n})w(\tau_{n}).$ (1)

The result of the summation, $F(t_{ref})$ in Equation 1, represents a
directional beam on the sky with a gain equivalent to the aggregate gain of
$N$ antennas. Coherence is achieved by referencing the signal
$f(t_{ref}-\tau_{n})$ received at antenna $n$ to a common phase centre $ref$
through the application of delay correction $w(\tau_{n})$. The delay
$\tau_{n}$ is a function of the desired pointing direction $u(\theta,\phi)$
and the position of the antenna $r_{n}(x,y,z)$ in the array with respect to
the phase center $ref$ as written in equation:

$\tau_{n}=\frac{u(\theta,\phi)\cdot r_{n}(x,y,z)}{c}$ (2)

where $c$ is the speed of light. The delay $\tau_{n}$ only represents the
geometric delay, any other delays in a system (e.g., instrumental delays) are
not included.

Beamforming can be implemented in multiple ways. Here we will only discuss
digital beamforming and assume that an initial stage in which the “coarse”
delay (the integer sample delay) is corrected for, leaving only “fine” delays
(sub-sample delays) remaining. This remaining fine delay may be corrected for
in either the time domain, through the application of a non-symmetric finite
impulse response (FIR) filter (see e.g. Frost, 1972), or in the frequency
domain following Fourier shift theorem. Additionally, these corrections may be
applied directly to the signals from the receptors or to the cross-
correlations of those signals, so-called “visibilities” (Thompson, 1999).

The visibility-based approach has used in Roy et al. (2018), where it is
described by the following expression:

$B(k,l)=\frac{1}{MN}\sum_{m=0}^{M-1}\sum_{n=0}^{N-1}V(m,n)e^{j2\pi\big{(}\frac{k}{M}m+\frac{l}{N}n\big{)}},$
(3)

where $B(k,l)$ is a coherent beam at sky position $(k,l)$, $M$ and $N$ are the
grid numbers of the axes in the UV plane. $V(m,n)$ is the visibility of
location $(m,n)$ in the UV plane. Typically, correlators produce visibilities
only every few seconds. While this is sufficient for imaging science, high-
time-resolution science such as pulsar searching and timing require
resolutions of the order of microseconds. For most arrays, particularly those
without redundant baselines, operating a correlator with microsecond dump
times would result in the production of more data than could reasonably be
processed.

When dealing directly with the signals from the receptors, compensation for a
delay $\tau$ in the time domain can be made by a phase shift of
$e^{-j\omega\tau}$ in the Fourier domain:

$\int_{-\infty}^{\infty}f(t-\tau)e^{-j\omega
t}dt=e^{-j\omega\tau}\int_{-\infty}^{\infty}f(t^{\prime})e^{-j\omega
t^{\prime}}dt^{\prime}\ ,$ (4)

where $t^{\prime}=t-\tau$. For broadband signals it is first necessary to
Fourier transform the voltages from each receptor such that the delays can
then be compensated in the form of a frequency-dependent phase shift
$e^{-j\omega\tau}$, often referred to as a “weight”. While the approaches in
the time domain and Fourier domain are analogous, delay compensation in the
Fourier domain is used in the majority of modern radio interferometers as it
allows for the reuse of the F–stage (Fourier transform) in an FX correlator
(Fourier transform followed by cross-correlation). Thus, this paper focuses on
beamforming applications achieved by phase shifting.

##  4 Multi-beam Observations

As noted in the introduction, multi-beam beamformers can greatly enhance the
scientific capabilities of an interferometer through increasing point-source
survey speed. However, to be effective in such surveys, care must be taken
during observation planning as the survey speed and completeness is dependent
on the ever-changing shape of the synthesised beams and the tiling of those
beams on the sky. In this section, we discuss the simulation of the
synthesised beam shape and the generation of efficient tiling patterns.

We can decompose the problem of efficient multi-beam observations into three
components:

  * •

Determination of the synthesised beam shape

  * •

Packing of the beams to generate a tiling

  * •

Time-dependent evolution of the tiling pattern

To solve this problem, we have developed
Mosaic111https://github.com/wchenastro/Beamforming, a pure Python software
package that implements tools to characterize the beam shape and generate
efficient tilings.

###  4.1 Beam shape simulation

Provided with an arbitrary set of antenna positions and a source position, the
Mosaic package determines the instantaneous UV coverage of the array and uses
a sparse discrete Fourier transform approach to recover the point spread
function (PSF) of the array with a user-defined spatial resolution and FoV.
The problem is simplified by assuming a monochromatic array. While this
assumption does not hold for modern wide-band interferometers, it speeds up
the process of PSF generation. Furthermore, the decision of which frequency to
optimise for an observation is often dictated by source and science goals. As
such, frequency selection is handled on a case-by-case basis. A natural
weighting scheme (equal weight per antenna) is assumed during the generation
of the PSF.

Due to the generally poor instantaneous UV coverage of interferometers, the
shape of the synthesised beam is often complex. As illustrated in Figure 1,
the region bounded by a certain response level may be irregular. However, for
the purposes of later tessellation of beams in a tiling, it is helpful to
approximate it as an ellipse. In this way, the beam shape is represented as a
pair of semi-major, $a$, and semi-minor, $b$, axis lengths and the orientation
of the beam, $\theta$, as shown in Figure 1. However, a single set of semi-
axes and angle neglects the variation of the beam shape at different response
levels. For this reason Mosaic produces a series of contours for the
synthesized beam at different response levels. For each response level the
beam is approximated by fitting an ellipse to the respective contour, using an
algorithm derived from Fitzgibbon et al. (1999). In the end, this set of
approximations is interpolated over to construct a smooth model of the beam
shape as a function of response level that can be used during tiling.

Figure 1: Beam shape approximation: The background shows a simulated PSF for a
MeerKAT observation using 54 equally weighted dishes. The orange ellipses
represent the approximations of the beam at 30%, 50% and 70% response levels
as determined by the modelling procedure (see text). The semi-major, $a$, and
semi-minor, $b$, axes and orientation angle, $\theta$, of the modelled beam at
the 50% response level are also indicated. The orientation angle is measured
from the east of the image plane. Black closed lines show the corresponding
true contours of the PSF at the 30%, 50%, 70% and 90% response levels.

###  4.2 Beam tiling

Figure 2: Beam shape variety and tiling: The PSF, fitted beam shape and tiling
configuration for the inner 44 antennas of the MeerKAT array when pointing at
zenith (top, with 405 beams) and near the horizon (bottom, with 409 beams).
The sky coverage of the tiling on the bottom is noticeably larger than the one
on top due to the shallower projection angle of the array when observing at
lower elevation.

With the beam shape approximated to an ellipse, an efficient tessellation can
be achieved through hexagonal packing. For the purposes of observation, we
generally optimise for sensitivity by tiling the beams around the array
boresight where the directional gain of the individual antennas is maximised.
This results in a circular bound to the tiling pattern. An optimal solution to
this bounded packing problem is beyond the scope of this work. Instead,
efficient packing is achieved by tiling the beams inside a circle (that
encapsulates the tiling in Figure 2) whose radius is estimated from the area
and number of the beams, then adjusting both the radius of the circle and the
position of the beams until the difference between the desired number of beams
and the number inside the radius is within a user-defined tolerance. As
different science goals often require a different uniformity of sensitivity,
the separation of the beams may be tuned by setting an “overlap” parameter.
This determines the power level at which neighbouring beams intersect. An
overlap of 50% implies that the beams intersect at their half-power points.
Two examples of generated PSFs for MeerKAT (discussed further in Section 5)
and the corresponding circular tiling pattern are shown in Figure 2.
Alternatively, a certain number of beams can be packed inside a boundary with
a fixed size by adjusting the amount of overlap. In this approach, a new beam
shape will be acquired from the beam model whenever the overlap ratio is
adjusted such that the sensitivity of the tiling is maximised within the
boundary. To facilitate different observational strategies, several shapes of
tiling were developed, including hexagon, ellipse, polygon and annulus. For
the case of annulus, different overlap ratios can be used in the “ring region”
and the central region to achieve variable beam covariance.

There are two important caveats to note here. First, we assume the desired
tiling covers a sufficiently small solid angle, such that the shape of all
beams in the tiling can be approximated by the boresight beam shape. Secondly,
we assume that the synthesised beam may always be reasonably approximated by
an ellipse at some response level. This assumption does not necessarily hold
for all array configurations and response levels. As such, when a specific
minimum sensitivity is required across the tiling pattern, it is necessary to
perform a priori simulations of the observation to determine a suitable
“overlap” parameter.

###  4.3 Tiling evolution

In the beam tiling process described above, the generated tiling is valid for
a single array configuration, frequency and epoch. When tracking positions
that are not in a fixed frame with respect to the array, there are two
important effects that alter the tiling. The first is the array projection,
whereby the instantaneous UV coverage of the array changes with the position
of the source, thus changing the PSF and therefore also the approximated
synthesised beam shape. The second is parallactic angle rotation whereby beams
with fixed celestial coordinates will orbit the array boresight due to the
rotation of the Earth. These effects are shown in Figure 3. This changing of
both beam shape and position results in tilings becoming inefficient over time
as the beams become more or less covariant and the sampling of the desired
tiling region becomes non-uniform. The magnitude of this effect can be
simulated. However, the metric by which one evaluates the efficiency of the
tiling is necessarily dependent on the goal of the observation. For example, a
targeted point source observation with a search region smaller than the tiling
pattern may optimise for a more uniform sensitivity, accepting large increases
in beam covariance during an observation as long as the region of interest is
covered. Alternatively, a blind point source survey may optimise for survey
speed by minimising the acceptable changes in beam covariance with time. For
the purpose of demonstrating the effects of tiling evolution, in this article,
we define the tiling efficiency as the fraction of the tiling area which is
neither oversampled nor undersampled. This idea is illustrated in the images
on the right of Figure 3. Here we treat all beams as their approximated
ellipses and sample the tiling pattern to find regions covered by only one
beam (shown in green), more than one beam (a proxy for excess covariance or
oversampling, shown in red), no beams (a proxy for undersampling, shown in
blue).

Figure 3: Overlap evolution through time: The images on the left are the
simulated sky power distribution inside the tiling. The images on the right
are the profile of the beams with their widths plotted at their 50% power
level. The top row shows the situation at the start of the observation with an
optimized tiling and an hour angle of 285∘. The middle row shows the situation
6 hours later with an hour angle of 15∘, large gaps exist between the beams.
The bottom row shows the situation 10 hours after the start of the observation
with an hour angle of 60∘ when the beams are overlapped with each other at a
high ratio while large gaps exist at the same time. These simulations are for
illustrative purposes and the scale would depend on the array configuration.

In practice, it is incumbent on the observer to simulate the evolution of the
tiling efficiency over the course of an observation to make decisions on if
and when re-tilings should occur. Re-tilings result in all beams changing
their tracking positions and thus may be problematic for observation types
that require unbroken integrations on specific positions such as coherent
pulsar searches. The rate of change of the tiling efficiency and its impact on
observations is explored in greater depth in Section 5.3. The techniques
described above are implemented in Mosaic for arbitrary arrays. Below we look
specifically at beamforming with the MeerKAT array and discuss Mosaic’s
deployment as part of the multi-beam beamformer.

##  5 Multi-beam observations with MeerKAT

###  5.1 The FBFUSE beamformer

To support time domain pulsar and fast transient discovery at MeerKAT, the
Max–Planck–Institute for Radio Astronomy has developed a multi-beam beamformer
capable of forming of the order of 1000 total power beams using the full
MeerKAT array. This beamformer, known as FBFUSE (Filterbanking Beamformer User
Supplied Equipment, Barr, 2018), provides the data products required by the
TRAPUM experiment (Stappers and Kramer, 2016) to perform target searches for
new pulsars, by the MeerTRAP project (Sanidas et al., 2018) to perform real-
time commensal searches for fast radio transients and by the MeerKAT Galactic
Plane Survey (MGPS) to perform a wide-field survey for pulsars in the Galactic
plane at L-band and S-band Kramer et al. (2016). The technical details of the
FBFUSE instrument will be addressed elsewhere. Here we will focus only on how
the Mosaic software is used as part of FBFUSE and the implications of the
effects described in Section 4 on observations.

The delay compensation chain for observations with FBFUSE is shown in Figure
4. Voltage signals sampled at individual antennas are first propagated to the
MeerKAT F-engines where channelisation is performed. The F-engines compensate
for the geometric delay to the source under observation through first a coarse
delay correction prior to channelisation and then a fine phase correction post
channelisation. The channelised voltages from the F-engines are then
transmitted via network to the FBFUSE instrument where complex gain correction
provided by the MeerKAT Science Processor sub-system is applied to correct for
the instrumental delays among the streams from all antennas at all frequencies
(depending on the telescope and observation configuration, this correction can
also be applied inside the F-engines). At this stage, a direct summation of
the signals from each antenna would result in a coherent beam at the telescope
phase centre. To produce tilings of off-boresight beams, FBFUSE invokes Mosaic
to generate both an efficient tiling of beams on the sky and the corresponding
delay polynomials required to re-phase the channelised voltage data to each
beam position.

Figure 4: Delay compensation chain for FBFUSE observations: The signals from
individual antennas first go through the F-engine where they are coarse delay
corrected, channelized using a polyphase filterbank and fine delay corrected.
After the F-engine, the signals are all phased with reference to the telescope
phase centre (usually the boresight pointing position). In FBFUSE a per-beam
differential delay compensation calculated by Mosaic is applied to the signal
to form coherent beams at sky positions defined by the tilings being used.
Figure 5: Beamforming procedure. Process units are in white boxes and the grey
boxes are inputs or outputs to or from the process units. The beam shape model
is a table of semi-axis and angles indexed by response level.

The tiling and delay pipeline used in Mosaic is shown in Figure 5 and is
described below:

  * •

Beam shape simulation: FBFUSE retrieves the observer requests and provides
Mosaic with the desired epoch of observation, the current MeerKAT array
configuration and the observing frequency. Using these, Mosaic simulates the
PSF at the array phase centre accordingly. A beam model is constructed by
fitting ellipses to the contours of the PSF at different response levels.

  * •

Tiling generation: FBFUSE retrieves one or more predefined tiling
configurations requested by the observer which include the number of beams in
each tiling, the shape of the tiling boundary, their overlap or the boundary
size and the central position of the tiling. As noted above, for an optimal
gain response the beams should be tiled around the array boresight where the
directional gain of the antennas is maximised. However, for more flexibility,
Mosaic supports the generation of an arbitrary number of off-boresight tilings
to support observations in which there are multiple extended sources of
interest within the telescope FoV. With the requested tiling configurations,
Mosaic derives the corresponding semi-axes and the orientation angle from the
beam model and generates a set of beam positions in equatorial coordinates. At
this stage, the observer may also specify independently tracking beams not in
a tiling to cover known point sources within the FoV.

  * •

Delay polynomial generation: For each of the beam coordinates $b_{i}$
generated in the preceding step, the geometric delay terms $d_{b_{i}}(t)$, are
calculated . Additionally, the geometric delay to the array phase centre,
$d_{\circ}(t)$, is calculated. As the data received by FBFUSE have already
undergone delay correction to the array phase centre, it is the differential
delay between each beam position and array phase centre, $\delta
d_{b_{i}}(t)=d_{b_{i}}(t)-d_{\circ}(t)$, that is required by FBFUSE to produce
coherent beams. The delay values, $\delta d_{b_{i}}(t)$, are calculated for
each beam and a first order polynomial is fit to the values. The validity time
of the delay polynomial depends on the maximum distance between the array
phase centre and a FBFUSE beam. In practice, FBFUSE periodically invokes
Mosaic to calculate new polynomials which are evaluated inside the beamformer
to produce weights.

###  5.2 Point source survey speed

The point-source survey speed achievable with beamformed observations with
MeerKAT is dependent on the array configuration, sensitivity of the individual
antennas, beamforming efficiency and the total number of beams that can be
produced. We estimate the survey speed figure of merit (FoMSS) using equation
(3) of Cordes (2009):

$FoMSS=B\frac{N_{\rm{beam}}\Omega_{\rm{beam}}}{N_{sa}}\left(\frac{f_{c}A_{e}}{mT_{\rm{sys}}}\right)^{2},$
(5)

where $B$ is the bandwidth of the receiver, $N_{\rm{beam}}$ is the number of
synthesized beams, $\Omega_{\rm{beam}}$ is the field of view of a single
synthesized beam, $N_{sa}$ is the number of sub arrays, $A_{e}$ is the
effective collecting area of the array, $f_{c}$ is the fraction of $A_{e}$
that is available, $m$ is the threshold of signal to noise ratio (S/N) for
detection and $T_{\rm{sys}}$ is the system temperature. Currently, the FBFUSE
beamformer may only be used in a single MeerKAT sub array making $N_{sa}=1$.

Figure 6 shows the $\rm{FoMSS}$ for multi-beam observations with MeerKAT under
different array configurations, numbers of beams and source elevations. Here
we consider a simple array definition based on the incremental inclusion of
dishes in order of increasing distance from the array centre. Initially, the
$\rm{FoMSS}$ values increase because of the addition of the core dishes and
the integrated FoV of the beams exceeding the FoV of the primary beam (here
defined as the FWHM of the primary beam at an arbitrary frequency). However,
as more dishes further away from the core are included, the beams become
smaller and eventually reach a turning point where they can no longer fill the
primary FoV. The array size that maximises the $\rm{FoMSS}$ for different
numbers of beams and source elevations is provided in Table 1. For a typical
observation using approximately 1000 beams, we find that the optimal number of
antennas to use for survey observations is between 33 and 42 depending on the
average elevation of observations.

Figure 6: Survey speed figure of merit ($\rm{FoMSS}$) as a function of array size when the source is at zenith (left) and at an elevation of 30∘ (right). Antennas are included in the array in order of increasing the distance from the array centre. The $\rm{FoMSS}$ is shown for four different maximum beam numbers (we note that although the current FBFUSE beamformer can only produce $\sim 1000$ beams for the full array, it can produce more for smaller arrays due to the lower computational cost of beamforming with fewer antennas). The maximum and minimum angular resolution of the simulated beam for each array size is shown in gray. Table 1: The array size (in number of antennas included radially from the array centre) required to maximise the survey speed figure of merit ($\rm{FoMSS}$) for different numbers of synthesised beams. | Elevation
---|---
$N_{\rm{beams}}$ |  90∘ |  60∘ |  45∘ |  30∘
512 |  25 |  26 |  28 |  32
1024 |  33 |  35 |  38 |  42
2048 |  42 |  43 |  46 |  49
4096 |  49 |  51 |  53 |  57

###  5.3 Tiling validity

Figure 7: Left: The tiling validity time as a function of starting hour angle
at various efficiency thresholds for a source that transits through the zenith
at MeerKAT. Right: The tiling validity time for an efficiency 0.8 tiling as a
function of starting hour angle for sources at a range of declinations.

For long duration observations, it may be necessary to re-tile the beams as
the current tiling becomes less efficient due to the changes in the beam shape
as illustrated in Figure 3. It is therefore useful to determine how long
before a tiling generated at the start of that observation drops below some
pre-defined efficiency. Here we simulated the beam shape evolution with time
for the full MeerKAT array to determine the validity time of a tiling at a
given declination.

Simulated observations start at timestamp $t_{0}$ with an optimized tiling.
The beam shape is then updated with an interval $\delta t$ (1 minute). At each
update, a new beam shape is generated for that epoch and the resultant beams
are positioned at the tiling coordinates from $t_{0}$. The tiling efficiency
(defined in Section 4.3) is then measured. If it exceeds the desired
efficiency, the simulation will continue. If the target efficiency is not met
anymore, the simulation is stopped and the time since $t_{0}$ is recorded as
the tiling validity time. The process then repeats with a new simulation
starting at time $t_{1}=t_{0}+\delta t$. This process is continued until the
source drops below 10∘ in elevation. The simulated validity time with
different efficiency thresholds is plotted in the left panel of Figure 7. This
shows the validity time of the tiling as a function of the hour angle for a
source that transits through the zenith at MeerKAT. The simulation shows that
the validity time has a sharp peak before the source reaches the zenith, then
gradually decreases for the rest of the simulation, with the peak of the
validity time occurring earlier for lower efficiency thresholds. The shape of
this curve can be partly seen as an artifact of the simulation, as we assume
that the optimal tiling is at the beginning of an observation. If instead, we
were to simulate observations in which the tiling was initially non-optimal
but transitioned into an optimal configuration with time the observed curve
would both be smoother and the validity time longer. However, the
determination of the optimal offset between the start of an observation and
the epoch of tiling is non-trivial and makes such a mode of operation not yet
practical for real MeerKAT observations.

The right hand panel of Figure 7 shows the tiling validity for a range of
source declinations. Here we see that declinations that result in lower-
elevation source transits have shorter validity times. This can be understood
by considering that at lower elevations the beams have a higher degree of
asymmetry and as such the rotation of these beams will have a larger effect on
the tiling efficiency.

###  5.4 A beamformed observation of 47 Tucanae

We carried out two observations using the MeerKAT MPIfR multi-beam system to
test and verify the beamforming and tiling system.

####  5.4.1 MeerKAT observations

At 06:00 UTC on May 2, 2020, a 4-hour observation of the globular cluster 47
Tucanae (NGC 104) was made. A total of 56 antennas were used with the majority
being in the MeerKAT core. Mosaic was invoked to produce a tiling of 264 beams
with an overlap ratio of 0.7. The tiling covered a region of approx. 0.45∘ in
R.A. and 0.14∘ in declination. Additionally there were 24 tracking beams
constantly pointing at 24 of the known pulsars in the cluster. The packing of
the tiling on the cluster is plotted in the central panel of Figure 8
overlaying an optical image of 47 Tucanae. Another 440-s observation on the
pulsar J1644-4559 was carried out at 11:00 UTC on Dec 1, 2020 with 54 antennas
and an overlap ratio of 0.996 to verify the beam shape.

####  5.4.2 Beamforming Efficiency

The first test was to check the efficiency of the beamforming process as it is
the foundation of other applications. If the weight calculation during the
beamforming process is incorrect or not accurate enough, the signal from the
known pulsars might not be detectable because the antenna data are combined
out of phase. To assess this situation, we folded the data of the tracking
beams using the corresponding ephemerides of each pulsar. The average profiles
for six of these pulsars are plotted in the left and right panels of Figure 8
with their positions labeled in the central panel. The average profiles shown
are a close match to those obtained by Freire (2000) and Ridolfi (2017).

Figure 8: Profiles of selected pulsars and the tiling pattern used for
observation of 47 Tucanae. In the central panel, the background is an optical
image of 47 Tucanae obtained from the STScI Digitized Sky Survey (Lasker,
1994), the ellipses are the tiling of the coherent beams at the start of the
observation and the white dots indicate the positions of the known pulsars in
the cluster. The average profiles in the left and right panels are folded from
the tracking beams on the pulsars whose locations are indicated by their
letter designation in the central panel.

To determine the efficiency of the beamforming process, we compared the S/N
between the incoherent beam and the coherent beams. An incoherent beam is
formed by square-law detecting the signal from each antenna and summing them
together. The uncorrelated noise of $N$ independent observation should
increase by $\sqrt{N}$ in the incoherent summing, thus the S/N of a incoherent
beam of $N$ antennas will increase by $\frac{N}{\sqrt{N}}=\sqrt{N}$. On the
other hand, the uncorrelated noise does not contribute significantly during a
coherent summing, assuming the noise has a zero mean. Thus for a coherent
beam, the S/N increases linearly with the number of antennas. So in principle,
the expected gain from the incoherent beam to the coherent beam for this
observation with 56 antennas should be $\sqrt{56}=7.48$.

We folded the incoherent beams and the coherent beams of the pulsar
J0024-7204C, J0024-7204D and J0024-7204E using
DSPSR222http://dspsr.sourceforge.net and compared their S/N listed in Table 2.
The S/N were measured using the pdmp command in the
PSRCHIVE333http://psrchive.sourceforge.net software package. The efficiency is
calculated as:

$\mathrm{Efficiency}=\frac{1}{\sqrt{N}}\frac{S/N_{\mathrm{coherent}}}{S/N_{\mathrm{incoherent}}}$
(6)

Table 2: The S/N of pulsars C, D and F in the coherent and incoherent beams and the derived beamforming efficiency. The efficiency is calculated using Equation 6. We note that the incoherent beam is more affected by RFI than the coherent beams which can lead to boosted beamforming efficiencies when using this method. | J0024-7204C | J0024-7204D | J0024-7204F
---|---|---|---
Incoherent | 82 | 25 | 35
Coherent | 642 | 177 | 270
Gain | 7.83 | 7.08 | 7.71
Efficiency | 105% | 94% | 103%

The incoherent beam is more susceptible to contamination by RFI and variations
in the system noise of individual antennas. For a coherent beam, these effects
are partially mitigated as they are added up out of phase during the coherent
summing (so called “phase-washing”). This effect can lead to the efficiency of
the beamforming being overestimated. Overall, the consistency of the gain
between the expectation and observation indicates that the calculation of the
weights is accurate enough to achieve high coherency.

####  5.4.3 Verification of the simulated beamshape

The second test was the verification of the simulated beam shape by comparing
it to the power distribution of the sky derived from the observation data of
PSR J1644-4559. First, we simulated the beam shape of the array pointed to the
source at the planned observation time. Then we observed the source with a
very high overlap ratio tiling to densely sample the PSF as illustrated in the
left panel of Figure 9. The beamformed data were folded and analysed to obtain
the S/N measurements of the pulsar in each beam. The observed beams are
represented by ellipses with colors normalized by their S/N and plotted in the
right panel of Figure 9 according to their sky position. The comparison shows
the power distribution of the sky is in high agreement with the simulated PSF
including the complex structure at the outer region of the beam shape. We note
that there is a small position offset of $\sim$0.796 arcsecond in the
beamformed data probably due to a minor phasing error.

Figure 9: Verification of the beam shape: The image on the left is the
simulated PSF covered by a tiling with the colour bar representing the
relative gain of the beam. An overlap ratio of 0.996 was used for the purpose
of supersampling the PSF. The image on the right is the power distribution of
the sky based on a real observation beamformed using the same tiling, array
configuration and boresight pointing. The ellipses represent the beams and the
color is normalized according to their S/N.

####  5.4.4 Tracking the change of the beam shape and the tiling

The third test performed was the verification of the change of the beam shape
and the tiling by tracking variations in S/N of 47 Tucanae pulsars in multiple
beams. There was no re-tiling during the 4-hour observation, so the efficiency
of the tiling was expected to vary due to the changes of the beam shape as a
function of time as discussed in Section 4.3 and illustrated in Figure 3. As a
result, the relative distances between a pulsar and the surrounding beams also
change which manifests in their S/N. By simulating the evolution of the
tiling, we can predict the trend of the change of the S/N of those beams and
verify them with the data.

To perform this test, the observation was divided into 4, 1-hour segments and
the surrounding beams of each pulsar were folded using the known ephemeris. To
mitigate temporal variations due to interstellar scintillation, the S/N of the
surrounding beam was normalised by the S/N of the corresponding tracking beam
pointing directly to the respective pulsar. The result for the surrounding
beams of pulsar J0024-7204Q is plotted in the top panel of Figure 10. The
simulated packing of the beams at the start and the end of the observation are
plotted in the bottom panels of Figure 10. The black dots in the low panels
indicate the known location of pulsar J0024-7204Q and the beams are rotating
clockwise with time.

The result from the observation is in overall agreement with the expectation
from the tiling simulation. The S/N of beams 95, 119, 117 and 97 changed as
expected with the changing distance to the pulsar as shown in the bottom
panels. However, We note that the first and second segments of beams 73 and
141 are not consistent with the expectation. This is likely due to two
factors. Firstly, these beams are the furthest from the pulsar and as
discussed in Section 4.1 the elliptical beam approximation is poor at lower
response levels. Secondly, the beam model here has only been determined for a
single observing frequency. It should be noted that even if it were averaged
over the band, to be representative it would require a weighted spectral
response based on the pulsar’s intrinsic spectral index and scintillation
during the observation.. Despite these two data points, the overall agreement
between the expectation and the observation indicates that the tiling is
correctly generated and implemented in the sky. It also suggests that the
fitted ellipse can largely represent the beam shape and provides a
verification of the evolutionary behaviour of the tiling shown in Figure 3.

Figure 10: The change of the observed S/N and beam packing for the surrounding
beams of PSR J0024-7204Q. In the top panel, the S/N of each beam is plotted
for each 1-hour segment of the observation. The S/N values are normalized by
the S/N of PSR J0024-7204Q in a tracking beam not shown here. The bottom left
and bottom right panels are the surrounding beams of the pulsar at the start
of the observation and the end of the observation, respectively. The black
dots in the bottom panels are known sky position of the pulsar, while the grey
dot is the result of the multi-beam localization for corresponding 1-hour
segment.

####  5.4.5 Localization of a point source using multiple beams

With the tiling verified, the position of a pulsar can be estimated using the
S/N of its detections in multiple beams. As a demonstration of this capability
we have estimated the position of the pulsars in 47 Tucanae using the Markov
Chain Monte Carlo method and compared them with the known pulsar positions. To
fit the position, we first assume that a pulsar is at an arbitrary position
$x$ inside multiple beams $N_{i}$, where $i$ is the index of the beam. If the
S/N of a beam centered at the pulsar is $R_{P}$, then at an ideal situation,
the S/N of the beam $N_{i}$ would be $R_{N_{i}}=R_{P}\times P_{N_{i}}(x)$
where $P_{N_{i}}(x)$ is the sensitivity of the beam $N_{i}$ at position $x$.
The values of $R_{N_{i}}$ can be measured on the observational data and the
function $P_{N_{i}}(x)$ can be constructed with a model of the sensitivity of
the beam. With those, the fitting algorithm generates samples of $x$ and
evaluates them for multiple beams. A similar method has been proposed by
(Obrocka, M. et al., 2015) which uses multiple tied-array beams and spectral
information to estimate the position of a transient source. We fitted the
positions of 6 pulsars with this method and their distances to the established
position are listed in Table 3. As an example, the fitted position for pulsar
J0024-7204Q is plotted as grey dots in the bottom panels of Figure 10 with the
black dot representing the known position from pulsar timing.

Table 3: The distance between the fitted location and the established
localization in units of arcseconds

Segment (Hour) Pulsar 1 2 3 4 J0024-7204C 0.77 3.03 3.79 1.71 J0024-7204D 1.27
3.19 2.26 4.00 J0024-7204E 4.95 3.80 3.12 4.64 J0024-7204F 1.60 1.42 3.08 4.62
J0024-7204I 0.67 1.28 1.37 1.13 J0024-7204Q 2.31 1.40 1.54 1.50

The localization of the pulsars with multiple beams is largely good with the
best segment within 1 arcsecond and the worst segment within 5 arcseconds for
the 6 pulsars that we tested. Larger offsets here are most likely due to the
limitations of the beam modelling combined with biases in the S/N estimation
routine and variations of the spectral response for each pulsar and segment.
Here, we modeled the gain response of the beam as a two-dimension Gaussian. It
is known that this approximation breaks down at lower response levels and as
such S/N estimates from beams further from the pulsar are subject to higher
uncertainties. Nevertheless, these results show that the localization of any
detection can be improved (within the size of the beam) even with a simple
two-dimension Gaussian approximation used here. It further verifies our
approximation of the beam shape and the tiling to a certain level, i.e., we
can use an ellipse as an approximation to the beam shape in most situations.
Regarding localization, a more complete approach would be to use a time and
frequency dependent PSF modeling which will be addressed in the forthcoming
Bezuidenhout et al. in prep.

##  6 Summary and Outlook

In the first part of the paper, we described a general beamforming approach
applicable to modern interferometers with a large number of receiving
elements. This enables wide field, high time and spatial resolution
observations with the sensitivity of all elements coherently combined. To
explore how multi-beam observations using this approach can be conducted, a
software package called Mosaic has been developed. The package provides tools
to simulate and model the beam shape, generate optimized tiling patterns, and
predict the evolution of the tiling through time with the choice of performing
a re-tile automatically when specific criteria are met.

In the second part of the paper, we discussed practical applications for the
aforementioned techniques by applying them to the MeerKAT telescope. It
described how beamforming is implemented on MeerKAT and its use during
observations. To demonstrate the planning of a multi-beam observation, the
validity times of the tilings in different array configurations and source
positions are provided.

Finally, we provide the result of real observations using the MPIfR-developed
FBFUSE beamformer. These demonstrate the capability of the system to observe
multiple pulsars simultaneously with the expected sensitivity and show a high
level of agreement between the simulated beam shape and the reconstructed beam
shape obtained from a high overlap observation. The consistency between the
predicted and observed S/N in each beam around the pulsar location, allowing
for arcsecond position determination, demonstrates that the complete system
from beamforming to PSF determination and tiling operates as expected for
MeerKAT.

The MeerKAT telescope has been using Mosaic and FBFUSE to carry out multi-beam
science observations since 2019. Three principal science programs rely on the
systems presented in this paper. The Transients and pulsars with MeerKAT
(TRAPUM) survey is a pulsar search targeting globular clusters, unassociated
FERMI LAT sources, nearby galaxies and miscellaneous high-energy sources. The
TRAPUM project seeks to discover millisecond and binary pulsar systems that
can be used to answer questions about fundamental physics, the pulsar emission
mechanism and stellar evolution. The survey uses various configurations
depending on the exact sources and science goals of each different target
class. Typically TRAPUM observations use between 277 and 480 beams depending
on the observing frequency used. For observations requiring the widest FoV,
the MeerKAT core is used, while for observations of targets with small extents
(e.g. globular clusters) the full array is used. Currently, TRAPUM operates at
L-band (856 – 1712 MHz) and U-band (544 – 1088 MHz) with plans to include
S-band observations when the receivers become available (Kramer et al., 2016).
Discoveries from the TRAPUM project are made public via the project
webpage444http://trapum.org/discoveries.html.

Complementing the TRAPUM project is the 3000-hour MPIfR Galactic Plane Survey
(MGPS). This project consists of three parts: a shallow L-band Galactic plane
survey, a deep, but limited latitude range, S-band Galactic plane survey and
an ultra-deep 200-hour survey of the region around Sgr A*. The MGPS survey
seeks to greatly increase the number of known pulsars in the Galactic field
with an emphasis on the discovery of relativistic binaries. Additionally, the
Galactic centre portion of the search has the strongest chance to detect a
pulsar in orbit around Sgr A*. The MGPS survey uses the MeerKAT core with 480
beams (Padmanabh et al. in prep.).

More TRAnsients and Pulsars (MeerTRAP) is a project to search for fast
transients commensally with other MeerKAT science programs. It performs real-
time, GPU-based, time-domain searches for short duration transients such as
FRBs and rotating radio transients (RRaTs). It is planned that upon the
detection of a new interesting source, the MeerTRAP project will provide a
trigger to the transient buffer implemented in the beamformer. This allows for
robust localisation and polarisation analysis for discovered transients. A
demonstration of the system has been made through observations of FRB 121102
(Caleb et al., 2020). Due to its nature as a commensal project, MeerTRAP has a
diverse array of configurations. Typically it uses the MeerKAT core with 480
or 864 beams depending on the configuration on the MeerKAT F-engines.

The upcoming SKA-mid (Swart and Dewdney, 2020) will be a similar array to
MeerKAT with an increased number of antennas and longer baselines. Therefore,
the projects mentioned above are demonstrating capabilities envisaged for SKA-
mid but at a smaller scale. Furthermore, the challenges and solutions that we
discussed in this paper or encounter in those projects could be applicable for
this next generation interferometric array.

##  7 Acknowledgment

We thank for the help from the commission group and committee of MeerKAT and
the help from the TRAPUM group. We thank for the help from Benjamin Hugo to
the development of the Mosaic and a useful conversation on localisation with
Vivek Venkatraman Krishnan. We also thank Tim Sprenger for the idea on
calculating the validity time of a tiling and Jason Wu for the help with the
observations. We thank the STScI digitized sky survey produced at the Space
Telescope Science Institute for the optical image of 47 Tucanae. The work of
this paper and the development of Mosaic was funded by BlackHoleCam, the
financial support by the European Research Council for the ERC Synergy Grant
BlackHoleCam (ERC-2013-SyG, Grant Agreement no. 610058) is gratefully
acknowledged. The development of FBFUSE is funded by the Max-Planck-
Gesellschaft. Benjamin W. Stappers acknowledges funding from the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme (Grant Agreement No. 694745). The MeerKAT telescope is
operated by the South African Radio Astronomy Observatory, which is a facility
of the National Research Foundation, an agency of the Department of Science
and Innovation.

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