Data reduction for a calorimetrically measured $^{163}\mathrm{Ho}$ spectrum of the ECHo-1k experiment

The electron capture in $^{163}\mathrm{Ho}$ experiment (ECHo) is designed to directly measure the effective electron neutrino mass by analysing the endpoint region of the $^{163}\mathrm{Ho}$ electron capture spectrum. We present a data reduction scheme for the analysis of high statistics data acquired with the first phase of the ECHo experiment, ECHo-1k, to reliably infer the energy of $^{163}\mathrm{Ho}$ events and discard triggered noise or pile-up events. On a first level, the raw data is filtered purely based on the trigger time information of the acquired signals. On a second level, the time profile of each triggered event is analysed to identify the signals corresponding to a single energy deposition in the detector. We demonstrate that events not belonging to this category are discarded with an efficiency above 99.8%, with a minimal loss of $^{163}\mathrm{Ho}$ events of about 0.7%. While the filter using the trigger time information is completely energy independent, a slight energy dependence of the filter based on the time profile is precisely characterised. This data reduction protocol will be important to minimise systematic errors in the analysis of the $^{163}\mathrm{Ho}$ spectrum for the determination of the effective electron neutrino mass.


Introduction
A promising approach for the direct determination of the effective electron neutrino mass m(ν e ) is the analysis of the endpoint region of a calorimetrically measured spectrum following the electron capture (EC) of 163 Ho. This method was first proposed in [1] and is presently pursued by two collaborations, ECHo [2] and HOLMES [3]. The EC of 163 Ho is characterised by Q EC =2.833 ± 0.030(stat) ± 0.015(sys) keV [4], which is a e-mail: robert.hammann@kip.uni-heidelberg.de b e-mail: arnulf.barth@kip.uni-heidelberg.de the available energy for the decay, and a half life of T 1/2 = 4570 ± 50 y [5].
In the EC of 163 Ho, a shell electron is captured by the nucleus, converting a proton into a neutron and emitting an electron neutrino. The daughter atom is left in an excited state 163 Dy * , which predominantly decays to the ground state 163 Dy via non-radiative processes. In first approximation, the excited state 163 Dy * is characterised by a vacancy of an inner shell electron and an additional electron in the 4f-shell. In this simplified model, the spectrum is given by six resonances centred at the binding energies of the orbitals of the captured electrons in the potential of the daughter nucleus: MI(3s 1/2 ), MII(3p 1/2 ), NI(4s 1/2 ), NII(4p 1/2 ), OI(5s 1/2 ), OII(5p 1/2 ) 1 . The amplitude of each resonance is defined by the phase space factor and the probability that an electron wave function in the given state overlaps with the nucleus. The phase space factor contains the information on the neutrino mass and gives rise to a cutoff at the endpoint energy E = Q EC − m(ν e ). The endpoint region is therefore most sensitive to a finite effective electron neutrino mass. Recently, a more accurate description of the calorimetrically measured 163 Ho spectrum has been developed. Besides the main resonances, it contains a number of structures that take into account electron scattering processes in the atom and excitations to the continuum [6,7].
In order to obtain sub-eV sensitivity on m(ν e ), one must measure energies below 3 keV with eV-precision. In addition, a good intrinsic time resolution ∼ 100 ns 1 Note that due to the low Q-value of this particular decay, the capture of electrons with principle quantum number n < 3 is kinematically forbidden. Moreover, an additional PI-line cannot be observed experimentally as the 6s 1/2 electrons move to the electron bands of the host material due to their weak binding to the 163 Ho atom. and a high statistics of more than 10 14 163 Ho events are essential [2]. For ECHo, metallic-magnetic calorimeters (MMCs) operated at temperatures below 30 mK [8] inside a dry dilution refrigerator 2 are used to meet these requirements. The particular type of MMC used for ECHo is characterised by a particle absorber that encloses the high-purity 163 Ho source. If an energy E is deposited in the absorber, its temperature rises with a time constant τ rise ∼ 100 ns. A paramagnetic Ag:Er temperature sensor, which is situated in an external static magnetic field and is thermally well coupled to the absorber, acts as a precise thermometer. The magnetisation of this sensor is temperature dependent. Consequently, a change in temperature causes a change of magnetic flux in a suitable pick-up coil. A flux-locked-loop dc-SQUID (direct current -superconducting quantum interference device) readout is then used to convert the change of flux into a change of voltage proportional to the initially deposited energy E. A gold thermal link made of several gold films with increasing width finally connects the detector to an on-chip thermal bath so that the initial temperature is restored. At the operating temperature of 20 mK, the recovery time is of the order of milliseconds. The decaying part of the temperature pulse can be described by a sum of exponential functions due to the step structure of the thermal link to the on-chip thermal bath. The rising part of the pulse can be affected by a reduced readout bandwidth, which effectively increases the sig- 2 Produced by BlueFors Cryogenics Oy, Arinatie 10, 00370 Helsinki, Finland.
nal rise time τ rise . For a DC-coupled signal, the time constants with their respective amplitudes fully specify the shape of a thermal pulse. AC coupling of the signal keeps the baseline offset at 0 V. This strongly modifies the signal shape as shown in fig. 1 (left).
The detector geometry used for ECHo is a double meander, which corresponds to two superconducting meander structures connected in parallel with the input coil of one dc-SQUID. On top of each meander, a paramagnetic sensor is fabricated. To polarise the spins in the sensor, a constant magnetic field is generated by a persistent current in the meander structures. Simultaneously, the meander structure serves as a readout coil to detect the magnetisation changes in the sensor. In such a gradiometric setup, the signal of a common change in temperature in the two sensors cancels out, which significantly reduces noise caused by global temperature fluctuations of the chip. On top of each sensor, a gold absorber with the dimensions 180 µm x 180 µm x 10 µm is fabricated. Each set comprising meander, sensor and absorber is referred to as one pixel and one gradiometer consisting of two pixels is referred to as one detector, which is read out by a two-stage SQUID setup [9]. Thus, each detector is associated to one readout channel (detector channel). Due to the opposite polarity of the screening current in the double meander structure of the detectors, the voltage signals from the two pixels of one detector have opposite signs. The triggered signals are separated into positive and negative polarity pulses based on the voltage slope after the trigger. Thus, signals from the two pixels of one detector can be distinguished.
In the ECHo-1k high statistics measurement (run 24 and run 25), two ECHo-1k chips [10,11] have been used. Each chip hosts 32 detectors implanted with 163 Ho, two double meanders to study the properties of nonimplanted pixels and two so-called temperature channels, which feature only one sensor and are therefore sensitive to temperature fluctuation of the substrate. In 7 of the 32 implanted detectors, only one pixel contains 163 Ho to allow for in-situ background measurements. Signals from a total of 68 pixels have been acquired over a period of five months, 58 of which are implanted with 163 Ho. The average activity per detector is approximately 1 Bq. The data reduction scheme discussed in this work has been developed to eliminate spurious events like triggered noise or pileup from these ECHo-1k datasets.
The signals of each detector channel are amplified by a room temperature SQUID electronics 3 , controlling the two-stage SQUID readout, and digitised by a 16channel analogue-to-digital converter (ADC) with 16-bit resolution and a maximum sampling rate of 125 MHz 4 [12]. To generate a trigger, a trapezoidal finite impulse response (FIR) filter is employed. The trigger threshold can be chosen individually for each detector channel and is usually set to be just above perceived noise levels. Once a signal is triggered, a time trace of 16384 voltage samples is saved, the first quarter of which is dedicated to pre-trigger samples. Three examples of saved ACcoupled 163 Ho traces are shown in fig. 1 (right). For a sampling rate of 125 MHz and an oversampling of 16, i.e. a difference between two samples of 16/125 MHz = 128 ns, the total saved time window of each trace is 2.097 15 ms. During this time period after a trigger, no further triggers from the same detector channel are accepted. For each trace, the timestamp of its trigger is saved. One can then calculate the time difference ∆t to the previously saved trace in any of the acquiring detector channels of one ECHo-1k chip, and the time difference to the previously saved trace within the same detector channel, ∆t ch .
The information extracted from the analysis of the timestamps is used for reliable and energy-independent data reduction. This is crucial to avoid distortions of the spectrum, particularly in the endpoint region. The shapes of the filtered traces are then analysed to remove remaining spurious signals. The method used is based on the chi-squared goodness of fit measure, which is calculated for each event following a template fit. 3  In the first part, we present different signal families that have been identified in our data. In the second part, methods to eliminate various spurious events are discussed and the performance of these algorithms applied to a subset of ECHo-1k data is evaluated in the last part.

Signal Families
The majority of triggered traces are 163 Ho events with an energy-independent pulse shape as shown in fig. 1 (right) and fig. 2a, and a statistically distributed ∆t ch depending on the activity of the particular detector channel. In addition, there are traces from various sources which, if not recognised and eliminated, can distort the spectrum. They can be divided into pileup originating from 163 Ho, and spurious signals from external sources.

163 Ho pileup
The pulse shape of a 163 Ho event can be distorted if a second event in the same detector occurs within a relatively short time interval ∆t ch . For a time difference larger than the time window ∆t ch > 2 ms, individual traces are triggered and saved for the two pulses as illustrated in fig. 3 (left). The pulse shape of the second trace is distorted by the tail of the previous pulse, as shown in fig. 2b. For ∆t ch smaller than a given value, which depends on the time profile of the signals, this distortion can result in an incorrect reconstruction of the amplitude. Events of this kind are referred to as "pileup-on-tail outside the time window" (POT). The distortions can be omitted by selecting traces with sufficiently high values of ∆t ch , as discussed in section 3.1.1.
If ∆t ch < 2 ms, only one trace is saved with a trigger time corresponding to the occurrence of the first event. For very short time differences ∆t ch ∼ τ rise , the pileup of two events with energies E 1 , E 2 cannot be distinguished from the trace of one event with an energy of E E 1 + E 2 . This unresolved pileup is examined in more detail in section 4.2.3 and can be taken into account statistically.
Events with ∆t ch < 1.57 ms but ∆t ch τ rise are referred to as "pileup-on-tail with both signals inside the time window" (PIT). For these events, the tail of the first pulse deviates strongly from the regular pulse shape (See fig. 2c). A template fit in which a reference pulse is scaled to the trace would provide a false amplitude. Identifying and discarding such events is possible by means of a larger χ 2 red value of the fit, which is described in section 3.2.

External Spurious Signals
Particle Background: Natural radioactivity and cosmic muons can produce events in the energy range of the 163 Ho spectrum. Not all these events can be distinguished from 163 Ho events by means of their pulse shape (see fig. 2d). Background suppression measures and a background model for the ECHo-1k setup are therefore of major importance to reliably analyse the endpoint region of the 163 Ho spectrum [13]. Coincident signals could arise from secondary particles generated by muons interacting in surrounding materials or from muons passing through a pixel and the substrate. Along this line, a search for coincident events among different detector channels allows to identify events part of these muon related events.
Mobile Phone Signal: We observed that mobile phone signals transmitted with the Global System for Mobile communication (GSM) [14] can couple into our readout system and generate triggered traces (see fig. 2e). The detailed underlying mechanism for this coupling is still under investigation. The time structure of a GSM signal is partitioned into time division multiple access (TDMA) frames with a duration of 120 ms/26 4.615 ms. Each TDMA frame consists of eight equal time slots, each of which can contain a burst of data. Normally, a user is assigned to one of these time slots, which can cause repeating signals to be triggered with a period of ∼ 4.615 ms, as illustrated in fig. 3 (right). In consideration of the respective guard periods one expects a burst duration of 0.5465 ms for a normal burst (i.e. digitised voice data) and 0.3210 ms for an access burst (i.e. communication to the base station).
Triggered Noise: In addition to the sources mentioned above, miscellaneous temporary electromagnetic spurious signals can couple into the readout chain and create a false triggered signal. One example for this are small fluctuations in the power grid. Typically, those signals are characterised by a quickly repeating time signature and an anomalous shape of the trace (see fig. 2f).

Data Reduction
Various methods have been studied to filter spurious events in microcalorimeters. The main objectives are mitigating nonlinearity of the detector output, reconstructing single events from pile-up events, lowering the threshold for unresolved pile-up, and detecting outliers. Most approaches are based on either (modified) optimal filtering techniques [15,16] or on principal component analysis [17][18][19][20][21]. While these methods show promising results, in this study we focus on arithmetically simple approaches as we aim for a fast online data reduction. For this, the developed algorithms have been tested offline with an available dataset first, and will be implemented into the data readout scheme in the future. The presented offline data reduction algorithm comprises two levels as illustrated in fig. 4. On the first level, only the trigger time information of the acquired raw traces is used to discard POT events and external spurious signals in an energy-independent way. On the second level, the remaining data filtered by the first level filter are further analysed based on their time profile to discard remaining spurious signals. A template pulse is automatically generated by averaging 163 Ho traces of the MI-line. Traces that deviate strongly from the template are then discarded.

Time Trigger Information Analysis
The time information filter is defined as the logical AND of four independent subfilters, each applied to the raw data. Thus, if a trace is discarded by at least one of the four subfilters, it is discarded by the time information filter. In the following, the aim and implementation of each of the subfilters is described. The holdoff and burst subfilters are performed channel by channel, with ∆t ch being analysed. The coincidence and GSM subfilters are done globally, analysing ∆t.

Holdoff Subfilter
The aim of the holdoff subfilter is to discard POT events. For this, traces are removed that fulfil The holdoff time t hold is fixed based on the time profile of a typical 163 Ho signal such that the distortion of a pulse with ∆t ch = t hold is sufficiently small to ensure correctly reconstructed amplitudes at all energies. The holdoff time is determined in dedicated characterisation measurements prior to the actual experiment run. For this purpose, traces are acquired over a large time window up to the point where the temperature pulse recovers its initial voltage value. This is done for both AC and DC coupled signals. This subfilter only removes the trace on the tail, i.e. the pulse occurring at a time interval ∆t ch < t hold after a previously triggered pulse in the same detector channel. For ECHo run 24 with AC-coupled signals, a value of t hold = 15 ms was determined.

Holdo Filter
Discard traces with ∆t ch < t hold

Burst Filter
Discard time intervals with abnormally high rate

Coincidence Filter
Discard traces with ∆t < t coinc

GSM Filter
Discard traces with ∆t associated to GSM pulse frequencies  Fig. 4 Schematic of the two-level offline data reduction algorithm described in the text

Burst Subfilter
In order to discard any traces from quickly repeating triggered noise, the burst subfilter identifies time intervals with an abnormally high trigger rate. The subfilter is applied channel by channel, since noise usually does not couple identically in all detector channels.
The timestamps of traces of each detector channel are binned with a bin width ∆t bin . The expected number of events from 163 Ho decay per bin is then given by where A ch is the 163 Ho activity in the corresponding detector channel known from detector characterisation. A bin that contains quickly repeating triggered noise will exhibit a number of counts that strongly exceeds the expected value N ch of the otherwise dominant 163 Ho events. The degree of deviation from the expected value can be expressed in terms of the statistical uncertainty of N ch , which for a Poissonian distribution is given by its standard deviation σ = N ch . The traces within a bin are discarded if the number of counts exceeds N ch + 4σ. If a bin fulfils this criterion, it is referred to as a seed bin. For the two neighbouring bins of a seed bin, the threshold for the bins to be discarded is lowered to N ch + 2σ. This ensures that no fragments of a burst are missed due to binning.
Two complementing burst subfilters are implemented, one optimised for faster bursts and the other for slower bursts. The difference lies in the way the bin width is defined. For fast bursts it is chosen such that N ch = 1, i.e.
This corresponds to the shortest bin width that can reasonably be defined. In order to be sensitive to noise triggered with a frequency down to f noise , the bin width of the second burst subfilter is defined in a way that f noise ∆t bin = 4σ. With the definition of σ and eq. (2), this condition is fulfilled for The burst subfilter is then defined as the logical AND of the decisions made with both methods. Thus, if a trace is discarded by at least one of the two methods, it is discarded by the burst subfilter.

Coincidence Subfilter
For an activity of 1 Bq per detector channel, which is typical for ECHo-1k, coincidence among different detector channels on a microsecond timescale due to 163 Ho has a low probability. Muon-induced events or certain electromagnetic signals in turn often cause triggered events in multiple detector channels at the same time. Thus, discarding coincident events altogether is an efficient way to reduce spurious signals. Traces that fulfil as well as the corresponding previous traces are considered coincident. The coincidence time t coinc can be defined by the time response of the signal. In the discussed datasets, the time response is governed by the gain bandwidth product (GBP) of the amplification circuit. One usually obtains an effective time resolution τ rise of a few hundred nanoseconds. For muon related events, ∆t < τ rise holds. However, for electromagnetic signals that couple into the readout scheme of multiple detector channels, time differences ∆t up to a few microseconds have been observed. Therefore, a conservative coincidence time of t coinc = 8 µs is used for ECHo run 24.   In all ten panels, the raw histogram prior to any filters is plotted in grey. In the two top panels, the histograms after application of the time information filter (TI filter) can be seen. The panels below present the histograms of traces discarded by the individual subfilters

GSM Subfilter
This subfilter is implemented to specifically reject triggered GSM phone signals. For this, characteristic ∆t values associated with GSM signals are defined. Besides integer multiples of the duration of a TDMA frame, this includes the burst duration of a normal burst and an access burst. The burst duration can appear in the data stream when the rising and falling edges of a burst are triggered in different detector channels. Traces with a relative ∆t within a ±20 µs interval around one of these characteristic ∆t values are discarded. In principle, there is an infinite number of characteristic time differences ∆t that can be associated when considering all integer multiples. In practice however, a maximum value of ∆t is defined according to the total activity of the chip such that the probability that two triggered GSM signals separated by ∆t are not interrupted by a 163 Ho signal is 10 %.

Application of the Time Information Filter
The first level filter is applied to a dataset acquired with 34 implanted pixels of one ECHo-1k chip during two days of run 24 with a total activity of ∼ 25 Bq. In the presented data reduction routine, the template fit as described in section 3.2.2 is only performed for data that passes the time information filter. For this twoday dataset however, amplitudes are obtained for all data in order to illustrate the working principle of the time information filter as well as to assess its efficiency (section 4.1).
In fig. 5, the number of acquired and discarded traces per minute (left) and the fit amplitudes of acquired and discarded traces (right) are shown for 18 detector channels of the two-day dataset acquired with an ECHo-1k chip. For both plots, the top panel shows the corresponding histograms after applying a time information filter consisting of all four subfilters. The lower panels show the histograms of discarded traces broken down by by the individual subfilters. In all ten panels, the histogram of acquired raw traces is shown in grey for comparison.  7). One can clearly see that the MI-line is the brightest. This holds for much higher signal heights as well, as highlighted by the 1-count limit in red. The number of bins (1000) is chosen simply for optical clarity and only slightly influences the position of the maximum terval drops below the average undisturbed value, while the undisturbed region is barely affected. During the period of a high count rate, the number of discarded traces increases strongly for all subfilters.
In the histograms of discarded fit amplitudes shown in fig. 5 (right, lower panels), one can see the reconstructed amplitudes of discarded background traces as well as a component of falsely discarded 163 Ho traces. The spectrum of traces discarded by the coincidence subfilter, which can be seen in the bottom panel, is a background spectrum with only few discarded 163 Ho traces. It is characterised by a strong increase of counts towards low fit amplitudes, particularly below the NIresonance. It is important to note that the fit amplitudes of background signals cannot necessarily be translated to an energy scale as is the case with 163 Ho signals.

Pulse Shape Analysis
The aim of the second level of data reduction is to recognise and eliminate PIT as well as time-uncorrelated noise traces. For this, a mean trace (template) is generated for each pixel. All traces that have passed the first filter will then undergo a template fit with the obtained template. The goodness of fit parameter χ 2 red is calculated, which provides a measure for how well each trace can be scaled to the template. This is used to define the second level filter. Counts / a.u.

Raw
After TI Filter After PS Filter Fit Fig. 7 Histogram of χ 2 red of a single detector channel. The plotted range contains 99.7 % of all raw events. The filtered region after the pulse shape filter still encompasses 97.6 % of all raw events. Also depicted is the skewed Gaussian fit from which the pulse shape filter is calculated

Automated Template Generation
It is apparent from fig. 1 (right) that the shapes of the traces from a single energy deposition in the detector are energy independent. Therefore, it is possible to build a discrimination scheme based on the deviation of traces from the general shape called the template. In order to process the vast amount of data acquired for the ECHo-1k experiment, an automated process for generating templates has been developed. To ensure a high reliability of the final pulse shape, the template is generated by averaging a large number of individual traces belonging to the MI-line of the 163 Ho spectrum. In fact, MI-events already have a very good figure-ofmerit F OM at the level of single events, defined as: where V and σ are the height of the template signal (the average of the 10 samples after the maximum) and the standard deviation of the noise of the pre-trigger samples (pre-trigger noise), respectively. A high F OM corresponds to a signal shape relatively undisturbed by noise. As an example, compare a pulse from the MIline (F OM ∼ 100 − 400) to a pulse from the OI-line (F OM ∼ 10 − 40), as shown in fig. 1.
To reach higher F OM values, multiple traces have to be averaged to form the template. By averaging N traces, the pre-trigger noise of the resulting template is reduced by a factor of N −1/2 . Ideally, the traces should already have a high signal height. In order to maximise the F OM of the template, the best approach is to select a region of the available spectrum with a high fraction of 163 Ho signals compared to traces from various other sources as detailed in section 2. This is the case for energies close to the main resonances of the 163 Ho spectrum. Also, selecting only traces from a single resonance as opposed to multiple ones ensures that F OM increases reliably.
The MI-resonance was found to be best suited for template generation based on the following approach: first, a histogram of the signal heights in V as provided by the acquisition software of the first few traces ( 10000) is generated, as shown in fig. 6 (left). Then, the brightness b is calculated per bin, defined by where I is the number of counts in each bin. The result can be seen in fig. 6 (right). Summing all traces of the region with the maximum b will result in an average trace with the maximum F OM . For the 163 Ho spectrum, the brightest line is the MI-line. Hence, as stated above, it is the ideal candidate for template generation. The signal height of the MI-line is located via a peak detection algorithm based on a continuous wavelet transform as implemented in the Python package SciPy [22], performed on b.
The last step is to iteratively read in traces with a signal height that is within 1 % tolerance of the mode of the MI-line in small batches of 200. One can then filter those traces with PIT and other defects by calculating their pairwise quadratic differences in a vectorised manner, discarding those that deviate from the median quadratic difference by a factor ≥ 2. The remaining traces can be averaged until an initially defined F OM for the template is reached. If the data is exhausted before reaching the intended F OM , this particular dataset is discarded from further analysis.

Template Fit Method
Once the average MI-signal is generated, a template fit for all the traces which survived the first level filter is performed. The measure used to determine how well the shape of the traces agrees with the template is the reduced chi-square, defined as: where A and O describe the amplitude and offset of the trace s respectively, θ is the template, and the sum runs over all f elements of s and θ. χ 2 red is then minimised with respect to A and O. The normalisation by 1/f (roughly the degrees of freedom of the fit) causes After all traces have been fit, a histogram of all χ 2 red is generated as shown in fig. 7, where in a typical measurement, 99.5 % of all traces are inside the region of 0 ≤ χ 2 red ≤ 5. A skewed Gaussian distribution is fit to the histogram and the pulse shape filter is defined such that all traces which lie outside the 99.73 %-region 5 of the skewed Gaussian are discarded.

Assessment of the Data Reduction Algorithm
In the following, the performance of the filters defined above is evaluated for a subset of ECHo-1k data. For the first level filter, the efficiencies to retain signal and reject background are estimated based on the two-day dataset acquired with an ECHo-1k chip with a total activity of ∼ 25 Bq as well as using simulated ∆t values. Hereinafter, the energy dependence of the second level filter is analysed based on simulated 163 Ho traces.

163 Ho Selection Efficiency
The fraction of 163 Ho events that are unaffected by the time information filter, which we will simply call signal efficiency, is estimated for each subfilter individually. For the holdoff subfilter, by definition, the signal efficiency is sig hold = 100 %. Even though not all discarded traces would cause a falsely reconstructed amplitude, it is crucial for them to be rejected in order to obtain an energy-independent filter. Hence, all traces discarded by this subfilter are considered a source of background.
For the GSM subfilter as well as for the coincidence subfilter, 163 Ho traces are randomly discarded if their time difference ∆t lies in a region that is associated with mobile phone signal or coincident events respectively. The fraction of 163 Ho events occurring within the time intervals related to GSM signals and the fraction of random coincidence of 163 Ho events are obtained by applying the subfilters on simulated data with values of ∆t distributed according to a total activity of A = 25 Bq. One finds that the fraction of 163 Ho events removed by applying the GSM subfilter is 5 % while in case of the coincidence subfilter the fraction of discarded 163 Ho traces amounts to 0.04 %. This corresponds to signal efficiencies of sig GSM = 95 % and sig coinc = 99.96 % respectively. As expected, it can be seen in fig. 5 (right) that for the two-day data set, the number of 163 Ho traces discarded by the GSM subfilter exceeds the number of 163 Ho traces discarded by the coincidence subfilter by more than two orders of magnitude.
For the burst subfilter, the effective off-time caused by the rejection of time intervals with abnormally high rates is determined for each detector channel. The number of falsely discarded 163 Ho traces can be estimated from the product of activity and effective off-time for each detector channel. For the two-day dataset, the fraction of effective off-time to acquisition time ranges from 0.1 % for low-noise detector channels to 1.5 % for detector channels with strong coupling of mobile phone signals. The total fraction of 163 Ho traces discarded by the burst subfilter is 0.7 % and thus sig burst = 99.3 %.

Background Rejection Efficiency
The efficiency to reject signals which could contribute as background to the 163 Ho spectrum is defined individually for the signal families specified in section 2. To reject POT events, the holdoff time is conservatively chosen such that the broadening of the spectral shape due to falsely reconstructed amplitude is negligible. Thus, a background rejection efficiency for POT of bkg POT = 100 % can be assumed.
To estimate the background rejection efficiency for mobile phone signals, we define a reference region in a parameter space with particularly high backgroundto-signal-ratio. The background rejection efficiency can then be estimated based on the fraction of events within this region that are discarded. A good way to separate GSM events from 163 Ho events is to plot the amplitude of the acquired traces against the amplitude of the preceding trace for all acquired traces in one detector channel. As indicated in fig. 8, 163 Ho events are mainly distributed in regions parallel to the main axes. Triggered noise was found to accumulate along the diagonal through the origin with unit slope. The correlation of fit amplitudes of subsequent mobile phone signals is not surprising, since triggered mobile phone signals are typically characterised repeating signals with the same shape. The reconstructed amplitudes of mobile phone signals are well below the M-resonances and in the continuum region between two resonances, the 163 Ho event rate is suppressed by several orders of magnitude compared to the region around the resonances. Thus, an ellipse-shaped reference region can be defined, centred between the NII-and OI-resonances, as drawn in fig. 8, which has a particularly high background-tosignal-ratio. In this way, traces contained within the ellipse can be considered a pure sample of background signals. For the two-day dataset, only 77 out of 44602 events within the ellipse are not discarded by a time information filter consisting of all four subfilters. The efficiency is estimated by the fraction of events within the ellipse that are discarded by the time information filter bkg phone = (99.826 ± 0.020) %. The error is obtained from the variance of a Binomial distribution 6 .
As for particle background, only signals initiated by atmospheric muons can be tackled with the time information filter. Such traces can be discarded if a coincident signal in multiple pixels is produced. The efficiency of rejecting muon-induced events by means of the coincidence subfilter can be estimated from an acquisition with an active muon veto installed around the dilution refrigerator. The background rejection efficiency for muon-induced events is the ratio of pixelpixel-veto coincidences and pixel-veto coincidences, i.e. the fraction of muon-induced signals that produce a signal in at least two pixels. In [13], a measurement with muon veto was described for 64 pixel-days. A total of 242 ± 20 pixel-veto coincidences and 194 ± 12 pixel-pixel-veto coincidences were measured. Thus, one can derive bkg muon = (80 ± 8) %.

Necessity of the Individual Subfilters
POT events are discarded efficiently by the holdoff subfilter and muon related background is tackled by the coincidence subfilter. Since these signal families are each addressed by only one subfilter, the use of both the holdoff subfilter and the coincidence subfilter is essential. This also implies that the combination of different subfilters does not improve the respective rejection efficiencies for traces from these signal families.
To evaluate the benefit of the additional use of a burst subfilter and a GSM subfilter to specifically reject mobile phone induced noise, the influence of these subfilters on the background rejection efficiency bkg phone is investigated. If instead of all four subfilters only the holdoff subfilter and the coincidence subfilter are applied, 226 instead of 77 out of a total of 44602 mobile phone signals within the ellipse are not discarded. If in addition to these two subfilters the burst subfilter is applied, 81 traces remain undiscarded. In the case of using the GSM subfilter in addition to the holdoff subfilter and the coincidence subfilter, 103 traces in the ellipse are not discarded.
Even though the improvement due to the additional subfilters seems to be minor for this dataset, the burst subfilter in particular should always be applied. The signal efficiency for this subfilter is already high and increases even further the lower the noise level of the acquisition. In addition, the burst subfilter is sensitive to abnormally high trigger rates that only occur in one detector channel, and even to rather low noise frequencies that cannot be resolved by any of the other subfilters. Applying a time information filter without the GSM subfilter, the background rejection efficiency bkg phone is still above 99.8 % and the signal efficiency is dominated by sig burst = 99.3 %. With a signal efficiency of only 95 %, the GSM subfilter is an expensive filter in terms of discarding good data. Furthermore, the reconstructed energy of GSM signals is well below Q EC and thus won't affect the spectral shape close to the endpoint. In the two-day dataset of ECHo-1k, the additional application of this particular subfilter shows no advantage over the sole use of the other three subfilters. In future runs, the coupling of GSM signals will be reduced by improving screening of the read out components. For analyses of the low energy part of the spectrum however, where background levels increase, as well as for acquisitions with high levels of triggered mobile phone signal, this subfilter can be of relevance.

Assessment of the Pulse Shape Analysis
To assess the pulse shape analysis, the template fit described in section 3.2 is performed on a set of simulated 163 Ho data. The aim is to find the sensitivity of identifying PIT events as a function of time difference and energies of two subsequent events. Traces of PIT as well as regular 163 Ho events are simulated with amplitudes, timestamps and polarities randomly drawn from corresponding distributions.

Simulation of pileup-on-tail with both signals inside the time window
For the simulation of signals with PIT, 10 7 events are generated, each with an amplitude of the triggered pulse A 1 , an amplitude of the subsequent pulse A 2 , a time difference to the subsequent pulse ∆t ch and a relative sign of the polarity to the subsequent pulse Π. The corresponding values are drawn randomly from the expected probability distributions of the parameters. For the two pulse amplitudes, this distribution is the theoretical 163 Ho spectrum [7] normalised by its area. The values of ∆t ch are drawn from an exponential distribution ∝ exp(−A∆t ch ) with activity A = 1 Bq. Integer multiples of 128 ns are allowed, which corresponds to the time difference between two sampled data points typically used for acquiring ECHo data, as described in section 1. For Π, a discrete distribution P (Π = −1) = P (Π = +1) = 0.5 is employed. Normal distributed noise N is generated with a constant F OM for a pulse height corresponding to the MI-line of the 163 Ho spectrum. In the following, F OM = 300 is used, which is a typical value for ECHo-1k data.
A simulated pileup trace P U (A 1 , A 2 , ∆t ch , Π) is then generated according to where θ(t shift ) is a template pulse (see section 3.2.1) shifted in time by t shift .
For a time difference larger than ∆t ch 1.57 ms, the rising edge of the subsequent pulse lies outside the time window and thus the pulse shape of the initial event is not affected. For this reason, the distribution of ∆t ch is truncated at ∆t ch = 10 ms for the simulation.
In total, 10 7 events are generated. The number of PIT events simulated with the truncated range of ∆t ch is equivalent to the number of PIT events from 1.01 · 10 9 events if no truncation would be applied. An exemplary path of decreasing ∆t ch for A 1 = A 2 = 1 and Π = +1 is indicated in red. The location of the maximum of the arc-shaped structures is determined by the amplitude of the pulse on the tail A 2 . b) Histogram of the fit amplitude of simulated events before and after applying the pulse shape filter. The large fraction of pileup is caused by the truncated ∆t ch distribution used for the simulation. The structure in the histogram of traces discarded by the pulse shape filter is discussed in the text. c) Fit amplitude vs. χ 2 red scatter plot for one detector channel of the two-day ECHo-1k dataset. The arc-shaped structures become apparent after applying the time information filter. d) Histogram of the fit amplitude of the ECHo-1k dataset with 18 detector channels where the endpoint region is blinded. The slight asymmetry of spikes around the main 163 Ho-lines is caused by detector channels with an asymmetric activity in the two pixels. The spike at fit amplitude 0 corresponds to triggered baselines. Note that the scale on the x-axis is the same for all plots ∆t ch ≤ 10 ms is 8.42 · 10 6 . Note that POT is not considered in this simulation, as these events are already sorted out by the holdoff subfilter as discussed in section 3.1.1.

Analysis of pileup-on-tail with both signals inside the time window
A template fit as described in section 3.2.2 is applied to the generated traces, thereby obtaining the fit amplitude and χ 2 red for each simulated event. The scatter plot of these fit parameters is shown in fig. 9a. For χ 2 red ∼ 1, the line structure of the 163 Ho spectrum is apparent with high abundances for fit amplitudes of ∼ 1.0 (MI-line), ∼ 0.9 (MII-line), ∼ 0.2 (NI-line), ∼ 0.16 (NII-line) and ∼ 0.025 (O-lines). For larger values of χ 2 red , arc-shaped structures that are centred around those amplitudes can be found. Three distinct groups of arc-shaped structures can be identified, culminating at χ 2 red ∼ 10, χ 2 red ∼ 400 and χ 2 red ∼ 10000. One finds that these groups correspond to PIT with amplitude A 2 corresponding to a 163 Ho event from the O-lines, N-lines and M-lines respectively, while the shift of the arc-shaped structures along the x-axis depends on the initial amplitude A 1 of the pileup event. The structures can further be understood when considering the influence of the time difference between the pulses ∆t ch . For illustrative purposes, the path of decreasing ∆t ch for fixed amplitudes A 1 = A 2 = 1 and relative sign of the polarities of Π = +1 is indicated in fig. 9a. For ∆t ch < 1.57 ms, the value of χ 2 red increases with decreasing ∆t ch and up to ∆t ch ∼ 0.55 ms the true amplitude of the triggered pulse is underestimated by an increasing amount. At ∆t ch ∼ 0.55 ms, the largest value of χ 2 red is reached. For further decreasing ∆t ch , the fit amplitude increases while χ 2 red decreases up to the point where ∆t ch = 0 ms. Here, a fit amplitude of A 1 +ΠA 2 2.0 and χ 2 red ∼ 1 is reached as it is expected for unresolved pileup of the two pulses. A corresponding mirrored structure arises from the same amplitudes with opposite relative sign Π = −1.
A simplified pulse shape filter that selects fitted traces with χ 2 red < 1.3 is used. Applying this filter to the simulated fit amplitudes yields a fairly clean theoretical 163 Ho spectrum (see fig. 9b upper panel orange), apart from a few outliers with fit amplitudes above 1.5 and below 0 that will be discussed in section 4.2.3. The region around the turning point in fig. 9b at ∆t ch ∼ 0.55 ms is densely populated. This increase in density gives rise to a spiky structure in the histogram of the fit amplitudes of traces that are discarded by the pulse shape filter (see fig. 9b  For the two-day dataset acquired with an ECHo-1k chip, the fit amplitude vs. χ 2 red scatter plot of one detector channel ( fig. 9c) and the histogram of fit amplitudes discarded by a pulse shape filter for 18 detector channels ( fig. 9d lower panel) show structures that have striking similarities to the ones found in the simulated data. For better comparison, the same simplified pulse shape filter applied to the simulated data is also applied to the ECHo-1k dataset. The arc-shaped structures described above become apparent in the scatter plot for the data after applying the time information filter. These in turn result in a similar structure of the histogram of fit amplitudes of traces discarded by the pulse shape filter. The most apparent difference between the histograms in fig. 9b and fig. 9d is the larger fraction of PIT events which arises from the truncated ∆t ch distribution used for the simulation. Furthermore, one can observe an asymmetry of spike pairs (e.g. fit amplitude of 0.9 and 1.1) in the histogram of the acquired data. For implanted detector channels, the activity in the two pixels is not identical, which yields to P (Π = +1) ≥ P (Π = −1). The probability that two consecutive triggers have the same polarity becomes larger for an increasing asymmetry of activity of the pixels. The asymmetry is maximal for detector channels, which only have one implanted pixel and thus P (Π = +1) = 1 and P (Π = −1) = 0.

Energy dependence of a pulse shape filter
In order to assess the energy dependence of a pulse shape filter, the histogram of true amplitudes A 1 of traces that are discarded by the filter is compared to the theoretical spectrum ( fig. 10). The ratio of the two histograms is shown in the lower panel together with the 1σ and 2σ error bands due to the Poisson error of the number of counts in each bin. A constant is fit to the ratio and, apart from one deviation of −2.20σ at an amplitude A 1 = 1.06, all ratios agree with the fit within the 2σ band. From this we can conclude that PIT events are discarded by a pulse shape filter in a fairly energy independent way. On a subdominant level, an energy-dependent distortion of the final spectrum arises from unresolved pileup, which in this context are pileup events that survive the pulse shape filter. In fig. 11, χ 2 red is plotted as a function of ∆t ch . The data points are coloured according to the amplitude of the pulse on the tail A 2 . Again, it becomes apparent that χ 2 red only depends on A 2 and ∆t ch , but not on A 1 . The horizontal bands indicated in fig. 9a correspond to the location of the plateaus of χ 2 red found for ∆t ch between ∼ 0.25 ms and ∼ 1.25 ms. For smaller ∆t ch , the value of χ 2 red steeply decreases towards χ 2 red = 1, as expected for ∆t ch = 0 ms. The inset in fig. 11 shows that the value of ∆t ch for which the events fulfil χ 2 red < 1.3 is larger the smaller the amplitude A 2 . These traces are considered good 163 Ho traces by a pulse shape filter and thus correspond to unresolved pileup. For 0.015 < A 2 < 0.050, i.e. OI-pulses on the tail, pileup is not recognised by the pulse shape filter for ∆t ch 10 µs while for 0.950 < A 2 < 1.500, i.e. MI-pulses and higher on the tail, the time resolution for pileup is of the order of ∆t ch ∼ 100 ns 7 , which is of the same order as the time difference between two samples of a trace of 128 ns. This energy-dependent characteristic has the effect that the unresolved pileup spectrum does not simply correspond to the autoconvolution of the 163 Ho spectrum as one would naively expect. Rather, we can infer from fig. 11 that the majority of unresolved pileup traces will feature small amplitudes A 2 and thus have a fit amplitude that deviates only slightly from their true amplitude. As a result, the acquired spectrum is only weakly distorted -mainly by means of a slight broadening of the resonances. The spectrum of the reconstructed amplitudes of unresolved pileup events is shown in the lower panel of fig. 12. In the upper panel, a corresponding scatter plot of fit amplitude vs. true amplitude A 1 is presented. As expected, the majority of events are distributed near a straight line with unitary slope through the origin. These data points correspond to barely distorted traces from O-line pulses on the tail. Further accumulations can be found on the diagonals shifted by ∼ ±0.2 (NI-pulse on the tail with Π = +1 (+) and Π = −1 (-)) and ∼ ±1 (MI-pulse on the tail). It can be seen that the outliers mentioned in section 4.2.2 with fit amplitudes above 1.5 and below 0 are concentrated near those shifted diagonals. The shape of the unresolved pileup spectrum is well understood and in particular no structure near the endpoint of the 163 Ho spectrum emerges. A total of 2801 unresolved pileup traces are found. The fraction of unresolved pileup for this simulation is f pu = 2.77 × 10 −6 , considering that the number of simulated PIT events is equivalent to the number of PIT events from 1.01 · 10 9 events without truncating ∆t ch . For comparison, the autoconvolution of the theoretical 163 Ho spectrum for a pileup fraction f pu = 3 × 10 −6 is superimposed in the lower panel of fig. 12. Unresolved pileup with an OI-line on the tail have similar rates in both spectra. However, structures with larger amplitudes on the tail are reduced by more than an order of magnitude, while those with barely altered fit amplitudes have a higher rate in the simulated spectrum.
This simulation is representative for the estimation of unresolved pileup in the high statistics spectrum of ECHo-1k.

Conclusions and Outlook
In the ECHo-1k high statistics measurement, 58 MMC pixels, each loaded with an average of about 0.5 Bq of 163 Ho, have been operated over several months in order to acquire more than 10 8 163 Ho events. This will allow to test the effective electron neutrino mass to a level of about 20 eV. To reach this sensitivity, a new data reduction scheme has been developed. The aim of this scheme is to efficiently remove signals which could act as a background for the 163 Ho spectrum, without sacrificing large fractions of 163 Ho events and to precisely characterise any energy dependence of the filters. We present a two-level data reduction scheme to obtain a clean signal from data acquired with ECHo-1k chips. The first level filter is purely based on the time information of traces. It is thus inherently energy independent. On a second level, the filtered data are further analysed by means of their deviation from a template pulse. The minor energy dependence due to unresolved pileup is well understood and can be modelled in an analysis of the 163 Ho spectrum. All implemented algorithms are designed such that they can be applied online.
After the data has been filtered by the two-level data reduction scheme, the recovered amplitudes are corrected for temperature fluctuations of the entire setup. The energies of the events are then obtained by identifying the major resonances of the 163 Ho spectrum and fitting their positions to the previously measured values with a polynomial function.
The methods discussed here can be adapted to be used for the next stages of the ECHo experiment. Future efforts will be directed towards resolving the energy of the first pulse in a PIT to maximise the signal yield of the second level filter. This is particularly important for a higher implanted activity per pixel, as envisaged in future phases of the ECHo experiment.