#  Nature-Inspired Intelligent $\alpha$-Fair Hybrid Precoding in Multiuser
Massive Multiple-Input Multiple-Output Systems

Asil Koc, Tho Le-Ngoc  This work was partially supported by InterDigital Inc.
and the Natural Sciences and Engineering Research Council of Canada (NSERC).
A. Koc and T. Le-Ngoc are with the Department of Electrical and Computer
Engineering, McGill University, Montréal, QC H3A 0E9, Canada (e-mail:
<EMAIL_ADDRESS>tho.le-ngoc@mcgill.ca). This work has been submitted
to the IEEE for possible publication.

###### Abstract

This paper proposes a novel nature-inspired $\alpha$-fair hybrid precoding
(NI-$\alpha$HP) technique for millimeter-wave multi-user massive multiple-
input multiple-output systems. Unlike the existing HP literature, we propose
to apply $\alpha$-fairness for maintaining various fairness expectations
(e.g., sum-rate maximization, proportional fairness, max-min fairness, etc.).
After developing the analog RF beamformer via slow time-varying angular
information, the digital baseband (BB) precoder is designed via the reduced-
dimensional effective channel matrix seen from the BB-stage. For the
$\alpha$-fairness, we derive the optimal digital BB precoder expression with a
set of parameters, where optimizing them is an NP-hard problem. Hence, we
efficiently optimize the parameters in the digital BB precoder via five
nature-inspired intelligent algorithms. Numerical results present that when
the sum-rate maximization is the target, the proposed NI-$\alpha$HP technique
greatly improves the sum-rate capacity and energy-efficiency performance
compared to other benchmarks. Moreover, NI-$\alpha$HP supports different
fairness expectations and reduces the rate gap among UEs by varying the
fairness level ($\alpha$).

###### Index Terms:

Massive MIMO, hybrid precoding, $\alpha$-fairness, nature-inspired
optimization, power allocation.

##  I Introduction

Fifth-generation (5G) cellular communication networks are rolling out
worldwide for supporting ever-growing mobile traffic rates and various
user/application demands. Massive multiple-input multiple-output (mMIMO) has
already become a key enabling technology in the 5G networks [1, 2, 3]. The
third generation partnership project (3GPP) standardizes the deployment of up
to 256 antennas at the base station (BS) in Release 17 [4]. Towards the next-
generation communication networks, the sixth-generation (6G), the expected
impacts of mMIMO systems are further extended with millimeter-wave (mmWave)
communications [5]. In addition to the extremely wide bandwidth at the mmWave
frequencies, the shorter wavelengths enable the utilization of larger antenna
arrays under the practical area requirement. Furthermore, the large antenna
arrays can combat the limited scattering mmWave propagation and focus the
signal energy in the desired direction via three-dimensional (3D) beamforming
[6]. Hence, the mmWave mMIMO technology brings a compelling potential for
various emerging applications (e.g., autonomous driving, healthcare, online
gaming, augmented/virtual reality (AR/VR), smart home/city, industrial
automation, etc.) [7].

###  I-A Literature Review

Regarding the multi-user (MU) downlink transmission, the precoding is an
essential signal processing technique for reliable communication quality. In
the conventional MIMO systems, the single-stage fully-digital precoding (FDP)
is widely studied thanks to the limited number of antennas [8]. Nevertheless,
FDP poses two critical issues for the mMIMO systems: (i) high power
consumption as well as hardware cost/complexity due to a dedicated power-
hungry RF chain per antenna, (ii) large channel estimation overhead size.
Hybrid precoding (HP) is proposed as a promising technique, which splits the
precoder into two stages as RF-stage and baseband(BB)-stage [9, 10, 11]. In
comparison to the large antenna array size, the HP architecture employs a
significantly low number of RF chains [12]. HP is also capable of reducing the
channel estimation overhead size, when the RF-stage is developed via slow
time-varying channel state information (CSI) (e.g., angle-of-departure (AoD),
channel covariance matrix) [13, 14, 15, 16]. For instance, eigen-beamforming
based HP (EBF-HP) proposed in [13] uses the channel covariance matrix in the
RF-stage design. EBF-HP is also investigated for various antenna array
geometries and low-resolution hardware components in [14]. Afterwards,
angular-based HP (AB-HP) is introduced to design the RF-stage via AoD
parameters in [15]. The authors present that AB-HP achieves a higher sum-rate
capacity than EBF-HP while closely approaching the sum-rate capacity of the
conventional FDP technique. Then, AB-HP is investigated for full-duplex
communications in [16], where the BS operates in full-duplex for the
simultaneous downlink and uplink transmission.

Fairness is principally an abstract sociopolitical concept implying justice,
equity, and impartiality in the distribution of finite resources among
users/clients in a given system [17]. When the technical systems are studied,
such as wireless communications, the fairness is attributed to the fair
allocation of the valuable resources (e.g., power and bandwidth) [18].

TABLE I: State-of-the-Art for Precoding Techniques in Massive MIMO Systems. Reference | Multi | Hybrid | Low CSI | Max-Min | Proportional | Sum-Rate | $\alpha$-Fairness
---|---|---|---|---|---|---|---
User | Precoding | Overhead | Fairness | Fairness | Maximization
[12] | ✓ | ✓ |  |  |  |  |
[13, 14, 15, 16] | ✓ | ✓ | ✓ |  |  |  |
[19, 20] | ✓ |  |  | ✓ |  |  |
[21] | ✓ | ✓ |  | ✓ |  |  |
[22] | ✓ |  |  |  | ✓ |  |
[23] | ✓ | ✓ |  |  | ✓ |  |
[24] | ✓ |  |  |  |  | ✓ |
[25, 26, 27] | ✓ | ✓ | ✓ |  |  | ✓ |
This Paper | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓

There are various qualitative fairness measures, such as max-min fairness [19,
20, 21], proportional fairness [22, 23], and sum-rate maximization [24, 27,
25, 26]. Particularly, [21] targets the maximization of minimum user rate
during the HP design assuming the availability of full-size CSI. However, in
general, the max-min fairness might lead to sum-rate capacity degradation in
the case of a weak user. On the contrary, [24, 27, 25, 26] aims to maximize
the sum-rate capacity in the MU-mMIMO systems, where FDP and HP are studied in
[24] and [27, 25, 26], respectively. Nonetheless, the sum-rate maximization
objective might result in poor performance for a few users, while enhancing
the overall system sum-rate capacity. To exploit the best of two worlds,
$\alpha$-fairness is an impressive concept to satisfy various fairness
expectations by adjusting a single non-negative scalar, denoted as $\alpha\geq
0$ [28]. The larger value of $\alpha$ signifies increased fairness among the
users. For instance, $\alpha=0$ indicates the sum-rate maximization objective,
whereas the proportional and max-min fairness are represented by $\alpha=1$
and $\alpha\to\infty$, respectively.

As a powerful component of artificial intelligence (AI), nature-inspired
optimization algorithms have recently gained considerable attention by means
of their success [29]. Unlike the traditional optimization techniques, the
nature-inspired algorithms enjoy the utilization of intelligent search agents,
which follow stochastic and deterministic components [30]. As it is inspired
by the nature, there are several ways to implement the characteristics of
search agents, such as particle swarm optimization (PSO) [29], grey wolf
optimization (GWO) [31], ant colony optimization (ACO) [32], cuckoo search
(CS) [30], firefly algorithm (FA) [30], etc. As stated by the no-free-lunch
theorem, there is no universally efficient algorithm valid for all types of
optimization problems [33]. In other words, the performance of each nature-
inspired algorithm heavily depends on the problem itself. For example, a
comprehensive comparison of various nature-inspired intelligent algorithms
(e.g., including GWO, PSO, CS) is presented in [34], where GWO achieves the
best performance on the location optimization of a drone BS. On the other
hand, a PSO-based power allocation (PSO-PA) algorithm along with AB-HP is
proposed for the MU-mMIMO systems in [25]. It is demonstrated that PSO-PA
tightly approaches the optimal PA scheme, when it is compared with the
computationally complex exhaustive search. Then, PSO-PA is used to develop a
supervised deep learning mechanism in [26], which makes both robust and faster
predictions for the allocated powers.

###  I-B Contributions

A new nature-inspired intelligent $\alpha$-fair HP (NI-$\alpha$HP) technique
is proposed for the Wave MU-mMIMO systems in this paper. Different from the
existing HP techniques in the literature, $\alpha$-fairness objective is
employed during the HP design for achieving various fairness requirements.
Table I summarizes a detailed comparison of this paper and existing precoding
techniques in the mMIMO systems.

The main contributions of this paper are summarized as:

  * •

Nature-Inspired $\alpha$-Fair Hybrid Precoding (NI-$\alpha$HP): We introduce a
two-stage NI-$\alpha$HP technique, where our main goals include performing
$\alpha$-fair resource allocation, reducing the number of RF chains, and
lowering the CSI overhead size. Analog RF beamformer, as the first-stage, is
developed via slow time-varying AoD information to reduce the CSI overhead
size and maximize the beamforming gain in the desired direction. Digital BB
precoder, as the second-stage, is constructed via the reduced-dimensional
effective channel matrix. According to the $\alpha$-fairness objective, we
obtain an optimal digital BB precoder expression including a set of NP-hard
parameters. Five nature-inspired intelligent algorithms (i.e., PSO, GWO, ACO,
CS, FA) are employed to optimize these parameters. Here, we aim to investigate
their performance and efficiency in our problem.

  * •

Low CSI Overhead & Hardware Cost/Complexity: The instantaneous channel
estimation overhead size is remarkably lowered by using the slow time-varying
AoD information at the RF-stage. Furthermore, the analog RF beamformer and
digital BB precoder are interconnected via a significantly low number of RF
chains compared to the large antenna array size. The numerical results show
that when the BS is equipped with $256$ antennas, the proposed NI-$\alpha$HP
technique is capable of reducing the number of RF chains and CSI overhead size
by $93.75\%$.

  * •

Sum-Rate & Energy-Efficiency: Regarding the sum-rate maximization objective,
we present that the proposed NI-$\alpha$HP technique greatly enhances the sum-
rate capacity compared to other HP benchmarks. Furthermore, NI-$\alpha$HP
achieves considerably higher energy-efficiency than the conventional FDP by
means of reduced hardware cost/complexity with a low number of RF chains.
Also, we observe that GWO converges faster and achieves higher capacity among
all five NI-$\alpha$HP techniques.

  * •

Fairness: Although the sum-rate maximization leads to a high rate gap among
the users, NI-$\alpha$HP successfully decreases the rate gap and satisfies
various fairness expectations by only adjusting a single scalar (i.e.,
fairness level $\alpha$). Also, Jain’s fairness index is employed as a
quantitative measure. We show that as the fairness level $\alpha$ increases,
NI-$\alpha$HP improves Jain’s fairness index.

###  I-C Organization

The rest of this paper is organized as follows. Section II introduces the
system model for the mmWave MU-mMIMO systems. Section III expresses the
problem formulation on $\alpha$-fairness. Afterwards, we propose five
NI-$\alpha$HP techniques in Section IV. Then, Section V presents the
comprehensive illustrative results. Finally, Section VI concludes this paper.

###  I-D Notation

Bold upper/lower case letters denote matrices/vectors.
$\left(\cdot\right)^{*}$, $\left(\cdot\right)^{T}$, $\left(\cdot\right)^{H}$,
$\left\|\cdot\right\|$, and $\left\|\cdot\right\|_{F}$ represent the complex
conjugate, the transpose, the conjugate transpose, the $2$-norm, and the
Frobenius norm, respectively. $\mathbf{I}_{K}$,
${\mathbb{E}}\left\\{\cdot\right\\}$, $\rm{tr}\left(\cdot\right)$, and
$\angle\left(\cdot\right)$ stand for $K\times K$ identity matrix, the
expectation operator, the trace operator, and the argument of a complex
number, respectively. ${\bf X}[m,n]$ denotes the element at the intersection
of $m^{th}$ row and $n^{th}$ column. $\mathbf{X}\otimes\mathbf{Y}$ and
$\mathbf{X}\odot\mathbf{Y}$ are respectively the Kronecker and Hadamard
product of two matrices $\mathbf{X}$ and $\mathbf{Y}$. We use ${{x}}\sim{\cal
C}{\cal N}\left({0,{\sigma}}\right)$, when ${x}$ is a complex Gaussian random
variable with zero-mean and variance ${\sigma}$. Also,
$x\sim\textrm{Ber}\left(\sigma\right)$ represents a Bernoulli random variable,
which equals $x=1$ with the probability of $\sigma\in\left[0,1\right]$,
otherwise, $x=0$.

##  II System Model

A single-cell MU-mMIMO system is considered for the downlink transmission as
illustrated in Fig. 1, where a base station (BS) with $M$ antennas serves $K$
single-antenna user equipments (UEs) clustered in $G$ groups. The BS employs a
uniform rectangular array (URA) with $M=M_{x}\times M_{y}$ antennas to enable
3D beamforming by exploiting azimuth and elevation angles, where $M_{x}$ and
$M_{y}$ denotes the number of antennas along $x$-axis and $y$-axis,
respectively [2]. Unlike the widely considered uniform linear array (ULA), URA
packs a large number of antennas on a two-dimensional (2D) grid and enhances
the area efficiency for the practical deployment of mMIMO systems [14, 2, 10].
Furthermore, we assume that there are $K_{g}$ UEs in the $g^{th}$ group with
$K=\sum_{g=1}^{G}K_{g}$.

According to the HP architecture illustrated in Fig. 1, the transmitted
downlink signal is defined as ${{\bf s}={\bf FBd}\in{\mathbb{C}^{M}}}$, where
${{\bf F}\in\mathbb{C}^{M\times N_{RF}}}$ is the analog RF beamformer matrix,
${{\bf B}=\left[{\bf b}_{1},\cdots,{\bf
b}_{K}\right]\in\mathbb{C}^{N_{RF}\times K}}$ is the digital BB precoder
matrix, ${\bf d}=\left[d_{1},\cdots,d_{K}\right]\in\mathbb{C}^{K}$ is the data
signal vector encoded by i.i.d. Gaussian codebook with the entries following
the distribution of $\mathcal{CN}\left(0,1\right)$ (i.e.,
$\mathbb{E}\big{\\{}{\bf dd}^{H}\big{\\}}={\bf I}_{K}$). Here, $N_{RF}$
denotes the number of RF chains for interconnecting the RF-stage and BB-stage.
It is important to highlight that $N_{RF}$ is chosen based on the following
condition $K\leq N_{RF}\ll M$ to support $K$ single-antenna UEs, while
remarkably reducing the hardware cost/complexity and power consumption in the
MU-mMIMO system with large antenna arrays. Additionally, the analog RF
beamformer is constructed via low-cost phase-shifters, which brings the
constant modulus (CM) constraint (i.e., $\big{|}{\bf
F}[{i,j}]\big{|}=\frac{1}{\sqrt{M}},\forall i,j,$) Furthermore, the
transmitted signal satisfies the maximum transmit power constraint of $P_{T}$
(i.e., $\mathbb{E}\big{\\{}\big{\|}{\bf s}\big{\|}^{2}\big{\\}}\leq P_{T}$).

Figure 1: MU-mMIMO systems with $\alpha$-fair hybrid precoding ($\alpha$HP).

The instantaneous downlink channel matrix is defined as ${\bf H}=\left[{\bf
h}_{1},\cdots,{\bf h}_{K}\right]^{T}\in\mathbb{C}^{K\times M}$ with ${\bf
h}_{k}\in\mathbb{C}^{M}$ as the $k^{th}$ UE channel vector. Different from the
sub-6 GHz frequency bands, the mmWave channels experience a limited scattering
propagation environment [6]. By employing the 3D geometry-based stochastic
channel model for mmWave communications [35, 36, 37], the $k^{th}$ UE channel
is given by:

${\bf{h}}_{k}=\sum_{l=1}^{L}{\tau^{-\eta}_{k_{l}}}{z_{k_{l}}}{\bm{\phi}}\big{(}{{\gamma_{x,k_{l}}},{\gamma_{y,k_{l}}}}\big{)}={{\bf{\Phi}}}_{k}^{T}{\bf{z}}_{k}\in\mathbb{C}^{M},$
(1)

where $L$ is the number of paths, $\eta$ is the path loss exponent,
$\tau_{k_{l}}$ and $z_{k_{l}}\sim\mathcal{CN}\big{(}0,{1}/{L}\big{)}$ are the
distance and complex path gain of $l^{th}$ path, respectively,
${\bm{\phi}}\big{(}\cdot,\cdot\big{)}\in\mathbb{C}^{M}$ is the phase response
vector,
$\gamma_{x,k_{l}}=\sin\left(\theta_{k_{l}}\right)\cos\left(\psi_{k_{l}}\right)\in\left[-1,1\right]$
and
$\gamma_{y,k_{l}}=\sin\left(\theta_{k_{l}}\right)\sin\left(\psi_{k_{l}}\right)\in\left[-1,1\right]$
are the coefficients reflecting the elevation AoD (EAoD) and azimuth AoD
(AAoD) for the corresponding path. Here, we define
$\theta_{k_{l}}\in\big{[}\theta_{k}-\delta_{k}^{\theta},\theta_{k}+\delta_{k}^{\theta}\big{]}$
as the EAoD with mean $\theta_{k}$ and spread $\delta_{k}^{\theta}$,
$\psi_{k_{l}}\in\big{[}\psi_{k}-\delta_{k}^{\psi},\psi_{k}+\delta_{k}^{\psi}\big{]}$
as the AAoD with mean $\psi_{k}$ and spread $\delta_{k}^{\psi}$. Based on the
URA structure, we define the phase response vector as [15]:

$\displaystyle{\bm{\phi}}\left({{\gamma_{x},\gamma_{y}}}\right)$
$\displaystyle=\big{[}{1,{e^{-j2\pi d{{\gamma_{x}}}}},\cdots,{e^{-j2\pi
d\left({{M_{x}}-1}\right){{\gamma_{x}}}}}}\big{]}^{T}$ (2)
$\displaystyle\otimes\big{[}{1,{e^{-j2\pi d{{\gamma_{y}}}}},\cdots,{e^{-j2\pi
d\left({{M_{y}}-1}\right){{\gamma_{y}}}}}}\big{]}^{T}\in\mathbb{C}^{M},$

where $d=0.5$ is the normalized half-wavelength distance between neighboring
antenna elements. As shown in (1), the instantaneous channel vector is
composed of two main components: (i) the fast time-varying path gain vector
${{\bf
z}_{k}=\big{[}\tau^{-\eta}_{k_{1}}z_{k_{1}},\cdots,\tau^{-\eta}_{k_{L}}z_{k_{K}}\big{]}^{T}\in\mathbb{C}^{L}}$,
(ii) slow time-varying phase response matrix
${{\bf{\Phi}}}_{k}\in\mathbb{C}^{L\times M}$ as a function of AoD information.

The received signal at the $k^{th}$ UE is written as follows:

$\displaystyle r_{k}$ $\displaystyle={\bf h}_{k}^{T}{\bf s}+n_{k}$ (3)
$\displaystyle=\underbrace{{\bf h}_{k}^{T}{\bf Fb}_{k}d_{k}}_{\textrm{Intended
Signal}}+\underbrace{\sum\nolimits_{u\neq k}^{K}{\bf h}_{k}^{T}{\bf
Fb}_{u}d_{u}}_{\textrm{Inter UE
Interference}}+\underbrace{n_{k}}_{\textrm{Noise}},$

where $n_{k}\sim\mathcal{CN}\big{(}0,\sigma_{n}^{2}\big{)}$ is the circularly
symmetric complex Gaussian noise. After some mathematical manipulations, the
instantaneous signal-to-interference-plus-noise-ratio (SINR) at the $k^{th}$
UE is obtained as:

$\displaystyle\textrm{SINR}_{k}\left({\bf F},{\bf B}\right)=\frac{\left|{\bf
h}_{k}^{T}{\bf Fb}_{k}\right|^{2}}{\sum_{u\neq k}^{K}\left|{\bf h}_{k}^{T}{\bf
Fb}_{u}\right|^{2}+\sigma_{n}^{2}}.$ (4)

Afterwards, as a function of the analog RF beamformer ${\bf F}$ and digital BB
precoder ${\bf B}$, the $k^{th}$ UE rate is calculated as:

$R_{k}\left({\bf F},{\bf B}\right)=\log_{2}\left(1+\frac{\left|{\bf
h}_{k}^{T}{\bf Fb}_{k}\right|^{2}}{\sum_{u\neq k}^{K}\left|{\bf h}_{k}^{T}{\bf
Fb}_{u}\right|^{2}+\sigma_{n}^{2}}\right)\textrm{ [bps/Hz]}.$ (5)

Finally, we find the sum-rate capacity in the MU-mMIMO systems as
$R_{\textrm{sum}}\left({\bf F},{\bf B}\right)=\sum_{k=1}^{K}R_{k}\left({\bf
F},{\bf B}\right)$.

##  III Problem Formulation on $\alpha$-Fairness

According to (4) and (5), we aim to jointly maximize the intended signal power
$\left|{\bf h}_{k}^{T}{\bf Fb}_{k}\right|^{2}$, $\forall k$, and mitigate the
inter UE interference power $\left|{\bf h}_{k}^{T}{\bf Fb}_{u}\right|^{2}$,
$\forall u\neq k$ in the HP design. Under the maximum transmit power
constraint of $P_{T}$, we have $\mathbb{E}\big{\\{}\big{\|}{\bf
s}\big{\|}^{2}\big{\\}}=\sum_{k=1}^{K}{\bf b}_{k}^{H}{\bf F}^{H}{\bf F}{\bf
b}_{k}=\sum_{k=1}^{K}p_{k}\leq P_{T}$, where $p_{k}={\bf b}_{k}^{H}{\bf
F}^{H}{\bf F}{\bf b}_{k}$ is the allocated power for the $k^{th}$ UE during
the HP design. However, while allocating the limited resources (e.g., power)
among the downlink UEs, there is an interesting trade-off between the sum-rate
maximization and fairness. For example, by defining $R_{\max}=\max_{k}R_{k}$
and $R_{\min}=\min_{k}R_{k}$, the sum-rate maximization objective only focuses
on optimizing $R_{\textrm{sum}}=\sum_{k=1}^{K}R_{k}$, which might cause unfair
UE rates with a large gap between $R_{\max}$ and $R_{\min}$. Moreover, it
might contradict the quality of service (QoS) requirement (e.g., achieving a
minimum rate for each UE). On the other hand, as an extreme opposite scenario,
the max-min fairness only targets maximizing $R_{\min}=\min_{k}R_{k}$, which
can enhance the QoS requirements. Nonetheless, it might lead to a considerable
degradation in the overall system capacity with a greatly reduced sum-rate
$R_{\textrm{sum}}$ [19].

For this purpose, we here propose to apply $\alpha$-fairness, which utilizes a
single scalar for enabling different UE fairness levels. In order to model
$\alpha$-fairness in the MU-mMIMO systems, we first define the following
utility function[18]:

$\mathcal{U}_{\alpha}\left(x\right)=\left\\{{\begin{array}[]{*{20}{l}}\frac{x^{1-\alpha}}{1-\alpha},&{\alpha\geq
0,\alpha\neq 1}\\\ \ln\left(x\right),&\alpha=1,\end{array}}\right.$ (6)

where $\alpha$ represents different fairness levels and $x\geq 0$ could be
considered as a given UE rate. By using (5) and (6), we formulate the
objective function on $\alpha$-fair HP ($\alpha$HP) design in the MU-mMIMO
systems as follows:

$\displaystyle\max_{{\bf F},{\bf B}}$
$\displaystyle\sum_{k=1}^{K}\mathcal{U}_{\alpha}\left(R_{k}\right),$ (7) s.t.
$\displaystyle C_{1}:~{}R_{k}=\log_{2}\left(1+\textrm{SINR}_{k}\right),$
$\displaystyle C_{2}:~{}\textrm{SINR}_{k}=\frac{\left|{\bf h}_{k}^{T}{\bf
Fb}_{k}\right|^{2}}{\sum_{u\neq k}^{K}\left|{\bf h}_{k}^{T}{\bf
Fb}_{u}\right|^{2}+\sigma_{n}^{2}},$ $\displaystyle
C_{3}:~{}\mathbb{E}\big{\\{}\big{\|}{\bf
s}\big{\|}_{2}^{2}\big{\\}}=\sum_{k=1}^{K}{\bf b}_{k}^{H}{\bf F}^{H}{\bf
F}{\bf b}_{k}=\sum_{k=1}^{K}p_{k}\leq P_{T},$ $\displaystyle
C_{4}:~{}p_{k}\geq 0,\forall k,$ $\displaystyle C_{5}:~{}\big{|}{\bf
F}[{i,j}]\big{|}=\frac{1}{\sqrt{M}},\forall i,j,$

where $C_{1}$ and $C_{2}$ refer the rate and SINR at the $k^{th}$ UE,
respectively, $C_{3}$ implies the total transmit power constraint, $C_{4}$
indicates the non-negative allocated power for each downlink UE, and $C_{5}$
expresses the CM constraint at the analog RF beamformer design due to the
utilization of low-cost phase-shifters. By adjusting the value of $\alpha$,
the above objective function addresses various fairness levels. To illustrate,
one can show that: (i) $\alpha=0$ is for the sum-rate maximization with
$\max\sum_{k=1}^{K}R_{k}$, (ii) $\alpha=1$ is for the proportional fairness
with $\max\sum_{k=1}^{K}\ln\left(R_{k}\right)$, (iii) $\alpha\to\infty$ is for
the max-min fairness with
$\max\left[\lim\limits_{\alpha\to\infty}\sum_{k=1}^{K}\frac{R_{k}^{1-\alpha}}{1-\alpha}\right]=\max\left[\min_{k}R_{k}\right]$,
(iv) other values of $\alpha$ are for different fairness levels.

However, the objective function given in (7) is a non-convex optimization
problem because of several reasons (e.g., the CM constraint [10], the
optimization variables interchangeably located in the numerator/denominator
[38, 39, 40]). For this reason, in Section IV, the proposed HP solution is
developed by applying various nature-inspired intelligent algorithms.

##  IV Nature-Inspired Intelligent $\alpha$-Fair
Hybrid Precoding (NI-$\alpha$HP)

In this section, we introduce the proposed nature-inspired intelligent
$\alpha$-fair hybrid precoding (NI-$\alpha$HP) technique for the MU-mMIMO
systems. Along with the aim of optimal $\alpha$-fair resource allocation, our
main objectives also include reducing the RF chain utilization and decreasing
the channel estimation overhead size. Hence, the proposed NI-$\alpha$HP
technique sequentially develops the analog RF beamformer ${{\bf
F}\in\mathbb{C}^{M\times N_{RF}}}$ and digital BB precoder ${{\bf
B}\in\mathbb{C}^{N_{RF}\times K}}$. First, we only employ the slow time-
varying AoD information111In addition to the conventional AoD estimation
techniques [41], the AoD parameters (e.g., mean and spread) can be acquired
via a deep learning and geospatial data-based offline estimation technique
presented in [42]. in the analog RF beamformer design to maximize the
beamforming gain in the intended direction, while lowering the instantaneous
CSI requirements for the MU-mMIMO systems. Second, we derive the optimal
parameterized digital BB precoder solution by using the reduced-size effective
CSI, where five nature-inspired optimization algorithms are applied to
optimize those parameters.

###  IV-A Analog RF Beamformer

When UEs are clustered in multiple geographical locations as illustrated in
Fig. 1, each UE group experiences similar AoD information [42]. Hence, we
design the analog RF beamformer matrix with $G$ blocks as follows:

${\bf F}=\left[{\bf F}_{1},\cdots,{\bf F}_{G}\right]\in\mathbb{C}^{M\times
N_{RF}},$ (8)

where ${\bf F}_{g}\in\mathbb{C}^{M\times N_{RF,g}}$ and $N_{RF,g}$ are the RF
beamformer and number of RF chains for the $g^{th}$ UE group, respectively,
with $N_{RF}=\sum\nolimits_{g=1}^{G}N_{RF,g}$. By using (2), the analog RF
beamformer is constructed via the unit-power steering vectors defined as
follows:

${\bf
e}\left(\gamma_{x},\gamma_{y}\right)=\frac{1}{\sqrt{M}}{\bm{\phi}}^{*}\left({{\gamma_{x},\gamma_{y}}}\right)\in\mathbb{C}^{M},\forall\gamma_{x},\gamma_{y}\in\left[-1,1\right],$
(9)

which satisfies ${\left\|{\bf
e}\left(\gamma_{x},\gamma_{y}\right)\right\|^{2}=1}$ and the CM constraint
(i.e., $C_{5}$ in (7)). Afterwards, we define the quantized angle-pairs as
${{\lambda^{x}_{m}}=-1+\frac{2m-1}{{{M_{x}}}}}$ for $m=1,\cdots,{M_{x}}$ and
${{\lambda^{y}_{n}}=-1+\frac{2n-1}{{{M_{y}}}}}$ for $n=1,\cdots,{M_{y}}$.
Here, we have $M$ possible $\left({\lambda^{x}_{m}},{\lambda^{y}_{n}}\right)$
angle-pairs, which enable us to cover the complete 3D elevation and azimuth
angular domain with the minimum number of steering vectors [15]. Moreover, the
quantized angle-pairs satisfy the orthogonality property (i.e., ${\bf
e}^{H}\big{(}\lambda^{x}_{m},\lambda^{y}_{n}\big{)}{\bf
e}\big{(}\lambda^{x}_{m^{\prime}},\lambda^{y}_{n^{\prime}}\big{)}=0,\forall\left[m,n\right]\neq\left[m^{\prime},n^{\prime}\right]$).

Based on the similarity of AoD information within each UE group, one can
define the AoD support of the $g^{th}$ group as:

$\mathcal{A}_{g}=\left[\begin{array}[]{*{20}{c}}\gamma_{x}=\sin\left(\theta\right)\cos\left(\psi\right)\\\
\gamma_{y}=\sin\left(\theta\right)\sin\left(\psi\right)\end{array}\right]\Big{|}\forall\theta\in{\bm{\theta}}_{g},\forall\psi\in{\bm{\psi}}_{g},\Big{.}$
(10)

where
$\bm{\theta}_{g}=\left[\theta_{g}-\delta_{g}^{\theta},\theta_{g}+\delta_{g}^{\theta}\right]$
and
$\bm{\psi}_{g}=\left[\psi_{g}-\delta_{g}^{\psi},\psi_{g}+\delta_{g}^{\psi}\right]$
are the boundaries EAoD and AAoD, respectively. Here, $\theta_{g}$ and
$\psi_{g}$ represent the mean EAoD and AAoD for the corresponding UE group
with their spread of $\delta_{g}^{\theta}$ and $\delta_{g}^{\psi}$,
respectively. By using (1), (2) and (10), the $g^{th}$ UE group channel matrix
can be rewritten as follows [13, 14, 15, 16]:

${\bf H}_{g}=\big{[}{\bf h}_{g_{1}},\cdots,{\bf h}_{g_{K_{g}}}\big{]}^{T}={\bf
Z}_{g}{\bf\Phi}_{g}\in\mathbb{C}^{K_{g}\times M},$ (11)

where ${\bf Z}_{g}=\big{[}{\bf z}_{g_{1}},\cdots,{\bf
z}_{g_{K_{g}}}\big{]}^{T}\in\mathbb{C}^{K_{g}\times L}$ is the fast time-
varying path gain matrix, ${\bf\Phi}_{g}\in\mathbb{C}^{L\times M}$ is the slow
time-varying phase response matrix, $g_{k}=k+\sum_{t=1}^{g-1}K_{t}$ is the UE
index. By using (8), (11), and the concatenated channel matrix as ${\bf
H}=\left[{\bf H}_{1}^{T},\cdots,{\bf
H}_{G}^{T}\right]^{T}\in\mathbb{C}^{K\times M}$, the effective channel matrix
seen from the BB-stage is obtained as:

$\bm{\mathcal{H}}={\bf H}{\bf
F}=\left[{\begin{array}[]{*{20}{c}}{{{\bf{H}}_{1}}{{\bf{F}}_{1}}}&{{{\bf{H}}_{1}}{{\bf{F}}_{2}}}&{\cdots}&{{{\bf{H}}_{1}}{{\bf{F}}_{G}}}\\\
{{{\bf{H}}_{2}}{{\bf{F}}_{1}}}&{{{\bf{H}}_{2}}{{\bf{F}}_{2}}}&{\cdots}&{{{\bf{H}}_{2}}{{\bf{F}}_{G}}}\\\
{\vdots}&{\vdots}&{\ddots}&{\vdots}\\\
{{{\bf{H}}_{G}}{{\bf{F}}_{1}}}&{{{\bf{H}}_{G}}{{\bf{F}}_{2}}}&{\cdots}&{{{\bf{H}}_{G}}{{\bf{F}}_{G}}}\\\
\end{array}}\right]\in\mathbb{C}^{K\times N_{RF}},$ (12)

where $\bm{\mathcal{H}}_{g}={\bf H}_{g}{\bf F}_{g}={\bf
Z}_{g}{\bf\Phi}_{g}{\bf F}_{g}\in\mathbb{C}^{K_{g}\times N_{RF,g}}$ is the
effective intended channel matrix for the $g^{th}$ UE group and ${\bf
H}_{t}{\bf F}_{g}={\bf Z}_{t}{\bf\Phi}_{t}{\bf
F}_{g}\in\mathbb{C}^{K_{t}\times N_{RF,g}}$ is the effective interference
channel matrix among the $g^{th}$ and $t^{th}$ UE group, $\forall t\neq g$.
Hence, we have two main goals for the RF beamformer design: (i) maximize the
beamforming gain towards the intended direction via constructing the columns
of ${\bf F}_{g}$ from the subspace spanned by ${\bf\Phi}_{g}$ (i.e.,
$\textrm{Span}\left({\bf
F}_{g}\right)\subset\textrm{Span}\left({\bf\Phi}_{g}\right)$), (ii) suppress
the inter-group interference via selecting the columns of ${\bf F}_{g}$
orthogonal to ${\bf\Phi}_{t}$, $\forall t\neq g$ (i.e.,
$\textrm{Span}\left({\bf F}_{g}\right)\subset\cup_{t\neq
g}\textrm{Null}\left({\bf\Phi}_{t}\right)$). In other words, the second goal
targets the approximate zero condition as:

${\bf\Phi}_{t}{\bf F}_{g}\approx 0,~{}\forall t\neq g.$ (13)

Both design goals can be addressed by constructing ${\bf F}_{g}$ via the
steering vector with the quantized angle-pairs covering the intended angular
support (i.e.,
${\left(\lambda_{m}^{x},\lambda_{n}^{y}\right)\in\mathcal{A}_{g}}$) and
excluding the other groups angular support (i.e.,
${\left(\lambda_{m}^{x},\lambda_{n}^{y}\right)\notin\mathcal{A}_{t}}$,
${\forall t\neq g}$). The corresponding quantized angle-pairs are found as
follows:

$\left(\hat{\lambda}_{m_{g}}^{x},\hat{\lambda}_{n_{g}}^{y}\right)\hskip
2.15277pt\Bigg{|}\hskip
2.15277pt\begin{array}[]{*{20}{l}}\gamma_{x}\in\bm{\lambda}_{m}^{x},\gamma_{y}\in\bm{\lambda}_{n}^{y},\left(\gamma_{x},\gamma_{y}\right)\in\mathcal{A}_{g},\\\
\left(\gamma_{x},\gamma_{y}\right)\notin\mathcal{A}_{t},\forall t\neq
g,\end{array}\Bigg{.}$ (14)

where
${\bm{\lambda}}^{x}_{m}=\big{[}{\lambda_{m}^{x}-\frac{1}{{{M_{x}}}},\lambda_{m}^{x}+\frac{1}{{{M_{x}}}}}\big{]}$
is the boundary of $\lambda_{m}^{x}$, and
${\bm{\lambda}}_{n}^{y}=\big{[}{\lambda_{n}^{y}-\frac{1}{{{M_{y}}}},\lambda_{n}^{y}+\frac{1}{{{M_{y}}}}}\big{]}$
is the boundary of $\lambda_{n}^{y}$. As shown proven in [15, eq. (16)], the
approximate zero condition given in (13) can be satisfied by building the
analog RF beamformer via the steering vector with the corresponding quantized
angle-pairs found in (14). Thus, as the antenna array size increases, one can
derive the following limit expression:

$\lim\limits_{M\to\infty}{\bf\Phi}_{t}{\bf
e}\big{(}\hat{\lambda}_{m_{g}}^{x},\hat{\lambda}_{n_{g}}^{y}\big{)}={\bf
0},~{}\forall t\neq g.$ (15)

By combining (13), (14), and (15), it is shown that when the analog RF
beamformer is designed with the corresponding quantized angle-pairs, the
inter-group interference can be suppressed. Hence, after finding $N_{RF,g}$
quantized angle-pairs covering $\mathcal{A}_{g}$ and excluding
$\mathcal{A}_{t}$ with $t\neq g$, the RF precoder for the $g^{th}$ group is
constructed as follows:

${\bf F}_{g}=\Big{[}{\bf
e}\big{(}\hat{\lambda}_{m_{1}}^{x},\hat{\lambda}_{n_{1}}^{y}\big{)},\cdots,{\bf
e}\big{(}\hat{\lambda}_{m_{N_{RF,g}}}^{x},\hat{\lambda}_{n_{N_{RF,g}}}^{y}\big{)}\Big{]}.$
(16)

Here, $N_{RF,g}$ also represents the number of orthogonal beams generated for
the $g^{th}$ group.

Finally, the analog RF beamformer ${\bf F}$ satisfying the CM constraint
(i.e., $C_{5}$ given in (7)) is obtained by substituting (16) into (8).
Furthermore, it is important to remark that the analog RF beamformer is a
unitary matrix (i.e., ${\bf F}^{H}{\bf F}={\bf I}_{N_{RF}}$).

Example 1: Fig. 2 illustrates 2D azimuth beampatterns, where there are only
$K=3$ UEs clustered in $G=3$ groups. We consider the mean AAoD of
$\psi_{1}=55^{\circ}$, $\psi_{2}=85^{\circ}$, $\psi_{3}=135^{\circ}$, and the
AAoD spread of
$\delta^{\psi}_{1}=\delta^{\psi}_{2}=\delta^{\psi}_{3}=2^{\circ}$.

(a) $M=16$ antennas.

(b) $M=64$ antennas.

Figure 2: 2D azimuth beampatterns with analog RF beamforming for each UE.

For the visualization perspective, we here consider the utilization of ULA
with $M=16$ and $M=64$ antennas in Fig. 2(a) and Fig. 2(b), respectively. By
substituting the given angle parameters into (14), a single beam is generated
for each UE in both scenarios. In other words, by substituting (9), (14), and
(16) into (8), the analog RF beamformer is obtained as ${\bf F}=\left[{\bf
f}_{1},{\bf f}_{2},{\bf f}_{3}\right]\in\mathbb{C}^{M\times 3}$ with ${\bf
f}_{k}={\bf
e}\big{(}\hat{\lambda}_{m_{k}}^{x},\hat{\lambda}_{n_{k}}^{y}\big{)}\in\mathbb{C}^{M}$
as the RF beamformer vector for the $k^{th}$ UE. When the number of antennas
is increased from $M=16$ to $M=64$, the array directivity improves from $12$
dBi to $18$ dBi by means of higher beamforming gain. Although the main beam
directions are orthogonal, the presence of considerable minor lobes might
cause the inter-group interference. As the array size increases, we observe
that the minor lobes towards undesired UE direction get smaller, which helps
to mitigate the inter-group interference as indicated in (15).

###  IV-B Digital BB Precoder

After deriving the analog RF beamformer ${\bf F}$, we here develop the optimal
digital BB precoder ${\bf B}=\left[{\bf b}_{1},\cdots,{\bf b}_{K}\right]$.
Hence, the $\alpha$HP objective function given in (7) is reformulated
accordingly:

$\displaystyle\max_{{\bf b}_{1},\cdots,{\bf b}_{K}}$
$\displaystyle\sum_{k=1}^{K}\mathcal{U}_{\alpha}\left(R_{k}\right),$ (17) s.t.
$\displaystyle C_{1}:~{}R_{k}=\log_{2}\left(1+\textrm{SINR}_{k}\right),$
$\displaystyle C_{2}:~{}\textrm{SINR}_{k}=\frac{\left|{\bf h}_{k}^{T}{\bf
Fb}_{k}\right|^{2}}{\sum_{u\neq k}^{K}\left|{\bf h}_{k}^{T}{\bf
Fb}_{u}\right|^{2}+\sigma_{n}^{2}}\geq\mu_{k},$ $\displaystyle
C_{3}:~{}\mathbb{E}\big{\\{}\big{\|}{\bf
s}\big{\|}_{2}^{2}\big{\\}}=\sum_{k=1}^{K}{\bf b}_{k}^{H}{\bf
b}_{k}=\sum_{k=1}^{K}p_{k}\leq P_{T},$ $\displaystyle C_{4}:~{}p_{k}\geq
0,\forall k,$

where $\mu_{1},\cdots,\mu_{K}$ is the set of given SINR targets maximizing the
objective function [43].

Proposition 1: The optimal digital BB precoder vector for the $k^{th}$ UE is
derived as:

${\bf{b}}_{k}=\underbrace{\sqrt{p_{k}}}_{{\textrm{Power}}}\underbrace{\frac{\left({\bf
I}_{N_{RF}}+\sum_{u=1}^{K}\frac{\beta_{u}}{\sigma^{2}_{n}}{\bf
A}_{u}\right)^{-1}{\bf F}^{H}{\bf h}_{k}^{*}}{\left\|\left({\bf
I}_{N_{RF}}+\sum_{u=1}^{K}\frac{\beta_{u}}{\sigma^{2}_{n}}{\bf
A}_{u}\right)^{-1}{\bf F}^{H}{\bf h}_{k}^{*}\right\|}}_{\textrm{Beamforming
Direction}}.$ (18)

where ${\bf{A}}_{k}={\bf F}^{H}{\bf h}_{k}^{*}{\bf h}_{k}^{T}{\bf F}$ and
${\sum_{k=1}^{K}p_{k}=\sum_{k=1}^{K}\beta_{k}=P_{T}}$. Here, $p_{k}$ and
$\beta_{k}$ are parameters to be optimized, however, their closed-form
solutions are not available due to the entangled optimization parameters.

Proof: Please see Appendix A.

Although we obtain the optimal parameterized digital BB precoder in a closed-
form, it is necessary to jointly optimize $p_{k}$ and $\beta_{k}$ for
$k=1,\cdots,K$. Thus, we apply nature-inspired intelligent algorithms to
optimize all $2K$ parameters. Particularly, $N_{S}$ search agents are randomly
placed in the $2K$ dimensional optimization space. During $Q$ iterations, the
search agents communicate and compete with each other for moving towards the
optimal solutions. The movement of each search agent generally depends on two
major factors: (i) deterministic (i.e., the experience on the observed best
solutions), (ii) stochastic (i.e., the tendency of random movement).

First, we define the normalized optimization parameters
$\hat{p}_{k}\in\left[0,1\right]$ and $\hat{\beta}_{k}\in\left[0,1\right]$ with
the normalization scalars of
$\varepsilon_{1}=\frac{P_{T}}{\sum_{k=1}^{K}\hat{p}_{k}}$ and
$\varepsilon_{2}=\frac{P_{T}}{\sum_{k=1}^{K}\hat{\beta}_{k}}$, respectively.
Afterwards, we present the $i^{th}$ search agent in the $q^{th}$ iteration as
follows:

${\bf
a}_{i}^{(q)}=\left[\hat{p}_{1,i}^{(q)},\cdots,\hat{p}_{K,i}^{(q)},\hat{\beta}_{1,i}^{(q)},\cdots,\hat{\beta}_{K,i}^{(q)}\right]^{T}\in\mathbb{R}^{2K},$
(19)

where $i=1,\cdots,N_{S}$ and $q=1,\cdots,Q$. The corresponding normalization
scalar is obtained as:

$\bm{\varepsilon}_{i,q}=\left[\begin{array}[]{*{20}{c}}{\varepsilon_{1,i}^{(q)}}\\\
{\varepsilon_{2,i}^{(q)}}\end{array}\right]=\left[\frac{P_{T}}{\sum_{k=1}^{K}\hat{p}_{k,i}^{(q)}},\frac{P_{T}}{\sum_{k=1}^{K}\hat{\beta}_{k,i}^{(q)}}\right]^{T}.$
(20)

By substituting (19) and (20) into (18), the digital BB precoder vector is
reformulated as a function of the search agent and normalization scalar as
follows:

${\bf{b}}_{k}\big{(}{\bf
a},\bm{\varepsilon}\big{)}=\underbrace{\sqrt{\varepsilon_{1}\hat{p}_{k}}}_{{\textrm{Power}}}\underbrace{\frac{\left({\bf
I}_{N_{RF}}+\sum_{u=1}^{K}\frac{\varepsilon_{2}\hat{\beta}_{u}}{\sigma^{2}_{n}}{\bf
A}_{u}\right)^{-1}{\bf F}^{H}{\bf h}_{k}^{*}}{\left\|\left({\bf
I}_{N_{RF}}+\sum_{u=1}^{K}\frac{\varepsilon_{2}\hat{\beta}_{u}}{\sigma^{2}_{n}}{\bf
A}_{u}\right)^{-1}{\bf F}^{H}{\bf h}_{k}^{*}\right\|}}_{\textrm{Beamforming
Direction}}.$ (21)

By combining (19), (20), and (21), the objective function defined in (17)
becomes equivalent to:

$\displaystyle\max_{{\bf a}}$
$\displaystyle\sum_{k=1}^{K}\mathcal{U}_{\alpha}\left(R_{k}\big{(}{\bf
a},\bm{\varepsilon}\big{)}\right),$ (22) s.t. $\displaystyle
C_{1}:~{}R_{k}\big{(}{\bf
a},\bm{\varepsilon}\big{)}=\log_{2}\left(1+\textrm{SINR}_{k}\big{(}{\bf
a},\bm{\varepsilon}\big{)}\right),$ $\displaystyle
C_{2}:~{}\textrm{SINR}_{k}\big{(}{\bf
a},\bm{\varepsilon}\big{)}=\frac{\left|{\bf h}_{k}^{T}{\bf Fb}_{k}\big{(}{\bf
a},\bm{\varepsilon}\big{)}\right|^{2}}{\sum_{u\neq k}^{K}\left|{\bf
h}_{k}^{T}{\bf Fb}_{u}\big{(}{\bf
a},\bm{\varepsilon}\big{)}\right|^{2}+\sigma_{n}^{2}},$ $\displaystyle
C_{6}:~{}{\bf
a}=\left[\hat{p}_{1},\cdots,\hat{p}_{K,},\hat{\beta}_{1},\cdots,\hat{\beta}_{K}\right]^{T}\in\left[0,1\right],$
$\displaystyle
C_{7}:~{}\bm{\varepsilon}=\left[\frac{P_{T}}{\sum_{k=1}^{K}\hat{p}_{k}},\frac{P_{T}}{\sum_{k=1}^{K}\hat{\beta}_{k}}\right]^{T},$

which aims to maximize the $\alpha$-fairness objective by finding the optimal
search agent (i.e., the optimal set of $p_{k}$ and $\beta_{k}$). Here, $C_{6}$
and $C_{7}$ refer to the total transmit power constraint. Through iterations,
we move each search agent from ${\bf a}_{i}^{(q)}$ to ${\bf a}_{i}^{(q+1)}$
with the aim of objective value maximization. After $Q$ iterations, the best
search agent is found as:

${\bf a}_{\textrm{best}}=\operatorname*{arg\,max}\limits_{{\bf
a}_{i}^{(q)},\forall
i,q}\sum_{k=1}^{K}\mathcal{U}_{\alpha}\left(R_{k}\big{(}{\bf
a}_{i}^{(q)},\bm{\varepsilon}_{i}^{(q)}\big{)}\right).$ (23)

Now, we propose to apply five nature-inspired intelligent algorithms to
describe how search agents move over iterations.

####  IV-B1 PSO-$\alpha$HP

Particle swarm optimization (PSO) is classified as a swarm intelligence
algorithm, which is inspired by the swarm behavior of animals [29]. In the
proposed PSO-$\alpha$HP technique, the position of each particle (i.e., search
agent) is updated as:

${\bf a}_{i}^{(q+1)}=\textrm{clip}\big{(}{\bf
a}_{i}^{(q)}+\bm{\Delta}_{i}^{(q)},\left[0,1\right]\big{)},$ (24)

where $\bm{\Delta}_{i}^{(q)}\in\mathbb{R}^{2K}$ denotes the velocity of the
$i^{th}$ particle in the $q^{th}$ iteration. Also, we define a clipping
function as
$\textrm{clip}\left(x,\left[a,b\right]\right)=\max\left(a,\min\left(x,b\right)\right)$
to keep the normalized optimization parameters within their predefined ranges.

Throughout iterations, when a particle finds a better position achieving
higher objective value than any of its previous positions, it is recorded as
its current best position. In the $q^{th}$ iteration, the velocity of each
particle is calculated based on three factors: (i) its current personal best
position ${\bf a}_{\textrm{best},i}^{(q)}$, (ii) the current global best
position among all particles ${\bf a}_{\textrm{gbest}}^{(q)}$, (iii)
randomness. Thus, the velocity in the $(q+1)^{th}$ iteration is updated as
follows:

$\displaystyle\bm{\Delta}_{i}^{(q+1)}$
$\displaystyle=\textrm{clip}\Big{(}\Big{.}\kappa_{q}^{\textrm{PSO}}\bm{\Delta}_{i}^{(q)}+{\bf
w}_{1}^{\textrm{PSO}}\odot\big{(}{\bf a}_{\textrm{gbest}}^{(q)}-{\bf
a}_{i}^{(q)}\big{)}$ (25) $\displaystyle+{\bf
w}_{2}^{\textrm{PSO}}\odot\big{(}{\bf a}_{\textrm{best},i}^{(q)}-{\bf
a}_{i}^{(q)}\big{)},\Big{.}\left[\kappa_{\Delta,\min}^{\textrm{PSO}},\kappa_{\Delta,\max}^{\textrm{PSO}}\right]\Big{)},$

where $\kappa_{q}^{\textrm{PSO}}=1-\frac{q-1}{Q}$ is a decreasing inertia
parameter for enhancing the exploitation towards the latest iterations,
$\kappa_{\Delta,\min}^{\textrm{PSO}}$ and
$\kappa_{\Delta,\max}^{\textrm{PSO}}$ are respectively the minimum and maximum
velocity parameters [30]. Also, ${\bf w}_{1}^{\textrm{PSO}}\in\mathbb{R}^{2K}$
and ${\bf w}_{2}^{\textrm{PSO}}\in\mathbb{R}^{2K}$ are random vectors with
uniformly distributed entries over $\left[0,2\right]$, which determine the
tendency of moving towards the current global and personal best positions to
explore the $2K$ dimensional optimization space. Particularly, the current
global and personal best positions in the ${(q+1)}^{th}$ iteration are
respectively found as:

$\displaystyle{\bf
a}_{\textrm{gbest}}^{(q+1)}=\operatorname*{arg\,max}\limits_{{{\bf
a}_{i}^{(q^{\prime})},}{\forall i,\forall q^{\prime}\leq q}}\hskip
4.30554pt\sum_{k=1}^{K}\mathcal{U}_{\alpha}\left(R_{k}\big{(}{\bf
a}_{i}^{(q^{\prime})},\bm{\varepsilon}_{i}^{(q^{\prime})}\big{)}\right),$ (26)
$\displaystyle{\bf
a}_{\textrm{best},i}^{(q+1)}=\operatorname*{arg\,max}\limits_{{{\bf
a}_{i}^{(q^{\prime})},}{\forall q^{\prime}\leq q}}\hskip
4.30554pt\sum_{k=1}^{K}\mathcal{U}_{\alpha}\left(R_{k}\big{(}{\bf
a}_{i}^{(q^{\prime})},\bm{\varepsilon}_{i}^{(q^{\prime})}\big{)}\right).$ (27)

After $Q$ iterations, PSO-$\alpha$HP finally finds the best position as ${\bf
a}_{\textrm{best}}={\bf a}_{\textrm{gbest}}^{(Q)}$, which is substituted into
(20) and (21) to construct the digital BB precoder. Algorithm 1 summarizes the
proposed PSO-$\alpha$HP technique for the optimal digital BB precoder design.

Algorithm 1 PSO-$\alpha$HP for optimal digital BB precoder

1:$\bm{\mathcal{H}}$, $K$, $P_{T}$, $Q$, $N_{S}$,
$\kappa_{\Delta,\max}^{\textrm{PSO}}$, $\kappa_{\Delta,\min}^{\textrm{PSO}}$.

2:Randomly initialize $N_{S}$ particles ${\bf a}_{i}^{(0)}$, $\forall
i=1:N_{S}$.

3:Set the initial personal best solution as ${\bf
a}_{\textrm{best},i}^{(0)}={\bf a}_{i}^{(0)}$.

4:Find the initial global best position ${\bf a}_{\textrm{gbest}}^{(0)}$ via
(26).

5:for $q=1:Q$, do

6: for $i=1:N_{S}$, do

7: Calculate velocity $\bm{\Delta}_{i}^{(q)}$ via (25).

8: Update the particle position ${\bf a}_{i}^{(q)}$ via (24).

9: Find the current personal best position ${\bf a}_{\textrm{best},i}^{(q)}$
via (27).

10: end for

11: Find the current global best position ${\bf a}_{\textrm{gbest}}^{(q)}$ via
(26).

12:end for

13:${\bf a}_{\textrm{best}}={\bf a}_{\textrm{gbest}}^{(Q)}$.

####  IV-B2 GWO-$\alpha$HP

Grey wolf optimization (GWO) mathematically implements the leadership
hierarchy and hunting mechanisms of the grey wolves, which are apex predators
at the top of food chain [31]. Unlike PSO, instead of moving towards its
personal best, each wolf (e.g., search agent) follows three best wolves
classified as alpha, beta, and delta. Specifically, the alpha wolf is
considered as the group leader, who is located at the global best position
with the highest objective value. Similarly, the beta wolf and delta wolf
represent the second and third global best positions, respectively. Based on
the variation in the objective values, the alpha, beta, and delta wolves might
be updated in every iteration. Hence, there is a stringent competition among
the three best wolves, which eventually benefits the increase of objective
value. In the proposed GWO-$\alpha$HP technique, the position of each wolf in
the $(q+1)^{th}$ iteration is found as222Similar to (24), we always apply the
clipping function in order to keep the search agents in $\left[0,1\right]$.
However, it is omitted for the brevity, when the search agent moves are
expressed in (28), (31), (32), (33) and (34).:

$\displaystyle{\bf a}_{i}^{(q+1)}$ $\displaystyle=\sum_{j=1}^{3}\frac{{\bf
a}_{\textrm{wolf},j}^{(q)}-{\bf w}_{j,1,q}^{\textrm{GWO}}\odot\left|{\bf
w}_{j,2}^{\textrm{GWO}}\odot{\bf a}_{\textrm{wolf},j}^{(q)}-{\bf
a}_{i}^{(q)}\right|}{3},$ (28)

where ${\bf a}_{\textrm{wolf},j}^{(q)}$ denotes the $j^{th}$ best wolf in the
$q^{th}$ iteration (i.e., $j=1$ for alpha, $j=2$ for beta, $j=3$ for delta),
${\bf w}_{j,1,q}^{\textrm{GWO}}$ and ${\bf w}_{j,2}^{\textrm{GWO}}$ are random
vectors with the uniformly distributed entries over
$\left[-\kappa_{q}^{\textrm{GWO}},+\kappa_{q}^{\textrm{GWO}}\right]$ and
$\left[0,2\right]$, respectively. Here,
$\kappa_{q}^{\textrm{GWO}}=2-\frac{2q-2}{Q}$ reduces from $2$ to $0$ for
switching from exploration to exploitation, respectively [31]. By using (23),
the alpha wolf in the $Q^{th}$ iteration is selected as the best search agent
as ${\bf a}_{\textrm{best}}={\bf a}_{\textrm{wolf},1}^{(Q)}$. The proposed
GWO-$\alpha$HP technique is presented in Algorithm 2.

Algorithm 2 GWO-$\alpha$HP for optimal digital BB precoder

1:$\bm{\mathcal{H}}$, $K$, $P_{T}$, $Q$, $N_{S}$.

2:Randomly initialize $N_{S}$ wolves ${\bf a}_{i}^{(0)}$, $\forall i=1:N_{S}$.

3:Find the initial three best position ${\bf a}_{\textrm{wolf},j}^{(0)}$,
$\forall j=1:3$.

4:for $q=1:Q$, do

5: for $i=1:N_{S}$, do

6: Update the wolf position ${\bf a}_{i}^{(q)}$ via (28).

7: end for

8: Find the current three best position ${\bf a}_{\textrm{wolf},j}^{(q)}$,
$\forall j=1:3$.

9:end for

10:${\bf a}_{\textrm{best}}={\bf a}_{\textrm{wolf},1}^{(Q)}$.

####  IV-B3 ACO-$\alpha$HP

Ant colony optimization (ACO) mimics the communication mechanism among the
ants while searching the food sources around their nest [32]. In an ant
colony, each ant (e.g., search agent) regularly seeks to find the food sources
in various directions. After discovering a food source, the ant marks the
corresponding path with an evaporating chemical called pheromone while
returning to the nest with food. The amount of pheromone deposited by each ant
varies based on the food quantity/quality, which is considered as the
objective value in the perspective of optimization. By the time, the ants
prefer moving in the direction of routes with dense pheromone levels instead
of entirely random search. This indirect communication mechanism motivates the
development of ACO algorithm. Although ACO is originally proposed for the
combinatorial optimization problems, it is also successfully adopted in the
continuous optimization problems [32].

For the mathematical foundation of the proposed ACO-$\alpha$HP technique,
$N_{S}$ ants discover a food source position ${\bf a}_{i}^{(q)}$ in every
iteration. Afterwards, only $\kappa_{1}^{\textrm{ACO}}$ of them with the
highest objective values are archived. Specifically, ${\bf
a}_{\textrm{ant},j}^{(q)}$ represents the $j^{th}$ best food source position
with $j=1,\cdots,\kappa_{1}^{\textrm{ACO}}$ discovered by the ant colony after
the $q^{th}$ iteration. By applying the Gaussian kernel, the pheromone level
for the $j^{th}$ food source is defined by [32]:

$\Upsilon_{j}^{\textrm{ACO}}=\frac{1}{\kappa_{1}^{\textrm{ACO}}\kappa_{2}^{\textrm{ACO}}\sqrt{2\pi}}e^{-\frac{1}{2}\frac{\left(j-1\right)^{2}}{\left(\kappa_{1}^{\textrm{ACO}}\kappa_{2}^{\textrm{ACO}}\right)^{2}}},$
(29)

where $\kappa_{2}^{\textrm{ACO}}$ is a parameter specifying the pheromone
level difference among the food sources. Based on the pheromone levels of a
set of $\kappa_{1}^{\textrm{ACO}}$ food sources, an ant prefers moving towards
the $j^{th}$ food source with the probability of:

$\mathcal{P}_{j}^{\textrm{ACO}}=\frac{\Upsilon_{j}^{\textrm{ACO}}}{\sum_{c=1}^{\kappa_{1}^{\textrm{ACO}}}~{}\Upsilon_{c}^{\textrm{ACO}}}\in\left[0,1\right].$
(30)

By using (29) and (30), one can show that when $\kappa_{2}^{\textrm{ACO}}$ is
small, the best food source position is highly preferred. Otherwise, for the
larger values of $\kappa_{2}^{\textrm{ACO}}$, the probability of choosing each
food source position becomes almost equivalent. Afterwards, the $v^{th}$
dimension of the $i^{th}$ ant position is calculated in the ${(q+1)^{th}}$
iteration as follows:

$\displaystyle{\bf a}_{i}^{(q+1)}\left[v\right]={\bf
a}_{\textrm{ant},j}^{(q)}\left[v\right]+w^{\textrm{ACO}}\kappa_{j,q,v}^{\textrm{ACO}},$
(31)

where $v=1,\cdots,2K$ represents the dimension index of an ant position, ${\bf
a}_{\textrm{ant},j}^{(q)}\left[v\right]$ denotes the randomly chosen $j^{th}$
food source position for the corresponding dimension via (30),
${w}^{\textrm{ACO}}$ is a random scalar following the distribution of
$\mathcal{N}\left(0,1\right)$,
$\kappa_{j,q,v}^{\textrm{ACO}}=\sum_{c=1}^{\kappa_{1}^{\textrm{ACO}}}{\big{|}{\bf
a}_{\textrm{ant},j}^{(q)}\left[v\right]-{\bf
a}_{\textrm{ant},c}^{(q)}\left[v\right]\big{|}}/\big{(}{\kappa_{1}^{\textrm{ACO}}-1}\big{)}$
expresses the average distance from the chosen food source position to the
others. According to the new objective values, $\kappa_{1}^{\textrm{ACO}}$
best food source positions are updated for the next iteration. After applying
this procedure for $Q$ iterations, we obtain the best position as ${\bf
a}_{\textrm{best}}={\bf a}_{\textrm{ant},1}^{(Q)}$. Finally, Algorithm 3
expresses the proposed ACO-$\alpha$HP technique.

Algorithm 3 ACO-$\alpha$HP for optimal digital BB precoder

1:$\bm{\mathcal{H}}$, $K$, $P_{T}$, $Q$, $N_{S}$, $\kappa_{1}^{\textrm{ACO}}$,
$\kappa_{2}^{\textrm{ACO}}$.

2:Randomly initialize $N_{S}$ ants ${\bf a}_{i}^{(0)}$, $\forall i=1:N_{S}$.

3:Find the initial $\kappa_{1}^{\textrm{ACO}}$ best position ${\bf
a}_{\textrm{ant},j}^{(0)}$, $\forall j=1:\kappa_{1}^{\textrm{ACO}}$.

4:Calculate $\mathcal{P}_{j}^{\textrm{ACO}}$ via (29) and (30), $\forall
j=1:\kappa_{1}^{\textrm{ACO}}$, as the probability of choosing the $j^{th}$
best food source.

5:for $q=1:Q$, do

6: for $i=1:N_{S}$, do

7: Randomly choose the $j^{th}$ best food source ${\bf
a}_{\textrm{ant},j}^{(q-1)}$.

8: Update the ant position ${\bf a}_{i}^{(q)}$ via (31).

9: end for

10: Find the current $\kappa_{1}^{\textrm{ACO}}$ best position ${\bf
a}_{\textrm{ant},j}^{(q)}$, $\forall j=1:\kappa_{1}^{\textrm{ACO}}$.

11:end for

12:${\bf a}_{\textrm{best}}={\bf a}_{\textrm{ant},1}^{(Q)}$.

####  IV-B4 CS-$\alpha$HP

Cuckoo search (CS) algorithm is inspired by the aggressive reproduction
strategy of the cuckoo birds [30]. Particularly, each cuckoo (e.g., search
agent) lays its own eggs to the nests of other cuckoos. Furthermore, it might
remove the eggs belonging to other cuckoos for increasing the survival
probability of its own eggs towards the next generations (i.e., iterations).
Here, the overall egg quality in a nest represents the objective value. Thus,
the CS algorithm includes three main steps in each iteration: (i) each cuckoo
randomly flies to a new nest following the Lévy flights and lays its eggs,
(ii) select either new or previous nest by comparing their egg quality, (iii)
the host cuckoo can replace the suspected eggs with a probability of
$\kappa_{1}^{\textrm{CS}}\in\left[0,1\right]$ as long as it improves the
overall egg quality.

In the proposed CS-$\alpha$HP technique, we first temporarily change the
$v^{th}$ dimension of the $i^{th}$ cuckoo position according to the Lévy
flight as follows [30]:

${\bf a}_{\textrm{CS}_{1},i}^{(q+1)}\left[v\right]={\bf
a}_{i}^{(q)}\left[v\right]+\frac{\kappa_{2}^{\textrm{CS}}}{\pi}\frac{\Gamma\left(\kappa_{2}^{\textrm{CS}}\right)\sin\left(\kappa_{2}^{\textrm{CS}}\frac{\pi}{2}\right)}{\left|w_{1}^{\textrm{CS}}\right|^{1+\kappa_{2}^{\textrm{CS}}}}$
(32)

where $\Gamma\left(\cdot\right)$ is the standard gamma function,
$\kappa_{2}^{\textrm{CS}}$ is a parameter for the Lévy exponent,
$w_{1}^{\textrm{CS}}=\frac{w_{2}^{\textrm{CS}}}{|w_{3}^{\textrm{CS}}|^{1/\kappa_{2}^{\textrm{CS}}}}$
is a random step size based on two Gaussian random variables
$w_{2}^{\textrm{CS}}\sim\mathcal{N}\bigg{(}0,\Big{[}\frac{\Gamma\left(1+\kappa_{2}^{\textrm{CS}}\right)}{\kappa_{2}^{\textrm{CS}}\Gamma\left((\kappa_{2}^{\textrm{CS}}+1)/2\right)}\frac{\sin\left(\kappa_{2}^{\textrm{CS}}\pi/2\right)}{2^{(\kappa_{2}^{\textrm{CS}}-1)/2}}\Big{]}^{1/\kappa_{2}^{\textrm{CS}}}\bigg{)}$
and $w_{3}^{\textrm{CS}}\sim\mathcal{N}\left(0,1\right)$. Afterwards, we
either select ${\bf a}_{\textrm{CS}_{1},i}^{(q+1)}$ or ${\bf a}_{i}^{(q)}$ by
comparing their objective values, then it is assigned to another temporary
position as ${\bf a}_{\textrm{CS}_{2},i}^{(q+1)}$. In the third and last step,
we apply a random replacement of the suspected eggs with the probability of
$\kappa_{1}^{\textrm{CS}}\in\left[0,1\right]$. Thus, the $i^{th}$ cuckoo
temporary position can be replaced with any randomly chosen $j^{th}$ cuckoo
temporary position as follows [30]:

${\bf
a}_{\textrm{CS}_{3},i}^{(q+1)}\left[v\right]=\left(1-w_{4}^{\textrm{CS}}\right){\bf
a}_{\textrm{CS}_{2},i}^{(q+1)}\left[v\right]+w_{4}^{\textrm{CS}}{\bf
a}_{\textrm{CS}_{2},j}^{(q+1)}\left[v\right],$ (33)

where $w_{4}^{\textrm{CS}}\sim\textrm{Ber}\left(\kappa_{1}^{CS}\right)$ is a
Bernoulli random variable (i.e., $w_{4}^{\textrm{CS}}=1$ with the probability
of $\kappa_{1}^{CS}$, otherwise, $w_{4}^{\textrm{CS}}=0$). Then, the best
temporary position among ${\bf a}_{\textrm{CS}_{1},i}^{(q+1)}$, ${\bf
a}_{\textrm{CS}_{2},i}^{(q+1)}$, and ${\bf a}_{\textrm{CS}_{3},i}^{(q+1)}$ is
transferred to the next iteration as ${\bf a}_{i}^{(q+1)}$. Finally, when we
complete $Q$ iterations, the proposed CS-$\alpha$HP finds the best position
${\bf a}_{\textrm{best}}$ in the last iteration via (22). The proposed
CS-$\alpha$HP technique is summarized in Algorithm 4.

Algorithm 4 CS-$\alpha$HP for optimal digital BB precoder

1:$\bm{\mathcal{H}}$, $K$, $P_{T}$, $Q$, $N_{S}$, $\kappa_{1}^{\textrm{CS}}$,
$\kappa_{2}^{\textrm{CS}}$.

2:Randomly initialize $N_{S}$ cuckoos ${\bf a}_{i}^{(0)}$, $\forall
i=1:N_{S}$.

3:for $q=1:Q$, do

4: for $i=1:N_{S}$, do

5: Calculate the first temporary position ${\bf a}_{\textrm{CS}_{1},i}^{(q)}$
via (32).

6: Select ${\bf a}_{\textrm{CS}_{1},i}^{(q)}$ or ${\bf a}_{i}^{(q-1)}$ based
on objective value, then

7: assign it to the second temporary position ${\bf
a}_{\textrm{CS}_{2},i}^{(q)}$.

8: end for

9: for $i=1:N_{S}$, do

10: Randomly choose a cuckoo’s second temporary

11: position as ${\bf a}_{\textrm{CS}_{2},j}^{(q)}$ with
$j\in\left\\{1,\cdots,N_{S}\right\\}$.

12: Calculate the third temporary position ${\bf a}_{\textrm{CS}_{3},i}^{(q)}$
via (33).

13: Update the cuckoo position ${\bf a}_{i}^{(q)}$ by selecting the best

14: of ${\bf a}_{\textrm{CS}_{1},i}^{(q)}$, ${\bf
a}_{\textrm{CS}_{2},i}^{(q)}$, and ${\bf a}_{\textrm{CS}_{3},i}^{(q)}$.

15: end for

16:end for

17:Find the best cuckoo ${\bf a}_{\textrm{best}}$ via (22).

18:${\bf a}_{\textrm{best}}$.

####  IV-B5 FA-$\alpha$HP

Firefly algorithm (FA) is motivated by the flashing light communication
strategy, where each firefly (i.e., search agent) aims to attract the mating
partners according to the brightness of its flashing light [30]. Here, the
objective value is described by the brightness. On the other hand, the
attractiveness of each firefly pair decays exponentially on the distance
between them. In every iteration, each firefly finds the best mating partner
among all combinations.

Unlike the previous four NI-$\alpha$HP techniques, the proposed FA-$\alpha$HP
only employs $\sqrt{N_{S}}$ search agents in order to have the same order of
computational complexity [30]. In other words, considering $\sqrt{N_{S}}$
fireflies and $\sqrt{N_{S}}$ mating opportunity for each firefly, we compute
$N_{S}=\sqrt{N_{S}}\times\sqrt{N_{S}}$ combinations for the possible firefly
positions per iteration. Therefore, the proposed FA-$\alpha$HP first computes
all possible positions for the $i^{th}$ firefly with respect to any mating
partner as follows:

${\bf a}_{\textrm{FA},i\to j}^{(q+1)}={\bf
a}_{i}^{(q)}+e^{-\kappa_{1}^{\textrm{FA}}\Upsilon_{i,j,q}^{\textrm{FA}}}\left[{\bf
a}_{j}^{(q)}-{\bf
a}_{i}^{(q)}\right]+\left[\kappa_{2}^{\textrm{FA}}\right]^{q}{\bf
w}^{\textrm{FA}},$ (34)

where $\forall i,j=1,\cdots,\sqrt{N_{S}}$ denotes the firefly index,
$\Upsilon_{i,j,q}^{\textrm{FA}}=\big{\|}{\bf a}_{i}^{(q)}-{\bf
a}_{j}^{(q)}\big{\|}^{2}$ indicates the distance between the corresponding
firefly pairs, ${\bf w}^{\textrm{FA}}$ is the random vector with the uniformly
distributed entries over $\left[-\frac{1}{2},+\frac{1}{2}\right]$,
$\kappa_{1}^{\textrm{FA}}$ and $\kappa_{2}^{\textrm{FA}}$ are the parameters
for the light absorption and randomness, respectively [30]. For the $i^{th}$
firefly, the best position with the highest objective value among all possible
${\bf a}_{\textrm{FA},i\to j}^{(q)}$ is assigned to ${\bf a}_{i}^{(q+1)}$.
This procedure is applied for each firefly through $Q$ iterations, then,
FA-$\alpha$HP finds the best position ${\bf a}_{\textrm{best}}$ via (22).
Algorithm 5 explains the proposed FA-$\alpha$HP technique for the optimal
digital BB precoder design.

Algorithm 5 FA-$\alpha$HP for optimal digital BB precoder

1:$\bm{\mathcal{H}}$, $K$, $P_{T}$, $Q$, $N_{S}$, $\kappa_{1}^{\textrm{FA}}$,
$\kappa_{2}^{\textrm{FA}}$.

2:Randomly initialize $\sqrt{N_{S}}$ fireflies ${\bf a}_{i}^{(0)}$, $\forall
i=1:\sqrt{N_{S}}$.

3:for $q=1:Q$, do

4: for $i=1:\sqrt{N_{S}}$, do

5: for $j=1:\sqrt{N_{S}}$, do

6: When the $j^{th}$ firefly is the mating partner of

7: the $i^{th}$ firefly, calculate ${\bf a}_{\textrm{FA},i\to j}^{(q)}$ via
(34).

8: end for

9: Update the firefly position ${\bf a}_{i}^{(q)}$ with the best of ${\bf
a}_{\textrm{FA},i\to j}^{(q)}$.

10: end for

11:end for

12:Find the best firefly ${\bf a}_{\textrm{best}}$ via (22).

13:${\bf a}_{\textrm{best}}$.

Algorithm 6 summarizes the proposed NI-$\alpha$HP technique, which
consecutively develops the analog RF beamformer ${\bf F}$ and digital BB
precoder ${\bf B}$. In addition to the $\alpha$-fairness, NI-$\alpha$HP
considerably reduces both the number of RF chains (i.e., from $M$ to
$N_{RF}$)333By applying the transfer block design in [15], NI-$\alpha$HP can
be also implemented via only $K$ RF chains by keeping the exactly same
performance.. and the channel estimation overhead size (i.e., from $M\times K$
to $N_{RF}\times K$).

Algorithm 6 Nature-inspired $\alpha$-fair hybrid precoding

— Analog RF Beamformer —

1:$M$, $G$, $\theta_{g}$, $\delta_{g}^{\theta}$, $\psi_{g}$,
$\delta_{g}^{\psi}$, $\forall g=1,\cdots,G$.

2:for $g=1:G$, do

3: Build angular support $\mathcal{A}_{g}$ via (10).

4: Find angle-pairs
$\big{(}\hat{\lambda}_{m_{g}}^{x},\hat{\lambda}_{n_{g}}^{y}\big{)}$ covering
$\mathcal{A}_{g}$ via (14).

5: Develop RF beamformer ${\bf F}_{g}$ via (16).

6:end for

7:${\bf F}=\left[{\bf F}_{1},\cdots,{\bf F}_{G}\right]$.

— Digital BB Precoder —

1:$\bm{\mathcal{H}}$, $K$, $P_{T}$.

2: Optimize ${\bf
a}=\big{[}\hat{p}_{1},\cdots,\hat{p}_{K},\hat{\beta}_{1},\cdots,\hat{\beta}_{K}\big{]}^{T}$
by applying PSO-$\alpha$HP, GWO-$\alpha$HP, ACO-$\alpha$HP, CS-$\alpha$HP, or
FA-$\alpha$HP. via Algorithm 1, 2, 3, 4, or 5, respectively.

3:Calculate scalars $\varepsilon_{1}=\frac{P_{T}}{\sum_{k=1}^{K}\hat{p}_{k}}$
and $\varepsilon_{2}=\frac{P_{T}}{\sum_{k=1}^{K}\hat{\beta}_{k}}$.

4:for $k=1:K$, do

5: Develop BB precoder vector ${\bf b}_{k}$ via (21).

6:end for

7:${\bf B}=\left[{\bf b}_{1},\cdots,{\bf b}_{K}\right]$.

##  V Illustrative Results

The sum-rate, energy-efficiency, and fairness performance evaluation of the
proposed NI-$\alpha$HP technique is presented throughout this section. Based
on the recent 3GPP Release 17 [4, 44, 45], the simulation setup for the MU-
mMIMO systems is summarized in Table II. On the other hand, Table III
expresses the hyper-parameters for each proposed NI-$\alpha$HP technique,
where GWO-$\alpha$HP does not have any hyper-parameters. Also, NI-$\alpha$HP
employs $N_{S}=100$ search agents through $Q=10$ iterations unless otherwise
stated.

The sum-rate capacity in the MU-mMIMO systems is calculated as
$R_{\textrm{sum}}=\sum_{k=1}^{K}R_{k}$ [bps/Hz]. Afterwards, the energy-
efficiency is obtained by taking the ratio of the sum-rate capacity and the
total transmit consumption as:

$\textrm{Energy-Efficiency}=\frac{R_{\textrm{sum}}}{P_{T}+N_{RF}\times
P_{RF}}\textrm{ [bps/Hz/W]},$ (35)

where $P_{RF}=250$ mW denotes the power consumption per RF chain [46]. The
number of RF chains utilized in the proposed NI-$\alpha$HP is provided in
Table II. On the other hand, the conventional single-stage FDP requires
$N_{RF}=M=256$ RF chains, which deteriorates the energy-efficiency.

As a quantitative fairness measure, we here adopt Jain’s fairness index444It
is important to note that a larger value of Jain’s fairness index indicates a
higher fairness level (e.g., Jain’s fairness index of $1$ implies absolute
fairness). based on each UE rate [18]:

$\textrm{Jain's Fairness
Index}=\frac{\left(\sum_{k=1}^{K}R_{k}\right)^{2}}{K\sum_{k=1}^{K}R_{k}^{2}}\in\left[0,1\right].$
(36)

TABLE II: Simulation parameters. # of antennas [4] | $M=16\times 16=256$
---|---
BS transmit power [44] |  $P_{T}=20$ dBm or $P_{T}=40$ dBm
Cell radius [4] | 100m
BS height |. UE height [45] | 10m |. 1.5m-2.5m
UE-BS horizontal distance | 10m – 100m
UE groups |  $G=2$ or $G=4$
UE per group | $K_{g}=\frac{K}{G}$
RF chains per group | $N_{RF,g}=\frac{N_{RF}}{G}=8$
Mean EAoD & AAoD |  $\theta_{g}=50^{\circ}$, $\psi_{g}=25^{\circ}+\frac{360^{\circ}}{G}\left(g-1\right)$
EAoD & AAoD spread [45] | $\delta_{g}^{\theta}=\delta_{g}^{\psi}=10^{\circ}$
Path loss exponent [44] | $\eta=3.76$
Noise PSD [44] |  $-174$ dBm/Hz
Channel bandwidth [44] |  $120$ kHz
# of paths[45] | $L=20$
# of network realizations | $5000$
TABLE III: Hyper-parameters for NI-$\alpha$HP. PSO-$\alpha$HP |  $\kappa_{\Delta,\max}^{\textrm{PSO}}=-\kappa_{\Delta,\min}^{\textrm{PSO}}=-0.2$ [25]
---|---
GWO-$\alpha$HP | -
ACO-$\alpha$HP |  $\kappa_{1}^{\textrm{ACO}}=10$, $\kappa_{2}^{\textrm{ACO}}=0.5$ [32]
CS-$\alpha$HP |  $\kappa_{1}^{\textrm{CS}}=0.25$, $\kappa_{2}^{\textrm{CS}}=1.5$ [30]
FA-$\alpha$HP |  $\kappa_{1}^{\textrm{FA}}=0.1$, $\kappa_{2}^{\textrm{FA}}=0.97$ [30]

###  V-A Benchmark on Sum-Rate Maximization

Fig. 3 plots the sum-rate and energy-efficiency performance of the proposed
PSO-$\alpha$HP technique versus the transmit power, where $K=10$ UEs are
clustered in $G=2$ groups. Here, we set $\alpha=0$ for sum-rate maximization
objective as shown in (6) and (22). For the benchmark, we consider the
conventional single-stage FDP [8] and various two-stage HP techniques such as
angular-based HP (AB-HP)[15], AB-HP with PSO-PA555AB-HP with PSO-PA develops
the BB precoder by concatenating two sub-blocks: (i) the well-known RZF, (ii)
PSO-PA optimizing $K$ power values. On the other hand, the proposed
PSO-$\alpha$HP has a single-shot solution for the BB precoder given in (21) by
jointly optimizing $2K$ parameters. [25], eigen-beamforming based HP (EBF-
HP)666EBF-HP employs both phase-shifter and variable-gain amplifiers in the
RF-stage, while the proposed NI-$\alpha$HP only uses the low-cost phase-
shifters. [13], non-orthogonal angle space based HP (NOAS-HP)777Unlike the
proposed NI-$\alpha$HP, NOAS-HP needs the fast time-varying full CSI in the RF
beamformer design [12]. Nevertheless, it serves as a benchmark. [12].
According to the simulation setup given in Table II, FDP requires $M=256$ RF
chains, whereas all HP schemes employ only $N_{RF}=16$ RF chains. Hence, it
indicates $93.75\%$ reduction in the hardware cost/complexity in the mMIMO
systems. The proposed NI-$\alpha$HP technique also lowers the channel
estimation overhead size by $93.75\%$ (i.e., it employs
$\bm{\mathcal{H}}\in\mathbb{C}^{10\times 16}$ instead of
${\bf{H}}\in\mathbb{C}^{10\times 256}$).

In Fig. 3(a), we first investigate the sum-rate performance, where all
benchmark schemes, except AB-HP with PSO-PA, apply equal PA. The numerical
results reveal that the proposed PSO-$\alpha$HP greatly enhances the sum-rate
capacity. To illustrate, when $P_{T}=30$ dBm, the sum-rate capacity for
PSO-$\alpha$HP is $108.9$ bps/Hz, which is approximately $10.4$ bps/Hz higher
than AB-HP with PSO-PA, $39.5$ bps/Hz higher than FDP, $44.6$ bps/Hz higher
than AB-HP, $48.7$ bps/Hz higher than NOAS-HP, $50.7$ bps/Hz higher than EBF-
HP. Moreover, the performance gap remains almost constant after $P_{T}=30$
dBm. On the other hand, it is seen that AB-HP outperforms both NOAS-HP and
EBF-HP with the performance gap of $1.25$ dB and $1.75$ dB, respectively, in
terms of the transmit power. Furthermore, AB-HP only experiences $1.5$ dB
degradation compared to the single-stage FDP.

In Fig. 3(b), the energy-efficiency curves are presented versus the transmit
power. By means of a significant reduction in the number of RF chains, all HP
schemes remarkably improve the energy-efficiency compared to the single-stage
FDP. Furthermore, the highest energy-efficiency is attained via the proposed
PSO-$\alpha$HP across all transmit power regimes. For example, the energy-
efficiency at $P_{T}=30$ dBm as $21.6$ bps/Hz/W for PSO-$\alpha$HP and $19.5$
bps/Hz/W for AB-HP with PSO-PA, however, it is only $1.1$ bps/Hz/W for FDP.

(a) Sum-rate

(b) Energy-efficiency

Figure 3: Sum-rate and energy-efficiency benchmark performance versus transmit
power ($\alpha=0$, $G=2$ groups, $K=10$ UEs).

Fig. 4 presents the sum-rate performance versus the number of iterations,
where the transmit power is ${P_{T}=40}$ dBm to serve either $K=2,4,6,8$ or
$10$ UEs clustered in $G=2$ groups.

Figure 4: Sum-rate benchmark performance versus number of iterations
($\alpha=0$, $G=2$ groups, $P_{T}=40$ dBm).

For the sum-rate maximization objective, the fairness level is kept as
$\alpha=0$ to compare PSO-$\alpha$HP and AB-HP with PSO-PA. Both techniques
are investigated up to $Q=20$ iterations, while $Q=0$ indicates the AB-HP with
equal PA. The promising numerical results show that even though PSO-$\alpha$HP
optimizes twice as many parameters as comparison to AB-HP with PSO-PA, it both
converges faster and achieves higher capacity, especially for the larger
number of UEs. For example, PSO-$\alpha$HP provides $3.5$ bps/Hz, $6.8$ bps/Hz
or $10.4$ bps/Hz higher sum-rate at $Q=10$ iterations, when there are $K=6$,
$8$ or $10$ UEs, respectively. Moreover, the sum-rate performance of
PSO-$\alpha$HP is approximately saturated within $Q=10$ iterations for
$K=6,8,10$ UEs, whereas AB-HP with PSO-PA requires more than $Q=20$
iterations. The main reason for this superior performance is the derived
optimal digital BB precoder expression given in (18). On the other hand, when
there are only $K=2$ or $4$ UEs, both techniques converge in the early
iterations. It is also important to highlight the sum-rate improvement
accomplished by the proposed PSO-$\alpha$HP. the AB-HP with equal PA (i.e.,
$Q=0$), PSO-$\alpha$HP improves the sum-rate capacity by $15.4\%$, $28.3\%$,
$36.1\%$, $41.6\%$, $47.7\%$ for $K=2,4,6,8,10$ UEs, respectively.
Attractively, the sum-rate capacity improvement increases for the larger
number of UEs.

###  V-B Performance Evaluation of NI-$\alpha$HP Techniques

Fig. 5 compares all five NI-$\alpha$HP techniques in regards to their sum-rate
performance versus the number of iterations, where $\alpha=0$ and $P_{T}=40$
dBm.

Figure 5: Sum-rate versus number of iterations ($P_{T}=40$ dBm and
$\alpha=0$).

Here, we serve either $K=6,10$ UEs in $G=2$ groups or $K=12,20$ UEs in $G=4$
groups. In every scenario, GWO-$\alpha$HP attains a higher sum-rate and
converges faster than all PSO-$\alpha$HP, ACO-$\alpha$HP, CS-$\alpha$HP and
FA-$\alpha$HP. For instance, the sum-rate performance of GWO-$\alpha$HP is
saturated around $Q=15$ iterations for $K=20$ UEs, while others require more
than $Q=20$ iterations. The intuitive explanation of these observations is
that GWO-$\alpha$HP diversifies the search agents and increases exploration in
the optimization space by asking them to simultaneously follow three best
solutions (i.e., alpha, beta, and delta wolves), which also brings a
beneficial competition. On the other hand, each search agent in PSO-$\alpha$HP
only follows the best solution achieved until the corresponding iteration,
which might severely damage the competition and make them stuck around a local
optima for long iterations. Similarly, each search agent in ACO-$\alpha$HP
tracks only one of the randomly chosen $\kappa_{1}^{\textrm{ACO}}=10$ best
solutions, which brings the competition among the best solutions and improves
the performance of ACO-$\alpha$HP in the later iterations (e.g.,
ACO-$\alpha$HP outperforms PSO-$\alpha$HP, as $Q$ increases). However,
compared to other four NI-$\alpha$HP techniques, ACO-$\alpha$HP experiences
slow improvements in the early iterations due to following only one of the
randomly chosen best solutions, which might be highly likely a local optimum.
For example, ACO-$\alpha$HP achieves a lower sum-rate performance for $Q<6$
and $Q<12$ iterations compared to PSO-$\alpha$HP, when there are $K=12$ and
$K=20$ UEs, respectively. On the other hand, we observe that both
CS-$\alpha$HP and FA-$\alpha$HP experience comparatively low sum-rate
improvements with respect to the other schemes. Additionally, by combining
Fig. 4 and Fig. 5, one can conclude that GWO-$\alpha$HP performs also better
than AB-HP with PSO-PA [25].

Table IV displays the runtime performance evaluation888Each NI-$\alpha$HP
scheme is implemented in MATLAB via a PC with Intel(R) Xeon(R) Gold 5220R CPU
@ 2.20 GHz and 96 GB RAM. for each NI-$\alpha$HP scheme. In addition to its
superior performance, GWO-$\alpha$HP has the shortest runtime among all five
schemes. On the contrary, CS-$\alpha$HP has the longest runtime in addition to
its inadequate sum-rate performance presented in Fig. 5. Thus, GWO-$\alpha$HP
is considered for the following $\alpha$-fairness illustrative results based
on its favorable performance.

TABLE IV: Runtime per iteration. |  $G=2$ groups |  $G=4$ groups
---|---|---
$6$ UEs |  $10$ UEs |  $12$ UEs |  $20$ UEs
PSO-$\alpha$HP | 4.0 msec | 4.4 msec | 6.4 msec | 7.6 msec
GWO-$\alpha$HP | 3.5 msec | 3.9 msec | 6.0 msec | 7.3 msec
ACO-$\alpha$HP | 3.7 msec | 4.2 msec | 6.2 msec | 7.5 msec
CS-$\alpha$HP | 7.8 msec | 8.3 msec | 11.7 msec | 14.5 msec
FA-$\alpha$HP | 4.3 msec | 4.7 msec | 6.9 msec | 8.3 msec

###  V-C $\alpha$-Fairness

(a) Sum-rate

(b) Jain’s fairness index

(c) Rate gap

Figure 6: Sum-rate, fairness, and rate-gap performance of GWO-$\alpha$HP
versus transmit power ($G=2$ groups, $K=2$ UEs).

(a) Jain’s fairness index

(b) Rate gap

Figure 7: Fairness and rate-gap performance of GWO-$\alpha$HP versus number of
UEs and fairness level ($G=2$ groups, $P_{T}=20$ dBm).

Fig. 6 investigates the proposed GWO-$\alpha$HP technique for various fairness
levels such as $\alpha=0,1,2,5,10,\infty$, where there are only $K=2$ UEs in
$G=2$ groups. As expressed in Section III, $\alpha=0$ is for the sum-rate
maximization, ${\alpha=1}$ indicates the proportional fairness, while
${\alpha\to\infty}$ implies the max-min fairness. Specifically, we illustrate
the sum-rate and Jain’s fairness index performance in Fig. 6(a) and Fig. 6(b),
respectively. By defining ${R_{\max}=\max_{k}R_{k}}$ and
${R_{\min}=\min_{k}R_{k}}$, Fig. 6(c) also plots the rate gap given by:

$\textrm{Rate Gap}=R_{\max}-R_{\min}\textrm{ [bps/Hz]}.$ (37)

In Fig. 6(a), we observe a sum-rate performance degradation for the higher
values of $\alpha$ as a result of increased fairness level. For example, the
sum-rate at $P_{T}=0$ dBm drops from $11.7$ bps/Hz to $6.7$ bps/Hz, when the
fairness level is increased from $\alpha=0$ to $\alpha\to\infty$. On the other
hand, as $\alpha$ increases, Jain’s fairness index given in (36) improves as
demonstrated in Fig. 6(b). For instance, the fairness index at $P_{T}=0$ dBm
is improved from $0.71$ to $0.81$ by switching from the sum-rate maximization
to the proportional fairness, while the max-min fairness achieves the fairness
index of $1.0$. The numerical results also reveal that the proposed
GWO-$\alpha$HP technique can address various fairness expectations by
adjusting $\alpha$. It is seen that GWO-$\alpha$HP even supports absolute
fairness (i.e., Jain’s fairness index of $1.0$). Furthermore, although the
sum-rate capacity linearly increases in the high transmit power regime as seen
in Fig. 6(a), the rate gap among $K=2$ UEs is almost saturated as shown in
Fig. 6(c). Hence, when the transmit power increases, Jain’s fairness index
also improves as presented in Fig. 6(b). Additionally, the rate gap decays for
the higher fairness levels and it even vanishes for $\alpha\to\infty$.

Fig. 7 illustrates Jain’s fairness index and rate gap contour plots versus the
number of UEs and the fairness level, where we set the transmit power
$P_{T}=20$ dBm to support either $K=2,4,6,8,10$ or $12$ UEs in $G=2$ groups.
Also, the fairness levels are selected as $\alpha=0,1,2,\cdots,20$. When
GWO-$\alpha$HP targets the sum-rate maximization objective function (i.e.,
$\alpha=0$), the rate gap remarkably increases for the larger number of UEs,
which also deteriorates Jain’s fairness index. To illustrate, when the number
of UEs is changed from $K=2$ to $K=12$, the rate gap enlarges from $6.6$
bps/Hz to $15.9$ bps/Hz and the fairness index sharply reduces from $0.91$ to
$0.66$, respectively. On the other hand, GWO-$\alpha$HP jointly reduces the
rate gap and improves Jain’s fairness index by selecting a higher fairness
level. For example, when the fairness level is set to $\alpha=20$ for $K=12$
UEs, the rate gap is calculated as only $4.8$ bps/Hz and Jain’s fairness index
is enhanced to $0.86$. As mentioned earlier, it is possible to adjust $\alpha$
based on the fairness index expectation. To illustrate, when the desired
Jain’s fairness index is at least $0.9$ for $K=8$ UEs, it is necessary to
choose $\alpha\geq 7$.

##  VI Conclusions

In this work, a novel nature-inspired $\alpha$-fair hybrid precoding
(NI-$\alpha$HP) technique has been proposed for the mmWave MU-mMIMO systems.
First, the analog RF beamformer has been developed via slow time-varying AoD
information for both reducing the channel estimation overhead size and
maximizing the beamforming gain in the desired direction. Second, the
effective channel seen from the BB-stage has been employed to design the
digital BB precoder. Based on the $\alpha$-fair resource allocation objective,
we have derived the optimal digital BB precoder expression, which includes a
set of NP-hard parameters. Afterwards, they have been effectively optimized
via applying five nature-inspired intelligent algorithms. Particularly, we
have introduced PSO-$\alpha$HP, GWO-$\alpha$HP, ACO-$\alpha$HP, CS-$\alpha$HP,
FA-$\alpha$HP. The numerical results reveal that under the sum-rate
maximization objective, the proposed NI-$\alpha$HP techniques outperform other
benchmark HP schemes in terms of both sum-rate and energy-efficiency
performance. Furthermore, we have observed that GWO-$\alpha$HP achieves a
higher sum-rate and converges faster among the five proposed NI-$\alpha$HP
techniques. Regarding the fairness perspective, NI-$\alpha$HP successfully
reduces the rate gap among UEs and satisfies various fairness expectations by
simply adjusting the fairness level ($\alpha$).

##  Appendix A Proof of Proposition 1

By using the $\alpha$-fairness utility function given in (6), we first write
the Lagrangian function of (17) for $\alpha\neq 1$ as:

$\displaystyle\mathcal{L}$ $\displaystyle\left({\bf b}_{1},\cdots,{\bf
b}_{K},\bar{\beta}_{0},\bar{\beta}_{1},\cdots,\bar{\beta}_{K}\right)$ (38)
$\displaystyle=\sum_{k=1}^{K}\log_{2}\left(1+\frac{\left|{\bf h}_{k}^{T}{\bf
Fb}_{k}\right|^{2}}{\sum_{u\neq k}^{K}\left|{\bf h}_{k}^{T}{\bf
Fb}_{u}\right|^{2}+\sigma_{n}^{2}}\right)^{1-\alpha}$
$\displaystyle+\bar{\beta}_{0}\left(\sum_{k=1}^{K}{\bf b}_{k}^{H}{\bf
b}_{k}-P_{T}\right)$
$\displaystyle+\sum_{k=1}^{K}\bar{\beta}_{k}\left(1-\frac{\left|{\bf
h}_{k}^{T}{\bf Fb}_{k}\right|^{2}}{\mu_{k}\sigma_{n}^{2}}+\frac{\sum_{u\neq
k}^{K}\left|{\bf h}_{k}^{T}{\bf Fb}_{u}\right|^{2}}{\sigma_{n}^{2}}\right),$

where $\bar{\beta}_{0}\geq 0$ and $\bar{\beta}_{k}\geq 0$ are the Lagrangian
multipliers associated with the transmit power constraint (i.e., $C_{3}$ in
(17)) and the $k^{th}$ UE SINR constraint (i.e., $C_{2}$ in (17)),
respectively. By applying the strong duality property [47], the dual function
is obtained as follows:

$\displaystyle\min_{{\bf b}_{1},\cdots,{\bf b}_{K}}\mathcal{L}$
$\displaystyle\left({\bf b}_{1},\cdots,{\bf
b}_{K},\bar{\beta}_{0},\bar{\beta}_{1},\cdots,\bar{\beta}_{K}\right)$ (39)
$\displaystyle=-\bar{\beta}_{0}P_{T}+\sum_{k=1}^{K}\bar{\beta}_{k}=0,$

which implies $\sum_{k=1}^{K}\beta_{k}=P_{T}$ with
$\beta_{k}=\frac{\bar{\beta}_{k}}{\bar{\beta}_{0}}$. In order to find the
optimal ${\bf b}_{k}$, we here exploit the KKT conditions. Thus, the gradient
of the Lagrangian with respect to ${\bf b}_{k}$ is set to zero:

$\displaystyle\frac{\partial\mathcal{L}\left({\bf b}_{1},\cdots,{\bf
b}_{K},\bar{\beta}_{0},\bar{\beta}_{1},\cdots,\bar{\beta}_{K}\right)}{\partial{\bf
b}_{k}}=0,~{}\forall k.$ (40)

By using (4) and (38), the Lagrangian function is expanded as
$\mathcal{L}\left({\bf b}_{1},\cdots,{\bf
b}_{K},\bar{\beta}_{0},\bar{\beta}_{1},\cdots,\bar{\beta}_{K}\right)=\mathcal{L}_{1}+\mathcal{L}_{2}+\mathcal{L}_{3}$
with
$\mathcal{L}_{1}=\sum_{u=1}^{K}\log_{2}\left(1+\textrm{SINR}_{u}\right)^{1-\alpha}$,
$\mathcal{L}_{2}=\bar{\beta}_{0}\left(\sum_{u=1}^{K}{\bf b}_{u}^{H}{\bf
b}_{u}-P_{T}\right)$ and
$\mathcal{L}_{3}=\sum_{u=1}^{K}\bar{\beta}_{u}\left(1-\frac{\left|{\bf
h}_{u}^{T}{\bf Fb}_{u}\right|^{2}}{\mu_{u}\sigma_{n}^{2}}+\frac{\sum_{i\neq
u}^{K}\left|{\bf h}_{u}^{T}{\bf Fb}_{i}\right|^{2}}{\sigma_{n}^{2}}\right)$.
After some mathematical derivations, we find the gradient of $\mathcal{L}_{1}$
as:

$\displaystyle\frac{\partial\mathcal{L}_{1}}{\partial{\bf b}_{k}}$
$\displaystyle=\frac{\partial}{\partial{\bf
b}_{k}}\left[\sum_{u=1}^{K}\log_{2}\left(1+\frac{\left|{\bf h}_{u}^{T}{\bf
Fb}_{u}\right|^{2}}{\sum_{i\neq u}^{K}\left|{\bf h}_{u}^{T}{\bf
Fb}_{i}\right|^{2}+\sigma_{n}^{2}}\right)^{1-\alpha}\right]$ (41)
$\displaystyle=\frac{\partial}{\partial{\bf
b}_{k}}\bigg{[}\log_{2}\left(1+\textrm{SINR}_{k}\right)^{1-\alpha}\bigg{.}$
$\displaystyle\bigg{.}+\sum_{u\neq
k}^{K}\log_{2}\left(1+\textrm{SINR}_{u}\right)^{1-\alpha}\bigg{]}$
$\displaystyle=\bigg{[}\frac{2}{\chi_{k}}\log_{2}\left(1+\textrm{SINR}_{k}\right)^{-\alpha}\Big{.}$
$\displaystyle\bigg{.}+\sum_{u\neq
k}^{K}\frac{2\times\textrm{SINR}_{u}}{\chi_{u}}{{\log}_{2}}{{\left({1+\textrm{SINR}_{u}}\right)}^{-\alpha}}\bigg{]}{\bf
A}_{k}{\bf b}_{k},$

where ${\bf{A}}_{k}={\bf F}^{H}{\bf h}_{k}^{*}{\bf h}_{k}^{T}{\bf F}$ and
$\chi_{k}=\frac{1-\alpha}{\ln\left(2\right)\left({\left\|{{\bf
h}_{k}^{T}\bf{FB}}\right\|^{2}+\sigma_{n}^{2}}\right)}$. Afterwards, the
gradient of $\mathcal{L}_{2}$ is obtained as follows:

$\frac{\partial\mathcal{L}_{2}}{\partial{\bf
b}_{k}}=\frac{\partial}{{\partial{\bf{b}}_{k}}}\left[{\bar{\beta}_{0}\left({\sum\limits_{u=1}^{K}{{\bf{b}}_{u}^{H}{{\bf{b}}_{u}}-{P_{T}}}}\right)}\right]=2\bar{\beta}_{0}{{\bf{b}}_{k}}.$
(42)

Moreover, the gradient of $\mathcal{L}_{3}$ is derived as:

$\displaystyle\frac{\partial\mathcal{L}_{3}}{\partial{\bf
b}_{k}}=\frac{\partial}{\partial{\bf
b}_{k}}\left[\sum_{u=1}^{K}\bar{\beta}_{u}\left(1-\frac{\left|{\bf
h}_{u}^{T}{\bf Fb}_{u}\right|^{2}}{\mu_{u}\sigma_{n}^{2}}+\frac{\sum_{i\neq
u}^{K}\left|{\bf h}_{u}^{T}{\bf Fb}_{i}\right|^{2}}{\sigma_{n}^{2}}\hskip
2.15277pt\right)\right]$ (43) $\displaystyle=\frac{\partial}{\partial{\bf
b}_{k}}\Bigg{[}\bar{\beta}_{k}\left(1-\frac{\left|{\bf h}_{k}^{T}{\bf
Fb}_{k}\right|^{2}}{\mu_{k}\sigma_{n}^{2}}+\frac{\sum_{i\neq k}^{K}\left|{\bf
h}_{k}^{T}{\bf Fb}_{i}\right|^{2}}{\sigma_{n}^{2}}\hskip
2.15277pt\right)\Bigg{.}$ $\displaystyle\Bigg{.}+\sum_{u\neq
k}^{K}\bar{\beta}_{u}\left(1-\frac{\left|{\bf h}_{u}^{T}{\bf
Fb}_{u}\right|^{2}}{\mu_{u}\sigma_{n}^{2}}+\frac{\left|{\bf h}_{u}^{T}{\bf
Fb}_{k}\right|^{2}}{\sigma_{n}^{2}}+\frac{\sum_{i\neq u,k}^{K}\left|{\bf
h}_{u}^{T}{\bf Fb}_{i}\right|^{2}}{\sigma_{n}^{2}}\hskip
2.15277pt\right)\Bigg{]}$
$\displaystyle=-\frac{2\bar{\beta}_{k}}{\mu_{k}\sigma_{n}^{2}}{\bf A}_{k}{\bf
b}_{k}+\sum_{u\neq k}^{K}\frac{2\bar{\beta}_{u}}{\sigma_{n}^{2}}{\bf
A}_{u}{\bf b}_{k}$
$\displaystyle=-\frac{2\bar{\beta}_{k}}{\sigma_{n}^{2}}\left(1+\frac{1}{\mu_{k}}\right){\bf
A}_{k}{\bf b}_{k}+\sum_{u=1}^{K}\frac{2\bar{\beta}_{u}}{\sigma_{n}^{2}}{\bf
A}_{u}{\bf b}_{k}.$

By substituting (38), (41), (42), and (43) into (40), the following expression
can be written:

$\displaystyle\frac{\partial\mathcal{L}\left({\bf b}_{1},\cdots,{\bf
b}_{K},\bar{\beta}_{0},\bar{\beta}_{1},\cdots,\bar{\beta}_{K}\right)}{\partial{\bf
b}_{k}}$ (44)
$\displaystyle=\bigg{[}-\frac{{\beta}_{k}}{\sigma_{n}^{2}}+\frac{1}{\chi_{k}\bar{\beta}_{0}}\log_{2}\left(1+\textrm{SINR}_{k}\right)^{-\alpha}\left(1+\frac{1}{\mu_{k}}\right)$
$\displaystyle+\sum_{u\neq
k}^{K}\frac{\textrm{SINR}_{u}}{\chi_{u}\bar{\beta}_{0}}{{\log}_{2}}{{\left({1+\textrm{SINR}_{u}}\right)}^{-\alpha}}\bigg{]}{\bf
h}_{k}^{T}{\bf{F}}{\bf b}_{k}{\bf F}^{H}{\bf h}_{k}^{*}$
$\displaystyle+\bigg{[}{\bf
I}_{N_{RF}}+\sum_{u=1}^{K}\frac{{\beta}_{u}}{\sigma_{n}^{2}}{\bf
A}_{u}\bigg{]}{\bf b}_{k}$ $\displaystyle=0$

where we utilize
$\frac{\partial\left(\mathcal{L}_{1}+\mathcal{L}_{2}+\mathcal{L}_{3}\right)}{\partial{\bf
b}_{k}}=\left({2\bar{\beta}_{0}}\right)^{-1}\frac{\partial\left(\mathcal{L}_{1}+\mathcal{L}_{2}+\mathcal{L}_{3}\right)}{\partial{\bf
b}_{k}}=0$ and ${\bf A}_{k}{\bf b}_{k}={\bf F}^{H}{\bf h}_{k}^{*}{\bf
h}_{k}^{T}{\bf{F}}{\bf b}_{k}={\bf h}_{k}^{T}{\bf{F}}{\bf b}_{k}{\bf
F}^{H}{\bf h}_{k}^{*}$. Afterwards, we derive the optimal digital BB precoder
vector for the $k^{th}$ UE as follows:

$\displaystyle{\bf b}_{k}$
$\displaystyle=\bigg{[}\frac{{\beta}_{k}}{\sigma_{n}^{2}}-\frac{1}{\chi_{k}\bar{\beta}_{0}}\log_{2}\left(1+\textrm{SINR}_{k}\right)^{-\alpha}\left(1+\frac{1}{\mu_{k}}\right)$
(45) $\displaystyle-\sum_{u\neq
k}^{K}\frac{\textrm{SINR}_{u}}{\chi_{u}\bar{\beta}_{0}}{{\log}_{2}}{{\left({1+\textrm{SINR}_{u}}\right)}^{-\alpha}}\bigg{]}{\bf
h}_{k}^{T}{\bf{F}}{\bf b}_{k}$ $\displaystyle\times\bigg{[}{\bf
I}_{N_{RF}}+\sum_{u=1}^{K}\frac{{\beta}_{u}}{\sigma_{n}^{2}}{\bf
A}_{u}\bigg{]}^{-1}{\bf F}^{H}{\bf h}_{k}^{*}.$

Then, one can define the allocated power for the $k^{th}$ UE as:

$\displaystyle{p_{k}}$ $\displaystyle={\bf b}_{k}^{H}{\bf b}_{k}$ (46)
$\displaystyle={\bigg{\|}\bigg{[}{\bf
I}_{N_{RF}}+\sum_{u=1}^{K}\frac{\beta_{u}}{\sigma^{2}_{n}}{\bf
A}_{u}\bigg{]}^{-1}{\bf F}^{H}{\bf h}_{k}^{*}\bigg{\|}_{2}^{2}}$
$\displaystyle\times\bigg{[}\frac{{\beta}_{k}}{\sigma_{n}^{2}}-\frac{1}{\chi_{k}\bar{\beta}_{0}}\log_{2}\left(1+\textrm{SINR}_{k}\right)^{-\alpha}\left(1+\frac{1}{\mu_{k}}\right)$
$\displaystyle-\sum_{u\neq
k}^{K}\frac{\textrm{SINR}_{u}}{\chi_{u}\bar{\beta}_{0}}{{\log}_{2}}{{\left({1+\textrm{SINR}_{u}}\right)}^{-\alpha}}\bigg{]}^{2}\left({\bf
h}_{k}^{T}{\bf{F}}{\bf b}_{k}\right)^{2}.$

Finally, by combining (45) and (46), the optimal digital BB precoder vector is
rewritten as follows:

$\displaystyle{\bf b}_{k}$
$\displaystyle=\underbrace{\sqrt{p_{k}}}_{{\textrm{Power}}}\underbrace{\frac{\left[{\bf
I}_{N_{RF}}+\sum_{u=1}^{K}\frac{\beta_{u}}{\sigma^{2}_{n}}{\bf
A}_{u}\right]^{-1}{\bf F}^{H}{\bf h}_{k}^{*}}{\left\|\left[{\bf
I}_{N_{RF}}+\sum_{u=1}^{K}\frac{\beta_{u}}{\sigma^{2}_{n}}{\bf
A}_{u}\right]^{-1}{\bf F}^{H}{\bf h}_{k}^{*}\right\|}}_{\textrm{Beamforming
Direction}},$ (47)

where $\sum_{k=1}^{K}p_{k}=P_{T}$ and $\sum_{k=1}^{K}\beta_{k}=P_{T}$. It is
important to remark that we find the closed-from expression for the optimal
digital BB precoder as shown in (47), however, finding the optimal values for
$p_{k}$ and $\beta_{k}$ is NP-hard due to the entangled optimization
parameters as expressed in (46) [38]. Moreover, one can easily show that (47)
is valid for all fairness levels including $\alpha=1$. Hence, this concludes
the proof of (18).

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|  Asil Koc (Graduate Student Member, IEEE) received the B.Sc. degree (Hons.)
in electronics and communication engineering, the M.Sc. degree (Hons.) in
telecommunication engineering from Istanbul Technical University, Istanbul,
Turkey, in 2015 and 2017, respectively. Currently, he is a Ph.D. candidate in
electrical engineering at McGill University, Montreal, Canada. From 2015 to
2017, he was a Research and Teaching Assistant with the Electronics and
Communication Engineering Department, Istanbul Technical University. Since
2017 and 2021, he has been a Teaching Assistant and Lecturer, respectively,
with the Electrical and Computer Engineering Department, McGill University.
His research interests include, but not limited to AI/ML based wireless
communications, massive MIMO, full-duplex, millimeter-wave/terahertz,
beamforming, index modulation, wireless power transfer, and cooperative
networks. He was a recipient of Erasmus Scholarship by European Union, McGill
Engineering Doctoral Award, IEEE ComSoc Student Travel Grant, Graduate
Research Enhancement and Travel Award by McGill University, and STARaCom
Collaborative Grant by the FRQNT.
---|---
|  Tho Le-Ngoc (Life Fellow, IEEE) received the B.Eng. degree in electrical
engineering, in 1976, the M.Eng. degree in microprocessor applications, in
1978, from McGill University, Montreal, and the Ph.D. degree in digital
communications, in 1983, from the University of Ottawa, Canada. From 1977 to
1982, he was with Spar Aerospace Ltd., Sainte-Anne-de-Bellevue, QC, Canada,
involved in the development and design of satellite communications systems.
From 1982 to 1985, he was with SRTelecom Inc., Saint-Laurent, QC, Canada,
where he developed the new point-to-multipoint DA-TDMA/TDM Subscriber Radio
System SR500. From 1985 to 2000, he was a Professor with the Department of
Electrical and Computer Engineering, Concordia University, Montreal. Since
2000, he has been with the Department of Electrical and Computer Engineering,
McGill University. His research interest includes broadband digital
communications. He is a Distinguished James McGill Professor, and a Fellow of
the Engineering Institute of Canada, the Canadian Academy of Engineering, and
the Royal Society of Canada. He was a recipient of the 2004 Canadian Award in
Telecommunications Research and the IEEE Canada Fessenden Award, in 2005.
---|---