# Learning Energy-Efficient Hardware Configurations for Massive MIMO
Beamforming

Hamed Hojatian, Zoubeir Mlika, Jérémy Nadal,
Jean-François Frigon, and François Leduc-Primeau This work was supported by
the Natural Sciences and Engineering Research Council of Canada (NSERC) (under
project ALLRP 566589-21) and InnovÉÉ (INNOV-R program) through the partnership
with Ericsson and ECCC. H. Hojatian, Zoubeir Mlika, J-F. Frigon, F. Leduc-
Primeau are with the Department of Electrical Engineering, Polytechnique
Montreal, Montreal, QC, H3T lJ4, Canada. J. Nadal is with the Department of
Mathematical and Electrical Engineering, IMT Atlantique, France. Corresponding
author: H. Hojatian (Emails<EMAIL_ADDRESS>
###### Abstract

Hybrid beamforming (HBF) and antenna selection are promising techniques for
improving the energy efficiency (EE) of massive multiple-input multiple-output
(mMIMO) systems. However, the transmitter architecture may contain several
parameters that need to be optimized, such as the power allocated to the
antennas and the connections between the antennas and the radio frequency
chains. Therefore, finding the optimal transmitter architecture requires
solving a non-convex mixed integer problem in a large search space. In this
paper, we consider the problem of maximizing the EE of fully digital precoder
(FDP) and hybrid beamforming (HBF) transmitters. First, we propose an energy
model for different beamforming structures. Then, based on the proposed energy
model, we develop an unsupervised deep learning method to maximize the EE by
designing the transmitter configuration for FDP and HBF. The proposed deep
neural networks can provide different trade-offs between spectral efficiency
and energy consumption while adapting to different numbers of active users.
Finally, to ensure that the proposed method can be implemented in practice, we
investigate the ability of the model to be trained exclusively using imperfect
channel state information (CSI), both for the input to the deep learning model
and for the calculation of the loss function. Simulation results show that the
proposed solutions can outperform conventional methods in terms of EE while
being trained with imperfect CSI. Furthermore, we show that the proposed
solutions are less complex and more robust to noise than conventional methods.

###### Index Terms:

Hybrid Beamforming, Subarray Hybrid Beamforming, Fully Digital Beamforming,
Deep Neural Network, Unsupervised Learning, Energy Efficiency.

##  I Introduction

Wireless communication has been revolutionized by massive MIMO (mMIMO)
technologies, which are already one of the key enabling technologies in the
fifth-generation (5G) of wireless networks thanks to their potential to
increase the transmission capacity through the deployment of large-scale
antenna arrays at the transmitter or receiver side [1]. As a result,
millimeter wave (mm-Wave) communications can be used at longer ranges, thus
greatly increasing the bandwidth available to wireless networks [2].

The conventional implementation of MIMO systems utilizes a dedicated radio
frequency (RF) chain for each antenna element. Even though this approach is
appropriate for common small-scale MIMO systems, it is inadvisable for mMIMO
systems equipped with a large number of antenna elements due to the high
production costs and power consumption associated with the RF circuitry.
Therefore, even though mMIMO is an important technology for future generations
of wireless networks, it still faces many technical challenges to improve its
energy efficiency (EE) and, to date, it remains a subject of ongoing research
[3]. In light of this, hybrid beamforming (HBF) and antenna selection are
proposed as an effective way to facilitate the implementation and to improve
the EE of mMIMO systems [4]. Indeed HBF reduces the number of RF chains and
digital-to-analog converters (DACs), helping to improve EE. Accordingly, for
better EE, HBF techniques are being examined for 5G cellular networks in the
mm-Wave frequency bands, and will likely also be found in sixth-generation
(6G) networks [5].

Different HBF structures have been proposed to achieve different trade-offs
between cost, energy consumption, and spectral efficiency (SE), which can be
grouped into three general categories, fully-connected HBF (FC-HBF) [6], fixed
subarray HBF (FSA-HBF) [7], and dynamic subarray HBF (DSA-HBF) [8]. Each
category has its advantages and limitations. FC-HBF offers flexibility but has
higher implementation complexity. FSA-HBF balances SE and complexity. Finally,
DSA-HBF provides adaptability but with additional design complexities. To
configure a HBF structure, one of the most prominent techniques consists in
minimizing the Euclidean distance between the desired fully digital precoder
(FDP) and its hybrid counterpart [9]. However, this technique requires
designing the FDP, which is computationally complex and not necessarily
energy-efficient. Furthermore, the number of possible HBF structures is
extremely large, making it complicated to find an optimal HBF structure.
Therefore, the question that arises is how to efficiently design the best HBF
structure in terms of energy consumption and SE. Towards answering this
question, our first step consists of proposing an accurate energy model that
finds the power consumption of each component in different beamforming
structures. Our second step involves applying machine learning-based
approaches to design the beamforming structure instead of using complex
optimization-based ones.

Thanks to the enormous success of deep learning (DL), in a wide variety of
engineering fields, deep neural networks (DNNs) have received significant
attention in recent years and have been widely applied to wireless
communication systems [10, 11]. Despite the fact that training DNNs to solve
wireless communication problems can be computationally intensive, it can take
place offline and only the trained DNN model will be used to make online
decisions, thus reducing the overall complexity. Different studies used DNNs
to address complex problems within the physical layer [12]. In supervised
learning approaches, the time spent during the data labeling procedure is not
negligible. In addition, this procedure must be performed each time a new
dataset is used for training. In reinforcement learning (RL) approaches, an
agent collects online data as it interacts with its environment in a trial-
and-error manner. In mMIMO systems, since the HBF action space is large, the
convergence of the RL model requires a large number of experiments. As a
consequence, unsupervised learning demonstrates superiority over supervised
learning and reinforcement learning in terms of its ability to autonomously
extract meaningful patterns and insights from large datasets without relying
on explicit labels or large training overhead.

To summarize, in this study, we aim to optimize the EE of mmWave mMIMO systems
by designing HBF structures and FDP using DL-based techniques. The problem
consists of jointly designing the transmitter configuration and beamforming
weights that maximize the EE. To accomplish this, we first propose an accurate
energy model that takes into account the power consumption of the different
components of the mMIMO system. Second, we propose an unsupervised deep
learning approach that incorporates two key components to design an energy-
efficient beamforming structure: (i) a novel loss function that considers
different trade-offs between SE, energy consumption, and active users, and
(ii) imperfect channel state information (CSI) during both the training and
inference phases.

###  I-A Related Works

In [13], the authors compared the EE of six different phase-shifter (PS)-based
and switch-based HBF structures. However, given the hardware available today,
the energy model in [13] overstates the power consumption of PSs, which makes
the conclusion unfair to PS-based approaches. Many studies are proposed in the
context of DL-aided HBF design and antenna selection algorithms [14, 15, 16,
17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. In particular, a received signal
strength indicator (RSSI)-based FC-HBF design implemented with supervised
learning is proposed in [14]. The authors of [15] suggested a supervised
learning approach for FC-HBF design under perfect CSI. Another form of
supervised learning is also proposed for the FSA-HBF design with perfect CSI
in [16]. The authors of [17] proposed a reinforcement learning (RL) approach
to design the HBF. However, they assumed that the CSI is known perfectly, and
due to the continuous action space, their method relies on deep deterministic
policy gradient (DDPG), which is computationally complex [27]. In the context
of unsupervised learning for the FC-HBF design, the authors in [19, 20]
presented a novel HBF design employing imperfect CSI for single base station
(BS) and cell-free mMIMO (CF-mMIMO), respectively. However, their approaches
are only for FC-HBF. In [21], the authors proposed an unsupervised learning
approach for HBF and antenna selection using a differentiable activation
function for 1-bit PSs. However, the main objective is to maximize the SE of
the mMIMO system, and the authors neither optimized the EE nor considered an
accurate energy model. In [22, 23, 24], the authors proposed a joint antenna
selection and precoding design with an iterative algorithm and a DL solution
to maximize the SE of multi-user multiple-antenna downlink systems. The
proposed ML approach assumes perfect CSI for the training data, does not
optimize the EE, and requires a complex iterative algorithm to generate the
training samples. In [25, 26], the authors proposed a supervised learning
approach to solve the antenna selection problem. However, the proposed method
only applies to FDP.

###  I-B Contributions

In this paper, we consider both FDP and HBF transmitters and develop new
unsupervised deep learning solutions that jointly design beamforming and
antenna selection while taking into account the power consumption and
insertion loss (IL) of all components. For FDP, the proposed solution designs
the FDP vectors along with the antenna selection solution, while for HBF,
thanks to a multi-tasking DNN, the proposed solution directly provides the
analog precoder (AP) and digital precoder (DP) with the power allocation among
the antennas. A preliminary version of this work was published in [28], where
we only considered maximizing the SE by designing the HBF for fixed and
dynamic HBF structures.

In summary, the contributions of this work are as follows:

  * •

We propose an accurate energy model for the FDP and HBF structures while
considering the latest state-of-the-art hardware solutions.

  * •

We propose an unsupervised deep learning solution robust against imperfect CSI
to find the optimal energy-efficient antenna selection for FDP and transmit
power allocation for HBF considering the proposed accurate energy model. Due
to the binary constraints of beamforming connections, our unsupervised deep
learning approach makes use of the Gumbel-Sigmoid technique inspired by
Gumbel-Softmax. The Gumbel-Sigmoid technique is designed such that it
considers the constraints of all components involved in the beamforming
connections.

  * •

We design an unsupervised loss function that takes into account the SE, the
energy consumption (EC) as well as the number of active users. Thanks to this
loss function, the proposed solution is flexible and can intelligently adjust
the power consumption according to the number of active users and can provide
an optimal trade-off between SE and EC.

  * •

We train the proposed unsupervised deep learning solution using imperfect CSI
for both the DNN input and the loss function computation. We also investigate
the noise tolerance of our approach by showing that imperfect inputs can be
beneficial and improve the EE of the mMIMO system.

  * •

The proposed solutions are evaluated in a realistic ray-tracing channel model
generated using a three-dimensional model of an urban environment to capture
the geometry-based characteristics of the channel. The simulation results show
that the proposed solution outperforms conventional solutions in terms of EE
with lower computational complexity, and can be adapted to achieve different
trade-offs between SE and EC.

###  I-C Paper Organization and Notation

The rest of the paper is organized as follows: In Section II, we present the
system setup followed by the baseline solutions and the channel model. The
proposed energy model for the different beamforming structures is provided in
Section III. Section IV presents the proposed energy-efficient unsupervised
learning solutions for HBF and FDP, including a discussion of the DNN
structure, the training phase, and the online phase. In Section V, we evaluate
the performance of the proposed algorithms by comparing them with state-of-
the-art solutions using a realistic ray-tracing channel model. Finally,
Section VI concludes the paper.

Matrices, vectors, and scalars are denoted by boldface uppercase, boldface
lowercase, and normal letters, respectively. The notations $(\cdot)^{\rm{H}}$,
$(\cdot)^{\rm T}$, $(\cdot)^{\dagger}$, $|\cdot|$, $\|\cdot\|_{F}$,
$\|\cdot\|_{2}$, $\Re[\cdot]$, $\Im[\cdot]$, $\mathbb{I}_{n}$, $\otimes$
denote Hermitian transpose, transpose, Moore-Penrose pseudoinverse, absolute
value, Frobenius norm, $\ell^{2}$-norm, real part, imaginary part, the
$n\times n$ identity matrix, and element-wise product, respectively.

##  II System Model and Baselines

Figure 1: Massive MIMO system model structure with one transmitter BS
employing HBF to serve a set of users.

Let us assume a time-division duplex (TDD) multi-user mMIMO system where
channel reciprocity is available such that the uplink channel estimate can be
used for the downlink transmission. The mMIMO system consists of a single BS
in a single-cell equipped with $N_{\sf{T}}$ antennas and $N_{\sf{RF}}$ RF
chains serving $N_{\sf{U}}$ single antenna users simultaneously as shown in
Figure 1. The DP is performed in the baseband and then the output signal goes
through the RF chains. Each RF chain is composed of a DAC, a low pass filter
(LPF), a local oscillator (LO) and a mixer, and is connected to the
$N_{\sf{T}}$ antennas. The resolution for all DACs and PSs are fixed
respectively to $b_{D}$ and $q$. The RF chains are connected to the antennas
through PSs. The network of these connections and PSs, known as AP, can be
tuned based on different HBF structures.

###  II-A Conventional Beamforming Structures

We first review the three conventional beamforming structures followed by
their non-DL design methods, which are used as baselines for our proposed
solutions. Connections between the RF chains and the antennas are represented
by a binary matrix $\boldsymbol{\Omega}$. For HBF,
$\boldsymbol{\Omega}=\boldsymbol{\Omega}_{\sf{HB}}\in\\{0,1\\}^{N_{\sf{T}}\times
N_{\sf{RF}}}$, and $[\boldsymbol{\Omega}_{\sf{HB}}]_{n,m}=1$ if antenna $n$ is
connected to RF chain $m$. For FDP,
$\boldsymbol{\Omega}=\boldsymbol{\Omega}_{\sf{FD}}$ is an $N_{\sf{T}}\times
N_{\sf{T}}$ diagonal binary matrix, with
$\boldsymbol{\Omega}_{\sf{FD}}=\text{diag}(\boldsymbol{\omega})$, where
$\boldsymbol{\omega}=[\omega_{1},\ldots,\omega_{N_{\sf{T}}}]$ and
$\omega_{n}=1$ if antenna $n$ is activated.

####  II-A1 Fully Digital Precoder (FDP)

In FDP, each antenna is connected to an RF chain through the circuit of DAC,
LPF, LO, and a mixer. The signal received by each user can be written as

$\mathbf{y}_{u}=\mathbf{h}_{u}^{\rm{H}}\sum_{u=1}^{N_{\sf{U}}}\mathbf{u}_{u}x_{u}+\boldsymbol{\eta},$
(1)

where $\mathbf{h}_{u}\in\mathbb{C}^{N_{\sf{T}}\times 1}$ stands for the
channel vector from the $N_{\sf{T}}$ antennas of the BS to the user index $u$,
$\mathbf{x}=[x_{1},\ldots,x_{u},\ldots,x_{N_{\sf{U}}}]$ is the matrix of
transmitted symbols for all users, normalized to
$\mathop{\mathbb{E}}[\mathbf{x}\mathbf{x}^{\rm{H}}]=\frac{1}{N_{\sf{U}}}\mathbb{I}_{N_{\sf{U}}}$,
$\boldsymbol{\eta}$ is the additive white gaussian noise (AWGN) with mean 0
and variance $\sigma^{2}$, and $\mathbf{u}_{u}$ denotes the precoder vector
for the $u^{\text{th}}$ user. The SE of the optimal fully digital precoder
(O-FDP),
$\mathbf{U}_{\sf{opt}}=[\mathbf{u}_{1},\ldots,\mathbf{u}_{u},\ldots,\mathbf{u}_{N_{\sf{U}}}]$
for single-antenna users is obtained by solving the following problem:

$\displaystyle\underset{\left\\{\mathbf{U}_{\sf{opt}}\right\\}}{\max}$
$\displaystyle R_{\sf{FDP}}(\mathbf{U}_{\sf{opt}})$ (2a) s. t.
$\displaystyle\sum_{u=1}^{N_{\sf{U}}}\mathbf{u}_{u}^{\rm{H}}\mathbf{u}_{u}\leq
P_{\sf{TX}}\,,$ (2b)

where
$R_{\sf{FDP}}(\mathbf{U}_{\sf{opt}})=\sum_{u=1}^{N_{\sf{U}}}\log_{2}(1+\text{SINR}(\mathbf{u}_{u}))\,$,
the signal-to-interference plus noise ratio (SINR) of the $u^{\text{th}}$ user
is given by

$\displaystyle\text{SINR}(\mathbf{u}_{u})=\frac{\big{|}\mathbf{h}^{\rm{H}}_{u}\mathbf{u}_{u}\big{|}^{2}}{\sum\limits_{\begin{subarray}{c}j=1\\\
j\neq
u\end{subarray}}^{N_{\sf{U}}}\big{|}\mathbf{h}^{\rm{H}}_{u}\mathbf{u}_{j}\big{|}^{2}+\sigma^{2}}\,,$
(3)

and $P_{\sf{TX}}$ is the normalized total transmit power constraint. The
baseline results presented in this paper are obtained by solving (2) based on
[29]. We refer to this approach as optimal FDP (O-FDP). For a given connection
matrix $\boldsymbol{\Omega}_{\sf{FD}}$, the FDP is given by
$\mathbf{U}=\boldsymbol{\Omega}_{\sf{FD}}\times\mathbf{U}_{\sf{opt}}$.

Figure 2: Example of connection matrices ($\boldsymbol{\Omega}_{\sf{HB}}$) of
some conventional HBF structures. Each green square indicates a connection.
Top: FC-HBF. Middle: FSA-HBF. Bottom: DSA-HBF.

####  II-A2 Fully Connected Hybrid Beamforming (FC-HBF)

In all HBF structures, we assume $N_{\sf{RF}}<<N_{\sf{T}}$. Regardless of the
chosen HBF structure, the signal received by each user can be written as

$\mathbf{y}_{u}=\mathbf{h}_{u}^{\rm{H}}\mathbf{A}\sum_{u=1}^{N_{\sf{U}}}\mathbf{w}_{u}x_{u}+\boldsymbol{\eta}\,.$
(4)

The HBF vectors consist of a DP,
$\mathbf{W}=[\mathbf{w}_{1},\ldots,\mathbf{w}_{u},\ldots,\mathbf{w}_{N_{\sf{U}}}]\in\mathbb{C}^{N_{\sf{RF}}\times
N_{\sf{U}}}$, and an AP, $\mathbf{A}\in\mathbb{C}^{N_{\sf{T}}\times
N_{\sf{RF}}}$. Since the AP is a combination of the PSs and combiners and it
depends on HBF structure and connection between the antennas and RF chains, we
define it as follows:

$\mathbf{A}=\mathbf{P}_{q}\otimes\boldsymbol{\Omega}_{\sf{HB}},$ (5)

where $\mathbf{P}_{q}\in\mathbb{C}^{N_{\sf{T}}\times N_{\sf{RF}}}$ is the
coefficient of the $q$ bits PS connecting the $n^{\text{th}}$ antenna and
$m^{\text{th}}$ RF chain, where $[\mathbf{P}_{q}]_{n,m}\in\\{e^{j2\pi
k/2^{q}}:k\in\\{1,\dots,2^{q}\\}\\}$. Therefore, the SE for a given HBF
$(\mathbf{A},\mathbf{W})$ is given by

$\displaystyle
R_{\sf{HBF}}(\mathbf{A},\mathbf{W})=\sum_{u=1}^{N_{\sf{U}}}\text{log}_{2}\bigl{(}1+\text{SINR}(\mathbf{A},\mathbf{w}_{u})\bigr{)}\,,$
(6)

and the SINR of the $u^{\text{th}}$ user can be expressed as

$\displaystyle\text{SINR}(\mathbf{A},\mathbf{w}_{u})$
$\displaystyle=\frac{\big{|}\mathbf{h}^{\rm{H}}_{u}\mathbf{A}\mathbf{w}_{u}\big{|}^{2}}{\sum\limits_{\begin{subarray}{c}j=1\\\
j\neq
u\end{subarray}}^{N_{\sf{U}}}\big{|}\mathbf{h}^{\rm{H}}_{u}\mathbf{A}\mathbf{w}_{j}\big{|}^{2}+\sigma^{2}}\,.$
(7)

In FC-HBF, we set
$\boldsymbol{\Omega}_{\sf{HB}}=\boldsymbol{\Omega}_{\sf{FC}}$, and all RF
chains are connected to all antennas through PSs, combiners, and power
amplifier (PA) as shown in Figure 2 (top), where the green boxes show the
connections. This structure thereby enables maximum design flexibility and
therefore requires a large number of PSs and combiners, which increase the
implementation cost and energy consumption. The AP of the FC-HBF can be
expressed according to (5) with

$[\boldsymbol{\Omega}_{\sf{HB}}]_{n,m}=1\quad\forall n,m\,,$ (8)
$[\mathbf{P}_{q}]_{n,m}\in\\{e^{j2\pi
k/2^{q}}:k\in\\{1,\dots,2^{q}\\}\\}\quad\forall n,m\,.$ (9)

Since all the antennas are connected to all the RF chains through a PS with
$q$-bit quantization, the feasible analog precoder for $n^{\text{th}}$ antenna
and $m^{\text{th}}$ RF chain is $[\mathbf{A}]_{n,m}\in\\{e^{j2\pi
k/2^{q}}:k\in\\{1,\dots,2^{q}\\}\\}$. Conventional HBF solutions either rely
on codebook-based solutions to limit the number of feasible solutions [30] or,
more rarely, use real-valued PSs [9]. The conventional approach consists of
first designing the O-FDP matrix in (2). Then, the AP and DP are designed in
such a way that the resulting precoders approximate $\mathbf{U}_{\sf{opt}}$ as
follows:

$\displaystyle\underset{\mathbf{A},\mathbf{W}}{\operatorname*{minimize}\;}$
$\displaystyle\big{\|}\mathbf{U}_{\sf{opt}}-\mathbf{A}\mathbf{W}\big{\|}^{2}_{F}$
(10a) s. t.
$\displaystyle\eqref{Eq:analog},\eqref{eq6},\eqref{eq6a},\|\mathbf{A}\mathbf{W}\|_{F}^{2}=N_{\sf{U}}.$
(10b)

We obtain the FC-HBF solution of (10) using “PE-AltMin” and “MO-AltMin”
proposed in [9].

####  II-A3 Subarray Hybrid Beamforming

Each antenna in a subarray structure is connected to only one RF chain through
a PS. Consequently, the total number of PSs is reduced to $N_{\sf{T}}$,
instead of $N_{\sf{T}}\times N_{\sf{RF}}$ in the FC-HBF. In the subarray HBF
structure, we consider two types of connection: (i) a structure equipped with
fixed connections, known as fixed subarray HBF (FSA-HBF), or (ii) a structure
equipped with dynamic connections, known as dynamic subarray HBF (DSA-HBF).
Examples of possible connection matrices for each case are shown in Figure 2.
The DSA-HBF structure enables the antennas and the RF chains to be dynamically
switched at each time interval in response to changing conditions. It was
shown that such a dynamic structure significantly enhances the SE of the
system by providing more degrees of freedom in the HBF design compared to a
FSA-HBF structure, and reduces the power consumption compared to the FC-HBF
structure [8]. Therefore, based on the general definition of the AP
($\mathbf{A}=\mathbf{P}_{q}\otimes\boldsymbol{\Omega}_{\sf{HB}}$), the
constraint on matrix $\boldsymbol{\Omega}_{\sf{HB}}$ for subarray HBF is given
by

$\sum_{m=1}^{N_{\sf{RF}}}[\boldsymbol{\Omega}_{\sf{HB}}]_{n,m}=1\quad\forall\
n\,.$ (11)

To find the precoder matrices for FSA-HBF, the general approach described in
(10) for FC-HBF can be used. For the DSA-HBF, the connection pattern
($\boldsymbol{\Omega}_{\sf{HB}}$) between the RF chains and the antennas is
dynamic and needs to be optimized, resulting in a large design space.

###  II-B Problem Definition

The main objective of this paper is to maximize the EE of the mMIMO system by
selecting the antennas and designing the BF structure. For the FDP case, the
problem consists of finding the precoder matrix $\mathbf{U}$ and antenna
selection $\boldsymbol{\Omega}_{\sf{FD}}=\text{diag}(\boldsymbol{\omega})$
that maximize the EE, while achieving a desired minimum average SE denoted as
$R_{\sf{d}}$. More formally, we seek to solve the following optimization
problem:

$\displaystyle\underset{\mathbf{U},\boldsymbol{\Omega}_{\sf{FD}}}{\text{maximize}}$
$\displaystyle
R_{\sf{FDP}}(\mathbf{U}\times\boldsymbol{\Omega}_{\sf{FD}})/P_{\text{FDP}}$
(12a) s. t.
$\displaystyle\sum_{u=1}^{N_{\sf{U}}}\mathbf{u}_{u}^{\rm{H}}\mathbf{u}_{u}\leq
P_{\sf{TX}}\,,$ (12d)
$\displaystyle[\boldsymbol{\Omega}_{\sf{FD}}]_{n,n}\in\\{0,1\\},\;\forall n,$
$\displaystyle R_{\sf{FDP}}(\mathbf{U}\times\boldsymbol{\Omega}_{\sf{FD}})\geq
N_{\sf{U}}R_{\sf{d}},$

where $P_{\text{FDP}}$ is the total power consumed by the BF components.

Similarly, the HBF design consists in finding the precoder matrices
$\mathbf{W}$ and $\mathbf{A}$ and the power allocation that maximizes the EE.
Therefore, we have the following optimization problem:

$\displaystyle\underset{\mathbf{A},\mathbf{W}}{\text{maximize}}$
$\displaystyle R_{\sf{HBF}}(\mathbf{A},\mathbf{W})/P_{\text{HBF}},$ (13a) s.t.
$\displaystyle\sum_{u=1}^{N_{\sf{U}}}\mathbf{w}_{u}^{\rm{H}}\mathbf{A}^{\rm{H}}\mathbf{A}\mathbf{w}_{u}\leq
P_{\sf{TX}}\,,$ (13e)
$\displaystyle\mathbf{A}=\mathbf{P}_{q}\otimes\boldsymbol{\Omega}_{\sf{HB}},$
$\displaystyle[\boldsymbol{\Omega}_{\sf{HB}}]_{n,m}\in\\{0,1\\},\;\forall
n,m,$ $\displaystyle R_{\sf{HBF}}(\mathbf{A},\mathbf{W})\geq
N_{\sf{U}}R_{\sf{d}},$

where $P_{\text{HBF}}$ is the total power consumed by the HBF transmitter, and
again $R_{\sf{d}}$ is the minimum average required SE. The power consumption
$P_{\text{FDP}}$ and $P_{\text{HBF}}$ will be described in detail in Section
III. In this paper, for simplicity, we consider a total power constraint for
the transmitter, where the power transmitted by each antenna is not
necessarily equal or limited to $P_{\sf{TX}}/N_{\sf{T}}$. It should be noted
that in HBF turning off an antenna is not necessarily corresponding to
deactivating an RF chain. On the contrary, since in FDP, each antenna is
connected to one RF chain, and the power consumed by RF chains is noticeable,
turning off the RF chains leads to the deactivation of the corresponding
antennas.

###  II-C Channel Model

The experiments presented in this paper are based on the generic deep learning
dataset for mm-Wave mMIMO systems (known as deepMIMO) [31], which provides a
channel vector $\mathbf{h}$ of length $N_{\sf{T}}$ for each user position on a
quantized grid. An $N_{\sf{T}}\times N_{\sf{U}}$ channel matrix entries in the
dataset are obtained by concatenating the $N_{\sf{U}}$ channel vectors
randomly selected from the available user positions of the considered area.

Since we consider TDD communication with channel reciprocity, the estimated
CSI in the uplink can be employed for downlink. However, due to channel
estimation errors, the downlink channel cannot be perfectly estimated. Thus,
to model the channel estimation error, the BS uses the minimum mean square
error such that the estimated channel matrix is given by [32]:

$\hat{\mathbf{H}}=\sqrt{1-\beta^{2}}\mathbf{H}+\beta\boldsymbol{\epsilon},$
(14)

where $\mathbf{H}=[\mathbf{h}_{1},...,\mathbf{h}_{N_{\sf{U}}}]^{\rm{T}}$ is
the actual channel matrix, the scaling coefficient $\beta\in[0,1]$ represents
the reliability of the estimate, and
$\boldsymbol{\epsilon}\sim\mathcal{N}(0,\sigma_{e}^{2})$ is an error matrix
modeled as a zero-mean Gaussian noise with variance $\sigma_{e}^{2}$. Unlike
previous DL-based studies, where perfect CSI is available during the training
of the DNN, in this work, we propose to use the imperfect CSI
($\hat{\mathbf{H}}$) not only as the input to the DNN but also to compute the
loss function during the training phase. In Section V, we further evaluate the
impact of the imperfect CSI by varying the value of $\beta$ and show that a
moderate level of imperfection in CSI can act as a regularizer for the DNN and
slightly improve the SE.

##  III Energy Model

In this section, we present an energy model for the different FDP and HBF
hardware configurations, considering both the direct energy consumption as
well as the energy consumption resulting from IL of each component.

Figure 3: General beamforming structure. (a) Hybrid beamforming structure. (b)
Fully digital precoder structure.

###  III-A General Beamforming Structure

We consider a regularity assumption where components of the same type have the
same input/output interface, i.e. their inputs and outputs are connected to
the same type and number of components. This assumption is generally true
because it eases the conception of generic circuits.

To better represent each HBF structure, we suggest a general template form as
shown in Figure 3 (a), where a given antenna is connected to a combiner having
$c\in\\{1,\ldots,N_{\sf{RF}}\\}$ inputs. Each input of a combiner is connected
to the output of a phase shifter. Then, each phase shifter is connected to an
RF chain through a switch. The number of switches is
$\psi\in\\{1,\ldots,N_{\sf{RF}}\\}$. As a result, the analog precoder can be
fully characterized by specifying the tuple $(\psi,c)$. For instance, for the
three conventional HBF structures that we discussed previously, we have:

  * •

$(N_{\sf{RF}},N_{\sf{RF}})$ for the FC-HBF structure. In the FC-HBF structure,
all the switches are connected (i.e., $\psi=N_{\sf{RF}}$), while the outputs
of all the PSs are combined before each antenna i.e., $c=N_{\sf{RF}}$.

  * •

$(N_{\sf{RF}},1)$ for the DSA-HBF structure. In DSA-HBF, only one switch can
be connected at each time interval, therefore $c=1$, while there are possible
connections for all the switches, thus $\psi=N_{\sf{RF}}$. It should be noted
that such configuration for switches works like a multiplexer. Thus, in a
practical system, the switches are replaced by a $\psi\times 1$ multiplexer.

  * •

$(1,1)$ for the FSA-HBF structure. In FSA-HBF, each antenna is only connected
to a fixed RF chain (i.e., $c=1$), while the connection is fixed i.e.,
$\psi=1$.

The hardware complexity of different beamforming techniques is compared in
Table II.

TABLE I: Energy Model Notations and Parameters Component | Notation $\langle c\rangle$ | Parameter $x$
---|---|---
Digital analog convert | $\sf{D}$ |  $\\#$ bits $b_{D}$
Low pass filter | $\sf{L}$ | -
Mixer | $\sf{M}$ | -
Local oscillator | $\sf{LO}$ | -
Switches (multiplexers) | $\boldsymbol{\Psi}$ |  $\\#$ inputs $\psi$
Phase shifters | $\boldsymbol{\Phi}$ | -
Combiners | $\sf{C}$ | -
Power amplifier | $\sf{PA}$ | -
TABLE II: Hardware Complexity Comparison

Beamforming | Hardware components
---|---
technique | RF chains | Antennas | PS | Combiners | Switches
FDP | $N_{\sf{T}}$ | $N_{\sf{T}}$ | - | - | -
FC-HBF | $N_{\sf{RF}}$ | $N_{\sf{T}}$ | $N_{\sf{RF}}N_{\sf{T}}$ | $N_{\sf{T}}$ | -
FSA-HBF | $N_{\sf{RF}}$ | $N_{\sf{T}}$ | $N_{\sf{T}}$ | - | -
DSA-HBF | $N_{\sf{RF}}$ | $N_{\sf{T}}$ | $N_{\sf{T}}$ | - | $N_{\sf{T}}$

###  III-B Energy Consumption Analysis

We now describe the energy consumption of each component, and we list the most
recent state-of-the-art hardware solutions. We consider components that are
suitable for operating in the frequency range of $20$-$40$ GHz.

A component of the set
$\\{\sf{D},\sf{L},\sf{M},\sf{LO},\boldsymbol{\Psi},\boldsymbol{\Phi},\sf{C},\sf{PA}\\}$
is denoted by $o$ and the correspondence between a component and its notation
is defined in Table I. We denote $\text{IL}_{o}$ as the insertion loss of
passive component $o$ and when $o$ depends on some parameter $x$, we use
$\text{IL}_{o}(x)$. The average power dissipated by the active component $o$
is denoted as $P_{o}$, or $P_{o}(x)$ if $o$ depends on the parameter $x$. See
Table I for the list of components and their parameters. Note that the power
dissipated by the wires is neglected and when $c=1$ there is no need for a
combiner (i.e., $\text{IL}_{\sf{C}}(1)=0\,\text{dB}$). Likewise, the switches
can be replaced with wires when $\psi=1$ or $\psi=c$, that is
$\text{IL}_{\boldsymbol{\Psi}}(1)=\text{IL}_{\boldsymbol{\Psi}}(c)=0\,\text{dB}$,
since all possible connections are always established.

In our energy model, we consider the possibility of turning off the RF chains
or antennas to save power. The $n^{\text{th}}$ antenna or the $m^{\text{th}}$
RF chain is turned off when the $n^{\text{th}}$ row or the $m^{\text{th}}$
column of the matrix $\boldsymbol{\Omega}$ is zero, respectively. Therefore,
we can define
$N_{\sf{T}}(\boldsymbol{\Omega})=\\{n:\sum_{m=1}^{N_{\sf{RF}}}[\mathbf{\Omega}]_{n,m}>0\\}$
and
$N_{\sf{RF}}(\boldsymbol{\Omega})=\\{m:\sum_{n=1}^{N_{\sf{T}}}[\mathbf{\Omega}]_{n,m}>0\\}$,
as the set of activated antennas and RF chains, respectively.

####  III-B1 RF Front-End

The RF front-end corresponds to the circuitry between the antenna and the DAC.
As shown in Figure 3 (b), for the FDP, this consists of low pass filters
(LPFs), mixers, local oscillators (LOs), switches, and power amplifiers (PAs).
On the other hand, in Figure 3 (a), the HBF requires a network of PS s,
splitters, and combiners in addition to the components described for the FDP.
Mixers, combiners, switches, and PSs are assumed to be passive devices that
introduce IL.

For the mixer, based on the recent solution in [33], we consider
$\text{IL}_{\sf{M}}=6.4$ dB. The IL of the PS and the combiner plays a key
role in designing energy-efficient HBF, especially for the FC-HBF, where all
the RF chains are connected to all the antennas through PS s and combiners. In
Table III, we list the IL s of PS s from some recent state-of-the-art
references. Based on this table, we choose $\text{IL}_{\boldsymbol{\Phi}}=3.7$
dB with $q=9.4$ bits resolution, and we assume $\text{IL}_{\sf{C}}=1.8$ dB
[34]. For DSA-HBF, the switches dynamically change the connections between the
RF chains and the antennas to improve the flexibility of the structure. Since
these ILs are in low power and they do not have a big impact on the final
power consumption, we assume $\text{IL}_{\boldsymbol{\Psi}}(\psi)=1.1$ dB for
the other values of $\psi$, by considering single pole single throw (SPST)
switch [35].

TABLE III: Passive Phase Shifter IL Comparison Reference | Year | Frequency | bits | $\mathrm{IL}_{\boldsymbol{\Phi}}$
---|---|---|---|---
|  | (GHz) | q | (dB)
[36] | 2019 | 24-28 | - | 5
[37] | 2018 | 28 | - | 4
[38] | 2018 | 22-36 | 3 | 5.6
[39] | 2018 | 28 | 9.4 | 3.7

Now, denoting by $P_{\mathsf{BB}}^{out}$ the output power of each RF chain,
the input power of the PA before the $n^{\text{th}}$ antenna for all
structures of the HBF (in mW) can be written as

$\displaystyle
P_{\sf{PA},\sf{HBF}}^{in,n}=\frac{P_{\mathsf{BB}}^{out}}{\text{IL}_{\sf{C}}\text{IL}_{\boldsymbol{\Psi}}(\psi)\text{IL}_{\boldsymbol{\Phi}}\text{IL}_{\sf{M}}}\sum_{m\in
N_{\sf{RF}}(\boldsymbol{\Omega})}\frac{[\boldsymbol{\Omega}_{\sf{HB}}]_{n,m}}{\sum_{n=1}^{N_{\sf{T}}}[\boldsymbol{\Omega}_{\sf{HB}}]_{n,m}},$
(15)

where $\text{IL}_{\boldsymbol{\Phi}}$ denotes the IL of PSs and IL values are
expressed in a linear scale. In the FC-HBF, where all the RF chains are
connected to the antennas ($\boldsymbol{\Omega}_{\sf{HB}}$ given in (8)), we
have $(\psi,c)=(N_{\sf{RF}},N_{\sf{RF}})$ and
$\text{IL}_{\boldsymbol{\Psi}}(\psi=c)=1$. For DSA-HBF with the structure of
($\psi$, c) = ($N_{\sf{RF}},1$) and the connection matrix
$\boldsymbol{\Omega}_{\sf{HB}}$ in (11), due to IL of switches, we have
$\text{IL}_{\boldsymbol{\Psi}}(\psi)~{}=~{}1.1$. In FSA-HBF that has a
structure $(\psi,c)=(1,1)$, there are neither combiners nor switches. As a
result, $\text{IL}_{\boldsymbol{\Psi}}(1)=1$, and $\text{IL}_{\sf{C}}=1$.
Similarly for the FDP, as shown in Figure 3 (b), the input power of the PA on
the $n^{\text{th}}$ antenna can be obtained as

$\displaystyle
P_{\sf{PA},\sf{FDP}}^{in,n}=\frac{P_{\mathsf{BB}}^{out}}{\text{IL}_{\sf{M}}}.$
(16)

Finally, the direct current (DC) power drawn by the $n^{\text{th}}$ active PA
$P_{\sf{PA}}$, can be written as

$P_{\sf{PA},\sf{BF}}^{\textrm{DC},n}=\frac{P_{\sf{TX}}^{n}-P_{\sf{PA},\sf{BF}}^{in,n}}{\alpha},$
(17)

where $\alpha$ is the power-added efficiency (PAE) of the linear power
amplifier (LPA), $P_{\sf{TX}}^{n}$ is the transmit power of the
$n^{\text{th}}$ antenna, and $\sf{BF}$ should be replaced with $\sf{HBF}$ or
$\sf{FDP}$ according to the chosen transmitter type. Based on the recent
solution for PA listed in [40, 41, 42], we consider an average PAE of
$\alpha=36$.

####  III-B2 Digital to Analog Converter

DACs are among the components having the largest power consumption in wireless
applications. The power consumed by a DAC ($P_{\sf{D}}$) is a linear function
of the sampling frequency ($f_{s}$) and the figure of merit
($\text{FoM}_{\sf{D}}$) of the converter, and grows exponentially with the
number of bits of resolution ($b_{D}$) as
$P_{\sf{D}}=\text{FoM}_{\sf{D}}\times f_{s}\times 2^{b_{D}}$ [43]. The
sampling frequencies for ultra wide-band applications are in the range of
$0.5$-$1$ GHz. It is shown in [43] that in terms of required signal-to-
quantization noise ratio (SQNR), FDP required $2$ bits less than HBF.
Therefore, we assume $b_{D}=4$ for FDP and $b_{D}=6$ for HBF, respectively.
Moreover, based on [44], we consider $\text{FoM}_{\sf{D}}=54.5$ fJ/conv.

####  III-B3 Low Pass Filter in TX

The output of the DACs will require analog LPF to reject spectral images and
maintain out-of-band emission limits. For an $m^{\prime}$-th order active LPF
with cutoff frequency $f_{c}$, the $\text{FoM}_{\sf{L}}$ is the power consumed
per pole per Hertz [45]. The power drawn by LPF is given by
$P_{\sf{L}}=\text{FoM}_{\sf{L}}\times f_{c}\times m^{\prime}$. Based on [45],
we assume a first order LPF with $f_{c}=500$MHz, and
$\text{FoM}_{\sf{L}}=1.4\;\text{mW/GHz}$. Furthermore, we define $P_{\sf{LO}}$
as the power consumed by the mixer from the LO and we consider
$P_{\sf{LO}}=10$ dBm [46].

####  III-B4 Total Energy Consumption

Now, putting it all together, the total power consumed by a given beamforming
structure can be written as follows:

$\displaystyle
P_{\sf{BF}}=\big{|}N_{\sf{RF}}(\boldsymbol{\Omega})\big{|}(P_{\sf{L}}+P_{\sf{LO}}+P_{\sf{D}}(b_{D}))+\sum_{n\in
N_{\sf{T}}(\boldsymbol{\Omega})}P_{\sf{PA},\sf{BF}}^{\textrm{DC},n},$ (18)

where $P_{\sf{PA},\sf{BF}}$ should be replaced with either
$P_{\sf{PA},\sf{HBF}}$ or $P_{\sf{PA},\sf{FDP}}$ according to the transmitter
type and
$\boldsymbol{\Omega}\in\\{\boldsymbol{\Omega}_{\sf{FD}},\boldsymbol{\Omega}_{\sf{HB}}\\}$.
In this paper, we focus on passive PS, but we note that active PS can be
easily considered in the model by setting $\text{IL}_{\boldsymbol{\Phi}}$ to 1
and adding the power consumption of all active PSs to (18). The energy
consumption $E_{\sf{BF}}$ can then be obtained with $E_{\sf{BF}}=T_{s}\times
P_{\sf{BF}}$, where $T_{s}$ is the duration of a symbol. When considering a
fixed symbol duration, minimizing the power consumption is equivalent to
minimizing the energy. Therefore, we evaluate the EE as b/s/Hz/W. It is
interesting to see that based on (15), considering passive PSs and combiners,
the power consumed by different HBF structures is similar since the IL of the
passive components is applied on the low power signals, before the PAs.
However, in terms of hardware complexity and cost, shown in Table II, the
subarray HBF is more efficient than FC-HBF.

In equations (6) and (18), we observe that both the SE and the EE are
influenced by the matrix $\boldsymbol{\Omega}$. This matrix defines the
connection between the RF chains and the antennas. Having more connections
results in higher SE as it increases beamforming flexibility. However, each
connection corresponds to the use of an RF chain in FDP, and in the case of
HBF, it involves a PS and a combiner, leading to increased costs and energy
consumption. This dependency makes the optimization problem in (13) difficult
to solve. Consequently, to address this issue, we propose a novel unsupervised
learning solution in the following sections. This approach aims to jointly
optimize both SE and EE.

##  IV Energy-Efficient Beamforming Driven By Deep Unsupervised Learning

In this section, we describe the unsupervised learning solution to design the
antenna selection and efficient HBF as well as FDP. We start by describing the
architecture of the proposed DNN in Section IV-A. Then, the proposed method is
divided into two phases: the training phase is described in Section IV-B, and
the online phase is described in Section IV-C.

Figure 4: $\text{DNN}_{\sf{core}}$ architecture which is identical for both
HBF and FDP.

###  IV-A Deep Neural Network Architecture

The input and the hidden layers of the proposed DNN architecture are common
for both the FDP and the HBF structures. However, the output layers are
different for each BF structure. We start by describing the architecture of
the input and the hidden layers denoted as $\text{DNN}_{\sf{core}}$ as shown
in Figure 4. Then in the following subsections, we describe the architectures
of the output layers of the HBF and the FDP.

The input of the DNN is given by the imperfect channel matrix
$\hat{\mathbf{H}}$ given in (14). To improve the representation learning, we
normalize the input to
$\bar{\mathbf{H}}=\hat{\mathbf{H}}/\|\hat{\mathbf{H}}\|_{F}^{2}$ such that
$\|\bar{\mathbf{H}}\|_{F}^{2}=1$. Then, we separate the real part
$\Re\\{\bar{\mathbf{H}}\\}$ and the imaginary part $\Im\\{\bar{\mathbf{H}}\\}$
of $\bar{\mathbf{H}}$ into two channels that are fed to the first
convolutional layer (CL). $\text{DNN}_{\sf{core}}$ consists of $2$ CLs
$16@N_{\sf{T}}\times N_{\sf{U}}$ where $16$ is the number of channels and
$N_{\sf{T}}\times N_{\sf{U}}$ is the dimension of each channel followed by $1$
CL $8@N_{\sf{T}}\times N_{\sf{U}}$. The kernel size is $3\times 3$ for all
CLs. The CLs are followed by $2$ fully-connected layers (FLs), each with
$1024$ neurons. The “Leaky ReLU” activation function and batch normalization
are used after all layers except for the output layers. This
$\text{DNN}_{\sf{core}}$ is then combined with different output layers to form
the HBF model, called efficient HBF network (E-HBF-Net), and the FDP model,
called efficient FDP network (E-FDP-Net). The models are relatively small. For
example, for $N_{\sf{T}}=64$, $N_{\sf{U}}=8$, the total number of parameters
including the output layers in E-HBF-Net is $5.8$M (with $N_{\sf{RF}}=8$), and
$4.9$M in E-FDP-Net. A detailed complexity analysis is presented in Section
V-D.

Figure 5: Proposed DNN architecture for (a) Hybrid beamforming, (b) Fully
digital precoder.

####  IV-A1 Output Layers for HBF

As shown in Figure 5 (a), we divide the output of the last FL into $4$
parallel layers. The first and second parallel layers, both of size
${N_{\sf{RF}}\times N_{\sf{U}}}$, generate the real and imaginary part of the
DP. The output of the third parallel layer generates the AP, thus its
dimension is $N_{\sf{RF}}\times N_{\sf{T}}$. The output of AP can also be
adapted to different PS resolutions. It is shown in [28] that using the
straight-through estimator (STE) technique, we are able to have different
numbers of quantization bits for the PSs. In this paper we again consider the
same approach for the output of the DNN dedicated for PS quantization in AP.
The fourth layer of size ${N_{\sf{RF}}\times N_{\sf{T}}}$ designs the matrix
$\boldsymbol{\Omega}_{\sf{HB}}$.

As we described before, $\boldsymbol{\Omega}_{\sf{HB}}$ must be a binary
matrix. Typically, this binary constraint requires using the “Sigmoid”
function during training and then, during the online phase, applying a
rounding technique to transform the real values into binary values. However,
we found that this approach does not lead to good results for unsupervised
learning, because the SE measured during training can be very different from
the actual SE measured during testing. To solve this problem, we propose to
use a differentiable approximation, called “Gumbel-Sigmoid” during training
inspired by the “Gumbel-Softmax” estimator [47]. The Gumbel-Softmax
approximation is a technique that allows sampling from a categorical
distribution during the forward pass of a neural network, by combining a re-
parameterization trick and a smooth relaxation. The connection between the RF
chains and the antennas can be represented using a categorical binary
distribution. Hence, defining $\pi_{n,m}$ as the probability that antenna $n$
is connected to the RF chain $m$, then we can form an $N_{\sf{T}}\times
N_{\sf{RF}}$ matrix $\boldsymbol{\Pi}$ that corresponds to the probability
states between antenna $n$ and the RF chain $m$. The Gumbel-Softmax function,
$G(\mathbf{\Pi})$, applied to each element of the matrix $\mathbf{\Pi}$ can
then be defined as

$\bar{\boldsymbol{\Omega}}_{\sf{HB}}=G(\boldsymbol{\Pi})=\frac{\exp((\log(\boldsymbol{\Pi})+\mathbf{g})/\tau)}{\exp((\log(\boldsymbol{\Pi})+\mathbf{g})/\tau)+\exp(\mathbf{g^{\prime}}/\tau)},$
(19)

where $\bar{\boldsymbol{\Omega}}_{\sf{HB}}$ is the output of the DNN, and
$\mathbf{g}$ and $\mathbf{g^{\prime}}$ are independent samples with zero mean
and unit variance, drawn from the Gumbel distribution. Note that the
$\exp(\cdot)$ and $\log(\cdot)$ functions are applied element-wise when taking
a matrix as input. The parameter $\tau$ is called the “Gumbel temperature”.
When $\tau\rightarrow 0$, $G(\boldsymbol{\Pi})$ tends to the categorical
distribution, but when $\tau\rightarrow\infty$, it converges to the uniform
distribution [47]. Therefore, there is a trade-off between small temperatures,
where sample vectors are close to one-hot but the variance of the gradient is
large, and large temperatures, where samples are more uniform but the variance
of the gradient is small. We thus consider $\tau$ as a hyper-parameter to be
optimized in our implementation.

####  IV-A2 Output Layers for FDP

The proposed architecture for FDP is shown in Figure 5 (b). We divide the
output layer into 3 parallel layers. The first two layers are dedicated to the
real and imaginary part of the FDP with dimension $N_{\sf{T}}\times
N_{\sf{U}}$. The third layer, similar to the one for HBF, designs the antenna
selection vector ($\boldsymbol{\omega}$) described in Section II-A1. Here
again, we use the Gumbel-Sigmoid described in (19) to obtain the binary
variables from $\boldsymbol{\omega}$. Let $\pi^{\prime}_{n}$ denotes the
probability of activating the $n$-th antenna and
$\boldsymbol{\pi}^{\prime}=[\pi^{\prime}_{1},\ldots,\pi^{\prime}_{N_{\sf{T}}}]$.
Then, we have
$\bar{\boldsymbol{\Omega}}_{\sf{FD}}=\text{diag}(\bar{\boldsymbol{\omega}})$,
where $\bar{\boldsymbol{\omega}}=G(\boldsymbol{\pi}^{\prime})$.

Figure 6: Training (left) and online (right) phases for efficient BF. The
outputs of the DNN depend on the BF structure (HBF, FDP).

###  IV-B Training Phase: Unsupervised Learning

In the training phase, thanks to unsupervised learning, the data samples
consist of only imperfect channel matrices without the need for labels. The
imperfect channel ($\hat{\mathbf{H}}$) is modeled as in (14) and it includes a
coefficient $\beta$ that determines the magnitude of the estimation error and
thus helps us study the impact of the estimation error of the channel on the
DNN training.

Although the approach to train the DNN is similar for E-HBF-Net and E-FDP-Net,
there are differences in their hardware configurations. Therefore, we first
present the common aspects shared by both DNN models and then proceed to
explain the parts specific to each model.

The objective of the proposed solutions is to design the beamforming
configuration to not only maximize the SE but also to minimize the EC while
being adaptive to the number of active users, i.e., when the number of active
users is small, it intelligently turns off part of the antennas since they
will no longer be needed. Consequently, it will reduce the EC. To achieve this
objective, we design the following unsupervised loss function to train the
DNN:

$\mathcal{L}_{\sf{BF}}=\gamma\mathcal{L}_{\sf{EC}}+\mathcal{L}_{\sf{AAS}},$
(20)

where the first term is related to EC and the second term is related to both
the SE and the active number of users and is called the adaptive antenna
selection (AAS) term. The hyper-parameter $\gamma$ is required to achieve
proper training convergence and should be tuned in the training phase. Each
term of the loss function is described in detail in the sequel.

EC term ($\mathcal{L}_{\sf{EC}}$): This term is introduced to add a penalty to
the total loss function to reduce EC. It is given as:

$\mathcal{L}_{\sf{EC}}=\bar{P}_{\sf{BF}},$ (21)

where $\bar{P}_{\sf{BF}}$ is the total power consumption for either HBF
($\bar{P}_{\sf{HBF}}$) or FDP ($\bar{P}_{\sf{FDP}}$) given in (18) as
discussed in Section III, which depends on
$\bar{\boldsymbol{\Omega}}\in\\{\bar{\boldsymbol{\Omega}}_{\sf{FD}},\bar{\boldsymbol{\Omega}}_{\sf{HB}}\\}$.
Thus, $\bar{\boldsymbol{\Omega}}_{\sf{BF}}$ affects both the SE as well as the
EC.

AAS term ($\mathcal{L}_{\sf{AAS}}$): This term of the loss function
$\mathcal{L}_{\sf{AAS}}$ is given by:

$\mathcal{L}_{\sf{AAS}}=\left(\frac{\bar{R}}{N_{\sf{U}}}-R_{\sf{d}}\right)^{2},$
(22)

where as discussed in Section II-B, parameter $R_{\sf{d}}$ denotes the desired
average SE value for each user, and $\bar{R}$ is either
$\bar{R}_{\sf{HBF}}(\bar{\mathbf{A}},\mathbf{W})$ for HBF or
$\bar{R}_{\sf{FDP}}(\mathbf{U}\times\bar{\boldsymbol{\Omega}}_{\sf{FD}})$ for
FDP. Thanks to the AAS term, the SE is forced to approach $R_{\sf{d}}$ while
parts of the antennas can be turned off to reduce the EC (according to the EC
term $\mathcal{L}_{\sf{EC}}$). As a result, the AAS term guarantees to consume
minimum power to satisfy an average desired SE ($R_{\sf{d}}$).

####  IV-B1 Efficient Hybrid Beamforming Network (E-HBF-Net)

To design an efficient HBF structure, a programmable switch is employed for
each connection ($N_{\sf{T}}\times N_{\sf{RF}}$) to find the best matrix
($\boldsymbol{\Omega}_{\sf{HB}}$) that maximizes the EE. As shown in the
“Training Phase” of Figure 6, the proposed DNN for HBF, E-HBF-Net, is
designing jointly the DP ($\mathbf{W}=\Re[\mathbf{W}]+i\Im[\mathbf{W}]$), the
PSs ($\bar{\mathbf{P}}_{q}$), and the connections matrix
($\bar{\boldsymbol{\Omega}}_{\sf{HB}}$) by employing the proposed “Gumbel
Sigmoid” function as in (19).

Obtaining $\bar{P}_{\sf{HBF}}$ requires computing (18) and thus we first need
to know the power consumed by the PAs. Thus, based on (17), we would need the
input and output power of the PAs. The output power of the PAs
$1,\ldots,N_{T}$ is given by

$\bar{\mathbf{p}}_{\sf{TX}}=\sum_{u=1}^{N_{\sf{U}}}|\bar{\mathbf{A}}\mathbf{w}_{u}|^{2},$
(23)

where
$\bar{\mathbf{p}}_{\sf{TX}}=[\bar{P}_{\sf{TX}}^{1},\ldots,\bar{P}_{\sf{TX}}^{N_{\sf{T}}}]^{\sf{T}}$
and $\bar{\mathbf{A}}$ and
$\mathbf{W}=[\mathbf{w}_{1},\ldots,\mathbf{w}_{N_{\sf{U}}}]$ are the AP and DP
outputs designed by the proposed DNN. Due to the total power constraint
assumed at the BS in (13e), we should normalize the power such
$\|\bar{\mathbf{A}}\mathbf{W}\|_{F}^{2}=\sum_{n=1}^{N_{\sf{T}}}\sum_{u=1}^{N_{\sf{U}}}|\bar{\mathbf{A}}\mathbf{w}_{u}|^{2}\leq
P_{\sf{TX}}$. To respect the inequality of the power constraint, we introduce
a new power threshold $\bar{P}_{\sf{TX}}$ that is a function of the connection
matrix as follows:

$\bar{P}_{\sf{TX}}=\sum_{\forall
m,n}[\bar{\boldsymbol{\Omega}}_{\sf{HB}}]_{n,m}P_{\sf{TX}}/(N_{\sf{RF}}N_{\sf{T}}).$
(24)

Therefore, the maximum transmitted power is limited to $P_{\sf{TX}}$ when all
connections are established
($[\bar{\boldsymbol{\Omega}}_{\sf{HB}}]_{n,m}=1\;\forall n,m$), while reducing
the number of connections reduces the transmit power. After power
normalization, we can obtain the input power of the PAs according to (15).
However, for the DNN loss function, we cannot have a sum over a dynamic set as
defined in (15). Therefore, we reformulate (15) as

$\displaystyle\bar{\mathbf{p}}_{\sf{PA},\sf{HBF}}^{in}$
$\displaystyle=\frac{P_{\mathsf{BB}}^{out}}{\text{IL}_{\sf{C}}\text{IL}_{\boldsymbol{\Psi}}(\psi)\text{IL}_{\boldsymbol{\Phi}}\text{IL}_{\sf{M}}}\bar{\boldsymbol{\Omega}}_{\sf{HB}}\text{diag}(\bar{\boldsymbol{\Omega}}_{\sf{HB}}^{\rm{T}}\boldsymbol{1}_{N_{\sf{T}}})^{\dagger}\boldsymbol{1}_{N_{\sf{RF}}},$
(25)

where
$\bar{\mathbf{p}}_{\sf{PA},\sf{HBF}}^{in}=[\bar{P}_{\sf{PA},\sf{HBF}}^{in,1},\ldots,\bar{P}_{\sf{PA},\sf{HBF}}^{in,N_{\sf{T}}}]$
is the vector of input power of the APs, $\bar{\boldsymbol{\Omega}}_{\sf{HB}}$
is the output of Gumbel-Sigmoid function for HBF, and $\boldsymbol{1}_{N}$
denotes the all-one column vector of size $N$. According to (17) and
(23)-(25), we can obtain $\bar{P}_{\sf{PA},\sf{HBF}}^{\textrm{DC},n}$.

To compute the power consumption of all activated RF chains as in (18), we
need to determine the number of activated RF chains (i.e.,
$N_{\sf{RF}}(\bar{\boldsymbol{\Omega}}_{\sf{HB}})$). However, finding
$N_{\sf{RF}}(\bar{\boldsymbol{\Omega}}_{\sf{HB}})$ again requires a summation
over a dynamic set and it is not appropriate for the loss function. As a
consequence, we use an alternative linear algebra formulation. First, we
compute the expectation over all antennas of each RF chain as
$\bar{\boldsymbol{\Omega}}_{\sf{HB}}^{\rm{T}}\boldsymbol{1}_{N_{\sf{T}}}/N_{\sf{T}}$.
Then, we find the expected number of activated RF chains as follows:

$\bar{N}_{\sf{RF}}(\bar{\boldsymbol{\Omega}}_{\sf{HB}})=\boldsymbol{1}_{N_{\sf{RF}}}^{\rm{T}}\bar{\boldsymbol{\Omega}}_{\sf{HB}}^{\rm{T}}\boldsymbol{1}_{N_{\sf{T}}}/N_{\sf{T}}.$
(26)

Finally, the total power consumption is given by

$\displaystyle\bar{P}_{\sf{HBF}}=\bar{N}_{\sf{RF}}(\bar{\boldsymbol{\Omega}}_{\sf{HB}})(P_{\sf{L}}+P_{\sf{LO}}+P_{\sf{D}}(b_{D}))+\sum_{n=1}^{N_{\sf{T}}}\bar{P}_{\sf{PA},\sf{HBF}}^{\textrm{DC},n},$
(27)

where $\bar{P}_{\sf{HBF}}$ is the power consumption terms that has been
employed in (21).

Algorithm 1 Efficient HBF (E-HBF-Net)

1:Input: $\bar{\mathbf{H}}$

2:Output: $\Re[\mathbf{W}]$, $\Im[\mathbf{W}]$, $\mathbf{P}_{q}$, and
$\bar{\boldsymbol{\Omega}}_{\sf{HB}}$

3:Hyper-Parameters:  $\gamma$, $R_{\sf{d}}$

4:for  _$i$ in epochs_ do

5: FeedForward E-HBF-Net.train()

6: $\mathbf{W}=\Re[\mathbf{W}]+i\Im[\mathbf{W}]$

7: $\mathbf{A}=\mathbf{P}_{q}\otimes\bar{\boldsymbol{\Omega}}_{\sf{HB}},$

8: $\bar{P}_{\sf{TX}}^{n}$, $\bar{P}_{\sf{PA},\sf{HBF}}^{in,n}$, and
$\bar{N}_{\sf{RF}}(\bar{\boldsymbol{\Omega}}_{\sf{HB}})$ in (23) in (25), and
(26)

9: compute $P_{\sf{HBF}}$ based on (27)

10: compute $R_{\sf{HBF}}(\bar{\mathbf{A}},\mathbf{W})$ based on (6)

11: Loss:
$\mathcal{L}_{\sf{BF}}=\gamma\mathcal{L}_{\sf{EC}}+\mathcal{L}_{\sf{AAS}}$

12: compute gradient over layers

13: update weights and biases with AdamW optimizer

14:Input: $\hat{\mathbf{H}}$

15:Output: $\mathbf{W}$, $\mathbf{P}_{q}$, and $\boldsymbol{\Omega}_{\sf{HB}}$

16:FeedForward E-HBF-Net.eval()

17:$\boldsymbol{\Omega}_{\sf{HB}}=\lfloor\bar{\boldsymbol{\Omega}}_{\sf{HB}}\rceil$

18:$\mathbf{W}=\Re[\mathbf{W}]+i\Im[\mathbf{W}]$

19:$\mathbf{A}=\mathbf{P}_{q}\otimes\boldsymbol{\Omega}_{\sf{HB}},$

Training Phase Online Phase

Algorithm 2 Efficient FDP (E-FDP-Net)

1:Input: $\hat{\mathbf{H}}$

2:Output: $\Re[\mathbf{U}]$, $\Im[\mathbf{U}]$, and
$\bar{\boldsymbol{\Omega}}_{\sf{FD}}$

3:Hyper-Parameters:  $\gamma$, $R_{\sf{d}}$

4:for  _$i$ in epochs_ do

5: FeedForward E-FDP-Net.train()

6:
$\bar{\boldsymbol{\Omega}}_{\sf{FD}}=\text{diag}(\bar{\boldsymbol{\omega}})$

7: compute $\bar{P}_{\sf{TX}}^{n}$ as (28)

8: $\bar{\mathbf{U}}=\Re[\mathbf{U}]+i\Im[\mathbf{U}]$

9: compute $P_{\sf{FDP}}$ based on (27)

10: compute
$R_{\sf{FDP}}(\boldsymbol{\bar{\Omega}}_{\sf{FD}}\times\mathbf{U})$ based on
(3)

11: Loss:
$\mathcal{L}_{\sf{BF}}=\gamma\mathcal{L}_{\sf{EC}}+\mathcal{L}_{\sf{AAS}}$

12: compute gradient over layers

13: update weights and biases with AdamW optimizer

14:Input: $\hat{\mathbf{H}}$

15:Output: $\mathbf{U}$, and $\boldsymbol{\Omega}_{\sf{FD}}$

16:FeedForward E-FDP-Net.eval()

17:$\boldsymbol{\Omega}_{\sf{FD}}=\text{diag}(\lfloor\bar{\boldsymbol{\omega}}\rceil)$

18:$\mathbf{U}=\Re[\mathbf{U}]+i\Im[\mathbf{U}]$

Training Phase Online Phase

####  IV-B2 Fully Digital Precoder (E-FDP-Net)

E-FDP-Net provides the precoder $\mathbf{U}=\Re[\mathbf{U}]+i\Im[\mathbf{U}]$
and the vector $\bar{\boldsymbol{\omega}}$ for antenna selection, where
$\bar{\boldsymbol{\Omega}}_{\sf{FD}}=\text{diag}(\bar{\boldsymbol{\omega}})$.
To evaluate the first term of the loss function detailed in (20), the total
power consumption of FDP ($\bar{P}_{\sf{FDP}}$) is required. Computing
$\bar{P}_{\sf{FDP}}$ for E-FDP-Net is simpler than HBF because in FDP each
antenna is connected to one RF chain. Consequently, the input power of each PA
is simply $P_{\sf{BB}}^{out}$. Similar to HBF, to respect the power constraint
for FDP, $\|\bar{\mathbf{U}}\|_{F}^{2}\leq P_{\sf{TX}}$, the output power
should be a function of
$\bar{\boldsymbol{\omega}}=[\bar{\omega}_{1},...,\bar{\omega}_{N_{\sf{T}}}]$.
As a consequence, we denote the output power of the $n^{\text{th}}$ antenna as

$\bar{P}_{\sf{TX}}^{n}=\sum_{u=1}^{N_{\sf{U}}}|[\mathbf{U}]_{n,u}|^{2}\bar{\omega}_{n}\,.$
(28)

Therefore, the power consumed by the PAs is given by (17). Finally, the power
consumed by the active RF chains is also easy to compute because the number of
active RF chains is given by
$\bar{N}_{\sf{RF}}(\bar{\boldsymbol{\Omega}}_{\sf{FD}})=\sum_{n=1}^{N_{\sf{T}}}\bar{\boldsymbol{\omega}}$.

###  IV-C Online Phase: Transmitting Data

Once the DNN has been trained, the online phase can start as shown in Figure 6
(right). In the online phase, the DNN input is only given by the imperfect
channel matrices $\hat{\mathbf{H}}$. In the online phase, like the training
phase, the outputs of the DNN: the AP ($\mathbf{P}_{q}$) and the DP
($\mathbf{W}$) in HBF and ($\mathbf{U}$) in FDP can be employed as is without
any further processing, which is not the case for the connection matrix
$\bar{\boldsymbol{\Omega}}$. Since the connection matrix
($\bar{\boldsymbol{\Omega}}_{\sf{HB}}$ in HBF or $\bar{\boldsymbol{\omega}}$
in FDP) should be binary, once it is output by the DNN in the online phase, it
requires binary quantization. To do so, we can use the element-wise round
function ($\lfloor\cdot\rceil$) on each element of the connection matrix as
follows:
$\boldsymbol{\Omega}_{\sf{HB}}=\lfloor\bar{\boldsymbol{\Omega}}_{\sf{HB}}\rceil$
for HBF and $\boldsymbol{\omega}=\lfloor\bar{\boldsymbol{\omega}}\rceil$, and
$\boldsymbol{\Omega}_{\sf{FD}}=\text{diag}(\boldsymbol{\omega})$ for FDP. The
output power of the $n^{\text{th}}$ antenna for E-HBF-Net is given by the
$n^{\text{th}}$ element of the power vector defined in (23) while for E-FDP-
Net it is given by (28). The two proposed DNN solutions, E-HBF-Net and E-FDP-
Net, are described in Algorithm 1, Algorithm 2, respectively.

##  V Performance Evaluation

In this section, the performance of the proposed DNN, implemented using the
PyTorch deep learning library, is numerically evaluated. The scenario “O1-$28$
GHz” of the deepMIMO channel model [31] is employed to generate the unlabeled
dataset (the channel coefficients $\mathbf{h}_{u}$ for user $u$) for the
training and testing. In the deepMIMO dataset [31], realistic channel
information is generated by applying ray-tracing methods to a three-
dimensional model of an urban environment to capture the geometry-based
characteristics, such as the correlation between the channels at different
locations, and the dependence on the materials of the various environmental
elements, among others. The parameters to generate the deepMIMO dataset are
shown in Table IV, where the channel model parameters active_user_first and
active_user_last are set to 1100 and 2200 respectively. The BS is equipped
with $N_{\sf{T}}=64$ antennas and $N_{\sf{RF}}=8$ RF chains with PSs serving
$N_{\sf{U}}=4$ users randomly located in a dedicated area (S1 in Figure 7).
Scenario “O1” consists of several users’ locations being randomly placed in
two streets surrounded by buildings. These two streets are orthogonal and
intersect in the middle of the considered area. The size of the DNN dataset is
set to $2\times 10^{6}$ samples, with $85$% of the samples used for the
training set and the remaining used to evaluate the performance. We used
“AdamW” as the DNN training optimizer. The hyper-parameters used in our DNN
model are listed in Table V. In addition, hyper-parameter $\tau$ known as the
Gumbel-Sigmoid temperature is set to $0.1$ and $0.5$ for E-HBF-Net and E-FDP-
Net, respectively, while the best value for hyper-parameter $\gamma$,
described in (20), depends on $R_{\sf{d}}$, and ranges from $\gamma=0.1$ for
$R_{\sf{d}}=1$, to $\gamma=0.005$ for $R_{\sf{d}}=8$. The training procedure
required $200$ epochs.

Figure 7: Illustration of the selected covered area from deepMIMO channel
model [31]. TABLE IV: Parameter selection for the deepMIMO channel model

System | Antennas
---|---
Parameter | Value |  Parameter | Value
scenario | “O1-28 GHz” |  num_ant_x | $1$
bandwidth |  $0.5$ GHz |  num_ant_y | $8$
num_OFDM | $512$ |  num_ant_z | $8$
num_paths | $2$ |  ant_spacing | $0.5$

TABLE V: Proposed DNN hyper-parameters

Parameter | Set Value
---|---
Mini-batch size | 1000
Initial learning rate | 0.005
ReduceLROnPlateau (factor) | 0.4
ReduceLROnPlateau (patience) | 10
Weight decay | $10^{-5}$
Dropout keep probability | .95
Kernel size | 3
Zero padding | 1
“$\epsilon$” in BatchNorm (1D & 2D) | $10^{-5}$

TABLE VI: Simulation parameters for the energy model

Parameter | Value | Parameter | Value
---|---|---|---
$\text{FoM}_{\sf{L}}$ | 1.4 mW/GHz | $\text{IL}_{\sf{C}}$ |  $1.8$ dB
$\text{FoM}_{\sf{D}}$ |  $54.5$ fJ/conv | $\text{IL}_{\sf{M}}$ |  $5.5$ dB
$f_{s}$ |  $0.5$ GHz | $\text{IL}_{\boldsymbol{\Phi}}$ |  $3.7$ dB
$f_{c}$ |  $500$ MHz | $\text{IL}_{\boldsymbol{\Psi}}$ |  $1$ dB
$P_{\sf{TX}}$ |  $40$ dBm | $P_{\sf{LO}}$ |  $10$ mW
$\alpha$ |  $36$% | $P_{\sf{L}}$ |  $0.7$ mW
$P_{\sf{BB}}^{out}$ |  $-5.6$ dBm | $b_{D}$ | 6 bits

###  V-A Spectral Efficiency and Power Consumption Analysis

Figure 8: Maximum spectral efficiency of E-FDP-Net and E-HBF-Net, compared to
other conventional approaches. System parameters are set to: $N_{\sf{U}}=4$,
$N_{\sf{RF}}=8$, $N_{\sf{T}}=64$.

We first verify the maximum SE that can be achieved by the proposed DNNs, when
they are trained without considering their power consumption, and compare them
with the baseline solutions presented in Section II-A. This maximal SE is
shown in Figure 8 when varying the noise power. Taking into account channel
attenuation, the average signal-to-noise ratio (SNR) ranges from $-7.8$ dB to
$22.2$ dB. To obtain the maximum SE, we set $\gamma=0$ so that the loss
function for E-HBF-Net and E-FDP-Net in (21) depends only on
$\mathcal{L}_{\sf{AS}}$ and we set $R_{\sf{d}}=15$ to have no constraint on
SE. On the one hand, the proposed E-FDP-Net gives a close-to-optimal
performance. On the other hand, E-HBF-Net, outperforms other conventional
solutions and is very close to E-FDP-Net performance. In the low-noise regime,
the SE of all solutions continues to increase. However, both E-HBF-Net and
E-FDP-Net outperform other conventional non-DL solutions in high SNR regimes.

Figure 9: Power required to achieve a given SE for the various transmitter
configurations. Idle $P_{\sf{BF}}$ is the power consumed by the BF structure
when $P_{\sf{TX}}=0$. The parameters are set to: $N_{\sf{U}}=4$,
$N_{\sf{T}}=64$, $N_{\sf{RF}}=8$, and $\sigma^{2}=-130$ dBm.

In Figure 9, we compare the power consumption of different BF hardware
configurations at a given SE. It is shown that by adjusting $R_{\sf{d}}$ for
E-FDP-Net and E-HBF-Net, different SE and power consumption trade-offs can be
obtained, where for each proposed technique we set $R_{\sf{d}}$ in
$\\{1,3,5,6,8\\}$. To cover a range of SE values, we also adjust the
transmitted power for the conventional methods by setting $P_{\sf{TX}}$ in
$\\{0.1,1,10\\}$W. We see that the optimal FDP and the proposed E-FDP-Net with
$R_{\sf{d}}=8$ achieve the best SE. However, they also consume the most power
because they require to activate all $N_{\sf{T}}$ RF chains. In this figure,
we see that when the desired SE parameter $R_{d}$ is reduced, both E-FDP-Net
and E-HBF-Net are able to reduce their power consumption. For example, when
$R_{\sf{d}}$ is decreased from $8$ to $5$ bits/s/Hz/user, the consumed power
for both E-FDP-Net and E-HBF-Net is reduced significantly ($64\%$ less for
E-FDP-Net and $68\%$ less for E-HDF-Net). By decreasing $R_{\sf{d}}$ further,
both the power consumption and the SE continue to decrease. Furthermore, we
see that E-FDP-Net and E-HBF-Net achieve must better energy efficiency than
the baseline approaches. For example, when $R_{\sf{d}}=6$, it can be seen that
E-HBF-Net achieves similar SE compared to FC-HBF solved with MO-AltMin, but
with almost $1.7$ times less consumed power. Further, the baseline solutions
exhibit a power floor, shown by red lines in the figure, that corresponds to
the power consumed by RF chains. When the transmit power of O-FDP and FC-HBF
is decreased to $P_{\sf{TX}}=1$W and $P_{\sf{TX}}=0.1$W, the SE is degraded
due to the lower transmit power. However, there is constant power consumption
for each beamforming technique due to the operation of RF chains. On the
contrary, E-FDP-Net and E-HBF-Net have the ability to reduce their power
consumption below these floors by adaptively turning off their RF chains.

Figure 10: The connection matrix
$\bar{\boldsymbol{\Omega}}_{\sf{FD}}=\text{diag}(\bar{\boldsymbol{\omega}})$
of E-FDP-Net for one input sample, and for different values of hyper-parameter
$R_{\sf{d}}$, where a blue square represents the value $1$ and a white one
represents the value $0$. System parameters are set to: $N_{\sf{U}}=4$,
$N_{\sf{T}}=64$, and $\sigma^{2}=-130$ dBm.

To illustrate how many antennas are activated by E-FDP-Net, we plot in Figure
10 the connection matrix $\bar{\boldsymbol{\Omega}}_{\sf{FD}}$ for one sample
of the test set, for different values of $R_{\sf{d}}$, where a blue square
represents the value $1$ and a white square represents the value $0$. It can
be seen that large values of $R_{\sf{d}}$ lead to more active antennas (and
thus more active RF chains), and thus to a higher power consumption.

Figure 11: The average value of the connection matrix
$\bar{\boldsymbol{\Omega}}_{\sf{HB}}$ of E-HBF-Net given for different values
of hyper-parameter $R_{\sf{d}}$, where the shade of each square represents the
range of values from $0$ (light) to $1$ (dark). System parameters are set to:
$N_{\sf{U}}=4$, $N_{\sf{T}}=64$, $N_{\sf{RF}}=8$, and $\sigma^{2}=-130$ dBm.

In Figure 11, we show the average value of
$\bar{\boldsymbol{\Omega}}_{\sf{HB}}$ over the inputs, for different values of
$R_{\sf{d}}$. When decreasing $R_{\sf{d}}$, the number of active antennas
(non-zero columns) remains constant, while the number of active RF chains
(non-zero rows) is reduced. This is because the power consumption of an
antenna depends on its transmit power, which can be adjusted, whereas RF
chains consume a fixed amount of power and must be turned off to save power.
It is interesting to see that with a lower value of $R_{\sf{d}}$, the E-HBF-
Net designs the connection matrix such that a small number of RF chains are
activated that are connected to several antennas, which helps to increase the
spatial multiplexing gain and degrees of freedom. Finally, we see that as
$R_{\sf{d}}$ increases, more antennas and more RF chains are activated, and
thus more power is used.

Figure 12: EE versus SE for the proposed E-HBF-Net and E-FDF-Net. The
parameters are set to: $N_{\sf{U}}=4$, $N_{\sf{T}}=64$, $N_{\sf{RF}}=8$, and
$\sigma^{2}=-130$ dBm.

Figure 12 presents the EE versus SE comparison for the proposed E-FDP-Net and
E-HBF-Net, with varying adjustments to $R_{\sf{d}}$. Notably, as SE decreases,
E-HBF-Net demonstrates superior EE performance compared to E-FDP-Net. This
outcome is attributed to the behavior of E-HBF-Net at lower SE values, where
it intelligently deactivates RF chains while keeping multiple antennas active.
Conversely, in E-FDP-Net, turning off an RF chain also turns off the
associated antenna. Consequently, E-HBF-Net excels in conserving energy while
simultaneously offering enhanced SE due to its higher flexibility.
Furthermore, as SE increases, E-HBF-Net maintains its efficiency advantage
over E-FDP-Net, although the performance gap between the two approaches
diminishes.

Figure 13: The normalized number of active RF chains, EE, and power
consumption of the proposed E-FDP-Net (left sub-plots), E-HBF-Net (right sub-
plots) versus different numbers of users. System parameters are set to:
$N_{\sf{RF}}=8$, $N_{\sf{T}}=64$, and $\sigma^{2}=-130$ dBm.

###  V-B Varying the Number of Users

To show the impact of antenna and RF chain selection when varying the number
of active users, we present Figure 13 for $R_{\sf{d}}=3$ and $R_{\sf{d}}=5$,
where the left-side sub-plots present E-FDP-Net and the right-side ones shows
E-HBF-Net. To improve the presentation we use the normalized number of active
RF chains ($\frac{N_{\sf{RF}}(\boldsymbol{\Omega})}{N_{\sf{RF}}}$), which in
the case of FDP is equal to the number of active antennas. In the proposed
solutions, we see that by increasing the number of active users, the DNN not
only activates more RF chains but also increases the transmitted power to meet
$R_{\sf{d}}$. Moreover, when $R_{\sf{d}}$ is small, the DNN requires a smaller
number of active RF chains while minimizing the transmitted power, thus
lowering power consumption and consequently increasing EE. Figure 13 shows
that the proposed DNN approaches are adaptive to the number of active users in
the network. That is, depending on the scenario, the DNN designs the
beamforming structures to adapt to the varying number of users in each
scenario. For instance, in a high-traffic scenario, when the number of active
users is large, the DNN will activate more antennas and RF chains to meet the
average SE. On the other hand, in a low-traffic scenario, when the number of
users is low, the DNN has no need to activate a large number of antennas and
RF chains, and thus can significantly increase its EE. Finally, we notice that
by controlling the value of $R_{\sf{d}}$, which depends on the application and
the objective of the service provider, the power consumption can be adjusted.

###  V-C Training with Imperfect CSI

Unlike other studies that assume perfect CSI for DNN training, in this work,
we employed imperfect CSI not only for the input of the DNN but also for the
computation of the loss function. The robustness of the proposed methods
against imperfect CSI is evaluated and compared to other non-DL methods in
Figure 14. Here we train the DNN with different $\beta$ in
$\\{0,0.1,0.2,0.3,0.4,0.5\\}$. It is clear that the SE performance decreases
as the value of $\beta$ increases. In particular, when $\beta$ increases from
$0$ to $0.5$, the SE performance for O-FDP degraded by $38\%$. For PE-AltMin,
the degradation is around $25\%$, whereas it is around $27\%$, for MO-AltMin.
The lowest degradation in terms of SE performance is achieved for E-HBF-Net
and E-FDP-Net, (e.g., the degradation is around $9\%$ and $11\%$,
respectively). Therefore, the proposed methods are more robust against
estimation errors. Moreover, the red lines in Figure 14 shows the ideal case
of perfect CSI when $\beta=0$. It is interesting to see that for a small
$\beta$ (i.e. $0.1$) the SE performance did not degrade, but in contrast, it
slightly improved in the online phase. This is due to the fact that training
with imperfect CSI can act as a regularization technique known as noise
injection in the machine learning literature and thus can improve the
generalization of the DNN in the online phase [48].

Figure 14: The SE comparison of the different BF solutions with different
values of $\beta$. System parameters are set to: $N_{\sf{U}}=4$,
$N_{\sf{RF}}=8$, $N_{\sf{T}}=64$, $\gamma=0$, $\sigma^{2}=-130$ dBm.

In Figure 15, we present the convergence of the training of the proposed
E-FDP-Net in terms of SE, power consumption, and EE, when $R_{\sf{d}}=3$ and
$N_{\sf{U}}=4$. We see in the top subplot that the DNN learns quickly to
design the connection matrix and the FDP to obtain an SE of
$N_{\sf{U}}R_{\sf{d}}=12$, i.e., after few epochs, the achieved SE for each
user is around $R_{\sf{d}}$. Then, while the SE target is respected, the DNN
learns to gradually reduce power consumption by turning off some RF chains
until it achieves the minimum power consumption as shown in the middle
subplot.

Figure 15: Training steps of E-FDP-Net. The parameters are set to:
$N_{\sf{U}}=4$, $N_{\sf{RF}}=8$, $N_{\sf{T}}=64$, and $R_{\sf{d}}=3$.

###  V-D Computational Complexity Analysis

To evaluate the computational complexity of the proposed DNNs, we derive the
analytical expression of the number of real multiplications (RM) and compare
it with other approaches. We assume that one complex multiplication (CM)
corresponds to $4$ RMs and that the $1$ complex division corresponds to $8$
RMs (assuming that the real division of $1$ is equal to $1$ RM). Only the
matrix multiplications and inversions are taken into consideration, the other
operations are considered negligible. A CM between a matrix of size $N\times
P$ and a matrix of size $P\times M$ requires $NMP$ CMs. To invert a square
matrix of size $N$, around $N^{3}/3$ CMs are required if the Gaussian
elimination algorithm is employed. Finally, we consider that the eigenvalues
of a square matrix of size $N$ are obtained using the Cholesky decomposition
[49], which requires approximately $4N^{3}$ RMs.

O-FDP requires
$4(2^{N_{\sf{U}}}-1)(2N_{\sf{U}}N_{\sf{T}}^{2}+N_{\sf{U}}^{2}N_{\sf{T}}+\frac{1}{3}N_{\sf{T}}^{3})$
RMs as described in [19]. In the specified scenario, we replicate the
implementation of SoTA algorithms. Our observations reveal that the PE-AltMin
algorithm typically achieves convergence within an average of
$\ell_{\sf{PE}}=15$ iterations. Given that the computation of the singular-
value decomposition of a $p\times q$ matrix necessitates approximately
$4p^{2}q+22q^{3}$ resource modules (RMs), we can formulate the total number of
RMs required for PE-AltMin as
$\ell_{\sf{PE}}(8N_{\sf{RF}}N_{\sf{U}}(N_{\sf{T}}+N_{\sf{U}})+22N_{\sf{RF}}^{3})$.
MO-AltMin has a much higher complexity than PE-AltMin [9]. MO-AltMin is
composed of a main loop that computes the DP, and of an inner loop applying
the “Conjugate Gradient” algorithm to find the HBF. In the main loop,
computing the DP requires $4N_{\sf{T}}N_{\sf{U}}N_{\sf{RF}}$ RMs, while in the
inner loop, the Kronecker product of a $N_{\sf{RF}}\times N_{\sf{T}}$ matrix
with a $N_{\sf{U}}\times N_{\sf{T}}$ matrix is computed, which requires
$4N_{\sf{T}}^{2}N_{\sf{U}}N_{\sf{RF}}$ RMs. Based on the defined scenario the
outer loop is repeated $\ell_{\sf{MO}}=2$ times while the inner loop is
repeated $\ell^{\prime}=30$ times, the total number of RMs used by MO-AltMin
is
$4\ell_{\sf{MO}}N_{\sf{T}}N_{\sf{U}}N_{\sf{RF}}\big{(}1+\ell^{\prime}N_{\sf{T}}\big{)}$.
To design the HBF, both PE-AltMin and MO-AltMin require designing the FDP as
discussed in (10), thus the complexity of obtaining the FDP should be added to
the complexity of PE-AltMin and MO-AltMin.

On the other hand, to compute the computational complexity of the DNN
approaches, we need to compute the number of parameters of the DNN
architectures. Both DNN architectures, E-HBF-Net and E-FDP-Net, have the same
$\text{DNN}_{\sf{core}}$ but their output layers are different due to
different output dimensions. The number of RMs in the $\text{DNN}_{\sf{core}}$
is calculated for each layer separately, then summed up. The width of the
$l^{\text{th}}$ FC and CL are respectively denoted as $f_{l}$ and $c_{l}$. The
number of multiplications required for $\text{DNN}_{\sf{core}}$ is
$\mathcal{M}(\text{DNN}_{\sf{core}})=(2c_{1}+c_{1}c_{2}+c_{2}c_{3}+c_{3}f_{1}/\kappa^{2})N_{\sf{T}}N_{\sf{U}}\kappa^{2}+f_{1}f_{2}$,
where $\kappa$ is the kernel size i.e. $\kappa=3$ [19]. Considering that for
E-HBF-Net there are $4$ output layers, one layer for the AP, two layers for
the DP, and one layer for the connection matrix, then the total number of
multiplications is
$\mathcal{M}(\text{DNN}_{\sf{core}})+f_{2}(N_{\sf{T}}N_{\sf{RF}}+2N_{\sf{U}}N_{\sf{RF}}+N_{\sf{T}}N_{\sf{RF}})$.
Likewise, for E-FDP-Net, the total number of multiplications is
$\mathcal{M}(\text{DNN}_{\sf{core}})+f_{2}(2N_{\sf{U}}N_{\sf{T}}+N_{\sf{T}})$.
Examples of the numerical values of these analytical expressions are shown in
Table VII. It can be seen that for HBF transmitters, E-HBF-Net reduces the
complexity by $38$% compared to the least complex conventional approach (PE-
AltMin), while for FDP transmitters, O-FDP is $1.5$ times more complex than
E-FDP-Net.

TABLE VII: Computational Complexity Comparison

Transmitter type | Method | # RMs ($\times 10^{6}$)
---|---|---
FDP | O-FDP | 7.27
HBF | PE-AltMin [9] | 8.48
HBF | MO-AltMin [9] | 38.7
FDP | E-FDP-Net | 4.69
HBF | E-HBF-Net | 5.21
($N_{\sf{U}}=4$, $N_{\sf{RF}}=8$ and $N_{\sf{T}}=64$)

##  VI Conclusion

In this paper, we studied the problem of antenna selection and beamforming
design in a massive multiple-input multiple-output (mMIMO) system with the
objective of maximizing energy efficiency (EE). First, we derived an accurate
energy model for the mMIMO system. Our proposed energy model takes into
account the transmit power as well as the power consumed by the hardware by
considering the insertion loss and the direct power consumption of different
components such as the combiners and the power amplifiers. Next, based on our
energy model, we designed unsupervised deep learning approaches to
intelligently and adaptively select the BF structures and the transmitting
antennas. Specifically, we proposed two deep neural networks models, called
E-HBF-Net and E-FDP-Net, for hybrid BF and for fully digital precoding,
respectively. Both DNNs optimize the EE of the mMIMO system by intelligently
selecting the transmitting antennas and choosing the precoding matrices for
HBF and FDP, which allows them to achieve significantly better EE than
conventional solutions. Simulation results confirm that the proposed DNNs can
adapt to the number of active users and that they provide different trade-offs
between SE and EC that can be controlled by tuning a hyper-parameter.
Furthermore, we show that the DNN models can be trained exclusively using
imperfect channel information (CSI), i.e., the imperfect CSI was used as input
to our DNN models as well as to compute the loss function during training.

## Acknowledgement

This work was supported by the Natural Sciences and Engineering Research
Council of Canada (NSERC) (under project ALLRP 566589-21) and InnovÉÉ (INNOV-R
program) through the partnership with Ericsson and ECCC.

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