# Enhanced ELM Based Channel Estimation for RIS-Assisted OFDM systems with
Insufficient
CP and Imperfect Hardware

Chaojin Qing, Li Wang, Lei Dong and Jiafan Wang This work is supported in part
by the Sichuan Science and Technology Program (Grant No. 2021JDRC0003), the
Major Special Funds of Science and Technology of Sichuan Science and
Technology Plan Project (Grant No. 19ZDZX0016 /2019YFG0395), the Demonstration
Project of Chengdu Major Science and Technology Application (Grant No.
2020-YF09-00048-SN), the Key Scientific Research Fund of Xihua University
(Grant No. Z1120941), and the Special Funds of Industry Development of Sichuan
Province (Grant No. zyf-2018-056).C. Qing, L. Wang and L. Dong are with the
School of Electrical Engineering and Electronic Information, Xihua University,
Chengdu, 610039, China (E-mail: qingchj@mail.xhu.edu.cn). J. Wang is with the
Synopsys Inc., 2025 NE Cornelius Pass Rd, Hillsboro, OR 97124, USA (E-mail:
jifanw@gmail.com).

###### Abstract

Reconfigurable intelligent surface (RIS)-assisted orthogonal frequency
division multiplexing (OFDM) systems have aroused extensive research interests
due to the controllable communication environment and the performance of
combating multi-path interference. However, as the premise of RIS-assisted
OFDM systems, the accuracy of channel estimation is severely degraded by the
increased possibility of insufficient cyclic prefix (CP) produced by extra
cascaded channels of RIS and the nonlinear distortion lead by imperfect
hardware. To address these issues, an enhanced extreme learning machine
(ELM)-based channel estimation (eELM-CE) is proposed in this letter to
facilitate accurate channel estimation. Based on the model-driven mode, least
square (LS) estimation is employed to highlight the initial linear features
for channel estimation. Then, according to the obtained initial features, an
enhanced ELM network is constructed to refine the channel estimation. In
particular, we start from the perspective of guiding it to recognize the
feature, and normalize the data after the network activation function to
enhance the ability of identifying non-linear factors. Experiment results show
that, compared with existing methods, the proposed method achieves a much
lower normalized mean square error (NMSE) given insufficient CP and imperfect
hardware. In addition, the simulation results indicate that the proposed
method possesses robustness against the parameter variations.

###### Index Terms:

Insufficient cyclic prefix (CP), reconfigurable intelligent surface (RIS),
extreme learning machine (ELM), channel estimation, nonlinear distortion.

##  I Introduction

Reconfigurable intelligent surface (RIS) has been recognized as a potential
technology for future sixth generation (6G) mobile communications [1], due to
its promising performances in spectrum efficiency (SE), energy efficiency
(EE), etc [2]. In particular, the wireless propagation environment is
manipulated by changing the phase shift [3], and this characteristic is
employed for “smart cities” in massive machine type communication (mMTC) [4].
To support the performance enhancement of wireless communication systems,
e.g., the SE and EE, accurate channel estimation is the premise of RIS systems
[5], [6]. However, the channel estimation in RIS-assisted wireless systems is
quite challenging due to the prohibitively high pilot overhead [1],
complicated cascaded channels [6] and passive surface elements [7], etc.
Especially, the premise of RIS, e.g., accurate channel estimation, is suffered
from multi-path interference. Given that the orthogonal frequency division
multiplexing (OFDM) is good at resisting multi-path interference [8], the RIS-
assisted OFDM systems are taken into account in this letter.

For RIS-assisted OFDM systems, existing researches, e.g., [9, 3, 2, 10], are
studied under two assumptions. First, the cyclic prefix (CP) length in RIS-
assisted OFDM systems is longer than the maximum delay spread. Second, perfect
hardware is applied in the systems. However, these two assumptions are not
common for practical application in RIS-assisted OFDM systems. For the first
assumption, namely, the sufficient CP, challenges come from the time-varying
channels and RIS-introduced extra paths. Specifically, adopting a sufficient
CP to deal with time-varying scenarios may consume massive spectrum resources
[11]. Given extra paths introduced by RIS, the possibility will increase that
the previously indiscernible path becomes a resolvable path, and this, in
turn, increases the possibility of insufficient CP [12]. Therefore, to save
valuable spectrum resources, the development of receivers with insufficient CP
is highly desired. In addition to the insufficient CP, imperfect hardware is
usually observed in most practical RIS-assisted OFDM systems [13]. For
example, high power amplifier (HPA), digital to analog converter (DAC), etc.,
these hardware inevitably causes nonlinear distortion in the real application
scenarios. For channel estimation, the nonlinear distortion destroys the
orthogonality of the training sequence and seriously degrades the estimation
accuracy [14]. Therefore, previous works, e.g., [3, 2] and [6], mainly based
on the previous two assumptions, could not work well in practical applications
given the scenarios of insufficient CP and nonlinear distortion.

To improve the channel estimation of RIS-assisted OFDM systems affected by
insufficient CP and imperfect hardware, we introduce an enhanced extreme
learning machine (ELM) network. Unlike the deep learning-based methods that
require complex parameter tuning and long training time [15], the proposed ELM
has the advantages of fast learning speed and performance robustness. We
enhance the ELM network by using hidden-layer standardization and propose an
enhanced ELM-based channel estimation (eELM-CE) method for RIS-assisted OFDM
systems. Different from the common standardization method, our standardization
is added before the activation function of hidden layer to ease feature
acquirement for ELM’s hidden layer output. In the proposed method, the
imperfect hardware and the insufficient CP are modeled as a nonlinear problem
and solved by exploiting the learning ability of the enhanced ELM. Generally,
the proposed method can be viewed as the model-driven mode [16] integrated
with the least square (LS) estimation and an enhanced ELM network, which
extracts initial features and refines the following channel estimation,
respectively. In contrast, [3, 2] are mainly data-driven, without the
cooperation of initial features. Simulation results show that the performance
of channel estimation is significantly improved by using our proposed eELM-CE
method. To the best of our knowledge, this is the first work that nonlinear
distortion with insufficient CP is taken into account in channel estimation
for RIS-assisted OFDM systems.

The remainder of this letter is structured as follows: In Section II, the
system model of RIS-assisted OFDM systems with insufficient CP and imperfect
hardware is presented. The proposed method, i.e., eELM-CE is presented in
Section III, and numerical results are given in Section IV. Finally, Section V
concludes our work.

Notations: Boldface upper case and lower case letters denote matrix and vector
respectively. ${\left(\cdot\right)^{T}}$ and $\mathbf{(\cdot)^{\dagger}}$
denote the transpose and matrix pseudo-inverse, respectively. $\odot$ is the
Hadamard product. ${\left\|\cdot\right\|_{2}}$ is the Euclidean norm.
${\mathbf{F}_{N}}$ is the normalized $N\times N$ Fourier transform matrix.
${{\bf{0}}_{N}}$ is an $N\times 1$ zero vector. $\mathds{E}(\cdot)$ is the
mathematical expectation. $\mathds{D}(\cdot)$ represents the variance.

##  II System Model

We consider a RIS-assisted OFDM system, which comprises one single-antenna
transmitter, one single-antenna receiver and one multi-element RIS. This
system employs $N$ subcarriers, and assumes the CP length
${L_{{\rm{\mathrm{CP}}}}}$ is shorter than the maximum delay spread $L$, i.e.,
${L_{{\rm{\mathrm{CP}}}}}<L$. The time-domain signal received at the receiver
is given by

${\bf{y}}={{\bf{\widetilde{X}}}_{c}}{\bf{h}}+{\bf{n}},$ (1)

where
${{\bf{\mathbf{\widetilde{X}}}}_{\mathrm{c}}}\in\mathbb{C}^{{\left({N+{L_{\mathrm{CP}}}+L-1}\right)\times
L}}$ is a cyclic matrix formed by the transmitted data,
$\mathbf{h}={\left[{{h_{1}},{h_{2}},\cdots,{h_{L}}}\right]^{T}}$ stands for
the composite channel impulse response (CIR) between the receiver and
transmitter, and
${\bf{n}}\in\mathbb{C}^{\left({N+{L_{\mathrm{CP}}}+L-1}\right)\times 1}$
denotes the circularly symmetric complex Gaussian (CSCG) distribution with
zero mean and variance ${\sigma^{2}}$. In (1), the first column of
${{\bf{\mathbf{\widetilde{X}}}}_{\mathrm{c}}}$ is given as

$\widetilde{\mathbf{x}}=[{{\widetilde{x}_{{-{L_{CP}}+1}}},\cdots,{\widetilde{x}_{0}}},{{\widetilde{x}_{1}},\cdots,{\widetilde{x}_{N}},}{{\bf{0}}_{L-1}}]^{T},$
(2)

where ${\widetilde{x}_{i}}$, $i={-{L_{\mathrm{CP}}}+1},\cdots,{{N}}$ denotes
the distorted data. That is, the transmitted data ${{x}_{i}}$ experiences the
nonlinear distortion caused by imperfect hardware to form
${\widetilde{x}_{i}}$, which is expressed as

${\widetilde{x}_{i}}=f_{\mathrm{dis}}\left({{x}_{i}}\right),$ (3)

where we use the function $f_{\mathrm{dis}}(\cdot)$ to describe the nonlinear
distortion caused by imperfect hardware[15].

The composite CIR between the receiver and transmitter is represented by

${\bf{h}}\;{\rm{=}}\;{{\bf{h}}_{\mathrm{TR}}}+{{\bf{H}}_{\mathrm{\mathrm{TRR}}}}\bm{\phi},$
(4)

where ${{\bf{h}}_{\mathrm{\mathrm{TR}}}}\in\mathbb{C}^{L\times 1}$ is the CIR
of the transmitter-receiver direct link,
${{\bf{H}}_{\mathrm{TRR}}}\in\mathbb{C}^{L\times M}$ is the equivalent
cascaded CIR of the reflecting link,
${{\bf{H}}_{\mathrm{\mathrm{TRR}}}}=\left[{{{\bf{h}}_{\mathrm{\mathrm{TRR}},1}},{{\bf{h}}_{\mathrm{\mathrm{TRR}},2}},\cdots,{{\bf{h}}_{\mathrm{\mathrm{TRR}},M}}}\right]$
is stacked by ${{\bf{h}}_{\mathrm{\mathrm{TRR}},m}}$ with $m=1,\cdots,M$, and
$\bm{\phi}\buildrel\Delta\over{=}{\left[{{\phi_{1}},{\phi_{2}},\cdots,{\phi_{M}}}\right]^{T}}$
means the phase-shift vector defined by
${\phi_{m}}={\beta_{m}}{e^{j{\varphi_{m}}}}$. Similar to [6], in order to
maximize the reflection power of the RIS and simplify the hardware design, we
fix ${\beta_{m}}=1$.

The equivalent cascaded CIR of the reflecting link is given by

${{\bf{h}}_{\mathrm{TRR},m}}={{\bf{h}}_{\mathrm{TR},m}}\odot{{\bf{h}}_{\mathrm{RR},m}},$
(5)

where ${{\bf{h}}_{\mathrm{TR},m}}\in\mathbb{C}^{L\times 1}$ and
${{\bf{h}}_{\mathrm{RR},m}}\in\mathbb{C}^{L\times 1}$ are the aggregated CIRs
of the transmitter-RIS link and RIS-receiver link associated with the $m$-th
sub-surface, respectively.

According to (1), (4) and (5), the equivalent baseband received signal in time
domain is rewritten by

${\bf{y}}={{\bf{\widetilde{X}}}_{c}}\left({{\bf{h}}_{{\rm{\mathrm{TR}}}}}+{\sum\limits_{m=1}^{M}({{{\bf{h}}_{{\rm{\mathrm{TR}}},m}}{\phi_{m}}\odot{{\bf{h}}_{{\rm{\mathrm{RR}}},m}}}})\right)+{\bf{n}}.$
(6)

With the received signal ${\bf{y}}$, we use the LS estimation to highlight the
initial estimation features for easing network learning. Then, an eELM-CE
method is proposed in Section III to improve the estimation accuracy for the
case of insufficient CP with imperfect hardware.

##  III Enhanced ELM-based Channel Estimation

###  III-A Estimation Preprocessing

With the received signal ${\bf{y}}$, we first remove its CP, and then employ
LS estimation to obtain the initial features of channel estimation in the
frequency domain. According to the transmission protocol proposed in [17], two
OFDM blocks are transmitted in each time slot. From [17], it is assumed that
the pilot tone, denoted as ${\bf{c}}$, and the transmitted data consist in the
first and second OFDM block, respectively. Hence, we transmit $M+1$ slots due
to the purpose to separate the direct link and reflecting link [6].

For LS estimation, the received pilot signal, denoted by
${\mathbf{Y}_{\mathrm{P}}}=\left[{{\mathbf{y}_{\mathrm{P},1}},{\mathbf{y}_{\mathrm{P},2}},\cdots,{\mathbf{y}_{\mathrm{P},M+1}}}\right]$,
is first picked out from the received signal
$\mathbf{Y}=\left[{{\mathbf{y}_{1}},{\mathbf{y}_{2}},\cdots,{\mathbf{y}_{M+1}}}\right]$.
Then, the received pilot signal ${{\bf{Y}}_{\mathrm{P}}}$ and transmitted
pilot tone
$\mathbf{C}=\left[{{\mathbf{c}_{1}},{\mathbf{c}_{2}},\cdots,{\mathbf{c}_{M+1}}}\right]$
are respectively transformed into frequency domain by using Discrete Fourier
Transform (DFT), which are expressed as

$\left\\{\begin{array}[]{l}{\\!\mathbf{\overline{Y}}_{\mathrm{p}}}={\mathbf{F}_{N}}{\mathbf{Y}_{\mathrm{p}}}\\\
\mathbf{\overline{C}}\;\>={\mathbf{F}_{N}}\mathbf{C}\end{array}\right.,$ (7)

where ${{\bf{\overline{Y}}}_{\mathrm{p}}}$ and ${\bf{\overline{C}}}$ represent
the received pilot signal and transmitted pilot in the frequency domain,
respectively. With ${{\bf{\overline{Y}}}_{\mathrm{p}}}$ and
${\bf{\overline{C}}}$, we employ LS estimation to obtain the initial channel
frequency response (CFR) of the composite channel, which is given by

${{{\widehat{H}}}_{{\mathrm{LS}}}{(m,n)}}=\frac{{{{{\overline{Y}}}_{\mathrm{p}}{(m,n)}}}}{{{{\overline{C}}}\,\,{(m,n)}}},m=1,\cdots,N;n=1,\cdots,M+1$
(8)

where $m$ represents the $m$-th row, $n$ represents the $n$-th column and
${{\bf{\widehat{H}}}_{\mathrm{LS}}}$ is the composite CFR between the receiver
and transmitter. Using the transmission protocol proposed in [6], the CFRs of
direct link and reflecting link are separated by

$\left[{{{\bf{\widehat{h}}}_{\mathrm{TR}}}\;{{\bf{\widehat{H}}}_{\mathrm{TRR}}}}\right]={{\bf{\widehat{H}}}_{\mathrm{LS}}}{\bm{\Theta}^{-1}},$
(9)

where ${{\bf{\widehat{h}}}_{\mathrm{TR}}}\in\mathbb{C}{{}^{N\times 1}}$
denotes the direct-link CFR,
${{\bf{\widehat{H}}}_{\mathrm{TRR}}}\in\mathbb{C}^{N\times M}$ is the
reflecting-link CFR composed of $M$ sub-surfaces reflection paths, and
${\bm{\Theta}}$ represents the RIS reflection matrix which is a full-rank
matrix [6]. From [6], ${\bm{\Theta}}$ is described as

${\bf{\Theta}}{\kern
1.0pt}{\rm{=}}\left[\begin{array}[]{l}\,1\quad\quad\,\,1\quad\quad\,\,\cdots\quad\quad\;\,\,\>1\\\
{\bm{\phi}^{\left(1\right)}}\quad{\bm{\phi}^{\left(2\right)}}\quad\;\cdots\quad\quad{\bm{\phi}^{\left({M+1}\right)}}\end{array}\right],$
(10)

where ${\bm{\phi}^{\left(i\right)}}$ is the RIS’s phase shift of the $i$-th
pilot symbol.

Based on the initial CFRs, i.e., ${{\bf{\widehat{h}}}_{\mathrm{TR}}}$ and
${{\bf{\widehat{H}}}_{\mathrm{TRR}}}$, an eELM-CE is proposed in the following
subsection (i.e., Section III-B) to refine the channel estimation.

Algorithm 1 eELM-CE method

Input: Separated CFRs
$\mathbf{\widehat{H}}\buildrel\Delta\over{=}[{{\mathbf{\widehat{h}}_{\mathrm{TR}}},{\mathbf{\widehat{h}}_{\mathrm{TRR},1}},}$
${\cdots,{\mathbf{\widehat{h}}_{\mathrm{TRR},M}}}]$.

Output: Refined CFRs
$\mathbf{\widetilde{H}}\buildrel\Delta\over{=}\left[{{\mathbf{\widetilde{h}}_{\mathrm{TR}}},{\mathbf{\widetilde{h}}_{\mathrm{TRR},1}},\cdots,{\mathbf{\widetilde{h}}_{\mathrm{TRR},M}}}\right]$.

Offline Training:

1: Initialize the weight $\mathbf{W}$ and bias $\mathbf{b}$.

2: for $j=1,...,M+1$ do

3: for $t=1,...,N_{\mathrm{d}}$ do

4: Normalize the samples $\mathbf{\widehat{h}}^{\left(t\right)}_{j}$ using Eq.
(11) to form $\mathbf{\widehat{h}}^{\left(t\right)}_{\mathrm{st},j}$.

5: Calculate the output ${\mathbf{o}^{\left(t\right)}_{j}}$ of the hidden
layer after ${\mathbf{\widehat{h}}^{\left(t\right)}_{\mathrm{st},j}}$ through
the activation function $\sigma\left(\cdot\right)$ using Eq. (12) and Eq. (13)
according to $\mathbf{W}$ and $\mathbf{b}$.

6: end for

7: Collect the output
${\bm{\mathrm{O}}_{j}}=\left[{\bm{\mathrm{o}}_{j}^{\left(1\right)},\cdots,\bm{\mathrm{o}}_{j}^{\left({{N_{d}}}\right)}}\right]$.

8: Use the known labels ${\mathbf{H}_{j}}$ and ${\mathbf{O}_{j}}$ to calculate
the hidden layer output weight ${\bm{\beta}_{j}}$ using Eq. (14).

9: end for Online Deployment:

10: Perform LS estimation to obtain ${{\bf{{\overline{h}}}}_{\mathrm{LS}}}$
using Eq. (8).

11: Separate the CFRs of the direct link
${{{\bf{{\overline{h}}}}_{\mathrm{TR}}}}$ and reflecting link
${{{\bf{{\overline{H}}}}_{\mathrm{TRR}}}}$ using Eq. (9).

12: Use the CFRs preprocessed to form
${{\mathbf{{{\overline{H}}}}}}\buildrel\Delta\over{=}[{{\mathbf{{\overline{h}}}_{\mathrm{TR}}}},$
${\mathbf{{\overline{h}}}_{\mathrm{TRR},1}},\cdots,$
${{\mathbf{{\overline{h}}}_{\mathrm{TRR},M}}}]$.

13: for $j=1,...,M+1$ do

14: Normalize ${{\mathbf{{{{\overline{h}}}}}}_{j}}$ using Eq. (11) to form
${{\mathbf{{{\overline{h}}}}}_{\mathrm{st},j}}$.

15: Feed ${{\mathbf{{{\overline{h}}}}}_{\mathrm{st},j}}$ into the trained ELM-
based network using Eq. (15) to obtain the refined CFR
${{\mathbf{{\widetilde{h}}}}_{j}}$.

16: end for

###  III-B Refining Estimation

The details of eELM-CE are presented in Algorithm 1. The processes of offline
training and online deployment are respectively given as follows.

####  III-B1 Offline Training

In the training phase of ELM-based network, the main task is to learn the
output weight. For ELM-based network, we need to train $M+1$ ELM networks, and
each ELM network training uses ${N_{d}}$ samples. With extensive experiments,
it is found that $N_{d}=10000$ training samples show a good enough learning
effect while no obvious improvement is noticed given more training samples.
$\mathbf{\widehat{h}}_{j}^{\left(t\right)}$ represents the CFRs of the $j$-th
ELM network, $j=1,...,M+1$, and the superscript $t$, $t=1,...,N_{d}$, is the
number of training sessions. We send the samples which are obtained using Eq.
(9) into the network to calculate the network output and then obtain the
output weight of hidden layer. In the algorithm, we obtain the samples and
labels through Eq. (8) and Eq. (9), and then normalize the samples, which is
expressed as

$\mathbf{\widehat{h}}^{\left(t\right)}_{\mathrm{st},j}=\frac{{\\!\\!\\!\mathbf{\widehat{h}}^{\left(t\right)}_{j}}}{{{{\left\|{\mathbf{\widehat{h}}^{\left(t\right)}_{j}}\right\|}_{2}}}}.$
(11)

It is worth noting that the standardization operation is added before the
activation function, which is given by

$\mathbf{y}={f_{\mathrm{std}}}\left({\mathbf{W}{\mathbf{\widehat{h}}^{\left(t\right)}_{\mathrm{st},j}}+\mathbf{b}}\right),$
(12)

where ${f_{\mathrm{std}}}\left(\cdot\right)$ denotes the standardization
operater with
${f_{\mathrm{std}}}\left(x\right)={x\mathord{\left/{\vphantom{x{\left({{{\mathds{E}}}\left({\sqrt{{\mathds{D}}\left(x\right)}}\right)}\right)}}}\right.\kern-1.2pt}{\left({\mathds{E}\left({\sqrt{\mathds{D}\left(x\right)}}\right)}\right)}}$.
In this letter, we collectively refer to the normalization in (11) and the
standardization operation in (12) as standardization. Then, the hidden layer
output matrix ${\mathbf{o}^{\left(t\right)}_{j}}$ is expressed as

${\mathbf{o}^{\left(t\right)}_{j}}=\sigma\left(\mathbf{y}\right),$ (13)

where $\sigma\left(\cdot\right)$ denotes the activation function such as
sigmoid, hyperbolic tangent (tanh), rectified linear units (ReLU) [15]. By
collecting
${\bm{\mathrm{O}}_{j}}=\left[{\bm{\mathrm{o}}_{j}^{\left(1\right)},\cdots,\bm{\mathrm{o}}_{j}^{\left({{N_{d}}}\right)}}\right]$,
the hidden layer output weight ${{\bf\bm{\beta}}_{j}}$ is obtained by

${{\bf\bm{\beta}}_{j}}={{\bf{H}}_{j}}{\bf{O}}_{j}^{\dagger}.$ (14)

####  III-B2 Online Deployment

For the online deployment, the LS estimation given in Eq. (8) is first
employed to obtain the initial features of composite CFRs. Then, we use Eq.
(9) to separate the CFRs of direct link ${\mathbf{\widehat{h}}_{\mathrm{TR}}}$
and reflecting link ${\mathbf{\widehat{H}}_{\mathrm{TRR}}}$. According to the
separated CFRs (e.g., ${\mathbf{\widehat{h}}_{\mathrm{TR}}}$ and
${\mathbf{\widehat{H}}_{\mathrm{TRR}}}$), we normalize the separated CFRs to
form ${{\mathbf{{{\overline{h}}}}}_{\mathrm{st},j}}$, i.e.,
${{{\bf{\bar{h}}}}_{\mathrm{st},j}}={{{{{\bf{\bar{h}}}}_{j}}}\mathord{\left/{\vphantom{{{{{\bf{\bar{h}}}}_{j}}}{\left({{{\left\|{{{{\bf{\bar{h}}}}_{j}}}\right\|}_{2}}}\right)}}}\right.\kern-1.2pt}{\left({{{\left\|{{{{\bf{\bar{h}}}}_{j}}}\right\|}_{2}}}\right)}}$.
And then the network output ${{\mathbf{\widetilde{h}}}_{j}}$ is obtained by
feeding ${{\mathbf{{{\overline{h}}}}}_{\mathrm{st},j}}$ into the trained ELM-
based network, which is given by

${\mathbf{\widetilde{h}}_{j}}={\bm{\beta}_{j}}\,.\sigma\left({{f_{\mathrm{std}}}\left({\mathbf{W}{{\mathbf{{{\overline{h}}}}}_{\mathrm{st},j}}+\mathbf{b}}\right)}\right).$
(15)

##  IV Simulation Results and Analysis

In this section, the normalized mean square error (NMSE) is applied as the
metric to compare the performance between the proposed method and the recent
novel methods, i.e., [2], [3] and [6] with the considerations of insufficient
CP and nonlinear distortion111We respectively employ the same architectures of
[2] and [3] for simulating their NMSEs. And the source code is available on
the authors link: https://github.com/meuseabe/deepChannelLearning4RIS. The
structure of this section is given as follows: The comparison of NMSE
performance is given in Section IV-A. Then, in Section IV-B, the robustness of
improvement with different parameters is discussed.

The basic parameters involved are listed below. The training sequence is
Zadoff-Chu sequence [6], $L=12$, $N=64$, ${L_{\mathrm{CP}}}=8$, and $M=8$. The
training and testing of the proposed method are carried out on a server with
Intel Xeon(R) E5-2620 CPU 2.1GHz$\times$16, and the results are obtained by
using MATLAB simulation on the server CPU due to the lack of a GPU solution.
And the offline training takes about 13 seconds, whereas the online deployment
of the proposed ELM network only needs 10 seconds. The frequency-selective
Rician fading channels are considered. In this letter, we consider the effects
of HPA, the nonlinear amplitude $A\left(x\right)$ and phase
$\Phi\left(x\right)$ are respectively adopted from

$A\left(x\right)=\frac{{{\alpha_{a}}x}}{{1+{\beta_{a}}{x^{2}}}}.{\rm{}}\Phi\left(x\right)=\frac{{{\alpha_{\phi}}{x^{2}}}}{{1+{\beta_{\phi}}{x^{2}}}},$
(16)

where ${\alpha_{a}}=1.96$, ${\beta_{a}}=0.99$, ${\alpha_{\phi}}=2.53$, and
${\beta_{\phi}}=2.82$ are considered in the simulations.

In addition, the intensity of nonlinear distortion is measured by error vector
magnitude (EVM), which is defined as

${\mathrm{EVM}\left(\%\right)=\sqrt{\frac{{\sum\limits_{n\in
N}{{{\left|{{{\widetilde{x}}_{n}}-{R_{n}}}\right|}^{2}}}}}{{\sum\limits_{n\in
N}{{{\left|{{R_{n}}}\right|}^{2}}}}}}},$ (17)

where ${\widetilde{x}_{n}}$ and ${R_{n}}$ are the distorted and undistorted
outputs of HPA given the same input, respectively. That is, for the same
input, ${\widetilde{x}_{n}}$ and ${R_{n}}$ are the outputs when HPA works in
the saturated region and linear region, respectively. Except for the
robustness analysis against EVMs, we set the basic value of EVM as ${55\%}$.

For simplicity, we use “Ref[2]”, “Ref[3]”, “Ref[6]” and “Prop” to denote the
reference method given in [2], [3], [6] and the proposed eELM-CE method under
nonlinear distortion with insufficient CP, respectively.

###  IV-A NMSE Performance Analysis

We add the nonlinear distortion and insufficient CP on the basis of [6], which
fully proves the effectiveness of the proposed method in terms of the NMSE
curves in Fig. 1. As shown in Fig. 1, the NMSE performance of “Ref[2]”,
“Ref[3]” and “Ref[6]” is much higher than that of “Prop”. It is noticed that
the proposed method has similar or better NMSE performance than “Ref[2]”,
“Ref[3]” and “Ref[6]” during all SNR range. This is mainly due to the
considerations of insufficient CP and the nonlinear distortion as well as the
application of enhanced ELM. Besides, different from the data-driven mode
adopted by “Ref[2]” and “Ref[3]”, the model-driven mode is employed by “Prop”
to exploit the advantages of both conventional algorithms and data-driven
mode. In addition, since improving the ELM network is the principal task, we
focus on the effectiveness of the proposed method, and the dedicated scenarios
in “Ref[2]” and “Ref[3]” are not explored.

Figure 1:  NMSE vs. SNR with insufficient CP.

###  IV-B Robustness Analysis

Usually, the NMSE performance of channel estimation is influenced by the
number of multi-path (i.e., $L$) and the degree of EVM. It is worth noting
that, besides the change of the impact parameter (i.e., $L$ and EVM), other
basic parameters remain the same as those given in Section IV.

Figure 2:  NMSE vs. SNR with different EVMs.

####  IV-B1 Robustness against EVM

EVM is usually used to measure the distortion intensity. To analyze the
robustness of the proposed method against different distortion intensities,
Fig. 2 plots the curves of NMSE with different EVMs (i.e., ${\rm{EVM}}=45\%$,
${\rm{EVM}}=55\%$ and ${\rm{EVM}}=65\%$). From Fig. 2, with the increase of
EVM, the NMSEs for all curves increase due to the rise of distortion
intensity. Besides, compared with “Ref[2]”, “Ref[3]” and “Ref[6]”, the “Prop”
method achieves the smallest NMSE for each given EVM. As a result, against the
impact of EVM, the “Prop” possesses its robustness for improving the
performance of NMSE.

Figure 3:  NMSE vs. SNR with different numbers of multi-path $L$.

####  IV-B2 Robustness against $L$

In this paragraph, explicitly refer to Fig. 3, it can be seen from the
simulations that as the degree of insufficient CP increases, the performance
of the NMSE deteriorates. For example, when ${L_{\mathrm{CP}}}=L-2$,
$\mathrm{SNR}=15\mathrm{dB}$, the reference methods [2], [3] and [6] start to
bend upward, but the performance is improved after using our proposed method,
which reflects the robustness of our proposed method against $L$.

##  V Conclusion

In this letter, a channel estimation method, eELM-CE is proposed to improve
the performance of the RIS-assisted OFDM system affected by the insufficient
CP and imperfect hardware. Based on the model-driven mode, LS estimation is
first employed as a feature extractor to extract the initial features. Then,
with the extracted initial features, an ELM network is developed to refine the
channel estimation. Noteworthily, the proposed ELM networks are enhanced by
using hidden-layer standardization to better capture the features of
distortion and thus enhance the performance. Extensive simulation results show
the effectiveness of the proposed method in relieving the influence of
insufficient CP and suppressing the nonlinear distortion. The robustness of
eELM-CE is also validated by its stable performance against parameter
variations. In this letter, the difficulty of obtaining desired labels is
simplified by generating them according to the existing channel model. In our
future works, we will consider the desired labels in real channel scenarios to
promote the application of ELM-based channel estimation in practical systems
(e.g., IoT, WLAN, etc).

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# Commissioning of the PADME experiment with a positron beam

P. Albicocco  R. Assiro  F. Bossi  P. Branchini  B. Buonomo  V. Capirossi  E.
Capitolo  C. Capoccia  A.P. Caricato  S. Ceravolo  G. Chiodini,11footnotetext:
Corresponding author G. Corradi  R. De Sangro  C. Di Giulio  D. Domenici  F.
Ferrarotto  S. Fiore  G. Finocchiaro  L.G Foggetta  A. Frankenthal  M.
Garattini  G. Georgiev  F. Giacchino,22footnotetext: presently at INFN sez.
Roma 2, via della Ricerca Scientifica 1, Roma A. Ghigo  P. Gianotti  F. Gontad
F. Iazzi  S. Ivanov  Sv. Ivanov  V. Kozhuharov  E. Leonardi  E. Long  M.
Martini  M. Martino  A. Miccoli  I. Oceano  F. Oliva  G.C. Organtini  F. Pinna
G. Piperno  C. Pinto  M. Raggi  F. Safai Tehrani  A. Saputi  I. Sarra  B.
Sciascia  R. Simeonov  T. Spadaro  S. Spagnolo  E. Spiriti  D. Tagnani  l.
Tsankov  C. Taruggi  P. Valente  A. Variola  E. Vilucchi

###### Abstract

The PADME experiment is designed to search for a hypothetical dark photon
$A^{\prime}$ produced in positron-electron annihilation using a bunched
positron beam at the Beam Test Facility of the INFN Laboratori Nazionali di
Frascati. The expected sensitivity to the $A^{\prime}$-photon mixing parameter
$\epsilon$ is 10-3, for $A^{\prime}$ mass $\leq$ 23.5 MeV/$c^{2}$ after
collecting $\sim 10^{13}$ positrons-on-target.

This paper presents the PADME detector status after commissioning in July
2019. In addition, the software algorithms employed to reconstruct physics
objects, such as photons and charged particles, and the calibration procedures
adopted are illustrated in detail. The results show that the experimental
apparatus reaches the design performance, and is able to identify and measure
standard electromagnetic processes, such as positron bremsstrahlung and
electron-positron annihilation into two photons.

##  1 Introduction

A new weakly-coupled $U_{D}$(1) hidden symmetry, with its corresponding dark
photon, has been proposed by several extensions of the Standard Model as a
possible explanation for the difficulty of detecting dark matter [1]. The
Positron Annihilation into Dark Matter Experiment (PADME), at the INFN
Laboratori Nazionali di Frascati (LNF) in Italy, searches for a possible dark
photon $A^{\prime}$ produced in the process $e^{+}e^{-}\rightarrow\gamma
A^{\prime}$, where the $A^{\prime}$ goes undetected [2]. A positron beam with
maximum energy of 550 MeV, produced and accelerated by the LNF LINAC, hits a
low-$Z$ target. The annihilation of a positron with an atomic electron in the
target can produce an $A^{\prime}$ with mass up to $\sim$ 23.5 MeV/$c^{2}$.
Under the assumption that the $A^{\prime}$ escapes detection, due to either
being long-lived or decaying predominantly via invisible modes, the production
signature is a single photon plus missing mass. PADME is the first experiment
to search for such a signal using this technique and would also be sensitive
to any new particle produced in $e^{+}e^{-}$ annihilation, including axion
like particles, dark Higgs, etc. This paper describes the apparatus and tools
developed to accomplish this program, and reports the achieved performance
during the early commissioning phase.

##  2 The BTF positron beams

The positron beam for the PADME experiment is provided by the LINAC of the LNF
DA$\Phi$NE complex. The LINAC can deliver bunched beams of electrons or
positrons with a maximum repetition rate of 50 Hz which can be sent, via a
dedicated beam line, to the experimental halls of the Beam Test Facility
(BTF).

For the PADME physics program, the LINAC must be operated in a dedicated mode
and with a specific setup of the BTF beam line: a bunch duration of hundreds
of ns, and an angular spread of few mrad. Two types of positron beam are
available at the BTF: a primary positron beam, produced at the LINAC positron
converter, or a secondary positron beam produced at a secondary target
installed on the BTF line [3][4].

For primary beams, positrons are generated at the LINAC positron converter
located downstream of the first accelerating sections and then transported to
the experimental halls. The bunch multiplicity can be varied between $10^{4}$
and $10^{10}$ particles/bunch with maximum energy of 550 MeV and energy spread
below 1%.

For secondary beams, a copper target is installed on the BTF transfer line.
The interaction of the primary electrons with the target produces a broad
distribution of particles both in energy and angle. These are then momentum-
selected and collimated, to produce a lower intensity beam, down to a single
particle per spill. In this configuration, the maximum energy available for
the positrons in the BTF halls is higher, since all the LINAC accelerating
sections are used, increasing the maximum energy by 220-250 MeV to a total of
770-800 MeV.

To obtain a higher beam energy, thus increasing the explorable $A^{\prime}$
mass range, a secondary positron beam of energy 545 MeV was used in the first
PADME runs of October and November 2018. The sensitivity obtained running with
this setup was discovered to be limited by the beam background in the
experiment. Therefore, only primary positron beam was used in the data-taking
runs of 2019 (this paper) and 2020 (runs with improved beam line) [5]. For
regulars runs, the primary beam energy was set at 490 MeV and 432.5 MeV,
respectively for the two periods, and not at 550 MeV, maximum energy, to avoid
LINAC instability at top energy.

##  3 The PADME detector

A sketch of the PADME setup is shown in Fig. 1. This layout is intended
primarily to measure the 4-momentum of the outgoing photons and of the
forward-emitted particles after the positrons interact with the target.
Assuming that the target electrons are at rest, the missing mass from
interactions with a single photon in the final state can be computed to search
for the $A^{\prime}$.

The main component of the PADME setup is a calorimeter consisting of two
systems: the Electromagnetic Calorimeter (ECal), which gives precise
measurements of the energy and impact point of all photons emitted in the
angular range 15-82 mrad with respect to the beam direction, and the Small-
Angle Calorimeter (SAC) which allows photons emitted below $\sim$15 mrad to be
vetoed. The SAC is most sensitive to background reactions such as
bremsstrahlung and $e^{+}e^{-}\rightarrow 3\gamma$ annihilation.

Any positron that does not interact in the target is directed towards a beam
monitor and the beam dump by a magnetic field, which also deflects secondary
charged particles towards a set of dedicated detectors. These detectors act
both as vetoes for background reactions and as spectrometers for forward
emitted particles.

Figure 1:  The PADME detector 3D CAD design (Left) and its schematic layout
from the top (Right). The positron beam enters from the right. The trajectory
of a non-interacting $e^{+}$ is drawn in black, and in black, the trajectories
corresponding to a $\gamma\gamma$/$\gamma A^{\prime}$ production in the
target. The particle trajectories from a bremsstrahlung interaction in the
target are drawn in red.

###  3.1 Active diamond target

A thin active diamond sensor not only provides the target for the annihilation
process, but also measures the XY profile of the beam and the beam
multiplicity, i.e. the number of positrons contained in each bunch. To have
the highest possible ratio of annihilation cross-section to bremsstrahlung
cross-section, the lowest possible atomic number Z is required. Out of a range
of solid materials considered for the target, diamond (Z = 6) was eventually
chosen because of its relatively straightforward manufacturing process. The
sensor has an active surface of 20$\times$20 mm2 and a thickness of 100
$\mu$m. Graphite strips with pitch 1 mm, including a 0.15 mm gap, were created
using a laser technique on both diamond faces [6]. Nineteen strips on one face
were oriented horizontally to measure the X view and nineteen strips on the
other face were oriented vertically to measure the Y view). Only 16 strips for
each views were connected to the electronic readout [7].

###  3.2 Electromagnetic calorimeter system

The ECal and the SAC were designed to study different types of interactions.
The main objective of the ECal is to detect photons produced in $e^{+}e^{-}$
annihilation. These include photon pairs coming from
$e^{+}e^{-}\rightarrow\gamma\gamma$ interactions as well as single photons
that may be produced along with the $A^{\prime}$ bosons (see Sec. 1). The ECal
is a high resolution detector composed of $616$ scintillating bismuth
germanate (BGO) crystals each of dimensions $21\times 21\times
230\,\leavevmode\nobreak\ \text{mm}^{3}$, arranged in a cylindrical shape and
readout by HZC XP1911 photomultipliers. It is placed $\sim$3.4 m downstream
from the target and the external radius is approximately $29\,\text{cm}$. The
resulting angular coverage ranges from $15.66\,\text{mrad}$ to
$82.11\,\text{mrad}$. The main component of its scintillation light has a
decay time of $\sim 300\,\text{ns}$. Thus, to avoid pile-up created by the
high rate of bremsstrahlung photons which are predominantly emitted in the
forward direction, the ECal was built with a hole of dimension 5$\times$5
crystals ($105\times 105\,\text{mm}^{2}$). The energy resolution of the ECal
was evaluated to be 2.62% at $490\,\text{MeV}$ [8].

The main purpose of the SAC is to veto photons from 3$\gamma$ annihilation and
bremsstrahlung ($e^{\pm}N\rightarrow e^{\pm}N\gamma$). It is placed behind the
central hole of the ECal and consists of a matrix of $25$ PbF2 crystals, each
measuring $30\times 30\times 140\leavevmode\nobreak\ \text{mm}^{3}$ and
coupled to Hamamatsu R13478UV photomultipliers. The readout signal from PbF2
crystals comes from Cherenkov light scintillation. The in-bunch bremsstrahlung
rate in this detector was expected to reach $\sim 10^{2}\,\text{MHz}$,
requiring a double-peak separation better than $5\,\text{ns}$. Typical signals
from the SAC are $2-3\,\text{ns}$ long, matching the requirement.

A beam test of a single SAC unit was performed at the BTF before the detector
was constructed. A single-particle electron beam with an energy between
$100\,\text{MeV}$ and $400\,\text{MeV}$ was fired onto the PbF2 crystal, to
study energy and time resolution, as well as the linearity of the detector
response. The measured energy resolution was below 20% across the entire
energy range, and the time resolution was 80 ps.

###  3.3 Charged particle detectors

Three stations of charged particle detectors are placed inside the vacuum
chamber of the experiment [9]. They are all composed of bars of polystyrene
plastic scintillator measuring $10\times 10\times 178\leavevmode\nobreak\
\text{mm}^{3}$, each equipped with a 1.2 mm2 diameter BCF-92 wavelength
shifter fibre (WLS fibre) along a longitudinal groove. The light from the WLS
fibre is detected by a $3\times 3$ mm2 Hamamatsu S13360 silicon
photomultiplier (SiPM). Four SiPMs are mounted on a custom-designed front-end
electronics card (FEE-card) providing a controllable high voltage. The card
also allows independent temperature and current monitoring for each channel.
Vacuum feed-through flanges ensure the communication and the signal
transmission between the FEE-cards and the custom designed controller modules.

Two of the charged particle detector stations, the positron veto (PVeto, 90
scintillator bars) and the electron veto (EVeto, 96 scintillator bars), are
located inside the dipole magnet and are read out from one side of the plastic
scintillators. The high energy positron veto (HEPVeto, 16 scintillator bars)
is placed downstream towards the positron beam exit window. Here the light
output is detected from both ends of each bar.

##  4 PADME data acquisition

###  4.1 Trigger and readout

The PADME data acquisition system [10] reads data from a total of 897 channels
with an acquisition rate of 50 Hz, defined by the LINAC operation frequency.
All channels are connected to a set of 29 CAEN V1742 [11] switched capacitor
ADC boards based on the DRS4 chip consisting of a 1024 cells switched
capacitor array (SCA) with a tunable 12-bit sampling rate. The relatively slow
signal produced by the BGO crystals and the target integral signal are sampled
at 1 GS/s, while the faster signals from the charged particle vetoes and the
small angle calorimeter are sampled at 2.5 GS/s. Two CAEN A3818 PCI Express
boards collect data from all ADC boards via 8 optical links, each with a
maximum throughput of 80 MB/s.

All ADC boards are interfaced with the main PADME Trigger and Timing System
(TTS). The TTS consists of a main trigger board and two trigger distribution
units. The trigger board can receive up to six NIM-level external triggers
(the most important being the BTF trigger from the LINAC, the cosmic ray
trigger for ECal calibration, and the random trigger).

The data acquisition process is organized into two stages: level 0 (L0), which
collects data from the ADC boards, and level 1 (L1), which merges all data
into a single event structure and produces the final data files.

With all sub-detectors enabled, optimal beam conditions, and the zero-
suppression algorithm set to flagging mode (i.e. no data are rejected) the
maximum data rate is $\sim$ 50 MB/s. All data produced by the PADME DAQ system
are written to a local disk buffer directly connected to the L1 servers. The
total size of the disk buffer is $\sim$ 38 TB, corresponding to approximately
ten days of continuous data-taking. The resulting RAW files are then copied
asynchronously to the off-line storage facilities of the experiment where the
reconstruction and analysis of the data are performed. Fig. 2 shows the PADME
DAQ system logical scheme.

Figure 2: Logical scheme of the PADME DAQ system.

###  4.2 Detector Control System

The PADME Detector Control System (DCS) is a standalone real-time threaded
application running on the DAQ server. It communicates with the different
hardware controllers and sensors of the experiment and continuously monitors
all relevant parameters. In case of channel current trips , the DCS can
recover it automatically. To guarantee that the DCS program is running
correctly, a dedicated daemon periodically checks its status and, in the case
that it has turned off, restarts it. The list of detector parameters to
monitor is passed to the DCS via a configurable text file that specifies the
IP address where each parameter can be read asynchronously via Ethernet. The
DCS also monitors the beam and ambient parameters.

The DCS continuously sends all collected information, along with respective
timestamps, to the Online Monitor. It also drives the alarm control system and
writes all monitored values to the local DCS database.

###  4.3 Online Monitor

Different monitoring tools have been developed to control the detector
operation and to check its stability during data-taking. The parameters to be
controlled fall into two categories: ambient parameters, and hardware settings
and detector responses. The two classes are managed by the different software
tools mentioned earlier: the DCS and the DAQ. Both generate data files that
are displayed via a web-based Online Monitor. The basic architecture of this
software is client-server and the communication is realized with the REST
protocol. The graphical characteristics of the plots and the values are
interpreted and graphed using the PlotlyJS library.

This system was used for the entire data collection period of the PADME
experiment showing excellent stability and scalability. The intrinsic
scalability of the system was a key factor in allowing geographically
distributed remote shifters to supervise the data acquisition during the 2020
run, when COVID-related travel restrictions prevented collaborators from being
able to travel to LNF to assist with data-taking.

###  4.4 Offline computing and data storage

To handle the data transfer from the DAQ system to the final storage area, a
Central Data Recording (CDR) software, named PadmeCDR, was developed. PadmeCDR
runs on a dedicated server connected with a 10 Gbps Ethernet link to the LNF
Tier2 infrastructure, part of the Worldwide LHC Computing Grid (WLCG), and
automatically transfers the DAQ data to the PADME tape storage area at the
tape library of the INFN CNAF national computing center [12].

The data are also automatically replicated on the local KLOE2 tape library at
LNF as a disaster-recovery safeguard. At the end of the second PADME data
taking, a total of $\sim$ 510 TB of data had been written to tape. The PADME
offline computing environment also includes PadmeProd, a custom software
package that handles the centralized production of simulated events and the
reconstruction of all the data, real and simulated, of the experiment using
dedicated WLCG resources located at LNF and CNAF. The distributed CVMFS file
system [13] was used to create a storage area at CNAF from which all WLCG
sites can access the PADME simulation and reconstruction software. A Virtual
Organization (VO), named vo.padme.org, was created to handle the WLCG
authorization procedures through the Virtual Organization Management System
(VOMS) at CNAF.

##  5 PADME offline software

Data processing and Monte Carlo (MC) simulation use a common software
framework. The data format and persistency mechanisms are based on the ROOT
[14] and Geant4 [15] libraries. The raw event consists of the full set of
digital signals recorded by each channel after zero suppression, along with
trigger information (see Sec. 4.1).

The reconstruction software for each detector converts the raw data into
collections of reconstructed hits, after applying decoding procedures based on
configuration data to map the electronic channels into the physical elements
of the PADME detector. Reconstructed hits contain information about position,
time and energy, corrected for alignment and calibration constants (as
described in Sec. 6). During reconstruction, hits are processed with a
clusterization logic that creates candidate objects to represent the signature
of particles in the detectors. In the ECal and SAC, these are electromagnetic
clusters from photons, while in the veto detectors they are clusters of
neighboring hits from short tracks of charged particles.

The data from the target are processed differently. The summary features of
the beam profile are the position of the charge centroid, the width of the
distribution in the X and Y view, and the calibrated number of positrons in
the bunch, inferred from the total charge released in the detector.

The output format of simulated events consists of collections of Geant4 hits
for each detector, with position, time, channel identifier and true energy
released in the active element of the detector, and the corresponding
collections of digitized information. It contains the detector channel hits
after some processing aiming at emulating the finite resolution of the
detectors and their readout electronics. The reconstruction software converts
the digits into collections of reconstructed hits using the same format as for
real data, by applying further grouping and smearing in order to best match
the detector response. Afterwards, the same clusterization and beam
reconstruction algorithms used for real data are also applied to simulated
events.

###  5.1 Physics processes and detector simulation

The PADME simulation feeds a description of the geometry and materials of the
experiment to Geant4 for the emulation of all particle interactions. This
includes, in addition to the active detectors with their inert support and
wrapping materials, the magnet; the vacuum chamber; the beam pipe, starting
from a window (made of beryllium in 2019 and Mylar in 2020) that separates the
PADME line from the upstream accelerator beam line; and the last bending
magnet of the beam before the target. A typical background simulated event
consists of approximately 2$\times 10^{4}$ beam positrons, with the time
structure of a typical beam bunch seen by PADME. The simulation accounts for
the main electromagnetic processes occurring in the interactions of beam
positrons in the target at PADME. These processes are bremsstrahlung, two-
photon production and Bhabha scattering. Simulating the beam line enables an
accurate reproduction of the main features of the beam-induced background.

For signal acceptance studies, single interactions corresponding to a specific
process are generated either by external Monte Carlo generators, in the case
of two- and three-photon interactions and other selected electromagnetic
processes, or by an internal generator, for example to simulate the production
of an invisible dark photon in association with a photon. The final-state
particles produced by the generator are injected into the PADME simulation in
a production vertex within the target volume and the propagation of one
positron of the beam pulse is stopped in the target. The resulting simulation
of the signal interaction is fully overlapped with the pile-up from background
electromagnetic interactions of the other beam particles.

###  5.2 Single- and multi-hit reconstruction

The front-end signals of all detectors are uniformly sampled inside a readout
window, when a trigger signal is received. This window is 1024 ns long for the
target and the ECal, and 409.6 ns long for the SAC and the vetoes (see Sec.
4.1). The ADC ranges from 0 to 4095 counts with a voltage range of [-0.5, 0.5]
V, for bipolar signals, and [0,1] V, for unipolar signals.

To reduce the amount of processed data, zero-suppression is applied to the
vetoes and the ECal, in which waveforms with standard deviation less than 5 mV
are not considered for hit reconstruction. In all detectors, a pedestal is
calculated on an event-by-event basis from the average of the first few
hundred sampled values, before the arrival of the trigger (the exact number
changes for each detector), and is then subtracted from each waveform. If a
negative polarity signal is obtained, a sign change is applied to the
pedestal-subtracted waveform. Finally, the voltage is converted to charge in
units of pC.

The hits are reconstructed from each ADC waveform using different algorithms
according to the detector characteristics. For the BGO crystals in the ECal,
overlapping hits in single channels were observed during commissioning, and
therefore a multi-hit reconstruction algorithm was developed based on a signal
template. Hit times and energy deposits are determined by evaluating the
derivative and the integral of the corresponding shifted and scaled templates,
respectively. To obtain the corresponding energy value a nominal conversion
factor of 15.3 pC/MeV (corresponding to 8 photoelectrons per keV and a PMT
nominal gain of 1.1$\times$104) is applied. The energy may be corrected for
fractional charge-loss due to the signal tail extending beyond the readout
window, and for signal saturation. An energy threshold of 1 MeV is applied to
the ECal hits.

The SAC crystals and the veto scintillating bars can have multiple hits per
event. The time of each hit is therefore determined using a 1D peak finder
algorithm based on the TSpectrum class of ROOT with a width of 0.8 ns for the
SAC and 2.4 ns for the vetoes. Each peak inside a fiducial time window is then
associated with a hit, the time of which is given by the peak position. The
SAC hit energy is calculated by integrating the charge over a time interval of
4 ns centered on the peak and converted to MeV with a nominal conversion
factor of 0.1 pC/MeV (corresponding to 2 photoelectrons per MeV and a PMT
nominal gain of 3.2$\times$105). The veto hit energy is obtained from the peak
maximum and converted into MeV with a nominal conversion factor of 25 mV/MeV,
using the fact that a minimum-ionizing particle (MIP) crossing a 1 cm
scintillator releases approximately 2 MeV of energy.

###  5.3 Photon and charged particle reconstruction

The identification of physics objects, such as photons and charged particles
entering a detector, is performed by clustering the hits which are close in
time and in space. Appropriate energy thresholds and time differences have
been defined for each detector (ECal, SAC, vetos) with MC simulation studies
and prototype tests. For each detector, the clustering algorithm starts from
the list of hits not associated with a previous cluster and with the most
energetic hit used as the seed of the new cluster. For the SAC and the vetoes,
hits are not associated with a cluster if there is at least one missing hit
between them and the seed along any direction. The energy of the cluster is
given by the sum of the energies of all participating hits and the time and
the position by the energy-weighted time and position of the hits. A detailed
description of the different clustering algorithms developed for the ECal can
be found in Ref. [16].

##  6 In-situ detector calibrations

The absolute calibration of detector response and its time alignment is a
vital step towards the reconstruction and measurement of physical phenomena.
Being an active element, the target also needs to be calibrated, which is of
crucial importance for the determination of the cross section of any physical
process.

A good energy calibration is fundamental for the ECal to precisely define the
missing mass in dark photon events (of a tenth of MeV), and for the SAC and
PVeto to match measured photons with the correct positron to detect
bremsstrahlung processes (of few tenths of MeV). Finally, a good time
calibration (of few ns) between the various sub-detectors is also needed to
correctly group particles coming from the same interaction.

###  6.1 Target calibration

The diamond target was calibrated by equalizing the response of each amplifier
and comparing the amount of charge collected by different strips. Each
amplifier is equipped with an internal charge injection circuit, which allows
the front-end to be tested and the gain to be measured. The charge injected is
shared between all the channels selected by an internal shift-register, making
channel-by-channel calibration possible. Calibration factors for each channel
were inserted in the calibration service of the PADME reconstruction software
to estimate the charge collected by the strips (Fig. 3, left).

The collected charge per bunch is one of the most important pieces of
information to extract, because it is expected to be proportional to the bunch
multiplicity. An absolute calibration of the diamond target was performed
using dedicated runs with different multiplicities of the positron beam as
measured by a monitoring lead-glass Cherenkov calorimeter available at the
BTF. At each calibration step, the beam was sent to the BTF calorimeter by
turning off the last bending magnet, after which the magnet was turned back
on, the beam was sent back to PADME and the target response was recorded.
Several runs at different bunch multiplicities were taken in order to cross-
calibrate the target. In Fig. 3, right, the average of the total charge
collected by the X and Y views is shown as a function of the multiplicity
measured by the BTF calorimeter. The slope of the linear fit was used to
define the absolute calibration factor to be used in the analysis software.

Figure 3:  Left: distribution of the charge gains of the target front-end
channels measured using the internal charge injection circuit. Right: active
diamond target total charge response to different positron bunch
multiplicities.

###  6.2 Charged veto calibration

Gain calibration constants of each channel are derived from data and then used
to perform offline corrections of any differences in the readout hardware of
the various detector elements. In the gain equalization of the charged veto
channels is important to apply a uniform threshold and to build clusters of
adjacent hits corresponding to MIPs crossing the detector with an angle, and
in the case of pile-up to distinguish between multiple charged particles
crossing.

The relative calibration constants were obtained by evaluating the
distribution of the hit energy for each electronic channel of the vetoes,
after setting a very low threshold for the hit reconstruction. A Gaussian fit
around the peak was used to estimate the most probable value of the pulse
height distribution averaged with respect to crossing angle and the impact
point of the MIP (Fig. 4, right). The relative calibration constant for each
channel was computed by scaling the most probable pulse height to 50 mV (Fig.
4, left) and applying a scale factor of 34 MeV/V, to obtain a peak energy of
1.7 MeV. This is the expected energy deposit of a MIP crossing 1.1 cm of
plastic scintillator.

Figure 4:  Left: distribution of relative gain of charged veto scintillating
bars measured with MIPs. Right: hit amplitude distribution of a positron veto
scintillating bar measured with positrons in time coincidence with the SAC
(bremsstrahlung positrons).

###  6.3 Time calibration

The time calibration of the front-end channels of downstream detectors
(calorimeters and vetoes) is necessary to prevent hits from different positron
interactions being assigned to the same cluster, which would result in a poor
measurement of energy and position.

A strong time correlation exists between adjacent channels of the same
detector, because of electromagnetic showers produced by charged particles
crossing more than one scintillating bar or crystal. A statistically
significant time correlation between channels of different detectors only
exists if they are produced by the same primary interaction. The data show
clear time correlations between each pair of detectors except the EVeto and
the SAC. This is probably due to the fact that the occupancy of the EVeto
arises from open-angle primary interactions, such as charged pair-production
and showers. Thus, there are few primary interactions expected to produce
signals in both the EVeto and the SAC. Because of the lack of correlation
between these two detectors, this distribution is not shown in Fig. 5.

The calibration is performed to determine the time offset t${}_{i+1,i}^{D}$
between channel i and the adjacent channel i+1 of detector D. Then, the time
offset t${}_{i+1}^{D}$ with respect to reference channel 0 is calculated by
summing the relative calibration constants t${}_{k+1,k}^{D}$ of all previous
channels k < i to align detector D in time. The final time alignment is
obtained by measuring the global time offset for each detector using all hit
correlations between PVeto - ECal, EVeto - ECal and SAC - PVeto. Fig. 5,
right, shows the time differences between clusters of different detectors
after the calibrations are applied. All distributions have a peak centered at
zero (when the physical object comes from the same interaction) above a
triangular uncorrelated background (from pile-up events and uncorrelated beam
background).

Time differences in detectors have two causes: variation in time-of-flight of
particles originating from the same interaction, and different cable lengths
or electronics module delays (see Fig. 5). To distinguish the contributions to
time differences observed in data, the same procedure was applied to simulated
events. This resulted in a second set of time calibration constants reflecting
only the difference in time-of-flight of particles, which was subsequently
applied to the reconstruction of data.

Figure 5: Distributions of time differences between different detector
channels after time calibration during data-taking with approximately 23000
positrons per bunch.

###  6.4 ECal calibration

Before mounting the calorimeter, all ECal scintillating units (SUs) were
calibrated with photons emitted by a 22Na radioactive source [8]. This was
done to determine the charge vs HV curve, to set the voltage of each unit to
produce the same gain. During all the data-taking, the HVs were set to have a
charge output of $15.3\,\mbox{pC/MeV}$. This value allows signals up to
$1\,\mbox{GeV}$ to be fully contained in the digitizer dynamic range.

To validate the 22Na calibration and assess and improve the channel
equalisation, cosmic ray events were used. During all the data-taking, the
cosmic ray trigger (see Sec. 4.1) was used to check the calorimeter response
to MIPs. Fig. 6, left, shows the distribution of the most probable values from
a Landau fit of all SUs charge distributions, resulting from three days of
cosmic rays at a rate of few Hz. The superimposed Gaussian fit shows that the
ECal response is uniform within ($10.99\pm 0.48)\%$. From the same dataset,
the detection efficiencies of the SUs were evaluated: $88.4\%$ of the units
showed an efficiency greater than $99.5\%$ [8].

Finally, a special run was performed with a single-positron beam of
$490\,\mbox{MeV}$ fired directly onto the calorimeter. A $5\times 5$ cluster
around the crystal with the largest energy deposit was considered and the
multi-hit ECal algorithm based on signal template applied. In Fig. 6, right,
the obtained energy spectrum is shown. The various multiplicity peaks are
clearly visible up to seven primary positrons entering the electromagnetic
calorimeter, corresponding to a total energy of 3.4 GeV. In normal running
conditions the number of hits per crystal per event does not exceed two.

Figure 6: Left: MPVs (Most Probable Values) distribution of all Ecal SUs. The
MPVs are obtained fitting each SU charge distribution with Landau curves.
Right: distribution of ECal $5\times 5$ cluster energies induced by a single-
positron beam of $490\,\mbox{MeV}$ energy and reconstructed with the ECal
multi-hit algorithm.

###  6.5 SAC calibration

The SAC is a Cherenkov-based detector, and therefore its calibration cannot be
performed with the same tools used to calibrate ECal crystals. The main
difference between the two detectors is in the light output: the light-yield
of PbF2 is only $\sim$ 2 photo-electrons/MeV, compared to $\sim$ 200 photo-
electrons/MeV for BGO. Hence, the calibration of the SAC was carried out using
the BTF positron beam and cosmic rays.

A first set of calibration constants was obtained using beams of single
positrons. By modifying the magnetic fields of the BTF beam line magnets, a
545 MeV single-positron beam was steered to strike the crystals of the SAC.
Since the dimensions of the SAC are larger than the ECal hole, only 9 out of
25 crystals were reachable by the beam.

In previous test beams, performed on single PbF2 units before detector
assembly, the number of photo-electrons and the photomultiplier gain [17] were
found to be $N_{p.e.}$ $\sim$ 2 and $G_{i}$ $\sim$ 3.2$\times$105,
respectively.

In data, the collected charge was evaluated with a Gaussian fit of the
signals. Combining this information with the parameters found during the test
beams, a set of calibration constants was obtained.

A second set of calibration constants was determined using cosmic rays. The
energy released in PbF2 by MIPs is 9.373 MeV/cm [18]. Since the short side of
a SAC unit is 3 cm long, the energy released by a cosmic ray crossing a
crystal perpendicularly is approximately 28.11 MeV. Only vertical cosmic rays
crossing all 5 crystals in a column were considered in this analysis. Charge
distributions for each crystal were fitted with a Landau curve, and the MPVs
from the fit were used to calculate a second set of calibration constants.

The two sets of calibration constants are plotted in Fig. 7, left. The results
of the two independent measurements are compatible within errors. From this
work, it was also seen that one of the central crystals (n. 12) was less
efficient than the others. The reason was identified in a bad optical contact
and was corrected before the data taking of 2020, which is not used for the
work presented in this paper. In Fig. 7, right, the cluster energy for the 9
central crystals is shown before and after the calibration. After calibration,
the position of the signal peak is shifted towards the energy of the beam (545
MeV) and the energy resolution improves from 11% to 6.7% [19].

Figure 7: Left: SAC calibration constants obtained from single-positron beam
measurements (orange) and from cosmic ray measurements (blue). Right: SAC
cluster energy distribution of a 545 MeV single-positron beam, before and
after calibration.

###  6.6 Momentum calibration

Thanks to the detailed map of the magnetic field, the veto calibration curve
of momentum vs position can be reliably extracted from simulations. From 50
mono-energetic simulated samples of single positrons, the distribution of
cluster z-positions is determined and the momentum calibration curve and
resolution are extracted from the average and width of the Gaussian fits. Fig.
8 shows the calibration points obtained from the simulations with the
following curve fit superimposed:

$p\,(z)=\frac{0.3B\left[{(z+z_{0})}^{2}+x^{2}\right]}{2x},$ (6.1)

where $p$ is the positron momentum, $z$ is the crossing point coordinate on
the veto detector and $x$ is the lateral distance of the veto detector from
the beam axis (192.5 mm). The magnetic field has been mapped precisely in 3D
by a dedicated survey, and the excitation curve was also measured in a subset
of points. Since the real magnetic field map is used in the simulations, the
fit parameters $z_{0}$ and $B$ of the above simple analytical model, which is
exact only for an ideal magnetic field $B$ constant for $z>-z_{0}$, provide an
effective length and a field that reproduce the calibration data points well.

Figure 8: Momentum of a positron emerging orthogonally from the target versus
impact position along z on PVeto (left) and HEPVeto (right) as obtained from
single-positron simulations with a magnetic field of 0.408 T.

##  7 Commissioning with primary positron beam

The data collected in July 2019 with a primary positron beam represented the
completion of the commissioning of the experiment. Two-photon annihilation
processes were clearly visible in the ECal, in addition to the bremsstrahlung
processes in the plot correlating PVeto and SAC signals. However,
bremsstrahlung events in which the photon goes into the ECal instead of the
SAC were only visible in the 2020 data taking with primary beam and the new
beam line.

In 2019, a primary positron beam of energy 490 MeV and bunch length of 150 ns
was used. It was focused to a spot of approximately 2-3 mm in X and Y
directions on the target, as can be seen in Fig. 9 showing the measured beam
profiles.

Figure 9: Beam profiles along the X (left) and Y (right) directions as
measured by the active diamond target during the July 2019 data-taking. Beam
energy and the total number of particles are indicated.

The typical positron multiplicity was $\sim$23,000 positrons/bunch (see Fig.
10), corresponding to a density of approximately 150 $e^{+}$/ns. This was
slightly above the limit imposed by the maximum acceptable event pile-up,
which was ascertained to be $\sim$100 $e^{+}$/ns, after a careful analysis
performed during beam commissioning for 2020 data taking.

Thanks to a remote controlled positioning motor, the target can be moved to
the “parked" position where it does not interact with the beam. Runs without
target were used to study the beam background arising from beam interactions
with different materials (seen by the PADME detectors). During the
commissioning phase the comparison of data with and without target was a
powerful tool to investigate the origin of the observed background. The two
datasets with and without target consist of 400,000 events each.

###  7.1 Detector occupancy

With the beam parameters previously mentioned, the distribution of the number
of clusters for each detector is shown in Fig. 10. The number of positrons
striking the PVeto is relatively high, while the number of electrons striking
the EVeto is very small. The rate of the photons in the two calorimeters are
very different, with ECal registering a much lower rate than SAC, as expected
from the hole in the centre of the ECal. The ECal photon flux is dominated by
beam background, while the SAC photon flux is mostly due to positron
bremsstrahlung in the target.

To better understand the occupancy of the detectors, in Fig. 11 the
distributions of the cluster sizes are shown. The ECal cluster size is quite
large, but a 5x5 matrix is adequate for the clusterization algorithm. The
average cluster size in the PVeto is similar to the EVeto but larger than that
of the HEPVeto. This reflects the position and angular orientation of the
experiment, since the HEPVeto can only be struck by positrons that are almost
orthogonal to the detector surface.

Figure 10: Left: distribution of the positron bunch multiplicity as measured
by the active diamond target. Right: distribution of the number of clusters
for the different detectors. The data correspond to a beam bunch multiplicity
of approximately 23,000 positrons per bunch.

Figure 11: Distribution of the cluster size for charged vetos and SAC (left)
and ECal (right) during data-taking with approximately 23,000 positrons per
bunch.

###  7.2 Photon spectra in the ECal

The energy distribution of the clusters reconstructed in the ECal has been
studied and compared with simulation (see Fig. 12). To evaluate the beam-
induced background, the spectrum measured in no-target runs is also shown. The
comparison with the simulation shows good agreement for energies above 30 MeV
while the low energy region is dominated by a background component that is
also observed in the data collected without target. The source for this
background is the large number of low-energy photons produced by unwanted
interactions of the beam with beamline and detector materials, a phenomenon
that was not predicted during the experimental design.

The most important reference process for PADME is the annihilation of a
positron with an electron in the target producing a pair of photons. The high
rate of bremsstrahlung photons in the forward direction and the coarse
granularity of the SAC prevent a clear identification of this process in this
detector. Therefore the search for $e^{+}e^{-}\rightarrow\gamma\gamma$
annihilation is performed using only the ECal detector, where the
bremsstrahlung background is quite suppressed with respect to the SAC. The
search exploits the two-photon time coincidence and the conservation of
momentum in the transverse plane. A simple event selection is implemented by
applying the following requirements, without any attempt to optimize the
acceptance or the ratio of signal yield to background:

  * •

Two ECal clusters in time coincidence $\Delta t=\mid
t_{Cl_{1}}-t_{Cl_{2}}\mid<3\ \rm{ns}$;

  * •

The energy-weighted position of the di-photon pair compatible with the beam
direction: $\mid CoG_{x}\mid<1\leavevmode\nobreak\ {\rm cm}$ and $\mid
CoG_{y}\mid<1\leavevmode\nobreak\ {\rm cm}$ 333The energy-weighted centroids
are defined as CoG${}_{x(y)}=\frac{x(y)_{cl_{1}}\cdot
E_{cl_{1}}+x(y)_{cl_{2}}\cdot E_{cl_{2}}}{E_{cl_{1}}+E_{cl_{2}}}$ where
$x(y)_{cl_{1(2)}}$ is the x(y) coordinate of the first (second) ECal cluster
and $E_{cl_{1(2)}}$ its energy.;

The distribution of the sum of the energies of the clusters that satisfy the
above requirements is shown in Fig. 12. The data show a well defined peak at
the beam energy, which is indicative of the annihilation process. The data
also seem well modeled by the simulation. The same distribution obtained for a
run without the target is superimposed. As expected in this distribution, the
annihilation peak is not present because annihilation can only occur in the
target. Furthermore, the smooth two-photon shoulder matches the shape of this
distribution well, indicating that it is almost completely due to beam
background coming from interactions of the beam with beam line infrastructure
and other materials.

Figure 12:  Energy of the ECal clusters (left) and energy sum of two ECal
clusters in time coincidence with clean-up cuts as explained in the text
(right). The data of a standard run with target (black points) are compared to
data of a dedicated run without target (blue points) and to simulation with
the 2019 beam line (red line). The distributions are normalized to the same
number of POT equalizing the distribution of out-of-time photons. The data and
simulation correspond to a beam bunch multiplicity of $\sim$23,000 positrons
per bunch.

###  7.3 Charged particle spectra

Charged particles coming from the interaction region with energy less than the
beam energy are bent towards the three veto detectors. In Fig. 13 the particle
profiles are shown as a function of scintillating bar number for a beam of
approximately 23,000 e+/bunch, with and without target. The normalization of
the run without target is obtained by imposing the same occupancy of the first
ten scintillator bars of PVeto, where no contributions from interactions in
the target are expected. The source of this low-energy charged particle
background was removed before 2020 data taking with a more focused beam and
larger beam-transport pipes.

For PVeto bars with high channel numbers (those furthest from the target), the
rate is very high and the detector is not able to provide precise information.
Here, the high pile-up probability and the large cluster size due to the angle
of incidence of the particles on the detector generate many hits in the same
bars that cannot be separated in time. In this region the hits are dominated
by bremsstrahlung interactions in the target and by primary positrons that
have slightly less energy than the beam core and that are bent at a larger
angle. This beam energy spread is mainly caused by the energy degradation of a
small fraction of positrons undergoing bremsstrahlung interactions in the
beryllium window separating the LINAC and the PADME vacuum. This effect was
significantly reduced with the new beam line installed in 2020.

The difference in rate with and without the target in the intermediate
scintillating bars, approximately between numbers 10 and 70, is due to the
contribution of hard bremsstrahlung interactions in the target [20]. This
corresponds to the momentum range between 50 MeV and 280 MeV, as seen in Fig.
13.

The HEPVeto detector acceptance partially overlaps that of the last bars of
the PVeto. Thus, the HEPVeto extends the positron veto acceptance by
overlapping the region of low efficiency of the PVeto. The HEPVeto rates with
and without target have a different shape for the first scintillating bars,
and a more similar shape in the last bars. This provides a measurement of the
real positron flux on the PVeto scintillator bars with high numbers. The
HEPVeto orientation is almost orthogonal to the positrons, and HEPVeto
clusters typically contain relatively few hits, avoiding the saturation
occurring in the PVeto. Consequently, the difference in rates of the HEPVeto
with and without target are an estimate of the number of positrons interacting
in the target.

As expected, the EVeto rates are much lower than that of the PVeto since
electrons must be created either by rather rare primary interactions or in
electromagnetic cascades. In addition, the EVeto occupancy is similar with and
without target, an indication that it is dominated by beam background
interactions.

The momentum calibration curves from Fig. 8 can be used to convert the
distribution of the charged particles’ position along the veto detectors into
positron and electron momentum spectra as shown in Fig. 13. Importantly, the
momentum is only accurate for forward particles emerging from the target with
low transverse momentum. In the same plot the difference between target and
no-target spectra is shown for PVeto and HEPVeto, where it is possible to
estimate the rate of positron bremsstrahlung from the target in the range from
50 MeV to 290 MeV, for the PVeto, and from 350 MeV to 390 MeV, for the
HEPVeto. This is important since it enables the calibration of the
experimental luminosity with another physical process in addition to two-
photon annihilation. The PVeto has a cutoff energy below 50 MeV and excessive
occupancy above 290 MeV, while the HEPVeto only begins to saturate above 390
MeV.

Figure 13: Left: charged particle profiles for positron, high energy positron
and electron vetos with and without target. Right: corresponding momentum
distributions with target data for all vetoes and with no-target data
subtracted for PVeto and HEPVeto. The data correspond to a beam multiplicity
of $\sim$23,000 positrons per bunch.

###  7.4 Photon and charged particle correlation

The full detection of positron bremsstrahlung processes occurring in the
target ($e^{+}Z\rightarrow e^{+}\gamma Z$) is relevant to perform high
statistics studies of the correlation between photons and charged particles,
which can be used to set the SAC energy scale and to measure photon detection
efficiency. Fig. 14 shows scatter plots of the sum of photon energy and
positron energy with respect to positron energy in a time coincidence window
of 1 ns. In the PVeto scatter plot the positron bremsstrahlung signal is
clearly visible. This is more difficult to observe in the HEPVeto because of
the lower relative energy resolution of the SAC for low energy photons, and to
the high beam background flux.

Figure 14: Left (right) Scatter plot of the sum of the SAC photon and PVeto
(HEPVeto) positron energies against PVeto positron energy evaluated in 1 ns
time-coincidence and for SAC photon energy above 100 MeV (50 MeV). The data
correspond to a beam multiplicity of $\sim$ 23,000 positrons per bunch.

##  8 Conclusions

PADME is the first fixed-target experiment searching for a dark photon signal
by computing the missing mass of single-photon final states resulting from the
annihilation of a pulsed positron beam with the electrons of a fixed target.
The collaboration collected several months’ worth of data during 2018 and 2019
with stable secondary and primary positron beams (Run I), which have been
analyzed to assess the experimental performance and the beam-induced
background level. Thanks to the detailed analysis performed on this first
dataset, a long and stable data-taking period with primary beam and a new beam
line was performed in 2020 (Run II). These Run II data have lower beam related
background and they are now being studied to produce the first physics results
and to establish final detector performances.

In this paper we presented an analysis of the data collected in July 2019 for
commissioning the old beam line with a primary beam, and discussed the
calibration procedures and the experimental status. Several detector
characteristics have been measured, such as acceptance and energy and time
resolutions. In addition, the simulation software tools, physics object
reconstruction algorithms, detector calibration methods and data quality
strategy have all been extensively documented.

The active diamond target has operated continuously and stably in vacuum since
September 2018. It fulfills its design goals of providing the beam position
resolution with better than 1 mm precision for 20,000 positrons/bunch and to
measure the integrated luminosity at the level of a few percent after cross-
calibration with an external electromagnetic calorimeter.

The electromagnetic calorimeter, which is made of BGO crystals, has excellent
energy resolution and an acceptance better than 99.7% for a 10 MeV mass dark
photon signal. In Run II a temperature-dependent single-crystal energy
calibration was implemented to improve and stabilize the energy resolution for
long periods data-taking.

The charged particle scintillating bars attained high efficiency in the
detection of MIPs, very good time resolution and good double-hit separation
capability. This level of performance was reached thanks to fast front-end
electronics, channel gain equalization, and a full multi-hit reconstruction
algorithm.

Finally, with a multi-hit reconstruction algorithm for the Small-Angle
Calorimeter, the PbF2 crystals proved to be efficient for the expected high
photon flux. In fact, each crystal was capable of detecting several tens of
photons inside a positron bunch of 150 ns length.

The two-photon annihilation process is clearly visible in data collected with
the primary positron beam in the high resolution electromagnetic calorimeter
after loose time and angular selections. This is the standard candle for the
PADME detector from which it is possible to demonstrate the spatial uniformity
of the detector and its stable response over time. The bremsstrahlung signal
can be seen in both datasets collected with primary and secondary positron
beams after accurate time-alignment of the electromagnetic calorimeter and the
positively charged particle veto.

The beam background observed in the data is caused by electromagnetic showers
induced by beam halo positrons striking the beam pipe or the finite apertures
of the magnet. These particle splashes produce lower energy charged particles
and photons with large angular spread in the PADME detectors. Another
important contribution to beam background comes from positrons radiating a
photon of several MeV while crossing the vacuum-separator beryllium window. To
handle this background it is necessary to develop efficient multi-hit
reconstruction algorithms for all detectors and to perform good time
calibrations.

The commissioning process was crucial to prove that the detector design goals
have been reached, to optimize the hardware and software tools, and to achieve
good agreement between simulation and data both for physics signals, such as
bremsstrahlung and two-photon annihilation, and for beam-induced backgrounds.
This activity was necessary to address the construction of the new beam line
and to establish the position of the new Mylar vacuum separation window for
Run II, replacing the beryllim window used in Run I.

##  9 Acknowledgments

We warmly thank the BTF and LINAC teams of LNF for providing an excellent
quality beam and for their full support during the data-taking period. This
work is partly supported by the Italian Ministry of Foreign Affairs and
International Cooperation (MAECI) under the grant PRG00226 (CUP
I86D16000060005), the BG-NSF KP-06-DO02/4 from 15.12.2020 as part of MUCCA,
CHIST-ERA-19-XAI-009, and TA-LNF as part of STRONG-2020 EU Grant Agreement
824093 projects.

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