# Deep Learning-based Codebook Design for Code-Domain Non-Orthogonal Multiple
Access Approaching Single-User Bit-Error Rate Performance

Minsig Han, Hanchang Seo, Ameha T. Abebe and Chung G. Kang,

###### Abstract

A general form of codebook design for code-domain non-orthogonal multiple
access (CD-NOMA) can be considered equivalent to an autoencoder (AE)-based
constellation design for multi-user multidimensional modulation (MU-MDM). Due
to a constrained design space for optimal constellation, e.g., fixed resource
mapping and equal power allocation to all codebooks, however, existing AE
architectures produce constellations with suboptimal bit-error-rate (BER)
performance. Accordingly, we propose a new architecture for MU-MDM AE and
underlying training methodology for joint optimization of resource mapping and
a constellation design with bit-to-symbol mapping, aiming at approaching the
BER performance of a single-user MDM (SU-MDM) AE model with the same spectral
efficiency. The core design of the proposed AE architecture is dense resource
mapping combined with the novel power allocation layer that normalizes the sum
of user codebook power across the entire resources. This globalizes the domain
of the constellation design by enabling flexible resource mapping and power
allocation. Furthermore, it allows the AE-based training to approach a global
optimal MU-MDM constellations for CD-NOMA. Extensive BER simulation results
demonstrate that the proposed design outperforms the existing CD-NOMA designs
while approaching the single-user BER performance achieved by the equivalent
SU-MDM AE within 0.3 dB over the additive white Gaussian noise channel.

###### Index Terms:

Deep learning (DL), autoencoder (AE), non-orthogonal multiple access (NOMA),
codebook (CB) design, multi-dimension modulation (MDM), multi-user
communication

##  I Introduction

Since the early days of 5G research and development, non-orthogonal multiple
access (NOMA) schemes have gained considerable attention owing to their
superior spectral efficiency, as compared with orthogonal multiple access
(OMA) [1]. The allows for sharing time/frequency resources among users by
separating them in either power or code domains. The power-domain NOMA (PD-
NOMA) utilizes superposition coding at the transmitter and successive
interference cancellation at the receiver by exploiting the channel gain
difference between multiplexed users [2]. In code-domain NOMA (CD-NOMA),
meanwhile, different users are allocated different codes and multiplexed over
the same time/frequency resources. Furthermore, variants of the NOMA schemes
have been proposed for grant-free access so as to reduce the access delay and
signaling overhead [3]-[5]. Because a user symbol is transmitted over multiple
complex dimensions (resources) in CD-NOMA, it inherently possesses both the
nature of the modulation and channel coding (diversity).

Sparse-code multiple access (SCMA) has been considered as a promising
candidate among the notable CD-NOMA schemes, owing to its simplicity and
superior performance [6]-[13]. It evenly distributes the number of users that
interfere with each other over some resources while exploiting constellation
rotations to maximize the Euclidean distance (ED) among different codebooks
(CBs). The sparse CBs in SCMA, as compared with the dense CBs, restrict the
decoding complexity of the message passing algorithm (MPA) to grow
exponentially with only a fraction of the number of superposed users, instead
of all the users. In general, all CD-NOMA schemes, including SCMA, achieve a
higher spectral efficiency than OMA at the expense of detection complexity.
Another CD-NOMA scheme that has gained considerable traction is pattern
division multiple access (PDMA) [14]. The CBs for PDMA are designed such that
there is a disparity in the transmit diversity order among superposed users
(layers). This enables a successive interference cancellation (SIC)-based
receiver to decode a user’s signal in the descending diversity order. It is
also worth mentioning that power domain (PD)-NOMA induces power disparity
among users. Furthermore, PD-NOMA is considered as a special case of CD-NOMA,
wherein a user’s symbol is transmitted over a single complex dimension. In
general, all of these NOMA schemes offer a clear performance advantage over
OMA. However, CD-NOMA takes a complicated optimization process for codeword
(CW) design along with high-complexity multi-user detection, making the
conventional and practical design schemes far from the optimum performance.

Following the seminal paper with deep learning (DL) by O’shea et al. [15], in
particular, an end-to-end optimized architecture that exploits the similarity
between a communication system and an autoencoder (AE) has gained considerable
attention [16]–[27]. In [15], the AE for the end-to-end learning of a
communication system is used to optimize a multi-dimensional modulation (MDM)
scheme for a single-user communication. The AE for single-user MDM (SU-MDM) in
[15], transmitting 4 bits over 7 dimensions, i.e., achieving the spectral
efficiency of 4/7 bits/dimension, has been shown to achieve the
indistinguishable block error rate (BLER) performance from a conventional
system employing a Hamming (7,4) code and maximum likelihood detection (MLD).
It is important to note that the AE in [15] was able to design a near-optimal
constellation in terms of BLER, provided that the AE training complexity is
affordable. Their loss function and hyperparameters of the AE were modified
for minimizing the bit-error-rate (BER) in [16]. It improved multi-dimensional
constellation shaping by increasing the minimum ED in a constellation with
better bit-to-symbol mapping. Furthermore, it is first shown in [27] that an
encoder-decoder pair with a near-optimum error performance can be equivalently
obtained by perfectly training a special AE, which however might not be
possible in practice. Accordingly, [27] proposes a practical AE-based channel
coding scheme that is well-suited for receivers with one-bit quantization.

Meanwhile, a superposition of multiuser symbols in CD-NOMA can be treated as
one of the signal points from a single constellation of SU-MDM with equivalent
spectral efficiency. However, from a multiple-access viewpoint, SU-MDM
corresponds to a virtual model for CD-NOMA wherein a user knows the other-
users’ simultaneous symbols a priori, i.e., by Genie aid. Given the optimized
constellation for a single user, it can be extended to the multi-user
scenario, in which signal points (codewords) in the multi-dimensional
modulation constellation can be divided into multiple subsets, each allocated
to the different users, while maintaining the same spectral density as in the
single-user scenario. Consequently, the multi-user version of AE which
specified different CB for each user can be considered with the same
principles as SU-MDM except for the absence of knowledge on the other-users’
symbols. Then in CD-NOMA, as individual constellations must be designed for
different users, it is referred to as multi-user MDM (MU-MDM). For example, an
AE-based SCMA was proposed in [17], generating an improved MU-MDM
constellation with sparse resource mapping, while realizing a low-complexity
deep neural network (DNN) decoder. It achieves a better BER performance than
the conventional SCMA. By contrast, dense code multiple access (DCMA) was
proposed in [18] by following the same AE-based design, now for dense resource
mapping.

It is important to note that the only difference in the objective of
constellation design for SU-MDM and MU-MDM with the same spectral efficiency
is that the former tries to maximize the minimum Euclidean distance (ED) of
the signal points while the latter tries to maximize the minimum ED of the
multi-user superposed signal points. Therefore, when both designs try to
transmit the same number of bits through the given signal dimension for the
same spectral efficiency, their optimal error performance (i.e., the optimal
sphere packing performance with MLD) should be the same. Such optimal error
performance can only be achieved if we can perfectly train the corresponding
AEs, i.e., fully and globally minimize the loss value [27]. However, such
perfect training has not been achieved in the existing literature for MU-MDM
design due to inherent and imposed constraints. In the existing MU-MDM
constellation designs [17]-[25], the structural constraints in MU-MDM AE
design make the performance of MU-MDM AE to be sub-optimal compared to
equivalent SU-MDM AE. The structural constraints in MU-MDM AE design as
compared to SU-MDM AE design can be summarized into three-fold. The first
constraint (referred to as Constraint A) is subject to independent encoder
network by input vector split for each user, not necessitating the Genie aid
and then, their signal superposition. The second constraint (referred to as
Constraint B) is subject to fixed resource mapping, e.g., sparse or dense
resource mapping. The last one (referred to as Constraint C) is based on power
control constraint with power normalization layer (PNL), which normalizes the
user’s CB power.

Generally, in solving the optimization problem including constellation
optimization, given all the conditions are same, the unconstrained (or
constraint-relaxed) optimization problem gives a better or equal performance
than the constrained one. Therefore, we want to relax the constraints
(equivalently, globalize the search space) for MU-MDM AE structure, aiming at
approaching the BER of an equivalent SU-MDM AE design which is not afflicted
by any structural constraints of Constraint A, Constraint B, and Constraint C.
However, as already described, Constraint A is an inherent (indispensable)
constraint in MU-MDM AE architecture. This is because a user’s symbols and the
corresponding CWs should be independent of other users’, not necessitating the
Genie aid. Thus, the proposed design focuses on relaxing the Constraint B and
Constraint C so as to maximize the degree-of-freedom (optimization search
space) in AE-based constellation optimization. Accordingly, despite numerous
AE-based constellation designs for MU-MDM [17]-[25], we argue that the
existing MU-MDM AE structures suffer from unnecessary constraints, e.g., fixed
resource mapping and equal power allocation to all CBs, which limits the
degrees-of-freedom for the CB design. For the relaxed constraints toward
global optimal solution, we propose PNL that normalizes a sum of the user’s CB
power across the entire resources. We highlight that, by combining the
proposed PNL with dense resource mapping, the MU-MDM AE architecture
globalizes the search space to allow AE to readily approach to a global
optimal MU-MDM constellation for CD-NOMA. In particular, this combination
globalizes the domain of the constellation design by enabling flexible
resource mapping and power allocation in multi-user CB design.

Furthermore, the existing AE-based MU-MDM design schemes in [17] and [20]-[21]
could not jointly optimize the signal constellation and its bit-to-symbol
mapping function. More specifically, they employed L2-norm or cross-entropy
with one-hot vector representation as their loss function, trying to minimize
the symbol error rate (SER), i.e., without considering a bit-to-symbol mapping
function. Moreover, we note that binary vector representation was employed for
the loss function to improve the BER performance of the constellation in [18],
[22]-[25]. However, the optimal design of the signal points and its bit-to-
symbol mapping varies with the signal-to-noise (SNR) level. Therefore, there
must be a means to minimize BER over a wide range of SNR levels. Thus, by
considering a loss function that simultaneously deals with the ED between
signal points and the Hamming distance (HD) between bits assigned to
neighboring signal points created through the superposition of multi-user
signal points, meaningful hyperparameters can be introduced to adjust the
relationship between the ED and HD of the constellation signal points for
achieving improved BER performance over a varying SNR range. In this paper, a
multi-user generalization of the hyperparameterized loss function and
associated two-step training process are proposed to improve a BER
performance.

To verify the performance improvement of the proposed MU-MDM design and
compare it with equivalent SU-MDM performance, there must be trainable AE
architectures for MU-MDM and SU-MDM. In order to handle the complexity of AE
architecture as growing exponentially with the number of users, we propose the
trainable architectures of deep learning-based designs for MU-MDM and baseline
SU-MDM, which can be equivalently compared to each other. They are implemented
to demonstrate that the proposed design for MU-MDM can approach the BER
performance of baseline SU-MDM design within 0.3 dB in the additive white
Gaussian noise (AWGN) channel, thus serving as the best existing design that
can be realized with its low-complex decoder.

The rest of the paper is organized as follows. In Section II, we present the
system model for CD-NOMA and a problem formulation for its CB design. The
limitations of existing studies are discussed thereafter. In Section III, we
present a baseline architecture for the AE-based SU-MDM system, which serves
as a lower BER performance bound for the proposed MU-MDM scheme subject to the
same order of spectral efficiency and complexity. Section IV presents the
proposed MU-MDM AE structure with different PNL levels and resource
utilization. The loss function of the proposed AE is also presented with the
underlying training method. Simulation results are presented and discussed in
Section V; these results indicate a near single-user BER performance when
using the proposed MU-MDM design. Finally, the concluding remarks are collated
in Section VI.

##  II System Model and Design Approach

###  II-A System Model for CD-NOMA

In the CD-NOMA system, $J$-user signals are superposed over $K$ orthogonal
resources, where $J>K$ and $J/K$ is referred to as an overloading factor. More
specifically, a user’s symbol is mapped to $N$ of the $K$ resources, wherein
if $N<K$, the CD-NOMA scheme is said to be sparse; otherwise (i.e., $N=K$), it
is dense. Fig. 1 details the bit-to-symbol and symbol-to-resource mapping
processes to generate the CWs of multiple users. Assuming $M$-ary symbols, a
vector of the transmitted bit sequence for user $j$ is represented as
${{\mathbf{b}}^{(j)}}\in{{\mathbb{B}}^{{{\log}_{2}}M}}$, which is mapped to an
$N$-dimensional symbol ${{\mathbf{c}}^{(j)}}\in{{\mathbb{C}}^{N}}$ using a
bit-to-symbol mapping function
$f_{b}^{(j)}:{{\mathbb{B}}^{{{\log}_{2}}M}}\to{{\mathbb{C}}^{N}}$,
$j=1,2,\cdots,J$.

For a multi-user modulation scheme that employs a different constellation for
different users, let ${{\mathsf{\pazocal{C}}}^{(j)}}$ denote a set of symbols
for user $j$, i.e.,
${{\mathsf{\pazocal{C}}}^{(j)}}=\\{\mathbf{c}_{1}^{(j)},\mathbf{c}_{2}^{(j)},\cdots,\mathbf{c}_{M}^{(j)}\\}$,
where $\mathbf{c}_{m}^{(j)}\in{{\mathbb{C}}^{N}}$ and $m=1,2,\cdots,M$.
Furthermore, a constellation for all users is defined as
$\mathsf{\pazocal{C}}=\\{{{\mathsf{\pazocal{C}}}^{(1)}},{{\mathsf{\pazocal{C}}}^{(2)}},\cdots,{{\mathsf{\pazocal{C}}}^{(J)}}\\}$.
In addition, the bit-to-symbol mapping functions for the constellation
$\mathsf{\pazocal{C}}$ are represented by the set
${{\mathsf{\pazocal{F}}}_{b}}=\\{f_{b}^{(1)},f_{b}^{(2)},\cdots,f_{b}^{(J)}\\}$.
Afterward, the $N$-dimensional symbol $\mathbf{c}^{(j)}$ for the $j$-th user
is mapped to a $K$-dimensional CW ${{\mathbf{s}}^{(j)}}\in{{\mathbb{C}}^{K}}$
using a mapping vector ${{\mathbf{f}}^{(j)}}\in{{\mathbb{B}}^{K}}$, which
determines the resources to be selected. More specifically, the $N$ resources
for $\mathbf{c}^{(j)}$ are selected by the non-zero elements of
${{\mathbf{f}}^{(j)}}\in{{\mathbb{B}}^{K}}$. As an example, consider
${{\mathbf{f}}^{(j)}}={{[1\text{ }0\text{ }1\text{ }0]}^{T}}$ for $K=4$ and
$N=2$. This indicates that a two-dimensional complex symbol of the $j$-th user
is mapped to the first and third resources. Therefore, a resource mapping
matrix for all users is constructed as
$\mathbf{F}=[{{\mathbf{f}}^{(1)}},{{\mathbf{f}}^{(2)}},\cdots,{{\mathbf{f}}^{(J)}}]\in{{\mathbb{B}}^{K\times
J}}$. Consequently, each user transmits a CW vector $\mathbf{s}^{(j)}$ with
its non-zero elements mapped to the resource elements determined by the $j$-th
user’s CB, which is denoted as
${{\mathsf{\pazocal{S}}}^{(j)}}=\\{\mathbf{s}_{1}^{(j)},\mathbf{s}_{2}^{(j)},\cdots,\mathbf{s}_{M}^{(j)}\\}\in{{\mathbb{C}}^{K}}$.
Given $J$, $K$, $N$, and $M$, a CB generation function can be represented as
$s\left(\mathsf{\pazocal{C}},{{\mathsf{\pazocal{F}}}_{b}},\mathbf{F};J,K,N,M\right)$,
which is an end-to-end function to map the input bit sequences of multiuser to
a set of multi-dimension CBs
$\mathsf{\pazocal{S}}=\\{{{\mathsf{\pazocal{S}}}^{(1)}},{{\mathsf{\pazocal{S}}}^{(2)}},\cdots,{{\mathsf{\pazocal{S}}}^{(J)}}\\}$.
Let ${{P}^{(j)}}$ denote the average CB power for the $j$-th user, that is,
average power of $M$ CWs in the user $j$’s CB, which is given as
${{P}^{(j)}}=(1/M)\cdot\sum\nolimits_{m=1}^{M}{\left\|\mathbf{s}_{m}^{(j)}\right\|_{2}^{2}}$.

Now, let us consider a superposition of the CWs from different users, which is
represented as
$\mathbf{x}\triangleq\sum\nolimits_{j=1}^{J}{{{\mathbf{s}}^{(j)}}}$. It must
be noted that $\mathbf{x}$ can take any of ${{M}^{J}}$ realizations, denoted
as $\mathsf{\pazocal{X}}$. The received signal is then expressed as
$\mathbf{y}=\mathbf{x}+\mathbf{n}$, where $\mathbf{n}\in{{\mathbb{C}}^{K}}$ is
a complex AWGN noise vector with
$\pazocal{C}\pazocal{N}(0,{{\sigma}^{2}}{{\mathbf{I}}_{K}})$ and
$\mathbf{x}\in\mathsf{\pazocal{X}}$. In the receiver, the realization
$\mathbf{x}$, i.e., superposed CWs from all the users, can be jointly detected
by the following multi-user maximum a posteriori probability detection:

$\displaystyle\mathbf{\hat{x}}=\underset{\mathbf{x}\in\mathsf{\pazocal{X}}}{\mathop{\arg\max}}\,p(\mathbf{x}|\mathbf{y}),$
(1)

where $p(\mathbf{x}|\mathbf{y})$ is the conditional probability of
$\mathbf{x}$, given $\mathbf{y}$. As observed in the system model, the CB of
the CD-NOMA system corresponds to the constellation for an MU-MDM system.
Therefore, in the following sections, the CWs in a CB are interchangeably
referred to as symbols or signal points in a MU-MDM constellation without loss
of generality.

Figure 1: System model for code-domain NOMA with $J$ users and $K$ resources

###  II-B Deep Learning-based MU-MDM Design: Problem Formulation

In this study, we consider a multi-user CB design problem to determine a
constellation set $\mathsf{\pazocal{C}}$, a bit-to-symbol mapping function set
${{\mathsf{\pazocal{F}}}_{b}}$, and a resource mapping matrix $\mathbf{F}$
that optimizes the performance of a multi-user CB generation function
$s\left(\mathsf{\pazocal{C}},{{\mathsf{\pazocal{F}}}_{b}},\mathbf{F};J,K,N,M\right)$
for CD-NOMA. It should be noted that the multi-user CB generation
$s\left(\mathsf{\pazocal{C}},{{\mathsf{\pazocal{F}}}_{b}},\mathbf{F};J,K,N,M\right)$
corresponds to the MU-MDM. In general, a BER, denoted as the function
${{\varepsilon}_{b}}(\cdot)$, is a typical performance metric for an MU-MDM
design. Then, an MU-MDM design problem can be comprehensively formulated
through the following optimization problem:

$({{\mathsf{\pazocal{C}}}^{*}},{{\mathsf{\pazocal{F}}}_{b}}^{*},{{\mathbf{F}}^{*}})=\underset{(\mathsf{\pazocal{C}},{{\mathsf{\pazocal{F}}}_{b}},\mathbf{F})}{\mathop{\arg\min}}\,{{\varepsilon}_{b}}\left(s\left(\mathsf{\pazocal{C}},{{\mathsf{\pazocal{F}}}_{b}},\mathbf{F};J,K,N,M\right)\right)\\\
$ $\displaystyle\text{ s}\text{.t}\text{.
}\frac{1}{J}\sum\nolimits_{j=1}^{J}{{{P}^{(j)}}}\leq P,$ (2)

where $P$ is the total power allocated for the $J$ superposed users.

Owing to the complexity of (2), the existing approaches attempt to find
$\mathsf{\pazocal{C}}$ for the given $\mathbf{F}$, that is, they fail to solve
the joint optimal solution for (2). In the existing approaches, specific
resource mapping rules are configured when $N$, $J$, $K$, and the detection
algorithm, are given. For example, in SCMA, ${{\mathbf{F}}^{*}}$ possesses the
property of keeping a minimum number of symbols overlapping the resources in
each dimension for the given degree of freedom ${{d}_{f}}=JN/K$ [7].
Furthermore, $N$ is set to be significantly lower than $K$ for sparse
$\mathbf{F}$, to ensure a low complexity MPA that warrants the detection
performance. Notably, this complexity for CD-NOMA is given as
$O({{M}^{{{d}_{f}}}})$, that is, ${{d}_{f}}<J$ for $N<<K$, which result in
lower complexity [6]. Therefore, the optimization problem (2) can be reduced
by replacing $\mathbf{F}$ with a handcrafted optimized matrix
${{\mathbf{F}}^{*}}$ while focusing on the constellation signal points and
bit-to-symbol mapping optimization alone. For a fixed matrix
${{\mathbf{F}}^{*}}$ and its sparsity level ${{N}^{*}}$, it is restated as

$({{\mathsf{\pazocal{C}}}^{*}},{{\mathsf{\pazocal{F}}}_{b}}^{*})=\underset{(\mathsf{\pazocal{C}},{{\mathsf{\pazocal{F}}}_{b}})}{\mathop{\arg\min}}\,{{\varepsilon}_{b}}\left(s\left(\mathsf{\pazocal{C}},{{\mathsf{\pazocal{F}}}_{b}},{{\mathbf{F}}^{*}};{{N}^{*}},J,K,M\right)\right)$
$\displaystyle\text{ s}\text{.t}\text{.
}\frac{1}{J}\sum\nolimits_{j=1}^{J}{{{P}^{(j)}}}\leq P.$ (3)

However, for DL-based approaches, we argue that the MPA receiver (or any other
conventional receiver) can be replaced with a neural network-based multi-user
decoder (NN-MUD) receiver, the complexity of which is no longer bounded by the
constraint $N$. Thus, it is thus unnecessary to impose any constraint on $N$,
as in the existing DL-based approaches, for example, a sparse
${{\mathbf{F}}^{*}}$ with $N<K$ for SCMA [17] or dense ${{\mathbf{F}}^{*}}$
with $N=K$, that is, all the entries of one [18]. Owing to the enormous
complexity of $({{M}^{J}})!$ possible combinations for a bit-to-symbol mapping
function$f_{b}^{(j)}:{{\mathbb{B}}^{{{\log}_{2}}M}}\to{{\mathbb{C}}^{N}}$,
only a signal point in the constellation design was explicitly considered in
both the conventional and DL-based approaches [10]–[13], [17]-[18], [20]–[25].
To avoid the complexity of dealing with both $\mathsf{\pazocal{C}}$ and
${{\mathsf{\pazocal{F}}}_{b}}$ for joint optimization, previous studies relied
on simple suboptimal approaches, such as a constellation rotation to the
mother constellation with a large minimum ED [10]-[13]. Furthermore, there has
been no attempt to consider both $\mathsf{\pazocal{C}}$ and
${{\mathsf{\pazocal{F}}}_{b}}$ simultaneously in DL-based MU-MDM design
approaches, which would require a new loss function to deal with the multi-
user interference for training.

To summarize, the existing MU-MDM design is focused solely on optimizing
$\mathsf{\pazocal{C}}$, while neglecting ${{\mathsf{\pazocal{F}}}_{b}}$ under
the given constraint of ${{\mathbf{F}}^{*}}$. This implies that there must
exist substantial room for reducing the BER, as long as
$\mathsf{\pazocal{C}}$, ${{\mathsf{\pazocal{F}}}_{b}}$, and $\mathbf{F}$ are
jointly optimized to solve (2). Therefore, we considered a more generalized
architecture for the MU-MDM AE, along with a more flexible power normalization
constraint, which allows for the exploitation of a full degree of freedom for
an MU-MDM constellation design. Furthermore, we utilized a hyperparameterized
loss function and its corresponding training procedure to optimize
${{\mathsf{\pazocal{F}}}_{b}}$ in an explicit manner.

##  III Baseline System: Autoencoder-based SU-MDM

As discussed earlier, SU-MDM AE can serve as a baseline model for assessing
the performance of the proposed MU-MDM design. We note that a generic
structure of AE for the design of MDM cannot be immediately extended to SU-
MDM, simply because it involves a large size of input vectors. In other words,
there must be another architecture that can handle the underlying complexity
of SU-MDM, while still providing a performance bound of MU-MDM design with the
same spectral efficiency. In this section, we discuss a new trainable
architecture for SU-MDM as a baseline model to be compared to MU-MDM. Toward
that end, we first present a brief introduction to channel AE, which can
represent a CD-NOMA system under design.

Figure 2: Basic structure of channel AE with corruption Figure 3: AE structure
for SU-MDM model

###  III-A Channel Autoencoder: Overview

As shown in Fig. 2, if the channel impairment is applied to the encoded data
in a hidden layer, it is termed as a channel AE [15]. The channel AE encodes
the input vector $\mathbf{r}$ through the encoder function $f(\cdot)$, and the
corruption is subsequently applied to the encoded data. Thereafter, the
decoder function $g(\cdot)$ is applied to the corrupted intermediate signal to
determine its estimate, denoted as $\mathbf{\hat{r}}$, which needs to be
similar (or identical in the perfect sense) to the input vector $\mathbf{r}$,
i.e., $\mathbf{\hat{r}}=g(f(\mathbf{r}))\approx\mathbf{r}$. Let
${{\theta}^{(e)}}$ and ${{\theta}^{(d)}}$ be the parameters of the two hidden
layer functions, $f(\cdot)$ and $g(\cdot)$, respectively. Given a loss
function $L(\cdot)$, the channel AE is trained to minimize it as follows:

$\displaystyle({{f}_{AE}}^{*},{{g}_{AE}}^{*})=\underset{(f,g)}{\mathop{\arg\min}}\,L\left(\mathbf{r},\mathbf{\hat{r}}\left|{{\theta}^{(f)}},{{\theta}^{(g)}}\right.\right).$
(4)

The loss function is minimized, as in (4), with an appropriate NN model and
hyperparameters. Despite of the corruption in the encoded data, an AE
reconstructs the vector $\mathbf{r}$, i.e., $\mathbf{\hat{r}}=\mathbf{r}$,
with a probability that approaches one.

In this study, we consider a channel AE in which corruption is given as AWGN
$\mathbf{n}$ on the encoded data. In this case, the reconstructed vector can
be represented as

$\displaystyle\mathbf{\hat{r}}=g\left(f\left(\mathbf{r};{{\theta}^{(f)}}\right)+\mathbf{n};{{\theta}^{(g)}}\right).$
(5)

As in many other practical AE designs, such as in [17] and [22] we also
consider the following L2-norm as a baseline loss function:

$\displaystyle{{L}_{2}}(\mathbf{r},\mathbf{\hat{r}})={{\left\|\mathbf{r}-\mathbf{\hat{r}}\right\|}_{2}}.$
(6)

Then, we establish the following proposition to justify our choice of the
baseline L2-norm loss function:

######  Proposition 1.

Assuming equally likely transmitted symbols over the AWGN channel, any AE with
parameters that are perfectly optimized by a loss function of
${{L}_{2}}(\mathbf{r},\mathbf{\hat{r}})={{\left\|\mathbf{r}-\mathbf{\hat{r}}\right\|}_{2}}$
yields the same encoder-decoder pair as that optimized for the minimum ED
detection in conventional communication systems for any input vector that
uniquely represents the transmitted bit sequence.

###### .

Proof. See Appendix A. ∎

In [27], it has already been proven that a channel AE with one-hot coding at
the input layer, a softmax activation function at the output layer, and
parameters that are optimized via the cross-entropy function can yield the
optimal encoder–decoder pair as the one in the conventional MLD for the linear
block code, provided it is trained perfectly. Note that L2-norm and cross-
entropy loss functions lead to the minimum Euclidean distance (MED)-based and
maximum likelihood (ML)-based detection, respectively. Following (A-5), ML-
based detection and the MED-based detection achieve the same SERs; thus, both
cross-entropy and L2-norm loss functions can interchangeably achieve an
optimal encoder-decoder pair that minimizes SER, as in many other existing
practical AWGN-based AE architectures [17]-[18], [20]-[25]. Proposition 1
implies that the input vector r does not necessarily need to be a long one-hot
input vector when it can be reduced to represent the same transmitted sequence
uniquely. In fact, it allows for constructing the equivalent AE with a reduced
order of training complexity, especially for SU-MDM, which will serve as a
baseline to design the MU-MDM with the same order of training complexity
subject to the same bandwidth efficiency.

(a) End-to-end configuration

(b) CW element generator for the $j$-th user

Figure 4: The AE structure for MU-MDM model with CW element-wise power
normalization

###  III-B AE-based SU-MDM

In this section, we consider a SU-MDM model with the same bandwidth efficiency
as MU-MDM, which transmits $J{{\log}_{2}}M$ bits in $K$-dimensional resources,
that is, it achieves the spectral efficiency of $J{{\log}_{2}}M/K$
bits/s/Hz/resource. It is considered to serve as a baseline system for
comparison with the proposed MU-MDM system subject to the same bandwidth
efficiency. In the existing AE structure for SU-MDM (for example, in [15] and
[16]), the input vector $\mathbf{b}$ is encoded to a one-hot vector with a
dimension of ${{M}^{J}}$ via a one-hot encoding function
$h:{{\mathbb{B}}^{{{\log}_{2}}{{M}^{J}}}}\to{{\mathbb{B}}^{{{M}^{J}}}}$. Let
${{\mathbf{r}}_{i}}\in{{\mathbb{B}}^{{{M}^{J}}}}$ denote one of the one-hot
vectors used to represent each input vector $\mathbf{b}$, which is defined as
the $i$-th column of the identity matrix ${{\mathbf{I}}_{{{M}^{J}}}}$, that
is,

$\displaystyle{{\mathbf{r}}_{i}}={{[\underbrace{0\cdots 0}_{(i-1)\text{ 0
}\\!\\!^{\prime}\\!\\!\text{ s}}\text{ }1\underbrace{0\cdots
0}_{({{M}^{J}}-i)\text{ 0 }\\!\\!^{\prime}\\!\\!\text{ s}}]}^{T}}.$ (7)

In other words, $\\{{{\mathbf{r}}_{i}}\\}$ can be used to represent all J
symbols uniquely.

It should be noted that the dimension of the aforementioned one-hot vector can
simply explode with $J$, making the NN excessively complex to converge. To
reduce the input dimension, multiple one-hot vectors, one for each symbol, can
be employed to represent the input vector $\mathbf{b}$. More specifically,
consider $J$ independent one-hot vectors, each with a dimension of $M$, which
can be concatenated to represent an ${{M}^{J}}$-dimensional one-hot vector.
Let ${{\mathbf{r}}^{(j)}}\in{{\mathbb{B}}^{M}}$ denote a one-hot vector to
indicate the $j$-th symbol, which is defined as
${{\mathbf{r}}^{(j)}}\in\left\\{\mathbf{r}_{1}^{(j)},\mathbf{r}_{2}^{(j)},\cdots,\mathbf{r}_{M}^{(j)}\right\\}$,
where $\mathbf{r}_{i}^{(j)}$ is given by the $i$-th column of the identity
matrix ${{\mathbf{I}}_{M}}$, that is,

$\displaystyle\mathbf{r}_{i}^{(j)}={{[\underbrace{0\cdots 0}_{(i-1)\text{ 0
}\\!\\!^{\prime}\\!\\!\text{ s}}\text{ }1\underbrace{0\cdots 0}_{(M-i)\text{ 0
}\\!\\!^{\prime}\\!\\!\text{ s}}]}^{T}}.$ (8)

The low-dimensional one-hot vectors are then concatenated as
$\mathbf{r}={{[{{({{\mathbf{r}}^{(1)}})}^{T}},{{({{\mathbf{r}}^{(2)}})}^{T}},\cdots,{{({{\mathbf{r}}^{(J)}})}^{T}}]}^{T}}$,
which can be uniquely mapped to a one-hot vector in (7). In fact, ${{M}^{J}}$
signal points in $2K$ dimensions, that is, real and imaginary components of
$K$ complex dimensions, are uniquely matched to ${{M}^{J}}$ realizations of
the bit sequences by the concatenated one-hot vectors with a total length of
$MJ$. Even if the input dimension has been reduced by (8), its performance can
still be maintained by Proposition 1.

Fig. 3 illustrates the overall architecture with the reduced input vectors,
which is used as a baseline system for comparison with the proposed MU-MDM
system in the next section. The numbers of nodes and layers on the encoder
sides are denoted as ${{D}^{({{f}_{SU}})}}$ and ${{L}^{({{f}_{SU}})}}$,
respectively. Similarly, the numbers of nodes and layers on the decoder sides
are denoted as ${{D}^{({{g}_{SU}})}}$ and ${{L}^{({{g}_{SU}})}}$,
respectively.

(a) Level-1 PNL

(b) Level-2 PNL

(c) Level-3 PNL

Figure 5: Power normalization layers for the proposed MU-MDM encoder

##  IV Channel Autoencoder for MU-MDM Design

###  IV-A Autoencoder Structure for MU-MDM Model

In this subsection, an AE structure for an MU-MDM design is generalized using
an extension of that for an SU-MDM from Section III; the same order of
training complexity is ensured subject to the same bandwidth efficiency. Fig.
4(a) shows the overall structure of the proposed AE for the MU-MDM design,
including the CW element generator in Fig. 4(b). There are two inherent
(indispensable) features in the AE structure for MU-MDM, which differentiate
it from that for SU-MDM. The first is the input vector split, where the one-
hot vector input
$\mathbf{r}\triangleq{{[{{({{\mathbf{r}}^{(1)}})}^{T}},{{({{\mathbf{r}}^{(2)}})}^{T}},\cdots,{{({{\mathbf{r}}^{(J)}})}^{T}}]}^{\text{T}}}$
associated with the input-bit sequence
$\mathbf{b}\triangleq{{[{{({{\mathbf{b}}^{(1)}})}^{T}},{{({{\mathbf{b}}^{(2)}})}^{T}},\cdots,{{({{\mathbf{b}}^{(J)}})}^{T}}]}^{T}}$
is compartmentalized into independent inputs, that is,
$\\{{{\mathbf{r}}^{(j)}}\\}_{j=1}^{J}$. Using an input vector split, user
$j$’s constellation
${{\mathsf{\pazocal{C}}}^{(j)}}=\\{\mathbf{c}_{1}^{(j)},\mathbf{c}_{2}^{(j)},\cdots,\mathbf{c}_{M}^{(j)}\\}$
is determined by the input vector ${{\mathbf{r}}^{(j)}}$ only and is
unaffected by the other user’s input vectors
$\\{{{\mathbf{r}}^{(j^{\prime})}}\\}_{j^{\prime}=1}^{J},j^{\prime}\neq j$. As
a result, user $j$’s constellation set has a size of $M$, as discussed for the
CD-NOMA system model in Section II-A. The second feature is an independent CW
generator ${{f}^{(j)}}$ for each user, which generates the multi-dimensional
CW over $K$ resources, as shown in Fig. 4(a). It should be noted that the CW
generator ${{f}^{(j)}}$ is a concatenation of $K$ CW element generators,
$\\{f_{k}^{(j)}\\}_{k=1}^{K}$, as given in Fig. 4(b).

Let ${{D}^{({{f}_{MU}})}}$ and ${{L}^{({{f}_{MU}})}}$ represent the
hyperparameters used to denote the numbers of nodes and layers in
$f_{k}^{(j)}$ on the encoder side, respectively. Similarly,
${{D}^{({{g}_{MU}})}}$ and ${{L}^{({{g}_{MU}})}}$ are hyperparameters that
denote the numbers of nodes and layers on the decoder side, respectively. In
the CW element generator $f_{k}^{(j)}$, there are two outputs for each
dimension: one for the real part and another for the imaginary part of the
complex signal points. The remaining the structure for the NN encoder can be
determined by resource mapping vector $\mathbf{F}$ and the PNL. If the
$(j,k)$-th element of $\mathbf{F}$ is 0, the PNL in $f_{k}^{(j)}$ forces the
normalized power to be zero; otherwise, power is normalized with a given power
constraint as prescribed in Section II-B.

Whenever one of the $M$ realizations for the one-hot sub-vector
$\mathbf{r}_{i}^{(j)}$ is applied to the CW element generator $f_{k}^{(j)}$,
it produces a complex signal point ${{\mathbf{s}}^{(j)}}$ in the $k$-th
dimension. The $K$ complex dimensional signal points from
$\\{f_{k}^{(j)}\\}_{k=1}^{K}$ that belong to each user are then superposed to
form a constellation over $K$ complex dimensions, that is,
$\mathbf{x}=\sum\nolimits_{j=1}^{J}{{{\mathbf{s}}^{(j)}}}$. For the corruption
layer, a complex AWGN corruption vector
$\mathbf{n}\sim\pazocal{C}\pazocal{N}(0,\sigma{{\mathbf{I}}_{K}})$ is then
added to yield an intermediate signal $\mathbf{y}$. The corruption level must
be carefully chosen because it determines the training and test performance of
the AE [17]. The superposed and corrupted multi-dimensional constellation is
fed into the NN-MUD, ${{g}_{MU}}(\cdot)$. The NN-MUD then produces an output
vector
$\mathbf{\hat{r}}={{[{{({{\mathbf{\hat{r}}}^{(1)}})}^{T}},{{({{\mathbf{\hat{r}}}^{(2)}})}^{T}},\cdots,{{({{\mathbf{\hat{r}}}^{(J)}})}^{T}}]}^{\text{T}}}\in{{\mathbb{B}}^{1\times(J\cdot
M)}}$ , which can be converted into the corresponding bit sequence,
$\mathbf{\hat{b}}={{[{{({{\mathbf{\hat{b}}}^{(1)}})}^{T}},{{({{\mathbf{\hat{b}}}^{(2)}})}^{T}},\cdots,{{({{\mathbf{\hat{b}}}^{(J)}})}^{T}}]}^{\text{T}}}\in{{\mathbb{B}}^{1\times
J{{\log}_{2}}M}}$. Given the AE structure, an appropriate loss function must
be considered to solve the optimization problem in (3).

Notably the input vector $\mathbf{r}$ with dimensions of $MJ$ result in
${{M}^{J}}$ different superposed signal points $\mathbf{x}$ in MU-MDM.
Moreover, the AE for the SU-MDM in Fig. 3 with ${{M}^{J}}$ different signal
points features the same input dimensions as that for the MU-MDM. This
confirms that both the SU-MDM and MU-MDM schemes have the same number of bits
per resource, that is, $(J{{\log}_{2}}M)/K$, while maintaining the same input
dimensionality. In addition, all the layers and nodes in the AEs for the SU-
MDM and MU-MDM are adjusted for a comparable number of training parameters;
this ensures that the SU-MDM and MU-MDM have the same order of training
complexity.

###  IV-B Power Normalization Layer for MU-MDM Design

Before discussing the different levels of PNL that can be applied to the AE,
we note that an input vector is divided into separate sub-vectors that are
independent of each other in the MU-MDM, as shown in Fig. 4(a). In this
structure, all the generated CWs are constrained to the same average CB power
in previous studies, i.e., [17] and [18]; therefore, this becomes a stringent
design constraint as compared to SU-MDM. To reduce the performance gap between
the MU-MDM and the baseline SU-MDM, we introduce a relaxed power normalization
among the power assigned to the different CBs in this study. It is also worth
mentioning that the relaxation in the power allocation to the different CBs
has some similarities with the design of the PDMA [14] and PD-NOMA [2]. To
evaluate the gain from the power constraint relaxation, we discuss three
different levels of PNL, referred to as Level-1, Level-2, and Level-3 PNL,
depending on the degree of freedom for the power allocation.

Level-1 PNL addresses a CW element-wise constraint, which requires each non-
zero element of a CW to be transmitted with the same power, as shown in Fig.
5(a). In this design, the CW element power normalization is given as follows:

$\displaystyle\left\|{{\mathbf{s}}^{(j)}}(k)\right\|_{2}^{2}=\left\\{\begin{matrix}\frac{{{P}_{j}}}{N}\text{
if }f_{k}^{(j)}=1\\\ 0\text{ otherwise}\\\ \end{matrix}\right.,\forall j,k,$
(9)

where ${{\mathbf{s}}^{(j)}}(k)$ represents the $k$-th row of entry of CW
vector ${{\mathbf{s}}^{(j)}}$. In Level-2 PNL, we consider CW-wise power
normalization, that is, each CW is set to a fixed power, as shown in Fig.
5(b). The CW-wise power normalization is then given as

$\displaystyle\left\|{{s}^{(j)}}\right\|_{2}^{2}={{P}^{(j)}},\forall j.$ (10)

It should be noted that this is less stringent than element-wise (Level-1)
power normalization, because each element (dimension) can be set to different
powers. Therefore, it has a more room for optimization of constellation shape
than element-wise optimization.

Lastly, in Level-3 PNL, the sum of the CW power is normalized, as shown in
Fig. 5(c). This allows the CBs assigned to different users to have different
powers, and they are only subject to a total sum constraint as follows:

$\displaystyle\sum\limits_{j=1}^{J}{\left\|{{\mathbf{s}}^{(j)}}\right\|_{2}^{2}}=\sum\limits_{j=1}^{J}{{{P}^{(j)}}}=JP.$
(11)

The Level-3 design involves the least stringent constraint, as opposed to (9)
and (10), which have the same average power for each user’s CB. This provides
additional degrees of freedom for the CB design, achieving a higher shaping
gain. In other words, the less stringent power constraint in Level-3
normalization enables the AE to increase the shaping gain, which, in turn,
maximizes the minimum ED among multi-dimensional constellations.

We note that the superior performance with a higher level of PNL has also been
demonstrated in conventional CB designs for SCMA [10], [13]. In Table I, the
different levels of PNLs are compared for the proposed and conventional design
approaches. In the very first CB design for SCMA in [7], signal points that
overlap in the same resource are differentiated by employing a rotated QPSK
constellation in each dimension of CW. Consequently, each of the overlapped
signal points in the CW element has the same power with a different phase as
in our Level-1 PNL. The CB in [10] exploits a layer-specific operation, such
as constellation rotation, with regard to the lattice-based mother
constellation. Thus, it allows for power imbalances across the dimensions of
CWs, resulting power differentiation among the colliding layers. However, its
shaping gain is still limited because its mother constellation forces each
user to have a fixed average CB power as in our Level-2 PNL. Recently in [13],
a more flexible form of power-imbalanced CB design was proposed to
“heuristically” optimize the average power allocated to different CBs. The BER
performance can be significantly improved, as in our Level-3 PNL, owing to its
higher multi-dimensional shaping gain, which is associated with a higher
degree of freedom by relaxing a power constraint.

However, besides the principle of power normalization, the proposed design
framework is significantly different from that in [13], because we exploit the
AE’s ability to optimize the multi-user multi-dimensional constellation in a
given relaxed PNL, while considering bit-to-symbol mapping for BER
minimization, which is be discussed in the following subsection.

Table I: Different Levels of CB Power Normalization: Comparison Conventional Designs |  DL-based Design |  Average CB Power |  Power Allocation to CW Eelements
---|---|---|---
Vanilla SCMA [7] |  Level-1: CW Element Power Normalization |  Same |  Same
Power-balanced CB Design for SCMA [10] |  Level-2: CW-wise Power Normalization |  Same |  Different
Power-imbalanced CB Design for SCMA [13] |  Level-3: Sum CW Power Normalization |  Different |  Different

###  IV-C Proposed Loss Function and Training Procedure for Joint Optimization

To explain the proposed loss function, we observe the factors governing the
BER in the arbitrary multi-dimensional constellation [28]. Let
${{d}_{H}}({{\mathbf{b}}^{(i)}},{{\mathbf{b}}^{(j)}})$ denote the HD between
${{\mathbf{b}}^{(i)}}$ and ${{\mathbf{b}}^{(j)}}$. It should be noted that a
pairwise symbol error probability (PEP) ${{p}_{s}}(\cdot)$ is determined by
the ED of the $i$-th and $j$-th CWs, which are given as
$f({{\mathbf{r}}^{(i)}})$ and $f({{\mathbf{r}}^{(j)}})$. Let
${{\varepsilon}_{b}}$ denote the average BER for arbitrary ${{M}^{J}}$ multi-
dimensional signal points, which can be computed as

$\displaystyle{{\varepsilon}_{b}}=\frac{1}{J{{\log}_{2}}M}\sum\limits_{i=1}^{{{M}^{J}}}{\sum\limits_{j\neq
i}{{{d}_{H}}({{\mathbf{b}}^{(i)}},{{\mathbf{b}}^{(j)}})\cdot{{p}_{s}}\left({{\mathbf{r}}^{(i)}}\to{{\mathbf{r}}^{(j)}}\right)}}.$
(12)

We first observe from (12) that the BER is governed by the multiplicative
relationship between the PEP of the CWs and its corresponding HD. The PEP is a
function of the SNR; therefore, the optimal multiplicative relationship
between ED and HD also depends on the SNR level. However, there is no explicit
expression that captures the multiplicative relationship in ED and HD to
minimize the BER in terms of the SNR level. Therefore, the BER in (12) cannot
be directly employed as an objective function for constellation design to
minimize the BER.

We note that the conventional design involves two separate steps: the first
step comprises a constellation design to minimize the CW error and the second
step comprises bit-to-symbol mapping to minimize the BER for the given
constellation, such as via Gray mapping. However, in the proposed MU-MDM AE
design, a loss function must be designed to jointly optimize the signal points
for the MU-MDM constellation and their bit-to-symbol mapping. Toward that end,
when a symbol has large HD from the adjacent symbols, a large L2-norm should
be applied to that symbol to increase the ED from those adjacent ones. In
other words, a larger L2-norm loss should be applied to symbols with a larger
Hamming error ${{d}_{H}}({{\mathbf{b}}^{(j)}},{{\mathbf{\hat{b}}}^{(j)}})$,
which corresponds to the HD between the input and output bit sequences.
Therefore, the loss function can be obtained by averaging the L2-norm for all
possible CWs, each weighted by the Hamming error.

However, as discussed previously, the appropriate amount of Hamming error
weighted with the L2-norm in minimizing BER depends on the SNR. To adjust the
weight based on the Hamming error such that a minimal BER can be maintained as
far as possible even under a varying SNR, a weight constant, denoted by
$\delta$, is introduced as a new hyperparameter to ensure that the design
function over a wide range of SNR levels. Subsequently, for a given Hamming
error, the following weight function with the hyperparameter $\delta$ is
defined for user $j$:

$\displaystyle
B({{\mathbf{b}}^{(j)}},{{\mathbf{\hat{b}}}^{(j)}})=\mu+\delta{{d}_{H}}({{\mathbf{b}}^{(j)}},{{\mathbf{\hat{b}}}^{(j)}}),$
(13)

where $\mu$ is a non-zero positive constant that prevents a loss function from
becoming zero in the correct detection case [16].

Thereafter, we attempt to minimize the average CW error, which is weighted by
(14). In other words, the following proposed loss function, denoted by
${{L}_{proposed}}$, can be defined:

$\displaystyle{{L}_{proposed}}=\frac{1}{J}\sum\limits_{j=1}^{J}{B({{\mathbf{b}}^{(j)}},{{{\mathbf{\hat{b}}}}^{(j)}})}\cdot{{\left\|{{\mathbf{r}}^{(j)}}-{{{\mathbf{\hat{r}}}}^{(j)}}\right\|}_{2}}.$
(14)

Note that ${{\theta}^{(e)}}$ and ${{\theta}^{(d)}}$ are updated to minimize
the loss function (14), in which the ED between CWs is mutually balanced with
the HD between the input and output sequences; thus, their combined effect can
be minimized. When $\delta$ approaches zero in (14), the loss function in (14)
converges to the ED-based L2-norm loss function, which is favorable for a high
SNR, and the Hamming error is neglected. However, a large value of $\delta$
deviates from the maximum ED constellation, which is favorable for a low SNR,
and the HD between adjacent signal points in the constellation is minimized.
This implies that there exists an appropriate value of the hyperparameter
$\delta$ for the given SNR level.

However, according to Proposition 1, deviations from the maximum ED
constellation caused by $\delta$ may induce further CW errors, i.e., it may
result in an unfavorable BER performance in the high SNR region. Thus, we
switch the loss function in (14) to the L2-norm after sufficient convergence
because the impact of Hamming weight in the loss function (14) is significant
during the initial stage of training [16]. After the bit-to-symbol mapping is
completed using the loss function (14), the effect of a constellation
deviation from $\delta$ can be reduced with the maximum ED constellation.
Therefore, the AE can be further trained using the L2-norm loss function after
sufficient convergence with the loss function (14). Two different loss
functions can be employed for the low and high SNR ranges; therefore, this
constitutes a two-step training process, which allows AE to achieve the
maximum ED constellation with near-optimal bit-to-symbol mapping. Once a
signal constellation is optimized to reduce the BER in Step 1, its performance
can be further improved by maximizing the ED between the bit-to-symbol-mapped
signal points in Step 2.

Figure 6: Proposed two-step training process

Two-step training process is depicted in Fig. 6. In Training Step 1, the
proposed NN-based AE is trained with a learning rate of ${{\eta}_{1}}$ until
both the training loss and the validation loss sufficiently converges within
${{T}_{1}}$ epochs (Step 1-A). After the loss hits a plateau, the learning
rate is reduced to ${{\eta}_{2}}$, i.e., ${{\eta}_{2}}<{{\eta}_{1}}$, which
allows for an increase in the convergence rate. After the network is fine-
tuned with ${{\eta}_{2}}$ over ${{T}_{2}}$ epochs (Step 1-B), Training Step 2
is initiated by switching the loss function to L2-norm. Subsequently, it is
trained over ${{T}_{3}}$ epochs until it converges. All the epochs,
${{T}_{1}}$, ${{T}_{2}}$, and ${{T}_{3}}$, can be set to be sufficiently large
in order to ensure convergence. However, the underlying convergence rate is
not critical in practice, simply because the proposed AE is trained offline;
only the trained encoder–decoder pair is exploited online.

##  V Simulation Results

In this section, the performance improvement achieved by proposed scheme, as
compared to existing conventional and DL-based constellation designs, is
demonstrated through extensive simulation results. To explain the BER
performance achieved by the proposed scheme, we also present the process of
hyperparameter optimization, while interpreting the constellation diagram
obtained by the current design.

###  V-A Simulation Setup

In our simulation, we consider a CD-NOMA system with $J=6$ users, $K=4$
resources, and a modulation order of $M=4$, as described in many existing
studies. We also consider both sparse ($N<K$) and dense ($N=K$) resource-
mapping matrices. First, for the sparse mapping case, we consider the
following resource mapping matrix:

$\displaystyle\mathbf{F}=\left[\begin{array}[]{*{35}{l}}0&1&1&0&0&1\\\
1&0&1&0&1&0\\\ 1&0&0&1&0&1\\\ 0&1&0&1&1&0\\\ \end{array}\right].$ (19)

For the dense mapping, we consider dense resource mapping matrix where all the
entries are one. Detailed AE structures corresponding to the sparse and dense
resource mapping matrices are described in Section IV-A.

We consider the offline training of the AE with the proposed structure and the
loss function. A set of output CBs,
${{\mathsf{\pazocal{S}}}^{(j)}}=\\{\mathbf{s}_{1}^{(j)},\mathbf{s}_{2}^{(j)},\cdots,\mathbf{s}_{J}^{(j)}\\}$,
is generated with $\\{{{f}^{(j)}}\\}_{j=1}^{J}$ for all possible input
vectors. Furthermore, we directly employ the trained decoder function
$g(\cdot)$ as an NN-MUD receiver, which achieves approximately the same BER
performance as the MPA receiver while incurring a much lower complexity
[17]-[19].

We generate a dataset with entire data classes for the dataset composition of
the offline training. A dataset of ${{M}^{J}}$ classes was constructed as
input vectors $\mathbf{r}\in{{\mathbb{B}}^{MJ}}$. Because random corruption is
added to the hidden layer, the data fed as the input to $g(\cdot)$ always
vary, even in the case where it is set to the same input vector. This prevents
the NN decoder from being overfitted to the dataset with a limited size; thus,
it is generalized to the unseen channel realizations. Offline training is
applied to obtain only the CB set $\pazocal{S}$ from the encoder function;
therefore, we are not concerned about the system complexity associated with
${{D}^{({{f}_{MU}})}}$ and ${{L}^{({{f}_{MU}})}}$. Therefore,
${{D}^{({{f}_{MU}})}}$ and ${{L}^{({{f}_{MU}})}}$ can be appropriately
selected so that NN can be neither overfitted nor underfitted in a given
training scenario. However, the receiver complexity of the NN-MUD is directly
governed by ${{D}^{({{g}_{MU}})}}$ and ${{L}^{({{g}_{MU}})}}$ in the actual
system. Therefore, they must be minimized as much as possible, while being
neither overfitted nor underfitted under a given training scenario.

###  V-B Hyperpatameter optimization

The rectified linear unit (ReLU) is preferred over the hyperbolic tangent
function (tanh) as an activation function for faster convergence, owing to a
very deep NN with millions of trainable parameters in the proposed AE.
However, zero-centered encoded data work better with using an activation
function with a zero-centered shape. In fact, the tanh function is more
appropriate for the constellation design, as shown in [16], [18]; its BER
performance is better than that obtained with the ReLU. This is also true for
the decoder side because the channel output is also composed of zero-centered
data. Therefore, in the proposed AE, tanh activation functions are employed
for both the encoder and the decoder. Moreover, a stochastic gradient descent
optimizer is used with the given learning rate.

The corruption level and weight constant $\delta$ for the loss function are
two important hyperparameters that need to be optimized to implement the
proposed AE network. The appropriate corruption level and $\delta$ depend on
various parameters, such as the number of users, number of resources, CB power
level of the user, and PNL level; therefore, extensive BER simulations with
the Level-3 PNL are performed to select the appropriate hyperparameters that
can minimize the BER under a general AWGN channel with various SNR level.
Meanwhile, we note that a common practice in the conventional CD-NOMA is to
design a constellation under AWGN channel and then, any channel can be
converted into the AWGN channel through channel estimation and multi-user
equalization [6]-[13]. We can also consider constellation rescaling scheme in
[30], which generalizes the proposed constellation to the Rayleigh channel.
However, generalization of the proposed constellation to the Rayleigh channel
will be beyond a scope of the current work.

The corruption level for each user is defined as the ratio of the AWGN power
to the average CB power, ${{\sigma}^{2}}/E[{{P}^{(j)}}]$. If this value is
excessively high, the AE fails to converge without learning the signal points.
By contrast, if this value is excessively low, the AE converges to a local
optimal point without fully exploring the globally optimal constellations;
thus, the AE is overfitted to the training dataset. This suggests that there
must exist an appropriate level of corruption, as discussed in [17]. In this
study, the appropriate corruption level in the MU-MDM design is identified as
$-$8 and $-$6 dB in the sparse and dense $\mathbf{F}$, respectively, based on
extensive BER simulations. In other words, the $-$8 dB and $-$6 dB corruption
levels generalize the low SNR region with fair convergence under the sparse
and dense settings, respectively. The appropriate corruption level in dense
CD-NOMA is slightly higher than that in sparse CD-NOMA, which is attributed to
the larger average inter-CW distance in the dense resource mapping. This
enables the for dense CD-NOMA to be more resilient against AWGN.

Figure 7: BER performance vs. weight constant $\delta$

In Fig. 7, we present the performance results that are used to select the
useful weight constant $\delta$ in the proposed loss function. It shows the
BER performance with the proposed loss function varying with respect to the
weight constant $\delta$ as a hyperparameter for the given SNR. Its
performance is compared with that obtained using the L2-norm loss function
with binary vector representation (represented by the dotted lines in the
graphs). It is observed that the proposed loss function generally outperforms
the L2-norm loss function for a particular weight constant range, i.e.,
$0.01\leq\delta\leq 0.05$, from the low to high SNR levels, thereby fulfilling
our design objectives. Therefore, in the performance analysis for the proposed
design with dense resource mapping and Level-3 PNL, the weight constant
$\delta$ is selected from this range, such as $\delta=0.05$. Detailed
hyperparameters for the DL used in the current design are summarized in Table
II.

Table II: Hyperparameters for deep learning: summary Hyperparameters |  Notation |  Value
---|---|---
Batch size |  ${N}_{batch}$ |  400
Learning Rate |  $\eta_{1}$ |  0.001
|  $\eta_{2}$ |  0.0001
Number of epochs |  $T_{1}$ |  8000
|  $T_{2}$ |  2000
|  $T_{3}$ |  1000
Corruption level |  ${{\sigma}^{2}}/E[{{P}^{(j)}}]$ |  $-$6 dB for dense and $-$8 dB for sparse
Weight constant in the proposed loss |  $\delta$ |  0.05
Number of nodes in the MU-MDM CW element generator |  ${{D}^{({{f}_{MU}})}}$ |  32
Number of layers in the MU-MDM CW element generator |  ${{L}^{({{f}_{MU}})}}$ |  6
Number of nodes in the MU-MDM decoder |  ${{D}^{({{g}_{MU}})}}$ |  512
Number of layers in the MU-MDM decoder |  ${{L}^{({{g}_{MU}})}}$ |  4
Number of nodes in the SU-MDM encoder |  ${{D}^{(f_{SU})}}$ |  256
Number of layers in the SU-MDM encoder |  ${{L}^{(f_{SU})}}$ |  6
Number of nodes in the SU-MDM decoder |  ${{D}^{(g_{SU})}}$ |  512
Number of layers in the SU-MDM decoder |  ${{L}^{(g_{SU})}}$ |  4
Figure 8: Constellation for sparse CD-NOMA with the sum power normalization
constraints

(a) DL-based SU-MDM

(b) Conventional SCMA

(c) DL-based MU-MDM

Figure 9: Comparison of superposed constellations for the different designs

###  V-C Constellation Diagrams

Fig. 8 shows an example of the constellation diagrams for sparse CD-NOMA using
F given in (15). These constellations are constructed using the proposed DL-
based MU-MDM with Level-3 PNL. There are two distinctive features in the
designed CBs, as compared with the conventional CBs. First, there are zero-
power signal points in the allocated resources, such as for users 2 and 4.
This implies that some users can use fewer resources than they are assigned,
depending on which constellation that each user attempts to transmit. Even if
the transmit diversity gain is reduced because some resources remain unused,
the inter-user distance between the signal points can be increased to minimize
the BER. Second, unlike the CB for the conventional SCMA, some constellations
only have two signal points; this also results in underutilization of the
resources. Notably, ${{\log}_{N}}M$ signal points per resource are sufficient
to uniquely distinguish $M$ signal points when N resources are utilized by a
user. Even if ${{\log}_{N}}M$ signal points per utilized resource are
sufficient, it is significantly complex to design CBs in which a subset of the
resources is utilized. However, DL allows for the learning of constellations
by varying the number of constellations per resource, while maximizing the
inter-constellation distance and minimizing the SER. Consequently, as shown in
Fig. 8, three different types of constellations are observed for the different
resources: $M$-symbol constellation, ${{\log}_{N}}M$-symbol constellation, and
constellation with zero-powered symbols.

Fig. 9 shows the constellations for the various designs before the AWGN
corruption is applied. Fig. 9(a) shows the DL-based SU-MDM constellation,
which includes ${{M}^{J}}$ signal points in each resource; this represents the
unique realization of the input vector. Although the constellation in Fig.
9(a) achieves good performance, it cannot be applied to MU-MDM constellations,
as discussed in Subsection IV-A. Fig. 9(b) and 9(c) show the conventional
SCMA-based superposed CB and the proposed DL-based superposed constellation,
respectively. The signal points in the superposed constellation correspond to
${{M}^{J}}$ possible realizations of the vector $\mathbf{x}$, as described in
Subsection II-A. It is straightforward for the CB of the conventional SCMA to
feature unique ${{M}^{{{d}_{f}}}}$ superposed constellations for each
resource. By contrast, the DL-based constellations exhibit a clustered form of
the superposed constellations. We observe that the inter-cluster constellation
distance is considerably larger than that for the conventional CBs; however,
the ED within the superposed constellation cluster is extremely small.

###  V-D Performance Analysis

(a) SU-MDM vs. MU-MDM

(b) Level-1 vs. Level-3 PNL

Figure 10: Comparison of BER performance: L2-norm vs. the proposed loss
function  Figure 11: Comparison of BER performance with different levels of
PNL: Dense vs. sparse resource mapping Figure 12: Comparison of BER
performance: The proposed MU-MDM vs. proposed SU-MDM

First of all, we investigate how our proposed loss function outperforms the
plain L2-norm loss function for the different setting in Fig. 10. Fig. 10(a)
presents a comparison of the BER performance between MU-MDM and SU-MDM with
the different loss functions, i.e., the L2-norm vs. the proposed one. As only
EDs among the different data sequences (signal points) are considered, BER
performance of L2-norm is optimized only in a ED sense but not in a bit-to-
symbol mapping sense. It is clearly shown here that that the proposed loss
function achieves 0.3 dB and 0.4 dB gain at a target BER of $10^{-3}$, as
compared to the L2-norm loss function, for both SU-MDM and MU-MDM,
respectively. In Fig. 10(b), furthermore, we investigate how effective the
proposed loss function works for MU-MDM with the different levels of PNL. Even
if we observe an obvious gain with the proposed loss function, its gain
decreases with Level-3 PNL, i.e., 0.7 dB to 0.4 dB. It implies that the
smaller the minimum ED between adjacent CWs, the more gain of increasing the
HD between them.

Fig. 11 presents a comparison of the BER performance for different levels of
PNL with sparse and dense resource mapping for the proposed MU-MDM. As far as
power-imbalance CB is considered with Level-3 PNL, however, dense resource
mapping will be a baseline structure that includes sparse resource mapping as
a special case. Whenever necessary, then, an option of Level-3 PNL along with
dense resource mapping would set the constellation power for specific
resources to zero, flexibly making the resource sparse as shown in Fig. 8.
That is why the performance of dense resource mapping would be always the same
or better than that of sparse resource mapping subject to the given level of
PNL. In Fig. 11, we observe that Level-2 PNL outperforms Level-1 by 1 dB at a
target BER of $10^{-3}$. Similarly, Level-3 outperforms Level-2 by 0.7 dB,
i.e., a combined gain of 1.7 dB over Level-1, for sparse MU-MDM. Similarly,
0.8 dB performance gain is observed for dense MU-MDM. In fact, the BER
performance gap between the dense and sparse CD-NOMA decreases with an
increase in the level of PNL, showing only 0.1 dB gap between them. In fact,
sparse CD-NOMA may be still comparably acceptable as long as Level-3 PNL is
employed under the proposed architecture. In Fig. 11, we can also note that
the BER performance gap between the dense and sparse CD-NOMA decreases with a
higher level of PNL. Interestingly, in Level-3 PNL, the performance gap
between sparse and dense CD-NOMA nearly disappears. This is attributed to a
less stringent power constraint in Level-3 PNL, which leads to more room for
optimizing the BER performance.

Fig. 12 presents a comparison of the BER performances between the proposed
designs for SU-MDM and MU-MDM. Our objective herein is to demonstrate that a
performance of the proposed design for MU-MDM approaches that for SU-MDM,
which serves as a lower-bound. Resorting to the conclusion from Fig. 10, we
just consider the best-performing design, which is a dense CD-NOMA with
Level-3 PNL, to be compared with SU-MDM. As shown therein, we observe that its
performance approaches that of SU-MDM, with approximately 0.3 dB only,
supporting our argument that the proposed AE architecture and the underlying
training framework for CD-NOMA, without knowing other-user symbols, approaches
the performance of SU-MDM with the same spectral efficiency. Meanwhile, we
would like to highlight that the Constraint A discussed in the Section I is
addressed in an implicit manner for the AE-based MU-MDM design as the AE sees
all possible combination of symbols from all users in the course of designing
the CBs. In particular, as the AE is trained for all the combination of the
symbols, i.e., $M^{J}=4096$ combinations, the CB design of each user is
implicitly interdependent via back propagation of the loss. It is to be noted
that SU-MDM can be considered as MU-MDM design with explicit dependence (among
users) in the symbols-to-CW mapping. However, in our MU-MDM case as the
interdependency is not explicit as compared to SU-MDM, we were only able to
approach the SU-MDM performance with 0.3 dB gap at $10^{-3}$ BER rather than
perfectly achieving it.

Furthermore, it is also compared with 8-QAM, which has the same bandwidth
efficiency, i.e., 3bps/Hz, as 4-ary CB ($M$ = 4) over four orthogonal
resources ($K$ = 4), overloaded with six users ($J$ = 6). The proposed design
outperforms 8-QAM, e.g., by 1.5 dB and 2.1 dB for non-rectangular 8-QAM and
rectangular 8-QAM, respectively. It can be translated into the maximum
possible overloading gain that can be achieved by CD-NOMA as compared to OMA.
Furthermore, in order to verify the current results, we also present a BER
curve for QPSK, which is referred to as “Single-user BER bound” in Fig. 12 It
corresponds to the BER of the same modulation scheme without overloading,
serving as an ultimate lower bound on all possible designs. It is clearly
shown that a performance of the proposed MU-MDM is now bounded by QPSK
performance with a gap of approximately 0.74 dB at a BER of $10^{-3}$.

Fig. 13 presents a comparison of BER performance between the proposed DL-based
design and conventional designs, which have a common feature of sparse
resource mapping. The conventional designs include the conventional SCMA [10],
[13] and DL-based SCMA [17]. First of all, we note that a performance of [17]
is mainly limited by non-optimized bit-to-symbol mapping (associated with a
loss function of L2-norm) and stringent power constraint (leve-1 PNL).
However, it outperforms the power-balanced CB design in [13], simply because
multi-user constellation can be further optimized with the AE-based design.
Meanwhile, we note that another conventional design of SCMA in [13] follows a
design discipline of allocating the different power for each CW, similar to
our Level-3 PNL. It is interesting to note that such a power-imbalanced SCMA
outperforms [17], even if its CB design relies just on a heuristic. It implies
how effective to allows for the different power among the CWs. We finally note
that the proposed design with Level-3 PNL outperforms the power-imbalanced CB
design in [13]. As discussed in Section IV-C, this is attributed to the joint
optimization capability of the DL-based design.

Figure 13: Comparison of BER performance: The proposed design vs. existing
designs

##  VI Conclusion

This paper proposed a new deep learning-architecture for autoencoder to design
a CD-NOMA scheme that aims at joint optimization of resource mapping and
codebook with bit-to-symbol mapping. The proposed deep learning-based
autoencoder learns a multi-user constellation, equivalently a codebook for CD-
NOMA. We also presented the trainable architectures of deep learning-based
designs for both SU-MDM and MU-MDM, which can be equivalently compared to each
other. Based on them, our implementation demonstrated that a BER performance
of our proposed architecture can approach single-user performance, which
serves as a lower BER bound on the performance of deep learning-based design.
The current achievement is mainly attributed to power-imbalanced codebook
design and flexible resource mapping, along with a hyperparameterized loss
function and corresponding training procedure to minimize the BER.

We note that both the existing and the proposed multi-user CBs are designed by
assuming an AWGN channel. However, in the case of a fading channel, the
inclusion of either conventional or NN-based equalizers at the receiver may
transform the fading channels to an AWGN channel. Particularly for the uplink,
as the user experiences different channels, a careful design of various
channel compensation methods, such as close-loop/open-loop power control,
should be an essential aspect of the proposed MU-MDM scheme in order to
achieve consistent performances, as in the AWGN channel. Therefore, an
interesting future research goal would be to investigate an NN-based multi-
user channel equalizer or power control schemes that can be matched by the
proposed scheme.

##  Appendix A Proof of Proposition 1

Let $\varepsilon(K,M)$ denote an SER, which is defined as

$\displaystyle\varepsilon(K,M)=p[g(\mathbf{y})\neq i|f(i)],$ (A-1)

for an equally likely symbol, indexed by $i=1,\cdots,M$. In conventional
communication theory, the encoder and decoder are designed to minimize the SER
by solving the following optimization problem:

$\displaystyle({{f}^{*}},{{g}^{*}})=\underset{f,g}{\mathop{\arg\min}}\,\varepsilon(K,M).$
(A-2)

However, in AE-based optimization, (4) is solved by replacing the loss
function with the L2-norm function, which is given as

$\displaystyle
L(\mathbf{r},\mathbf{\hat{r}})=L(\mathbf{r},g(f(\mathbf{r})))={{\left\|\mathbf{r}-g(f(\mathbf{r}))\right\|}_{2}}.$
(A-3)

for an input vector $\mathbf{r}$ and its estimated
$\mathbf{\hat{r}=}g(f(\mathbf{r}))$ with
$\dim(\mathbf{r})=dim(\mathbf{\hat{r}})\geq{{\log}_{2}}M$. Furthermore, the
encoder–decoder pair from the AE that was optimized by L2-norm in (A-3) is
given as:

$\displaystyle(f_{AE}^{{{L}_{2}}},g_{AE}^{{{L}_{2}}})=\underset{(f,g)}{\mathop{\arg\min}}\,{{L}_{2}}\left(\mathbf{r},\mathbf{\hat{r}}\left|{{\theta}^{(f)}},{{\theta}^{(g)}}\right.\right).$
(A-4)

It is demonstrated in [27] that MLD minimizes SER, which is determined by the
$K$, $M$, and SNR levels. Furthermore, as all $M$ symbols are transmitted with
an equal likelihood, and AWGN is assumed; therefore ML-based detection and the
minimum ED (MED)-based detection achieve the same SERs [29]. Therefore,

$\displaystyle{{\varepsilon}_{MED}}(K,M)={{\varepsilon}_{ML}}(K,M)=\underset{f,g}{\mathop{\min}}\,\varepsilon(K,M),$
(A-5)

where ${{\varepsilon}_{MED}}(K,M)$ denotes the SER with the minimum ED
detection. Furthermore, in conventional communication theory, the optimal
encoder–decoder pair that achieves ${{\varepsilon}_{MED}}(K,M)$ can be
designed by solving following the optimization problem:

$\displaystyle(f_{MED}^{*},g_{MED}^{*})=$
$\displaystyle\underset{(f,g)}{\mathop{arg\min}}\,p\left[{{\left\|{{\mathbf{r}}_{i}}-g(f({{\mathbf{r}}_{i}}))\right\|}_{2}}>{{\left\|{{\mathbf{r}}_{j}}-g(f({{\mathbf{r}}_{i}}))\right\|}_{2}}|f({{\mathbf{r}}_{i}})\right],\forall
i,j\neq i,$

for the $k$-th symbol ${{\mathbf{r}}_{k}}$.

If the AE is perfectly trained with the L2-norm, (A-4) for the L2-norm loss
function in (A-3) results in

$\displaystyle{{\left\|{{\mathbf{r}}_{i}}-g_{AE}^{{{L}_{2}}}(f_{AE}^{{{L}_{2}}}({{\mathbf{r}}_{i}}))\right\|}_{2}}\to
0.$ (A-7)

It should be noted that the optimal encoder-decoder pair for minimum ED
detection in (A-6) can be achieved when (A-7) goes to zero. Therefore, the
optimal encoder-decoder pair in (A-6) can be achieved by globally minimize
(A-4). Proposition 1 is thus established through the following conclusion:

$\displaystyle(f_{AE}^{{{L}_{2}}},g_{AE}^{{{L}_{2}}})=(f_{MED}^{*},g_{MED}^{*}).$
(A-8)

###### .

∎

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# Comment on “Macroscopic surface charges from microscopic simulations” [J.
Chem. Phys. 153, 164709 (2020)]

Zhonghan Hu<EMAIL_ADDRESS>Qingdao Institute for Theoretical and
Computational Sciences (QiTCS), Shandong University, Qingdao, 266237, P. R.
China  Institute of Frontier and Interdisciplinary Science, Shandong
University, Qingdao, 266237, P. R. China

Sayer and Cox (hereafter referred as the authors) recently presented a method
to simulate a slab of electrolyte confined between walls composed of $n+1$
layers of charged surfaces, depicted in Fig.1 (b) of ref.Sayer_Cox2020 , under
the 3-dimensional periodic boundary condition (PBC). The authors strongly
suggested that both $H_{\rm YB}$, which is the Yeh-Berkowitz (YB) method that
adds a correction term to the usual Ewald sum with the tinfoil boundary
condition (e3dtf)Yeh_Berkowitz1999 , and the mirrored slab geometry are
unsuitable for modeling those systems. In our opinion, however, both YB method
and the mirrored slab geometry can be used to achieve the goal of ref.[1], as
already explained in a previous publicationPan_Hu2019 . Besides, all methods
that satisfy the symmetry-preserving mean-field (SPMF) conditionPan_Hu2019 ;
Hu2014spmf  work well at an arbitrarily large value of $n$, not just $n=1$, as
pointed out incorrectly by the authors. At last, while the slab system by
itself is of interest to study, the authors’ assertion of its physical
relevance is completely wrong — the corresponding Coulomb system in the
absence of an external electric field does not really exist in the framework
of statistical mechanics.

I shall first simply repeat the theory in ref.Pan_Hu2019  for the present case
of electrolyte (dielectric constant $\varepsilon_{r}=\infty$) to illustrate
the artifacts produced by e3dtf in slab geometry. Physics at the level of high
school states that a pair of infinitely large walls with the surface charge
densities $\pm\sigma$ separated at a finite distance of $d$ produces an
electric field of $\sigma/\varepsilon_{0}$ inside the wall, and zero electric
field outside the wall as a consequence of the $1/r$ Coulomb interaction. When
such a pair of walls is simulated under the PBC of length $L$, the electric
field produced by the corresponding e3dtf potential, which is modified from
$1/r$, is instead $\sigma(L-d)/(\varepsilon_{0}L)$ inside, and $-\sigma
d/(\varepsilon_{0}L)$ outside (see bottom panel of Fig.1 in ref.Pan_Hu2019 ).
In the slab system depicted in Fig.1 (b) of ref.Sayer_Cox2020 , there are
$(n+1)/2$ pairs of fixed walls ($n+1$ solid lines), each of which is separated
at a distance of $R$ with the surface charge densities $\mp\sigma_{0}$, and
one pair of walls composed of mobile ions ($2$ dash lines) with the surface
charge densities $\pm\sigma^{\rm macro}$ separated at $L-nR-\Delta$ after the
left dash line at $x$ is PBC-transformed to the right at $L+x$ to obtain a
continuous region of bulk electrolyte as done in Fig. 1 of ref.Pan_Hu2019 .
The continuous bulk electrolyte therefore locates outside the $(n+1)/2$ pairs
of fixed walls, and simultaneously inside the pair of mobile walls. At
equilibrium, it experiences no net electrostatic forces on average:

$-\frac{n+1}{2}\frac{\sigma_{0}}{\varepsilon_{0}}\frac{R}{L}+\frac{\sigma^{\rm
macro}}{\varepsilon_{0}}\frac{nR+\Delta}{L}=0.$ (1)

Relying on the fact that $\Delta$ depends weakly on the exerted electric
field, which is explained in Fig. 2 of ref.Pan_Hu2019 , the derived
compensating charge as a function of $n$,

$\sigma^{\rm macro}=\frac{\sigma_{0}}{2}\frac{n+1}{n+\Delta/R},$ (2)

therefore explains successfully the simulation data in ref.Sayer_Cox2020  as
shown by Table 1 and Fig. 1. A more elegant theory handling the general case
of multiple dielectric fluids with arbitrary values of $\varepsilon_{r}$
($1\leqslant\varepsilon_{r}\leqslant\infty$) has been presented extensively in
ref.Pan_Hu2019  to explain the otherwise not well explained simulation data of
ref.Zhang_Sprik2016  (see Fig. 4 in ref.Pan_Hu2019  and Fig. 8 in
ref.Zhang_Sprik2016 ) where the finite-field method, appreciated deeply now by
the authors, was originally presented.

n |  3 |  5 |  7 |  11 |  23
---|---|---|---|---|---
$A\sigma^{\rm macro}$ |  7.34 |  7.47 |  7.60 |  7.74 |  7.87
$\Delta({\rm nm})$ |  0.22 |  0.23 |  0.23 |  0.23 |  0.23
Table 1: Simulation setup, $\sigma^{\rm macro}$ extracted from Fig. 2 of
ref.Sayer_Cox2020 , and $\Delta$ derived from Eq. (2) with $R=0.1628$ nm

Figure 1: Spurious compensating charges as a function of $n$ in the presence
of no external field given the smallest (black solid line) and largest (red
dash line) possible values of $\Delta$. Data points (blue filled triangles)
extracted from Fig.2 in ref.Sayer_Cox2020  are listed in Table 1.

The existence of the nonzero compensating charges violating the basic argument
of high school physics reflects essentially the spurious effect of the e3dtf
potential for the slab geometry. Of course, there will be no compensating
charges at all once the periodicity in the $z$ direction is removed and a
formally exact Ewald2D potential or its SPMF approximation is used
subsequently (e.g. refs.Parry1975 ; Pan_Hu2014 ; Pan_Hu2019 ). Indeed, the
mentioned YB method is an example of SPMF approximation to Ewald2D by adding a
term corresponding to a planar limit of $k\to 0$ term of the usual Ewald3D
sumHu2014ib ; Pan_Hu2017 ,

$-\frac{1}{2\varepsilon_{0}V}\sum_{i<j}q_{i}q_{j}z_{ij}^{2}=\frac{1}{2\varepsilon_{0}V}\left(\sum_{j}q_{j}z_{j}\right)^{2}=\frac{VP^{2}}{2\varepsilon_{0}},$
(3)

where $V$ is the volume of the primary unit cell and $P$ is the density of
total dipole moment in the $z$ direction. $z_{ij}^{2}$ in the present setup
depicted in Fig. 1(b) of ref.Sayer_Cox2020  is instead interpreted as the non-
periodic counterpart of the minimum image convention (see Fig.5 of
ref.Pan_Hu2019 ).

The above discussion for the case of no external field immediately suggest
that, in the presence of an applied external electric field, there are two
routes to reach any amount of desired compensating charge $\sigma_{\rm
desired}$ (e.g. $\sigma_{0}/2$ or $\sigma_{0}/4$) in simulation: either by
simply applying any SPMF-condition satisfied method (e.g. refs.Parry1975 ;
Yeh_Berkowitz1999 ; Zhang_Sprik2016 ; Hu2014spmf ; Pan_Hu2019 ) or applying a
SPMF-condition not satisfied method (like the usual e3dtf) but paying enough
attention to the actual response and the spurious effect as in Eq. (1). For
the former, the electric field applied simply reads

$E_{\rm external}^{\rm spmf}+\frac{\sigma_{\rm desired}}{\varepsilon_{0}}=0.$
(4)

A special case of the above equation would be using the YB correction method —
Eq. (3) plus the usual e3dtf under the electric field of $E_{\rm
external}^{\rm spmf}$ — which is fully equivalent to setting $E_{\rm
external}^{\rm spmf}=D$ in eq. (4) of ref. Sayer_Cox2020  up to a constant of
$D^{2}V/(8\pi)$ and a prefactor of $1/(4\pi\varepsilon_{0})$. $D$ used by the
authors is actually an external field ! Any method in this category would be
either the exact Ewald2D or its SPMF approximation. For the latter, paying
enough attention to the spurious effect as in Eq. (1) yields

$E_{\rm external}^{\rm
e3dtf}-\frac{n+1}{2}\frac{\sigma_{0}}{\varepsilon_{0}}\frac{R}{L}+\frac{\sigma_{\rm
desired}}{\varepsilon_{0}}\frac{nR+\Delta}{L}=0,$ (5)

which however involves the static limit of the SPMF approximationPan_Hu2019 ;
Yi_Hu2017mf . In a word, as discussed already in ref.Pan_Hu2019 , there are
plenty of SPMF approximations or the static limits, that account for the
desired responses of mobile charges, including the one appreciated, the one
pointed out incorrectly, those denied, and many others not mentioned by the
authors.

For the mirror slab geometry depicted in Fig. 1(c) of ref.Sayer_Cox2020  with
or without an external field, two routes exist again. For the former with YB
method, because of the mirror slab and the large empty space, the minimum
image convention is the same as its non-periodic counterpart, much less
attention should be paid in the employment of $z_{ij}$Pan_Hu2019 . For the
latter case of the static limit of SPMF, the second term of Eq. (5) will
cancel the contribution from its mirror ( $\sigma_{0}$ versus $-\sigma_{0}$).
Of course, in the presence of an external electric field, the exposed surface
and its mirror face two oppositely charged “compensating” layers respectively,
one is with cations, the other with anions. Either or both would be of
particular interest.

In contrast to what $D=E_{\rm external}^{\rm spmf}$ and the equivalence
between the developed method in ref.Sayer_Cox2020  and the SPMF approximation
clearly state — the slab system in fact responds to an applied external
electric field, the authors, however, mysteriously regarded their Hamiltonian
as a realization of a macroscopic crystal placed inside the electrolyte in the
absence of any applied external electric field (see bottom of Fig.1(a) in
ref.Sayer_Cox2020 ). Such an assertion of the physical relevance is simply
inconsistent with the framework of statistical mechanics. To be more precise,
let ${\mathbf{R}}_{1},{\mathbf{R}}_{2},\cdots{\mathbf{R}}_{N}$ and
${\mathbf{r}}_{1},{\mathbf{r}}_{2},\cdots,{\mathbf{r}}_{M}$ specify the
positions of immobile atoms in the crystal and the degrees of freedom of
mobile particles (ions and atoms in water molecules) respectively, with both
$M$ and $N$ on the order of $10^{23}$, with the use of Coulomb potential $1/r$
and any short-range bonded and non-bonded interactions in the absence of an
external electric field, one just can not write down a Hamiltonian with its
corresponding partition function producing compensating surface charge
densities (e.g. $\pm\sigma_{0}/2$ ) — nothing to do with interpreting the
experiment, truly or falsely — there is just no such a Hamiltonian as a
function of the $\sim 10^{23}$ specific variables in the framework of
statistic mechanics.

I would like to thank Xuhui Huang, Chu Li, and Zhuo Liu at HKUST for
stimulating discussions and the support from NSFC (grant no.21873037).

## References

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