# ADMM-based Detector for Large-scale MIMO Code-domain NOMA Systems

Vinjamoori Vikas1, Kuntal Deka1, Sanjeev Sharma2, and A. Rajesh1
1Indian Institute of Technology Guwahati, India, 2Indian Institute of
Technology (BHU) Varanasi, India

###### Abstract

Large-scale multi-input multi-output (MIMO) code domain non-orthogonal
multiple access (CD-NOMA) techniques are one of the potential candidates to
address the next-generation wireless needs such as massive connectivity, and
high reliability. This work focuses on two primary CD-NOMA techniques: sparse-
code multiple access (SCMA) and dense-code multiple access (DCMA). One of the
primary challenges in implementing MIMO-CD-NOMA systems is designing the
optimal detector with affordable computation cost and complexity. This paper
proposes an iterative linear detector based on the alternating direction
method of multipliers (ADMM). First, the maximum likelihood (ML) detection
problem is converted into a sharing optimization problem. The set constraint
in the ML detection problem is relaxed into the box constraint sharing
problem. An alternative variable is introduced via the penalty term, which
compensates for the loss incurred by the constraint relaxation. The system
models, i.e., the relation between the input signal and the received signal,
are reformulated so that the proposed sharing optimization problem can be
readily applied. The ADMM is a robust algorithm to solve the sharing problem
in a distributed manner. The proposed detector leverages the distributive
nature to reduce per-iteration cost and time. An ADMM-based linear detector is
designed for three MIMO-CD-NOMA systems: single input multi output CD-NOMA
(SIMO-CD-NOMA), spatial multiplexing CD-NOMA (SMX-CD-NOMA), and spatial
modulated CD-NOMA (SM-CD-NOMA). The impact of various system parameters and
ADMM parameters on computational complexity and symbol error rate (SER) has
been thoroughly examined through extensive Monte Carlo simulations.

###### Index Terms:

Code domain-NOMA (CD-NOMA), dense code multiple access (DCMA), sparse code
multiple access (SCMA), alternating direction method of multipliers (ADMM),
single input multi output (SIMO), spatial multiplexing MIMO (SMX-MIMO),
spatial modulation MIMO (SM-MIMO).

##  I Introduction

###  I-A Motivation

The existing literature extensively studies two types of code domain non-
orthogonal multiple access (CD-NOMA) systems. The first type is sparse coded
CD-NOMA systems, which utilize low-density spreading sequence design (e.g.,
low-density signatures (LDS)) or sparse codebook design (e.g., sparse code
multiple access (SCMA)) [1]. The second type is densely coded CD-NOMA systems,
which utilize dense spreading sequence design (e.g., overloaded code-division
multiple access (CDMA)) or dense codebook design (e.g., dense code multiple
access (DCMA)) [2]. Throughout the paper, dense codebook-based NOMA is
referred to as DCMA, and dense spreading sequence-based NOMA is referred to as
overloaded CDMA.

SCMA and DCMA are critical enabling technologies to improve spectral
efficiency and massive connectivity by overloading multiple user equipment’s
(UEs) data on a single resource element (RE). These techniques exploit the
codebooks’ multi-dimensional constellation (MDC) coding gain. Spreading
sequence-based NOMA (LDS, overloaded CDMA) does not benefit from the coding
gain of the MDC. Thus, codebook-based NOMA techniques offer substantial error
rate performance gains over the spreading sequence-based NOMA.

Multi-input multi-output (MIMO) technology is another crucial enabler in
improving spectral efficiency and system reliability [3]. CD-NOMA systems
using a MIMO system (MIMO-CD-NOMA) offer even greater spectral efficiency,
reliability, and massive connectivity improvements. This paper considers three
types of uplink (UL) MIMO-CD-NOMA systems: (1) Single-input multi-output
(SIMO) aided CD-NOMA, where each UE has a single antenna [4]; (2) Spatial
multiplexing-CD-NOMA (SMX-CD-NOMA), where each UE is equipped with multiple
transmit antennas [5]; and (3) Spatial modulated CD-NOMA (SM-CD-NOMA), where
each UE activates a single transmit antenna out of multiple transmit antennas
at the UE [6]. The primary challenge in implementing MIMO-CD-NOMA systems lies
in designing a multiuser signal detector that offers excellent performance
while maintaining affordable complexity.

The message passing algorithm (MPA) is usually used in SCMA detection [7, 8].
However, the MPA for MIMO-SCMA signal detection [9, 10] needs to improve its
computational complexity. MPA detection over the MIMO-SCMA system uses an
extended factor graph due to the additional antennas at the UEs and the BS
[11]. The MPA exhibits exponential complexity with the codebook size ($M$),
the number of UEs ($J$), and the number of transmit antennas. Thus, MPA
becomes impractical for highly overloaded SCMA systems over large-scale MIMO
systems. Further, MPA often faces convergence issues when the factor graph
contains cycles. To overcome the limited diversity issues (due to sparsity) in
SCMA, researchers have recently started designing the DCMA systems [2].
However, MPA demands sparsity for the accurate detection of codewords. DCMA
codewords do not hold sparsity properties. Due to enormous short cycles in the
DCMA factor graph, MPA is no longer a suitable detector for DCMA systems.
Thus, an alternate detection algorithm must be developed to achieve the full
diversity offered by DCMA with minimal detection complexity.

The generalized sphere decoder (GSD) is used as a detector for a spread
sequence-based DCMA system by considering a single antenna at both the base
station (BS) and the UE [2]. The GSD works on principles of the tree search
algorithm. Its complexity depends on the number of tree nodes and floating
point operations (FLOPs) at each node. The number of tree nodes will increase
multi-fold in the codebook-based DCMA systems. Moreover, the complexity of GSD
increases rapidly for large-scale MIMO DCMA systems. So, it is not an
efficient detector for large-scale MIMO CD-NOMA systems. Designing a low-
complexity multiuser detector (MUD) for large-scale MIMO-aided CD-NOMA systems
with higher overloading factors ($\lambda$) and modulation order/codebook-size
($M$) is becoming essential. The existing research studies mainly nonlinear
detectors (MPA and sphere decoder) for CD-NOMA systems [7, 12]. On the other
hand, minimum mean square error (MMSE) and zeros forcing (ZF) are commonly
used linear detectors in standalone MIMO systems[13]. These detectors are
simple to implement and they exhibit low computational complexity over
nonlinear detectors. However, these detectors show poor symbol error rate
(SER) performance over CD-NOMA systems with high $\lambda$ values. In
addition, they fail to detect the CD-NOMA signals as $M$ increases. Thus,
designing an efficient detector with low complexity is mandatory to fully
unfurl the benefits of MIMO CD-NOMA systems with a practically viable
approach. Further, the detector must perform better than the existing linear
detectors for large-scale CD-NOMA parameters ($\lambda$, $M$).

###  I-B Related prior works

Recently, the ADMM has been widely used to solve convex and non-convex
problems in a distributed manner[14]. Glowinski and Marrocco first proposed
the ADMM in the mid-1970s. Later on, Boyd et al. rigorously discussed various
complex optimization problems that can be solved using ADMM in their tutorial
[14]. The ADMM is formed by the composition of dual ascent and the method of
multipliers. The dual decomposition properties of dual ascent make the ADMM
solve in a distributed manner. ADMM has superior convergence properties and
often is able to converge without the requirement of strict convexity. A
comprehensive treatment of ADMM and its advancements are detailed in [15].

The ADMM is extensively applied to solve the linear programming (LP) problem
in low-density parity-check code’s (LDPC) decoding[16, 17]. The authors here
have also shown that the ADMM significantly reduces the complexity of the
decoder compared to the belief propagation (BP) methods. The MUD problem of UL
grant-free NOMA is solved by using ADMM [18]. The ADMM-based infinity norm
(ADMIN) iterative linear detector is proposed for massive MIMO systems[19].
The authors have also proposed VLSI architecture for the ADMIN detector. The
results show that ADMIN outperforms all the linear detectors in terms of SER
performance and low hardware cost compared to the nonlinear BP-based
detectors. Similarly, an ADMM-based QAM signal detector for massive MIMO
systems is proposed [20]. A distributed penalty sharing (DPS) ADMM method is
designed to convert the maximum-likelihood (ML) detection problem into sharing
optimization problem. This method shows good performance and complexity trade-
offs for massive MIMO systems. The authors in [20, 19] show that the ADMM-
based detector performs MMSE equalization in the first iteration. The
performance of the ADMM improves over MMSE as iterations progress.

###  I-C Contributions:

This paper proposes an ADMM-based iterative linear detector that solves the
MUD problem of large-scale MIMO-aided CD-NOMA systems. It exhibits the best
trade-off between the SER performance and computational complexity for SCMA
and DCMA systems. We have formulated the highly computationally complex ML
detection problem of large-scale MIMO-CD-NOMA systems as the non-convex
distributed optimization problem. To the best of the authors’ knowledge, this
is the first attempt to convert the ML detection problem of CD-NOMA into a
sharing problem. Further, the ADMM approach is applied to solve the proposed
sharing problem using distributed optimization. The proposed ADMM-based
detector is guaranteed to improve the error rate performance over linear
detectors with lower computational complexity than nonlinear detectors. The
main contributions of this work are listed below:

  * •

The ML MUD problem is nondeterministic polynomial-time hard (NP-hard) and can
be solved using exhaustive search [21]. However, the complexity of the
exhaustive search increases exponentially as the number of UEs and codebook
size increase. The ML problem is transformed into a non-convex sharing
optimization problem to overcome this challenge. An efficient distributed
optimization method is then applied to solve the proposed sharing problem.

  * •

The CD-NOMA system models, i.e., the relationships between the input signal
and received signal, are reformulated so that the sharing ADMM algorithm can
be readily applied for detection. For SIMO-CD-NOMA, the signal received by
multiple antennas at BS is considered as a single observation vector for
detection. For SMX-CD-NOMA and SM-CD-NOMA, resource-wise processing of the
received signal is carried out during ADMM detection.

  * •

A low-complexity iterative linear detector is designed via the ADMM approach
to solve the sharing optimization problem for large-scale MIMO-CD-NOMA
systems. ADMM solves the sharing optimization problem through a distribution
optimization framework. Indeed, distributive nature of sharing ADMM allows
parallel processing in multiuser detection, reducing per-iteration
computational time.

  * •

The proposed ADMM-based detector is applied to large-scale MIMO (SIMO, SMX-
MIMO, and SM-MIMO) aided CD-NOMA systems. Thus, the proposed single ADMM-based
MUD is capable of solving the detection problem of various CD-NOMA systems.

  * •

The complexity of the proposed ADMM-based detector is independent of the CD-
NOMA system’s codebook size/modulation order ($M$), unlike the conventional
MPA detector. Thus, it can be applied to large-size codebooks. The impact of
the different MIMO-CD-NOMA system parameters is also thoroughly examined. The
ADMM-based detector exhibits polynomial complexity with all other parameters,
such as the numbers of antennas, UEs, and REs.

  * •

The comparison of the error rate performance and receiver complexity of the
proposed ADMM-based detector with all other conventional detectors is
thoroughly examined. Comprehensive Monte Carlo simulations indicate that the
ADMM-based detector substantially reduces the computational complexity while
maintaining an error performance comparable to the MPA in SCMA detection. For
spread sequence-based DCMA, the ADMM gives superior performance than GSD.

###  I-D Organization:

The paper is organized as follows. The preliminaries of the existing CD-NOMA
system models and sharing ADMM problem are discussed in Section II. The
proposed system models and ADMM-based detection for UL MIMO-CD-NOMA systems
are provided in Section III. Section IV presents a detailed computational
complexity analysis. The simulation results for different MIMO-CD-NOMA systems
are discussed in Section V. Section VI concludes the paper and mentions future
scopes.

Notations: Lower case, bold lower case, and bold upper case letters denote
scalars, vectors, and matrices, respectively. $(\cdot)^{T}$ and $(\cdot)^{H}$
denote transpose and Hermitian transpose, respectively. $\|(\cdot)\|$ denotes
the Euclidean norm of a vector. $\prod_{[-\alpha,\alpha]}(\cdot)$ denotes the
Euclidean projection onto the interval $[-\alpha,\alpha]$.
$\langle\cdot,\cdot\rangle$ denotes the inner product, and
${\text{Re}}(\cdot)$ denotes the real part of a complex variable.
$\mathbb{R}^{n}$ and $\mathbb{C}^{n}$ denote $n$-dimensional real and complex
vector spaces, respectively. $\mathcal{CN}(0,\sigma^{2})$ denotes complex
Gaussian distribution with zero mean and variance $\sigma^{2}$.

##  II Preliminaries

This section mainly discusses the existing CD-NOMA system models and steps to
solve the sharing optimization problem via the ADMM approach. The CD-NOMA
techniques can be broadly divided into sparse-coded and dense-coded NOMA. Low-
density signature (LDS) [22] and SCMA [23] belong to the first group, whereas
overloaded CDMA [2] and DCMA [2] belong to the second group. LDS and
overloaded CDMA are sequence-based CD-NOMA techniques where mapping and
spreading are performed separately. LDS utilizes sparse sequences, while
overloaded CDMA employs dense sequences. These techniques suffer from error
performance loss due to limited coding gain. On the other hand, SCMA and DCMA
are codebook-based CD-NOMA techniques. In these systems, the data of each UE
is mapped to a multi-dimensional codeword. SCMA utilizes sparse codewords,
while DCMA applies dense codewords. These techniques offer error performance
benefits through the use of MDCs [24].

###  II-A SCMA system model

SCMA is a sparse codebook-based NOMA technique. Each UE’s data is sent in the
form of sparse codewords. Consider a CD-NOMA system having $J$ UEs accessing
$K$ REs, where $J>K$ ensures the overloading nature of SCMA. Each UE in the
SCMA system has access to $d_{\mathrm{v}}<K$ active REs among $K$ REs. Thus,
$d_{\mathrm{v}}$ non-zero elements are present in the $K$-dimensional sparse
codewords. The SCMA encoder maps each UE bitstream to a sparse codeword
$\mathbf{x}^{K\times 1}$ in a pre-designed SCMA codebook $\mathcal{X}^{K\times
M}$ of the respective UE. The sparse nature of SCMA codewords enforces
$d_{\mathrm{f}}<J$ number of overlapping UEs on each RE. The diversity order
of SCMA is essentially limited to $d_{\mathrm{v}}$, i.e., far less than the
maximum possible diversity order $K$. For example, the codebook structure for
150 % overloaded SCMA system ($d_{\mathrm{v}}=2$, and $d_{\mathrm{f}}=3$) is
shown in Fig. 1(a). The received signal vector is given by

$\displaystyle\mathbf{r}=\sum_{j=1}^{J}\text{diag}(\mathbf{h}_{j})\mathbf{x}_{j}+\mathbf{w},$
(1)

where

  * •

$\mathbf{w}\sim\mathcal{CN}(0,\sigma^{2}\textbf{I}_{K})$ denotes the additive
white Gaussian noise (AWGN) at the receiver.

  * •

$\mathbf{h}_{j}\sim\mathcal{CN}(0,\textbf{I}_{K})$ denotes the Rayleigh fading
channel vector for the $j$th UE.

  * •

$\text{diag}({\mathbf{h}}=\left[h[1],\ldots,h[k],\ldots,h[K]\right]^{T})$
denotes diagonal matrix and $h[k]$ is the $k$th diagonal element.

  * •

$\mathbf{x}_{j}$ is the $K$-dimensional codeword of $j$th UE.

(a) SCMA codebook structure

(b) DCMA codebook structure

Figure 1: CD-NOMA codebook structure for $J=6$ and $K=4$.

###  II-B DCMA system model

The error performance of SCMA suffers from limited diversity order. This
limitation of SCMA can be overcome by using dense codebooks in the DCMA system
[2]. Each UE in the DCMA system has access to all $K$ REs. No zero entries are
present in the $K$-dimensional dense codewords. Therefore, the DCMA system
exploits the full diversity of multi-dimensional codewords. For example, the
codebook structure of 150 % overloaded DCMA system is shown in Fig. 1(b). The
received signal vector for the DCMA system is similar to that of the SCMA
system, as described in (1). In addition to the codebook-based DCMA system,
the spread sequence-based DCMA can be designed with non-orthogonal spreading
sequences. The idea is similar to the overloaded CDMA systems. The number of
UEs is larger than the length of the spreading sequence. Uni-modular spreading
sequences have been used in this paper to achieve full diversity of the dense
sequences [2].

###  II-C Shared ADMM problem

This subsection introduces the fundamental concepts of solving sharing
problems using ADMM within a distributed optimization framework. A novel
method, referred to as ADMM, was proposed to address both convex and non-
convex sharing optimization problems [14]. The subsequent subsection delves
into the details and ideas behind the sharing ADMM problem. The ADMM is formed
by combining the superior properties of dual ascent and the method of
multipliers. This combination ensures the robustness of ADMM [14]. The generic
sharing problem is

$\displaystyle\min\quad\sum_{i=1}^{N}f_{i}(\mathbf{x}_{i})+g\left(\sum_{i=1}^{N}\mathbf{x}_{i}\right),$
(2)

where $\mathbf{x}_{i}\in\mathbb{R}^{n},i=1,\ldots,N$, the associated local
cost function
$f_{i}(\mathbf{x}_{i})(f_{i}:\mathbb{R}^{n}\rightarrow\mathbb{R})$ of
subsystem $i$ is handled by processor $i$, and $g$
$(g:\mathbb{R}^{n}\rightarrow\mathbb{R})$ is the shared objective, whose
argument is the sum of $N$ variables. Each variable $\mathbf{x}_{i}$ is
involved in minimizing the individual cost $f_{i}(\mathbf{x}_{i})$, as well as
the shared objective $g\left(\sum_{i=1}^{N}\mathbf{x}_{i}\right)$. The sharing
problem can be converted into an ADMM problem by introducing an alternative
variable, $\mathbf{z}_{i}\in\mathbb{R}^{n}$. Thus, the cost function is
minimized over $\mathbf{x}_{i}$ and $\mathbf{z}_{i}$, alternatively. The
problem (2) is converted to the following problem:

$\displaystyle\min\quad$
$\displaystyle\sum_{i=1}^{N}f_{i}(\mathbf{x}_{i})+g\left(\sum_{i=1}^{N}\mathbf{z}_{i}\right)$
(3) s.t. $\displaystyle\mathbf{x}_{i}-\mathbf{z}_{i}=0\quad i=1,\ldots,N.$

The augmented Lagrangian function for (3) can be written as

$\displaystyle\mathcal{L}\left(\\{\mathbf{x}_{i},\mathbf{z}_{i},\mathbf{y}_{i}\\}_{i=1}^{N}\right)=\sum_{i=1}^{N}f_{i}(\mathbf{x}_{i})+g\left(\sum_{i=1}^{N}\mathbf{z}_{i}\right)+\sum_{i=1}^{N}\langle\mathbf{x}_{i}-\mathbf{z}_{i},\mathbf{y}_{i}\rangle+\frac{\rho}{2}\sum_{i=1}^{N}\|\mathbf{x}_{i}-\mathbf{z}_{i}\|_{2}^{2}$
(4)

where $\mathbf{x}_{i},\mathbf{z}_{i}$ are called primal variables,
$\mathbf{y}_{i}\in\mathbb{R}^{n}$ is the Lagrangian variable and $\rho>0$ is
called the penalty parameter. The function in (4) can be minimized by the ADMM
steps [14]

$\displaystyle\mathbf{x}_{i}^{t+1}$
$\displaystyle:=\operatorname*{argmin}_{\mathbf{x}_{i}}\bigg{(}f_{i}(\mathbf{x}_{i})+\frac{\rho}{2}\|\mathbf{x}_{i}-\mathbf{z}_{i}^{t}+\mathbf{u}_{i}^{t}\|_{2}^{2}\bigg{)}$
(5) $\displaystyle\mathbf{z}_{i}^{t+1}$
$\displaystyle:=\operatorname*{argmin}_{\mathbf{z}_{i}}g\left(\sum_{i=1}^{N}\mathbf{z}_{i}\right)+\frac{\rho}{2}\sum_{i=1}^{N}\|\mathbf{z}_{i}-\mathbf{u}_{i}^{t}-\mathbf{x}_{i}^{t+1}\|_{2}^{2}$
(6) $\displaystyle\mathbf{u}_{i}^{t+1}$
$\displaystyle:=\mathbf{u}_{i}^{t}+\mathbf{x}_{i}^{t+1}-\mathbf{z}_{i}^{t+1}$
(7)

where $\mathbf{u}_{i}$ is called a dual variable (scaled version of Lagrangian
variable, $\mathbf{u}_{i}=\frac{\mathbf{y}_{i}}{\rho}$).
The problems (5) and (7) can be solved in parallel for $i=1,\ldots,N$. Let
$\mathbf{q}_{i}=\mathbf{u}_{i}^{t}+\mathbf{x}_{i}^{t+1}$. Then (6) can be
rewritten as

$\displaystyle\min\quad$ $\displaystyle
g\left(\sum_{i=1}^{N}\mathbf{z}_{i}\right)+\frac{\rho}{2}\sum_{i=1}^{N}\|\mathbf{z}_{i}-\mathbf{q}_{i}\|_{2}^{2}$
(8) s.t
$\displaystyle\bar{\mathbf{z}}=\frac{1}{N}\sum_{i=1}^{N}\mathbf{z}_{i}$

with fixed variable $\bar{\mathbf{z}}\in\mathbb{R}^{n}.$ The problem (8) has
the following solution

$\displaystyle\mathbf{z}_{i}=\mathbf{q}_{i}+\bar{\mathbf{z}}-\bar{\mathbf{q}},\quad\text{where}\quad\bar{\mathbf{q}}=\frac{1}{N}\sum_{i=1}^{N}\mathbf{q}_{i}=\bar{\mathbf{u}}^{t}+\bar{\mathbf{x}}^{t+1},$
(9) $\bar{\mathbf{u}}^{t}=\frac{1}{N}\sum_{i=1}^{N}\mathbf{u}_{i}^{t},\hskip
5.69054pt\text{and}\hskip
5.69054pt\bar{\mathbf{x}}^{t+1}=\frac{1}{N}\sum_{i=1}^{N}\mathbf{x}_{i}^{t+1}.$

From (8) and (9), the $\bar{\mathbf{z}}$-update step can be simplified to the
following unconstrained problem:

$\displaystyle\min\quad
g(N\bar{\mathbf{z}})+\frac{\rho}{2}\sum_{i=1}^{N}\|\bar{\mathbf{z}}-\bar{\mathbf{q}}|.$
(10)

Substituting (9) for $\mathbf{z}_{i}^{t+1}$ into (7), the
$\bar{\mathbf{u}}$-update step is

$\displaystyle\mathbf{u}_{i}^{t+1}=\bar{\mathbf{u}}^{t}+\bar{\mathbf{x}}^{t+1}-\bar{\mathbf{z}}^{t+1}$
(11)

The dual variables $\\{\mathbf{u}_{i}^{t}\\}_{i=1}^{N}$ are equal and can be
replaced by a single variable $\mathbf{u}^{t}$. The ADMM steps are simplified
as follows

$\displaystyle\mathbf{x}_{i}^{t+1}$
$\displaystyle:=\operatorname*{argmin}_{\mathbf{x}_{i}}\bigg{(}f_{i}(\mathbf{x}_{i})+\frac{\rho}{2}\|\mathbf{x}_{i}-\mathbf{x}_{i}^{t}+\bar{\mathbf{x}}^{t}-\bar{\mathbf{z}}^{t}+u^{t}\|_{2}^{2}\bigg{)}$
(12) $\displaystyle\bar{\mathbf{z}}^{t+1}$
$\displaystyle:=\operatorname*{argmin}_{\bar{\mathbf{z}}}\bigg{(}g(N\bar{\mathbf{z}})+\frac{N\rho}{2}\|\bar{\mathbf{z}}-u^{t}-\bar{\mathbf{x}}^{t+1}\|_{2}^{2}\bigg{)}$
(13) $\displaystyle\mathbf{u}^{t+1}$
$\displaystyle:=\mathbf{u}^{t}+\bar{\mathbf{x}}^{t+1}-\bar{\mathbf{z}}^{t+1}$
(14)

The original sharing problem (2) is decomposed into a three-step iterative
optimization problem. The problem (12) can be solved in parallel for
$i=1,\ldots,N$. The step (13) solves the shared objective, and (14) is the
dual variable update step.

##  III  ADMM-based detection for UL CD-NOMA systems

This section focuses on reformulating the relation between the input and
received signals of three multi-antenna CD-NOMA system models. These models
are 1. SIMO-CD-NOMA, 2. Spatial multiplexing CD-NOMA (SMX-CD-NOMA), and 3.
Spatial modulated CD-NOMA (SM-CD-NOMA). The reformulated models are readily
applied for sharing ADMM-based detection process. These models, along with the
derivation of the ADMM steps, are described in the following.

###  III-A SIMO CD-NOMA

Consider a UL scenario with the BS equipped with $N_{\mathrm{r}}$ antennas and
each UE having a single antenna. The idea is to exploit the spatial diversity
gain offered by multiple receive antennas at BS [25]. The error rate
performance of the CD-NOMA system with $N_{\mathrm{r}}$ antennas at the
receiver is significantly improved [4] due to higher diversity gain. The
$K\times 1$ observation vector at $n_{\mathrm{r}}$th BS antenna can be
expressed as

$\displaystyle\mathbf{r}^{(n_{\mathrm{r}})}=\sum_{j=1}^{J}\text{diag}\left(\mathbf{h}_{j}^{(n_{\mathrm{r}})}\right)\mathbf{x}_{j}+\mathbf{w}^{(n_{\mathrm{r}})},\hskip
5.69054ptn_{\rm r}=1,\ldots,N_{\rm r}$

where

  * •

$\mathbf{w}^{(n_{\mathrm{r}})}\sim\mathcal{CN}(0,\sigma^{2}\mathbf{I}_{K})$,
$K\times 1$ vector assumed to be independent and identically distributed
(i.i.d.), AWGN at the $n_{\mathrm{r}}$th BS antenna.

  * •

$\mathbf{h}_{j}^{(n_{\mathrm{r}})}\sim\mathcal{CN}(0,\mathbf{I}_{K})$,
$K\times 1$ vector assumed to be i.i.d. complex Gaussian random vector,
denotes Rayleigh fading channel coefficients between the $j$th UE and
$n_{\mathrm{r}}$th BS antenna.
$\mathbf{h}_{j}^{(n_{\mathrm{r}})}=\left[h[1]\ldots h[k]\ldots
h[K]\right]^{T}$ and
$\text{diag}\left(\mathbf{h}_{j}^{(n_{\mathrm{r}})}\right)$ denotes diagonal
matrix with $h[k]$ being the $k$th diagonal element.

  * •

$\mathbf{x}_{j}=\left[x_{j}[1]\ldots x_{j}[k]\ldots x_{j}[K]\right]^{T}$ is a
codeword from the $j$th UE’s codebook, $\mathcal{X}_{j}^{K\times M}$.

The overall $KN_{\mathrm{r}}\times 1$ observation vector at the BS for the
SIMO CD-NOMA system is given by

$\displaystyle\mathbf{r}=\mathbf{H}\mathbf{x}_{\rm{mu}}+\mathbf{w},$ (15)

where

  * •

$\mathbf{r}=[\mathbf{r}^{(1)^{T}}\ldots\mathbf{r}^{(n_{\mathrm{r}})^{T}},\ldots,\mathbf{r}^{(N_{\mathrm{r}})^{T}}]^{T}$.

  * •

$\mathbf{w}\sim\mathcal{CN}(0,\sigma^{2}\mathbf{I}_{KN_{\mathrm{r}}}),N_{\mathrm{r}}K\times
1$ vector denotes the AWGN vector at the BS.

  * •

$\mathbf{x}_{\rm{mu}}$ is the $JN_{\mathrm{e}}\times 1$ multi-user
concatenated transmitted signal.
$\mathbf{x}_{\rm{mu}}=[x_{1}[1]\ldots x_{1}[N_{\mathrm{e}}]\ldots
x_{j}[1]\ldots x_{j}[N_{\mathrm{e}}]\ldots x_{J}[1]\ldots
x_{J}[N_{\mathrm{e}}]]^{T}$, where $N_{\mathrm{e}}$ is the number of nonzero
elements in codeword. We have $N_{\mathrm{e}}=K$ and
$N_{\mathrm{e}}=d_{\mathrm{v}}$ for DCMA and SCMA, respectively. The
transmitted signal $\mathbf{x}_{\rm{mu}}$ can be rewritten to facilitate the
sharing-based detection problem in Section III-D i.e.,
$\mathbf{x}_{\rm{mu}}=\sum_{j=1}^{J}\mathbf{x}_{0j}$. The variable
$\mathbf{x}_{0j}=[0\ldots 0\hskip 5.69054ptx_{j}[1]\hskip
5.69054ptx_{j}[2]\ldots x_{j}[N_{\mathrm{e}}]\hskip 5.69054pt0\ldots 0]^{T}$
represents the $j$th UE codeword.

  * •

The channel matrix for DCMA system of size $KN_{\mathrm{r}}\times JK$ is given
by

$\displaystyle\mathbf{H}=\begin{bmatrix}\text{diag}(\mathbf{h}_{1}^{(1)})&\ldots&\text{diag}(\mathbf{h}_{j}^{(1)})&\ldots&\text{diag}(\mathbf{h}_{J}^{(1)})\\\
\text{diag}(\mathbf{h}_{1}^{(2)})&\ldots&\text{diag}(\mathbf{h}_{j}^{(2)})&\ldots&\text{diag}(\mathbf{h}_{J}^{(2)})\\\
\vdots&\ldots&\vdots&\ldots&\vdots\\\
\text{diag}(\mathbf{h}_{1}^{(N_{\mathrm{r}})})&\ldots&\text{diag}(\mathbf{h}_{j}^{(N_{\mathrm{r}})})&\ldots&\text{diag}(\mathbf{h}_{J}^{(N_{\mathrm{r}})})\end{bmatrix}.$

  * •

The channel matrix for SCMA system of size $KN_{\mathrm{r}}\times
Jd_{\mathrm{v}}$ is given by

$\displaystyle\mathbf{H}=\begin{bmatrix}\overline{\text{diag}(\mathbf{h}_{1}^{(1)})}&\ldots&\overline{\text{diag}(\mathbf{h}_{j}^{(1)})}&\ldots&\overline{\text{diag}(\mathbf{h}_{J}^{(1)})}\\\
\overline{\text{diag}(\mathbf{h}_{1}^{(2)})}&\ldots&\overline{\text{diag}(\mathbf{h}_{j}^{(2)})}&\ldots&\overline{\text{diag}(\mathbf{h}_{J}^{(2)})}\\\
\vdots&\ldots&\vdots&\ldots&\vdots\\\
\overline{\text{diag}(\mathbf{h}_{1}^{(N_{\mathrm{r}})})}&\ldots&\overline{\text{diag}(\mathbf{h}_{j}^{(N_{\mathrm{r}})})}&\ldots&\overline{\text{diag}(\mathbf{h}_{J}^{(N_{\mathrm{r}})})}\end{bmatrix}.$

where $\overline{\text{diag}\left(\mathbf{h}_{j}^{(N_{\mathrm{r}})}\right)}$
is $K\times d_{\mathrm{v}}$ matrix after removing the columns in
$\text{diag}(\mathbf{h}_{j}^{(N_{\mathrm{r}})})$ corresponding to zero
elements of $j$th UE codeword $\mathbf{x}_{j}$ (inactive REs of $j$th UE).

######  Example 1.

Consider the factor graph matrix for the 150 % overloading SCMA system with
$K=4,J=6,d_{\mathrm{f}}=3,d_{\mathrm{v}}=2$:

$\displaystyle\mathbf{F}=\begin{bmatrix}1&0&1&0&1&0\\\ 0&1&1&0&0&1\\\
1&0&0&1&0&1\\\ 0&1&0&1&1&0\end{bmatrix}.$ (16)

For the $1$st UE and the $n_{\mathrm{r}}$th receiving antenna, the channel
matrix is given below:

$\overline{\rm{diag}(\mathbf{h}_{1}^{(n_{\mathrm{r}})})}=\left[\begin{tabular}[]{cccccc}$h_{1}^{(n_{\mathrm{r}})}[1]$&0\\\
0&0\\\ 0&$h_{1}^{(n_{\mathrm{r}})}[3]$\\\ 0&0\par\end{tabular}\right].$

The transmitted signal from all UEs is given by

$\displaystyle{{\bf{x}}_{{\rm{mu}}}}={\left[{\underbrace{{x_{1}}[1]\;\;{x_{1}}[3]}_{{\rm{UE}}\;1}\;\;\underbrace{{x_{2}}[2]\;\;{x_{2}}[4]}_{{\rm{UE}}\;2}\;\;\underbrace{{x_{3}}[1]\;\;{x_{3}}[2]\;}_{{\rm{UE}}\;3}\;\underbrace{{x_{4}}[3]\;\;{x_{4}}[4]}_{{\rm{UE}}\;4}\;\;\underbrace{{x_{5}}[1]\;\;{x_{5}}[4]\;}_{{\rm{UE}}\;5}\;\underbrace{{x_{6}}[2]\;\;{x_{6}}[3]}_{{\rm{UE}}\;6}}\right]^{T}}.$
(17)

The 3rd UE codeword $\mathbf{x}_{03}$ in ${{\bf{x}}_{{\rm{mu}}}}$ is given by

$\displaystyle{\mathbf{x}_{03}}={\left[{\underbrace{{0}\;\;{0}}_{{\rm{UE}}\;1}\;\;\underbrace{{0}\;\;{0}}_{{\rm{UE}}\;2}\;\;\underbrace{{x_{3}}[1]\;\;{x_{3}}[2]\;}_{{\rm{UE}}\;3}\;\underbrace{{0}\;\;{0}}_{{\rm{UE}}\;4}\;\;\underbrace{{0}\;\;{0}\;}_{{\rm{UE}}\;5}\;\underbrace{{0}\;\;{0}}_{{\rm{UE}}\;6}}\right]^{T}}.$
(18)

The variables $\\{\mathbf{x}_{0j}\\}_{j=1}^{J}$ are similarly represented as
$\mathbf{x}_{03}$. Note that, the transmitted signal $\mathbf{x}_{\rm{mu}}$
can be written as, $\mathbf{x}_{\rm{mu}}=\sum_{j=1}^{6}\mathbf{x}_{0j}$. ∎

###  III-B SMX-CD-NOMA

The spectral efficiency of the UL CD-NOMA system is further enhanced by
placing multiple antennas at the transmitter [5]. Multiple antennas at both
transmitter and receiver of the CD-NOMA system are considered in this section.
Multiple antennas at the transmitter exploit multiplexing gain offered by
spatially multiplexing several data streams onto the MIMO channel. Consider a
scenario where each UE is equipped with $N_{\mathrm{t}}$ transmitting
antennas, and the BS is equipped with $N_{\mathrm{r}}$ receiving antennas. The
total input data at each UE, $N_{\mathrm{t}}\text{log}_{2}(M)$ bits, are
divided into $N_{\mathrm{t}}$ parallel data streams. Each data stream is fed
to the CD-NOMA encoder, as shown in Fig. 2. Note that in the SMX-CD-NOMA
system, all the UE antennas transmit data simultaneously. The main challenge
of SMX-CD-NOMA in practical implementation is the computational complexity at
both transmitter and receiver. Further, detecting the data transmitted from
multiple antennas of multiple UEs is a complex operation. The proposed method
performs resource-wise processing at the BS via the ADMM algorithm, as shown
in Fig. 3.

Figure 2: SMX-CD-NOMA system model in the UL.

The received signal at the BS over the $k$th RE is given by

$\displaystyle\mathbf{r}_{k}=\mathbf{H}_{k}\mathbf{x}_{\rm{smx},k}+\mathbf{w}_{k},$
(19)

where

  * •

$\mathbf{r}_{k}=[r_{k}^{1}\ldots r_{k}^{n_{\mathrm{r}}}\ldots
r_{k}^{N_{\mathrm{r}}}]^{T}$, $N_{\mathrm{r}}\times 1$ observation vector.

  * •

$\mathbf{w}_{k}\sim\mathcal{CN}(0,\sigma^{2}\mathbf{I}_{N_{\mathrm{r}}}),N_{\mathrm{r}}\times
1$ vector denotes the AWGN at the BS over $k$th RE.

  * •

$\mathbf{x}_{{\rm{smx}},k}$ is the $N_{\mathrm{u}}N_{\mathrm{t}}\times 1$
transmitted vector on $k$th RE.
$\mathbf{x}_{{\rm{smx}},k}=[\mathbf{x}_{1,k}^{T}\ldots\mathbf{x}_{j,k}^{T}\ldots\mathbf{x}_{N_{\mathrm{u}},k}^{T}]^{T}$,
and each $\mathbf{x}_{j,k}=[x_{j,k,1}\ldots x_{j,k,n_{\mathrm{t}}}\ldots
x_{j,k,N_{\mathrm{t}}}]^{T}$ is $N_{\mathrm{t}}\times 1$ vector corresponding
to $j$th UE. Each $x_{j,k,n_{\mathrm{t}}}\in\mathbf{x}_{j}^{n_{\mathrm{t}}}$
is the symbol transmitted from $n_{\mathrm{t}}$th antenna of $j$th UE over
$k$th RE, where $\mathbf{x}_{j}^{n_{\mathrm{t}}}$ is the codeword of $j$th UE
transmitted from $n_{\mathrm{t}}$th antenna. The set $\zeta_{k}$ represents
the set of UEs overlapping on $k$th RE given by

$\zeta_{k}=\\{j:\mathbf{x}_{j}[k]\neq 0;1\leq j\leq J\\},\hskip
5.69054pt\text{and}\hskip 5.69054pt|\zeta_{k}|=N_{\mathrm{u}},$

where $N_{\mathrm{u}}=d_{\mathrm{f}}$ and $N_{\mathrm{u}}=J$ for SCMA and
DCMA, respectively.

The transmitted signal $\mathbf{x}_{{\rm{smx}},k}$ can be rewritten to
formulate the sharing-based detection problem in Section III-D i.e.,
$\mathbf{x}_{{\rm{smx}},k}=\sum_{j=1}^{N_{\mathrm{u}}}\mathbf{x}_{0j,k}$. The
variable $\mathbf{x}_{0j,k}=[0\ldots 0\hskip 5.69054ptx_{j,k,1}\hskip
5.69054ptx_{j,k,2}\ldots x_{j,k,N_{\rm t}}\hskip 5.69054pt0\ldots 0]^{T}$ is
the $N_{\mathrm{u}}N_{\mathrm{t}}\times 1$ vector, and represents the $j$th
overlapped UE on $k$th RE. The nonzero elements in $\mathbf{x}_{0j,k}$ are
symbols transmitted from $N_{\rm t}$ antennas.

  * •

$\mathbf{H}_{k}$ is the $N_{\mathrm{r}}\times N_{\mathrm{u}}N_{\mathrm{t}}$
matrix given by

$\mathbf{H}_{k}=[\mathbf{H}_{1,k}\ldots\mathbf{H}_{j,k}\ldots\mathbf{H}_{N_{\mathrm{u}},k}]$

where $\mathbf{H}_{j,k}$ represents the $j$th UE MIMO channel matrix of size
$N_{\mathrm{r}}\times N_{\mathrm{t}}$.

Figure 3: Processing at $k$th RE of SMX-CD-NOMA system.

######  Example 2.

Consider $N_{\mathrm{t}}=2$, and according to the factor graph matrix given in
(16). The transmitted signal over the first RE in the SCMA system is

$\mathbf{x}_{\rm{smx},1}=\begin{bmatrix}\mathbf{x}_{1,1}^{T}&\mathbf{x}_{3,1}^{T}&\mathbf{x}_{5,1}^{T}\end{bmatrix}^{T}.$
(20)

For DCMA, due to the dense structure of codebooks, all $J$ UEs overlap on each
RE [2]. The transmitted signal over the first RE is given by

$\mathbf{x}_{\rm{smx},1}=\begin{bmatrix}\mathbf{x}_{1,1}^{T}&\mathbf{x}_{2,1}^{T}&\ldots&\mathbf{x}_{5,1}^{T}&\mathbf{x}_{6,1}^{T}\end{bmatrix}^{T},$
(21)

where each $\mathbf{x}_{j,1}$ is a $2\times 1$ ($N_{\mathrm{t}}\times 1$)
vector given by

$\mathbf{x}_{j,1}=\begin{bmatrix}x_{j,1,1}&x_{j,1,2}\end{bmatrix}^{T}.$

The transmitted signal over each remaining RE
($\mathbf{x}_{{\rm{smx}},k},k=2,\ldots,K$) has a similar representation as in
(20) and (21) for SCMA and DCMA, respectively.

For SCMA, $\mathbf{x}_{03,1}$ represents the 3rd UE’s symbols as given by

$\displaystyle{\mathbf{x}_{03,1}}={\left[{\underbrace{{0}\;\;{0}}_{{\rm{UE}}\;1}\;\;\underbrace{{x_{3,1,1}}\;\;{x_{3,1,2}}\;}_{{\rm{UE}}\;3}\;\;\underbrace{{0}\;\;{0}\;}_{{\rm{UE}}\;5}}\right]^{T}}.$
(22)

For DCMA, $\mathbf{x}_{03,1}$ can be written as

$\displaystyle{\mathbf{x}_{03,1}}={\left[{\underbrace{{0}\;\;{0}}_{{\rm{UE}}\;1}\;\;\underbrace{{0}\;\;{0}}_{{\rm{UE}}\;2}\;\;\underbrace{{x_{3,1,1}}\;\;{x_{3,1,2}}\;}_{{\rm{UE}}\;3}\;\underbrace{{0}\;\;{0}}_{{\rm{UE}}\;4}\;\;\underbrace{{0}\;\;{0}\;}_{{\rm{UE}}\;5}\;\underbrace{{0}\;\;{0}}_{{\rm{UE}}\;6}}\right]^{T}}.$
(23)

The variables $\\{\mathbf{x}_{0j,k}\\}_{k=1}^{K}$ are similarly represented as
$\mathbf{x}_{03,1}$. The transmitted signal $\mathbf{x}_{{\rm{smx}},k}$ can be
written as, $\mathbf{x}_{{\rm{smx}},k}=\sum_{j=1}^{3}\mathbf{x}_{0j,k}$ and
$\mathbf{x}_{{\rm{smx}},k}=\sum_{j=1}^{6}\mathbf{x}_{0j,k}$ for SCMA and DCMA,
respectively. ∎

###  III-C SM-CD-NOMA

Due to the requirement of multiple radio frequency (RF) chains, the SMX-CD-
NOMA system is not affordable for various applications. Further, inter-channel
interference is the main limitation of this system. Spatial modulation (SM)
MIMO systems are promising technology to overcome the limitations of SMX MIMO
systems. SM for single-UE communications is well studied in the literature
[26]. Due to the demands of future wireless networks, it is important to study
and analyze SM in multiuser scenarios. In [27], the authors studied and
analyzed SM sparse CDMA. SM-aided CD-NOMA (SM-CD-NOMA) is guaranteed to
improve the spectral efficiency of the CD-NOMA with feasible complexity at
both transmitter and receiver [6, 28]. Fig. 4 shows the system model for the
SM-CD-NOMA system. Each UE is equipped with $N_{\mathrm{t}}$ transmitting
antennas, in which only one antenna is active at any time, as shown in Fig. 4.
The active antenna index of the $j$th UE is denoted by $n_{\rm a}^{j}$. All
other antennas remain silent in that particular time slot. Thus, the active
antenna index is a spatial modulation symbol to transmit extra information
bits. The information bit stream $\mathbf{b}_{j}$ of the $j$th UE, is split
into two parts [$\mathbf{b}_{j\mathrm{a}}$,$\mathbf{b}_{j\mathrm{c}}$] having
$\text{log}_{2}(N_{\mathrm{t}})$ and $\text{log}_{2}(M)$ bits, respectively.
Each UE transmits an overall
$\text{log}_{2}(N_{\mathrm{t}})+\text{log}_{2}(M)$ bits from the active
antenna.

Figure 4:  SM-CD-NOMA system model in the UL.

The observation vector over the $k$th RE at the BS is an $N_{\mathrm{r}}\times
1$ vector given by

$\displaystyle\mathbf{r}_{k}=\mathbf{H}_{k}\mathbf{x}_{\text{sm},k}+\mathbf{w}_{k},$
(24)

where

  * •

$\mathbf{r}_{k},\mathbf{w}_{k}$, and $\mathbf{H}_{k}$ have the similar forms
as in SMX-CD-NOMA for both DCMA and SCMA systems.

  * •

$\mathbf{x}_{\text{sm},k}$ is the $N_{\mathrm{u}}N_{\mathrm{t}}\times 1$
vector
$\mathbf{x}_{\text{sm},k}=[\mathbf{x}_{1,k}^{T}\ldots\mathbf{x}_{j,k}^{T}\ldots\mathbf{x}_{N_{\mathrm{u}},k}^{T}]^{T}$,
and each $\mathbf{x}_{j,k}=[0\ldots x_{j,k,n_{\rm a}^{j}}\ldots 0]^{T}$ is
$N_{\mathrm{t}}\times 1$ vector corresponding to $j$th UE. The nonzero element
$x_{j,k,n_{\rm a}^{j}}$ is the symbol transmitted from $n_{\rm a}^{j}$th
active antenna. The transmitted signal $\mathbf{x}_{\text{sm},k}$ can be
rewritten to formulate the sharing-based detection problem in Section III-D
i.e.,
$\mathbf{x}_{\text{sm},k}=\sum_{j=1}^{N_{\mathrm{u}}}\mathbf{x}_{0j,k}^{n_{\rm
a}^{j}}$. The variable $\mathbf{x}_{0j,k}^{n_{\rm a}^{j}}=[0\ldots
0\ldots\hskip 5.69054ptx_{j,k,n_{\rm a}^{j}}\hskip 5.69054pt0\ldots\hskip
5.69054pt0\ldots 0]^{T}$ represents the $j$th overlapped UE on $k$th RE. The
nonzero element in $\mathbf{x}_{0j,k}^{n_{\rm a}^{j}}$ is the symbol
transmitted from $n_{\rm a}^{j}$th active antenna.

Fig. 5 depicts the $k$th RE processing at the BS through the extended factor
graph. The thick and dotted lines correspond to the active and inactive
antennas, respectively.

Figure 5: Processing at $k$th RE of SM-CD-NOMA system.

######  Example 3.

Consider $N_{\mathrm{t}}=2$, and the factor graph matrix given in (16).
Suppose the active antenna indices of the $6$-users are {2,1,2,2,1,1}. The
structure of transmit codewords from $J$ UEs for SM-CD-NOMA is as follows
[28]:

$\displaystyle\mathbf{x}_{\rm{tx}}=\begin{bmatrix}\mathbf{0}_{K\times
1}&\mathbf{x}_{2_{K\times 1}}^{1}&\mathbf{0}_{K\times 1}&\mathbf{0}_{K\times
1}&\mathbf{x}_{5_{K\times 1}}^{1}&\mathbf{x}_{6_{K\times 1}}^{1}\\\
\mathbf{x}_{1_{K\times 1}}^{2}&\mathbf{0}_{K\times 1}&\mathbf{x}_{3_{K\times
1}}^{2}&\mathbf{x}_{4_{K\times 1}}^{2}&\mathbf{0}_{K\times
1}&\mathbf{0}_{K\times 1}\\\ \end{bmatrix}_{N_{\mathrm{t}}K\times J=8\times
6}.$ (25)

Each $\mathbf{x}_{j_{{}_{K\times 1}}}^{n_{\rm a}^{j}}$ in
$\mathbf{x}_{\rm{tx}}$ represents the transmitted codeword from $j$th UE, and
each column of $\mathbf{x}_{\text{tx}}$ carries information about active
transmit antenna ($n_{\rm a}^{j}$) as well as the codeword. In (25),
$\mathbf{0}_{K\times 1}$ indicates zero power transmitted by a deactivated
antenna. To facilitate the resource-wise processing at the receiver via ADMM,
the transmitted signal over the first RE in the SCMA system is modeled as

$\mathbf{x}_{\rm{sm},1}=\begin{bmatrix}\mathbf{x}_{1,1}^{T}&\mathbf{x}_{3,1}^{T}&\mathbf{x}_{5,1}^{T}\hskip
8.53581pt\end{bmatrix}^{T}.$ (26)

The transmitted signal over the first RE in the DCMA system is modeled as

$\mathbf{x}_{\rm{sm},1}=\begin{bmatrix}\mathbf{x}_{1,1}^{T}&\mathbf{x}_{2,1}^{T}&\ldots&\mathbf{x}_{5,1}^{T}&\mathbf{x}_{6,1}^{T}\hskip
8.53581pt\end{bmatrix}^{T},$ (27)

where $\mathbf{x}_{1,1}=[0\hskip 5.69054ptx_{1,1,2}]^{T}$,
$\mathbf{x}_{2,1}=[x_{2,1,1}\hskip 5.69054pt0]^{T}$,
$\mathbf{x}_{3,1}=[0\hskip 5.69054ptx_{3,1,2}]^{T}$,
$\mathbf{x}_{4,1}=[0\hskip 5.69054ptx_{4,1,2}]^{T}$,
$\mathbf{x}_{5,1}=[x_{5,1,1}\hskip 5.69054pt0]^{T}$, and
$\mathbf{x}_{6,1}=[x_{6,1,1}\hskip 5.69054pt0]^{T}$. The transmitted signals
over the remaining REs ($\mathbf{x}_{{\rm{sm}},k},k=2,\ldots,K$ ) have similar
representations as (26) and (27) for SCMA and DCMA, respectively. For SCMA,
$\mathbf{x}_{03,1}^{2}$ represents the symbol transmitted from 2nd active
antenna of 3rd UE on 1st RE. $\mathbf{x}_{03,1}^{2}$ can be written as

$\displaystyle{\mathbf{x}_{03,1}^{2}}={\left[{\underbrace{{0}\;\;{0}}_{{\rm{UE}}\;1}\;\;\underbrace{{0}\;\;{x_{3,1,2}}\;}_{{\rm{UE}}\;3}\;\;\underbrace{{0}\;\;{0}\;}_{{\rm{UE}}\;5}}\right]^{T}}.$
(28)

For DCMA, $\mathbf{x}_{03,1}^{2}$ can be written as

$\displaystyle{\mathbf{x}_{03,1}^{2}}={\left[{\underbrace{{0}\;\;{0}}_{{\rm{UE}}\;1}\;\;\underbrace{{0}\;\;{0}}_{{\rm{UE}}\;2}\;\;\underbrace{{0}\;\;{x_{3,1,2}}\;}_{{\rm{UE}}\;3}\;\underbrace{{0}\;\;{0}}_{{\rm{UE}}\;4}\;\;\underbrace{{0}\;\;{0}\;}_{{\rm{UE}}\;5}\;\underbrace{{0}\;\;{0}}_{{\rm{UE}}\;6}}\right]^{T}}.$
(29)

The variables $\\{\mathbf{x}_{0j}^{n_{\rm a}^{j}}\\}_{k=1}^{K}$ are similarly
represented as $\mathbf{x}_{03,1}^{2}$. The transmitted signal
$\mathbf{x}_{{\rm{sm}},k}$ can be written as,
$\mathbf{x}_{{\rm{sm}},k}=\sum_{j=1}^{3}\mathbf{x}_{0j,k}^{n_{\rm a}^{j}}$ and
$\mathbf{x}_{{\rm{sm}},k}=\sum_{j=1}^{6}\mathbf{x}_{0j,k}^{n_{\rm a}^{j}}$ for
SCMA and DCMA, respectively ∎

Observe from equations (15), (19), and (24) that the system models of three
MIMO-CD-NOMA systems are similar to those of the conventional MIMO system
model. This model is given by

$\displaystyle\mathbf{r}=\mathbf{H}\mathbf{x}+\mathbf{w},\leavevmode\nobreak\
\mathbf{x}\in\mathcal{X}^{N_{\mathrm{t}}},$ (30)

where $\mathcal{X}$ is the signal constellation, $\mathbf{r}$ is the
$N_{\mathrm{r}}\times 1$ observation vector, $\mathbf{H}$ is
$N_{\mathrm{r}}\times N_{\mathrm{t}}$ channel matrix
($N_{\mathrm{r}}>N_{\mathrm{t}}$), $\mathbf{x}$ is $N_{\mathrm{t}}\times 1$
transmitted vector and $\mathbf{w}$ is i.i.d. AWGN vector with each component
being distributed as $\mathcal{CN}(0,\sigma^{2})$. The ML detection problem
can be formulated as

$\displaystyle\text{min}_{\mathbf{x}\in\mathcal{X}^{N_{\rm
t}}}\|\mathbf{r}-\mathbf{H}\mathbf{x}\|^{2}.$

The ML detection problem for SIMO CD-NOMA system in (15) is given by

$\displaystyle\min_{\mathbf{x}_{\rm{mu}}\in\mathcal{X}_{\text{mu}}^{JN_{\rm
e}}}\quad\|\mathbf{r}-\mathbf{H}\mathbf{x}_{\rm{mu}}\|^{2},$ (31)

where $\mathcal{X}_{\text{mu}}^{JN_{\rm e}}$ denotes the multi-user signal
constellation and it consists of $J$-UEs concatenated codewords. However,
solving ML detection problems, in general, is NP-hard. The exhaustive search
method can be used to solve the ML detection problem. However, the exhaustive
search is exponentially complex as per $\mathcal{O}(M^{J})$ [13] and thus it
is not feasible. The ML detection problem (31) can be solved with a polynomial
complexity using a distributed optimization framework. The above problem needs
to be converted into a sharing problem that can be solved using distributed
optimization methods. In the next section, we apply an efficient method based
on the ADMM algorithm to solve the sharing-based detection problem in a
distributed manner.

###  III-D Large-scale UL multi-antenna CD-NOMA detection via ADMM

This section discusses the design of ADMM-based MUD for two CD-NOMA
techniques, i.e., SCMA and DCMA. The ADMM-based detector is applied to three
different MIMO systems, namely, SIMO-CD-NOMA, spatial multiplexed CD-NOMA
(SMX-CD-NOMA), and spatially modulated CD-NOMA (SM-CD-NOMA). The implications
of the proposed method are discussed in subsequent sections.

####  III-D1 SIMO CD-NOMA

The ML detection problem in (31) can be converted into a sharing problem.
Further, this problem can be solved in a distributed manner via the ADMM
approach, as discussed in Section II-C. ADMM allows parallel processing in MUD
problems, guaranteeing a minimal computational time at the receiver. Recall
from Section III-A, the transmitted signal $\mathbf{x}_{\rm{mu}}$ is given by
$\mathbf{x}_{\rm{mu}}=\sum_{j=1}^{J}\mathbf{x}_{0j}$. The problem (31) can be
rewritten as

$\displaystyle\min_{\mathbf{x}_{0j}}\quad$
$\displaystyle\|\mathbf{r}-\mathbf{H}\left(\sum_{j=1}^{J}\mathbf{x}_{0j}\right)\|^{2}$
(32) s.t
$\displaystyle\mathbf{x}_{0j}\in\mathcal{X}_{\text{mu}}^{JN_{\mathrm{e}}}\quad
j=1,\ldots,J.$

Here, for the channel matrix $\mathbf{H}^{N_{\mathrm{r}}K\times
JN_{\mathrm{e}}}$, we consider $N_{\mathrm{r}}K>JN_{\mathrm{e}}$. Each entry’s
real and imaginary parts in $\mathbf{x}_{0j}$ are restricted by set
constraints. The set constraint must be converted into an interval constraint
to convert (32) into a sharing problem. Box constraint relaxation (BCR) is
used to relax the set constraints of each entry in the codeword[20]. Hence,
each entry’s real and imaginary parts in the $j$th UE codeword belong to
$[-\alpha_{j},\alpha_{j}]$ and $[-\beta_{j},\beta_{j}]$. After relaxation,
each element in the $j$th UE codebook is defined as
$\tilde{\mathcal{X}}_{j}=\\{x_{j}=x_{jR}+ix_{jI}|x_{jR}\in[-\alpha_{j},\alpha_{j}],x_{jI}\in[-\beta_{j},\beta_{j}]\\}$,
$\alpha_{j}=\max|{\mathbb{R}(\mathcal{X}_{j})}|,\beta_{j}=\max|{\mathcal{I}(\mathcal{X}_{j})}|$.
The highly complex MIMO-CD-NOMA ML detection problem in (32) is now ready to
be converted into a non-convex distributed optimization problem. However, the
constraint relaxation degrades the detection performance due to the losses
introduced by the interval constraints. The losses can be compensated by
adding a set of quadratic penalty functions
$\sum_{j=1}^{J}\frac{\gamma_{j}}{2}\|\mathbf{x}_{0j}\|_{2}^{2}$ to the
objective function where $\gamma_{j}\geq 0$ is a penalty parameter. The
penalty functions are selected so that each variable, $\mathbf{x}_{0j}$, in
the penalty term minimizes the individual penalty and the shared objective.
The added penalty term makes the solution as sparse as possible. However, the
sparse vectors with specific numbers ($d_{\mathrm{v}}$ for SCMA and $K$ for
DCMA) of non-zero entries only will minimize the shared objective function.
The favorable solutions are the sparse vectors with $d_{\mathrm{v}}$ and $K$
non-zero elements for SCMA and DCMA, respectively. The sharing ADMM problem
can be written as

$\displaystyle\min_{\mathbf{x}_{0j}}\quad$
$\displaystyle\|\mathbf{r}-\mathbf{H}\left(\sum_{j=1}^{J}\mathbf{x}_{0j}\right)\|^{2}+\sum_{j=1}^{J}\frac{\gamma_{j}}{2}\|\mathbf{x}_{0j}\|_{2}^{2}$
(33) s.t
$\displaystyle\mathbf{x}_{0j}\in\tilde{\mathcal{X}}_{\text{mu}}^{JN_{\mathrm{e}}},\quad
j=1,\ldots,J.$

The problem in (33) is similar to the sharing ADMM problem in (2). Proceeding
similarly as in Section II-C, (33) can be written as

$\displaystyle\min_{\mathbf{x}_{0j},\mathbf{z}_{0j}}\quad$
$\displaystyle\|\mathbf{r}-\mathbf{H}\left(\sum_{j=1}^{J}\mathbf{x}_{0j}\right)\|^{2}+\sum_{j=1}^{J}\frac{\gamma_{j}}{2}\|\mathbf{z}_{0j}\|_{2}^{2}$
(34) s.t
$\displaystyle\mathbf{x}_{0j}=\mathbf{z}_{0j},\quad\mathbf{x}_{0j}\in\tilde{\mathcal{X}}_{\text{mu}}^{JN_{\mathrm{e}}},\quad\forall
j=1,\ldots,J$

For the formulation of the ADMM steps, the augmented Lagrangian function for
the problem (34) is considered as shown below:

$\displaystyle\mathcal{L}\left(\\{\mathbf{x}_{0j},\mathbf{z}_{0j},\textbf{y}_{j}\\}_{j=1}^{J}\right)=\|\textbf{r}-\mathbf{H}\left(\sum_{j=1}^{J}\mathbf{x}_{0j}\right)\|_{2}^{2}+\sum_{j=1}^{J}\frac{\gamma_{j}}{2}\|\mathbf{z}_{0j}\|_{2}^{2}$
(35)
$\displaystyle+\sum_{j=1}^{J}{\text{Re}}\langle\mathbf{x}_{0j}-\mathbf{z}_{0j},\textbf{y}_{j}\rangle+\frac{\rho}{2}\sum_{j=1}^{J}\|\mathbf{x}_{0j}-\mathbf{z}_{0j}\|_{2}^{2}$

where $\textbf{y}_{j}\in\mathbb{C}^{JN_{\mathrm{e}}}$ is Lagrangian multiplier
of the $j$th UE. Letting $\mathbf{u}_{j}=\frac{\textbf{y}_{j}}{\rho}$, the
scaled form of ADMM steps are as follows [14] :

$\displaystyle\mathbf{z}_{0j}^{t+1}:=$
$\displaystyle\operatorname*{argmin}_{\mathbf{z}_{0j}\in\tilde{\mathcal{X}}_{\text{mu}}^{JN_{\rm
e}}}\frac{\gamma_{j}}{2}\|\mathbf{z}_{0j}\|_{2}^{2}+\frac{\rho}{2}\|\mathbf{z}_{0j}-\mathbf{x}_{0j}^{t}+\mathbf{u}_{j}^{t}\|_{2}^{2}$
(36) $\displaystyle\mathbf{x}_{0j}^{t+1}:=$
$\displaystyle\operatorname*{argmin}_{\mathbf{x}_{0j}\in\tilde{\mathcal{X}}_{\text{mu}}^{JN_{\rm
e}}}\|\mathbf{r}-\mathbf{H}\left(\sum_{j=1}^{J}\mathbf{x}_{0j}\right)\|^{2}+\frac{\rho}{2}\sum_{j=1}^{J}\|\mathbf{z}_{0j}^{t+1}-\mathbf{x}_{0j}+\mathbf{u}_{j}^{k}\|_{2}^{2}$
(37) $\displaystyle\mathbf{u}_{j}^{t+1}:=$
$\displaystyle\mathbf{u}_{j}^{k}+(\mathbf{z}_{0j}^{t+1}-\mathbf{x}_{0j}^{t+1})$
(38)

Algorithm 1 SIMO CD-NOMA detection via sharing ADMM

1:Input: Noise variance ($N_{0}$), average codebook energy ($E_{\mathrm{s}}$)
and initialize
$\\{\mathbf{z}_{0j}\\}_{j=1}^{J},\bar{\mathbf{x}}_{0},\mathbf{u},\bar{\mathbf{z}}_{0}$
with zero vectors.

2:Output: $\bar{\mathbf{x}}_{0}^{(T)}$, where $T$ is the maximum number of
iterations.

3:From (39) and (40), obtain optimal solutions (42) and (43), respectively.

4:Preprocessing

5:$\rho=N_{0}/E_{\mathrm{s}}$

6:$\mathbf{G}=(\mathbf{H}^{H}\mathbf{H}J+\rho\mathbf{I})^{-1}$

7:$\mathbf{M}=\mathbf{H}^{H}\mathbf{r}$

8:for $t=1,2,,\ldots T$ do

9: Step:1 Update $\\{\mathbf{z}_{0j}^{t+1}\\}_{j=1}^{J}$ in parallel via (42)
and compute $\bar{\mathbf{x}_{1}}$.

10: Step:2

11: Update $\bar{\mathbf{x}}_{0}^{t+1}$ via (43).

12: Update $\mathbf{u}^{t+1}$ via (41)

13:end for

14:Compute $\hat{\mathbf{x}}_{\text{mu}}=J\bar{\mathbf{x}}_{0}^{(T)}$.

15:After extracting each UE codeword from $\hat{\mathbf{x}}_{\text{mu}}$,
apply the MED rule according to (44).

The variables $\mathbf{z}_{0j}$ and $\mathbf{u}_{j}$ can be updated
independently in parallel for each $j=1,\ldots,J$. Following a similar
sequence of steps as discussed in Section II-C, the steps in (36), (37), and
(38) are simplified as

$\displaystyle\mathbf{z}_{0j}^{t+1}:=$
$\displaystyle\operatorname*{argmin}_{\mathbf{z}_{0j}\in\tilde{\mathcal{X}}_{\text{mu}}^{JN_{\rm
e}}}\frac{\gamma_{j}}{2}\|\mathbf{z}_{0j}\|_{2}^{2}+\frac{\rho}{2}\|\mathbf{z}_{0j}-\mathbf{z}_{0j}^{t}-\bar{\mathbf{x}}_{0}^{t}+\mathbf{u}^{t}+\bar{\mathbf{z}}_{0}^{t}\|_{2}^{2}$
(39) $\displaystyle\bar{\mathbf{x}}_{0}^{t+1}:=$
$\displaystyle\operatorname*{argmin}_{\bar{\mathbf{x}}_{0}}\|\mathbf{r}-J\mathbf{H}\bar{\mathbf{x}}_{0}\|_{2}^{2}+\frac{\rho
J}{2}\|\bar{\mathbf{x}}_{0}-\bar{\mathbf{x}}_{1}^{t+1}-\mathbf{u}^{t}\|_{2}^{2}$
(40) $\displaystyle\mathbf{u}^{t+1}:=$
$\displaystyle\mathbf{u}^{t}+\bar{\mathbf{z}}_{0}^{t+1}-\bar{\mathbf{x}}_{0}^{t+1}$
(41)

where $\bar{\mathbf{x}}_{0}=\frac{1}{J}\sum_{j=1}^{J}\mathbf{x}_{0j}$ and
$\bar{\mathbf{z}}_{0}=\frac{1}{J}\sum_{j=1}^{J}\mathbf{z}_{0j}$. The solutions
obtained by solving (39) and (40) are as follows

$\displaystyle\mathbf{z}_{0j}^{t+1}$
$\displaystyle=\prod_{[-\alpha_{j},\alpha_{j}]\&[-\beta_{j},\beta_{j}]}\frac{\rho}{(\rho+\gamma_{j})}(\mathbf{z}_{0j}^{t}+\bar{\mathbf{x}}_{0}^{t}-\mathbf{u}^{t}-\bar{\mathbf{z}}_{0}^{t}),\forall
j=1,\ldots,J.$ (42) $\displaystyle\bar{\mathbf{x}}_{0}^{t+1}$
$\displaystyle=(\mathbf{H}^{H}\mathbf{H}J+\rho\mathbf{I})^{-1}(\mathbf{H}^{H}\mathbf{r}+\rho(\bar{\mathbf{z}}_{0}^{t+1}+\mathbf{u}^{t})),$
(43)

where $\prod_{[-\alpha_{j},\alpha_{j}]\&[-\beta_{j},\beta_{j}]}(.)$ denotes
the projection of the real part of each entry of the vector onto
$[-\alpha_{j},\alpha_{j}]$ and imaginary part of each entry of the vector onto
$[-\beta_{j},\beta_{j}]$. The step in (42) can be solved in parallel, for
$j=1,\ldots,J$. $\mathbf{I}$ is identity matrix of size $Jd_{\mathrm{v}}\times
Jd_{\mathrm{v}}$ and $JK\times JK$ for SCMA and DCMA, respectively. From the
definition of $\bar{\mathbf{x}}_{0}$ given above, the estimated multi-user
codeword is given by, $\hat{\mathbf{x}}_{\text{mu}}=J\bar{\mathbf{x}}_{0}$.
Each UE’s concatenated sparse codeword $\hat{\mathbf{x}}_{0j}$ can be
extracted from the $\hat{\mathbf{x}}_{\text{mu}}$ . Let
$\tilde{\mathbf{x}}_{0j}$ be the $j$th UE codeword after removing zeros from
$\hat{\mathbf{x}}_{0j}$. The minimum Euclidean distance (MED) rule is applied
to detect the transmitted codeword index corresponding to each UE,

$\displaystyle\hat{p}_{j}=\operatorname*{argmin}_{p}\|\tilde{\mathbf{x}}_{0j}-\textbf{x}_{j,p}\|,\quad
j=1,\ldots,J$ (44)

where $\textbf{x}_{j,p}$ represents $p$th codeword of the $j$th UE codebook
$\textbf{x}_{j,p}\in\mathcal{X}_{j}^{K\times M}$. Note that for SCMA, the
zeros from $\textbf{x}_{j,p}$ are removed to find $\hat{p}_{j}$. Algorithm 1
details the steps for detection.

####  III-D2 SMX-CD-NOMA

The detector of SMX-CD-NOMA needs to detect the signals transmitted from
$N_{\rm t}$ antennas of $J$ UEs. Here, the detection is more complex than the
SIMO case. The resource-wise processing is adapted to simplify the ADMM-based
detection problem. Recall from Section III-B, that the SMX codeword is given
by, $\mathbf{x}_{{\rm{smx}},k}=\sum_{j=1}^{N_{\mathrm{u}}}\mathbf{x}_{0j,k}$.
The ADMM processing first estimates the resource-wise spatial multiplexed
transmitted vector $\hat{\mathbf{x}}_{\text{smx},k}$, for $k=1,\ldots,K$.
Then, the transmitted codeword indices are detected via the MED rule. The ML
detection problem is converted into sharing problem as follows

Algorithm 2 SMX CD-NOMA detection via sharing ADMM

1:Input: $N_{0},E_{\mathrm{s}}$ and initialize
$\\{\mathbf{z}_{0j,k}\\}_{j=1}^{N_{u}},\bar{\mathbf{x}}_{0,k},\mathbf{u},\bar{\mathbf{z}}_{0,k}$
with zero vectors.

2:Output: $\bar{\mathbf{x}}_{0}^{(T)}$.

3:From (47) and (48), obtain optimal solutions (49) and (50) respectively.

4:Preprocessing

5:$\rho=N_{0}/E_{\mathrm{s}}$

6:$\mathbf{G}=(\mathbf{H}_{k}^{H}\mathbf{H}_{k}J+\rho\mathbf{I})^{-1}$

7:$\mathbf{M}=\mathbf{H}_{k}^{H}\mathbf{r}_{k}$

8:for $t=1,2,,\ldots T$ do

9: Step:1 Update $\\{\mathbf{z}_{0j,k}^{t+1}\\}_{j=1}^{J}$ in parallel via
(49) and compute $\bar{\mathbf{x}_{1}}$.

10: Step:2

11: Update $\bar{\mathbf{x}}_{0,k}^{t+1}$ via (50).

12: Update $\mathbf{u}^{t+1}$ via (48)

13:end for

14:Compute $\hat{\mathbf{x}}_{\text{smx},k}=J\bar{\mathbf{x}}_{0,k}^{(T)}$
(for DCMA), and
$\hat{\mathbf{x}}_{\text{smx},k}=d_{\mathrm{f}}\bar{\mathbf{x}}_{0,k}^{(T)}$
(for SCMA).

15:After extracting $\hat{\mathbf{x}}_{j,N_{\mathrm{t}}}$, apply the MED rule
according to (52).

$\displaystyle\min_{\mathbf{x}_{0j,k}}\quad$
$\displaystyle\|\mathbf{r}_{k}-\mathbf{H}_{k}(\sum_{j=1}^{N_{\mathrm{u}}}\mathbf{x}_{0j,k})\|_{2}^{2}+\sum_{j=1}^{N_{\mathrm{u}}}\frac{\gamma_{j}}{2}\|\mathbf{x}_{0j,k}\|_{2}^{2}$
(45) s.t.
$\displaystyle\mathbf{x}_{0j,k}\in\tilde{\mathcal{X}}_{j,\text{smx}}^{N_{\mathrm{u}}N_{\mathrm{t}}}$

where $\gamma_{j}\geq 0$ is the penalty parameter. By introducing alternative
variable $\mathbf{z}_{0j,k}=\mathbf{x}_{0j,k}$, the ADMM steps for (45) are as
follows:

$\displaystyle\mathbf{z}_{0j,k}^{t+1}:=$
$\displaystyle\operatorname*{argmin}_{\mathbf{z}_{0j,k}\in\tilde{\mathcal{X}}_{j,\text{smx}}^{N_{\mathrm{u}}N_{\mathrm{t}}}}\frac{\gamma_{j}}{2}\|\mathbf{z}_{0j,k}\|_{2}^{2}+\frac{\rho}{2}\|\mathbf{z}_{0j,k}-\mathbf{x}_{0j,k}^{t}+\mathbf{u}_{j}^{t}\|_{2}^{2}$
(46) $\displaystyle\mathbf{x}_{0j,k}^{t+1}:=$
$\displaystyle\operatorname*{argmin}_{\mathbf{x}_{0j,k}}\|\mathbf{r}_{k}-\mathbf{H}_{k}\left(\sum_{j=1}^{N_{\mathrm{u}}}\mathbf{x}_{0j,k}\right)\|_{2}^{2}+\frac{\rho}{2}\sum_{j=1}^{J}\|\mathbf{z}_{0j,k}^{t+1}-\mathbf{x}_{0j,k}+\mathbf{u}_{j}^{k}\|_{2}^{2}$
(47) $\displaystyle\mathbf{u}_{j}^{t+1}:=$
$\displaystyle\mathbf{u}_{j}^{k}+\left(\mathbf{z}_{0j,k}^{t+1}-\mathbf{x}_{0j,k}^{t+1}\right).$
(48)

The solutions for (46) and (47) are as follows:

$\displaystyle\mathbf{z}_{0j,k}^{t+1}=$
$\displaystyle\prod_{[-\alpha_{j},\alpha_{j}]\&[-\beta_{j},\beta_{j}]}\frac{\rho}{(\rho+\gamma_{j})}(\mathbf{z}_{0j,k}^{t}+\bar{\mathbf{x}}_{0,k}^{t}-\mathbf{u}^{t}-\bar{\mathbf{z}}_{0,k}^{t})$
(49) $\displaystyle\bar{\mathbf{x}}_{0,k}^{t+1}=$
$\displaystyle(\mathbf{H}_{k}^{H}\mathbf{H}_{k}N_{\mathrm{u}}+\rho\mathbf{I})^{-1}(\mathbf{H}_{k}^{H}\mathbf{r}_{k}+\rho(\bar{\mathbf{z}}_{0,k}^{t+1}+\mathbf{u}^{t}))$
(50)

where $\mathbf{I}$ is identity matrix of size
$N_{\mathrm{u}}N_{\mathrm{t}}\times N_{\mathrm{u}}N_{\mathrm{t}}$ and the
definitions of $\bar{\mathbf{x}}_{0,k}$ and $\bar{\mathbf{z}}_{0,k}$ are given
below

$\displaystyle\bar{\mathbf{x}}_{0,k}=$
$\displaystyle\frac{1}{N_{\mathrm{u}}}\sum_{j=1}^{N_{\mathrm{u}}}\mathbf{x}_{0j,k}\quad\quad\bar{\mathbf{z}}_{0,k}=\frac{1}{N_{\mathrm{u}}}\sum_{j=1}^{N_{\mathrm{u}}}\mathbf{z}_{0j,k}.$
(51)

From the definition of $\bar{\mathbf{x}}_{0,k}$, the estimated codeword
corresponding to the $k$th RE is given by
$\hat{\mathbf{x}}_{\text{smx},k}=N_{\mathrm{u}}\bar{\mathbf{x}}_{0,k}$. The
sparse vector $\hat{\mathbf{x}}_{0j,k}$ corresponding to $j$th UE can be
extracted from $\hat{\mathbf{x}}_{\text{smx},k}$. Let
$\tilde{\mathbf{x}}_{j,k}$ be a $N_{\mathrm{t}}\times 1$ vector formed after
removing the zeros from sparse vector $\hat{\mathbf{x}}_{0j,k}$. Let
$\hat{\mathbf{x}}_{j}^{n_{\mathrm{t}}}$ be the estimated codeword of $j$th UE
corresponding to $n_{\mathrm{t}}$th transmit antenna.
$\hat{\mathbf{x}}_{j}^{n_{\mathrm{t}}}$ is formed after resource-wise
processing is finished. MED rule is applied to detect the transmitted codeword
index corresponding to each UE and transmit antenna as follows:

$\displaystyle\hat{p}_{j}^{n_{\mathrm{t}}}=\operatorname*{argmin}_{p}\|\hat{\mathbf{x}}_{\mathrm{j}}^{n_{\mathrm{t}}}-\mathbf{x}_{j,p}\|,\quad
j=1,\ldots,J,\hskip 5.69054ptn_{\mathrm{t}}=1,\ldots,N_{\mathrm{t}}.$ (52)

where $\mathbf{x}_{j,p}$ represents the $p$th codeword of the $j$th UE
codebook $\mathcal{X}_{j}^{K\times M}$. The detailed ADMM-based detection
procedure is given in Algorithm 2.

####  III-D3 SM-CD-NOMA

This section discusses the formulation of the ADMM-based detection problem for
the SM-CD-NOMA system. The modulation happens in both the signal domain and
the spatial domain. The SM-CD-NOMA system transmits information on the
codeword and active antenna indices simultaneously. The receiver needs to
estimate these two quantities. The variable $\mathbf{x}_{\text{sm},k}$ from
Section III-A, is given by
$\mathbf{x}_{\text{sm},k}=\sum_{j=1}^{N_{\mathrm{u}}}\mathbf{x}_{0j,k}^{n_{\rm
a}^{j}}$. The proposed method estimates the resource-wise spatial modulated
transmitted vector $\hat{\mathbf{x}}_{\text{sm},k}$ for $k=1,\ldots,K$ via
ADMM processing. Then, the $L$1-norm rule is applied to detect the antenna
index, and the MED rule is applied to detect the transmitted codeword index.
The sharing problem of the SM-CD-NOMA system is similar to the problem in
(45). Following the steps (46), (47), and (48), the sharing ADMM problem of
SM-CD-NOMA can be solved. The solutions obtained for the sharing ADMM problem
are similar to (49) and (50) and are repeated here for clarity:

$\displaystyle\mathbf{z}_{0j,\text{sm},k}^{t+1}=$
$\displaystyle\prod_{[-\alpha_{j},\alpha_{j}]\&[-\beta_{j},\beta_{j}]}\frac{\rho}{(\rho+\gamma_{j})}(\mathbf{z}_{0j,\text{sm},k}^{t}+\bar{\mathbf{x}}_{0,\text{sm},k}^{t}-\mathbf{u}^{t}-\bar{\mathbf{z}}_{0,\text{sm},k}^{t})$
(53) $\displaystyle\bar{\mathbf{x}}_{0,\text{sm},k}^{t+1}=$
$\displaystyle(\mathbf{H}_{k}^{H}\mathbf{H}_{k}N_{\mathrm{u}}+\rho\mathbf{I})^{-1}(\mathbf{H}_{k}^{H}\mathbf{r}_{k}+\rho(\bar{\mathbf{z}}_{0,\text{sm},k}^{t+1}+\mathbf{u}^{t})).$
(54)

The definitions of $\bar{\mathbf{x}}_{0,\text{sm},k}$ and
$\bar{\mathbf{z}}_{0,\text{sm},k}$ are similar to (51). The estimated SM
codeword corresponding to $k$th RE is
$\hat{\mathbf{x}}_{\text{sm},k}=N_{\mathrm{u}}\bar{\mathbf{x}}_{0,\text{sm},k}$.
After resource-wise processing, the structure of the transmitted codewords for
$J$ UEs can be retrieved. For Example 3, the detected codewords for $6$ UEs is
given by

$\displaystyle\hat{\mathbf{x}}_{\text{tx}}=\begin{bmatrix}\hat{\mathbf{x}}_{1}^{1}&\hat{\mathbf{x}}_{2}^{1}&\hat{\mathbf{x}}_{3}^{1}&\hat{\mathbf{x}}_{4}^{1}&\hat{\mathbf{x}}_{5}^{1}&\hat{\mathbf{x}}_{6}^{1}\\\
\hat{\mathbf{x}}_{1}^{2}&\hat{\mathbf{x}}_{2}^{2}&\hat{\mathbf{x}}_{3}^{2}&\hat{\mathbf{x}}_{4}^{2}&\hat{\mathbf{x}}_{5}^{2}&\hat{\mathbf{x}}_{6}^{2}\\\
\end{bmatrix}_{N_{\mathrm{t}}K\times J(8\times 6)}$

where $\hat{\mathbf{x}}_{j}^{n_{\mathrm{t}}}$ indicates the estimated codeword
of $j$th UE over the $n_{\mathrm{t}}$th transmit antenna. In the presence of
noise, the detected active transmit antenna index is given by

$\displaystyle\hat{n}_{\rm
a}^{j}=\operatorname*{argmax}_{n_{\mathrm{t}}}(\|\hat{\mathbf{x}}_{j}^{\mathrm{n}_{\mathrm{t}}}\|_{1}),\quad\forall
j$ (55)

where $\|(\cdot)\|_{1}$ denotes the $L$1-norm of $(\cdot)$. Let
$\hat{\mathbf{x}}_{j}^{\hat{n}_{\rm a}^{j}}$ be the estimated codeword of the
$j$th UE transmitted from $\hat{n}_{\rm a}^{j}$ active antenna. The MED rule
is applied to detect the codeword index corresponding to each UE

$\displaystyle\hat{p}_{j}=\operatorname*{argmin}_{p}\|\hat{\mathbf{x}}_{j}^{\hat{n}_{a}^{j}}-\mathbf{x}_{j,p}\|,\quad\forall
j.$ (56)

Similar steps of Algorithm 2 are followed in the detection of the SM-CD-NOMA
system.

##  IV Computational Complexity

This section analyses the computational complexity of the SIMO-CD-NOMA system.
The detection algorithm’s computational complexity determines its practical
viability. The number of FLOPs (mainly complex multiplications) is a useful
metric to analyze the complexity of the detector [12]. The number of
calculations in the proposed CD-NOMA detection via ADMM consists of two parts.
Part-1 is iteration-independent (Pre-processing) steps, described in lines 6
and 7 in Algorithm 1, and Part-2 is iteration-dependent steps, from lines 9 to
12. The calculations in Part-1 are performed only once, i.e., before the ADMM
iterations. In the SIMO-CD-NOMA system, Part-1 contains three steps of
calculations such as $\mathbf{H}^{H}\mathbf{H}$,
$(\mathbf{H}^{H}\mathbf{H}+\rho\mathbf{I})^{-1}$, and
$\mathbf{H}^{H}\mathbf{r}$. The size of $\mathbf{H}$ is $N_{\mathrm{r}}K\times
Jd_{\mathrm{v}}$ and $N_{\mathrm{r}}K\times JK$ for SCMA and DCMA,
respectively. Further, the numbers of FLOPs required to perform these three
steps over the SCMA system are
$(N_{\mathrm{r}}K)(Jd_{\mathrm{v}})^{2},(Jd_{\mathrm{v}})^{3}$ and
$(N_{\mathrm{r}}K)(Jd_{\mathrm{v}})$. The numbers of FLOPs required to perform
the same steps over the DCMA system are $(N_{\mathrm{r}}K)(JK)^{2},(JK)^{3}$,
and $(N_{\mathrm{r}}K)(JK)$. The calculations in Part-2 need to be repeated in
every iteration and contain mainly two steps. For SCMA system, these steps
involve scalar multiplication of $Jd_{\mathrm{v}}\times 1$ vector for $J$ UEs
parallelly in (42), multiplication of $Jd_{\mathrm{v}}\times Jd_{\mathrm{v}}$
matrix with $Jd_{\mathrm{v}}\times 1$ and scalar multiplication with
$Jd_{\mathrm{v}}\times 1$ vector in (43), in total
$J^{2}d_{\mathrm{v}}+(Jd_{\mathrm{v}})^{2}+Jd_{\mathrm{v}}$ FLOPs
approximately. For DCMA system, $(J^{2}K+(JK)^{2}+JK)$ FLOPs are required. The
approximate total computational cost to implement the ADMM-based detector over
the SIMO-SCMA system and SIMO-DCMA system are
$(N_{\mathrm{r}}K)(Jd_{\mathrm{v}})^{2}+(Jd_{\mathrm{v}})^{3}+(N_{\mathrm{r}}K)(Jd_{\mathrm{v}})+T(J^{2}d_{\mathrm{v}}+(Jd_{\mathrm{v}})^{2}+Jd_{\mathrm{v}})$
and
$(N_{\mathrm{r}}K)(JK)^{2}+(JK)^{3}+(N_{\mathrm{r}}K)(JK)+T(J^{2}K+(JK)^{2}+JK)$
respectively. TABLE I compares the total complexity of the sharing ADMM-based
detection problem with other known detection schemes. In TABLE I, NA stands
for ‘Not applicable’.

MPA is widely used for SCMA systems, and its complexity is exponential with
$M$ and $d_{\mathrm{f}}$, as mentioned in TABLE I [12]. The computational
burden will grow further for large-scale SCMA systems. Further, due to
exponential complexity, MPA is not feasible in large-scale MIMO systems.
Additionally, the lack of sparsity in DCMA makes MPA impractical for
detection.

A single tree search (STS) based soft-in soft-out (SISO) GSD [29] is applied
to detect the signals of a spreading sequence-based DCMA system, also known as
an overloaded CDMA system [2]. GSD performs key pre-processing steps to
convert the rank deficient system into full rank one [30]. These steps include
$\mathbf{H}^{H}\mathbf{H}$ in
$(\mathbf{Q}=\mathbf{H}^{H}\mathbf{H}+\lambda\mathbf{I}_{J})$, Cholesky
decomposition $\mathbf{Q}=\mathbf{D}^{H}\mathbf{D}$, and
$(\mathbf{H}\mathbf{D}^{-1})^{H}\mathbf{r}$. These steps require
$(N_{\mathrm{r}}K)J^{2}$, $J^{3}/3$, and
$(J^{3}+(N_{\mathrm{r}}K)J^{2}+JN_{\mathrm{r}}K)$ FLOPs, respectively. The
rank-deficient linear system is thus converted into a full-rank one, and
standard sphere decoding (SD) can be readily applied. The SD performs pre-
processing steps, including QR decomposition and $\mathbf{Q}^{H}\mathbf{r}$.
Further, the numbers of FLOPs to perform these steps are $J^{3}$ and $2J^{2}$,
respectively. The major complexity of the SD lies in the tree search algorithm
[31]. The expected complexity of SD is
$E_{\text{FLOPS}}=\sum_{j=1}^{J}f_{p}(j)N_{j}$, where $N_{j}$ is the average
number of nodes visited in level-$j$ of the tree and $f_{p}(j)$ is the number
of FLOPs needed in level-$j$. It is given in [31], $f_{p}(j)=2j+11$, and
$N_{j}$ is roughly cubic in the number $J$ of unknowns to be solved. Fig. 6
depicts the computational complexity comparison of various detection schemes
for different CD-NOMA systems with
$N_{\mathrm{r}}=4,K=4,J=6,d_{\mathrm{v}}=2$, and $M=4$. It can be observed
that the complexity of ADMM-based detector is almost equal to that of the low-
complexity MMSE detector.

Table I: Computational complexity of different detectors for SIMO CD-NOMA systems Detector | Spreading sequence based | Codebook based
---|---|---
| LDS |  DCMA |  SCMA |  DCMA
MPA | NA |  NA |  $(Kd_{\mathrm{f}}^{2}M^{d_{\mathrm{f}}}N_{\mathrm{r}}+Nd_{\mathrm{f}}Md_{\mathrm{v}})T$ |  NA
GSD | NA |  $(N_{\mathrm{r}}K)J^{2}+J^{3}/3+(J^{3}+(N_{\mathrm{r}}K)J^{2}+JN_{\mathrm{r}}K)$ \+ $J^{3}+2J^{2}+E_{\text{FLOPS}}$ |  NA |  NA
MMSE | NA |  $(N_{\mathrm{r}}K)J^{2}+J^{3}+(N_{\mathrm{r}}K)J$ |  $(N_{\mathrm{r}}K)(Jd_{\mathrm{v}})^{2}+(Jd_{\mathrm{v}})^{3}+(N_{\mathrm{r}}K)(Jd_{\mathrm{v}})$ |  $(N_{\mathrm{r}}K)(JK)^{2}+(JK)^{3}+(N_{\mathrm{r}}K)(JK)$
Sharing ADMM | NA |  $(N_{\mathrm{r}}K)J^{2}+J^{3}$\+ $(N_{\mathrm{r}}K)J+T(2J^{2}+J)$ |  $N_{\mathrm{r}}K(Jd_{\mathrm{v}})^{2}+(Jd_{\mathrm{v}})^{3}$\+ $(N_{\mathrm{r}}K)(Jd_{\mathrm{v}})+T(J^{2}d_{\mathrm{v}}+(Jd_{\mathrm{v}})^{2}+Jd_{\mathrm{v}})$ |  $N_{\mathrm{r}}K(JK)^{2}+(JK)^{3}+(N_{\mathrm{r}}K)(JK)$ $+T(J^{2}K+(JK)^{2}+JK)$
SCMADCMASpreading sequence based
DCMA$0$$5{,}000$$10{,}000$$15{,}000$$20{,}000$$25{,}000$Number of
FLOPSADMMMPAMMSEGSD Figure 6: Computational complexity comparison of various
detectors.

##  V Simulations and Discussions

This section presents wide-ranging simulation results of various MIMO-CD-NOMA
systems, including SER performance and selection of ADMM parameters
$\left(T,\\{\gamma\\}_{j=1}^{J}\right)$. The simulations are performed for
various MIMO-CD-NOMA systems described in Section III with varying parameters
$\lambda,M,N_{\mathrm{r}}$, and $N_{\mathrm{t}}$. Further, the performance of
the ADMM-based detector is compared with that of the MPA detector with ten
iterations ($T=10$) in the case of the SCMA system. The DCMA and overloaded
CDMA systems’ detection is carried out using the MMSE and the GSD detectors,
respectively. The codebooks designed in [32] and [2] are considered, for SCMA
and DCMA, respectively. Table II further details the simulation parameters.
The sample simulation codes for this work are available at
https://github.com/vikas2020-del/ADMM-based-detector-for-NOMA.

Parameter | Value
---|---
Modulation order ($M$) | $4,8,16$
Number of UEs ($J$) | $6,10$
Number of resources ($K$) | $4,5$
Number of transmit antenna ($N_{\mathrm{t}}$) | $2,4,8$
Number of receive antenna ($N_{\mathrm{r}}$) | $4,8,32,64$
Overloading factor ($\lambda$) | $150\leavevmode\nobreak\ \%,200\leavevmode\nobreak\ \%$
Detectors | ADMM, MPA, MMSE, GSD
Number of iterations ($T$) | 30 (ADMM), 10 (MPA)
Table II: Simulation parameters.

###  V-A SER performance of SIMO CD-NOMA system

Fig. 7(a) and Fig. 7(b) show the SER performance of SIMO-SCMA system for
$150\leavevmode\nobreak\ \%$ and $200\leavevmode\nobreak\ \%$ overloading,
respectively.

$-4$$-2$$0$$2$$4$$10^{-6}$$10^{-5}$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$E_{b}/N_{0}$
(dB)SER$N_{\mathrm{r}}$=16, $M$=8$N_{\mathrm{r}}$=4, $M$=4$N_{\mathrm{r}}$=4,
$M$=8$N_{\mathrm{r}}$=8, $M$=8$N_{\mathrm{r}}$=8, $M$=4$N_{\mathrm{r}}$=4,
$M$=4$N_{\mathrm{r}}$=8, $M$=4$\lambda=150\leavevmode\nobreak\ \%$

(a) SER vs. SNR performance for $\lambda=150\leavevmode\nobreak\ \%$.

$-5$$0$$5$$10$$15$$10^{-7}$$10^{-6}$$10^{-5}$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$E_{b}/N_{0}$
(dB)SER$N_{\mathrm{r}}=4,M=4$$N_{\mathrm{r}}=8,M=4$$N_{\mathrm{r}}=8,M=8$$N_{\mathrm{r}}=4,M=8$$N_{\mathrm{r}}=16,M=8$$N_{\mathrm{r}}=4,M=4$$N_{\mathrm{r}}=8,M=4$$\lambda=200\leavevmode\nobreak\
\%$

(b) SER vs. SNR performance for $\lambda=200\leavevmode\nobreak\ \%$.

Figure 7: SER performance SIMO SCMA system. Solid and dotted lines denote the
ADMM and MMSE performance, respectively.

Each curve represents the SER performance for a specific number
$N_{\mathrm{r}}$ of the receive antennas at the BS and codebook size $M$. Fig.
7(a) and Fig. 7(b) show that the SER performance improves as $N_{\mathrm{r}}$
increases. Thus, the proposed ADMM-based detector is able to exploit the
diversity gain provided by the SIMO-CD-NOMA system. However, for $M=8$, the
SER performance is slightly worse than $M=4$ due to the inevitable decrease in
the minimum Euclidean and minimum product distance [32]. Fig. 7(a) and Fig.
7(b) also compare the SER performance of the MMSE and ADMM-based detectors.
For a small-scale SCMA system ($M=4$, $\lambda=150\leavevmode\nobreak\ \%$),
the performance of the MMSE detector is close to that of the ADMM-based
detector. However, for $\lambda=200\leavevmode\nobreak\ \%$, the MMSE
performance is inferior to ADMM. Observe from (15) that as the number $J$ of
UEs increases, the number $Jd_{v}$ of unknowns approaches the number $KN_{r}$
of observations. The MMSE performance degrades when the $KN_{r}$ to $Jd_{v}$
ratio approaches unity. ADMM provides approximately 2 dB SNR gain at
$\text{SER}=10^{-3}$ over the MMSE detector for 200 % overloaded SCMA system,
as observed from Fig. 7(b). Fig. 8(a) and Fig. 8(b) show the SER performance
of $M=4$ and $M=16$ overloaded SIMO-DCMA systems, respectively. Fig. 8(a)
compares the SER performance of the MMSE and ADMM-based detectors. ADMM
provides approximately 2 dB SNR gain at $\text{SER}=10^{-3}$ over the MMSE
detector for the 200 % overloaded DCMA system, as observed from Fig. 8(a).
Therefore, ADMM is a more suitable detector compared with MMSE for 200 %
overloaded CD-NOMA systems. Furthermore, simulation attempts indicate that
MMSE cannot detect the CD-NOMA signals for $M=8$ and $M=16$ size codebooks.
Therefore, these plots are not shown in Fig. 7 and Fig. 8.

$-4$$-2$$0$$2$$4$$6$$8$$10$$10^{-6}$$10^{-5}$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$E_{\mathrm{b}}/N_{0}$(dB)SER$200\leavevmode\nobreak\
\%$$150\leavevmode\nobreak\ \%$$150\leavevmode\nobreak\
\%$$150\leavevmode\nobreak\ \%$$200\leavevmode\nobreak\
\%$$N_{\mathrm{r}}$=$8$$N_{\mathrm{r}}$=$16$

(a) Codebook size $M=4$.

$0$$5$$10$$15$$20$$10^{-7}$$10^{-6}$$10^{-5}$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$E_{\mathrm{b}}/N_{0}$
(dB)SER$200\leavevmode\nobreak\ \%$$150\leavevmode\nobreak\
\%$$150\leavevmode\nobreak\ \%$$N_{\mathrm{r}}=16$$N_{\mathrm{r}}$=$8$

(b) Codebook size $M=16$.

Figure 8: SER performance of SIMO DCMA system for 150 % and 200 % overloading.
Solid and dotted lines denote the ADMM and MMSE performance, respectively.

###  V-B SER performance of SMX CD-NOMA systems

Fig. 9(a) and Fig. 9(b) show the SER performance of SMX-SCMA system for
$150\leavevmode\nobreak\ \%$ and $200\leavevmode\nobreak\ \%$ overloading,
respectively. Observe from Fig. 9(a) and Fig. 9(b), that the higher the value
of $N_{\mathrm{r}}$, the better is the SER performance. As the value of
$N_{\mathrm{t}}$ increases, $N_{\mathrm{r}}$ is increased to maintain
$N_{\mathrm{r}}>d_{\mathrm{f}}N_{\mathrm{t}}$ in (19). The improvement in the
SER performance is maximum for $N_{\mathrm{r}}=64$ case due to more
observations at the BS. This improvement comes at the cost of a marginal
computational complexity increment (TABLE I) in ADMM detection. Fig. 9(c)
depicts the SER performance of SMX-DCMA systems. For SCMA, $N_{\rm u}=d_{\rm
f}$ UEs overlap on each RE. On the other hand, for DCMA $N_{\rm
u}=J\;(J>d_{\rm f})$ UEs overlap on each RE, as shown in Fig. 1(b). This
significant overlapping increases the number of unknowns in a DCMA system
compared to the SCMA one, as observed in (19). To maintain the number of
observations higher than the number of unknowns
($N_{\mathrm{r}}>JN_{\mathrm{t}}$), DCMA requires larger $N_{\mathrm{r}}$ at
the BS as compared to SCMA for a fixed $N_{\mathrm{t}}$. As a result, the
computational complexity of DCMA detection significantly increases. The tree
search-based SD algorithms are highly complex in codebook-based DCMA, as
explained in Section IV. Further, the ADMM-based detector shows similar
complexity as that of MMSE, as shown in Fig. 6. Therefore, compared with MMSE,
the ADMM-based detector performs well with almost the same complexity.

$-15$$-10$$-5$$0$$5$$10$$15$$10^{-6}$$10^{-4}$$10^{-2}$$10^{0}$$E_{\mathrm{b}}/N_{0}$
(dB)SER$N_{\mathrm{r}}=16,N_{\mathrm{t}}=4$$N_{\mathrm{r}}=16,N_{\mathrm{t}}=4$$N_{\mathrm{r}}=8,N_{\mathrm{t}}=2$$N_{\mathrm{r}}=8,N_{\mathrm{t}}=2$$N_{\mathrm{r}}=64,N_{\mathrm{t}}=8$$N_{\mathrm{r}}=32,N_{\mathrm{t}}=8$$\lambda$=$150\leavevmode\nobreak\
\%$$M$=$4$$M$=$8$

(a) SMX-SCMA with $\lambda=150\leavevmode\nobreak\ \%$

$-10$$0$$10$$20$$10^{-7}$$10^{-5}$$10^{-3}$$10^{-1}$$10^{1}$$E_{\mathrm{b}}/N_{0}$
(dB)SER$N_{\mathrm{r}}$=8, $N_{\mathrm{t}}$=2$N_{\mathrm{r}}$=8,
$N_{\mathrm{t}}$=2$N_{\mathrm{r}}$=16, $N_{\mathrm{t}}$=4$N_{\mathrm{r}}$=16,
$N_{\mathrm{t}}$=4$N_{\mathrm{r}}$=32, $N_{\mathrm{t}}$=8$N_{\mathrm{r}}$=64,
$N_{\mathrm{t}}$=8$\lambda$=$200\leavevmode\nobreak\ \%$$M$=$4$$M$=$8$

(b) SMX-SCMA with $\lambda=200\leavevmode\nobreak\ \%$

$-15$$-10$$-5$$0$$5$$10$$15$$10^{-6}$$10^{-5}$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$E_{\mathrm{b}}/N_{0}$
(dB)SER$N_{\mathrm{r}}=16,\lambda=150\leavevmode\nobreak\
\%$$N_{\mathrm{r}}=32,\lambda=200\leavevmode\nobreak\
\%$$N_{\mathrm{r}}=32,\lambda=200\leavevmode\nobreak\
\%$$N_{\mathrm{r}}=16,\lambda=150\leavevmode\nobreak\
\%$$N_{\mathrm{t}}$=$2$$M$=$4$$M$=$16$

(c) SMX-DCMA

Figure 9: SER vs. SNR performance of SMX-CD-NOMA system by varying
$N_{\mathrm{t}},N_{\mathrm{r}},M$ using the ADMM-based detector.

###  V-C SER performance of SM CD-NOMA systems

Fig. 10(a) and Fig. 10(b) show the SER performance of SM-SCMA system for
$150\leavevmode\nobreak\ \%$ and $200\leavevmode\nobreak\ \%$ overloading,
respectively. The performance of the ADMM-based detector mainly depends on
$N_{\mathrm{t}},N_{\mathrm{r}}$, and $M$, as we have seen in the previous
subsections. The $M=4$ size codebook exhibits superior performance over $M=8$
due to its better distance properties (Euclidean distance and product
distance). In SM-CD-NOMA systems, each UE’s codeword contains the active
antenna and codeword index information of that particular UE, as given in
(25). Fig. 10(a) and Fig. 10(b) depict the efficiency of ADMM in detecting the
information both in the signal domain and space domain. Note that the
codewords in (25) contain zero-column vectors to indicate zero power
transmission from the inactive antenna. As a result, the number of
observations goes below the number of unknowns
($N_{\mathrm{r}}<N_{\mathrm{u}}N_{\mathrm{t}}$). Thus, it leads to performance
loss for an ADMM-based detector. However, increasing the number of
observations ($N_{\mathrm{r}}$) at the BS can compensate for this loss.
Therefore, the performance of the ADMM-based detector can be maintained by a
marginal increment in the computational complexity as given in TABLE I. The
observations mentioned above for SMX-DCMA and SM-SCMA systems apply to SM-DCMA
systems, as shown in Fig. 10(c). Further,
$N_{\mathrm{r}}=32,\lambda=200\leavevmode\nobreak\ \%$ provides significantly
better performance than $N_{\mathrm{r}}=16,\lambda=150\leavevmode\nobreak\
\%$, as the former has more observations ($N_{\rm r}$) than the latter. So the
ADMM-based detector is more appropriate for highly overloaded DCMA systems
equipped with large-scale MIMO parameters.

$-15$$-10$$-5$$0$$5$$10$$15$$20$$10^{-6}$$10^{-5}$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$E_{\mathrm{b}}/N_{0}$
(dB)SER$N_{\mathrm{r}}=32,N_{\mathrm{t}}=8$$N_{\mathrm{r}}=16,N_{\mathrm{t}}=4$$N_{\mathrm{r}}=16,N_{\mathrm{t}}=4$$N_{\mathrm{r}}=32,N_{\mathrm{t}}=8$$N_{\mathrm{r}}=64,N_{\mathrm{t}}=8$$\lambda$=$150\leavevmode\nobreak\
\%$$M$=$4$$M$=$8$

(a) SM-SCMA with $\lambda=150\leavevmode\nobreak\ \%$

$-10$$0$$10$$20$$30$$10^{-7}$$10^{-6}$$10^{-5}$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$E_{\mathrm{b}}/N_{0}$
(dB)SER$N_{\mathrm{t}}$=16, $N_{\mathrm{t}}$=4$N_{\mathrm{r}}$=32,
$N_{\mathrm{t}}$=8$N_{\mathrm{r}}$=16, $N_{\mathrm{t}}$=4$N_{\mathrm{r}}$=32,
$N_{\mathrm{t}}$=8$N_{\mathrm{r}}$=64,
$N_{\mathrm{t}}$=8$\lambda$=$200\leavevmode\nobreak\ \%$$M$=$4$$M$=$8$

(b) SM-SCMA with $\lambda=200\leavevmode\nobreak\ \%$

$-15$$-10$$-5$$0$$5$$10$$15$$20$$10^{-6}$$10^{-5}$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$E_{\mathrm{b}}/N_{0}$
(dB)SER$N_{\mathrm{r}}$=16, $\lambda=150\leavevmode\nobreak\
\%$$N_{\mathrm{r}}$=16, $\lambda=150\leavevmode\nobreak\
\%$$N_{\mathrm{r}}$=32, $\lambda=200\leavevmode\nobreak\
\%$$N_{\mathrm{r}}$=32, $\lambda=200\leavevmode\nobreak\
\%$$M$=$4$$M=16$$N_{\mathrm{t}}$=$2$

(c) SM-DCMA

Figure 10: SER vs. SNR performance for SM-MIMO CD-NOMA system using the
proposed ADMM-based detector.

###  V-D SER performance: ADMM vs. MPA

Fig. 11(a) shows the SER performance comparison between the ADMM and MPA
detectors for the SIMO-SCMA system. Existing research shows that the MPA is a
near-optimal detection with high computational complexity [7]. MPA gives a
maximum SNR gain of around 2.5 dB over ADMM at $\text{SER}=10^{-2}$ for
$N_{\mathrm{r}}=4$ and $M=4$, as shown in Fig. 11(a). As the modulation order
$M$ increases, the performance of ADMM becomes close to MPA, as observed for
$N_{\mathrm{r}}=4,M=8$ in Fig. 11(a). Therefore, for $M=8$, codebook distance
properties influence MPA and ADMM-based detectors similarly. Further, for
$N_{\mathrm{r}}=8,M=8$, the ADMM-based detector shows approximately 2 dB SNR
gain at $\text{SER}=10^{-3}$ over the MPA detector. The increase in the number
of observations significantly improves the ADMM performance compared to MPA
for large-size codebooks (i.e. with large $M$).

$-4$$-2$$0$$2$$4$$10^{-6}$$10^{-5}$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$E_{\mathrm{b}}/N_{0}$
(dB)SER$N_{\mathrm{r}}=4,M=4$, ADMM$N_{\mathrm{r}}=4,M=8$,
ADMM$N_{\mathrm{r}}=8,M=8$, ADMM$N_{\mathrm{r}}=8,M=4$,
ADMM$N_{\mathrm{r}}=4,M=4$, MPA$N_{\mathrm{r}}=4,M=8$,
MPA$N_{\mathrm{r}}=8,M=8$, MPA$N_{\mathrm{r}}=8,M=4$, MPA

(a) SIMO SCMA: ADMM vs. MPA with $\lambda$=$150\leavevmode\nobreak\ \%$

$-4$$-2$$0$$2$$4$$10^{-7}$$10^{-6}$$10^{-5}$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$E_{\mathrm{b}}/N_{0}$
(dB)SER$N_{\mathrm{r}}=8,M=16$, GSD$N_{\mathrm{r}}=8,M=16$,
ADMM$N_{\mathrm{r}}=4,M=4$, ADMM$N_{\mathrm{r}}=4,M=4$,
GSD$\lambda$=$150\leavevmode\nobreak\ \%$

(b) SIMO Overloaded CDMA: ADMM vs. GSD

Figure 11: Comparison of SER performance using the proposed ADMM and the
conventional MPA and GSD detectors.

$-4$$-2$$0$$2$$10^{-5}$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$E_{\mathrm{b}}/N_{0}$
(dB)SERe=0 %e=5 %e=10 %e=20 %$\lambda$=$150\leavevmode\nobreak\ \%$

(a) SIMO SCMA system with $N_{\mathrm{r}}=8,M=4$

$-4$$-2$$0$$2$$4$$10^{-5}$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$E_{\mathrm{b}}/N_{0}$
(dB)SERe=0 %e=5 %e=10 %e=20 %$\lambda$=$150\leavevmode\nobreak\ \%$

(b) SIMO-DCMA system with $N_{\mathrm{r}}=8,M=4$

Figure 12: SER performance of CD-NOMA system in imperfect CSI for
$e=0\leavevmode\nobreak\ \%,e=5\leavevmode\nobreak\
\%,e=10\leavevmode\nobreak\ \%,e=20\leavevmode\nobreak\ \%$.

###  V-E SER performance: ADMM vs. GSD

Fig. 11(b) shows the SER performance comparison between ADMM and GSD detectors
for the SIMO-overloaded CDMA system. Observe from Fig. 11(b) that, at low SNR
regions, GSD outperforms the ADMM with a maximum of around 2 dB SNR gain at
SER=$10^{-4}$. As the SNR increases, GSD performance degrades compared to
ADMM. Unlike GSD, as $M$ increases, ADMM performance is enhanced by doubling
the number of received antennas as shown in Fig. 11(b). Note that ADMM
performance improves as the number of observations increases.

###  V-F SER performance: Imperfect channel state information (CSI)

All the above simulations are analyzed by considering perfect CSI at the
receiver. In practical scenarios, obtaining perfect CSI at the receiver is not
feasible due to channel estimation errors (CEEs). Thus, the proposed
detector’s performance in the presence of CEEs is an important metric to show
its practical feasibility. An imperfect channel can be modeled as [33]

$\displaystyle\hat{\mathbf{H}}=\mathbf{H}+e\ \mathbf{\Omega}$ (57)

where $\mathbf{\Omega}$ is the CEE and is considered to be uncorrelated with
$\mathbf{H}$. The entries of $\mathbf{\Omega}$ are independent and i.i.d.
complex Gaussian random variables with zero mean and unit variance, i.e.,
$\mathcal{CN}(0,1)$. The quantity $e$ in (57) determines the variance of the
CEE. Fig. 12 depicts the performance of the proposed ADMM-based detector for
the imperfect CSI with $e=0\leavevmode\nobreak\ \%,e=5\leavevmode\nobreak\
\%,e=10\leavevmode\nobreak\ \%$, and $e=20\leavevmode\nobreak\ \%$. The
simulations for SCMA and DCMA systems are shown in Fig. 12(a) and Fig. 12(b),
respectively. Observe that the impact of the CEE on the ADMM-based detector’s
performance is minimal. Therefore, the ADMM-based detector can be applied in
imperfect CSI scenarios.

###  V-G Convergence of ADMM and selection of parameters

The convergence analysis of the ADMM-based detector in the DCMA system is
performed based on the Monte Carlo simulations. Fig. 13(a) shows the impact of
ADMM iterations on the SER performance. The proposed detector exploits the
iterative nature of ADMM to converge.

$5$$10$$15$$20$$25$$10^{-7}$$10^{-6}$$10^{-5}$$10^{-4}$Number of iterations
($T$)SER$\frac{E_{\mathrm{b}}}{N_{0}}$=15dB$\frac{E_{\mathrm{b}}}{N_{0}}$=18dB$\frac{E_{\mathrm{b}}}{N_{0}}$=20dB

(a) Iterations vs. SER with $\lambda=150\leavevmode\nobreak\
\%,M=4,N_{\mathrm{r}}=4$

$0$$50$$100$$150$$200$$250$$300$$10^{-8}$$10^{-7}$$10^{-6}$$10^{-5}$$10^{-4}$$\gamma$SER$\frac{E_{\mathrm{b}}}{N_{0}}$=15dB$\frac{E_{\mathrm{b}}}{N_{0}}$=18dB$\frac{E_{\mathrm{b}}}{N_{0}}$=20dB

(b) $\gamma$ vs. SER with $\lambda=150\leavevmode\nobreak\
\%,M=4,N_{\mathrm{r}}=4$

Figure 13:  Impact of $T$ and $\gamma$ on the ADMM-based detector’s SER
performance.

The SER performance improves as the number of iterations ($T$) increases. The
improvement in SER becomes marginal after a certain number of iterations. For
all considered $\frac{E_{\mathrm{b}}}{N_{0}}$ values, ADMM converges after 15
iterations, as shown in Fig. 13(a). The penalty parameters
$\\{\gamma_{j}\\}_{j=1}^{J}$ and $\rho$ are selected to maximize the SER
performance. The augmented Lagrangian parameter $\rho$ is selected as the
reciprocal of SNR [19]. The impact of $\gamma$ on SER performance in the ADMM-
based detector is analyzed through Monte Carlo simulations, and the results
are given in Fig. 13(b). For low values of $\gamma$, the ADMM performance
degrades, and for large values, the variation in SER becomes inconsequential.
When $\gamma$ is close to zero, the impact of penalty terms is nullified.
Thus, the alternative variables in the penalty function lose importance in
minimizing the objective function. Observe from the Fig. 13(b) that the ADMM
performs better in the range $\\{\gamma_{j}\\}_{j=1}^{J}\in[50\hskip
5.69054pt100]$.

##  VI Conclusions and Future Scopes

This paper proposed new system models and a low-complexity iterative linear
detector for large-scale MIMO-CD-NOMA systems. The optimal ML detection is
converted into a sharing problem, which is efficiently solved via the ADMM
algorithm using distributed optimization framework. The proposed ADMM-based
detector enables the detection of large-scale MIMO-CD-NOMA systems with high
overloading factors ($\lambda$) and modulation orders ($M$) while maintaining
low complexity. By leveraging the proposed ADMM, the CD-NOMA systems achieve
significantly increased connectivity. Further, the impact of ADMM parameters,
such as the number of iterations and penalty parameters, is analyzed.
Exhaustive simulation results are presented to validate the effectiveness of
the proposed ADMM-based detector when compared to conventional detectors such
as MPA, GSD, and MMSE detectors. In addition, the results demonstrate that the
ADMM-based detector offers excellent performance with low complexity across
various CD-NOMA system variants. This paper has explored the concept of ADMM
detection specifically for uncoded CD-NOMA systems. Designing a soft decision
detector using the ADMM algorithm for coded CD-NOMA systems is an interesting
future work.

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