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Resource Allocation Based PAPR Analysis in Uplink SCMA-OFDM systems

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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 (SC’s). 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 SC’s 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 SC’s. Further, the SCMA codebook dictates that an UL user can only transmit on a subset of the available SC’s. 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.
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Date of publication xxxx 00, 0000, date of current version xxxx 00, 0000.
Digital Object Identifier 10.1109/ACCESS.2019.DOI
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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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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S/P converter
N-pt IDFT
P/S converter
U1U2U3U4U5U6
SC1
SC2
SC3
SC4
Z=20, N=4, NB=5
(L codewords)
Y0
Y1
YZ1
D/A
Converter
y[0]
y[1]
y[Z1]
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
Z1
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) =
Z1
X
k=0
akexp(j2π(fc+kf)t)
= exp(j2πfct)
Z1
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] =
Z1
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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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 (N1)
SCs in each SCMA block will be such that they are equally
spread apart and can be determined as
Z
N+ (N1)µmax =Z. (8)
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A. Rajasekaran et al.: Resource Allocation Based PAPR Analysis in Uplink SCMA-OFDM Systems
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 va3pa65 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 i1to Zdo
if imod N= 1 then
for j1to (Z/N)do
if pa(j)6=then
pa(j)va(i);
break;
else
nimod N
if n= 0 then
nN
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.
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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...bZ1
m],0mR1,
bn
m=en
m,0nZ1.(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),0mR1.(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 L1modulation
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 Ndvnull 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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