# Evidence of Ultra-faint Radio Frequency Interference in Deep 21 cm Epoch of
Reionization Power Spectra with the Murchison Widefield Array

Michael J. Wilensky  Jodrell Bank Centre for Astrophysics, University of
Manchester, Manchester M13 9PL, UK  Astronomy Unit, Queen Mary University of
London, Mile End Road, London, E1 4NS, United Kingdom  Department of Physics,
University of Washington, Seattle, WA 98195, USA
<EMAIL_ADDRESS>Miguel F. Morales  Department of Physics,
University of Washington, Seattle, WA 98195, USA  Dark Universe Science
Center, University of Washington, Seattle, 98195, USA  Bryna J. Hazelton
Department of Physics, University of Washington, Seattle, WA 98195, USA
eScience Institute, University of Washington, Seattle, WA 98195, USA  Pyxie L.
Star  Department of Physics, University of Washington, Seattle, WA 98195, USA
Nichole Barry  International Centre for Radio Astronomy Research, Curtin
University, Perth, WA 6845, Australia  ARC Centre of Excellence for All Sky
Astrophysics in 3 Dimensions (ASTRO 3D)  Ruby Byrne  Astronomy Department,
California Institute of Technology, 1200 E California Blvd, Pasadena, CA,
91125, USA  Department of Physics, University of Washington, Seattle, WA
98195, USA  C. H. Jordan  International Centre for Radio Astronomy Research,
Curtin University, Perth, WA 6845, Australia  ARC Centre of Excellence for All
Sky Astrophysics in 3 Dimensions (ASTRO 3D)  Daniel C. Jacobs  Arizona State
University, School of Earth and Space Exploration, Tempe, AZ, 85008, USA
Jonathan C. Pober  Department of Physics, Brown University, Providence, RI
02912, USA  C. M. Trott  International Centre for Radio Astronomy Research,
Curtin University, Perth, WA 6845, Australia  ARC Centre of Excellence for All
Sky Astrophysics in 3 Dimensions (ASTRO 3D)

###### Abstract

We present deep upper limits from the 2014 Murchison Widefield Array (MWA)
Phase I observing season, with a particular emphasis on identifying the
spectral fingerprints of extremely faint radio frequency interference (RFI)
contamination in the 21 cm power spectra (PS). After meticulous RFI excision
involving a combination of the SSINS RFI flagger and a series of PS-based
jackknife tests, our lowest upper limit on the Epoch of Reionization (EoR) 21
cm PS signal is $\Delta^{2}\leq 1.61\cdot 10^{4}\text{mK}^{2}$ at
$k=0.258\text{hMpc}^{-1}$ at a redshift of 7.1 using 14.7 hours of data. By
leveraging our understanding of how even fainter RFI is likely to contaminate
the EoR PS, we are able to identify ultra-faint RFI signals in the cylindrical
PS. Surprisingly this signature is most obvious in PS formed with less than an
hour of data, but is potentially subdominant to other systematics in multiple-
hour integrations. Since the total RFI budget in a PS detection is quite
strict, this nontrivial integration behavior suggests a need to more
realistically model coherently integrated ultra-faint RFI in PS measurements
so that its potential contribution to a future detection can be diagnosed.

##  1 Introduction

The Epoch of Reionization (EoR) is one of few periods in the Universe’s
history that is still largely unconstrained. Since the neutral Hydrogen 21-cm
line can directly track the remaining neutral Hydrogen at a given redshift,
high redshift intensity maps of 21-cm radiation will allow us to directly map
the reionization process. For reviews, see Furlanetto et al. (2006), Morales &
Wyithe (2010), and Pritchard & Loeb (2012). A first step towards understanding
cosmic reionization is to measure the power spectrum of redshifted 21-cm
brightness temperature fluctuations, which is a particularly natural task for
radio interferometers. Examples of interferometers that have produced upper
limits on the reionization power spectrum signal are the Giant Metrewave Radio
Telescope (GMRT; Paciga et al. (2013)), the LOw Frequency ARray (LOFAR; van
Haarlem et al. (2013)), the Murchison Widefield Array Phase I (MWA Phase I;
Tingay et al. (2013)) and Phase II (MWA Phase II; Wayth et al. (2018)), the
Precision Array for Probing the Epoch of Reionization (PAPER; Parsons et al.
(2010)), and the Hydrogen Epoch of Reionization Array (HERA; DeBoer et al.
(2017)). Lessons learned from analysis of early results have been specifically
incorporated into newer generations of EoR-focused telescopes, such as the
Hydrogen Epoch of Reionization Array, and the MWA Phase II. As the knowledge
of 21-cm power spectrum analysis has grown, upper limits at various redshifts
have gradually improved (Beardsley et al., 2016; Patil et al., 2017; Barry et
al., 2019b; Li et al., 2019; Mertens et al., 2020; Trott et al., 2020; Rahimi
et al., 2021; Abdurashidova et al., 2022; HERA Collaboration et al., 2023).

Theoretical models of the reionization signal suggest that it is 4-5 orders of
magnitude fainter than typical astrophysical radio sources. Due to the extreme
dynamic range requirements of these experiments, a myriad of data analysis
challenges exist. Most prominently, power spectrum analyses require exquisite
understanding of the spectral variations in the data in order to make a high
quality measurement. Such variations have numerous origins: incomplete or
inaccurate calibration models (Barry et al., 2016; Patil et al., 2016; Ewall-
Wice et al., 2017; Byrne et al., 2019; Dillon et al., 2020; Byrne et al.,
2021), intrinsic instrument bandpass response (Ewall-Wice et al., 2016a; Trott
& Wayth, 2016; Barry et al., 2019a; Li et al., 2019; Fagnoni et al., 2021),
internal cable reflections (Ewall-Wice et al., 2016b; Kern et al., 2019,
2020), and digital nonlinearities (Benkevitch et al., 2016). Additionally,
failing to model refraction of incoming radiation through the ionosphere can
significantly disrupt measurements relying solely on direction-independent
calibration (Morales & Matejek, 2009; Jordan et al., 2017; Trott et al.,
2018). These effects can be mitigated using direction-dependent calibration
techniques (Vedantham & Koopmans, 2016; Hurley-Walker & Hancock, 2018),
however such techniques can sometimes induce signal loss (Mouri Sardarabadi &
Koopmans, 2019; Mevius et al., 2022).

Even if other instrumental effects can be handled, an interferometer is
naturally chromatic because interferometric baselines measure different
angular wave modes at different frequencies. This gives rise to a well-studied
wedge-shaped contamination in cylindrical power spectra known as the
foreground wedge (Datta et al., 2010; Trott et al., 2012; Morales et al.,
2012; Vedantham et al., 2012; Hazelton et al., 2013). Generally the spectral
resolution of typical reionization-focused instruments allows for sampling of
larger line-of-sight wave modes compared to perpendicular wave modes, where
sampling of the latter is determined by baseline lengths and orientations.
Consequently, there is only a narrow region in this space that is both outside
the wedge and accessible by typical baselines that is free of foreground
contamination. However, systematic effects may still contaminate it. This area
is known as the EoR window (Liu et al., 2014a, b).

An exhaustive review of systematic effects is outside the scope of this work.
We refer the reader to the comprehensive review of data analysis in 21-cm
cosmology presented in Liu & Shaw (2020). The purpose of this work is to
present an upper limit of the cosmological 21-cm power spectrum signal derived
from the second observing season of the MWA Phase I. Of primary importance in
the analysis leading to this limit was the nature of a particular systematic
effect ubiquitous to all radio experiments called radio frequency interference
(RFI), considered here as any non-astrophysical radio signal that is observed
by the telescope. Typical RFI sources in EoR and cosmic dawn experiments are
anthropogenic, such as frequency modulated (FM) radio broadcasts, digital
television broadcasts (DTV; Wilensky et al. (2019)), and ORBCOMM satellite
transmissions (Line et al., 2018). In concept, RFI poses a unique threat to
EoR measurements in that most sources demonstrate sharp spectral variation
over the band used for an EoR power spectrum measurement. Wilensky et al.
(2020) found that the contamination in the power spectrum produced by as
little as 1 mJy apparent RFI flux can be comparable to the EoR signal on some
wave modes. Bright RFI can be easily identified and removed in the raw
visibilities. This process is called “flagging.” A profusion of flagging
algorithms exist in the radio astronomy literature. Two that have been applied
to MWA phase I data are AOFlagger (Offringa et al., 2015) and SSINS (Wilensky
et al., 2019). In Barry et al. (2019b), it was found that removing
observations identified by SSINS to contain RFI that went undetected by
previous excision methods made a measurable difference in the power spectrum
upper limit, particularly for the most highly affected observations.

In our analysis, we find it valuable to distinguish between different types of
RFI based on the level of sensitivity required to statistically identify it.
We separate it into four categories:

  1. I

Visibility level: RFI that can be detected in the visibilities of a single
baseline, where most 21-cm EoR analyses do their flagging.

  2. II

Full array: RFI that requires the sensitivity of the full array (combined
coherently or incoherently). More analyses are moving towards flagging at this
level, e.g. Barry et al. (2019b); Li et al. (2019); Rahimi et al. (2021),
Abdurashidova et al. (2022); HERA Collaboration et al. (2023), and this one.
The SSINS algorithm, used in the aforementioned MWA limits and this work, is
designed to find full array RFI.111In previous literature where SSINS has been
used, its class of RFI excision capability has been described as “ultra-
faint,” whereas here we use the term ultra-faint to refer to RFI fainter than
what SSINS detects.

  3. III

Ultra-faint: RFI that requires coherent averaging of some significant subset
of data in consideration for an upper limit, but significantly less than the
entire volume of data. It may also require the full array sensitivity in order
to detect it. In this work we observe this in the Fourier domain, i.e. in
power spectra, when coherently combining approximately 2-5% of the full
analysis data set.

  4. IV

Unexcisable: RFI that is only detectable when coherently combining an
extremely large subset of the data in a full-array manner, such that exclusion
of the affected data would significantly alter the sensitivity of the analysis
(e.g. half the analysis volume). We propose this type of RFI as a distinct
possibility that has not been ruled out, but do not claim to have necessarily
observed it in this analysis.

Selection State | Amount of Data Remaining (hours)
---|---
Initial pointing selection | 98.56
Ionospheric Cut | 39.98
Full Array RFI Cut (Absolved Observation cut) | 18.39
Final RFI Cut (Wall of Shame cut) | 14.68
Table 1: Amount of data remaining after each selection listed in the left-hand
column.

In this work we show that RFI yet fainter than that detected by SSINS (ultra-
faint RFI) exists within MWA observations, and its effect on the power
spectrum measurement is observable when less than an hour of data is
coherently integrated. We accomplish this by splitting the data into subsets
that are either highly likely or highly unlikely to contain this ultra-faint
RFI, and comparing their cylindrical power spectra. We confirm that the
distinct signature is indeed associated with the RFI by constructing power
spectra with and without RFI flags applied, and noting an enhancement of the
signature when RFI flags are not applied. We also note, however, that this
signature is not necessarily observed in our deepest integrations, suggesting
that RFI may have favorable integration properties for EoR measurements. This
leaves the status of RFI in EoR science unclear, since it may yet appear at a
level fainter than our current integrations, but brighter than the
cosmological 21-cm signal.

This paper is laid out as follows. In §2 we describe the instrument and data
acquisition, pre-processing, calibration, and power spectrum estimation
pipeline. From the flagged and subsequently calibrated data, we derive
statistics about the RFI content and conduct a jackknife test to determine the
effect of ultra-faint RFI on power spectrum measurements, presented in §§3-4.
In §5, we show our final power spectrum upper limit, and in §6 we present our
conclusions.

##  2 Instrument and Observation Description

Here we briefly describe the properties of the MWA that are relevant to our
analysis and provide some details about the choice of observational
parameters. For a more thorough description of the MWA Phase I, see Tingay et
al. (2013). We also describe some preliminary data cuts based on metrics that
have been used similarly in previous MWA upper limits.

###  2.1 Instrument and Observations

To construct this upper limit, we used data from the second season of the MWA
Phase I. This is a radio array located in the radio-quiet Murchison Radio
Observatory in Western Australia. The Phase I configuration correlates 128
pseudorandomly located receiving elements to form interferometric
visibilities. Each receiving element of the MWA is a square tile of 16
crossed-dipole antennas that have been beamformed to produce a single voltage
signal. Using analog delays between individual dipoles within a tile, we can
coarsely point the tile to different locations on the sky.

We select MWA observations of the “EoR0” field, which is centered on a right
ascension of 0 hours and a declination of $-27$ degrees. Throughout each
observing night, the array is repointed roughly every 30 minutes in order to
track this field. This field has been studied in previous MWA EoR limits from
the first observing season (Beardsley et al., 2016; Barry et al., 2019b), as
well as in other seasons, array configurations, and observing bands (Li et
al., 2019; Trott et al., 2020). We focus on five pointings centered on zenith
as the field transits from East to West. This is based on analysis in
Beardsley et al. (2016), where it was found that the galaxy’s presence in the
array sidelobes produces extreme contamination in the EoR window. This
analysis was reconfirmed with a similar quality metric using the Phase II
configuration in Li et al. (2019).

The voltage stream of each tile is channelized by a two-stage polyphase filter
bank (PFB; Prabu et al. (2015); McSweeney et al. (2020)). This divides the
30.72 MHz observing band into 24 coarse channels of width 1.28 MHz, each of
which is divided into 32 fine channels of width 40 kHz. The observing band for
this analysis extends from 167.1-197.8 MHz. Each observation is 112 seconds in
length, with an integration cadence of 2 seconds.

In total we analyze 3168 observations from the second season of the MWA Phase
I, which amounts to 98.56 hours of data. However, the vast majority of these
are removed from the final limit by a strict ionospheric cut and subsequent
RFI-based cuts. We describe these cuts in significantly more detail in the
following sections and summarize the results in Table 1. In brief, about 1/3
of the observations have less-than-excellent ionospheric quality, another 1/4
could not have its ionospheric quality collected (usually a sign of poor
observational quality), and the rest were cut strictly based on inferred RFI
content. Among observations whose ionospheric quality could be gathered, we do
not find any obvious correlation between total RFI occupancy and the magnitude
of the ionospheric metric. We do observe a slight excess of observations with
high RFI occupancy among observations whose ionospheric metric could not be
gathered, however this only accounts for a tiny minority of such observations.
It could be that excess RFI is causing the failure in the metric gathering in
some instances (e.g. one night of extremely bad RFI exists within this
subset), however there exist numerous examples within the data for high-
occupancy observations with collectable ionospheric metrics, so this
hypothetical relationship is tenuously supported at best.

Previous limits using a highly similar analysis pipeline on the MWA Phase I
analyzed the first season of data (Beardsley et al., 2016; Barry et al.,
2019b). We expected the second season of data may be better for power spectrum
measurements due to the the correction of the “digital gain jump” from the
first season as well as reduced digital nonlinearities. Both of these effects
arise in the receiver chain. The digital gain jump is caused by an interaction
between non-linearities in the digital signal chain (arising from
quantization) and the choice of digital multipliers (called “digital gains”)
used to flatten the bandpass between the receivers and the correlator (Prabu
et al., 2015). The digital gains are removed (by dividing out by their value)
after correlation, but in some cases the digital non-linearities of the signal
chain make it impossible to perfectly remove the digital gains. In the first
season of MWA data, the digital gains that were chosen resulted in a single
large jump at 188.16 MHz. This large jump interacted strongly with the digital
non-linearities and as a result there was a residual discontinuity (called the
“digital gain jump”) in the data even after the digital gains correction. Data
above the digital gain jump were excluded in Barry et al. (2019b). For the
second season, the digital gains were chosen to have much smaller changes
across the band, resulting in significant improvements in the bandpass
smoothness after the digital gains correction. In addition we correct for the
remaining digital non-linearities using a Van Vleck correction (Benkevitch et
al., 2016), explained more in §2.3. See Barry (2018) for a detailed
description and analysis of these effects. Finally, we will likely benefit in
future analyses by combining multiple seasons of data, and therefore stand to
gain from analyzing multiple seasons with a similar pipeline.

Figure 1: Flowchart of most salient steps in the analysis of this data set.
The top row of nodes shows the major analysis steps. Analysis stages are color
coded as follows; Blue: Data Acquisition, Cyan: Pre-processing, Yellow: RFI
Analysis, Green: Calibration, Imaging, and Power Spectrum Estimation. The
lower flowchart shows each major analysis step in more detail and how the
outputs of each step feed into subsequent steps.

###  2.2 Analysis Overview

As with all 21-cm power spectrum estimation pipelines, ours involves several
analysis steps. A flowchart describing the process from data acquisition to
power spectrum estimation is shown in Figure 1. Before calibration, imaging,
coherent averaging, and power spectrum estimation, the raw visibilities must
be pre-processed (cyan nodes in Figure 1). Compared to previous MWA limits
that use the same calibration, imaging, and power spectrum software as this
analysis (Beardsley et al., 2016; Barry et al., 2019b; Li et al., 2019), we
use an enhanced pre-processing pipeline. We describe specific enhancements to
the pipeline in the next subsection.

Pre-processing involves several corrections to the raw visibilities. The
phases of the visibilities are adjusted according to the differences in cable
length between the tiles and receivers, and also to specify a phase center on
the sky. Since the alteration to the bandpass resulting from the PFB is
theoretically known, this shape and the digital gains for each coarse channel
are divided out. Due to quantization nonlinearities, this does not fully
correct the bandpass, and residual frequency structure is left over. We
address nonlinearities due to quantization error during pre-processing using a
Van Vleck correction (Benkevitch et al., 2016), which in general corrects for
quantization artefacts by solving an integral equation that relates the
analytic correlation coefficient to the output of a digital correlator. This
is a new technique in our pipeline, and we describe it in more detail in §2.3.
We also flag RFI using Sky Subtracted Incoherent Noise
Spectra222https://github.com/mwilensky768/SSINS (SSINS; Wilensky et al.
(2019)), as well as 80 kHz around each coarse channel edge. Finally, we
downsample to 80 kHz frequency resolution to save disk space and processing
time.

The pre-processed visibilities are then calibrated, gridded, and imaged (at
each frequency) in HEALPix format (Górski et al., 2005) using Fast Holographic
Deconvolution333https://github.com/EoRImaging/FHD (FHD; Sullivan et al.
(2012); Barry et al. (2019a)). The calibration strategy is almost identical to
that used for Barry et al. (2019b). However, we use the autocorrelations to
establish a bandpass shape in the manner of Li et al. (2019), rather than how
they were used in Barry et al. (2019b). Cable reflection systematics are
handled during calibration by fitting the amplitude, phase, and delay in a
hyperresolved Fourier basis (Barry et al., 2019a). We use a modified gridding
kernel, as described in Barry et al. (2019a, b). We also grid model
visibilities generated from the GLEAM catalog (Hurley-Walker et al., 2017),
allowing us to form “dirty” (just the data), model, and residual (data minus
model) HEALPix cubes, which are downsampled to 160 kHz frequency resolution.
We can also propagate weights and variance information about the measurement
by gridding as if each visibility is equal to 1 using the beam and the square
of the beam, respectively. HEALPix cubes from multiple observations can then
be coherently averaged before passage to Error Propagated Power Spectrum with
InterLeaved Observed Noise444https://github.com/EoRImaging/eppsilon
($\varepsilon$ppsilon; Barry et al. (2019a)), where power spectrum estimates
are formed. We form separate cubes for the even and odd time integrations
within each observation, so that $\varepsilon$ppsilon can form a power
spectrum estimate without a thermal noise bias. Important features of
$\varepsilon$ppsilon include the use of a generalized Lomb-Scargle periodogram
as well as multiple forms of noise metrics, described in Barry et al. (2019a).

The in-situ simulation capabilities of FHD allow for robust tests of signal
loss. By simulating visibilities for a fiducial EoR signal and estimating a
power spectrum from them under various conditions, Barry et al. (2019b)
largely verifies that there is no appreciable signal loss in the power
spectrum estimation pipeline used in this work. More specifically, in the
absence of direction-independent calibration errors (note there is no
direction-dependent calibration in the pipeline, an effect known to cause
signal loss in some instances), we are able to recover an input EoR signal in
an in situ simulation i.e. residuals from gridding and other effects are
beneath the expected signal. This and our end-to-end error propagation allows
us to confidently place upper limits on the EoR power spectrum signal in our
measured power spectra.

We also selected data based on ionospheric quality using a metric described in
Jordan et al. (2017). This metric is based on a combination of the median
ionospheric offset as well as a PCA-based measure of offset isotropy to
determine ionospheric quality. Lower scores indicate less active ionospheres.
We perform a relatively harsh cut, removing all observations from the analysis
with an ionospheric quality assurance metric of 5 or greater, cf. Trott et al.
(2020), which amounted to 33.26 hours. We also cut any observation for which
the ionospheric metric could not be gathered, of which there were 25.32 hours.
This usually results from a failure of the observation to calibrate in the MWA
Real-Time System (RTS; Mitchell et al. (2008)), which is a requirement for
assessing the ionosphere. In some instances, this can indicate poor
observational quality. Since we wanted to isolate the effect of RFI on power
spectrum measurements, we resolved to remove such observations outright, thus
avoiding a potential source of confusion. Combined, these cuts remove 58.58 of
the 98.56 hours of data from the original selection that was based solely on
field, observing band, and pointing. We show the result of this cut in Figure
2.

Figure 2: Scatter plot of observations included in the analysis in only the
central five pointings (left), and after an ionospheric quality cut (right).
Vertical axis shows the date of the observations (blue dots), and horizontal
axis is the Local Sidereal Time.

###  2.3 Pipeline Enhancements

The majority of the pipeline enhancements in this analysis are related to RFI
flagging and analysis of RFI statistics to inform data cuts. We briefly
describe our flagging enhancements and changes here, and devote §3 to
describing the statistical analysis. We also describe our digital nonlinearity
correction and calibration enhancements here.

In Barry et al. (2019b), a major improvement to the limit came from excluding
observations that contained DTV RFI. These observations were identified by
SSINS, which is designed to identify full-array RFI (§1). SSINS is described
in full detail in Wilensky et al. (2019), but we also briefly describe it
here. As implemented in this work, it begins by time differencing the
uncalibrated visibilities of a single 2-minute observation at the 2-second
integration cadence, which subtracts out all slowly varying components such as
the cosmological and astrophysical signals. It then incoherently averages
these visibility differences (averages their amplitudes, discarding their
phases) over all the baselines to produce a single dynamic spectrum. In the
absence of RFI, this remaining spectrum should consist of Gaussian noise.
Assuming weak stationarity (unchanging mean and covariance function), the mean
of this Gaussian at each frequency is determined by a time-average of each
channel. Due to the properties of the thermal noise in the visibility, the
standard deviation of the noise at this stage is proportional to its mean, so
we can obtain a $z$-score for each sample in this way. We then use an
iterative match filter to search for pre-determined shapes within the spectrum
that belong to known sources of interference. This can be thought of in a
Bayesian setting as generating flags using the maximum a posteriori decision
rule amongst a small dictionary of rectangular signals (top-hat power spectral
density) of a given strength. The resulting flags in this spectrum are
propagated across all baselines and in time according to the visibilities that
formed the corresponding differences.

While SSINS is capable of producing flags at the time and frequency resolution
of the visibilities, the actual RFI flags from SSINS were not applied in the
Barry et al. (2019b) limit. Rather, observations with known RFI contaminants
were excluded from the analysis altogether, including data within those
observations that were not classified as contaminated by SSINS. In this
analysis, we apply the SSINS flags to the data, extending them in frequency so
as to avoid excess EoR window power generated by chromatic flags (Offringa et
al., 2019; Ewall-Wice et al., 2021; Wilensky et al., 2022a). Actually applying
the SSINS flags could allow for a measurement with reduced thermal uncertainty
compared to Barry et al. (2019b) since it may allow for more data to be used.
However, in §4 we analyze power spectra with these flags applied at small
integration depths and resolve to fully remove any observations identified as
containing RFI in developing the final limit, as was done in Barry et al.
(2019b).

In this limit, we do not apply AOFlagger flags, unlike every other MWA EoR
limit so far. Upon examining the outputs of each flagging pipeline, we found a
modest number of RFI events that were caught by AOFlagger and not by the pre-
extended SSINS flags. After frequency extension, many of these events were
coincidentally excised. Additionally, some samples that were flagged by
AOFlagger and not by SSINS may be false positives that result from AOFlagger’s
morphological detection algorithm, which can overextend flags for broad
features (Offringa et al., 2012). The AOFlagger flags have a small random
false-positive rate (percent-level, depending on the night; Offringa et al.,
2015) that appears uniformly distributed throughout the times and frequencies
of the data. If we extend these flags in frequency to avoid excess power in
the window, we incur massive data loss due to the false positives. In
principle some of these data could be recovered by implementing some of the
strategies in Offringa et al. (2019), but we did not do this. Since we cannot
generically extend the AOFlagger flags in frequency without causing massive
data loss, we opt to entirely forego them at the possible expense of a few
extra false negatives in the entire season.

We apply a Van Vleck correction for the nonlinear distortion present in the
raw visibilities during pre-processing.555For implementation details, see
007_Van_Vleck_A.pdf and 007_Van_Vleck_B.pdf in
https://github.com/EoRImaging/Memos The MWA digital signal pipeline involves
several stages of quantization. The Van Vleck correction employed in this work
corrects the nonlinearity associated with the final round of quantization.
Corrections for other quantization stages are forthcoming. The actual
correction involves solving an integral equation that relates the quantized
and analog visibilities. This equation has no closed-form inverse for the MWA
bit depth. In order to generate fast solutions, the equation is approximately
inverted using Chebyshev polynomials. The Van Vleck correction reduces
calibration errors due to spectral structure from the digital nonlinearities
that cannot be handled by a calibration that assumes a linear gain.

Most of calibration proceeds identically to Barry et al. (2019b). To speed
calibration convergence by roughly a factor of 2 relative to what was used in
Barry et al. (2019b), we employ a Kalman filter during calibration. Within
iterations of the calibration solver, a new guess is determined based on a
weighted combination of the previous guess and the current one. The filter is
a Bayesian optimization technique where a prediction about the likely location
of the solution guides the weighting of the two guesses so as to speed up
calibration (as opposed to e.g. a strictly gradient-based approach). The 2013
MWA data had a digital gain jump approximately 2/3 the way through the EoR
highband, which required calibration to operate separately on the sections of
the band to either side of this discontinuity. The data we selected in 2014
did not require separating the band into two pieces for calibration since the
digital gain jump has been corrected in data acquisition. For estimation of
the bandpass, we use the autocorrelations in the style of Li et al. (2019),
described thoroughly in §5 of that reference. This method avoids pitfalls
associated with using a particular reference antenna, which may cause
idiosyncrasies of that antenna (e.g. imperfect cable reflection fitting) to be
imprinted on the other tiles.

##  3 Analysis of RFI Statistics

We perform an occupancy analysis based on the SSINS flags. We find several
interesting features about the time dependence of the RFI occupancy. We also
use these flags in order to study the brightness statistics of RFI after
calibration. We then use these time and brightness properties in order to
conduct jackknife tests with cylindrical power spectra. We find that there is
a definite signature of ultra-faint RFI in the cylindrical power spectrum that
is most visible in sub-hour length integrations. The effect on the power
spectrum is the main topic of §4, however we comment briefly here on possible
reasons for an enhanced RFI signature in the power spectrum at certain
integration depths.

Our sub-hour integrations tend to consist of observations from the same night
or relatively few nights, and oftentimes observations that are contiguous with
one another. If these contiguous observations are flagged due to a common RFI
reflector, then it is possible that the RFI signal might integrate coherently
for those observations. For example, if an airplane moving due N-S is
reflecting DTV interference, then the E-W baselines in particular will observe
no fringing of this source and thus average coherently. In longer
integrations, we are likely to combine more RFI sources since we include more
nights of data. If different RFI sources tend to average incoherently, then
deep integrations may dilute the RFI signal and therefore produce smaller
power spectrum contamination (Wilensky et al., 2020). Our observations
therefore seem to support the idea that independent RFI sources tend to
average incoherently. This is our leading hypothesis as to why we seem to see
more contamination in sub-hour integrations compared to longer ones.

###  3.1 Summary of Flagging Settings

The match-filter in SSINS can be programmed to search for occupants that span
arbitrary contiguous frequency ranges in the observing band. We generally
reserve this feature for physically or empirically motivated frequency ranges,
since searching every possible frequency range creates massive computational
overhead. The most common occupants for this observing band are digital
television and whole-band “streaks” of uncertain origin (Wilensky et al.,
2019).666The uncertainty stems from the fact that the contaminants are not
band-limited, and can therefore come from a number of hypothetical sources.
Some occurrences have been clearly associated with extremely bright ORBCOMM
(Wilensky, 2021), but this only accounts for a small fraction of instances.
Hereafter, we will refer to these whole-band streaks as “broadband” RFI. Based
on the Western Australian digital television allocations, we search for four
7-MHz-wide DTV signals, all adjacent, starting at 174 MHz; these are
designated as channels 6-9. Channel 9 begins at 195 MHz, so we only observe
2.8 MHz of its allocated bandwidth. We also search for narrowband occupants
and broadband RFI.

Recall from §2.3 that the SSINS $z$-scores are calculated from the 2-minute
time-average of incoherently baseline-averaged visibility differences, and
reflect a probability under a null hypothesis for pure thermal noise to
achieve a deviation from the mean at least as large as the data’s. Under that
null hypothesis, we expect less than 1 datum with $z$-score greater than about
4.2 in a single observation’s SSINS. For all shapes other than broadband RFI,
we use a significance threshold of 5, i.e. a (potentially frequency-averaged)
sample whose $z$-score is $5$ or greater is considered as contaminated. For
broadband RFI, we use a significance threshold of 10. We use a greater
significance threshold for broadband RFI since the data exhibit lightly
nonstationary thermal noise as a result of the receiver refrigeration cycle,
which results in a significant number of false broadband detections at
$z$-scores less than $10$ (Wilensky, 2021).

The SSINS match filter proceeds iteratively in that when it detects an RFI
event, it excises that data and recalculates the $z$-scores of the remaining
data before searching for the next strongest RFI event. Before recalculating
the $z$-scores, we flag a frequency channel for the entire observation if its
occupancy is greater than 60% and extend flags in frequency. This combination
means that if any RFI occupant is found for 60% of an observation, the entire
observation is flagged. We summarize these settings in Table 2.

Shape Label | Frequencies (MHz) | Significance Threshold
---|---|---
TV6 | 174-181 | 5
TV7 | 181-188 | 5
TV8 | 188-195 | 5
TV9 | 195-202 (197.8) | 5
Narrowband | Various | 5
Broadband | 167.1-197.8 | 10
Table 2: Association of shapes in the match filter, their frequencies, and the
significance thresholds used in the filter. The TV9 shape is allocated all the
way to 202 MHz, but our observing band cuts off at 197.8 MHz. Narrowband
shapes are 1 fine channel wide and allowed anywhere in the band.

###  3.2 RFI Occupancy Analysis

Any time data is excised by the SSINS match filter, the time, frequencies, and
shape label of the most recently identified RFI event are recorded. We analyze
this flagging metadata to understand the RFI content of the data set. First we
discuss a few important caveats about this flagging metadata related to
misclassification, which are explored in substantially more detail in Wilensky
(2021). Then we present the occupancy results.

####  3.2.1 Flagging Nuances

Below we list a few nuances about the SSINS flags that are important to
understand when interpreting trends within the flagging results.

  1. 1.

Prioritizing excision over classification: While SSINS does associate each RFI
event with a label, there can be misclassifications that produce identical
flags, and we do not formally check for this in every possible instance. For
example, 3-4 DTV events occurring simultaneously may be classified as a
broadband event since combinations of DTV events are not checked in the match
filter. This will produce the same flags after frequency extension. We judge
the rate of this misclassification to be low based on hand-grading the SSINS
flags for each observation (prior to frequency extension).

  2. 2.

Assumption of stationary noise: The $z$-scores in SSINS are calculated
assuming weak stationarity (specifically constant mean and variance in this
case). When bright sources enter or exit sensitive parts of the beam, the
variance of the thermal noise can change significantly on timescales
associated with the beam crossing time, which is insignificant for the 2
minute observations in this work. In the MWA Phase I, there are 16 receivers,
each of which are responsible for digitizing and channelizing the voltage
signals of 8 tiles. A temperature-dependent gain in the amplifiers (Barry,
2018) causes smooth temporal variations in the noise variance on time scales
associated with the receiver refrigeration cycle (a few minutes; Wilensky,
2021). Since SSINS flags are generated on uncalibrated data, this cycle tends
to impart a gradient in the $z$-scores that is minuscule but coherent in the
SSINS over the entire observing band. The coherency leads to an overreporting
of broadband events at a given significance. We counter this by increasing the
significance threshold for broadband events. We expect this produces an
average false positive rate of less than 2%.

  3. 3.

Insufficient time samples: In heavily contaminated observations, having fewer
time integrations with which to estimate the mean per frequency and
polarization prevents the $z$-scores from being approximately Gaussian. What
we generally find is that the $z$-scores calculated by SSINS for simulated
stationary Gaussian noise are more concentrated than a standard normal random
variable, which can lead to false negatives. For this reason, we completely
flag all observations where greater than 60% of integrations have been
flagged.

####  3.2.2 Occupancy Results

In this section we discuss the RFI occupancy of the data set as reported by
SSINS. We find that the overall distribution of RFI occupancy is relatively
unaffected by the ionospheric cut, and therefore only show distributions
before the ionospheric cut so that we have a larger sample size.

As an example of fairly typical flagging results for moderately bright DTV
contamination, we show Figure 3. This shows the raw SSINS and its $z$-scores
for a 112-second observation in the data set. There is an initial flagging
mask that flags the edges of the coarse channels as well as the first
integration. This latter flagging set is a holdover from previous analyses,
where the correlator would occasionally produce oddities in the data at the
very beginning of the observation. We did not check whether this behavior was
still present, and flagged the beginning of each observation just in case. The
DTV contamination is most obvious in the mean-subtracted version, where it is
clear that it lasts at least 20 seconds. The $z$-scores shown are based on a
time-average of the unflagged data, which biases the $z$-scores low for the
rest of the integrations in the channels occupied by the DTV. Since the
flagger recalculates $z$-scores after each event is identified (Wilensky et
al., 2019), this initial bias does not lead to extra false detections. The
middle row shows the results of the SSINS match filter, indicating that the
event lasted roughly 40 seconds. In imaging experiments for DTV events that
appear in SSINS similarly to these, we find that they are usually some sort of
aircraft reflection often appearing in the second southern sidelobe of the
beam. The bottom row shows flags after frequency extension. We use these
extended flags to develop the flagging time series used in the statistical
analysis in this section.

Figure 3: Left: Raw SSINS of a 112-second observation at various stages of
flagging. Right: Mean-subtracted SSINS. Top: Data with initial flagging mask.
Middle: Additional flagging mask after match filtering. Bottom: Flags after
frequency extension.

In Figure 4, we show a histogram of the TV occupancies for all of the
112-second observations before the ionospheric cut. We observe that channels
6, 7, and 8 have very similar occupancy distributions, while channel 9 is much
rarer to observe, appearing in only 3.5% of observations. Examining
broadcasting stations in Western Australia, we find that channels 6, 7, and 8
are used relatively equally, while channel 9 is used less often. Since we only
observe 2.8 MHz of channel 9’s allocation, SSINS is also less sensitive to its
presence.

Figure 4: Histogram of DTV occupancies per observation, before the ionospheric
cut. Since observations with more than 60% occupancy are entirely flagged,
there is a cutoff in the histogram at 0.6 and a buildup at 1. Only a small
fraction of the allocation for TV9 is in the observing band, so the SSINS
match filter is less sensitive to it. Data about TV broadcasting stations also
suggests this allocation may be used less often.

Since the majority of the observing band is allocated for DTV, we expect very
few narrowband events, which is consistent with what we observe. Curiously,
when we do observe narrowband events that are clearly true positives, they are
often within the DTV allocation. Some of these events are extremely bright,
and may be a result of out-of-band RFI clipping the ADC or causing
intermodulation products. Alternatively, there could be locally generated RFI.

Since we suspect a significant number of the RFI events recorded by SSINS are
due to reflections from aircraft (Wilensky et al., 2019), we seek to
understand the timelike properties of the RFI occupancy. First, we notice that
the different TV channels are sometimes flagged simultaneously, and flags tend
to appear in clusters within a night. By constructing the correlation function
of the time series of flags belonging to different channels, we sometimes
observe that flags for different TV channels are strongly correlated within a
night, however not all nights have such correlations. Not all broadcasting
stations in Western Australia transmit at the same frequencies, and there is
probably a range of aircraft trajectories that intercept these various
stations. Furthermore, direct reception of the signals is possible. For
instance, there is a night of times with extremely high TV occupancy on August
27, clearly visible in the occupancy scatter in Figure 5. The SSINS of this
night show extremely strong, persistent DTV interference. Since the RTS could
not calibrate these observations, no ionospheric metric could be gathered to
assess the possibility of long-range ionospheric-based reception such as
“sporadic-E” propagation. This could be a tropospheric ducting event allowing
for long range direct reception of the RFI (Sokolowski et al., 2016), however
we have not cross-referenced any sort of weather database or performed any
other type of analysis that allows us to understand the exact origin of this
extreme event. Combining all of the aforementioned properties, we expect that
we sometimes observe different DTV signals simultaneously, but not always. The
flags reflect this in their two-point statistics and this expectation is
corroborated by manual inspection of the SSINS.

Figure 5: Total Occupancy scatter plot after extending flags across frequency,
which roughly doubles the total occupancy in the data set. Each marker
represents a 2-minute observation, where its horizontal position is its
starting sidereal time and its vertical position is its beginning civil time.
Clusters of flags are apparent.

In Figure 5, we show the total occupancy of each observation with the flags
extended. Here, the clustering behavior of the flags is more apparent. To also
illustrate this clustering, we show a histogram of interarrival times between
SSINS flags (time elapsed between flagged integrations) in Figure 6. We do not
include the night with strong, persistent interference. We also show two
models with parameters estimated from the data: a Poisson process based on the
mean flagging rate and a Markov model based on the observed transition rates
between flagged and unflagged data. Over 80% of the mass of this histogram is
contained at the sample spacing. If we model each night as its own Poisson
process (i.e. each night has a different false positive rate, and the flags
are dominated by false positives), Poisson mixture model fails to predict the
abundance of mass at the sample spacing and also decays much faster than the
data, which means it underpredicts long gaps between flagged data. This
suggests clustered flagging with a range of gap sizes rather than a simple
mixture of Poisson processes. If we perform a Kolomogorov-Smirnov test, we
obtain a test statistic value of about 0.7, which corresponds to the
discrepancy at short interrarival times, highlighting the Poisson model’s
failure to predict clumps. This corresponds to a $p$-value that is basically 0
in floating point precision, i.e. there is almost no probability that the
empirical CDF of a collection of random samples from the Poisson mixture model
would appear at least as discrepant as the data.

To probe the clumping properties of the flags, we examine the transition
probabilities in the time series. What we find is that unflagged data is
almost always followed by more unflagged data at a rate of 98%, while flagged
data is followed by more flagged data at a rate of 80%. Since the flag
occupancy is relatively low ($\sim 10\%$), this manifests as long runs of
unflagged data with small-medium sized clumps of flagged data. We find that
such clumps can be modeled using a discrete time Markov chain with transition
rates matched to the data. Each night we generate a time series using that
night’s transition rates, and then we histogram the resulting interarrival
times. Specifically, we begin with an initial seed state, $x_{0}$, which takes
the value 1 or 0 (1 indicates that it represents a flagged datum, while 0
indicates otherwise). We then sample from a Bernoulli distribution with
probability of success equal to $P(X_{1}=x_{0}|X_{0}=x_{0})$. For example, if
$x_{0}=1$ and the Bernoulli sample is a success, then $x_{1}=1$ as well,
otherwise $x_{1}=0$. We then repeat this process to generate the chain for
each night. While the realization of the season of interarrival times is
noisy, it appears to emulate the interarrival time distribution. This
quantitatively supports that the apparent clumping of the flags is indeed
real.

Figure 6: Top: Histogram of flag interarrival times over all nights for the
broadcast SSINS flags. The “Poisson Mixture Comparison” (green) is a
hyperexponential distribution where each component has consistent mean
interarrival time as each night of data (and is weighted according to the
amount of data in the corresponding night). The Markov chain histogram is
built from Markov chain realizations of each night with consistent transition
probabilities as the night of RFI flags they are built from. Interarrival
times from the Markov chain realizations are roughly consistent with those
from the RFI flag time series, whereas the Poisson mixture model tends to
underpredict short and long interarrival times, thereby failing to predict
clumps of flags. Bottom: Cumulative density for what is shown in the top plot.
The data and Markov chain cumulative densities are inconsistent with a Poisson
model.

The RFI flag time series for each night is not entirely consistent with the
Markov chain; they differ in their two point statistics. The Markov chain is
memoryless, and so any given realization generally does not produce an
autocorrelation function with strong peaks away from zero lag (difference in
time between samples). On the other hand, many of the RFI flags exhibit
discernible peaks at a range of lags, though some nights with comparatively
few flags appear somewhat consistent with the Markov realization. We show an
example in Figure 7. This suggests that flags from SSINS are not entirely
random point processes with only short-range correlations. These additional
peaks in the autocorrelation function may occur when reflectors transit
different sidelobes of the beam, or may suggest multiple reflectors with some
lag between them such as multiple aircraft on the same flight path. The
additional peaks also appear to have a characteristic width from night-to-
night, possibly indicating a typical clump size related to the speed of
reflector transit through the beam.

Figure 7: Autocorrelation of RFI flag time series for a particular night
(blue) along with the autocorrelation function for a Markov chain realization
(transparent orange). Bottom Panel: Residual of the top panel. The Markov
chain realization is clearly missing the peaks at lags between 0 and 50
minutes, meaning that the Markov model does not fully capture the timelike
properties of the RFI events, and so a more detailed model is required in
future efforts.

It is clear from the transition probabilities, interarrival times, and two-
point statistics of the RFI flag time series that RFI events tend to
temporally cluster. This motivates a particular style of jackknife test, where
observations with no RFI flags are compared to observations that contained
some RFI flags. The logic is that an observation with RFI flags is more likely
to contain RFI just beneath the sensitivity of SSINS than observations with no
flags at all. In the next section, we develop a series of jackknife tests
based on the brightness of the flagged emitters with this physically motivated
assumption in mind.

###  3.3 RFI Brightness Analysis

The excess contamination in the power spectrum from RFI responds quadratically
to its flux. Since SSINS uses incoherently averaged time-differenced
visibilities, it is not immediately apparent how to exactly relate the
brightness in the SSINS to the brightness in the visibilities. However, the
amplitude of a visibility difference can be no more than the sum of the
brightnesses of the two samples that were differenced. In more detail, if the
two samples are out of phase by exactly $\pi$, this upper bound is achieved.
However, if the difference in phase and amplitude is small, then the amplitude
of the difference will also be small. This relationship may not be constant
over the set of baselines, however the samples identified by SSINS will fall
more often in the former case than the latter with the exception of
extraordinarily bright RFI (i.e. SSINS will select for large phase or
amplitude differences). Therefore, we can use the SSINS amplitudes as a proxy
for the brightness in the visibilities.

Figure 8: Histograms of SSINS amplitudes and z-scores for the entire season,
separated by flagged and unflagged data. These $z$-scores are calculated per
fine 40 kHz channel using the time-average of 112s of data as a reference. The
horizontal axis is a hybrid linear-logarithmic scale, where the boundary
between scales is demarcated by the black vertical lines. The z-scores of the
unflagged samples (bottom orange) are highly Gaussian, as expected from the
assumptions about the thermal noise. The flagged samples have a highly non-
Gaussian z-score distribution. Their brightness distribution has significant
overlap with the unflagged brightness distribution, and cannot be separated
solely by drawing an amplitude cut. Figure 9: Same as Figure 8, but with
brightnesses and z-scores calculated by doing the SSINS sub-band sum that is
employed in the match filter over the TV7 frequencies. Interestingly, the
unflagged and flagged brightness samples appear bimodal. Examination of this
plot on a per-pointing basis shows that the mode above 40 Jy belongs to the
eastern-most pointing in the data, while the other mode is a combination of
all other pointings.

In Figure 8, we show histograms of SSINS amplitudes as well as z-scores for
the entire season, separated by those samples identified as contaminated
(flagged) and those that were not (unflagged). In the top panel we show SSINS
amplitudes. Due to the central limit theorem, we expect that the per-frequency
distribution of the purely thermal samples in the SSINS follow a Gaussian
distribution. However, when all frequencies and many observations are combined
in mixture, we observe a highly non-Gaussian distribution. In the bottom
panel, the unflagged z-scores, which are calculated per frequency and per
observation, are highly Gaussian, reflecting the statistical assumptions
underlying the SSINS pipeline. The unflagged samples have brightness between
33 and 52 Jy. On the other hand, the flagged samples have amplitudes that
range from 30 to 450 Jy. The z-scores of the flagged samples exhibit a highly
non-Gaussian distribution.

Many of the flagged z-scores are far below the significance threshold. This
occurs when the sample was flagged as a part of a shape in the match filter.
These shapes are determined from a combination of official allocations from
the Australian Communications and Media
Authority777https://www.acma.gov.au/publications/2004-04/guide/technical-
planning-parameters-and-methods-terrestrial-broadcasting-tpps and empirical
verification of the presence of such RFI shapes in our data (Offringa et al.,
2015; Barry, 2018; Wilensky et al., 2019). Some shapes are significantly
broader than others. Broader RFI can be found at a lower average flux density
compared to narrower shapes due to the fact that broad clusters of positive
(or negative) outliers would exist only with vanishing probability in a purely
thermal SSINS. Thus, the $z$-scores at the single channel level can be very
low when their corresponding measurements are identified as part of a broad
shape.

These data are perhaps better visualized in Figure 9, where we show the same
quantities as in Figure 8, but emulate the match filter process by summing
over frequency and histogramming the maximum absolute deviation across the
polarization axis of the averaged scores. The top panel now appears obviously
bimodal for both the flagged and unflagged data. Remaking this plot per
pointing shows that the mode above 40 Jy comes from the eastern-most pointing
we selected (designated -2 in this work), while the other mode is a
combination of the other four pointings. The thermal background in the bottom
panel now takes the form of an extreme value distribution for a Gaussian of
sample length equal to 4 (number of polarizations), and the recalculated
flagged z-scores are offset from zero. Many still exist below the significance
threshold. This happens because we flag uncalibrated data, but these data have
been calibrated before being histogrammed. This can slightly change the
$z$-score calculation since calibration corrects relative amplitude variations
between different tiles, ultimately affecting the time dependence of the SSINS
and therefore the $z$-scores (Wilensky, 2021).

We argue that these are still likely to be true positives. The frequency-
summing step in the SSINS match filter reduces the sample size in
consideration by the number of frequency channels in the observation. This
means that substantially fewer extreme values are expected in sub-band sums,
and so a significance threshold of 5, which was set based on the sample size
before summing, gives a wider margin of outliers compared to what is expected
from the thermal background. More importantly, we find that observations with
only these low significance values are still correlated with excess EoR window
power compared to their clean counterparts. We show these in the next section
by constructing jackknife tests designed to investigate the possibility of
residual RFI in observations flagged by SSINS.

##  4 Power Spectrum Subintegrations: Evidence of Ultra-faint RFI

The level to which residual RFI contaminates a power spectrum is determined by
its apparent brightness and frequency structure. The apparent brightness is
determined by the transmitter brightness, the circumstances of propagation,
and the relative position of the telescope primary beam. The larger population
centers of Geraldton and Perth are located to the Southwest and South of the
array, respectively. Either of these locations may take in flight from
countries Northwest of Australia, which would put reflectors in a more
sensitive part of a Westerly pointed beam. Smaller cities and towns are also
located due West, such as Carnarvon and Denham. Since transmitters tend to be
located close to population centers, we expect that RFI contamination should
generally increase when the telescope is pointed more to the West.

To probe these different variables, we establish a series of jackknife tests
where we group observations according to different RFI-related variables and
examine their integrated power spectra. Some observations in the data set have
no flags reported by SSINS, while some are reported to contain RFI. In some of
our tests, we purposefully disable flags before calculating power spectra. It
is standard to refer to data that has been identified as suspicious as
“flagged” and data that has not been flagged as “unflagged.” The latter term
has a particularly confusing linguistic structure that seems to imply “undoing
flags that were already present” but more often refers to data that was found
to be without a need for flagging i.e. deemed non-suspicious after a rigorous
inspection (and indeed this is how we meant it in §3). Since we need to make
use of both senses of the word “unflagged,” we instead use the following
nomenclature in this section when categorizing observations based on their
SSINS:

  * •

Pure: Observations for which SSINS found no RFI.

  * •

Absolved: Observations for which SSINS found RFI, and we have applied the
flags so that the contaminated data has been excised.

  * •

Repentant: Observations for which SSINS found RFI, but for investigative
purposes we have elected not to use the calculated flags so that the
contamination is still present.

Because the EoR power spectrum is sensitive to very faint RFI contamination we
want to test whether the data immediately adjacent in time to detected RFI is
also contaminated but at a brightness below SSINS’ sensitivity. To make this
comparison, we compare the power spectra of absolved observations (RFI
detected and excised to the best of our ability) to those of LST-matched pure
observations with no detected RFI. At this stage, we only consider
observations that pass the ionospheric cut. We jackknife by dividing the
absolved observations into integration subsets separated by RFI shape reported
by the match filter (narrowband, broadband, or any TV channel), telescope
pointing (integers from -2 to +2), and reported z-scores of the RFI events
(bins with edges 3, 5, 10, 100, 1000, 10000). For each absolved observation in
a subset, we find a matching pure observation at a similar sidereal time. We
then integrate these subsets, form power spectra, and examine them side by
side.

Figure 10: Top: East-West polarized power spectra for pure observations over
the five pointings present in the data. The array points more Westerly towards
the right of the figure. The number of observations in each integration is
annotated at the bottom of the figure. The depth of integration ranges by
about a factor of 2.5. All of these power spectra appear noise-dominated in
the EoR window (region above the solid black line) in that there is a roughly
even speckling of positive and negative data. Bottom: Corresponding absolved
power spectra, where any RFI type is included so long as the observation has
RFI z-scores that lie between 10 and 100. We see a signal-like component in
the EoR window. Rather than an even speckling of positive and negative data,
we see consistent, smoothly varying positive power up to and occasionally past
the first coarse band harmonic. This suggests residual RFI that was uncaught
by SSINS. This contamination is generally worse in the later pointings that
point more West. There is no clear correlation between integration depth and
contamination levels.

This produced 160 power spectra per polarization (80 pure/absolved pairs) over
the 1285 observations that passed the ionospheric cut. Before examining our
jackknife axes, we examined each power spectrum individually, paying special
attention to the EoR window where we expect RFI contamination to be most
obvious if at all present. Of these 160 observation sets, we identified 46
observation sets, 32 absolved and 14 pure, that demonstrated clear excess
power in the lower left corner of the EoR window. The excess power is “signal-
like” in the sense that it appears to vary smoothly as a function of
$k_{\perp}$ and $k_{\parallel}$ until it seemingly becomes noise dominated at
very high $k_{\parallel}$. This feature can vary in power anywhere from
$10^{9}$ to $10^{11}$ $($ mK2 $h^{-3}$ Mpc3 $)$, with stronger features
generally appearing in shallower integrations. Since most of these integration
subsets are relatively short (approximately 1 hour or less), a signal-like
component in the window is surprising and indicative of systematic
contamination.

Figure 11: Top, middle: Cumulative density functions for the absolute value of
the inverse-noise weighted spherical power neglecting the foreground wedge and
coarse band harmonics. The top row, which shows measurements from absolved
observations, appears ordered by pointing from East to West for most of the
domain. The pure observations are not as clearly ordered or distinct from one
another. Neither data set looks consistent with pure noise, however. Bottom:
Cumulative density function for the difference between corresponding inverse-
noise weighted spherical power of the top two rows (absolved minus pure). The
preference of the distributions to lie towards the right of the dotted curve,
which reflects a null hypothesis, shows that absolved observations generally
produce power spectra that are more positively outlying than their pure
counterparts.

This feature almost exclusively appeared in power spectra made from only E-W
polarized dipoles and has similar morphology across all identified sets. For
the 14 pure observation sets we identified, 12 of their absolved counterparts
were also identified in the 46 identified sets (that is, 24 of the 46 sets
were LST-matched pure/absolved pairs). The power is usually substantially
worse in the absolved sets. The significantly larger severity and prevalence
of this feature in the absolved sets suggests that this signature is likely to
be residual RFI unidentified by SSINS.888Recall from §3.2 that we flag all
frequencies whenever we identify RFI at any frequency, so this cannot be
excess power from spectral variations introduced by flagging.

Figure 12: Similar power spectrum jackknife test as Figure 10, except this
time comparing only the +2 (most Westerly) pointing for RFI of any type in the
annotated z-score bins at the top of the figure. This comparison demonstrates
two features that were a common among our 2d power spectrum jackknife tests.
First, excess power is more noticeable for observations in which the
identified contaminants had relatively low SSINS z-scores. Since the number of
contributing observations ranges from 1 to 32, there is significant variation
in the noise levels of each integration, which makes the significance of
excess power from residual RFI harder to discern from this Figure alone.
Figure 13 makes this claim more evident. Second, the +2 pointing appears to be
the most obviously contaminated pointing, and contamination can sometimes even
be seen in the pure observations. For example, the corresponding pure
integration for the power spectra whose SSINS $z$-scores fall between 10 and
100 as well as the one for $z$-scores between 3 and 5 appear to have moderate
excess window contamination, though less than their absolved counterparts.

To further assess the probability of this hypothesis, we co-examined
pure/absolved pairs across axes of our predetermined subsets. If the severity
or prevalence of the effect is enhanced for choices of RFI parameters that
should enhance them, then the evidence for the RFI hypothesis is increased.

###  4.1 Power Spectrum Jackknife Test Analysis

In Figures 10-15, we show several representative examples of such comparisons
across each of our jackknife axes. In each comparison, we hold two of the
three jackknife axes fixed and compare across the third (e.g. Figure 10 holds
$z$-score and RFI shape fixed and lets pointing vary). While some comparisons
clearly support our premise, others are more anomalous though not
inexplicable. We show one more test in Figure 16 in which we compare the
absolved and repentant forms of a single DTV-affected observation set (i.e. we
make power spectra with and without flags for the same observations), and note
an obvious enhancement of the contamination in the repentant power spectrum.
The enhancement has the same morphology as the feature we identify in our
other jackknife tests, even those with more dubious results. Overall we find
the results of these tests to be highly compelling and supportive of the idea
that this power spectrum feature is the signature of ultra-faint RFI. In what
follows, we walk through the specific comparisons associated with each figure
in more detail.

Figure 13: Cumulative distribution functions in the same style as Figure 11,
but this time collated by the $z$-score of the events found by SSINS. In this
case, the absolved observations appear to fall into two classes. The
observations with extremely bright events do not appear to produce
significantly outlying power spectra, whereas observations with only faint or
moderately bright RFI appear to produce more strongly outlying power spectra.
The pure power spectra also seem divided in this way, although the distinction
is less obvious. Since observations with lower SSINS $z$-scores are more
plentiful, this trend that is present in both the top and middle panels may
indicate a systematic that integrates somewhat coherently over the time scales
used in these jackknife tests. We again see that absolved power spectra are
more often positively outlying than pure ones.

####  4.1.1 Pointing Test

In general, we found that this excess power becomes more noticeable as the
array points more Westerly. As an example, we show a sequence of pointings for
integrations over observations with any RFI type in them whose z-scores were
all between 10 and 100 in Figure 10. The excess window power suggests that
absolved observations possess residual RFI uncaught by SSINS, i.e. ultra-faint
RFI, that makes a noticeable effect on the power spectrum measurement. In the
full set of integrated power spectra, we see the pointing trend occur
regardless of RFI type, and in most z-score bands. We also note that 13 of the
14 pure subsets we initially identified as containing the excess window
contamination all came from the 2 most Western pointings. Supposing this is
caused by ultra-faint RFI, we expect more prevalence for Western pointings
physically since there are more transmitter sites and population centers
towards the West of the array than towards the East, and so pointing the
telescope beam more West will put associated RFI events in a more sensitive
location in the beam.

Since showing every integration comparison available to us in the manner of
Figure 10 would be impractical, we show a summary statistic in Figure 11 to
demonstrate our claimed trends. For each integration subset, we take the
spherically averaged power spectra, excluding contributions from the
foreground wedge and coarse band harmonics, and divide the power at each wave
mode by its noise level as propagated by $\varepsilon$ppsilon. This choice of
weighting allows us to more easily compare integrations with varying amounts
of contributing observations.

For the top two panels, we calculate the empirical cumulative distribution
function for the absolute value of this quantity, collated by pointing and
whether or not the observation set is pure or absolved. In other words, for
each distribution function, the pointing and pure/absolved status is fixed,
and the brightness within the SSINS and RFI type are unrestricted. We expect
that the noise is, at least approximately, zero-mean and Gaussian distributed
(Wilensky et al., 2023). Therefore if the data were strongly noise-dominated,
we would expect this distribution to resemble a folded Standard Normal
distribution (i.e. a normal distribution with mean zero and standard deviation
1, folded over the vertical axis since we are looking at the absolute value).
What we observe is that neither the pure nor absolved observation sets appear
noise-dominated; deviations from zero are greater than expected under that
hypothesis. Importantly, the absolved data (top row) are clearly ordered by
pointing from East to West for weighted power values greater than about 1
(though zenith and +1 track each other almost perfectly). The pure data, on
the other hand, are not clearly ordered by pointing until the very highest
power values. In summary, absolved power spectra from more Westerly pointed
data appear consistently more outlying in the EoR window than spectra from
more Easterly pointed data, and power spectra from pure data do not show this
as obviously or for as great a range of powers.

In the bottom panel, we compute the (signed) difference between the inverse-
noise-weighted spherical power between corresponding absolved and pure power
spectra and form the cumulative distribution functions collated in the same
way as the top two panels, but without taking the absolute value. In this
case, we would expect that if the pure and absolved sets were biased by the
same amount relative to the noise (potentially indicating similar residual
systematics, though this comparison is complicated), then the difference would
be noise-like and appear normally distributed with standard deviation equal to
$\sqrt{2}$. We find in this panel that the mass of the data is almost
exclusively to the right of this null hypothesis, except in the -2 pointing,
indicating that absolved power spectra are generally more positively outlying
than pure power spectra. Thus, there is a clear statistical difference between
the pure and absolved data, and we suggest that this difference is caused by
residual RFI.

We generally find that the excess power is far more noticeable in the East-
West polarization than the North-South. We know from inspection of the SSINS
that RFI is often seen as unpolarized, i.e. roughly equal strength in all four
instrumental polarizations. In some specific instances it can appear
polarized. Usually if it appears polarized, it appears stronger in the East-
West dipoles. Occasionally it will appear stronger in the North-South dipoles.
The apparent polarization of the RFI source is a combination of its broadcast
polarization, some propagation effects, and perhaps most importantly, its
geometric location relative to the array.

Since a dipole has no sensitivity along its axis, RFI from the Southern or
Northern horizon will appear to the array as if it is East-West polarized, and
vice-versa for the Western and Eastern horizons. There are a number of digital
television transmitter sites in Western Australia, and therefore a range of
possible propagation directions for DTV RFI. The strongest transmitters are in
Perth, which is to the South and slightly West. However, direct reception is
extraordinarily unlikely due to the extreme remoteness of the MWA. It is more
likely that reflections off of aircraft or other transient phenomena make up
the bulk of DTV receptions. For example, the largest population center near
the MWA is Perth, and most flights that could reflect DTV to the MWA are
probably headed towards there (or perhaps towards Geraldton, which is slightly
Southwest of the array). Some of these reflections occur when the aircraft is
near its destination, and so the RFI source will appear in the Southern
sidelobes, e.g. Wilensky et al. (2019). Since this is also where the strongest
transmitters are, we expect more noticeable power spectrum contamination in
the East-West polarization.

####  4.1.2 SSINS $z$-score Test

In Figure 12, we show power spectra in the +2 pointing, which is the most
commonly contaminated pointing in our jackknife tests, separated by the SSINS
z-scores of the identified events without discriminating between RFI types.
Interestingly, we see that integrations of observations with high SSINS
z-scores do not display the characteristic excess power in the window.
However, this may be due to the fact that such high z-scores are relatively
rare, and that there is significant variation in the noise-levels between
different members of the jackknife test. It may be that the noise must be
integrated down before an obvious RFI signal appears in the window. We also
remark that all integrations with RFI z-scores below 100 have a corresponding
power spectrum made from pure observations that demonstrates some excess power
in the window, though not as severely as the power spectrum for absolved
observations. This suggests the presence of false negatives in the SSINS
pipeline that can have a significant effect on the power spectrum measurement.

Figure 14: A power spectrum jackknife test for zenith-pointed observations,
where spectra are separated by RFI type. These all have RFI events with
z-scores between 10 and 100. We find that the region of contamination seems
unrelated to the type of RFI. This may be due to multiple RFI types cohabiting
these observations.

To summarize the jackknife tests across the SSINS $z$-score axis, we show
another collection of cumulative distribution functions in Figure 13. This is
the same data as in Figure 11, but it has now been collated according to the
$z$-score of the events that were found by SSINS. We find that brighter events
in the SSINS do not correlate with stronger outliers in the EoR window. It
seems that, within the absolved sets, observations with faint to moderately
bright events in the SSINS tend to produce the most strongly outlying power
spectra. Interestingly, the LST-matched pure observations also seem to at
least weakly display this trend. Since there are generally more observations
with fainter events, the power spectra corresponding to the fainter $z$-score
bins usually have more integration time. One might expect this behavior in the
presence of a systematic that averages coherently at moderate integration
depths. While the bottom panel of Figure 13 shows that window power is
generally more outlying in the absolved power spectra, we speculate all of the
observations are affected by a residual systematic that averages coherently in
sub-hour length integrations. In a test discussed later in this section, we
find that the window contamination displayed in the 2d power spectra so far is
enhanced when RFI flags are turned off. Given the similarity of contamination
between the two rows of Figure 12, we suspect that this coherently integrating
systematic is indeed residual RFI that is generally worse in the absolved
observations but still present in the pure observations.

Figure 15: Power spectrum jackknife test for observations classified as
containing broadband interference. Excess window power takes similar shape as
RFI events in Figure 14.

####  4.1.3 RFI Shape Test

From the theoretical discussion in Wilensky et al. (2020), we expect that
different RFI types might have different power spectrum contamination shapes.
In Figure 14, we show a jackknife test over RFI shape as determined by SSINS.
In general, we find no obvious power spectrum difference among the shapes,
contrary to expectation. For instance, we expect narrowband contamination to
be approximately constant as a function of $k_{\parallel}$. Even in more
dramatic cases with substantially more noticeable residual interference than
is shown in Figure 14, the narrowband RFI does not produce a noticeable
constant pattern. If we compare to Wilensky et al. (2020), we see that
narrowband RFI power spectra at a given flux density is generally lower
compared to their DTV counterparts, but also that it is constant in
$k_{\parallel}$. Since the noise levels are approximately constant in
$k_{\parallel}$, we expect that if this were only due to narrowband
contamination, then the entire EoR window would show clear contamination.

Figure 16: A power spectrum test in which two separate power spectra were made
for the same observations, once with flags applied and once without flags
applied. In the difference plot, blue bins indicate power is higher without
flags. We find that applying the SSINS flags removes power from the window in
the exact shape as seen in other Absolved power spectra. Additionally, power
is removed in the region of the wedge corresponding to the sidelobes, which is
where we observe DTV sources in our images. The clean LST-matched observations
are shown on the far left as a reference.

If there is residual RFI in this power spectrum, it is not narrow in
frequency. For this to be true, it must be something broader that SSINS
pathologically misses altogether. It could be a relatively stationary source
of DTV interference, such as a distant reflector that is coincidentally moving
towards the array or nearly so. This reflector may be simultaneously
reflecting multiple RFI signals, some of which are narrow in frequency and
identified by SSINS. Alternatively, while the RFI appears narrow in frequency
in the SSINS, it is possible that the signal has structure outside of the
frequencies where most of its power is concentrated. This would lead to non-
constant power spectrum contamination, but would require frequency sidebands
bright enough to produce such structure, yet faint enough that SSINS cannot
flag it as a broad shape. All of our narrowband subintegrations contained less
than 16 minutes of data, and all subintegrations that showed window
contamination appeared similarly to the narrowband panel of Figure 14, but had
just 2-4 minutes of contributing data. This was the only type of RFI that
showed any obvious smooth window contamination at such a shallow integration
depth. Since so little narrowband interference is observed, it is hard to
speculate confidently about its effect on the power spectrum measurements.
However, given that this particular shape does seem associated with absolved
observations that have narrowband RFI, we are inclined to think this is some
sort of RFI effect.

Since the broadband events in the jackknife test shown in 14 showed no obvious
excess power, we show a power spectrum for a group of absolved observations in
the -1 pointing classified as containing broadband interference flagged by
SSINS, along with the LST-matched observations in Figure 15. These z-scores
were between 10 and 100, indicating that the interference was moderately
bright in the SSINS compared to the thermal background. The excess window
power is similar to that for other shapes shown in Figure 14, but much
fainter. We show this example to point out that we do see some excess power
associated with broadband events. However due to the ambiguous nature of
broadband events, as well as the distinct possibility that other RFI may be
present in addition to the broadband events caught by SSINS, it is difficult
to know if broadband interference in general poses as significant a problem as
other types of RFI. We discuss these two points in more detail in what
follows.

It is nontrivial to theorize about the expected contamination of broadband
interference due to the fact that it could come from several different types
of sources. For instance, we have evidence that it could be the sidebands of
bright ORBCOMM interference, 40-70 MHz away from the central frequency. The
ORBCOMM signal would need to be understood in extreme detail since these
sidebands are so far from the allocation. If it were approximately flat-
spectrum over the observing band, then it would give no excess power in the
window, similar to the foregrounds. Therefore, supposing these signals are
indeed responsible for the excess power, there must be some nontrivial
structure in the sidebands of these signals. A related possibility is that the
sheer brightness of an RFI source has caused clipping or other nonlinearities
in the receiver chain. Another broadband emitter is lightning, which produces
emissions over a range of radiofrequency scales (Vine, 1987). Some theoretical
studies suggest that certain types of lightning events can produce relatively
flat emissions over the bands considered in this work (Luque, 2017; Shi et
al., 2019), although Luque (2017) shows some nontrivial structure in this
range. Observations of lightning emission at VHF radio frequencies are
numerous (Hare et al., 2018, 2020; Pu et al., 2021; Sterpka et al., 2021;
Scholten et al., 2022). In summary, since the particular signal structure
behind the broadband events in the MWA SSINS is as of yet undetermined, we
find it difficult to hypothesize about the expected shape of power spectrum
contamination.

The similarity in contamination between the different shapes could also be due
to other factors. First, as noted in §3.2, SSINS is more optimized for
flagging than classification, and the occasional misclassification does occur.
Second, it is possible that a given reflector is reflecting multiple different
transmissions simultaneously or with a short gap between them. This means that
any given observation classified as containing some type of interference does
not exclude it from containing another type of interference. In other words,
there is some amount of overlap between the subsets shown in Figure 14, and
this may be responsible for the similarity in contamination shape.

####  4.1.4 Flags On/Off Test

To more directly investigate the effects of applying the SSINS flags, we
examine an integration in which residual RFI was present in absolved
observations by forming the power spectrum using the observations in their
repentant form and then subtracting the absolved power spectrum. Note that we
are using the same observations in each case, and only choosing whether or not
to apply the flags. We show the result in Figure 16. While the change in the
power spectrum is slight, when we plot the difference we see a clear signature
in the window matching the shape of excess power that we have seen so far. The
flags produce a 35% difference in data volume, meaning that the noise levels
are appreciably different. However, the power difference varies smoothly and
is positive for a significant number of adjacent wave modes, indicating the
effect is not due strictly to a difference in noise levels. We also see that
the flags removed power in the region of the wedge between the dashed and
solid lines. The corresponding LST-match of this observation set is shown on
the far left of the figure to allow for a comparison in the style of the
previous jackknife tests.

The dashed and solid lines in the wedge represent the extent in
$k_{\parallel}$ to which sources at the edge of the primary beam and at the
horizon throw power due to the chromatic point-spread-function. The region
between these two lines corresponds to the sidelobes. In this example, we
chose a handful of observations from the +1 pointing that were identified as
containing DTV interference. We expect that a DTV source has a smooth
component that appears in the power spectrum wedge, and a spectrally sharp
component that throws power into the higher $k_{\parallel}$ modes. This power
spectrum suggests that the RFI sources are consistently located in our
sidelobes, corroborating the imaging experiments in Wilensky et al. (2019).

####  4.1.5 Discussion

These jackknife tests show that there is very likely a class of ultra-faint
RFI uncaught by our current excision methods, and that this RFI will bias
power spectrum measurements for a wide swath of spherical modes. The severity
of this bias in a deep integration compared to its appearance in these
jackknife tests depends on the integration properties of the RFI. We expect
RFI will not average coherently over many hours of data, but we cannot
reliably speculate about the strength of this bias since we do not have a
model of unexcisable RFI. However, since we have identified subsets of the
data that contain ultra-faint RFI beyond what is identified during pre-
processing, we can at least compute the cleanest power spectrum upper limit
possible with this data set and examine whether our efforts are likely to have
made a difference.

##  5 Final Power Spectrum and Upper Limits

Before extracting an upper limit on the 21-cm EoR power spectrum signal, we
perform a final reduction based on the results of the jackknife tests. These
final eliminations are based on a qualitative assessment of the EoR window in
the power spectrum of various integrated subsets. Since our aim is to make a
power spectrum measurement, we perform these cuts conservatively so as to
avoid selection bias as much as possible.

###  5.1 Final Cuts

First, we remove all absolved observations. This is motivated by the fact that
a substantial number of power spectra involving only absolved observations
expressed excess window power compared to their pure, LST-matched
counterparts. Furthermore, the shape of this excess power matches the shape of
the power spectrum difference in the jackknife test shown in Figure 16. We
term this shape as “the RFI footprint.” Finally, the idea that residual faint
RFI might temporally neighbour SSINS-identified samples is motivated both by
physical considerations and the occupancy study in 3.2. We did not remove any
pure observations, even if they were immediately adjacent to absolved ones.
This leaves 591 observations, which is about 18.4 hours of data.

Figure 17: Deep 2d power spectra from three different sets of data, all of
which were deemed uncontaminated by SSINS. As with all other 2d power spectra
shown in this work, we show the “dirty” power spectrum i.e. without model
subtraction. The Wall of Shame set shows a clear RFI footprint in the spectrum
made from the E-W dipoles, but appears noise-limited in the one from the N-S
dipoles. On the other hand, the Limit Set power spectra appear similarly
regardless of which polarization is used. Specifically both the E-W and N-S
power spectra appear systematically dominated below the first coarse band
harmonic in the limit set, in contrast to the Wall of Shame set and the
majority of shorter power spectrum integrations we inspected. A key difference
between this contamination and the RFI footprint is that it does not extend to
large $k_{\perp}$. The coherent average of the two performs to expectation in
that adding the Wall of Shame set does not strongly affect the appearance of
the power spectrum from the N-S data, and imprints the characteristic RFI
footprint in the spectrum made from the E-W data.

Next, we note that some integrations involving only pure observations also
display this excess window power, although this is substantially less common
than in their absolved counterparts. Hoping to remove as much RFI as possible
before forming an upper limit, we perform cuts on these observations based on
manual inspection of window power. From the integration subsets that were
already constructed for the previous stages of the jackknife test, we remove
all integrations with fewer than 20 observations that have obvious excess
window power. We then reintegrate the remaining observations, separating them
roughly by pointing and day, such that there are between 12 and 20
observations (24-40 minutes) per integration. This integration depth is based
on balancing the interference-to-noise ratio in the EoR window with the desire
to be as fine-grained as possible in removing observations with this metric.
In other words, we want to use the smallest integration subset possible that
allows us to identify ultra-faint RFI. From these new integration subsets, we
again remove any whose resulting power spectra show obvious excess window
contamination. Since we are wary of selection bias, we attempt to be as
conservative as possible during this step. We show the power spectra of all
integrations that are removed during this stage in Appendix A. This removes a
further 119 observations, which we designate as the “Wall of Shame” set, and
leaves 472 observations (14.7 hours), which we call the “Limit Set.” We also
note that no observations from the +2 (most Westerly) pointing survive this
test.

###  5.2 Deep Cylindrica Power Spectra

We show deep 2d power spectra from each of the three sets (wall of shame,
limit set, and coherent average of the two) calculated over the entire
observing band in Figure 17. The power spectrum estimated from the East-West
dipole data for the Wall of Shame set displays the characteristic RFI
footprint. On the other hand, the power spectrum made using only the North-
South dipole data appears noise-limited in the window. Thus, we see that even
when we deeply integrate observations known to have RFI contamination, the
footprint appears polarized. This can be understood by considering that the
brightest DTV transmitters in Western Australia are nearly due South of the
array. Since the North-South dipoles are not sensitive to the Southern
horizon, we generally expect the RFI flux to be stronger in the East-West
dipoles than the North-South ones.

In contrast, the Limit Set displays power spectra that look very similar
between the different dipoles. This is remarkable in the first instance since
previous power spectrum upper limits with the MWA are preferentially deeper
from the North-South dipoles compared to the East-West ones. It is possible
that this long-standing preference is related to the difference in RFI content
between dipoles. We could investigate this claim more rigorously by extending
these RFI mitigation techniques to previously analyzed data that express the
preference. Both polarizations appear to have a systematic source of
contamination in the lower-left corner of the EoR window. What distinguishes
this systematic from the characteristic RFI footprint is that it does not
extend to higher $k_{\perp}$ modes. Furthermore, it is relatively equal in
power across the polarizations, which would be peculiar for RFI given the Wall
of Shame power spectra. We suggest that this is possibly a different
systematic effect that is unaccounted for in this analysis. Note that when we
coherently average the Limit Set and Wall of Shame, we observe the
contamination extend to higher $k_{\perp}$ in the power spectrum from E-W
data, but not in the one from N-S data.

As a counterargument, we remark that there are many other DTV transmission
sites in Western Australia than the aforementioned ones South of the array,
distributed in varying directions. Most of these transmission sites are
significantly less powerful than the one in Perth, some by multiple orders of
magnitude. Since this is a complex scattering problem, it is difficult to
deduce exactly how bright each transmitter should appear. However, some basic
scaling arguments suggest that the RFI flux should still be dominated by the
Southern transmitters. Therefore, it is possible that this could be extremely
faint RFI from the dimmer transmitters, while the flux from the brighter
transmitters was removed by our repeated selections. However, for this to be
true, there would need to be an explanation for why the longer baselines
(higher $k_{\perp}$ modes) display less power from these hypothetical RFI
events. None of the transmitters are close enough for regular direct
reception, and therefore the nature of the scatter would have to be different
such that the long baselines do not respond as strongly to the RFI. This is
physically unusual, since the scatterers are likely to be the same class of
object regardless of where the transmitter is. One possibility is that the
closer transmitters tend to scatter closer to the array. If the scatterer is
sufficiently close, it might appear less point-like and thus the longer
baselines might not respond as strongly to the interference. However, this
would be an unusual coincidence and is therefore less likely to explain a
longitudinal trend in a large data set.

Yet another possibility is locally generated RFI within the array e.g. from
electronics. This type of RFI, depending on its source, might be seen
longitudinally in a large data set and may appear roughly equally in the
different dipole arms depending on its location. Furthermore, if it is close
enough, we might expect it to preferentially affect shorter baselines compared
to longer ones, and we might also expect it to preferentially affect some
pointings compared to others. However, locally generated RFI is often bright
and therefore eliminated early in the data analysis. This would have to be a
particularly pernicious form of ultra-faint locally generated RFI. We cannot
rule out this hypothesis from this analysis alone. Examining the near-field
RFI environment is an important topic of ongoing investigation within the MWA
collaboration, however we consider it outside the scope of this work.

###  5.3 Spherical Power Spectrum Upper Limits

We report the final limits using the 472-observation set. We also integrate
the 119 observation set separately to examine how the residual RFI integrates
in the power spectrum. Finally, we make an integration with the entire set of
591 observations to see how this final cut affects the power spectrum. We draw
limits for three different redshifts: 6.5, 6.8, and 7.1. These correspond to
three different bands, each of which is exactly half the 30.72 MHz bandwidth
of the MWA EoR highband. They are centered 1/4, 1/2, and 3/4 the way through
the band. Since we use a Blackman-Harris taper, the effective bandwidth of
each measurement is 7.68 MHz i.e. 1/4 of the band (Harris, 1978). This makes
the measurements roughly independent.999More specifically, these frequency
bands have 50% overlap, and the minimum 4-term Blackman-Harris window function
has a 50% overlap correlation of 3.8% (Harris, 1978). We display the spherical
power spectra along with the 2-$\sigma$ upper limits in Figure 18. Our lowest
upper limit that we report is $\Delta^{2}\leq 1.61\cdot 10^{4}\text{mK}^{2}$
at $k=0.258\text{hMpc}^{-1}$ using the East-West polarized dipoles at a
redshift of 7.1.

Figure 18: Spherical power spectra for the Limit Set, along with $2\sigma$
upper limits, a fiducial theory model for the EoR signal whose implementation
is described in Barry et al. (2019b). In short, the astrophysical constraints
from Park et al. (2019) are used in combination with 21cmmc (Greig & Mesinger,
2015) to generate samples from a probability distribution of 21-cm power
spectra. The fiducial model and theoretical confidence interval are then
calculated from these samples (brown solid and dashed lines, respectively). We
notice that the range of wave modes between the coarse band harmonics are
almost all noise-dominated. This could be a direct consequence of the rigorous
RFI cuts we have employed. Compare to limits at the same redshifts in Li et
al. (2019).

###  5.4 Discussion

These limits are calculated assuming a positive-defnite signal-like component
and thermal noise, as in Appendix A of Li et al. (2019). No distinction is
made between systematic contributions to the signal-like term and the
cosmological signal of interest. With information about the strength of
various contaminating effects, one could produce an upper limit that takes
these into account. For example, if we estimated an RFI excision algorithm to
be effective down to some nominal flux level and had a statistical model of
the remaining RFI sources e.g. Offringa et al. (2013), we could incorporate
our uncertainty about this systematic into the upper limit calculation. In
general, this is frustrated by the fact that we must extrapolate about RFI we
did not detect based on RFI we detected. This is necessarily dependent on the
flagging strategy. For SSINS this may be particularly complicated, since flags
are generated on time-differenced data and therefore undetected RFI may belong
to an entirely different class not necessarily distinguished by total
brightness. One way to build up this model is to complement an existing RFI
excision strategy with the following imaging strategy.

While some RFI might be stationary within an observation, most RFI will not be
stable in celestial coordinates from observation to observation, particularly
if those observations are significantly lagged with respect to civil time.
Therefore, one could difference images of appropriate observations and look
for strongly outlying pixels. This could be done at varying integration depths
for relatively cheap computational cost within the current pipeline. In other
words, this could be used to find full-array RFI within an observation, and
then again at a deeper integration depth to find ultra-faint RFI. This would
simultaneously allow for RFI identification and localization of the RFI
sources, which will be important for developing models of the RFI statistics
of the observatory. These models can ultimately be used for determining the
depth to which RFI excision needs to be performed. In fact, a similar strategy
for RFI identification was implemented in Prabu et al. (2020) and further
tested with the Engineering Development Array in Tingay et al. (2020), though
only at a few seconds of integration depth rather than combining multiple
observations. Thus, there is some promise that such a method could work,
although clearly some experimentation would be required to determine exactly
how to implement it in deeper integrations and between different observations.

In addition to time differencing strategies, we can imagine incorporating
significantly more specific prior information regarding the RFI signals than
is currently used by common flaggers. The details about reflector
trajectories, details about the signal properties that are known a priori by
the transmitting parties, and an atmospheric model could be combined to
produce an accurate physical model of a received RFI signal. For example,
Prabu et al. (2022) demonstrates that RFI detection can be greatly enhanced by
two different methods that incorporate more specific prior information about
the RFI. The first method, called “shift-stacking,” coherently averages in
image space over a region expected to be occupied by a known reflector. The
second method implements a post-correlation refocusing of the MWA to a
distance at which we expect a reflector. In brief, at these distances, there
is still non-negligible curvature in the wave fronts of the reflected signal,
and the visibility phase can be adjusted to account for this. One could
imagine checking for catalogs of reflectors that ought to be present in the
data using methods like these, or conversely, using methods like these to help
build such catalogs.

A clear challenge with a physical modeling approach is the construction of a
complete model, particularly when different propagation phenomena may produce
spectral distortions in the RFI signal. In general, programming a flagger to
identify a particular spectral distortion can implicitly make it less
sensitive to other types of spectral distortion unless each class of
distortion is included in the model.101010For example, the match filter for a
decreasing spectral tilt will be less sensitive to distortion that produces an
increasing spectral tilt. However, assuming such a model can be made, this
will not only enhance flagging performance but also allow us to more
accurately predict statistical properties about RFI that was left unflagged.
This ultimately will allow for more rigorous uncertainty estimates on power
spectrum measurements since there will be an explicit model for unmitigated
systematics.

We observe that the range of wave modes between the first and second coarse
band harmonics is noise-limited in most of the presented spherical power
spectra. This suggests that adding more high-quality data would probably
improve the limit. Interestingly, if we evaluate the upper limit using the
coherent average of the Wall of Shame and Limit Set, we see that it degrades
the limit systematically between the first and second coarse band harmonics
(which was not the region of the EoR window used for the final cut) for
redshifts 6.5 and 6.8 in the East-West polarization. The effect is slight, but
significantly consistent across the stated bins. The lowest limit of any
spherical mode that we observed, which we do not report, occurs in this
coherent average below the first coarse band harmonic. We do not report that
limit since the coherent average was known to be RFI-contaminated before we
constructed it.

##  6 Conclusion

In this work, we analyzed a season of data from the MWA phase I with a
particular focus on the presence and potential effect of RFI in 21-cm power
spectrum measurements. The goal was to examine to what extent ultra-faint RFI
exists within deep power spectrum integrations despite the deployment of
proven mitigation methods, as well as to see whether this ultra-faint RFI
contamination was likely to deteriorate constraints on the 21-cm cosmic
reionization signal.

This RFI analysis, in combination with other quality metrics, ultimately
resulted in the decision to cut about 85% of the data from this season to
produce the best upper limit. Even in the absence of an ionospheric quality
cut, we still would have cut at least 54% of the data based on RFI content
alone. This level of data loss is catastrophic from a scientific standpoint.
Starting from just 2.5 hours per night that we select due to contamination
from the galaxy at other sidereal times, it would take dozens of seasons of
data of the same quality, instrumental sensitivity, and cut decisions to reach
a confident detection. While a more sensitive instrument such as the SKA might
need fewer high-quality observing hours to reach the same detection
confidence, we expect that the global RFI environment is worsening with the
addition of extremely large satellite constellations in low Earth orbit, and
so data cuts of this style may be even more severe for future experiments.

From our RFI analysis, we identified a host of RFI that evaded post-
correlation detection by observing a residual RFI footprint in measured
cylindrical power spectra that is enhanced when RFI flags are turned off.
Residual RFI within observations already known to contain RFI is motivated by
the physical properties of RFI reception, which we know to be largely due to
transient scatterers. We found the timelike properties of the RFI flags to be
consistent with this physical picture. We also found that the RFI footprint
was most prevalent the further West the array was pointed. This was true to
such an extent that after all data cuts based on independent pre-power-
spectrum metrics, all remaining subintegrations in the most Westerly pointed
data showed the RFI footprint. The footprint also appeared preferentially in
power spectra made from only the East-West aligned dipoles as compared to
those made from the North-South aligned dipoles. These two observations are
consistent with the fact that the brightest DTV transmitters are nearly due
South of the array, where the North-South dipoles are less sensitive, and
offset slightly West.

We performed a deep integration, cutting all observations identified as
contaminated by any amount as well as any subintegrations that clearly
presented the RFI footprint. The resulting power spectra appeared consistent
between the polarizations, and did not have an obvious RFI footprint, although
some systematic contamination is visible in the lower-left corner of the EoR
window. This similarity between the polarizations is a new feature of this
limit compared to previous ones made with the MWA, and may be a result of the
enhanced RFI mitigation. When we only use the subintegrations known to possess
an RFI footprint, we obtain power spectra that are systematically limited in
the East-West polarized data, but not in the North-South polarized data, and
the presence of a footprint is more obvious. Finally, when we coherently
average all the data together, we see systematic dominance in both
polarizations, but the systematic dominance only extends to larger
perpendicular wave modes in the East-West polarized data. This suggests that
ultra-faint RFI is unlikely to disappear through dilution in a coherent
average, and by corollary, that the systematic domination in the EoR window of
the limit set may in fact be something other than RFI. However, since (a) the
behavior of RFI in coherent averaging schemes is poorly understood, (b) the
RFI budget for EoR detection is strict, and (c) we can willfully construct
relatively deep integrations with noticeable RFI footprints, we ultimately
suggest solving the problem of modeling ultra-faint RFI in deep power spectrum
integrations.

We use the subset of data least likely to be contaminated in order to set
upper limits on the cosmological 21-cm signal. In total, about 85% of the
initial data selection was not used for the upper limit. Our deepest upper
limit was $\Delta^{2}\leq 1.61\cdot 10^{4}\text{mK}^{2}$ at
$k=0.258\text{hMpc}^{-1}$ and $z=7.1$. While this is not the lowest upper
limit set with MWA Phase I, we remark that these limits are noise dominated in
the majority of modes higher than the first coarse band harmonic, not
including the harmonics themselves. Since RFI produces significant power to
extremely high line-of-sight wave modes, we suggest that this prominence of
noise-dominated bins at high $k$ (in excess of previous limits at similar
depths) may be a consequence of our extremely thorough RFI cuts. We note that
this required cutting about 2/3 of the data spared by the ionospheric cut.
Since the RFI environment is unlikely to improve in future seasons, obtaining
enough high-quality, RFI-free data for a sensitive EoR measurement appears a
daunting task. We therefore highlight not only a need to improve RFI
mitigation, but to do so in a data-sparing way.

##  7 Acknowledgements

We acknowledge Jacob Burba, Hugh Garsden, Steven Tingay, and Zheng Zhang for
helpful comments. We thank the development teams of pyuvdata, FHD, and
$\varepsilon$ppsilon which enabled this work. This work was directly supported
by NSF grants AST-1643011, AST- 1613855, OAC-1835421, AST-1506024,
AST-1907777, AST-2106510, AST-2107538, AST-2205471, and AST-2228990. This
project has received funding from the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innovation programme (grant
agreement No. 948764). We acknowledge the University of Washington eScience
Institute and UW Research Computing Azure Cloud Credits program for providing
computational resources.

This scientific work makes use of the Murchison Radio- astronomy Observatory,
operated by CSIRO. This scientific work uses data obtained from Inyarrimanha
Ilgari Bundara / the Murchison Radio-astronomy Observatory. We acknowledge the
Wajarri Yamaji People as the Traditional Owners and native title holders of
the Observatory site. Establishment of CSIRO’s Murchison Radio-astronomy
Observatory is an initiative of the Australian Government, with support from
the Government of Western Australia and the Science and Industry Endowment
Fund. Support for the operation of the MWA is provided by the Australian
Government (NCRIS), under a contract to Curtin University administered by
Astronomy Australia Limited. This work was supported by resources provided by
the Pawsey Supercomputing Research Centre with funding from the Australian
Government and the Government of Western Australia.

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##  Appendix A Power Spectrum Wall of Shame

We show all power spectra that were used to remove observations from the 21-cm
power spectrum upper limit in that were not determined to contain RFI using
only SSINS. We show them in no particular order, along with the number of
observations present in each integration. Since the excess window power is
almost exclusively observed in the East-West polarization, we show only power
spectra in that polarization. There is some overlap between the lists,
resulting in an observation count higher than 119, which is the number of
observations finally removed from the limit calculation.

Figure 19: All subintegrations that were determined to possess clear
indications of ultra-faint RFI contamination in the EoR window (but found to
be unoccupied by SSINS) that were subsequently removed from the limit
integration.