# Reduced-resolution beamforming: lowering the computational cost
for pulsar and technosignature surveys

D. C. Price  International Centre for Radio Astronomy Research, Curtin
University, Bentley, WA 6102, Australia  [

(TBD; TBD; TBD)

###### Abstract

In radio astronomy, the science output of a telescope is often limited by
computational resources. This is especially true for transient and
technosignature surveys that need to search high-resolution data across a
large parameter space. The tremendous data volumes produced by modern radio
array telescopes exacerbate these processing challenges. Here, we introduce a
‘reduced-resolution’ beamforming approach to alleviate downstream processing
requirements. Our approach, based on post-correlation beamforming, allows
sensitivity to be traded against the number of beams needed to cover a given
survey area. Using the MeerKAT and Murchison Widefield Array telescopes as
examples, we show that survey speed can be vastly increased, and downstream
signal processing requirements vastly decreased, if a moderate sacrifice to
sensitivity is allowed. We show the reduced-resolution beamforming technique
is intimately related to standard techniques used in synthesis imaging. We
suggest that reduced-resolution beamforming should be considered to ease data
processing challenges in current and planned searches; further, reduced-
resolution beamforming may provide a path computationally-expensive search
strategies previously considered infeasible.

###### doi:

10.1017/pasa.2020.32

###### keywords:

radio telescopes; astronomical techniques; pulsars; technosignatures; search
for extraterrestrial intelligence (SETI)

Square Kilometre Array Observatory, Bentley, WA 6102, Australia D. C.
<EMAIL_ADDRESS>TBD

##  1 Introduction

Extracting science results from astronomy datasets can often be a
computationally-demanding process. Supercomputers have become vital tools in
modern astronomy—unfortunately to the point that energy consumption of
supercomputers and research infrastructure dominate astronomy’s carbon
footprint (Stevens et al., 2020; Portegies Zwart, 2020). As well as the
ecological impact, access to computing resources is limited by the technology
and funding available. Observational astronomers must work within these
boundaries to unravel the scientific mysteries and marvels hiding in their
data.

In radio astronomy, one of the most computationally-expensive exercises is
searching for pulsars and other fast transients. In order to detect a new
pulsar with unknown characteristics, one must search across a range of
different periods, dedispersion trials, and pulse widths (see Chapter 6,
Lorimer & Kramer, 2004). Searches are particularly challenging when using
radio array telescopes, as a large number of beams must be formed to cover a
survey area, and the search process must be repeated on each beam.
Nevertheless, there are active pulsar searches on several radio array
telescopes (e.g. Sanidas et al., 2019; Chen et al., 2021; Singh et al., 2023;
Bhat et al., 2023a). Fast Radio Burst (FRB) searches (e.g. Law et al., 2015;
Bailes et al., 2017; Bannister et al., 2017; Ng et al., 2017) must also search
through multiple dispersion and pulse width trials, but are (somewhat) more
manageable as they do not apply a periodicity search.

Another computationally-demanding avenue is the search for extraterrestrial
intelligence (SETI), which seeks to detect ‘technosignatures’ from
extraterrestrial intelligence as a proxy for intelligent life. Indeed, the
SETI@Home project (Anderson et al., 2002) was a pioneer in developing
distributed computing in order to attain sufficient compute resources for data
analysis. At one stage, the SETI@Home network—peaking at over 5.2 m
volunteers—constituted the largest supercomputer on the planet.

Decades later, thanks to exponential ‘Moore’s law’ scaling of compute
capacity, the Breakthrough Listen project (Worden et al., 2017; Isaacson et
al., 2017) are able to search petabytes of data for technosignatures with
modest server clusters located at the observatory (MacMahon et al., 2018;
Price et al., 2018; Lebofsky et al., 2019), making the distributed computing
approach unnecessary. Even so, technosignature searches using radio arrays,
such as the Breakthrough Listen program on the MeerKAT telescope (Czech et
al., 2021) and the recently-announced COSMIC111COSMIC: Commensal Open-Source
Multimode Interferometer Cluster Search for Extraterrestrial Intelligence.
https://science.nrao.edu/facilities/vla/observing/cosmic-seti instrument on
the Karl G. Jansky Very Large Array, are orders of magnitude more challenging
than previous single-dish searches, motivating new search techniques.

With a radio array, when a desired search strategy is limited by the available
compute resources, it is often advantageous to sacrifice sensitivity and
imaging fidelity against instantaneous field of view. As beam width scales
with $\sim\lambda/D$, where $D$ is the longest baseline in the array and
$\lambda$ is wavelength, the number of beams required to cover a given survey
area scales with $D^{2}$. As such, pulsar searches tend to only use the ‘core’
part of an antenna array, and longer baselines are dropped. For example, in
the MeerKAT TRAPUM222TRAPUM: Transients and Pulsars with MeerKAT survey, only
the central 37–41 antennas from the 64-antenna array are regularly used,
corresponding to 57–64% of the peak sensitivity (Chen et al., 2021). The
LOTASS333LOTASS: LOFAR Tied-Array All-sky Survey survey used the dense central
‘Superterp’ core of LOFAR to balance field of view against sensitivity.
Similarly, the Murchison Widefield Array (MWA) SMART444SMART: Southern-sky
Rapid Two-Metre pulsar survey survey (Bhat et al., 2023a) uses the ‘compact’
MWA configuration of 128 tiles within $\sim$800 m.

The computational challenges for pulsar surveys are exacerbated at low
frequencies, as the number of dispersion trials (DMs) required to correct for
dispersive delays due to propagation through the interstellar medium scales
with $\lambda^{2}$. The LOTASS survey searched across 10120 dispersion measure
(DM) trials, which required 30M CPU-core hours on the Cartesius supercomputer
to process data up to Jan 2019 (Sanidas et al., 2019). Based on first-pass
benchmarks, the SMART survey will require $\sim$60M CPU-core hours to process
$\sim$93 hr of observational data using the OzStar supercomputer (Bhat et al.,
2023a). As the SMART team anticipate receiving $\sim$0.5–0.6M CPU-core hours
per semester, they are in the process of optimizing and accelerating their
search pipeline to ensure processing is possible.

In this paper, we introduce a ‘reduced-resolution’ beamforming technique that
allows for sensitivity to be traded off against sky coverage. Our reduced-
resolution beamforming approach, applied after inter-antenna correlation,
down-weights baselines longer than a specified distance. This per-baseline
approach results in appreciably larger sensitivity than simply discarding all
antennas outside of a core region. Reduced-resolution beamforming effectively
covers a desired survey area with a smaller number of beams, for a moderate
decrease in survey speed. By doing so, the downstream compute requirements for
a given survey can be greatly reduced.

This paper is organized as follows. In Section 2, we give an overview of post-
correlation beamforming, introduce a compact tensor notation, and highlight
the relationship between a power beam and a pixel within a synthesis image.
Reduced-resolution beamforming is introduced in Section 3, along with survey
speed and computational cost metrics. In Section 4, we consider the
application of reduced-resolution beamforming to the SMART survey; in Section
5 we argue that the survey speed for MeerKAT pulsar searches could be
increased dramatically by reduced-resolution beamforming, without increasing
the number of beams formed. We conclude with a short discussion in Section 6.

##  2 Post-correlation beamforming

Figure 1: Pathways toward generating power beams—or equally, images—from
antenna voltages. Equivalent tasks are color-coded; dashed lines represent
hybrid architectures.

Tied-array (coherent) beamforming is a common signal processing operation used
to combine the voltage-level outputs from an array of $N$ radio telescopes. A
tied-array of $N$ identical telescopes increases sensitivity by a factor of
$N$ over a single telescope, but this comes at the expense of a smaller field
of view. Multiple tied-array beams can be formed improve the instantaneous
field of view of the array; however, storing and processing multiple beams can
be challenging.

An alternative approach, known as incoherent beamforming, sums the output
power from each telescope. An incoherent beam retains the field of view, but
the sensitivity only improves as $\sqrt{N}$. For computationally-expensive
algorithms—such as those required in pulsar, fast radio burst, and
technosignature searches—it is often infeasible to form tied-array beams
across the instrument’s entire field of view. On the other hand, the
incoherent beam may not have enough sensitivity nor spatial resolution to be
scientifically interesting.

A beamformed voltage beam from $P$ antennas is given by the sum of weights
$w=Ae^{i\theta}$ with a voltage stream $v(t)$:

$b(t)=\sum_{p=1}^{P}w_{p}v_{p}(t)$ (1)

The power beam, B, is the beam after squaring and averaging in time. Denoting
time average with angled brackets $\langle\rangle$, the power beam is given by

$\displaystyle B$ $\displaystyle=\langle b(t)b^{*}(t)\rangle\ $ (2)
$\displaystyle=\left\langle\left(\sum_{p=1}^{P}w_{p}v_{p}(t)\right)\left(\sum_{q=1}^{P}w_{q}v_{q}(t)\right)^{*}\right\rangle$
(3) $\displaystyle=\sum_{p=1}^{P}\sum_{q=1}^{P}w_{p}\langle
v_{p}(t)v^{*}_{q}(t)\rangle w^{*}_{q}$ (4)

The quantity $\langle v_{p}(t)v^{*}_{q}(t)\rangle$ can be treated as a
$(P\times P)$ matrix V. This is known as the visibility matrix, and is the
fundamental data product produced by a radio interferometer. Equation 4 can
alternatively be written in terms of V and a weight vector w:

$B=\textbf{w}\,\textbf{V}\,\textbf{w}^{H}$ (5)

where $\textbf{w}^{H}$ is the weight vector’s Hermitian conjugate.

There are therefore two paths toward generating a power beam, which we refer
to as pre-correlation beamforming and post-correlation beamforming. These
paths are shown diagramatically in Figure 1. The figure also shows optional
gridding steps, which allows multiple regularly-spaced beams to be formed
using a 2-dimensional Fast Fourier Transform (2D FFT) if the input data are
regularly-gridded. The dashed lines in Figure 1 represent hybrid
archetictures, e.g. arrays like the MWA and LOFAR where beamforming is
performed on a subset of antennas within a ‘station’, and then the station
tied-array beams are correlated.

###  2.1 Direct imaging approaches

With the advent of large-$N$ telescopes, there is increased interest in
‘direct imaging’ instruments which follow the pre-correlation gridded path of
Figure 1. In one approach, antennas are physically arranged on a grid, then a
spatial 2D FFT is applied to form images (Tegmark & Zaldarriaga, 2009). This
approach is extended by the Modular Optimal Frequency Fourier (MOFF) imaging
technique. In the MOFF approach, antennas do not need to be physically placed
on a grid; rather, voltages are gridded electronically based on the aperture
illumation of consituent antennas (Morales, 2011; Thyagarajan et al., 2017;
Kent et al., 2019). Note that a visibility matrix can be retrieved from a grid
of beams via an inverse 2D FFT (Tegmark & Zaldarriaga, 2009; Foster et al.,
2014), which is shown in Figure 1 as a dashed line between pathways.

Here, we focus on non-gridded methods, but note that equivalent tradeoffs
between resolution and sensitivity could be leveraged with these alternative
architectures.

###  2.2 Tensor formalism

Eq. 5 can be written in an equivalent compact form using summation notation:

$B=\textbf{w}\,\textbf{V}\,\textbf{w}^{H}\equiv\
\textbf{w}^{p}\,\textbf{V}_{pq}\,(\textbf{w}^{*})^{q}=\textbf{W}^{pq}\textbf{V}_{pq}$
(6)

In this notation, summation is implied over all indices that appear in an
upper and lower index (* represents complex conjugation). So, we sum across
indexes $p$ and $q$ (summation indices representing antennas) which returns a
single value $B$. The quantity
$\textbf{W}_{pq}=\textbf{w}_{p}\textbf{w}^{*}_{q}$ is equivalent the ($P\times
P$) matrix formed from the outer product $\textbf{w}\textbf{w}^{H}$.

The visibility matrix can itself be written in summation notation, instead of
using $\langle\rangle$ brackets, as

$\textbf{V}_{pq}=\textbf{v}^{t}_{p}(\textbf{v}^{*})_{qt}$ (7)

where $t$ represents time.

These bold-face quantities are often referred to as data tensors. Tensors can
simply be considered as multi-dimensional arrays, or generalizations of a
matrix into higher dimensions. Here, we will follow the nomenclature used in
the TensorFlow software package555https://www.tensorflow.org/guide/tensor. As
we are using data tensors as a computational tool, we do not need to worry
about other uses of tensors found in physics and mathematics, such as their
interpretation as mappings between vector spaces, so we do not define a tensor
metric here. We note that tensors have indeed been used previously in
calibration and imaging (e.g. Smirnov, 2011; Price & Smirnov, 2015;
Thekkeppattu et al., 2024).

Whenever a pair of tensors have matching upper and lower indices, entries are
summed across all matching indices; this is known as a pairwise contraction.
The order of pairwise contractions for a chain of tensors is known as the
contraction path. The computational cost to evaluate a tensor chain depends on
the contraction path; finding the optimal path is an NP-hard problem that can
quickly become intractable as the number of tensors increases.

###  2.3 Computing tensor contractions

Numerous software packages for performing tensor contractions are available.
We highlight that the Python Numpy package (Harris et al., 2020) includes a
flexible ‘one-liner’ einsum method for computing contractions on Numpy arrays.
An equivalent einsum method exists in the Cupy software package (Okuta et al.,
2017), which offloads computations to a graphics processor unit (GPU). If
compiled against the NVIDIA cuTENSOR
library666https://docs.nvidia.com/cuda/cutensor/index.html, highly-optimized
tensor cores on the GPU may be target; these offer orders-of-magnitude greater
energy efficiency than regular GPU cores.

A highly-optimized correlation code that uses tensor cores is detailed in
Romein (2021), and a complex general matrix multiply (cGEMM) code has been
developed for use for beamforming with a phased array
feed777https://gitlab.mpcdf.mpg.de/nesser/tc_cgemm/-/tree/master/. Together,
these two codes could be used to compute Equation 6, and the einsum method
provides a straightforward path for prototyping and developing post-
correlation beamforming techniques.

###  2.4 Multiple beams and frequency channels

We can extend Equation. 6 with extra indices (subscripts $l$ and $m$) to
represent a grid of $(L\times M)$ beams on the sky, and across $F$ frequency
channels (subscript $\nu$):

$\textbf{B}_{lm\nu}=\left(\textbf{w}_{lmp\nu}\textbf{v}^{pt}_{\nu}\right)\left(\textbf{w}^{*}_{lmq\nu}(\textbf{v}^{*})^{q}_{\nu
t}\right)$ (8)

That is, we sum across indexes $p$ and $q$ (summation indices), and output an
N-dimensional array with indices $(l,m,\nu)$. We can also write Equation. 6 as

$\textbf{B}_{lm\nu}=\textbf{W}^{pq}_{lm\nu}\,\textbf{V}_{pq\nu}.$ (9)

If power beams are evaluated at a series of timesteps, we may write
$\textbf{B}_{lm\nu}=\textbf{B}(t)_{lm\nu}$ as a function of time.

$\textbf{B}(t)_{lm\nu}=\textbf{W}^{pq}_{lm\nu}\,\textbf{V}(t)_{pq\nu}.$ (10)

There are two key differences between Equation. 8 and Equation. 11. The first
is that the tensor $\textbf{W}_{pqlm\nu}$ has $(P^{2}\times L\times M\times
F)$ entries, but $\textbf{w}_{plm\nu}$ has only $(P\times L\times M\times F)$
entries. The second is that Equation 8 requires three pairwise contractions to
compute, while Equation 11 is only one pairwise contraction—two contractions
have already been performed to produce $\textbf{W}_{lmpq\nu}$ and
$\textbf{V}_{pq\nu}$.

To extend to a polarization-aware version, we may simply add a subscript $x$:

$\textbf{B}(t)_{lm\nu x}=\textbf{W}^{pq}_{lm\nu x}\,\textbf{V}(t)_{pq\nu x}.$
(11)

where $x$ represents a set of four polarization coherency measurements (e.g.
XX*, XY*, YX*, YY* for a linearly-polarized dual-polarization antenna).

###  2.5 Comparison to interferometric imaging

In synthesis imaging, an image $I(l,m)$ is created by evaluating

$I(l,m)=\frac{1}{M}\sum_{m=1}^{M}V(u_{k},v_{k})e^{2\pi(u_{k}l+v_{k}m)},$ (12)

where $M$ is the number of baselines, and $(u_{k},v_{k})$ are coordinates
relating to the baseline $k$ between a ($p$, $q$) antenna pair (Eq. 7-3,
Briggs et al., 1999). This equation can written as a tensor contraction

$\displaystyle\textbf{I}_{lm}$
$\displaystyle=\textbf{W}^{k}_{lm}\textbf{V}_{k}$ (13)
$\displaystyle\textbf{W}_{klm}$
$\displaystyle=\frac{1}{M}exp\left(-i2\pi(u_{k}l+v_{k}m)\right),$ (14)

which by splitting $k=pq$ can be rewritten as

$\displaystyle\textbf{I}_{lm}$
$\displaystyle=\textbf{W}^{pq}_{lm}\,\textbf{V}_{pq},$ (15)
$\displaystyle\textbf{W}_{pqlm}$ $\displaystyle=\frac{1}{M}exp\left(2\pi
i(u_{pq}l+v_{pq}m)\right).$ (16)

Comparison of Eq. 5 and Eq. 16 reveals that
$\textbf{I}_{pq}\equiv\textbf{B}_{pq}$. That is: each pixel in an image is
equivalent to a power beam, or alternatively, an image can be created out of a
grid of power beams. Note that in synthesis imaging, autocorrelations are
generally not included; i.e. the weights tensor $\textbf{W}_{pq}=0$ if $p=q$.
From a power beam interpretation, this is equivalent to subtracting the
incoherently summed beam from the tied-array power beam (Roy et al., 2018).

###  2.6 Historical perspective

While the interpretation of image pixels as power beams is not commonly
discussed, the link between imaging and beamforming is fundamental to radio
astronomy and was leveraged by early interferometers, including the Mills
Cross (Mills et al., 1958). However, for the most part, the historical
development of pulsar search pipelines has run parallel to high-fidelity
synthesis imaging systems, leading to disparate science communities and
techniques.

Since their discovery in 1967 by Jocelyn Bell Burnell, a majority of pulsars
have been discovered using single-dish instruments, such as the Parkes
Murriyang, Arecibo, Lovell, and Green Bank telescopes (Manchester et al.,
2005). As such, most observing techniques and knowledge within the pulsar
community is derived from single-dish approaches. Pulsar search codes such as
PRESTO888https://github.com/scottransom/presto (Ransom, 2011) were also
designed for single-dish telescopes, and are, in general, incompatible with
interferometric data products.

In contrast, radio interferometers have predominantly been used for long-
exposure synthesis imaging. Earth-rotation synthesis, popularized by the
Cambridge One-Mile Radio Telescope (Ryle, 1962), combines data across many
hours of observation to improve imaging fidelity. Instruments were not
designed to output high time resolution data products; rather, they were
designed to integrate data for as long as possible to minimize output data
rates. Despite their lack of time resolution, pulsars have been detected in
synthesis images via other characteristics; exceptionally, the first
millisecond pulsar was detected in synthesis images as a source with an
anomalous spectrum (Backer et al., 1982).

The quest to discover new pulsars has nonetheless led pulsar astronomers
toward radio arrays. Modern interferometers have good sensitivity, a large
field of view, and better localization capability; on a technical level, tied-
array beamforming is increasingly feasible. Fast imaging techniques—where each
pixel is treated as a power beam—are have been used for searches of
millisecond transient pulses (Law et al., 2015, 2018), as have FFT-based
beamforming approaches (Ng et al., 2017), but are yet to be eagerly adopted in
pulsar periodicity searches.

Techosignature searches have followed a similar trajectory. Frank Drake
performed the first SETI search using the Tatel single-dish telescope in 1961
(Drake, 1961); single-dish telescopes remained the primary technology used
until the construction of the Allen Telescope Array (Welch et al., 2009). Many
technosignature experiments have targeted nearby stars and have sought to
maximize frequency coverage (e.g. Isaacson et al., 2017), hence field of view
has not historically been a science driver. But more ambitious surveys of
nearby star targets (Czech et al., 2021) motivate faster survey speed, and
there is a growing movement toward non-targeted widefield technosignature
searches (Houston et al., 2021).

##  3 Reduced-resolution beamforming

One tactic used to improve sky coverage of a tied-array beamformer is to only
use antennas within a ‘core’ region. Via Equation 8, selecting a core region
is equivalent to setting weights $\textbf{w}_{lmp\nu}$ to zero for any antenna
$p$ that is above a cutoff distance from a reference antenna (i.e. per-antenna
filtering). To first order, the full-width at half-maximum (FWHM) of the
resultant tied-array beam will scale with $\lambda/d_{\rm{max}}$.

Post-correlation beamforming allows the user to set weights between any
antenna pair ($\textbf{W}_{pq}$), allowing per-baseline filtering (this is, in
fact, the standard approach in synthesis imaging). The per-baseline approach
means that all baselines below a given threshold distance can be included in
the power beam—including short baselines outside the core. It follows that a
per-baseline filtering approach (only possible with post-correlation
beamforming) will always yield a more sensitive power beam than a per-antenna
approach.

We refer to post-correlation beamforming schemes that set weights to zero for
any baseline above a cutoff distance as reduced-resolution beamforming.
Controlling the cutoff distance essentially allows for sensitivity to be
traded off against beam width. For simplicity, we will only consider weights
with magnitude 0 (discard baseline) or 1 (keep baseline); but note there are
myriad weighting schemes used in synthesis imaging that could be considered
(see Chapter 10, Thompson et al., 2017).

Figure 2: Reduced-resolution beam formed from the MWA compact configuration.
The full-resolution beam is shown in grey; a reduced-resolution beam for a
100-m maximum baseline is shown in red.

Figure 2 shows an example of reduced-resolution beamforming, as applied to
Murchison Widefield Array 128-tile Compact configuration (Wayth et al., 2018).
The grey line shows the beam pattern (known as point spread function, or dirty
beam, in synthesis imaging) for a zenith-pointed tied-array beam. The red line
shows the resulting beam pattern using reduced-resolution beamforming, with a
$d_{\rm{max}}=100$ m cutoff. The reduced-resolution beam is wider and
smoother, but has a lower power gain, which corresponds to lower sensitivity.
Note Figure 2 doesn’t account for the tile beam pattern.

###  3.1 Sensitivity

From the radiometer equation, the expected thermal noise (in Jy units) of a
beam formed via post-correlation beamforming can be written as

$\Delta S_{\rm{post-x}}=\frac{k_{B}T_{\rm{sys}}}{A_{e}\sqrt{\Delta\nu\tau
N_{\rm{pol}}(2N_{\rm{baselines}}+N_{\rm{ant}})}}$ (17)

where we introduce the following quantities:

$T_{\rm{sys}}$ | System temperature
---|---
$k_{B}$ | Boltzmann’s constant
$\Delta\nu$ | Bandwidth
$A_{\rm{ant}}$ | Effective area of an antenna element
$\tau$ | Integration time
$N_{\rm{ant}}$ | Number of antennas
$N_{\rm{pol}}$ | Number of polarizations
$N_{\rm{baselines}}$ | Number of baselines in array

Equation 17 differs slightly from the standard interferometer radiometer
equation (Eq. 9.26, Wilson et al., 2013) as it includes a $N_{\rm{ant}}$ term
to account for autocorrelations. Noting that

$\displaystyle N_{\rm{baselines}}$
$\displaystyle=N_{\rm{ant}}(N_{\rm{ant}}-1)/2$ (18) $\displaystyle
N_{\rm{ant}}^{2}$ $\displaystyle=2N_{\rm{baselines}}+N_{\rm{ant}}$ (19)

one finds that Equation 17 reduces to the tied-array radiometer equation if
all baselines are included

$\Delta
S_{\rm{tied}}=\frac{2k_{B}T_{\rm{sys}}}{A_{\rm{ant}}N_{\rm{ant}}\sqrt{\Delta\nu\tau
N_{\rm{pol}}}}.$ (20)

Additionally, if cross-correlation terms are not included (i.e.
$N_{\rm{baselines}}=0$), then we retrieve the radiometer equation for an
incoherent beam

$\Delta
S_{\rm{incoherent}}=\frac{2k_{B}T_{\rm{sys}}}{A_{\rm{ant}}\sqrt{\Delta\nu\tau
N_{\rm{ant}}N_{\rm{pol}}}}.$ (21)

If we choose a subset of baselines, based on maximum length, the resulting
noise will lie between the ideal $\Delta S_{\rm{tied}}$ thermal noise (all
baselines included) and the incoherent $\Delta S_{\rm{incoherent}}$ noise
(only autocorrelations included). Put another way, reduced-resolution
beamforming provides access to resolution and sensitivity regimes that lie
between that of incoherent and coherent beamforming techniques.

###  3.2 Fractional sensitivity

Consider the case where a maximum baseline length $d$ is imposed on an array
with longest baseline $D$. For a tied array beamformer, the sensitivity can be
treated as a function of baseline length, $\Delta S_{\rm{coherent}}(d)$
(Equation 20), with the number of antennas in the sub-array is
$N^{\prime}_{\rm{ant}}(d)$. For a post-correlation beamformer (Equation 17),
the thermal noise level reached $\Delta S_{\rm{post-x}}(d)$ depends upon
$N^{\prime}_{\rm{baselines}}(d)$.

It is useful to define a fractional sensitivity factor, $f_{S}$, that relates
the sensitivity of the selected subarray to the full array:

$\Delta S(d)=\Delta S(D)\big{/}f_{S}.$ (22)

For a reduced-resolution beam with all autocorrelations,
$N^{\prime}_{\rm{ant}}=N_{\rm{ant}}$, and

$f_{S}=\frac{\Delta S_{\rm{post-x}}(D)}{\Delta
S_{\rm{post-x}}(d)}=\frac{\sqrt{2N^{\prime}_{\rm{baselines}}(d)+N_{\rm{ant}}}}{N_{\rm{ant}}}$
(23)

whereas for a tied-array beam

$f_{S}=\frac{\Delta S_{\rm{tied}}(D)}{\Delta
S_{\rm{tied}}(d)}=\frac{N^{\prime}_{\rm{ant}}(d)}{N_{\rm{ant}}}$ (24)

and $N^{\prime}_{\rm{ant}}\leq N_{\rm{ant}}$.

Figure 3: Comparison of EDA2 fractional sensitivity for a tied-array beam
(red) and a post-x beamformed beam (black), as a function of maximum baseline
length $d_{\rm{max}}$. The post-x approach allows short baselines to be
included, boosting sensitivity.

Figure 3 shows the fractional sensitivity for a reduced-resolution beam and
tied-array beam as a function of maximum baseline length, using the
EDA2999EDA2: Engineering Development Array 2 array as an example (Wayth et
al., 2021). The reduced-resolution beam always has a larger fractional
sensitivity than a tied-array beam from a sub-selection of antennas. This fact
is a key motivation for reduced-resolution beamforming.

The slope of $\Delta S_{\rm{post-x}}(d)$ depends upon the antenna
configuration. Figure 4 considers the application of reduced-resolution
beamforming to three radio arrays: the 64-antenna MeerKAT telescope (Camilo,
2018), the 128-tile MWA (in compact configuration, Wayth et al., 2018), and
the EDA2 (Wayth et al., 2021). The top panels show the antenna layout for each
array, and the middle panels show baseline distribution histograms as a
function of baseline length. The lower panel shows the fractional sensitivity
for each array as a function of maximum baseline length (solid black line),
and the number of beams required to fill the primary field of view, assuming
the beam width scales as $\lambda/d$ (dashed red line).

A key takeaway from Figure 4 is that all telescopes reach a reasonable
fractional sensitivity ($\sim 0.5$) with an order of magnitude fewer beams to
fill their field of view than the full array. In the coming sections, we will
explore how this finding could be applied on these telescopes.

(a) MeerKAT

(b) MWA (compact configuration)

(c) EDA2 (SKA-low pathfinder)

Figure 4: Sensitivity analysis for reduced resolution beamforming applied to
MeerKAT, the MWA (compact configuration) and the EDA2. The top panels show
antenna location, and the middle panels show histograms of baseline lengths
for each telescope. The bottom panel shows the fractional sensitivity for
reduced resolution beamforming, as a function of maximum baseline length
(black lines). The minimum sensitivity is equivalent to incoherent
beamforming, and maximum sensitivity is equivalent to standard beamforming.
The number of beams required to cover each telescope’s field of view scales
with $d_{\rm{max}}^{2}$ (dashed red lines).

###  3.3 Survey speed

Survey speed is a useful metric for optimizing the performance of an array.
The point-source survey speed figure of merit (PFoM) relates to the time taken
to survey a field with solid angle $\Omega_{\rm{survey}}$ to a required
thermal noise level in Jy, $\Delta S_{\rm{survey}}$. Following Equation 3 of
Cordes
(2009)101010https://www.skatelescope.org/uploaded/26865_109_Memo_Cordes.pdf,
we may define a PFoM modified in a similar fashion to Chen et al. (2021) to
account for number of beams and maximum baseline length.

The fraction of the maximum PFoM (all antennas, beams covering instrumental
FoV) reached with a reduced-resolution beamforming system is

$f_{\rm{PFoM}}=\frac{{\rm{PFoM}}(d)}{\rm{PFoM_{full}}}=f^{2}_{S}\times
f_{\rm{FoV}}$ (25)

where the fractional coverage of the instrument’s field of view
($\Omega_{\rm{full}}$), depends on the number of beams formed:

$f_{\rm{FoV}}=\frac{\Omega_{\rm{FoV}}(d)}{\Omega_{\rm{full}}}\approx\frac{N_{\rm{beam}}}{\Omega_{\rm{full}}}\left(\frac{\lambda}{d}\right)^{2},$
(26)

assuming resolution follows the $\lambda/d$ rule of thumb. However, the actual
beam width depends upon both the array configuration and the beam’s
declination in a complex fashion. Many radio arrays have decreasing antenna
density for longer baselines, for which $\lambda/d$ underestimates the beam
width. Nevertheless, if the number of beams is fixed, we expect the fractional
field of view to decrease rapidly as $d$ increases.

###  3.4 Computational cost

The approximate computational cost of a correlator and multi-pixel beamformer
can be written in terms of complex-multiply accumulate operations per second
(CMACs/s) as

$\displaystyle{\rm{C_{X}}}$
$\displaystyle=\left(N_{\rm{ant}}N_{\rm{pol}}(N_{\rm{ant}}N_{\rm{pol}}+1)\right)\times\Delta\nu$
(27) $\displaystyle{\rm{C_{B}}}$
$\displaystyle=\left(N_{\rm{beam}}\,N_{\rm{ant}}\,N_{\rm{pol}}\right)\times\Delta\nu$
(28)

where $\Delta\nu$ is the bandwidth of data processed. To form post-correlation
beams from the correlator output requires an additional step, with
computational cost

${\rm{C_{pxb}}}=f_{\rm{baselines}}\times\left(N_{\rm{pxb}}N^{2}_{\rm{ant}}N^{2}_{\rm{pol}}\right)\times\frac{\Delta\nu}{N_{\rm{int}}}$
(29)

where $N_{\rm{int}}$ is the number of time samples summed during integration,
$f_{\rm{baselines}}$ is the fraction of baselines included, and $N_{\rm{pxb}}$
is the number of post-correlation beams. Note that the cost of FFT-based
imaging scales proportionally to $N_{\rm{pxb}}{\rm{log}}_{2}N_{\rm{pxb}}$, and
for most arrays111111Noting $N_{\rm{pxb}}\equiv N_{\rm{pix}}\times
N_{\rm{pix}}$, where $N_{\rm{pix}}$ is image width in pixels.

${\rm{log}}_{2}N_{\rm{pxb}}\ll N_{\rm{ant}}^{2},$ (30)

so we stress that the PXB approach is generally far more computationally
expensive than the 2D-FFT approach employed in imaging pipelines. That said,
for fair comparison with computationally expensive searches—where only a few
beams can be processed on any one compute server—we do not compare against
FFT-based imaging any further in this article.

Based on Equations 28–29, tied-array beamforming is more computationally
expensive if $N_{\rm{beams}}\lessapprox N_{\rm{pxb}}$ and/or if the correlator
does not integrate many time samples, i.e. $N_{\rm{int}}$ is large.

This conclusion is consistent with Roy et al. (2018), which considers post-
correlation beamforming techniques for pulsar studies with the Giant Metrewave
Radio Telescope. Roy et al. (2018) also concludes that post-correlation
beamforming is computationally cheaper for a large number of beams at low time
resolution, and regular beamforming is cheaper for a small number of high
time-resolution beams.

As with imaging, the cost of post-correlation beamforming (Eq 29) can be
reduced significantly if visibilities are gridded and a 2D FFT is used (Briggs
et al., 1999). However, the FFT approach is not be suitable for a small number
of beams and/or wide-field imaging; further, when distributing processing
tasks across a supercomputer, it may be advantageous to have only a small
number of beams per node. As such, we present values for the non-gridded
approach.

Table 1: MWA SMART survey parameters for a single zenith pointing. $N_{\rm{ant}}$ | 128 | Number of antennas
---|---|---
$N_{\rm{pol}}$ | 2 | Number of polarizations
$N_{\rm{chan}}$ | 3072 | Number of channels
$\Omega_{\rm{FoV}}$ | 610 | Field of view
$N_{\rm{beam}}$ | 6100 | Number of beams to tile FoV
$\Delta\nu$ | 30.72 | Bandwidth (MHz)
$\tau$ | 100 | Integration time ($\mu$s)
$t_{\rm{obs}}$ | 80 | Observation time (min)

##  4 Application to MWA SMART

SMART (Bhat et al., 2023a) is a pulsar and fast transient search project on
the MWA, using the 128-tile compact configuration (Wayth et al., 2018). SMART
aims to conduct a pulsar search across the full sky below 30∘ declination, to
a limiting sensitivity (10$\sigma$) of 2–3 mJy. Observations began in 2018,
and are approximate 75% complete (Bhat et al., 2023b). The SMART project
stores voltage-level data products for each of 128 tiles within the MWA
compact configuration, which allows beamforming and/or correlation to be
performed offline. SMART consists of 70 pointings of 80-minute duration (93 hr
total), which corresponds to $\sim 3$ PB of data. These data are stored on the
Pawsey Banksia object store121212https://pawsey.org.au/systems/banksia/.

SMART uses a multi-pixel tied-array beamformer (Swainston et al., 2022), which
is used to form up to 20 beams at once (per compute node). Each pointing
requires $\sim 6000$ beams to cover the full field of view. The pulsar search
is performed on power beam data (Stokes-I). On the OzSTAR
supercomputer131313https://supercomputing.swin.edu.au/, beamforming 10 minutes
of data takes 2 kSU (service units), and postprocessing search tasks take 25
kSU (1 kSU is equivalent to 1000 CPU-core hours, or 125 GPU-core hours).
Extrapolating from these numbers, assuming processing scales linearly with
observation time, processing all 70 pointings for the full 80 minutes would
require $\sim 1120$ kSU for beamforming, and $\sim 14,000$ kSU for post-
processing.

###  4.1 Trading sensitivity against compute requirements

The post-processing requirements for SMART scale proportionally with the
number of beams searched. By applying reduced-resolution beamforming, the
SMART survey volume can be searched with fewer beams, potentially improving
computational requirements by an order of magnitude for a moderate drop in
sensitivity. However, any improvement in downstream processing costs must be
offset against any increases in beamforming costs. Using Equations 28–29, and
the SMART survey parameters (Table 1), the computational costs are:

$\displaystyle{\rm{C_{X}}}$ $\displaystyle=16.2\ {\rm{TCMAC/s}}$ (31)
$\displaystyle{\rm{C_{B}}}$ $\displaystyle=767.6\ {\rm{TCMAC/s}}.$ (32)

The comparable computational cost for reduced-resolution beamforming
($C_{\rm{PXB}}$) as a function of fractional sensitivity $f_{s}$ is shown in
Table 2. The computational cost is considerable, as the integration time
$\tau=100\mu$s only allows $N_{\rm{int}}=2$. For such a large number of beams,
computational cost could be reduced significantly by using a 2D FFT imaging
approach. Regardless, Selecting $f_{s}=0.5$ yields $C_{\rm{PXB}}=1146$ TMAC/s,
i.e. roughly 1.5$\times$ the cost of tied-array beamforming, and the current
SMART tiling approach could be maintained.

Table 2: Computational requirements and corresponding sensitivity limits for MWA reduced-resolution beamforming, for increasing fractional sensitivity.  $f_{s}$ | $d_{\rm{max}}$ | $N_{\rm{beam}}$ | $N_{\rm{baseline}}$ | $C_{\rm{PXB}}$ |  $\Delta S$ (10$\sigma$)
---|---|---|---|---|---
| (m) |  |  | (TCMAC/s) | (mJy)
0.2 | 17.4 | 16 | 389 | 12 | 10.0–15.0
0.3 | 30.8 | 53 | 810 | 84 | 6.7–10.0
0.4 | 48.3 | 130 | 1353 | 343 | 5.0–7.5
0.5 | 70.4 | 277 | 2121 | 1146 | 4.0–6.0
0.6 | 123.2 | 851 | 2970 | 4931 | 3.3–5.0
0.7 | 154.0 | 1330 | 4084 | 10596 | 2.9–4.3
0.8 | 180.3 | 1825 | 5368 | 19112 | 2.5–3.8
0.9 | 211.1 | 2501 | 6643 | 32412 | 2.2–3.3

While the beamforming cost is higher, the number of beams to search drops from
6100 to 277. It follows that post-processing requirements will drop by a
factor of 22$\times$, while sensitivity drops only by a factor of 2$\times$.
We extrapolate that the post-processing requirements on OzSTAR would drop from
$\sim 14,000$ kSU to $\sim 636$ kSU. The SMART team anticipates securing
500–600 kSU per annum (Bhat et al., 2023a), meaning it is plausible that the
entire survey could be processed within a year if reduced-resolution
beamforming is adopted (instead of an implausible 22 years).

The processing requirements for SMART may also be alleviated by planned
improvements to the performance of their pulsar search pipeline. The SMART
pipeline currently uses PRESTO, a CPU-only code written primarily in ANSI C
(Ransom, 2011). GPU search codes, such as astroacclerate (Armour et al., 2020)
and peasoup (Barr, 2020), are hoped to offer an order-of-magnitude
improvement. For example, the astroacclerate Fourier domain acceleration
search is up to $8\times$ faster than its PRESTO equivalent (Adámek et al.,
2020); recent work to exploit half-precision data types may offer an
additional $1.6\times$ speedup (White et al., 2023). Alleviating I/O
bottlenecks, such as disk read speed, may also improve efficiency. Regardless,
any speedup from pipeline optimization is complementary to the reduced-
resolution beamforming approach.

###  4.2 Application to technosignature searches

A narrowband technosignature search of SMART data has been proposed, which
would be the first all-sky technosignature search in the Southern Hemisphere.
As such, as SMART technosignature search would place some of the most
stringent constraints on the prevalence of putative engineered transmitters in
the Galaxy. It is argued that all-sky SETI searches at low frequency are one
of the most compelling methods to detect evidence of technologically-capable
life beyond Earth (Garrett et al., 2017; Houston et al., 2021). Narrowband
technosignature searches do not require high time resolution, which decreases
the computational cost of post-correlation beamforming. For example,
decreasing the time resolution from $100\,\mu$s to 1 s decreases
$C_{\rm{PXB}}$ (Equation 29) by a factor of 10,000.

However, narrowband technosignature searches require high frequency
resolution, meaning that large data volumes must be stored in memory on which
a large FFT can be performed. This will limit the number of beams that can be
processed at any time. As with the pulsar search, the post-processing
requirements for narrowband technosignatures scale linearly with the number of
beams. Similar tradeoffs between sensitivity and number of beams are thus well
motivated for technosignature surveys.

##  5 Application to MeerKAT: optimizing survey speed

The TRAPUM program (Chen et al., 2021) is a tied-array pulsar search program
on the MeerKAT 64-antenna array (Camilo, 2018). Pulsar searches are conducted
using up to $\sim 1000$ power beams, formed from a tied-array beamforming
approach (Barr, 2018). An incoherent beam may also be formed, and the maximum
number of beams can increase to 4096 if a smaller subset of antennas is
beamformed.

Figure 5:  Comparison of MeerKAT survey speed with reduced resolution
beamforming and tied-array beamforming approaches.

TRAPUM uses a point-source figure of merit (PFoM) modified for MeerKAT to
optimize survey speed (Equation 4 Chen et al., 2021), from which the optimal
number of antennas is found to be between 37 and 41, depending on the altitude
observed; this corresponds to a $\sim 1$ km maximum baseline, and 58–64%
fractional sensitivity. If reduced-resolution beamforming was applied, the
corresponding fractional sensitivity would be 71–75% for the same baseline
cutoff. The extra baselines would also slightly improve the beam sidelobe
response by filling in the $uv$-plane.

Based on $N_{\rm{ant}}=64$, and time integration used in the TRAPUM search
mode ($\tau=76\mu$s, $N_{\rm{int}}=32$), the computational cost of reduced-
resolution beamforming yields ${\rm{C_{PXB}=4C_{B}}}$. Memory requirements may
also be higher, which could limit the number of reduced-resolution beams that
could be formed.

Figure 5 shows a comparison of survey speed against maximum baseline length
for the MeerKAT array. If tied-array beamforming is used, the maximum baseline
length corresponds to the size of the ‘core’ region of antennas used. For
reduced-resolution beamforming, the maximum baseline length corresponds to the
subset of baselines from the entire array. Note the PFoM for the tied-array
beam is comparable to Figure 6 Chen et al. (2021). The PFoM is consistently
higher for reduced-resolution beamforming. For $N_{\rm{beam}}=1024$, the
maximum PFoM peaks 3.9$\times$ higher for the reduced-resolution approach.

However, taking into account the instrument’s capabilities, we conclude that
post-correlation beamforming is unlikely to yield a higher PFoM. TRAPUM
processes data streams in real time, and the beamformer is computationally-
bound. So, the increase in computational cost (${\rm{C_{PXB}=4C_{B}}}$)
outweighs the benefit of the 3.9$\times$ PFoM increase: 4$\times$ fewer beams
could be formed. Nevertheless, the approach may be useful for other projects
on MeerKAT where the downstream search algorithms are computationally
expensive.

##  6 Conclusions

This article has introduced reduced-resolution beamforming: a post-correlation
beamforming approach that allows sensitivity to be traded against the number
of beams needed to cover a survey area. Reduced-resolution beamforming offers
an alternative to tied-array beamforming of a core region of antennas. The
sensitivity of a reduced-resolution beam is always higher than tied-array
beamforming of a core antenna region.

There are two pathways toward forming power beams: pre-correlation
beamforming, in which voltages are summed, squared then averaged, and post-
correlation beamforming, in which antenna pairs are cross-correlated before
summation and averaging (Figure 1). Our approach relies on the latter, in
which per-baseline weights can be applied in lieu of per-antenna weights.

In Section 2 we introduced a tensor formalism to highlight the difference
between pathways of Figure 1, and showed that a pixel within an
interferometric image is equivalent to a power beam. Tensors have become
standard data structures within commonly-used software packages such as Numpy,
and offer compact notation; given these advantages, we encourage the use of
tensor formalisms for further research on beamforming approaches.

There are two main advantages of reduced-resolution beamforming. Firstly, by
decreasing the number of beams needed to cover a survey area, the downstream
processing requirements to search the beams is lower: in Section 4 we show by
applying reduced-resolution beamforming to the SMART survey, the same sky
volume could be covered with 22$\times$ fewer beams, while retaining 50%
fractional sensitivity. Secondly, for a fixed number of beams, the survey
speed of the telescope may be increased: in Section 5 we argue that the PFoM
for the TRAPUM search mode on the MeerKAT telescope could be increased by
3.9$\times$ by if reduced-resolution beamforming was adopted, albeit at the
expense of increased computational requirements for beamforming.

While reduced-resolution beamforming demands a sensitivity tradeoff, it could
be offset by improved downstream processing. For example, in a pulsar search,
more dispersion measure or acceleration trials may become feasible if fewer
beams need to be searched, improving final signal-to-noise of candidates. One
could also consider using reduced-resolution beamforming as a first-pass
approach to find candidates above a less stringent threshold (e.g. 5$\sigma$),
then form tied-array beams to follow-up candidates, setting a more stringent
threshold (e.g. 10$\sigma$) for validation or rejection.

Nevertheless, there are limitations of the approach. Reduced-resolution
beamforming produces power beams, so cannot be applied in cases where voltage
beams are needed (e.g. coherent dedispersion). For science cases where
confusion noise is a relevant concern, such as continuum imaging, reduced-
resolution beamforming is not appropriate. Another disadvantage is that the
decrease in angular resolution means that any transient object found will have
a comparatively poor localization, although this could be improved using
intra-beam methods (Obrocka et al., 2015). Finer localization of candidate
objects could also be done by reprocessing/reobserving at full resolution,
then using known parameters (e.g. DM, period, pulse width) to narrow the
search space. Finally, the computational requirements for post-correlation
beamforming may become unfeasibly large for arrays with a large number of
antennas (Equation 29). Memory requirements for storing the correlation matrix
and beamforming weights may also limit the number of beams that a single
compute node could process.

All things considered, we conclude that reduced-resolution beamforming is best
suited to science cases where downstream processing requirements dominate the
processing budget, such as searches for pulsar, technosignatures and fast
transients. We encourage experimental application and verification of the
approach to ease the data processing challenges faced by current and planned
searches.

D.C. Price thanks E. Barr, R. Wayth, N. Thyagarajan and the SMART
collaboration for their comments.

#### Software

Numpy (Harris et al., 2020), Matplotlib (Hunter, 2007), Pandas (Wes McKinney,
2010).

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