figurec

# Optimum filter-based analysis for the characterization of a high-resolution
magnetic microcalorimeter towards the DELight experiment

Francesco Toschi<EMAIL_ADDRESS>Institute for Astroparticle Physics
(IAP), Karlsruhe Institute of Technology (KIT), 76131 Karlsruhe, Germany
Kirchhoff-Institute for Physics, Heidelberg University, Im Neuenheimer Feld
227, 69120 Heidelberg, Germany  Benedikt Maier  Institute of Experimental
Particle Physics (ETP), Karlsruhe Institute of Technology (KIT), 76131
Karlsruhe, Germany  Greta Heine  Institute of Experimental Particle Physics
(ETP), Karlsruhe Institute of Technology (KIT), 76131 Karlsruhe, Germany
Torben Ferber  Institute of Experimental Particle Physics (ETP), Karlsruhe
Institute of Technology (KIT), 76131 Karlsruhe, Germany  Sebastian Kempf
Institute of Micro- and Nanoelectronic Systems (IMS), Karlsruhe Institute of
Technology (KIT), 76131 Karlsruhe, Germany  Institute for Data Processing and
Electronics (IPE), Karlsruhe Institute of Technology (KIT), 76131 Karlsruhe,
Germany  Markus Klute  Institute of Experimental Particle Physics (ETP),
Karlsruhe Institute of Technology (KIT), 76131 Karlsruhe, Germany  Belina von
Krosigk  Kirchhoff-Institute for Physics, Heidelberg University, Im
Neuenheimer Feld 227, 69120 Heidelberg, Germany  Institute for Astroparticle
Physics (IAP), Karlsruhe Institute of Technology (KIT), 76131 Karlsruhe,
Germany

###### Abstract

Ultra-sensitive cryogenic calorimeters have become a favored technology with
widespread application where eV-scale energy resolutions are needed. In this
article, we characterize the performance of an X-ray magnetic microcalorimeter
(MMC) using a ${}^{55}\mathrm{Fe}$ source. Employing an optimum filter-based
amplitude estimation and energy reconstruction, we demonstrate that an
unprecedented FWHM resolution of $\Delta E_{\mathrm{FWHM}}=\left($1.25\pm\pm
0.17$\text{\scriptsize{(stat)}}^{+0.05}_{-0.07}\text{\scriptsize{(syst)}}\right)\unit{}$
can be achieved. We also derive the best possible resolution and discuss
limiting factors affecting the measurement. The analysis pipeline for the MMC
data developed in this paper is furthermore an important step for the
realization of the proposed superfluid helium-based experiment DELight, which
will search for direct interaction of dark matter with masses below
$100\text{\,}\mathrm{M}\mathrm{eV}\mathrm{/}\text{$c$}\mathrm{{}^{2}}$.

##  I Introduction

Over the past decade, the field of cryogenic calorimetry has significantly
evolved, yielding detectors with eV-scale energy thresholds and resolutions
[1, 2, 3, 4, 5]. These detectors have a wide range of applications across
various scientific fields due to their exceptional sensitivity to minute
energy depositions. Some of their key applications include X-, gamma-ray and
mass spectroscopy [6, 7, 8], quantum information processing [9], neutrino
research [10, 11], and searches for dark matter (DM) particles. In particular,
the low energy threshold of cryogenic detectors allows the investigation of DM
parameter space which has been inaccessible in the past, i.e., the parameter
space of light dark matter (LDM) candidates with masses notably below the
typical GeV- to TeV-scale of weakly interacting massive particles (WIMPs)
[12]. A breakthrough in the energy resolution of cryogenic detectors was
achieved by the SuperCDMS collaboration in 2018, which reached a resolution of
$11\text{\,}\mathrm{eV}$ employing transition edge sensors (TES) [13].
Nowadays, the best resolution for a cryogenic detector-based experiment is
reached by the CRESST collaboration, with a baseline energy resolution of
around $1.4\text{\,}\mathrm{eV}$ employing TES [14].

Among the many cryogenic detectors currently available, magnetic
microcalorimeters (MMCs) are a particularly promising technology as they
provide not only an outstanding energy resolution, but also a large dynamic
range and an excellent linearity [1]. This work presents the analysis based on
the optimum filter (OF) of the calibration data acquired using the MMC
discussed in [15], which was optimized for the detection of soft and tender
X-rays. The presented analysis improves the previous full-width half-maximum
(FWHM) resolution of $1.8\text{\,}\mathrm{eV}$ obtained using a template
fitting [16] to $\Delta E_{\mathrm{FWHM}}=\left($1.25\pm\pm
0.17$\text{\scriptsize{(stat)}}^{+0.05}_{-0.07}\text{\scriptsize{(syst)}}\right)\unit{}$
for X-rays of energy $5.9\text{\,}\mathrm{keV}$. The MMC remarkable
performance sets it apart as the top-performing energy-dispersive gram-scale
cryogenic detector, despite the limitation coming from the thermal
fluctuations of the experimental setup.

Since MMCs are also particularly suited for the development of large-area
detectors, the future superfluid ${}^{4}\mathrm{He}$-based DELight experiment
for direct search for LDM will deploy this technology [17]. Although the
expected resolution will be lower than the one achieved for these X-ray
microcalorimeters because of the larger absorber area, the analysis and
processing framework developed for this X-ray MMC lays the foundation for
future data processing and analysis in the context of DELight.

This paper is organized as follows. The MMC detector is briefly described in
Section II, followed by an overview of the experimental setup and the detector
operation in Section III. Section IV details the analysis of the MMC data
leading to the final results. The main findings are summarized in Section V.

##  II Magnetic microcalorimeter

The operation of an MMC involves the conversion of an energy deposition
$\delta E$ into a variation in the magnetic flux. The interaction primarily
occurs in an absorber which is thermally coupled to a paramagnetic temperature
sensor that is situated in a weak external magnetic field. Upon energy
deposition, the temperature of the sensor increases by $\delta T=\delta
E/C_{\mathrm{tot}}$, where $C_{\mathrm{tot}}$ is the total heat capacity of
the system. According to Curie’s law, this leads to a change in magnetization
$\delta M\propto\partial M/\partial T\cdot\delta T$, which causes a variation
of the magnetic flux $\delta\Phi\propto\delta M$. A direct current
superconducting quantum interference device (dc-SQUID) converts this small
change in magnetic flux $\delta\Phi$ into a voltage change.

The data used in this work were acquired using the second variant MMC
prototype as described by M. Krantz _et al._ in [15]. This MMC integrates the
Ag:Er paramagnetic temperature sensor directly into the dc-SQUID loop to
enhance the magnetic flux coupling and minimize the influence of SQUID noise
on the achievable energy resolution [1]. The SQUID consists of two meander-
shaped coils, which are connected in parallel. One of these superconducting
coils forms the SQUID loop, interrupted by two Josephson tunnel junctions,
while the other coil below it generates the necessary bias magnetic field. The
absorber is made of electroplated gold with dimensions 150 x 150 x 3,
providing absorption probabilities of $98\text{\,}\mathrm{\char 37\relax}$ for
X-rays below $5\text{\,}\mathrm{keV}$ and around $50\text{\,}\mathrm{\char
37\relax}$ up to $10\text{\,}\mathrm{keV}$. To minimize athermal phonon escape
into the substrate, the absorber is designed in a “tetrapod geometry”,
extending over the sensor. This increases the path available for phonons to
thermalize, while reducing the volume with a direct line of sight to the
sensor.

##  III Experimental setup and operation

The setup used to characterize the MMC was directly mounted to the mixing
chamber platform of a $\ch{{}^{3}He}/\ch{{}^{4}He}$ dilution refrigerator
which was operated at its base temperature of about $7\text{\,}\mathrm{mK}$,
without any active stabilization. A schematic cross section view of the setup
is shown in Fig. 1.

Figure 1: Schematic view of the experimental setup for the calibration of the
MMC using a ${}^{55}\mathrm{Fe}$ source.

The MMC and its readout board were mounted on a copper case enclosed by an
aluminum cover that works as a superconducting magnetic shield at the
operational temperature. The Al cover features a small slit which allows
X-rays from a nearby calibration source to reach the detector. As the presence
of this opening hinders the shielding potential of the cover, an additional
long Al shield is placed above it to reduce the leakage of external magnetic
fields. The slit is aligned with a $15\text{\,}\mathrm{\SIUnitSymbolMicro m}$
thick gold collimator placed right above the MMC which avoids X-rays to hit
the detector substrate, which would lead to temperature fluctuations and
spurious signals. The ${}^{55}\mathrm{Fe}$ X-ray source is mounted on a brass
holder and placed right above the extended Al shield, around
$5\text{\,}\mathrm{cm}$ from the MMC.

The iron source is contained in a stainless steel case with a
$200\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thick beryllium window. The
${}^{55}\mathrm{Fe}$ isotope undergoes electron capture into
${}^{55}\mathrm{Mn}$, leaving a vacancy typically in the K-shell that is
filled by an electron from a higher energetic shell, producing a cascade of
X-rays and/or Auger electrons. Due to its low energy, the emitted radiation
coming from de-excitation into non-K shells is blocked by the beryllium
window. The highly energetic X-rays that reach the MMC arise from the K-shell
vacancy being filled either by an L-shell electron, known as
$\mathrm{K}_{\alpha}$ transition, or by an M-shell electron,
$\mathrm{K}_{\beta}$ transition. The $\mathrm{K}_{\alpha}$ spectral line is
further split due to the $LS$-coupling of the electrons into
$\mathrm{K}_{\alpha 1}$ ($2p_{3/2}\rightarrow 1s_{1/2}$) with an energy of
$5.899\text{\,}\mathrm{keV}$ and natural width $2.47\text{\,}\mathrm{eV}$, and
$\mathrm{K}_{\alpha 2}$ ($2p_{1/2}\rightarrow 1s_{1/2}$) with an energy of
$5.888\text{\,}\mathrm{keV}$ and natural width $2.92\text{\,}\mathrm{eV}$
[18]. The $\mathrm{K}_{\beta}$, although affected as well by the
$LS$-coupling, does not present a similar fine-structure splitting. Its X-ray
energy distribution peaks at $6.49\text{\,}\mathrm{keV}$ with a natural width
of $2.97\text{\,}\mathrm{eV}$. The energy spectrum of these characteristic
X-rays is further described in Section IV.3.

The ${}^{55}\mathrm{Fe}$ source was used to calibrate the MMC described in
Section II. The voltage response of its coupled dc-SQUID was read out by a
custom-made multichannel data acquisition (DAQ) system discussed in [19]. It
provided a maximum sampling rate of $125\text{\,}\mathrm{MHz}$ with a 16-bit
voltage resolution and an independent trigger for each channel. The
acquisition was triggered using a constant fraction discriminator, as the
signals are expected to have different amplitudes but same rise time. The high
sampling rate of the DAQ allowed for a precise trigger time, while the
acquired data, known as traces, were stored with a downsampling factor of 32
to reduce the necessary disk space. With a total length of 32768 samples, a
single trace has a duration of about $8.4\text{\,}\mathrm{ms}$ with the
triggering time slightly after $1\text{\,}\mathrm{ms}$. For this reason the
baseline of each trace is obtained from the first 4000 samples (around
$1\text{\,}\mathrm{ms}$). The calibration lasted around $50\text{\,}\min$ and
a subset of acquired traces is shown in Fig. 2.

Figure 2:  Small subset of 50 traces from the ${}^{55}\mathrm{Fe}$ calibration
run. A total of 48 overlapping black traces are from X-rays interacting in the
absorber and they are considered good events. The remaining two traces
correspond to X-rays interacting directly in the paramagnetic sensor (magenta)
and pile-up of events in the absorber (dark blue). The first 4000 samples
(gray selection) are used to determine the trace baseline.

##  IV  Data analysis

###  IV.1  Event selection

A set of essential selection criteria is applied to the data in order to
discard pile-up events, traces acquired during unstable conditions and events
happening directly in the sensor or its substrate (see Fig. 2). These consist
of selections of the signal rise-time, the mean value of the entire trace, and
a loose selection on the chi-square from fitting a simple template to the
trace in the time domain. Additionally, to ensure detector stability, further
criteria are applied to the standard deviation of the baseline and its offset,
which comes from a linear fit. The level of the baseline is an indicator for
the temperature of the detector right before an X-ray photon hits the sensor.
To study the effect of temperature variations, we therefore consider an
additional selection on the mean value of the baseline, together with the
aforementioned selections on its standard deviation and offset. When applied,
we refer to the dataset as “tight selection” with an acceptance of
$75\text{\,}\mathrm{\char 37\relax}$ and characterized by a more stable
detector temperature. In case the selection is not applied, we refer to it as
“loose selection”.

###  IV.2 Optimum filtering and energy reconstruction

To achieve the highest possible resolution, the energy is reconstructed using
an estimator based on the optimum filter (OF) [20, 21]. This consists of a
finite-impulse-response filter which minimizes a $\chi^{2}$ value in the
frequency domain, $\nu$. This value is defined as

$\chi^{2}=\sum_{\nu}\frac{\left[S(\nu)-aA(\nu)\right]^{2}}{J(\nu)},$ (1)

where $S(\nu)$ is the signal, $A(\nu)$ is the finite-size signal template, and
$J(\nu)$ is the power spectrum density (PSD), all in the frequency domain. The
amplitude obtained by minimizing $\chi^{2}$ is

$a=\frac{\sum_{\nu}S^{\ast}(\nu)A(\nu)/J(\nu)}{\sum_{\nu}|A(\nu)|^{2}/J(\nu)}.$
(2)

The employed OF allows for the time shift of the signal with respect to the
template, possible by considering a time delay $t_{0}$ in the time domain
signal $A(t-t_{0})$, which translates into a phase in frequency domain
$e^{-2\pi it_{0}\nu}A(\nu)$. The phase-shifted template is included in Eq. 2
and the delay $t_{0}$ that maximizes $a$ is chosen, as it is possible to show
that it also minimizes $\chi^{2}$ [22].

The OF algorithm requires the PSD of the noise $J(\nu)$ and the signal
template $A(\nu)$, which are obtained via an iterative process. Initially, a
set of traces with similar height is selected and averaged to obtain a simple
template of the signal. This is then matched to the traces in the time domain
through template fitting, and the resulting value of $\chi_{\mathrm{TF}}^{2}$
is employed as an additional selection. In the absence of baseline data, the
noise is computed by subtracting the template from the traces. To eliminate
artifacts arising from the steeply rising edge, the noise PSD is calculated
using the average of the latter half of each trace, which corresponds to
approximately $4\text{\,}\mathrm{ms}$ of the falling edge. The obtained PSD is
thus defined for frequency bins $\Delta f$ twice the size required for the
acquired traces: the finer binning is recovered through linear interpolation.
The noise PSD is shown in Fig. 3.

Figure 3:  Amplitude of the template from the $\mathrm{K}_{\alpha}$ energy
region (dark blue, left y-axis) and noise PSD (magenta, right y-axis).

The OF fit is applied to the traces using the simple template described above
and the interpolated PSD. The resulting OF amplitudes show two distinct
populations corresponding to the $\mathrm{K}_{\alpha}$ and
$\mathrm{K}_{\beta}$ lines. On average the former exhibits a reduced
$\chi^{2}$ roughly $2\text{\,}\mathrm{\char 37\relax}$ lower than the latter.
This difference can be attributed to the fact that the simple template was
produced using signals from $\mathrm{K}_{\alpha}$, while slight variations in
the signal shapes between the two energy regions are observed. Distinct
templates are thus defined for each population by averaging the traces chosen
from the $\mathrm{K}_{\alpha}$ and $\mathrm{K}_{\beta}$ lines. The traces are
then processed using both templates and the same PSD, resulting in two
different filters $\mathrm{OF}_{\alpha}$ and $\mathrm{OF}_{\beta}$. The
maximum relative difference between the amplitudes yielded by them remains
below $4\text{\times}{10}^{-5}$, an effect that is neglected for the
estimation of systematic uncertainties. A similar difference is obtained when
comparing the templates derived from traces selected using the loose and tight
selections. No trace showed a preference for a non-null time shift, excluding
jitter effects of the order of the time bin of $256\text{\,}\mathrm{ns}$. The
$\mathrm{K}_{\alpha}$ and $\mathrm{K}_{\beta}$ templates obtained from the
loose selection are used in the following. An example of a typical raw trace
with the OF-matched signal is shown in Fig. 4.

Figure 4:  Typical raw trace (black solid) and template scaled with the
amplitude from an optimum filter-based fit (red dashed). The bottom panels
shows the residuals, defined as the difference between raw trace and scaled
template.

Over the course of the calibration, the OF amplitudes steadily decreased by
$0.04\text{\,}\mathrm{\char 37\relax}$, as shown in Fig. 5.

Figure 5:  Distribution of the fine-splitting structure of the
$\mathrm{K}_{\alpha}$ energy spectrum as a function of time. The red dashed
line indicates the time-dependent correction function.

While the exact nature of this effect remains uncertain, it is likely
attributed to a temperature drift either of the detector itself or of the
associated electronics outside the cryostat. The amplitude decrease is well
described by a first-order polynomial, which is fitted to both the
$\mathrm{OF}_{\alpha}$ and $\mathrm{OF}_{\beta}$ amplitudes of the
$\mathrm{K}_{\alpha}$ line. The obtained linear relations are used to scale
the OF amplitudes and account for their time-dependence. For the rest of this
work, “cOF” amplitudes indicates the corrected amplitudes coming from
$\mathrm{OF}_{\alpha}$ when considering $\mathrm{K}_{\alpha}$ events, and from
$\mathrm{OF}_{\beta}$ when considering $\mathrm{K}_{\beta}$ events. Efforts
were made to incorporate temperature information from the baseline mean by
implementing a two-dimensional correction including both baseline mean and
time, but with no improvement compared to the time-only correction.

###  IV.3 Event spectrum and energy resolution

The spectral shape of the characteristic X-rays from $\mathrm{K}_{\alpha}$ and
$\mathrm{K}_{\beta}$ lines of ${}^{55}\mathrm{Fe}$ was studied in detail in
[18, 23]. They are described as the superposition of Voigt distributions
$\mathcal{V}\left(x\right)$, consisting of a Lorentzian with central energy
$E_{0}$ and natural width $\Gamma$ which is convoluted with a Gaussian
detector response of variance $\sigma_{E}^{2}$. This variance is correlated to
the energy resolution of the detector, defined as the full-width half-maximum
(FWHM) $\Delta E_{\mathrm{FWHM}}=2\sqrt{2\ln{2}}\,\sigma_{E}$. Each Voigt
distribution is multiplied by an error function $\Phi\left(x\right)$ modeling
the remaining contribution of the athermal phonons escaping into the substrate
of the MMC before thermalizing, leading to a partial energy loss. The
probability density function (pdf) for the spectral lines is thus defined as

$\displaystyle
f^{\mathrm{esc}}_{\alpha/\beta}\left(E\,|\,\sigma_{E},\varepsilon_{\mathrm{esc}}\right)=\sum_{i}^{\alpha/\beta}k_{i}$
$\displaystyle\mathcal{V}\left(E\,|\,E_{0}^{(i)},\Gamma^{(i)},\sigma_{E}\right)$
(3) $\displaystyle\times$
$\displaystyle\Phi\left(\varepsilon_{\mathrm{esc}}\left(E-E_{0}^{(i)}\right)\right),$
(4)

where $k_{i}$ is the relative intensity of the different contributions and
$\varepsilon_{\mathrm{esc}}$ describes the impact of the escaping phonons. The
values describing the natural shape of the X-rays are taken from [18], where
the $\mathrm{K}_{\alpha}$ line is corrected after private communication with
the author, as in [16]. Both the scenarios including the escaping loss
($f^{\mathrm{esc}}_{\alpha/\beta}\left(E\,|\,\sigma_{E},\varepsilon_{\mathrm{esc}}\right)$)
and excluding it ($f_{\alpha/\beta}\left(E\,|\,\sigma_{E}\right)$) are
considered in this work and their pdfs are plotted in Fig. 6.

Figure 6:  Probability distribution function of $\mathrm{K}_{\alpha}$ and
$\mathrm{K}_{\beta}$ lines assuming an energy resolution of
$2\text{\,}\mathrm{eV}$. Both the model including and excluding the impact of
athermal phonons escaping the sensor are shown.

The calibration function correlating the cOF amplitude $A_{\mathrm{cOF}}$ to
the energy is described by a second-order polynomial with no constant
component:

$E=p_{1}\cdot A_{\mathrm{cOF}}^{2}+p_{2}\cdot A_{\mathrm{cOF}},$ (5)

as suggested by a fit of the amplitude peak positions.

The $A_{\mathrm{cOF}}$ spectrum from the ${}^{55}\mathrm{Fe}$ calibration is
fitted only for its $\mathrm{K}_{\alpha}$ component. This is accomplished
through the chi-square minimization for binned data, where the chi-square
value is referred to as $\chi^{2}_{\alpha}$. This is done both for the model
including the athermal phonon escape $a\cdot
f^{\mathrm{esc}}_{\alpha}\left(E\,|\,\sigma_{E},\varepsilon_{\mathrm{esc}}\right)$
and for the model excluding it $a\cdot
f_{\alpha}\left(E\,|\,\sigma_{E}\right)$, where $a$ is the scaling factor. The
minimization returns the best estimators of the free parameters $a$,
$\sigma_{E}$, $p_{1}$, $p_{2}$ and, when included,
$\varepsilon_{\mathrm{esc}}$. The $\chi^{2}_{\alpha}$ is minimized using the
iminuit software tool [24]. Due to the multidimensional and stochastic nature
of the problem, different bin sizes and initial parameters are considered.
This results in a distribution of best estimators for four different
configurations, differentiated by the model used for the fit (including or
excluding the athermal phonon escape) and by the thermal fluctuation selection
(tight or loose). The energy spectrum in the energy range of the
$\mathrm{K}_{\beta}$ line is reconstructed for each set of estimated
parameters under the assumption of non-varying energy resolution and athermal
phonon escape contribution, when included. The chi-square in the
$\mathrm{K}_{\beta}$ region, $\chi^{2}_{\beta}$, is used as an additional
quality information about the performance of the individual fitting results.

The value of $\chi^{2}_{\alpha}$ indicates the goodness-of-fit of the
considered model in the fit region, ie., $\mathrm{K}_{\alpha}$. The
distribution for the four different configurations is shown in Fig. 7, where a
quality criterion on the goodness-of-fit in the $\mathrm{K}_{\beta}$ region is
applied by requiring $\chi^{2}_{\beta}<1.2$.

Figure 7:  Distribution of the $\chi^{2}_{\alpha}$ for the loose (left panel)
and tight (right panel) selections on the temperature fluctuations. Both model
including the athermal phonon escape (magenta) and excluding it (dark blue)
are shown.

The introduction of the factor modeling the escape of athermal phonons does
not improve the quality of the fit, while a tighter selection on the
temperature fluctuations clearly does. Consequently, the performance of the
detector is evaluated considering the dataset with the tight selection
criterion applied and without including an escaping factor in the model. This
mitigates the temperature fluctuations that ultimately increase the measured
energy resolution $\sigma_{E}$.

Figure 8:  Energy resolution $\Delta E_{\mathrm{FWHM}}$ as from chi-square
minimization and its statistical uncertainty as a function of the
corresponding $\chi^{2}_{\alpha}$ (black circles). The systematic uncertainty
is indicated by the blue shaded area. The best fit value with the lowest
$\chi^{2}_{\alpha}$ is reported as the MMC energy resolution and it is
indicated by the blue horizontal line. Figure 9:  Measured spectrum of the (a)
$\mathrm{K}_{\alpha}$ and (b) $\mathrm{K}_{\beta}$ lines from the
${}^{55}\mathrm{Fe}$ calibration source. The dashed red line is the result
from the fitting procedure with the lowest $\chi^{2}_{\alpha}$, while the
filled area is the natural line shape assuming an ideal detector resolution.
The residuals are shown in the bottom panels in units of standard deviation
$\sigma$, were the 1-, 2-, and 3-sigma bands are shown as shaded areas.

The best-performing fit results are selected by requiring
$\chi^{2}_{\alpha}<1.5$ and $\chi^{2}_{\beta}<1.2$. This selection returns a
distribution of energy resolutions which is shown in Fig. 8 as a function of
$\chi^{2}_{\alpha}$. The value returning the minimum $\chi^{2}_{\alpha}$ is
considered the energy resolution with the statistical uncertainty coming from
the fit. The systematic uncertainty is estimated by considering the highest
and lowest value of the FWHM in the selected population. The final value is
$\Delta E_{\mathrm{FWHM}}=\left($1.25\pm\pm
0.17$\text{\scriptsize{(stat)}}^{+0.05}_{-0.07}\text{\scriptsize{(syst)}}\right)\unit{}$.
The large statistical uncertainty arises from the small value of the energy
resolution compared to the natural width of the $\mathrm{K}_{\alpha 1}$ line
of $\Delta E_{\alpha 1}=2.47~{}\mathrm{eV}$ [18]. The fitted
$\mathrm{K}_{\alpha}$ and $\mathrm{K}_{\beta}$ spectra are shown in Fig. 9
together with the their natural shapes, obtained by requiring $\Delta
E_{\mathrm{FWHM}}=0$. When ignoring the impact of temperature fluctuations and
including the escaping athermal phonon component, the energy resolution
increases to $\Delta E_{\mathrm{FWHM}}=\left($1.35\pm\pm
0.15$\text{\scriptsize{(stat)}}^{+0.03}_{-0.09}\text{\scriptsize{(syst)}}\right)\unit{}$.
This is the most conservative scenario for the energy resolution, which should
be considered to have this as upper value, although this scenario is
disfavored by the fit results when comparing the models with and without
athermal escaping contribution. The energy resolution is compared to the
baseline resolution of the OF, given by:

$\Delta
E_{\mathrm{FWHM}}^{\mathrm{OF}}=2\sqrt{2\ln{2}}\left(\sum_{\nu}\frac{|A(\nu)|^{2}}{J(\nu)}\right)^{-0.5},$
(6)

where the factor $|A(\nu)|^{2}/J(\nu)$ is known as noise-equivalent power
[21]. This value corresponds to the expected resolution for a detector in
thermal equilibrium and it represents the best resolution attainable with the
OF given the noise condition of the detector. The baseline resolution obtained
using the information shown in Fig. 3 corresponds to approximately
$1.1\text{\,}\mathrm{eV}$, confirming the good performance of this analysis.

##  V Conclusion

MMCs stand out as one of the most promising technologies within the fast-
growing field of cryogenic detectors, owing to their excellent energy
resolution, large dynamic range and good linearity [1]. This work presented
the analysis of ${}^{55}\mathrm{Fe}$ calibration data acquired with the MMC
detector discussed in [15], directly integrating the paramagnetic sensor into
the dc-SQUID loop and improving the calorimeter design. The amplitudes of the
acquired pulses are extracted using an optimum filter-based algorithm and they
are used to reconstruct the energy of the event. After applying quality
selections and a time-dependent correction, the obtained energy spectrum is
fitted to the natural shape of the $\mathrm{K}_{\alpha}$ spectral line
convoluted with a Gaussian energy resolution. The calibration parameters
obtained from the fit are used to reconstruct the $\mathrm{K}_{\beta}$ energy
region of the spectrum and its goodness-of-fit is used to validate the
results. A systematic study of this fit procedure returns a best-fit
resolution of $\smash{\Delta E_{\mathrm{FWHM}}=\left($1.25\pm\pm
0.17$\text{\scriptsize{(stat)}}^{+0.05}_{-0.07}\text{\scriptsize{(syst)}}\right)\unit{}}$,
which is to our knowledge the best resolving power ever achieved with an
energy-dispersive detector in this energy range. The analysis improves the
resolution by $30\text{\,}\mathrm{\char 37\relax}$ upon the previous template
fitting-based analysis [16]. Notably, the analysis also shows that the
introduction of a factor to account for the escape of athermal phonons does
not improve the final result. However, the energy resolution deteriorates with
a looser selection on the baseline mean, which serves as a proxy for the
temperature of the detector. This suggests that the energy resolution of the
MMC is limited by the temperature fluctuations: the calorimeter could
potentially reach sub-eV resolution with the implementation of a proper
temperature stabilization. In addition to presenting the world-leading energy
resolution achieved by the X-ray MMC, this paper serves as the foundational
work for the analysis and processing framework of the MMC-based DELight dark
matter experiment.

###### Acknowledgements.

It is a great pleasure to thank (in alphabetical order) Klaus Eitel, Christian
Enss, Felix Kahlhoefer, Sebastian Lindemann, Marc Schumann, Kathrin Valerius,
Matthew Wilson and Alexander Zaytsev for discussions, and Klaus Eitel, Kathrin
Valerius and Sebastian Lindemann for feedback and comments on earlier versions
of the manuscript. We further thank Matthew Wilson and Alexander Zaytsev for
the help in the development of the software used for the processing of the MMC
traces. This work was supported by the Deutsche Forschungsgemeinschaft (DFG)
through the Emmy Noether Grant No. 420484612, and by the Alexander von
Humboldt Foundation.

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# Lightweight UAV-based Measurement System for Air-to-Ground Channels at 28
GHz

Vasilii Semkin1, Seongjoon Kang2, Jaakko Haarla3, William Xia2, Ismo
Huhtinen1, Giovanni Geraci4,
Angel Lozano4, Giuseppe Loianno2, Marco Mezzavilla2, and Sundeep Rangan2 1VTT
Technical Research Centre of Finland<EMAIL_ADDRESS> 2NYU Tandon
School of Engineering, Brooklyn, NY, USA  3Aalto University, School of
Electrical Engineering, Finland  4Univ. Pompeu Fabra, Barcelona, Spain

###### Abstract

Wireless communication at millimeter wave frequencies has attracted
considerable attention for the delivery of high-bit-rate connectivity to
unmanned aerial vehicles (UAVs). However, conducting the channel measurements
necessary to assess communication at these frequencies has been challenging
due to the severe payload and power restrictions in commercial UAVs. This work
presents a novel lightweight (approximately 1.3 kg) channel measurement system
at 28 GHz installed on a commercially available UAV. A ground transmitter
equipped with a horn antenna conveys sounding signals to a UAV equipped with a
lightweight spectrum analyzer. We demonstrate that the measurements can be
highly influenced by the antenna pattern as shaped by the UAV’s frame. A
calibration procedure is presented to correct for the resulting angular
variations in antenna gain. The measurement setup is then validated on real
flights from an airstrip at distances in excess of 300 m.

##  I Introduction

There is growing interest in the use of unmamnned aerial vehicles (UAVs) for
many connectivity applications where the mobile nature of the UAVs is
advantageous as compared with fixed transceivers [1, 2, 3]. Indeed, the
applicability of UAVs for communication purposes has enormous potential, both
recreational and in support of the operation of a variety of industries and
businesses. In particular, because UAVs can fly over any type of terrain or
obstacles, including traffic jams or blocked streets, they can ensure seamless
communication to and from users [4]. Also, integration with autonomous driving
cars or first-responders can provide additional safety mechanisms in complex
environments [5].

The millimeter wave (mmWave) range is an inviting realm for UAV communication
because of the enormous bandwidth availability and the possibility of line-of-
sight (LOS) transmissions [6]. As with any other technology, the design and
eventual deployment of UAV communication at mmWave frequencies requires proper
channel models, yet the environments and operating conditions of UAVs largely
differ from their terrestrial counterparts and hence the corresponding models
are not readily applicable. Custom models are necessary and devising these
models, in turn, requires large sets of empirical measurements.

Unfortunately, actual measurement of air-to-ground (A2G) channels in the
mmWave band has been tremendously difficult, primarily due to the strict
payload, size, and power limitations on commercially available UAVs combined
with the massive form factor of the channel sounders typically employed in
terrestrial applications. The paucity of A2G measurements at mmWave
frequencies has been one of most significant barriers to deployment of aerial
systems at these frequencies.

Various studies of aerial channels at mmWave frequencies are available in the
literature [7, 8, 9, 10, 11, 12, 13]. However, just a few of them rely on low-
altitude platforms that can be representative of UAVs at mmWave frequencies.
In [14], the authors provide an analysis of the small-scale temporal and
spatial characteristics of A2G radio channels by means of simulation with ray
tracing software. In [15], the authors present A2G channel measurements in the
L-band ($m$970 MHz) and C-band ($m$5GHz), conducted from an aircraft flying at
altitudes of 580 m and 900 m. Path loss models for the LTE bands are presented
in [16] and [17]. A recent study [18] reports path loss measurements at
several frequencies between 1 and 24GHz. However, there are only a handful of
contributions that focus on real-world mmWave aerial measurements, meaning
28GHz and higher. The measurements most similar to ours are those reported for
a street-canyon scenario [19] at 27 and 39 GHz, based on which power azimuth
and elevation profiles are derived.

To overcome the challenges of mmWave A2G measurements, we present a
lightweight setup characterized by:

  * •

The use of a directional horn antenna at a ground transmitter, thereby
avoiding the power consumption limitations at the UAV and capitalizing on
reciprocity to measure the A2G channel from the ground.

  * •

The deployment of a lightweight (282 g) spectrum analyzer connected to an
Intel processor stick (68 g) that enables accurate channel measurements with
minimal payload.

A relevant insight of our work is the observation that a main source of errors
is the uncertainty in the position of the UAV. Small variations in angle can
cause significant differences in antenna gain and, subsequently, on the path
loss, meaning that an error of several meters in the UAV position leads to
offsets of several dB in the path loss.

To validate the measurements, experiments were conducted at 28GHz111While 28
GHz is formally not part of the mmWave band (30 GHz to 300 GHz), it is close
enough that it is typically subsumed therein. in an open space—a dedicated
airfield—where the ground-truth path loss is known (Friis’ law). Preliminary
experiments demonstrate that we can obtain accurate measurements, opening the
door to extensive measurements in more complex environments. Also, similar
equipment can be utilized at even higher frequencies and the work can further
serve to calibrate ray tracing simulations [12, 20].

The structure of this article is as follows. Section II presents an overview
of the measurement setup and a description of the environment. Section III
introduces preliminary A2G channel measurements and describes the path loss
model that they give rise to. Finally, Section IV concludes the paper,
highlighting possible issues in A2G measurements and discussing plans for
subsequent work.

##  II Measurement Equipment

Our setup was developed at VTT,
Finland.222https://www.vttresearch.com/en/ourservices/millimetre-wave-
technologies The measurements were conducted at 28 GHz by a system composed of
two separate nodes: (_i_) the transmitter, positioned on the ground, and
(_ii_) the receiver, mounted on the UAV [21]. The tests were conducted at the
Nummela airport, in Finland.

###  II-A Transmitter

The signal generator is an Anritsu MG3697C and the transmitter features an ETS
Lindgren 3116 horn antenna with a maximum measured gain of 14.7 dBi. This
transmitting antenna was placed on a tripod and an elevation angle of 15∘ was
maintained throughout the measurements. The transmit power was set to +22 dBm.
The location and the coordinates of the transmitter are presented in Fig. 1.

###  II-B Receiver

The UAV on which the receiver was installed is a DJI M210 RTK. A compact Intel
compute stick PC and an Anritsu spectrum analyzer MS2760A-0100 were mounted on
a custom 3D printed structure below the UAV. The spectrum analyzer is also a
small-form-factor device with a weights of 282 gr. The equipment was secured
with heavy duty velcro straps. The PC was powered by a 2S Li-Po battery
through a voltage regulator. Batteries with different capacities, ranging
between 2200 and 3000 mAh, were used in the measurement campaign, allowing for
the PC to run for 30–40 minutes at an outside temperature of 2∘C. (Batteries
with higher capacity could also be featured since the voltage regulator takes
up to 40V as an input.) The batteries were chosen to keep the weight of the
overall system relatively low. The spectrum analyzer, an external GPS—to
record the flight path—and a super-bright LED were connected to the PC via
USB-hub; the purpose of the LED was to verify whether the PC was indeed
recording during the flight.

Another 4S Li-Po battery acted fed the power amplifier. The receive antenna,
omnidirectional in azimuth and having a measured gain of 2.1dBi, was installed
on the same 3D printed fixture. In addition, a secondary GPS device was
installed on the UAV for more accurate flight recording. The measurement
equipment was distributed to balance the UAV and avoid interference between
different parts of the system and to ensure that as many GPS satellites as
possible were visible. The portable PC was shielded with copper tape.

The total weight of the equipment was 1.27kg (see Table I), which is below the
maximum payload of 1.5kg that the UAV is able to carry with standard TB50
batteries. The weight of the UAV is 3.8kg while the maximum takeoff weight is
6.14kg. The block diagram of the receiver and a photograph of the UAV can be
found in Fig. 2.

Figure 1: Aerial photograph of the measurement location with specified route
and coordinates.

(a) Receiver block diagram. (b) Photograph of the UAV carrying the measurement
equipment.

Figure 2: UAV-based measurement setup for power measurements at 28 GHz. TABLE I: Equipment installed on the UAV. No | Equipment | Weight (kg) | Model | Notes
---|---|---|---|---
1 | Spectrum analyzer | 0.282 | Anritsu MS2760A-0100 | Frequency range 9 kHz-110 GHz
2 | Omni antenna 28 GHz | 0.068 | Eravant SAO-2734030345-28-S1 | 3 dBi gain
3 | Amplifier 26-40 GHz | 0.060 | ERZ-LNA-2600-4000-50-2.5 | Signal gain 50 dB, Power supply 15V
4 | RF cable | 0.012 |  | Flexible cable 50 cm (up to 67 GHz)
5 | Intel USB stick | 0.068 | Core m3 4 GB 64 GB Flash | Shielded with copper tape
6 | Power cables | 0.124 |  | Used for PC, USB hub, etc.
7 | Battery for the PC | 0.128 | 2S Li-Po 2200mAh | Any battery from 2S to 6S can be used
8 | Battery for the amplifier | 0.224 | 4S Li-Po 4000 mAh | 4S and larger batteries only
9 | 3D printed fixtures | 0.106 |  | Printed with PLA material
10 | GPS antenna | 0.046 | Globalsat BU-353S4 | Data from this GPS is not used
11 | USB hub | 0.038 |  | Used to connect spectrum analyzer to PC + periphery devices
12 | Voltage regulator (x2) | 0.068 | DFR0379 | Adjustable voltage output 4-40V
13 | External GPS device | 0.046 | Polar Vantage M | Fixed on the UAV’s arm
| Total | 1.270 |  |

##  III Preliminary Results

###  III-A Methodology

In this section, we present the early results gathered at 28GHz and the
measured radiation patterns of the antennas that were utilized. The
measurements were conducted at Nummela airport, right above an unused runway
as shown in Fig. 1. The 11-way point mission indicated in Fig. 3 was uploaded
prior to the flights and used for each measurement. The UAV would take off in
automatic mode, reach the starting point and hover for 30 s, then fly on
towards the specified second location and hover thereon for another 30 s, and
so on. Once the waypoint mission was complete, the UAV would turn and fly back
to the starting point, where it would finally land. Usually, the next flight
would be conducted at a different altitude.

The GPS recordings for two flights at respective heights of 15 and 30 m are
compared with the corresponding planned waypoint mission in Fig. 3. Some
drifting can be observed in the UAV trajectories with respect to the planned
route because of small uncertainties in the GPS receiver and, especially, due
to deviations cause by the wind. The higher altitude flight exhibits a larger
deviation from the planned route on account of the more pronounced drifting
induced by the wind. When the UAV reaches a desired measurement position, it
hovers and tries to maintain that position according to the uploaded GPS
coordinates. However, with an increasing height, the wind becomes stronger and
these more pronounced shifts from the planned position are observed.

For the data processing, the recorded GPS data was employed and received
amplitude values were computed using distances extracted from the GPS data.
The spectrum analyzer recorded 9 samples per second and saved them on the PC,
leading to 270 samples per UAV position. During the post-processing, samples
taken while the UAV was in motion towards the starting point, between
waypoints, and during ascending/descending maneuvers in-between flights, were
all discarded. The amplitudes of the received signals at each measurement
location were averaged to distill the large-scale channel behavior.

Figure 3: Flight route according to the GPS compared with the initially
planned route on the XY-plane (height not shown).

###  III-B Antenna Radiation Patterns

The transmitter and receiver antenna patterns should be carefully determined
and taken into account for every calculation. A horn antenna by ETS Lindgren
was utilized at the ground transmitter while an Eravant omnidirectional
antenna was mounted on the UAV. Antenna measurements were performed in the
anechoic chamber of Aalto University, Finland.

The measured horn antenna radiation pattern, which exhibits a typical shape
and a maximum antenna gain of 14.7 dB, is depicted in Fig. 4.

Figure 4: Measured radiation pattern of the horn antenna installed on the
ground transmitter.  Figure 5: Setup for radiation pattern measurements of
omnidirectional antenna located on the UAV.

(a) Omnidirectional antenna pattern.

(b) Radiation pattern of the omnidirectional antenna located on the UAV.

Figure 6: Measured 3D radiation patterns of omnidirectional antenna in free
space and installed on DJI M210 as in the measurement setup presented in Fig.
2. Bottom view.

Distinct measurements of the omnidirectional antenna were taken in the
anechoic chamber: first, in isolation, and then mounted on the UAV. In the
latter case, all equipment was also installed (see Fig. 5) to mimic the actual
flight conditions. Fig. 6 presents the two measurements. As can be appreciated
in Fig. 6(a), the isolated antenna does exhibit a highly uniform radiation
pattern on the azimuth plane. However, as seen in Fig. 6(b), once the antenna
is mounted on the UAV, its radiation pattern is severely affected by prominent
parts of the UAV’s own carbon-fiber hull. For example, the decrease in gain at
an azimuth angle of $225^{\circ}$ directly corresponds to the position of the
gimbal under the UAV (recall Fig. 5) and indicates that protruding parts of
the UAV can affect the antenna radiation patterns. The landing skid of the UAV
also has an effect on the antenna pattern. It can be noticed that, at
$10^{\circ}$, the pattern is more affected than at $190^{\circ}$, even though
the bottom of the UAV is symmetrical. This is due to the close location of the
antenna to the right-side landing gear.

The obtained patterns (Figs. 4 and 6) are processed by interpolating the data
over a mesh grid of azimuth and elevation angles, and used in subsequent
calculations.

###  III-C Path Loss

The measurement results obtained from two flight at 15 and 30 m were processed
and the ensuing path loss model is presented next.

To process all the parameters and extract final path loss values, we
considered the distance vectors between the ground transmitter and the UAV
position where the samples were taken. Next, we computed the 3D distances and
the azimuth and elevation angles from the distance vectors. Taking into
account the transmitter uptilt, we obtained local angles and the ensuing
antenna gains.

Having measured all the losses in the cables, we finally obtained the path
loss as

$L(d)=L(d_{0})+10\alpha\log_{10}(d)+\beta,$ (1)

where $d$ is the 3D distance between transmitter and receiver, $d_{0}=1$ m is
a reference distance, $\alpha$ is the path loss exponent, and $\beta$ accounts
for all the losses.

The measured power at $d_{0}$ was 15.4 dB. The measured cable losses between
the signal generator and the transmit antenna were 10 dB, the losses in the
cable between spectrum analyzer and amplifier were 2.5 dB, and the amplifier
gain was 55 dB at 28 GHz. The path loss exponent came out as $\alpha=2.2$
while $\beta=41.6$ dB. The path loss exponent value is a bit higher than that
one for free space environment, which is equal to 2, as stated in [22].
Several reasons may cause this difference: i) changes in the yaw while
maintaining nose direction and roll angles while hovering which could vary due
to strong wind, ii) inaccuracy in the recorded GPS coordinates - this will
heavily affect the gain values which should be used in the post-processing,
and iii) possibly slightly lower gain of the amplifier due to discharging
battery during the measurements, thus leading to smaller amplification. It
should be also noted, that we assumed constant height of the UAV during the
measurements, however, it was verified that inaccuracy can be in the order of
several meters and larger than in the horizontal plane.

Figure 7: Measured path loss values obtained during all flights, and linear
fit.

##  IV Conclusion and discussion

This work has presented an overview of the UAV-based measurement setup at
mmWave frequencies and shown some preliminary results obtained in the open-air
field at 28 GHz. Based on the obtained data, a path loss exponent equal to 2.2
was calculated. According to the recorded GPS coordinates of the flights, we
verified that there is some drifting of the UAV relative to the planned route.
In addition, the antenna patterns were measured and the corresponding antenna
gain values were taken into account in the path loss calculations. The
measurements performed in this work are relatively simple, but, at the same
time, they are extremely time consuming due to many technical and logistical
issues. The main purpose of this work was to test the equipment and identify
issues that might arise in A2G empirical channel characterizations at mmWave
frequencies.

Several sources of errors were pinpointed that should be carefully considered
when similar measurement campaigns are executed:

  * •

Positioning accuracy of the UAV. Conventional GPS receivers may not be
suitable due to potential inaccuracies of several meters. Specifically, any
major imprecision in the UAV height should be corrected. One possible solution
is utilizing a real-time-kinematic positioning system with the centimeter-
accuracy level.

  * •

Effect of the UAV’s hull on the antenna radiation pattern. As it was shown in
this work, the antenna performance can be affected by a particular UAV and
antenna location. Therefore, it is important to estimate the effect of the
UAV’s frame on the antenna patterns.

  * •

Verifying yaw and roll angles when the UAV is hovering, and pitch angles when
the measurements are conducted in flight. This will help in identifying the
correct angles for the antenna patterns.

  * •

Wobbling of the UAV due to rotating motors. This may be not so crucial for
amplitude measurements, yet it would become critical if full coherent
measurements are of interest [23].

For subsequent measurements we are planning to utilize a real-time kinematic
system, which will provide higher level of accuracy in terms of positioning of
the UAV. We are planning to study wobbling effect on the RF system performance
in more details. Our ongoing measurement campaign includes similar
measurements at 60 GHz. Next, these measurements will be carried out in urban
and sub-urban environments and A2G models will be derived. The results of this
research work will enable realizing novel applications and deploying UAV-based
communication systems in cities in the nearest future.

## Acknowledgment

The work of V. Semkin is supported in part by the Academy of Finland. The work
of W. Xia, S. Rangan, M. Mezzavilla, and G. Loianno is supported by NSF grants
1302336, 1564142, 1547332, and 1824434, NIST, SRC, and the industrial
affiliates of NYU WIRELESS. The work of A. Lozano and G. Geraci is supported
by ERC grant 694974, by MINECO’s Project RTI2018-101040, by ICREA, and by the
Junior Leader Fellowship Program from “la Caixa”.

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