![FIR Finite impulse response
GML Global maximum likelihood detector
IoT Internet-of-Things
IoV Internet of Vehicles
IrLDS Irregular low-density spreading
ITS Intelligent transportation system
JSG Joint sparse graph
LDPC Low-density parity-check
LDS Low-density spreading
LDSM Low-density superposition modulation
LDSMA Low-density spreading multiple access
LLR Log-likelihood ratio
LRS Large random spreading
LTE Long-term evolution
M2M Machine-to-machine
MAI Multiple-access interference
MAP Maximum a posteriori detector
MC Multicarrier
MF Matched filter
MIMO Multiple-input multiple-output
ML Maximum-likelihood
MLCM Multilevel coded modulation
MMSE Minimum mean-square estimation
mMTC Massive machine-type-communications
mmWave Millimeter-wave
MPA Message passing algorithm
MS Mobile station
MUD Multiuser detection
MUSA Multiuser shared access
MWBE Maximum-Welch-bound-equality
NOMA Non-orthogonal multiple access
OFDM Orthogonal frequency-division multiplexing
OMA Orthogonal multiple access
PDA Probabilistic data association
PDMA Pattern division multiple access
PEG Progressive edge-growth
PIC Parallel interference cancellation
QAM Quadrature amplitude modulation
QLSS Quasi-large sparse sequence
QPP Quadratic permutation polynomial
QPSK Quadrature phase-shift keying
RE Resource element
SCDMA Sparse code-division multiple-access
SCMA Sparse code multiple access
SD Sphere-decoding algorithm
SEP Symbol error probability
SER Symbol error rate
SFBC Space-frequency block codes
SIC Successive interference cancellation
SINR Signal-to-interference-noise ratio
SISO Soft-input soft-output
SM Spatial modulation
SNR Signal-to-noise ratio
SVE Spreading vector extension
TCM Trellis-coded modulation
TDMA Time-division multiple access
TSC Total squared correlation
TTCM Turbo trellis-coded modulation
UD Uniquely decodable
uRLLC Ultra-reliable low-latency communications
VA Viterbi algorithm
WBE Welch-bound-equality
ZF Zero-forcing filter
I. INTRODUCTION
HIgh spectral- and power-efficiency, massive connec-
tivity and low latency are among the requirements
for next generation communications and these require-
ments are expected to increase in the future, as researchers
turn their efforts towards sixth generation (6G) wire-
less communications. Enhanced mobile broadband (EMB),
ultra-reliable low-latency communications (uRLLC) and
massive machine-type communication (mMTC) support a
suite of compelling applications driving these require-
ments. Massive multiple-input multiple-output (MIMO),
non-orthogonal multiple access (NOMA) and millimeter-
wave (mmWave) communications constitute promising tech-
niques of addressing these stringent requirements [1].
In the previous generations spanning from 1G to 4G,
the multiple access schemes were exclusively character-
ized by orthogonal multiple access (OMA) techniques,
where users are assigned unique, user-specific resources
in either frequency- (frequency-division multiple access
(FDMA)), time- (time-division multiple access (TDMA))
or code-domain (code-division multiple access (CDMA)).
However, the multiple access scheme of 5G is required
to support a wide range of use cases, including a mas-
sive number of low-power Internet-of-Things (IoT) de-
vices, device-to-device (D2D) communications, device-to-
everything (D2E), the Internet of Vehicles (IoV), as well
as seamless machine-to-machine (M2M) communications
[2]–[6]. The mMTC mode includes, for example, e-health
services, smart cities/villages, e-farms, and intelligent trans-
portation systems (ITS) [7], [8]. They require improved
connectivity compared to previous generations of wireless
communications.
Supporting a large number of users communicating over
a common channel may not be readily achievable by OMA
techniques due to presence of multiple-access interference
(MAI) in rank-deficient systems, where the number of users
is higher than that of the resource blocks. To meet the
demand of increased bandwidth efficiency in synchronous
CDMA, a dense spreading NOMA CDMA concept was
introduced in [9], which can support many more users for
a given code length compared to traditional CDMA. A
number of signature designs have been conceived [10]–
[12], where low cross-correlation sequence sets are designed
to minimize the overall MAI, which allows more users to
simultaneously access the common channel. This in turn
results in increased spectral efficiency.
Using low cross-correlation sequence sets might not be
the best design policy for highly rank-deficient systems. One
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![of the important design criteria in such rank-deficient sys-
tems is for the code set to be uniquely decodable (UD) [9].
By definition, the UD codes can be unambiguously decoded
in a noiseless channel using linear recursive decoders [13].
Low-complexity linear decoders were introduced for these
UD code sets using either binary {0, 1}, or antipodal {±1},
or alternatively ternary {0, ±1} chips in [14]–[16]. Although
these code set designs attain a substantial increase in system
capacity even with the aid of low-complexity detectors, they
only perform well for synchronous transmission over non-
dispersive fading channels, such as additive white Gaussian
noise (AWGN) channels. To satisfy the UD criterion, all
the users have to rely on accurate transmit power control
so that their signals are received with equal power. In prac-
tice, the wireless transmission channel exhibits, numerous
impairments, such as frequency-selective multipath fading,
and unequal received power. Another limitation of linear
decoders is that they do not produce soft output decisions
required by the channel decoders.
To combat the MAI at a reasonable cost, many researchers
have proposed the construction of sparsely structured se-
quences for multiple access so as to take advantage of
efficient sparse signal processing, relying for example on
the message passing algorithm (MPA) for reducing to
reducing the complexity of multiuser detection (MUD).
These challenges can be addressed by the introduction of
sparse spreading based NOMA techniques, which can be
categorized into power-domain NOMA (PDM-NOMA) [1],
[17]–[20] and code-domain NOMA (CDM-NOMA) [21].
A few of the strong contenders of CDM-NOMA are low-
density spreading aided CDMA (LDS-CDMA) [22], low-
density spreading assisted orthogonal frequency-division
multiplexing (LDS-OFDM) [23], sparse code multiple ac-
cess (SCMA) [24], [25], irregular LDS (IrLDS), pattern
division multiple access (PDMA) [26] and multi-user shared
access (MUSA) [27]. The LDS can be considered a special
case of SCMA, which may also be characterized by sparse
codebooks, each of which can be expressed as the Kro-
necker product of a sparse sequence denoted by sj, and a
constellation set of order M. Specifically, we have:
Xj = [sjβ1, sjβ2, . . . , sjβM ], (1)
where {β1, β2, . . . , βM } indicates a constellation set. Hence,
the rank of the users’ LDS codebooks, Xj, is equal to
one. However, this is not the case for the users’ SCMA
codebooks. The rank of SCMA codebooks is higher than
one and it is equal to the number of non-zero values
in the SCMA waveforms. The comparison between direct
sequence CDMA (DS-CDMA), multicarrier CDMA (MC-
CDMA), LDS-OFDM and SCMA is illustrated in Fig. 1.
Readers are referred to surveys of SCMA [28] and signature-
based NOMA [29] for further reading.
CDM-NOMA offers flexible resource element (RE) al-
location where the sparsity may be flexibly configured for
handling time-variant user-loads. It performs well in terms
of handling the MAI imposed by rank-deficient systems
FIGURE 1. Multiple access technique comparisons.
and has low-complexity receivers compared to conventional
dense spreading based CDMA. LDS, may also be appropri-
ate for IoT communications [21] and it is also considered
as a potential candidate for the uplink of mMTC [21].
There have been various criteria for the optimization of
sparse spreading based NOMA [30]–[43], which maps the
signals of users to REs in a sparse manner, whilst relying
on the constellation shaping of non-zero entries [31], [32],
and accurate power allocation for each spreading sequence
[44]. The RE mapping methods can be broadly divided into
two types; a) regular RE mapping, where the spreading
densities of all users are the same, as in LDS-OFDM and b)
irregular RE mapping, where the densities are non-identical,
as in IrLDS and PDMA. Constellation shaping can be
categorized into a) widely studied constellations {0, 1} [31],
binary phase-shift keying (BPSK) [45], quadrature phase-
shift keying (QPSK) [39], quadrature amplitude modulation
(QAM) [39], etc., b) two-dimensional constellation bounded
in unit circle [32], c) or any other constellations. The power
allocation of each spreading sequence can be divided into
two classes a) equal power, b) unequal power among all
users.
A. RELATED LITERATURE
Spreading sequences of the low-density type containing
many zeros were first introduced in [30] supporting low-
complexity MUD. The introduction of cyclically shifted
LDS design [30] allows maximum-likelihood (ML) detec-
tion to be carried out by computationally efficient methods,
such as the Viterbi algorithm (VA) when BPSK modulation
is used. It is widely recognized that finding the ML solution
is generally NP-hard [46]. Various sub-optimal solutions can
be applied such as sphere-decoding (SD) [47], probabilistic
data association (PDA) [48], decision-feedback methods
[49] etc. The problem, however, becomes more difficult
if the system is rank-deficient. The complexity of the
decoding process is crucial with the advent of iterative turbo
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![detection, the so-called turbo MUD algorithm approximates
the complex optimum joint detection scheme by iteratively
exchanging soft decision variables between the multiuser de-
tector and single-user soft-input soft-output (SISO) channel
decoders. Based on this idea Hoshyar et al. [31] showed that
iterative decoding is necessary for fully exploiting the LDS
structure. To further exploit the lower complexity of iterative
detection, sparse spreading sequences were conceived [22],
[31], [32]. The family of low-density parity-check (LDPC)
codes has been shown to be attractive due to its capacity-
approaching capability and decoding simplicity, when using
the MPA. This is why, Hoshyar et al. [31] proposed an LDS
structure based on LDPC codes, where the user’s symbol
are arranged in such a way that the interference seen by
each user at each chip is different. Explicitly, the specific
choice of the non-zero entries is in perfect harmony with the
particular choice of the LDPC indicator matrix that defines
the structure of the LDS code matrix. As a further advance,
a near-optimum chip-level SISO iterative MUD is developed
in [22] for the LDS structure for transmission over AWGN
channels. It was shown to yield promising performance
for rank-deficient systems, especially, for BPSK modulation
[22], where the emphasis was on the MUD structure, rather
than on design of spreading sequences having particular
structure, which were found by simple trial and error under
a unit amplitude constraint. In contrast to [22] a structured
approach focusing on the design of spreading sequences
was proposed by Van de Beek and Popović [32] based on
the LDPC indicator matrix. In general, signatures having a
unity scalar magnitude are designed by maximizing their
minimum distance. Moreover, Van de Beek and Popović
advocated the so-called Latin-rectangular mappings, where
not only the non-zero elements of each row are distinct,
but also those in each column, because they are capable of
significantly outperforming a randomly generated signature
matrix, as a benefit of their high minimum distance.
It is widely recognized that the global search based maxi-
mum likelihood (GML) detector approaches the single-user
bit-error-rate (BER), at high signal-to-noise ratios (SNR),
when using long random spreading (LRS) sequence based
CDMA [50]. Inspired by the LRS-CDMA concept, Sun and
Xiao [45], [50] proposed the so-called quasi-large sparse
sequence (QLSS) - CDMA concept by replacing the dense
sequences of QLRS-CDMA by sparse sequences.
The specific constructions of LDS signatures found in
[31], [32] have been inspired by classic LDPC code designs
in order to facilitate the employment of the MPA algorithm.
Safavi et al. [33] considered schemes, where the spread-
ing and mapping to conventional QAM constellations are
performed separately. Their proposed recursive matrix con-
struction has been optimized for maximizing the Euclidean
distance.
Apart from the fact that the multiple access sequences
play a key role in NOMA for supporting low-complexity
detection, they determine the achievable sum rate. Qi et
al. [34] analyze the sparsity of the sum capacity-achieving
sequences and propose a beneficial construction method
with the aid of classic frame theory [51]. The particular low-
density spreading sequence design that maximizes the sum
rate based on frame theory for complex zero-mean Gaussian
random variables is presented in [34], where each row
has almost the same number of non-zero entries, forming
a nearly regular sparse spreading sequences. In contrast
to this design, Yu et al. [40] proposed the simultaneous
optimization of the RE assignment and power allocation
among REs, where the users employing the same radio
resource have different channel gains. It was achieved by
first formulating a sum-rate optimization problem subject to
practical sparsity and power constraints.
In 2017, Qi et al. [37] formulated an optimization prob-
lem for specifically designing the sparsity of spreading se-
quences, while maximizing the efficiency of NOMA subject
to the maximum tolerable symbol error rate (SER) as well
as to the affordable detection complexity. Another challenge
is the construction of the sparse matrix that optimizes the
performance of the MPA detector, since there are no closed-
form expressions for characterizing the detection perfor-
mance of MPA for sparse sequences. Despite this short-
coming, Qi et al. [35] proposed a systematic technique for
constructing the sparse sequences relying on a hierarchical
method with the objective of optimizing the performance of
MPA for BPSK modulation. For the given SNR and target
factor graph girth, the algorithm produces the optimum
sparsity. Based on the optimum sparsity and the minimum
girth, the algorithm directly produces the position of non-
zero entries in the matrix. Lastly, the particular values of
non-zero entries are determined by specifically maximizing
the minimum distance. Wang et al. [42] took a step further
by combining multicarrier (MC) LDS and channel coding
schemes into a joint sparse factor graph and quantified the
average BER based on the mean and variance of the soft
information distribution obtained. Explicitly, through their
theoretical analysis, the average BER has been derived based
on the mean and variance of the soft information distribution
at the output of the joint sparse factor graph. The proposed
design produces the optimal degree distribution of LDS
spreading capable of approaching the theoretical capacity
in terms of SNR.
The optimization of sparse matrices is typically carried
out by assuming to have Gaussian input signal, which is
suboptimal, for practical discrete constellations. Xiao et
al. [52] proposed a codebook design for multicarrier-low-
density spreading aided multiple access (MC-LDSCMA)
based on the maximization of the minimal user rate for
practical finite alphabet signalling.
Another LDS signature spreading vector extension (LDS-
SVE) method is introduced by Zhang et al. [38] for up-
link OFDM systems. Compared to LDS-OFDM, LDS-SVE
jointly transforms and spreads a pair of modulated symbols
across four subcarriers. This is achieved upon multiplying
the real and imaginary parts of two modulated symbols by
a transformation matrix, which is optimized by minimizing
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![the single-user BER.
LDS designs that are based on the sparseness of the LDPC
parity check matrix [31], [32] are typically considered as
having a regular parity check matrix. By contrast, Jiang
and Wu et al. [36] proposed a low-density superposition
modulation (LDSM) scheme that is based on an irregular
parity check matrix, which provides both diversity and
coding gains, hence improving both the overall average
performance as well as the cell-edge performance. The
progressive edge-growth (PEG) algorithm is utilized to con-
struct the LDSM matrix. A compelling systematic technique
of designing the degree distribution of the LDSM signature
matrices is proposed by Lu and Jiang in [43], which is based
on the powerful extrinsic information transfer (EXIT) chart
tool and the so-called bare-bone particle swarm optimization
(BBPSO) algorithm they optimize the degree distribution
of LDSM signature matrices. Their EXIT chart analysis
in rightfully characterizes the resultant design. Similarly,
Zhang et al. in [41] proposed a pair of sparse superposition
matrices.
Similar to the minimum distance criterion based LDS
code design of [35] developed for BPSK modulation, Song
et al. address the maximization of the minimum Euclidean
distance for QAM constellations in [39]. More explicitly,
signature matrices having factor graphs exhibiting very
few short cycles and large superposed signal constellation
distances are designed by Song et al. In short, for a given
factor graph structure the algorithm produces the optimal
signature matrix associated with the maximum LDS code
distance. The LDS code set of Song et al., which are
detected both by the MPA and the ML detector, exhibit an
excellent performance.
By expanding the traditional direct sequence CDMA to
NOMA, Liu et al. [53] developed a cyclic shift based mul-
tiple access scheme, where the in-phase and quadruature-
phase channels are used for transmitting the data and pilots,
respectively. In contrast to conventional SCMA, which is
based on geometric shaping design, Jiang and Wang [54]
combine both geometric and probability-based for increas-
ing the channel capacity and reducing the BER. As a
benefit of using a sparse spreading matrix, low-complexity
iterative MUD can be employed. Song et al. [55] propose
super-sparse on-off division multiple access using spreading
waveforms based on idling. On the other hand, Ye et al.
[56] resort to using a deep multi-task learning technique
for optimizing an end-to-end NOMA system. As a further
development, Xie et al. [57] design constellations for non-
coherent reception of the signals arriving from multiple
users, and reduce the SER simultaneously. Combinatorial
structures relying on the so-called balanced incomplete
block design have also been widely studies in the context
of LDPC constructions. The designs of Lan et al. [58] used
as sparse codes have better interference properties, hence
they provide higher user/bandwidth efficiency and have the
flexibility of creating variable code rates. Motivated by this
fact, Wu et al. [59] proposed LDS designs based on Steiner
codes [60], whose incidence matrix conviniently supports
superposition based multiuser communications. By using
algebraic code construction methods, Liu et al. in [44]
proposed power-imbalanced LDS designs of the non-zero
entries for a given factor graph with the aid of Eisenstein
integersi
.
According to the above construction designs and studies,
the sparsity of spreading sequences, significantly influences
the performance of MUD due to its crucial impact on
the MAI characteristics. Since the performance analysis of
finite-size multiuser systems is mathematically intractable,
the large-system limit based analysis was provided in [61]–
[63]. By deriving a probability density model for the non-
zero entries of sparse spreading sequences, a method based
on statistical mechanics was proposed to analyze the optimal
detection performance in [63] and the spectral efficiency of
the scheme in [61].
The theoretical analysis of LDS systems in the presence
of flat fading channel in terms of their spectral efficiency
relying on both linear and non-linear optimum receivers
(such as maximum a posteriori detector) was carried out
in the large-system limit in [64]. Furthermore, the channel
capacities of SCMA and low-density spreading multiple ac-
cess (LDSMA) schemes are analyzed in [65] and compared
to that of the Gaussian multiple access channel imposing
random phase rotations and fast fading. The performance
advantage of LDSMA, which exploits the high degree of
flexibility of subcarrier allocation, has been demonstrated
in [66]. The results showed that the diversity gain attained
improves the link-level performance in terms of the achiev-
able block error rate (BLER).
The capacity region of uncoded LDS schemes communi-
cating over a multiple access channel is analysed in [77].
However, low-complexity of MPA decoding of LDS as a
multiple access technique has a lower capacity than suc-
cessive decoding [78]. For the coded LDS multiple access
channel the mutual information transfer characteristics of
turbo MUD applied to LDS-OFDM is studied using EXIT
charts in [79].
The rigorous information-theoretic analysis of infinite
graphs showed that having a regular user-to-RE allocation is
advantageous [80]. However, increasing the pattern matrix
dimensionality results in a significantly increased detection
complexity. Moreover, the rigorous closed-form analytical
expression of the spectral efficiency of regular sparse se-
quence based NOMA relying on optimum decoding in
terms of spectral efficiency is derived in [81], for Gaussian
signaling over non-fading channels in the asymptotic large-
system limit.
The LDS concept was applied in various attractive com-
munication systems. As an example, Hoshyar et al. [67] and
Al-Imari et al. [82] used LDS structures for spreading the
iEisenstein integers, also known as Eulerian integers, are complex num-
bers of the form z = a + bω, where a, b ∈ N and ω = −1+i
√
3
2
= e
2πi
3
constitute a primitive cube root of unity.
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![2003
2021
Choi [30] proposes a cyclically shifted LDS for multicarrier systems has been proposed to exploit the
trade-off between the receiver complexity and performance improvement.
2004
Hoshyar et al. [31] conceive LDS structure based on LDPC codes.
2006
Sun [45] introduces the quasi-large sparse sequence - CDMA based on randomly generated sparse vectors.
2008
Van de Beek and Popović [32] propose LDS structure based on LDPC indicator-matrix tailored to the
belief-propagation detector.
2009
Safavi et al. [33] propose new concept for LDS design based on ultra low-density spread signatures.
2016
Qi et al. [34] introduce LDS sequence design that maximizes sum rate of the system and sequences
sparsity based on frame theory.
2017
Qi et al. [37] extend design of the sparsity of LDS that maximizes the efficiency of NOMA system.
2017
Qi et al. [35] conceive a systematic scheme to construct the sparse sequences in a hierarchical way with
the aim of optimizing the performance of MPA for BPSK modulation.
2017
Xiao et al. [52] propose novel design method based on the maximization of the minimal user rate with the
finite alphabet inputs based on minimizing single user mutual information.
2017
Zhang et al. [38] introduce LDS signature vector extension jointly transforms and spreads two modulated
symbols onto twice the subcarriers.
2017
Jiang and Wu [36] propose a novel low-density superposition modulation design with the sparser and
irregular check matrix..
2017
Song et al. [39] introduce an optimal signature matrix with the systems of a two-dimensional quadrature
amplitude modulation.
2017
Zhang et al. [41] present a design of two sparse superposition matrices for 150% and 200% overloaded
LDSM scheme.
2018
Wu et al. [59] propose a NOMA design based on STS.
2018
Yu et al. [40] conceive an optimal sparse RE mapping patterns via sum-rate optimization problem subject
to sparsity and power constraints.
2018
Wang et al. [42] propose to optimize the degree distribution of the joint sparse factor graph by leveraging
the differential evolution method.
2019
Liu et al. [44] conceive new density design for LDS based on Eisenstein integers.
2019
Lu and Jiang [43] introduce the optimization problem for degree distribution of LDSM signature matrix.
2019
Liu et al. [53] propose an identical code cyclic shift code for downlink DS-CDMA to enable multiple
access using only one spreading code.
2020
Jiang and Wang [54] conceive waveform design based on geometric shaping and probabilistic shaping.
2020 Ye et al. [56] present a constellation shape using deep learning techniques.
2020
Song et al. [55] propose very low-complexity on-off division multiple access scheme for NOMA systems.
2020
Xie et al. [57] introduce a joint multi-user isometric constellation design is proposed to find constellations
that enable non-coherent reception and reduce SER.
2020
FIGURE 2. Timeline of LDS design contribution.
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![2009
2021
Hoshyar et al. [67] propose LDS-OFDM is introduced as an uplink multicarrier multiple access scheme.
2010 Li and Hanly [68] introduce a novel MC-CDMA system, where random sparse signatures are deployed in
the frequency domain.
2014
Suraweera et al. [69] conceive a distributed beamforming for sparsely-spread MC-CDMA using sum-
product algorithm.
2017
Fontana da Silva et al. [70] present an Alamouti SFBC scheme for a simple MIMO LDS-OFDM system.
2017
Liu et al. [71] propose a SM-SCDMA scheme is proposed to support a high normalized user-load in
uplink communications.
2018
Wen et al. [72] introduce joint sparse graph for FBMC is proposed to combine single graphs of LDS,
LDWM, and LDPC codes.
2018
Osamura et al. [73] propose to mitigate multi-user interference, the codeword of each user is randomly
punctured and the punctured bits are replaced by idle slots.
2018
Denno et al. [74] introduce a low density signature based multiple access with phase only adaptive
precoding for increasing network throughput.
2019 Zhao et al. [75] present a joint design of the energy interleaver and the constellation rotation-based
modulator in the symbol-block level by constructively superimposing the symbols.
2019
Özyurt and Kucur [76] propose a low-complexity multiple access method based on coordinate interleaving.
2020
FIGURE 3. Timeline of LDS applied in applications.
TABLE 1. Contrasting our novel contributions to the state-of-the-art.
Contributions This work [56] [53] [40] [39] [35] [34] [33] [32]
Minimum Euclidean distance X X X
Gaussian margin X
Sum Capacity X X
Maximum SINR per user X
Adaptive to number of users X X X X X X X X X
Joint RE and constellation shaping X X X X X X
BER approach single user at K/L = 2 X
symbols across the frequency domain, hence their technique
was termed as LDS-OFDM. Li and Hanly [68] and Li
et al. [83] introduced MC-CDMA for downlink commu-
nication, where sparse random signatures are deployed in
the frequency domain. A power-efficient non-linear transmit
precoder weight optimization problem is formulated, while
satisfying the maximum tolerable symbol error probability
(SEP) targets at the mobile stations (MSs). Suraweera et al.
[69] approached this problem by conceiving a distributed
linear beamforming technique for the multicell MC-CDMA
downlink by using the sum-product algorithm for detecting
the sparse signatures. LDS spreading has also been proposed
for MIMO systems by da Silva et al. [70]. Their proposed
technique relies on Alamouti’s space-frequency block codes
(SFBC) conceived for low-complexity MIMO LDS-OFDM
systems. As a parallel development in MIMO systems, spa-
tial modulation (SM) has drawn a lot of research attention
in recent years. Liu et al. [71] have proposed a sparse code-
division multiple-access (SCDMA) scheme for supporting
a high normalized user-load in uplink communications.
Recently, filter-bank multicarrier (FBMC) transceivers have
drawn a lot of attention as a benefit of circumventing
several OFDM drawbacks. Wen et al. [72] designed a LDS-
FBMC scheme, which applies LDS for constructing FBMC
signals. Additionally, a joint sparse graph (JSG) based
FBMC transceiver termed as JSG-FBMC was proposed for
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![combining the single graphs of LDS, a low-density weight
matrix, and LDPC codes, which represent popular NOMA,
multicarrier modulation and channel coding techniques,
respectively. Osamura et al. [73] proposed a new multi-
user scheme for mitigating the multi-user interference, in
which the codeword of each user is randomly punctured and
the punctured bits represent idle slots, hence, only a small
random set of users are active at each time. This constraint
imposed on the number of concurrent users significantly
reduces the multi-user detection complexity. As a further
advance, Denno et al. [74] proposed ‘phase-only’ based
transmit precoding in support of multiple user terminals
having a single antenna.
As a further advance, Zhao et al. [75] designed an energy
interleaver and constellation rotation-based modulator by
exploiting the NOMA concept for improving the energy
transfer efficiency of wirelessly powered systems. Similarly,
Özyurt and Kucur [76] designed a low-complexity multiple
access method for single-antenna nodes by exploiting the
concept of signal space diversity by relying on the power-
domain NOMA philosophy for reducing both the BER and
the number of SIC iterations.
B. CONTRIBUTION
Compared to the design of conventional dense spreading
sequences for classic CDMA, designing the LDS sequences
for NOMA systems is more complicated, since the design
should be implemented under the sparsity constraint of
the signature matrix. In the literature, there is a paucity
of optimal signature matrix designs exhibiting maximum
minimum code distance.
Against this background, we study a range of differ-
ent distance metrics and the properties of a sophisticated
signature matrix. Table 1 boldly and explicitly contrasts
the novelty of our design to the family of state-of-the-art
LDS code set designs. Explicitly, our new contributions are
summarized as follows:
(1) We propose a novel iterative LDS design algorithm
for maximizing the signal-to-interference-noise ratio
(SINR) of each individual user of interest which jointly
maps the user-signals to REs in a sparse manner and
applies constellation shaping tp tje non-zero entries.
(2) We demonstrate that the code sets having the highest
minimum distance are also optimal in terms of the BER
criterion for transmission over Gaussian channels. Fur-
thermore, when the code sets have the same minimum
distance, those associated with higher average Gaussian
separability tend to exhibit better BER performance.
We show that our improved LDS code set outper-
forms the existing LDS designs in terms of its BER
performance for BPSK and 4QAM transmissions over
AWGN, non-dispersive and frequency-selective fading
channels.
(3) Moreover, we design both a minimum mean-square
estimation (MMSE) based and a parallel interference
cancellation (PIC) (MMSE-PIC) aided detector [84],
both which exhibit a comparable BER performance to
that of the high-complexity ML detector.
The rest of the paper is organized as follows. In Section
II, we discuss the system model, followed by the specific
properties and design criteria of the spreading codes in
Section III. Our improved iterative LDS sequence design is
presented in Section IV, followed by our detection method
proposed for AWGN, non-dispersive and frequency selective
fading channels in Section V. After illustrating our simu-
lation results in Section VIII, our conclusion and design
guidelines are drawn in Section IX.
The following notations are used in this paper. All
boldface lower case letters indicate column vectors and
upper case letters indicate matrices, ()T
denotes transpose
operation, sgn denotes the sign function, |.| is the scalar
magnitude, || · ||p denotes `p norm, || · || , || · ||2 is vector
norm and E{·} denotes expected value.
II. SYSTEM MODEL
First of all, perfect chip synchronization among all the
transmitters is assumed. This provides the best-case estimate
of the performance of what is in reality a fully asynchronous
system, which only requires chip synchronization between
the source transmitter and the target receiver. The spreading
sequence ck ∈ CL×1
is considered to be s-sparse, when s
coefficients are non-zero and (L−s) are zeros, with the non-
zero coefficients located in Ik ⊂ {1, 2, ..., L}. In the scope
of LDS design ck can be considered sparse if the cardinality
of non-zero entries obeys |Ik| ≤ L/2. However, the sparsity
metric is also discussed further in the next section.
A. AWGN CHANNEL
We assume that the data stream is partitioned into length-Q
subsequences, bk , [bk,1, bk,2, . . . , bk,Q], of k-th user bits
bk,i ∈ {0, 1}, for 1 ≤ j ≤ Q. The modulator maps each
subsequence bk to a symbol xk from the M-ary symbol
alphabet Xk = {xk,1, xk,2, . . . , xk,M }, where, xk,m ∈ C
corresponds to the bit pattern bk(m) = [bm
k,1, bm
k,2, . . . , bm
k,Q]
and M = 2Q
. Let the modulator’s bijective mapping ψk,
representing the binary-to-symbol conversion of user k be
defined as
ψk : bk(m) ∈ {0, 1}Q×1
7→ am ∈ Xk, ∀m, (2)
and vice versa, its inverse operation be represented by
bk(m) = ψ−1
k (am), bm
k,i = ai
m, where ai
m denotes the i-th
bit of the binary vector ψ−1
k (am). Then, the users’ symbols
are multiplexed after spreading them using the LDS codes.
Mathematically, we can formulate the system model as
y =
K
X
k=1
ckdkxk + n
= CDx + n, (3)
where K is the number of the users, dk is the k-th
user’s amplitude, xk ∈ Xk is the k-th user’s symbol
to be transmitted from the constellation alphabet, Xk ,
8 VOLUME 4, 2016](https://image.slidesharecdn.com/ldsieeeaccess2020lajosfinaldraft1-250215201455-18753086/85/LDS_IEEE_Access2020_Lajos_Final_Draft-1-pdf-8-320.jpg)
![C = [c1, c2, . . . , cK] ∈ CL×K
is the column-normalized
LDS code matrix, ||ck|| = 1 for 1 ≤ k ≤ K, n ∈ CL×1
is an L-dimensional complex-valued AWGN vector with
variance of σ2
and D is a diagonal matrix hosting the users’
amplitude, which is given as
D =
d1 0 · · · 0
0 d2 0
.
.
.
.
.
. 0
... 0
0 . . . 0 dk
. (4)
We assume that the constellation alphabet of each user
is identical, i.e., Xk = X, ∀k and the cardinality of the
constellation is M = |X|. The block diagram of the LDS
transmitter is shown in Fig. 4. Note that for the AWGN
channel, we assume that hk = 1 for 1 ≤ k ≤ K.
B. NON-DISPERSIVE FADING CHANNEL
A channel is said to exhibit flat or non-dispersive Rayleigh
fading if the coherence bandwidth of the channel is higher
than the bandwidth of the signal. In this case, all of
the received multipath components arrive within a delay
that is much smaller than the symbol duration; where the
symbol is defined as one chip in the case of LDS spread
signals. These channel coefficients are circularly symmetric
complex Gaussian random variables with zero mean and unit
variance. The magnitudes of the channel gains are Rayleigh
distributed. The model of the flat or non-dispersive fading
channel can be represented as
y =
K
X
k=1
ckhkdkxk + n
= CHDx + n, (5)
where hk is the k-th user’s channel coefficient and H is a
diagonal matrix with channel coefficients as shown below,
H =
h1 0 · · · 0
0 h2 0
.
.
.
.
.
. 0
... 0
0 . . . 0 hk
. (6)
The block diagram of the transmitter model of an LDS
system in non-dispersive fading channels is shown in Fig.
4.
C. FREQUENCY-SELECTIVE FADING CHANNEL
A channel is said to exhibit frequency-selective fading, if
the coherence bandwidth of the channel is lower than the
bandwidth of the signal. In other words, it occurs whenever
the received multipath components of a symbol extend be-
yond the symbol’s time duration. The multipath channel can
be modeled by a tap delay line based finite impulse response
(FIR) filter of length Lp [85]. The system model for uplink
communication over the frequency-selective fading channel
can be written as
y =
K
X
k=1
hk ∗ (ckdkxk) + n
=
K
X
k=1
H̄kckdkxk + n, (7)
where hk = 1
√
Lp
[hk,1, hk,2, . . . , hk,Lp
]T
is the k-th user’s
channel impulse response (CIR), ∗ is the convolution
operator and H̄k is a channel matrix with the size of
(L + Lp − 1) × L and is expressed as,
H̄ =
hk,Lp
0 · · · · · · · · · · · · 0
.
.
.
...
...
...
...
...
.
.
.
hk,2 · · · hk,Lp 0 · · · · · · 0
hk,1 hk,2 · · · hk,Lp 0 · · · 0
0 hk,1 hk,2 · · · hk,Lp
· · · 0
.
.
.
...
...
...
...
...
.
.
.
0 . . . 0 hk,1 hk,2 · · · hk,Lp
0 · · · · · · 0 hk,1 · · · hk,Lp−1
.
.
.
...
...
...
...
...
.
.
.
0 · · · · · · · · · · · · · · · hk,1
.
(8)
mapping
b1 d1·x1
+
u1
mapping
b2
u2
mapping
bK
uK
n
y
AWGN
·
·
·
·
·
·
c1
d2·x2
c2
dk·xk
ck
h1
h2
hk
FIGURE 4. Transmitter of an LDS system communicating over non-dispersive
fading channels.
The Gaussian random variables hk,i where i =
1, 2, . . . , Lp have a zero mean and unit variance, while the
factor 1
√
Lp
ensures that the channel gain experienced by the
transmitted signal on average is unity. Here, we assume that
the CIRs between users are independent from one another.
The block diagram of the transmitter model of an LDS
system for uplink communication over frequency-selective
fading channel is shown in Fig. 5.
III. CODE PROPERTIES AND DESIGN CRITERIA
In this section, we first present some of the distance metrics
that will be used in the development of an iterative algorithm
with the intention of finding the improved LDS signature
VOLUME 4, 2016 9](https://image.slidesharecdn.com/ldsieeeaccess2020lajosfinaldraft1-250215201455-18753086/85/LDS_IEEE_Access2020_Lajos_Final_Draft-1-pdf-9-320.jpg)
![sets. Given the channel and receiver design specifics, the
overall system performance is determined by the specific
selection of the user signature set. One of the signature set
metrics of interest is the minimum distance. The larger the
distance, the better the performance in terms of BER. We
recall that the minimum distance for the BPSK constellation
X = {±1} is 2 ii
.
mapping
b1 d1·x1
h1
+
u1
mapping
b2
u2
h2
mapping
bK
uK
hk
n
y
AWGN
·
·
·
·
·
·
c1
c1
d2·x2
c2
c2
dk·xk
ck
ck
FIGURE 5. Transmitter of an LDS system communicating over
frequency-selective fading channels.
A. DISTANCE METRICS
Definition III.1. The Euclidean distance of two L-
dimensional vectors yi and yj for i 6= j is given by
dE(yi, yj) = ||yi − yj||2, (9)
where yi = Cxi, yj = Cxj, xi, xj ∈ XK×1
and xi 6= xj.
The minimum distance of the received vectors for a given
code set can be formulated by
dE,min(C) = argmin
xi,xj ∈XK×1
/
∈{0}K×1
yi=Cxi,yj =Cxj
dE(yi, yj). (10)
Theorem 1. Let C ∈ CL×K
represent the set of all distinct
sparse normalized column LDS matrices. Then dE,min(C)
is equal to 2 when X = {±1}.
Proof. Assume that cT
i cj = 0 for all Ii = Ij, i 6= j.
Let dE,min(C) = dE(yn, ym), where yn = Cxn and
ym = Cxm. The difference vector y = yn − ym =
C(xn −xm) = Cx̄ must have one non-zero element x̄t 6= 0,
xn,t 6= xm,t, and L−1 zeros x̄z = 0, xn,z = xm,z for z 6= t
to achieve dE,min. Then x̄t can only be 2 or −2, since we
have xn,t, xm,t ∈ {±1}. Therefore, the Euclidean distance
obeys ||y|| = ||Cx̄|| = ||2ct|| = 2||ct|| = 2.
Definition III.2. The product distance of two L-
dimensional vectors yi and yj for i 6= j is expressed by
dP (yi, yj) =
Y
t∈Ii,j
|yi,t − yj,t|, (11)
iiIn signal space representation with the constellation points ±
p
Eb/Tb
the minimum distance is expressed as 2
p
Eb/Tb.
where yi,t−yj,t 6= 0 for all t ∈ Ii,j ⊂ {1, ..., L}. Let dP,min
be the minimum product distance of the code set C.
Definition III.3. The Manhattan distance [86] of two L-
dimensional vectors yi and yj for i 6= j is defined as
dM (yi, yj) = ||yi − yj||1. (12)
Let dM,min be the minimum Manhattan distance of the code
set C.
B. CODE PROPERTIES
Another signature set metric of interest includes the total
squared correlation, which can be linked to the MAI power
associated with a code set.
Definition III.4. The total squared correlation (TSC) of C
is the sum of the squared magnitudes of all inner products
between signatures, which is expressed as
TSC(C) =
K
X
i=1
K
X
j=1
|cH
i cj|2
. (13)
Let us denote, the number of bits per symbol xk of user
k by ρk. Then, there exists a K-dimensional capacity region
Φ ⊂ RK×1
for which each set of the number of bits/symbol
also termed as the rate ρ = (ρ1, . . . , ρK) within this region
can be achieved, while maintaining an infinitesimally low
BER for every user, provided that the codeword length tends
to infinity. In particular, the sum capacity Csum over Φ is
defined as
Csum = max
ρ∈Φ
K
X
k
ρk. (14)
Definition III.5. The total sum capacity Csum(C, γ) [87]–
[89] in bits/symbol, defined as the maximum possible sum of
the users’ transmission rates attained, while still maintaining
reliable reception of the signatures in an AWGN channel is
expressed as
Csum(C, γ) = log2 |IL + γCPCH
|, (15)
where we have P = DE{xxH
}DH
, γ is the received SNR
of each user’s signaliii
and IL is the (L×L)-element identity
matrix.
Definition III.6. The root-mean-square (RMS) cross-
correlation and the maximum cross-correlation amplitude
are expressed as
Irms(C) =
v
u
u
t 1
K(K − 1)
K
X
i=1
K
X
j6=i
|cH
i cj|2, (16)
Imax(C) = max
1≤i<j≤K
|cH
i cj|. (17)
iiiHere we assume identical received SNR for all user’s signals.
10 VOLUME 4, 2016](https://image.slidesharecdn.com/ldsieeeaccess2020lajosfinaldraft1-250215201455-18753086/85/LDS_IEEE_Access2020_Lajos_Final_Draft-1-pdf-10-320.jpg)
![Lemma 1. The Welch Lower Bound [90] for any code set
C, with L ≤ K, is expressed as
Irms(C) ≥
s
K − L
(K − 1)L
, (18)
with equality if and only if
PK
i=1 cicH
i = K
L IL. Further-
more,
Imax(C) ≥
s
K − L
(K − 1)L
, (19)
with equality if and only if
|cH
i cj| =
s
K − L
(K − 1)L
∀i 6= j. (20)
The detailed proof of a well-known performance index
that assesses the cross-correlation of the code matrix can
be found in [90]. The spreading sequence C constitutes
a Welch-bound-equality (WBE) and/or a maximum-Welch-
bound-equality (MWBE) code matrix, when equality is
satisfied in (18). Then Irms meets the Welch bound and/or
(19) meets the Welch bound on Imax. Since the MWBE is a
stricter bound than the WBE, a MWBE code matrix is said
to be a WBE matrix, but not vice versa. The Welch bound
(19) is tight for smaller values of K, but becomes quite loose
for larger K. It is a challenge to find C associated with an
arbitrary L and K that can satisfy the Welch bound on Imax,
(19). As an example, it is widely recognized that there is no
C that satisfies the Welch bound on Imax when K > L2
in
the complex case, C ∈ CL×K
, or when K > L(L + 1)/2
in the real case C ∈ RL×K
. Note that the expressions for
the WBE and MWBE bounds for s-sparse matrices C are
a bit different from the ones defined in (18) and (19).
Definition III.7. Let all of the users be defined as U =
{1, 2, . . . , K}. The k-th symbol is considered to be Gaussian
separable [48], if for all small variances, σ2
d → 0, we have
cH
k R−1
k ck >
X
j∈U−k
|cH
k R−1
k cj|, (21)
where
Rk =
X
j∈U−k
cjcH
j + σ2
dIL, (22)
and the parameters
∆k = cH
k R−1
k ck −
X
j∈U−k
|cH
k R−1
k cj|, (23)
∆ave(C) =
1
K
K
X
k=1
∆k, (24)
are called the Gaussian margin and average Gaussian margin
of the matrix C, respectively.
Linear detectors rely on a decision-boundary partition-
ing the composite multiuser signal-space into subspaces
uniquely and unambiguously identified by the users’ sig-
natures. Therefore the existence of these hyperplanes that
partition the projection subspace of the binary user signals
into two sets for each user in the absence of channel noise,
is a prerequisite for a high performance. This geometri-
cal perspective allows us to formally state a separability
criterion for linear detectors. As for this linear classifier,
upon assuming that the underlying classes follow a Gaussian
distribution, it was shown in [48] the optimal ML decision
relies on this hyperplane which partitions the decision-space
into a pair of L-dimensional subspaces. Therefore, the linear
decision rule for user K is said to be Gaussian separable, if
the probability of error tends to zero when the noise variance
tends to zero.
Definition III.8. There are many metrics of vector sparsity,
as described in [91], but we will define the general Hoyer
sparseness measure of a vector ci based on the relationship
between the `m and `n norms as follows,
Sm,n(ci) =
L(1/m)
L(1/n) − ||ci||m
||ci||n
L(1/m)
L(1/n) − 1
. (25)
The average sparseness of a matrix C can be expressed as,
Sm,n,ave(C) =
1
K
K
X
k=1
Sm,n(ck). (26)
Interesting special cases are those, when m = 1, n = 2,
which are known as the Hoyer sparseness measure [91] and
m = 1, n = ∞.
IV. PROPOSED LDS CODE DESIGN
In the following section, we describe the proposed iterative
algorithm, which is used for designing the LDS code matrix
C. For the sake of simplicity let us assume that dk = 1 for
1 ≤ k ≤ K and rewrite (3) as
y = ckxk +
K
X
i6=k
cixi + n, (27)
= ckxk + ik + n, (28)
where y ∈ CL×1
and ik ∈ CL×1
denotes the colored
interference imposed by the other users, when the autocor-
relation matrix is given by R0
k , E{ikiH
k }. Let us define the
overall perturbation, gk = ik +n, and the autocorrelation as
Rk , E{gkgH
k } = R0
k + σ2
IL. The detection of the infor-
mation bit of user k can be achieved via max-SINR filtering
(or, equivalently, min-TSC filtering, linear MMSE filtering).
The filter that exhibits the maximum output SINR for user-k
is a scaled version of e.g., wSINR,k(ck) , R−1
k ck. Then
the corresponding maximum post-filtering SINR output of
the filter wSINR,k is given by
SINR(ck) =
E
n
|wH
SINR,k(ckxk)|2
o
E
n
|wH
SINR,k(ik + n)|2
o (29)
= cH
k Qkck, (30)
VOLUME 4, 2016 11](https://image.slidesharecdn.com/ldsieeeaccess2020lajosfinaldraft1-250215201455-18753086/85/LDS_IEEE_Access2020_Lajos_Final_Draft-1-pdf-11-320.jpg)
![where Qk , R−1
k . Our objective is to find the specific s-
sparse complex signature ck that maximizes (29), namely:
c
(s)
k,maxSINR = argmax
c∈CL×1
,||c||=1
|Ik|=s
cH
Qkc. (31)
The superscript (s) indicates that c
(s)
k,maxSINR is s-sparse
with |Ik| = s. To tackle the problem, we now propose
to relax the sparseness constraint of (31) and proceed by
solving the following problem instead,
ck,maxSINR = argmax
c∈CL×1,||c||=1
cH
Qkc. (32)
Let {qk,1, qk,2, · · · , qk,L} be the L eigenvectors of Qk with
corresponding eigenvalues λk,1 ≥ λk,2 ≥ · · · ≥ λk,L. The
sequence c that maximizes (32) is well known and it is
equal to the eigenvector that corresponds to the maximum
eigenvalue of the matrix Qk, i.e.,
ck,maxSINR = argmax
c∈CL×1,||c||=1
cH
Qkc = qk,1. (33)
Alternatively, we can design the code set C based on the
TSC criterion that is defined in Section III. Let us now
demonstrate the iterative method used for minimizing the
TSC(C). We rewrite (13) as
TSC(C) =
K
X
i6=k
K
X
j6=k
|cH
i cj|2
+ |cH
k ck|2
+ 2
K
X
i6=k
|cH
k ci|2
= TSC(C[k]) + 1 + 2cH
k
K
X
i6=k
cicH
i
ck
= TSC(C[k]) + 1 + 2cH
k R̄kck, (34)
where C[k] denotes the preexisting code set except for the
k-th column of the code set C and R̄k =
PK
i6=k cicH
i
denotes the autocorrelation of the matrix C[k], respectively.
It becomes clear from (34) that the conditional minimization
of TSC(C) with respect to ck for fixed (min-TSC-valued)
TSC(C[k]) reduces to
c
(s)
k,minT SC = argmin
c∈CL×1
,||c||=1
|Ik|=s
cH
R̄kc. (35)
The sequence c that minimizes the relaxed problem (35)
is well known and it is equal to the eigenvector, rk,υ, that
corresponds to the minimum non-zero eigenvalue υ of the
matrix R̄k, i.e.,
ck,minT SC = argmin
c∈CL×1,||c||=1
cH
R̄kc = rk,υ. (36)
Similarly, since our construction of spreading codes is
restricted to unit energy, i.e., cH
k ck = 1, the underlying
problem of minimizing (34) does not change, if we add
2σ2
cH
k ck and subtract 2σ2
from (34) to obtain
TSC(C) = TSC(C[k]) + 1 + 2cH
k Rkck − 2σ2
, (37)
where Rk is defined in (22). The normalized MMSE filter
for user k [92], is
sk,MMSE =
R−1
k ck
(cH
k R−1
k ck)1/2
. (38)
Therefore, the results of (36), (38) can be used in Step 7
of the iterative construction of our proposed LDS design
algorithm, as shown in Table 2. In other words, instead of
computing qk,1, we compute rk,υ or sk,MMSE. We are now
ready to present our proposed algorithm, which is shown in
Table 2,
TABLE 2.
LDS design algorithm
Input: L, K, δ, σd, s-sparse
1: Initialize : ∆0
ave ← 0, ← 0.2, 0 ← 102;
2: while ∆0
ave δ
3: Initialize : Civ;
4: while 0
5: C0 ← C
6: for k ∈ {1, . . . , K}
7: Compute qk,1 in (33), or (36), or (38)
8: ck ← q
[n]
k,1
9: 0 ← ||C − C0||F
10: Compute ∆0
ave(C) in (24)
Output: C
where q
[n]
k,1 denotes the s-sparse vector with only those s
elements of qk,1 that have the largest absolute values. The
scalar values δ and σd are design parameters, respectively.
Note that the non-zero complex values of the sparse columns
ci are not necessarily unimodular, i.e., ci = {ci,j ∈ C :
|ci,j| = 1, j ∈ Ii}, for 1 ≤ i ≤ K, [93]. Unimodular s-
sparse vectors have equal Hoyer sparsity, i.e., S1,2(ci) =
L(1/2)
−s(1/2)
L(1/2)−1
.
A. CONVERGENCE OF THE ALGORITHM
The proposed algorithm converges to a locally optimal
solution with an s-sparse matrix C that has one of the
specific structures A, where we have Ai,j = {|ci,j|0 : 1 ≤
i ≤ L, 1 ≤ j ≤ K}. For a size of 4 × 6 and for the random
initialization of C the algorithm converges to one of the
three structures (e.g., A1
, A2
and A3
), which is described
as follows,
A1
=
1 0 1 0 0 1
1 0 1 0 0 1
0 1 0 1 1 0
0 1 0 1 1 0
, (39)
A2
=
0 1 1 0 0 1
1 0 0 1 1 0
0 1 1 0 0 1
1 0 0 1 1 0
, (40)
ivRandomly generate C ∈ CL×K , where ||ci|| = 1, ∀ 1 ≤ i ≤ K.
12 VOLUME 4, 2016](https://image.slidesharecdn.com/ldsieeeaccess2020lajosfinaldraft1-250215201455-18753086/85/LDS_IEEE_Access2020_Lajos_Final_Draft-1-pdf-12-320.jpg)
![A3
=
1 1 1 0 0 0
0 0 0 1 1 1
0 0 0 1 1 1
1 1 1 0 0 0
, (41)
where the order of the columns of all of the three structures
can be arbitrarily ordered and not necessarily as shown
above. The output structure of the matrix of the algorithm
depends mainly on the initialization of the matrix C. Specif-
ically, if we arbitrarily initialize a 4×6 matrix that possesses
one of the structures (e.g., A1
, A2
and A3
), the output of
the algorithm will have the same structure as that of the
initialization matrix. Therefore, to speed up the convergence
of the algorithm in Table 2 we may choose to initialize the
matrix C with one of the known structures.
B. UPWARD SCALING DESIGN FOR LDS
The algorithm proposed in Table 2 may potentially be con-
sidered for an upward scaling design for a given optimum
LDS sequence. The underlying requirement is to develop
a subset of the sparse matrix with the given optimal LDS
code set for ensuring that the resultant LDS code set still
maintains optimality. Appending spreading codes to a given
LDS set may require complete redesign/reassignment of
the resultant LDS code set. Mathematically, we wish to
design an s-sparse code set Ck0
+1
K = [ck0+1, ck0+2, . . . , cK]
matrix, where ci = {ci,j ∈ C : j ∈ Ii}, for i ∈ K =
{k0
+1, k0
+2, . . . , K} that can be appended to a given code
set C1
k0 = [c1, c2, . . . , ck0 ], where ci = {ci,j ∈ C : j ∈ Ii},
for i ∈ K0
= {1, 2, . . . , k0
} to result in an improved
LDS code matrix C = [C1
k0 Ck0
+1
K ]. The only constraint
imposed on the proposed algorithm while generating the
LDS sequence is that the input matrix C1
k0 must obey
one of the convergent structures discussed in Section IV-A
above. Therefore, the resultant algorithm can be applied
for designing s-sparse spreading codes that are appended
to an input optimal LDS sequence with the aid of a small
modification.
A =
1 0 0 1 0 0 1 0 0
1 0 0 1 0 0 0 0 0
0 1 0 0 1 0 0 1 0
0 1 0 0 1 0 0 0 1
0 0 1 0 0 1 0 0 0
0 0 1 0 0 1 0 0 0
.
In Step 3, we randomly initialize Ck0
+1
K and form C =
[C1
k0 Ck0
+1
K ] instead of randomly initializing C. In Step 6, we
use k ∈ K instead of k ∈ {1, 2, . . . , K} as shown in Table
2. To illustrate one of the sample LDS outputs of size 6×9
from the modified algorithm, we first arbitrarily generate an
orthogonal 2-sparse 6 × 6 matrix, which belongs to one of
the convergent structure discussed in Section IV-A and use
that as an input to the modified algorithm. The resultant LDS
sequence structure is shown in Fig. 42. Observe that that in
Fig. 42 the C7
9 matrix is 1-sparse and since the columns have
unit energy, the elements are simply 1. We will characterize
the performance of such code sets in our simulations. The
proposed receiver is presented in the next section.
C. COMPLEXITY OF THE PROPOSED LDS
CONSTRUCTION
The main complexity contribution of the proposed algorithm
is associated with updating qk,1 either in (33), or in (36),
or alternatively in (38). Direct calculations of R−1
k for
each user is expensive. However, with the aid of effi-
cient numerical techniques such as the Sherman-Morrison-
Woodbury formula, we can compute its inverse, i.e., R−1
k ,
at a complexity order of O(K2
). This process is repeated
K times for obtaining all K users’ LDS waveforms. On the
other hand, the number of iterations in the ‘while’ loops in
lines 4 and 2 depends on the thresholds and δ, respectively.
The smaller the thresholds the longer it takes to complete
the process. With our design parameters, the number of
iterations on average was about 3 and 10 for the ‘while’
loops in lines 4 and 2, respectively. Therefore, the overall
complexity is O(c · K3
) = O(K3
) where c is a constant,
e.g., c = 3 · 10 = 30.
V. MULTIUSER DETECTION
It is widely recognized that obtaining the ML solution is
generally NP-hard [46]. Various suboptimal low-complexity
detection techniques have already been proposed for con-
ventional dense spreading based CDMA systems. These
suboptimal approaches can be classified into two categories:
linear and non-linear MUDs. Linear MUDs include among
others, matched filtering (MF), MMSE, and zero-forcing
(ZF) based schemes. In a non-linear successive interference
cancellation aided detector the interference is first estimated
and then it is subtracted from the received signal before
detection. The cancellation process can then be carried out
either successively (SIC) [94], or in parallel (PIC) [95]–
[97]. In non-linear iterative detectors [98]–[102], PDA [48]
aims for suppressing the MAI in each iteration in order to
improve the overall error performance. Suboptimal so-called
polynomial-time detectors that are based on the geometric
approach are studied in [103], [104].
In comparison to dense CDMA, sparse CDMA or LDS
is capable of substantially reducing the computational com-
plexity of MUDs. This is a benefit of the sparse nature
of LDS sequences that enables both the MPA and belief
propagation (BP) algorithms to be applied at a lower com-
plexity than the optimum MUD. In terms of reducing the
complexity of the MPA algorithm even further without much
performance erosion, Du et al. [105] proposed a detection
scheme based on a dynamic factor graph by exploiting the
channel state information. Another solution conceived by
Tian et al. [106] reduces the complexity by restricting the
search region of the superimposed multiuser constellation
to a quadrant-like part of it. Razavi et al. [107] proposed
a beneficial receiver component activation scheduling for
iterative MUD in order to reduce its complexity by utilizing
the LDPC codes for an LDS-OFDM system.
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![The relationship between the optimal performance and the
performance achieved by iterative BP has been established
by Guo and Wang in [108] in the CDMA context. Their
study demonstrated that for about a hundred users, the
theoretical performance limit of large systems is approached
as a result of the central-limit theorem. Those studies are
normally performed under the assumption of a large system,
where both the number of users and the spreading factor
tend towards infinity, while their ratio is kept constant.
As an example, Takeuchi et al. [109] characterized the
family of BP receivers via density evolution (DE) in the
dense limit after assuming the large-system limit. In those
studies the specific way the MPA is implemented played
a significant role. The user’s data detection based on the
MPA and on the optimal ML detection using turbo-style
processing is reported by Razavi et al. [23]. In contrast
to this, it is shown in [110] that a joint detection and
decoding approach based on an optimised sparse graph of
the multiuser channel and the LDPC codes outperforms the
iterative receiver of LDS-OFDM systems. Wen and Su [111]
showed both numerically and analytically that the JSG-
CDMA, which combines multiple access using LDS-CDMA
and LDPC forward error correcting techniques, attains a
satisfactory performance under rank-deficient conditions and
outperforms conventional CDMA, LDS-CDMA as well as
iterative detection aided LDS-CDMA. The detailed analysis
is presented in [112].
Nevertheless, the BP and MPA detection methods still
have exponential by increased computational complexity as
a function of the number of users. The trade-off between
the computational complexity and bandwidth efficiency at
different user-load is studied by Raymond in [113]. Near-
suboptimal detectors tend to strike a compelling perfor-
mance as complexity trade-off compared to an MPA detec-
tor. Therefore, by taking full advantage of the LDS scheme,
which has a lower MAI than dense CDMA systems, we
consider an attractive low-complexity detector, which is
based on the MMSE criterion and on PIC (MMSE-PIC)
based detection [84] that has an even lower complexity than
the MPA based detector, whilst achieving the same spectral
efficiency. Fantuz and D’Amours [84] showed that the BER
performance of MMSE-PIC is very close to that of the MPA
detector for LDS systems communicating over AWGN, non-
dispersive and frequency selective fading channels.
A. MMSE-PIC DETECTOR
The MMSE-PIC detector of Fig. 6 is constituted by a benefi-
cial amalgam of the MMSE and PIC detectors which will be
characterized for transmission over AWGN, non-dispersive
and frequency-selective fading channels, respectively. For
the sake of simplicity, the derivation of the detector is
provided for BPSK and 4QAM, constellations of X =
{−1, +1} and X = {−1 − j, −1 + j, +1 − j, +1 + j}/
√
2.
However, it should be noted that similar derivations can
be readily provided for higher-order constellations, such as
8QAM, 16QAM, 32QAM, etc.
1) AWGN Channel
The despreading is performed by multiplying the received
vector in (3) by the LDS code as follows,
r = CH
y = RDx + CH
n, (42)
where we have r ∈ CK×1
and the correlation matrix obeys
R = CH
C ∈ CK×K
. The optimal receiver achieves the
minimum probability of error Pr(x 6= b
x) for each symbol
vector x, which is arranged by estimating b
x upon maximiz-
ing the a posteriori probability (APP) Pr(x|r)’s given the
observed despread sequence r, which is formulated as
b
x = argmax
x∈XK×1
Pr(x|r). (43)
This decision criterion is commonly referred to as the MAP
[114] algorithm. It is widely known that the MAP detector
has an exponentially increased complexity by the number of
users K, which makes its application somewhat unrealistic
even for moderate values of K. In practice it is more
convenient to work with log-likelihood ratios (LLRs) than
with probabilities. The LLRs for each symbol am, where
am ∈ X for 1 ≤ m ≤ M, of the k-th user can be written
as
Λk(am) = log
P
x∈Aam
xk
Pr(r|x)Pr(x)
P
x̄∈Aam
xk
Pr(r|x̄)Pr(x̄)
, (44)
with Aam
xk
⊂ Ax, Aam
xk
⊂ Ax representing the set of all
symbol vectors x ∈ Ax in which we have xk = am and
xk 6= am for the k-th user. Furthermore, we have Ax =
XK×1
and
Pr(r|x) =
1
πK|Σ|
exp[−fH
(x)Σ−1
f(x)], (45)
where f : x 7→ r − RCx represents a linear mapping of
RC = RD, the covariance matrix obey Σ = σ2
CH
C and
|Σ| denotes the determinant of Σ. If we assume that all
symbol vectors have the same probability distribution of
Pr(x) = 1/MK
, then the log-sum approximation of (44)
can be expressed as
Λk(am) ≈ min
x∈Aam
xk
||Σ− 1
2 f(x)||2
− min
x̄∈Aam
xk
||Σ− 1
2 f(x̄)||2
. (46)
The computational complexity is increased exponentially
versus the number of users K because the LLRs in (46)
are calculated jointly for all the users hence requiring the
computation of MK
norm values. By contrast, the popular
family of minimum mean square error detectors minimize
the error-variance between the transmitted symbol and the
filtered signal at the user level and they are more desirable
in terms of complexity. Therefore, the per-user LLRs are
computed separately. After MMSE filtering, our goal is to
estimate the users’ symbols independently. Therefore, the
MMSE detector’s action can be expressed in this form
u = WMMSEr ∈ CK×1
, (47)
where u represents the decision variables after the MMSE
detector. The MMSE filter, weight-matrix WMMSE ∈ CK×K
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![is found by minimizing the mean-square error between the
estimated symbols and the true transmitted symbol x, which
is expressed as
WMMSE = argmin
W∈CK×K
E{|x − Wr|2
}. (48)
Under the reasonable assumption that each user’s symbols
are independent and identically distributed (i.i.d.) with unit
energy, when we have E{xxH
} = IK, the solution of (48)
is given by
WMMSE = RH
C RCRH
C + Σ
†
, (49)
where (·)†
denotes the Moore-Penrose pseudoinverse op-
eration [115]. The MMSE decision variables u are then
processed to obtain the log likelihood ratios. The MMSE
decision variable for the k-th user can be written as
uk = wkr
= wkRk
Cxk +
K
X
i=1
i6=k
wkRi
Cxi + wkCn
= βk,kxk +
K
X
i=1
i6=k
βk,ixi + wkCn, (50)
where wk ∈ C1×K
is the k-th row vector of WMMSE,
Rk
C ∈ CK×1
is the k-th column of RC, xk ∈ X is the
k-th symbol of the vector x and βk,i = wkRi
C ∈ C. Since
the direct evaluation of Pr(xk = am|u) is computationally
prohibitive, the PDA detector attempts to estimate it by
using the Gaussian - “forcing” idea of [116] by approxi-
mating Pr[xk = am|u, {p(j)}∀j6=k], that can serve as the
updated value for pm(k). The vector p(k) is associated to
xk, whose m-th element pm(k), is the current estimate of a
posteriori probability of xk = am. In contrast to the PDA
detector, MMSE detector attempts to estimate it by making a
reasonable assumption on conceiving the a priori probability
distribution of Pr(xk), namely that it is i.i.d. having an
expected value of unity. If we model the residual MAI after
the MMSE detector by a complex Gaussian random variable
which is independent of the noise, then uk is Gaussian as
well. Let
αm(k) = −
|uk − βk,kam|2
σ2
k
, (51)
where we have σ2
k =
PK
i=1
i6=k
|βk,i|2
E{|xi|2
} + wkΣwH
k ,
E{|xi|2
} = 1 since the xi values are i.i.d. random variable.
Provided that all the transmitted symbols have identical a
priori probabilities, the a posteriori symbol probability is
given by
Pr(xk = am|uk) =
Pr(uk|xk = am)Pr(xk = am)
P
am∈X Pr(uk|xk = am)Pr(xk = am)
=
exp[αm(k)]
P
j exp[αj(k)]
, (52)
where we have Pr(uk|xk = am) = 1
πσ2
k
exp[αm(k)]. The
a posteriori probabilities of the symbols am can also be
expressed in terms of their LLR’s as follows,
ΛMMSE
k (am) = log
Pr(xk = am|uk)
Pr(xk 6= am|uk)
= log
exp[αm(k)]
P
j6=m exp[αj(k)]
. (53)
Furthermore, to simplify the avaluation of (53), the log-sum
approximation can be used:
ΛMMSE
k (am) ≈ max log exp[αm(k)]
− max
j6=m
log exp[αj(k)]
≈ αm(k) − max
j6=m
αj(k). (54)
In the case of BPSK, (53) simplifies to:
ΛMMSE
k (a1) = log
exp[α1(k)]
exp[α2(k)]
= α1(k) − α2(k)
=
2βk,kuk
σ2
k
, (55)
where a1 = +1 and a2 = −1. Note that the a posteriori
probability Pr(xk = am|uk) in (52) can be expressed in
terms of the LLRs of (53) as follows,
Pr[xk = am|ΛMMSE
k (am)] =
1
2
(1 + tanh[
1
2
ΛMMSE
k (am)]).
(56)
In practice, binary channel decoders require bit-level LLRs.
Even though Gray-coding is used for QAM, which imposes
correlation, for simplicity we assume the independence of
the bits. Hence, if we assume the coded bits to be i.i.d., the
log-likelihood ratio of a bit bk,i can be formulated as,
ΛMMSE
k (bi) = log
Pr(bk,i = 1|uk)
Pr(bk,i = 0|uk)
= log
P
aj ∈X1
i
Pr(xk = aj|uk)
P
aj ∈X0
i
Pr(xk = aj|uk)
= log
P
j|aj ∈X1
i
exp[αj(k)]
P
j|aj ∈X0
i
exp[αj(k)]
, (57)
where bk,i represents the i-th bit of the symbol xk, Xλ
i =
{aj ∈ X|b(j) = ψ−1
(aj), bj
i = λ}, and λ = {1, 0}. Note
that the probability of having bk,i = 1 can be expressed in
terms of ΛMMSE
k (bi) as:
Pr(bk,i = 1|uk) =
exp[ΛMMSE
k (bi)]
1 + exp[ΛMMSE
k (bi)]
. (58)
The complexity of (57) can be reduced by using the log-sum
approximation, which is expressed as
ΛMMSE
k (bi) ≈ max
j|aj ∈X1
i
log exp[αj(k)]
− max
j|aj ∈X0
i
log exp[αj(k)]
≈ max
j|aj ∈X1
i
αj(k) − max
j|aj ∈X0
i
αj(k). (59)
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![y
Single-
user
detector
Split real
and
imaginary
parts
MMSE
detector
Estimate
symbols
PIC
detector
r rR u
x̂
x̅
y
Single-
user
detector
Split real
and
imaginary
parts
MMSE
detector
Estimate
symbols
PIC
detector
r rR u
x̂
x̅
FIGURE 6. Block diagram of MMSE-PIC detector [84].
Given the bit-level LLRs ΛMMSE
k (bi), the a posteriori proba-
bilities can be expressed as follows,
Pr(xk = aj|ΛMMSE
k (b)) =
Q
Y
i=1
Pr[bk,i = ai
j|ΛMMSE
k (bi)],
where we have:
Pr(bk,i = λ|ΛMMSE
k (bi)) =
1
2
(1 + b̃k,itanh[
1
2
ΛMMSE
k (bi)]),
(60)
and
b̃k,i =
(
+1, if bk,i = 1
−1, if bk,i = 0
, (61)
while ΛMMSE
k (b) = [ΛMMSE
k (b1), . . . , ΛMMSE
k (bQ)]T
. The
MMSE-PIC algorithm approximates the estimates x̄k of the
transmitted symbols xk of user k by its mean value, which
is formulated as,
x̄k = E{xk}
=
X
aj ∈X
aj · Pr[xk = aj|ΛMMSE
k (aj)]
=
X
aj ∈X
aj · Pr[xk = aj|ΛMMSE
k (b)], (62)
for k = 1, . . . , K. Alternatively, the soft-decision of the
estimates of x̄k can be expressed as
x̄k = argmax
aj ∈X
Pr[xk = aj|ΛMMSE
k (aj)]. (63)
In case of QAM, (62) can be expressed in terms of ΛMMSE
k (aj)
as
x̄k =
1
2
√
2
(−tanh[
1
2
ΛMMSE
k (a1)] − tanh[
1
2
ΛMMSE
k (a2)]
+tanh[
1
2
ΛMMSE
k (a3)] + tanh[
1
2
ΛMMSE
k (a4)]
+j(−tanh[
1
2
ΛMMSE
k (a1)] + tanh[
1
2
ΛMMSE
k (a2)]
−tanh[
1
2
ΛMMSE
k (a3)] + tanh[
1
2
ΛMMSE
k (a4)])),
where a1 = {−1 − j}/
√
2, a2 = {−1 + j}/
√
2, a3 =
{+1−j}/
√
2, and a4 = {+1+j}/
√
2. In terms of ΛMMSE
k (b),
(62) can be expressed as
x̄k =
1
√
2
(tanh[
1
2
ΛMMSE
k (b1)] + jtanh[
1
2
ΛMMSE
k (b2)]), (64)
and the estimates of the transmitted symbol vector x of all
users can be written as
x =
1
√
2
(tanh[
1
2
ΛMMSE
(b1)] + jtanh[
1
2
ΛMMSE
(b2)]), (65)
where ΛMMSE
(bη) = [ΛMMSE
1 (bη), . . . , ΛMMSE
K (bη)]T
and η ∈
{1, 2}. In case of BPSK, (62) can be expressed in terms of
ΛMMSE
k (a) as
x̄k = tanh[
1
2
ΛMMSE
k (a)], (66)
where a = {−1, +1} and the estimates of the transmitted
symbol vector x of all users can be written as
x = tanh[
1
2
ΛMMSE
(a)], (67)
where ΛMMSE
(a) = [ΛMMSE
1 (a), . . . , ΛMMSE
K (a)]T
.
The PIC stage of the detector produces the final decision
variables according to
uPIC,k = uk −
K
X
i6=k
RC,k,ixi, (68)
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![where RC,k,i is the element in the k-th row and i-th column
of RC, while uk is the k-th element of u. The estimates
in (68) can be used as a priori probabilities for the channel
decoder. If no channel coding is used, then hard-decision
detection can be employed, which is formulated as
x̂k = argmin
aj ∈X
||uPIC,k − aj||2
, ∀k. (69)
In case of BPSK, the hard-decision bits may be then
estimated by
x̂ = sgn ({uPIC}), (70)
where we have uPIC = [uPIC,1, uPIC,2, . . . , uPIC,K]T
. After
computing uPIC,k using x in (68), we can recompute the
LLRs in (53) by using the uPIC,k values instead of uk
obtained after the MMSE filter for improving the detection
performance.
In order to take advantage of the potential diversity gain
of the multi-dimensional signal space, we will exploit it by
converting the Cartesian product of the complex plane to
the real space, which will double the number of dimen-
sions. Since detection of complex symbols (e.g., QAM) is
equivalent to estimating the real and the imaginary parts of
the complex symbols in parallel, this simplifies the detection
process and reduces the decoding complexity as well. We
then split the vectors and matrices in (42) into their real and
imaginary components, as follows:
rR = RRxR + CRnR, (71)
where we have rR ∈ R2K×1
, RR ∈ R2K×2K
, CR ∈
R2K×2L
, nR ∈ R2L×1
and the subscript R, {} and ={}
represent the real domain as well as the real and imaginary
parts of a complex number,
rR =
{r}
={r}
, xR =
{x}
={x}
, RR =
{RD} − ={RD}
={RD} {RD}
,
CR =
{CH
} − ={CH
}
={CH
} {CH
}
and
nR =
{n}
={n}
,
respectively. We treat the elements of xR as independent
multivariate random variables, where the i-th element, xR,i,
is a member of one of two possible sets,
xR,i ∈
(
{xk = am|am ∈ X}, i ∈ [1, K]
={xk = am|am ∈ X}, i ∈ [K + 1, 2K],
(72)
where k ∈ {i, i−K}. The noise nR has the variance matrix
of ΣR = σ2
2 CRCH
R . Note that for BPSK transmission x ∈
{±1}K×1
is real-valued, which results in its imaginary part
being a zero vector. Then (71) can be simplified to:
rR =
{RD}
={RD}
x +
{CH
} − ={CH
}
={CH
} {CH
}
{n}
={n}
. (73)
The separation of the real and imaginary parts provides
an extra dimension for the detector in order to have a
better estimate of each user’s symbol. Therefore, the MMSE
detector can be expressed as
u = WMMSErR ∈ R2K×1
, (74)
where the MMSE filter, WMMSE ∈ R2K×2K
, is found by
minimizing the mean-square error between the estimated
symbols and the true transmitted symbol xR, which is
expressed as
WMMSE = argmin
W∈R2K×2K
E{||xR − WrR||2
}. (75)
The solution of (75) is given by
WMMSE = RT
R RRRT
R + ΣR
†
. (76)
Note that in case of BPSK, we have WMMSE ∈ RK×2K
,
u ∈ RK×1
and xR = x. The MMSE decision variable for
the i-th element can be written as
ui = wirR
= wiRi
RxR,i +
2K
X
j=1
j6=i
wiRj
RxR,j + wiCRnR
= βi,ixR,i +
2K
X
j=1
j6=i
βi,jxR,j + wiCRnR, (77)
where wi ∈ R1×2K
is the i-th row vector of WMMSE, Ri
R ∈
R2K×1
is the i-th column of RR, and βi,j = wiRj
R ∈ R.
Expression (51) in the real domain can be expressed as
αm(i) = −
(ui − βi,iam)2
2σ2
i
, (78)
where am ∈ XR for 1 ≤ m ≤ 2M, XR = {{X}, ={X}},
σ2
i =
P2K
j=1
i6=i
β2
i,jE{x2
R,j} + wiΣRwT
i and E{x2
R,j} = 1
since the xR,js are i.i.d. random variables. Based on (78),
the LLRs for each symbol am, defined as ΛMMSE
i (am) =
log(Pr(xR,i = am|ui)/Pr(xR,i 6= am|ui), can be calcu-
lated by (53) as in the complex formulation scenario. All
the other LLRs and a posterior probabilities are computed in
a similar way to the complex formulation case, except that
now we have to perform for 1 ≤ i ≤ 2K elements and am ∈
XR, 1 ≤ m ≤ 2M symbols with the exception of the BPSK
case. In the PIC stage of (68), we substitute RR,j,i instead
of RC,k,i. In the case of QAM, the decision variable for
user k can be computed as uPIC,k = uR,PIC,k + juR,PIC,k+K
and the hard-decision is given by x̂k = x̂R,k + jx̂R,k+K,
for 1 ≤ k ≤ K.
2) Non-dispersive Fading Channel
In addition to the despreading operation the decision vari-
ables uks are multiplied by the corresponding channel gains
as follows,
r̃k = h∗
kcH
k y
= |hk|2
dkxk +
K
X
i=1,i6=k
Rk,ih∗
khidixi + h∗
kcH
k n, (79)
VOLUME 4, 2016 17](https://image.slidesharecdn.com/ldsieeeaccess2020lajosfinaldraft1-250215201455-18753086/85/LDS_IEEE_Access2020_Lajos_Final_Draft-1-pdf-17-320.jpg)
![encoder
b1
+
n
y
AWGN
·
·
·
·
·
·
interleaver
d1·x1
c1
u1
modulator
encoder
b2
interleaver
d2·x2
c2
u2
modulator
encoder
bK
interleaver
dK·xK
cK
uK
modulator
FIGURE 7. Block diagram of our BICM transmitter.
where the superscript ∗
denotes the complex conjugate. The
vector of decision variables can be expressed as
r̃ = HH
CH
y
= HH
CH
CHDx + HH
CH
n
= HH
RHDx + HH
CH
n. (80)
The vectors and matrices in (80) are then split into real and
imaginary components, as shown below:
r̃R = R̃RxR + C̃RnR, (81)
where
r̃R =
{r̃}
={r̃}
, R̃R =
{HH
RHD} − ={HH
RHD}
={HH
RHD} {HH
RHD}
,
C̃R =
{HH
CH
} − ={HH
CH
}
={HH
CH
} {HH
CH
}
,
respectively. The PIC-MMSE detector design for non-
dispersive fading channel is very similar to that of the
AWGN channel, except that the transmitted signal is sub-
jected to the complex-valued gains. Nonetheless, the differ-
ence is that the correlation matrix R̃ and the LDS sequence
matrix C̃ are defined above, as opposed to the correlation
matrix R and LDS sequence matrix C used for AWGN
channel.
3) Frequency-Selective Fading Channel
There has been extensive research on LDS and/or SCMA
systems communicating over AWGN [22], [32]–[34], [39],
[41] and non-dispersive fading channels [25], [35], [36],
[40]. Most of the studies are dedicated to frequency-selective
channels relying on LDS-OFDM [67], or MC-CDMA [83].
LDS-OFDM is eminently suitable for frequency-selective
channels, since its subcarriers bandwidth is narrower than
the channels coherence bandwidth [84]. Traditional CDMA
tends to mitigate the multipath effects by using RAKE
receivers [117], [118]. A whole suite of fading-mitigation
techniques were conceived in Hanzo et al. [119] ; Hanzo
et al. [120]. By contrast, here we employ a transmit pre-
coding scheme for overcoming the multipath channel effect
as proposed by Fantuz and D’Amours, which is detailed
in [84]. Briefly, this transmit precoding scheme exploits
the knowledge of the CIR for transforming the multipath
channel into a single-path non-dispersive channel. More
explicitly, it transforms (7) to (5), which is equivalent to over
non-dispersive Rayleigh fading channel model. Therefore,
the MMSE-PIC detector derived for non-dispersive fading
channels can be directly applied to frequency-selective chan-
nels with the aid of the transmit precoding scheme of [84].
B. PDA DETECTOR
The PDA [116], [121] has been widely applied by low-
complexity design alternative of the optimal maximum a
posteriori (MAP) symbol decoders/detectors, as a benefit
of its near-optimal detection performance in rank-deficient
CDMA systems [48], [116]. Explicitly, its complexity in-
creases no faster than O(K3
). The PDA detector was
originally conceived in 2001 for CDMA [116] and its
generalized version [122] can be directly applied to our
LDS system designed for BPSK and QAM transmissions.
In the case of QAM, Yang et al. [123] presented a unified
bit-based PDA detection approach, which transforms a high-
order rectangular QAM based multiuser system into a BPSK
multiuser system. By contrast, in [124] an SCMA scheme
is converted to a BPSK modulated CDMA system. More
explicitly, we can convert (3) into a BPSK system as follows,
y = CDWb + n (82)
= Qb + n, (83)
18 VOLUME 4, 2016](https://image.slidesharecdn.com/ldsieeeaccess2020lajosfinaldraft1-250215201455-18753086/85/LDS_IEEE_Access2020_Lajos_Final_Draft-1-pdf-18-320.jpg)
![y
+
_
+
_
Λ1(b1,i) λ1(b1,i)
+ λ1(b1,n) Λ2(b1,n) λ2(b1,n)
+
SISO
channel decoder
interleaver
deinterleaver
λ2(b1,i)
u1
+
_
+
_
Λ1(b2,i) λ1(b2,i)
+ λ1(b2,n) Λ2(b2,n) λ2(b2,n)
+
SISO
channel decoder
interleaver
deinterleaver
λ2(b2,i)
u2
+
_
+
_
Λ1(bK,i) λ1(bK,i)
+ λ1(bK,n) Λ2(bK,n) λ2(bK,n)
+
SISO
channel decoder
interleaver
deinterleaver
λ2(bK,i)
uK
·
·
·
·
·
·
·
·
·
SISO
multiuser
detector
·
·
·
FIGURE 8. Block diagram of our iterative turbo MUD [99].
where we have W = IK ⊗ sT
, Q = CDW, s = [j, 1]T
and
⊗ is a Kronecker operator. In SectionVIII we will present
the BER performance of PDA detectors for transmission
over AWGN, non-dispersive, and frequency-selective chan-
nels.
VI. CHANNEL ENCODING
In his groundbreaking work [125] Shannon beautifully laid
out the fundamental limit of communications known as
channel capacity. However, the early design of communica-
tion systems has focused on separate modulation and error
correcting codes. Yet the solution to the problem of increas-
ing the transmission rate without bandwidth expansion is
to use a high-order constellation transmitting with spectral
efficiency η where 1 ≤ η ≤ log2|X| bits/symbol. Shannon
also introduced [125] the idea of combining coding with
nonbinary modulation using high-order constellations, for
coded modulation (CM) [125]. In this context we emphasize
that both the specific choice of coding as well as the
mapping of coded bits to constellation points is influential
in terms of determining the attainable performance.
The first practical CM scheme, namely the so-called
multilevel coded modulation (MLCM) arrangement was
introduced by Imai and Hirakawa in 1977 [126], [127].
Then in 1982 Unberboeck and Csajka developed the so-
called the trellis-coded modulation (TCM) scheme that was
specifically designed for increasing the Euclidean distance
(ED) between the transmitted codewords because this is the
most important criterion, when communicating over AWGN
channels [128]. Later, CM inspired by the turbo principle
has led to the so-called turbo trellis-coded modulation
(TTCM) concept [129], [130].
Another important technique, namely the so-called bit-
interleaved coded modulation (BICM) was conceived by
Zhavi for fading channels [131]. Although BICM is inferior
to TCM in terms of its ED, it outperforms TCM for
transmission over fading channels as a benefit of its diversity
gain. The original motivation of involving bit-interleavers
was to improve the performance for transmission over
fast-fading channels, because for such fading channels,
the most important parameter of the code is its diversity
gain rather than its ED. Morever, BICM exhibited very
good performance for transmission over AWGN channels
as well. This is the primary reason why BICM gained
interest among researchers, but it also exhibits substantial
flexibility in terms of its code design. In contrast to both
TCM and TTCM, where the coding rate of n/(n + 1) must
be carefully matched to the modulation constellation, BICM
allows the constellation and the encoder to be designed more
independently. The block diagram of a BICM transmitter
is shown in Fig. 7 for a bk-bit QAM scheme, while the
matching SISO turbo multiuser detector [99] portrayed in
Fig. 8.
VII. COMPLEXITY OF DETECTORS
The computational complexity of the existing MMSE-PIC,
MPA, and PDA algorithms is compared in Table 3.
TABLE 3. Computational Complexity Comparison
Algorithms Complexity Main procedures
MMSE-PIC O(K2) multiplication, addition
MPA O(Mdf ) multiplication, addition
PDA O(K3) multiplication, addition
In the MMSE-PIC detector the MMSE filter, which
requires matrix inversion, does not have to update the filter
for every signaling interval when transmitting over AWGN
channels, since there are no changes in the channel condi-
tions. Explicitly, it can be computed before communications,
given the prior knowledge of the spreading sequence of each
user and the noise variance. By contrast, for non-dispersive
VOLUME 4, 2016 19](https://image.slidesharecdn.com/ldsieeeaccess2020lajosfinaldraft1-250215201455-18753086/85/LDS_IEEE_Access2020_Lajos_Final_Draft-1-pdf-19-320.jpg)
![TABLE 4. LDS spreading code set coefficients for 4 × 6
a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11
C 1 ej60◦
ej120◦
C4 1 ej30◦
ej60◦
C3 1 ej0.143π ej0.202π ej0.313π ej0.574π ej0.377π ej0.394π ej0.267π ej0.308π
Cp 0.223 0.975 0.975 0.223 0.519 0.855 0.855 0.519 0.339 0.941 0.941 0.339
ej0.841π e−j0.455π ej0.455π ej0.159π e−j0.024π e−j0.943π ej0.943π e−j0.976π e−j0.964π 1 ej0.036π 1
C2 0.646 0.764 0.679 0.734 0.852 0.524 0.729 0.684 0.743 0.670 0.594 0.829
e−j0.073π 1 ej0.655π e−j0.004π e−j0.639π ej0.027π e−j0.740π 1 ej0.590π 1 e−j0.010π ej0.024π
Note that a0 coefficient of CP
can be read as a0 = 0.223ej0.841π
.
and frequency-selective channels, it needs updating of the
correlation matrix, as and when the channel gain changes.
To avoid the regular recomputation of the filter coefficients,
adaptive algorithms may be used, such as the recursive least
squares or least mean squares techniques [132] for directly
updating the inverse. Additionally, both the MPA and the
PDA algorithms will also require some additional processing
to adapt to the channel conditions [84].
In contrast to the MMSE-PIC, the MPA does not need to
perform any matrix inversion, but its complexity increases
exponential by both with the size of the symbol alphabet M
and number of non-zero positions of the spreading wave-
form df [133]. Finally, the PDA requires matrix inversion,
but fortunately this can be carried out quite efficiently with
the aid of the Sherman–Morrison–Woodbury formula at an
overall complexity order of O(K3
) [116].
VIII. COMPARISONS WITH OTHER LDS DESIGNS
In this section, we evaluate the performance of the proposed
LDS code sequences generated by the algorithm of Table 2
for the LDS sequence designs of sizes 4 × 6 and 6 × 9.
A. UNCODED LDS
Simulations are performed for transmission over the com-
plex AWGN channel using an identical transmission power
for each user, whilst relying on unit-energy LDS sequences
and no channel encoding. In the first experiment, we com-
pare the LDS code matrices, of (84) and (85) that are
generated by our proposed algorithm to the code matrices
derived in [32], and shown in (87) and (88). The proposed
algorithm is ran using the following parameters L = 4,
K = 6, δ = 1.9, σ2
d = 0.5 and s = 2. The initialization of
the matrix C was performed, as discussed in Section IV-A,
which results in a code set described as follows,
Cp
=
a0 0 a4 0 0 a10
a1 0 a5 0 0 a11
0 a2 0 a6 a8 0
0 a3 0 a7 a9 0
, (84)
where all the corresponding coefficients ai are described in
Table 4. We run again the algorithm, but this time with the
random initialization of the matrix C with δ = 1.7, which
outputs the following code set,
C2
=
0 a2 a4 0 0 a10
a0 0 0 a6 a8 0
0 a3 a5 0 0 a11
a1 0 0 a7 a9 0
, (85)
where all the corresponding coefficients ai are described in
Table 4. For fair comparison, we take the existing LDS code
sets presented in [32], [39], and [35] and label them as C,
C3
, and C4
, which are then normalized as follows,
C =
1
√
2
a0 a1 a2 0 0 0
a0 0 0 a1 a2 0
0 a0 0 a1 0 a2
0 0 a0 0 a1 a2
, (86)
C3
=
1
√
2
a0 a1 a2 0 0 0
a0 0 0 a3 a5 0
0 a1 0 a4 0 a7
0 0 a2 0 a6 a8
, (87)
C4
=
1
√
2
a0 a1 a2 0 0 0
a0 0 0 a1 a2 0
0 a0 0 a1 0 a2
0 0 a0 0 a1 a2
, (88)
where all the corresponding coefficients ai derived for each
code set are described in Table 4. The properties of the
matrices in our comparisons are summarized at a glance in
Table 5.
TABLE 5. Comparisons (C4×6)
Metric C Cp
C2 C3 C4
dE,min 2.00 2.00 2.00 1.83 1.47
∆ave 1.49 1.90 1.75 1.10 1.16
dP,min 2.00 0.76 0.04 0.11 0.29
dM,min 2.83 2.40 2.38 2.83 2.83
Csum 4.95 5.23 5.29 4.99 4.99
S1,2,ave 0.59 0.72 0.60 0.59 0.59
MWBE No No Yes No No
As seen in Fig. 9, Cp
outperforms the other candidates
(e.g., C, C3
and C4
), when ML detection is used. Although
our proposed matrices, Cp
and C2
, have the same dmin,
they have a higher value of ∆ave compared to C, C3
and
C4
. Furthermore, C2
is considered to be a MWBE matrix,
whereas all the other candidates are not. However, the code
set Cp
exhibits better BER performance than C2
.
Therefore, we surmise that the BER performance depends
not only on the minimum distance (e.g., dmin), but also
on the average Gaussian separability margin ∆ave. We also
note that the average sparsity of our proposed matrix Cp
defined in (26) is higher than that of its counterparts, which
is shown in bold in Table 5. In the case of the code sets
having dimensions of 6 × 9, we illustrate the code sets
generated by Table 2 Cp
, C3
, C4
and C5
. The initialization
20 VOLUME 4, 2016](https://image.slidesharecdn.com/ldsieeeaccess2020lajosfinaldraft1-250215201455-18753086/85/LDS_IEEE_Access2020_Lajos_Final_Draft-1-pdf-20-320.jpg)
![of matrix C is performed as discussed in Section IV-A using
δ = 1.8, which results in the following code set,
Cp
=
a0 a2 0 0 0 0 0 a12 0
a1 a3 0 0 0 0 a12 0 0
0 0 a4 a6 0 0 0 0 0
0 0 a5 a7 0 0 0 0 0
0 0 0 0 a8 a10 0 0 0
0 0 0 0 a9 a11 0 0 a12
, (89)
Note in our simulations, ‘off-line’ computation is assumed,
however ‘on-line’ computation can be performed upon any
changes such as channel conditions, L and s-sparseness, etc.
FIGURE 9. Uncoded BPSK case comparisons of C4×6 with labels LDS [32],
LDS3 [39], LDS4 [35].
We run the algorithm once again, but this time with random
initialization of the matrix C, in conjunction with δ = 1.7,
δ = 1.65 and δ = 1.65, which outputs the following code
sets,
C3
=
a0 0 0 a6 0 0 a12 0 0
a1 0 0 a7 0 0 a13 0 0
0 a2 0 0 a8 0 0 a14 0
0 a3 0 0 a9 0 0 a15 0
0 0 a4 0 0 a10 0 0 a16
0 0 a5 0 0 a11 0 0 a17
, (90)
C4
=
a0 0 0 a6 0 0 a12 0 0
0 a2 0 0 a8 0 0 a14 0
0 0 a4 0 0 a10 0 0 a16
a1 0 0 a7 0 0 a13 0 0
0 a3 0 0 a9 0 0 a15 0
0 0 a5 0 0 a11 0 0 a17
, (91)
C5
=
a0 0 0 a6 0 0 a12 0 0
0 a2 0 0 a8 0 0 a14 0
a1 0 0 a7 0 0 a13 0 0
0 a3 0 0 a9 0 0 a15 0
0 0 a4 0 0 a10 0 0 a16
0 0 a5 0 0 a11 0 0 a17
, (92)
where all the corresponding coefficients ai for each code
set are described in Table 6. For comparison we consider
the existing LDS sequence sets proposed in [32] and label
them as C and C2
, which are then normalized as follows,
C =
1
√
2
0 0 a2 0 0 0 a1 0 a0
0 a2 0 a1 0 0 a0 0 0
0 0 a1 a0 0 a2 0 0 0
a0 0 0 0 a1 0 0 0 a2
a2 0 0 0 0 a0 0 a1 0
0 a1 0 0 a0 0 0 a2 0
, (93)
C2
=
1
√
2
0 0 a0 0 0 0 a1 0 a2
0 a0 0 a1 0 0 a2 0 0
0 0 a0 a1 0 a2 0 0 0
a0 0 0 0 a1 0 0 0 a2
a0 0 0 0 0 a1 0 a2 0
0 a0 0 0 a1 0 0 a2 0
, (94)
where all the corresponding coefficients ai for each code
set are described in Table 6. Table 7 shows the comparison
metric of all the LDS sequences. Similar to the case of
the 4 × 6 code set, observe in Fig. 10 that the proposed
Cp
outperforms other LDS sequences in terms of its BER
performance.
FIGURE 10. Uncoded BPSK case comparisons of C6×9 code sets with [32]
labeled as LDS.
We again observe that the average Gaussian separability
value, ∆ave, and the average sparsity, S1,2,ave, are higher
for the matrix Cp
compared the other matrices. In order
to further characterize the performance, we performed sim-
ulations using channel encoding, as discussed in the next
section.
B. CODED LDS
Compared to the LDS designs conceived in [32], [35], [39],
the sum rate Csum of our proposed codes is higher by about
0.30 bits per channel use. Hence it is expected that there is
a channel code for our proposed LDS sequences that can
produce a higher coded sum rate than those advocated in
[32], [35], [39].
Therefore, to illustrate this hypothesis, we performed sim-
ulations using LDPC, turbo and polar encoding to compare
VOLUME 4, 2016 21](https://image.slidesharecdn.com/ldsieeeaccess2020lajosfinaldraft1-250215201455-18753086/85/LDS_IEEE_Access2020_Lajos_Final_Draft-1-pdf-21-320.jpg)
![TABLE 6. LDS spreading code set coefficients for 6 × 9
a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 a16 a17
C 1 ej60◦
ej120◦
C2
1 ej60◦
ej120◦
Cp 0.951 0.310 0.310 0.951 0.473 0.881 0.881 0.473 0.783 0.622 0.622 0.783 1.000
−0.450 −0.682 0.682 −0.550 −0.870 −0.941 0.941 −0.130 0.416 −0.367 0.367 0.584 0.000
C3 0.875 0.484 0.600 0.800 0.862 0.510 0.658 0.753 0.841 0.541 0.664 0.748 0.547 0.837 0.658 0.753 0.562 0.827
−0.234 0.000 −0.778 0.000 −0.441 0.000 0.440 0.000 −0.130 0.000 0.888 0.000 −0.853 0.000 0.532 0.000 0.189 0.000
C4 0.351 0.937 0.785 0.620 0.865 0.502 0.865 0.502 0.615 0.789 0.419 0.908 0.793 0.609 0.711 0.704 0.760 0.651
−1.000 0.000 0.046 −0.981 0.595 0.000 −0.448 −0.020 0.400 0.030 −0.818 0.000 0.340 0.000 −0.300 0.000 −0.140 0.000
C5 0.317 0.949 0.953 0.302 0.743 0.670 0.780 0.626 0.500 0.866 0.710 0.705 0.890 0.457 0.583 0.812 0.667 0.745
0.023 1.000 −0.197 0.000 0.187 1.000 0.706 1.000 0.362 0.000 −0.145 0.000 −0.482 −1.000 −0.841 0.000 0.527 0.000
Note that a0 and a12 coefficients of CP
can be interpreted as a0 = 0.951e−j0.450π
and a12 = 1.000.
our proposed LDS sequences to the ones advocated in [32],
[35], [39].
TABLE 7. Comparisons (C6×9)
Metric C Cp
C2 C3 C4 C5
dE,min 2.00 2.00 2.00 2.00 1.81 1.60
∆ave 1.34 1.88 1.34 1.76 1.62 1.64
dP,min 2.00 1.10 2.00 0.03 0.26 0.02
dM,min 2.83 2.00 2.83 2.37 2.13 1.85
Csum 7.35 7.75 7.39 7.93 7.93 7.93
S1,2,ave 0.71 0.84 0.71 0.73 0.74 0.75
MWBE No No No No Yes Yes
We used three different error control codes. The first is a
custom semi-random parity check matrix generator for the
LDPC code as described in [134]. The second, we used the
long-term evolution (LTE) turbo code described in [135].
-4 -2 0 2 4 6 8
Eb
/N0
(dB)
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
Average
BER
LDS MMSE-PIC
LDS proposed MMSE-PIC
LDS2 MMSE-PIC
LDS3 MMSE-PIC
LDS4 MMSE-PIC
Single User
Shannon Limit
FIGURE 11. BPSK with LDPC encoding comparisons of C4×6 code sets with
labels LDS [32], LDS3 [39], LDS4 [35].
The third we used the polar code for which we calculated the
Bhattacharyya parameters for the bit channel construction
method as described in [136]. The construction of the LTE
turbo interleaver is based on the quadratic permutation
polynomial (QPP) scheme of [135]. For all three channel
encoding cases we used a code rate of 1/3 with input
message block lengths of 320 bits and the encoded code
length of 972 for both the LDPC and LTE turbo codes.
Furthermore, we used 340 input, and 1024 encoded code bits
for polar coding. All of the output codewords the channel
coders are then interleaved as in the BICM scheme discussed
in Section VI.
-4 -2 0 2 4 6 8
Eb
/N0
(dB)
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
Average
BER
LDS MMSE-PIC
LDS proposed MMSE-PIC
LDS2 MMSE-PIC
LDS3 MMSE-PIC
LDS4 MMSE-PIC
LDS5 MMSE-PIC
Single User
Shannon Limit
FIGURE 12. BPSK with LDPC encoding comparisons of C6×9 code sets with
[32] labeled as LDS.
The BER performance of the LDS sequences shown in Figs.
11 - 14 for BPSK modulation shows that our proposed LDS
code sets outperform the ones proposed in [32], [35], [39]
for these coded cases. Thus trend is more clear when using
LDPC encoding, as shown in Figs. 11 and 12 rather than
LTE turbo encoding, shown in Figs. 13 and 14. In addition
to the MMSE-PIC detector we applied both PDA [116] and
SISO MMSE [99] detectors for BPSK modulation, which
are characterized in Figs. 15 and 16. Our propsoed LDS
based scheme outperforms the code set of [32] in terms of
its BER performance for both the PDA and SISO MMSE
detectors.
The complex PDA detector [122] was adopted for QAM,
is characterized in Figs. 17-20. The SCMA scheme associ-
ated with a factor graph of 4×6 and M = 4 is compared to
the LDS spreading matrix of size 4×6 using 4QAM in Figs.
18-20. We observe that the proposed LDS outperforms the
SCMA arrangement using an MPA detector and the LDS of
[32]. Similar results are also presented in Figs. 21 and 22
for bit-based PDA detection in [123].
22 VOLUME 4, 2016](https://image.slidesharecdn.com/ldsieeeaccess2020lajosfinaldraft1-250215201455-18753086/85/LDS_IEEE_Access2020_Lajos_Final_Draft-1-pdf-22-320.jpg)
![C. LDS CODE SETS FOR 200% OVERLOAD FACTOR
In this section we evaluate the BER performance of our LDS
sets for a normalized load factor of β = K/L = 2. More
explicitly, we have constructed 4 × 8, 6 × 12 and 8 × 16
LDS code sets using our proposed algorithm presented in
Table 2.
-4 -3 -2 -1 0 1 2 3 4 5
Eb
/N0
(dB)
10
-6
10
-4
10
-2
10
0
Average
BER
LDS MMSE-PIC
LDS proposed MMSE-PIC
LDS2
MMSE-PIC
LDS3
MMSE-PIC
LDS
4
MMSE-PIC
Single User
Shannon Limit
FIGURE 13. BPSK with turbo encoding comparisons of C4×6 code sets with
labels LDS [32], LDS3 [39], LDS4 [35].
The resultant column vectors have only two non-zero values.
For comparison purposes, we also included LDS sets asso-
ciated with β = K/L = 2 from the designs found in [22]
and [34]. The minimum Euclidean distance for the 4 × 8,
and 6 × 12 LDS code sets of [22] are 1.17 and 1.43, whilst
for the proposed sets they are 2.0, respectively.
-4 -3 -2 -1 0 1 2 3 4
Eb
/N0
(dB)
10-6
10
-5
10-4
10
-3
10
-2
10
-1
Average
BER
LDS MMSE-PIC
LDS proposed MMSE-PIC
LDS
2
MMSE-PIC
LDS
3
MMSE-PIC
LDS
4
MMSE-PIC
LDS5
MMSE-PIC
Single User
Shannon Limit
FIGURE 14. BPSK with turbo encoding comparisons of C6×9 code sets with
[32] labeled as LDS.
Similarly, the average Gaussian separability margins for the
4 × 8, 6 × 12, and 8 × 16 LDS code sets of [22] are 0.96,
0.0 and 1.48, whilst for the proposed sets they are 1.79, 1.6
and 1.66, respectively.
The reason why the LDS code sets in [22] and [34] are
selected as the benchmarks is because the BER performance
of other LDS candidates is similar for the 200% normalized
load factor scenarios.
-4 -3 -2 -1 0 1 2 3
Eb
/N0
(dB)
10
-7
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
Average
BER
LDS SISO MMSE
LDS Proposed SISO MMSE
LDS PDA
LDS proposed PDA
Single User
Shannon Limit
FIGURE 15. BPSK with turbo encoding comparisons of C4×6 code sets with
[32] labeled as LDS.
Observe in Figs. 23, 27, 29 and Figs. 26, 28, 30 that our
proposed LDS code sets designed both for uncoded and
turbo coded scenarios outperform the LDS code sets of [22]
and [34]. At the BER of 10−3
there is about 1 − 2 dB SNR
gain for the uncoded scenarios and a slighter smaller SNR
gain is observed for coded scenarios.
-4 -3 -2 -1 0 1 2 3 4 5
Eb
/N0
(dB)
10
-6
10-5
10
-4
10
-3
10-2
10
-1
Average
BER
LDS SISO MMSE
LDS Proposed SISO MMSE
LDS PDA
LDS proposed PDA
Single User
Shannon Limit
FIGURE 16. BPSK with polar encoding comparisons of C4×6 code sets with
[32] labeled as LDS.
As seen in Figs. 23, 26-31, the proposed LDS code sets
tend to approach the single-user BER. This is achieved as
a benefit of the diversity gain obtained when splitting the
complex vectors and matrices into real and imaginary parts,
as discussed in the context of (73). For the BPSK case, our
complex LDS matrix C of size L × K is transformed into
CR of size 2L×K after splitting it into real and imaginary
VOLUME 4, 2016 23](https://image.slidesharecdn.com/ldsieeeaccess2020lajosfinaldraft1-250215201455-18753086/85/LDS_IEEE_Access2020_Lajos_Final_Draft-1-pdf-23-320.jpg)
![parts. In order to have an orthogonal matrix CR, the number
of users should be K = 2L, hence we have β = 2.
-4 -2 0 2 4 6
Eb
/N0
(dB)
10
-6
10
-5
10
-4
10
-3
10-2
10-1
10
0
Average
BER
LDS PDA1
LDS proposed PDA1
Single User
Shannon Limit
FIGURE 17. QAM with LDPC encoding, iteration number = 1, comparisons of
C4×6 code sets with [32] labeled as LDS.
The proposed LDS construction seen in Table 2 provides
an LDS matrix, so that when we convert it into its real
and imaginary parts, the resultant CR matrix becomes near-
orthogonal. This explains the reason for having a near-
single-user BER performance for the proposed LDS, but
we also observe that the BER performance deteriorates
dramatically for scenarios of K 2L. On the other hand
in contrast to BPSK, for QAM signaling, the LDS matrix
C is converted into CR of size 2L × 2K after the real and
imaginary parts are split.
-4 -3 -2 -1 0 1 2 3 4 5 6
Eb
/N0
(dB)
10-6
10-5
10
-4
10-3
10-2
10
-1
10
0
Average
BER
LDS PDA
1
LDS proposed PDA
1
SCMA MPA
Single User
Shannon Limit
FIGURE 18. QAM with LDPC encoding, iteration number = 5, comparisons of
C4×6 code sets with [32] labeled as LDS.
Therefore, CR cannot be orthogonal, unless we have L =
K, as verified by our simulations.
Furthermore, the orthogonality of C̃R in BPSK signalling
can be further degraded, when no transmitter precoding
is utilized for transmission over frequency-selective fading
channels. The performance difference of LDS codes over
non-dispersive and frequency-selective fading channels are
portrayed in Figs. 32 and 33. In our simulations, we assumed
Lp = 7 for the frequency-selective fading channel.
-4 -3 -2 -1 0 1 2 3
Eb
/N0
(dB)
10
-6
10-5
10
-4
10
-3
10-2
10-1
10
0
Average
BER
LDS PDA1
LDS proposed PDA1
SCMA MPA
Single User
Shannon Limit
FIGURE 19. QAM with turbo encoding, iteration number = 1, comparisons of
C4×6 code sets with [32] labeled as LDS.
As for future research, we have to perform a detailed
stochastic analysis of the legitimate LDS code sets for
complex signal constellations to find the specific sets, which
have the best performance. We also have to study the direct
optimization of the matrix CR in real domain in order to
achieve near-orthogonality, instead of optimizing C in the
complex domain under the constraint of keeping the non-
zero locations of the real and imaginary parts identical.
-4 -3 -2 -1 0 1 2 3 4
Eb
/N0
(dB)
10-6
10-5
10
-4
10-3
10-2
10
-1
10
0
Average
BER
LDS PDA1
LDS proposed PDA1
SCMA MPA
Single User
Shannon Limit
FIGURE 20. QAM with polar encoding, iteration number = 1, comparisons of
C4×6 code sets with [32] labeled as LDS.
As for QAM, we can investigate the design of matrices for
the rank-deficient scenarios of K L in the real domain
instead of the complex domain. Furthermore, we have to
conceive LDS designs for ensuring that the resultant C̃R is
near-orthogonal even under fading channels.
24 VOLUME 4, 2016](https://image.slidesharecdn.com/ldsieeeaccess2020lajosfinaldraft1-250215201455-18753086/85/LDS_IEEE_Access2020_Lajos_Final_Draft-1-pdf-24-320.jpg)
![D. SPECTRAL EFFICIENCY
One of the key performance metrics of LDS spreading
code design is the resultant spectral efficiency, ηLDS(C, γ)
(bits/s/Hz), which can be expressed as a function of either
the SNR, γ or of the energy per bit Eb/No.
-4 -3 -2 -1 0 1 2 3 4 5 6
Eb
/N0
(dB)
10
-5
10
-4
10
-3
10-2
10
-1
10
0
Average
BER
LDS PDA2
LDS proposed PDA2
Single User
Shannon Limit
FIGURE 21. QAM with LDPC encoding, iteration number = 5, comparisons of
C4×6 code sets with [32] labeled as LDS.
The spectral efficiency ηLDS(C, γ) is defined as the
maximum mutual information between the symbol vector
x and the observed L-dimensional vector y in (3) for a
given C over distributions of x normalized to L.
-4 -3 -2 -1 0 1 2 3 4
Eb
/N0
(dB)
10-6
10-5
10
-4
10
-3
10-2
10
-1
10
0
Average
BER
LDS PDA2
LDS proposed PDA
2
SCMA MPA
Single User
Shannon Limit
FIGURE 22. QAM with turbo encoding, iteration number = 1, comparisons of
C4×6 code sets with [32] labeled as LDS.
Under the constraint of E{xxH
} = EsIL, the opti-
mum detection for a given LDS C may be achieved; for
a Gaussian distributed x the resultant spectral efficiency
ηLDS(C, γ) can be expressed by [10]
ηLDS(C, γ) =
Csum(C, γ)
L
=
1
L
log2 |IL + γCDDH
CH
|,
(95)
where Es denotes energy per symbol, NoIL is the noise
covariance and the per-symbol SNR γ is given by [137]
γ =
1
K E{|x|2
}
1
L E{|n|2}
=
1
K EbNb
1
L NoL
=
1
β
Eb
No
Nb
L
=
1
β
Eb
No
ηLDS,
(96)
-2 0 2 4 6 8
Eb
/N0
(dB)
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
Average
BER
LDS MMSE-PIC
LDS proposed MMSE-PIC
Single User
Shannon Limit
FIGURE 23. Uncoded BPSK transmission, C4×8 code sets with [22] labeled
as LDS.
where we have E{|x|2
} = NbEb, E{|n|2
} = LNo, Nb
denotes the number of bits encoded in x for a capacity-
achieving system, and Nb/L, which is expressed in bits per
dimension, represents the spectral efficiency of (95). Since
L denotes the number of complex dimensions in our system,
ηLDS(C, γ), can be interpreted as the maximum number of
bits per each complex dimension.
-4 -2 0 2 4 6 8
SNR (dB)
0.5
1
1.5
2
2.5
3
3.5
4
Spectral
efficiency
(bits/s/Hz)
LDS 200%
LDS 200% proposed
LDS 4x6
LDS 4x6 proposed
LDS 6x9
LDS 6x9 proposed
Unrestricted bound 200%
Unrestricted bound 150%
MMSE bound 200%
MMSE bound 150%
2 2.5 3
1.8
2
2.2
FIGURE 24. Spectral efficiency vs SNR of the AWGN channel for BPSK.
An upper bound on ηLDS(C, γ) can be considered as the
spectral efficiency, when the LDS spreading sequence has a
length of L = 1. This is equivalent of a K-user Gaussian
multiple access channel and its spectral efficiency in the case
of the average-energy-constraint is given by log2 (1 + γdtot)
bits/s/Hz per chip [87], where dtot = c0
1c0H
1 = · · · =
VOLUME 4, 2016 25](https://image.slidesharecdn.com/ldsieeeaccess2020lajosfinaldraft1-250215201455-18753086/85/LDS_IEEE_Access2020_Lajos_Final_Draft-1-pdf-25-320.jpg)
![c0
Lc0H
L , and c0
i are row vectors of CD. We can show that
ηLDS(C, γ) is indeed capable of achieving the upper bound
even when L 1.
-2 -1 0 1 2 3 4 5 6 7
Eb
/N0
(dB)
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Spectral
efficiency
(bits/s/Hz)
LDS 200%
LDS 200% proposed
LDS 4x6
LDS 4x6 proposed
LDS 6x9
LDS 6x9 proposed
Unrestricted bound
MMSE bound
2.6 2.8 3 3.2 3.4
2
2.2
2.4
2.6
2.8
FIGURE 25. Spectral efficiency vs Eb/No of the AWGN channel for BPSK.
Proposition 1. Let C be an LDS spreading matrix with D
being the diagonal energy-constraint matrix and K L.
Then, we have:
ηLDS(C, γ) ≤ log2 (1 + γdtot). (97)
-4 -2 0 2 4 6 8
Eb
/N0
(dB)
10-5
10
-4
10
-3
10
-2
10
-1
Average
BER
LDS MMSE-PIC
LDS proposed MMSE-PIC
Single User
Shannon Limit
FIGURE 26. Turbo coded BPSK transmission, C4×8 code sets with [22]
labeled as LDS.
The necessary and sufficient condition of attaining the
spectral efficiency upper bound of the system dispensing
with spreading when the LDS signature waveforms are WBE
sequences, is that of satisfying the condition CDDH
CH
=
dtotIL [87].
Proof. The proposition can be proved by first applying
Hadamard’s inequality [138] to the determinant in (95),
yielding:
|IL + γdtotIL| ≤
L
Y
i=1
(1 + γdtot). (98)
Indeed the above expression satisfies the condition of equal-
ity, since the determinant is a diagonal matrix. Then using
Jensen’s inequality, we have [138]:
log2 |IL + γdtotIL| = log2
L
Y
i=1
(1 + γdtot)
=
L
X
i=1
log2 (1 + γdtot)
= L log2 (1 + γdtot). (99)
-2 0 2 4 6 8 10
Eb
/N0
(dB)
10
-6
10
-5
10
-4
10
-3
10
-2
10-1
10
0
Average
BER
LDS MMSE-PIC
LDS proposed MMSE-PIC
Single User
Shannon Limit
FIGURE 27. Uncoded BPSK transmission, C6×12 code sets with [22] labeled
as LDS.
Since in our design the columns of C are of unit-length
and D = IK, the Frobenius norm of CD can be written as
||CD||F =
K
X
k=1
cH
k ck =
L
X
i=1
c0
ic0H
i = K. (100)
-4 -2 0 2 4 6
Eb
/N0
(dB)
10-6
10-5
10
-4
10
-3
10-2
10-1
100
Average
BER
LDS MMSE-PIC
LDS proposed MMSE-PIC
Single User
Shannon Limit
FIGURE 28. Turbo coded BPSK transmission, C6×12 code sets with [22]
labeled as LDS.
26 VOLUME 4, 2016](https://image.slidesharecdn.com/ldsieeeaccess2020lajosfinaldraft1-250215201455-18753086/85/LDS_IEEE_Access2020_Lajos_Final_Draft-1-pdf-26-320.jpg)
![-2 0 2 4 6 8 10
E
b
/N
0
(dB)
10
-5
10-4
10
-3
10
-2
10
-1
Average
BER
LDS MMSE-PIC
LDS proposed MMSE-PIC
Single User
Shannon Limit
FIGURE 29. Uncoded BPSK transmission, C8×16 code sets with [22] labeled
as LDS.
Therefore, we have dtot = K/L as c0
1c0H
1 = · · · =
c0
Lc0H
L . Note that if the K users do not have equal average-
input-energy constraints, i.e., DDH
6= d0
IL, it is generally
hard to design an LDS code set that maximizes ηLDS(C, γ)
in Proposition 1.
-4 -3 -2 -1 0 1 2 3 4 5
Eb
/N0
(dB)
10-8
10
-6
10
-4
10
-2
10
0
Average
BER
LDS MMSE-PIC
LDS proposed MMSE-PIC
Single User
Shannon Limit
FIGURE 30. Turbo coded BPSK transmission, C8×16 code sets with [22]
labeled as LDS.
In all our BER performance plots, the information rates
for LDS 4 × 6, 6 × 9, 4 × 8, 6 × 12 and 8 × 16 in case of
BPSK are ηLDS = Nb/L = 1.5, ηLDS = 1.5, ηLDS = 2,
ηLDS = 2, and ηLDS = 2 bits/s/Hz, respectively. Therefore,
the corresponding unrestricted Shannon limits are calculated
by using the upper bound log2 (1 + γβ) (97) as Eb/No =
(2ηLDS
−1)/ηLDS, Eb/No = 1.219(0.86dB) and Eb/No =
1.5 (1.76dB) for ηLDS = 1.5 and ηLDS = 2, respectively.
In case of 4QAM (2 bits per symbol) the corresponding
ηLDS is multiplied by 2 and for a channel coding rate of
1/3 by 1/3.
-4 -2 0 2 4 6 8
E
b
/N
0
(dB)
10-4
10
-3
10
-2
10
-1
Average
BER
LDS MMSE-PIC
LDS proposed MMSE-PIC
Single User
Shannon Limit
FIGURE 31. Turbo coded BPSK transmission, C8×16 code sets with [34]
labeled as LDS.
In addition to analysing the spectral efficiency of the
optimal detection, a range of linear detectors, such as the
single-user MF (SUMF), ZF, MMSE are derived in [64].
The spectral efficiency of these multiple access channels is
given by [64]:
Rsumf
lds (β, γ) = Rzf
lds(β, γ) = Rmmse
lds (β, γ)
= β
X
k≥0
βk
exp (−β)
k!
log2
1 +
γ
kγ + 1
0 2 4 6 8 10 12
Eb
/N0
(dB)
10-8
10-6
10
-4
10
-2
10
0
Average
BER
LDS MMSE-PIC
LDS proposed MMSE-PIC
Single User
FIGURE 32. Turbo coded BPSK transmission over non-dispersive fading
chananel, C8×16 code sets with [22] labeled as LDS.
where ! denotes factorial. The proposed LDS design ap-
proaches the upper bound of the spectral efficiency, as
shown in Figs. 24 and 25 for the case of optimal detection.
On average there is a 0.2 bits/s/Hz gap between our pro-
posed scheme and the existing state-of-the-art LDS designs.
According to Proposition 1 the proposed LDSs are WBE
sequences or exhibit WBE-like properties, as they approach
the upper bound. Having LDS code sets exhibiting optimal
VOLUME 4, 2016 27](https://image.slidesharecdn.com/ldsieeeaccess2020lajosfinaldraft1-250215201455-18753086/85/LDS_IEEE_Access2020_Lajos_Final_Draft-1-pdf-27-320.jpg)
![spectral efficiency inspires us to design low-complexity
detectors such as the MMSE-PIC arrangement, which is
capable of operating even beyond a normalized load factor
of 200%.
0 1 2 3 4 5 6 7 8
Eb
/N0
(dB)
10
-5
10
-4
10
-3
10
-2
10
-1
Average
BER
LDS MMSE-PIC
LDS proposed MMSE-PIC
Single User
FIGURE 33. Turbo coded BPSK transmission over frequency-selective fading
channel, C8×16 code sets with [22] labeled as LDS.
IX. CONCLUSION AND DESIGN GUIDELINES
In this paper, we have provided a comprehensive literature
review of LDS construction designs by considering the
most recent developments. Both the design and application
of LDS code sets have been described in Tables 2 and
3, respectively. Widely used design criteria conceived for
developing the LDS matrices have also been presented.
Moreover, we conceived an improved LDS sequence design
based on the Gaussian separability criterion. We demon-
strated that achieving the best BER performance depends
not only on the minimum distance, but also on the average
Gaussian separability margin.
Based on that criterion, we developed an iterative al-
gorithm that is based on maximizing the SINR of each
individual user of interest, which converges to the desired
solution. We select the optimum candidates having the
highest minimum distance and those associated with the
highest average Gaussian separability, which perform well
along with channel coding.
Our proposed LDS code set outperforms the existing LDS
designs both for BPSK and 4QAM transmission in terms of
its BER. We elaborate a little further on the design guide-
lines associated with the proposed algorithm and presented
in Table 2. More explicitly, as portrayed in Figs. 9 and 10,
our code design, conceived, for 4×6 and 6×9 constructions
provides some, modest, power gain compared to other code
designs without any increase in computational complexity
when using our codes. A compelling BER performance
is shown for the size of K = 2L. Our conclusion is
that the Gaussian separability margin has to be considered
when comparing code sets with equal minimum Euclidean
distance or TSC properties. We can summarize our design
guidelines as follows:
(1) For a given transmission channel, we have to determine
the number of users K, the length L of the waveform
sequence, the grade of sparseness, as well as the
parameters δ and σd, which are obtained heuristically,
as discussed in Section IV.
(2) The proposed designs jointly map the signals of the
users to REs in a sparse manner, they perform con-
stellation shaping and judiciously allocate the power
to each spreading sequence.
(3) The proposed algorithm is iterative, hence whenever
there is a change in the channel conditions and/or the
number of users K, we can re-run our algorithm to
produce new LDS codes. However, if for some reason
one should avoid adapting to the channel environment,
we suggest to use the average noise variance associated
with the maximum number of users. If less users
are present, using a subset of the LDS code sets is
recommended.
Furthermore, we also proposed a low-complexity minimum
mean-square estimation and parallel interference cancella-
tion aided detector, which exhibited a comparable BER
performance to that of ML detection. The MMSE-PIC
algorithm has however much lower complexity than the
MPA. In our future research we will conceive LDS designs
for higher-order constellations for transmission over disper-
sive fading channels and the radical direct minimum BER
optimization criterion of [139].
REFERENCES
[1] L. Dai, B. Wang, Z. Ding, Z. Wang, S. Chen, and L. Hanzo, “A survey of
non-orthogonal multiple access for 5G,” IEEE Commun. Surveys Tuts.,
vol. 20, no. 3, pp. 2294–2323, thirdquarter 2018.
[2] C. Bockelmann, N. K. Pratas, G. Wunder, S. Saur, M. Navarro, D. Gre-
goratti, G. Vivier, E. De Carvalho, Y. Ji, C. Stefanovic, P. Popovski,
Q. Wang, M. Schellmann, E. Kosmatos, P. Demestichas, M. Raceala-
Motoc, P. Jung, S. Stanczak, and A. Dekorsy, “Towards massive con-
nectivity support for scalable mMTC communications in 5G networks,”
IEEE Access, vol. 6, pp. 28 969–28 992, May 2018.
[3] A. Ghosh, A. Maeder, M. Baker, and D. Chandramouli, “5G evolution: A
view on 5G cellular technology beyond 3GPP release 15,” IEEE Access,
vol. 7, pp. 127 639–127 651, Sep. 2019.
[4] J. Ding, M. Nemati, C. Ranaweera, and J. Choi, “IoT connectivity tech-
nologies and applications: A survey,” IEEE Access, vol. 8, pp. 67 646–
67 673, Apr. 2020.
[5] L. Chettri and R. Bera, “A comprehensive survey on internet of things
(IoT) toward 5g wireless systems,” IEEE Internet of Things Journal,
vol. 7, pp. 16–32, Jan. 2020.
[6] C. Kalalas and J. Alonso-Zarate, “Massive connectivity in 5G and be-
yond: Technical enablers for the energy and automotive verticals,” in
Proc. 6G Wireless Summit (6G SUMMIT), Levi, Finland, May 2020, pp.
1–5.
[7] H. Kulhandjian, “A visible light communications framework for intel-
ligent transportation systems,” Mineta Transportation Institute Publica-
tions, Tech. Rep., Project No. 1911, August 2020.
[8] R. M. Marè, C. E. Cugnasca, C. L. Marte, and G. Gentile, “Intelligent
transport systems and visible light communication applications: An
overview,” in Proc. of IEEE International Conf. on Intelligent Trans-
portation Systems (ITSC), Rio de Janeiro, Brazil, Nov. 2016, pp. 2101–
2106.
[9] H. Sari, F. Vanhaverbeke, and M. Moeneclaey, “Extending the capacity
of multiple access channels,” IEEE Commun. Mag., vol. 38, no. 1, pp.
74–82, Jan. 2000.
28 VOLUME 4, 2016](https://image.slidesharecdn.com/ldsieeeaccess2020lajosfinaldraft1-250215201455-18753086/85/LDS_IEEE_Access2020_Lajos_Final_Draft-1-pdf-28-320.jpg)
![[10] S. Verdu and S. Shamai, “Spectral efficiency of CDMA with random
spreading,” IEEE Trans. Inf. Theory, vol. 45, no. 2, pp. 622–640, Mar.
1999.
[11] H. Sari, F. Vanhaverbeke, and M. Moeneclaey, “Multiple access using
two sets of orthogonal signal waveforms,” IEEE Commun. Lett., vol. 4,
no. 1, pp. 4–6, Jan. 2000.
[12] F. Vanhaverbeke, M. Moeneclaey, and H. Sari, “DS/CDMA with two
sets of orthogonal spreading sequences and iterative detection,” IEEE
Commun. Lett., vol. 4, no. 9, pp. 289–291, Sep. 2000.
[13] M. Kulhandjian and D. A. Pados, “Uniquely decodable code-division
via augmented Sylvester-Hadamard matrices,” in Proc. IEEE Wireless
Commun. and Network. Conf. (WCNC), Paris, France, Apr. 2012, pp.
359–363.
[14] M. Kulhandjian, C. D’Amours, and H. Kulhandjian, “Uniquely decod-
able ternary codes for synchronous CDMA systems,” in Proc. IEEE Pers.,
Indoor, Mobile Radio Conf. (PIMRC), Bologna, Italy, Sep. 2018, pp. 1–6.
[15] M. Kulhandjian, C. D’Amours, H. Kulhandjian, H. Yanikomeroglu,
D. A. Pados, and G. Khachatrian, “Fast decoder for overloaded uniquely
decodable synchronous CDMA,” arXiv:1806.03958 [eess.SP], Jun.
2018. [Online]. Available: https://arxiv.org/abs/1806.03958
[16] M. Kulhandjian, C. D’Amours, H. Kulhandjian, H. Yanikomeroglu,
and G. Khachatrian, “Fast decoder for overloaded uniquely decodable
synchronous optical CDMA,” in Proc. IEEE Wireless Commun. and
Network. Conf. (WCNC), Marrakech, Morocco, Apr. 2019, pp. 1–7.
[17] M. Basharat, W. Ejaz, M. Naeem, A. Masood Khattak, and A. Anpalagan,
“A survey and taxonomy on nonorthogonal multiple-access schemes for
5G networks,” Trans. on Emerging Telecommun. Technologies, vol. 29,
no. 1, p. e3202, Jun. 2017.
[18] Q. Wang, R. Zhang, L. Yang, and L. Hanzo, “Non-orthogonal multiple
access: A unified perspective,” IEEE Wireless Commun., vol. 25, no. 2,
pp. 10–16, Apr. 2018.
[19] N. Ye, H. Han, L. Zhao, and A.-H. Wang, “Uplink nonorthogonal
multiple access technologies toward 5G: A survey,” Wireless Commun.
and Mobile Comp., vol. 2018, no. article ID 6187580, pp. 1–26, Jun.
2018.
[20] M. B. Shahab, R. Abbas, M. Shirvanimoghaddam, and S. J.
Johnson, “Grant-free non-orthogonal multiple access for IoT: A
survey,” arXiv:1910.06529 [eess.SP], Oct. 2019. [Online]. Available:
https://arxiv.org/abs/1910.06529
[21] L. Dai, B. Wang, Y. Yuan, S. Han, C. I, and Z. Wang, “Non-orthogonal
multiple access for 5G: solutions, challenges, opportunities, and future
research trends,” IEEE Commun. Mag., vol. 53, no. 9, pp. 74–81, Sep.
2015.
[22] R. Hoshyar, F. P. Wathan, and R. Tafazolli, “Novel low-density signature
for synchronous CDMA systems over AWGN channel,” IEEE Trans.
Signal Process., vol. 56, no. 4, pp. 1616–1626, Apr. 2008.
[23] R. Razavi, M. AL-Imari, M. A. Imran, R. Hoshyar, and D. Chen, “On
receiver design for uplink low density signature OFDM (LDS-OFDM),”
IEEE Trans. Commun., vol. 60, no. 11, pp. 3499–3508, Nov. 2012.
[24] H. Nikopour and H. Baligh, “Sparse code multiple access,” in Proc. IEEE
Pers., Indoor, Mobile Radio Conf. (PIMRC), London, U.K., Sep. 2013,
pp. 332–336.
[25] L. Li, Z. Ma, P. Z. Fan, and L. Hanzo, “High-dimensional codebook
design for the SCMA down link,” IEEE Trans. on Vehic. Tech., vol. 67,
no. 10, pp. 10 118–10 122, Oct. 2018.
[26] S. Chen, B. Ren, Q. Gao, S. Kang, S. Sun, and K. Niu, “Pattern
division multiple access - a novel nonorthogonal multiple access for fifth-
generation radio networks,” IEEE Trans. on Vehic. Tech., vol. 66, no. 4,
pp. 3185–3196, Apr. 2017.
[27] Z. Yuan, G. Yu, W. Li, Y. Yuan, X. Wang, and J. Xu, “Multi-user shared
access for internet of things,” in Proc. IEEE Veh. Technol. Conf. (VTC
Spring), Nanjing, China, May 2016, pp. 1–5.
[28] M. Vameghestahbanati, I. D. Marsland, R. H. Gohary, and
H. Yanikomeroglu, “Multidimensional constellations for uplink SCMA
systems—a comparative study,” IEEE Commun. Surveys Tuts., vol. 21,
no. 3, pp. 2169–2194, thirdquarter 2019.
[29] M. Mohammadkarimi, M. A. Raza, and O. A. Dobre, “Signature-based
nonorthogonal massive multiple access for future wireless networks:
Uplink massive connectivity for machine-type communications,” IEEE
Veh. Technol. Magazine, vol. 13, no. 4, pp. 40–50, Dec. 2018.
[30] J. Choi, “Low density spreading for multicarrier systems,” in Proc.
IEEE Int. Symp. Spread Spectr. Techn. Appl. Programm. Book Abstracts
(ISSSTA), Sydney, Australia, Aug. 2004, pp. 575–578.
[31] R. Hoshyar, F. P. Wathan, and R. Tafazolli, “Novel low-density signature
structure for synchronous DS-CDMA systems,” in Proc. IEEE Global
Telecommun. Conf. (GLOBECOM), San Francisco, U.S.A., Nov. 2006,
pp. 1–5.
[32] J. van de Beek and B. M. Popović, “Multiple access with low-density
signatures,” in Proc. IEEE Global Telecommun. Conf. (GLOBECOM),
Honolulu, U.S.A., Nov. 2009, pp. 1–6.
[33] A. R. Safavi, A. G. Perotti, and B. M. Popović, “Ultra low density spread
transmission,” IEEE Commun. Lett., vol. 20, no. 7, pp. 1373–1376, Jul.
2016.
[34] T. Qi, W. Feng, and Y. Wang, “Optimal sequences for non-orthogonal
multiple access: A sparsity maximization perspective,” IEEE Commun.
Lett., vol. 21, no. 3, pp. 636–639, Mar. 2017.
[35] T. Qi, W. Feng, Y. Chen, and Y. Wang, “When NOMA meets sparse sig-
nal processing: Asymptotic performance analysis and optimal sequence
design,” IEEE Access, vol. 5, pp. 18 516–18 525, Jul. 2017.
[36] C. Jiang and Z. Wu, “A novel uplink NOMA scheme based on low
density superposition modulation,” in Proc. IEEE Veh. Technol. Conf.
(VTC-Fall), Toronto, Canada, Sep. 2017, pp. 1–5.
[37] T. Qi, W. Feng, and Y. Wang, “On the sparsity of spreading sequences for
NOMA with reliability guarantee and detection complexity limitation,”
in Proc. IEEE Veh. Technol. Conf. (VTC-Spring), Sydney, Australia, Jun.
2017, pp. 1–6.
[38] J. Zhang, X. Wang, X. Yang, and H. Zhou, “Low density spreading
signature vector extension (LDS-SVE) for uplink multiple access,” in
Proc. IEEE Veh. Technol. Conf. (VTC-Fall), Toronto, Canada, Sep. 2017,
pp. 1–5.
[39] G. Song, X. Wang, and J. Cheng, “Signature design of sparsely spread
code division multiple access based on superposed constellation distance
analysis,” IEEE Access, vol. 5, pp. 23 809–23 821, Oct. 2017.
[40] H. Yu, Z. Fei, N. Yang, and N. Ye, “Optimal design of resource element
mapping for sparse spreading non-orthogonal multiple access,” IEEE
Wireless Commun. Lett., vol. 7, no. 5, pp. 744–747, Oct. 2018.
[41] K. Zhang, J. Xu, C. Yin, W. Tan, and C. Li, “A sparse superposition
multiple access scheme and performance analysis for 5G wireless com-
munication systems,” in Proc. IEEE/CIC Int. Conf. on Commun. in China
(ICCC) Workshops, Kansas City, U.S.A., May 2018, pp. 90–95.
[42] H. Wang, K. Xiao, B. Xia, and J. Wang, “Performance analysis and
optimization for the LDPC-coded multi-carrier LDS system,” in Proc.
IEEE Veh. Technol. Conf. (VTC-Spring), Kuala Lumpur, Malaysia, Apr.
2019, pp. 1–5.
[43] K. Lu and C. Jiang, “Optimized low density superposition modulation
for 5G mobile multimedia wireless networks,” IEEE Access, vol. 7, pp.
174 227–174 235, Dec. 2019.
[44] Z. Liu, P. Xiao, and Z. Mheich, “Power-imbalanced low-density signa-
tures (LDS) from Eisenstein numbers,” in Proc. IEEE VTS Asia Pacific
Wireless Commun. Symp. (APWCS), Singapore, Aug. 2019, pp. 1–5.
[45] Y. Sun, “Quasi-large sparse-sequence CDMA: Approach to single-user
bound by linearly-complex LAS detector,” in Proc. IEEE Annual Conf.
on Inf. Sci. and Systems, Princeton, U.S.A., Mar. 2008, pp. 320–323.
[46] R. Lupas and S. Verdu, “Linear multiuser detectors for synchronous code-
division multiple-access channels,” IEEE Trans. Inf. Theory, vol. 35,
no. 1, pp. 123–136, Jan. 1989.
[47] B. Hassibi and H. Vikalo, “On the sphere-decoding algorithm I. Expected
complexity,” IEEE Trans. Signal Processing, vol. 53, no. 8, pp. 2806–
2818, Aug. 2005.
[48] G. Romano, F. Palmieri, P. K. Willett, and D. Mattera, “Separability and
gain control for overloaded CDMA,” in Proc. of the Conf. on Inf. Sci. and
Sys. (CISS), Princeton, NJ, USA, Mar. 2005, pp. 1–6.
[49] M. K. Varanasi, “Decision feedback multiuser detection: a systematic
approach,” IEEE Trans. Inf. Theory, vol. 45, no. 1, pp. 219–240, Jan.
1999.
[50] Y. Sun and J. Xiao, “Multicode sparse-sequence CDMA: Approach to
optimum performance by linearly complex wslas detectors,” Wireless
Pers. Commun., vol. 71, no. 2, pp. 1049–1056, Jul. 2013.
[51] P. G. Casazza, A. Heinecke, F. Krahmer, and G. Kutyniok, “Optimally
sparse frames,” IEEE Trans. Inf. Theory, vol. 57, no. 11, pp. 7279–7287,
Nov. 2011.
[52] K. Xiao, B. Xia, Z. Chen, J. Wang, D. Chen, and S. Ma, “On optimiz-
ing multicarrier-low-density codebook for GMAC with finite alphabet
inputs,” IEEE Commun. Lett., vol. 21, no. 8, pp. 1811–1814, Aug. 2017.
[53] X. Liu, H. Chen, M. Peng, and F. Yang, “Identical code cyclic shift
multiple access—a bridge between CDMA and NOMA,” IEEE Trans.
on Vehic. Tech., vol. 69, no. 3, pp. 2878–2890, Mar. 2020.
VOLUME 4, 2016 29](https://image.slidesharecdn.com/ldsieeeaccess2020lajosfinaldraft1-250215201455-18753086/85/LDS_IEEE_Access2020_Lajos_Final_Draft-1-pdf-29-320.jpg)
![[54] C. Jiang and Y. Wang, “An uplink SCMA codebook design combining
probabilistic shaping and geometric shaping,” IEEE Access, vol. 8, pp.
76 726–76 736, Apr. 2020.
[55] G. Song, K. Cai, Y. Chi, J. Guo, and J. Cheng, “Super-sparse on-off
division multiple access: Replacing repetition with idling,” IEEE Trans.
Commun., vol. 68, no. 4, pp. 2251–2263, Apr. 2020.
[56] N. Ye, X. Li, H. Yu, L. Zhao, W. Liu, and X. Hou, “DeepNOMA: A
unified framework for NOMA using deep multi-task learning,” IEEE
Trans. Commun., vol. 19, no. 4, pp. 2208–2225, Apr. 2020.
[57] H. Xie, W. Xu, H. Q. Ngo, and B. Li, “Non-coherent massive MIMO sys-
tems: A constellation design approach,” IEEE Trans. Wireless Commun.,
vol. 19, no. 6, pp. 3812–3825, Jun. 2020.
[58] L. Lan, Y. Y. Tai, S. Lin, B. Memari, and B. Honary, “New constructions
of quasi-cyclic LDPC codes based on special classes of BIBD’s for the
AWGN and binary erasure channels,” IEEE Trans. Commun., vol. 56,
no. 1, pp. 39–48, Jan. 2008.
[59] Y. Wu, E. Attang, and G. E. Atkin, “A novel noma design based on Steiner
system,” in Proc. IEEE Int. Conf. Electro/Inf. Technol. (EIT), Rochester,
U.S.A., May 2018, pp. 0846–0850.
[60] J. H. Dinitz and J. C. Colbourn, Handbook of Combinatorial Designs,
2nd ed. Chapman Hall, CRC, 2007.
[61] M. Yoshida and T. Tanaka, “Analysis of sparsely-spread CDMA via
statistical mechanics,” in Proc. IEEE Int. Symp. Inf. Theory (ISIT),
Seattle, U.S.A., Jul. 2006, pp. 2378–2382.
[62] A. Montanari and D. Tse, “Analysis of belief propagation for non-linear
problems: The example of CDMA (or: How to prove Tanaka’s formula),”
in Proc. IEEE Inf. Theory Workshop (ITW), Punta del Este, Uruguay, Mar.
2006, pp. 160–164.
[63] J. Raymond and D. Saad, “Sparsely spread CDMA—a statistical
mechanics-based analysis,” IEEE Trans. Inf. Theory, vol. 40, no. 41, pp.
12 315–12 333, Sep. 2007.
[64] M. T. P. Le, G. C. Ferrante, T. Q. S. Quek, and M. Di Benedetto,
“Fundamental limits of low-density spreading NOMA with fading,” IEEE
Trans. Wireless Commun., vol. 17, no. 7, pp. 4648–4659, Jul. 2018.
[65] S. Chen, K. Peng, Y. Zhang, and J. Song, “A comparative study on the
capacity of SCMA and LDSMA,” in Proc. Int. Wireless Commun. Mobile
Computing Conf. (IWCMC), Limassol, Cyprus, Jun. 2018, pp. 1099–
1103.
[66] M. AL-Imari, M. A. Imran, R. Tafazolli, and D. Chen, “Performance eval-
uation of low density spreading multiple access,” in Proc. Int. Wireless
Commun. and Mobile Computing Conf. (IWCMC), Limassol, Cyprus,
Aug. 2012, pp. 383–388.
[67] R. Hoshyar, R. Razavi, and M. Al-Imari, “LDS-OFDM an efficient
multiple access technique,” in Proc. IEEE Veh. Technol. Conf. (VTC
Spring), Taipei, Taiwan, May 2010, pp. 1–5.
[68] M. Li and S. V. Hanly, “Precoding optimization for the sparse MC-
CDMA downlink communication,” in Proc. IEEE Int. Conf. on Commun.
(ICC), Syndney, Australia, Jun. 2014, pp. 2106–2111.
[69] N. Suraweera, S. V. Hanly, and P. Whiting, “Distributed beamforming for
the multicell sparsely-spread MC-CDMA downlink,” in Proc. IEEE Veh.
Technol. Conf. (VTC-Spring), Syndney, Australia, Jun. 2017, pp. 1–5.
[70] B. Fontana da Silva, C. A. A. Meza, D. Silva, and B. F. Uchôa-Filho,
“Exploiting spatial diversity in overloaded MIMO LDS-OFDM multiple
access systems,” in Proc. IEEE 9th Latin-American Conf. on Commun.
(LATINCOM), Guatemala City, Guatemala, Nov. 2017, pp. 1–6.
[71] Y. Liu, L. Yang, and L. Hanzo, “Spatial modulation aided sparse code-
division multiple access,” IEEE Trans. Wireless Commun., vol. 17, no. 3,
pp. 1474–1487, Mar. 2018.
[72] L. Wen, P. Xiao, R. Razavi, M. A. Imran, M. Al-Imari, A. Maaref, and
J. Lei, “Joint sparse graph for FBMC/OQAM systems,” IEEE Trans. Veh.
Technol., vol. 67, no. 7, pp. 6098–6112, Jul. 2018.
[73] A. Osamura, G. Song, J. Cheng, and K. Cai, “Sparse multiple access and
code design with near channel capacity performance,” in Proc. Int. Symp.
Inf. Theory and Its Applic. (ISITA), Singapore, Oct. 2018, pp. 139–143.
[74] S. Denno, R. Sasaki, and Y. Hou, “Low density signature based packet
access with phase only adaptive precoding,” in Proc. IEEE Veh. Technol.
Conf. (VTC-Spring), Kuala Lumpur, Malaysia, Apr. 2019, pp. 1–5.
[75] Y. Zhao, J. Hu, Z. Ding, and K. Yang, “Joint interleaver and modulation
design for multi-user SWIPT-NOMA,” IEEE Trans. Commun., vol. 67,
no. 10, pp. 7288–7301, Oct. 2019.
[76] S. Özyurt and O. Kucur, “Quadrature NOMA: A low-complexity multiple
access technique with coordinate interleaving,” IEEE Wireless Commun.
Lett., vol. 9, no. 9, pp. 1452–1456, Sep. 2020.
[77] R. Razavi, R. Hoshyar, M. A. Imran, and Y. Wang, “Information theoretic
analysis of LDS scheme,” IEEE Commun. Lett., vol. 15, no. 8, pp. 798–
800, Aug. 2011.
[78] M. K. Varanasi and T. Guess, “Optimum decision feedback multiuser
equalization with successive decoding achieves the total capacity of the
Gaussian multiple-access channel,” in Proc. Asilomar Conf. on Signals,
Systems and Computers, vol. 2, Pacific Grove, CA, U.S.A., Nov. 1997,
pp. 1405–1409.
[79] R. Razavi, M. Ali Imran, and R. Tafazolli, “EXIT chart analysis for
turbo LDS-OFDM receivers,” in Proc. Int. Wireless Commun. and Mobile
Computing Conf. (IWCMC), Istanbul, Turkey, Jul. 2011, pp. 354–358.
[80] O. Shental, B. M. Zaidel, and S. S. Shitz, “Low-density code-domain
NOMA: Better be regular,” in Proc. IEEE Int. Symp. Inf. Theory (ISIT),
Aachen, Germany, Jun. 2017, pp. 2628–2632.
[81] B. M. Zaidel, O. Shental, and S. S. Shitz, “Sparse NOMA: A closed-form
characterization,” in Proc. IEEE Int. Symp. Inf. Theory (ISIT), Colorado,
U.S.A., Jun. 2018, pp. 1106–1110.
[82] M. AL-Imari, M. A. Imran, and R. Tafazolli, “Low density spreading
for next generation multicarrier cellular systems,” in Proc. Int. Conf. on
Future Commun. Networks, Baghdad, Iraq, Apr. 2012, pp. 52–57.
[83] M. Li, C. Liu, and S. V. Hanly, “Precoding for the sparsely spread
MC-CDMA downlink with discrete-alphabet inputs,” IEEE Trans. Veh.
Technol., vol. 66, no. 2, pp. 1116–1129, Feb. 2017.
[84] M. Fantuz and C. D’Amours, “Low complexity PIC-MMSE detector
for LDS systems,” in Proc. IEEE Canadian Conf. on Electrical and
Computer Engineering (CECE), Edmonton, Canada, May 2019.
[85] R. Price and P. E. Green, “A communication technique for multipath
channels,” Proc. IRE, vol. 46, no. 3, pp. 555–570, Mar. 1958.
[86] S. Craw, “Manhattan Distance,” in Encyclopedia of Machine Learning,
C. Sammut and G. I. Webb, Eds. Boston, MA: Springer US, 2010, pp.
639–639.
[87] M. Rupf and J. L. Massey, “Optimum sequence multisets for syn-
chronous code-division multiple-access channels,” IEEE Trans. Inf. The-
ory, vol. 40, no. 4, pp. 1261–1266, Jul. 1994.
[88] P. Viswanath and V. Anantharam, “Optimal sequences and sum capacity
of synchronous CDMA systems,” IEEE Trans. Inf. Theory, vol. 45, no. 6,
pp. 1984–1991, Sep. 1999.
[89] F. Vanhaverbeke and M. Moeneclaey, “Sum capacity of the
OCDMA/OCDMA signature sequence set,” IEEE Commun. Lett.,
vol. 6, no. 8, pp. 340–342, Aug. 2002.
[90] L. Welch, “Lower bounds on the maximum cross correlation of signals
(Corresp.),” IEEE Trans. Inf. Theory, vol. 20, no. 3, pp. 397–399, May
1974.
[91] N. Hurley and S. Rickard, “Comparing Measures of Sparsity,” IEEE
Trans. Inf. Theory, vol. 55, no. 10, pp. 4723–4741, Oct. 2009.
[92] S. Ulukus and R. D. Yates, “Iterative construction of optimum signature
sequence sets in synchronous CDMA systems,” IEEE Trans. Inf. Theory,
vol. 47, no. 5, pp. 1989–1998, Jul. 2001.
[93] M. Soltanalian and P. Stoica, “Designing unimodular codes via quadratic
optimization,” IEEE Trans. on Signal Proc., vol. 62, no. 5, pp. 1221–
1234, Mar. 2014.
[94] M. Kobayashi, J. Boutros, and G. Caire, “Successive interference can-
cellation with SISO decoding and EM channel estimation,” IEEE J. Sel.
Areas Commun., vol. 19, no. 8, pp. 1450–1460, Aug. 2001.
[95] M. K. Varanasi and B. Aazhang, “Multistage detection in asynchronous
code-division multiple-access communications,” IEEE Trans. Commun.,
vol. 38, no. 4, pp. 509–519, Apr. 1990.
[96] G. Xue, J. Weng, T. Le-Ngoc, and S. Tahar, “Adaptive multistage parallel
interference cancellation for CDMA,” IEEE J. Sel. Areas Commun.,
vol. 17, no. 10, pp. 1815–1827, Oct. 1999.
[97] D. Guo, L. K. Rasmussen, S. Sun, and T. J. Lim, “A matrix-algebraic
approach to linear parallel interference cancellation in CDMA,” IEEE
Trans. Commun., vol. 48, no. 1, pp. 152–161, Jan. 2000.
[98] M. C. Reed, C. B. Schlegel, P. D. Alexander, and J. A. Asenstorfer,
“Iterative multiuser detection for CDMA with FEC: near-single-user
performance,” IEEE Trans. Commun., vol. 46, no. 12, pp. 1693–1699,
Dec. 1998.
[99] X. Wang and H. V. Poor, “Iterative (turbo) soft interference cancellation
and decoding for coded CDMA,” IEEE Trans. Commun., vol. 47, no. 7,
pp. 1046–1061, Jul. 1999.
[100] S. Morosi, R. Fantacci, E. Del Re, and A. Chiassai, “Design of turbo-
MUD receivers for overloaded CDMA systems by density evolution
technique,” IEEE Trans. Wireless Commun., vol. 6, no. 10, pp. 3552–
3557, Oct. 2007.
30 VOLUME 4, 2016](https://image.slidesharecdn.com/ldsieeeaccess2020lajosfinaldraft1-250215201455-18753086/85/LDS_IEEE_Access2020_Lajos_Final_Draft-1-pdf-30-320.jpg)
![[101] P. Kumar and S. Chakrabarti, “An analytical model of iterative inter-
ference cancellation receiver for orthogonal/orthogonal overloaded DS-
CDMA system,” Int. J. Wirel. Inf. Netw., vol. 17, no. 1-2, pp. 64–72, Jun.
2010.
[102] S. Sasipriya and C. S. Ravichandran, “An analytical model of iterative
interference cancellation receiver for orthogonal/orthogonal overloaded
DS-CDMA system,” Telecommun. Syst., vol. 56, no. 4, pp. 509–518, Aug.
2014.
[103] A. Najkha, C. Roland, and E. Boutillon, “A near-optimal multiuser
detector for MC-CDMA systems using geometrical approach,” in Proc.
IEEE Int. Conf. on Acoust., Speech, and Sig. Proc. (ICASSP), vol. 3,
Philadelphia, U.S.A., Mar. 2005, pp. 877–880.
[104] G. Manglani and A. K. Chaturvedi, “Application of computational geom-
etry to multiuser detection in CDMA,” IEEE Trans. Commun., vol. 54,
no. 2, pp. 204–207, Feb. 2006.
[105] Y. Du, B. Dong, P. Gao, Z. Chen, J. Fang, and S. Wang, “Low-
complexity LDS-CDMA detection based on dynamic factor graph,” in
Proc. IEEE Global Telecommun. Conf. Workshops (GC Wkshps), Wash-
ington, U.S.A., Dec. 2016, pp. 1–6.
[106] L. Tian, J. Zhong, M. Zhao, and L. Wen, “An optimized superposition
constellation and region-restricted MPA detector for LDS system,” in
Proc. IEEE Int. Conf. on Commun. (ICC), Kuala Lumpur, Malaysia, May
2016, pp. 1–5.
[107] L. Wen, R. Razavi, P. Xiao, and M. A. Imran, “Fast convergence and
reduced complexity receiver design for LDS-OFDM system,” in Proc.
IEEE Pers., Indoor, Mobile Radio Conf. (PIMRC), Washington DC,
U.S.A., Sep. 2014, pp. 918–922.
[108] D. Guo and C. Wang, “Multiuser detection of sparsely spread CDMA,”
IEEE Journal on Selected Areas in Commun., vol. 26, no. 3, pp. 421–431,
Apr. 2008.
[109] K. Takeuchi, T. Tanaka, and T. Kawabata, “Performance improvement of
iterative multiuser detection for large sparsely spread CDMA systems by
spatial coupling,” IEEE Trans. Inf. Theory, vol. 61, no. 4, pp. 1557–9654,
Apr. 2015.
[110] L. Wen, R. Razavi, M. A. Imran, and P. Xiao, “Design of joint sparse
graph for OFDM system,” IEEE Trans. on Wireless Commun., vol. 15,
no. 4, pp. 1823–1836, Apr. 2015.
[111] L. Wen and M. Su, “Joint sparse graph over GF(q) for code division
multiple access systems,” IET Communications, vol. 9, no. 5, pp. 707–
718, Mar. 2015.
[112] L. Wen, J. Lei, and M. Su, “Improved algorithm for joint detection and
decoding on the joint sparse graph for CDMA systems,” IET Communi-
cations, vol. 10, no. 3, pp. 336–345, Feb. 2016.
[113] J. Raymond, “Optimal sparse CDMA detection at high load,” in Proc. Int.
Symp. Modeling Optim. Mobile, Ad Hoc, Wireless Netw. (WiOpt), Seoul,
South Korea, Jun. 2009, pp. 1–5.
[114] J. G. Proakis, Digital Communications, 5th ed. McGraw Hill, 2007.
[115] G. H. Golub and C. F. Van Loan, Matrix Computations, 4th ed. The
Johns Hopkins University Press, 2013.
[116] J. Luo, K. R. Pattipati, and P. K. Willett, “Near-optimal multiuser detec-
tion in synchronous CDMA using probabilistic data association,” IEEE
Commun. Lett., vol. 5, no. 9, pp. 361–366, Sep. 2001.
[117] G. L. Turin, “Introduction to spread-spectrum antimultipath techniques
and their application to urban digital radio,” Proc. IEEE, vol. 68, no. 3,
pp. 328–353, Mar. 1980.
[118] C. D’Amours, M. Moher, and A. Yongacoglu, “Comparison of pilot
symbol-assisted and differentially detected BPSK for DS-CDMA sys-
tems employing RAKE receivers in Rayleigh fading channels,” IEEE
Trans. on Vehic. Tech., vol. 47, no. 4, pp. 1258–1267, Nov. 1998.
[119] L. Hanzo, M. Münster, B. Choi, and T. Keller, OFDM and MC-CDMA
for Broadband Multi-User Communications, WLANs and Broadcasting.
John Wiley Sons, 2003.
[120] L. Hanzo, L.-L. Yang, E.-L. Kuan, and K. Yen, Single- and Multi-Carrier
DS-CDMA: Multi-User Detection, Space-Time Spreading, Synchronisa-
tion, Networking and Standards. Wiley-IEEE Press, 2003.
[121] M. Kulhandjian, E. Bedeer, H. Kulhandjian, C. D’Amours, and
H. Yanikomeroglu, “Low-complexity detection for faster-than-Nyquist
signaling based on probabilistic data association,” IEEE Commun. Lett.,
pp. 1–6, Dec. 2019.
[122] D. Pham, K. R. Pattipati, P. K. Willett, and J. Luo, “A generalized
probabilistic data association detector for multiple antenna systems,”
IEEE Commun. Lett., vol. 8, no. 4, pp. 205–207, Apr. 2004.
[123] S. Yang, T. Lv, R. G. Maunder, and L. Hanzo, “Unified bit-based
probabilistic data association aided MIMO detection for high-order QAM
constellations,” IEEE Trans. on Vehic. Tech., vol. 60, no. 3, pp. 981–991,
Mar. 2011.
[124] M. Kulhandjian and C. D’Amours, “Design of permutation-based sparse
code multiple access system,” in Proc. IEEE Pers., Indoor, Mobile Radio
Conf. (PIMRC), Montreal, Canada, Oct. 2017, pp. 1031–1035.
[125] C. E. Shannon, “A mathematical theory of communication,” The Bell
Syst. Tech. Journal, vol. 27, no. 4, pp. 623–656, Oct. 1948.
[126] H. Imai and S. Hirakawa, “A new multilevel coding method using error-
correcting codes,” IEEE Trans. Inf. Theory, vol. 23, no. 3, pp. 371–377,
May 1977.
[127] ——, “Correction to ’A new multilevel coding method using error-
correcting codes’,” IEEE Trans. Inf. Theory, vol. 23, no. 6, pp. 784–784,
Nov. 1977.
[128] G. Ungerboeck, “Channel coding with multilevel/phase signals,” IEEE
Trans. Inf. Theory, vol. 28, no. 1, pp. 55–67, Jan. 1982.
[129] ——, “Trellis-coded modulation with redundant signal sets Part I: Intro-
duction,” IEEE Commun. Mag., vol. 25, no. 2, pp. 5–11, Feb. 1987.
[130] ——, “Trellis-coded modulation with redundant signal sets Part II: State
of the art,” IEEE Commun. Mag., vol. 25, no. 2, pp. 12–21, Feb. 1987.
[131] E. Zehavi, “8-PSK trellis codes for a Rayleigh channel,” IEEE Trans.
Commun., vol. 40, no. 5, pp. 873–884, May 1992.
[132] B. N. Thanh, V. Krishnamurthy, and R. J. Evans, “Detection-aided
recursive least squares adaptive multiuser detection in DS/CDMA,” IEEE
Signal Proc. Lett., vol. 9, no. 8, pp. 229–232, Aug. 2002.
[133] Y. Du, B. Dong, Z. Chen, J. Fang, and X. Wang, “A fast convergence
multiuser detection scheme for uplink SCMA systems,” IEEE Wireless
Commun. Lett., vol. 5, no. 4, pp. 388–391, Aug. 2016.
[134] Li Ping, W. K. Leung, and Nam Phamdo, “Low density parity check
codes with semi-random parity check matrix,” Electronics Letters,
vol. 35, no. 1, pp. 38–39, Jan. 1999.
[135] H. Zarrinkoub, Understanding LTE with MATLAB: From Mathematical
Modeling to Simulation and Prototyping, 1st ed. Wiley Publishing,
2014.
[136] H. Vangala, E. Viterbo, and Y. Hong, “A comparative study of polar
code constructions for the AWGN channel,” arXiv:1501.02473 [cs.IT],
Jan. 2015. [Online]. Available: https://arxiv.org/abs/1501.02473
[137] S. Verdu, “Spectral efficiency in the wideband regime,” IEEE Trans. Inf.
Theory, vol. 48, no. 6, pp. 1319–1343, Jun. 2002.
[138] T. M. Cover and J. A. Thomas, Elements of Information Theory. USA:
Wiley-Interscience, 2006.
[139] Sheng Chen, A. K. Samingan, B. Mulgrew, and L. Hanzo, “Adaptive
minimum-BER linear multiuser detection for DS-CDMA signals in mul-
tipath channels,” IEEE Trans. on Signal Proc., vol. 49, no. 6, pp. 1240–
1247, Jun. 2001.
MICHEL KULHANDJIAN (M’18-SM’20) re-
ceived his M.S. and Ph.D. degrees in Electrical
Engineer from the State University of New York
at Buffalo in 2007 and 2012, respectively. He had
previously received his B.S. degree in Electronics
Engineering and Computer Science (Minor), with
“Summa Cum Laude” from the American Univer-
sity in Cairo (AUC) in 2005. He was employed
at Alcatel-Lucent, in Ottawa, Ontario, in 2012. In
the same year he was appointed as a Research
Associate at EION Inc. He received Natural Science and Engineering
Research Council of Canada (NSERC) Industrial RD Fellowship (IRDF).
He is currently a Research Scientist at the School of Electrical Engineering
and Computer Science at the University of Ottawa. He is also employed
as a senior embedded software engineer at L3Harris Technologies.
His research interests include wireless multiple access communications,
adaptive coded modulation, waveform design for overloaded code-division
multiplexing applications, channel coding, space-time coding, adaptive
multiuser detection, statistical signal processing, covert communications,
spread-spectrum steganography and steganalysis. He has served as a guest
editor for Journal of Sensor and Actuator Networks (JSON). He actively
serves as member of Technical Program Committee (TPC) of IEEE WCNC,
IEEE GLOBECOM, IEEE ICC, and IEEE VTC.
VOLUME 4, 2016 31](https://image.slidesharecdn.com/ldsieeeaccess2020lajosfinaldraft1-250215201455-18753086/85/LDS_IEEE_Access2020_Lajos_Final_Draft-1-pdf-31-320.jpg)
LDS_IEEE_Access2020_Lajos_Final_Draft (1).pdf
- 1. Date of publication xxxx 00, 0000, date of current version xxxx 00, 0000. Digital Object Identifier 10.1109/ACCESS.2019.DOI Low-Density Spreading Codes for NOMA Systems and a Gaussian Separability Based Design MICHEL KULHANDJIAN1 , Senior Member, IEEE, HOVANNES KULHANDJIAN2 , Senior Member, IEEE, CLAUDE D’AMOURS1 , Member, IEEE and LAJOS HANZO3 , Fellow, IEEE 1 School of Electrical Engineering and Computer Science, University of Ottawa, Ottawa, Ontario, K1N 6N5, Canada (e-mail: [email protected],[email protected]) 2 Department of Electrical and Computer Engineering, California State University, Fresno, Fresno, CA 93740, U.S.A. (e-mail: [email protected]) 3 Electronics and Computer Science, University of Southampton, Southampton, SO17 1BJ, U.K. (e-mail: [email protected]) Corresponding author: Michel Kulhandjian (e-mail: [email protected]). L. Hanzo would like to acknowledge the financial support of the Engineering and Physical Sciences Research Council projects EP/N004558/1, EP/P034284/1, EP/P034284/1, EP/P003990/1 (COALESCE), of the Royal Society’s Global Challenges Research Fund Grant as well as of the European Research Council’s Advanced Fellow Grant QuantCom. This project was partially funded by C. D’Amours’ NSERC Discovery Grant. ABSTRACT Improved low-density spreading (LDS) code designs based on the Gaussian separability criterion are conceived. We show that the bit-error-rate (BER) hinges not only on the minimum distance criterion, but also on the average Gaussian separability margin. If two code sets have the same minimum distance, the code set having the highest Gaussian separability margin will lead to a lower BER. Based on the latter criterion, we develop an iterative algorithm that converges to the best known solution having the lowest BER. Our improved LDS code set outperforms the existing LDS designs in terms of its BER performance both for binary phase-shift keying (BPSK) and for quadrature amplitude modulation (QAM) transmissions. Furthermore, we use an appallingly low-complexity minimum mean-square estimation (MMSE) and parallel interference cancellation (PIC) (MMSE-PIC) technique, which is shown to have comparable BER performance to the potentially excessive-complexity maximum-likelihood (ML) detector. This MMSE-PIC algorithm has a much lower computational complexity than the message passing algorithm (MPA)a . aCode sets for MPA are designed similar to low-density parity-check (LDPC) codes to avoid cycles and to increase girth of the Tanner graph, code sets that are “optimal” for MMSE-PIC might not be optimal for MPA. INDEX TERMS Non-orthogonal multiple-access (NOMA), low-density spreading signatures (LDS), sparse-code multiple-access (SCMA). NOMENCLATURE 5G The fifth generation 6G The sixth generation APP A posteriori probability AWGN Additive white Gaussian noise BBPSO Bare-bone particle swarm optimization BER Bit-error-rate BICM Bit-interleaved coded modulation BLER Block error rate BP Believe propagation BPSK Binary phase-shift keying CDMA Code-division multiple access CIR Channel impulse response CM Coded modulation D2D Device-to-device D2E Device-to-everything DE Domain equalization ED Euclidean distance EMB Enhanced mobile broadband EXIT Extrinsic information transfer FBMC Filter-bank multicarrier FDMA Frequency division multiple access VOLUME 4, 2016 1
- 2. FIR Finite impulse response GML Global maximum likelihood detector IoT Internet-of-Things IoV Internet of Vehicles IrLDS Irregular low-density spreading ITS Intelligent transportation system JSG Joint sparse graph LDPC Low-density parity-check LDS Low-density spreading LDSM Low-density superposition modulation LDSMA Low-density spreading multiple access LLR Log-likelihood ratio LRS Large random spreading LTE Long-term evolution M2M Machine-to-machine MAI Multiple-access interference MAP Maximum a posteriori detector MC Multicarrier MF Matched filter MIMO Multiple-input multiple-output ML Maximum-likelihood MLCM Multilevel coded modulation MMSE Minimum mean-square estimation mMTC Massive machine-type-communications mmWave Millimeter-wave MPA Message passing algorithm MS Mobile station MUD Multiuser detection MUSA Multiuser shared access MWBE Maximum-Welch-bound-equality NOMA Non-orthogonal multiple access OFDM Orthogonal frequency-division multiplexing OMA Orthogonal multiple access PDA Probabilistic data association PDMA Pattern division multiple access PEG Progressive edge-growth PIC Parallel interference cancellation QAM Quadrature amplitude modulation QLSS Quasi-large sparse sequence QPP Quadratic permutation polynomial QPSK Quadrature phase-shift keying RE Resource element SCDMA Sparse code-division multiple-access SCMA Sparse code multiple access SD Sphere-decoding algorithm SEP Symbol error probability SER Symbol error rate SFBC Space-frequency block codes SIC Successive interference cancellation SINR Signal-to-interference-noise ratio SISO Soft-input soft-output SM Spatial modulation SNR Signal-to-noise ratio SVE Spreading vector extension TCM Trellis-coded modulation TDMA Time-division multiple access TSC Total squared correlation TTCM Turbo trellis-coded modulation UD Uniquely decodable uRLLC Ultra-reliable low-latency communications VA Viterbi algorithm WBE Welch-bound-equality ZF Zero-forcing filter I. INTRODUCTION HIgh spectral- and power-efficiency, massive connec- tivity and low latency are among the requirements for next generation communications and these require- ments are expected to increase in the future, as researchers turn their efforts towards sixth generation (6G) wire- less communications. Enhanced mobile broadband (EMB), ultra-reliable low-latency communications (uRLLC) and massive machine-type communication (mMTC) support a suite of compelling applications driving these require- ments. Massive multiple-input multiple-output (MIMO), non-orthogonal multiple access (NOMA) and millimeter- wave (mmWave) communications constitute promising tech- niques of addressing these stringent requirements [1]. In the previous generations spanning from 1G to 4G, the multiple access schemes were exclusively character- ized by orthogonal multiple access (OMA) techniques, where users are assigned unique, user-specific resources in either frequency- (frequency-division multiple access (FDMA)), time- (time-division multiple access (TDMA)) or code-domain (code-division multiple access (CDMA)). However, the multiple access scheme of 5G is required to support a wide range of use cases, including a mas- sive number of low-power Internet-of-Things (IoT) de- vices, device-to-device (D2D) communications, device-to- everything (D2E), the Internet of Vehicles (IoV), as well as seamless machine-to-machine (M2M) communications [2]–[6]. The mMTC mode includes, for example, e-health services, smart cities/villages, e-farms, and intelligent trans- portation systems (ITS) [7], [8]. They require improved connectivity compared to previous generations of wireless communications. Supporting a large number of users communicating over a common channel may not be readily achievable by OMA techniques due to presence of multiple-access interference (MAI) in rank-deficient systems, where the number of users is higher than that of the resource blocks. To meet the demand of increased bandwidth efficiency in synchronous CDMA, a dense spreading NOMA CDMA concept was introduced in [9], which can support many more users for a given code length compared to traditional CDMA. A number of signature designs have been conceived [10]– [12], where low cross-correlation sequence sets are designed to minimize the overall MAI, which allows more users to simultaneously access the common channel. This in turn results in increased spectral efficiency. Using low cross-correlation sequence sets might not be the best design policy for highly rank-deficient systems. One 2 VOLUME 4, 2016
- 3. of the important design criteria in such rank-deficient sys- tems is for the code set to be uniquely decodable (UD) [9]. By definition, the UD codes can be unambiguously decoded in a noiseless channel using linear recursive decoders [13]. Low-complexity linear decoders were introduced for these UD code sets using either binary {0, 1}, or antipodal {±1}, or alternatively ternary {0, ±1} chips in [14]–[16]. Although these code set designs attain a substantial increase in system capacity even with the aid of low-complexity detectors, they only perform well for synchronous transmission over non- dispersive fading channels, such as additive white Gaussian noise (AWGN) channels. To satisfy the UD criterion, all the users have to rely on accurate transmit power control so that their signals are received with equal power. In prac- tice, the wireless transmission channel exhibits, numerous impairments, such as frequency-selective multipath fading, and unequal received power. Another limitation of linear decoders is that they do not produce soft output decisions required by the channel decoders. To combat the MAI at a reasonable cost, many researchers have proposed the construction of sparsely structured se- quences for multiple access so as to take advantage of efficient sparse signal processing, relying for example on the message passing algorithm (MPA) for reducing to reducing the complexity of multiuser detection (MUD). These challenges can be addressed by the introduction of sparse spreading based NOMA techniques, which can be categorized into power-domain NOMA (PDM-NOMA) [1], [17]–[20] and code-domain NOMA (CDM-NOMA) [21]. A few of the strong contenders of CDM-NOMA are low- density spreading aided CDMA (LDS-CDMA) [22], low- density spreading assisted orthogonal frequency-division multiplexing (LDS-OFDM) [23], sparse code multiple ac- cess (SCMA) [24], [25], irregular LDS (IrLDS), pattern division multiple access (PDMA) [26] and multi-user shared access (MUSA) [27]. The LDS can be considered a special case of SCMA, which may also be characterized by sparse codebooks, each of which can be expressed as the Kro- necker product of a sparse sequence denoted by sj, and a constellation set of order M. Specifically, we have: Xj = [sjβ1, sjβ2, . . . , sjβM ], (1) where {β1, β2, . . . , βM } indicates a constellation set. Hence, the rank of the users’ LDS codebooks, Xj, is equal to one. However, this is not the case for the users’ SCMA codebooks. The rank of SCMA codebooks is higher than one and it is equal to the number of non-zero values in the SCMA waveforms. The comparison between direct sequence CDMA (DS-CDMA), multicarrier CDMA (MC- CDMA), LDS-OFDM and SCMA is illustrated in Fig. 1. Readers are referred to surveys of SCMA [28] and signature- based NOMA [29] for further reading. CDM-NOMA offers flexible resource element (RE) al- location where the sparsity may be flexibly configured for handling time-variant user-loads. It performs well in terms of handling the MAI imposed by rank-deficient systems FIGURE 1. Multiple access technique comparisons. and has low-complexity receivers compared to conventional dense spreading based CDMA. LDS, may also be appropri- ate for IoT communications [21] and it is also considered as a potential candidate for the uplink of mMTC [21]. There have been various criteria for the optimization of sparse spreading based NOMA [30]–[43], which maps the signals of users to REs in a sparse manner, whilst relying on the constellation shaping of non-zero entries [31], [32], and accurate power allocation for each spreading sequence [44]. The RE mapping methods can be broadly divided into two types; a) regular RE mapping, where the spreading densities of all users are the same, as in LDS-OFDM and b) irregular RE mapping, where the densities are non-identical, as in IrLDS and PDMA. Constellation shaping can be categorized into a) widely studied constellations {0, 1} [31], binary phase-shift keying (BPSK) [45], quadrature phase- shift keying (QPSK) [39], quadrature amplitude modulation (QAM) [39], etc., b) two-dimensional constellation bounded in unit circle [32], c) or any other constellations. The power allocation of each spreading sequence can be divided into two classes a) equal power, b) unequal power among all users. A. RELATED LITERATURE Spreading sequences of the low-density type containing many zeros were first introduced in [30] supporting low- complexity MUD. The introduction of cyclically shifted LDS design [30] allows maximum-likelihood (ML) detec- tion to be carried out by computationally efficient methods, such as the Viterbi algorithm (VA) when BPSK modulation is used. It is widely recognized that finding the ML solution is generally NP-hard [46]. Various sub-optimal solutions can be applied such as sphere-decoding (SD) [47], probabilistic data association (PDA) [48], decision-feedback methods [49] etc. The problem, however, becomes more difficult if the system is rank-deficient. The complexity of the decoding process is crucial with the advent of iterative turbo VOLUME 4, 2016 3
- 4. detection, the so-called turbo MUD algorithm approximates the complex optimum joint detection scheme by iteratively exchanging soft decision variables between the multiuser de- tector and single-user soft-input soft-output (SISO) channel decoders. Based on this idea Hoshyar et al. [31] showed that iterative decoding is necessary for fully exploiting the LDS structure. To further exploit the lower complexity of iterative detection, sparse spreading sequences were conceived [22], [31], [32]. The family of low-density parity-check (LDPC) codes has been shown to be attractive due to its capacity- approaching capability and decoding simplicity, when using the MPA. This is why, Hoshyar et al. [31] proposed an LDS structure based on LDPC codes, where the user’s symbol are arranged in such a way that the interference seen by each user at each chip is different. Explicitly, the specific choice of the non-zero entries is in perfect harmony with the particular choice of the LDPC indicator matrix that defines the structure of the LDS code matrix. As a further advance, a near-optimum chip-level SISO iterative MUD is developed in [22] for the LDS structure for transmission over AWGN channels. It was shown to yield promising performance for rank-deficient systems, especially, for BPSK modulation [22], where the emphasis was on the MUD structure, rather than on design of spreading sequences having particular structure, which were found by simple trial and error under a unit amplitude constraint. In contrast to [22] a structured approach focusing on the design of spreading sequences was proposed by Van de Beek and Popović [32] based on the LDPC indicator matrix. In general, signatures having a unity scalar magnitude are designed by maximizing their minimum distance. Moreover, Van de Beek and Popović advocated the so-called Latin-rectangular mappings, where not only the non-zero elements of each row are distinct, but also those in each column, because they are capable of significantly outperforming a randomly generated signature matrix, as a benefit of their high minimum distance. It is widely recognized that the global search based maxi- mum likelihood (GML) detector approaches the single-user bit-error-rate (BER), at high signal-to-noise ratios (SNR), when using long random spreading (LRS) sequence based CDMA [50]. Inspired by the LRS-CDMA concept, Sun and Xiao [45], [50] proposed the so-called quasi-large sparse sequence (QLSS) - CDMA concept by replacing the dense sequences of QLRS-CDMA by sparse sequences. The specific constructions of LDS signatures found in [31], [32] have been inspired by classic LDPC code designs in order to facilitate the employment of the MPA algorithm. Safavi et al. [33] considered schemes, where the spread- ing and mapping to conventional QAM constellations are performed separately. Their proposed recursive matrix con- struction has been optimized for maximizing the Euclidean distance. Apart from the fact that the multiple access sequences play a key role in NOMA for supporting low-complexity detection, they determine the achievable sum rate. Qi et al. [34] analyze the sparsity of the sum capacity-achieving sequences and propose a beneficial construction method with the aid of classic frame theory [51]. The particular low- density spreading sequence design that maximizes the sum rate based on frame theory for complex zero-mean Gaussian random variables is presented in [34], where each row has almost the same number of non-zero entries, forming a nearly regular sparse spreading sequences. In contrast to this design, Yu et al. [40] proposed the simultaneous optimization of the RE assignment and power allocation among REs, where the users employing the same radio resource have different channel gains. It was achieved by first formulating a sum-rate optimization problem subject to practical sparsity and power constraints. In 2017, Qi et al. [37] formulated an optimization prob- lem for specifically designing the sparsity of spreading se- quences, while maximizing the efficiency of NOMA subject to the maximum tolerable symbol error rate (SER) as well as to the affordable detection complexity. Another challenge is the construction of the sparse matrix that optimizes the performance of the MPA detector, since there are no closed- form expressions for characterizing the detection perfor- mance of MPA for sparse sequences. Despite this short- coming, Qi et al. [35] proposed a systematic technique for constructing the sparse sequences relying on a hierarchical method with the objective of optimizing the performance of MPA for BPSK modulation. For the given SNR and target factor graph girth, the algorithm produces the optimum sparsity. Based on the optimum sparsity and the minimum girth, the algorithm directly produces the position of non- zero entries in the matrix. Lastly, the particular values of non-zero entries are determined by specifically maximizing the minimum distance. Wang et al. [42] took a step further by combining multicarrier (MC) LDS and channel coding schemes into a joint sparse factor graph and quantified the average BER based on the mean and variance of the soft information distribution obtained. Explicitly, through their theoretical analysis, the average BER has been derived based on the mean and variance of the soft information distribution at the output of the joint sparse factor graph. The proposed design produces the optimal degree distribution of LDS spreading capable of approaching the theoretical capacity in terms of SNR. The optimization of sparse matrices is typically carried out by assuming to have Gaussian input signal, which is suboptimal, for practical discrete constellations. Xiao et al. [52] proposed a codebook design for multicarrier-low- density spreading aided multiple access (MC-LDSCMA) based on the maximization of the minimal user rate for practical finite alphabet signalling. Another LDS signature spreading vector extension (LDS- SVE) method is introduced by Zhang et al. [38] for up- link OFDM systems. Compared to LDS-OFDM, LDS-SVE jointly transforms and spreads a pair of modulated symbols across four subcarriers. This is achieved upon multiplying the real and imaginary parts of two modulated symbols by a transformation matrix, which is optimized by minimizing 4 VOLUME 4, 2016
- 5. the single-user BER. LDS designs that are based on the sparseness of the LDPC parity check matrix [31], [32] are typically considered as having a regular parity check matrix. By contrast, Jiang and Wu et al. [36] proposed a low-density superposition modulation (LDSM) scheme that is based on an irregular parity check matrix, which provides both diversity and coding gains, hence improving both the overall average performance as well as the cell-edge performance. The progressive edge-growth (PEG) algorithm is utilized to con- struct the LDSM matrix. A compelling systematic technique of designing the degree distribution of the LDSM signature matrices is proposed by Lu and Jiang in [43], which is based on the powerful extrinsic information transfer (EXIT) chart tool and the so-called bare-bone particle swarm optimization (BBPSO) algorithm they optimize the degree distribution of LDSM signature matrices. Their EXIT chart analysis in rightfully characterizes the resultant design. Similarly, Zhang et al. in [41] proposed a pair of sparse superposition matrices. Similar to the minimum distance criterion based LDS code design of [35] developed for BPSK modulation, Song et al. address the maximization of the minimum Euclidean distance for QAM constellations in [39]. More explicitly, signature matrices having factor graphs exhibiting very few short cycles and large superposed signal constellation distances are designed by Song et al. In short, for a given factor graph structure the algorithm produces the optimal signature matrix associated with the maximum LDS code distance. The LDS code set of Song et al., which are detected both by the MPA and the ML detector, exhibit an excellent performance. By expanding the traditional direct sequence CDMA to NOMA, Liu et al. [53] developed a cyclic shift based mul- tiple access scheme, where the in-phase and quadruature- phase channels are used for transmitting the data and pilots, respectively. In contrast to conventional SCMA, which is based on geometric shaping design, Jiang and Wang [54] combine both geometric and probability-based for increas- ing the channel capacity and reducing the BER. As a benefit of using a sparse spreading matrix, low-complexity iterative MUD can be employed. Song et al. [55] propose super-sparse on-off division multiple access using spreading waveforms based on idling. On the other hand, Ye et al. [56] resort to using a deep multi-task learning technique for optimizing an end-to-end NOMA system. As a further development, Xie et al. [57] design constellations for non- coherent reception of the signals arriving from multiple users, and reduce the SER simultaneously. Combinatorial structures relying on the so-called balanced incomplete block design have also been widely studies in the context of LDPC constructions. The designs of Lan et al. [58] used as sparse codes have better interference properties, hence they provide higher user/bandwidth efficiency and have the flexibility of creating variable code rates. Motivated by this fact, Wu et al. [59] proposed LDS designs based on Steiner codes [60], whose incidence matrix conviniently supports superposition based multiuser communications. By using algebraic code construction methods, Liu et al. in [44] proposed power-imbalanced LDS designs of the non-zero entries for a given factor graph with the aid of Eisenstein integersi . According to the above construction designs and studies, the sparsity of spreading sequences, significantly influences the performance of MUD due to its crucial impact on the MAI characteristics. Since the performance analysis of finite-size multiuser systems is mathematically intractable, the large-system limit based analysis was provided in [61]– [63]. By deriving a probability density model for the non- zero entries of sparse spreading sequences, a method based on statistical mechanics was proposed to analyze the optimal detection performance in [63] and the spectral efficiency of the scheme in [61]. The theoretical analysis of LDS systems in the presence of flat fading channel in terms of their spectral efficiency relying on both linear and non-linear optimum receivers (such as maximum a posteriori detector) was carried out in the large-system limit in [64]. Furthermore, the channel capacities of SCMA and low-density spreading multiple ac- cess (LDSMA) schemes are analyzed in [65] and compared to that of the Gaussian multiple access channel imposing random phase rotations and fast fading. The performance advantage of LDSMA, which exploits the high degree of flexibility of subcarrier allocation, has been demonstrated in [66]. The results showed that the diversity gain attained improves the link-level performance in terms of the achiev- able block error rate (BLER). The capacity region of uncoded LDS schemes communi- cating over a multiple access channel is analysed in [77]. However, low-complexity of MPA decoding of LDS as a multiple access technique has a lower capacity than suc- cessive decoding [78]. For the coded LDS multiple access channel the mutual information transfer characteristics of turbo MUD applied to LDS-OFDM is studied using EXIT charts in [79]. The rigorous information-theoretic analysis of infinite graphs showed that having a regular user-to-RE allocation is advantageous [80]. However, increasing the pattern matrix dimensionality results in a significantly increased detection complexity. Moreover, the rigorous closed-form analytical expression of the spectral efficiency of regular sparse se- quence based NOMA relying on optimum decoding in terms of spectral efficiency is derived in [81], for Gaussian signaling over non-fading channels in the asymptotic large- system limit. The LDS concept was applied in various attractive com- munication systems. As an example, Hoshyar et al. [67] and Al-Imari et al. [82] used LDS structures for spreading the iEisenstein integers, also known as Eulerian integers, are complex num- bers of the form z = a + bω, where a, b ∈ N and ω = −1+i √ 3 2 = e 2πi 3 constitute a primitive cube root of unity. VOLUME 4, 2016 5
- 6. 2003 2021 Choi [30] proposes a cyclically shifted LDS for multicarrier systems has been proposed to exploit the trade-off between the receiver complexity and performance improvement. 2004 Hoshyar et al. [31] conceive LDS structure based on LDPC codes. 2006 Sun [45] introduces the quasi-large sparse sequence - CDMA based on randomly generated sparse vectors. 2008 Van de Beek and Popović [32] propose LDS structure based on LDPC indicator-matrix tailored to the belief-propagation detector. 2009 Safavi et al. [33] propose new concept for LDS design based on ultra low-density spread signatures. 2016 Qi et al. [34] introduce LDS sequence design that maximizes sum rate of the system and sequences sparsity based on frame theory. 2017 Qi et al. [37] extend design of the sparsity of LDS that maximizes the efficiency of NOMA system. 2017 Qi et al. [35] conceive a systematic scheme to construct the sparse sequences in a hierarchical way with the aim of optimizing the performance of MPA for BPSK modulation. 2017 Xiao et al. [52] propose novel design method based on the maximization of the minimal user rate with the finite alphabet inputs based on minimizing single user mutual information. 2017 Zhang et al. [38] introduce LDS signature vector extension jointly transforms and spreads two modulated symbols onto twice the subcarriers. 2017 Jiang and Wu [36] propose a novel low-density superposition modulation design with the sparser and irregular check matrix.. 2017 Song et al. [39] introduce an optimal signature matrix with the systems of a two-dimensional quadrature amplitude modulation. 2017 Zhang et al. [41] present a design of two sparse superposition matrices for 150% and 200% overloaded LDSM scheme. 2018 Wu et al. [59] propose a NOMA design based on STS. 2018 Yu et al. [40] conceive an optimal sparse RE mapping patterns via sum-rate optimization problem subject to sparsity and power constraints. 2018 Wang et al. [42] propose to optimize the degree distribution of the joint sparse factor graph by leveraging the differential evolution method. 2019 Liu et al. [44] conceive new density design for LDS based on Eisenstein integers. 2019 Lu and Jiang [43] introduce the optimization problem for degree distribution of LDSM signature matrix. 2019 Liu et al. [53] propose an identical code cyclic shift code for downlink DS-CDMA to enable multiple access using only one spreading code. 2020 Jiang and Wang [54] conceive waveform design based on geometric shaping and probabilistic shaping. 2020 Ye et al. [56] present a constellation shape using deep learning techniques. 2020 Song et al. [55] propose very low-complexity on-off division multiple access scheme for NOMA systems. 2020 Xie et al. [57] introduce a joint multi-user isometric constellation design is proposed to find constellations that enable non-coherent reception and reduce SER. 2020 FIGURE 2. Timeline of LDS design contribution. 6 VOLUME 4, 2016
- 7. 2009 2021 Hoshyar et al. [67] propose LDS-OFDM is introduced as an uplink multicarrier multiple access scheme. 2010 Li and Hanly [68] introduce a novel MC-CDMA system, where random sparse signatures are deployed in the frequency domain. 2014 Suraweera et al. [69] conceive a distributed beamforming for sparsely-spread MC-CDMA using sum- product algorithm. 2017 Fontana da Silva et al. [70] present an Alamouti SFBC scheme for a simple MIMO LDS-OFDM system. 2017 Liu et al. [71] propose a SM-SCDMA scheme is proposed to support a high normalized user-load in uplink communications. 2018 Wen et al. [72] introduce joint sparse graph for FBMC is proposed to combine single graphs of LDS, LDWM, and LDPC codes. 2018 Osamura et al. [73] propose to mitigate multi-user interference, the codeword of each user is randomly punctured and the punctured bits are replaced by idle slots. 2018 Denno et al. [74] introduce a low density signature based multiple access with phase only adaptive precoding for increasing network throughput. 2019 Zhao et al. [75] present a joint design of the energy interleaver and the constellation rotation-based modulator in the symbol-block level by constructively superimposing the symbols. 2019 Özyurt and Kucur [76] propose a low-complexity multiple access method based on coordinate interleaving. 2020 FIGURE 3. Timeline of LDS applied in applications. TABLE 1. Contrasting our novel contributions to the state-of-the-art. Contributions This work [56] [53] [40] [39] [35] [34] [33] [32] Minimum Euclidean distance X X X Gaussian margin X Sum Capacity X X Maximum SINR per user X Adaptive to number of users X X X X X X X X X Joint RE and constellation shaping X X X X X X BER approach single user at K/L = 2 X symbols across the frequency domain, hence their technique was termed as LDS-OFDM. Li and Hanly [68] and Li et al. [83] introduced MC-CDMA for downlink commu- nication, where sparse random signatures are deployed in the frequency domain. A power-efficient non-linear transmit precoder weight optimization problem is formulated, while satisfying the maximum tolerable symbol error probability (SEP) targets at the mobile stations (MSs). Suraweera et al. [69] approached this problem by conceiving a distributed linear beamforming technique for the multicell MC-CDMA downlink by using the sum-product algorithm for detecting the sparse signatures. LDS spreading has also been proposed for MIMO systems by da Silva et al. [70]. Their proposed technique relies on Alamouti’s space-frequency block codes (SFBC) conceived for low-complexity MIMO LDS-OFDM systems. As a parallel development in MIMO systems, spa- tial modulation (SM) has drawn a lot of research attention in recent years. Liu et al. [71] have proposed a sparse code- division multiple-access (SCDMA) scheme for supporting a high normalized user-load in uplink communications. Recently, filter-bank multicarrier (FBMC) transceivers have drawn a lot of attention as a benefit of circumventing several OFDM drawbacks. Wen et al. [72] designed a LDS- FBMC scheme, which applies LDS for constructing FBMC signals. Additionally, a joint sparse graph (JSG) based FBMC transceiver termed as JSG-FBMC was proposed for VOLUME 4, 2016 7
- 8. combining the single graphs of LDS, a low-density weight matrix, and LDPC codes, which represent popular NOMA, multicarrier modulation and channel coding techniques, respectively. Osamura et al. [73] proposed a new multi- user scheme for mitigating the multi-user interference, in which the codeword of each user is randomly punctured and the punctured bits represent idle slots, hence, only a small random set of users are active at each time. This constraint imposed on the number of concurrent users significantly reduces the multi-user detection complexity. As a further advance, Denno et al. [74] proposed ‘phase-only’ based transmit precoding in support of multiple user terminals having a single antenna. As a further advance, Zhao et al. [75] designed an energy interleaver and constellation rotation-based modulator by exploiting the NOMA concept for improving the energy transfer efficiency of wirelessly powered systems. Similarly, Özyurt and Kucur [76] designed a low-complexity multiple access method for single-antenna nodes by exploiting the concept of signal space diversity by relying on the power- domain NOMA philosophy for reducing both the BER and the number of SIC iterations. B. CONTRIBUTION Compared to the design of conventional dense spreading sequences for classic CDMA, designing the LDS sequences for NOMA systems is more complicated, since the design should be implemented under the sparsity constraint of the signature matrix. In the literature, there is a paucity of optimal signature matrix designs exhibiting maximum minimum code distance. Against this background, we study a range of differ- ent distance metrics and the properties of a sophisticated signature matrix. Table 1 boldly and explicitly contrasts the novelty of our design to the family of state-of-the-art LDS code set designs. Explicitly, our new contributions are summarized as follows: (1) We propose a novel iterative LDS design algorithm for maximizing the signal-to-interference-noise ratio (SINR) of each individual user of interest which jointly maps the user-signals to REs in a sparse manner and applies constellation shaping tp tje non-zero entries. (2) We demonstrate that the code sets having the highest minimum distance are also optimal in terms of the BER criterion for transmission over Gaussian channels. Fur- thermore, when the code sets have the same minimum distance, those associated with higher average Gaussian separability tend to exhibit better BER performance. We show that our improved LDS code set outper- forms the existing LDS designs in terms of its BER performance for BPSK and 4QAM transmissions over AWGN, non-dispersive and frequency-selective fading channels. (3) Moreover, we design both a minimum mean-square estimation (MMSE) based and a parallel interference cancellation (PIC) (MMSE-PIC) aided detector [84], both which exhibit a comparable BER performance to that of the high-complexity ML detector. The rest of the paper is organized as follows. In Section II, we discuss the system model, followed by the specific properties and design criteria of the spreading codes in Section III. Our improved iterative LDS sequence design is presented in Section IV, followed by our detection method proposed for AWGN, non-dispersive and frequency selective fading channels in Section V. After illustrating our simu- lation results in Section VIII, our conclusion and design guidelines are drawn in Section IX. The following notations are used in this paper. All boldface lower case letters indicate column vectors and upper case letters indicate matrices, ()T denotes transpose operation, sgn denotes the sign function, |.| is the scalar magnitude, || · ||p denotes `p norm, || · || , || · ||2 is vector norm and E{·} denotes expected value. II. SYSTEM MODEL First of all, perfect chip synchronization among all the transmitters is assumed. This provides the best-case estimate of the performance of what is in reality a fully asynchronous system, which only requires chip synchronization between the source transmitter and the target receiver. The spreading sequence ck ∈ CL×1 is considered to be s-sparse, when s coefficients are non-zero and (L−s) are zeros, with the non- zero coefficients located in Ik ⊂ {1, 2, ..., L}. In the scope of LDS design ck can be considered sparse if the cardinality of non-zero entries obeys |Ik| ≤ L/2. However, the sparsity metric is also discussed further in the next section. A. AWGN CHANNEL We assume that the data stream is partitioned into length-Q subsequences, bk , [bk,1, bk,2, . . . , bk,Q], of k-th user bits bk,i ∈ {0, 1}, for 1 ≤ j ≤ Q. The modulator maps each subsequence bk to a symbol xk from the M-ary symbol alphabet Xk = {xk,1, xk,2, . . . , xk,M }, where, xk,m ∈ C corresponds to the bit pattern bk(m) = [bm k,1, bm k,2, . . . , bm k,Q] and M = 2Q . Let the modulator’s bijective mapping ψk, representing the binary-to-symbol conversion of user k be defined as ψk : bk(m) ∈ {0, 1}Q×1 7→ am ∈ Xk, ∀m, (2) and vice versa, its inverse operation be represented by bk(m) = ψ−1 k (am), bm k,i = ai m, where ai m denotes the i-th bit of the binary vector ψ−1 k (am). Then, the users’ symbols are multiplexed after spreading them using the LDS codes. Mathematically, we can formulate the system model as y = K X k=1 ckdkxk + n = CDx + n, (3) where K is the number of the users, dk is the k-th user’s amplitude, xk ∈ Xk is the k-th user’s symbol to be transmitted from the constellation alphabet, Xk , 8 VOLUME 4, 2016
- 9. C = [c1, c2, . . . , cK] ∈ CL×K is the column-normalized LDS code matrix, ||ck|| = 1 for 1 ≤ k ≤ K, n ∈ CL×1 is an L-dimensional complex-valued AWGN vector with variance of σ2 and D is a diagonal matrix hosting the users’ amplitude, which is given as D = d1 0 · · · 0 0 d2 0 . . . . . . 0 ... 0 0 . . . 0 dk . (4) We assume that the constellation alphabet of each user is identical, i.e., Xk = X, ∀k and the cardinality of the constellation is M = |X|. The block diagram of the LDS transmitter is shown in Fig. 4. Note that for the AWGN channel, we assume that hk = 1 for 1 ≤ k ≤ K. B. NON-DISPERSIVE FADING CHANNEL A channel is said to exhibit flat or non-dispersive Rayleigh fading if the coherence bandwidth of the channel is higher than the bandwidth of the signal. In this case, all of the received multipath components arrive within a delay that is much smaller than the symbol duration; where the symbol is defined as one chip in the case of LDS spread signals. These channel coefficients are circularly symmetric complex Gaussian random variables with zero mean and unit variance. The magnitudes of the channel gains are Rayleigh distributed. The model of the flat or non-dispersive fading channel can be represented as y = K X k=1 ckhkdkxk + n = CHDx + n, (5) where hk is the k-th user’s channel coefficient and H is a diagonal matrix with channel coefficients as shown below, H = h1 0 · · · 0 0 h2 0 . . . . . . 0 ... 0 0 . . . 0 hk . (6) The block diagram of the transmitter model of an LDS system in non-dispersive fading channels is shown in Fig. 4. C. FREQUENCY-SELECTIVE FADING CHANNEL A channel is said to exhibit frequency-selective fading, if the coherence bandwidth of the channel is lower than the bandwidth of the signal. In other words, it occurs whenever the received multipath components of a symbol extend be- yond the symbol’s time duration. The multipath channel can be modeled by a tap delay line based finite impulse response (FIR) filter of length Lp [85]. The system model for uplink communication over the frequency-selective fading channel can be written as y = K X k=1 hk ∗ (ckdkxk) + n = K X k=1 H̄kckdkxk + n, (7) where hk = 1 √ Lp [hk,1, hk,2, . . . , hk,Lp ]T is the k-th user’s channel impulse response (CIR), ∗ is the convolution operator and H̄k is a channel matrix with the size of (L + Lp − 1) × L and is expressed as, H̄ = hk,Lp 0 · · · · · · · · · · · · 0 . . . ... ... ... ... ... . . . hk,2 · · · hk,Lp 0 · · · · · · 0 hk,1 hk,2 · · · hk,Lp 0 · · · 0 0 hk,1 hk,2 · · · hk,Lp · · · 0 . . . ... ... ... ... ... . . . 0 . . . 0 hk,1 hk,2 · · · hk,Lp 0 · · · · · · 0 hk,1 · · · hk,Lp−1 . . . ... ... ... ... ... . . . 0 · · · · · · · · · · · · · · · hk,1 . (8) mapping b1 d1·x1 + u1 mapping b2 u2 mapping bK uK n y AWGN · · · · · · c1 d2·x2 c2 dk·xk ck h1 h2 hk FIGURE 4. Transmitter of an LDS system communicating over non-dispersive fading channels. The Gaussian random variables hk,i where i = 1, 2, . . . , Lp have a zero mean and unit variance, while the factor 1 √ Lp ensures that the channel gain experienced by the transmitted signal on average is unity. Here, we assume that the CIRs between users are independent from one another. The block diagram of the transmitter model of an LDS system for uplink communication over frequency-selective fading channel is shown in Fig. 5. III. CODE PROPERTIES AND DESIGN CRITERIA In this section, we first present some of the distance metrics that will be used in the development of an iterative algorithm with the intention of finding the improved LDS signature VOLUME 4, 2016 9
- 10. sets. Given the channel and receiver design specifics, the overall system performance is determined by the specific selection of the user signature set. One of the signature set metrics of interest is the minimum distance. The larger the distance, the better the performance in terms of BER. We recall that the minimum distance for the BPSK constellation X = {±1} is 2 ii . mapping b1 d1·x1 h1 + u1 mapping b2 u2 h2 mapping bK uK hk n y AWGN · · · · · · c1 c1 d2·x2 c2 c2 dk·xk ck ck FIGURE 5. Transmitter of an LDS system communicating over frequency-selective fading channels. A. DISTANCE METRICS Definition III.1. The Euclidean distance of two L- dimensional vectors yi and yj for i 6= j is given by dE(yi, yj) = ||yi − yj||2, (9) where yi = Cxi, yj = Cxj, xi, xj ∈ XK×1 and xi 6= xj. The minimum distance of the received vectors for a given code set can be formulated by dE,min(C) = argmin xi,xj ∈XK×1 / ∈{0}K×1 yi=Cxi,yj =Cxj dE(yi, yj). (10) Theorem 1. Let C ∈ CL×K represent the set of all distinct sparse normalized column LDS matrices. Then dE,min(C) is equal to 2 when X = {±1}. Proof. Assume that cT i cj = 0 for all Ii = Ij, i 6= j. Let dE,min(C) = dE(yn, ym), where yn = Cxn and ym = Cxm. The difference vector y = yn − ym = C(xn −xm) = Cx̄ must have one non-zero element x̄t 6= 0, xn,t 6= xm,t, and L−1 zeros x̄z = 0, xn,z = xm,z for z 6= t to achieve dE,min. Then x̄t can only be 2 or −2, since we have xn,t, xm,t ∈ {±1}. Therefore, the Euclidean distance obeys ||y|| = ||Cx̄|| = ||2ct|| = 2||ct|| = 2. Definition III.2. The product distance of two L- dimensional vectors yi and yj for i 6= j is expressed by dP (yi, yj) = Y t∈Ii,j |yi,t − yj,t|, (11) iiIn signal space representation with the constellation points ± p Eb/Tb the minimum distance is expressed as 2 p Eb/Tb. where yi,t−yj,t 6= 0 for all t ∈ Ii,j ⊂ {1, ..., L}. Let dP,min be the minimum product distance of the code set C. Definition III.3. The Manhattan distance [86] of two L- dimensional vectors yi and yj for i 6= j is defined as dM (yi, yj) = ||yi − yj||1. (12) Let dM,min be the minimum Manhattan distance of the code set C. B. CODE PROPERTIES Another signature set metric of interest includes the total squared correlation, which can be linked to the MAI power associated with a code set. Definition III.4. The total squared correlation (TSC) of C is the sum of the squared magnitudes of all inner products between signatures, which is expressed as TSC(C) = K X i=1 K X j=1 |cH i cj|2 . (13) Let us denote, the number of bits per symbol xk of user k by ρk. Then, there exists a K-dimensional capacity region Φ ⊂ RK×1 for which each set of the number of bits/symbol also termed as the rate ρ = (ρ1, . . . , ρK) within this region can be achieved, while maintaining an infinitesimally low BER for every user, provided that the codeword length tends to infinity. In particular, the sum capacity Csum over Φ is defined as Csum = max ρ∈Φ K X k ρk. (14) Definition III.5. The total sum capacity Csum(C, γ) [87]– [89] in bits/symbol, defined as the maximum possible sum of the users’ transmission rates attained, while still maintaining reliable reception of the signatures in an AWGN channel is expressed as Csum(C, γ) = log2 |IL + γCPCH |, (15) where we have P = DE{xxH }DH , γ is the received SNR of each user’s signaliii and IL is the (L×L)-element identity matrix. Definition III.6. The root-mean-square (RMS) cross- correlation and the maximum cross-correlation amplitude are expressed as Irms(C) = v u u t 1 K(K − 1) K X i=1 K X j6=i |cH i cj|2, (16) Imax(C) = max 1≤i<j≤K |cH i cj|. (17) iiiHere we assume identical received SNR for all user’s signals. 10 VOLUME 4, 2016
- 11. Lemma 1. The Welch Lower Bound [90] for any code set C, with L ≤ K, is expressed as Irms(C) ≥ s K − L (K − 1)L , (18) with equality if and only if PK i=1 cicH i = K L IL. Further- more, Imax(C) ≥ s K − L (K − 1)L , (19) with equality if and only if |cH i cj| = s K − L (K − 1)L ∀i 6= j. (20) The detailed proof of a well-known performance index that assesses the cross-correlation of the code matrix can be found in [90]. The spreading sequence C constitutes a Welch-bound-equality (WBE) and/or a maximum-Welch- bound-equality (MWBE) code matrix, when equality is satisfied in (18). Then Irms meets the Welch bound and/or (19) meets the Welch bound on Imax. Since the MWBE is a stricter bound than the WBE, a MWBE code matrix is said to be a WBE matrix, but not vice versa. The Welch bound (19) is tight for smaller values of K, but becomes quite loose for larger K. It is a challenge to find C associated with an arbitrary L and K that can satisfy the Welch bound on Imax, (19). As an example, it is widely recognized that there is no C that satisfies the Welch bound on Imax when K > L2 in the complex case, C ∈ CL×K , or when K > L(L + 1)/2 in the real case C ∈ RL×K . Note that the expressions for the WBE and MWBE bounds for s-sparse matrices C are a bit different from the ones defined in (18) and (19). Definition III.7. Let all of the users be defined as U = {1, 2, . . . , K}. The k-th symbol is considered to be Gaussian separable [48], if for all small variances, σ2 d → 0, we have cH k R−1 k ck > X j∈U−k |cH k R−1 k cj|, (21) where Rk = X j∈U−k cjcH j + σ2 dIL, (22) and the parameters ∆k = cH k R−1 k ck − X j∈U−k |cH k R−1 k cj|, (23) ∆ave(C) = 1 K K X k=1 ∆k, (24) are called the Gaussian margin and average Gaussian margin of the matrix C, respectively. Linear detectors rely on a decision-boundary partition- ing the composite multiuser signal-space into subspaces uniquely and unambiguously identified by the users’ sig- natures. Therefore the existence of these hyperplanes that partition the projection subspace of the binary user signals into two sets for each user in the absence of channel noise, is a prerequisite for a high performance. This geometri- cal perspective allows us to formally state a separability criterion for linear detectors. As for this linear classifier, upon assuming that the underlying classes follow a Gaussian distribution, it was shown in [48] the optimal ML decision relies on this hyperplane which partitions the decision-space into a pair of L-dimensional subspaces. Therefore, the linear decision rule for user K is said to be Gaussian separable, if the probability of error tends to zero when the noise variance tends to zero. Definition III.8. There are many metrics of vector sparsity, as described in [91], but we will define the general Hoyer sparseness measure of a vector ci based on the relationship between the `m and `n norms as follows, Sm,n(ci) = L(1/m) L(1/n) − ||ci||m ||ci||n L(1/m) L(1/n) − 1 . (25) The average sparseness of a matrix C can be expressed as, Sm,n,ave(C) = 1 K K X k=1 Sm,n(ck). (26) Interesting special cases are those, when m = 1, n = 2, which are known as the Hoyer sparseness measure [91] and m = 1, n = ∞. IV. PROPOSED LDS CODE DESIGN In the following section, we describe the proposed iterative algorithm, which is used for designing the LDS code matrix C. For the sake of simplicity let us assume that dk = 1 for 1 ≤ k ≤ K and rewrite (3) as y = ckxk + K X i6=k cixi + n, (27) = ckxk + ik + n, (28) where y ∈ CL×1 and ik ∈ CL×1 denotes the colored interference imposed by the other users, when the autocor- relation matrix is given by R0 k , E{ikiH k }. Let us define the overall perturbation, gk = ik +n, and the autocorrelation as Rk , E{gkgH k } = R0 k + σ2 IL. The detection of the infor- mation bit of user k can be achieved via max-SINR filtering (or, equivalently, min-TSC filtering, linear MMSE filtering). The filter that exhibits the maximum output SINR for user-k is a scaled version of e.g., wSINR,k(ck) , R−1 k ck. Then the corresponding maximum post-filtering SINR output of the filter wSINR,k is given by SINR(ck) = E n |wH SINR,k(ckxk)|2 o E n |wH SINR,k(ik + n)|2 o (29) = cH k Qkck, (30) VOLUME 4, 2016 11
- 12. where Qk , R−1 k . Our objective is to find the specific s- sparse complex signature ck that maximizes (29), namely: c (s) k,maxSINR = argmax c∈CL×1 ,||c||=1 |Ik|=s cH Qkc. (31) The superscript (s) indicates that c (s) k,maxSINR is s-sparse with |Ik| = s. To tackle the problem, we now propose to relax the sparseness constraint of (31) and proceed by solving the following problem instead, ck,maxSINR = argmax c∈CL×1,||c||=1 cH Qkc. (32) Let {qk,1, qk,2, · · · , qk,L} be the L eigenvectors of Qk with corresponding eigenvalues λk,1 ≥ λk,2 ≥ · · · ≥ λk,L. The sequence c that maximizes (32) is well known and it is equal to the eigenvector that corresponds to the maximum eigenvalue of the matrix Qk, i.e., ck,maxSINR = argmax c∈CL×1,||c||=1 cH Qkc = qk,1. (33) Alternatively, we can design the code set C based on the TSC criterion that is defined in Section III. Let us now demonstrate the iterative method used for minimizing the TSC(C). We rewrite (13) as TSC(C) = K X i6=k K X j6=k |cH i cj|2 + |cH k ck|2 + 2 K X i6=k |cH k ci|2 = TSC(C[k]) + 1 + 2cH k K X i6=k cicH i ck = TSC(C[k]) + 1 + 2cH k R̄kck, (34) where C[k] denotes the preexisting code set except for the k-th column of the code set C and R̄k = PK i6=k cicH i denotes the autocorrelation of the matrix C[k], respectively. It becomes clear from (34) that the conditional minimization of TSC(C) with respect to ck for fixed (min-TSC-valued) TSC(C[k]) reduces to c (s) k,minT SC = argmin c∈CL×1 ,||c||=1 |Ik|=s cH R̄kc. (35) The sequence c that minimizes the relaxed problem (35) is well known and it is equal to the eigenvector, rk,υ, that corresponds to the minimum non-zero eigenvalue υ of the matrix R̄k, i.e., ck,minT SC = argmin c∈CL×1,||c||=1 cH R̄kc = rk,υ. (36) Similarly, since our construction of spreading codes is restricted to unit energy, i.e., cH k ck = 1, the underlying problem of minimizing (34) does not change, if we add 2σ2 cH k ck and subtract 2σ2 from (34) to obtain TSC(C) = TSC(C[k]) + 1 + 2cH k Rkck − 2σ2 , (37) where Rk is defined in (22). The normalized MMSE filter for user k [92], is sk,MMSE = R−1 k ck (cH k R−1 k ck)1/2 . (38) Therefore, the results of (36), (38) can be used in Step 7 of the iterative construction of our proposed LDS design algorithm, as shown in Table 2. In other words, instead of computing qk,1, we compute rk,υ or sk,MMSE. We are now ready to present our proposed algorithm, which is shown in Table 2, TABLE 2. LDS design algorithm Input: L, K, δ, σd, s-sparse 1: Initialize : ∆0 ave ← 0, ← 0.2, 0 ← 102; 2: while ∆0 ave δ 3: Initialize : Civ; 4: while 0 5: C0 ← C 6: for k ∈ {1, . . . , K} 7: Compute qk,1 in (33), or (36), or (38) 8: ck ← q [n] k,1 9: 0 ← ||C − C0||F 10: Compute ∆0 ave(C) in (24) Output: C where q [n] k,1 denotes the s-sparse vector with only those s elements of qk,1 that have the largest absolute values. The scalar values δ and σd are design parameters, respectively. Note that the non-zero complex values of the sparse columns ci are not necessarily unimodular, i.e., ci = {ci,j ∈ C : |ci,j| = 1, j ∈ Ii}, for 1 ≤ i ≤ K, [93]. Unimodular s- sparse vectors have equal Hoyer sparsity, i.e., S1,2(ci) = L(1/2) −s(1/2) L(1/2)−1 . A. CONVERGENCE OF THE ALGORITHM The proposed algorithm converges to a locally optimal solution with an s-sparse matrix C that has one of the specific structures A, where we have Ai,j = {|ci,j|0 : 1 ≤ i ≤ L, 1 ≤ j ≤ K}. For a size of 4 × 6 and for the random initialization of C the algorithm converges to one of the three structures (e.g., A1 , A2 and A3 ), which is described as follows, A1 = 1 0 1 0 0 1 1 0 1 0 0 1 0 1 0 1 1 0 0 1 0 1 1 0 , (39) A2 = 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 , (40) ivRandomly generate C ∈ CL×K , where ||ci|| = 1, ∀ 1 ≤ i ≤ K. 12 VOLUME 4, 2016
- 13. A3 = 1 1 1 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 1 1 1 0 0 0 , (41) where the order of the columns of all of the three structures can be arbitrarily ordered and not necessarily as shown above. The output structure of the matrix of the algorithm depends mainly on the initialization of the matrix C. Specif- ically, if we arbitrarily initialize a 4×6 matrix that possesses one of the structures (e.g., A1 , A2 and A3 ), the output of the algorithm will have the same structure as that of the initialization matrix. Therefore, to speed up the convergence of the algorithm in Table 2 we may choose to initialize the matrix C with one of the known structures. B. UPWARD SCALING DESIGN FOR LDS The algorithm proposed in Table 2 may potentially be con- sidered for an upward scaling design for a given optimum LDS sequence. The underlying requirement is to develop a subset of the sparse matrix with the given optimal LDS code set for ensuring that the resultant LDS code set still maintains optimality. Appending spreading codes to a given LDS set may require complete redesign/reassignment of the resultant LDS code set. Mathematically, we wish to design an s-sparse code set Ck0 +1 K = [ck0+1, ck0+2, . . . , cK] matrix, where ci = {ci,j ∈ C : j ∈ Ii}, for i ∈ K = {k0 +1, k0 +2, . . . , K} that can be appended to a given code set C1 k0 = [c1, c2, . . . , ck0 ], where ci = {ci,j ∈ C : j ∈ Ii}, for i ∈ K0 = {1, 2, . . . , k0 } to result in an improved LDS code matrix C = [C1 k0 Ck0 +1 K ]. The only constraint imposed on the proposed algorithm while generating the LDS sequence is that the input matrix C1 k0 must obey one of the convergent structures discussed in Section IV-A above. Therefore, the resultant algorithm can be applied for designing s-sparse spreading codes that are appended to an input optimal LDS sequence with the aid of a small modification. A = 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 . In Step 3, we randomly initialize Ck0 +1 K and form C = [C1 k0 Ck0 +1 K ] instead of randomly initializing C. In Step 6, we use k ∈ K instead of k ∈ {1, 2, . . . , K} as shown in Table 2. To illustrate one of the sample LDS outputs of size 6×9 from the modified algorithm, we first arbitrarily generate an orthogonal 2-sparse 6 × 6 matrix, which belongs to one of the convergent structure discussed in Section IV-A and use that as an input to the modified algorithm. The resultant LDS sequence structure is shown in Fig. 42. Observe that that in Fig. 42 the C7 9 matrix is 1-sparse and since the columns have unit energy, the elements are simply 1. We will characterize the performance of such code sets in our simulations. The proposed receiver is presented in the next section. C. COMPLEXITY OF THE PROPOSED LDS CONSTRUCTION The main complexity contribution of the proposed algorithm is associated with updating qk,1 either in (33), or in (36), or alternatively in (38). Direct calculations of R−1 k for each user is expensive. However, with the aid of effi- cient numerical techniques such as the Sherman-Morrison- Woodbury formula, we can compute its inverse, i.e., R−1 k , at a complexity order of O(K2 ). This process is repeated K times for obtaining all K users’ LDS waveforms. On the other hand, the number of iterations in the ‘while’ loops in lines 4 and 2 depends on the thresholds and δ, respectively. The smaller the thresholds the longer it takes to complete the process. With our design parameters, the number of iterations on average was about 3 and 10 for the ‘while’ loops in lines 4 and 2, respectively. Therefore, the overall complexity is O(c · K3 ) = O(K3 ) where c is a constant, e.g., c = 3 · 10 = 30. V. MULTIUSER DETECTION It is widely recognized that obtaining the ML solution is generally NP-hard [46]. Various suboptimal low-complexity detection techniques have already been proposed for con- ventional dense spreading based CDMA systems. These suboptimal approaches can be classified into two categories: linear and non-linear MUDs. Linear MUDs include among others, matched filtering (MF), MMSE, and zero-forcing (ZF) based schemes. In a non-linear successive interference cancellation aided detector the interference is first estimated and then it is subtracted from the received signal before detection. The cancellation process can then be carried out either successively (SIC) [94], or in parallel (PIC) [95]– [97]. In non-linear iterative detectors [98]–[102], PDA [48] aims for suppressing the MAI in each iteration in order to improve the overall error performance. Suboptimal so-called polynomial-time detectors that are based on the geometric approach are studied in [103], [104]. In comparison to dense CDMA, sparse CDMA or LDS is capable of substantially reducing the computational com- plexity of MUDs. This is a benefit of the sparse nature of LDS sequences that enables both the MPA and belief propagation (BP) algorithms to be applied at a lower com- plexity than the optimum MUD. In terms of reducing the complexity of the MPA algorithm even further without much performance erosion, Du et al. [105] proposed a detection scheme based on a dynamic factor graph by exploiting the channel state information. Another solution conceived by Tian et al. [106] reduces the complexity by restricting the search region of the superimposed multiuser constellation to a quadrant-like part of it. Razavi et al. [107] proposed a beneficial receiver component activation scheduling for iterative MUD in order to reduce its complexity by utilizing the LDPC codes for an LDS-OFDM system. VOLUME 4, 2016 13
- 14. The relationship between the optimal performance and the performance achieved by iterative BP has been established by Guo and Wang in [108] in the CDMA context. Their study demonstrated that for about a hundred users, the theoretical performance limit of large systems is approached as a result of the central-limit theorem. Those studies are normally performed under the assumption of a large system, where both the number of users and the spreading factor tend towards infinity, while their ratio is kept constant. As an example, Takeuchi et al. [109] characterized the family of BP receivers via density evolution (DE) in the dense limit after assuming the large-system limit. In those studies the specific way the MPA is implemented played a significant role. The user’s data detection based on the MPA and on the optimal ML detection using turbo-style processing is reported by Razavi et al. [23]. In contrast to this, it is shown in [110] that a joint detection and decoding approach based on an optimised sparse graph of the multiuser channel and the LDPC codes outperforms the iterative receiver of LDS-OFDM systems. Wen and Su [111] showed both numerically and analytically that the JSG- CDMA, which combines multiple access using LDS-CDMA and LDPC forward error correcting techniques, attains a satisfactory performance under rank-deficient conditions and outperforms conventional CDMA, LDS-CDMA as well as iterative detection aided LDS-CDMA. The detailed analysis is presented in [112]. Nevertheless, the BP and MPA detection methods still have exponential by increased computational complexity as a function of the number of users. The trade-off between the computational complexity and bandwidth efficiency at different user-load is studied by Raymond in [113]. Near- suboptimal detectors tend to strike a compelling perfor- mance as complexity trade-off compared to an MPA detec- tor. Therefore, by taking full advantage of the LDS scheme, which has a lower MAI than dense CDMA systems, we consider an attractive low-complexity detector, which is based on the MMSE criterion and on PIC (MMSE-PIC) based detection [84] that has an even lower complexity than the MPA based detector, whilst achieving the same spectral efficiency. Fantuz and D’Amours [84] showed that the BER performance of MMSE-PIC is very close to that of the MPA detector for LDS systems communicating over AWGN, non- dispersive and frequency selective fading channels. A. MMSE-PIC DETECTOR The MMSE-PIC detector of Fig. 6 is constituted by a benefi- cial amalgam of the MMSE and PIC detectors which will be characterized for transmission over AWGN, non-dispersive and frequency-selective fading channels, respectively. For the sake of simplicity, the derivation of the detector is provided for BPSK and 4QAM, constellations of X = {−1, +1} and X = {−1 − j, −1 + j, +1 − j, +1 + j}/ √ 2. However, it should be noted that similar derivations can be readily provided for higher-order constellations, such as 8QAM, 16QAM, 32QAM, etc. 1) AWGN Channel The despreading is performed by multiplying the received vector in (3) by the LDS code as follows, r = CH y = RDx + CH n, (42) where we have r ∈ CK×1 and the correlation matrix obeys R = CH C ∈ CK×K . The optimal receiver achieves the minimum probability of error Pr(x 6= b x) for each symbol vector x, which is arranged by estimating b x upon maximiz- ing the a posteriori probability (APP) Pr(x|r)’s given the observed despread sequence r, which is formulated as b x = argmax x∈XK×1 Pr(x|r). (43) This decision criterion is commonly referred to as the MAP [114] algorithm. It is widely known that the MAP detector has an exponentially increased complexity by the number of users K, which makes its application somewhat unrealistic even for moderate values of K. In practice it is more convenient to work with log-likelihood ratios (LLRs) than with probabilities. The LLRs for each symbol am, where am ∈ X for 1 ≤ m ≤ M, of the k-th user can be written as Λk(am) = log P x∈Aam xk Pr(r|x)Pr(x) P x̄∈Aam xk Pr(r|x̄)Pr(x̄) , (44) with Aam xk ⊂ Ax, Aam xk ⊂ Ax representing the set of all symbol vectors x ∈ Ax in which we have xk = am and xk 6= am for the k-th user. Furthermore, we have Ax = XK×1 and Pr(r|x) = 1 πK|Σ| exp[−fH (x)Σ−1 f(x)], (45) where f : x 7→ r − RCx represents a linear mapping of RC = RD, the covariance matrix obey Σ = σ2 CH C and |Σ| denotes the determinant of Σ. If we assume that all symbol vectors have the same probability distribution of Pr(x) = 1/MK , then the log-sum approximation of (44) can be expressed as Λk(am) ≈ min x∈Aam xk ||Σ− 1 2 f(x)||2 − min x̄∈Aam xk ||Σ− 1 2 f(x̄)||2 . (46) The computational complexity is increased exponentially versus the number of users K because the LLRs in (46) are calculated jointly for all the users hence requiring the computation of MK norm values. By contrast, the popular family of minimum mean square error detectors minimize the error-variance between the transmitted symbol and the filtered signal at the user level and they are more desirable in terms of complexity. Therefore, the per-user LLRs are computed separately. After MMSE filtering, our goal is to estimate the users’ symbols independently. Therefore, the MMSE detector’s action can be expressed in this form u = WMMSEr ∈ CK×1 , (47) where u represents the decision variables after the MMSE detector. The MMSE filter, weight-matrix WMMSE ∈ CK×K 14 VOLUME 4, 2016
- 15. is found by minimizing the mean-square error between the estimated symbols and the true transmitted symbol x, which is expressed as WMMSE = argmin W∈CK×K E{|x − Wr|2 }. (48) Under the reasonable assumption that each user’s symbols are independent and identically distributed (i.i.d.) with unit energy, when we have E{xxH } = IK, the solution of (48) is given by WMMSE = RH C RCRH C + Σ † , (49) where (·)† denotes the Moore-Penrose pseudoinverse op- eration [115]. The MMSE decision variables u are then processed to obtain the log likelihood ratios. The MMSE decision variable for the k-th user can be written as uk = wkr = wkRk Cxk + K X i=1 i6=k wkRi Cxi + wkCn = βk,kxk + K X i=1 i6=k βk,ixi + wkCn, (50) where wk ∈ C1×K is the k-th row vector of WMMSE, Rk C ∈ CK×1 is the k-th column of RC, xk ∈ X is the k-th symbol of the vector x and βk,i = wkRi C ∈ C. Since the direct evaluation of Pr(xk = am|u) is computationally prohibitive, the PDA detector attempts to estimate it by using the Gaussian - “forcing” idea of [116] by approxi- mating Pr[xk = am|u, {p(j)}∀j6=k], that can serve as the updated value for pm(k). The vector p(k) is associated to xk, whose m-th element pm(k), is the current estimate of a posteriori probability of xk = am. In contrast to the PDA detector, MMSE detector attempts to estimate it by making a reasonable assumption on conceiving the a priori probability distribution of Pr(xk), namely that it is i.i.d. having an expected value of unity. If we model the residual MAI after the MMSE detector by a complex Gaussian random variable which is independent of the noise, then uk is Gaussian as well. Let αm(k) = − |uk − βk,kam|2 σ2 k , (51) where we have σ2 k = PK i=1 i6=k |βk,i|2 E{|xi|2 } + wkΣwH k , E{|xi|2 } = 1 since the xi values are i.i.d. random variable. Provided that all the transmitted symbols have identical a priori probabilities, the a posteriori symbol probability is given by Pr(xk = am|uk) = Pr(uk|xk = am)Pr(xk = am) P am∈X Pr(uk|xk = am)Pr(xk = am) = exp[αm(k)] P j exp[αj(k)] , (52) where we have Pr(uk|xk = am) = 1 πσ2 k exp[αm(k)]. The a posteriori probabilities of the symbols am can also be expressed in terms of their LLR’s as follows, ΛMMSE k (am) = log Pr(xk = am|uk) Pr(xk 6= am|uk) = log exp[αm(k)] P j6=m exp[αj(k)] . (53) Furthermore, to simplify the avaluation of (53), the log-sum approximation can be used: ΛMMSE k (am) ≈ max log exp[αm(k)] − max j6=m log exp[αj(k)] ≈ αm(k) − max j6=m αj(k). (54) In the case of BPSK, (53) simplifies to: ΛMMSE k (a1) = log exp[α1(k)] exp[α2(k)] = α1(k) − α2(k) = 2βk,kuk σ2 k , (55) where a1 = +1 and a2 = −1. Note that the a posteriori probability Pr(xk = am|uk) in (52) can be expressed in terms of the LLRs of (53) as follows, Pr[xk = am|ΛMMSE k (am)] = 1 2 (1 + tanh[ 1 2 ΛMMSE k (am)]). (56) In practice, binary channel decoders require bit-level LLRs. Even though Gray-coding is used for QAM, which imposes correlation, for simplicity we assume the independence of the bits. Hence, if we assume the coded bits to be i.i.d., the log-likelihood ratio of a bit bk,i can be formulated as, ΛMMSE k (bi) = log Pr(bk,i = 1|uk) Pr(bk,i = 0|uk) = log P aj ∈X1 i Pr(xk = aj|uk) P aj ∈X0 i Pr(xk = aj|uk) = log P j|aj ∈X1 i exp[αj(k)] P j|aj ∈X0 i exp[αj(k)] , (57) where bk,i represents the i-th bit of the symbol xk, Xλ i = {aj ∈ X|b(j) = ψ−1 (aj), bj i = λ}, and λ = {1, 0}. Note that the probability of having bk,i = 1 can be expressed in terms of ΛMMSE k (bi) as: Pr(bk,i = 1|uk) = exp[ΛMMSE k (bi)] 1 + exp[ΛMMSE k (bi)] . (58) The complexity of (57) can be reduced by using the log-sum approximation, which is expressed as ΛMMSE k (bi) ≈ max j|aj ∈X1 i log exp[αj(k)] − max j|aj ∈X0 i log exp[αj(k)] ≈ max j|aj ∈X1 i αj(k) − max j|aj ∈X0 i αj(k). (59) VOLUME 4, 2016 15
- 16. y Single- user detector Split real and imaginary parts MMSE detector Estimate symbols PIC detector r rR u x̂ x̅ y Single- user detector Split real and imaginary parts MMSE detector Estimate symbols PIC detector r rR u x̂ x̅ FIGURE 6. Block diagram of MMSE-PIC detector [84]. Given the bit-level LLRs ΛMMSE k (bi), the a posteriori proba- bilities can be expressed as follows, Pr(xk = aj|ΛMMSE k (b)) = Q Y i=1 Pr[bk,i = ai j|ΛMMSE k (bi)], where we have: Pr(bk,i = λ|ΛMMSE k (bi)) = 1 2 (1 + b̃k,itanh[ 1 2 ΛMMSE k (bi)]), (60) and b̃k,i = ( +1, if bk,i = 1 −1, if bk,i = 0 , (61) while ΛMMSE k (b) = [ΛMMSE k (b1), . . . , ΛMMSE k (bQ)]T . The MMSE-PIC algorithm approximates the estimates x̄k of the transmitted symbols xk of user k by its mean value, which is formulated as, x̄k = E{xk} = X aj ∈X aj · Pr[xk = aj|ΛMMSE k (aj)] = X aj ∈X aj · Pr[xk = aj|ΛMMSE k (b)], (62) for k = 1, . . . , K. Alternatively, the soft-decision of the estimates of x̄k can be expressed as x̄k = argmax aj ∈X Pr[xk = aj|ΛMMSE k (aj)]. (63) In case of QAM, (62) can be expressed in terms of ΛMMSE k (aj) as x̄k = 1 2 √ 2 (−tanh[ 1 2 ΛMMSE k (a1)] − tanh[ 1 2 ΛMMSE k (a2)] +tanh[ 1 2 ΛMMSE k (a3)] + tanh[ 1 2 ΛMMSE k (a4)] +j(−tanh[ 1 2 ΛMMSE k (a1)] + tanh[ 1 2 ΛMMSE k (a2)] −tanh[ 1 2 ΛMMSE k (a3)] + tanh[ 1 2 ΛMMSE k (a4)])), where a1 = {−1 − j}/ √ 2, a2 = {−1 + j}/ √ 2, a3 = {+1−j}/ √ 2, and a4 = {+1+j}/ √ 2. In terms of ΛMMSE k (b), (62) can be expressed as x̄k = 1 √ 2 (tanh[ 1 2 ΛMMSE k (b1)] + jtanh[ 1 2 ΛMMSE k (b2)]), (64) and the estimates of the transmitted symbol vector x of all users can be written as x = 1 √ 2 (tanh[ 1 2 ΛMMSE (b1)] + jtanh[ 1 2 ΛMMSE (b2)]), (65) where ΛMMSE (bη) = [ΛMMSE 1 (bη), . . . , ΛMMSE K (bη)]T and η ∈ {1, 2}. In case of BPSK, (62) can be expressed in terms of ΛMMSE k (a) as x̄k = tanh[ 1 2 ΛMMSE k (a)], (66) where a = {−1, +1} and the estimates of the transmitted symbol vector x of all users can be written as x = tanh[ 1 2 ΛMMSE (a)], (67) where ΛMMSE (a) = [ΛMMSE 1 (a), . . . , ΛMMSE K (a)]T . The PIC stage of the detector produces the final decision variables according to uPIC,k = uk − K X i6=k RC,k,ixi, (68) 16 VOLUME 4, 2016
- 17. where RC,k,i is the element in the k-th row and i-th column of RC, while uk is the k-th element of u. The estimates in (68) can be used as a priori probabilities for the channel decoder. If no channel coding is used, then hard-decision detection can be employed, which is formulated as x̂k = argmin aj ∈X ||uPIC,k − aj||2 , ∀k. (69) In case of BPSK, the hard-decision bits may be then estimated by x̂ = sgn ({uPIC}), (70) where we have uPIC = [uPIC,1, uPIC,2, . . . , uPIC,K]T . After computing uPIC,k using x in (68), we can recompute the LLRs in (53) by using the uPIC,k values instead of uk obtained after the MMSE filter for improving the detection performance. In order to take advantage of the potential diversity gain of the multi-dimensional signal space, we will exploit it by converting the Cartesian product of the complex plane to the real space, which will double the number of dimen- sions. Since detection of complex symbols (e.g., QAM) is equivalent to estimating the real and the imaginary parts of the complex symbols in parallel, this simplifies the detection process and reduces the decoding complexity as well. We then split the vectors and matrices in (42) into their real and imaginary components, as follows: rR = RRxR + CRnR, (71) where we have rR ∈ R2K×1 , RR ∈ R2K×2K , CR ∈ R2K×2L , nR ∈ R2L×1 and the subscript R, {} and ={} represent the real domain as well as the real and imaginary parts of a complex number, rR = {r} ={r} , xR = {x} ={x} , RR = {RD} − ={RD} ={RD} {RD} , CR = {CH } − ={CH } ={CH } {CH } and nR = {n} ={n} , respectively. We treat the elements of xR as independent multivariate random variables, where the i-th element, xR,i, is a member of one of two possible sets, xR,i ∈ ( {xk = am|am ∈ X}, i ∈ [1, K] ={xk = am|am ∈ X}, i ∈ [K + 1, 2K], (72) where k ∈ {i, i−K}. The noise nR has the variance matrix of ΣR = σ2 2 CRCH R . Note that for BPSK transmission x ∈ {±1}K×1 is real-valued, which results in its imaginary part being a zero vector. Then (71) can be simplified to: rR = {RD} ={RD} x + {CH } − ={CH } ={CH } {CH } {n} ={n} . (73) The separation of the real and imaginary parts provides an extra dimension for the detector in order to have a better estimate of each user’s symbol. Therefore, the MMSE detector can be expressed as u = WMMSErR ∈ R2K×1 , (74) where the MMSE filter, WMMSE ∈ R2K×2K , is found by minimizing the mean-square error between the estimated symbols and the true transmitted symbol xR, which is expressed as WMMSE = argmin W∈R2K×2K E{||xR − WrR||2 }. (75) The solution of (75) is given by WMMSE = RT R RRRT R + ΣR † . (76) Note that in case of BPSK, we have WMMSE ∈ RK×2K , u ∈ RK×1 and xR = x. The MMSE decision variable for the i-th element can be written as ui = wirR = wiRi RxR,i + 2K X j=1 j6=i wiRj RxR,j + wiCRnR = βi,ixR,i + 2K X j=1 j6=i βi,jxR,j + wiCRnR, (77) where wi ∈ R1×2K is the i-th row vector of WMMSE, Ri R ∈ R2K×1 is the i-th column of RR, and βi,j = wiRj R ∈ R. Expression (51) in the real domain can be expressed as αm(i) = − (ui − βi,iam)2 2σ2 i , (78) where am ∈ XR for 1 ≤ m ≤ 2M, XR = {{X}, ={X}}, σ2 i = P2K j=1 i6=i β2 i,jE{x2 R,j} + wiΣRwT i and E{x2 R,j} = 1 since the xR,js are i.i.d. random variables. Based on (78), the LLRs for each symbol am, defined as ΛMMSE i (am) = log(Pr(xR,i = am|ui)/Pr(xR,i 6= am|ui), can be calcu- lated by (53) as in the complex formulation scenario. All the other LLRs and a posterior probabilities are computed in a similar way to the complex formulation case, except that now we have to perform for 1 ≤ i ≤ 2K elements and am ∈ XR, 1 ≤ m ≤ 2M symbols with the exception of the BPSK case. In the PIC stage of (68), we substitute RR,j,i instead of RC,k,i. In the case of QAM, the decision variable for user k can be computed as uPIC,k = uR,PIC,k + juR,PIC,k+K and the hard-decision is given by x̂k = x̂R,k + jx̂R,k+K, for 1 ≤ k ≤ K. 2) Non-dispersive Fading Channel In addition to the despreading operation the decision vari- ables uks are multiplied by the corresponding channel gains as follows, r̃k = h∗ kcH k y = |hk|2 dkxk + K X i=1,i6=k Rk,ih∗ khidixi + h∗ kcH k n, (79) VOLUME 4, 2016 17
- 18. encoder b1 + n y AWGN · · · · · · interleaver d1·x1 c1 u1 modulator encoder b2 interleaver d2·x2 c2 u2 modulator encoder bK interleaver dK·xK cK uK modulator FIGURE 7. Block diagram of our BICM transmitter. where the superscript ∗ denotes the complex conjugate. The vector of decision variables can be expressed as r̃ = HH CH y = HH CH CHDx + HH CH n = HH RHDx + HH CH n. (80) The vectors and matrices in (80) are then split into real and imaginary components, as shown below: r̃R = R̃RxR + C̃RnR, (81) where r̃R = {r̃} ={r̃} , R̃R = {HH RHD} − ={HH RHD} ={HH RHD} {HH RHD} , C̃R = {HH CH } − ={HH CH } ={HH CH } {HH CH } , respectively. The PIC-MMSE detector design for non- dispersive fading channel is very similar to that of the AWGN channel, except that the transmitted signal is sub- jected to the complex-valued gains. Nonetheless, the differ- ence is that the correlation matrix R̃ and the LDS sequence matrix C̃ are defined above, as opposed to the correlation matrix R and LDS sequence matrix C used for AWGN channel. 3) Frequency-Selective Fading Channel There has been extensive research on LDS and/or SCMA systems communicating over AWGN [22], [32]–[34], [39], [41] and non-dispersive fading channels [25], [35], [36], [40]. Most of the studies are dedicated to frequency-selective channels relying on LDS-OFDM [67], or MC-CDMA [83]. LDS-OFDM is eminently suitable for frequency-selective channels, since its subcarriers bandwidth is narrower than the channels coherence bandwidth [84]. Traditional CDMA tends to mitigate the multipath effects by using RAKE receivers [117], [118]. A whole suite of fading-mitigation techniques were conceived in Hanzo et al. [119] ; Hanzo et al. [120]. By contrast, here we employ a transmit pre- coding scheme for overcoming the multipath channel effect as proposed by Fantuz and D’Amours, which is detailed in [84]. Briefly, this transmit precoding scheme exploits the knowledge of the CIR for transforming the multipath channel into a single-path non-dispersive channel. More explicitly, it transforms (7) to (5), which is equivalent to over non-dispersive Rayleigh fading channel model. Therefore, the MMSE-PIC detector derived for non-dispersive fading channels can be directly applied to frequency-selective chan- nels with the aid of the transmit precoding scheme of [84]. B. PDA DETECTOR The PDA [116], [121] has been widely applied by low- complexity design alternative of the optimal maximum a posteriori (MAP) symbol decoders/detectors, as a benefit of its near-optimal detection performance in rank-deficient CDMA systems [48], [116]. Explicitly, its complexity in- creases no faster than O(K3 ). The PDA detector was originally conceived in 2001 for CDMA [116] and its generalized version [122] can be directly applied to our LDS system designed for BPSK and QAM transmissions. In the case of QAM, Yang et al. [123] presented a unified bit-based PDA detection approach, which transforms a high- order rectangular QAM based multiuser system into a BPSK multiuser system. By contrast, in [124] an SCMA scheme is converted to a BPSK modulated CDMA system. More explicitly, we can convert (3) into a BPSK system as follows, y = CDWb + n (82) = Qb + n, (83) 18 VOLUME 4, 2016
- 19. y + _ + _ Λ1(b1,i) λ1(b1,i) + λ1(b1,n) Λ2(b1,n) λ2(b1,n) + SISO channel decoder interleaver deinterleaver λ2(b1,i) u1 + _ + _ Λ1(b2,i) λ1(b2,i) + λ1(b2,n) Λ2(b2,n) λ2(b2,n) + SISO channel decoder interleaver deinterleaver λ2(b2,i) u2 + _ + _ Λ1(bK,i) λ1(bK,i) + λ1(bK,n) Λ2(bK,n) λ2(bK,n) + SISO channel decoder interleaver deinterleaver λ2(bK,i) uK · · · · · · · · · SISO multiuser detector · · · FIGURE 8. Block diagram of our iterative turbo MUD [99]. where we have W = IK ⊗ sT , Q = CDW, s = [j, 1]T and ⊗ is a Kronecker operator. In SectionVIII we will present the BER performance of PDA detectors for transmission over AWGN, non-dispersive, and frequency-selective chan- nels. VI. CHANNEL ENCODING In his groundbreaking work [125] Shannon beautifully laid out the fundamental limit of communications known as channel capacity. However, the early design of communica- tion systems has focused on separate modulation and error correcting codes. Yet the solution to the problem of increas- ing the transmission rate without bandwidth expansion is to use a high-order constellation transmitting with spectral efficiency η where 1 ≤ η ≤ log2|X| bits/symbol. Shannon also introduced [125] the idea of combining coding with nonbinary modulation using high-order constellations, for coded modulation (CM) [125]. In this context we emphasize that both the specific choice of coding as well as the mapping of coded bits to constellation points is influential in terms of determining the attainable performance. The first practical CM scheme, namely the so-called multilevel coded modulation (MLCM) arrangement was introduced by Imai and Hirakawa in 1977 [126], [127]. Then in 1982 Unberboeck and Csajka developed the so- called the trellis-coded modulation (TCM) scheme that was specifically designed for increasing the Euclidean distance (ED) between the transmitted codewords because this is the most important criterion, when communicating over AWGN channels [128]. Later, CM inspired by the turbo principle has led to the so-called turbo trellis-coded modulation (TTCM) concept [129], [130]. Another important technique, namely the so-called bit- interleaved coded modulation (BICM) was conceived by Zhavi for fading channels [131]. Although BICM is inferior to TCM in terms of its ED, it outperforms TCM for transmission over fading channels as a benefit of its diversity gain. The original motivation of involving bit-interleavers was to improve the performance for transmission over fast-fading channels, because for such fading channels, the most important parameter of the code is its diversity gain rather than its ED. Morever, BICM exhibited very good performance for transmission over AWGN channels as well. This is the primary reason why BICM gained interest among researchers, but it also exhibits substantial flexibility in terms of its code design. In contrast to both TCM and TTCM, where the coding rate of n/(n + 1) must be carefully matched to the modulation constellation, BICM allows the constellation and the encoder to be designed more independently. The block diagram of a BICM transmitter is shown in Fig. 7 for a bk-bit QAM scheme, while the matching SISO turbo multiuser detector [99] portrayed in Fig. 8. VII. COMPLEXITY OF DETECTORS The computational complexity of the existing MMSE-PIC, MPA, and PDA algorithms is compared in Table 3. TABLE 3. Computational Complexity Comparison Algorithms Complexity Main procedures MMSE-PIC O(K2) multiplication, addition MPA O(Mdf ) multiplication, addition PDA O(K3) multiplication, addition In the MMSE-PIC detector the MMSE filter, which requires matrix inversion, does not have to update the filter for every signaling interval when transmitting over AWGN channels, since there are no changes in the channel condi- tions. Explicitly, it can be computed before communications, given the prior knowledge of the spreading sequence of each user and the noise variance. By contrast, for non-dispersive VOLUME 4, 2016 19
- 20. TABLE 4. LDS spreading code set coefficients for 4 × 6 a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 C 1 ej60◦ ej120◦ C4 1 ej30◦ ej60◦ C3 1 ej0.143π ej0.202π ej0.313π ej0.574π ej0.377π ej0.394π ej0.267π ej0.308π Cp 0.223 0.975 0.975 0.223 0.519 0.855 0.855 0.519 0.339 0.941 0.941 0.339 ej0.841π e−j0.455π ej0.455π ej0.159π e−j0.024π e−j0.943π ej0.943π e−j0.976π e−j0.964π 1 ej0.036π 1 C2 0.646 0.764 0.679 0.734 0.852 0.524 0.729 0.684 0.743 0.670 0.594 0.829 e−j0.073π 1 ej0.655π e−j0.004π e−j0.639π ej0.027π e−j0.740π 1 ej0.590π 1 e−j0.010π ej0.024π Note that a0 coefficient of CP can be read as a0 = 0.223ej0.841π . and frequency-selective channels, it needs updating of the correlation matrix, as and when the channel gain changes. To avoid the regular recomputation of the filter coefficients, adaptive algorithms may be used, such as the recursive least squares or least mean squares techniques [132] for directly updating the inverse. Additionally, both the MPA and the PDA algorithms will also require some additional processing to adapt to the channel conditions [84]. In contrast to the MMSE-PIC, the MPA does not need to perform any matrix inversion, but its complexity increases exponential by both with the size of the symbol alphabet M and number of non-zero positions of the spreading wave- form df [133]. Finally, the PDA requires matrix inversion, but fortunately this can be carried out quite efficiently with the aid of the Sherman–Morrison–Woodbury formula at an overall complexity order of O(K3 ) [116]. VIII. COMPARISONS WITH OTHER LDS DESIGNS In this section, we evaluate the performance of the proposed LDS code sequences generated by the algorithm of Table 2 for the LDS sequence designs of sizes 4 × 6 and 6 × 9. A. UNCODED LDS Simulations are performed for transmission over the com- plex AWGN channel using an identical transmission power for each user, whilst relying on unit-energy LDS sequences and no channel encoding. In the first experiment, we com- pare the LDS code matrices, of (84) and (85) that are generated by our proposed algorithm to the code matrices derived in [32], and shown in (87) and (88). The proposed algorithm is ran using the following parameters L = 4, K = 6, δ = 1.9, σ2 d = 0.5 and s = 2. The initialization of the matrix C was performed, as discussed in Section IV-A, which results in a code set described as follows, Cp = a0 0 a4 0 0 a10 a1 0 a5 0 0 a11 0 a2 0 a6 a8 0 0 a3 0 a7 a9 0 , (84) where all the corresponding coefficients ai are described in Table 4. We run again the algorithm, but this time with the random initialization of the matrix C with δ = 1.7, which outputs the following code set, C2 = 0 a2 a4 0 0 a10 a0 0 0 a6 a8 0 0 a3 a5 0 0 a11 a1 0 0 a7 a9 0 , (85) where all the corresponding coefficients ai are described in Table 4. For fair comparison, we take the existing LDS code sets presented in [32], [39], and [35] and label them as C, C3 , and C4 , which are then normalized as follows, C = 1 √ 2 a0 a1 a2 0 0 0 a0 0 0 a1 a2 0 0 a0 0 a1 0 a2 0 0 a0 0 a1 a2 , (86) C3 = 1 √ 2 a0 a1 a2 0 0 0 a0 0 0 a3 a5 0 0 a1 0 a4 0 a7 0 0 a2 0 a6 a8 , (87) C4 = 1 √ 2 a0 a1 a2 0 0 0 a0 0 0 a1 a2 0 0 a0 0 a1 0 a2 0 0 a0 0 a1 a2 , (88) where all the corresponding coefficients ai derived for each code set are described in Table 4. The properties of the matrices in our comparisons are summarized at a glance in Table 5. TABLE 5. Comparisons (C4×6) Metric C Cp C2 C3 C4 dE,min 2.00 2.00 2.00 1.83 1.47 ∆ave 1.49 1.90 1.75 1.10 1.16 dP,min 2.00 0.76 0.04 0.11 0.29 dM,min 2.83 2.40 2.38 2.83 2.83 Csum 4.95 5.23 5.29 4.99 4.99 S1,2,ave 0.59 0.72 0.60 0.59 0.59 MWBE No No Yes No No As seen in Fig. 9, Cp outperforms the other candidates (e.g., C, C3 and C4 ), when ML detection is used. Although our proposed matrices, Cp and C2 , have the same dmin, they have a higher value of ∆ave compared to C, C3 and C4 . Furthermore, C2 is considered to be a MWBE matrix, whereas all the other candidates are not. However, the code set Cp exhibits better BER performance than C2 . Therefore, we surmise that the BER performance depends not only on the minimum distance (e.g., dmin), but also on the average Gaussian separability margin ∆ave. We also note that the average sparsity of our proposed matrix Cp defined in (26) is higher than that of its counterparts, which is shown in bold in Table 5. In the case of the code sets having dimensions of 6 × 9, we illustrate the code sets generated by Table 2 Cp , C3 , C4 and C5 . The initialization 20 VOLUME 4, 2016
- 21. of matrix C is performed as discussed in Section IV-A using δ = 1.8, which results in the following code set, Cp = a0 a2 0 0 0 0 0 a12 0 a1 a3 0 0 0 0 a12 0 0 0 0 a4 a6 0 0 0 0 0 0 0 a5 a7 0 0 0 0 0 0 0 0 0 a8 a10 0 0 0 0 0 0 0 a9 a11 0 0 a12 , (89) Note in our simulations, ‘off-line’ computation is assumed, however ‘on-line’ computation can be performed upon any changes such as channel conditions, L and s-sparseness, etc. FIGURE 9. Uncoded BPSK case comparisons of C4×6 with labels LDS [32], LDS3 [39], LDS4 [35]. We run the algorithm once again, but this time with random initialization of the matrix C, in conjunction with δ = 1.7, δ = 1.65 and δ = 1.65, which outputs the following code sets, C3 = a0 0 0 a6 0 0 a12 0 0 a1 0 0 a7 0 0 a13 0 0 0 a2 0 0 a8 0 0 a14 0 0 a3 0 0 a9 0 0 a15 0 0 0 a4 0 0 a10 0 0 a16 0 0 a5 0 0 a11 0 0 a17 , (90) C4 = a0 0 0 a6 0 0 a12 0 0 0 a2 0 0 a8 0 0 a14 0 0 0 a4 0 0 a10 0 0 a16 a1 0 0 a7 0 0 a13 0 0 0 a3 0 0 a9 0 0 a15 0 0 0 a5 0 0 a11 0 0 a17 , (91) C5 = a0 0 0 a6 0 0 a12 0 0 0 a2 0 0 a8 0 0 a14 0 a1 0 0 a7 0 0 a13 0 0 0 a3 0 0 a9 0 0 a15 0 0 0 a4 0 0 a10 0 0 a16 0 0 a5 0 0 a11 0 0 a17 , (92) where all the corresponding coefficients ai for each code set are described in Table 6. For comparison we consider the existing LDS sequence sets proposed in [32] and label them as C and C2 , which are then normalized as follows, C = 1 √ 2 0 0 a2 0 0 0 a1 0 a0 0 a2 0 a1 0 0 a0 0 0 0 0 a1 a0 0 a2 0 0 0 a0 0 0 0 a1 0 0 0 a2 a2 0 0 0 0 a0 0 a1 0 0 a1 0 0 a0 0 0 a2 0 , (93) C2 = 1 √ 2 0 0 a0 0 0 0 a1 0 a2 0 a0 0 a1 0 0 a2 0 0 0 0 a0 a1 0 a2 0 0 0 a0 0 0 0 a1 0 0 0 a2 a0 0 0 0 0 a1 0 a2 0 0 a0 0 0 a1 0 0 a2 0 , (94) where all the corresponding coefficients ai for each code set are described in Table 6. Table 7 shows the comparison metric of all the LDS sequences. Similar to the case of the 4 × 6 code set, observe in Fig. 10 that the proposed Cp outperforms other LDS sequences in terms of its BER performance. FIGURE 10. Uncoded BPSK case comparisons of C6×9 code sets with [32] labeled as LDS. We again observe that the average Gaussian separability value, ∆ave, and the average sparsity, S1,2,ave, are higher for the matrix Cp compared the other matrices. In order to further characterize the performance, we performed sim- ulations using channel encoding, as discussed in the next section. B. CODED LDS Compared to the LDS designs conceived in [32], [35], [39], the sum rate Csum of our proposed codes is higher by about 0.30 bits per channel use. Hence it is expected that there is a channel code for our proposed LDS sequences that can produce a higher coded sum rate than those advocated in [32], [35], [39]. Therefore, to illustrate this hypothesis, we performed sim- ulations using LDPC, turbo and polar encoding to compare VOLUME 4, 2016 21
- 22. TABLE 6. LDS spreading code set coefficients for 6 × 9 a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 a16 a17 C 1 ej60◦ ej120◦ C2 1 ej60◦ ej120◦ Cp 0.951 0.310 0.310 0.951 0.473 0.881 0.881 0.473 0.783 0.622 0.622 0.783 1.000 −0.450 −0.682 0.682 −0.550 −0.870 −0.941 0.941 −0.130 0.416 −0.367 0.367 0.584 0.000 C3 0.875 0.484 0.600 0.800 0.862 0.510 0.658 0.753 0.841 0.541 0.664 0.748 0.547 0.837 0.658 0.753 0.562 0.827 −0.234 0.000 −0.778 0.000 −0.441 0.000 0.440 0.000 −0.130 0.000 0.888 0.000 −0.853 0.000 0.532 0.000 0.189 0.000 C4 0.351 0.937 0.785 0.620 0.865 0.502 0.865 0.502 0.615 0.789 0.419 0.908 0.793 0.609 0.711 0.704 0.760 0.651 −1.000 0.000 0.046 −0.981 0.595 0.000 −0.448 −0.020 0.400 0.030 −0.818 0.000 0.340 0.000 −0.300 0.000 −0.140 0.000 C5 0.317 0.949 0.953 0.302 0.743 0.670 0.780 0.626 0.500 0.866 0.710 0.705 0.890 0.457 0.583 0.812 0.667 0.745 0.023 1.000 −0.197 0.000 0.187 1.000 0.706 1.000 0.362 0.000 −0.145 0.000 −0.482 −1.000 −0.841 0.000 0.527 0.000 Note that a0 and a12 coefficients of CP can be interpreted as a0 = 0.951e−j0.450π and a12 = 1.000. our proposed LDS sequences to the ones advocated in [32], [35], [39]. TABLE 7. Comparisons (C6×9) Metric C Cp C2 C3 C4 C5 dE,min 2.00 2.00 2.00 2.00 1.81 1.60 ∆ave 1.34 1.88 1.34 1.76 1.62 1.64 dP,min 2.00 1.10 2.00 0.03 0.26 0.02 dM,min 2.83 2.00 2.83 2.37 2.13 1.85 Csum 7.35 7.75 7.39 7.93 7.93 7.93 S1,2,ave 0.71 0.84 0.71 0.73 0.74 0.75 MWBE No No No No Yes Yes We used three different error control codes. The first is a custom semi-random parity check matrix generator for the LDPC code as described in [134]. The second, we used the long-term evolution (LTE) turbo code described in [135]. -4 -2 0 2 4 6 8 Eb /N0 (dB) 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 Average BER LDS MMSE-PIC LDS proposed MMSE-PIC LDS2 MMSE-PIC LDS3 MMSE-PIC LDS4 MMSE-PIC Single User Shannon Limit FIGURE 11. BPSK with LDPC encoding comparisons of C4×6 code sets with labels LDS [32], LDS3 [39], LDS4 [35]. The third we used the polar code for which we calculated the Bhattacharyya parameters for the bit channel construction method as described in [136]. The construction of the LTE turbo interleaver is based on the quadratic permutation polynomial (QPP) scheme of [135]. For all three channel encoding cases we used a code rate of 1/3 with input message block lengths of 320 bits and the encoded code length of 972 for both the LDPC and LTE turbo codes. Furthermore, we used 340 input, and 1024 encoded code bits for polar coding. All of the output codewords the channel coders are then interleaved as in the BICM scheme discussed in Section VI. -4 -2 0 2 4 6 8 Eb /N0 (dB) 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 Average BER LDS MMSE-PIC LDS proposed MMSE-PIC LDS2 MMSE-PIC LDS3 MMSE-PIC LDS4 MMSE-PIC LDS5 MMSE-PIC Single User Shannon Limit FIGURE 12. BPSK with LDPC encoding comparisons of C6×9 code sets with [32] labeled as LDS. The BER performance of the LDS sequences shown in Figs. 11 - 14 for BPSK modulation shows that our proposed LDS code sets outperform the ones proposed in [32], [35], [39] for these coded cases. Thus trend is more clear when using LDPC encoding, as shown in Figs. 11 and 12 rather than LTE turbo encoding, shown in Figs. 13 and 14. In addition to the MMSE-PIC detector we applied both PDA [116] and SISO MMSE [99] detectors for BPSK modulation, which are characterized in Figs. 15 and 16. Our propsoed LDS based scheme outperforms the code set of [32] in terms of its BER performance for both the PDA and SISO MMSE detectors. The complex PDA detector [122] was adopted for QAM, is characterized in Figs. 17-20. The SCMA scheme associ- ated with a factor graph of 4×6 and M = 4 is compared to the LDS spreading matrix of size 4×6 using 4QAM in Figs. 18-20. We observe that the proposed LDS outperforms the SCMA arrangement using an MPA detector and the LDS of [32]. Similar results are also presented in Figs. 21 and 22 for bit-based PDA detection in [123]. 22 VOLUME 4, 2016
- 23. C. LDS CODE SETS FOR 200% OVERLOAD FACTOR In this section we evaluate the BER performance of our LDS sets for a normalized load factor of β = K/L = 2. More explicitly, we have constructed 4 × 8, 6 × 12 and 8 × 16 LDS code sets using our proposed algorithm presented in Table 2. -4 -3 -2 -1 0 1 2 3 4 5 Eb /N0 (dB) 10 -6 10 -4 10 -2 10 0 Average BER LDS MMSE-PIC LDS proposed MMSE-PIC LDS2 MMSE-PIC LDS3 MMSE-PIC LDS 4 MMSE-PIC Single User Shannon Limit FIGURE 13. BPSK with turbo encoding comparisons of C4×6 code sets with labels LDS [32], LDS3 [39], LDS4 [35]. The resultant column vectors have only two non-zero values. For comparison purposes, we also included LDS sets asso- ciated with β = K/L = 2 from the designs found in [22] and [34]. The minimum Euclidean distance for the 4 × 8, and 6 × 12 LDS code sets of [22] are 1.17 and 1.43, whilst for the proposed sets they are 2.0, respectively. -4 -3 -2 -1 0 1 2 3 4 Eb /N0 (dB) 10-6 10 -5 10-4 10 -3 10 -2 10 -1 Average BER LDS MMSE-PIC LDS proposed MMSE-PIC LDS 2 MMSE-PIC LDS 3 MMSE-PIC LDS 4 MMSE-PIC LDS5 MMSE-PIC Single User Shannon Limit FIGURE 14. BPSK with turbo encoding comparisons of C6×9 code sets with [32] labeled as LDS. Similarly, the average Gaussian separability margins for the 4 × 8, 6 × 12, and 8 × 16 LDS code sets of [22] are 0.96, 0.0 and 1.48, whilst for the proposed sets they are 1.79, 1.6 and 1.66, respectively. The reason why the LDS code sets in [22] and [34] are selected as the benchmarks is because the BER performance of other LDS candidates is similar for the 200% normalized load factor scenarios. -4 -3 -2 -1 0 1 2 3 Eb /N0 (dB) 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 Average BER LDS SISO MMSE LDS Proposed SISO MMSE LDS PDA LDS proposed PDA Single User Shannon Limit FIGURE 15. BPSK with turbo encoding comparisons of C4×6 code sets with [32] labeled as LDS. Observe in Figs. 23, 27, 29 and Figs. 26, 28, 30 that our proposed LDS code sets designed both for uncoded and turbo coded scenarios outperform the LDS code sets of [22] and [34]. At the BER of 10−3 there is about 1 − 2 dB SNR gain for the uncoded scenarios and a slighter smaller SNR gain is observed for coded scenarios. -4 -3 -2 -1 0 1 2 3 4 5 Eb /N0 (dB) 10 -6 10-5 10 -4 10 -3 10-2 10 -1 Average BER LDS SISO MMSE LDS Proposed SISO MMSE LDS PDA LDS proposed PDA Single User Shannon Limit FIGURE 16. BPSK with polar encoding comparisons of C4×6 code sets with [32] labeled as LDS. As seen in Figs. 23, 26-31, the proposed LDS code sets tend to approach the single-user BER. This is achieved as a benefit of the diversity gain obtained when splitting the complex vectors and matrices into real and imaginary parts, as discussed in the context of (73). For the BPSK case, our complex LDS matrix C of size L × K is transformed into CR of size 2L×K after splitting it into real and imaginary VOLUME 4, 2016 23
- 24. parts. In order to have an orthogonal matrix CR, the number of users should be K = 2L, hence we have β = 2. -4 -2 0 2 4 6 Eb /N0 (dB) 10 -6 10 -5 10 -4 10 -3 10-2 10-1 10 0 Average BER LDS PDA1 LDS proposed PDA1 Single User Shannon Limit FIGURE 17. QAM with LDPC encoding, iteration number = 1, comparisons of C4×6 code sets with [32] labeled as LDS. The proposed LDS construction seen in Table 2 provides an LDS matrix, so that when we convert it into its real and imaginary parts, the resultant CR matrix becomes near- orthogonal. This explains the reason for having a near- single-user BER performance for the proposed LDS, but we also observe that the BER performance deteriorates dramatically for scenarios of K 2L. On the other hand in contrast to BPSK, for QAM signaling, the LDS matrix C is converted into CR of size 2L × 2K after the real and imaginary parts are split. -4 -3 -2 -1 0 1 2 3 4 5 6 Eb /N0 (dB) 10-6 10-5 10 -4 10-3 10-2 10 -1 10 0 Average BER LDS PDA 1 LDS proposed PDA 1 SCMA MPA Single User Shannon Limit FIGURE 18. QAM with LDPC encoding, iteration number = 5, comparisons of C4×6 code sets with [32] labeled as LDS. Therefore, CR cannot be orthogonal, unless we have L = K, as verified by our simulations. Furthermore, the orthogonality of C̃R in BPSK signalling can be further degraded, when no transmitter precoding is utilized for transmission over frequency-selective fading channels. The performance difference of LDS codes over non-dispersive and frequency-selective fading channels are portrayed in Figs. 32 and 33. In our simulations, we assumed Lp = 7 for the frequency-selective fading channel. -4 -3 -2 -1 0 1 2 3 Eb /N0 (dB) 10 -6 10-5 10 -4 10 -3 10-2 10-1 10 0 Average BER LDS PDA1 LDS proposed PDA1 SCMA MPA Single User Shannon Limit FIGURE 19. QAM with turbo encoding, iteration number = 1, comparisons of C4×6 code sets with [32] labeled as LDS. As for future research, we have to perform a detailed stochastic analysis of the legitimate LDS code sets for complex signal constellations to find the specific sets, which have the best performance. We also have to study the direct optimization of the matrix CR in real domain in order to achieve near-orthogonality, instead of optimizing C in the complex domain under the constraint of keeping the non- zero locations of the real and imaginary parts identical. -4 -3 -2 -1 0 1 2 3 4 Eb /N0 (dB) 10-6 10-5 10 -4 10-3 10-2 10 -1 10 0 Average BER LDS PDA1 LDS proposed PDA1 SCMA MPA Single User Shannon Limit FIGURE 20. QAM with polar encoding, iteration number = 1, comparisons of C4×6 code sets with [32] labeled as LDS. As for QAM, we can investigate the design of matrices for the rank-deficient scenarios of K L in the real domain instead of the complex domain. Furthermore, we have to conceive LDS designs for ensuring that the resultant C̃R is near-orthogonal even under fading channels. 24 VOLUME 4, 2016
- 25. D. SPECTRAL EFFICIENCY One of the key performance metrics of LDS spreading code design is the resultant spectral efficiency, ηLDS(C, γ) (bits/s/Hz), which can be expressed as a function of either the SNR, γ or of the energy per bit Eb/No. -4 -3 -2 -1 0 1 2 3 4 5 6 Eb /N0 (dB) 10 -5 10 -4 10 -3 10-2 10 -1 10 0 Average BER LDS PDA2 LDS proposed PDA2 Single User Shannon Limit FIGURE 21. QAM with LDPC encoding, iteration number = 5, comparisons of C4×6 code sets with [32] labeled as LDS. The spectral efficiency ηLDS(C, γ) is defined as the maximum mutual information between the symbol vector x and the observed L-dimensional vector y in (3) for a given C over distributions of x normalized to L. -4 -3 -2 -1 0 1 2 3 4 Eb /N0 (dB) 10-6 10-5 10 -4 10 -3 10-2 10 -1 10 0 Average BER LDS PDA2 LDS proposed PDA 2 SCMA MPA Single User Shannon Limit FIGURE 22. QAM with turbo encoding, iteration number = 1, comparisons of C4×6 code sets with [32] labeled as LDS. Under the constraint of E{xxH } = EsIL, the opti- mum detection for a given LDS C may be achieved; for a Gaussian distributed x the resultant spectral efficiency ηLDS(C, γ) can be expressed by [10] ηLDS(C, γ) = Csum(C, γ) L = 1 L log2 |IL + γCDDH CH |, (95) where Es denotes energy per symbol, NoIL is the noise covariance and the per-symbol SNR γ is given by [137] γ = 1 K E{|x|2 } 1 L E{|n|2} = 1 K EbNb 1 L NoL = 1 β Eb No Nb L = 1 β Eb No ηLDS, (96) -2 0 2 4 6 8 Eb /N0 (dB) 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 Average BER LDS MMSE-PIC LDS proposed MMSE-PIC Single User Shannon Limit FIGURE 23. Uncoded BPSK transmission, C4×8 code sets with [22] labeled as LDS. where we have E{|x|2 } = NbEb, E{|n|2 } = LNo, Nb denotes the number of bits encoded in x for a capacity- achieving system, and Nb/L, which is expressed in bits per dimension, represents the spectral efficiency of (95). Since L denotes the number of complex dimensions in our system, ηLDS(C, γ), can be interpreted as the maximum number of bits per each complex dimension. -4 -2 0 2 4 6 8 SNR (dB) 0.5 1 1.5 2 2.5 3 3.5 4 Spectral efficiency (bits/s/Hz) LDS 200% LDS 200% proposed LDS 4x6 LDS 4x6 proposed LDS 6x9 LDS 6x9 proposed Unrestricted bound 200% Unrestricted bound 150% MMSE bound 200% MMSE bound 150% 2 2.5 3 1.8 2 2.2 FIGURE 24. Spectral efficiency vs SNR of the AWGN channel for BPSK. An upper bound on ηLDS(C, γ) can be considered as the spectral efficiency, when the LDS spreading sequence has a length of L = 1. This is equivalent of a K-user Gaussian multiple access channel and its spectral efficiency in the case of the average-energy-constraint is given by log2 (1 + γdtot) bits/s/Hz per chip [87], where dtot = c0 1c0H 1 = · · · = VOLUME 4, 2016 25
- 26. c0 Lc0H L , and c0 i are row vectors of CD. We can show that ηLDS(C, γ) is indeed capable of achieving the upper bound even when L 1. -2 -1 0 1 2 3 4 5 6 7 Eb /N0 (dB) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Spectral efficiency (bits/s/Hz) LDS 200% LDS 200% proposed LDS 4x6 LDS 4x6 proposed LDS 6x9 LDS 6x9 proposed Unrestricted bound MMSE bound 2.6 2.8 3 3.2 3.4 2 2.2 2.4 2.6 2.8 FIGURE 25. Spectral efficiency vs Eb/No of the AWGN channel for BPSK. Proposition 1. Let C be an LDS spreading matrix with D being the diagonal energy-constraint matrix and K L. Then, we have: ηLDS(C, γ) ≤ log2 (1 + γdtot). (97) -4 -2 0 2 4 6 8 Eb /N0 (dB) 10-5 10 -4 10 -3 10 -2 10 -1 Average BER LDS MMSE-PIC LDS proposed MMSE-PIC Single User Shannon Limit FIGURE 26. Turbo coded BPSK transmission, C4×8 code sets with [22] labeled as LDS. The necessary and sufficient condition of attaining the spectral efficiency upper bound of the system dispensing with spreading when the LDS signature waveforms are WBE sequences, is that of satisfying the condition CDDH CH = dtotIL [87]. Proof. The proposition can be proved by first applying Hadamard’s inequality [138] to the determinant in (95), yielding: |IL + γdtotIL| ≤ L Y i=1 (1 + γdtot). (98) Indeed the above expression satisfies the condition of equal- ity, since the determinant is a diagonal matrix. Then using Jensen’s inequality, we have [138]: log2 |IL + γdtotIL| = log2 L Y i=1 (1 + γdtot) = L X i=1 log2 (1 + γdtot) = L log2 (1 + γdtot). (99) -2 0 2 4 6 8 10 Eb /N0 (dB) 10 -6 10 -5 10 -4 10 -3 10 -2 10-1 10 0 Average BER LDS MMSE-PIC LDS proposed MMSE-PIC Single User Shannon Limit FIGURE 27. Uncoded BPSK transmission, C6×12 code sets with [22] labeled as LDS. Since in our design the columns of C are of unit-length and D = IK, the Frobenius norm of CD can be written as ||CD||F = K X k=1 cH k ck = L X i=1 c0 ic0H i = K. (100) -4 -2 0 2 4 6 Eb /N0 (dB) 10-6 10-5 10 -4 10 -3 10-2 10-1 100 Average BER LDS MMSE-PIC LDS proposed MMSE-PIC Single User Shannon Limit FIGURE 28. Turbo coded BPSK transmission, C6×12 code sets with [22] labeled as LDS. 26 VOLUME 4, 2016
- 27. -2 0 2 4 6 8 10 E b /N 0 (dB) 10 -5 10-4 10 -3 10 -2 10 -1 Average BER LDS MMSE-PIC LDS proposed MMSE-PIC Single User Shannon Limit FIGURE 29. Uncoded BPSK transmission, C8×16 code sets with [22] labeled as LDS. Therefore, we have dtot = K/L as c0 1c0H 1 = · · · = c0 Lc0H L . Note that if the K users do not have equal average- input-energy constraints, i.e., DDH 6= d0 IL, it is generally hard to design an LDS code set that maximizes ηLDS(C, γ) in Proposition 1. -4 -3 -2 -1 0 1 2 3 4 5 Eb /N0 (dB) 10-8 10 -6 10 -4 10 -2 10 0 Average BER LDS MMSE-PIC LDS proposed MMSE-PIC Single User Shannon Limit FIGURE 30. Turbo coded BPSK transmission, C8×16 code sets with [22] labeled as LDS. In all our BER performance plots, the information rates for LDS 4 × 6, 6 × 9, 4 × 8, 6 × 12 and 8 × 16 in case of BPSK are ηLDS = Nb/L = 1.5, ηLDS = 1.5, ηLDS = 2, ηLDS = 2, and ηLDS = 2 bits/s/Hz, respectively. Therefore, the corresponding unrestricted Shannon limits are calculated by using the upper bound log2 (1 + γβ) (97) as Eb/No = (2ηLDS −1)/ηLDS, Eb/No = 1.219(0.86dB) and Eb/No = 1.5 (1.76dB) for ηLDS = 1.5 and ηLDS = 2, respectively. In case of 4QAM (2 bits per symbol) the corresponding ηLDS is multiplied by 2 and for a channel coding rate of 1/3 by 1/3. -4 -2 0 2 4 6 8 E b /N 0 (dB) 10-4 10 -3 10 -2 10 -1 Average BER LDS MMSE-PIC LDS proposed MMSE-PIC Single User Shannon Limit FIGURE 31. Turbo coded BPSK transmission, C8×16 code sets with [34] labeled as LDS. In addition to analysing the spectral efficiency of the optimal detection, a range of linear detectors, such as the single-user MF (SUMF), ZF, MMSE are derived in [64]. The spectral efficiency of these multiple access channels is given by [64]: Rsumf lds (β, γ) = Rzf lds(β, γ) = Rmmse lds (β, γ) = β X k≥0 βk exp (−β) k! log2 1 + γ kγ + 1 0 2 4 6 8 10 12 Eb /N0 (dB) 10-8 10-6 10 -4 10 -2 10 0 Average BER LDS MMSE-PIC LDS proposed MMSE-PIC Single User FIGURE 32. Turbo coded BPSK transmission over non-dispersive fading chananel, C8×16 code sets with [22] labeled as LDS. where ! denotes factorial. The proposed LDS design ap- proaches the upper bound of the spectral efficiency, as shown in Figs. 24 and 25 for the case of optimal detection. On average there is a 0.2 bits/s/Hz gap between our pro- posed scheme and the existing state-of-the-art LDS designs. According to Proposition 1 the proposed LDSs are WBE sequences or exhibit WBE-like properties, as they approach the upper bound. Having LDS code sets exhibiting optimal VOLUME 4, 2016 27
- 28. spectral efficiency inspires us to design low-complexity detectors such as the MMSE-PIC arrangement, which is capable of operating even beyond a normalized load factor of 200%. 0 1 2 3 4 5 6 7 8 Eb /N0 (dB) 10 -5 10 -4 10 -3 10 -2 10 -1 Average BER LDS MMSE-PIC LDS proposed MMSE-PIC Single User FIGURE 33. Turbo coded BPSK transmission over frequency-selective fading channel, C8×16 code sets with [22] labeled as LDS. IX. CONCLUSION AND DESIGN GUIDELINES In this paper, we have provided a comprehensive literature review of LDS construction designs by considering the most recent developments. Both the design and application of LDS code sets have been described in Tables 2 and 3, respectively. Widely used design criteria conceived for developing the LDS matrices have also been presented. Moreover, we conceived an improved LDS sequence design based on the Gaussian separability criterion. We demon- strated that achieving the best BER performance depends not only on the minimum distance, but also on the average Gaussian separability margin. Based on that criterion, we developed an iterative al- gorithm that is based on maximizing the SINR of each individual user of interest, which converges to the desired solution. We select the optimum candidates having the highest minimum distance and those associated with the highest average Gaussian separability, which perform well along with channel coding. Our proposed LDS code set outperforms the existing LDS designs both for BPSK and 4QAM transmission in terms of its BER. We elaborate a little further on the design guide- lines associated with the proposed algorithm and presented in Table 2. More explicitly, as portrayed in Figs. 9 and 10, our code design, conceived, for 4×6 and 6×9 constructions provides some, modest, power gain compared to other code designs without any increase in computational complexity when using our codes. A compelling BER performance is shown for the size of K = 2L. Our conclusion is that the Gaussian separability margin has to be considered when comparing code sets with equal minimum Euclidean distance or TSC properties. We can summarize our design guidelines as follows: (1) For a given transmission channel, we have to determine the number of users K, the length L of the waveform sequence, the grade of sparseness, as well as the parameters δ and σd, which are obtained heuristically, as discussed in Section IV. (2) The proposed designs jointly map the signals of the users to REs in a sparse manner, they perform con- stellation shaping and judiciously allocate the power to each spreading sequence. (3) The proposed algorithm is iterative, hence whenever there is a change in the channel conditions and/or the number of users K, we can re-run our algorithm to produce new LDS codes. However, if for some reason one should avoid adapting to the channel environment, we suggest to use the average noise variance associated with the maximum number of users. If less users are present, using a subset of the LDS code sets is recommended. Furthermore, we also proposed a low-complexity minimum mean-square estimation and parallel interference cancella- tion aided detector, which exhibited a comparable BER performance to that of ML detection. The MMSE-PIC algorithm has however much lower complexity than the MPA. In our future research we will conceive LDS designs for higher-order constellations for transmission over disper- sive fading channels and the radical direct minimum BER optimization criterion of [139]. REFERENCES [1] L. Dai, B. Wang, Z. Ding, Z. Wang, S. Chen, and L. Hanzo, “A survey of non-orthogonal multiple access for 5G,” IEEE Commun. Surveys Tuts., vol. 20, no. 3, pp. 2294–2323, thirdquarter 2018. [2] C. Bockelmann, N. K. Pratas, G. Wunder, S. Saur, M. Navarro, D. Gre- goratti, G. Vivier, E. De Carvalho, Y. Ji, C. Stefanovic, P. Popovski, Q. Wang, M. Schellmann, E. Kosmatos, P. Demestichas, M. Raceala- Motoc, P. Jung, S. Stanczak, and A. Dekorsy, “Towards massive con- nectivity support for scalable mMTC communications in 5G networks,” IEEE Access, vol. 6, pp. 28 969–28 992, May 2018. [3] A. Ghosh, A. Maeder, M. Baker, and D. Chandramouli, “5G evolution: A view on 5G cellular technology beyond 3GPP release 15,” IEEE Access, vol. 7, pp. 127 639–127 651, Sep. 2019. [4] J. Ding, M. Nemati, C. Ranaweera, and J. Choi, “IoT connectivity tech- nologies and applications: A survey,” IEEE Access, vol. 8, pp. 67 646– 67 673, Apr. 2020. [5] L. Chettri and R. Bera, “A comprehensive survey on internet of things (IoT) toward 5g wireless systems,” IEEE Internet of Things Journal, vol. 7, pp. 16–32, Jan. 2020. [6] C. Kalalas and J. Alonso-Zarate, “Massive connectivity in 5G and be- yond: Technical enablers for the energy and automotive verticals,” in Proc. 6G Wireless Summit (6G SUMMIT), Levi, Finland, May 2020, pp. 1–5. [7] H. Kulhandjian, “A visible light communications framework for intel- ligent transportation systems,” Mineta Transportation Institute Publica- tions, Tech. Rep., Project No. 1911, August 2020. [8] R. M. Marè, C. E. Cugnasca, C. L. Marte, and G. Gentile, “Intelligent transport systems and visible light communication applications: An overview,” in Proc. of IEEE International Conf. on Intelligent Trans- portation Systems (ITSC), Rio de Janeiro, Brazil, Nov. 2016, pp. 2101– 2106. [9] H. Sari, F. Vanhaverbeke, and M. Moeneclaey, “Extending the capacity of multiple access channels,” IEEE Commun. Mag., vol. 38, no. 1, pp. 74–82, Jan. 2000. 28 VOLUME 4, 2016
- 29. [10] S. Verdu and S. Shamai, “Spectral efficiency of CDMA with random spreading,” IEEE Trans. Inf. Theory, vol. 45, no. 2, pp. 622–640, Mar. 1999. [11] H. Sari, F. Vanhaverbeke, and M. Moeneclaey, “Multiple access using two sets of orthogonal signal waveforms,” IEEE Commun. Lett., vol. 4, no. 1, pp. 4–6, Jan. 2000. [12] F. Vanhaverbeke, M. Moeneclaey, and H. Sari, “DS/CDMA with two sets of orthogonal spreading sequences and iterative detection,” IEEE Commun. Lett., vol. 4, no. 9, pp. 289–291, Sep. 2000. [13] M. Kulhandjian and D. A. Pados, “Uniquely decodable code-division via augmented Sylvester-Hadamard matrices,” in Proc. IEEE Wireless Commun. and Network. Conf. (WCNC), Paris, France, Apr. 2012, pp. 359–363. [14] M. Kulhandjian, C. D’Amours, and H. Kulhandjian, “Uniquely decod- able ternary codes for synchronous CDMA systems,” in Proc. IEEE Pers., Indoor, Mobile Radio Conf. (PIMRC), Bologna, Italy, Sep. 2018, pp. 1–6. [15] M. Kulhandjian, C. D’Amours, H. Kulhandjian, H. Yanikomeroglu, D. A. Pados, and G. Khachatrian, “Fast decoder for overloaded uniquely decodable synchronous CDMA,” arXiv:1806.03958 [eess.SP], Jun. 2018. [Online]. Available: https://arxiv.org/abs/1806.03958 [16] M. Kulhandjian, C. D’Amours, H. Kulhandjian, H. Yanikomeroglu, and G. Khachatrian, “Fast decoder for overloaded uniquely decodable synchronous optical CDMA,” in Proc. IEEE Wireless Commun. and Network. Conf. (WCNC), Marrakech, Morocco, Apr. 2019, pp. 1–7. [17] M. Basharat, W. Ejaz, M. Naeem, A. Masood Khattak, and A. Anpalagan, “A survey and taxonomy on nonorthogonal multiple-access schemes for 5G networks,” Trans. on Emerging Telecommun. Technologies, vol. 29, no. 1, p. e3202, Jun. 2017. [18] Q. Wang, R. Zhang, L. Yang, and L. Hanzo, “Non-orthogonal multiple access: A unified perspective,” IEEE Wireless Commun., vol. 25, no. 2, pp. 10–16, Apr. 2018. [19] N. Ye, H. Han, L. Zhao, and A.-H. Wang, “Uplink nonorthogonal multiple access technologies toward 5G: A survey,” Wireless Commun. and Mobile Comp., vol. 2018, no. article ID 6187580, pp. 1–26, Jun. 2018. [20] M. B. Shahab, R. Abbas, M. Shirvanimoghaddam, and S. J. Johnson, “Grant-free non-orthogonal multiple access for IoT: A survey,” arXiv:1910.06529 [eess.SP], Oct. 2019. [Online]. Available: https://arxiv.org/abs/1910.06529 [21] L. Dai, B. Wang, Y. Yuan, S. Han, C. I, and Z. Wang, “Non-orthogonal multiple access for 5G: solutions, challenges, opportunities, and future research trends,” IEEE Commun. Mag., vol. 53, no. 9, pp. 74–81, Sep. 2015. [22] R. Hoshyar, F. P. Wathan, and R. Tafazolli, “Novel low-density signature for synchronous CDMA systems over AWGN channel,” IEEE Trans. Signal Process., vol. 56, no. 4, pp. 1616–1626, Apr. 2008. [23] R. Razavi, M. AL-Imari, M. A. Imran, R. Hoshyar, and D. Chen, “On receiver design for uplink low density signature OFDM (LDS-OFDM),” IEEE Trans. Commun., vol. 60, no. 11, pp. 3499–3508, Nov. 2012. [24] H. Nikopour and H. Baligh, “Sparse code multiple access,” in Proc. IEEE Pers., Indoor, Mobile Radio Conf. (PIMRC), London, U.K., Sep. 2013, pp. 332–336. [25] L. Li, Z. Ma, P. Z. Fan, and L. Hanzo, “High-dimensional codebook design for the SCMA down link,” IEEE Trans. on Vehic. Tech., vol. 67, no. 10, pp. 10 118–10 122, Oct. 2018. [26] S. Chen, B. Ren, Q. Gao, S. Kang, S. Sun, and K. Niu, “Pattern division multiple access - a novel nonorthogonal multiple access for fifth- generation radio networks,” IEEE Trans. on Vehic. Tech., vol. 66, no. 4, pp. 3185–3196, Apr. 2017. [27] Z. Yuan, G. Yu, W. Li, Y. Yuan, X. Wang, and J. Xu, “Multi-user shared access for internet of things,” in Proc. IEEE Veh. Technol. Conf. (VTC Spring), Nanjing, China, May 2016, pp. 1–5. [28] M. Vameghestahbanati, I. D. Marsland, R. H. Gohary, and H. Yanikomeroglu, “Multidimensional constellations for uplink SCMA systems—a comparative study,” IEEE Commun. Surveys Tuts., vol. 21, no. 3, pp. 2169–2194, thirdquarter 2019. [29] M. Mohammadkarimi, M. A. Raza, and O. A. Dobre, “Signature-based nonorthogonal massive multiple access for future wireless networks: Uplink massive connectivity for machine-type communications,” IEEE Veh. Technol. Magazine, vol. 13, no. 4, pp. 40–50, Dec. 2018. [30] J. Choi, “Low density spreading for multicarrier systems,” in Proc. IEEE Int. Symp. Spread Spectr. Techn. Appl. Programm. Book Abstracts (ISSSTA), Sydney, Australia, Aug. 2004, pp. 575–578. [31] R. Hoshyar, F. P. Wathan, and R. Tafazolli, “Novel low-density signature structure for synchronous DS-CDMA systems,” in Proc. IEEE Global Telecommun. Conf. (GLOBECOM), San Francisco, U.S.A., Nov. 2006, pp. 1–5. [32] J. van de Beek and B. M. Popović, “Multiple access with low-density signatures,” in Proc. IEEE Global Telecommun. Conf. (GLOBECOM), Honolulu, U.S.A., Nov. 2009, pp. 1–6. [33] A. R. Safavi, A. G. Perotti, and B. M. Popović, “Ultra low density spread transmission,” IEEE Commun. Lett., vol. 20, no. 7, pp. 1373–1376, Jul. 2016. [34] T. Qi, W. Feng, and Y. Wang, “Optimal sequences for non-orthogonal multiple access: A sparsity maximization perspective,” IEEE Commun. Lett., vol. 21, no. 3, pp. 636–639, Mar. 2017. [35] T. Qi, W. Feng, Y. Chen, and Y. Wang, “When NOMA meets sparse sig- nal processing: Asymptotic performance analysis and optimal sequence design,” IEEE Access, vol. 5, pp. 18 516–18 525, Jul. 2017. [36] C. Jiang and Z. Wu, “A novel uplink NOMA scheme based on low density superposition modulation,” in Proc. IEEE Veh. Technol. Conf. (VTC-Fall), Toronto, Canada, Sep. 2017, pp. 1–5. [37] T. Qi, W. Feng, and Y. Wang, “On the sparsity of spreading sequences for NOMA with reliability guarantee and detection complexity limitation,” in Proc. IEEE Veh. Technol. Conf. (VTC-Spring), Sydney, Australia, Jun. 2017, pp. 1–6. [38] J. Zhang, X. Wang, X. Yang, and H. Zhou, “Low density spreading signature vector extension (LDS-SVE) for uplink multiple access,” in Proc. IEEE Veh. Technol. Conf. (VTC-Fall), Toronto, Canada, Sep. 2017, pp. 1–5. [39] G. Song, X. Wang, and J. Cheng, “Signature design of sparsely spread code division multiple access based on superposed constellation distance analysis,” IEEE Access, vol. 5, pp. 23 809–23 821, Oct. 2017. [40] H. Yu, Z. Fei, N. Yang, and N. Ye, “Optimal design of resource element mapping for sparse spreading non-orthogonal multiple access,” IEEE Wireless Commun. Lett., vol. 7, no. 5, pp. 744–747, Oct. 2018. [41] K. Zhang, J. Xu, C. Yin, W. Tan, and C. Li, “A sparse superposition multiple access scheme and performance analysis for 5G wireless com- munication systems,” in Proc. IEEE/CIC Int. Conf. on Commun. in China (ICCC) Workshops, Kansas City, U.S.A., May 2018, pp. 90–95. [42] H. Wang, K. Xiao, B. Xia, and J. Wang, “Performance analysis and optimization for the LDPC-coded multi-carrier LDS system,” in Proc. IEEE Veh. Technol. Conf. (VTC-Spring), Kuala Lumpur, Malaysia, Apr. 2019, pp. 1–5. [43] K. Lu and C. Jiang, “Optimized low density superposition modulation for 5G mobile multimedia wireless networks,” IEEE Access, vol. 7, pp. 174 227–174 235, Dec. 2019. [44] Z. Liu, P. Xiao, and Z. Mheich, “Power-imbalanced low-density signa- tures (LDS) from Eisenstein numbers,” in Proc. IEEE VTS Asia Pacific Wireless Commun. Symp. (APWCS), Singapore, Aug. 2019, pp. 1–5. [45] Y. Sun, “Quasi-large sparse-sequence CDMA: Approach to single-user bound by linearly-complex LAS detector,” in Proc. IEEE Annual Conf. on Inf. Sci. and Systems, Princeton, U.S.A., Mar. 2008, pp. 320–323. [46] R. Lupas and S. Verdu, “Linear multiuser detectors for synchronous code- division multiple-access channels,” IEEE Trans. Inf. Theory, vol. 35, no. 1, pp. 123–136, Jan. 1989. [47] B. Hassibi and H. Vikalo, “On the sphere-decoding algorithm I. Expected complexity,” IEEE Trans. Signal Processing, vol. 53, no. 8, pp. 2806– 2818, Aug. 2005. [48] G. Romano, F. Palmieri, P. K. Willett, and D. Mattera, “Separability and gain control for overloaded CDMA,” in Proc. of the Conf. on Inf. Sci. and Sys. (CISS), Princeton, NJ, USA, Mar. 2005, pp. 1–6. [49] M. K. Varanasi, “Decision feedback multiuser detection: a systematic approach,” IEEE Trans. Inf. Theory, vol. 45, no. 1, pp. 219–240, Jan. 1999. [50] Y. Sun and J. Xiao, “Multicode sparse-sequence CDMA: Approach to optimum performance by linearly complex wslas detectors,” Wireless Pers. Commun., vol. 71, no. 2, pp. 1049–1056, Jul. 2013. [51] P. G. Casazza, A. Heinecke, F. Krahmer, and G. Kutyniok, “Optimally sparse frames,” IEEE Trans. Inf. Theory, vol. 57, no. 11, pp. 7279–7287, Nov. 2011. [52] K. Xiao, B. Xia, Z. Chen, J. Wang, D. Chen, and S. Ma, “On optimiz- ing multicarrier-low-density codebook for GMAC with finite alphabet inputs,” IEEE Commun. Lett., vol. 21, no. 8, pp. 1811–1814, Aug. 2017. [53] X. Liu, H. Chen, M. Peng, and F. Yang, “Identical code cyclic shift multiple access—a bridge between CDMA and NOMA,” IEEE Trans. on Vehic. Tech., vol. 69, no. 3, pp. 2878–2890, Mar. 2020. VOLUME 4, 2016 29
- 30. [54] C. Jiang and Y. Wang, “An uplink SCMA codebook design combining probabilistic shaping and geometric shaping,” IEEE Access, vol. 8, pp. 76 726–76 736, Apr. 2020. [55] G. Song, K. Cai, Y. Chi, J. Guo, and J. Cheng, “Super-sparse on-off division multiple access: Replacing repetition with idling,” IEEE Trans. Commun., vol. 68, no. 4, pp. 2251–2263, Apr. 2020. [56] N. Ye, X. Li, H. Yu, L. Zhao, W. Liu, and X. Hou, “DeepNOMA: A unified framework for NOMA using deep multi-task learning,” IEEE Trans. Commun., vol. 19, no. 4, pp. 2208–2225, Apr. 2020. [57] H. Xie, W. Xu, H. Q. Ngo, and B. Li, “Non-coherent massive MIMO sys- tems: A constellation design approach,” IEEE Trans. Wireless Commun., vol. 19, no. 6, pp. 3812–3825, Jun. 2020. [58] L. Lan, Y. Y. Tai, S. Lin, B. Memari, and B. Honary, “New constructions of quasi-cyclic LDPC codes based on special classes of BIBD’s for the AWGN and binary erasure channels,” IEEE Trans. Commun., vol. 56, no. 1, pp. 39–48, Jan. 2008. [59] Y. Wu, E. Attang, and G. E. Atkin, “A novel noma design based on Steiner system,” in Proc. IEEE Int. Conf. Electro/Inf. Technol. (EIT), Rochester, U.S.A., May 2018, pp. 0846–0850. [60] J. H. Dinitz and J. C. Colbourn, Handbook of Combinatorial Designs, 2nd ed. Chapman Hall, CRC, 2007. [61] M. Yoshida and T. Tanaka, “Analysis of sparsely-spread CDMA via statistical mechanics,” in Proc. IEEE Int. Symp. Inf. Theory (ISIT), Seattle, U.S.A., Jul. 2006, pp. 2378–2382. [62] A. Montanari and D. Tse, “Analysis of belief propagation for non-linear problems: The example of CDMA (or: How to prove Tanaka’s formula),” in Proc. IEEE Inf. Theory Workshop (ITW), Punta del Este, Uruguay, Mar. 2006, pp. 160–164. [63] J. Raymond and D. Saad, “Sparsely spread CDMA—a statistical mechanics-based analysis,” IEEE Trans. Inf. Theory, vol. 40, no. 41, pp. 12 315–12 333, Sep. 2007. [64] M. T. P. Le, G. C. Ferrante, T. Q. S. Quek, and M. Di Benedetto, “Fundamental limits of low-density spreading NOMA with fading,” IEEE Trans. Wireless Commun., vol. 17, no. 7, pp. 4648–4659, Jul. 2018. [65] S. Chen, K. Peng, Y. Zhang, and J. Song, “A comparative study on the capacity of SCMA and LDSMA,” in Proc. Int. Wireless Commun. Mobile Computing Conf. (IWCMC), Limassol, Cyprus, Jun. 2018, pp. 1099– 1103. [66] M. AL-Imari, M. A. Imran, R. Tafazolli, and D. Chen, “Performance eval- uation of low density spreading multiple access,” in Proc. Int. Wireless Commun. and Mobile Computing Conf. (IWCMC), Limassol, Cyprus, Aug. 2012, pp. 383–388. [67] R. Hoshyar, R. Razavi, and M. Al-Imari, “LDS-OFDM an efficient multiple access technique,” in Proc. IEEE Veh. Technol. Conf. (VTC Spring), Taipei, Taiwan, May 2010, pp. 1–5. [68] M. Li and S. V. Hanly, “Precoding optimization for the sparse MC- CDMA downlink communication,” in Proc. IEEE Int. Conf. on Commun. (ICC), Syndney, Australia, Jun. 2014, pp. 2106–2111. [69] N. Suraweera, S. V. Hanly, and P. Whiting, “Distributed beamforming for the multicell sparsely-spread MC-CDMA downlink,” in Proc. IEEE Veh. Technol. Conf. (VTC-Spring), Syndney, Australia, Jun. 2017, pp. 1–5. [70] B. Fontana da Silva, C. A. A. Meza, D. Silva, and B. F. Uchôa-Filho, “Exploiting spatial diversity in overloaded MIMO LDS-OFDM multiple access systems,” in Proc. IEEE 9th Latin-American Conf. on Commun. (LATINCOM), Guatemala City, Guatemala, Nov. 2017, pp. 1–6. [71] Y. Liu, L. Yang, and L. Hanzo, “Spatial modulation aided sparse code- division multiple access,” IEEE Trans. Wireless Commun., vol. 17, no. 3, pp. 1474–1487, Mar. 2018. [72] L. Wen, P. Xiao, R. Razavi, M. A. Imran, M. Al-Imari, A. Maaref, and J. Lei, “Joint sparse graph for FBMC/OQAM systems,” IEEE Trans. Veh. Technol., vol. 67, no. 7, pp. 6098–6112, Jul. 2018. [73] A. Osamura, G. Song, J. Cheng, and K. Cai, “Sparse multiple access and code design with near channel capacity performance,” in Proc. Int. Symp. Inf. Theory and Its Applic. (ISITA), Singapore, Oct. 2018, pp. 139–143. [74] S. Denno, R. Sasaki, and Y. Hou, “Low density signature based packet access with phase only adaptive precoding,” in Proc. IEEE Veh. Technol. Conf. (VTC-Spring), Kuala Lumpur, Malaysia, Apr. 2019, pp. 1–5. [75] Y. Zhao, J. Hu, Z. Ding, and K. Yang, “Joint interleaver and modulation design for multi-user SWIPT-NOMA,” IEEE Trans. Commun., vol. 67, no. 10, pp. 7288–7301, Oct. 2019. [76] S. Özyurt and O. Kucur, “Quadrature NOMA: A low-complexity multiple access technique with coordinate interleaving,” IEEE Wireless Commun. Lett., vol. 9, no. 9, pp. 1452–1456, Sep. 2020. [77] R. Razavi, R. Hoshyar, M. A. Imran, and Y. Wang, “Information theoretic analysis of LDS scheme,” IEEE Commun. Lett., vol. 15, no. 8, pp. 798– 800, Aug. 2011. [78] M. K. Varanasi and T. Guess, “Optimum decision feedback multiuser equalization with successive decoding achieves the total capacity of the Gaussian multiple-access channel,” in Proc. Asilomar Conf. on Signals, Systems and Computers, vol. 2, Pacific Grove, CA, U.S.A., Nov. 1997, pp. 1405–1409. [79] R. Razavi, M. Ali Imran, and R. Tafazolli, “EXIT chart analysis for turbo LDS-OFDM receivers,” in Proc. Int. Wireless Commun. and Mobile Computing Conf. (IWCMC), Istanbul, Turkey, Jul. 2011, pp. 354–358. [80] O. Shental, B. M. Zaidel, and S. S. Shitz, “Low-density code-domain NOMA: Better be regular,” in Proc. IEEE Int. Symp. Inf. Theory (ISIT), Aachen, Germany, Jun. 2017, pp. 2628–2632. [81] B. M. Zaidel, O. Shental, and S. S. Shitz, “Sparse NOMA: A closed-form characterization,” in Proc. IEEE Int. Symp. Inf. Theory (ISIT), Colorado, U.S.A., Jun. 2018, pp. 1106–1110. [82] M. AL-Imari, M. A. Imran, and R. Tafazolli, “Low density spreading for next generation multicarrier cellular systems,” in Proc. Int. Conf. on Future Commun. Networks, Baghdad, Iraq, Apr. 2012, pp. 52–57. [83] M. Li, C. Liu, and S. V. Hanly, “Precoding for the sparsely spread MC-CDMA downlink with discrete-alphabet inputs,” IEEE Trans. Veh. Technol., vol. 66, no. 2, pp. 1116–1129, Feb. 2017. [84] M. Fantuz and C. D’Amours, “Low complexity PIC-MMSE detector for LDS systems,” in Proc. IEEE Canadian Conf. on Electrical and Computer Engineering (CECE), Edmonton, Canada, May 2019. [85] R. Price and P. E. Green, “A communication technique for multipath channels,” Proc. IRE, vol. 46, no. 3, pp. 555–570, Mar. 1958. [86] S. Craw, “Manhattan Distance,” in Encyclopedia of Machine Learning, C. Sammut and G. I. Webb, Eds. Boston, MA: Springer US, 2010, pp. 639–639. [87] M. Rupf and J. L. Massey, “Optimum sequence multisets for syn- chronous code-division multiple-access channels,” IEEE Trans. Inf. The- ory, vol. 40, no. 4, pp. 1261–1266, Jul. 1994. [88] P. Viswanath and V. Anantharam, “Optimal sequences and sum capacity of synchronous CDMA systems,” IEEE Trans. Inf. Theory, vol. 45, no. 6, pp. 1984–1991, Sep. 1999. [89] F. Vanhaverbeke and M. Moeneclaey, “Sum capacity of the OCDMA/OCDMA signature sequence set,” IEEE Commun. Lett., vol. 6, no. 8, pp. 340–342, Aug. 2002. [90] L. Welch, “Lower bounds on the maximum cross correlation of signals (Corresp.),” IEEE Trans. Inf. Theory, vol. 20, no. 3, pp. 397–399, May 1974. [91] N. Hurley and S. Rickard, “Comparing Measures of Sparsity,” IEEE Trans. Inf. Theory, vol. 55, no. 10, pp. 4723–4741, Oct. 2009. [92] S. Ulukus and R. D. Yates, “Iterative construction of optimum signature sequence sets in synchronous CDMA systems,” IEEE Trans. Inf. Theory, vol. 47, no. 5, pp. 1989–1998, Jul. 2001. [93] M. Soltanalian and P. Stoica, “Designing unimodular codes via quadratic optimization,” IEEE Trans. on Signal Proc., vol. 62, no. 5, pp. 1221– 1234, Mar. 2014. [94] M. Kobayashi, J. Boutros, and G. Caire, “Successive interference can- cellation with SISO decoding and EM channel estimation,” IEEE J. Sel. Areas Commun., vol. 19, no. 8, pp. 1450–1460, Aug. 2001. [95] M. K. Varanasi and B. Aazhang, “Multistage detection in asynchronous code-division multiple-access communications,” IEEE Trans. Commun., vol. 38, no. 4, pp. 509–519, Apr. 1990. [96] G. Xue, J. Weng, T. Le-Ngoc, and S. Tahar, “Adaptive multistage parallel interference cancellation for CDMA,” IEEE J. Sel. Areas Commun., vol. 17, no. 10, pp. 1815–1827, Oct. 1999. [97] D. Guo, L. K. Rasmussen, S. Sun, and T. J. Lim, “A matrix-algebraic approach to linear parallel interference cancellation in CDMA,” IEEE Trans. Commun., vol. 48, no. 1, pp. 152–161, Jan. 2000. [98] M. C. Reed, C. B. Schlegel, P. D. Alexander, and J. A. Asenstorfer, “Iterative multiuser detection for CDMA with FEC: near-single-user performance,” IEEE Trans. Commun., vol. 46, no. 12, pp. 1693–1699, Dec. 1998. [99] X. Wang and H. V. Poor, “Iterative (turbo) soft interference cancellation and decoding for coded CDMA,” IEEE Trans. Commun., vol. 47, no. 7, pp. 1046–1061, Jul. 1999. [100] S. Morosi, R. Fantacci, E. Del Re, and A. Chiassai, “Design of turbo- MUD receivers for overloaded CDMA systems by density evolution technique,” IEEE Trans. Wireless Commun., vol. 6, no. 10, pp. 3552– 3557, Oct. 2007. 30 VOLUME 4, 2016
- 31. [101] P. Kumar and S. Chakrabarti, “An analytical model of iterative inter- ference cancellation receiver for orthogonal/orthogonal overloaded DS- CDMA system,” Int. J. Wirel. Inf. Netw., vol. 17, no. 1-2, pp. 64–72, Jun. 2010. [102] S. Sasipriya and C. S. Ravichandran, “An analytical model of iterative interference cancellation receiver for orthogonal/orthogonal overloaded DS-CDMA system,” Telecommun. Syst., vol. 56, no. 4, pp. 509–518, Aug. 2014. [103] A. Najkha, C. Roland, and E. Boutillon, “A near-optimal multiuser detector for MC-CDMA systems using geometrical approach,” in Proc. IEEE Int. Conf. on Acoust., Speech, and Sig. Proc. (ICASSP), vol. 3, Philadelphia, U.S.A., Mar. 2005, pp. 877–880. [104] G. Manglani and A. K. Chaturvedi, “Application of computational geom- etry to multiuser detection in CDMA,” IEEE Trans. Commun., vol. 54, no. 2, pp. 204–207, Feb. 2006. [105] Y. Du, B. Dong, P. Gao, Z. Chen, J. Fang, and S. Wang, “Low- complexity LDS-CDMA detection based on dynamic factor graph,” in Proc. IEEE Global Telecommun. Conf. Workshops (GC Wkshps), Wash- ington, U.S.A., Dec. 2016, pp. 1–6. [106] L. Tian, J. Zhong, M. Zhao, and L. Wen, “An optimized superposition constellation and region-restricted MPA detector for LDS system,” in Proc. IEEE Int. Conf. on Commun. (ICC), Kuala Lumpur, Malaysia, May 2016, pp. 1–5. [107] L. Wen, R. Razavi, P. Xiao, and M. A. Imran, “Fast convergence and reduced complexity receiver design for LDS-OFDM system,” in Proc. IEEE Pers., Indoor, Mobile Radio Conf. (PIMRC), Washington DC, U.S.A., Sep. 2014, pp. 918–922. [108] D. Guo and C. Wang, “Multiuser detection of sparsely spread CDMA,” IEEE Journal on Selected Areas in Commun., vol. 26, no. 3, pp. 421–431, Apr. 2008. [109] K. Takeuchi, T. Tanaka, and T. Kawabata, “Performance improvement of iterative multiuser detection for large sparsely spread CDMA systems by spatial coupling,” IEEE Trans. Inf. Theory, vol. 61, no. 4, pp. 1557–9654, Apr. 2015. [110] L. Wen, R. Razavi, M. A. Imran, and P. Xiao, “Design of joint sparse graph for OFDM system,” IEEE Trans. on Wireless Commun., vol. 15, no. 4, pp. 1823–1836, Apr. 2015. [111] L. Wen and M. Su, “Joint sparse graph over GF(q) for code division multiple access systems,” IET Communications, vol. 9, no. 5, pp. 707– 718, Mar. 2015. [112] L. Wen, J. Lei, and M. Su, “Improved algorithm for joint detection and decoding on the joint sparse graph for CDMA systems,” IET Communi- cations, vol. 10, no. 3, pp. 336–345, Feb. 2016. [113] J. Raymond, “Optimal sparse CDMA detection at high load,” in Proc. Int. Symp. Modeling Optim. Mobile, Ad Hoc, Wireless Netw. (WiOpt), Seoul, South Korea, Jun. 2009, pp. 1–5. [114] J. G. Proakis, Digital Communications, 5th ed. McGraw Hill, 2007. [115] G. H. Golub and C. F. Van Loan, Matrix Computations, 4th ed. The Johns Hopkins University Press, 2013. [116] J. Luo, K. R. Pattipati, and P. K. Willett, “Near-optimal multiuser detec- tion in synchronous CDMA using probabilistic data association,” IEEE Commun. Lett., vol. 5, no. 9, pp. 361–366, Sep. 2001. [117] G. L. Turin, “Introduction to spread-spectrum antimultipath techniques and their application to urban digital radio,” Proc. IEEE, vol. 68, no. 3, pp. 328–353, Mar. 1980. [118] C. D’Amours, M. Moher, and A. Yongacoglu, “Comparison of pilot symbol-assisted and differentially detected BPSK for DS-CDMA sys- tems employing RAKE receivers in Rayleigh fading channels,” IEEE Trans. on Vehic. Tech., vol. 47, no. 4, pp. 1258–1267, Nov. 1998. [119] L. Hanzo, M. Münster, B. Choi, and T. Keller, OFDM and MC-CDMA for Broadband Multi-User Communications, WLANs and Broadcasting. John Wiley Sons, 2003. [120] L. Hanzo, L.-L. Yang, E.-L. Kuan, and K. Yen, Single- and Multi-Carrier DS-CDMA: Multi-User Detection, Space-Time Spreading, Synchronisa- tion, Networking and Standards. Wiley-IEEE Press, 2003. [121] M. Kulhandjian, E. Bedeer, H. Kulhandjian, C. D’Amours, and H. Yanikomeroglu, “Low-complexity detection for faster-than-Nyquist signaling based on probabilistic data association,” IEEE Commun. Lett., pp. 1–6, Dec. 2019. [122] D. Pham, K. R. Pattipati, P. K. Willett, and J. Luo, “A generalized probabilistic data association detector for multiple antenna systems,” IEEE Commun. Lett., vol. 8, no. 4, pp. 205–207, Apr. 2004. [123] S. Yang, T. Lv, R. G. Maunder, and L. Hanzo, “Unified bit-based probabilistic data association aided MIMO detection for high-order QAM constellations,” IEEE Trans. on Vehic. Tech., vol. 60, no. 3, pp. 981–991, Mar. 2011. [124] M. Kulhandjian and C. D’Amours, “Design of permutation-based sparse code multiple access system,” in Proc. IEEE Pers., Indoor, Mobile Radio Conf. (PIMRC), Montreal, Canada, Oct. 2017, pp. 1031–1035. [125] C. E. Shannon, “A mathematical theory of communication,” The Bell Syst. Tech. Journal, vol. 27, no. 4, pp. 623–656, Oct. 1948. [126] H. Imai and S. Hirakawa, “A new multilevel coding method using error- correcting codes,” IEEE Trans. Inf. Theory, vol. 23, no. 3, pp. 371–377, May 1977. [127] ——, “Correction to ’A new multilevel coding method using error- correcting codes’,” IEEE Trans. Inf. Theory, vol. 23, no. 6, pp. 784–784, Nov. 1977. [128] G. Ungerboeck, “Channel coding with multilevel/phase signals,” IEEE Trans. Inf. Theory, vol. 28, no. 1, pp. 55–67, Jan. 1982. [129] ——, “Trellis-coded modulation with redundant signal sets Part I: Intro- duction,” IEEE Commun. Mag., vol. 25, no. 2, pp. 5–11, Feb. 1987. [130] ——, “Trellis-coded modulation with redundant signal sets Part II: State of the art,” IEEE Commun. Mag., vol. 25, no. 2, pp. 12–21, Feb. 1987. [131] E. Zehavi, “8-PSK trellis codes for a Rayleigh channel,” IEEE Trans. Commun., vol. 40, no. 5, pp. 873–884, May 1992. [132] B. N. Thanh, V. Krishnamurthy, and R. J. Evans, “Detection-aided recursive least squares adaptive multiuser detection in DS/CDMA,” IEEE Signal Proc. Lett., vol. 9, no. 8, pp. 229–232, Aug. 2002. [133] Y. Du, B. Dong, Z. Chen, J. Fang, and X. Wang, “A fast convergence multiuser detection scheme for uplink SCMA systems,” IEEE Wireless Commun. Lett., vol. 5, no. 4, pp. 388–391, Aug. 2016. [134] Li Ping, W. K. Leung, and Nam Phamdo, “Low density parity check codes with semi-random parity check matrix,” Electronics Letters, vol. 35, no. 1, pp. 38–39, Jan. 1999. [135] H. Zarrinkoub, Understanding LTE with MATLAB: From Mathematical Modeling to Simulation and Prototyping, 1st ed. Wiley Publishing, 2014. [136] H. Vangala, E. Viterbo, and Y. Hong, “A comparative study of polar code constructions for the AWGN channel,” arXiv:1501.02473 [cs.IT], Jan. 2015. [Online]. Available: https://arxiv.org/abs/1501.02473 [137] S. Verdu, “Spectral efficiency in the wideband regime,” IEEE Trans. Inf. Theory, vol. 48, no. 6, pp. 1319–1343, Jun. 2002. [138] T. M. Cover and J. A. Thomas, Elements of Information Theory. USA: Wiley-Interscience, 2006. [139] Sheng Chen, A. K. Samingan, B. Mulgrew, and L. Hanzo, “Adaptive minimum-BER linear multiuser detection for DS-CDMA signals in mul- tipath channels,” IEEE Trans. on Signal Proc., vol. 49, no. 6, pp. 1240– 1247, Jun. 2001. MICHEL KULHANDJIAN (M’18-SM’20) re- ceived his M.S. and Ph.D. degrees in Electrical Engineer from the State University of New York at Buffalo in 2007 and 2012, respectively. He had previously received his B.S. degree in Electronics Engineering and Computer Science (Minor), with “Summa Cum Laude” from the American Univer- sity in Cairo (AUC) in 2005. He was employed at Alcatel-Lucent, in Ottawa, Ontario, in 2012. In the same year he was appointed as a Research Associate at EION Inc. He received Natural Science and Engineering Research Council of Canada (NSERC) Industrial RD Fellowship (IRDF). He is currently a Research Scientist at the School of Electrical Engineering and Computer Science at the University of Ottawa. He is also employed as a senior embedded software engineer at L3Harris Technologies. His research interests include wireless multiple access communications, adaptive coded modulation, waveform design for overloaded code-division multiplexing applications, channel coding, space-time coding, adaptive multiuser detection, statistical signal processing, covert communications, spread-spectrum steganography and steganalysis. He has served as a guest editor for Journal of Sensor and Actuator Networks (JSON). He actively serves as member of Technical Program Committee (TPC) of IEEE WCNC, IEEE GLOBECOM, IEEE ICC, and IEEE VTC. VOLUME 4, 2016 31
- 32. HOVANNES KULHANDJIAN (S’14-M’15- SM’20) received the B.S. degree (magna cum laude) in electronics engineering from The Amer- ican University in Cairo, Cairo, Egypt, in 2008, and the M.S. and Ph.D. degrees in electrical engineering from the State University of New York at Buffalo, Buffalo, NY, USA, in 2010 and 2014, respectively. From December 2014 to July 2015, he was an Associate Research Engineer with the Department of Electrical and Computer Engineering, Northeastern University, Boston, MA, USA. He is currently an Assistant Professor with the Department of Electrical and Computer Engineering, California State University, Fresno, Fresno, CA, USA. His current research interests include wireless communications and networking, with applications to underwater acoustic communications, visible light communications and applied machine learning. He has served as a guest editor for IEEE Access - Special Section Journal on Underwater Wireless Communications and Networking. He has also served as a Session Co- Chair for IEEE UComms 2020, Session Chair for ACM WUWNet 2019. He actively serves as a member of the Technical Program Committee for ACM and IEEE conferences such as IEEE GLOBECOM 2015-2020, UComms 2020, PIMRC 2020, WD 2019, ACM WUWNet 2019, ICC 2015-2018, among others. CLAUDE D’AMOURS received the degrees of B.A.Sc, M.A.Sc. and Ph.D. in Electrical Engi- neering from the University of Ottawa in 1990, 1992 and 1995 respectively. In 1992 he was employed as a Systems Engineer at Calian Com- munications Ltd. In 1995 he joined the Com- munications Research Centre in Ottawa, Ontario, Canada, as a Systems Engineer. Later in 1995, he joined the Department of Electrical and Computer Engineering at the Royal Military College of Canada in Kingston, Ontario, Canada, as an Assistant Professor. He joined the School of Information Technology and Engineering (SITE), which has since been renamed as the School of Electrical Engineering and Computer Science (EECS), at the University of Ottawa as an Assistant Professor in 1999. From 2007-2011, he served as Vice Dean of Undergraduate Studies for the Faculty of Engineering and has been serving as the Director of the School of EECS at the University of Ottawa since 2013. His research interests are in physical layer technologies for wireless communications systems, notably in multiple access techniques and interference cancella- tion. LAJOS HANZO (M’91-SM’92-F’04) (http://www-mobile.ecs.soton.ac.uk, https://en.wikipedia.org/wiki/Lajos_Hanzo) (FIEEE’04, Fellow of the Royal Academy of En- gineering F(REng), of the IET and of EURASIP), received his Master degree and Doctorate in 1976 and 1983, respectively from the Technical Uni- versity (TU) of Budapest. He was also awarded the Doctor of Sciences (DSc) degree by the University of Southampton (2004) and Honorary Doctorates by the TU of Budapest (2009) and by the University of Edinburgh (2015). He is a Foreign Member of the Hungarian Academy of Sciences and a former Editor-in-Chief of the IEEE Press. He has served several terms as Governor of both IEEE ComSoc and of VTS. He has published 1900+ contributions at IEEE Xplore, 19 Wiley-IEEE Press books and has helped the fast-track career of 123 PhD students. Over 40 of them are Professors at various stages of their careers in academia and many of them are leading scientists in the wireless industry. 32 VOLUME 4, 2016