# Deep Learning Aided Beamforming for Downlink Non Orthogonal Multiple Access
Systems

Georgios Konstantopoulos, Georgios A. Ropokis, Yves Louët G. Konstantopoulos
is with CentraleSupélec, Campus Rennes, 35510, Cesson Sevigne, France and the
Institute of Electronics and Telecommunications of Rennes
(e-mail:georgios.konstantopoulos@centralesupelec.fr).G. A. Ropokis is with
CentraleSupélec, Campus Rennes, 35510, Cesson Sevigne, France and the
Institute of Electronics and Telecommunications of Rennes
(e-mail:georgios.ropokis@centralesupelec.fr).Y. Louët is with CentraleSupélec,
Campus Rennes, 35510, Cesson Sevigne, France and the Institute of Electronics
and Telecommunications of Rennes (e-mail:Yves.Louet@centralesupelec.fr).This
paper was submitted and is under review by IEEE Transactions on Machine
Learning in Communications and Networking Journal (TMLCN).

###### Abstract

We investigate the problem of optimal beamformer design for the downlink of
Multi Input Single Output (MISO) Non-Orthogonal Multiple Access (NOMA). In
more detail, focusing on the two-user scenario, we first derive a closed from
expression for the Bit Error Rate (BER) experienced by both user. Using the
derived expression, in an effort to introduce fairness in our system design,
we introduce the problem of optimal, with respect to minimizing the maximum of
the BER values experienced by the two users, beamforming and propose a Machine
Learning (ML) based solution for this problem. Finally, we conduct simulations
which allow us to verify that our proposed algorithm outperforms other
existing benchmarks as well as that in a variety of cases, it may result to
BER performance close to the one obtained by the use of time consuming
constrained optimization methods, such as to solve the given optimization
problem.

###### Index Terms:

Bit Error Rate, Deep Learning, Machine Learning, MISO, Neural Networks, NOMA,
QAM

##  I Introduction

One of the main novelties of fifth generation (5G) and beyond 5G communication
systems is the use of novel Medium Access Schemes (MAC) such as Non Orthogonal
Multiple Access [1, 2]. The key concept introduced by NOMA, and more
specifically of power domain NOMA is the concurrent access of spectral
resources by multiple users and the use of successive interference
cancellation such as to retrieve the transmitted signals at the intended
receiver(s) [3]. Following such an approach, not only can we increase the
total transmit rate of our system, but also achieve low latency, massive
connectivity and spectral efficiency.

Due to the above important advantages of NOMA, in the open technical
literature, several different approaches exist for optimizing the parameters
and the design of NOMA communication techniques (see for example [4, 5, 6]).
Specifically, in [4] we find a study of NOMA in terms of power allocation, in
[5] in terms of outage probability and in [6] in terms of user pairing.
Subsequently, in [7, 8, 9] we find studies on multiple antenna NOMA, with [7]
emphasizing on the review of multiple-antenna aided NOMA structures and the
resource management of MIMO-NOMA, [8] exploring a multi-user MIMO-NOMA scheme
and focusing on the clustering of NOMA users and the power allocation in terms
of spectral efficiency and [9] that consider a power minimization problem in a
MISO-NOMA scheme, based on an energy harvesting scenario.

However, in their majority, the above works fail to tackle the most severe
limiting factor of NOMA, i.e., that fact that it may result in high error
rates for the weak user. Motivated by this in this work we focus on optimizing
the design of NOMA multi-antenna downlink. In more detail, we focus on the
case that users are grouped in pairs in order to access the channel and seek
to solve the optimal beamforming problem such as to maximize the fairness,
experienced by the two users concurrently accessing the channel. To this end,
the measure which we use in order to express fairness is the maximum BER of
the two users served concurrently. Besides the fact that this metric
introduces fairness in terms of the reliability experienced by two users, it
also manages to tackle the problem of high error rates, discussed earlier. For
this reason, the same metric has also been considered in different research
works [10, 11, 12, 13, 14].

While the performance metric used in this work, i.e., the maximum BER provides
as a meaningful way to optimize the fairness and reliability of our system, as
it will become evident in later sections, it does not allow to express the
optimal beamforming solution in closed form and requires solving a non-convex
optimization problem, i.e., the beamforming problem, in real time. To overcome
the complications introduced by such an approach, in this work, we propose the
use of Machine Learning (ML) techniques, and more particular Deep Learning
(DL) such as to learn the association between observed Channel State
Information and optimal beamforming solution. Our choice is motivated by the
fact that DL has shown competitive, with respect to conventional optimization
methods, performance results when used for solving optimization problems in
communications systems. In more detail, in the recent literature DL has been
successfully applied for the solving routing optimization [15], detection and
interference managing [16, 17] and for channel estimation [18] problems.
Furthermore, DL has also been in order to solve power allocation problems in
multiple antenna systems. For example, in [19, 20, 21, 22, 23, 24] DL is used
in order to provide power allocation solutions which maximize the minimum
capacity of the users [19], maximize the spectral efficiency (SE) [20, 21],
eliminate interference [22, 23] and optimize the reliability of NOMA systems
[24].

Motivated by the aforementioned studies and considerations, in this paper we
examine a general MISO-NOMA system and conduct a comprehensive study in order
to optimize in in terms of BER fairness. We base our optimization on the use
of DL techniques. The main contributions of this paper are summarized as
follows.

  * •

We introduce the fairness optimization problem between two users in a MISO-
NOMA downlink system, subject to a total power consumption constrain. That is,
that we try to optimize the downlink beamformer design in order to minimize
the maximum BER between the two users. To do so, we also propose a novel
beamforming structure that is based on an orthonormal basis.

  * •

We derive the closed form BER expression for each user in the MISO-NOMA
scenario. In our system we consider Quadrature Amplitude Modulation (QAM) to
be used for the transmission of the user symbols. These expressions are used
afterwards for the solution of the optimization problem and the beamformers’
design.

  * •

We propose a learning algorithm that is based on the use of DL. We conduct
extensive simulations, provide results and a performance analysis of the
proposed DL-based method. We demonstrate the efficiency and robustness of the
proposed scheme by comparing it with other widely used beamforming techniques.

The rest of the paper is organized as follows. In Sec.II the system model is
presented, as well as the formulation of our problem. The BER analysis of our
system is presented in Sec.III. In Sec.IV the proposed solution algorithm is
introduced, and in Sec.V the simulation results are presented. Finally, in
Sec.VI there are the conclusions of our work, as well as the future ideas
around it.

##  II System Model

We focus on the downlink of a single cell NOMA-based system, where users are
grouped in pairs of two in order to access the channel concurrently. As shown
in Fig. 1, the Base Station (BS) of our cell is equipped with $N_{t}$ antennas
and the two users being served concurrently have single-antenna User Equipment
(UE). We use notation $U_{n}$ for the $n$-th user, $n=1,2$, and we assume that
$U_{n}$ uses Quadrature Amplitude Modulation (QAM), with bits being mapped to
QAM symbols with the use of Gray coding. Moreover, we use notation $M_{n}$ for
the modulation order of the constellation used by $U_{n},n=1,2$.

###  II-A Operation at the transmitter side (BS)

Let $s_{n}$ be the QAM symbol intended for $U_{n},n=1,2$ during a given symbol
period, and $A_{n,I},A_{n,Q}$ the in-phase and quadrature components of
$s_{n}$ respectively. It then follows $A_{n,I}$ and $A_{n,Q}$ take values of
the form:

$A_{n,I}=\pm(2k_{n}-1),\;\;\textrm{with}\;\;k_{n}=-\frac{\sqrt{M_{n}}}{2}+1,...\;,\frac{\sqrt{M_{n}}}{2},$
(1)

and:

$A_{n,Q}=\pm(2l_{n}-1),\;\;\textrm{with}\;\;l_{n}=-\frac{\sqrt{M_{n}}}{2}+1,...\;,\frac{\sqrt{M_{n}}}{2},$
(2)

and the transmitted QAM symbol is expressed as:

$s_{n}=\frac{A_{n,I}}{\sqrt{E_{n}}}+j\frac{A_{n,Q}}{\sqrt{E_{n}}},$ (3)

where $E_{n}=\frac{2}{3}(M_{n}-1),n=1,2$ are normalization constants used such
as to ensure that the average power of the constellation used by the two users
is equal to one [25]. We also note that the total transmit power of the BS is
normalized to unity.

In order to properly exploit the multiple antennas at the BS, the BS applies
beamforming on the signal of the $n$-th user prior to transmitting it. As a
result, the signal which is actually transmitted is expressed as:

$\mathbf{x}=\mathbf{w}_{1}s_{1}+\mathbf{w}_{2}s_{2}$ (4)

where $\mathbf{w}_{n}\in\mathbb{C}^{N_{t}}$ the complex beamforming vector for
user $U_{n}$, where $\left\|\mathbf{w}_{n}\right\|_{2}=1,n=1,2$. Given the
above signal model at the transmitter side, in what follows, we describe the
operation at the receiver side.

###  II-B Operation at the receiver side (users)

Given that both users employ single-antenna terminals, the signal at $U_{n}$
is expressed as:

$y_{n}=\mathbf{h}_{n}^{H}\mathbf{x}+\nu_{n}.$ (5)

where $\nu_{n}\sim\mathcal{CN}\left(0,N_{0}\mathbf{I}\right)$ is the additive
white Gaussian noise at the $n$-th user, with $N_{0}$ being the noise variance
and $\mathbf{I}$ the identity matrix. Moreover, vector
$\mathbf{h}_{n}\in\mathbb{C}^{N_{t}}$ is the channel vector formed between the
BS and $U_{n}$ which we model it using a combination of pathloss and
independent identically distributed (i.i.d.) Rayleigh fading channel. As a
result, $\mathbf{h}_{n}$ is expressed as:

$\mathbf{h}_{n}=\sqrt{\beta_{n}}\mathbf{\tilde{h}}_{n}$ (6)

where $\mathbf{\tilde{h}}_{n}\sim\mathcal{CN}(\mathbf{0},\mathbf{I})$
represents the small-scale fading and $\beta_{n}$ represents the large-scale
fading that can be formulated as

$\beta_{n}=10^{\frac{\textrm{PL}_{n}}{10}}$ (7)

with $\textrm{PL}_{n}$ the path loss expressed in dB as in [26].

As far as message decoding is concerned, we will assume that
$\left\|\mathbf{h}_{1}\right\|_{2}\leq\left\|\mathbf{h}_{2}\right\|_{2}$ and
that, as in [27, 28] the user with the weakest channel (i.e., $U_{1}$)
directly decodes the signal intended for it. In more detail the decoding
process at the two users is summarized as follows.

Figure 1: System model.

####  II-B1 Decoding at $U_{1}$

Combining (4) and (5) the signal received by $U_{1}$ can be written as:

$y_{1}=\mathbf{h}_{1}^{H}\mathbf{w}_{1}s_{1}+\mathbf{h}_{1}^{H}\mathbf{w}_{2}s_{2}+\nu_{1}.$
(8)

As a result, assuming perfect knowledge of the channel responses and the
beamformers applied by the BS, $U_{1}$ starts by applying phase correction,
such as to correct the rotation caused by the channel to its constellation.
This phase correction involves multiplying $y_{1}$ with a complex exponential
of phase selected to be opposite to the phase of the complex quantity
$\mathbf{h}_{1}^{H}\mathbf{w}_{1}$. In more detail, introducing terms

$\theta_{1,j}=\angle\mathbf{h}_{1}^{H}\mathbf{w}_{j},\;\;j=1,2,$ (9)

phase correction at $U_{1}$ involves multiplying $y_{1}$ with
$e^{-j\theta_{1,1}}$. The resulting signal is then expressed as:

$\tilde{y}_{1}=|\mathbf{h}_{1}^{H}\mathbf{w}_{1}|s_{1}+|\mathbf{h}_{1}^{H}\mathbf{w}_{2}|e^{j\left(\theta_{1,2}-\theta_{1,1}\right)}s_{2}+\tilde{\nu}_{1}$
(10)

where $\tilde{\nu}_{1}=\nu_{1}e^{-j\theta_{1,1}}$. Since no interference
cancellation is applied at $U_{1}$, the detection of $s_{1}$ at $U_{1}$ is
based on $\tilde{y}_{1}$. Moreover, for the detection process we consider the
use of a maximum likelihood (ML) detector and assuming perfect knowledge of
the channel coefficeint $\hat{\mathbf{h}}_{1}=\mathbf{h}_{1}$. The detection
decision is then expressed as:

$\hat{s}_{1}=\arg\min\limits_{s_{1}}\bigg{|}\tilde{y}_{1}-|\mathbf{h}_{1}^{H}\mathbf{w}_{1}|s_{1}\bigg{|}^{2}.$
(11)

####  II-B2 Decoding at $U_{2}$

Using again (4) and (5), the received signal at $U_{2}$ can be written as

$y_{2}=\mathbf{h}_{2}^{H}\mathbf{w}_{1}s_{1}+\mathbf{h}_{2}^{H}\mathbf{w}_{2}s_{2}+\nu_{2}.$
(12)

Using $y_{2}$, $U_{2}$ first applies SIC in order to decode and remove the
signal of $U_{1}$ from the received one. Following that, it decodes the signal
that was intended for it. In order to describe this process with a
mathematical formalism, let us introduce the variables
$\theta_{2,j}=\angle\mathbf{h}_{2}^{H}\mathbf{w}_{j},\;\;j=1,2$. We can then
express the signal used for decoding $U_{2}$ as:

$\tilde{y}_{2}=|\mathbf{h}_{2}^{H}\mathbf{w}_{1}|e^{j(\theta_{2,1}-\theta_{2,2})}s_{1}+|\mathbf{h}_{2}^{H}\mathbf{w}_{2}|s_{2}+\tilde{\nu}_{2},$
(13)

with $\tilde{\nu}_{2}=\nu_{2}e^{-j\theta_{2,2}}$. In the following theorem, we
explain the transformation of (13) with the use of (18).

As we saw in previous section, all NOMA users but the first, need to apply SIC
on the received signal in order to be able to decode their own symbols.
Specifically, in the two case scenario that we explore, the second user will
apply SIC to remove the first user’s signal before decoding his own. The
detection of $U_{1}$’s signal at $U_{2}$ can be presented as

$s^{\prime}_{1}=\arg\min\limits_{s_{1}}\bigg{|}\tilde{y}_{2}-|\mathbf{h}_{2}^{H}\mathbf{w}_{1}|e^{j(\theta_{2,1}-\theta_{2,2})}s_{1}\bigg{|}^{2}.$
(14)

The next step of the SIC procedure will be to remove the estimated $U_{1}$’s
signal from the received one and afterwards detect the symbol that was
intended for the second user. We can combine these to steps as in the
following equation

$\hat{s}_{2}=\arg\min\limits_{s_{2}}\bigg{|}\tilde{y}_{2}-|\mathbf{h}_{2}^{H}\mathbf{w}_{1}|e^{j(\theta_{2,1}-\theta_{2,2})}s^{\prime}_{1}-|\mathbf{h}_{2}^{H}\mathbf{w}_{2}|s_{2}\bigg{|}^{2},$
(15)

where $s_{1}^{\prime}$ is again the estimated symbol o user $U_{1}$. The
appearance of $s_{1}^{\prime}$ in the detector of $U_{2}$’s signal highlights
that the probability of bit error of the second user is tightly connected with
the estimation of $s_{1}^{\prime}$ during the SIC.

###  II-C The considered optimization problem

Working on the system model introduced above, in this paper we seek to
optimize the downlink beamformer design at the BS such as to maximize the
fairness or our system, subject to a total power consumption constraint. To
this end, we set as objective of our system design to minimize the maximum BER
of the two users. In more detail, the system design which we consider is based
on solving the following optimization problem:

$\begin{split}{}&{}\textrm{minimize}_{\mathbf{w}_{1},\mathbf{w}_{2}}\\{\max\\{P_{e}^{(1)},P_{e}^{(2)}\\}\\}\\\
&\textrm{subject
to:}\;\;\left\|\mathbf{w}_{1}\right\|^{2}+\left\|\mathbf{w}_{2}\right\|^{2}\leq
1,\end{split}$ (16)

where $P_{e}^{(n)},n=1,2,$ is the BER experienced by $U_{n}$. The reason for
choosing a BER-focused perfomance metric is the fact that such a metric allows
to capture one of the most important characteristic of a communication
system/standard, which is the modulation scheme. Therefore, such a metric can
optimize the performance of true deployments. This is not necessarily the case
when rate-based metrics, such as the sum rate, are used, which cannot capture
real network characteristics (such as the modulation scheme) and potential
errors in the SIC process.

In the following sections, we discuss the process of solving problem (16).
This process also includes the derivation of closed form expressions for the
BER of the two users.

##  III BER analysis of our system

In order to express the BER expressions for our users in convenient closed
forms, we firs propose a methodology for describing the beamformers
$\mathbf{w}_{n},n=1,2$. As it will become apparent in later parts of our
analysis, our approach for describing the beamformers allows not only to
derive a closed form expression for the BER values of the two users, but also
to simplify the process of optimal beamformer design.

###  III-A Beamformer structure

Following the approach introduced in [29] and [30] we choose to to express
beamfomers $\mathbf{w}_{1}$ and $\mathbf{w}_{2}$ with the aid of an
orthonormal basis $\\{\mathbf{u}_{1},...,\mathbf{u}_{N}\\}$ for
$\mathcal{C}^{N_{t}\times 1}$, where we have selected vectors $\mathbf{u_{1}}$
and $\mathbf{u_{2}}$ as follows:

$\mathbf{u_{1}}=\frac{\mathbf{h_{1}}}{\lVert\mathbf{h_{1}}\rVert},\;\;\;\;\;\mathbf{u_{2}}=\frac{(\mathbf{I}-\mathbf{u_{1}u_{1}^{H}})\mathbf{h_{2}}}{\lVert(\mathbf{I}-\mathbf{u_{1}u_{1}^{H}})\mathbf{h_{2}}\rVert}$
(17)

As a result, beamformers $\mathbf{w}_{n},n=1,2,$ can then be written in the
form

$\mathbf{w}_{1}=\sum_{i=1}^{N}\rho_{i}e^{j\theta_{i}}\mathbf{u}_{i},\;\;\textrm{and}\;\;\mathbf{w}_{2}=\sum_{i=1}^{N}\delta_{i}e^{j\phi_{i}}\mathbf{u}_{i}.$
(18)

where $\rho_{i}$ and $\delta_{i}\in\mathbb{R}^{+},i=1,\ldots,N_{t}$, and
$\theta_{i},\phi_{i}\in[0,2\pi),i=1,\ldots,N_{t}$. Using this representation
for the beamformers, we now present our novel BER expressions, starting wit
$U_{1}$.

###  III-B BER performance at $U_{1}$

We start by exploiting the beamformer structure in (18) such as express the
signal $\tilde{y}_{1}$ in a more convenient form. This is done with the help
of the following theorem. The result of this process is summarized with the
help of the following theorem.

###### Theorem 1

The decision variable $\tilde{y}_{1}$ can be equivalently written as:

$\tilde{y}_{1}=\rho_{1}\lVert\mathbf{h}_{1}\rVert
s_{1}+\delta_{1}e^{j(\tau_{1})}\lVert\mathbf{h}_{1}\rVert s_{2}+\nu_{1}$ (19)

where $\tau_{1}=\phi_{1}-\theta_{1}$. Assuming that the AWGN is circularly
symmetric, then the probability density function of the noise will be the same
before and after the phase correction in each user. Subsequently, $\nu_{n}$
can be used instead of $\tilde{\nu}_{n}$.

###### Proof:

Using the abovementioned orthonormal basis, we obtain:

$\mathbf{h}_{1}^{H}\mathbf{u}_{1}=\lVert\mathbf{h_{1}}\rVert,\;\;\textrm{and}\;\;\mathbf{h}_{1}^{H}\mathbf{u}_{m}=\lVert\mathbf{h}_{1}\rVert\mathbf{u}^{H}_{1}\mathbf{u}_{m},$
(20)

for $m=2,\ldots,K$. Hence, by substituting (18) in (10) and exploiting (20) as
well as the fact that $\mathbf{u}_{1}^{H}\mathbf{u}_{m}=0$, for $m\neq 1$ the
form described by the theorem for $\tilde{y}_{1}$ is obtained. ∎

Using (19), the ML decoder in (11) is written as:

$\hat{s}_{1}=\arg\min\limits_{s_{1}}\bigg{|}\tilde{y}_{1}-\rho_{1}\lVert\mathbf{h}_{1}\rVert
s_{1}\bigg{|}^{2},$ (21)

and the BER of $U_{1}$ can then be calculated with the help of the following
theorem.

In what follows, using (21), we derive a closed form expression for the BER of
$U_{1}$. Let $P_{b,i}^{(1)}$ be the probability of an error on the $i-th$ bit
of a symbol of $U_{1}$. As discussed in (e.g. [31]) one can then express the
average conditional BER for $U_{1}$ as:

$P_{e}^{(1)}=\frac{2}{\log_{2}\sqrt{M_{1}}}\sum_{i=1}^{\log_{2}\sqrt{M_{1}}}P_{b,i}^{(1)}.$
(22)

and the BER expression can be derived by finding the probability of error for
each one of the transmit bits which constitute a QAM symbol, separately. Let
us therefore focus on the $i$-th bit of a symbol,
$i\in\left\\{1,2,\ldots,\log_{2}M_{1}\right\\}$. Below we can find the
expression that gives the closed form for $P_{b,i}^{(1)}$.

###### Theorem 2

Assuming Gray coding, the BER for the $i$-th bit of a QAM symbol is given as:

$\begin{split}{}&{}P_{b,i}^{(1)}=\frac{1}{M_{2}\sqrt{M_{1}}}\times\sum_{k=0}^{L_{1,1}}\sum_{l=0}^{\Lambda_{2}}\sum_{m=0}^{\Lambda_{2}}\Bigg{[}D_{1}(i,k)\cdot\\\
&\;\;\;\;\;\;\;\;\;\;\;Q\bigg{(}g_{1}\bigg{(}2k+1,2l-\sqrt{M_{2}}+1,2m-\sqrt{M_{2}}+1\bigg{)}\bigg{)}\Bigg{]},\\\
&\textrm{where}\;\;D_{1}(i,k)=(-1)^{\lambda_{i,k,1}}\bigg{(}2^{i-1}-\bigg{\lfloor}\frac{k2^{i-1}}{\sqrt{M_{1}}}+\frac{1}{2}\bigg{\rfloor}\bigg{)},\\\
&\;\;\;\;\;\;\;\;\;\;\;L_{1,1}=(1-2^{-i})\sqrt{M_{1}}-1,\\\
&\;\;\;\;\;\;\;\;\;\;\;\lambda_{i,k,1}=\bigg{\lfloor}\frac{k2^{i-1}}{\sqrt{M_{1}}}\bigg{\rfloor},\;\;\Lambda_{n}=\sqrt{M_{n}}-1\\\
&\textrm{and}\\\
&g_{1}\left(a,b,c\right)=\frac{\frac{a\cdot\rho_{1}\lVert\mathbf{h}_{1}\rVert}{\sqrt{E_{1}}}+\frac{\delta_{1}\lVert\mathbf{h}_{1}\rVert}{\sqrt{E_{2}}}\bigg{(}b\cdot\cos\tau_{1}+c\cdot\sin\tau_{1}\bigg{)}}{\sqrt{N_{0}/2}},\\\
&\end{split}$ (23)

with $Q(\cdot)$ being the $Q$-function.

###### Proof:

As explained in [31], we notice that due to Gray coding, demodulation of the
first $\log_{2}\sqrt{M_{1}}$ bits of $U_{1}$ can be done using only the real
part of $\tilde{y}_{1}$ and the demodulation of the last
$\log_{2}\sqrt{M_{1}}$ bits can be done using the imaginary part of
$\tilde{y}_{1}$. As a result, the interference experienced during demodulation
of any of the bits of $U_{1}$ carried by the real part of $\tilde{y}_{1}$ is
equal to:

$I_{real,1}=\Re\left\\{\delta_{1}e^{j(\tau_{1})}\lVert\mathbf{h}_{1}\rVert
s_{2}+\tilde{\nu}_{1}\right\\}$ (24)

and the interference experienced during demodulation of any of the bits of
$U_{1}$ carried by the imaginary part of $\tilde{y}_{1}$ is equal to:

$I_{imag,1}=\Im\left\\{\delta_{1}e^{j(\tau_{1})}\lVert\mathbf{h}_{1}\rVert
s_{2}+\tilde{\nu}_{1}\right\\}.$ (25)

Unlike [31], the existing interference of (24) and (25) includes interference
both from the real and the imaginary parts of $e^{j(\tau_{1})}$ and $s_{2}$.
Thus, extending the BER expression from SISO to MISO systems is not a
straightforward procedure. In our analysis, we take into consideration these
interference properties, and we come up with (23), which includes them with
the use of the triple sum expression and $g_{1}$. Expressions $D_{1},L_{1,1}$
and $\Lambda_{n}$ shape expression $P_{b,i}^{(1)}$, as they indicate the
number of $Q$-functions that will appear for each modulation pair ($M_{1}$,
$M_{2}$). Further details on the derivation can be found in Appendix A where
this process is further explained for the example of 4-QAM constellation use
by both users. ∎

###  III-C BER performance at $U_{2}$

###### Theorem 3

The decision variable $\tilde{y}_{2}$ can be equivalently written as:

$\begin{split}{}&{}\tilde{y}_{2}=\big{(}\rho_{1}\big{|}\mathbf{h}_{2}^{H}\mathbf{u}_{1}\big{|}+\rho_{2}\big{|}\mathbf{h}_{2}^{H}\mathbf{u}_{2}\big{|}\big{)}s_{1}\\\
&\;\;\;\;+\big{|}\delta_{1}e^{j\phi_{1}}\mathbf{h}_{2}^{H}\mathbf{u}_{1}+\delta_{2}e^{j\phi_{2}}\mathbf{h}_{2}^{H}\mathbf{u}_{2}\big{|}s_{2}+\tilde{\nu}_{2}.\end{split}$
(26)

###### Proof:

For the second user we need to satisfy some properties. First of all, in the
second user’s case, we can exploit the property of the basis which suggests
that $h_{2}^{H}\mathbf{u}_{m}$ is equal to zero for $m=3,...,N$. Additionally,
we have mentioned before that the second user has to extract the part of the
signal intended for the first user, through the procedure of SIC, and then
decode his own part.Thus, initially (13) will be transformed with the use of
(18) as

$\begin{split}{}&{}\tilde{y}_{2}=\big{(}\rho_{1}e^{j\theta_{1}}\mathbf{h}_{2}^{H}\mathbf{u}_{1}+\rho_{2}e^{j\theta_{2}}\mathbf{h}_{2}^{H}\mathbf{u}_{2}\big{)}s_{1}\cdot
e^{-j\hat{\phi}}\\\
&\;\;\;\;+\big{|}\delta_{1}e^{j\phi_{1}}\mathbf{h}_{2}^{H}\mathbf{u}_{1}+\delta_{2}e^{j\phi_{2}}\mathbf{h}_{2}^{H}\mathbf{u}_{2}\big{|}s_{2}+\tilde{\nu}_{2},\end{split}$
(27)

where
$\hat{\phi}=\angle\big{(}\delta_{1}e^{j\phi_{1}}\mathbf{h}_{2}^{H}\mathbf{u}_{1}+\delta_{2}e^{j\phi_{2}}\mathbf{h}_{2}^{H}\mathbf{u}_{2}\big{)}$.
Since we want to maximize the probabilities of correct SIC, the two components
of the beamformer concerning symbol $s_{1}$ should be aligned as follows

$\frac{e^{j\theta_{1}}\mathbf{h}_{2}^{H}\mathbf{u}_{1}}{|\mathbf{h}_{2}^{H}\mathbf{u}_{1}|}=\frac{e^{j\theta_{2}}\mathbf{h}_{2}^{H}\mathbf{u}_{2}}{|\mathbf{h}_{2}^{H}\mathbf{u}_{2}|}=e^{j\hat{\phi}}.$
(28)

By exploiting the aforementioned property and after setting the phase
parameters $\theta_{1},\theta_{2}$ in such a way that they satisfy (28), we
can obtain $\tilde{y}_{2}$ as in (26). ∎

As mentioned before, with the use of the detector in $U_{2}$ we want to decode
and remove $U_{1}$’s symbol from the received signal and then decode the
symbol intended for $U_{2}$. Consequently, with the help of (15) and (26) the
detector for $U_{2}$’s signal will be

$\begin{split}{}&{}\hat{s}_{2}=\arg\min\limits_{s_{2}}\bigg{|}\tilde{y}_{2}-\big{(}\rho_{1}\big{|}\mathbf{h}_{2}^{H}\mathbf{u}_{1}\big{|}+\rho_{2}\big{|}\mathbf{h}_{2}^{H}\mathbf{u}_{2}\big{|}\big{)}s^{\prime}_{1}\\\
&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;-\big{|}\delta_{1}e^{j\phi_{1}}\mathbf{h}_{2}^{H}\mathbf{u}_{1}+\delta_{2}e^{j\phi_{2}}\mathbf{h}_{2}^{H}\mathbf{u}_{2}\big{|}s_{2}\bigg{|}^{2},\end{split}$
(29)

where is the estimated $U_{1}$’s signal at the second user and can be
presented as

$s^{\prime}_{1}=\arg\min\limits_{s_{1}}\bigg{|}\tilde{y}_{2}-\big{(}\rho_{1}\big{|}\mathbf{h}_{2}^{H}\mathbf{u}_{1}\big{|}+\rho_{2}\big{|}\mathbf{h}_{2}^{H}\mathbf{u}_{2}\big{|}\big{)}s_{1}\bigg{|}^{2}.$
(30)

Based on the nature of QAM constellations and the use of Gray coding of the
symbols at each user, a pattern emerges, similar to the one of $U_{1}$, that
dictates the division of the bits of each user into two equal groups, each
consisting of $u_{2}$ bits and the two groups have the same BER. Consequently,
the average conditional BER expression for $U_{2}$ will be:

$P_{e}^{(2)}=\frac{2}{\log_{2}\sqrt{M_{2}}}\sum_{i=1}^{\log_{2}\sqrt{M_{2}}}{P_{b,i}^{(2)}}.$
(31)

Again, the derivation of this BER expression can be done by finding the
probability of error for each one of the transmitted bits which constitute a
QAM symbol, separately. By focusing on the $i$-th bit of a symbol, where
$i\in\left\\{1,2,\ldots,\log_{2}M_{2}\right\\}$, the expression for
$P_{b,i}^{(1)}$ can be given from the following theorem.

###### Theorem 4

Assuming Gray coding, the BER for the $i$-th bit of a QAM symbol is given as:

$\begin{split}{}&{}P_{b,i}^{(2)}=\frac{1}{\sqrt{M_{1}\cdot M_{2}}}\times\\\
&\;\;\;\;\;\;\;\;\;\bigg{(}\sum_{k=0}^{L_{1,2}}\sum_{l=0}^{2\Lambda_{1}}SD_{2}(i,k)D_{3}(i,l)Q\bigg{(}g_{2,i}^{+}\bigg{(}l,2k+1\bigg{)}\bigg{)}\\\
&\;\;\;\;\;-\sum_{k=0}^{L_{1,2}}\sum_{l=1}^{2\Lambda_{1}}SD_{2}(i,k)D_{3}(i,l)Q\bigg{(}g_{2,i}^{-}\bigg{(}l,2k+1\bigg{)}\bigg{)}\bigg{)}\\\
&\textrm{where}\;\;S=(-1)^{\big{\lfloor}\frac{l2^{(\log_{2}\sqrt{M_{2}})+i-1}}{\sqrt{M_{2}}}\big{\rfloor}+\lambda_{i,k,2}},\\\
&\;\;\;\;\;\;\;\;\;\;\Lambda_{n}=\sqrt{M_{n}}-1,\;\;D_{2}(i,k)=2^{i-1}-\bigg{\lfloor}\frac{k2^{i-1}}{\sqrt{M_{2}}}+\frac{1}{2}\bigg{\rfloor},\\\
&\;\;\;\;\;\;\;\;\;\;D_{3}(i,l)=2^{\log_{2}\sqrt{M_{1}}}-\bigg{\lfloor}\frac{l}{2^{1-(i-1)\log_{2}(\sqrt{M_{1}}-1)}}+\frac{1}{2}\bigg{\rfloor},\\\
&\;\;\;\;\;\;\;\;\;\;L_{1,2}=(1-2^{-i})\sqrt{M_{2}}-1,\;\;\lambda_{i,k,2}=\bigg{\lfloor}\frac{k2^{i-1}}{\sqrt{M_{2}}}\bigg{\rfloor},\\\
&\textrm{and}\\\
&g_{2,i}^{\pm}\left(a,b\right)=\frac{a\frac{\big{(}\rho_{1}\big{|}\mathbf{h}_{2}^{H}\mathbf{u}_{1}\big{|}+\rho_{2}\big{|}\mathbf{h}_{2}^{H}\mathbf{u}_{2}\big{|}\big{)}}{\sqrt{E_{1}}}\pm
b\frac{\big{|}\delta_{1}e^{j\phi_{1}}\mathbf{h}_{2}^{H}\mathbf{u}_{1}+\delta_{2}e^{j\phi_{2}}\mathbf{h}_{2}^{H}\mathbf{u}_{2}\big{|}}{\sqrt{E_{2}}}}{\sqrt{N_{0}/2}}\end{split}$
(32)

with $Q(\cdot)$ being the $Q$-function.

Figure 2: Average BER using $M_{1}=M_{2}$ for $U_{1}$ and $U_{2}$.

###### Proof:

As mentioned before, the probability of error for $U_{2}$ depends on
$s^{\prime}_{1}$ which is the estimation of $U_{1}$’s symbol during the SIC.
In [31] it is explained thoroughly how the error probability of each bit of
$U_{2}$’s symbol is connected only with specific bits of the first users
estimated symbol. Briefly, the bits of $U_{1}$’s symbol can be divided into
two equal groups based on the nature of interference that they cause on the
received signal. The first group can be the set of symbols that introduce
interference at the real part of the received signal and the second group can
contain the bits that introduce interference at the imaginary part of the
signal. Each group of $U_{1}$’s bits can affect $\log_{2}\sqrt{M_{1}}$ bits of
the second users symbol. Thus, the interference that is experienced during the
demodulation of the $s_{2}$ symbol can be expressed as

$I_{real,2}=\Re\left\\{\big{(}\rho_{1}\big{|}\mathbf{h}_{2}^{H}\mathbf{u}_{1}\big{|}+\rho_{2}\big{|}\mathbf{h}_{2}^{H}\mathbf{u}_{2}\big{|}\big{)}s^{\prime}_{1}+\tilde{\nu}_{2}\right\\},$
(33)

for the bits of $s_{2}$ that are carried by the real part of $\tilde{y}_{2}$,
and as

$I_{imag,2}=\Im\left\\{\big{(}\rho_{1}\big{|}\mathbf{h}_{2}^{H}\mathbf{u}_{1}\big{|}+\rho_{2}\big{|}\mathbf{h}_{2}^{H}\mathbf{u}_{2}\big{|}\big{)}s^{\prime}_{1}+\tilde{\nu}_{2}\right\\}$
(34)

for the bits of $s_{2}$ that are carried by the imaginary part of
$\tilde{y}_{2}$. Expressions $S,D_{2},D_{3},L_{1,2}$ and $\Lambda_{n}$ define
the number of $Q$-functions that will appear in each part of $P_{b,i}^{(2)}$
expression. In Appendix A we present the case of $M_{1}=M_{2}=4$ for QAM
constellations in detail. Additionally, in Fig. 2 we can observe the
analytical and the simulated BER of user $U_{1}$ for the aforementioned case.

∎

###  III-D BER optimization problem.

To ensure correct transmission and prevent any overlap among users’ symbols,
it is necessary for the beamformer parameters to adhere to the following
constraints. As we can observe in [32] the following rate has to be satisfied

$\bigg{|}\frac{\rho_{1}}{\delta_{1}(\cos(\tau_{1})-\sin(\tau_{1}))\cdot\Lambda_{2}}\bigg{|}>Y_{1},$
(35)

where $Y_{1}=\frac{M_{1}-1}{M_{2}-1}\cdot(\sqrt{M_{2}}-1)^{2}$.

Likewise, by following the analysis of the previous expressions, we reach to
the conclusion that in order for the SIC to be reliable, the following
expression also has to be valid

$\frac{\rho_{1}\big{|}\mathbf{h}_{2}^{H}\mathbf{u}_{1}\big{|}+\rho_{2}\big{|}\mathbf{h}_{2}^{H}\mathbf{u}_{2}\big{|}}{\big{|}\delta_{1}e^{j\phi_{1}}\mathbf{h}_{2}^{H}\mathbf{u}_{1}+\delta_{2}e^{j\phi_{2}}\mathbf{h}_{2}^{H}\mathbf{u}_{2}\big{|}}>Y_{1}.$
(36)

In addition to these restrictions, we have to consider that there are
limitations on the maximum transmit power of our system. By normalizing the
transmit power of the constraints in (16) to unity, we derive the following
expression

$\rho_{1}^{2}+\rho_{2}^{2}+\delta_{1}^{2}+\delta_{2}^{2}\leq 1.$ (37)

Based on this derived framework, the search for beamforming vectors that
provide the lowest possible maximum BER between the two users (16), comes down
to finding the optimal $\rho_{i},\delta_{i},\tau_{i},\phi_{i}$ parameters and
can be formulated as

$\begin{split}{}&{}\operatorname*{arg\,min}_{\rho_{1},\rho_{2},\delta_{1},\delta_{2},\tau_{1},\phi_{1},\phi_{2}}\\{\max\\{P_{e}^{(1)},P_{e}^{(2)}\\}\\}\\\
&\textrm{subject
to:}\;\;\eqref{constr1},\;\eqref{constr2},\;\eqref{constr3}.\end{split}$ (38)

It has to be noted that the objective function of our problem (38), is to find
the maximum between two $Q$-function sums. It is known that the $Q$-function
is neither convex nor concave, thus the problem can not be solved with
conventional convex optimization techniques like the ones found in [33].
Therefore, in an effort to create a fast and robust system that can predict
the optimal beamformer that achieves fairness between the users in terms of
conditional BER, we apply Machine Learning techniques. Specifically, we use
Neural Networks in order to learn the non-linear correlation that exists
between the channel state information of the two users and the characteristics
of their beamformers.

##  IV Proposed Learning Algorithm

In this section, we introduce the proposed learning algorithm that will be
used for the selection of the beamforming parameters of each user. Our
approach is based on using neural networks to learn the non-linear correlation
mentioned above. In what follows, we separately discuss the several steps
which are related to the design and application of our algorithm.

###  IV-A Generation of training/CSI data

In order to create the training data which are necessary for the training
process, we generate random realizations of the channels formed between the
Base Station (BS) and the two users, using the statistical channel model
presented in (6). Such an approach is also followed in [19, 20, 21, 23, 34,
35, 36] for the purposes of designing ML-based resource allocation and signal
processing algorithms. Following such an approach, allows us to apply our ML
based methodology in the absence of real channel data. Moreover, in practice
the Rayleigh fading channel model has been found to accurately model
wireless/mobile propagation channels [37].

###  IV-B Solving the optimal beamforming problem for the training data

In order for our algorithm to learn how to infer beamforming decisions, the
optimal beamformer decision for each one of the training CSI data is required.
As a result for each one of the training data instances we need to solve the
corresponding instance of problem (38). However, as already explained
optimally solving problem (38) is not straightforward mainly due to the fact
that the cost function is not convex. As a result, the application of any
standard iterative algorithm for solving problem (38) is not guaranteed to
provide the globally optimal solution. To overcome this problem, we solve
problem (38) using a constraint optimization algorithm. More specifically, we
use MATLAB’s optimization toolbox and the corresponding functions for
constrained nonlinear multivariable problems. For each CSI instance, we apply
the constrained optimization procedure from 20 random initial points and we
keep the response that minimizes the maximum BER among the two users. This
way, we create pairs of channel realizations and optimal beamforming vectors
that constitute the inputs and the outputs of the neural network respectively.
This proposed method cannot ensure that the global minimum will be found
universally, but is able to choose the best solution out of several local
minima.

Figure 3: Architecture of the used Neural Network. In this figure we present
the architecture of the proposed NN and give information about the shape of
the input layers, the size and the activation functions of the hidden layers
and the linear outputs that are formed by the two branches that exit the last
hidden layer.

###  IV-C Selecting appropriate Neural Network architectures

The literature contains multiple proposals for solving the power allocation
problem using deep neural networks (DNN). First, we can see in [19] the use of
deep convolutional neural networks to perform min-max power allocation in a
cell-free massive MIMO system. The use of convolutional layers is also met in
[20], where the use of a residual network that consists of multiple
convolutional and fully connected layers is used for power allocation in a
multi-cell massive MIMO setup. The use of residual networks (ResNet) are
commonly used in image recognition problems [38]. Specifically, in [39] a
ResNet is used in order to predict angles, similar to the ones in our problem.
Nevertheless, all these works use tend to use convolutional networks in order
to exploit the spatial information hidden in the multidimensional data that
they use as inputs. A characteristic that is not met in our work.

Another commonly used architecture of neural networks for power allocation is
the fully connected one. In [23] a fully connected DNN is introduced with the
aim of managing interference by approximating the WMMSE algorithm. This scheme
uses as inputs of the DNN the magnitude of the channel realization and it
achieves very good levels of performance. Additionally, feed forward neural
networks with fully connected layers are used in [21, 34] and use vectors of
user positions or shadowing coefficients in order to extract the optimal power
allocation parameters.

Based on the previous analysis and according to the universal function
approximation property [40, 41], our problem (38) can be solved with the use
of a DNN. The model that we consider to be the most suitable for our case is a
fully connected deep neural network. It consists of 6 hidden layers, the
designated sizes of which are noted in Fig.3. All hidden layers will use the
Rectified Linear Unit (ReLU) activation function. In order to decrease the
network’s complexity and processing time, the architecture is partly shared
between the two tasks. However, at the final stage, it is divided into two
branches and forms the two outputs, as we can observe in Fig.3. The first
branch of the output includes the four parameters that take values in the
range $[0,1]$ (i.e. $\rho_{1},\rho_{2},\delta_{1},\delta_{2}$) and the second
branch, includes the three angles that take values in the range $[0,2\pi)$
(i.e. $\tau_{1},\phi_{1},\phi_{2}$). Although the prediction of all these
outputs is a regression problem, due to their different nature and value
range, we propose the separation of the last layer into two independent
compartments. This technique is used for the prediction of heterogeneous
outputs from the same data as we can see in [39]. Both of NN outputs will use
linear activation functions. Since the task that we are trying to accomplish
is a regression problem, the mean absolute error (MAE) loss function is
preferred by both outputs. Subsequently, the overall loss function of the
neural network will be a weighted sum of the two individual losses. The
algorithm that is used for the training of the neural network is the ADAM
algorithm with Nesterov’s momentum [42, 43]. The training continues for
maximum of 200 epochs, unless it meets the predetermined training conditions,
where it stops automatically. Finally, the input of the DNN will consist of
the magnitude and the angles of the dot products between the channel
realizations of the two NOMA users and the vectors of the orthonormal basis,
as presented in Fig.3. We note that we ignore some parts of the available
information, such as the angles of
$\langle\mathbf{h}_{1}^{H},\mathbf{u}_{1}\rangle$ and
$\langle\mathbf{h}_{2}^{H},\mathbf{u}_{2}\rangle$ because they are equal to
zero, due to the properties of the orthonormal basis. We use $\xi$ as an
equalization factor, so that all the input components belong to similar orders
of magnitude. Thus, the neural network will have the same input size for every
different antenna scheme, with dimensions $7\times 1$. This property allows as
to train a single neural network for all the different antenna schemes that we
consider. It also enables the opportunity to use the same neural network in
case that a new antenna scheme appears, just by merging the data from all the
antennas schemes and retraining the same neural network.

Although we suggest a specific DNN structure, it is important to note that
discovering the optimal neural network architecture can be viewed as a
separate optimization problem that requires additional research. Based on the
approximation and estimation bounds for NNs presented in [44] and the works of
[41] in deep learning techniques, we experimented with various architectures
of fully connected networks, aiming to find the best one in terms of MAE. The
range of the fully connected NNs that we tested varies from two to six layers
and consists of different numbers of neurons. During our tests, we also used
various activation functions for the hidden layers, as well as for the output
layer and different training algorithms. We also examined the use of two
separate neural networks, one for each output and we compared these results
with our proposed layout.

##  V Simulation & Numerical results

In order to evaluate the performance of the proposed learning algorithms we
considered a scenario where a multi-antenna BS communicates with two users
following the channel model presented in (6). Note that the specific channel
model is part of 3GPP standard channel models [45]. As a result, it can be
considered a good option for generating channel instances which preserve some
of the properties of real channel data measurements: The remaining parameters
of our simulation driven performance evaluation are presented in Table I.
Concerning user placement, for our purposes, we have considered the case that
$U_{1}$ is always located at a distance between 600 and 650 meters from the BS
at each realization, and $U_{2}$ at a distance between 350 and 400 meters.
Moreover, we have assumed that the modulation order of $U_{1}$ and $U_{2}$ is
the same and is equal to $M=M_{1}=M_{2}=4$, while the noise variance was also
assumed to be the same for both users and equal to $N_{0}\times B$, where
$N_{0}$ the noise density and $B$ the system bandwidth, both given in Table I.
For this particular simulation scenario, we have investigated the performance
of our proposed algorithm, and compared it with commonly used beamformers,
which serve as benchmarks for our algorithm. These beamformers are briefly
presented in what follows.

TABLE I: Model parameters used in simulations Parameter | Value
---|---
Carrier frequency | 2GHz
System Bandwidth | 10MHz
Noise power density | -174 dBm/Hz
Path-loss in dB | PL = 128.1 + 37.6$\log_{10}$d, d in km
Transmit power | 100mW
$U_{1}$ distance | 600m - 650m
$U_{2}$ distance | 350m - 400m
$M_{1},M_{2}$ | 4

###  V-A The considered benchmarks

The beamforming algorithms which were compared, in terms of performance, to
our proposed algorithm, were the following.

####  V-A1 Maximum Ratio Transmission (MRT)

In the case of MRT [46], the beamforming vector applied on the data of user
$U_{k}$ is expressed as:

$\mathbf{w}_{k}=\frac{\mathbf{h}_{k}}{||\mathbf{h}_{k}||}.$ (39)

MRT beamforming aims at maximizing the SNR of the signal $s_{k}$ at user
$U_{k}$. The main disadvantage of this approach is that it does not take into
account the interference caused to the weak user, which can be large.

####  V-A2 Zero Forcing Beamforming (ZFBF)

ZFBF aims to create beamforming vectors that cause zero interference to other
users [47]. To do so, the beamforming vectors that this technique creates are
orthogonal to the channel vectors of all the users, but the one that transmits
with it, as we observe in the following equations.

$\mathbf{h}_{1}^{H}\mathbf{w}_{2}=0\;\;,\;\;\mathbf{h}_{2}^{H}\mathbf{w}_{1}=0.$
(40)

The two aforementioned techniques will be used as follows. First, we will use
MRT for both users, then the ZFBF, again for both users and finally, we will
use combinations of these two choices between the users. That is, MRT for the
first user and ZFBF for the second one and vice versa.

Figure 4: Comparison of the maximum BER of the two users for various
beamforming techniques. Figure 5: Histogram of the maximum BER of the two
users. Figure 6: Histogram of the maximum BER of the two users - zoomed.

In Fig.4 we can observe the average BER of all these four techniques,
alongside with the results of the neural network and of the CO for the
different antenna schemes. These BER values are the result of the averaging of
thousand of different channel realizations for each antenna scheme. As we can
see, CO outperforms all other techniques for a low number of antennas. We can
see that ZFBF takes a steep increase in performance in terms of BER as the
number of antennas increases. Next, the NN, improves its BER performance as
the number of antennas increases. The rest of the techniques seem to perform
poorly in all antenna cases.

Looking back on the ZFBF technique, we would like to make some comments. First
of all, ZFBF as we use it can not be considered to be a NOMA technique, since
it completely eliminates the interference of each user in the receiving signal
of the other one. Nevertheless, in the figures that we present, ZFBF seems to
be a good choice for a beamforming technique. Indeed, it seems to be a good
solution, without needing the computational time of constraint optimization in
order to find a good solution. We can also see, that as the number of
transmitting antennas increases, the gap between ZFBF and any other technique
increases. This is happening because ZFBF is based on the orthogonal
relationship of the beamformers with the channel vectors of the other users.
This property is enhanced as the number of transmitting antennas is increasing
compared to the number of users and therefore the number of receive antennas
since this increase offers more degrees of freedom at the beamforming vector.
However, in multiple antenna NOMA systems, many user pairs or clusters
coexist. In these cases, ZFBF is not used as a beamforming technique for each
user of each cluster. Usually, it is used as the first step of the beamforming
design in order to reduce inter-cluster interference. But again, in most of
these cases, the number of users and receive antennas surpasses the number of
transmit antennas. Thus, the zero - interference properties of ZFBF cannot be
achieved even with ZFBF that tackles inter-cluster interference and therefore,
this technique is considered suboptimal [8, 48].

We have to note that, although from the average BER of each scheme we can
judge each technique, this judgment can be sometimes distorted since this
representation technique is based on the mean values of the sizes and not on
their distribution. A more accurate image can be seen with the use of
histograms. In Fig.5 and Fig. 6 we can observe the histograms that present the
maximum BER of the two users based on the beamforming technique used each
time. The results that were used in order to generate these histograms
consider a plethora of randomly positioned user pairs, as well as different
antenna schemes (2,3,4 or 5 antennas at the BS). First of all we have to
notice again the superiority of CO, NN and ZFBF against all other techniques.
On the other hand, we have to notice that the judgment for the three other
techniques is not the same as before. We can see that the poor average values
that are presented in Fig.4 are not a result of the universally poor
performance of these techniques. On the contrary, a big portion of their uses
produces low maximum BERs, lower even that $10^{-5}$ as we can see in Fig.6.
More specifically, the most characteristic case is the one of MRT, in which
maximum BERs are mainly concentrated around very low or extremely high values.
Similar behavior we also notice while we use the combinations of MRT and ZFBF.
In these cases, more than half of the tested cases produce maximum BERs lower
than $10^{-5}$, but a large number of them produces maximum BERs of the
highest value. Going back to CO, NN and ZFBF, we can observe that the
conclusions that we extract are different from the ones when looking to Fig.4.
We can see that the difference in performance between these three techniques
is not that big. CO has the most cases of maximum BER under 0.05 in Fig.5 and
the biggest amount of cases under $10^{-5}$ as we can see in Fig.6. The neural
network beamforming technique comes second in front of ZFBF in these metrics,
something strange based on the results of Fig.4. A good explanation for these
difference can be the existance of outliers in the results of CO and NN when
compared with ZFBF. That is, that these two techniques have an extremely small
number of cases, in which the maximum BER is very high, affecting the mean
values of their BERs.

Figure 7: Performance comparison for the 3-antenna scheme between CO,
universal NN and NN trained only with data from the 3-antenna scheme.

To conclude, CO can find suitable beamforming vectors by computing the
parameters of the beamforming structure that we present in (17) and (18). With
the help of the neural network that we proposed, these parameters can be
learned and used to create beamforming vectors in any MISO-NOMA system that is
based on user fairness. However, we need to emphasize on the time that these
last two techniques (i.e. CO, NN) need in order to provide these results. The
calculation of the optimal beamformer in our case is the task of constrained
optimization with multiple random initialization points in a non-convex seven-
dimensional space. In combination with the fact that the BER expressions
consist of multiple sums, of which the derivatives have to be calculated, we
understand that finding of the optimal solution through constrained
optimization is a procedure that requires a lot of computational time. On the
other hand, the use of pre-trained NN with size such as in our case is a
robust procedure with very low complexity, that needs negligible time in order
to provide responses. The robustness and efficiency of our NN based system can
also be seen in Fig.7. Our algorithm enables the NN to be additionally trained
with data from various antenna schemes. This is possible thanks to the use of
the inner products between the channel vectors and the orthogonal basis
vectors as inputs of the NN. With this technique, we can train a single NN for
any number of antennas and use it universally with great results. In Fig.7 we
can see that the use of the same NN for all the available antenna schemes does
not affect at all the maximum BER performance of the system when compared with
the NN that is trained only on the data from the 3-antenna scheme. On the
contrary, it performs if anything equally well, and in our case it outperforms
the single scheme trained NN, since it results to a few more cases in BER
lower than $10^{-6}$.

##  VI Conclusion

In this paper, the closed-form expression of the BER for two users was
expanded for the scenario of MISO-NOMA over Rayleigh channels. A new method of
providing beamformers that grant fairness between the users was proposed, with
the use of very simple and agile neural networks. This is a promising method
that provides good results rapidly and with capabilities of easy scale for
other use cases. We also saw that our system works successfully when different
number of antennas used from the transmitter. A quality that brings the cost
of training and deploying a system like this to minimum, as it is not
necessary to use a different NN pair for each case. Finally, some extensions
that we could explore in future works, are the scenarios with more users, as
well as scenarios with use of real-time data.

## Acknowledgment

This work was carried out within the framework of the French collaborative
project ”Covera5Ge” supported by DGA and whose partners are CentraleSupélec,
ENENSYS Technologies and Siradel.

##  Appendix A Example $M_{1}=M_{2}=4$

In this appendix we present the expressions of the conditional BER for the two
users in the case of multiple antenna at BS (MISO - NOMA). We note that both
users use a 4-QAM modulation, i.e. $M_{1}=M_{2}=4$.

###  A-A $U_{1}$ Expression

In this section, we present the expressions for the conditional BER, as an
average of the probabilities of error for each bit of user $U_{1}$. First of
all, the index representation of each bit will be $b_{1i}$, where $i$ is the
index of the first user’s bits, with $i\in\\{1,2,...,B_{1}\\}$ and
$B_{1}=\log_{2}M_{1}$. If we take as an example the case of $M_{1}=M_{2}=4$,
we can create a constellation diagram such as in Fig. 8, where the $B_{1}=2$
first bits will represent $U_{1}$’s symbols and the rest will represent
$U_{2}$’s ones.

Figure 8: Constellation diagram of a NOMA symbol, (4-QAM modulation for both
users).

It should be mentioned that the use of Gray coding allows the appearance of
patterns, quite useful in the analysis of the multiple antenna scenario.
Specifically, due to Gray coding, the BER expression is affected only by the
real part of the interference and the noise for the first $B_{1}/2$ bits,
while it is affected by the imaginary part of them for the remaining $B_{1}/2$
bits of the first user, as we can notice in Fig. 8. This phenomenon, in
combination with the fact that $U_{1}$’s bits remain unchanged inside each
quadrant, generates two properties that can be exploited for the benefit of
the detection of $\hat{s}_{1}$ in (21). Initially, the $x$ and $y$ axes can be
used as the decision boundaries of the two groups of $U_{1}$’s bits,
respectively. Secondarily, the computation of the distances between all the
feasible symbols and $\hat{s}_{1}$ can be done, either by choosing only 4
symmetrical symbols from the constellation, as seen clearly in Fig. 8. Due to
these properties, the probability of error for the first $B_{1}/2$ bits is
equal with probability of error with the second $B_{1}/2$ bits, with the only
difference to be the decision boundary that we consider. Consequently, in our
case the probability of error for $b_{11}$ is equal to the probability of
error for $b_{12}$. Thus, the expression of BER for $U_{1}$ in the case of
$M_{1}=M_{2}=4$ is

$\begin{split}{}&{}P_{e}^{(1)}=\frac{1}{4}\bigg{[}Q(g_{1}\left(1,-1,-1\right))+Q(g_{1}\left(1,-1,1\right))\\\
&\;\;\;\;\;\;\;\;\;\;\;\;+Q(g_{1}\left(1,1,-1\right))+Q(g_{1}\left(1,1,1\right))\bigg{]}.\end{split}$
(41)

with $g_{1}$ as defined in (23).

###  A-B $U_{2}$ Expression

Previously, we explained the connection between the bits of the first users
symbol that are estimated during the SIC and the bits of the $U_{2}$ symbol.
Since both users’ constellation use Gray coding, we know that the probability
of error for the $b_{21}$ and $b_{22}$ bits are equal. Thus, the average
conditional BER expression of the second user for $M_{1}=M_{2}=4$ can be
written as

$\begin{split}{}&{}P_{e}^{(2)}=\frac{1}{2}\bigg{[}2Q(g^{+}_{2}(0,1))-Q(g^{+}_{2}(1,1))+Q(g^{+}_{2}(2,1))\\\
&\;\;\;\;\;\;\;\;\;\;\;+Q(g^{-}_{2}(1,1))-Q(g^{-}_{2}(2,1))\bigg{]},\end{split}$
(42)

with $g_{2,i}^{\pm}$ as defined in (32).

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