#  Effect of color reconnection and hadronic re-scattering on underlying
events in pp collisions at LHC energies.

Krishna<EMAIL_ADDRESS>and Sadhana
<EMAIL_ADDRESS>
Indian Institute of Science Education and Research Thiruvananthapuram, Vithura
695551, India
Department of Physics, Indian Institute of Technology Bombay, Mumbai 400076,
India

###### Abstract

Underlying events dominate most of the hadronic activity in p$-$p collisions
and are spanned from perturbative to non-perturbative QCD, having a
sensitivity ranging from the multi-scale to very low-x scale physics. A
detailed understanding of such events plays a crucial role in the accurate
understanding of Standard Model ()SM and Beyond Standard Model physics. The
underlying event activities has been studied within the framework of Pythia 8
Monte Carlo model, considering the underlying events observables mean charged
particle multiplicity density , $\langle d^{2}N/d\eta d\phi\rangle$ and mean
scalar $p_{T}$ sum, $\langle d^{2}\sum p_{T}/d\eta d\phi\rangle$ as a function
of leading charged particle in towards, away, and transverse region of p$-$p
collisions at $\sqrt{s}$ = 2.76, 7 and 13 TeV. The towards, away, and
transverse regions have been defined on an azimuthal plane relative to leading
particle in p$-$p collisions. The energy dependence of underlying events and
their activities in the central and forward region has also been studied. The
effect of hadronic re-scattering, color reconnection, and rope hadronization
mechanism implemented in Pythia 8 has been studied in details to gain insight
into the different processes contributing to underlying events in soft sector.

##  1 Introduction

A quintessential proton-proton(p$-$p) collision can be sub-divided into two
components: the primary hard partonic scattering and other associated activity
collectively termed as underlying events, UE. The cross-section of
interactions involved in primary hard-scattering processes can be determined
using perturbative QCD (pQCD) calculations as they involve sufficiently large
momentum transfer and, therefore the strong interaction coupling constant is
small. On the contrary, the production mechanism of underlying events occur at
a lower momentum scale. In order to understand the recent Standard Model
measurements at LHC and search for physical phenomena beyond Standard Model
(BSM), it is crucial to have a detailed understanding of the underlying events
in p$-$p collisions [2, 3].
Experimentally, the underlying events can not be distinguished from the
primary hard scattering process on an event by event basis, and one can not
uniquely determine the origin of the final state hadrons. Therefore, the
observables generally chosen to study the underlying events receive
contributions from both hard scattering and underlying events. However, one
can utilize the topology of hadron-hadron collisions to understand the
accompanying interactions in p$-$p collisions apart from the initial hard
scattering one [4, 5].

The traditional approach had been used to study the observables sensitive to
UE on an event by event basis. In this approach, the leading particle
(particle with highest $p_{T}$) is used to segment the $\eta-\phi$ space into
three distinct regions based on the azimuthal angular difference
($\Delta\Phi$) relative to the leading particle as shown in Figure 1. The
leading particle acts as a proxy for the main flow of hard-scattering process
[2]. The azimuthal angular difference is defined as,
$\Delta\Phi=|\Phi-\Phi_{L}|$, where $\Phi$ is the azimuthal angle of an
outgoing charged particle in an event and $\Phi_{L}$ is the azimuthal angle of
the charged particle having highest transverse momentum ($p_{T}^{lead}$) in
the event. The toward region of $\eta-\Phi$ space is defined as
$|\Delta\Phi|<60^{\circ}$ and the away region is defined as
$|\Delta\Phi|>120^{\circ}$. The two transverse regions defined as
$60^{\circ}<\Delta\Phi<120^{\circ}$ and $60^{\circ}<-\Delta\Phi<120^{\circ}$
are referred to as transverse 1 and transverse 2 regions. The transverse
region is a combination of the transverse 1 and transverse 2 regions [2]. The
towards and away regions are dominated by particles produced in hard
processes, while the transverse region is more sensitive to underlying events.
In this work, an attempt has been made to study the underlying events through
two observables, namely, the mean charged particle multiplicity density,
$\langle d^{2}N/d\eta d\phi\rangle$ and mean scalar $p_{T}$ sum, $\langle
d^{2}\sum p_{T}/d\eta d\phi\rangle$ in the three different regions for p$-$p
collisions at $\sqrt{s}$ = 2.76, 7 and 13 TeV. The energy dependence of
underlying events has been explored using Pythia 8 Monte Carlo model in the
central region with $|\eta|<$ 2.5 and forward region with -6.6 $<$ $\eta$ $<$
-5.2 [6]. The effect of hadronic re-scattering, color reconnection, and rope
hadronization mechanism have been explored to shed light on the underlying
events.

Figure 1: The topology of p$-$p collision where the azimuthal plane is
segmented into towards, away and transverse regions in an event defined by
$\Delta\Phi$ angle relative to leading particle direction [cms2015event,
aaltonen2015study].

##  2 PYTHIA 8 Monte Carlo Generator

In this present work, Pythia 8.3 [7] has been used for generating events, and
ROOT [8] framework has been used to analyse the monte-carlo data. Pythia 8 is
a Monte Carlo event generator for high-energy particle collisions to study
collider physics. The hadronization framework is based on Lund string model.
It assumes that a string can be formed due to the colour field between two
interacting quarks, and string potential energy linearly rises with the
distance between two quarks. At some point, due to stretched string potential
energy, it will break into hadrons. and the partons emanate from the same
vertex as one can anticipate for $e^{+}-e^{-}$ collisions. However, protons
are composite objects and their overlap in hadronic collisions can lead to
many partonic interactions in the region of overlap. This constitutes the MPI
framework in Pythia. These fragmenting strings can overlap with each other and
the number of primary hadrons produced can be very high. In such a scenario,
there is a high probability that the hadrons interact with each other (
elastically or inelastically) around the region surrounding primary
scattering. These hadronic re-scatterings can modify some of the event
properties and therefore it will be worthwhile to study the effect of hadronic
re-scattering on underlying event observables. The default parameters values
given in Pythia 8.3 have been used in this work for hadronic re-scattering
model implementation [9]. Color reconnection is a microscopic mechanism that
describes the interaction between colour fields. In this mechanism, the final
partons are connected by colour strings in a way as to reduce the total string
length to the minimum. One string that connects two partons follows their
endpoint movements, resulting in a common boost of string fragmentation [10,
11]. Three different color reconnection mechanisms are implemented. The MPI
based model connects the partons of a lower $p_{T}$ MPI system with one from a
higher $p_{T}$ MPI system to reduce the total string length. In the QCD based
model, reconnection happens if and only if it reduces the total string length
and the string potential energy. One can form different quark junctions due to
reconnection of hadronizing strings iand QCD colour rules are used to
determine the reconnection probability. The Gluon move based model assumes
that partons can move from one place to other to reduce the total string
length.
Rope hadronization model is an extension of the Lund string model which is
significant when many overlapping strings are present. These strings act
coherently to form stronger ropes which would then be hadronized with larger,
effective string tension. Due to rope formation, a smaller number of
q$\overline{q}$ is needed to break the rope but having an effective string
tension. There is reduction in the rope tension when new q$\overline{q}$ pair
is produced in the process [12, 13]. Pythia 8 rope hadronization model
describes the interaction between these strings by two mechanisms implemented
as follows :
$\bullet$ String Shoving : The model allow nearby strings to shove each other
with an interaction potential derived from the colour superconductor analogy.
$\bullet$ Flavor Ropes :The model assumes formation of ropes between strings
overlapping in a dense environment which is hadronized with larger, effective
string tension .

##  3 Results and Discussion

The observables used to study the underlying events are defined as follows :
$\bullet$ Mean charged particle multiplicity density, $\langle d^{2}N/d\eta
d\phi\rangle$ : defined as the number of charged particles per unit pseudo-
rapidity ($\Delta\eta$) per unit azimuthal angular difference ($\Delta\phi$).
$\bullet$ Mean charged particle scalar $p_{T}$ sum, $\langle d^{2}\sum
p_{T}/d\eta d\phi\rangle$ : defined as the $p_{T}$ sum of charged particles
per unit pseudo-rapidity ($\delta\eta$) per unit azimuthal angular difference
($\delta\phi$) [2, 14].
The transverse region is further subdivided into TransMAX and TransMIN  region
based on either transverse 1 or transverse 2 regions having the largest or
smallest number of charged particles or mean scalar $p_{T}$ sum. The area
factor is $\delta\phi=2\pi/3$ for towards, away and transverse region, while
for TransMAX and TransMIN region $\delta\phi=\pi/3$. The particles are
selected to have $p_{T}$ $>$ 0.5 GeV/c, with $|\eta|\leq$ 2.5 .The underlying
event activities in forward region have been explored for $-6.6<\eta<-5.2$.

Figure 2: $\langle d^{2}N/d\eta d\phi\rangle$ as a function of $p_{T}^{lead}$
for towards, away, and transverse regions in p$-$p collisions at $\sqrt{s}$ =
13 TeV regions with(and without) the effect of hadronic re-scattering.

Figure 3:  $\langle d^{2}\sum p_{T}/d\eta d\phi\rangle$as a function of
$p_{T}^{lead}$ for towards, away, and transverse regions in p$-$p collisions
at $\sqrt{s}$ = 13 TeV regions with(and without) the effect of hadronic re-
scattering.

Figure 2 and 3 shows the variation of $\langle d^{2}N/d\eta d\phi\rangle$ and
$\langle d^{2}\sum p_{T}/d\eta d\phi\rangle$ as a function of $p_{T}^{lead}$
for p$-$p collisions at $\sqrt{s}$ = 13 TeV. The Pythia 8 predictions are
compared with the ATLAS measurements [14]. Recently, the hadronic re-
scattering has been introduced in Pythia 8 to study the effect of re-
scattering of final state hadrons on final observables [9]. The figures also
depict the predictions with (and without) hadronic re-scattering. The general
shape of the evolution is similar for the three regions i.e. it exhibits a
characteristic rapid rise for low values of $p_{T}^{lead}$ with an abrupt
change around 4.5 GeV/c value of $p_{T}^{lead}$. In the transverse region, the
curve almost saturates showing a plateau like behavior while for the other two
regions, it shows an increasing trend. This point of transition is mostly
attributed to reduction in impact parameter of p$-$p interactions and hence a
change from soft to hard scattering regime. Therefore, one can see a continued
activity from hard processes in towards and away region as the $p_{T}^{lead}$
increases while the transverse region is least affected. This is consistent
with the pedestal effect where the overlap between colliding protons is
complete and any further growth seen is related to hard processes or
contamination from the same rather than more MPI scattering. As the toward
region includes the leading charged particle, the multiplicity is slightly
lower compared to away region as there is less energy available for additional
particle production. The increase of the $\sum p_{T}$ densities in the toward
and away regions indicates towards the partitioning of the total energy in
each region in terms of charged particle production. The toward and away
regions are the active regions compared to the transverse for higher values of
$p_{T}^{lead}$. The effect of hadronic re-scattering for the $\langle
d^{2}N/d\eta d\phi\rangle$ is negligible at low $p_{T}^{lead}$ while for
higher values it describes the data better in towards and away region. In the
transverse region, the hadronic-rescattering over predicts the measured data.
There is no effect of the hadronic re-scattering for $\langle d^{2}\sum
p_{T}/d\eta d\phi\rangle$.

Figure 4: $\langle d^{2}N/d\eta d\phi\rangle$ as a function of $p_{T}^{lead}$
for TransMAX and TransMIN regions in p$-$p collisions at $\sqrt{s}$ = 13 TeV
regions with(and without) the effect of hadronic re-scattering.

Figure 5:  $\langle d^{2}\sum p_{T}/d\eta d\phi\rangle$as a function of
$p_{T}^{lead}$ for TransMAX and TransMIN regions in p$-$p collisions at
$\sqrt{s}$ = 13 TeV regions with(and without) the effect of hadronic re-
scattering.

Figure 4 and 5 shows $\langle d^{2}N/d\eta d\phi\rangle$ and $\langle
d^{2}\sum p_{T}/d\eta d\phi\rangle$ as a function of $p_{T}^{lead}$ in
TransMAX and TransMIN regions. The TransMIN region is dominated by pedestal
effect while the TransMAX region receives contributions from MPIs as well as
contaminations from hard-scattering processes. One can observe an increase of
charged particle densities in TransMAX region with $p_{T}^{lead}$ as it
receives contributions from hard processes and the hadronic re-scattering
slightly over predicts the measured data. Similarly, for mean $\sum p_{T}$
densities, one can observe a slightly increasing trend in activity with
$p_{T}^{lead}$ for TransMAX region while it shows no activity in the TransMIN
region with no effect of hadronic re-scattering in either of the regions.

The hadronic activity has been also studied in forward regions (higher $\eta$
regions) where the contaminations from hard scattering processes are expected
to be minimum and all the three regions are therefore sensitive to underlying
events. Figure 6 and 7 shows $\langle d^{2}N/d\eta d\phi\rangle$ and $\langle
d^{2}\sum p_{T}/d\eta d\phi\rangle$ as a function of $p_{T}^{lead}$ in the
forward region with $-6.6<\eta<-5.2$ . It can be observed that at low
$p_{T}^{lead}$, there is a sharp rise in both $\langle d^{2}N/d\eta
d\phi\rangle$ and $\langle d^{2}\sum p_{T}/d\eta d\phi\rangle$ with
$p_{T}^{lead}$ in towards, away, and transverse regions due to increased
multi-partonic interactions but $\sim$ 4.0 GeV/c onwards, both the observables
saturates in all the three regions. This interesting observation of the
plateau in towards and away regions is in contrast with the observation at
central region. This indicates the near absence of contributions from hard
-scattering processes in forward regions[6]. The effect of hadronic re-
scattering is similar, as discussed earlier, in the towards, away, and
transverse regions in the forward pseudo-rapidity region as observed in
central regions. Figure 8 and 9 shows the same in TransMAX and TransMIN
regions at $\sqrt{s}$ = 13 TeV and the trend is similar to that observed in
central regions.

Figure 6: $\langle d^{2}N/d\eta d\phi\rangle$ as a function of $p_{T}^{lead}$
for towards, away, and transverse regions in p$-$p collisions at $\sqrt{s}$ =
13 TeV for -6.6 $<\eta<$ -5.2 with(and without) the effect of hadronic re-
scattering.

Figure 7:  $\langle d^{2}\sum p_{T}/d\eta d\phi\rangle$as a function of
$p_{T}^{lead}$ for towards, away, and transverse regions in p$-$p collisions
at $\sqrt{s}$ = 13 TeV for -6.6 $<\eta<$ -5.2 with(and without) the effect of
hadronic re-scattering.

Figure 8: $\langle d^{2}N/d\eta d\phi\rangle$ as a function of $p_{T}^{lead}$
for TransMAX and TransMIN regions in p$-$p collisions at $\sqrt{s}$ = 13 TeV
for -6.6 $<\eta<$ -5.2 with(and without) the effect of hadronic re-scattering.

Figure 9:  $\langle d^{2}\sum p_{T}/d\eta d\phi\rangle$ as a function of
$p_{T}^{lead}$ for TransMAX and TransMIN regions in p$-$p collisions at
$\sqrt{s}$ = 13 TeV for -6.6 $<\eta<$ -5.2 with(and without) the effect of
hadronic re-scattering.

The energy dependence of the underlying events in p$-$p collisions have been
studied at centre of mass-energy, $\sqrt{s}$ = 2.76, 7 and 13 TeV and have
been shown in Figure 10 and 11. One can observe that there is a strong rise in
mean charged particle multiplicity and mean $\sum p_{T}$ densities with
increasing $\sqrt{s}$ due to increased partonic event activities. It is also
observed that on going from 2.76 TeV to 13 TeV, the mean charge multiplicity
and mean scalar $p_{T}$ sum is almost doubled, which implies that the multi-
partonic interactions (MPI) activity grows more with centre of mass-energy[15,
16]. It was also shown in reference [16] that the dependence on beam energy
becomes vanishingly small as the charged particle densities are scaled by the
relative rise in multiplicity.

Figure 10:  $\langle d^{2}N/d\eta d\phi\rangle$ as a function of
$p_{T}^{lead}$ for towards, away, and transverse regions for p$-$p collisions
at $\sqrt{s}$ = 2.76, 7 and 13 TeV

Figure 11:  $\langle d^{2}\sum p_{T}/d\eta d\phi\rangle$ as a function
$p_{T}^{lead}$ for towards, away, and transverse regions for for p$-$p
collisions at $\sqrt{s}$ = 2.76, 7 and 13 TeV.

The additional studies carried on effect of color reconnections can be
observed in Figure 12 and 13. The color reconnection mechanism plays a crucial
role in the underlying events observables as it governs the interactions
between the partons of different $p_{T}$ scales before the hadronization
process. It modifies the particle production in the events with a large number
of multi-partonic interactions. It can be seen that the estimations with color
reconnections are in good agreement with the measured data compared to the
events without color reconnections which over-predict the data emphasizing on
the importance of color reconnection mechanism [17]. Furthermore, it shows the
comparison of three different modes of color reconnection(CR ) : MPI-based,
QCD-based and gluon-move based model implemented in Pythia 8.

Figure 12:  $\langle d^{2}N/d\eta d\phi\rangle$ as a function of
$p_{T}^{lead}$ for towards, away, and transverse regions at $\sqrt{s}$ = 13
TeV for different modes of color reconnections .

Figure 13:  $\langle d^{2}\sum p_{T}/d\eta d\phi\rangle$ as a function of
$p_{T}^{lead}$ for towards, away, and transverse regions at $\sqrt{s}$ = 13
TeV for different modes of color reconnections .

The effect of rope hadronization with QCD-based color reconnection model on
charged particle multiplicity density and mean scalar sum $p_{T}$ has been
shown as a function of $p_{T}^{lead}$ at $\sqrt{s}$ = 13 TeV for towards,
away, and transverse region in Figure 14 and 15 for the string shoving and
with flavor ropes . The rope hadronization mechanism describes the interaction
of the overlapping strings produced in a small transverse area. The effect is
more pronounced in high multiplicity events and forms The string shoving
mechanism refers to the pushing action of nearby strings before hadronization
and can increase the multiplicity due to gluon excitations. One observes an
increase in mean charged particle multiplicity density for all three regions
as an effect of the string shoving and it describes the data better in the
transverse region. The switching of flavor rope mechanism enables the nearby
strings to interact with each other to form color ropes. These ropes due to
increased string tension preferably hadronize to massive particles. Therefore,
there is a slight decrease in multiplicity density. There is negligible effect
of string shoving and flavour ropes on mean scalar $p_{T}$ sum.
These studies together with the predictions of other models like Herwig7 [18,
19] and Sherpa[20] can act as baseline to understand the processes at partonic
and hadronic level to understand the contribution of underlying events in
future experimental studies.

Figure 14: $\langle d^{2}N/d\eta d\phi\rangle$ as a function of $p_{T}^{lead}$
for rope hadronization with flavor ropes and string shoving.

Figure 15:  $\langle d^{2}\sum p_{T}/d\eta d\phi\rangle$ as a function of
$p_{T}^{lead}$ for rope hadronization with flavor ropes and string shoving.

##  4 Summary

The underlying event activities have been studied in p$-$p collisions at
$\sqrt{s}$ = 2.76, 7 and 13 TeV in central and forward regions with $|\eta|<$
2.5 and -6.6 $<\eta<$ -5.2 respectively, using the Pythia 8 event generator.
The hadronic activity due to underlying events has been studied by segmenting
the azimuthal plane in towards, away, and transverse regions. The transverse
region is sensitive to underlying event activity, while the towards and away
regions receive dominant contributions from the primary hard scattering. The
observables, mean charged particle multiplicity density , $\langle
d^{2}N/d\eta d\phi\rangle$ and mean scalar $p_{T}$ sum, $\langle d^{2}\sum
p_{T}/d\eta d\phi\rangle$ were investigated as a function of leading
$p_{T}^{lead}$. The effect of hadronic re-scattering, different mode of color
reconnections and rope hadronization has also been studied. This study can act
as a baseline for fine tuning of event generators focusing on underlying
events.

##  5 Acknowledgements

The authors would like to thank the Department of Science and Technology
(DST), India for supporting the present work.

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Send correspondence to M. Weber: E-mail<EMAIL_ADDRESS>
# Using raytracing to derive the expected performance of STELLA’s SES-VIS
spectrograph

Michael Weber  Leibniz-Institut for Astrophysics Potsdam (AIP), An der
Sternwarte 16, D–14482 Potsdam, Germany  Klaus G. Strassmeier  Leibniz-
Institut for Astrophysics Potsdam (AIP), An der Sternwarte 16, D–14482
Potsdam, Germany  Manfred Woche  Leibniz-Institut for Astrophysics Potsdam
(AIP), An der Sternwarte 16, D–14482 Potsdam, Germany  Ilya Ilyin  Leibniz-
Institut for Astrophysics Potsdam (AIP), An der Sternwarte 16, D–14482
Potsdam, Germany  Arto Järvinen  Leibniz-Institut for Astrophysics Potsdam
(AIP), An der Sternwarte 16, D–14482 Potsdam, Germany

###### Abstract

The visual STELLA échelle spectrograph (SES-VIS) is a new instrument for the
STELLA-II telescope at the Izaña observatory on Tenerife. Together with the
original SES spectrograph - which will still be used in the near IR - and a
new H&K-optimized spectrograph, which is currently in the design phase, it
will extend the capabilities of STELLA with the follow up of planetary
candidates from space missions (TESS, PLATO2). SES-VIS is optimized for
precise radial velocity determinations and long term stability. We have
developed a ZEMAX based software package to create simulated spectra, which
are then extracted using our new data reduction package developed for the
PEPSI spectrograph. The focus in this paper has been put on calibration
spectra, and the full range of available calibration sources (flat field, Th-
Ar, and Fabry-Pérot etalon), which can be compared to actual commissioning
data once they are available. Furthermore we tested for the effect of changes
of the environmental parameters to the wavelength calibration precision.

###### keywords:

Instrumentation: spectrographs – Observatory: robotic – Techniques: radial
velocities – Techniques: spectroscopic

##  1 INTRODUCTION

To measure precise radial velocities with a spectrograph essentially means to
measure the position of the edges of bright and dark features on the CCD. This
is most obvious for calibration spectra, which are either line-spectra from
elements like Thorium, or artificial line spectra from a laser comb or a
stabilized Fabry-Pérot etalon (FPE). These edges can be determined
approximately (depending on contrast) with an accuracy of 10% of the width of
a pixel, which is in case of a spectrograph with a resolving power of 5$\times
10^{4}$ approximately 100 m s-1. To reach the goal of m s-1 accuracy, one
needs many lines, in this case about 1$\times 10^{4}$. For calibration light
sources, if light sources like ThAr are not sufficient any more, laser combs
or stabilized Fabry-Pérot etalons that can be matched to the spectrograph
parameters are the solution. For science observations, the only way to
increase the number of lines is the increase of the wavelength range, and to
some extent for slowly rotating stars the increase of spectral resolution. The
actual factor defining the radial velocity precision is the slope of the
spectral lines.

Hatzes & Cochran[1] used this fact to derive a formula for the expected radial
velocity precision in terms of spectral resolution, signal/noise ratio and
wavelength coverage:

$\sigma=1.45\times 10^{9}(S/N)^{-1}R^{-1}B^{-1/2}(m/s)$ (1)

where $S/N$ is the signal-to-noise ratio of the spectrum, $R$ the spectral
resolution $\lambda/\Delta\lambda$, and $B$ the spectral coverage in Å. For
the present case of SES-VIS with the parameters listed in Sect. 3, the
expected precision from this formula would only depend on $S/N$ and would vary
between 1 m/s for a S/N of 600 and 6 m/s for a S/N of 100.

A more stringent formula for the expected radial velocity precision has been
derived by Bouchy et al. (2001)[2], taking the spectral type and the
broadening of the star into account in addition to S/N and spectral
resolution. A good recent summary of the various approaches has been written
by Figueira (2018)[3].

##  2 FABRY-PEROT UNIT

Figure 1:  Design of the Fabry-Pérot etalon (FPE) unit of SES-VIS. Top:
Assembly drawing of the FPE unit at the left, an image of the assembled unit
during lab testing on the right. Fiber input and output is located at the top,
the etalon is located in the center of the unit. Bottom: One order of the FPE-
spectrum recorded with a badly aligned laboratory échelle spectrograph.

Since a laser-comb calibration light source was out of reach when we started
this project, we decided to build a calibration source using a Fabry-Pérot
etalon, similar to the one used in PEPSI. We used two off-axis parabolic
mirrors (OAPs), one to collimate the f/3 output of an optical fiber fed with
halogen light, the other to re-focus the beam onto a fiber which feeds the
calibration light path of the spectrograph. For the first OAP we used a high-
quality ($\lambda$/10) Zerodur mirror with broadband silver coating to make
sure the etalon is well illuminated. The second OAP was manufactured out of
solid aluminum to match the properties of the first OAP. The etalon has a
cavity size of 3.08 mm and a beam size of 25 mm, resulting in an effective
finesse (neglecting the aperture finesse of the feed fiber) of 23 to 25 at our
wavelength range of 470 nm to 690 nm.

The unit was built as a vacuum-compatible tube-structure (see Fig, 1 top),
which will be put inside the vacuum vessel of SES-VIS to keep the spectrum
stable. A prototype was assembled at the AIP, aligned, and we used a small
laboratory spectrograph to test the functionality of the unit (see Fig, 1
bottom). Unfortunately, alignment problems of the laboratory spectrograph
hindered us to make detailed analysis of the performance of the etalon.

In total we will get about 175 FPE lines per order in all 42 orders of SES-
VIS, resulting in about 7000 lines in total. Assuming 100 m s-1accuracy per
line, the calibration accuracy will achieve about 1 m s-1, matching the best-
case stellar radial velocity sensitivity derived above. In order to keep the
wavelengths of the FPE-peaks constant to better than 1 m s-1, the pressure
stability inside the FPE must be better than 0.01 mbar.

##  3 THE SES-VIS SPECTROGRAPH

The new SES-VIS spectrograph was designed for the solar telescope GREGOR and
known as GANS[4]. It uses a classical white pupil optical design. Its design
goals were to measure precise radial velocities, it is therefore contained
inside a vacuum vessel for stability, which again is surrounded by an actively
temperature-controlled enclosure (Fig. 2). We use a R4-échelle grating, with
orders 89 - 130 for wavelengths from 468 nm to 691 nm. In order to detect
possible shifts between wavelength calibration exposures, a spectrum through a
FPE is put alongside all calibration and science exposures (Fig. 3). More
details can be found in the presentation from SPIE 2018[5].

Figure 2:  System design of the SES-VIS spectrograph. The innermost part is
the vacuum vessel with a diameter of 500 mm, surrounded by two insulated
enclosures. The volume between these two enclosures is actively kept at a
constant temperature. The whole unit is located in a commercially air-
conditioned room.

##  4 SIMULATED SPECTRA

Figure 3:  Small excerpt of one two-dimensional spectrum from our simulations:
Next to the science spectrum, which is double-humped due to the image slicer,
is the spectrum of the FPE with its evenly distributed peaks.

Using the optics design program Zemax we used the as-built optical design of
SES-VIS to compute 2D-échelle spectra from standard one-dimensional spectra[6,
7]. Recent versions of Zemax OpticStudio feature a well documented programming
interface, which allows - among other things - to create standalone python
programs that use OpticStudio to trace rays. The python program reads in the
model spectrum and the order list of GANS specifying start and end wavelength
for each spectral order that fits on the detector surface. For each spectral
order, the flux is divided into small wavelength chunks (matching the sampling
of the spectrograph), and each of these chunks is propagated through the
spectrograph. The relative intensity is preserved by scaling the propagated
rays with the incoming intensity. The rays that end up at the detector surface
are assigned to their respective pixel, and the 2D-échelle image is obtained
after going through the complete order list. We use a fairly high number of
rays (10000) per sampling element (one pixel in dispersion direction), which
are constructed by uniformly distributing them across the field (the optical
fibre diameter) and pupil of the spectrograph.

Our primary spectrum for all further analysis was a model spectrum of the sun.
We used the solar atmosphere model from MARCS[8], a line list from VALD3[9]
and the Turbospectrum[10] code to compute spectra the model spectrum of the
sun. After the model spectrum was broadened to account for rotation and
macroturbulence, it became our reference spectrum for further analysis.

To analyze the results we used our data reduction package SDS4PEPSI[11],
extracting the spectral orders from for both the science spectrum as well as
the FPE-spectrum. For the purpose of data reduction we constructed flat-field
images from one model with a constant intensity, and wavelength reference
images using a high-resolution PEPSI[11] ThAr spectrum (see Fig. 3).

During the analysis of our data it became clear that we need to add features
to our data reduction package. Currently it is not possible to supply a
reference line list on the fly, as would be necessary to use the positions of
the FPE-peaks to set the wavelengths of the FPE-spectrum. We overcame this
limitations by using a ThAr-Spectrum in place of the FPE reference spectrum.
We will also measure this during commissioning of the spectrograph, but during
routine operations only light from the FPE-unit can be placed simultaneously
onto the detector. It is also not possible currently to use optimal
extraction[12], since this fails on the FPE spectrum but can not be turned off
selectively for individual slices, and our simultaneous calibration spectrum
is treated as an extra slice. Solutions for these problems are under way and
will be ready for lab-commissioning.

##  5 RESULTS

Figure 4:  Example graph from the cross-correlation procedure: The science
spectrum is first cross-correlated with a template using a standard routine,
re-binning all points to the same velocity scale (top left). In a second step
a cross-correlation is performed changing only the sampling of the template
(top right). The bottom panel shows the science spectrum and the shifted and
spun up template spectrum.

We created a set of simulated spectra to test the data analysis software, and
to see up to what quality of the vacuum the radial velocity can be derived at
a nominal precision of about 1 m s-1, which is roughly our design goal since
at that resolution and wavelength range it is the best precision one can
obtain from a well-behaved stellar spectrum (see Sect. 1 above). Combinations
are written as ThAR/FPE, meaning that ThAr-light was used for the science
orders, and FPE-light was used for the calibration orders. This set consisted
of flat field spectra in all combinations (Flat/zero, zero/Flat, Flat/Flat),
ThAr-calibration spectra (ThAr/ThAr and Thar/FPE), and s series of Sun/FPE
spectra at pressure settings inside the vacuum vessel between 0.01 and 100
mbar.

We plan to run the vessel at a pressure of 10-4 mbar, but want to know at what
level we would need to start pumping in order to keep up our wavelength
stability. The FPE is located in the same vacuum vessel, and a calibration
accuracy of 1 m s-1 demands a pressure stability of better than 0.01 mbar,
otherwise changes in the FPE-peak wavelengths would dictate the precision of
the simultaneous calibration.

All spectra are extracted using standard extraction, the science and
calibration spectra are treated as two slices and are extracted together, and
the wavelength solution is obtained using both spectra simultaneously also.
The spectra are then separated out into two sets of spectra, science spectra
and calibration spectra, normalized using per-order spline-fits, and merged
into single one-dimensional spectra. For simplicity we did not make a global
fit to set the continuum, nor did we iterate the continuum to match the order-
overlaps, thus small continuum-errors are introduced during this procedure.

Figure 5:  Result from cross-correlating the extracted solar spectrum and the
FPE spectrum with its corresponding model, at various pressure settings. A
relative drift sets in at about 1 mbar.

As our data reduction software is not yet capable of directly measuring the
stability of the FPE-peaks’ positions, we independently derived the global
shift of the solar spectrum using the extracted science spectra, and of the
FPE-spectrum using the extracted calibration spectra. We used the same spectra
as cross-correlation templates as we used as input for ray tracing. No
additional noise was added to the images, the number of rays traced
corresponds to a S/N of about 100 per pixel.

Cross-correlation was carried out by a multi-purpose python script developed
for merged STELLA and PEPSI spectra. The template can optionally be spun up to
an arbitrary rotational velocity, and macro-turbulence and instrumental
broadening can be folded in. Wavelength-regions of interest can be defined
that leave out regions which are heavily affected by telluric lines, or to
leave out (or focus on) very strong lines like Balmer-lines or Na-D, but here
the whole spectral region was used. In a first step both template and science
spectrum are re-sampled to a constant velocity step size and a regular
correlation routine is used to obtain a first approximation of the peak
position (top left of Fig. 4). In a second step, the natural sampling of the
science spectrum is preserved, and the template spectrum is evaluated at these
sampling wavelengths. We then repeat this procedure for 5000 different
velocity shifts, offset each by 1 m s-1centered around the peak derived in the
first step, and derive the final offset by a fit to this function (top right
of Fig. 4). In the bottom of Fig. 4 the science spectrum is shown along with
the shifted and broadened template spectrum. Naturally, since both are derived
for the identical model spectrum, they are very similar, but a small waviness
in the continuum setting is visible as well as systematic differences around
strong lines.

Now, in the end, we finally get to compare the offsets derived from both the
science spectra as well as the calibration spectra with respect to the vacuum-
vessel’s pressure (Fig. 5). The offsets stay constant up to 0.1 mbar, and only
an insignificant difference is visible at 1 mbar. From here onwards, the
differences are much bigger than the desired accuracy of the spectrograph, as
was expected.

This experiment shows that we are on the safe side with our planned pressure
level at 10-4 mbar. The spectrograph itself tolerates two orders of magnitudes
more than the FPE, which leaves some room for effects like residual
temperature variations which are not covered here since we can not simulate
them during lab-commissioning due to the lack of space for the thermal
enclosure. And the extra margin of two orders of magnitude from our planned
vacuum level of 10-4 mbar to a tolerable 0.01 mbar suggests that we can
refrain from pumping during the nights.

###### Acknowledgements.

The STELLA facility is funded by the Science and Culture Ministry of the
German State of Brandenburg (MWFK) and the German Federal Ministry for
Education and Research (BMBF), and is operated by the AIP jointly with the
IAC.

## References

  * [1] Hatzes, A. P. and Cochran, W. D., “Spectrograph Requirements for Precise Radial Velocity Measurements,” in [European Southern Observatory Conference and Workshop Proceedings ], Ulrich, M.-H., ed., European Southern Observatory Conference and Workshop Proceedings 40, 275 (1992).
  * [2] Bouchy, F., Pepe, F., and Queloz, D., “Fundamental photon noise limit to radial velocity measurements,” A&A 374, 733–739 (2001).
  * [3] Figueira, P., “Deriving High-Precision Radial Velocities,” Asteroseismology and Exoplanets: Listening to the Stars and Searching for New Worlds 49, 181 (2018).
  * [4] Järvinen, A. S., Weber, M., Strassmeier, K. G., Granzer, T., Woche, M., Bauer, S. M., Bittner, W., Weingrill, J., and Sablowski, D. P., “GANS: a nighttime spectrograph for the GREGOR solar telescope,” in [Ground-based and Airborne Instrumentation for Astronomy VII ], Evans, C. J., Simard, L., and Takami, H., eds., Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series 10702, 107026L (2018).
  * [5] Weber, M., Strassmeier, K. G., Woche, M., Ilyin, I., and Järvinen, A., “Radial velocity accuracy prediction of the GREGOR at night spectrograph based on simulated spectra,” in [Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series ], Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series 10705, 107051E (2018).
  * [6] Barnes, S. I., “Echmod: a matlab toolbox for modeling high resolution échelle spectrographs,” Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series 8550, 8550 – 8550 – 8 (2012).
  * [7] Gibson, S. R. and Wishnow, E. H., “A method for generating a synthetic spectrum within Zemax,” in [Modeling, Systems Engineering, and Project Management for Astronomy VI ], Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series 9911, 99112D (2016).
  * [8] Gustafsson, B., Edvardsson, B., Eriksson, K., Jørgensen, U. G., Nordlund, Å., and Plez, B., “A grid of MARCS model atmospheres for late-type stars. I. Methods and general properties,” A&A 486, 951–970 (2008).
  * [9] Ryabchikova, T., Piskunov, N., Kurucz, R. L., Stempels, H. C., Heiter, U., Pakhomov, Y., and Barklem, P. S., “A major upgrade of the VALD database,” Phys. Scr. 90(5), 054005 (2015).
  * [10] Alvarez, R. and Plez, B., “Near-infrared narrow-band photometry of M-giant and Mira stars: models meet observations,” A&A 330, 1109–1119 (1998).
  * [11] Strassmeier, K. G., Ilyin, I., and Steffen, M., “PEPSI deep spectra. I. The Sun-as-a-star,” A&A 612, A44 (2018).
  * [12] Horne, K., “An optimal extraction algorithm for CCD spectroscopy,” Publications of the Astronomical Society of the Pacific 98, 609 (1986).


The A-channel is the multiple access channel in which users noiselessly transmit codewords with entries in a $q$-ary alphabet and the receiver observes the set of different transmitted symbols but not their multiplicity.

The unsourced A channel is a variation of the A channel above in which all users pick the transmitted codeword from a common codebook.

In this setting, the receiver's objective is to identify a list of transmitted codewords, while the identity of the transmitter is immaterial.

Motivated by the machine-type communication and the coded compress sensing, we consider the unsourced A channel with erasures (UACE), in which symbols transmitted by a user are erased with a given iid probability.

For the UACE, we focus on the design of a code that produces the desired list of active codewords despite the erasures of the transmitted symbols.

For this problem, we design the linked-loop code (LLC): in this code a number of parity bits are used in each transmission to link the last $L$ symbols in a tail-biting manner, meaning that the last symbols of the transmission are linked to the first ones.

The decoding then occurs in a two-phase manner: in the first phase, the codewords that do not suffer any erasures are decoded.

In the second decoding phase, LLC attempt to recover the erased symbols using the available parities.

Given the cyclical structure of the code,

We compare the performance of the LLC with other codes in the literature and argue for the effectiveness of the construction.

§ INTRODUCTION

The A channel with erasures is defined by the input/output relationship

Y = ⋃_k ∈[K]  δ_E_k (x_k)

where $x_{k} \in \Xcal$ with $|\Xcal|=Q$,

$E_{k} \stackrel{\text{iid}}{\sim} \Bcal(p_e)$,  and

x       E = 1

∅    E = 0.

Here $\Bcal(p)$ indicates the Bernoulli distribution with parameter $p\in [0,1]$.

In other words, the A channel with erasures of (<ref>) is the channel  model in which each user $k \in [K]$ transmits a symbol from the alphabet $\Xcal$ where $\Xcal$ is a set of cardinality $Q$.

This symbol is correctly communicated at the receiver with probability $1-p_e$ and is otherwise erased.

Motivated by the short-packet random access wireless scenario, we consider the scenario in which the receiver is interested only in knowing which codewords have been transmitted but not by which user.

This paradigm is useful as it removes the need for user enrolment, which is too taxing when the payload is short.

We refer to this transmission setting as the unsourced A channel with erasures (UACE).

In this paper, we (i) introduce the UACE and (ii) design the first coding scheme for recovering erased symbols which we refer to as the linked-loop code (LLC). (iii) The performance of the LLC is evaluated and compared to other schemes available in the literature.

§.§ Literature Review

The A channel was originally introduced in <cit.> to model a multiple-access channel model where frequency division is used for transmission but where the intensity of the signal over each frequency is not available.

The authors of <cit.> determine the mutual information of the A channel under uniform inputs.

Its limit when $K$ and $Q$ tend to infinity but its ratio $\lambda = K/Q$ is fixed was studied in

The mutual information with uniform inputs in the sparse limit of $K,Q \goes \infty$  with fixed ratio $(\log K)/Q$
was computed in <cit.> and it was shown that in this limit the
mutual information grows proportional to $\log Q$.

More recently, the A channel has found renewed interest in the literature due to the emergence of short-packet machine-type communication and the IoT paradigm.

Consider the scenario in which a large number of devices wishes  to communicate a small payload to a central server.

When the payload is so short, the customary identify-then-transmit procedure  in which each user uses a control channel to obtain a network resource becomes inefficient.

On the other hand, freely accessing the medium causes  collisions among the transmissions that can only be resolved through a feedback signal, which also requires too much coordination.

An alternative paradigm to these two approaches is that of the unsourced random access (URA) channel <cit.>.

In the URA channel, users transmit using the same codebook so that  the receiver can only recover a list of transmitted codewords up to permutation.

The A channel played an important role for codes design in URA as it allows the transmission block-length to be partitioned in multiple sections.

This is attained as follows <cit.>.

Over each section, a linear code is used to encode information and parity bits.

The parity bits are obtained from a code that allows recovery over the A channel.

At decoding, each section is decoded independently and a list of possible transmitted messages is produced for each section.

The parity bits from the A channel are then used to link the information bits from the same user across sections.

In <cit.>, a coding scheme for AWGN URA termed coded
compressed sensing (CCS) was introduced.

The A channel code constructed for this purpose was termed tree code. The flexibility of this code construction
allowed it to be extended to different channel models. Several follow-up works on URA have further elaborated on this approach <cit.>

More recently, following the renewed interest in these applications of the A channel, the authors of <cit.> present finite-blocklength achievability bounds for the unsourced A-channel.

§.§ Contribution

In this paper, we provide the following contributions:

* Sec. <ref> : We introduce the UACE model, define the information transmission problem for this model, and define relevant performance metrics.

* Sec. <ref> : We introduce the linked-loop code
(LLC) as a first code specifically designed for the UACE. Encoding and decoding procedures are describei.  The code parameters are also discussed.

* Sec. <ref> : We review codes in the literature that are compatible to the UACE setting, although not specifically designed for this channel: (i) the tree code of <cit.>, and the (ii) belief propagation approach of <cit.>.

These two codes are used for numerical comparison of performance in Sec. <ref>.

* Sec. <ref> :  We present numerical simulations that provide insight into the performance of the LLC in Sec. <ref> and provide a comparison the codes in Sec. <ref>

Finally, Sec. <ref> concludes the paper.

Notation: In the following, lowercase boldface letters (eg. $\zv$) are used for row vectors and uppercase boldface letters (e.g., $\Mv$) designate matrices.

We also adopt the short-hand notation

$[N] \triangleq \{0, \ldots, N-1\}$.

The $p$-norm of the vector $\xv$ is indicated as $\| \xv\|_p$.

Random variables are indicated with uppercase letters, e.i. $N$.

Calligraphic scripts are used to denote sets (e.g., $\Acal$) and $|\Acal|$ is used to denote its cardinality.

$\rb{GF}_2$ indicates the Galois field of size 2.

Given two row vectors $\vv$ and $\wv$, we denote their concatenation as $\vv.\wv$.

With a certain abuse of notation, we sometimes interchange the notation for (random) vectors and sets when this is clear from the context. For instance in (<ref>) and similar expressions, we indicate the channel output as a random vector, rather than a set. This is done mostly to simplify the notation.

The code for the numerical evaluations of this paper is provided online at <https://github.com/williamzheng0711/URAC23>.

§ CHANNEL MODEL

The UACE is a variation of the unsoured A channel from <cit.> that also encompasses symbol erasures.

In the (perfect) unsourced A channel of <cit.>, the channel output at the $l^{\rm th}$ channel use  is defined as

Y_l = ⋃_k ∈[K] x_l,k,

where $x_{l,k} \in \Qcal$.

When transmission takes place over $L$ channel uses,

with $\Yv$ we indicate the $L$-tuple $\Yv=[ {Y}_0,\ldots,Y_{L-1}]$ for $l \in [L]$.

$\xv_k$ is similarly defined as  $\xv_k=[ x_{0,k},\ldots,x_{L-1,k} ]$.

The UACE is the variation of the model in (<ref>) defined in (<ref>).

We return to (<ref>) to note that the erasure makes the cardinality of each $Y_l$, $|Y_l|$, is a random variable which is less than $K$ due to both collisions and erasure. Here collision denotes two distinct users $i\neq j$ have their $x_{l,i}=x_{l,j} \in \Xcal$.

Note that the UACE in (<ref>) is parametrized by (i) the number of users – $K$, (ii) the cardinality of the input set – $Q$, and (iii) the erasure probability – $p_e$.

§.§ Binary vector representation

When describing coding schemes, it is often convenient to operate over $\rb{GF}_2$: this is the case also for the UACE.

For this reason, we introduce here  a specific notation for the case in which $Q=2^J$ for some $J \in \Nbb$.

For this case we let $\Xcal = [2]^J$ and the binary vector representation of the UACE

in (<ref>) is expressed as

_l = ⋃_k ∈[K]  δ_E_l,k v_k(l).

After $L$ uses of the channel, what we obtain that  again be expressed as an $L$-tuple  as

=(_0,_2, …,_L-1).

In the following, it shall be convenient to represent the concatenation of the $L$ input binary vectors, each comprised of $J$ bits as a single row vector, that is

v_k = v_k(0)v_k(2) …v_k(L-1).

In this case, we have that the binary representation of the input of user $k$ is  a binary vector $\vv_k$ of dimension $N=JL$.

In the following we shall use the binary representation in Sec. <ref>.

§.§ Coding for the UACE

Next, let us introduce a series of rather standard definitions for coding over a UACE.

Let us consider the problem in which each user $k$ wishes to transmit a payload of $B$ bits in $L$ channel uses.

Assume further that each payload $\Wv=\{\wv_k\}_{k \in [K]}$ is drawn uniformly at random from $[2]^B$.

Let us refer to $\wv \in \Wv$ as an active payload, while $\wv \notin \Wv$ as inactive.

Finally, if two users randomly select the same payload, let us refer to this event as a payload collision.

A code of block-length $L$ and payload length $B$ for the UACE with parameters $(K,Q,p_e)$ is defined as a (possibly) random encoding mapping to produce $L$ channel input at each user $k \in [K]$ to encode their message $\wv_k$, defined as

_k = f_enc _k ,

where $\wv_k \in [2]^B$ and $\xv_k \in \Xcal^L$,
and a (possibly) random decoding mapping which produces a list of transmitted payloads from the channel output, that is

f_dec( ),

with  $|\widehat{W}|=\Kh\leq K$.

In other words, the encoding function maps the  message of user $k$, $\wv_k$, which is comprised of $B$ bits into the $L$-tuple of channel inputs $\xv_k$.

Upon observing the $L$ channel outputs, $\Yv$, the decoding function produces a list of payloads that are likely to be active.

The list of possibly active payloads produced by the decoding function has length $K'$.

Note that the rate of the code in Def. <ref> is obtained as $R = B/N$.

Also note that in a UACE each encoder $k \in [K]$ utilizes the same encoding function: this is in contrast with the classic A channel or the multiple access channel, in which each user utilizes  a different encoding function.

Payload probability of error

Given a code with parameters $(N,B)$ for a  UACE with parameters $(K,Q,p_e)$, we define the

* payload dropping probability (PDP) as

P_PDP = 1 K  ∑_k ∈[K] _k ∉() | _k ∈,

* the payload hallucination probability (PAP)

P_PAP = 1 K'  ∑_k' ∈[K'] _k' ∉| _k'  ∈() ,

where the probability in (<ref>) and (<ref>) is taken over (i)

the choice of the payload at each user which is selected uniformly at random over $[2]^B$,  (ii) the realization of the erasure sequence $\{E_k(l)\}_{k \in [K]}$, and (iii) the possible randomness in the encoding and decoding function.

More informally, the PDP indicates the probability that a payload is declared inactive when it was actually active, while the PAP indicates the probability that a payload is declared as active when it was indeed inactive.

The PDP and PAP are somewhat analogous to the type I and type II errors, respectively, in the hypothesis testing literature.

Birthday problem
Payload collision is analogous to the classic birthday problem,  that is the probability of a collision is similar to the probability that two people in a certain group share the same birthday.

In the parameter regimes which we investigate, the probability of payload collision is orders of magnitude smaller that the PDP and PAP of Def. <ref>.
For this reason, we generally do not discuss this source of performance degradation.

Following the notation of <cit.>, we next describe a binary linear code for the UACE as follows.

Here the binary vector representation of Sec. <ref> will be particularly useful.

§.§ Linear codes

Binary linear codes are a special case of the class of codes in Def. <ref> in which the encoding function of (<ref>) is a matrix multiplication over $\rb{GF}_2$.

More precisely, a binary linear code is defined as a set of $L$ parity check  matrices $\{\Gv_l\}_{l \in [L]}$  which specifies how the parity bits for section $l \in [L]$ are generated from the payload $\wv$.

More formally, let $\{m(l)\}_{l\in [L]}$ with $\sum_{l \in [L]} m(l) = B$ indicate the number of information bits allocated to section $l \in [L]$. Again, we define $\mv = [m(0),m(1),\ldots, m(L-1)]$

Next, partition the payload $\wv$ into $L$ vectors, each of dimension $m(l)$ so that

= (0) . (1) . ….(L-1),

with $\wv(l) \in [2]^{m(l)}$ for all $l\in [L]$.

Let the parity bits in section $l$ be indicated as $\pv_k(l) \in \mathbb{F}_2^{p(l)}$, with its length $p(l)=J-m(l)$.

The parity bits are obtained from the payload as

(l)     =   _l

= ∑_l' ∈[L] (l') _l',l,

where $\Gv_{l} \in \mathbb{F}_2^{ B \times (J-m(l))}$.

When considering the partitioning of the payload as in (<ref>), one naturally obtains a partitioning of the $\Gv_{l}$ in sub-matrices  as in (<ref>) where $\Gv_{l,l'} \in \mathbb{F}_2^{m_{l'} \times (J - m(l))}$.

Finally, the channel input is obtained as

(l) = (l).(l).

In the section above, we have dropped the subscript $k$ as in (<ref>). This is to simplify the notation as well as stress the fact that the encoding is the same at all users $k \in [K]$.

§.§ Motivation and Further comments

The UACE  finds its application in the URA channel as in <cit.> in which AMP decoding returns a list of estimated active payload in each section $[L]$.

In its original analysis, the authors of <cit.> assume that this list contains all active payloads plus some hallucinated payloads.

The UACE allows one tackle the scenario in which the AMP decoder outputs a list that misses some of the active payloads.

The UACE, does not capture the full complexity of a stitching code for CCS as it does not consider both insertions and deletions.

The unsourced A channel with insertions and deletions is left for future research.

Finally, another interesting variation is the one in which  the erasures $E_{l,k}$  are correlated in $k$.

This model would be relevant in the case of an unsourced MAC channel with fading.

§ LINKED-LOOP CODE (LLC)

When introducing binary linear codes in Sec.  <ref> we only discussed the encoding operation but not the decoding one.

Let us briefly discuss the inherent difficulty in correctly identifying active payloads so as to minimize PDP and PAP before introducing the proposed approach.

For simplicity, let us consider the regime in which

each user experiences either no erasures or only one erasure across the $L$ sections.

The decoder can decode an active payload as follows.

First (i) pick one of the list entries in section $0$, call it $\Lcal_0$
(ii) extract the information bits from this section and check which entries in the list of  section $1$ are parity consistent with  the information bits in section $0$. Generally speaking, there can be a number of parity-consistent entries, so add these entries in a list, let's call this list $\Lcal_1$. (iii) at the next iteration, use the information bits in $\Lcal_0$ and every entry in $\Lcal_1$ to extract a list of parity-consistent entires in section $2$, call this $\Lcal_2$. (iv) Repeat (iii) until the last section $L-1$.

Following <cit.>, let us refer to this operation of  “linking” parity-consistent list entries across multiple sections as stitching. Also call the set of stitched elements in each section a path. A path starts with an entry (referred as "root") in the list of section $0$ then stitch some parity consistent element from the list of section $1$ and so forth. A graphical representation of this process is provided in Fig. <ref>.

The stitching of elements across sections

From the discussion above, it is clear that a computationally efficient linear code for the UACE is one in which the information bits used to generate the parity bits only cover a small number of sections so that the number of stitched elements across lists $\Lcal_l$ does not grow exponentially as the stitching process continues.

In practice, it is more efficient to link all paths starting from the root entries and proceed sequentially, section after section. This is because parity-consistency for all paths can be formulated as a single matrix multiplication.

We refer to this decoding phase as phase 1 in the following.

Next, let us discuss a decoding approach when only one erasure is encountered.

When an erasure occurs for some certain user

, the decoder is not able to insert the correct element in one of the lists, let's say $\Lcal_l$.

Consequently, the number of surviving paths becomes zero for some section $l'$ with $l \leq l' \leq L$.

This means that after all initial paths have been linked, some of the elements of the root do not have a complete path of length $L$.

For these remaining codewords, a phase 2 of decoding is needed.

In phase $2$, we consider only  non-terminating paths in phase $1$. For these paths, we introduce a special symbol, $\na$,
that enables each path to have a one-section loss tolerance. After all other present sections have been stitched to some path, we calculate the lost section's information bits by reverse-engineering (<ref>).

Having discussed the properties of a good  approach above, we shall now detail the LLC construction, encoding, and decoding.

§.§ Construction

The LLC is a binary linear code for the UACE defined by: (i) a memory depth $M$,  and (ii)  a set of $M$ parity matrices $\{\Gv_r
% \in \mathbb{F}_2^{m \times (J-m)}
\}_{r = 1}^{M}$.

§.§ Encoding

The LLC has a fixed number of information and parity bits in each section, i.e.

$m(l)=m$ for some $m>0$.

Additionally, the $J-m$ parity bits depend on the information bits of the $M$ previous sections, in a ring manner, that is

(l)     =  (<ref>)

= ∑_0 < l-l'   mod   L ≤M  (l') _l',l

= ∑_r =1^M ( l-r   mod   L) _r.

where in (<ref>) we have set $\Gv_{l',l}=\Gv_{r}$ for $r = l-l' \, {\rm mod} \, L$ and $1\leq r\leq M$.

For example, if $L=16$ and $M=2$, the first three sets of parity bits are obtained as:

(0)     =  (14)_1 + (15)_0

(1)     =  (15)_1 + (0)_0

(2)     =  (0)_1 + (1)_0.

§.§ Decoding

Following the discussion above, let us consider a two phase decoding algorithm:

* phase 1: the receiver identifies and decode the active payloads which suffer no erasures in any of the $L$ sections.

* phase 2: the receiver attempts to identify and recover the active payloads which suffer erasure on only one section.

2WZ: complete here. add a nice picture

§.§.§ Phase 1

In phase 1, we do the following for each entry $x_0 \in \bold{y}_0$ in list of section 0. Note that only if a path have stitched section $0, \ldots ,M-1$, we can for the first time calculate some parity, that is $\bold{p}(M)$. Hence we will keep all possible path up to section $M-1$ in our
$$\Lcal_{M-1}^1(x_0)=\{[x_0, \ldots, x_{M-1}] |x_l \in \bold{y}_l, \forall 1\leq l \leq M-1 \}.$$

We update to $\Lcal_M^{1}(x_0)$ by stitching each path in $\Lcal_{M-1}^1(x_0)$ with parity consistent entries in section $M$ and discarding paths that do not admit a stitching. Repeat this process until we get $\Lcal_{L-1}^1(x_0)$.

For each path $\in \Lcal_{L-1}^1(x_0)$, we check the parity consistency of section $[M]$, since we did not check them when we stitched them on, to obtain $\Lcal^{1}(x_0)$ by dropping inconsistent paths.

Where inconsistent path refers to a path having some section known to be inconsistent according to (<ref>). All information bits of the elements in $\Lcal^1\coloneqq \bigcup_{x_0 \in \yv_0} \Lcal^1(x_0)$ will be extracted as decoded messages, constituting a set $\bold{\widehat{W}}_1$.

still have to check

§.§.§ Phase 2

For phase 2, we only choose those $x_{0}^\times \in \bold{y}_0$ such that $x_{0}^\times$ does not give rise to any decoded message in phase 1 to be roots.

After we've fixed a root $x_{0}^\times$, we then proceed almost the same as in phase 1.

The only difference is that we have a one-section erasure tolerance. That is for all $l\in[L-2]$, when we update our $\Lcal_l^2(x_{0}^\times)$ to $\Lcal_{l+1}^2(x_{0}^\times)$, for any path $\in \Lcal_l^2(x_{0}^\times)$, if that path has not contain an $\na$, we make a copy of that path and stitch an $\na$ onto its section $l+1$, no matter this path has a parity consistent entry in the list of section $l+1$ or not. This new path with $\na$ will also be an element of $\Lcal_{l+1}^2(x_{0}^\times)$. Keep doing this until we eventually end up with $\Lcal_{L-1}^2(x_{0}^\times)$.

Then we prune $\Lcal_{L-1}^2(x_{0}^\times)$ to $\Lcal^2(x_{0}^\times)$ by discarding all inconsistent paths of $\Lcal_{L-1}^2(x_{0}^\times)$ and calculate the lost section's information bits. Similar to phase 1, all elements of

$$\Lcal^2\coloneqq \bigcup_{x_{0}^\times \in \yv_0} \Lcal^2(x_{0}^\times)$$

will be extracted their information bits and those resultant messages forms a set $\widehat{\Wv}_2$. Finally,

$$\widehat{\Wv} \coloneqq \widehat{\Wv}_1 \cup \widehat{\Wv}_2$$
is our set of decoded messages.

Now let us see in detail how a payload suffer only one erasure is recovered. Suppose a codeword $\vv=\vv(0). \ldots .\vv(L-1)$ has its section $l \in [L]$ erased by UACE, while all other sections are present.

In this case, we must have an element

$$[\vv(0),\vv(1),\ldots,\underbrace{\na}_{\text{section $l$}},\ldots,\vv(L-1)] \in \Lcal^2(\vv(0)) \subset \Lcal^2$$

after phase 2.
If we can uniquely determine $\bold{w}(l)$, the information portion of $\vv(l)$, then we are done. The following $M$ equations deduced by reverse-engineering (<ref>) enable us to do so. That is
$\forall \iota : 0< (\iota - l) \mod{L} \leq M
% \in \{l+M' {\rm mod} L : M' \in [1,M], M' \in \mathbb{N}^+\}

\begin{equation}
\bold{w}(l)\bold{G}_{l,\iota}=\bold{p}(\iota )-\sum_{\substack{l' \neq l \\ l':0 < (\iota-l') {\rm mod} L \leq M} }\bold{w}(l')\bold{G}_{(\iota-l') {\rm mod} L}.
\label{eq:recover erasure}
\end{equation}

We see that it suffice to let $m\leq (J-m)$ and all matrix of $\{ \bold{G}_{r}\in \mathbb{F}_2^{m \times (J-m)} \}_{r=1}^M$ have full rank. In this example, though $\wv(l)$ is uniquely determined using any one of $M$ equations in (<ref>), checking if all $M$ equaions give the same $\bold{w}(l)$ reduces PAP in general cases.

still have to check

Pseudo code of the decoding algorithm of LLC still have to check

Input is of form $\Yv=(\yv_0 \ldots \yv_{L-1})$

$\forall l \in [L]$, $|\yv_l| \leq K;
% \bold{G}_{l_1,l_2} $ are full rank and fat.

\State $^1 ∅$; $_1 ∅$

\ForEach{$x_0 \in \yv_0$} \tikzmark{top}
\State $^1(x_0) findNoErasurePathsWithRoot(x_0)$
\State $^1 ^1 ∪^1(x_0)$
\EndFor

\State $_1 extractInfoBits(^1)$

\State $^2 ∅$; $_2 ∅$

\ForEach{$x_0^\times \in \yv_0$ doesn't lead to any element in $\widehat{\Wv}_1$}
\State $^2(x_0^×) findOneErasurePathsWithRoot(x_0^×)$
\State $^2 ^2 ∪^2(x_0^×)$
\EndFor

\State $_2 extractInfoBits(^2)$

\State $ _1 ∪_2$

\State \Return $$

% \If{$N$ is even}
%     \State $X \gets X \times X$
%     \State $N \gets \frac{N}{2}$  \Comment{This is a comment}
% \ElsIf{$N$ is odd}
%     \State $y \gets y \times X$
%     \State $N \gets N - 1$
% \EndIf

\end{algorithmic}
\end{algorithm}

\subsection{Further comments}

\begin{remark}
\sr{say no phase >1, but is trivial}
\end{remark}

\begin{itemize}
\item mention extension to more than one erasure
\end{itemize}

\section{ Related Results}
\label{sec:Related Results}
In our numerical simulations, we shall compare the performance of the LLC with the tree code from \cite{amalladinne2020coded} and the belief propagation code from \cite{amalladinne2021unsourced}.
We again stress that both these codes were not originally designed for the UACE model and thus their performance cannot be properly compared with that of the proposed approach.
We present these schemes solely because they provide the only available comparison of performance in the literature.
%at the moment

\subsection{Tree code}
The tree code of \cite{amalladinne2020coded} is a binary linear code in which the parity bits $p(l)$ increase at each section.
In decoding, it is assumed that $Y$ in \eqref{eq:channel error} contains only symbol insertions, as discussed in
Sec. \ref{sec:Motivation and Further comments}.
For this reason, the decoding algorithm is substantially equivalent to phase $1$ in \eqref{sec:Phase 1}, with the difference that the number of parity bits increases with the section number, thus allowing for a more efficient pruning of surviving paths.

\sr{wz: figure}
\begin{figure}[t!]
\centering
\input{figures/llc_vs_tc_vs_ldpc_performance}
\caption{Changes of $P_{\text{PAP}}$ and $P_{\text{PDP}}$ of LLC, Tree code and LDPC code under fixed $K=100$ and increasing $p_e$.}
\label{fig:acc_rate1}
\end{figure}

\begin{figure}[t!]
\centering
\input{figures/only_llc_changing_kpe}
\caption{Changes of $P_{\text{PAP}}$ and $P_{\text{PDP}}$ of LLC under changing $K$ and $p_e$.}
\label{fig:acc_rate1}
\end{figure}

\subsection{Belief Propagation Code}
Another feasible coding scheme for the UACE is the belief propagation code of \cite{amalladinne2021unsourced}.
In this code, each section is interpreted as a symbol in a non-binary LDPC code: accordingly, sections are classified in information sections and parity sections.
Parity sections are obtained from information sections as in a classic LDPC code.
In decoding, the FFT method for decoding non-binary LDPC codes is used to approximate belief propagation between parity sections and information sections.

\begin{figure}
\centering
\include{figures/ldpc_factor_graph}
\caption{\textcolor{blue}{update caption later}}
\label{fig:ldpc_factor_graph}
\end{figure}

\sr{WZ: add the figures from Jamison}

\section{Simulation Results}
\label{sec:Simulation Results}
In this section, we study the performance of the linked-loop code on the  A-channel with erasure with probability $p_e$, for realistic parameters of codes attuned to practical applications. We also provide a comparison with the existing method called tree code found in the []. We consider a system with $K_a ∈[50, 200]$ active users, each having $B=128$ bits of information to transmit. These bits are divided into $L=16$ sub-blocks, each of length $J$. For the tree code, the specific value for $J$ leads to different discriminating power and was chosen carefully. While for LLC, the value for $J$ is determined by rate only.

\begin{table}
\footnotesize
\centering
\vspace{0.04in}\caption{Relevant parameters}
%{This table contains key parameters encountered}
\label{tab:DNN parameters}
\begin{tabular}{|c|c|}
\hline
Notation & Parameter Description \\ \hline
%$K_a$ & Number of active users  \\ \hline
$B$ & Message length in bits \\ \hline
$L$ & Number of coded sub-blocks per round \\\hline
$J$ & Length of coded sub-blocks  \\ \hline
$N$ & Length of coded $B$-bit message, $N=LJ$ \\ \hline
$M$ & Number of previous sub-blocks used to generate parity bits\\ \hline
$m(l)$ & Number of information bits in $l$-th sub-block \\ \hline
$p(l)$ & Number of parity bits in $l$-th sub-block, $p(l) = J-m(l)$ \\ \hline
\end{tabular}
\end{table}

\sr{Some pictures here. }

\begin{figure}
\centering
\include{figures/llc_vs_tc_performance}
\caption{LLC vs TC Performance}
\label{fig:llc_vs_tc_performance}
\end{figure}

\section{Conclusion}
\label{sec:Conclusion}

\newpage
\bibliographystyle{plain}
\bibliography{IEEEabrv,A_channel}
\end{document}