#  Frequency Hopping Sequence Design and Demodulator Allocation Strategies for
LR-FHSS

Diego Maldonado1, Megumi Kaneko2, Juan A. Fraire13, Oana Iova1, Alexandre
Guitton4, Hervé Rivano1  1Univ Lyon, INSA Lyon, Inria, CITI, F-69621
Villeurbanne, France
2National Institute of Informatics, 101-8430, Tokyo, Japan
3Saarland University, Saarland Informatics Campus E1 3, 66123 Saarbrücken,
Germany
4Université Clermont-Auvergne, CNRS, Clermont-Auvergne-INP, LIMOS, 63000
Clermont-Ferrand, France

###### Abstract

The accelerating growth of the Internet of Things (IoT) and its integration
with Low-Earth Orbit (LEO) satellites demands efficient and reliable
communication protocols. Among these, the Long-Range Frequency Hopping Spread
Spectrum (LR-FHSS) modulation, tailored for LEO satellite IoT communications,
has seen growing interest. This paper deepens into Frequency Hopping Sequence
(FHS) mechanisms within LR-FHSS, spotlighting the potential of leveraging
Wide-Gap sequences. Additionally, we introduce three novel demodulator
allocation strategies, namely, “Early-Decode,” “Early-Drop,” and “Early-
Header-Drop,” aimed at optimizing the utilization of gateway decoding
resources. Our research further validates these findings with extensive
simulations, offering a comprehensive look into the future potential of LR-
FHSS in IoT settings.

###### Index Terms:

LR-FHSS, Frequency Hopping Spread Spectrum, Hamming Correlation, Optimal
Sequences

##  I Introduction

The rapid emergence of the Internet of Things (IoT) has led to a technological
revolution, allowing various devices to connect and facilitating seamless
communication and data exchange [nguyen20216g]. In this context, Low-Earth
Orbit (LEO) satellites have become instrumental in extending the coverage of
various IoT applications, where terrestrial infrastructures are unsuitable due
to the challenging geographical locations [fraire2022space]. These include the
environmental and ecological observation of oceans, poles, expansive forests,
and large natural parks [de2015satellite]. In these cases, LEO satellites are
more cost-effective than geostationary satellites largely because of their
closer proximity to Earth, reducing transmission delays and power
requirements.

As the number of IoT devices surges, the demand for efficient and reliable
communication protocols that address terrestrial and direct-to-space access
(a.k.a. Direct-to-Satellite IoT [fraire2019direct]) escalates. One solution
that gained substantial attention is Long-range (LoRa) modulation, which
stands out for its low power consumption, expansive coverage, and scalability.
LoRa has been successfully ported to the satellite context with increasing
interest from the academic and industrial communities (e.g., FossaSat,
LacunaSat).

In this context, Semtech introduced a new modulation technique called Long-
Range Frequency Hopping Spread Spectrum (LR-FHSS) [alliance2018lorawan].
Specifically designed for LEO satellite IoT communications, LR-FHSS ensures
connectivity between low-power terrestrial devices and satellites. Moreover,
LR-FHSS is compatible with traditional LoRa modulation, streamlining its
adoption and integration with terrestrial IoT infrastructures. Indeed,
existing LoRaWAN gateways can readily support this modulation with a simple
firmware update. Employing various narrow-band channels and a Frequency
Hopping technique, it dynamically changes channels based on a Frequency
Hopping Sequence (FHS) to evade interference.

Although the introduction of LR-FHSS is relatively recent, it has swiftly
garnered attention from the research community. Areas of focus include
performance evaluation [ullah2021analysis, boquet2021lr], outage probabilities
[maleki2022d2d, maleki2022outage], headerless recovery [fraire2023recovering],
and transceiver design [jung2023transceiver]. Yet, two pivotal topics await
further research and refinement: the sequence determining the frequency hop
pattern and the concurrent methodology for allocating demodulator processors
to multiple incoming frames.

##### Sequence Design

The FHS is a technique commonly employed in diverse communication systems to
enhance both security and the reliability of wireless transmissions
[bao2016frequency]. These sequences quickly alternate the carrier frequency
during data transmission, effectively neutralizing interference and jamming
threats [ge2009optimal]. A major strength of FHS is its ability to disperse
the transmitted frame over an extensive frequency range. This makes it notably
challenging for potential adversaries to intercept or interrupt the ongoing
communication [peng2004lower]. The introduction of Wide-Gap FHS (WGFHS) is a
recent advancement in this domain. This innovative sequence ensures a
guaranteed minimum gap between successive frequency hops and boasts optimal
attributes in terms of Hamming correlation [li2019new]. What makes WGFHS
particularly noteworthy is its compliance with standard requirements set by
regional regulators for maintaining gaps between adjacent frequency channels,
which is precisely the case of LR-FHSS. This adheres to regulations and
fortifies the anti-jamming capabilities of FHS systems [huaqing2010design,
bin1997one, guan2014generation]. Despite the promising potential of this new
FHS construction in the context of LR-FHSS modulation, to the authors’ best
knowledge, it remains an under-explored topic in existing literature.

##### Demodulator Allocation

Existing LoRaWAN gateways, primarily powered by Semtech’s SX1301 chip
[semtech2017sx1301], are equipped to concurrently demodulate several LoRa
signals as long as they operate on unique channels or different Spreading
Factors (SF). This SX1301 chip, a standard in most LoRaWAN gateways, houses
eight distinct demodulator chips. It continuously scans for preambles across
all channels and SFs, and upon detection, the integrated packet arbiter
decides which of the eight demodulators should demodulate the corresponding
frame. However, the limited count of eight demodulators can be a potential
bottleneck, potentially impairing the network’s overall performance
[sorensen2019analysis, dalela2019lorawan, magrin2020study]. Addressing this
limitation, authors in [guitton2020improving, guitton2022multi] proposed
better algorithms to harness these demodulators in individual gateway
configurations. The precise workings of the SX1301’s packet arbiter for LoRa
remain undocumented, and the architecture of LR-FHSS remains unknown to the
community. Indeed, LR-FHSS will likely rely on a more advanced chipset
specifically designed for demodulating FHS signals. The simultaneous
demodulation capacity of these newer chipsets is still unclear, but estimates
suggest that accommodating up to 1000 might be plausible [applicationnote].
Nevertheless, no official statement has confirmed this capability. To the
authors’ knowledge, there has been no proposed demodulator allocation
technique for LR-FHSS thus far.

###  I-A Contributions

This paper delves into the sophisticated mechanisms of FHS, mainly as they
apply to modern LR-FHSS systems. Our in-depth examination shows how FHS is
generated for LR-FHSS modulation, particularly emphasizing the Wide-Gap
property. This property emerges as a superior candidate, outclassing current
LR-FHSS driver implementations in certain scenarios. Through this, we equip
researchers and engineers with valuable knowledge to explore alternative FHS
sequences, aiming to enhance device-to-gateway extraction rates in LR-FHSS
systems.

Our second major contribution introduces three pioneering strategies for
demodulator allocation designed to amplify the scalability of LR-FHSS-based
networks. At the heart of these strategies is the ingenious use of
redundancies inherent in the payload’s convolutional encoding and decoding
processes. The first, termed “Early-Decode,” frees up demodulators once
they’ve gathered sufficient convolutional blocks for frame decoding. The
second, “Early-Drop,” proactively dismisses a packet if a certain number of
its blocks fail, liberating the processor for subsequent demodulation.
Finally, an “Early-Header-Drop” exploits the relevance of LR-FHSS header
information.

The detailed contributions made in this paper are:

  * •

Comprehensive FHS Review: We thoroughly review various types of FHS, paving
the way for a deeper understanding of their properties and implications in LR-
FHSS systems.

  * •

Wide-Gap FHS for LR-FHSS: By investigating the Wide-Gap FHS concept, we
propose an alternative FHS construction optimized for the specifics of LR-FHSS
modulation.

  * •

Demodulator Optimization: Three novel approaches - the Early-Decode, Early-
Drop, and Early-Header-Drop techniques - are introduced, tailored to amplify
the decoding capacities of LR-FHSS demodulators, thereby maximizing their
utility.

  * •

Simulation-Driven Validation: We put our propositions in value with an
expansive simulation campaign, shedding light on our proposed techniques’
practical benefits and potential implications for LR-FHSS systems.

The remainder of the paper is organized as follows. Section II presents the
system model, covering the network architecture and LR-FHSS modulation.
Section III offers an in-depth look at this study’s foundational topics. Our
proposed methods, encompassing FHS family constructions and algorithms, are
delineated in Section IV. Section V showcases and discusses our simulation
results. Conclusions and avenues for future research are drawn in Section VI.

##  II LR-FHSS Modulation Overview

LR-FHSS is a unique modulation technique that facilitates rapid frequency
hopping within a single frame (e.g., frame splitting). Among the modulations
endorsed for LoRaWAN devices, it is accompanied by LoRa and FSK
[Sornin2021-ue]. Primarily tailored for uplink communications, LR-FHSS proves
particularly efficient when energy-limited devices transmit to satellites
stationed in Low Earth Orbit (LEO). Nonetheless, LoRa is still the modulation
for downlink transmissions. Significantly, LoRaWAN gateways can be
effortlessly updated via firmware to decode LR-FHSS frames, all while
preserving the undisturbed reception and decoding of LoRa frames.
Significantly, the model gateway design empowers a gateway to monitor all
accessible sub-channels concurrently.

LR-FHSS utilizes Gaussian Minimum-Shift Keying (GMSK) modulation within its
sub-channels, prized for its exceptional spectral and energy efficiency. By
encoding data via distinct frequency shifts and employing a Gaussian filter,
GMSK effectively reduces the sidebands emerging from the modulation process.
This trait aligns perfectly with the needs of narrow-band communications.
Notably, GMSK’s prominence is evident in its adoption in systems like the
traditional Global System for Mobile (GSM) and various satellite communication
frameworks.

LR-FHSS segments the frequency spectrum into a series of contiguous Occupied
Channel Widths (OCWs). Each OCW is further partitioned into multiple Occupied
Bandwidths (OBWs) [boquet2021lr]. As illustrated in Fig. 2, an individual OBW
acts as a physical narrow-band sub-carrier with a bandwidth set at 488 Hz. To
align with regional regulations stipulating the minimum spacing between sub-
carriers in the frequency hopping sequence, OBWs are systematically arranged
into grids. These grids play a pivotal role in LR-FHSS: a hopping sequence
within any given frame is restricted to using OBWs residing within the same
grid. Grids comprise non-sequential OBWs spaced apart by a mandated minimum
distance, which is subject to change based on regional regulatory standards. A
summary of the LR-FHSS modulation parameters for chosen channels is presented
in Table I. Details on the OCW channel bandwidth, OBW channel tally, and the
count of grids endorsed by the LR-FHSS modulation can be gleaned from
[applicationnote] and are concisely laid out in Table II.

TABLE I: LR-FHSS Modulation Parameters [Sornin2021-ue] Region | EU863-870 MHz | US902-928
---|---|---
LoRaWAN DR | DR8 | DR9 | DR10 | DR11 | DR5 | DR6
OCW [kHz] | 137 | 336 | 1523
OBW [Hz] | 488
Grid spacing [kHz] | 3.9 | 25.4
FH channels | 35 | 86 | 60
OBW channels | 280 (8x35) | 688 (8x86) | 3120 (52x60)
Coding Rate | 1/3 | 2/3 | 1/3 | 2/3 | 1/3 | 2/3
Header Replicas | 3 | 2 | 3 | 2 | 3 | 2
Max. $L$ [bytes] | 58 | 123 | 58 | 123 | 58 | 133
Max. $N_{F}$ | 31 | 32 | 31 | 32 | 31 | 34
TABLE II: LR-FHSS Channel Parameters [applicationnote] OCW Channel Bandwidth [kHz] | OBW channels | Hopping Grids
---|---|---
| 3.9 kHz
---
spacing
| 25.4 kHz
---
spacing
39.06 | 80 | 10 (1) | x
85.94 | 176 | 22 (1) | x
136.70 | 280 | 35 (1) | x
183.59 | 376 | 47 (1) | x
335.94 | 688 | 86 (3) | x
386.72 | 792 | 99 (3) | x
710.90 | 1480 | 185 (4) | x
722.66 | x | 28 (1)
761.70 | 1584 | 198 (4) | x
773.44 | x | 30 (1)
1523.4 | 3120 | 390 (5) | 60 (2)
1574.2 | 3224 | 403 (5) | 62 (2)

Frame transmission initiates with the broadcast of multiple header copies,
each lasting 233 ms in airtime, followed by the segmented payload, with each
segment enduring 102.4 ms in airtime. These segments are relayed over varied
OBWs within a single grid. Every header copy conveys the index of the chosen
frequency hopping sequence. Depending on the chosen data rate scheme, every
copy is reiterated two or three times to ensure reliable reception. The
plethora of sequence options virtually eliminates the possibility of multiple
frames adopting an identical hopping sequence. Payload encoding is designed to
keep data integrity intact even with substantial fragment loss [boquet2021lr].
The LoRaWAN standard prescribes two LR-FHSS configurations:

  * •

Robust Configuration: Triply transmitted headers, coupled with a payload
encoded at CR = 1/3, meaning successful frame recovery demands a minimum of
33% fragment decoding.

  * •

Fast Configuration: Headers are transmitted twice, with payload encoding set
at CR = 2/3. Here, at least 67% of fragments need decoding for successful
frame recovery.

Despite its novelty, LR-FHSS technology has rapidly gained traction in
research. Primarily analytical or simulation-based, initial studies have been
foundational in illuminating its capabilities. Boquet et al. [boquet2021lr]
pioneered assessing LR-FHSS performance, spotlighting its potential to boost
network capacity by up to two orders of magnitude over LoRa. Ullah et al.
[ullah2021analysis] furthered this exploration through simulations,
pinpointing header losses as the predominant factor in LR-FHSS frame loss.
Subsequently, Maleki et al. [maleki2022d2d] provided an analytical perspective
on the outage probability using a comprehensive channel model, an effort they
broadened in [maleki2022outage] to encompass outage probabilities within DtS-
IoT networks. Innovations in headerless recovery techniques
[fraire2023recovering] and the intricacies of transceiver design
[jung2023transceiver] have recently been published.

###  II-A LR-FHSS in DtS-IoT

The architecture of a Direct to Satellite (DtS) IoT network is illustrated in
Fig. 1. In this setup, a satellite, orbiting at a Low Earth Orbit (LEO)
roughly 500 km above the Earth, acts as the gateway for terrestrial IoT
devices. This satellite moves at a speed of about 7.5 km/s and completes an
orbit around the Earth in roughly 90 minutes. Within its coverage area, the
satellite communicates with a cluster of IoT nodes on the ground.

Figure 1: Data Frames along a LEO satellite pass [vogelgesang2021uplink]

Figure 2: Division into OCWs, OBWs, and grids for 1 OCW of 137 kHz of the
European 868 MHz frequency [fraire2023recovering]

In the research conducted by authors in [vogelgesang2021uplink], a
communication protocol based on LoRaWAN Class B, as defined in the LoRaWAN
Specification by the LoRa Alliance [alliance2018lorawan], was put forth. Each
communication frame initiates with a downlink phase, during which the gateway
signals its presence by broadcasting a beacon. This is followed by an uplink
window, wherein the nodes send transmissions to the satellite. Traditional
LoRa modulation is employed for the downlink [applicationnote]. In contrast,
the novel LR-FHSS modulation is exclusively utilized for the uplink
[alliance2018lorawan], an aspect impacted by its collision rate affecting
scalability. In such a setup, a satellite’s coverage area might encompass
thousands of nodes, all of which could attempt to communicate with the gateway
within a single frame.

The free space channel model is especially fitting for LR-FHSS in satellite
communications due to outer space’s open and unobstructed nature. Contrary to
the cluttered terrestrial environments, space provides a clear signal path,
validating the model’s straightforwardness as an effective means of
characterizing signal interactions between satellites and terrestrial stations
[vogelgesang2021uplink]. Such a channel becomes even more pertinent when
factoring in the vast distances involved, which are not typically accounted
for in standard urban and suburban IoT settings [gonzalez2021analysis].
Furthermore, frequency-selective fading is negligible, given the narrow-band
channels in LR-FHSS.

In isolated gateway deployment contexts, like satellite communications or
sparsely populated rural areas, LR-FHSS has demonstrated superior network
capacity [ullah2021analysis, boquet2021lr]. Conversely, LoRa exhibits
comparable, if not superior, performance in densely populated urban regions.
For both modulation types, the overarching constraint is the capacity of
Digital Signal Processing (DSP). Conventional LoRa gateways have up to 8 DSPs
for concurrent decoding [magrin2019thorough]. Present implementations suggest
that the Uplink capacity of LR-FHSS per Gateway per Day stands at 700,000
packets [applicationnote].

In this context, we will delve into the intricate methodology of allocating
demodulator processors to manage a simultaneous influx of frames in Section
IV. In this section, we also introduce a novel adaptation tailored for the LR-
FHSS hopping sequence design. This adaptation’s comprehensive background is
first provided in the next Section III.

##  III Frequency Hopping Sequences

Frequency Hopping Sequences (FHSs) are the backbone of FHSS communication
systems like LR-FHSS, dictating the rhythm and sequence of frequency
transitions. In the realm of FHSS, there’s a swift and systematic variation in
the transmission frequency, oscillating across a broad spectrum of frequencies
designated within a specific band. The FHS essentially choreographs the
sequence and timing of these frequency shifts. Often, these sequences spring
from mathematical algorithms or are birthed from pseudo-random number
generators. The hallmark of these sequences is their unpredictability and non-
repetitive nature, ensuring that potential adversaries find it challenging to
predict the hopping pattern, thereby safeguarding the transmission from
interception or jamming attempts [Torrieri2018]. The inception of FHS is
motivated by the twin goals of fortifying security and enhancing the
reliability of wireless communication infrastructures. The agility with which
FHSS systems transition between frequencies protects against interference from
contemporaneous devices operating on overlapping frequency bands. This
dynamism renders FHSS systems considerably more resilient against threats like
jamming and eavesdropping compared to their fixed-frequency counterparts
[Torrieri2018].

For this investigation, we define the set of accessible frequency channels in
an FHSS system as $\mathcal{F}=\\{f_{0},f_{1},\cdots,f_{\ell-1}\\}$, where
each $f_{i}$ represents a specific frequency and $\ell$ stands for a positive
integer value.

A nuanced characteristic often sought in FHS is the attribute of a wide gap. A
wide-gap FHS (WGFHS) is characterized by a sequence wherein consecutive
frequencies possess a substantial gap or distance between them [li2019new,
li2021constructions]. In the case of LR-FHSS, the concept of grids is a means
to achieve such a wide gap property.

Consider $X$ as an FHS spanning over $\mathcal{F}$ with a period denoted by
$L$. The smallest gap or distance $e$ between successive frequencies is
delineated as:

$e=\min_{0\leq i\leq L-2}\\{|x_{i+1}-x_{i}|,|x_{L-1}-x_{0}|\\}.$ (1)

In situations where $e>0$, the sequence $X$ is classified as a WGFHS,
possessing a minimum gap of $e$.

###  III-A Hamming Correlation Metrics

Hamming Correlation provides a quantifiable metric to determine the similarity
or correlation between two FHSs [sarwate1994reed]. Given a family
$\mathcal{S}$ comprising $M$ FHSs over $\mathcal{F}$, each FHS within this
family is considered to be periodic with a specified length (or period) $L$.
Additionally, it is presumed that the FHS does not contain repetitive
frequencies, necessitating that $L\leq\ell$. With $X=\\{x_{i}\\}$ and
$Y=\\{y_{j}\\}$ belonging to $\mathcal{S}$, the Hamming Correlation between
$X$ and $Y$ at an offset $\tau$ is formulated as:

$H(X,Y;\tau)=\sum_{i=0}^{L-1}h(x_{i},y_{i+\tau\mod L}),\hskip
14.22636pt0\leq\tau\leq L-1,$ (2)

where the function $h$ is defined as:

$h(f_{i},f_{j})=\begin{cases}1,&\text{if}\ f_{i}=f_{j}\\\
0,&\text{otherwise}.\end{cases}$ (3)

In the above expressions, $\tau$ denotes the temporal displacement, and the
term $i+\tau\mod L$ signifies taking the modulo $L$ after adding $i$ and
$\tau$ to ensure the result remains within the bounds of the sequence length.

####  III-A1 Cross-Correlation and Auto-Correlation

The Hamming Correlation defined earlier can be bifurcated into two main types
based on the sequences being compared:

  * •

Auto-Correlation (AC): This represents the correlation of a sequence with
itself, i.e., $X=Y$. The shift value $\tau=0$ is not interesting for AC since
it would lead to a perfect match. The maximum AC across all possible non-zero
shifts can be expressed as:

$H_{\mathrm{max}}(X)=\max_{1\leq\tau\leq L-1}H(X,X;\tau),$ (4)

  * •

Cross-Correlation (CC): This quantifies the similarity between two different
sequences, i.e., $X\neq Y$. The maximum CC across all possible shifts,
including zero, is articulated as follows:

$H_{\mathrm{max}}(X,Y)=\max_{0\leq\tau\leq L-1}H(X,Y;\tau).$ (5)

Further, the mean of the maximum AC values for every sequence in $\mathcal{S}$
is denoted as:

$\bar{H}_{\mathrm{max}}(\mathcal{S})=\frac{1}{M}\sum_{X\in\mathcal{S}}H_{\mathrm{max}}(X),$
(6)

Meanwhile, the mean of the maximum CC values, considering every possible
unique pair of sequences within $\mathcal{S}$, is represented as:

$\bar{H}_{\mathrm{max}}(\mathcal{S},\mathcal{S})=\frac{1}{M(M-1)}\sum_{X,Y\in\mathcal{S},X\neq
Y}H_{\mathrm{max}}(X,Y).$ (7)

These metrics provide valuable insights into the distinctiveness and
similarity of FHSs within the family $\mathcal{S}$, vital for evaluating
system performance and security.

####  III-A2 Average CC and AC

Expanding upon the metrics we previously discussed, another valuable metric
emerges: the average Hamming Correlation. While we have previously inspected
the maximum correlation values to understand the peaks in similarity or
identity between sequences, it’s also insightful to consider their average
behavior across all shifts. The average Hamming Correlation for a given
sequence $X$ when compared with itself (Auto-Correlation) over all shifts
$\tau$ is described by:

$H_{\mathrm{avg}}(X)=\frac{1}{L-1}\sum_{X\in S,1\leq\tau\leq L-1}H(X,X;\tau),$
(8)

Conversely, when comparing sequence $X$ to another sequence $Y$ (Cross-
Correlation), the average can be computed using:

$H_{\mathrm{avg}}(X,Y)=\frac{1}{L}\sum_{X,Y\in S,X\neq Y,0\leq\tau\leq
L-1}H(X,Y;\tau).$ (9)

Diving deeper, the mean of these average Auto-Correlation values for every
sequence in $\mathcal{S}$ is:

$\bar{H}_{\mathrm{avg}}(\mathcal{S})=\frac{1}{M}\sum_{X\in\mathcal{S}}H_{\mathrm{avg}}(X),$
(10)

On the other hand, the mean of the average Cross-Correlation values, when
considering every possible unique pair of sequences within $\mathcal{S}$, is
represented as:

$\bar{H}_{\mathrm{avg}}(\mathcal{S},\mathcal{S})=\frac{1}{M(M-1)}\sum_{X,Y\in\mathcal{S},X\neq
Y}H_{\mathrm{avg}}(X,Y).$ (11)

Together, these metrics—both maximum and average—offer a comprehensive
understanding of the relationships and characteristics of sequences within the
set $\mathcal{S}$, crucial for optimizing and ensuring the reliability of FHSS
communication systems.

####  III-A3 Desired Hamming Correlation Properties

In the domain of FHSS communication, the essence of effective and reliable
communication boils down to how individual FHSs interact. Precisely, we desire
each FHS to exhibit minimal similarity with itself over various shifts—this
self-similarity measure is termed auto-correlation. Similarly, the similarity
between any pair of distinct FHSs—denoted as cross-correlation—should also be
minimized. These measures ensure resistance to interference, jamming, and
unauthorized interception, resulting in enhanced communication reliability.

The seminal work of Lempel and Greenberger [lempel1974families] provided
theoretical bounds for this minimal similarity. Specifically, for any FHS $X$
with a period $L$ across a frequency set $\mathcal{F}$, they proposed that the
auto-correlation $H_{\mathrm{max}}(X)$ cannot go below a certain threshold,
mathematically represented by:

$H_{\mathrm{max}}(X)\geq\biggl{\lceil}\frac{(L-\epsilon)(L+\epsilon-\ell)}{\ell(L-1)}\biggr{\rceil},$
(12)

where the parameter $\epsilon$ is defined as the difference between $L$ and
the largest multiple of $\ell$ less than $L$.

Based on this bound, Lempel and Greenberger pioneered one of the first
construction techniques to design FHSs that optimize the Hamming Correlation
metric. Their approach leveraged the properties of an $m$-sequence—a specific
type of pseudo-random binary sequence. Their contributions did not just stop
at individual sequences; they also established bounds for pairs of FHSs
constrained by the metric:

$M(X,Y)=\max\\{H_{\mathrm{max}}(X),H_{\mathrm{max}}(Y),H_{\mathrm{max}}(X,Y)\\},$
(13)

where this bound was expressed as:

$M(X,Y)\geq p^{n-k}.$ (14)

Such findings opened doors to the concept of optimal FHS families. An FHS
family $\mathcal{S}$ achieves ”optimality” if every distinct pair of sequences
within it satisfies the equality in Eq. (14). Further extending their initial
construction approach, Lempel and Greenberger proposed a second technique
based on $m$-sequences, targeting the generation of these optimal FHS
families.

However, the domain has since seen further enhancements. In a subsequent study
by Li and colleagues [li2019new], these bounds were extended to accommodate
FHSs with an additional wide-gap property, leading to a new threshold defined
as:

$H_{\mathrm{max}}(X)\geq\biggl{\lceil}\frac{(L-\epsilon)(L+\epsilon-\ell)}{\ell(L-3)}\biggr{\rceil}.$
(15)

Then, $X$ is said to be an optimal WGFHS if it achieves the equal sign in Eq.
(15). In [li2019new], the author also extends the bound for FHS families found
in [peng2004lower].

Next, we describe the benchmark methods for constructing FHSs: the general
$m$-sequences, a WGFHS from [li2021constructions], and one specific to LR-
FHSS.

###  III-B Construction Techniques

####  III-B1 Constructing Maximum Length Sequences ($m$-sequences)

At the heart of many sequence generation techniques lies the Linear Feedback
Shift Register (LFSR), a mechanism where the input is derived from a linear
combination of prior states. The essential characteristics of an LFSR include:
i) Generator Polynomial, which dictates the recurrence relation for the
sequence generation, and ii) Initial State Vector, often termed the state
vector or seed, is an $m$-bit binary integer that initializes the sequence
generation process.

Two primary configurations of LFSRs are recognized: the Fibonacci and the
Galois, with references for their implementations given in
[goresky2012algebraic]. For illustrative purposes, consider Fig. 3. It
showcases a typical Galois LFSR setup with the following: i) $m$ registers;
ii) an initial state, denoted as $h^{0}$, is represented in the figure as $h$;
and iii) a generator polynomial, $q$, which carries binary coefficients
enumerated as $\\{q_{m},\cdots,q_{1}\\}$.

The progression from one state to the next, say from $h^{0}$ to $h^{1}$, is
mathematically represented by:

$h_{i}^{1}=h_{i+1}^{0}+q_{i+1}h_{0}^{0},\hskip 8.53581pt0\leq i<m-1,$ (16)
$h_{m-1}^{1}=q_{m}h_{0}^{0}=h_{0}^{0}.$ (17)

It’s a standard practice to ensure $q_{m}\neq 0$ as outlined in Eq. (17).
Using the formula in Eq. (16), the subsequent state, $h^{2}$, derives its
value from $h^{1}$.

Now, focusing on the output, the sequence, denoted as $s$, emerges as a
pseudo-random binary sequence and is expressed as
$s=\\{h_{0}^{0},h_{0}^{1},h_{0}^{2},\cdots\\}$. Given that the register has a
capped number of potential states (precisely $2^{m}$), it’s inevitable to
circle back and repeat states. Yet, with a primitive polynomial governing the
LFSR, the resultant sequence, $s$, mimics randomness and boasts an extensive
cycle length. Such sequences earn the title Maximum Length Sequence or simply
$m$-sequence. Notably, their period stands at $2^{m}-1$ [Torrieri2018].

Figure 3: Galois Linear Feedback Shift Register

####  III-B2 Constructing WGFHS with Li-Fan’s Approach

In the research article [li2021constructions], two distinct methodologies for
constructing WGFHS families are presented.

The first construction methodology (theorem 1) yields an optimal WGFHS with a
period $L=2\ell$. The fundamental parameters are two positive integers, $\ell$
and $d$, with the constraints $1<d<\ell/2$ and both $gcd(\ell,d)$ and
$gcd(\ell,d+1)$ equating to 1. Here, $gcd()$ symbolizes the greatest common
divisor. Two sequential structures, $s=\\{s_{i}\\}_{i=0}^{i=\ell-1}$ and
$t=\\{t_{i}\\}_{i=0}^{i=\ell-1}$, are crafted based on the formulas:

$\displaystyle s_{i}$ $\displaystyle=(i\cdot d)_{\ell},\hskip 8.53581pt0\leq
i<\ell,$ $\displaystyle t_{i}$ $\displaystyle=(i\cdot(d+1)+1)_{\ell},\hskip
8.53581pt0\leq i<\ell.$

In these equations, the term $(\cdot)_{\ell}$ represents the smallest
nonnegative residue under the modulus $\ell$. Now, by amalgamating sequences
$s$ and $t$, a new sequence $\mathbf{S}$ emerges with a length $L=2\ell$. This
sequence, $\mathbf{S}$, stands out as an optimal WGFHS over the domain
$\mathcal{F}$ with a size $\ell$. Its characteristics include $H(X)=2$ and a
minimal gap of $e=d-1$.

The second construction methodology (theorem 2) differs in its outcome. It
produces an optimal WGFHS with a longer period, $L=3\ell$, over the domain
$\mathcal{F}$ with the same size $\ell$. The sequence’s defining features
include $H(X)=3$ and a minimum gap of $e=d-1$.

####  III-B3 Constructing LR-FHSS Sequences wit Semtech’s Driver

The construction technique for FHSS sequences tailored for LR-FHSS devices, as
found in an official LoRa repository on Github111SX126X radio driver:
https://github.com/Lora-net/sx126x_driver, diverges from the method detailed
in [boquet2021lr]. Rather than relying on a hash function, it harnesses the
potential of $m$-sequences. This methodology is illustrated in Fig. 4.

Figure 4: LR-FHSS Device’s FHS Generation Scheme Using the Driver

Table III provides an exhaustive breakdown of the LR-FHSS sequence generator’s
configurations assumed in the driver. Five distinct configurations arise based
on the number of grids they support, as further elaborated in Table II (each
configuration is marked by a specific case in brackets). It’s worth noting
that for cases 1 and 2, a 6-register LFSR is employed. Yet, with this
configuration, only six primitive polynomials exist, meaning the FHS families
produced for LR-FHSS devices in these scenarios are limited to 384 sequences.

TABLE III: LR-FHSS FHS Construction Parameters case | 1 | 2 | 3 | 4 | 5
---|---|---|---|---|---
initial state | 6 | 56 | 6
LFSR size | 6 | 7 | 8 | 9
polynomial bits | 3 | 2 | 1 | 0
polynomial set |  | {33,45,48,
---
51,54,57}
| {65,68,
---
71,72}
{142,149} | {264}
seed bits | 6 | 7 | 8 | 9
total seeds | 64 | 128 | 256 | 512
FHS family size | 384 | 512

The generator block for the $m$-sequence, positioned at the top of the figure,
signifies an LFSR comprising $m$ registers. The primitive polynomial,
instrumental in the $m$-sequence’s creation, is encoded within the $x$ most
significant bits of the 9-bit Hopping Sequence field conveyed in the Header,
where $x$ can take values from the set $\\{0,1,2,3\\}$. The modulation
parameters inform the determination of the initial state. At the bottom, a
constant xoring seed is coded into the remaining $y=9-x$ least significant
bits of the 9-bit Hopping Sequence field. The LR-FHSS sequence is derived by
performing a bit-wise exclusive OR (often referred to as xoring, synonymous
with modulo 2 addition) between each component of the generated $m$-sequence
and the xoring seed. The sequence’s length hinges on the necessary hops to
encapsulate the payload data and header replicas. Furthermore, if the
resulting values after the operation surpass the count of supported grids in a
given scenario, they are omitted. Consequently, the next element in the
sequence is selected. This selective process trims down the LR-FHSS sequence’s
length from the original $m$-sequence length (i.e., $2^{m}-1$) to the exact
count of the grids.

As a result, the driver family is composed of the $M=384$ possible sequences
that can be generated by the LR-FHSS Driver method III-B3 for data rates DR8
and DR9, all of length $L=31$.

##  IV Proposed Methods for LR-FHSS

This section introduces sequences crafted by harnessing the WGFHS principle
delineated in Section III-B. These sequences are tailored to align with the
stipulations of the LR-FHSS regulations. Additionally, we put forward a range
of strategies designed to enhance the decoding proficiency of the LoRa
demodulators stationed in the LEO satellite.

###  IV-A FHS Families for LR-FHSS

This section delves into the construction of FHS families tailored for the LR-
FHSS protocol. The families constructed can be categorized based on the design
principles derived from Lempel-Greenberger (see Section III-B1) and Li-Fan
(see Section III-B2).

####  IV-A1 Lempel-Greenberger FHS Families

The Lempel-Greenberger FHS construction involves selecting parameters
$\\{p,k,n\\}$ and appropriate primitive polynomials available for LFSR of a
given order. This method facilitates the generation of $M=p^{k}$ sequences of
length $q=p^{n}-1$.

  * •

Example 1: lem-green Family:

    * –

Parameters: $p=2,k=5,n=5$

    * –

Primitive Polynomial: One of the 6 primitive polynomials available for LFSR of
order 5.

    * –

Outcome: 32 sequences of length 31.

  * •

Example 2: lem-green-2x Family:

    * –

Composition: Merge of the lem-green family and a second family.

    * –

Parameters: The same set as in Example 1 but with a different primitive
polynomial.

####  IV-A2 Li-Fan FHS Families

The Li-Fan FHS construction provides families based on the methods of the
$2\ell$ or $3\ell$. Depending on the chosen parameters, families might need
further splitting or merging to attain desired frequency values and lengths.

  * •

Example 1: li-fan-2l Family:

    * –

Method: $2\ell$-method.

    * –

Parameters: $\ell=281,d=8$

    * –

Construction Specifics: Single FHS of length $2\ell=562$, further split into
18 sequences of length 31.

  * •

Example 2: li-fan-2l-4x Family:

    * –

Composition: Merged family from li-fan-2l and three other families.

    * –

Additional Parameters: $\ell=277,\ell=283,\ell=287$ all with $d=8$.

    * –

Adjustments: Frequency values above 280 are discarded, and the minimum gap is
adjusted to 8.

  * •

Example 3: li-fan-3l Family:

    * –

Method: $3\ell$ FHS construction.

    * –

Parameters: Same as the li-fan-2l family.

  * •

Example 4: li-fan-3l-4x Family:

    * –

Composition and method: Analogous to the li-fan-2l-4x family.

By employing the above design principles and methods, we tailored FHS families
that are optimized for the LR-FHSS protocol. These are crucial as they lay the
foundation for enhancing the decoding proficiency of LoRa demodulators, a
significant step forward for satellite-based communications.

###  IV-B Early Drop and Early Decode mechanisms for LR-FHSS

The satellite-based LR-FHSS gateway’s primary functionality can be
conceptualized as follows:

  * •

Transmission Receipt: Upon the LoRa gateway’s reception of a transmission from
a satellite, an available demodulator springs into action. Its core
responsibilities encompass collision monitoring and the capture of headers and
any fragmented data in the transmission.

  * •

Post-reception Analysis: After completing the reception process, the
demodulator enters an assessment mode. Here, it gauges the health of the
packet. Essential parameters evaluated during this phase include the
successful decoding of at least one header and the receipt of the minimum
necessary fragments for the intended payload decoding, ensuring they are free
from collisions.

  * •

Demodulator Availability: In instances where no demodulators are on standby or
accessible, the transmission in question is sidestepped. As a result, the
corresponding packet faces discarding.

This structured approach streamlines the gateway’s operation, ensuring
efficiency and minimal data loss.

Given the encoding structure of the payload, there’s potential to decode it
successfully, even if only a portion of the fragments are available. This
suggests an opportunity to utilize each demodulator’s capacity better. To
further bolster decoding efficiency, we recommend the adoption of the
subsequent mechanisms:

####  IV-B1 Early Drop

The “Early Drop” mechanism allows a demodulator to preemptively terminate the
reception of a packet when it ascertains that achieving the minimum fragment
count needed for successful decoding is no longer possible. This strategy
saves valuable processing time and increases the efficiency of the satellite’s
demodulation resources. To elucidate the concept, consider a payload
consisting of 13 fragments. Depending on the encoding scheme:

  * •

Code Rate 1 (CR1): A minimum of 5 fragments is essential for successful
decoding.

  * •

Code Rate 2 (CR2): 9 fragments are required.

Figure 5 delineates these configurations, where each fragment corresponds to a
frequency channel. Red highlights represent collisions (unsuccessfully
received fragments), while the green ones signify successfully received
fragments.

In a CR1 scenario, after recognizing the 9th collision, it becomes clear that
only a maximum of 4 successful fragments can be accumulated, which is
inadequate for decoding. Recognizing this, the demodulator can promptly
categorize the payload as ’collided’ and free itself for other tasks.
Analogously, for CR2, the demodulator would adopt a similar course of action
after the 5th collision. The steps involved in this process are detailed in
Algorithm 1.

Figure 5: Illustration of Early Drop for a 13-fragment payload

1

2while  _fragments are being received_ do

3 collidedFrags $\leftarrow$ number of collided fragments

4 threshold $\leftarrow$ minimum fragments necessary for decoding

5 if  _collidedFrags $>$ threshold_ then

6 increment collidedPayloads

7 terminate current reception

8 Return

9 end if

10

11 end while

12

Algorithm 1 Early Drop

####  IV-B2 Early Decode

The “Early Decode” mechanism enables the demodulator to commence the decoding
process once it has collected the minimum number of fragments required for
successful payload decoding. This optimization lets the demodulator quickly
process and release packets, enhancing throughput and efficiency. Using the
same fragment configurations detailed above:

  * •

Code Rate 1 (CR1): The decoding process initiates immediately after the
successful reception of the 5th fragment.

  * •

Code Rate 2 (CR2): Decoding begins after the successful receipt of the 9th
fragment.

Fig. 6 showcases these configurations with various fragment collision
statuses. Once the required number of fragments is received successfully, the
demodulator has adequate data to decode the payload. It can then ready itself
for the reception of subsequent packets. Algorithm 2 offers an in-depth
procedural description of the Early Decode mechanism.

Figure 6: Representation of Early Decode for a 13-fragment payload

1

2validFrags $\leftarrow$ number of successful fragments

3threshold $\leftarrow$ minimum fragments for decoding

4if  _validFrags $\geq$ threshold_ then

5

6 collidedHdrs $\leftarrow$ number of collided headers

7 if  _collidedHdrs $<$ headerReplicas_ then

8 decodedPackets $\leftarrow$ decodedPackets +1

9 end if

10

11 decodedPayloads $\leftarrow$ decodedPayloads +1

12 clear current reception

13 Return

14 end if

Algorithm 2 Early Decode

####  IV-B3 Early Header Drop

In our extended gateway model, we introduce the “Early Header Drop” feature,
aiming to portray an LR-FHSS gateway’s operation accurately. The importance of
a header in packet communication cannot be overstated, as it carries the FHS
information crucial for decoding the corresponding packet. Thus, if a header
isn’t successfully decoded, the processor remains uninformed about the payload
fragment sequence sequence. In such situations, the gateway’s logical course
of action would be to release the demodulator, enabling it to track subsequent
packets.

While our research primarily focuses on assessing FHS performance within LR-
FHSS networks, evaluating the “Early Header Drop” mechanism is largely
excluded from most tests in Section V. Still, the study of decoding payloads
without their associated headers holds considerable potential and significance
in LR-FHSS research, as noted in [fraire2023recovering]. For a comprehensive
understanding of the “Early Header Drop” mechanism, refer to Algorithm 3.

1

2validFrags $\leftarrow$ count of successfully received fragments

3threshold $\leftarrow$ minimum fragments required for decoding

4if  _validFrags $\geq$ threshold_ then

5 collidedHdrs $\leftarrow$ count of collided headers

6 if  _collidedHdrs $<$ headerReplicas_ then

7 decodedPackets $\leftarrow$ decodedPackets +1

8 end if

9

10 increment decodedPayloads

11 terminate current reception

12 Return

13 end if

Algorithm 3 Early Header Drop

##  V Results and Analysis

###  V-A Simulation Settings and Parameters

We constructed a simplified but representative simulation model to assess the
performance implications of various FHS families in an LR-FHSS network. This
model encapsulates fragments’ and headers’ time-slotted communication dynamics
within finite frequency channels.

####  V-A1 Simulation Model Overview

Each fragment in the simulation persists for $g$ time slots, where $g$ denotes
the model’s granularity. Correspondingly, headers last for
$\bigl{\lfloor}7g/3\bigr{\rfloor}$ slots. Throughout the simulation duration,
$N$ nodes engage in transmitting a packet of fixed length $L$, broken down
into fragments. These transmissions are initiated at random time slots. The
choice of an FHS for each node is randomized from the available FHS family.
The gateway employed in the model is equipped with $p$ demodulators, essential
for decoding the packets.

A collision is deemed to have occurred if two or more fragments or headers
share at least one time slot on an identical frequency channel. Payloads
receive a ’collided’ designation if their collided fragments breach a certain
threshold, contingent on the CR utilized for encoding. Additionally, a
parameter determines the header’s collision tolerance, defining the maximum
collided slots permissible for a successful decode. Our simulator’s codebase
is publicly accessible on Github222LR-FHSS simulator:
https://github.com/diegomm6/lr-fhss_seq-families.

####  V-A2 Simulation Configuration

The simulations were primed using the parameters outlined in Table IV. With a
fragment count of 31 and a time slot granularity of 6, the maximum sequence
duration arises during CR1 encoding (factoring in 3 header repetitions),
culminating in a transmission span of 228 time slots. In contrast, CR2-encoded
transmissions wrap up in 214 time slots. We chose the simulation time as a
fourfold duration of the longest sequence, totaling 912 time slots. Diego:
this section has no units. This is confusing. Are we talking about seconds?

TABLE IV: Simulation Configuration Parameter | Value
---|---
Number of repetitions | 10
Simulation time | 912 [slots]
OCW channels | 7
OBW channels | 280
Payload size | 31 [fragments]
Fragment size | 6 [slots]
Header size | 14 [slots]

The ratio of fragment-to-header durations in our simulation, which stands at
$14/6\approx 2.33$, is a close reflection of real-world values approximated at
$233/102.4\approx 2.27$.

####  V-A3 Parameters Under Study

Table V delineates the parameters that underwent scrutiny in our research,
each having two feasible values. Both the Early Decode and Early Drop features
can be toggled on or off individually. Our control parameter, the node count,
is the transmission tally since each node transmits once within the simulation
timeframe. Across different scenarios in the ensuing sections, the node (or
transmission) count oscillates between 10 and 10,000.

TABLE V: Study Parameters Parameter | Possible values
---|---
Coding Rate | cr1, cr2
FHS family | lifan, driver
Number of demodulators | 100, 1000
Early Decode | True, False
Early Drop | True, False
Header Drop | True, False
Header tolerance in slots | 0, 4

In their work, Boquet et al. [boquet2021lr] pointed out that the FHS
generation method in LR-FHSS pivots around a hash function. Drawing
inspiration from this methodology, we formulated a hash family for FHS. This
curated FHS family encompasses $M=384$ families, each with a length of $L=31$.

Diego: you do not mention header drop and tolerance. These need to be
discussed clearly here.

###  V-B Preliminary Frequency Hopping Sequence Analysis

####  V-B1 Hamming Distance Analysis

The summary of FHS families can be found in Table VI, where they are
distinguished based on their size $M$ and the family’s maximum Hamming Cross
Correlation, denoted as $\bar{H}_{\mathrm{max}}(S,S)$, as formalized in Eq.
(7). The Hamming Correlation of the driver and lifan FHS families are assessed
in relation to the FHS length. For the construction of the driver family as
detailed in Section III-B3, its family size remains steady at 384 FHSs.
Conversely, the size of the lifan family fluctuates based on the sequence
length. Diego: we need an interpretation of the table: what does it ’says’?

TABLE VI: FHS family size and correlation comparison family | size | $\bar{H}_{\mathrm{max}}(S,S)$
---|---|---
lem-green | 32 | 0.939
lem-green-2x | 64 | 2.108
li-fan-2l | 18 | 0.480
li-fan-2l-4x | 71 | 1.769
li-fan-3l | 26 | 0.704
li-fan-3l-4x | 107 | 1.566
hash | 384 | 0.407
driver | 384 | 0.416

The family maximum Hamming Correlation, denoted by
$\bar{H}_{\mathrm{max}}(S,S)$, as defined in Eq. (11), is illustrated in
Figure 7a for both the lifan and the driver families. The correlation is
assessed against increasing sequence lengths. For shorter sequences spanning
up to 10 hops, the lifan family exhibits a reduced correlation compared to the
driver family. However, as the sequence length extends beyond this range, the
driver family takes precedence in terms of lower correlation. It’s noteworthy
that even though the lifan FHS is designed to yield an optimal sequence, this
property is guaranteed only for the singular sequence generated. This explains
the observed degradation in correlation performance when this primary sequence
is partitioned into more minor sequences to constitute an LR-FHSS-compatible
FHS family.

Fig. 7b presents the family average Hamming Correlation, denoted by
$\bar{H}_{\mathrm{avg}}(S,S)$, based on Eq. (7). This correlation is
demonstrated for both driver and lifan families, each formulated under three
distinct configuration parameters, as detailed in Table I. For the
configurations in question:

  * •

The EU137 configuration:

    * –

Maximum FHS length: 35 hops

  * •

The EU336 configuration:

    * –

Maximum FHS length: 86 hops

  * •

The US1523 configuration:

    * –

Maximum FHS length: 60 hops

Interestingly, across all these configurations, the lifan FHS families
consistently outperform their driver counterparts, showcasing superior
correlation metrics.

(a) Maximum Hamming Correlation

(b) Average Hamming Correlation

Figure 7: Maximum and average Hamming Correlation for different LR-FHSS FHS
families (lifan and driver).

####  V-B2 Collision Rate Analysis

Figure 8: Relative collision rate for different LR-FHSS FHS families,
considering no granularity, no headers, and unlimited decoding capacity.

For an initial assessment, we used a preliminary test to study the collision
rate at the payload level for different FHS families, as detailed in Table VI.
This model is more basic than the one discussed above, encompassing fewer
parameters.

In this design, the concept of granularity was eliminated, which meant each
fragment was equivalent to a single time slot. Throughout the simulation, each
node was responsible for transmitting a single payload consisting of 31
fragments devoid of headers. The total time allocated for simulation was
fourfold the time required for payload transmission, summing up to 124 time
slots. Another distinguishing feature of this model was its lack of a decoding
threshold for simultaneous receptions.

The core aim of this basic model was to analyze the FHS’s correlation
performance under conditions of multiple concurrent transmissions.

Fig. 8 presents the relative collision rate, which is compared against the
baseline established by the driver family. Observations highlighted that the
li-fan families exhibited superior performance compared to the reference point
after reaching a specific number of nodes. This trend was consistent for both
Coding Rates. However, the lem-green family initially demonstrated a better
performance than the driver family up to a certain number of transmissions but
later faced a decline, especially for CR2.

Given the insights derived from this preliminary evaluation, we selected the
li-fan-2l family for the upcoming comprehensive simulation campaigns.

###  V-C Early Drop and Early Decode Analysis

(a) CR1

(b) CR2

Figure 9: Total decoded payloads fragments with CR1 and CR2. Diego: payload
fragments, right?

Building on our selection of the lifan family, our analysis proceeds with the
comprehensive model detailed in Section IV. Under this framework, a packet can
experience one of three outcomes:

  1. 1.

Discarded: This occurs when no demodulators are available.

  2. 2.

Successful: A demodulator successfully decodes the payload.

  3. 3.

Collided: The payload fails to be decoded as the minimum number of fragments
is unmet.

Although headers are not essential for payload decoding in this scenario,
their presence still influences collision dynamics.

Upon activating the Early Drop and Early Decode mechanisms, we assessed the
decoding proficiency by gauging the decoded payloads for each base demodulator
model. As Fig. 9a demonstrates, the graph plots the cumulative decoded
payloads against the total packets transmitted using CR1. The comparison is
multifaceted:

  * •

Families Comparison: The solid lines represent the lifan family, while the
dashed lines depict the driver family.

  * •

Decoding Mechanisms: The symbol $l$ marks the results derived from the Early
Drop and Early Decode mechanisms, tagged as earlydd, which are set against the
baseline demodulator model.

  * •

Demodulators Count: The data segregates between setups with 100 demodulators
(depicted in shades of orange and blue) and 1000 demodulators (showcased in
shades of green and purple). For reference, the unlimited decoding capacity
scenario is illustrated with the label infp in red.

This multifactorial examination provides a comprehensive understanding of how
different configurations influence the decoding potential under the stipulated
conditions.

With the configuration of 100 demodulators (100p), activating the earlydd
mechanism brings about a significant enhancement in decoding performance for
both the lifan and driver families. Specifically, the number of successfully
decoded payloads can see an increase of up to 100%, effectively doubling the
decoding output. However, when the demodulator count is raised to 1000
(1000p), the benefits of the earlydd mechanism become less pronounced. Even
so, the decoding capability curve is nudged closer to that of the model with
infinite decoding capacity. A notable observation is that, under this
configuration, the lifan family witnesses an improvement of up to 50% in
successfully decoded payloads when the total transmitted packet count hovers
around 4000.

When contrasting the performance of the lifan and driver FHS families, the
driver FHS consistently outperforms the lifan up to a specific threshold of
transmissions. This superiority becomes even more pronounced when operating
with 1000 demodulators (1000p). Beyond this threshold, however, a transition
from the driver FHS to the lifan FHS has the potential to yield better network
throughput.

As illustrated in figure 9b for the case using CR2, the earlydd mechanism
consistently enhances the decoding capacity for both the lifan and driver FHS
families. The improvement is particularly significant when utilizing 100
demodulators (100p), but it tapers off as the channel nears saturation. The
benefits are less pronounced when employing 1000 demodulators (1000p).

A comparative evaluation of the lifan and driver families reveals a
considerable advantage for the lifan family. This superiority is especially
evident in the 1000p setup, where the decoding capacity of the lifan family
surpasses that of the driver family by a remarkable 170%. While the driver
family does exhibit marginally superior performance under lighter network
loads, the disparity at higher loads suggests that the lifan family is the
optimal choice when CR2 is in use.

###  V-D Packet versus Payload Analysis

(a) 100 demodulators

(b) 1000 demodulators

Figure 10: Total decoded data volume with 100 and 1000 demodulators with
Early-Drop and Early-Decode enabled.

This section delves into a comparison of packet decoding versus payload
decoding. While the latter was the primary focus of the preceding section, the
emphasis here shifts to packet decoding. Packet decoding implies that at least
one header replica is received in addition to a successful payload decode. For
the context of this analysis, both the Early Drop and Early Decode mechanisms
remain activated. Our metric of interest is the total decoded data,
represented in kilobytes, compared against the total data sent. This approach
offers a balanced comparison between CR1 and CR2. Notably, despite requiring
the same number of hops, CR1 encodes less data than CR2. Thus, this metric
facilitates an unbiased assessment of their performance relative to the data
throughput they deliver.

####  V-D1 100 Processor Analysis

The distinction between packet and payload decoding is brought to the fore in
Fig. 10a. The lifan and driver FHS families are portrayed with different color
codes: orange and green for lifan, and blue and purple for driver. A further
distinction is made between CR1 (depicted in shades of orange and blue) and
CR2 (represented in shades of green and purple). Dashed lines signify packet
(pckt) decoding, while solid lines represent payload (pyld) decoding.

The observation from the figure confirms the intuitive expectation: the rate
of packet decoding, which requires both a successful payload and at least one
header replica, is invariably lower than the payload-only decoding. This is
depicted by the consistently higher solid lines (pyld) compared to their
dashed counterparts (pckt) across the graph.

When considering CR1, this decrease in the packet decoding performance
relative to payload decoding also manifests in the competitive dynamics
between the lifan and driver families. Here, the lifan family fails to
outperform the driver family across the entirety of the depicted range.

Turning our attention to CR2, a similar trend unfolds. Although lifan had
previously displayed a higher efficiency, this distinction diminishes when
pitted against driver in the context of packet decoding. The relative
performance advantage of lifan observed in the payload-only decoding scenario
wanes, cementing the assertion that the introduction of the header replica
decoding requirement poses a challenge to both families, altering their
relative efficiencies.

####  V-D2 1000 Processor Analysis

With the enhanced processing capacity depicted in Figure 10b, we observe
distinct trends compared to the prior analysis with 100p.

The increased processing capability does not shield the lifan family from the
added complexity of pckt decoding, as the family remains noticeably more
affected than when only considering pyld decoding. For CR1, the performance
gap between the two families is quite pronounced. The driver family outpaces
the lifan by a substantial margin, decoding up to 60% more data. This
demonstrates the driver’s efficiency and highlights the challenges lifan
encounters when packet decoding is in play, even with enhanced processing
capability.

For CR2, the dynamics between the two families become more intricate. The
driver family retains its dominance over a broader range of transmissions
before lifan starts to take the lead. This extension of the driver’s dominant
zone underscores the impact of packet decoding on the lifan’s performance,
even in the context of CR2.

Taking a step back to compare the broader impact of the coding rates, it’s
evident that CR2 offers superior throughput compared to CR1. This is
particularly true when the available OCW channels are kept constant for both
coding rates. The data, therefore, lends itself to the idea that, in terms of
raw throughput and irrespective of the FHS family in question, CR2 presents a
more efficient encoding choice.

###  V-E Header Drop Analysis

(a) 100 demodulators

(b) 1000 demodulators

Figure 11: Total decoded data with 100 and 1000 demodulators, Early-Drop and
Early-Decode. Diego: fix legend

Incorporating the Early Header Drop mechanism brings about a significant shift
in the dynamics of the decoding process, casting a spotlight on the vital role
headers play in LR-FHSS networks.

In Fig. 11a, which details the outcomes with 100p, a noteworthy observation is
the proximity of the hdrdrp (header drop) curves to the pyld (payload drop)
curves Diego: please check I correctly understand what hdrdrp and pyld means.
Expand if needed. This indicates that when relieved from payloads whose
headers weren’t successfully received, the demodulators can maintain a
decoding rate almost as efficiently as if headers were non-essential. This
suggests that in more constrained processing environments, the Early Header
Drop mechanism assists in focusing resources on more probable decoding
successes.

However, the dynamics change as the processing power is ramped up in Fig. 11b.
Here, the hdrdrp curves converge with the pckt curves. The vast gap between
the performance of the hdrdrp and pyld signifies missed decoding
opportunities. The divergence between these curves indicates that with more
processing resources, there’s substantial potential for successfully decoding
payloads even when headers are missing. This observation underscores the
limitation of being overly reliant on headers in LR-FHSS systems.

Given the contrast observed between the hdrdrp and pyld results in the 1000p
scenario, the findings hint at the prospective advantages of a headerless
decoding mechanisms [fraire2023recovering]. If LR-FHSS networks were to
develop a reliable method of decoding payloads without needing headers, it
could substantially amplify the effective throughput and overall performance.
With its apparent untapped potential, this direction warrants further research
and could radically change LR-FHSS network methodologies.

###  V-F Header Tolerance

(a) 100 demodulators

(b) 1000 demodulators

Figure 12: Total decoded data volume with 100 and 1000 demodulators with
Early-Drop, Early-Decode, Header Drop, and Header Tolerance enabled of 4 time
slots.

This section offers an extended perspective on the decoding dynamics by
introducing a “collision tolerance” for the header to properly mode its more
robust modulation. By permitting a certain number of collisions without
deeming the header as “failed”, the dynamics of packet reception and decoding
stand to gain a significant shift.

In Fig. 12a for the 100p scenario, the inclusion of the Header Tolerance
(hdrtol) displays the immediate benefits of the flexibility offered by this
model. While both FHS families witness an improvement, the lifan family reaps
a more pronounced uptick in decoding performance. For CR2, even though the
driver family starts with a performance edge over lifan, this dominance is
short-lived. Beyond the 60 Kb transmission threshold, lifan surges ahead with
its decoding capacity, outperforming the driver by as much as half. Similarly,
in the CR1 scenario, the earlier observations from Fig. 9a come to mind, where
lifan eventually surpasses the driver in decoding performance after a specific
transmission volume.

Scaling the processing power to 1000p, as represented in Fig. 12b, the
benefits of the Header Tolerance model become even more conspicuous. The
resilience against header collisions boosts the decoding capacities for both
families, but once again, the lifan stands out. The CR2 results reiterate the
shift in dominance, with lifan pulling far ahead of the driver family,
especially after a specific transmission volume. The differential reaches a
staggering 100%, highlighting the merits of the lifan family in this specific
scenario. The CR1 results echo a similar outcome, marking a clear comeback for
lifan over driver with increasing transmission counts.

This analysis accentuates the importance of adaptability in network protocols
and how slight adjustments, such as allowing for some collision tolerance in
headers, can have outsized impacts on performance metrics. It underscores the
potential of optimizing network parameters to adapt to specific requirements,
thus enhancing overall network efficiency.

##  VI Conclusions

The burgeoning integration of IoT with LEO satellites necessitates the
evolution of robust and efficient communication protocols. Our in-depth
exploration into LR-FHSS modulation, mainly focusing on the design of
Frequency Hopping Sequences (FHS) and efficient demodulator allocation
strategies, has highlighted the significant promise of Wide-Gap sequences for
LEO satellite IoT communications. Our primary findings and contributions are
summarized as follows:

##### Wide-Gap Sequence Design

The study underscored the efficacy of the lifan FHS family over traditional
sequences, especially in scenarios with higher demodulator capacities. This
establishes the lifan family as a viable choice for optimizing LR-FHSS
performance in satellite communications.

##### Demodulator Allocation Strategies

Introducing the “Early-Decode,” “Early-Drop,” and “Early-Header-Drop”
strategies showcases a paradigm shift in optimizing gateway decoding
resources. The efficacy of these strategies was particularly evident when
observing the performance boost experienced by the lifan family upon their
implementation.

##### Collision Tolerance Analysis

A novel aspect of our research was the analysis of collision tolerance in
header decoding. Allowing a certain degree of collision tolerance, instead of
marking a header as ’failed’ due to a single collision, accentuated the
model’s flexibility benefits. Such tolerance increased decoding performance,
notably for the lifan FHS family.

##### Future Potential of LR-FHSS

The comparative analysis between lifan and driver families, especially when
adopting our novel demodulator strategies, underscores the untapped potential
of LR-FHSS in IoT settings. The significant performance gains observed pave
the way for more adaptive, efficient, and robust IoT communications,
especially in LEO satellite environments.

The strategies and observations presented in this paper not only cement the
viability of LR-FHSS for next-generation IoT communications but also emphasize
the need for continual optimization in sequence design and resource
allocation. The realm of LEO satellite IoT communications is on the precipice
of a revolution, and with the insights gleaned from this study, we are better
equipped to navigate its vast potential.

##### Future Work

TBD

## Acknowledgment

This research has received support from the EU’s H2020 R&D program under the
Marie Skłodowska-Curie grant agreement No 101008233 (MISSION project) and the
French National Research Agency (ANR) ANR-22-CE25-0014-01.


# CE$\nu$NS at the European Spallation Source

A. Simón

###### Abstract

The recent detection of coherent elastic neutrino-nucleus scattering
(CE$\nu$NS) opens the possibility to use neutrinos to explore physics beyond
standard model, with small-size detectors. However, the CE$\nu$NS process
generates signals at the few keV level, requiring very sensitive detector
technologies. The European Spallation Source (ESS) has been identified as an
optimal source of low energy neutrinos, offering an opportunity for a
definitive exploration of all phenomenological applications of CE$\nu$NS.

A number of different detector approaches are currently under development for
deployment at ESS. These next-generation technologies will be able to observe
the process with lower energy threshold and better energy resolution than
current detectors. The combination of their observations will allow for a
complete phenomenological exploitation of the CE$\nu$NS signal. In particular,
these measurements will not be statistically-limited, a result of the large
neutrino flux expected at the ESS.

The main projects currently being developed to detect the CE$\nu$NS at the ESS
are presented: the GanESS project which will use high pressure gas TPC filled
with different noble gases; the C${}^{\circ}\\!$sI project, which employs
cryogenic undoped CsI crystals; and p-type point contact HPGe detectors.

##  1 Introduction

Predicted in the 70s, coherent elastic neutrino-nucleus scattering (CE$\nu$NS)
was only first measured just a few years ago [1]. In this process low energy
neutrinos (few tens of MeV) interact coherently with an atomic nucleus as
whole, through the weak current channel, as long as the coherent condition |Q|
< 1/R is satisfied, with |Q| being the momentum transfer and R the radius of
the nucleus. As a result of the low value of the weak mixing angle for
protons, the coupling ends up being effectively proportional to the squared
number of neutrons (N2) in the target nucleus. The process is of utmost
relevance to deepen our knowledge of a large number of neutrino properties.
For example, current measurements of this interaction have already been used
in nuclear structure studies to improve bounds on Non-Standard neutrino
Interactions. Improved CE$\nu$NS measurements can also be used to better
determine the neutrino electromagnetic properties, constrain the weak mixing
angle and test both dark matter and sterile neutrinos models. Finally,
CE$\nu$NS can be used to monitor nuclear proliferation thus honing the
detection techniques of this process may potentially lead to technological
applications. The most intense neutron spallation source in the coming years,
the European Spallation Source (ESS), will produce $\sim$2.53 $\times$ 1023
neutrinos per year, almost a factor 10 improvement over the Spallation Neutron
Source, where CE$\nu$NS was first detected. The ESS will provide first protons
on target at lower energy in 2025 with the goal to increase it to 5 MW and 2
GeV in the following years. As described in [2, 3], the combination of compact
detectors and the large neutrino flux produced by the ESS will allow CE$\nu$NS
measurements only limited by few percent systematics and not by statistical
fluctuations. Three of the detection techniques described in [2, 3] are funded
and currently in active development for deployment in the coming years: noble
gases time projection chamber, cryogenic pure CsI and p-type high purity Ge
detectors. Moreover, the combination of the first two offer a uniquely
interesting approach as their response to CE$\nu$NS is mostly identical given
their similarity in nuclear structure, but their technologies and expected
systematics are nothing alike. Consequently, the simultaneous use of these
detectors will serve to reinforce, or refute, any hint of unexpected
observations.

##  2 GanESS: Noble gas time projection chamber.

The GanESS experiment, illustrated in Fig. 1, will use of a high-pressure
noble gas time projection chamber to measure CE$\nu$NS at ESS. The technique,
which was developed for neutrinoless double beta searches, offers several
interesting aspects for the detection of the process. First, it allows to
operate with different nuclei with minimal modifications of the system.
Concretely it will operate with Ar, Kr and Xe, providing the first ever
CE$\nu$NS observations in Kr and Xe. Operation with different gases allows to
explore different regions of the Non-Standar Interactions (NSI) of the
neutrino-quarks phase space. A combined analysis of the data from different
nuclei with same systematics can further reduce such allowed region improving
total sensitivity to NSI [2, 3]. Second, amplification with
electroluminiscence is widely used in particle physics and can easily produce
several hundreds of photons per electron. A detector with a moderate optical
coverage can thus be sensitive to a single free electron crossing the
amplification region allowing to detect tiny energy depositions in the
detector, as low as $\sim$25 eVee.

Gaseous detectors weren’t considered for CE$\nu$NS detection because their
relatively low density (in comparison with solid scintillators or liquid phase
detectors) could limit the statistics in a moderate size detector.
Fortunately, the strong neutrino flux at the ESS solves this issue allowing
full exploitation of the advantages of the noble gas TPC technology.
Importantly, gaseous detectors avoids the liquid-gas interface of dual phase
detectors, where trapped electrons temporarily accumulate and end up producing
a background signal increasing the detection threshold.

Figure 1: Left: Design of the GanESS experimentL: a symmetric TPC with the
cathode at the center. Light is detected with PMTs at each side of the
chamber. Right: A close-up of GaP’s time projection chamber.

This technique is extraordinarily promising for CE$\nu$NS detection albeit
characterization of the response to few-keV nuclear recoils will be necessary.
With this goal, we are currently comissioning GaP, a small prototype capable
of operating up to 50 bar. GaP will serve to fully evaluate the low energy
response of the technique, with a strong focus on measuring the quenching
factor for the different noble gases that will later be used at GanESS. The
GaP detector, shown on Fig. 1, is a 2 cm drift TPC currently equipped with 7
photomultiplier tubes near the amplification region. It recently started
operations operating with 9.5 bar of Ar. The drift region is being enlarged to
10 cm to maximize the sensitive volume. Once characterization of Ar is
complete, Xe and Kr will follow.

##  3 C${}^{\circ}\\!$sI: cryogenic pure CsI.

The C${}^{\circ}\\!$sI detector is a natural evolution from the first
CE$\nu$NS measurement (CsI[Na] at SNS [1]). Instead of sodium-doped crystals,
it will use pure CsI crystals cryogenically operated. Concretely, 7 crystals
of 7$\times$7$\times$32 cm will be employed, adding up to $\sim$ 50 kg of
mass.

The reason for using undoped CsI comes from the fact that the light yield of
such crystals increases enormously at cold temperatures (see Fig. 2),
providing a two-fold increase on the light production when compared to CsI[Na]
crystals. This light increase is fully exploited by wavelength shifting the
emission of cryogenic CsI ($\sim$350 nm) to $\sim$600 nm with nanostructured
organosilicon luminophores (NOL) and having the light read with a large area
avalanche photodiode (LAAPD), with a quantum efficiency of $\sim$80% at such
wavelength.

Figure 2: Left: Energy threshold of a few c.c. undoped CsI crystal operated at
liquid nitrogen temperature. Light is detected with a LAAAPD following
wavelength shifting with NOL-9. Right: energy spectrum from nuclear recoils
induced in such crystal by 88Y/Be and 124Sb/Be photoneutron sources. Recoils
from 124Sb/Be have a maximum recoil energy of $\sim$0.7 keVnr.

The combination of light yield, wavelength shifter and high quantum efficiency
pushes down the detection threshold well below the keVee. In fact, a threshold
slightly above 60 eV has been already achieved in a small (few c.c.)
demonstrator at University of Chicago, with sub-keV nuclear recoils observed
for the first time in a CsI crystal (Fig. 2). A dependance of the detection
threshold with the operation temperature was observed, caused by variations in
the LAAPD gain. Good temperature stability will be required to maximize the
performance of the upcoming C${}^{\circ}\\!$sI detector.

##  4 P-type HPGe detectors.

Germanium detectors are being broadly used to detect CE$\nu$NS from reactor
neutrinos and they are just as interesting for CE$\nu$NS from spallation
sources. At least two p-type point contact high purity germanium detectors,
each with $\sim$ 3 kg mass, will be deployed in the ESS for that purpose. One
was used in the first observation of CE$\nu$NS in reactor neutrinos [4] and
demonstrated a threshold of $\sim$180 eVee. A series of improvements have been
identified and are currently under development to push the threshold below 100
eVee. A new detector is currently under construction, building on the
experience of the Dresden detector, aiming at a threshold as low as 80 eVee.

Existing discrepancies in the quenching factor of Ge need to be addressed to
fully understand the CE$\nu$NS signal. These will be clarified in the coming
months using thermal neutron capture. The gamma-induced recoils from this
process will allow to measure the germanium quenching factor in the few
hundred eV range.

## References

  * [1] D. Akimov et al. [COHERENT], Science 357 (2017) no.6356, 1123-1126 doi:10.1126/science.aao0990 [arXiv:1708.01294 [nucl-ex]].
  * [2] D. Baxter et al., JHEP 02 (2020), 123 doi:10.1007/JHEP02(2020)123 [arXiv:1911.00762 [physics.ins-det]].
  * [3] H. Abele et al., Phys. Reports 1023 (2023), 1-84 doi.org/10.1016/j.physrep.2023.06.001 [arXiv:2211.10396 [hep-ex]].
  * [4] J. Colaresi, J. I. Collar, T. W. Hossbach, C. M. Lewis and K. M. Yocum, Phys. Rev. Lett. 129 (2022) no.21, 211802 doi:10.1103/PhysRevLett.129.211802 [arXiv:2202.09672 [hep-ex]].