1 Dual-Mode Time Domain Multiplexed Chirp Spread Spectrum Ali Waqar Azim Ahmad Bazzi Mahrukh Fatima Raed Shubair Marwa Chaﬁi

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Dual-Mode Time Domain Multiplexed Chirp Spread Spectrum

Ali Waqar Azim, Ahmad Bazzi, Mahrukh Fatima, Raed Shubair, Marwa Chaﬁi

Abstract—We propose a dual-mode (DM) time domain mul-

tiplexed (TDM) chirp spread spectrum (CSS) modulation for

spectral and energy-efﬁcient low-power wide-area networks (LP-

WANs). DM-CSS modulation that uses both the even and odd

cyclic time shifts has been proposed for LPWANs to achieve

noteworthy performance improvement over classical counter-

parts. However, its spectral efﬁciency (SE) is half of the in-

phase and quadrature (IQ)-TDM-CSS scheme that employs IQ

components with both up and down chirps, resulting in a SE

that is four times relative to Long Range (LoRa) modulation.

Nevertheless, the IQ-TDM-CSS scheme only allows coherent

detection. Furthermore, it is also sensitive to carrier frequency

and phase offsets, making it less practical for low-cost battery-

powered LPWANs for Internet-of-Things (IoT) applications. DM-

CSS uses either an up-chirp or a down-chirp. DM-TDM-CSS

consists of two chirped symbols that are multiplexed in the

time domain. One of these symbols consisting of even and odd

frequency shifts (FSs) is chirped using an up-chirp. The second

chirped symbol also consists of even and odd FSs, but they

are chirped using a down-chirp. It shall be demonstrated that

DM-TDM-CSS attains a maximum achievable SE close to IQ-

TDM-CSS while also allowing both coherent and non-coherent

detection. Additionally, unlike IQ-TDM-CSS, DM-TDM-CSS is

robust against carrier frequency and phase offsets.

Index Terms—LoRa, chirp spread spectrum, IoT.

I. INTRODUCTION

THE rudimentary idea for Internet-of-Thing (IoT) ap-

plications is to communicate between battery-powered

devices/sensors by consuming low power to extend the battery

lifetime of the terminals. In this regard, low-power wide-

area networks (LPWANs) are of utmost signiﬁcance. One of

the emerging technologies of LPWANs is the Long Range

(LoRa) wide-area network (LoRaWAN), which uses LoRa as

the physical layer modulation scheme.

LoRa is a proprietary derivative of chirp spread spectrum

(CSS) modulation, developed by the Semtech corporation, ca-

pable of trading off sensitivity with data rates for ﬁxed channel

bandwidths [1], [2]. Even though the Semtech corporation

never published the details of LoRa modulation, Vangelista in

[3] has provided a comprehensive theoretical explanation with

an optimal low-complexity detection process based on discrete

Fourier transform (DFT). The scalable parameter of the LoRa

modulation is the spreading factor,λ, where λ=J6,12K.λ

is, in fact, equal to the number of bits that a LoRa symbol can

Ali Waqar Azim is with Department of Telecommunication Engineer-

ing, 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: {ah-

mad.bazzi,raed.shubair}@nyu.edu).

Mahrukh Fatima is with National Defence University, Islamabad, Pakistan

(email: mahrukh@ndu.edu.pk).

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).

transmit. Moreover, LoRa symbols are deﬁned using Mcyclic

time shifts of the chirp, which are the information-bearing

elements. These cyclic time shifts correspond to frequency

shifts (FSs) of the complex conjugate of the chirp signal, i.e.,

down-chirp signal; thus, LoRa can be regarded as FS chirp

modulation, where λ= log2(M).

Besides the broad adoption and numerous beneﬁts of LoRa,

one of the limiting factors is that it achieves low achievable

rates in all three bands it utilizes. Several recent studies pro-

posed spectral-efﬁcient CSS modulation schemes as possible

alternatives to LoRa. The waveform design of these CSS alter-

natives is comprehensively elucidated in [4]. It is noteworthy

that these CSS variants can have different properties, e.g.,

some possess constant envelope properties and are single chirp,

whereas others use multiple chirps in the symbol structure and

do not retain constant envelope properties. Possessing a con-

stant envelope is desirable; however, the schemes possessing

a constant envelope generally have low spectral efﬁciencies,

which could be an inﬂuential limiting factor.

Another aspect to consider for the CSS alternatives to LoRa

is that they can achieve different maximum achievable spectral

efﬁciencies. Moreover, most LoRa alternatives aim to improve

spectral efﬁciency (SE), energy efﬁciency (EE), or both. Some

of the most promising recently proposed alternatives to LoRa

are in-phase and quadrature (IQ)-CSS [5], slope-shift keying

interleaved chirp spreading LoRa (SSK-ICS-LoRa) [6], dual-

mode CSS [7], and the time domain multiplexed (TDM)

schemes [8], such as TDM-CSS and in-phase and quadrature

(IQ)-TDM-CSS. It is accentuated that here we only mention

a subset of energy-efﬁcient CSS modulations available in the

literature. Numerous other CSS schemes exist in the state-of-

the-art; interested readers are referred to [4] for more details.

IQ-CSS is another multiple chirp modulation that encodes

information bits on both in-phase and quadrature components

of the chirp signal. SSK-ICS-LoRa uses up chirps, down

chirps, interleaved up-chirps, and interleaved down chirps to

expand the symbol set and hence can carry two additional

bits per symbol relative to LoRa. DM-CSS simultaneously

multiplexes even and odd chirp symbols, use phase shifts (PSs)

of 0and πradians for these even and odd chirp symbols, and

employs either up-chirp or down-chirp signal. In TDM-CSS,

two chirped symbols are multiplexed in the time domain, each

having a different chirp slope, i.e., one (chirped) symbol is

attained using an up-chirp, whereas the other one is attained

using a down-chirp. The fundamental idea of IQ-TDM-CSS is

similar to that of TDM-CSS; however, unlike TDM-CSS, IQ-

TDM-CSS uses both the IQ components of the un-chirped

symbols. It may be noticed that the SE of the DM-CSS

and TDM schemes is higher than that of SSK-ICS-LoRa and

classical LoRa. Note that if LoRa transmits λbits per symbol,

then SSK-ICS-LoRa, IQ-CSS, TDM-CSS, DM-CSS, and IQ-

arXiv:2210.04094v1 [eess.SP] 8 Oct 2022

TDM, respectively, transmit λ+ 2,2λ,2λ,2λ+ 1 and 4λbits

per symbol of the same duration.

Nevertheless, all these schemes also have some noteworthy

shortcomings.IQ-CSS is very sensitive to the carrier frequency

offset, and its maximum achievable SE is less than that of

DM-CSS and IQ-TDM-CSS. Besides being capable of both

coherent and non-coherent detection and providing improved

EE, SSK-ICS-LoRa does not signiﬁcantly improve the SE

relative to LoRa and its other counterparts. TDM-CSS symbols

can also be detected coherently and non-coherently; however,

its maximum achievable SE is lesser than DM-CSS and IQ-

TDM-CSS. Besides being sensitive to carrier frequency offset,

for IQ-TDM-CSS, only highly complex coherent detection is

possible. DM-CSS offers improved EE relative to LoRa and

allows both coherent and non-coherent detection; however,

the use of PSs allows the non-coherent detection to be only

feasible if the channel phase rotation is less than π/2, making

it less practical. Furthermore, its maximum achievable SE is

less than that of IQ-TDM-CSS.

In this work, we propose the DM-TDM-CSS scheme that

can achieve almost the same maximum achievable SE as IQ-

TDM-CSS and eliminate its shortcomings. In other words,

both coherent and non-coherent detection can be applied for

DM-TDM-CSS and is more robust against carrier frequency

and phase offsets. In DM-TDM-CSS, we amalgamate the

precepts of a modiﬁed version of DM-CSS and TDM-CSS.

To be more precise, like DM-CSS, we use the even and the

odd FSs; however, unlike DM-CSS, we do not use any PSs for

the even and the odd FSs, allowing more practical coherent

detection. Moreover, like TDM-CSS, two chirped symbols are

multiplexed in the time domain. Both un-chirped symbols have

unique even and odd FSs; however, one is chirped using an up-

chirp and the other with a down-chirp. Unlike DM-CSS, which

uses either up-chirp or down-chirp symbols for chirping the

un-chirped symbol, DM-TDM-CSS uses both simultaneously.

The symbols structure of DM-TDM-CSS allows for achieving

a maximum SE, which is only 4bits less than that of IQ-

TDM-CSS.

The contributions of this work can be summarized as

follows:

1) We propose the DM-TDM-CSS scheme as an alternative

to state-of-the-art CSS schemes (including LoRa). DM-

TDM-CSS inherits the advantageous properties of both

DM-CSS and TDM-CSS while avoiding the limitations

of both schemes. Thus, the resulting scheme is not only

energy and spectral-efﬁcient but is also robust against

the carrier frequency and phase offsets.

2) We comprehensively explain the transceiver design of

the proposed DM-TDM-CSS. The waveform generation

and the coherent and non-coherent detection mecha-

nisms are elaborately presented.

3) We mathematically determine if the DM-TDM-CSS

symbols are orthogonal to each other or not. It shall be

demonstrated that the even and odd FSs in the up-chirp

symbol cause interference with the even and odd FSs in

the down-chirp symbol and vice versa. This interference

among the time domain multiplexed symbols results in

non-orthogonal DM-TDM-CSS symbols.

4) Through mathematical analysis, we estimate the inter-

ference caused by the two TDM chirped symbols at the

receiver. The results shall afﬁrm the conclusions of the

orthogonality analysis that the two TDM symbols cause

interference among each other.

5) We evaluate the performance of DM-TDM-CSS con-

sidering different performance metrics: (i) SE versus

required signal-to-noise ratio (SNR) per bit for a tar-

get bit error rate (BER); (ii) BER performance in an

additive white Gaussian noise (AWGN) and a fading

channel; and (iii) BER performance considering phase

and frequency offsets.

6) We provide closed-form expressions on the interference

terms on both the up/down chirped symbols. We com-

pute the signal-to-interference ratio (SIR) expressions

thanks to the closed-form expressions. We show that the

interference vanishes by increasing λ.

II. PRELIMINARIES

A. Basic Deﬁnitions

In this section, we have provided brief deﬁnitions of these

parameters for the clarity of the readers.

1) Bandwidth: Bandwidth, Bis the range of frequencies

in Hertz (Hz) occupied by a CSS symbol. Bis divided into

Mfrequencies, where the separation between two adjacent

frequencies is ∆fHz; therefore, B=M∆fHz.

2) Symbol Period: Symbol period, Tsis the time in which

one CSS symbol, occupying bandwidth, B, can be transmitted.

Tsis linked to ∆fas ∆f=1/Ts.

3) Bit Rate: Bit rate Rin bits/s is the number of bits that

can be transmitted in Ts.

4) Spectral Efﬁciency: It is the achievable Rper Band is

given as η=R/B.

5) Energy Efﬁciency: It is the required SNR for correct bit

detection at a given BER.

B. System Model

Without loss of generality, we consider a chirped symbol

in CSS modulation composed of two components: (i) an un-

chirped symbol and (ii) a spreading symbol that spreads the

information in the bandwidth, B. The un-chirped symbol is

a pure sinusoid when only one FS kis activated, or it can

be a combination of multiple sinusoids in case multiple FSs

are used. When the un-chirped symbol is spread, the FS(s)

have an injective mapping to cyclic time-shift(s). Moreover,

the spreading symbol can have different slope rates [9], [10].

We consider that the occupied bandwidth is B=M/Tsthat

corresponds to the availability of MFSs implying that k∈

[0, M −1]. In the discrete time, we denote the CSS (chirped)

symbol consisting of Msamples by s(n) = g(n)cγ(n),

for n=J0, M −1K, where g(n)is the un-chirped sym-

bol, and cγ(n)is the spreading symbol given as cγ(n) =

exp jπ

Mγn2, where j2=−1and γis the slope rate.

Based on the type of CSS modulation, g(n)can have different

symbol structures. Moreover, when γ= 1, the spreading

symbol corresponds to up-chirp, i.e., cu(n) = exp jπ

Mn2.

Conversely, if γ=−1, the spreading symbol is a down-

chirp denoted as cd(n) = exp −jπ

Mn2. Typically, an up-

chirp symbol is used to spread the information in most CSS

modulations.

λ1bits bi2de

ke,1

exp j2π

Mke,1nfe,1(n)

λ2bits bi2de

ko,1

exp j2π

Mko,1nfo,1(n)

P×

exp jπ

Mn2

λ3bits bi2de

ke,2

exp j2π

Mke,2nfe,2(n)

λ4bits bi2de

ko,2

exp j2π

Mko,2nfo,2(n)

P×

exp −jπ

Mn2

g1(n)

g2(n)

cu(n)

cd(n)

s1(n)

s2(n)

s(n)

Fig. 1: DM-TDM-CSS transmitter architecture.

The discrete-time baseband received symbol is:

y(n) = hs(n) + w(n),(1)

where his the complex channel gain, and w(n)are the AWGN

samples having single-sided noise power spectral density of

N0, and noise variance of σ2

n=N0B. It may be noticed

that in LPWANs, CSS symbols maintain a narrow bandwidth

of 500 kHz or smaller. Therefore, a ﬂat fading channel can

have a constant attenuation over the entire B. Thus, it can be

considered equal to unity if channel state information (CSI) is

known in the simplest of cases.

III. DUAL-MODE TIME DOMAIN MULTIPLEXED CHIRP

SPREAD SPECTRUM

A. Transmission

The transmitter architecture of DM-TDM-CSS is provided

in Fig. 1. In DM-TDM-CSS, two chirped symbols are mul-

tiplexed in the time domain; therefore, two different un-

chirped symbols are needed. For each un-chirped symbol,

one even and one odd frequency is activated. Consider that

for each un-chirped symbol, Mfrequencies are available.

Among these Mfrequencies, M/2frequencies are even and

M/2frequencies are odd. The even activated frequencies for

these un-chirped symbols are ke,1and ke,2, whereas the odd

activated frequencies are ko,1and ko,2. Note that the even

and odd frequencies are identiﬁed by indexes ˜

ke= 2keand

ko= 2ko+ 1, where ke∈[0,M/2−1] and ko∈[0,M/2−1].

ke,1and ko,1are determined after binary-to-decimal (bi2de)

conversion of bit sequences of lengths λ1=λ−1and

λ2=λ−1, respectively. On the other hand, ke,2and ko,2after

bi2de conversion of bit sequences having lengths λ3=λ−1

and λ4=λ−1, respectively.

The ﬁrst un-chirped symbol, g1(n), is composed of two

sinusoids. The ﬁrst sinusoid, fe,1(n), has an even activated

frequency, ke,1, whereas the second sinusoid, fo,1(n), has an

odd activated frequency, ko,1. Then, g1(n)is given as:

g1(n) = fe,1(n) + fo,1(n)

= exp j2π

Mke,1n+ exp j2π

Mko,1n.(2)

Similarly, the second un-chirped symbol, g2(n), also con-

sists of even frequency, ke,2, and odd frequency, ko,2, activated

sinusoids, fe,2(n), and fo,2(n).g2(n)is given as:

g2(n) = fe,2(n) + fo,2(n)

= exp j2π

Mke,2n+ exp j2π

Mko,2n.(3)

The next step is to spread the un-chirped symbols, g1(n),

and g2(n).g1(n)is then spread using an up-chirp, cu(n),

whereas g2(n)is spread using a down-chirp, cd(n)resulting

in s1(n)and s2(n), i.e.,

s1(n) = g1(n)cu(n)

= exp njπ

M2ke,1n+n2o+exp njπ

M2ko,1n+n2o,

(4)

and

s2(n) = g2(n)cd(n)

= exp njπ

M2ke,2n−n2o+exp njπ

M2ko,2n−n2o.

(5)

Afterward, these two chirped symbols, s1(n)and s2(n),

are multiplexed in the time domain resulting in s(n), which

is given as:

s(n) = s1(n) + s2(n)

= exp njπ

M2ke,1n+n2o+exp njπ

M2ko,1n+n2o

+ exp njπ

M2ke,2n−n2o+exp njπ

M2ko,2n−n2o.

(6)

The symbol energy of the DM-TDM-CSS symbol is Es=

1/MPM−1

n=0 |s(n)|2.

B. Detection

This section presents coherent and non-coherent detection

mechanisms for DM-TDM-CSS received symbols, y(n). For

clarity of exposition, we consider the following vectorial

representations, y= [y(0), y(1),··· , y(M−1)]T, and s=

[s(0), s(1),··· , s(M−1)]T, where [·]Tis the transpose op-

erator.

1) Coherent Detection: The coherent detector achitecture

for DM-TDM-CSS is illustrated in Fig. 2. The coherent

detection of DM-TDM-CSS symbols involves the estimation

of the FSs of the un-chirped symbols, ke,1,ke,2,ko,1, and ko,2.

Assuming that his known at the receiver and the transmit

symbols are equiprobable, the coherent detection dictates to

maximize the probability of receiving ywhen swas sent given

h, i.e., prob (y|s, h). The likelihood function, prob (y|s, h)is

given as:

prob (y|s, h) = 1

2πσ2

nM

exp ky−hsk2

2σ2

n

=ρexp <{hy, hsi}

σ2

n,

(7)

where k · k2evaluates Euclidean norm, <{·} determines the

real component of a complex-valued argument, and

ρ=1

2πσ2

nM

exp −kyk2+khsk2

2σ2

n.

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1Dual-ModeTimeDomainMultiplexedChirpSpreadSpectrumAliWaqarAzim,AhmadBazzi,MahrukhFatima,RaedShubair,MarwaChaiAbstractWeproposeadual-mode(DM)timedomainmul-tiplexed(TDM)chirpspreadspectrum(CSS)modulationforspectralandenergy-efcientlow-powerwide-areanetworks(LP-WANs).DM-CSSmodulationthatusesboththee...

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