# Chirp Spread Spectrum-based Waveform Design and Detection Mechanisms for
LPWAN-based IoT – A Survey

Ali Waqar Azim, Ahmad Bazzi, Raed Shubair, Marwa Chafii Ali Waqar Azim is with
Department of Telecommunication Engineering, University of Engineering and
Technology, Taxila, Pakistan (email: aliwaqarazim@gmail.com).Ahmad Bazzi and
Raed Shubair is with with Engineering Division, New York University (NYU) Abu
Dhabi, 129188, UAE (email: {ahmad.bazzi,raed.shubair}@nyu.edu).Marwa Chafii is
with Engineering Division, New York University (NYU) Abu Dhabi, 129188, UAE
and NYU WIRELESS, NYU Tandon School of Engineering, Brooklyn, 11201, NY, USA
(email: marwa.chafii@nyu.edu).

###### Abstract

The Long Range protocol, commonly referred to as LoRa, is a widely adopted and
highly regarded method of utilizing chirp spread spectrum (CSS) techniques at
the physical (PHY) layer to facilitate low-power wide-area network (LPWAN)
connectivity. By tailoring the spreading factors, LoRa can achieve a diverse
array of spectral and energy efficiency (EE) levels, making it amenable to a
plethora of Internet-of-Things (IoT) applications that rely on LPWAN
infrastructure. However, a primary drawback of LoRa is its relatively low data
transfer rate. Despite this, there has been a dearth of research dedicated to
enhancing the data transfer capabilities of LoRa until recently, when a
plethora of CSS-based PHY layer alternatives to LoRa for LPWANs was proposed.
This survey, for the first time, presents a comprehensive examination of the
waveform design of these CSS-based PHY layer alternatives, which have been
proposed between $2019$ and $2022$. A total of fifteen alternatives to LoRa
are analyzed and compared. Other surveys on LoRa focus on topics such as LoRa
networking, the deployment of LoRa in massive IoT networks, and LoRa
architectural considerations. In contrast, this study delves deeply into the
waveform design of alternatives to LoRa. The CSS schemes studied in this study
are classified into three categories: single chirp, multiple chirps, and
multiple chirps with index modulation, based on the number of activated
frequency shifts activated for un-chirped symbols. The transceiver
architecture of these schemes is thoroughly explicated. Additionally, we
propose coherent/non-coherent detection mechanisms for specific schemes that
have not been previously documented in the literature. We also provide some
key insights and recommendations based on the performance of the schemes. The
performance of the schemes is evaluated based on metrics such as EE, spectral
efficiency, the bit-error-rate (BER) in additive white Gaussian noise, and BER
in the presence of phase and frequency offsets. Finally, we highlight some
open research issues and future research directions in this field.

###### Index Terms:

Waveform design, LoRa, chirp spread spectrum, IoT.

##  I Introduction

The advent of Low-Power Wide-Area Networks (LPWANs) represents a significant
technological advancement for the Internet-of-Things (IoT). These networks
serve as a conduit for vast quantities of intelligent devices may gain access
to the centralized infrastructure of the internet. Through the utilization of
LPWANs, it becomes possible to achieve ubiquitous connectivity between an
extensive array of sensors, thereby facilitating data collection and promoting
cooperation among them. This, in turn, enables the proliferation of various
intelligent IoT application development, paving the way for the realization of
an innovative way of living characterized by the interconnectedness and remote
control of everyday devices via the internet. As such, it is anticipated that
IoT devices will become an ever-present reality in the imminent future [1].

The design of waveforms represents a fundamental aspect of the realization of
pervasive connectivity in IoT. This is because the implementation of an IoT
network necessitates the connection of a vast number of devices, and the
achievement of low-power consumption through the optimization of waveform
design is of paramount importance. This is particularly relevant in light of
the fact that the nodes within an IoT network are typically battery-powered
terminals, rendering regular maintenance of each node a challenging and
unwieldy task. Based on the transmission frequency, the battery life of these
nodes is typically targeted to be five to ten years. To this end, various
LPWAN standards have been proposed, utilizing both licensed and unlicensed
frequency bands, such as the industrial, scientific, and medical (ISM) bands.
One notable example of a standard that utilizes a licensed band is Narrowband-
IoT (NB-IoT), as proposed by the 3rd Generation Partnership Project (3GPP)
[2]. An alternative license-free standard is LoRa Wide-Area Network (LoRaWAN),
which has gained widespread deployment compared to other contenders such as
Sigfox and Ingenu [3]. This study focuses specifically on LoRaWAN due to its
prevalent implementation [4].

LoRaWAN protocol employs LoRa for modulation and demodulation. LoRa, a
proprietary technique developed by Semtech, has emerged as a highly promising
physical (PHY) layer method for achieving both low-power consumption and wide-
scale connectivity over extended distances, as evidenced by various studies
such as [5, 6, 7, 8, 9]. Although the specifics of LoRa’s design remain
largely undisclosed, it is widely acknowledged that its waveform design is a
derivative of the chirp spread spectrum (CSS) technique. Recent efforts have
been made to theoretically interpret the properties of LoRa, with notable
examples including [7, 10, 4]. In particular, Vangelista in [7] describes LoRa
as a frequency-shift chirp modulation, in which there is a frequency-shift
keying (FSK) component that is the un-chirped symbol and a spreading component
that spreads the bandwidth of the un-chirped symbol in the entire bandwidth.
The frequency spreading of LoRa makes it resilient to Doppler shift. It is
also inherently robust against doubly dispersive channels while offering high
sensitivity in detecting potentially weak signals while consuming extremely
low power, as demonstrated in studies such as [11, 12]. Furthermore, LoRa’s
constant envelope property allows for the use of low-cost devices such as
power-efficient non-linear amplifiers, as highlighted in studies such as [13,
14, 15].

LoRa operates within the ISM band and utilizes three different bandwidths;
$125$ kHz, $250$ kHz, and $500$ kHz. The bit rate of LoRa is determined by the
parameter known as the spreading factor (SF), which varies between $6$ and
$12$. By utilizing different SFs and code rates, a wide range of throughputs
can be achieved with LoRa, making it a versatile option for a diverse array of
applications, ranging from smart cities to agriculture, as observed in studies
such as [13]. However, the achievable bit rate/spectral efficiency (SE) of
LoRa is relatively low, which may present a significant limitation. To
overcome the limitation of low achievable SE, LoRa modems can also be equipped
with an FSK transceiver to double the throughput. However, this added feature
can be costly and may increase power consumption due to the requirement for
additional computational resources.

To alleviate the limitations inherent in LoRa, various PHY layer variants of
CSS have been proposed in the literature as direct competitors to LoRa. The
foremost objectives of these schemes are to either enhance spectral efficiency
(SE) or to improve energy efficiency (EE). These improvements are directly
linked to the waveform design of these schemes. It is noteworthy that the
waveform design and properties of CSS alternatives to LoRa vary significantly;
for example, certain schemes incorporate a phase-shift (PS) to the chirp (as
exemplified in the work of PSK-LoRa [16]), while other designs employ varying
chirp rates (CRs) (as demonstrated in SSK-LoRa [17] and DCRK-LoRa [18]), as
well as the index modulation (IM) paradigm (as represented by FSCSS-IM [19]
and IQCIM [20]), among others. While a limited number of CSS schemes for
LPWAN-based IoT have been proposed in the literature in recent years, to the
best of our knowledge, a study consolidating the waveform design of these CSS
PHY layer alternatives to LoRa has yet to be made available in the literature.
This survey comprehensively provides the waveform design of several CSS
alternatives to LoRa proposed in the literature from $2019$ to $2022$. Table I
provides a list of acronyms that are used in this study.

TABLE I: List of acronyms used in this study. Acronym | Meaning
---|---
ADR | Adaptive Data Rate
AWGN | Additive White Gaussian Noise
BER | Bit-Error Rate
CR | Chirp Rate
CSI | Channel State Information
CSS | Chirp Spread Spectrum
DCRK-LoRaS | Discrete CR Keying LoRa
DFRC | Dual Function Radar Communications
DFT | Discrete Fourier Transform
DM-CSS | Dual-Mode CSS
DO-CSS | Dual Orthogonal CSS
EE | Energy Efficiency
E-LoRa | Extended-LoRa
ePSK-CSS | Enhanced Phase-Shift Keying CSS
FAP | Frequency Activation Pattern
FF | Fundamental Frequency
FS | Frequency-Shift
FSCM | Frequency-Shift Chirp Modulation
FSCSS-IM | Frequency-Shift CSS with IM
FSK | Frequency-Shift Keying
GCSS | Group-based CSS
IM | Index Modulation
ICS-LORa | Interleaved Chirp Spreading-LoRa
IoT | Internet-of-Things
IQ-CIM | In-phase and Quadrature Chirp IM
IQ-CSS | In-phase and Quadrature CSS
ISM | Industrial, Scientific and Medical
IQ-TDM-CSS | In-phase and Quadrature
| Time Domain Multiplexed CSS
I/Q | In-phase/Quadrature
LoRa | Long Range
LoRaWAN | LoRa Wide-Area Network
LPWAN | Low-Power Wide-Area Networks
MAC | Media Access Control
ML | Maximum Likelihood
NB-IoT | Narrowband-IoT
OCDM | Orthogonal Chirp-Division
| Multiplexing
PAPR | Peak-to-average Power Ratio
PHY layer | Physical layer
PS | Phase-Shift
PSK-LoRa | Phase-Shift Keying LoRa
QoS | Quality of Service
SE | Spectral efficiency
SF | Spreading Factor
SNR | Signal-to-noise Ratio
SSK-LoRa | Slope-Shift Keying LoRa
SSK-ICS-LoRa | Slope-Shift Keying Interleaved
| Chirp Spreading LoRa
TDM-CSS | Time Domain Multiplexed CSS
UAV | Unmanned Ariel Vehicle

###  I-A LoRa Surveys

Amid a plethora of literature extant pertaining to LoRa, various surveys have
been conducted to assess a multitude of facets of the technology, including
but not limited to: the PHY and data link layer performance, the various MAC
protocols employed, deployment considerations, implementation strategies,
simulation testbeds, and field performance evaluations, etc. Additionally,
certain studies have sought to examine the potential applications of LoRa
within the domain of smart cities and agriculture, etc. Even with the
abundance of literature pertaining to LoRA, a dearth of surveys exists that
delves explicitly into the waveform design of PHY layer CSS-based alternatives
to LoRa. As such, we have established a set of inclusion and exclusion
criteria, as detailed in Table II, to guide our examination of the relevant
literature in the area.

TABLE II: Inclusion exclusion criteria for LoRa-based surveys. Inclusion Criteria | Exclusion Criteria
---|---
1\. Written in English | 1\. Written in language
language | other than English.
2\. Must be published |  2\. Published before $2016$.
between $2016$ and $2022$. |
3\. Keywords: LPWANs, | 3\. Presents an incomplete
LoRa or LoRaWAN | study.
appear in title or abstract. |
4\. Published in journals |
conference proceedings, |
symposiums or workshops. |
5\. Preprints in arXiv. |
6\. Provides a complete |
study on the topic. |
TABLE III: A comparison between this study and relevant surveys on LoRa available in the literature. Year | Survey | PHY Waveform | Detection Mechanisms | Performance | Recommendations
---|---|---|---|---|---
|  | Design | Principles | Evaluation | on Waveform Design
$2016$ | [21] | ✗ | ✗ | ✗ | ✗
$2016$ | [14] | ✗ | ✗ | ✗ | ✗
$2017$ | [22] | ✗ | ✗ | ✗ | ✗
$2017$ | [23] | ✗ | ✗ | ✗ | ✗
$2017$ | [9] | ✗ | ✗ | ✗ | ✗
$2017$ | [24] | ✗ | ✗ | ✓(only LoRa) | ✗
$2017$ | [25] | ✗ | ✗ | ✗ | ✗
$2017$ | [26] | ✗ | ✗ | ✗ | ✗
$2018$ | [27] | ✓(only LoRa) | ✗ | ✓(only LoRa) | ✗
$2018$ | [28] | ✗ | ✗ | ✗ | ✗
$2019$ | [13] | ✗ | ✗ | ✗ | ✗
$2019$ | [29] | ✗ | ✗ | ✗ | ✗
$2019$ | [30] | ✗ | ✗ | ✗ | ✗
$2019$ | [31] | ✗ | ✗ | ✗ | ✗
$2019$ | [32] | ✗ | ✗ | ✗ | ✗
$2020$ | [33] | ✗ | ✗ | ✗ | ✗
$2020$ | [34] | ✗ | ✗ | ✗ | ✗
$2020$ | [35] | ✗ | ✗ | ✗ | ✗
$2020$ | [36] | ✗ | ✗ | ✗ | ✗
$2020$ | [37] | ✗ | ✗ | ✗ | ✗
$2020$ | [38] | ✗ | ✗ | ✗ | ✗
$2020$ | [39] | ✗ | ✗ | ✗ | ✗
$2021$ | [40] | ✗ | ✗ | ✗ | ✗
$2021$ | [41] | ✓(only LoRa) | ✗ | ✗ | ✗
$2021$ | [42] | ✗ | ✗ | ✗ | ✗
$2021$ | [43] | ✗ | ✗ | ✗ | ✗
$2022$ | [44] | ✗ | ✗ | ✗ | ✗
$2022$ | [45] | ✗ | ✗ | ✗ | ✗
$2022$ | [46] | ✗ | ✗ | ✗ | ✗
$2022$ | [47] | ✗ | ✗ | ✗ | ✗
$2022$ | [48] | ✗ | ✗ | ✗ | ✗
$2022$ | [49] | ✗ | ✗ | ✗ | ✗
$2022$ | [50] | ✗ | ✗ | ✗ | ✗
$2022$ | [51] | ✗ | ✗ | ✗ | ✗
$2022$ | [52] | ✗ | ✗ | ✗ | ✗
$2023$ | This study | ✓ | ✓ | ✓ | ✓

The present study aims to comprehensively examine the PHY layer waveform
design of alternatives to LoRa based on CSS and proposed specifically for
LPWANs. In consonance with the parameters detailed in Table II, Table III
enumerates a selection of surveys chosen for comparative analysis. The primary
objective of this study is to expound upon the various mechanisms of
detection, contrast the performance of these alternatives across a plethora of
scenarios, and proffer suggestions for improvement as well as potential areas
for further research. The comparison of these chosen surveys with the present
study, as evaluated against the aforementioned four criteria, is exhibited in
Table III. It is worth noting that, with a few exceptions, most other surveys
do not delve into the intricacies of the PHY layer waveform design of CSS-
based alternatives to LoRa for LPWANs in great depth and the ones which do;
they only consider the state-of-the-art LoRa.

TABLE IV: Summary of related surveys and comparative works. Survey | Short Description
---|---
[21] | Provides an overview of LPWANs, and discusses its advantages for applications related to smart cities.
[14] | Provides an overview of LoRa, and evaluates the PHY and data link layer performance.
[22] | Compares the strength and limitations of different LoRa/LoRaWAN test beds.
[23] | Compares NB-IoT and LoRa considering different performance metrics.
[9] | Surveys the design and techniques employed in LPWANs offering wide-area coverage.
[24] | Evaluates the performance of LoRa considering IoT requirements.
[25] | Surveys different LPWAN solutions in context of machine-to-machine communications.
[26] | Compares research and industrial states of NB-IoT and LoRa considering different metrics.
[27] | Consolidates LoRa and LoRaWAN research considering PHY and network layer aspects.
[28] | Identify LoRa applications, research trends, and practical applications of LoRa in IoT deployment.
[13] | Surveys technical challenges in LoRa networks and their possible solutions.
[29] | Explores LoRa application fields and integration of edge computation capability in IoT.
[30] | Investigates NB-IoT and LoRa in the context of challenges preventing LPWANs from widespread adoption.
[31] | Provides an overview of LoRaWAN, including MAC protocols and its architecture.
[32] | Studies the use of LoRa in developing smart cities.
[33] | Provides a review of the multi-hop proposals for LoRaWAN.
[34] | Surveys ADR algorithms for LoRaWAN.
[35] | Reviews LoRaWAN architecture, applications, security concerns, and possible countermeasures.
[36] | Reviews LoRa and LoRaWAN, and discusses applying ultra-dense network concept on LPWAN.
[37] | Reviews the methods for image transfer via LoRa infrastructure.
[38] | Reviews use cases requiring confirmed traffic and investigate LoRaWAN confirmed traffic.
[39] | Surveys LoRa performance under different scenarios and difficulties in its implementation.
[40] | Surveys the available tools for simulating LoRa networks in NS-3 network simulator.
[41] | Presents an overview of LoRa PHY layer, MAC protocols, deployment, and end device transmission settings.
[42] | Provides experimental performance evaluation of ADR enhancements in a mobile node scenario.
[43] | Studies the real-time deployments of UAV-based LoRa network.
[44] | Compares simulation tools for investigating and analyzing LoRa/LoRaWAN network performance.
[45] | Provides an overview on LoRa, insights for building energy-efficient IoT infrastructures, and IoT devices.
[46] | Categorizes and compares LoRa networking techniques.
[47] | Surveys LoRa from a systematic perspective.
[48] | Surveys the scalability challenges/solutions to assist LoRaWAN deployment in massive IoT networks.
[49] | Provides an overview of LoRaWAN, discusses open-source simulation tools, and evaluates their performance.
[50] | Surveys LoRa-based solutions in smart agriculture.
[51] | Provides an overview of simulators supported by cloud infrastructure and discusses data-related operations.
[52] | Surveys LoRaWAN protocols considering EE perspective.

The lack of literature on the PHY layer waveform design of CSS-based protocols
for LPWANs is demonstrated by Table III. To furnish a more exhaustive
understanding of the target areas of the selected surveys per the criteria
outlined in Table II, Table IV elaborates on this subject matter by providing
a succinct synopsis for the reader’s edification.

The authors in [21] provide an overview of sub-GHz LPWANs and explore various
techniques for providing connectivity in IoT environments, highlighting the
advantages of such approaches over extant paradigms in terms of efficiency,
effectiveness, and architectural design, specifically in the context of smart
cities applications. The authors in [14] presents a general overview of LoRa
and undertake an in-depth analysis of its functional components, utilizing
both field tests and simulations to evaluate the performance of the PHY and
data link layers, subsequently proposing potential solutions to enhance
performance. In [22], the authors conduct a retrospective analysis of prior
work related to LoRa and LoRaWAN testbeds, identifying flaws and strengths by
comparing the testbeds and further recognizing the limitations of LoRaWANs.

The authors in [23] present a comprehensive survey of NB-IoT and LoRa as
efficient solutions for connecting devices in LPWANs and ascertaining that
LoRa has advantages in terms of battery lifetime, capacity, and cost, while
licensed NB-IoT offers QoS, latency, reliability, and range benefits. In [9],
the authors review the design goals and techniques that LPWANs employ to offer
wide-area coverage to low-power devices at the expense of low data rates and
further analyze several emerging LPWA technologies, the standardization
activities carried out by different standards development organizations and
the industrial consortia built around individual LPWA technologies. In [24], a
thorough evaluation of LoRa is conducted by examining the obstacles impeding
IoT development, identifying the challenges associated with such obstacles,
and subsequently elucidation the need for solutions to address these problems.
Additionally, the authors of this study also perform a comprehensive analysis
of the LoRaWAN communication protocol architecture requirements and LoRa
modulation performance.

Similarly, in [25], the authors survey the benefits and drawbacks of LPWAN
machine-to-machine solutions, as well as the corresponding deployment
strategies, and identify areas of knowledge deficiency through the
presentation of a summary of research directions aimed at improving the
performance of surveyed low power and long-range machine-to-machine
communication technologies. Like [23], [26] focuses on providing a
comprehensive and comparative survey study of the current research and
industrial states of NB-IoT and LoRa, with a particular emphasis on analyzing
their power efficiency, capacity, QoS, reliability, and range of coverage.
[27] offers an overview of research work on LoRa and LoRaWAN, published
between $2015$ and September $2018$, as retrievable through both the Google
Scholar and IEEE Explore databases. This work evaluates the performance of
LoRa and LoRaWAN, considering security and reliability mechanisms, PHY layer
aspects, and network layer aspects. Furthermore, the authors of this study
also proffer possible improvements and extensions to the standard.

In [28], Saari et al. survey the most recent trends in research and practical
applications of LoRa and also provide recommendations to fully exploit the
advantages of LoRa-based technologies in the development of IoT systems and
solutions. [13] surveys technical challenges in LoRa deployment and inspires
work on improving the performance of LoRa networks and enabling more practical
deployments. In [29], the authors explore and analyze different application
fields which use LoRa and investigate potential improvement opportunities and
considerations. Furthermore, the authors also propose a generic architecture
to integrate edge computation capability in IoT-based applications for
enhanced performance. [30] compares competing candidates of LPWANs, such as
NB-IoT and LoRa, in terms of technical fundamentals and large-scale deployment
potential. The authors also identify several challenges that prevent LPWAN
technologies from moving from theory to widespread practice.

[31] provides an overview of LoRaWAN, including MAC protocols and its
architecture. In [32], Andrade and Yoo identify the technological barriers
that hamper the development of solutions, find possible future trends that
could exist in the context of smart cities and IoT, and understand how the
industry and academy could exploit them. [33] reviews the state-of-the-art
multi-hop proposals for LoRaWAN. by providing comparative analysis and
classification, considering technical characteristics, intermediate devices
function, and network topologies. Moreover, the authors also discuss open
issues and future directions to realize the full potential of multi-hop
networking for LPWANs. [34] survey ADR solutions as an optimization approach
to improve throughput, EE and scalability in LoRaWANs.

In [35], Noura et al. provide a comprehensive analysis of the LoRaWAN
architecture, including its various applications and associated security
concerns. They also outline countermeasures to address existing
vulnerabilities and prevent potential attacks on the LoRaWAN protocol.
Similarly, [36] conducts a literature review on LoRa and advances in the
LoRaWAN protocol, highlighting the challenges faced by the technology and
exploring the potential for utilizing ultra-dense network concepts in the
context of low-power wide area networks. [37] evaluates the various methods
for image transfer via LoRa infrastructure, noting the limitations of each
approach and identifying key challenges that must be addressed to achieve
reliable image transfer over a LoRa network. In [38], Marais et al. examine
the use cases that necessitate confirmed traffic in LoRaWAN, surveying recent
literature on the topic and discussing the mechanisms by which confirmed
traffic is implemented. They also identify and propose solutions to open
research challenges that must be overcome to enhance the feasibility of
confirmed traffic.

[39] provides an overview of the performance of LoRa in different scenarios,
as well as an examination of the implementation hindrances of this technology.
Additionally, [40] presents a survey of the available tools for simulating
LoRa networks in the NS-3 network simulator, detailing their main features and
limitations. Finally, in reference [41], Gkotsiopoulos et al. present an
overview of LoRa-based networks, analyzing their behavior and categorizing
them based on their technological breakthroughs, in terms of various
performance determinants, including PHY layer characteristics, deployment and
hardware features, end device transmission settings, LoRa MAC protocols, and
application requirements. The literature reviewed in [42] evaluates the
advancements in ADR and presents the experimental results of their
implementation in a mobile node scenario.

The study conducted in [43] critically examines the practical deployment of
UAV-based LoRa communication networks, explicitly emphasizing the
communication setup. In [44], the authors present a comprehensive comparative
analysis of various LPWAN technologies, including NB-IoT, SigFox, Telensa,
Ingenu, and LoRa/LoRaWAN, providing a technical overview of LoRa/LoRaWAN by
highlighting its key features, opportunities, open issues, and simulation
tools for evaluating network performance. [45] offers a tutorial on the LoRa
standard and examines existing solutions, hot topics, and future insights for
the construction of energy-efficient IoT infrastructures and devices, with a
particular focus on LoRa/LoRaWAN EE across the PHY, MAC, and network layers.
[46] compares and categorizes state-of-the-art LoRa networking techniques.
[47] summarizes research focusing on the performance analysis and security
vulnerabilities of LoRa networks and potential countermeasures.

[48] investigates scalability challenges in LoRaWAN and the current state-of-
the-art solutions in the PHY and MAC layers, summarizing existing literature
solutions and proposing potential enhancements for future research. [49]
provides a chronological survey of available IoT simulation tools using a
systemic approach and presents an overview of LoRa/LoRaWAN technology,
including its architecture, transmission parameters, device classes, and
simulation tools, and evaluates LoRaWAN performance in terms of Packet
Delivery Ratio, memory usage, execution time, and the number of collisions.
[50] surveys LoRa-based solutions for irrigation systems, plantation and crop
monitoring, tree monitoring, and livestock monitoring, analyzing their
scalability, interoperability, network architecture, and energy efficiency.
[51] offers a comparative overview of cloud-based simulators for LPWANs. In
[52], the authors identify and discuss critical aspects and research
challenges in designing a LoRaWAN communication protocol from an EE
perspective and propose a GreenLoRaWAN communication protocol that prioritizes
EE, robustness, and scalability.

###  I-B Motivation and Contributions

It is evident that there exists a lack of surveys that consolidates the
various PHY layer CSS-based alternatives to LoRa for LPWANs (cf. Table IV). As
such, our primary objective is to thoroughly explicate a comparative analysis
of the CSS-based waveform designs proposed in the literature between $2019$
and $2022$ as a viable alternative to LoRa. This study is distinct from other
surveys on the topic in that it diverges from the conventional approach of
providing a general overview of LoRa technology, focusing on networking,
deployment, architecture, and applications.

The contributions of this comparative study are summarized as follows:

  * •

We classify all the CSS schemes between $2019$ and $2022$ into three distinct
taxonomies based on the number of FSs activated for the un-chirped symbol.
These three taxonomies are (i) single chirp (SC); (ii) multiple chirp (MC);
and (iii) MC with IM (MC-IM). This method of characterization has yet to be
previously done in the literature. Each taxonomy has unique advantages and
disadvantages, which shall be elucidated in the subsequent analysis.

  * •

Within the purview of SC, MC, and MC-IM taxonomies, we expound upon the
various design methodologies utilized in implementing CSS-based schemes for
IoT based on LPWANs. These design methodologies are the fundamental tenets of
various CSS modulations. Furthermore, these methodologies furnish guidance for
enhancing the efficacy of the protocols in terms of various performance
metrics.

  * •

We present a comprehensive analysis of coherent and non-coherent detection
principles from a waveform design standpoint. To the best of our knowledge,
these principles have yet to be explicitly expounded upon in the literature.

  * •

Thereafter, we furnish a comprehensive analysis of the PHY waveform design of
each CSS-based scheme for LPWAN, in which the time domain symbol structure, as
well as the coherent and non-coherent detection mechanisms (where applicable),
are also expounded upon. Furthermore, for the first time, we also present the
coherent detection mechanisms for four of the fifteen schemes included in the
study, which have yet to be documented in the extant literature. Additionally,
for the sake of clarity and ease of implementation, we also provide the block
diagram of the transceivers of these schemes. We have recreated the
transceiver block diagrams to conform with the notations utilized throughout
this study.

  * •

Based on the waveform design of each scheme, we proffer several critical
insights concerning the ease of implementation, potential performance
limitations, and issues inherent in the waveform design, among other
considerations.

  * •

Following a systematic examination of the examined CSS schemes for LPWANs in
terms of SE, a categorical designation of the schemes into six distinct groups
is established based on the maximum attainable SE. Subsequently, an
investigation is conducted to assess the performance of the CSS-based schemes
within each grouping concerning the SE and EE trade-off and bit error rate
(BER) performance in an additive white Gaussian noise (AWGN) channel.
Furthermore, an analysis of the BER performance is performed, accounting for
phase and frequency offsets inherent in low-cost devices utilized in LPWANs.

  * •

In this study, certain open research issues about the waveform design of CSS
schemes were brought to attention. We also propose a framework to improve the
performance of the schemes. The framework proposed in this study will likely
catalyze the emergence of new CSS schemes. Furthermore, we intend this work to
stimulate further exploration and investigation into this study area.

  * •

In conclusion, a thorough examination of the various CSS schemes’ pros and
cons was conducted, utilizing the previously established performance metrics.
A comparative analysis is performed to identify the most advantageous CSS
schemes within each grouping. Specific attention is given to those CSS schemes
that may serve as viable alternatives to LoRa in different environmental and
operational scenarios. Note that the evaluation of the CSS schemes’ relative
merits and shortcomings is conducted by comparing and contrasting them based
on previously discussed performance metrics.

To encourage reproducible research and to facilitate the replication of the
results presented in this study, public versions of the MATLAB codes for all
the schemes discussed in this study will be made available.

###  I-C Paper Organization

The overall structure of this survey is illustrated in Fig. 1. The rest of the
manuscript is organized in the following manner: In Section II, we commence by
presenting the system model adopted in this study, wherein we subsequently
categorize the schemes in SC, MC and MC with IM and offer some salient
observations from these categories concerning waveform design. Within the
context of SC, MC, and MC-IM, various design methodologies are elucidated with
regards to their specific characteristics and constraints. Furthermore,
Section II also endeavors to provide a comprehensive explication of the
coherent and non-coherent CSS detection principles. The waveform design and
detection mechanisms of the CSS schemes for IoT based on LPWANs are expounded
in Section III, along with some pertinent observations regarding different
design aspects. In Section IV, we undertake a performance comparison of the
studied CSS schemes in terms of SE, SE/EE trade-off, BER performance in the
AWGN channel, and BER performance taking into account phase and frequency
offsets. In Section V, we delve into the individual merits and limitations of
the investigated CSS schemes and propose alternative CSS schemes to LoRa,
which may exhibit superior performance under certain conditions. In Section
VI, we provide open research issues and future research directions. Finally,
in Section VII, we present the conclusions drawn from this study.

Figure 1: Section names and the main components of the conducted study.

##  II System Model and CSS Schemes Categorization

This section presents a generalized system model as the theoretical foundation
for our analysis. Subsequently, we classify the CSS schemes under examination
into three distinct categories based on the number of activated FSs for the
un-chirped symbol, with these categories being SC, MC and MC-IM.

###  II-A System Model

Figure 2: Generalized system model adopted in this study.

In this study, we adopt a generalized system model as in [7] and [10] without
loss of generality as illustrated in Fig. 2. Specifically, we consider a
discrete-time chirp symbol, $s(n)$, for $n\in\llbracket 0,M-1\rrbracket$,
where $M$ is the number of samples of duration $T_{\mathrm{c}}$. The sample
rate, $T_{\mathrm{c}}$ corresponds to a sampling frequency of
$F_{\mathrm{c}}=\nicefrac{{M}}{{T_{\mathrm{c}}}}=B$; thus, each discrete-time
chirp symbol is fully represented by $M$ samples taken at the rate
$F_{\mathrm{c}}$. By definition, $s(n)$ is the product of two components,
namely (i) an un-chirped symbol, $f(n)$, and (ii) a spreading/chirp symbol,
$c(n)$; therefore, $s(n)=f(n)c(n)$, where $c(n)$ serves to spread the
information in the bandwidth, $B=\nicefrac{{M}}{{T_{\mathrm{s}}}}$, where
$T_{\mathrm{s}}$ is the symbol duration. Moreover, when $f(n)$ is chirped
using $c(n)$, the FS(s) has an injective mapping to a unique cyclic time-
shift(s) of the chirped signal, $s(n)$.

A total of $M$ FSs are available for activation in the un-chirped symbol,
where one or multiple FSs can be activated. It is assumed that the FS is the
information-bearing element at this point. The FS(s) are chosen from the set
$[0,M-1]$. It is worth noting that, in addition to the FS(s), other components
of the transmit symbol structure, such as the PS, the use of multiple CRs, and
interleaving on the chirped symbol, etc. can also be utilized to convey
information. These parameters provide added flexibility in incorporating more
bits into the CSS symbols. As such, in the subsequent sections of this study,
we shall undertake a comprehensive examination of different design aspects
related to CSS-based schemes for LPWANs.

In LPWANs, the CSS symbols are characterized by a limited frequency bandwidth
of $500$ kHz or less. As such, it is appropriate to posit that the channel
through which these symbols propagate exhibits flat fading behavior, wherein
the attenuation experienced by the signal is constant across the entire
bandwidth, $B$. The discrete-time baseband received symbol is mathematically
represented as:

$y(n)=s(n)+w(n),$ (1)

for $n=\llbracket 0,M-1\rrbracket$, where w(n) represents the samples of AWGN,
which possess a single-sided noise power spectral density of $N_{0}$ and a
noise variance of $\sigma_{n}^{2}=N_{0}B$. In this formulation, the channel
attenuation is assumed to be unity and, as such, is not explicitly present in
eq. (1). This is due to the assumption that a priori knowledge of the CSI is
available; thus, the channel does not need to be explicitly represented.

###  II-B CSS Schemes Taxonomy

In the context of LPWAN-based IoT, the utilization of CSS schemes necessitates
the presence of one or multiple FSs within the un-chirped symbol.
Consequently, the initial characterization of CSS schemes is predicated on the
number and configuration of activated FSs, categorizing three distinct
taxonomies: SC, MC, and MC-IM, as previously alluded to. It is imperative to
reiterate that the scope of the present analysis is restricted to those CSS
schemes specifically devised for LPWANs, wherein not all available FSs are
activated for the un-chirped symbols, as in OCDM, etc.

The previously enumerated three taxonomies exhibit distinctions in the
methodology employed to attain an un-chirped symbol. We consider that a total
of $M$ available FSs for activation.

####  II-B1 SC CSS Schemes

In the SC schemes, only a single FS is activated for the un-chirped symbol.
Then, the un-chirped symbol in SC schemes is mathematically represented as
follows:

${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}f(n)=\exp\left\\{j\frac{2\pi}{M}kn\right\\},}$
(2)

where $k\in\left[0,M-1\right]$ denotes the a activated FS and $n=\llbracket
0,M-1\rrbracket$. It is underlined that $k$ is obtained by converting the
input bit sequence from its binary representation to its equivalent decimal
form.

From a waveform design standpoint, the SC schemes boast a plethora of
advantageous characteristics. These include: (i) high EE due to the activation
of a single FS; (ii) constant envelope for the chirped symbol, which is
beneficial for the utilization of low-cost components within LPWANs; (iii)
simple waveform design, implementation, forthright detection mechanisms, and
maintenance; (iv) improved robustness to interference and phase and frequency
offsets; (v) lower latency resulting from reduced complexity in terms of
signal processing; (vi) more cost-effective as they require fewer components
and less complex signal processing.

Despite the plethora of advantageous characteristics enumerated above, one of
the most substantial drawbacks of these SC systems is their relatively low
achievable SE, which precludes the widespread utilization of these schemes,
particularly for high data-rate IoT applications.

####  II-B2 MC CSS Schemes

In MC schemes, multiple FSs are activated in the un-chirped symbol rather than
activating a single FS. Assuming that a total of $q>1$ FSs are activated, the
un-chirped symbol, $f(n)$, for MC schemes can be represented as:

${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}f(n)=\sum_{q}\exp\left\\{j\frac{2\pi}{M}k_{q}n\right\\},}$
(3)

where $k_{q}\in\left[0,M-1\right]$ are the activated FS, such that $k_{q}\neq
k_{p}$ for $q\neq p$. From the standpoint of waveform design, it is typical to
select a value of $q$ that is relatively diminutive, such as $2$ or maybe $3$.
This is because augmenting the value of $q$ results in a decrease in the EE
metric (at the cost of an increase in SE), rendering it inapplicable for
LPWANs. The implementation of activating $q$ FSs can be accomplished through
various techniques. A potential strategy includes partitioning the total
number of FSs, $M$, into $q$ subsets, with each subset comprising
$\nicefrac{{M}}{{q}}$ FSs. Subsequently, one FS from each designated subset is
activated in correspondence to the specific input bit sequence.

MC schemes also offer numerous advantages, such as (i) higher SE relative to
SC CSS schemes; (ii) negligible decrease in EE due to activation of additional
FS(s); (iii) multiple FSs can be used to implement diversity; and (iv)
acceptable robustness against the phase and frequency offsets.

The utilization of MC schemes has its limitations. One such limitation is the
potential for non-constant envelope symbols resulting from the activation of
multiple FSs, necessitating the utilization of costly equipment. Additionally,
MC schemes often exhibit a higher PAPR than SC schemes, thereby exacerbating
the non-linearities of different system components. Furthermore, the design of
transceivers for MC systems is often complicated by the need for complex
signal-processing algorithms to facilitate the detection of multiple active
FSs. Finally, MC schemes may not possess the same robustness against phase and
frequency offsets as SC modulations.

####  II-B3 MC-IM CSS Schemes

In MC-IM, multiple FSs are simultaneously activated following a predefined
frequency activation pattern (FAP). The information bits are encoded within
the FAP rather than in the individual FSs activated within the un-chirped
symbol. This method typically utilizes two to three FSs, as activating a
higher number may negatively impact EE and increase system complexity. The set
of available FSs is represented as
$\boldsymbol{\Omega}=\left\\{0,1,\cdots,M-1\right\\}$, where $\varsigma$ FSs
are to be activated. The FAP, denoted as
$\boldsymbol{\sigma}=\\{\sigma_{1},\sigma_{2},\cdots,\sigma_{\varsigma}\\}\in\boldsymbol{\Omega}$,
is determined through binary-to-decimal (bi2de) conversion on
$\lfloor\log_{2}\binom{M}{\varsigma}\rfloor$ bits. The resulting un-chirped
symbol in MC-IM is then formulated as a function of the FAP as:

${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}f(n)=\sum_{k\in\boldsymbol{\sigma}}\exp\left\\{j\frac{2\pi}{M}kn\right\\}.}$
(4)

The utilization of MC-IM schemes not only confers the benefits inherent to MC
schemes but also potentially yields superior SE via incorporating additional
bits in the FAPs. However, the employment of MC-IM schemes also incurs added
complexity, as detecting FAPs necessitates sophisticated signal processing
algorithms, thereby exacerbating latency and power consumption.

####  II-B4 Takeaways

It is unequivocally apparent that each taxonomic classification of modulation
schemes possesses its own set of inherent limitations. Specifically, SC
schemes are optimal from a waveform design perspective as they afford the
utilization of low-cost equipment, are comparatively simple to implement and
maintain and possess minimal signal processing requirements. However, such
schemes’ maximum achievable SE is limited, rendering them ill-suited for
applications requiring high data rates. Conversely, MC schemes exhibit an
improvement in SE relative to SC schemes but are accompanied by the added
complexity of signal processing and other associated issues. Furthermore, MC-
IM schemes, being a relatively nascent concept, necessitates a comprehensive
analysis to optimize various parameters to enhance their overall performance.

In the practical implementation of LPWANs for IoT applications, LoRa is
commonly employed, as it belongs to the class of SC schemes and is one of the
first CSS-based schemes proposed for LPWANS. However, with the advent of CSS-
based schemes from other taxonomies, it is possible that schemes other than
LoRa, belonging to different taxonomies, exhibit superior performance under
varying circumstances. Consequently, a comprehensive analysis from a practical
implementation standpoint is imperative. From a theoretical perspective, SC
schemes can be employed in scenarios where low-cost equipment is compulsory,
different offsets are anticipated, and SE is not the paramount concern.
Conversely, MC and MC-IM schemes can be implemented in scenarios where a
higher SE is required, and the channel is relatively clean. At the same time,
equipment cost should not be a critical factor.

###  II-C CSS-based Schemes for LPWANs

A plethora of modulation schemes predicated on using CSS has been put forth in
the literature. However, the focus of this inquiry is precisely on those CSS
modulation schemes proposed explicitly for implementation within the framework
of LPWANs to facilitate IoT applications. To ensure clarity and concision, we
outline the parameters for the inclusion and exclusion of the various
modulation schemes in Table V that will be the subject of study in the context
of this comparative study.

TABLE V: Inclusion exclusion criteria for the schemes compared in this study. Inclusion Criteria | Exclusion Criteria
---|---
1\. Written in English | 1\. Written in language
language | other than English.
2\. Must be published |  2\. Published before $2016$
between $2016$ and $2022$. |  and after $2022$.
3\. Proposed for low-data | 3\. Proposed for other
rate LPWANs for IoT. | applications requiring
applications. | high data rate, such as
| DFRC, etc.
4\. Published in journals | 4\. Preprints in arXiv.
conference proceedings, |
symposiums or workshops. |
TABLE VI: CSS schemes fulfilling the inclusion criteria to be discussed in this study. Ref. | Name | Time | Taxonomy
---|---|---|---
[12, 7] | LoRa/FSCM | $2015/2017$ | SC
[53] | ICS-LoRa | $2019$ | SC
[54] | E-LoRa | $2019$ | SC
[16] | PSK-LoRa | $2019$ | SC
[55] | DO-CSS | $2019$ | MC
[17] | SSK-LoRa | $2020$ | SC
[56] | IQ-CSS | $2020$ | MC
[18] | DCRK-LoRa | $2021$ | SC
[19] | FSCSS-IM | $2021$ | MC-IM
[20] | IQ-CIM | $2021$ | MC-IM
[57] | SSK-ICS-LoRa | $2022$ | SC
[58] | ePSK-CSS | $2022$ | MC
[59] | GCSS | $2022$ | MC
[60] | TDM-CSS | $2022$ | MC
[60] | IQ-TDM-CSS | $2022$ | MC
[61] | DM-CSS | $2022$ | MC

Following the inclusion criteria established for this study, a chronologically
ordered listing of schemes and their respective categorization can be observed
in Table VI. It is worth noting that various CSS-based schemes, such as those
detailed in references [62, 63, 64, 65, 66, 67, 68, 69, 70, 71], exist within
the literature. However, these schemes do not adhere to the inclusion criteria
established for the current analysis. It is noteworthy that protocols
belonging to the same taxonomical classification may exhibit different design
methodologies from the perspective of waveform design. Thus, a thorough
comprehension of these disparate methodologies is essential, as expounded upon
in the subsequent sections of this study.

Following the established parameters of inclusion and exclusion criteria and
the taxonomic categorizations of the CSS schemes, Fig. 3 depicts a visual
representation of the schemes within the purview of our comparative
investigation, as well as those schemes which fall outside the scope of our
examination.

Figure 3: CSS schemes analyzed in this study and their categorization based on
the taxonomy.

###  II-D Design Methodologies

Figure 4: Design methodologies for the CSS schemes to attain different
waveform designs. Figure 5: System Model for different design methodologies.

This section provides a comprehensive exposition regarding the various design
methodologies that can be implemented for the waveform design pertaining to
LPWANs. It is to be noted that these design methodologies pertain exclusively
to the un-chirped symbols, $f(n)$ unless explicitly indicated otherwise. We
shall categorically specify the methods in question concerned with the chirped
symbols, $s(n)$. These methodologies can be explained with the help of Fig. 5,
where dashed and shaded blocks depict different design methodologies that can
be employed. Moreover, different design methodologies that can be applied to
attain different waveform designs are illustrated in Fig. 4.

####  II-D1 Activation of Frequency Shift(s)

A rudimentary design methodology commonly involves the deployment of a single
FS (as depicted in eq. (2) ) or multiple FSs (as depicted in eqs. (3) and (4))
for the un-chirped symbol. Schemes adhering to other design methodologies also
active either a single FS or multiple FSs. Thus, activating FS(s) is
fundamental for all the CSS schemes. Of the CSS schemes analyzed in this
comparative examination, LoRa activates a single FS, DO-CSS and GCSS adhere to
activating multiple FSs. Furthermore, FCSS-IM activates a multiple of FSs and
falls under the categorization of MC-IM taxonomy.

####  II-D2 Interleaving the Chirped Symbol

Another means to enhance the design of the LoRa waveform is to perform
interleaving of the chirped symbol, $s(n)$ denoted as $\Pi\left[s(n)\right]$.
It is imperative to acknowledge that the un-chirped symbol is similar to those
utilized in systems classified under the SC or MC taxonomies. In a rudimentary
manner, the chirped symbol is either interleaved or not, resulting in the ICS-
LoRa scheme, which conveys one extra bit compared to conventional LoRa.
However, the diversification of interleaving patterns can be explored, as
evidenced in [72], resulting in a further increase in the transmitted bits.
Nevertheless, it is crucial to note that an increase in the interleaving
patterns shall lead to an augmentation in low correlation patterns, thereby
yielding a detrimental effect on the scheme’s performance. Thus, a reasonable
selection of interleave patterns is imperative, with preference given to
patterns exhibiting high correlation values.

####  II-D3 Using Phase Shift(s)

An alternate strategy for the waveform design of CSS systems is the
integration of PS(s) into the un-chirped symbol, in addition to FS(s). This
creates a composite symbol, $f(n)\varphi$, which can then be chirped utilizing
a spreading symbol, $c(n)$. This design approach showcases a multitude of
advantages and several drawbacks. On the one hand, it can enhance the SE and
EE of the CSS scheme considerably. On the other hand, its performance can be
significantly degraded in PO, a common occurrence in low-cost devices utilized
in LPWANs. Furthermore, the utilization of PS mandates a coherent detection
mechanism, as opposed to the de-chirped received symbol’s FS, which may be
detected through non-coherent means. Nevertheless, if only two PSs are used,
non-coherent detection is also possible. However, deploying a limited number
of PSs would constrict the enhancement in SE to a lower level. The SC scheme
which uses PS is PSK-LoRa, while the MC approach using PSs is ePSK-LoRa.

####  II-D4 Chirp Rates Variation

An alternate design paradigm involves varying CRs within the spreading symbol,
which was previously denoted as $c(n)$. The spreading symbol can be
represented in a more generalized manner as follows:

${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}c_{\mathrm{c}}(n)=\exp\left\\{j\frac{\pi}{M}\gamma_{\mathrm{c}}n^{2}\right\\},}$
(5)

where $\gamma_{\mathrm{c}}$ is the CR, $\mathrm{c}$ is the indexing variable
whose values are $\mathrm{c}=\llbracket 0,M_{\mathrm{c}}-1\rrbracket$, and
$M_{\mathrm{c}}$ is the number of different CRs that can used. In the sequel
of the article, if $\gamma_{\mathrm{c}}=1$, we will refer to the spreading
symbol as an up-chirp symbol, $c_{\mathrm{u}}(n)$.

In this methodology, the chirped symbol is obtained by multiplying the un-
chirped symbol with a spreading symbol with varying chirp rates, i.e.,
$f(n)c_{\mathrm{c}}(n)$. In this particular design paradigm, the methodology
necessitates only an alteration of the CR. At the same time, the un-chirped
symbol is the same as in SC or MC schemes. The utilization of varying CRs
affords a higher EE and SE than methodologies that rely solely on using a
single CR. However, we must note that the detection complexity of these
schemes increases linearly with the number of CRs utilized. The SC schemes
that adopt this methodology include DCRK-LoRa and SSK-LoRa; however, it is
worth mentioning that this methodology can also be easily adapted for
implementation in MC schemes.

####  II-D5 Chirp Rates Variation and Interleaving the Chirped Symbol

The design methodology under consideration involves the integration of two
previous methodologies, namely the utilization of varying CRs and the
interleaving of the resulting chirped symbol. The chirped symbol, $s(n)$, is
obtained via the multiplication of the un-chirped symbol, $f(n)$ and the
spreading symbol, $c_{\mathrm{c}}$, i.e., $s(n)=f(n)c_{\mathrm{c}}(n)$,
following which it is subjected to interleaving, i.e., $\Pi\left[s(n)\right]$.
This design paradigm elevates the SE by incorporating additional information
bits into the CRs and interleaved symbols. The scheme belonging to the SC
taxonomy of the CSS schemes, which adheres to this methodology, is SSK-ICS-
LoRa.

####  II-D6 Using Phase Shift(s) and Varying Chirp Rates

This methodology synergistically incorporates FS(s) with integrating PS(s) and
varying CRs; thus, the chirped symbol is obtained as $s(n)=f(n)\varphi
c_{\mathrm{c}}(n)$. This design methodology affords a substantial degree of
operational latitude in waveform design, as it allows to alter PS(s) and vary
the CRs. The eventual outcome of this approach is a noteworthy improvement in
SE and an enhancement in EE. Nevertheless, the utilization of diverse design
methodologies within this method heightens the complexity of detection.
Furthermore, it is only possible to employ coherent detection when the number
of PS(s) is substantial. The CSS scheme employing this methodology is DM-CSS,
which falls within the MC taxonomy. However, it is important to note that in
DM-CSS, only two PS(s) are utilized; thus, non-coherent detection remains a
viable option.

####  II-D7 Time Domain Multiplexing of Chirped Symbols

TDM design methodology results in a MC CSS scheme as it multiplexed different
chirped symbols. Let’s consider two distinct chirped signals, denoted as
$s_{1}(n)$ and $s_{2}(n)$; then, through the implementation of TDM, the
resulting transmitted chirp signal is $s(n)=s_{1}(n)+s_{2}(n)$. It is
imperative to note that the underlying un-chirped signals, $f_{1}(n)$ and
$f_{2}(n)$, within $s_{1}(n)$ and $s_{2}(n)$, respectively, can be obtained
via a single FS or multiple FSs. The spreading symbols, $c_{1}(n)$ and
$c_{2}(n)$, use varying CRs to facilitate a straightforward detection process.
This design approach dramatically enhances the scheme’s SE. However, it is
essential to acknowledge that the scheme’s complexity increases proportionally
with the number of multiplexed symbols. Additionally, it is worth mentioning
that TDM-based schemes do not possess a constant envelope characteristic. In
the existing literature, TDM-CSS and IQ-TDM-CSS are among the MC schemes
incorporating TDM design methodology.

####  II-D8 Use of In-phase and Quadrature Components

This methodology exploits the complex nature inherent within the un-chirped
signal, $f(n)$, by inducing simultaneous activation of FS in both in-phase and
quadrature components. As a result, a substantial enhancement in SE is
observed for schemes that adopt this methodology. Despite its advantages, we
must note that schemes that employ this methodology are highly vulnerable to
phase and frequency shifts, thus rendering its practicality relatively
limited. Furthermore, in most scenarios, the non-coherent detection mechanism
of these schemes is not possible or is notably complex. We can implement this
methodology across CSS schemes that belong to all taxonomies, with E-LoRa
exemplifying its utilization in the SC category through the utilization of I/Q
components. Other CSS schemes that belong to the MC and MC-IM taxonomies, such
as IQ-CSS and IQ-CIM, respectively, also adopt this methodology. IQ-TDM-CSS
also exploits the I/Q components; however, in this study, we classify it as a
TDM approach.

####  II-D9 Takeaways

Each methodology used for the waveform design has distinct advantages and
drawbacks. As a general trend, such methodologies do not exhibit any
discernible impact on the inherent attributes of the SC, MC, and MC-IM
taxonomies. For instance, they do not alter envelope characteristics; SC-based
approaches maintain a constant envelope, while MC and MC-IM schemes exhibit a
non-constant envelope. In addition, these design methodologies are not
predisposed to create any complications regarding coherent or non-coherent
detection, except for utilizing PS(s) and I/Q components that mandate the
utilization of a coherent detection mechanism. Note that these design
methodologies serve as the means by which efficient waveforms can be generated
for LPWANs, with the SE metric commonly being associated with the taxonomy and
not the waveform design methodology.

Utilizing specific design methodologies can result in a substantial effect on
the operational efficiency of resulting waveform designs. For instance, the
utilization of PS(s) and I/Q components renders the configurations vulnerable
to phase and frequency offsets, thus rendering these design methodologies less
viable from a practicality standpoint.

TABLE VII: Differentiation of SC CSS schemes based on design methodologies. FS | Interleaving | PS | CR | CR Variation | PS and CR | TDM | I/Q
---|---|---|---|---|---|---|---
|  |  | Variation | and Interleaving | Variation |  | Components
LoRa | ICS-LoRa | PSK-LoRa | SSK-LoRa | SSK-ICS-LoRa | - | Not Applicable | E-LoRa
|  |  | DCRK-LoRa |  |  |  |
TABLE VIII: Differentiation of MC CSS schemes based on design methodologies. FS | Interleaving | PS | CR | CR Variation | PS and CR | TDM | I/Q
---|---|---|---|---|---|---|---
|  |  | Variation | and Interleaving | Variation |  | Components
DO-CSS | - | ePSK-CSS | - | - | DM-CSS | TDM-CSS | IQ-CSS
GCSS |  |  |  |  |  | IQ-TDM-CSS |
TABLE IX: Differentiation of MC-IM CSS schemes based on design methodologies. FS | Interleaving | PS | CR | CR Variation | PS and CR | TDM | I/Q
---|---|---|---|---|---|---|---
|  |  | Variation | and Interleaving | Variation |  | Components
FSCSS-IM | - | - | - | - | - | - | IQ-CIM
Figure 6: Graphical representation of how different SC schemes are derived
from LoRa using different design methodologies. Figure 7: Graphical
representation of how different MC schemes are derived from LoRa using
different design methodologies.

The design methodologies applicable to SC, MC, and MC-IM taxonomies are
presented in Tables VII, VIII, and IX, respectively. Given that both SC and MC
schemes are derived from classical LoRa, the graphical representation of their
relationship with LoRa based on different design methodologies is illustrated
in Figs. 6 and 7, respectively. A cursory examination reveals that certain
design methodologies are exclusively pertinent to a specific taxonomy, as
demonstrated by the fact that we can utilize all design methodologies in SC
schemes except the methodology that incorporates both PS and CRs variation.
Additionally, literature has yet to be documented exploring MC schemes
incorporating interleaving, CRs variation, and a combination of interleaving
and CR variation. The existing MC-IM schemes in the scholarly corpus either
employ FSs or harness I/Q components. Thus, the differentiation serves as a
foundation for future research by illuminating the design methodologies that
remain unexplored in this field.

###  II-E Detection Principles

Within this purview, we delve into the different detection mechanisms we can
leverage for CSS schemes. Coherent detection is implemented when there exists
a priori information regarding the CSI; in contrast, non-coherent detection is
exercised when CSI information is not accessible. Additionally, when the
magnitude of the PS vector exceeds a cardinality of $2$ and when the I/Q
components are utilized, then coherent detection becomes an imperative
requirement. It is to be noted that, despite its efficacy, coherent detection
bears a relatively elevated level of complexity, rendering it less practical
to execute in cost-sensitive LPWANs.

####  II-E1 Coherent Detection Principles

The fundamental principle of coherent detection rests on the assumption that
all the information-bearing elements are to be estimated. Thus, in its most
rudimentary form, coherent detection primarily focuses on estimating the FS,
i.e., $k$ (cf. eq. (2)). Nevertheless, in other CSS schemes, it may be
necessary to extend the estimation procedure to detect other information-
bearing elements, such as PS(s) and the like. Nonetheless, the coherent
detection process can effectively estimate all these parameters by
concurrently optimizing the Likelihood Function across all the viable
orthonormal bases.

The coherent detection process dictates to maximize the probability of
receiving $\boldsymbol{y}=\left[y(0),y(1),\cdots,y(M-1)\right]^{\mathrm{T}}$
given $\boldsymbol{s}=\left[s(0),s(1),\cdots,s(M-1)\right]^{\mathrm{T}}$,
i.e., $\mathrm{prob}\left(\boldsymbol{y}|\boldsymbol{s}\right)$. Considering
that all the possible symbols are equiprobable, the likelihood function,
$\mathrm{prob}\left(\boldsymbol{y}|\boldsymbol{s}\right)$ adheres to the
Gaussian probability density function as follows:

$\begin{split}\mathrm{prob}\left(\boldsymbol{y}|\boldsymbol{s}\right)&=\left(\frac{1}{2\pi\sigma_{n}^{2}}\right)^{M}\exp\left\\{\frac{\|\boldsymbol{y}-\boldsymbol{s}\|^{2}}{2\sigma_{n}^{2}}\right\\}\\\
&=\mathrm{C_{st}}\exp\left\\{\frac{\Re\left\\{\langle\boldsymbol{y},\boldsymbol{s}\rangle\right\\}}{\sigma_{n}^{2}}\right\\},\end{split}$
(6)

where $\|\cdot\|^{2}$ evaluates Euclidean norm, the real component of a
complex argument is determined through $\Re\\{\cdot\\}$, the inner product is
evaluated via $\langle\cdot,\cdot\rangle$, and the constant term is defined as
$\mathrm{C_{st}}=({1}/{2\pi\sigma_{n}^{2}})^{M}\exp\left\\{-(\|\boldsymbol{y}\|^{2}+\|\boldsymbol{s}\|^{2})/2\sigma_{n}^{2}\right\\}$.
Consequently, the coherent detection problem is reduced to the following
simplified form:

$\hat{k}_{\mathrm{coh}}=\mathrm{arg}\max_{k}~{}\mathrm{prob}\left(\boldsymbol{y}|\boldsymbol{s}\right)=\mathrm{arg}\max_{k}~{}\Re\left\\{\langle\boldsymbol{y},\boldsymbol{s}\rangle\right\\}.$
(7)

Eq. (7) conveys that with the utilization of coherent detection, the
calculation of the inner product between the received signal vector,
$\boldsymbol{y}$, and all feasible transmit symbols must be performed. This
computation is computationally intensive. However, because the transmit symbol
is represented as $s(n)=f(n)c_{\mathrm{u}}(n)$, the simplification of
$\Re\left\\{\langle\boldsymbol{y},\boldsymbol{s}\rangle\right\\}$ can be
achieved as follows:

$\begin{split}\Re\left\\{\langle\boldsymbol{y},\boldsymbol{s}\rangle\right\\}&=\Re\left\\{\sum_{n=0}^{M-1}y(n)\overline{s}(n)\right\\}\\\
&=\Re\left\\{\sum_{n=0}^{M-1}y(n)\overline{f}(n)c_{\mathrm{d}}(n)\right\\}\\\
&=\Re\left\\{\sum_{n=0}^{M-1}r(n)\overline{f}(n)\right\\}\\\
&=\Re\left\\{\sum_{n=0}^{M-1}r(n)\exp\left\\{-j\frac{2\pi}{M}nk\right\\}\right\\}\\\
&=\Re\left\\{R(k)\right\\},\end{split}$ (8)

where $\overline{(\cdot)}$ is the conjugate operator,
$r(n)=y(n)c_{\mathrm{d}}(n)$, and $R(k)$ is the DFT of $r(n)$ evaluated at
$k$th index. Here,
$c_{\mathrm{d}}(n)=\exp\left\\{-j\frac{\pi}{M}n^{2}\right\\}$ is the down-
chirp which is the conjugate of $c_{\mathrm{u}}(n)$. Now, the detection
problem eq. (8) in its simplified form becomes:

$\hat{k}_{\mathrm{coh}}=\mathrm{arg}\max_{k}~{}\Re\left\\{R(k)\right\\}.$ (9)

In practice, the process commences with the correlation of the received
signal, $y(n)$ with $c_{\mathrm{d}}(n)$, yielding the $r(n)$. Subsequently,
the DFT of $r(n)$ is computed, resulting in $R(k)$. The estimation of the
transmitted FS, represented as $\hat{k}$, is acquired by taking the real
argument of $R(k)$. It is noteworthy that for coherent detection, only the
real component of $R(k)$ is deemed pertinent, leading to the neglect of all
noise present in the imaginary components.

It is imperative to emphasize that eq. (9) is only applicable to specific CSS
designs, such as LoRa, and may not hold for other CSS configurations. The
inner product of $\langle\boldsymbol{y},\boldsymbol{s}\rangle$ may diverge
greatly based on the waveform design of the transmitted symbol. Hence, in
subsequent discussions on applying coherent detection to CSS schemes, we will
deduce the evaluation of $\langle\boldsymbol{y},\boldsymbol{s}\rangle$ based
on the specific signal structure of the transmitted symbol of each CSS scheme
considered in this study.

####  II-E2 Non-Coherent Detection Principles

According to previous discourse, in the absence of a priori knowledge of CSI,
non-coherent detection may be leveraged to ascertain the information-bearing
element, assumed to be the transmit FS, $k$. This non-coherent detection
methodology is more feasible due to its comparatively reduced computational
demands. Thus, it is a viable option for implementation in LPWANs where
constraints such as limited computational power and the cost-effectiveness of
components are paramount considerations. Hence, through the utilization of
non-coherent detection, we can determine the transmit FS, $k$, as:

$\begin{split}\hat{k}_{\mathrm{non-
coh}}&=\mathrm{arg}\max_{k}~{}\left|\sum_{n=0}^{M-1}y(n)\overline{s}(n)\right|\\\
&=\mathrm{arg}\max_{k}~{}\left|R(k)\right|,\end{split}$ (10)

where $|\cdot|$ is the absolute operator, and the second equality holds owing
to eq. (8).

In the context of non-coherent detection-based estimation of the transmit FS,
we commence by performing a DFT of the de-chirped received signal, $r(n)$, to
yield $R(k)$. Subsequently, the argument of the absolute maximum of $R(k)$ is
computed, resulting in the index corresponding to the highest peak. Notably,
in contrast to coherent detection, non-coherent detection incorporates the
effect of both real and imaginary noise components. Consequently, it is
anticipated that in an AWGN channel, the performance of coherent detection
would surpass that of non-coherent detection. It is also noteworthy that while
a coherent detector enables the correction of phase rotation induced by the
channel, non-coherent detection cannot mitigate the random phase rotation
imposed by the channel.

####  II-E3 Takeaways

Coherent detection is a formidable technique from a waveform design
perspective, exhibiting high performance in ideal operating conditions.
However, its applicability in low-power and low-cost applications is
questionable given the requirement of prior knowledge of the CSI, increased
computational complexity, and heightened sensitivity to phase and frequency
offsets. In contrast, non-coherent detection constitutes a feasible and
computationally tractable mechanism. Yet, it may fall short of providing the
performance and robustness attainable through coherent detection in scenarios
with more stringent requirements.

##  III CSS-based PHY Layer Waveform Design and Detection Principles

In the sequel, the purview of our examination is limited to the waveform
design and detection mechanisms pertaining to uncoded variants of CSS-based
PHY layer to maintain simplicity and clarity. Investigating the performance of
these schemes incorporating appropriate coding is a matter of future
exploration. It should be noted that the CSS schemes analyzed in this study,
as enumerated in Table VI, are depicted based on specific taxonomical
classifications.

###  III-A Single Carrier CSS Schemes

####  III-A1 LoRa

LoRa is a proprietary SC CSS scheme developed by the Semtech corporation,
which allows for the trade-off of sensitivity and data rates for fixed channel
bandwidths. Although the Semtech corporation has not publicly disclosed the
specifics of the LoRa modulation scheme, [7], and [10] have provided a
comprehensive theoretical overview of the LoRa modulation scheme, including an
optimal low-complexity detection process based on DFT. A scalable parameter,
SF, characterizes the LoRa modulation scheme, denoted by $\lambda$, and is
defined in the interval $\lambda=\llbracket 6,12\rrbracket$. In actuality,
$\lambda$ equals the number of bits transmitted by a LoRa symbol of duration
$T_{\mathrm{s}}$. Based on $\lambda$, a total of $M$ distinct orthogonal
symbols can be realized for LoRa, defined through $M$ distinct cyclic time
shifts of the chirp signal. These cyclic time shifts correspond to FSs of the
complex conjugate of the chirp signal, i.e., the down-chirp signal. It is
imperative to note that LoRa operates at the PHY layer and is agnostic to
higher-layer implementations, allowing for its coexistence with other network
architectures.

##### Transmission

Figure 8: LoRa transmitter architecture.

The transmitter architecture of LoRa is depicted in Fig. 8. Consider that
there are $M$ FSs within the bandwidth, $B$. The activated FS, denoted by $k$,
is determined through the bi2de (binary-to-decimal) conversion of incoming
$\lambda$ bits. Utilizing $k$, a discrete-time un-chirped symbol, $f(n)$, is
generated as outlined in eq. (2). This un-chirped symbol is then subjected to
spreading through the application of an up-chirp symbol, $c_{\mathrm{u}}(n)$,
resulting in the generation of the LoRa signal, $s_{\mathrm{LoRa}}(n)$. The
discrete-time chirped signal, $s_{\mathrm{LoRa}}(n)$, can be represented as:

$s_{\mathrm{LoRa}}(n)=f(n)c_{\mathrm{u}}(n)=\exp\left\\{j\frac{\pi}{M}n(2k+n)\right\\}.$
(11)

It is imperative to note that the normalization factors have been omitted for
notational simplicity. The energy of a LoRa symbol is
$E_{\mathrm{s}}=\nicefrac{{1}}{{M}}\sum_{n=0}^{M-1}|s_{\mathrm{LoRa}}(n)|^{2}=1$.

##### Detection

Figure 9: Coherent detector architecture for LoRa.

The received LoRa symbol can be detected by the coherent or non-coherent
method. The implementation of a coherent detector for the LoRa symbol is
depicted in Fig. 9. Coherent detection entails computing the inner product of
the received symbol vector, represented as $\boldsymbol{y}$, and the pre-
defined transmit symbol vector,
$\boldsymbol{s}\mathrm{LoRa}=\left[{s}\mathrm{LoRa}(0),{s}\mathrm{LoRa}(1),\cdots,{s}\mathrm{LoRa}(M-1)\right]^{\mathrm{T}}$,
i.e., the dot product
$\langle\boldsymbol{y},\boldsymbol{s}_{\mathrm{LoRa}}\rangle$ is computed as:

$\langle\boldsymbol{y},\boldsymbol{s}_{\mathrm{LoRa}}\rangle=\sum_{n=0}^{M-1}y(n)\overline{s}_{\mathrm{LoRa}}=R(k),$
(12)

where the realization of equation (12) is achieved by adhering to the
procedure delineated in eq. (8). Subsequently, by employing the principle of
coherent detection as outlined in eq. (9), the estimation of the received
symbol’s FS, $\hat{k}$, can be ascertained as follows:

$\hat{k}=\mathrm{arg}\max_{k}~{}\Re\left\\{R(k)\right\\}.$ (13)

Therefore, for the implementation of coherent detection, the process begins by
performing a de-chirping of the received symbol, $y(n)$, using a down-chirp,
$c_{\mathrm{d}}(n)$, resulting in $r(n)=y(n)c_{\mathrm{d}}(n)$ which serves as
the received equivalent of $f(n)$. Thereafter, the output of the de-chirping
process, $r(n)$, is subjected to a DFT evaluation, which yields $R(k)$, for
$k\in\llbracket 0,M-1\rrbracket$. It is worth highlighting that the detection
problem for LoRa is equivalent to that presented in eq. (9).

Figure 10: Non-coherent detector architecture for LoRa.

The non-coherent detector utilized in the LoRa is depicted in Fig. 10. After
the evaluation of $R(k)$ in a manner congruent to the evaluation procedure
utilized in the coherent detection method, the estimation of the FS, denoted
as $\hat{k}$, is determined through the implementation of the non-coherent
detection principle, as outlined in equation (10) as:

$\hat{k}=\mathrm{arg}\max_{k}~{}\left|R(k)\right|.$ (14)

After the utilization of the decimal-to-binary conversion function (de2bi) to
obtain the binary form of $\hat{k}$, the bit sequence, $\hat{\lambda}$,
comprising the estimate of the transmitted bit sequence, is determined.

##### Takeaways

LoRa has a simplistic and versatile waveform design, in which SFs may be
adjusted to comply with specific application requisites, such as the intended
coverage range, data transfer rate, and energy expenditure. Moreover, the SC
nature of LoRa fortifies its advantageous features in the context of low-cost
implementation in LPWANs. LoRa’s simple waveform architecture is conducive to
sustaining satisfactory performance levels under varying channel conditions.
LoRa detection mechanisms are notably facile to realize, with non-coherent
detection manifesting comparatively lower complexity vis-à-vis coherent
detection. Coherent detection yields superior performance outcomes by virtue
of its exclusion of noise on the imaginary constituents of the received
frequency-domain un-chirped symbol. Conversely, non-coherent detection takes
both the real and imaginary noise components of the received frequency-domain
un-chirped symbol into account, culminating in comparatively inferior
performance. Nonetheless, a small number of bits per symbol transmitted by
LoRa precludes its attractiveness as an adaptive waveform choice that can
cater to a wide range of applications requiring higher data rates.
Furthermore, in LoRa’s waveform design, not all the parameters that can
improve the performance have been targeted; therefore, there is considerable
room for enhancement in the waveform design.

####  III-A2 Interleaved Chirp-Spreading (ICS)-LoRa

The first PHY layer-based approach aimed at elevating SE relative to the LoRa
is ICS-LoRa, as outlined in the seminal work presented in [53]. This scheme
belongs to the SC taxonomy and leverages interleaved chirped symbols as a
waveform design methodology. The coexistence of LoRa and ICS-LoRa was studied
in [73], concluding that a LoRa network can support both protocols
simultaneously. [74] explored the utilization of ICS-LoRa as a parallel
logical network to augment the capacity of a LoRa network. The number of
transmitted bits in ICS-LoRa is augmented by including both the interleaved
version of the LoRa symbol and the conventional LoRa symbol. However, this
increases one bit per transmitted symbol of duration $T_{\mathrm{s}}$, thereby
yielding $\lambda+1$ bits transmitted per ICS-LoRa symbol.

Figure 11: ICS-LoRa transmitter architecture.

##### Transmission

The implementation architecture of the ICS-LoRa transmitter is graphically
represented in Fig. 11. Adhering to the LoRa protocol, the process commences
by attain the FS, $k$ by converting $\lambda$ bits using a bi2de converter.
Subsequently, the output, $f(n)$, is generated using $k$ following the eq.
(2). The derived function, $f(n)$, is then up-chirped through using
$c_{\mathrm{u}}(n)$, resulting in the LoRa transmit symbol,
$s_{\mathrm{LoRa}}(n)$, as per eq. (11). The binary input,
$\lambda_{\mathrm{I}}=1$, is utilized to determine the value of
$\beta_{\mathrm{I}}$ which decides whether to transmit the conventional LoRa
symbol, $s_{\mathrm{LoRa}}(n)$, or its interleaved equivalent, i.e.,
$\Pi\left[s_{\mathrm{LoRa}}(n)\right]$, where $\Pi[\cdot]$ represents the
interleaving operation. The value of $\beta_{\mathrm{I}}$ is obtained post-
conversion via the bi2de converter and is restricted to the binary set
$\left\\{0,1\right\\}$, as indicated by $\beta_{\mathrm{I}}=\llbracket
0,2^{\lambda_{\mathrm{I}}}-1\rrbracket$. The discrete-time ICS-LoRa transmit
symbol is given by:

$s_{\mathrm{ICS}}(n)=\begin{dcases}s_{\mathrm{LoRa}}(n)&\quad\beta_{\mathrm{I}}=0\\\
\Pi\left[s_{\mathrm{LoRa}}(n)\right]&\quad\beta_{\mathrm{I}}=1\end{dcases}.$
(15)

The interleave function, as shown in Fig. 2, performs a shuffling operation on
the $s_{\mathrm{LoRa}}(n)$ of length $M$, whereby the 2nd and 3rd quartered
segments are rearranged in a permutated manner, as outlined below:

$\Pi\left[s_{\mathrm{LoRa}}(n)\right]=\begin{dcases}s_{\mathrm{LoRa}}(n)&\quad
n=\llbracket 0,\nicefrac{{M}}{{4}}\rrbracket\\\
s_{\mathrm{LoRa}}(n+\nicefrac{{M}}{{4}})&\quad
n=\llbracket\nicefrac{{M}}{{4}},\nicefrac{{M}}{{2}}\rrbracket\\\
s_{\mathrm{LoRa}}(n-\nicefrac{{M}}{{4}})&\quad
n=\llbracket\nicefrac{{M}}{{2}},\nicefrac{{3M}}{{4}}\rrbracket\\\
s_{\mathrm{LoRa}}(n)&\quad n=\llbracket\nicefrac{{3M}}{{4}},M\rrbracket\\\
\end{dcases}.$ (16)

The symbol energy of a ICS-LoRa symbol is
$E_{\mathrm{s}}=\nicefrac{{1}}{{M}}\sum_{n=0}^{M-1}|s_{\mathrm{ICS}}(n)|^{2}=1$.

##### Detection

Figure 12: Coherent detector architecture for ICS-LoRa.

In [53], the authors proffered only a non-coherent method for detecting the
ICS-LoRa symbol. However, it is also feasible to employ a coherent detection
approach for symbol detection in ICS-LoRa. In this work, we extend the scope
of detection in ICS-LoRa by presenting both coherent and non-coherent
detection mechanisms.

Fig. 12 depicts the coherent detection architecture for ICS-LoRa. As specified
by eq. (15), it is evident that LoRa is a particular instance of ICS-LoRa.
Consequently, the inner product calculation of the received symbol vector,
represented by $\boldsymbol{y}$, and the transmit symbol vector,
$\boldsymbol{s}_{\mathrm{ICS}}=\left[s_{\mathrm{ICS}}(0),s_{\mathrm{ICS}}(1),\cdots,s_{\mathrm{ICS}}(M-1)\right]^{\mathrm{T}}$,
i.e., $\langle\boldsymbol{y},\boldsymbol{s}_{\mathrm{ICS}}\rangle$, must be
computed for two distinct scenarios, namely, (i) when the conventional LoRa
symbol is transmitted, i.e., $\beta_{\mathrm{I}}=0$, and (ii) the interleaved
LoRa symbol is transmitted, i.e., $\beta_{\mathrm{I}}=1$. For the former
scenario (i), the inner product term
$\langle\boldsymbol{y},\boldsymbol{s}_{\mathrm{ICS}}\rangle$ evaluates to:

$\begin{split}\langle\boldsymbol{y},\boldsymbol{s}_{\mathrm{ICS}}\rangle&=\sum_{n=0}^{M-1}y(n)\overline{s}_{\mathrm{ICS}}\\\
&=\sum_{n=0}^{M-1}y(n)\overline{f}(n)c_{\mathrm{d}}(n)\\\
&=\sum_{n=0}^{M-1}r_{1}(n)\overline{f}(n)\\\ &=R_{1}(k),\end{split}$ (17)

where $r_{1}(n)=y(n)c_{\mathrm{d}}(n)$. Conversely, when the interleaved LoRa
is transmitted, then
$\langle\boldsymbol{y},\boldsymbol{s}_{\mathrm{ICS}}\rangle$ yields:

$\begin{split}\langle\boldsymbol{y},\boldsymbol{s}_{\mathrm{ICS}}\rangle&=\sum_{n=0}^{M-1}y(n)\overline{s}_{\mathrm{ICS}}\\\
&=\sum_{n=0}^{M-1}\Pi\left[y(n)\right]\overline{f}(n)c_{\mathrm{d}}(n)\\\
&=\sum_{n=0}^{M-1}r_{2}(n)\overline{f}(n)\\\ &=R_{2}(k),\end{split}$ (18)

where $r_{2}(n)=\Pi\left[y(n)\right]c_{\mathrm{d}}(n)$.

As can be deduced from eqs. (17) and (18), the received symbol, $y(n)$ and its
interleaved counterpart, $\Pi\left[y(n)\right]$, undergo de-chirping utilizing
$c_{\mathrm{d}}(n)$ leading to $r_{1}(n)=y(n)c_{\mathrm{d}}(n)$ and
$r_{2}(n)=\Pi\left[y(n)\right]c_{\mathrm{d}}(n)$. Subsequently, the DFT is
computed for both $r_{1}(n)$ and $r_{2}(n)$, yielding $R_{1}(k)$ and
$R_{2}(k)$, respectively. The subsequent step is to identify whether a
conventional LoRa symbol or an interleaved LoRa symbol was transmitted. This
is determined through the calculation of two parameters; namely,
$\kappa_{1}^{\mathrm{coh}}=\max~{}\Re\left\\{R_{1}(k)\right\\}$ and
$\kappa_{2}^{\mathrm{coh}}=\max~{}\Re\left\\{R_{2}(k)\right\\}$ in the
interleave decision block. These parameters then facilitate the determination
of $\hat{\beta}_{\mathrm{I}}$ as:

$\hat{\beta}_{\mathrm{I}}=\begin{dcases}0&\quad\kappa_{1}^{\mathrm{coh}}>\kappa_{2}^{\mathrm{coh}}\\\
1&\quad\kappa_{1}^{\mathrm{coh}}<\kappa_{2}^{\mathrm{coh}}\\\ \end{dcases},$
(19)

which facilitates the discernment of the bit sequence,
$\hat{\lambda}_{\mathrm{I}}$, signifying the interleaved information in the
transmit symbol post transformation by de2bi.

Subsequently, the determination of the utilization of either $R_{1}(k)$ or
$R_{2}(k)$ in the identification of $\hat{k}$ is made using
$\kappa_{1}^{\mathrm{coh}}$ and $\kappa_{2}^{\mathrm{coh}}$, as follows:

$R(k)=\begin{dcases}R_{1}(k)&\quad\kappa_{1}^{\mathrm{coh}}>\kappa_{2}^{\mathrm{coh}}\\\
R_{2}(k)&\quad\kappa_{1}^{\mathrm{coh}}<\kappa_{2}^{\mathrm{coh}}\\\
\end{dcases}.$ (20)

Finally, using the coherent detection, the FS $\hat{k}$ is evaluated as:

$\hat{k}=\mathrm{arg}\max_{k}~{}\Re\left\\{R(k)\right\\},$ (21)

that determines $\hat{\lambda}$ after de2bi conversion.

Figure 13: Non-coherent detector architecture for ICS-LoRa.

The non-coherent detector configuration for ICS-LoRa is depicted in Fig. 13. A
distinction between non-coherent detection and coherent detection is evident
in two key aspects. The first differentiation arises in the interleave
decision block, where, instead of utilizing $\kappa_{1}^{\mathrm{coh}}$ and
$\kappa_{2}^{\mathrm{coh}}$, the utilization of $\kappa_{1}^{\mathrm{non-
coh}}=\max~{}|R_{1}(k)|$, and $\kappa_{2}^{\mathrm{non-
coh}}=\max~{}|R_{2}(k)|$ is employed in the determination of
$\hat{\lambda}_{\mathrm{I}}$ and $R(k)$. Secondly, a deviation is noted in
identifying the FS, executed using the following criterion:

$\hat{k}=\mathrm{arg}\max_{k}~{}\left|R(k)\right|.$ (22)

After bi2de conversion of $\hat{k}$, the transmitted bit sequence,
$\hat{\lambda}$ is attained.

##### Takeaways

ICS-LoRa, a technological advancement over conventional LoRa, offers the
capability of transmitting one additional bit, albeit the enhancement in the
number of transmitted bits is marginal. ICS-LoRa’s inherent superiority lies
in its improved EE, necessitating less energy per bit to correctly detect bits
for the target BER. Like LoRa, ICS-LoRa is endowed with a constant envelop,
facilitating low-cost implementation and augmenting the robustness of waveform
design. Conversely, the detection mechanisms of ICS-LoRa are relatively
intricate compared to LoRa, given the compulsion to discern interleaved
versions of LoRa symbols. Additionally, ICS-LoRa symbols lack orthogonality
with LoRa symbols because of the use of interleaved symbols, thereby
engendering high correlation peaks approximately around the correct DFT index,
which may contribute to BER degradation. Although the waveform design allows
for a larger number of interleaving patterns, augmenting them diminishes the
cross-correlation, which may ultimately lead to performance deterioration.

####  III-A3 Extended (E)-LoRa

The E-LoRa technique represents a PHY layer strategy that endeavors to augment
the number of bits transmitted per symbol compared to conventional LoRa,
through the utilization of I/Q components. Unlike the standard LoRa scenario
in which the symbol $f(n)$, as depicted in eq. (2), restricts the information
transmission solely to the in-phase components, E-LoRa harnesses the benefits
of both I/Q components, thereby enabling the encoding of an additional bit per
symbol. Hence, the total number of bits transmitted per symbol of duration
$T_{\mathrm{s}}$ in E-LoRa is $\lambda+1$.

##### Transmission

Figure 14: E-LoRa transmitter architecture.

The architecture of the E-LoRa transmitter is depicted in Fig. 14. The
transmitter utilizes a classical LoRa modulation technique, wherein the FS,
$k$, is ascertained based on the bi2de conversion of $\lambda$ bits. $k$ is
then employed to determine the discrete time signal $f(n)$ as expressed in
equation (2). Subsequently, the I/Q component encoding of the un-chirped
symbol, represented as $\tilde{f}(n)$, is achieved using
$\lambda_{\mathrm{IQ}}$, which yields $\beta_{\mathrm{IQ}}$ that determines
the encoding of the information into either the in-phase or the quadrature
component as follows:

$\tilde{f}(n)=\begin{dcases}f(n)&\quad\beta_{\mathrm{IQ}}=0\\\
jf(n)&\quad\beta_{\mathrm{IQ}}=1\end{dcases}.$ (23)

Afterwards, $\tilde{f}(n)$ is chirped using an up-chirp, $c_{\mathrm{u}}(n)$
yielding $s_{\mathrm{E-LoRa}}(n)=\tilde{f}(n)c_{\mathrm{u}}(n)$, that is
mathematically given as:

$s_{\mathrm{E-LoRa}}(n)=\begin{dcases}\exp\left\\{j\frac{\pi}{M}n(2k+n)\right\\}&\beta_{\mathrm{IQ}}=0\\\
j\exp\left\\{j\frac{\pi}{M}n(2k+n)\right\\}&\beta_{\mathrm{IQ}}=1\end{dcases}.$
(24)

The symbol energy of E-LoRa is
$E_{\mathrm{s}}=\nicefrac{{1}}{{M}}\sum_{n=0}^{M-1}|s_{\mathrm{E-LoRa}}(n)|^{2}=1$.

##### Detection

Figure 15: Coherent detector architecture for E-LoRa.

The analysis of the E-LoRa symbol structure reveals that the information is
conveyed through the phase of the un-chirped symbol, thus necessitating the
use of coherent detection as the only viable option for retrieving said
transmitted information. The architecture of the coherent detector in E-LoRa
is illustrated in Fig. 15. It is noteworthy that the utilization of either the
in-phase or quadrature un-chirped symbols in E-LoRa results in varying inner
product calculations between the received symbol vector, represented by
$\boldsymbol{y}$, and the transmit symbol vector, represented by
$\boldsymbol{s}_{\mathrm{E-LoRa}}=\left[{s}_{\mathrm{E-LoRa}}(0),{s}_{\mathrm{E-LoRa}}(1),\cdots,{s}_{\mathrm{E-LoRa}}(M-1)\right]^{\mathrm{T}}$,
as quantified by
$\langle\boldsymbol{y},\boldsymbol{s}_{\mathrm{E-LoRa}}\rangle$. Hence, a
comprehensive evaluation of
$\langle\boldsymbol{y},\boldsymbol{s}_{\mathrm{E-LoRa}}\rangle$ is performed
to consider both cases. It is worth mentioning that when the bit encoding is
performed through the in-phase components, the inner product calculation,
$\langle\boldsymbol{y},\boldsymbol{s}_{\mathrm{E-LoRa}}\rangle$, is similar to
LoRa; thus we have:

$\begin{split}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\langle\boldsymbol{y},\boldsymbol{s}_{\mathrm{E-LoRa}}\rangle=R(k).}\end{split}$
(25)

In contrast, when the quadrature component of the un-chirped symbol is used,
the result of the inner product,
$\langle\boldsymbol{y},\boldsymbol{s}_{\mathrm{E-LoRa}}\rangle$, is as
follows:

$\begin{split}\langle\boldsymbol{y},\boldsymbol{s}_{\mathrm{E-LoRa}}\rangle&=\sum_{n=0}^{M-1}y(n)\overline{s}_{\mathrm{E-LoRa}}(n)\\\
&=-j\sum_{n=0}^{M-1}r(n){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\overline{\tilde{f}}(n)}\\\
&=-jR(k).\end{split}$ (26)

When the real argument of eqs. (25) and (26) is evaluated, we attain the
following:

$\Re\left\\{\langle\boldsymbol{y},\boldsymbol{s}_{\mathrm{E-LoRa}}\rangle\right\\}=\begin{dcases}\Re\left\\{R(k)\right\\}&\beta_{\mathrm{IQ}}=0\\\
\Im\left\\{R(k)\right\\}&\beta_{\mathrm{IQ}}=1\end{dcases}.$ (27)

Therefore, in the context of coherent detection, the received symbol $y(n)$
undergoes de-chirping using $c_{\mathrm{d}}(n)$ to yield $r(n)$. Subsequently,
the DFT of $r(n)$ results in $R(k)$. As per eq. (23), the information is
transmitted through either the in-phase component or the quadrature component
of the un-chirped symbol. Moreover, eq. (28) dictates that the separation of
the I/Q components of $R(k)$ is required, which is performed by the I/Q
separator block, resulting in $\Re\left\\{R(k)\right\\}$ and
$\Im\left\\{R(k)\right\\}$. The subsequent evaluation of
$\kappa_{1}=\max~{}\Re\left\\{R(k)\right\\}$ and
$\kappa_{2}=\max~{}\Im\left\\{R(k)\right\\}$ in the phase evaluator block then
determines $\hat{\beta}_{\mathrm{IQ}}$ based on the following criterion:

$\hat{\beta}_{\mathrm{IQ}}=\begin{dcases}0&\kappa_{1}>\kappa_{2}\\\
1&\kappa_{1}<\kappa_{2}\end{dcases}.$ (28)

The parameter $\hat{\lambda}\mathrm{IQ}$ is determined via the de2bi
conversion of $\hat{\beta}_{\mathrm{IQ}}$. The phase evaluator block plays a
crucial role in this process, as it facilitates determining the information
regarding the presence or absence of a $\nicefrac{{\pi}}{{2}}$ rotation
introduced by the transmitter based on the value of $\beta_{\mathrm{IQ}}$ .

Moreover, based on the values of $\kappa_{1}$ and $\kappa_{2}$, $\tilde{R}(k)$
which is required for further evaluation is obtained as:

$\tilde{R}(k)=\begin{dcases}\Re\left\\{R(k)\right\\}&\kappa_{1}>\kappa_{2}\\\
\Im\left\\{R(k)\right\\}&\kappa_{1}<\kappa_{2}\end{dcases}.$ (29)

Subsequently, the FS, $\hat{k}$ is identified as:

$\hat{k}=\mathrm{arg}\max_{k}~{}\tilde{R}(k).$ (30)

Then $\hat{\lambda}$ is determined after de2bi conversion of $\hat{k}$.

##### Takeaways

E-LoRa represents a pioneering technique that leverages I/Q components as a
design methodology for waveform design leading to a SC scheme with a constant
envelop. This methodology offers an incremental addition of one extra bit per
transmitted symbol compared to LoRa, because of the use of the in-phase or
quadrature component. Notwithstanding, deploying the I/Q component as a design
methodology is not conducive to non-coherent detection. Consequently, only
more intricate coherent detection may be employed, reducing its
practicability. Furthermore, utilizing I/Q components precipitates
vulnerability for the waveform design resilience against potential offsets
that may manifest in low-cost devices utilized in LPWANs.

####  III-A4 Phase-Shift Keying (PSK)-LoRa

PSK-LoRa waveform design employs the PSs methodology to encode more bits in
the transmit symbol. It has been established that all the previously explored
schemes can transmit one additional bit per symbol with respect to the LoRa
protocol. However, it is to be noted that PSK-LoRa stands distinct in its
ability to transmit a higher number of additional bits compared to its LoRa
counterpart. The information transmission in LoRa occurs via the FS of the un-
chirped symbol. Conversely, the PSK-LoRa uses PSK alphabets to transmit
additional information in the un-chirped signal’s PS [16]. PSK-LoRa is an SC
scheme, as only one FS is activated, akin to other SC schemes. The amount of
additional bits transmitted by the PSK-LoRa symbol is contingent upon the
cardinality of the PSK alphabets, $M_{\varphi}$. As a result, in addition to
the $\lambda$ bits transmitted via the FS, the FS imparts an additional
$\log_{2}(M_{\varphi})$ bits, yielding a total of
$\lambda+\log_{2}(M_{\varphi})$ bits per symbol of duration $T_{\mathrm{s}}$.

Figure 16: PSK-LoRa transmitter architecture.

##### Transmission

The transmitter configuration of the PSK-LoRa is depicted in Fig. 16. Within
the PSK-LoRa framework, the FS, $k$, is determined by $\lambda$ bits, which
enables obtaining the un-chirped symbol, $f(n)$ (cf. eq. (2)). Furthermore, an
additional $\lambda_{\mathrm{PS}}=\log_{2}(M_{\varphi})$ bits are utilized to
determine the PS parameter, $p=\llbracket
0,2^{\lambda_{\mathrm{PS}}}-1\rrbracket$, as indicated by the following
equation:

$\varphi=\exp\left\\{\frac{j2\pi}{2^{\lambda_{\mathrm{PS}}}}p\right\\}.$ (31)

The resulting discrete time symbol chirped PSK-LoRa symbol is given as:

$\begin{split}s_{\mathrm{PSK}}(n)&=f(n)\varphi c_{\mathrm{u}}(n)\\\
&=\exp\left\\{j\pi\left(\frac{2}{M}kn+\frac{2p}{2^{\lambda_{\mathrm{PS}}}}+\frac{n^{2}}{M}\right)\right\\}.\end{split}$
(32)

The symbol energy of PSK-LoRa is
$E_{\mathrm{s}}=\nicefrac{{1}}{{M}}\sum_{n=0}^{M-1}|s_{\mathrm{PSK}}(n)|^{2}=1$.

##### Detection

Like E-LoRa, PSK-LoRa encodes information within the symbol phase, thereby
rendering it infeasible to establish a non-coherent detector as it would
result in the loss of the information in the PSs. Consequently, only two
alternatives exist, that are, coherent detection and the co-called semi-
coherent detection. In the case of coherent detection, the ML criterion is
utilized to determine both the FS and the PS. On the other hand, in semi-
coherent detection, the FS is determined non-coherently, while the PS is
estimated using the ML criterion.

Figure 17: Coherent detector architecture for PSK-LoRa.

The coherent detector of PSK-LoRa is depicted in Fig. 17. The process of
coherent detection entails the calculation of the inner product between the
received symbol vector, represented by $\boldsymbol{y}$, and the transmit
symbol vector, denoted by $\boldsymbol{s}_{\mathrm{PSK}}$. The latter can be
expressed in vectorial form as
$\boldsymbol{s}_{\mathrm{PSK}}=\left[{s}_{\mathrm{PSK}}(0),{s}_{\mathrm{PSK}}(1),\cdots,{s}_{\mathrm{PSK}}(M-1)\right]^{\mathrm{T}}$.
The evaluation of the inner product between the two vectors, $\boldsymbol{y}$
and $\boldsymbol{s}_{\mathrm{PSK}}$ yields:

$\begin{split}\langle\boldsymbol{y},\boldsymbol{s}_{\mathrm{PSK}}\rangle&=\sum_{n=0}^{N-1}y(n)s_{\mathrm{PSK}}(n)\\\
&=\overline{\varphi}\sum_{n=0}^{N-1}y(n)\overline{f}(n)c_{\mathrm{d}}(n)\\\
&=\overline{\varphi}R(k).\end{split}$ (33)

Then FS, $\hat{k}$, and the PSs, $\hat{p}$ are jointly evaluated using
coherent detector as:

$\hat{k},\hat{p}=\mathrm{arg}\max_{k,p}~{}\Re\left\\{\overline{\varphi}R(k)\right\\}$
(34)

The process of coherent detection commences with the de-chirping of the
received symbol, $y(n)$, using the down-chirp symbol, $c_{\mathrm{d}}(n)$,
resulting in $r(n)$. The next step involves the calculation of the DFT of
$r(n)$, resulting in $R(k)$. Subsequently, the detection problem, as described
in eq. (34), is solved by multiplying $R(k)$ with the conjugates of all the
possible PSs. This operation yields the estimated values of both $\hat{k}$ and
$\hat{p}$. The final step entails the conversion of the estimated $\hat{k}$
and $\hat{p}$ from decimal to binary representation, resulting in the bit
sequences, $\hat{\lambda}$ and $\hat{\lambda}_{\mathrm{PS}}$, respectively.

Figure 18: Semi-Coherent detector architecture for PSK-LoRa.

Fig. 18 illustrates the semi-coherent detector for PSK-LoRa. In semi-coherent
detection, upon the evaluation of $R(k)$, an estimation of $\hat{k}$ is made
through a non-coherent approach as:

$\hat{k}=\mathrm{arg}\max_{k}~{}\left|R(k)\right|$ (35)

which then determines the bit sequence , $\hat{\lambda}$ after de2bi
conversion.

It is imperative to recognize that the PSs cannot be determined via the non-
coherent detection process described in eq. (35). Nonetheless, it is also
important to acknowledge that the activated FS contains the PS information as
well. Therefore, upon achieving the estimated value of $\hat{k}$, the PS can
be quantified by utilizing the $\hat{k}$th index within $R(k)$, i.e.,
$R(\hat{k})$ as:

$\hat{p}=\mathrm{angle}\left\\{R(\hat{k})\right\\},$ (36)

where $\mathrm{angle}\\{\cdot\\}$ constitutes the ML criterion employed for
evaluating the PSs. Finally, the estimated value of the symbol PS. Finally,
the estimated value of the symbol PS denoted as $\hat{p}$ is employed to
calculate the bit sequence, possessing the estimated value of
$\hat{\lambda}_{\mathrm{PS}}$.

##### Takeaways

PSK-LoRa is the first SC constant envelop scheme that confers an increase of
over one transmitted bit per symbol over LoRa by deploying PSK alphabets. The
increase in the bit depends on the cardinality of the PSK alphabets; the
higher the cardinality, the greater the number of transmitted bits.
Nonetheless, this augmentation necessitates a trade-off in increased detection
complexity that scales proportionally to the PSK alphabets’ cardinality. Given
that the waveform design features PSs, their detection is feasible only
through ML criteria. The coherent detection procedure employs the ML criterion
to identify the FS and the PS. On the other hand, non-coherent detection
cannot be utilized due to the loss of phase information. Consequently, semi-
coherent detection must be employed, in which the FS is detected in a non-
coherent manner, whereas PS detection relies on ML criteria. Additionally, the
deployment of phase information increases the waveform design’s vulnerability
to frequency and phase offsets.

####  III-A5 Slope-Shift Keying (SSK)-LoRa

SSK-LoRa implements varying CRs to augment the symbol cardinality compared to
the conventional LoRa approach. LoRa exclusively employs an up-chirp to spread
the un-chirped symbol. In contrast, the SSK-LoRa utilizes both up-chirp and
down-chirp, resulting in the ability to transmit an additional bit per symbol
in addition to $\lambda$.

Figure 19: SSK-LoRa transmitter architecture.

##### Transmission

The transmitter architecture of the SSK-LoRa is depicted in Fig. 19. Like
LoRa, $\lambda$ bits are utilized to ascertain the FS, $k$, which yields the
corresponding un-chirped symbol, $f(n)$. Furthermore, an additional bit,
$\lambda_{\mathrm{s}}=1$ determines the chirping rate index,
$\beta_{\mathrm{s}}=\llbracket
0,2^{\lambda_{\mathrm{s}}}-1\rrbracket=\left\\{0,1\right\\}$. This index, in
turn, identifies the CR, $\gamma_{\mathrm{s}}$, with $\mathrm{s}=\llbracket
0,1\rrbracket$ as:

$\gamma_{\mathrm{s}}=\begin{dcases}1&\quad\beta_{\mathrm{s}}=0\\\
-1&\quad\beta_{\mathrm{s}}=1\end{dcases}.$ (37)

Subsequently, based on $\gamma_{\mathrm{s}}$, the spreading symbol
$c_{\mathrm{s}}(n)$ is determined as:

$c_{\mathrm{s}}(n)=\begin{dcases}c_{\mathrm{u}}(n)&\quad\gamma_{\mathrm{s}}=1\\\
c_{\mathrm{d}}(n)&\quad\gamma_{\mathrm{s}}=-1\end{dcases}.$ (38)

$f(n)$ undergoes a process of multiplication with the spreading symbol,
$c_{\mathrm{s}}(n)$, thereby yielding the discrete-time representation of the
chirped SSK-LoRa symbol, $s_{\mathrm{SSK}}(n)$, which is expressed as follows:

$\begin{split}s_{\mathrm{SSK}}(n)&=f(n)c_{\mathrm{s}}(n)\\\
&=\begin{dcases}\exp\left\\{j\frac{\pi}{M}n(2k+n)\right\\}&\gamma_{\mathrm{s}}=1\\\
\exp\left\\{j\frac{\pi}{M}n(2k-n)\right\\}&\gamma_{\mathrm{s}}=-1\end{dcases}.\end{split}$
(39)

The symbol energy of SSK-LoRa symbol is
$E_{\mathrm{s}}=\nicefrac{{1}}{{M}}\sum_{n=0}^{M-1}|s_{\mathrm{SSK}}(n)|^{2}=1$.

##### Detection

Figure 20: Coherent detector architecture for SSK-LoRa.

The coherent detector for SSK-LoRa is illustrated in Fig. 20. The goal is to
determine the inner product between the received symbol vector,
$\boldsymbol{y}$, and the transmit SSK-LoRa symbol vector,
$\boldsymbol{s}_{\mathrm{SSK}}=\left[{s}_{\mathrm{SSK}}(0),{s}_{\mathrm{SSK}}(1),\cdots,{s}_{\mathrm{SSK}}(M-1)\right]^{\mathrm{T}}$,
i.e., to evaluate
$\langle\boldsymbol{y},\boldsymbol{s}_{\mathrm{SSK}}\rangle$. It is necessary
to consider two distinct scenarios in this evaluation process: (i) the
generation of the transmit symbol occurs through the utilization of the up-
chirp, i.e., ${s}_{\mathrm{SSK}}=f(n)c_{\mathrm{u}}(n)$, and (ii) the transmit
symbol is attained using the down-chirp, i.e.,
${s}_{\mathrm{SSK}}=f(n)c_{\mathrm{d}}(n)$. For the first scenario, the
calculation of the inner product is conducted as follows:

$\begin{split}\langle\boldsymbol{y},\boldsymbol{s}_{\mathrm{SSK}}\rangle&=\sum_{n=0}^{M-1}y(n)\overline{s}_{\mathrm{SSK}}(n)\\\
&=\sum_{n=0}^{M-1}y(n)\overline{f}(n)c_{\mathrm{d}}(n)\\\
&=\sum_{n=0}^{M-1}r_{1}(n)\overline{f}(n)\\\ &=R_{1}(k),\end{split}$ (40)

where $r_{1}(n)=y(n)c_{\mathrm{d}}(n)$ and $R_{1}(k)$ is the DFT of
$r_{1}(n)$. On the other hand, for the second scenario, the inner product
yields:

$\begin{split}\langle\boldsymbol{y},\boldsymbol{s}_{\mathrm{SSK}}\rangle&=\sum_{n=0}^{M-1}y(n)\overline{s}_{\mathrm{SSK}}(n)\\\
&=\sum_{n=0}^{M-1}y(n)\overline{f}(n)c_{\mathrm{u}}(n)\\\
&=\sum_{n=0}^{M-1}r_{2}(n)\overline{f}(n)\\\ &=R_{2}(k),\end{split}$ (41)

where $r_{2}(n)=y(n)c_{\mathrm{u}}(n)$ and $R_{2}(k)$ is the DFT of
$r_{2}(n)$.

Consequently, in the context of coherent detection, after calculating
$R_{1}(k)$ and $R_{2}(k)$, the subsequent stage entails determining whether an
up-chirp or a down-chirp was used at the transmitter. This identification is
accomplished by computing the parameters
$\kappa_{1}^{\mathrm{coh}}=\max~{}\Re\left\\{R_{1}(k)\right\\}$ and
$\kappa_{2}^{\mathrm{coh}}=\max~{}\Re\left\\{R_{2}(k)\right\\}$. Subsequently,
the CR estimation module derives the determines $\hat{\beta}_{\mathrm{s}}$,
based on these parameters as follows:

$\hat{\beta}_{\mathrm{s}}=\begin{dcases}0&\kappa_{1}^{\mathrm{coh}}>\kappa_{2}^{\mathrm{coh}}\\\
1&\kappa_{1}^{\mathrm{coh}}<\kappa_{2}^{\mathrm{coh}}\end{dcases}.$ (42)

Once the spreading symbol is identified, $\kappa_{1}^{\mathrm{coh}}$ and
$\kappa_{2}^{\mathrm{coh}}$ are then also used to determine whether the
transmitted FS belongs to $R_{1}(k)$ or $R_{2}(k)$ as follows:

$R(k)=\begin{dcases}R_{1}(k)&\kappa_{1}^{\mathrm{coh}}>\kappa_{2}^{\mathrm{coh}}\\\
R_{2}(k)&\kappa_{1}^{\mathrm{coh}}<\kappa_{2}^{\mathrm{coh}}\end{dcases}.$
(43)

Then, using the coherent detection, the FS is identified as:

$\hat{k}=\mathrm{arg}\max_{k}~{}\Re\left\\{R(k)\right\\}.$ (44)

Lastly, $\hat{k}$ and $\hat{\beta}_{\mathrm{s}}$ determine the bit sequences
of lengths $\hat{\lambda}$ and $\hat{\lambda}_{\mathrm{s}}$, respectively
after de2bi conversion.

Figure 21: Non-coherent detector architecture for SSK-LoRa.

The non-coherent detector configuration for SSK-LoRa is presented in Fig. 21.
In contradistinction to the coherent detector, the non-coherent detector
utilizes $\kappa_{1}^{\mathrm{non-coh}}=\max\left|R_{1}(k)\right|$ and
$\kappa_{2}^{\mathrm{non-coh}}=\max\left|R_{2}(k)\right|$ for estimating
$\hat{\beta}_{\mathrm{s}}$ and $R(k)$. $\hat{\beta}_{\mathrm{s}}$ is
determined similarly to eq. (42) but using $\kappa_{1}^{\mathrm{non-coh}}$ and
$\kappa_{1}^{\mathrm{non-coh}}$ instead of $\kappa_{1}^{\mathrm{coh}}$ an
$\kappa_{1}^{\mathrm{coh}}$. Furthermore, the detection in the transmit FS,
$\hat{k}$ is performed as follows:

$\hat{k}=\mathrm{arg}\max_{k}~{}\left|R(k)\right|.$ (45)

The bit sequence comprising of $\hat{\lambda}$ and
$\hat{\lambda}_{\mathrm{s}}$ bits are determined after de2bi conversion of
$\hat{k}$ and $\hat{\beta}_{\mathrm{s}}$.

##### Takeaways

SSK-LoRa is a constant envelope SC scheme that boasts superior performance
characteristics compared to other schemes, such as ICS-LoRa and E-LoRa. This
scheme can transmit an extra bit per symbol relative to LoRa, which is only
marginal. SSK-LoRa’s waveform properties are also highly robust and provide
enhanced EE. Compared to ICS-LoRa, SSK-LoRa utilizes two different CRs, which
have a lesser impact on cross-correlation properties, resulting in improved
BER performance. Furthermore, by not using I/Q components, SSK-LoRa increases
its robustness against frequency and phase offsets. Despite these benefits,
the detection complexity of the coherent and non-coherent detectors for SSK-
LoRa is higher than that of LoRa. This is due to the additional computation of
a DFT and related processing. Thus, while SSK-LoRa offers enhanced performance
characteristics, it comes at the cost of increased computational complexity.

####  III-A6 Discrete Chirp Rate Keying (DCRK)-LoRa

In the SSK-LoRa, two complementary chirp signals, i.e., an up-chirp and a
down-chirp, are utilized to achieve a transmission rate of one additional bit
per symbol as compared to the conventional LoRa. The extension of this
framework through the use of multiple CRs (which generally exceeds two)
results in a higher encoding of bits per symbol. This scheme is referred to as
the DCRK-LoRa, as described in [18], and is a generalization of the SSK-LoRa.
In the DCRK-LoRa, a total of $M_{\mathrm{c}}$ distinct CRs can be utilized,
thereby enabling the transmission of $\lambda+\log_{2}(M_{\mathrm{c}})$ bits
per symbol of duration $T_{\mathrm{s}}$.

Figure 22: DCRK-LoRa transmitter architecture.

##### Transmission

The transmitter architecture of the DCRK-LoRa is depicted in Fig. 22. Within
the DCRK-LoRa framework, the value of the FS parameter, $k$, is calculated
through the bi2de conversion of $\lambda$ bits. Subsequently, the un-chirped
symbol, $f(n)$, is derived, as expressed in eq. (2). In addition to the
$\lambda$ bits, $\lambda_{\mathrm{c}}=\log_{2}(M_{\mathrm{c}})$ bits are
utilized to determine the CR, $\gamma_{\mathrm{c}}$. The bi2de conversion of
$\lambda_{\mathrm{c}}$ results in $\beta_{\mathrm{c}}=\llbracket
0,2^{\lambda_{\mathrm{c}}}-1\rrbracket$ with a cardinality of
$M_{\mathrm{c}}$, which is employed to determine $\gamma_{\mathrm{c}}$, which
assumes $M_{\mathrm{c}}$ non-zero integer values through $\beta_{\mathrm{c}}$
as follows:

$\gamma_{\mathrm{c}}=\begin{dcases}-\frac{M_{\mathrm{c}}}{2}&\quad\beta_{\mathrm{c}}=0\\\
~{}~{}~{}\cdot&\quad~{}~{}~{}\cdot\\\ ~{}~{}~{}\cdot&\quad~{}~{}~{}\cdot\\\
~{}~{}~{}\cdot&\quad~{}~{}~{}\cdot\\\
-1&\quad\beta_{\mathrm{c}}=\frac{2^{\lambda_{\mathrm{c}}-1}}{2}\\\
1&\quad\beta_{\mathrm{c}}=\frac{2^{\lambda_{\mathrm{c}}-1}}{2}+1\\\
~{}~{}~{}\cdot&\quad~{}~{}~{}\cdot\\\ ~{}~{}~{}\cdot&\quad~{}~{}~{}\cdot\\\
~{}~{}~{}\cdot&\quad~{}~{}~{}\cdot\\\
\frac{M_{\mathrm{c}}}{2}&\quad\beta_{\mathrm{c}}=2^{\lambda_{\mathrm{c}}}-1\end{dcases},$
(46)

where $\mathrm{c}$ is the indexing variable whose values are
$\mathrm{c}=\llbracket 0,M_{\mathrm{c}}-1\rrbracket$. Once the CR
$\gamma_{\mathrm{c}}$ is determined, the spreading symbol $c_{\mathrm{c}}(n)$
is ascertained as in eq. (5).

Afterwards, $f(n)$ is chirped using $c_{\mathrm{c}}(n)$, resulting in DCRK-
LoRa discrete-time symbol, $s_{\mathrm{DCRK}}(n)$ as:

$s_{\mathrm{DCRK}}(n)=f(n)c_{\mathrm{c}}(n)=\exp\left\\{j\frac{\pi}{M}n\left(2k+\gamma_{\mathrm{c}}n\right)\right\\}.$
(47)

The symbol energy of DCRK-LoRa is equal to
$E_{\mathrm{s}}=\nicefrac{{1}}{{M}}\sum_{n=0}^{M-1}|s_{\mathrm{DCRK}}(n)|^{2}=1$.

##### Detection

Figure 23: Coherent detector architecture for DCRK-LoRa.

The coherent detector for DCRK-LoRa is depicted in Fig. 23. The coherent
detection process comprises two key elements: (i) the determination of the FS
of the un-chirped symbol, denoted by $\hat{k}$; and (ii) identification of the
CR, represented by $\hat{\gamma}\mathrm{c}$. The proposed coherent detector
for DCRK-LoRa leverages the inner product of the received signal,
$\boldsymbol{y}$, and the DCRK-LoRa transmit symbol,
$\boldsymbol{s}_{\mathrm{DCRK}}=\left[{s}_{\mathrm{DCRK}}(0),{s}_{\mathrm{DCRK}}(1),\cdots,{s}_{\mathrm{DCRK}}(M-1)\right]^{\mathrm{T}}$.
The inner product is calculated as
$\langle\boldsymbol{y},\boldsymbol{s}_{\mathrm{DCRK}}\rangle$, yields:

$\begin{split}\langle\boldsymbol{y},\boldsymbol{s}_{\mathrm{DCRK}}\rangle&=\sum_{n=0}^{M-1}y(n)\overline{s}_{\mathrm{DCRK}}(n)\\\
&=\sum_{n=0}^{M-1}y(n)\overline{f}(n)\overline{c}_{\mathrm{c}}(n)\\\
&=\sum_{n=0}^{M-1}r_{\mathrm{c}}(n)\overline{f}(n)\\\
&=R_{\mathrm{c}}(k).\end{split}$ (48)

whereby, $r_{\mathrm{c}}(n)=y(n)\overline{c}_{\mathrm{c}}(n)$. The evaluation
of the CR requires the calculation of the CR considering all feasible CRs that
the transmitter can utilize, as stated in eq. (48). To determine the CR,
$\kappa_{\mathrm{c}}^{\mathrm{coh}}=\max~{}\Re\left\\{R_{\mathrm{c}}(k)\right\\}$
is evaluated. Then, $\hat{\beta}_{\mathrm{c}}$ is attained as:

$\hat{\beta}_{\mathrm{c}}=\mathrm{arg}\max\limits_{\mathrm{c}}~{}\kappa_{\mathrm{c}}^{\mathrm{coh}}.$
(49)

Then, using $\hat{\beta}_{\mathrm{c}}$, the bit information in the CR, i.e.,
$\hat{\lambda}_{\mathrm{c}}$ is attained. Moreover, the DFT outputs
corresponding to $\hat{\beta}_{\mathrm{c}}$, i.e. $R_{\mathrm{c}}(k)$
determines the FS of the un-chirped symbol as:

$\hat{k}=\mathrm{arg}\max\limits_{k}~{}\Re\left\\{R_{\mathrm{c}}(k)\right\\}.$
(50)

The bit sequence of length $\hat{\lambda}$ is determined after bi2de
conversion of $\hat{k}$.

Figure 24: Non-coherent detector architecture for DCRK-LoRa.

Similarly to the non-coherent detection of SSK-LoRa, the evaluation of the CR
parameter, $\hat{\beta}_{\mathrm{c}}$ in DCRK-LoRa non-coherent detection is
achieved by computing $\kappa_{\mathrm{c}}^{\mathrm{non-
coh}}=\max\left|R_{\mathrm{c}}(k)\right|$. The determination of the FS,
$\hat{k}$, is also performed in a non-coherent manner as:

$\hat{k}=\mathrm{arg}\max\limits_{k}~{}\left|R_{\mathrm{c}}(k)\right|.$ (51)

$\hat{\beta}_{\mathrm{c}}$ and $\hat{k}$ then determine the transmitted bits
of lengths $\hat{\lambda}_{\mathrm{c}}$ and $\hat{\lambda}$, respectively,
after de2bi conversion.

##### Takeaways

The waveform designs of DCRK-LoRa and SSK-LoRa demonstrate certain
similarities. While SSK-LoRa utilizes only two CRs, DCRK-LoRa presents a more
adaptable approach by permitting any number of CRs. In addition, DCRK-LoRa
also permits transmitting a higher number of bits per symbol, which is
directly proportional to the number of CRs employed. DCRK-CSS exhibits SC
constant envelope characteristics, contributing to a resilient waveform
design. Nonetheless, certain limitations exist within the waveform design of
DCRK-LoRa. Firstly, as the number of CRs increases, the computational
complexity of the detector also increases proportionally. This is a noteworthy
limitation, as while it allows for a significant increase in the number of
transmitted bits per symbol, the power consumption rises due to the amplified
complexity. Secondly, while the correlation between chirped symbols with
various CRs is low, they are not orthogonal, resulting in decreased intrinsic
interference with minimal performance degradation, which can be further
reduced by increasing $\lambda$.

####  III-A7 SSK-ICS-LoRa

The SSK-ICS-LoRa protocol, as its name implies, integrates SSK-LoRa and ICS-
LoRa [57]. Furthermore, it is categorically placed within the SC taxonomy,
incorporating interleaving techniques and varying CRs to facilitate the
waveform design. It is pertinent to note that the SSK-LoRa enhances the number
of bits per symbol by $1$ relative to LoRa, by incorporating both up-chirp and
down-chirp symbols. On the other hand, the ICS-LoRa protocol also transmits
$1$ additional bit per symbol relative to LoRa by employing both up-chirped
symbols and their interleaved counterparts. Through the amalgamation of SSK-
LoRa and ICS-LoRa, SSK-ICS-LoRa leverages the utilization of up-chirp symbols,
down-chirp symbols, and interleaved counterparts, thereby elevating the number
of transmitted bits per symbol by $2$ with respect to LoRa.

##### Transmission

Figure 25: SSK-ICS-LoRa transmitter architecture.

The transmitter architecture of SSK-ICS-LoRa is depicted in Fig. 25. Like
LoRa, the number of bits, denoted by $\lambda$, determines the FS, $k$, which
is then utilized to obtain the un-chirped symbol, $f(n)$. Further, the
conversion of $\lambda_{\mathrm{s}}=1$ bit via bi2de results in
$\beta_{\mathrm{s}}=\left\\{0,1\right\\}$, which serves to identify the CR,
$\gamma_{\mathrm{s}}=\left\\{-1,1\right\\}$. It is noteworthy that SSK-ICS-
LoRa utilizes only two distinct CRs, similar to SSK-LoRa. Once
$\gamma_{\mathrm{s}}$ is established, the chirp symbol, $c_{\mathrm{s}}(n)$ is
obtained (cf. eq. 5). The chirped symbol, $s_{\mathrm{SSK}}(n)$ is then
computed by the multiplication of $f(n)$ and $c_{\mathrm{s}}(n)$ (cf. eq. 39).
We can observe that $s_{\mathrm{SSK}}(n)$ is identical to the SSK-LoRa
symbol.Upon determination of $s_{\mathrm{SSK}}(n)$, the value of
$\lambda_{\mathrm{I}}$ serves to identify whether $s_{\mathrm{SSK}}(n)$ is to
be interleaved or not. After bi2de conversion of $\lambda_{\mathrm{I}}=1$ bit,
another parameter, $\beta_{\mathrm{I}}$, is determined, which indicates
whether the transmitter will transmit the SSK-LoRa symbol,
$s_{\mathrm{SSK}}(n)$ or its interleaved version, represented as
$\Pi\left[s_{\mathrm{SSK}}(n)\right]$. The discrete-time SSK-ICS-LoRa transmit
symbol is defined as:

${s}_{\mathrm{SSK-
ICS}}(n)=\begin{dcases}{s}_{\mathrm{SSK}}(n)&\quad\beta_{\mathrm{I}}=0\\\
\Pi\left[{s}_{\mathrm{SSK}}(n)\right]&\quad\beta_{\mathrm{I}}=1\end{dcases}.$
(52)

The symbol energy of SSK-ICS-LoRa is
$E_{\mathrm{s}}=\nicefrac{{1}}{{M}}\sum_{n=0}^{M-1}|{s}_{\mathrm{SSK-
ICS}}(n)|^{2}=1$.

##### Detection

Figure 26: Coherent detector architecture for SSK-ICS-LoRa.

The coherent detection mechanism employed in SSK-ICS-LoRa, as depicted in Fig.
26, encompasses four distinct possibilities concerning the transmission
symbol. These possibilities can be classified as (i) utilization of the up-
chirp symbol for spreading without interleaving, indicated by the binary value
of the parameters, ${\beta_{\mathrm{s}},\beta_{\mathrm{I}}}=\\{0,0\\}$; (ii)
deployment of the down-chirp symbol for spreading without interleaving,
indicated by the binary value of the parameters,
${\beta_{\mathrm{s}},\beta_{\mathrm{I}}}=\\{1,0\\}$; (iii) utilization of the
up-chirp symbol for spreading with interleaving, indicated by the binary value
of the parameters, ${\beta_{\mathrm{s}},\beta_{\mathrm{I}}}=\\{0,1\\}$; (iv)
deployment of the down-chirp symbol for spreading with interleaving, indicated
by the binary value of the parameters,
${\beta_{\mathrm{s}},\beta_{\mathrm{I}}}=\\{1,1\\}$. In all cases, the inner
product, $\langle\boldsymbol{y},\boldsymbol{s}_{\mathrm{SSK-ICS}}\rangle$,
between the received symbol, $\boldsymbol{y}$, and the SSK-ICS-LoRa transmit
symbol, $\boldsymbol{s}_{\mathrm{SSK-ICS}}=\left[s_{\mathrm{SSK-
ICS}}(0),s_{\mathrm{SSK-ICS}}(1),\cdots,s_{\mathrm{SSK-ICS}}(M-1)\right]^{T}$,
are distinctive. For the scenario of
${\beta_{\mathrm{s}},\beta_{\mathrm{I}}}={0,0}$, the evaluation of
$\langle\boldsymbol{y},\boldsymbol{s}_{\mathrm{SSK-ICS}}\rangle$ is as
follows:

$\begin{split}\langle\boldsymbol{y},\boldsymbol{s}_{\mathrm{SSK-
ICS}}\rangle&=\sum_{n=0}^{M-1}y(n)\overline{s}_{\mathrm{SSK-ICS}}(n)\\\
&=\sum_{n=0}^{M-1}y(n)\overline{f}(n)c_{\mathrm{d}}(n)\\\
&=\sum_{n=0}^{M-1}r_{1}(n)\overline{f}(n)\\\ &=R_{1}(k),\end{split}$ (53)

where $R_{1}(k)$ is the DFT of $r_{1}(n)=y(n)c_{\mathrm{d}}(n)$. For
$\\{\beta_{\mathrm{s}},\beta_{\mathrm{I}}\\}=\\{1,0\\}$,
$\langle\boldsymbol{y},\boldsymbol{s}_{\mathrm{SSK-ICS}}\rangle$ evaluates to:

$\begin{split}\langle\boldsymbol{y},\boldsymbol{s}_{\mathrm{SSK-
ICS}}\rangle&=\sum_{n=0}^{M-1}y(n)\overline{s}_{\mathrm{SSK-ICS}}(n)\\\
&=\sum_{n=0}^{M-1}y(n)\overline{f}(n)c_{\mathrm{u}}(n)\\\
&=\sum_{n=0}^{M-1}r_{2}(n)\overline{f}(n)\\\ &=R_{2}(k),\end{split}$ (54)

where $R_{2}(k)$ is the DFT of $r_{2}(n)=y(n)c_{\mathrm{u}}(n)$. Considering
up-chirp symbol for spreading with interleaving, i.e.,
$\\{\beta_{\mathrm{s}},\beta_{\mathrm{I}}\\}=\\{0,1\\}$, then
$\langle\boldsymbol{y},\boldsymbol{s}_{\mathrm{SSK-ICS}}\rangle$ results in:

$\begin{split}\langle\boldsymbol{y},\boldsymbol{s}_{\mathrm{SSK-
ICS}}\rangle&=\sum_{n=0}^{M-1}y(n)\overline{s}_{\mathrm{SSK-ICS}}(n)\\\
&=\sum_{n=0}^{M-1}\Pi\left[y(n)\right]\overline{f}(n)c_{\mathrm{d}}(n)\\\
&=\sum_{n=0}^{M-1}r_{3}(n)\overline{f}(n)\\\ &=R_{3}(k),\end{split}$ (55)

where $R_{3}(k)$ is the DFT of $r_{3}(n)=\Pi[y(n)]c_{\mathrm{d}}(n)$. Lastly,
considering $\\{\beta_{\mathrm{s}},\beta_{\mathrm{I}}\\}=\\{1,1\\}$, the inner
product yields:

$\begin{split}\langle\boldsymbol{y},\boldsymbol{s}_{\mathrm{SSK-
ICS}}\rangle&=\sum_{n=0}^{M-1}y(n)\overline{s}_{\mathrm{SSK-ICS}}(n)\\\
&=\sum_{n=0}^{M-1}\Pi\left[y(n)\right]\overline{f}(n)c_{\mathrm{u}}(n)\\\
&=\sum_{n=0}^{M-1}r_{4}(n)\overline{f}(n)\\\ &=R_{4}(k),\end{split}$ (56)

where $R_{4}(k)$ is the DFT of $r_{4}(n)=\Pi[y(n)]c_{\mathrm{u}}(n)$. The
subsequent procedure entails utilizing the parameters $R_{1}(k)$, $R_{2}(k)$,
$R_{3}(k)$, and $R_{4}(k)$ as specified in equations (53), (54), (55) and
(56), respectively to determine whether the transmitter employed an up-chirp
or down-chirp and the presence of interleaving in the transmission of the
chirped symbol. In this regard, the parameters,
$\kappa_{2}^{\mathrm{coh}}=\max~{}\Re\left\\{R_{2}(k)\right\\}$,
$\kappa_{3}^{\mathrm{coh}}=\max~{}\Re\left\\{R_{3}(k)\right\\}$, and
$\kappa_{4}^{\mathrm{coh}}=\max~{}\Re\left\\{R_{4}(k)\right\\}$ are
calculated. Subsequently, based on these values, the spreading and
interleaving parameters, $\hat{\beta}_{\mathrm{s}}$ and
$\hat{\beta}_{\mathrm{I}}$ are ascertained as:

$\hat{\beta}_{\mathrm{s}}=\begin{dcases}0&\quad\max\\{\kappa_{1}^{\mathrm{coh}},\kappa_{3}^{\mathrm{coh}}\\}>\max\\{\kappa_{2}^{\mathrm{coh}},\kappa_{4}^{\mathrm{coh}}\\}\\\
1&\quad\max\\{\kappa_{2}^{\mathrm{coh}},\kappa_{4}^{\mathrm{coh}}\\}>\max\\{\kappa_{1}^{\mathrm{coh}},\kappa_{3}^{\mathrm{coh}}\\}\end{dcases},$
(57)

and

$\hat{\beta}_{\mathrm{I}}=\begin{dcases}0&\quad\max\\{\kappa_{1}^{\mathrm{coh}},\kappa_{2}^{\mathrm{coh}}\\}>\max\\{\kappa_{3}^{\mathrm{coh}},\kappa_{4}^{\mathrm{coh}}\\}\\\
1&\quad\max\\{\kappa_{3}^{\mathrm{coh}},\kappa_{4}^{\mathrm{coh}}\\}>\max\\{\kappa_{1}^{\mathrm{coh}},\kappa_{2}^{\mathrm{coh}}\\}\end{dcases},$
(58)

respectively. Subsequently, $\hat{\beta}_{\mathrm{s}}$ and
$\hat{\beta}_{\mathrm{I}}$ determine $\hat{\lambda}_{\mathrm{s}}$ and
$\hat{\lambda}_{\mathrm{I}}$, respectively. After the evaluation of
$\hat{\beta}_{\mathrm{s}}$ and $\hat{\beta}_{\mathrm{I}}$, the FS, $\hat{k}$
is identified coherently as:

$\hat{k}=\mathrm{arg}\max_{k}~{}\Re\left\\{R(k)\right\\},$ (59)

where

$R(k)=\begin{dcases}R_{1}(k)&\quad\\{\hat{\beta}_{\mathrm{s}},\hat{\beta}_{\mathrm{I}}\\}=\left\\{0,0\right\\}\\\
R_{2}(k)&\quad\\{\hat{\beta}_{\mathrm{s}},\hat{\beta}_{\mathrm{I}}\\}=\left\\{1,0\right\\}\\\
R_{3}(k)&\quad\\{\hat{\beta}_{\mathrm{s}},\hat{\beta}_{\mathrm{I}}\\}=\left\\{0,1\right\\}\\\
R_{4}(k)&\quad\\{\hat{\beta}_{\mathrm{s}},\hat{\beta}_{\mathrm{I}}\\}=\left\\{1,1\right\\}\\\
\end{dcases},$ (60)

The transmit bit sequence consisting of $\hat{\lambda}$ bits is determined
after de2bi conversion of $\hat{k}$.

Figure 27: Non-coherent detector architecture for SSK-ICS-LoRa.

The non-coherent detector for SSK-ICS-LoRa is illustrated in Fig. 27. Upon
evaluating $R_{1}(k)$, $R_{2}(k)$, $R_{3}(k)$, and $R_{4}(k)$, the following
differences can be observed between the coherent and non-coherent detection
mechanisms. Firstly, the interleaving and CR decision on the estimations of
$\hat{\beta}_{\mathrm{s}}$ and $\hat{\beta}_{\mathrm{I}}$ is executed based on
the parameters calculated via the computation of $\kappa_{1}^{\mathrm{non-
coh}}=\max\left|R_{1}(k)\right|$, $\kappa_{2}^{\mathrm{non-
coh}}=\max\left|R_{2}(k)\right|$, $\kappa_{3}^{\mathrm{non-
coh}}=\max\left|R_{3}(k)\right|$, and $\kappa_{4}^{\mathrm{non-
coh}}=\max\left|R_{4}(k)\right|$. Secondly, the non-coherent determination of
the FS, $\hat{k}$, is performed in a non-coherent manner as:

$\hat{k}=\mathrm{arg}\max_{k}~{}\left|R(k)\right|,$ (61)

where $R(k)$ is same as in eq. (60). $\hat{k}$ is then used to determine
$\hat{\lambda}$ after de2bi conversion.

##### Takeaways

SSK-ICS-LoRa features a constant envelope and is capable of transmitting two
additional bits per symbol in comparison to LoRa. The proficient waveform
design properties of SSK-ICS-LoRa, including constant envelope
characteristics, among others, results in the development of robustness
against various channel conditions, offsets, and an improvement in energy
efficiency. Nevertheless, detection complexity is notably high as the detector
is compelled to identify and discriminate between up-chirp symbols, down-chirp
symbols, and their interleaved counterparts. Consequently, the SSK-ICS-LoRa
receiver must undertake a minimum of four DFT computations, in contrast to one
DFT computation for LoRa or two DFT computations for SSK-LoRa or ICS-LoRa.
Additionally, the high cross-correlation between the chirped symbol and its
interleaved version in ICS-LoRa results in the degradation of the bit error
rate and loss of orthogonality. Similarly, the utilization of two diverse
coding rates also results in a loss of orthogonality for SSK-LoRa.
Consequently, the discrete domain in SSK-ICS-LoRa’s different symbols is also
not orthogonal, leading to a few performance limitations.

###  III-B Multiple Carrier CSS Schemes

####  III-B1 Dual-Orthogonal Chirp Spread Spectrum (DO-CSS)

One of the pioneering CSS schemes belonging to MC taxonomy for LPWANs is DO-
CSS conceptualized by Vangelista, and Cattapan [55]. The waveform design
approach adopted by DO-CSS mirrors that of the LoRa in that it relies on
activating only the FSs. However, unlike LoRa, which activates a single FS,
DO-CSS involves the activation of two FSs, with one FS representing an even
frequency and the other denoting an odd frequency. Given that each FS
transmits $\lambda-1$ bits, the activation of both FSs results in the
transmission of a total of $2\lambda-2$ bits per DO-CSS symbol of duration
$T_{\mathrm{s}}$. This simultaneous activation of two FSs corresponds to the
multiplexing of two chirp symbols that exhibit different cyclic time shifts.

Figure 28: DO-CSS transmitter architecture.

##### Transmission

The transmitter configuration of the DO-CSS is depicted in Fig. 28. In DO-CSS,
the available $M$ FSs within bandwidth $B$ are split into two distinct
subsets, i.e., an even subset and an odd subset, each consisting of
$\nicefrac{{M}}{{2}}$ FSs. Then, the respective even and odd FSs,
$k_{\mathrm{e}}$ and $k_{\mathrm{o}}$, as specified by
$\lambda_{\mathrm{e}}=\log_{2}(\nicefrac{{M}}{{2}})=\lambda-1$ bits for the
even subset and $\lambda_{\mathrm{o}}=\log_{2}(\nicefrac{{M}}{{2}})=\lambda-1$
bits for the odd subset are attained. Finally, the un-chirped symbols
corresponding to the specified even and odd FSs, $k_{\mathrm{e}}$ and
$k_{\mathrm{o}}$, are computed as:

$f_{1}(n)=\exp\left\\{j\frac{2\pi}{M}k_{\mathrm{e}}n\right\\},$ (62)

and

$f_{2}(n)=\exp\left\\{j\frac{2\pi}{M}k_{\mathrm{o}}n\right\\},$ (63)

respectively. Subsequently, $f_{1}(n)$ and $f_{2}(n)$ are added resulting in
$f(n)$, that is given as:

$\begin{split}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}f(n)=f_{1}(n)+f_{2}(n)=\exp\left\\{j\frac{2\pi}{M}k_{\mathrm{e}}n\right\\}+\exp\left\\{j\frac{2\pi}{M}k_{\mathrm{o}}n\right\\}.}\end{split}$
(64)

Subsequently, $f(n)$ is multiplied with an up-chirp symbol,
$c_{\mathrm{u}}(n)$ resulting in discrete time chirped DO-CSS symbol
$s_{\mathrm{DO}}(n)$, that is given as:

$\begin{split}s_{\mathrm{DO}}(n)&=f(n)c_{\mathrm{u}}(n)\\\
&=\exp\left\\{j\frac{\pi}{M}n\left(2k_{\mathrm{e}}+n\right)\right\\}+\exp\left\\{j\frac{\pi}{M}n\left(2k_{\mathrm{o}}+n\right)\right\\}.\end{split}$
(65)

The symbol energy of DO-CSS symbol is
$E_{\mathrm{s}}=\nicefrac{{1}}{{M}}\sum_{n=0}^{M-1}|s_{\mathrm{DO}}(n)|^{2}=2$.

##### Detection

In [55], the authors propose implementing a non-coherent detection for DO-CSS.
However, in the present study, a more comprehensive examination is conducted,
including the non-coherent detection mechanism and a study of the coherent
detection applied to the DO-CSS framework.

Figure 29: Coherent detector architecture for DO-CSS.

The inner product of the received symbol, $\boldsymbol{y}$ and the DO-CSS
transmit symbol,
$\boldsymbol{s}_{\mathrm{DO}}=\left[{s}_{\mathrm{DO}}(0),{s}_{\mathrm{DO}}(1),\cdots,{s}_{\mathrm{DO}}(M-1)\right]^{\mathrm{T}}$,
i.e., $\langle\boldsymbol{y},\boldsymbol{s}_{\mathrm{DO}}\rangle$ is given as:

$\begin{split}\langle\boldsymbol{y},\boldsymbol{s}_{\mathrm{DO}}\rangle&=\sum_{n=0}^{M-1}y(n)\overline{s}_{\mathrm{DO}}(n)\\\
&=\sum_{n=0}^{M-1}y(n)\overline{f}(n)c_{\mathrm{d}}(n)\\\
&=\sum_{n=0}^{M-1}r(n)\left\\{\overline{f}_{1}(n)+\overline{f}_{2}(n)\right\\}\\\
&=R(k_{\mathrm{e}})+R(k_{\mathrm{o}}).\end{split}$ (66)

Then, employing eq. (66), the even and odd activated FS are evaluated as:

$\hat{k}_{\mathrm{e}},\hat{k}_{\mathrm{o}}=\mathrm{arg}\max_{k_{\mathrm{e}},k_{\mathrm{o}}}~{}\Re\left\\{R(k_{\mathrm{e}})+R(k_{\mathrm{o}})\right\\}.$
(67)

The DFT outputs, $R(k_{\mathrm{e}})$ and $R(k_{\mathrm{o}})$, respectively
correspond to the DFT results obtained at the even and odd frequency indexes.
The evaluation of both the estimated parameters, namely,
$\hat{k}_{\mathrm{e}}$ and $\hat{k}_{\mathrm{o}}$, are jointly carried out
following eq. (67). However, the coherent detection problem encapsulated
within eq. (67) can be solved such that the separate identification of both
the parameters, i.e., $\hat{k}_{\mathrm{e}}$ and $\hat{k}_{\mathrm{o}}$, can
be achieved as follows:

$\hat{k}_{\mathrm{e}}=\mathrm{arg}\max_{k_{\mathrm{e}}}~{}\Re\left\\{R(k_{\mathrm{e}})\right\\},$
(68)

and

$\hat{k}_{\mathrm{o}}=\mathrm{arg}\max_{k_{\mathrm{o}}}~{}\Re\left\\{R(k_{\mathrm{o}})\right\\},$
(69)

respectively. The coherence-based detection architecture depicted in Fig. 29
elucidates the capacity for disjoint identification of $\hat{k}_{\mathrm{e}}$
and $\hat{k}_{\mathrm{o}}$. The coherent detection entails the de-chirping of
the received symbol, $y(n)$, using a down-chirp, $c_{\mathrm{d}}(n)$, which
results in $r(n)$. After that, the DFT of $r(n)$, i.e., $R(k)$, is computed.
The even/odd separator function then isolates the even and odd frequency
components of $R(k)$, resulting in $R(k_{\mathrm{e}})$ and
$R(k_{\mathrm{o}})$, respectively. Utilizing the decision criteria specified
in eqs. (68) and (69), the even and odd activated frequency indexes,
$\hat{k}_{\mathrm{e}}$ and $\hat{k}_{\mathrm{o}}$, are separately identified.
Finally, the bit sequences, comprised of the corresponding
$\hat{\lambda}_{\mathrm{e}}$ and $\hat{\lambda}_{\mathrm{o}}$ bits, are
determined, utilizing $\hat{k}_{\mathrm{e}}$ and $\hat{k}_{\mathrm{o}}$ as
inputs.

Figure 30: Non-coherent detector architecture for DO-CSS.

The non-coherent detector architecture utilized for DO-CSS is depicted in
reference Fig. 30. With regards to the coherent detector, the distinctive
aspect lies in the fact that post-acquisition of $R(k_{\mathrm{e}})$ and
$R(k_{\mathrm{o}})$, the activation of even and odd FSs is determined as:

$\hat{k}_{\mathrm{e}}=\mathrm{arg}\max_{k_{\mathrm{e}}}\left|R(k_{\mathrm{e}})\right|$
(70)

and

$\hat{k}_{\mathrm{o}}=\mathrm{arg}\max_{k_{\mathrm{o}}}\left|R(k_{\mathrm{o}})\right|,$
(71)

Afterwards, using $\hat{k}_{\mathrm{e}}$ and $\hat{k}_{\mathrm{o}}$, the bit
sequences having respective lengths of $\hat{\lambda}_{\mathrm{e}}$ and
$\hat{\lambda}_{\mathrm{o}}$ are ascertained.

##### Takeaways

DO-CSS is a MC CSS scheme that enhances LoRa, a scheme from SC taxonomy, by
activating two FSs instead of one. While multiple approaches can be used to
activate its two FSs, DO-CSS involves the activation of an even FS and an odd
FS. DO-CSS features the simplest waveform design of any MC scheme. Its
principal benefit over LoRa is an almost twofold increase in the number of
transmitted bits, which is a substantial gain. Nonetheless, the DO-CSS
waveform design analysis reveals that its discrete-time symbol lacks a
constant envelope and may exhibit a high PAPR, mainly depending on the two
activated FSs. This factor could result in implementation issues since low-
cost components utilized in LPWANs may have a limited linear range of
operation. Although only non-coherent detection is suggested in the seminal
work, both coherent and non-coherent detection of the DO-CSS symbol is
possible. Moreover, if both the even and odd FSs are detected in a disjointed
manner, the computation complexity of detection is similar to that of LoRa.

####  III-B2 In-Phase and Quadrature (IQ)-CSS

IQ-CSS scheme is a MC approach that leverages the utilization of both I/Q
components to encode the information bits onto the chirp signal [56]. IQ-CSS
takes advantage of the orthogonality between sine and cosine waveforms,
thereby enabling the concurrent transmission of two data symbols. In contrast
to E-LoRa, where the information is modulated onto either the in-phase or the
quadrature component, IQ-CSS implements simultaneous encoding onto both the
in-phase and quadrature components. In IQ-CSS, one FS generates the in-phase
component of the un-chirped symbol. In contrast, another FS generates the
quadrature component of the un-chirped symbol. The capability of transmitting
information through both the in-phase and quadrature components, each having a
capacity of transmitting $\lambda$ bits, results in the total number of bits
transmitted per IQ-CSS symbol being double that of conventional LoRa, i.e.,
$2\lambda$.

Figure 31: IQ-CSS transmitter architecture.

##### Transmission

The graphical representation of the transmitter configuration in IQ-CSS is
presented in Fig. 31. Within the framework of IQ-CSS, the transmission of a
total of $2\lambda$ bits takes place, wherein half of the bits are allocated
towards determining $k_{i}$, and the remaining half is earmarked to determine
$k_{q}$. Subsequently, the generation of in-phase and quadrature un-chirped
symbols, $f_{1}(n)$ and $f_{2}(n)$, respectively, transpires through the
utilization of $k_{i}$ and $k_{q}$ as follows:

$f_{1}(n)=\exp\left\\{j\frac{2\pi}{M}k_{i}n\right\\},$ (72)

and

$f_{2}(n)=\exp\left\\{j\frac{2\pi}{M}k_{q}n\right\\}.$ (73)

Subsequently, the un-chirped in-phase symbol $f_{1}(n)$ undergoes a
multiplication with the up-chirp symbol, $c_{\mathrm{u}}(n)$, resulting in the
chirped in-phase symbol $s_{1}(n)$. On the other hand, the un-chirped
quadrature symbol $f_{2}(n)$ is subjected to a phase rotation of
$\nicefrac{{\pi}}{{2}}$ before its multiplication with the
$c_{\mathrm{u}}(n)$; thus yielding the chirped quadrature symbol $s_{2}(n)$,
where the chirped in-phase and quadrature symbols are given as:

$s_{1}(n)=f_{1}(n)c_{\mathrm{u}}=\exp\left\\{j\frac{\pi}{M}n\left(2k_{i}+n\right)\right\\},$
(74)

and

$s_{2}(n)=jf_{2}(n)c_{\mathrm{u}}=j\exp\left\\{j\frac{\pi}{M}n\left(2k_{q}+n\right)\right\\}.$
(75)

Lastly, both $s_{1}(n)$ and $s_{2}(n)$ are added to attain the discrete-time
IQ-CSS symbol, which is given as:

$\begin{split}&s_{\mathrm{IQ}}(n)=s_{1}(n)+s_{2}(n)\\\
&~{}~{}=\exp\left\\{j\frac{\pi}{M}n\left(2k_{i}+n\right)\right\\}+j\exp\left\\{j\frac{\pi}{M}n\left((2k_{q}+n\right)\right\\}\end{split}$