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Resource Allocation Based PAPR

Analysis in Uplink SCMA-OFDM Systems

ADITYA S. RAJASEKARAN1,2, (Member, IEEE), MONIROSHARIEH VAMEGHESTAHBANATI1,

MOHAMMAD FARSI3, HALIM YANIKOMEROGLU1, (Fellow, IEEE), AND HAMID SAEEDI3,

(Member, IEEE)

1The authors are all with the Department of Systems and Computer Engineering, Carleton University, Ottawa, ON K1S 5B6, Canada (email:

aditya.rajasekaran@carleton.ca; mvamegh@sce.carleton.ca; halim@sce.carleton.ca).

2Aditya Rajasekaran is also with Ericsson Canada Inc, Ottawa, ON K2K 2V6, Canada

3The authors are with the Department of Electrical and Computer Engineering, Tarbiat Modares University, Tehran 14115-111, Iran

Corresponding author: Aditya Rajasekaran (email: aditya.rajasekaran@carleton.ca).

This work was supported in part by Ericsson Canada Inc. and in part by a Discovery Grant of the Natural Sciences and Engineering

Research Council of Canada.

ABSTRACT Sparse code multiple access (SCMA) is a non-orthogonal multiple access (NOMA) uplink

solution that overloads resource elements (RE’s) with more than one user. Given the success of orthogonal

frequency division multiplexing (OFDM) systems, SCMA will likely be deployed as a multiple access

scheme over OFDM, called an SCMA-OFDM system. One of the major challenges with OFDM systems

is the high peak-to-average power ratio (PAPR) problem, which is typically studied through the PAPR

statistics for a system with a large number of independently modulated sub-carriers (SCs). In the context

of SCMA systems, the PAPR problem has been studied before through the SCMA codebook design for

certain narrowband scenarios, applicable more for low-rate users. However, we show that for high-rate

users in wideband systems, it is more meaningful to study the PAPR statistics. In this paper, we highlight

some novel aspects to the PAPR statistics for SCMA-OFDM systems that is different from the vast body

of existing PAPR literature in the context of traditional OFDM systems. The main difference lies in the

fact that the SCs are not independently modulated in SCMA-OFDM systems. Instead, the SCMA codebook

uses multi-dimensional constellations, leading to a statistical dependency between the data carrying SCs.

Further, the SCMA codebook dictates that an UL user can only transmit on a subset of the available SCs.

We highlight the joint effect of the two major factors that inﬂuence the PAPR statistics - the phase bias

in the multi-dimensional constellation design along with the resource allocation strategy. By considering

such a cross-layer systematization perspective, we motivate the fact that PAPR reduction can be achieved

through the setting of static conﬁguration parameters. Compared to the class of PAPR reduction techniques

in the OFDM literature that rely on multiple signalling and probabilistic techniques, these gains come with

no computational overhead. In this paper, we also examine these PAPR reduction techniques and their

applicability to SCMA-OFDM systems.

INDEX TERMS Sparse code multiple access (SCMA), peak-to-average power ratio (PAPR), orthogonal

frequency division multiplexing (OFDM), sub-carrier (SC), uplink (UL), selective mapping (SLM), inter-

leaving (IL).

I. INTRODUCTION

Non-orthogonal multiple access (NOMA) solutions are being

actively studied to address the massive connectivity require-

ments for 5G and beyond 5G (B5G) communication systems

[1]. The sparse code multiple access (SCMA), proposed in

[2], is one such NOMA scheme that has received a lot

of attention particularly for the uplink (UL) direction [3].

SCMA will likely be used as a multiple access scheme over

orthogonal frequency division multiplexing (OFDM), which

is referred to as an SCMA-OFDM system [4], [5]. In SCMA-

OFDM systems, the orthogonal OFDM sub-carriers (SCs)

are the resource elements (RE’s) over which the SCMA code-

words are spread. Traditional OFDM systems that indepen-

dently modulate the individual SCs are known to create large

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A. Rajasekaran et al.: Resource Allocation Based PAPR Analysis in Uplink SCMA-OFDM Systems

power peaks compared to the average power, resulting in

the well known peak-to-average power ratio (PAPR) problem

[6]. A high PAPR means that the power ampliﬁer needs to

operate in an inefﬁcient region to avoid power leakage, which

in turn affects the battery-life of the transmitting UL end-user

device. The article in [7] highlights the high PAPR problem

in OFDM systems while also describing why operators are

unlikely to opt for a NOMA scheme that does away with

OFDM entirely. Hence, studying the PAPR problem in an

OFDM system that uses a NOMA scheme like SCMA is

an important problem for 5G and beyond communication

systems as highlighted by [7].

In the context of traditional OFDM systems that indepen-

dently modulate a large number of SCs, the PAPR attained

is a random quantity, since it depends on the sequence

of complex-valued constellation points transmitted in the

OFDM SCs along with the symbol rate. The PAPR can then

be analysed in terms of its maximum theoretically attainable

value based on the constellation scheme used to modulate

the individual SCs. Alternatively, the PAPR can be studied

in terms of its statistics using the complementary cumulative

distribution function (CCDF), often referred to as the PAPR

statistics [8]. It has been shown that when the number of SCs

is sufﬁciently large, the maximum theoretically attainable

PAPR value occurs with negligible probability and the PAPR

statistics offer more meaningful insights [8]–[11]. Since

traditional OFDM systems typically involve independently

modulated SCs, the PAPR statistics have been characterized

with this assumption in several studies [9], [11], [12]. How-

ever, using SCMA as a multiple access scheme over OFDM

means that the individual SCs are not independently mod-

ulated; thus motivating the need to characterize the PAPR

analysis speciﬁcally for SCMA-OFDM systems.

In the SCMA construct, each user maps its incoming bits to

a multi-dimensional modulation symbol coded over multiple

RE’s, which is termed as a codeword. Each modulation sym-

bol has its own codeword and together they form a codebook

that is ideally unique to a user. In this way, the users are

separated by their unique codebooks. The SCMA codebook

design problem involves the multi-dimensional constella-

tion design [3] and several such constellations have been

proposed in the literature [13]–[17]. This SCMA codebook

design provides an additional degree of freedom to the PAPR

problem in SCMA based systems [18] and corresponding

multi-dimensional constellation schemes that minimize the

PAPR that can theoretically be attained have been proposed

in [19]–[22]. However, we show in this paper that like with

traditional OFDM systems, this maximum theoretically at-

tainable PAPR based on the codebook design is a meaningful

metric only for low-rate users or in narrow-band systems.

For high rate users, i.e., when a larger number of modulation

symbols are transmitted in the same OFDM symbol duration,

it is more meaningful to study the PAPR statistics. While the

SCMA paradigm is often discussed for massive connectivity

deployment involving low-rate IoT devices [18], SCMA can

just as easily be used for traditional wireless devices and

other high-rate users in 5G and B5G networks [4], [5], [23].

The characteristics of the SCMA codebook that affect the

PAPR statistics are different from those studied to date in

the SCMA literature to the best of our knowledge, and is the

focus of this paper.

Since SCMA employs multi-dimensional constellations,

each modulation symbol is transmitted over multiple indi-

vidual SCs. Thus, the transmitted OFDM SCs are not in-

dependently modulated like they are in traditional OFDM

systems. The multi-dimensional constellation design dictates

what is transmitted in the individual SCs. The PAPR statistics

in SCMA-OFDM systems will thus reﬂect this dependency

between the modulated SCs. For instance, if the SCMA code-

word consists of constellation points all of the same phase

and the SCs over which they are transmitted are contiguous,

it will have a detrimental effect on the PAPR statistics. Al-

ternatively, an SCMA scheme where the codewords contain

constellation points of opposite phases and transmitted over

contiguous SCs is likely to have a positive effect on the PAPR

statistics. However, the SCs carrying SCMA codewords do

not necessarily have to be contiguous SCs. In other words, the

SCs that carry the codewords can be located anywhere in the

frequency spectrum. Hence, the joint impact of the SCMA

modulation scheme along with the OFDM SC placement

must be considered when studying the PAPR statistics for

high-rate SCMA-OFDM users.

In this paper, we highlight these novel aspects to the PAPR

analysis for SCMA-OFDM systems that is different from

the vast body of existing PAPR literature in the context of

traditional OFDM systems. We highlight two major factors

that inﬂuence the PAPR statistics - the phase bias in the multi-

dimensional constellation design along with the resource

allocation strategy1. By considering such a cross-layer sys-

tematization perspective, we motivate the fact that signiﬁcant

PAPR reduction can be achieved through the setting of static

conﬁguration parameters. Such gains could come without

any additional overhead to the system that is typically in-

troduced by most multiple signalling based PAPR reduction

techniques that work with the assumption of independently

modulated SCs, i.e., no a-priori knowledge of any statistical

dependencies in the transmitted signal [6].

Further, in the context of traditional OFDM systems, nu-

merous techniques that improve the PAPR statistics have

been proposed, as captured by the surveys in [6], [24], [25].

Broadly, there are two categories of such PAPR reduction

techniques: signal distortion based techniques and multiple

signalling and probabilistic based techniques. Signal distor-

tion techniques like clipping distort the transmitted signal

by not transmitting any power peaks above a certain thresh-

old [26]. Alternatively, PAPR reduction techniques based

on multiple signalling generate a set of candidate signals

every OFDM symbol and transmit the signal with the least

PAPR. The set of candidate signals are generated through

1The placement of the SCs that carry an SCMA codeword is basically

a resource allocation problem. In this paper, we use the terms resource

allocation and SC placement interchangeably.

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operations like phase changes [27] or interleaving [28] on the

original data set. There is a signiﬁcant complexity overhead

in generating these extra candidate signals as well as some

throughput loss since sidelink information about the opera-

tions performed on the data set needs to be transmitted to the

receiver [6].

In [4] and [5], the class of signal distortion techniques is

studied in the context of high-rate SCMA-OFDM systems.

Speciﬁcally, these two papers investigate the challenge of

allowing the SCMA receiver to cope with the distortions

introduced by signal clipping at the transmitter. However,

the class of multiple signalling and probabilistic techniques

has not been thoroughly examined in the context of SCMA-

OFDM systems to the best of the authors knowledge. As we

discuss in this paper, some of the techniques that involve

constellation shaping [29], [30] cannot easily lend itself to

SCMA systems because it affects the SCMA constellation

design. However, other multiple signalling techniques such

as selective mapping (SLM) [27], partial transmit sequences

(PTS) [31] and interleaving (IL) [28] can be tailored to meet

the constraints of an SCMA-OFDM system. In this paper,

we discuss what adaptations are needed to these well known

PAPR reduction techniques to make them work in SCMA-

OFDM systems. Moreover, in traditional OFDM systems

where each SC is independently modulated, until the SCs to

be transmitted in an OFDM symbol are known, there is no

way to know which SC allocation strategy results in the least

PAPR. Hence, a certain number of permutations are tried

dynamically every OFDM symbol and the one with the least

PAPR is transmitted. However, with SCMA-OFDM systems,

we can exploit the statistics known in advance through the

novel aspects we present in this paper to reduce or even elim-

inate the complexity and sidelink information overhead typ-

ically incurred by these PAPR reduction techniques. Further,

it is worth mentioning that these PAPR reduction techniques

have also been recently investigated in other non-SCMA

based NOMA systems [32]–[35], but these are beyond the

scope of this paper.

The contributions of this paper can then be summarized as:

•We highlight the two main factors that impact the PAPR

statistics in SCMA-OFDM systems as a result of the de-

pendency between data carrying SCs - the phase bias in

the SCMA constellation design and the accompanying

resource allocation strategy.

•We show that such a resource allocation based PAPR

analysis allows for PAPR gains through the setting of

static conﬁguration parameters that does not incur any

computational overhead. Such gains are not possible in

traditional OFDM systems that individually modulate

the SCs.

•Finally, we analyse the class of PAPR reduction tech-

niques based on multiple signalling and probabilistic

techniques in the context of SCMA-OFDM systems. We

then compare the static PAPR gains from the resource

allocation based strategies with the gains from these

well known PAPR reduction techniques and offer some

insights into how they can be used together to improve

the PAPR while minimizing complexity and throughput

loss.

The rest of this paper is organized as follows. Section

II describes the uplink SCMA-OFDM system model. Sec-

tion III provides a detailed comparison between the PAPR

analysis in traditional OFDM systems and SCMA-OFDM

systems, highlighting the key differences between the two.

In particular, Section III-B describes the novel aspects that

make the PAPR statistics different in SCMA-OFDM systems.

Section IV then describes resource allocation strategies that

impact the PAPR statistics and in Section V, we provide

simulation results. Section VI examines the PAPR reduction

techniques based on multiple signalling in the context of

SCMA-OFDM systems. Finally, the conclusion with future

research directions is presented in Section VII.

II. SYSTEM MODEL: UPLINK SCMA-OFDM

TRANSMISSION

Consider an SCMA-OFDM system where the total band-

width is comprised of ZOFDM SCs. The ZSCs are uni-

formly divided into SCMA blocks of size N. Therefore, a

total of NB=Z/N SCMA blocks can be conﬁgured in

the system. Within one SCMA block, Kusers share the N

SCs such that N < K . Due to the sparse overloading of

SCMA, each user is assigned to only dv<< N SCs. If the

SCMA block is fully loaded, each user is assigned a unique

combination of dvSCs and

K=N

dv.(1)

In an M-point signal constellation, each NM= log2Mbits

for each user is sent over the dvSCs. A user-to-SC binary

allocation matrix Sof dimensions N×Kdictates which dv

SCs are assigned to which users. Every row in Srepresents

a SC, while every column represents a user. For example, for

N= 4,dv= 2 and K= 6, a sample user-to-SC allocation

matrix is

S=

101010

011001

100101

010110

.(2)

As the ﬁrst and third positions in the ﬁrst column of Sin (2)

are non-zero, we can say the ﬁrst user has allocation “1010”

in one SCMA block. In other words, user 1 is assigned to the

ﬁrst and the third SCs. Similarly, the second user is assigned

the second and fourth SC in the SCMA block, i.e., it has

allocation “0101” and so on. Since this is a fully loaded

system, all unique combinations of dv= 2 SCs are covered

in the Kcolumns of S.

We consider a modulation symbol to be the representation

of log2Mbits, e.g., when M= 4, the modulation symbols

are ‘00’, ‘01’, ‘10’ and ‘11’. Each user can send L(L≤

NB) modulation symbols over Ldifferent SCMA blocks in

the same OFDM symbol duration. For simplicity, we assume

each user has the same user allocation in each of these L

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A. Rajasekaran et al.: Resource Allocation Based PAPR Analysis in Uplink SCMA-OFDM Systems

S/P converter

N-pt IDFT

P/S converter

U1U2U3U4U5U6

SC1

SC2

SC3

SC4

Z=20, N=4, NB=5

(L codewords)

Y0

Y1

YZ−1

D/A

Converter

y[0]

y[1]

y[Z−1]

y(t)

y[n]: Discrete time samples of signal

FIGURE 1: The transmitter for User-1 in an SCMA-OFDM

system where 6 users share 4 SCs, with 2 non-zero SCs

allocated to each user.

blocks. For instance, for the user-to-SC allocation matrix in

(2), user 1is assigned to the allocation “1010” over all the L

blocks.

In an SCMA-OFDM system with Kusers, each symbol

of the kth user is mapped to a dv-dimensional complex

constellation ˜xk= (˜x1,k, ..., ˜xdv,k)T, that is selected from

the columns of a dv×Mmatrix called Xk. As in [3],

for uplink transmission, we assume that the constellation

scheme used is the same for all Kusers. Hence, when

describing the transmitter for one user in an SCMA-OFDM

system, the index kcan be dropped and the dv-dimensional

SCMA constellation scheme for the user can be represented

by X. Each column represents what the user transmits for

the m={1, .., M }symbol, i.e., X= (x1, ..., xM) and

xm= (x1,m, ..., xdv,m )T.

In an SCMA-OFDM system, the transmitter maps Lsets

of NMbits to Lmodulation symbols, based on X. These

Lmodulation symbols will be carried over the L×dvSCs

assigned to it and the user is required to leave the other SCs

in the system as null SCs, i.e., the user does not transmit

anything on these null SCs. Let Yidenote the complex

constellation point transmitted in SCiin the system. Yiwill

either be null if SCiis not assigned to the user, or else

Yiwill contain xj˜xm,j {1, .., dv}, for the symbol m

transmitted in SCi. An inverse fast Fourier transform (IFFT)

based implementation of an OFDM transmitter uses the input

on these SCs to generate the discrete time domain samples of

the signal, y[n], as follows:

y[n] = 1

√Z

Z−1

X

i=0

Yiej2πin

Z.(3)

Fig. 1 illustrates an example of the transmitter of an

SCMA-OFDM system. In the top-left part of the ﬁgure,

one SCMA block with K= 6,N= 4,dv= 2 with

the user-to-SC allocation matrix Sfrom (2) is depicted.

The users are labelled from U1through to U6and the SCs

from SC1through to SC4. The coloured boxes indicate that

(a) 4-LDS constellation.

(b) 4-Bao constellation.

(c) 4-OPP constellation.

FIGURE 2: The 4-point SCMA constellations used in this

study.

the corresponding SC is assigned to the user from S. The

IFFT based implementation of the transmitter for U1is then

illustrated. Since, from (2), U1has allocation “1010”, it uses

the ﬁrst and third SC in every SCMA block it transmits on.

In this example, U1transmits one modulation symbol in each

of the available SCMA blocks, i.e., L= 5. This corresponds

to the user using 5blocks with 2SCs in each block, which

constitutes a total of 10 SCs.

The M-point SCMA modulation scheme2determines how

NMbits of user data are mapped to the dvSCs allocated to a

user. The known multi-dimensional constellation schemes in

the literature are outlined in detail in the survey in [3]. In Fig.

2, we show multi-dimensional SCMA constellation schemes

when M= 4 and dv= 2. The ﬁrst two constellations

are named 4-LDS and 4-Bao respectively, which follows the

same naming convention as the authors in [3] for consistency.

The third, named 4-OPP, is a new constellation we will

introduce in Section III-B.

2In this paper, we use the terms SCMA modulation scheme and SCMA

multi-dimensional constellation interchangeably.

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III. PAPR IN TRADITIONAL OFDM SYSTEMS VS. PAPR

IN SCMA-OFDM SYSTEMS

In this section, we present the key differences between study-

ing the PAPR in traditional OFDM systems that indepen-

dently modulate the SCs vs. in an SCMA-OFDM system.

As we discussed in the introduction in Section I, the PAPR

attained during an OFDM signal transmission is a random

quantity that can be analysed in terms of the maximum

theoretically attainable PAPR in a given OFDM symbol

duration or through a statistical characterization of the PAPR,

called the PAPR statistics. In traditional OFDM systems, the

maximum theoretically attainable PAPR can be determined

through the knowledge of the constellation scheme used to

independently modulate each SC, e.g., M-QAM. In SCMA-

OFDM systems, the theoretically attainable PAPR can be

determined in a similar way but must account for the multi-

dimensional constellation scheme in use. However, in the

vast body of PAPR literature for OFDM systems, it has been

shown that when the number of SCs is sufﬁciently large, i.e,

high-rate users, the maximum theoretically attainable PAPR

value occurs with next to negligible probability and the PAPR

statistics offer more meaningful insights [8]–[11]. The same

is true for an SCMA-OFDM transmitter that transmits over

a large number of SCs in the same OFDM symbol dura-

tion, i.e., high-rate users. However, the discussion on PAPR

statistics for SCMA-OFDM systems is different because of

the statistical dependency between SCs. In what follows,

we discuss how the combination of the multi-dimensional

modulation scheme along with the accompanying resource

allocation strategy impacts the PAPR statistics in such high-

rate SCMA-OFDM systems.

A. PAPR IN OFDM SYSTEMS

An OFDM signal, y(t), that is generated from individually

modulated subcarriers transmitted in the same OFDM sym-

bol duration can be represented as

y(t) =

Z−1

X

k=0

akexp(j2π(fc+k∆f)t)

= exp(j2πfct)

Z−1

X

k=0

akexp(j2πkt/Ts),

(4)

where akis the complex-valued constellation point transmit-

ted in SC k,fcis the centre frequency of SC kand Zis the

total number of SCs in the system. If y(t)is sampled at a rate

of Z/Ts, i.e., every sample is taken at multiples of Ts/Z, the

discrete time version for the baseband part of y(t)from (4)

can be expressed as

y[n] =

Z−1

X

k=0

akexp(j2πkn/Z).(5)

When the transmitted OFDM signal is generated from

independently modulated SCs, the non-constant envelope

creates large instantaneous peaks in the signal. These power

0 500 1000 1500 2000

-3

-2

-1

0

1

2

3

FIGURE 3: Illustration of the power peaks produced when

summing sinusoids of evenly spaced frequencies as is the

case in an OFDM signal.

peaks occur when the individual signals align in phase, which

can be much larger than the average power of the transmitted

signal and results in a high PAPR. This is illustrated in Fig. 3,

where four sinusoids with equal subcarrier spacing are added

together and the resultant signal has large power peaks. While

PAPR applies to the continuous time transmitted signal,

studies have shown that if over-sampled with a sufﬁciently

high ratio [36], PAPR can be accurately calculated from the

discrete time samples y[n]from (5) as follows:

PAPR (dB) = 10 log10 max(|y[n]|2)

E(|y[n]|2).(6)

The PAPR attained is a random quantity, since it depends

on the sequence of complex-valued constellation points

transmitted in the ZSCs along with the symbol rate. If ak

is selected from a QAM constellation of Mpoints of equal

magnitude, e.g., 4-QAM, then the maximum theoretically

attainable PAPR is Z[10]. This is because when all the

SCs add coherently, the instantaneous power is Z2, while the

average power to transmit ZSCs of unit energy signals each

is Z. However, it has been shown that when Zis sufﬁciently

large, this theoretically attainable PAPR value occurs with

negligible probability and the PAPR statistics offer more

meaningful insights [8]–[11]. For example, with Z= 32,4-

ary modulation and an OFDM symbol duration of 100 µs,

the authors in [9] showed that the theoretically attainable

PAPR occurs every 3.7 million years. For an OFDM system

with ZSCs, MZunique symbol sequences and thus MZ

unique OFDM waveforms per block can be generated [8].

Some of these waveforms will have low PAPR and some will

have a higher PAPR value. Since traditional OFDM systems

typically involve independently modulated SCs, the PAPR

statistics have been characterized with this assumption in sev-

eral studies [9], [11], [12]. Since the ZSCs are individually

modulated, and if Zis large, the central limit theorem dictates

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A. Rajasekaran et al.: Resource Allocation Based PAPR Analysis in Uplink SCMA-OFDM Systems

that the real and imaginary parts of the transmitted OFDM

signal can be modelled by Gaussian random processes. As a

result, the overall envelope of the transmitted signal follows

a Rayleigh distribution [11].

B. PAPR IN SCMA-OFDM SYSTEMS

When Lmodulation symbols, carried over LSCMA blocks,

are transmitted in the same OFDM symbol duration, the

maximum theoretically attainable PAPR of the transmitted

signal can be computed from (6). Let Xm,max denote the

maximum possible instantaneous peak to transmit symbol

m. Thus, Xm,max is the sum of the amplitudes of all dv

dimensions of xm. Let Xmax denote the maximum possible

instantaneous peak from the constellation scheme, i.e., the

peak which occurs when the modulation symbol that contains

the maximum peak is transmitted. The maximum attainable

peak from transmitting Lsymbols is achieved when the

symbol corresponding to Xmax is transmitted on all SCMA

blocks and each of the SCs line up in phase. Also, let PX,m

represent the power required to transmit xmand PX,avg rep-

resent the average power of transmitting a symbol from the

constellation. The maximum possible PAPR value, calculated

per OFDM symbol duration, is computed as follows:

PAPR (dB) = 10 log10 |L×Xmax |2

L×PX,avg ,

where

Xm,max =

dv

X

i=1

xi,m,∀m {1..M }

Xmax = max(Xm,max)

PX,avg =PM

m=1 Pdv

i=1 x2

i,m

M.

(7)

If L= 1, we can consider that as the constellation PAPR.

It is clear that this constellation PAPR value is determined

entirely from the design of the SCMA multi-dimensional

constellation. For example, for the constellation schemes

shown in Fig. 2, Xmax = 1.4,PX,avg = 1 and so the

constellation PAPR is 3.04 dB. In this context, codebook

designs that minimize this theoretically attainable PAPR have

been proposed [19]–[22]. For example, in [19], a low-PAPR

codebook that minimizes the number of projections, i.e., non-

zero dimensions in the constellation scheme, is proposed.

If a zero dB PAPR constellation design is required, i.e., 0

dB constellation PAPR when L= 1, then each modulation

symbol should be coded with the same amplitude on only one

of the dvSCs. The constellation design approach for this is

outlined in Appendix A.

We described in Section III-A that for traditional OFDM

systems, when the number of modulated SCs is large, the

theoretically attainable PAPR occurs with negligible proba-

bility and the PAPR statistics are more meaningful to study.

An SCMA-OFDM user transmits Lmodulation symbols per

OFDM symbol duration over L×dvSCs. Hence, when

Lis large, the PAPR statistics become more meaningful to

study in an SCMA-OFDM system. Since Lcorresponds to

the number of modulation symbols transmitted per OFDM

symbol duration, we can equate a large value of Lwith high-

rate users. Hence, the PAPR statistics should be investigated

for such high-rate users. The SCMA codebook inﬂuences the

PAPR statistics in ways that are different from the PAPR

perspective studied in SCMA systems for low-rate users in

[19]–[22]. Further, the characterization of the PAPR statistics

is different from traditional OFDM systems for the reasons

we discuss next.

The PAPR statistics in SCMA-OFDM systems are differ-

ent from traditional OFDM systems because of two main fac-

tors. Firstly, the presence of null SCs in the codebooks means

that each user transmits on only a small fraction of the total

available SCs. Secondly, the data carrying SCs are not inde-

pendently modulated. One modulation symbol dictates what

is transmitted in dvSCs. Hence, these dvSCs are dependent

in the statistical sense. As we discussed in the system model

in Section II, it is the multi-dimensional constellation used in

the SCMA codebook that determines what the user transmits

on each of the dvSCs. This creates a statistical dependency

between these dvSCs, since they are collectively determined

by the choice of one modulation symbol. This dependency af-

fects the PAPR statistics through the level of phase bias in the

constellation design. To illustrate this concept of phase bias

in an SCMA multi-dimensional constellation, we use two of

the known multi-dimensional constellations from the SCMA

literature when M= 4 and dv= 2, named 4-LDS and 4-

Bao respectively [3]. To these, we introduce another SCMA

constellation, namely 4-OPP. These three constellations are

depicted in Fig. 2 and all have the same constellation points

in each dimension, but are combined differently to form the

codewords for the respective modulation symbols. The 4-

LDS scheme repeats the same constellation point over all the

dimensions in which they are coded. This means that all the

SCs carrying an LDS modulation symbol are guaranteed to

have the same phase. In 4-Bao, two symbols, “00” and “11”,

are coded with the same point in both SCs (i.e., same phase),

while the other two symbols, “01” and “10”, are coded with

constellation points of exactly opposite phase. This means if

the symbols “00” and “11” are transmitted, a phase bias of

having two SCs with the same phase guaranteed will occur.

On the other hand, when “01” and “10” are transmitted, the

opposite bias is introduced. Given each of the four symbols

are equally likely to be transmitted, we can expect these two

opposing bias effects to statistically cancel each other out. 4-

OPP on the other hand has the opposite effect of LDS and

introduces a guaranteed 180° phase difference between the

two constellation points for each and every symbol.

The impact that such a phase bias in the constellation

scheme has on the PAPR statistics comes from the a-priori

statistical dependency it introduces between transmitted SCs

in the same OFDM symbol. This is a feature that is very

different from traditional OFDM systems where the SCs are

independently modulated. However, whether the phase bias

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has a positive or negative effect on the PAPR statistics com-

pared to independently modulating the SCs is inﬂuenced by

whether these dependent SCs are of similar or very different

centre frequencies in the spectrum. For example, when 4-

LDS with the phase bias of having constellation points all of

the same phase is transmitted over dvstatistically dependent

SCs that are near contiguous in the spectrum, it will have

a detrimental effect on the PAPR statistics. Alternatively,

when 4-OPP that contains a phase bias of having constel-

lation points of exactly opposite phases is transmitted over

contiguous SCs, it is likely to have a positive effect on the

PAPR statistics. These are just a simple consequence of the

way the signals in OFDM SCs add up to form the equivalent

OFDM signal, as illustrated earlier in Fig. 3. However, the

SCs carrying SCMA codewords do not necessarily have to

be contiguous SCs. It depends on which SCs are allocated

to the SCMA blocks assigned to the user. Hence, we refer to

this as the SC allocation or resource allocation strategy. In

what follows, we ﬁrst describe different resource allocation

strategies that could impact the PAPR statistics in Section

IV and then analyze the joint impact of the phase bias in

the constellation design and the accompanying SC allocation

strategy on the PAPR statistics in Section V.

IV. RESOURCE ALLOCATION SCHEMES

As we discussed in Section III-B, the placement of the depen-

dent SCs in an SCMA-OFDM system inﬂuences the PAPR

statistics. The placement of the dependent SCs is determined

by how the SCs in the spectrum are assigned to carry the

SCMA blocks, i.e., the resource allocation strategy. While

any number of such resource allocation strategies can be

considered, we focus on two resource allocations which rep-

resent the two extremes in terms of frequency separation, i.e.,

the spacing between the dependent SCs. In the illustration

shown in the system model in Fig. 1, every Ncontiguous SCs

is grouped into an SCMA block. We term this as the regular

allocation. However, if the grouping of every Nconsecutive

SCs to form an SCMA block is considered as a virtual view of

the system SCs, it can be mapped in any way to the physical

OFDM SCs. In the second resource allocation strategy, we

separate the individual SCs that make up an SCMA block by

as much as possible. Since separating the SCs that carry the

SCMA codeword equates to providing frequency diversity

between the dvSCs, we call this the diversity allocation.

An illustration of this mapping for a system with 16 SCs

is shown in Fig. 4. The virtual allocation, depicted as va

in the ﬁgure, is the same as the regular allocation. The

physical allocation, represented as pa in the ﬁgure, represents

the diversity allocation. The algorithm for constructing this

diversity allocation is discussed next.

A. CONSTRUCTING THE DIVERSITY ALLOCATION

SCHEME

Attaining frequency diversity translates to providing as much

separation as possible between the non-zero dimensions over

which an SCMA modulation symbol is transmitted. Since

2

14

36

58

710

911 12 13 14 15 16

5

1913 6

210 14 7

3 11 15 8

4 12 16

f2

f1f4

f3f6

f5f8

f7f10

f9f11 f12 f13 f14 f15 f16

va:

pa:

SC's in the system:

FIGURE 4: Illustration of the diversity allocation mapping from

the virtual view of SCMA blocks (also the regular allocation)

to the physical OFDM SCs.

every pair of SCs in an SCMA block belong to some user’s

allocation in user-to-SC allocation matrix Sas shown in

(2), we seek some guaranteed minimum level of separation

between every pair of SCs that belong to an SCMA block.

Let the OFDM SCs in the system be indexed as

{f1,f2,...,fZ}, with the SC spacing between any two SCs de-

noted by ∆f. If {f1,...,fN} constitutes the ﬁrst SCMA block,

{fN+1,...,f2N+1 } constitutes the second SCMA block and so

on, it is termed as the “regular allocation”. On the other hand,

in the diversity SC allocation scheme, we distribute NSCs to

each SCMA block such that there is a minimum number of

SCs that separate any pair of SCs in an SCMA block. In order

to deﬁne this diversity allocation scheme, we will treat the

regular allocation as the virtual allocation of SCs to SCMA

blocks and deﬁne a mapping from the virtual allocation to the

physical allocation of SCs in the system. We thus deﬁne two

Z-dimensional vectors, va and pa, to represent the virtual

and physical allocation of SCs in the system, respectively.

The goal of this diversity scheme is to provide a mapping

from va →pa such that in the pa vector, every pair of SCs

in an SCMA block is separated by at least a certain number

of SCs. An example of this mapping was illustrated in Fig. 4

for a system with Z= 16 and N= 4.

We ﬁrst seek to concretely determine this minimum level

of SC separation that can be attained. Let µrepresent the

minimum number of SCs that separate any pair of SCs in an

SCMA block. The number of SCMA blocks is NB=Z/N.

If the ﬁrst SC in every block from va is mapped contiguously

to the ﬁrst available index in pa, the last block starts at index

NB. For the example in Fig. 4, indexes {1,5,9,13} represent

the ﬁrst index of each block in the virtual allocation, which

are placed contiguously at the start of the physical allocation.

Clearly, the ﬁrst index of the last block, i.e., index 13 from

va, gets placed at index NB= 4 in pa. The last available SC

index in the system is Z, so µmax for the remaining (N−1)

SCs in each SCMA block will be such that they are equally

spread apart and can be determined as

Z

N+ (N−1)µmax =Z. (8)

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Solving (8), we get µmax =Z/N =NB, which is the

maximum amount of SC spacing we can guarantee to any

pair of SCs in every SCMA block in the system. Algorithm

1 then describes the mapping from va →pa such that in the

pa vector, every pair of SCs in an SCMA block is separated

by at least µmax SCs.

The algorithm iterates through each of the SCs in va but

operates on an SCMA block by SCMA block basis. When

it detects the start of a new SCMA block in va, it takes the

ﬁrst SC in the block and assigns it to the smallest available

index in pa. Every subsequent SC in the block from va is

then placed µmax SCs apart. This is done for all the NB

blocks in the system. For example, if Z= 128 and K= 4,

then µmax = 32. The algorithm starts from va1which is

assigned to pa1. The SC in va2will then be placed 32 SCs

apart at pa33. Similarly, allocations va3→pa65 and va4→

pa97 are made. Now, va5represents the start of a new SCMA

block and is hence assigned to pa2, the smallest available

index since pa1is used. Again, the remaining SCs in this

block are placed 32 SCs apart starting from index 2and the

process repeats for the remaining blocks. The ﬁnal SCMA

block will have its ﬁrst SC placed at pa32 and ﬁnal SC at the

last available index at pa128.

Returning to the example in Fig. 4, it also depicts how

different users get a different level of SC separation depend-

ing on their user allocation, from the user-to-SC matrix in

(2). A user with allocation “1100” has a spacing of four

SCs between the coded dimensions while the user with

allocation “1001” gets a much larger separation of twelve

SCs. However, all users are assured a separation of at least

µmax between their coded dimensions.

Algorithm 1: Proposed diversity-based SC mapping

Input : va →a size Zvector representing the virtual

allocation where every Nconsecutive

entities represent an SCMA block

Output: pa →a size Zvector representing the

physical allocation of SCs, where at least

Z/N SCs separate the entities of an SCMA

block.

initialize pa ← ∞ (all elements);

for i←1to Zdo

if imod N= 1 then

for j←1to (Z/N)do

if pa(j)6=∞then

pa(j)←va(i);

break;

else

n←imod N

if n= 0 then

n←N

pa(j+n∗(Z/N)) ←va(i)

V. SIMULATION RESULTS

We describe the joint impact of the SCMA constellation

design and SC allocation on the PAPR statistics of high-rate

users with the help of the MATLAB simulations presented

in Fig. 5. All experiments were run for a large number of

OFDM symbols, in the order of 104. The total number of SCs

in the system, Z= 128, are divided into NB= 32 SCMA

blocks of N= 4 SCs each with L= 32. Also included in the

results in Fig. 5 is a simulation run for randomly generated

independent 4-QAM constellation points transmitted only on

the data carrying SCs assigned to the user under test from

the user-to-SC allocation matrix Sin (2). Note that this is

different from 4-LDS, because the SCs are being individually

modulated. We illustrate the regular scheme for the user with

allocation “0011” from matrix S, while the diversity scheme

for the users with allocations “1001” and “0011” for the

reasons we outline next.

Since each user is allocated a different set of dvSCs

per SCMA block to transmit on (deﬁned from matrix S),

the PAPR statistics of each user will not necessarily be the

same. With the regular scheme, in our illustration, an SCMA

block is comprised of N= 4 contiguous SCs. Hence, any

combination of dv= 2 SCs comprises SCs of similar centre

frequencies. Thus, any one of the possible user allocations,

e.g., “0011”, is representative of the PAPR performance for

all users. On the other hand, with the diversity scheme, differ-

ent user allocations experience different levels of frequency

separation. For instance, referring to the example in Fig. 4,

the user with allocation “1001” has the ﬁrst and fourth SC in

the virtual allocation which are separated the furthest, while

the user with “1001” has the third and fourth SC in the virtual

allocation which is separated the least. Thus, the user with

allocation “1001” represents the maximum SC separation

scenario while “0011” corresponds to the user having the

minimum SC separation.

We see from the results in Fig. 5 that with the regular

allocation scheme, 4-OPP outperforms 4-LDS. With 4-OPP,

we are placing dv= 2 constellation points of equal magni-

tude but opposite phase in two near-contiguous SCs. While

with LDS, we have the guaranteed placement of two points

with the same amplitude and same phase in near contiguous

SCs, i.e., SCs of similar centre frequencies. The sum of two

sinusoids of similar frequencies will line up for higher peaks

if they start at the same phase. However, with the diversity

scheme, we see the trend shifts. With the “0011” allocation

that provides the minimum frequency separation, we start

to see 4-LDS and 4-OPP behave similarly, while with the

maximum separation “1001” user allocation, the results are

the exact opposite of the regular scheme. Increasing the SC

separation between the dvSCs, means that we are adding

sinusoids of increasingly different frequencies to generate the

OFDM signal. As the frequency separation becomes large

enough, the simulations show the modulation scheme biased

to have both dimensions start at the same phase, i.e., 4-LDS,

generates better PAPR statistics.

The complete change in the order from Fig. 5a to Fig.

8VOLUME x, 2019

A. Rajasekaran et al.: Resource Allocation Based PAPR Analysis in Uplink SCMA-OFDM Systems

PAPR

(a) Regular SC allocation scheme, user allocation =

“0011”.

PAPR

(b) Diversity SC allocation scheme, user allocation =

“0011”.

PAPR

(c) Diversity SC allocation scheme, user allocation =

“1001”.

FIGURE 5: Comparing 4-LDS, 4-Bao and 4-OPP with differ-

ent SC allocation schemes. For illustration, the curve when

independent 4-QAM symbols (non-SCMA symbols) are trans-

mitted in the same data carrying SCs is included.

5c highlights why it is important to study the PAPR statis-

tics as the joint effect of the SCMA modulation scheme

and the corresponding SC allocation strategy. Hence, when

comparing the PAPR performance of different SCMA mod-

ulation schemes from the literature [3], it is not sufﬁcient

to conclude that one scheme outperforms the other. The

modulation schemes have to be analyzed in conjunction with

the associated SC allocation strategy to fully understand their

impact on the PAPR statistics. Since the modulation scheme

is a physical layer design parameter while the SC alloca-

tion comes from the layer-2 resource allocation strategy, the

PAPR problem for high-rate users in SCMA-OFDM systems

should be studied as a cross-layer systematization problem.

As we showed in Section III-B, this is in contrast to the low-

rate users with a small value of Lwhere it is sufﬁcient to

analyse the PAPR purely from the layer-1 perspective of the

SCMA multi-dimensional constellation design.

Further, as seen in Fig. 5, with each SC allocation, 4-Bao

performs similar to just placing random 4-QAM points in the

data carrying SCs. This is because it contains an equal mix of

same and opposite phase bias among its constellation points,

so the PAPR statistics reﬂect that it is no different from

independently modulating the data carrying SCs. However,

the SCMA codebook still plays an important role in the PAPR

statistics for this 4-Bao scheme, even though there is no phase

bias in the constellation. That is because only a subset of

the SCs are being modulated with data carrying complex

constellation points and the SC allocation strategy determines

which are the data carrying SCs and which are the null SCs.

As we can see, there is a nearly 3 dB performance difference

between 4-Bao with regular scheme and the diversity scheme

for “1001” allocation in Fig. 5a and Fig. 5c, respectively.

From the existing PAPR literature on OFDM systems, it

is known that swapping the location of data carrying and

reserved null SCs can lead to signiﬁcant PAPR reduction

[37], [38]. It is the same observation we make here, except

that the null SCs are determined by the SCMA codebook and

the associated SC placement strategy. This highlights the fact

that we can attain a better PAPR performance for any SCMA

constellation through the SC placement strategy.

It is clear from the results in Fig. 5 that we should adopt

a SC allocation strategy that shufﬂes around the SCs if we

have a modulation scheme with the phase bias of having

constellation points of the same phase, while we should use

a contiguous SC allocation strategy for a scheme that has a

phase bias of having constellation points of opposite phase.

For the 4-LDS and 4-Bao schemes, which are schemes from

the existing SCMA literature, we summarize the observations

for these constellations in Fig. 6. We see that for a desired

SCMA scheme, an appropriate SC allocation strategy based

on the phase bias in the chosen SCMA scheme can be

selected to reduce the PAPR or even vice-versa, i.e., for

a desired SC allocation strategy, an SCMA scheme with

favourable phase bias characteristics to reduce the PAPR

can be selected. These results are important because the

PAPR gains achieved between different conﬁgurations are

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A. Rajasekaran et al.: Resource Allocation Based PAPR Analysis in Uplink SCMA-OFDM Systems

a result of static conﬁguration parameters, i.e., they are

conﬁgured one-time on setup and come with no additional

computational overhead. This is made possible by the SCMA

codebook that introduces dependency between the transmit-

ted SCs in the system. These statistical dependencies can

be exploited to achieve PAPR reduction in a static manner,

not possible in traditional OFDM systems that individually

modulate the SCs. We discuss these opportunities for PAPR

reduction in detail next in Section VI.

PAPR

FIGURE 6: Comparing the PAPR statistics with different

choices of SCMA modulation schemes and SC allocation

strategies. These options are static conﬁguration parameters

that impact the PAPR statistics in SCMA-OFDM systems.

VI. EXPLOITING STATISTICAL DEPENDENCY IN PAPR

REDUCTION SCHEMES BASED ON MULTIPLE

SIGNALLING

In this section, we investigate how the novel aspects to the

analysis of PAPR statistics in SCMA-OFDM systems, dis-

cussed in Section III-B, impact the class of PAPR reduction

techniques based on multiple signalling and probabilistic

techniques [6]. The general idea with these PAPR reduction

techniques is to generate a set of candidate signals every

OFDM symbol and transmit the signal with the least PAPR.

These techniques are information lossless, since they do

not distort the transmitted signal. However, they come with

the complexity overhead of generating the set of candidate

signals every OFDM symbol as opposed to just one signal.

They also incur a throughput loss due to the need to transmit

sidelink information, not ideal for overloaded NOMA sys-

tems. When these PAPR reduction techniques are used in

traditional OFDM systems, since each SC is independently

modulated, there is no advance knowledge of any statistical

dependencies between the SCs to exploit. Hence, the set of

candidate signals can only be generated after the information

sequence in that OFDM symbol is known. However, with

SCMA-OFDM systems, the statistical dependency between

the transmitted SCs can be exploited for PAPR reduction in

conjunction with these well established techniques. We show

(a) SLM

(b) Interleaving (IL)

FIGURE 7: Block diagrams highlighting the PAPR reduction

techniques of SLM and IL that are described in the context of

SCMA-OFDM systems.

that for a given level of PAPR reduction, the overhead in-

curred by these multiple signalling techniques can be reduced

or even eliminated in some scenarios.

The PAPR reduction techniques described in [6] under the

class of multiple signalling and probabilistic techniques all

assume that the SCs in the system are independently mod-

ulated with QAM symbols. With SCMA-OFDM systems,

some of these techniques can be applied with some modiﬁca-

tions to satisfy the SCMA constraints while some techniques

cannot be easily extended to the SCMA-OFDM paradigm.

For example, techniques that involve constellation shaping

[29], [30] or tone injection cannot easily lend itself to SCMA

systems because it affects the SCMA constellation design.

SCMA constellations are designed with a number of criteria

[3] that will be affected by the constellation shaping and

is beyond the scope of the discussion here. Similarly, tech-

niques that involve using null SCs such as tone reservation

[39] or the dynamic swapping of data and null SCs [37], [38]

is difﬁcult to extend to the SCMA-OFDM paradigm. This

is because the null SCs are an integral part of every SCMA

block and cannot be rearranged randomly for PAPR reduction

purposes. However, other multiple signalling techniques such

as selective mapping (SLM) [27], partial transmit sequences

(PTS) [31] and interleaving (IL) [28] can be tailored to

meet the constraints of an SCMA-OFDM system. The block

diagrams for these three techniques are depicted in Fig.7 and

in the discussion that follows, we focus on how they can be

adapted to SCMA-based systems.

In the SLM technique used in traditional OFDM systems, a

set of candidate OFDM symbols are generated that represent

exactly the same information. The signal with the least PAPR

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is then transmitted. The set of candidate signals is generated

by multiplying the original data carried in the SCs for that

symbol with Rdifferent sets of phase factors

bm= [b0

mb1

m...bZ−1

m],0≤m≤R−1,

bn

m=ejθn

m,0≤n≤Z−1.(9)

After the inverse discrete Fourier transform (IDFT) block,

this multiplication generates Rsequences in time domain

and the one with the least PAPR is transmitted. Sidelink

information about the phase factor is sent to the receiver to

indicate which set of phase sequences were used, so that

the receiver can undo the multiplication and regenerate the

original data. The side link information is log2Rbits long,

since we only have to identify which sequence was used. The

set of possible sequences are known to both the transmitter

and receiver. Additionally, there is signiﬁcant complexity

introduced by the extra IDFT operations every symbol that

scales linearly with R[6].

SLM can be applied to SCMA-OFDM systems because

the typical SCMA constellation design process used in the

literature allows for random user-speciﬁc rotations to be

performed without affecting the error rate performance in

the uplink [3]. While it is a sub-optimal approach to SCMA

constellation design to ﬁnd the mother constellation and user-

speciﬁc rotations separately, it is by far the most widely used

approach in the literature [3]. Further, as shown in [3], in

the uplink, the user-speciﬁc rotations designed as part of the

constellation design process lose meaning due to the fact

that different users experience different fading channels. As

a result, the user-speciﬁc rotations in the UL SCMA systems

can be designed for PAPR reduction purposes instead. These

random user speciﬁc rotations translate to a random phase be-

ing multiplied to the dvdimensions of the SCMA codeword.

However, the dvdimensions of the SCMA codeword cannot

each be multiplied by their own phase factor, as doing so

would destruct the SCMA. Hence, this additional constraint

needs to be placed when generating the set of phase factors

that make up the phase sequences. In other words, the set

of candidate phase sequences should be generated such that

each set of dvSCs is assigned a phase factor, rather than

each SC being assigned its own phase factor. The set of

phase sequences generated for a Z-SC OFDM system in

(9) needs to then be modiﬁed to only generate a list of

Lphase factors. These phase factors are multiplied by the

original data sequence to generate a set of Rdifferent OFDM

symbols, ym,∀m= 0, .., R −1, and the signal with the least

PAPR is transmitted as follows:

yˆm= arg min

m

PAPR(ym),0≤m≤R−1.(10)

In Fig. 8, we run MATLAB simulations for the 4-Bao

scheme with both the regular and diversity-based SC allo-

cation schemes. We run with R= 2 and R= 4, which corre-

sponds to one and two bits of additional sidelink information

respectively. With M= 4, that corresponds to one SCMA

PAPR

FIGURE 8: Comparing the PAPR reduction achieved with

SLM in an SCMA-OFDM system with 4-Bao constellation and

different SC allocation strategies and different values of Rfor

the SLM reduction.

block of transmission reserved for sidelink information. This

means there is a throughput loss from Lto L−1modulation

symbols per OFDM symbol duration. Additionally, there is

a computational complexity overhead that is higher when

R= 4 compared to when R= 2. We can see that for this

4-Bao SCMA constellation, the PAPR reduction achieved

with R= 2 for the diversity scheme is the same as that

achieved with no PAPR reduction using the regular alloca-

tion. Similarly, the PAPR reduction achieved with R= 4

in the diversity scheme is achieved with R= 2 using the

regular scheme. The key takeaway message here is that the

statistical dependency introduced by the SCMA codebook

between certain SCs transmitted in an OFDM symbol can

be exploited to achieve PAPR reduction gains through the

setting of static conﬁguration parameters such as the SCMA

constellation scheme, SC allocation strategy, SCMA block

dimensions like Nand dvetc. Such gains are not possible in

traditional OFDM systems where the SCs are independently

modulated and so there is no advance knowledge of the

statistics to exploit.

Another multiple signalling technique called partial trans-

mit sequences (PTS) follows a similar idea to SLM. In PTS,

the ZSCs are divided into disjoint sub-blocks and the IDFT

of each block is taken. Different phase factors are multiplied

to these IDFT outputs, i.e., to the time-domain data, and once

again the OFDM signal with the least PAPR is transmitted.

When applied to SCMA-OFDM systems, as long as every

dvSCs that make up an SCMA block are contained in

the same sub-block, none of the SCMA related constraints

are violated. Hence, a logical split for these disjoint sub-

blocks would be along the SCMA blocks. The computational

overhead involved to generate the candidate signals scales

with the number of sub-blocks and is larger than that incurred

with SLM [6]. Like with SLM, there is also the sidelink

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A. Rajasekaran et al.: Resource Allocation Based PAPR Analysis in Uplink SCMA-OFDM Systems

information required which results in a throughput penalty.

Simulation results for PTS are not shown here as they are

very similar to the observations from SLM, where the static

conﬁguration gains from the choice of SC allocation strategy

compares with the PAPR reduction gains from PTS when R

is small.

Interleaving is another probabilistic technique for PAPR

reduction commonly used in OFDM systems. In interleaving,

the idea is once again to create a set of data block candi-

dates and select the block with the least PAPR to transmit.

Compared to SLM, in this method, an interleaver block is

used instead of phase sequences. Interleaver is a device which

reorders the entries of a block of length Zin a speciﬁc order.

Similar to SLM and PTS, there is overhead in computing

the IFFT of the different interleaved sequences and also the

receiver needs sidelink information to de-interleave the re-

ceived data block. When applied to SCMA-OFDM systems,

we can only interleave the Lmodulation symbols with each

other. In other words, there are Lmodulation symbols to be

transmitted in an OFDM symbol, and they can be transmitted

on any of the LSCMA blocks. However, the dvSCs within

a block that contains the codeword for a modulation symbol

must remain within the same SCMA block and the interleaver

cannot reorder these SCs. With these constraints, even after

applying an interleaver block, the PAPR statistics are still

subject to the joint effect of the modulation scheme phase

bias and SC placement that was described in Section III-B.

This is illustrated by the simulation results in Fig. 9 where

the gains attained by interleaving in some conﬁgurations are

small and do not compare with the PAPR reduction gains

from interleaving in traditional OFDM literature [28].

PAPR

FIGURE 9: Comparing the PAPR reduction achieved with

interleaving in an SCMA-OFDM system with 4-Bao constel-

lation and different SC allocation strategies.

VII. CONCLUSION AND FUTURE WORK

In this paper, we showed that optimizing the SCMA code-

book design to just have a low constellation PAPR, for ex-

ample through low-projection codebooks, is only applicable

for low-rate users. For high-rate UL SCMA-OFDM users,

the PAPR statistics should be considered. Unlike traditional

OFDM systems that independently modulate the SCs, the

PAPR statistics of SCMA-OFDM systems are inﬂuenced by

the joint impact of the phase bias in the multi-dimensional

modulation scheme and the placement of the SCs that carry

the SCMA codewords. Through simulations, we showed the

joint impact of these two factors on the PAPR statistics.

The PAPR performance difference observed was a result of

static conﬁguration parameters, allowing for PAPR reduction

opportunities without incurring any computational overhead.

PAPR reduction techniques based on multiple signalling

were investigated in the context of SCMA-OFDM systems.

We showed that techniques such as SLM, PTS and IL can

be adapted to SCMA-OFDM systems with certain modiﬁ-

cations. However, the complexity and sidelink information

overhead can be reduced by ﬁrst tuning the static conﬁgura-

tion parameters to be favourable to a low PAPR.

In future work, the statistical dependencies between the

transmitted SCs in an SCMA-OFDM system can be further

characterized. For example, a metric to capture the level of

phase bias in an SCMA-OFDM constellation can be derived.

In this way, the SCMA constellation design process can aim

to maximize this metric, in order to be exploited later with

the appropriate SC allocation scheme for PAPR reduction.

Further, the impact of the SCMA conﬁguration parameters

like dvand Ncan be studied. We would expect the level

of statistical dependency to grow as dvincreases. PAPR

reduction techniques like SLM and PTS can also be enhanced

to exploit these statistical dependencies. For small values of

L, there are a limited number of possible sequences and so

all possible combinations can be tried beforehand to ﬁnd the

favourable sequences. An interleaver algorithm can then be

developed to quickly match a favourable sequence from a

PAPR perspective with minimum computational overhead.

The ﬁndings in this paper can be used in many interesting

cross-layer systematization problems to include PAPR con-

siderations. For example, in [3], [40], [41], frequency diver-

sity gains in terms of error rate performance were demon-

strated. This could be coupled with selecting a scheme like

LDS that has the phase bias for accompanying PAPR gains

for high-rate users. Further, 5G and beyond communication

systems are intended to support a wide variety of use-cases

with users of very different requirements. This includes the

level of PAPR tolerance among the users in the system. For

instance, we saw that with the diversity scheme, some users

are offered more SC separation than others, which impacts

the PAPR statistics. Users with the least PAPR tolerance can

be assigned the most favourable SC separation.

.

APPENDIX A ZERO DB PAPR CONSTELLATION DESIGN

A zero dB constellation PAPR means zero dB PAPR for the

transmission of one modulation symbol, i.e., L= 1. Such

a constellation scheme requires each modulation symbol to

be coded on only one of the dvSCs. The Ndimensional

12 VOLUME x, 2019

A. Rajasekaran et al.: Resource Allocation Based PAPR Analysis in Uplink SCMA-OFDM Systems

(a) 4-0PAPR constellation.

(b) 16-0PAPR constellation.

FIGURE 10: Proposed M-0PAPR scheme that has zero dB constellation PAPR.

codewords already allocate N−dvnull dimensions, but the

zero dB PAPR constraint means that out of the remaining

dvdimensions, only one can be non-zero. Further, the mag-

nitude of each of these symbol points should also be con-

stant. For an M-point constellation, this translates to having

Nd=M/dvpoints per dimension evenly spaced around

a circle. To normalize the energy of the constellation, this

needs to be the unit circle. Therefore, we have an Nd-PSK

constellation shape in the dvdimensions. The Msymbols

are then assigned such that each symbol gets exactly one

non-zero constellation point in any one dimension. However,

which symbol is assigned to which constellation point is not

important from a PAPR perspective. Algorithm 2 illustrates

the construction of this constellation, that we call M-0PAPR

because of the zero dB constellation PAPR property. The 4-

point and 16-point version, namely 4-0PAPR and 16-0PAPR,

are shown in Fig. 10a and 10b, respectively.

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Algorithm 2: M

M

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