# Low PAPR MIMO-OFDM Design Based on Convolutional Autoencoder

Yara Huleihel, and Haim H. Permuter

###### Abstract

An enhanced framework for peak-to-average power ratio ($\mathsf{PAPR}$)
reduction and waveform design for Multiple-Input-Multiple-Output
($\mathsf{MIMO}$) orthogonal frequency-division multiplexing ($\mathsf{OFDM}$)
systems, based on a convolutional-autoencoder ($\mathsf{CAE}$) architecture,
is presented. The end-to-end learning-based autoencoder ($\mathsf{AE}$) for
communication networks represents the network by an encoder and decoder, where
in between, the learned latent representation goes through a physical
communication channel. We introduce a joint learning scheme based on projected
gradient descent iteration to optimize the spectral mask behavior and MIMO
detection under the influence of a non-linear high power amplifier
($\mathsf{HPA}$) and a multipath fading channel. The offered efficient
implementation novel waveform design technique utilizes only a single
$\mathsf{PAPR}$ reduction block for all antennas. It is throughput-lossless,
as no side information is required at the decoder. Performance is analyzed by
examining the bit error rate ($\mathsf{BER}$), the $\mathsf{PAPR}$, and the
spectral response and compared with classical $\mathsf{PAPR}$ reduction
$\mathsf{MIMO}$ detector methods on 5G simulated data. The suggested system
exhibits competitive performance when considering all optimization criteria
simultaneously. We apply gradual loss learning for multi-objective
optimization and show empirically that a single trained model covers the tasks
of $\mathsf{PAPR}$ reduction, spectrum design, and $\mathsf{MIMO}$ detection
together over a wide range of SNR levels.

###### Index Terms:

Deep learning, Autoencoder, Multiple-Input-Multiple-Output, Orthogonal
frequency-division multiplexing, Peak-to-average power ratio, Wireless signal
processing.

##  I Introduction

The Multiple-input-multiple-output ($\mathsf{MIMO}$) scheme is a widely used
technique for enhancing channel capacity and transmission reliability, thanks
to the diversity and multiplexing gains. Orthogonal frequency division
multiplexing ($\mathsf{OFDM}$) is a waveform design method known for providing
high bandwidth efficiency, high throughput, simple equalization in wireless
transmission, and efficient hardware implementation. For these reasons, it has
been adopted as a standard technology in various wireless communication
systems, such as WiFi, 4G, and 5G standards for wireless communications.
Nonetheless, significant drawbacks of the $\mathsf{OFDM}$ multi-carrier system
appear in the form of adjacent channel power ratio ($\mathsf{ACPR}$)
limitations and the tendency to produce signals with a high peak-to-average
power ratio ($\mathsf{PAPR}$) in the time-domain, since many subcarrier
components are added via a fast Fourier transform ($\mathsf{FFT}$) operation.
The contribution of each subcarrier to the total power is dynamic, which makes
the total power highly variable. In particular, the high $\mathsf{PAPR}$ in
$\mathsf{MIMO}$-$\mathsf{OFDM}$ systems is exacerbated as the number of
antennas increases [1]. The demand for higher energy efficiency as well as
reduced power consumption is expected to increase for future radio systems
[2]. Moreover, future communication system design and equipment are expected
to be more compatible with machine learning ($\mathsf{ML}$) implementation
requirements, for example, allowing learning in the field to make some design
choices [2]. Accordingly, waveform designs using $\mathsf{ML}$ techniques are
becoming significantly attractive.

A high power amplifier ($\mathsf{HPA}$) is required to provide enough output
power for reliable communication. In practical systems, the $\mathsf{HPA}$ is
not linear and distorts the transmitted signal. As a result, severe non-linear
signal distortions are found when these high $\mathsf{PAPR}$ signals pass
through the non-linear $\mathsf{HPA}$. The resulting signal exhibits spectral
regrowth in the form of in-band signal distortions and out-of-band radiation
[3], and the bit error rate ($\mathsf{BER}$) increases. Hence, it is crucial
to develop $\mathsf{PAPR}$ reduction techniques for
$\mathsf{MIMO}$-$\mathsf{OFDM}$ systems to increase their efficiency in
handling large data streams and to reduce their error rates. Training and
applying the $\mathsf{PAPR}$ reduction block to each antenna individually
exacerbates the complexity, cost, and redundancy in proportion to the number
of transmit antennas in the system. Instead, in this work, a single
$\mathsf{PAPR}$ reduction block jointly operates on all antenna
$\mathsf{OFDM}$ sequences, and it is designed according to the maximum
$\mathsf{PAPR}$ value of all antenna sequences.

A central difficulty of the multiple transmitter (TX) and receiver (RX)
antenna structure is posed by the need for joint detection of the data symbols
sent by each transmitter. Unfortunately, the optimal $\mathsf{MIMO}$ detection
solution imposes an NP-hard problem on the receiver. Consequently, various
sub-optimal yet feasible detection algorithms have been proposed. Other than
classical model-driven solutions, an increasing effort has been dedicated to
$\mathsf{ML}$, and, specifically, deep learning ($\mathsf{DL}$) based
techniques to solve the $\mathsf{MIMO}$ detection problem, and more generally,
various wireless communication tasks.

The design of $\mathsf{OFDM}$ waveform signals aims to simultaneously achieve
a high data rate, high spectral efficiency (measured by the $\mathsf{ACPR}$),
and low computational complexity [4, 5]. This design is highly affected by the
non-linear effects of the $\mathsf{HPA}$. While keeping the $\mathsf{PAPR}$
level low is favorable, it is of particular importance to have acceptable
signal spectral behavior and $\mathsf{BER}$, which are often referred to as
_waveform design_. In order to fulfill that, this work suggests an overall
communication network multi-objective optimization, such that the transmitter,
$\mathsf{HPA}$, channel, and the receiver, are represented as a single
optimization block. Instead of separately optimizing different components of
the transmitter and the receiver, an end-to-end convolutional-autoencoder
($\mathsf{CAE}$) learning model is proposed. This end-to-end optimization
block is presented as a constrained optimization problem where the transmitted
signal estimation is the objective, and the $\mathsf{PAPR}$ and
$\mathsf{ACPR}$ requirements are the constraints. $\mathsf{MIMO}$ detection
over multiple channel realizations is performed as a part of the end-to-end
joint optimization model, utilizing an iterative approach based on
convolutional layers, and a gradual loss learning approach. We evaluate the
performance of our algorithms over both additive white Gaussian noise
($\mathsf{AWGN}$) and 3rd Generation Partnership Project (3GPP) fading
channels [6]. By analyzing the BER, PAPR, and spectrum performance, we show
that the proposed end-to-end learning approach can integrate different
communication network blocks to balance those performance objectives
successfully. We show that the suggested scheme is able to achieve better
spectral performance for higher $\mathsf{HPA}$ efficiency operation. Various
$\mathsf{OFDM}$ $\mathsf{PAPR}$ reduction techniques have been proposed in the
literature, as well as for $\mathsf{MIMO}$ detection. Generally, these
techniques can be categorized into _model-driven_ and _data-driven_
techniques. The first category refers to standard approaches in classical
communications theory, while the second relies on recently developed
approaches based on $\mathsf{ML}$ techniques. The following subsections review
different earlier solutions for the above-mentioned problems.

###  I-A Classical Approaches (Model Driven) for MIMO Detection

Many $\mathsf{MIMO}$ detection algorithms have been developed over the years.
The maximum likelihood estimation ($\mathsf{MLE}$) solution is optimal for the
joint detection of transmitted symbols in a $\mathsf{MIMO}$ system. However,
its exponential computational and time complexity (due to exhaustive searches
over all possible transmitted signals) render it infeasible when the number of
transmitters and the modulation order are high. An example of suboptimal high
accuracy non-linear detection algorithms are those based on sphere decoding
(SD) [7], but they become computationally expensive as the number of antennas
grows. The general idea is based on a lattice search for a solution in an
iterative manner, and the accuracy/complexity ratio strongly depends on the
value chosen for the radius parameter. More advanced detectors include the
successive interference cancellation (SIC) based detectors [8] and the semi-
definite relaxation detectors [9].

###  I-B Machine-Learning-Based Schemes (Data Driven) for MIMO Detection

The motivation for DL-based detectors is to enhance the performance of
classical model-driven detection algorithms by learning, from the training
data set, an optimized mapping of the received signals onto the transmitted
symbols. In [10, 11], an iteration-based algorithm for implementing a receiver
for $\mathsf{MIMO}$ detection was suggested. One of the highlights of the
presented model’s framework is that it enables training through different
random communication channel realizations. In [12], a model-based algorithm
was suggested, where a classical SD algorithm was integrated with a neural
network ($\mathsf{NN}$) that was trained to optimize the selection of the
initial radius. In [13], a neural detector-based transformer architecture
implements a recurrent estimation scheme by learning an iterative decoding
algorithm.

In [14, 15, 16] an $\mathsf{AE}$ was offered to design a physical layer in
which DL-based CSI encoding was suggested for different scenarios to achieve
lower $\mathsf{BER}$ together with better robustness to the wireless channel
characteristics. Thorough surveys and analysis are presented in [17].

###  I-C Classical Approaches (Model Driven) for PAPR Reduction

$\mathsf{PAPR}$ reduction schemes are roughly classified into three
categories. The signal distortion category consists of techniques such as
clipping and filtering ($\mathsf{CF}$) [18, 19], which limit the peak envelope
of the input signal in the time domain to a predetermined value. The multiple
signaling probabilistic category includes methods such as selective mapping
($\mathsf{SLM}$) [19, 20], partial transmit sequence ($\mathsf{PTS}$) [20],
ton reservation and ton injection [21], and constellation shaping [22]. The
main principle of $\mathsf{SLM}$ is to generate different candidates for each
$\mathsf{OFDM}$ block by multiplying the symbols vector with a set of
different pseudo-random sequences and choosing the candidate with the lowest
$\mathsf{PAPR}$. The third category is the coding technique category [21, 23],
attempting to reduce the occurrence probability of the same phase signals.

Earlier schemes were mainly developed for single-antenna systems. Extended
works which applied the single-antenna $\mathsf{PAPR}$ reduction schemes on
each antenna of the $\mathsf{MIMO}$ configuration separately are found in,
e.g., [24], but those required considerable computations, cost, and
complexity. Model-driven approaches to simultaneously reduce $\mathsf{PAPR}$
over all antennas were also proposed. In [25], instead of applying
$\mathsf{SLM}$ to each antenna, the sequence with the highest $\mathsf{PAPR}$
over all transmit antennas was selected. Usually, $\mathsf{SLM}$ and
$\mathsf{PTS}$ methods demand side information (SI) to be sent to the receiver
along with each transmitted data block for retrieving the original data. The
need for SI requires extra bandwidth overhead, and the incorrect detection of
the SI bits over the channel will lead to significant degradation in the
$\mathsf{BER}$ performance of the receiver in the
$\mathsf{MIMO}$-$\mathsf{OFDM}$ system.

###  I-D Deep-Learning-Based Schemes (Data Driven) for PAPR Reduction

In recent years much research has been dedicated to applying $\mathsf{DL}$
techniques in the design and optimization of wireless communication networks,
e.g., [26, 4, 11]. Several papers have proposed $\mathsf{DL}$ methods to
handle $\mathsf{PAPR}$ reduction. For example, the authors of [27, 28], added
a $\mathsf{NN}$ to reduce the complexity of the active constellation scheme,
followed by $\mathsf{CF}$. In [29, 30] the authors present an $\mathsf{AE}$
solution for $\mathsf{PAPR}$ reduction, while minimizing the $\mathsf{BER}$
degradation. In [31] a $\mathsf{CAE}$ was suggested for the implementation of
an end-to-end $\mathsf{SISO-OFDM}$ communication network that simultaneously
reduces the $\mathsf{PAPR}$ and reconstructs the transmitted symbols, while
keeping acceptable spectral requirements. Another learning-based approach,
which considers the reduction of the $\mathsf{PAPR}$ and $\mathsf{ACPR}$
together with the maximization of the achievable information rate for a
single-carrier waveform above multipath channels, was proposed in [32]. The
authors in [33] proposed a deep $\mathsf{NN}$ combined with $\mathsf{SLM}$ to
mitigate the high $\mathsf{PAPR}$ issue of $\mathsf{OFDM}$ signal types.

All of the above papers consider a $\mathsf{SISO}$ network. A $\mathsf{PAPR}$
reduction scheme assisted by $\mathsf{DL}$ for a
$\mathsf{MIMO}$-$\mathsf{OFDM}$ system was suggested in [34]. The authors
apply selective tone reservation [35] on each antenna separately and then
apply unused beam reservation [36] on all antennas together. An
$\mathsf{ML}$-based method for approximating the optimal tabular
hyperparameters required for using selective tone reservation and unused beam
reservation was suggested.

###  I-E Main Contributions

Some of the aforementioned $\mathsf{PAPR}$ reduction approaches suffer from
in-band interference, out-of-band distortions, and high computational
complexity. Moreover, published $\mathsf{ML}$-based solutions mostly handle
single antenna scenarios. Those who deal with $\mathsf{PAPR}$ reduction for
$\mathsf{MIMO}$ systems use $\mathsf{ML}$ only for the $\mathsf{PAPR}$
reduction block and not for the end-to-end network implementation. This paper
aims to handle the $\mathsf{PAPR}$ problem in $\mathsf{MIMO}$ systems as an
integral part of a waveform design objective. In particular, we design a
communication system that simultaneously achieves $\mathsf{PAPR}$ reduction,
acceptable spectral behavior of the PA’s output, and good $\mathsf{BER}$
performance. The suggested end-to-end network aims to resolve the
$\mathsf{MIMO}$ detection problem as a part of the other mentioned objectives.
To the best of our knowledge, this approach is new. Novelties we introduce
include using a $\mathsf{CAE}$ combined with a gradual loss learning technique
to handle the multi-objective optimization of the network, and adding the
effect of the $\mathsf{HPA}$ on an integrated end-to-end $\mathsf{MIMO}$
communication system. We present an iterative $\mathsf{MIMO}$ detection
algorithm integrated into transmitter-receiver end-to-end communication system
joint optimization. We demonstrate our algorithm’s results on 5G
$\mathsf{MIMO}$-$\mathsf{OFDM}$ Matlab toolbox simulated data, and we compare
our method with classical methods for $\mathsf{PAPR}$ reduction and waveform
design, and show competitive results for all the objectives mentioned above.
The proposed algorithm offers performance improvement for future wireless
communication systems. We show that our model provides competitive
$\mathsf{PAPR}$ reduction, waveform design, and detection results.

The rest of this paper is structured as follows. In Section II, the problem is
defined and formulated separately for $\mathsf{MIMO}$ detection, and for
$\mathsf{PAPR}$ reduction as a part of the $\mathsf{MIMO}$-$\mathsf{OFDM}$
system. We then present the proposed DL-based system architecture for the
multi-objective optimization and explain the training procedure in Section
III. Section IV provides detailed numerical simulation results and insights.
Finally, Section V gives concluding remarks.

##  II Notation and Problem Definition

In this section, we introduce the notation and the problem definition.

###  II-A Notation

Throughout this paper, we use the following notations. The set of real numbers
is denoted by $\mathbb{R}$, while the set of complex numbers is denoted by
$\mathbb{C}$. Random variables will be denoted by capital letters, and their
realizations will be denoted by lower-case letters, e.g., $X$ and $x$,
respectively. Calligraphic letters denote sets, e.g., $\mathcal{X}$. We use
the notation $X^{n}$ to denote the random vector $(X_{1},X_{2},\dots,X_{n})$
and $x^{n}$ to denote the realization of such a random vector. The expectation
operator is denoted by $\mathbb{E}\left[\cdot\right]$. $(\cdot)^{*}$,
$(\cdot)^{{\dagger}}$ denote the conjugate, and pseudo-inverse operators,
respectively.

###  II-B Problem Definition

In this section, we describe mathematically each part of the integrated
problem of $\mathsf{MIMO}$ detection together with $\mathsf{PAPR}$ reduction
and spectrum constraints. First, we give a brief introduction to the end-to-
end setup used in our system.

####  II-B1 MIMO detection model

Let us assume a standard $\mathsf{MIMO}$-$\mathsf{OFDM}$ system with $N_{t}$
transmit antennas and $N_{r}$ receive antennas. Transmission is considered
over a memoryless complex-valued channel model, while assuming frequency
flatness and slow fading. A $\mathsf{MIMO}$ system can be modeled by the
following complex baseband model:

$\displaystyle\boldsymbol{y}=\boldsymbol{H}\boldsymbol{x}+\boldsymbol{n},\vspace{-0.5cm}$
(1)

where $\boldsymbol{x}\in\mathbb{C}^{N_{t}}$ is the transmitted complex symbol
vector drawn from a finite discrete constellation of size $|\mathcal{M}|$,
$\boldsymbol{H}\in\mathbb{C}^{N_{r}\times N_{t}}$ is the complex baseband
channel matrix that is related to a specific subcarrier,
$\boldsymbol{n}\in\mathbb{C}^{N_{r}}$ is complex background $\mathsf{AWGN}$
seen at the receiver, and $\boldsymbol{y}\in\mathbb{C}^{N_{r}}$ is the
received complex vector resulting from the propagation of the transmitted
symbols through the channel contaminated by $\mathsf{AWGN}$.

As the proposed implementation is based on a real-valued $\mathsf{NN}$ model
determined by the $\mathsf{DL}$ Pytorch library, (1) is expressed with real
values by splitting and concatenating each signal into its real and imaginary
parts:

$\displaystyle\boldsymbol{x}=\begin{bmatrix}Re\\{\boldsymbol{x}\\}\\\
Im\\{\boldsymbol{x}\\}\\\ \end{bmatrix},\
\boldsymbol{y}=\begin{bmatrix}Re\\{\boldsymbol{y}\\}\\\
Im\\{\boldsymbol{y}\\}\\\ \end{bmatrix},\
\boldsymbol{n}=\begin{bmatrix}Re\\{\boldsymbol{n}\\}\\\
Im\\{\boldsymbol{n}\\}\\\
\end{bmatrix},\boldsymbol{H}=\begin{bmatrix}Re\\{\boldsymbol{H}\\}&-Im\\{\boldsymbol{H}\\}\\\
Im\\{\boldsymbol{H}\\}&Re\\{\boldsymbol{H}\\}\\\ \end{bmatrix}.$ (2)

In the $\mathsf{MIMO}$ detection problem, the objective is to detect the
transmitted symbols, $\boldsymbol{x}$, given the received data
$\boldsymbol{y}$. The optimal solution for the $\mathsf{MIMO}$ detection of
the transmitted symbols problem defined above is given by the $\mathsf{MLE}$
algorithm, that is,

$\displaystyle\hat{\boldsymbol{x}}_{mle}=\operatorname*{arg\,min}_{\boldsymbol{x}\in\mathcal{X}^{N_{t}}}||\boldsymbol{y}-\boldsymbol{H}\boldsymbol{x}||^{2},$
(3)

where $\mathcal{X}$ denotes the set of possible transmitted symbols (i.e.,
signal constellation). The solution of (3) requires an exhaustive search over
all $|\mathcal{M}|^{N_{t}}$ possible transmitted vectors. Therefore, it is
infeasible for an actual implementation where large-scale $\mathsf{MIMO}$
setups and/or a large constellation are in use.

####  II-B2 PAPR problem in MIMO-OFDM

In an OFDM system with ${N}$ complex orthogonal subcarriers, the discrete-time
transmitted $\mathsf{OFDM}$ signal at the $n_{t}$ antenna, is given by the
inverse discrete Fourier transform ($\mathsf{IDFT}$):

$\displaystyle{x}_{n_{t},n}=\frac{1}{\sqrt{{N}}}\sum_{k=0}^{{N}-1}{X}_{n_{t},k}e^{j\frac{2\pi}{{L}{N}}kn},\;\;0\leq
n\leq{L}{N}-1,\;\;1\leq n_{t}\leq N_{t},$ (4)

where $\\{{X}_{n_{t},k}\\}_{k=0}^{{N}-1}$ are random input symbols per
antenna, modulated by a finite constellation, and ${L}\geq 1$ is the over-
sampling factor (${L}=1$ is the Nyquist sampling rate). As shown in [21],
oversampling by a factor of four results in a good approximation of the
continuous-time $\mathsf{PAPR}$ of complex $\mathsf{OFDM}$ signals. The
discussed problem considers non-linear $\mathsf{HPA}$s at each of the $N_{t}$
TX branches. We assume that the $\mathsf{HPA}$s in all branches have the same
non-linear characteristic, which is a reasonable assumption, considering
current wireless $\mathsf{MIMO}$ systems. Also, in a discrete implementation,
the same $\mathsf{HPA}$s are usually used.

The $\mathsf{PAPR}$ of the transmitted signal in (4) is defined as the ratio
between the maximum peak power and the average power of the $\mathsf{OFDM}$
signal. Specifically, the $\mathsf{PAPR}$ at the $n_{t}$-th transmit antenna
is defined by:

$\displaystyle\mathsf{PAPR_{n_{t}}}\triangleq\frac{\max_{0\leq
n\leq{LN}-1}|{x_{n_{t},n}}|^{2}}{\mathbb{E}|{x_{n_{t},n}}|^{2}}.$ (5)

For the entire $\mathsf{MIMO}$-$\mathsf{OFDM}$ system, the $\mathsf{PAPR}$
reduction method we use will consider the maximum $\mathsf{PAPR}$ among all
$N_{t}$ transmit antennas, as the same PA model is used in all branches:

$\displaystyle\mathsf{PAPR_{MIMO-OFDM}}=\max_{1\leq n_{t}\leq
N_{t}}{\mathsf{PAPR_{n_{t}}}}.$ (6)

As $\mathsf{HPA}$ non-linearity causes spectral regrowth, an important
assessment for the spectral purity of the system is the $\mathsf{ACPR}$
criterion, which is the ratio between the power of the adjacent channel and
the power of the main channel. Following [6], we define it as

$\displaystyle\mathsf{ACPR}\triangleq\frac{\max{\left(\int_{\mathsf{BW}/2}^{3\mathsf{BW}/2}\mathsf{P_{ss}}(f)\;\mathrm{d}f,\int_{-3\mathsf{BW}/2}^{\mathsf{BW}/2}\mathsf{P_{ss}}(f)\;\mathrm{d}f\right)}}{\int_{-\mathsf{BW}/2}^{\mathsf{BW}/2}\mathsf{P_{ss}}(f)\;\mathrm{d}f},$
(7)

where $\mathsf{P_{ss}}(\cdot)$ is the power spectral density ($\mathsf{PSD}$)
of the signal at the $\mathsf{HPA}$’s output, and $\mathsf{BW}$ is the primary
channel bandwidth, which is assumed to be equal to the data signal bandwidth.

A block diagram of the communication system model is shown in Fig. 1.

Figure 1: General system model diagram.

Specifically, the encoder and filter blocks mitigate the $\mathsf{PAPR}$
effect and design the waveform to comply with predefined spectral mask
requirements. For example, the encoder block can model a clipping operation,
while the filter can be a standard band-pass filter ($\mathsf{BPF}$). The
filtered signal ${x}_{n}^{\mathsf{F}}$ is amplified by a non-linear
$\mathsf{HPA}$. The amplified signal,
${x}_{n}^{\mathsf{P}}=\mathrm{G}({x}_{n}^{\mathsf{F}})$, is transmitted
through a fading channel with $\mathsf{AWGN}$. The channel decoder receives
the noisy signal and attempts to reconstruct and detect the transmitted
signal. For model-driven approaches, a classical detection algorithm, e.g.,
$\mathsf{MLE}$, is applied for detecting the estimated symbol denoted by
$\hat{{X}}_{k}$.

The role of the $\mathsf{HPA}$ is to convert the low-level transmission signal
to a high power signal, capable of driving the antenna at the desired power
level. The $\mathsf{HPA}$ has to operate close to its saturation region for
maximal power efficiency. If the $\mathsf{HPA}$ exceeds the saturation point
and enters the non-linear area of operation, the output signal becomes non-
linear. Accordingly, to operate the amplifier only in the linear region, we
need to make sure that the amplifier operates at a power level that is lower
than the saturation point. This is achieved by down-scaling the input signal
by an input back-off ($\mathsf{IBO}$) factor. The drawback of adding the
$\mathsf{IBO}$ attenuation is that the output power decreases, which makes the
$\mathsf{HPA}$ power-inefficient.

Figure 2: RAPP HPA output versus input signal for different smoothness $p$
values.

There are several commonly used models for the non-linearity of an
$\mathsf{HPA}$. Here, we will focus on the RAPP behavioral amplifier model
[37], which is very accurate for solid-state-power amplifiers. The model’s
AM/AM conversion is given by

$\displaystyle\mathsf{G}({A_{\mathsf{in}}})=v\cdot{A_{\mathsf{in}}}\cdot\left(1+\left(\frac{v{A}_{\mathsf{in}}}{{A}_{0}}\right)^{2p}\right)^{-\frac{1}{2p}\vspace{-0.9cm}},$
(8)

where ${A_{in}}$ is the input amplitude, ${A}_{0}$ is the limiting output
amplitude, $v$ is the small signal gain, ${p}$ is a smoothness parameter
controlling the transition from the linear region to the saturation region,
and $\mathsf{G}({A})$ is the output amplitude. Figure 2 shows RAPP
$\mathsf{HPA}$ outputs versus input for several smoothing factor values.

##  III Proposed Waveform Design Structure

In this section, we describe our multi-objective optimization $\mathsf{CAE}$
model architecture. Motivated by research evidence of powerful learning
ability, under the same conditions of the $\mathsf{MIMO}$-$\mathsf{OFDM}$
examined structure, it is expected that the proposed $\mathsf{CAE}$ model will
achieve good enough performance to be compared with classical $\mathsf{PAPR}$
reduction methods combined with the $\mathsf{MLE}$ detector. We first briefly
discuss the general $\mathsf{CAE}$ concept. Then, we describe our algorithm
building blocks, and the joint $\mathsf{PAPR}$ reduction, spectral design, and
detection operation in detail. The motivation and structure of the iteration-
based decoder with regard to handling the $\mathsf{MIMO}$ detection problem as
a part of the $\mathsf{CAE}$ network will be explained. The proposed
architecture in Fig. 3 is then elaborated, including the Bussgang’s non-
linearity compensation theorem, followed by a description of the gradual
learning process. Last, the training procedure of the $\mathsf{CAE}$ multi-
objective optimization network operation will be described.

(a) Conv-AE overall scheme.

(b) PAPR reduction block - Encoder scheme $f(x)$.

(c) Detection block - Decoder scheme $g(x)$.

Figure 3: Structure of the proposed conv-AE.

Fig. 3(a) shows the overall end-to-end communication network structure
implemented by a $\mathsf{CAE}$ model, where the encoder and the decoder are
the trainable blocks. Each mentioned loss component is calculated using the
operations appearing in the red blocks in the figure. It can be noticed that
there are two $\mathsf{PAPR}$ calculation-based loss components, as will be
detailed later, that helped achieve improved spectral behavior and
$\mathsf{BER}$ results. Fig. 3(b) describes the encoder structure. It is
constructed of 2D convolutional layers to be compatible with the input example
dimensions, a fully connected layer, and a power normalization layer at the
output. We also added a skip connection to improve stability and performance.
Fig. 3(c) shows the decoder structure. It is also implemented with 2D
convolutional layers. To be able to optimize the network under different
communication channel realizations, we used an iterative-based solution to the
MIMO detection problem. We emphasize that the presented $\mathsf{CAE}$ model,
once trained on a single training set, presents competitive results for the
discussed multi-objective optimization problem in the presence of a wide range
of noise power levels, without any knowledge of the SNR level.

###  III-A Convolutional Autoencoder (CAE)

The proposed implementation uses an $\mathsf{AE}$ learning system based on a
convolutional neural network ($\mathsf{CNN}$). The general structure of an
$\mathsf{AE}$ consists of two main blocks: the encoder $f(\boldsymbol{x})$ and
the decoder $g(\boldsymbol{x})$, where $\boldsymbol{x}$ is the input data. The
$\mathsf{AE}$ is trained to minimize a certain joint loss function, which we
denote by $\mathcal{L}(\boldsymbol{x},g(f(\boldsymbol{x})))$. An end-to-end
communication system can be interpreted as an $\mathsf{AE}$ in which the
encoder and the decoder are part of the transmitter and the receiver,
respectively, and can jointly optimize them through an end-to-end learning
procedure.

$\mathsf{CNN}$s are widely used for feature extraction and pattern recognition
in $\mathsf{ML}$ models. Compared with a fully connected ($\mathsf{FC}$)
network, a CNN has significantly fewer connections between adjacent layers,
and thus fewer parameters and weights to train, resulting in lower complexity
and much faster training.

###  III-B Iteration-Based Model Motivation

An iterative approach based on convolutional layers, was adopted to implement
the decoder. The $\mathsf{MIM0}$ detector was designed to solve the
$\mathsf{MLE}$ optimization (3) using the projected gradient descent approach,
where the input to the detector is a linear combination of features instead of
the channel output directly. Utilizing such detectors significantly improved
the detection results when various channel realizations were integrated. The
detector input feature selection was based on the following projected gradient
decent approach used to optimize (9):

$\displaystyle\vspace{-0.4cm}||\boldsymbol{y}-H\boldsymbol{x}||^{2}.\vspace{-0.7cm}$
(9)

As shown in [10, 11], such iterations are updated by

$\displaystyle\hat{\boldsymbol{x}}_{k+1}$
$\displaystyle=P_{c}\left[\hat{\boldsymbol{x}}_{k}-\delta_{k}\frac{\partial||\boldsymbol{y}-\boldsymbol{H}\boldsymbol{x}||^{2}}{\partial\boldsymbol{x}}\biggr{\rvert}_{\mathbf{\boldsymbol{x}=\boldsymbol{\hat{x}}_{k}}}\right]=P_{c}\left[\hat{\boldsymbol{x}}_{k}-\delta_{k}\left(\boldsymbol{H}^{H}\boldsymbol{y}-\boldsymbol{H}^{H}\boldsymbol{H}\hat{\boldsymbol{x}}_{k}\right)\right],$
(10)

where $\hat{\boldsymbol{x}}_{k}$ is the objective estimation in the $k$’th
iteration, $\boldsymbol{H}^{H}$ indicates the conjugate transpose operation
over the channel matrix $\boldsymbol{H}$, $\delta_{k}$ is the step size, and
$P_{c}$ is the non-linear projection operator. The above can be adapted to a
deep $\mathsf{NN}$ iterative solution by the following linear combination
form:

$\displaystyle\vspace{-0.2cm}\hat{\boldsymbol{x}}_{k+1}=P_{c}(\hat{\boldsymbol{x}}_{k}+\delta_{1k}\boldsymbol{H}^{H}\boldsymbol{y}+\delta_{2k}\boldsymbol{H}^{H}\boldsymbol{H}\hat{\boldsymbol{x}}_{k}),\vspace{-0.2cm}$
(11)

where $\delta_{1k}$ and $\delta_{2k}$ are learned hyper-parameters to be
optimized. One of the motivations for using the described iterative decoder
approach was a former work published in [10, 11]. In our work, it is designed
as part of the joint encoder-decoder multi-task optimization. Also, it was
implemented with convolutional layers, that enabled better computational and
performance capabilities for the joint optimization end-to-end system, and the
per subcarrier/antenna alternating analysis. A softmax layer is added at the
output of the $\mathsf{CAE}$ to generate probabilistic outputs. In the
following section, we provide the complete detection procedure.

###  III-C Proposed CAE Architecture

This section introduces the suggested $\mathsf{CAE}$ learning network
implementation of the $\mathsf{MIMO}$-$\mathsf{OFDM}$ system for the multi-
objective optimization task. We consider a $\mathsf{MIMO}$-$\mathsf{OFDM}$
scheme with $\mathsf{N_{t}}$ transmit antennas and $\mathsf{N_{r}}$ receive
antennas, where the $\mathsf{OFDM}$ is of order $\mathsf{K}$. The input is
represented by a matrix in the frequency domain, i.e.

$\displaystyle\underline{\mathbf{X}}=\begin{pmatrix}X^{(1)}(1)&X^{(1)}(2)&\cdots&X^{(1)}(K)\\\
X^{(2)}(1)&X^{(2)}(2)&\cdots&X^{(2)}(K)\\\ \vdots&\vdots&\ddots&\vdots\\\
X^{(N_{t})}(1)&X^{(N_{t})}(2)&\cdots&X^{(N_{t})}(K)\end{pmatrix},$ (12)

where, for any $\mathsf{n_{t}}\in{[1,N_{t}]}$ and $k\in{[1,K]}$,
$X^{(n_{t})}(k)$ is a M-QAM constellation complex-valued symbol.

In Fig. 3(a), we illustrate the general structure of the end-to-end
communication network implemented by the CAE configuration. Specifically, we
consider a transmitter that takes the two-dimensional matrix
$\underline{\mathbf{X}}$ as an input. The transmitter’s output goes through a
$\mathsf{MIMO}$ channel, together with $\mathsf{AWGN}$. Finally, the noisy
channel outputs are fed into a receiver to estimate $\underline{\mathbf{X}}$.
The operations within the transmitter and the receiver are described below.

  * •

Transmitter: the input signal $\underline{\mathbf{X}}$ is zero-padded on the
subcarriers’ dimension and converted to the time domain via an $\mathsf{IFFT}$
applied on each of the transmitter branches, outputting
$\\{{x}_{n}\\}_{n=0}^{{{LN}-1}}$. These symbols serve as the input to the
encoder, which acts as a $\mathsf{PAPR}$ reduction block, followed by a
$\mathsf{BPF}$ filter for optimizing the spectral behavior by reducing the
out-of-band radiation. Its frequency response is a rectangular window with the
same bandwidth as $X^{(n_{t})}_{k}$. Then, a predefined $\mathsf{IBO}$ is
applied just before the signal is amplified by the $\mathsf{HPA}$.

  * •

Receiver: the distorted $\mathsf{OFDM}$ symbols are divided by an $\alpha$
factor to compensate for the non-linear distortions, as will be detailed in
the following. Finally, the proposed $\mathsf{CAE}$ decoder reconstructs and
detects the estimated $\mathsf{MIMO}$-$\mathsf{OFDM}$ transmitted signals.

The encoder comprises three convolutional layers, and the decoder is composed
of iterative construction of convolutional layers. Each convolutional layer is
followed by a non-linear activation function and batch normalization [38], and
then a fully connected layer. In addition, a residual connection is added to
the encoder block, which sums (element-wise) the input to the second
convolutional layer and the output of the third convolutional layer. It turns
out that this modification improves the overall performance of the suggested
scheme significantly. The intuition is that adding another path for data to
reach the latter parts of the $\mathsf{NN}$ makes it easier to optimize the
mapping [39]. Furthermore, the encoder has a power normalization layer, which
ensures that the transmitted signal meets the power constraints of unit
average energy per $\mathsf{OFDM}$ symbol. This way, the intended SNR is
maintained. We tested several activation functions, including sigmoid,
rectified linear unit ($\mathsf{RELU}$), Gaussian error linear unit
($\mathsf{GELU}$), and scaled exponential linear unit ($\mathsf{SELU}$) [40].
Empirically, it was found that $\mathsf{SELU}$ activation provides the best
results for our $\mathsf{CAE}$ scheme.

As illustrated in Fig. 3(a) and Fig. 3(b), since the encoder is responsible
for the $\mathsf{PAPR}$ reduction, which is calculated per $\mathsf{OFDM}$
symbol, we start with per antenna analysis, where each antenna is treated
separately. A one-dimensional kernel per TX branch handles this. The encoder
architecture can be described by the following:

$\displaystyle f(\boldsymbol{x})$
$\displaystyle=\rho_{L_{f}}\left(\biggr{\rvert}\boldsymbol{W}^{f}_{L_{f}}\left(\rho_{L_{f}-1}\left(...\left(\rho_{1}\left(\biggr{\rvert}\boldsymbol{W}^{f}_{1}\boldsymbol{x}+\boldsymbol{b}^{f}_{1}\biggr{\rvert}_{bnorm}\right)\right)\right)...\right)\right.$
$\displaystyle\left.\ \ \ \ \ \ \ \ \
+\boldsymbol{b}^{f}_{L_{f}}\biggr{\rvert}_{bnorm}+\left(\biggr{\rvert}\boldsymbol{W}^{f}_{1}\boldsymbol{x}+\boldsymbol{b}^{f}_{1}\biggr{\rvert}_{bnorm}\right)\right),$
(13)

where $L_{f}$ is the number of the encoder’s convolutional layers,
$\boldsymbol{W}_{i}^{f}$, and $\boldsymbol{b}_{i}^{f}$ are the encoder’s
weight matrix and bias vector, respectively, for the $i$’th layer, with size
determined as a part of the network design. $\rho_{i}(\cdot)$ is the
activation function of the $i$’th layer, and $bnorm$ means the layer passes
through a batch normalization.

The next part of this process applies the non-linear $\mathsf{HPA}$s on each
TX branch of the transmitter time domain signals, each composed of all
subcarriers. The signal is then converted via $\mathsf{FFT}$ to the frequency
domain, and the zero-unpadding block removes the out-of-band samples.
Afterward, frequency domain analysis is performed on each subcarrier
transmitted through all $N_{t}$ antennas. Each subcarrier is transmitted
through its related complex baseband channel described by a $\left(N_{r}\times
N_{t}\right)$ matrix, and the $\mathsf{AWGN}$ is added as well.

To continue with the per subcarrier analysis on the receiver side, we need to
overcome the non-linearity of the $\mathsf{HPA}$. To that end, we compensate
the receiver input signal by applying an attenuation factor represented by
$\alpha$. Bussgang’s decomposition theorem [41] states that if a zero-mean
Gaussian signal passes through a memoryless non-linear device, then the
output-input cross-correlation function is proportional to the input
autocovariance. Accordingly, the value of $\alpha$ is chosen to minimize the
variance of the non-linear signal distortions, such that it is attempted that
the transmitted signal in each transmitter branch is linearly separated, and
thus represented as a sum of the signal and distortion. It can be shown that

$\displaystyle\mathsf{\alpha}=\frac{\mathbb{E}\left({x_{n}^{\mathsf{F}}}{\overline{x}_{n}^{\mathsf{P}}}\right)}{\mathbb{E}\left(|{x_{n}^{\mathsf{F}}}|^{2}\right)},$
(14)

where ${{x}_{n}^{\mathsf{P}}}$ is the complex output signal of the PA, and
${\overline{x}_{n}^{\mathsf{P}}}$ is its complex conjugate. By assuming that
the $\mathsf{PSD}$ of the in-band distortion is approximately flat [42] in the
frequency domain, the output signal of the $\mathsf{HPA}$ on the $k$-th
subcarrier can then be expressed as

$\displaystyle X^{\mathsf{P}}(k)=\alpha(k)X^{\mathsf{F}}(k)+D(k),$ (15)

where $D(k)$ is the non-linear distortion on the $k$-th subcarrier. The same
model is assumed for all $\mathsf{PA}$s; therefore, it can be concluded that
$\mathsf{\alpha_{n_{t}}}=\mathsf{\alpha}$.

At the $\mathsf{MIMO}$ decoder, Fig. 3(c), the per subcarrier analysis is
continued, meaning that different subcarriers of the same RX branch will not
be mixed. To generalize our end-to-end structure, we modified the 1D kernel to
a 2D kernel at the decoder part. It also helped better reconstruct the signal
after the encoder layers. As explained in the previous sub-section, we use an
iterative procedure to implement the decoder, which is designed for signal
reconstruction and detection. A general mathematical description of one
iteration, $k$, of the presented decoder is given by

$\displaystyle\begin{split}\boldsymbol{d}_{k}&=\begin{pmatrix}\boldsymbol{\hat{x}}_{k-1},\delta_{1k}\boldsymbol{H}^{H}\boldsymbol{y},\delta_{2k}\boldsymbol{H}^{H}\boldsymbol{H}\hat{\boldsymbol{x}}_{k-1}\end{pmatrix}\\\
\boldsymbol{\hat{x}}_{k}&=g(\boldsymbol{d}_{k})\\\
&=\rho_{L_{g},k}\left(\biggr{\rvert}\boldsymbol{W}^{g}_{L_{g},k}\left(\rho_{L_{g}-1,k}\left(...\left(\rho_{1,k}\left(\biggr{\rvert}\boldsymbol{W}^{g}_{1,k}\boldsymbol{d}_{k}^{T}+\boldsymbol{b}^{g}_{1,k}\biggr{\rvert}_{bnorm}\right)\right)\right)...\right)+\boldsymbol{b}^{g}_{L_{g},k}\biggr{\rvert}_{bnorm}\right),\end{split}$

where $L_{g}$, $\boldsymbol{W}_{i}^{g}$, and $\boldsymbol{b}_{i}^{g}$, have
the same definitions as described for the encoder’s block, only that these
apply for the decoder. The decoder’s input features vector,
$\boldsymbol{d}_{k}$, was initialized by randomizing a prediction
$\hat{\boldsymbol{x}}_{0}$. Initialization by zeros resulted in performance
degradation.

###  III-D Training of the CAE Network

We train a single $\mathsf{CAE}$ model for all tested $\mathsf{SNR}$ values.
We use the AdamW optimizer [43] that runs back-propagation to optimize the
model during training. This optimizer is designed to improve gradients when
$\text{L}_{2}$ regularization is used. Our loss function is set to solve the
constrained optimization problem by handling three objectives: accurate signal
reconstruction (minimal $\mathsf{BER}$), minimal $\mathsf{PAPR}$, and
acceptable $\mathsf{ACPR}$.

We solve this constrained optimization problem by recasting it as an
unconstrained problem by constructing the Lagrangian function and augmenting
the objective function with a quadratic penalty term [44]. The augmented
Lagrangian ($\mathsf{AL}$) combines the Lagrangian formulation with a weighted
quadratic penalty function. The general $\mathsf{AL}$ for an inequality-
constrained problem can be described by

$\displaystyle\vspace{-0.4cm}\mathcal{F}_{\rho^{k}}(\mathbf{x},\lambda^{k})=f(\mathbf{x})+\lambda_{1}^{k}c_{1}(\mathbf{x})+\frac{1}{2}\rho_{1}^{k}\left\|c_{1}(\mathbf{x})\right\|_{2}^{2}+\frac{1}{2\rho_{2}^{k}}\left\\{\left[\max\\{0,\lambda_{2}^{k}+\rho_{2}^{k}c_{2}(\mathbf{x})\\}\right]^{2}-(\lambda_{2}^{k})^{2}\right\\},$
(16)

where $f$ denotes the objective function,
$\rho^{k}\triangleq(\rho_{1}^{k},\rho_{2}^{k})$ are positive penalty
parameters, $\lambda^{k}\triangleq(\lambda_{1}^{k},\lambda_{2}^{k})$ are the
Lagrangian multipliers, the $c_{1}$-involved expressions handle the equality
constraint, and $c_{2}$ is for the inequality constraint. Equation (16)
considers the elimination of a slack variable $s\geq 0$ that was introduced in
the representation of the inequality constraint to transform it into a relaxed
equality constraint. As suggested in [45], the minimizer
$s=\max\\{0,c_{2}(x)-\lambda_{2}\frac{1}{\rho_{2}}\\}$ was used. $k$ is the
iteration number for updating the Lagrangian multipliers and penalty term,
according to the following rule derived by the dual ascent method [44],

$\displaystyle x^{k+1}$
$\displaystyle:=\operatorname*{arg\,min}_{x}\mathcal{F}_{\rho^{k}}(\mathbf{x},\lambda^{k})\vspace{-0.4cm}$
(17) $\displaystyle\lambda_{1}^{k+1}$
$\displaystyle:=\lambda_{1}^{k}+\rho_{1}^{k}c_{1}(\mathbf{x}^{k+1})\vspace{-0.2cm}$
(18) $\displaystyle\lambda_{2}^{k+1}$
$\displaystyle:=\max\\{0,\lambda_{2}^{k}+\rho_{2}^{k}c_{2}(\mathbf{x}^{k+1})\\}.\vspace{-0.2cm}$
(19)

We saw better convergence and more stable results for different BO values by
adding the quadratic penalty function and adaptively updating the multipliers
instead of keeping them constant. Since adaptive penalty parameter update was
not beneficial for the examined cases, it was added as a fixed hyperparameter.

Following the above-described general inequality constraint optimization
problem, the formulation of our loss function based on the appropriate
objective and constraints, represented by four loss components
$\mathcal{L}_{1}$, $\mathcal{L}_{2a}$, $\mathcal{L}_{2b}$, and
$\mathcal{L}_{3}$, is

$\displaystyle\vspace{-0.3cm}\mathcal{L}(\mathbb{x},\hat{\mathbb{x}},\lambda_{2a}^{k},\lambda_{2b}^{k},\lambda_{3}^{k})$
$\displaystyle=\mathcal{L}_{1}(\mathbb{x},\hat{\mathbb{x}})+\lambda_{2a}^{k}\mathcal{L}_{2a}(\mathbb{x})+\frac{\rho_{2a}}{2}\left\|\mathcal{L}_{2a}(\mathbb{x})\right\|_{2}^{2}$
$\displaystyle+\lambda_{2b}^{k}\mathcal{L}_{2b}(\mathbb{x})+\frac{\rho_{2b}}{2}\left\|\mathcal{L}_{2b}(\mathbb{x})\right\|_{2}^{2}+\frac{1}{2\rho_{3}}\left\\{\left[\max\\{0,\lambda_{3}^{k}+\rho_{3}\mathcal{L}_{3}(\mathbb{x})\\}\right]^{2}-(\lambda_{3}^{k})^{2}\right\\},$
(20)

where $\lambda_{2a}$, $\lambda_{2b}$, $\lambda_{3}$, $\rho_{2a}$, $\rho_{2b}$,
and $\rho_{3}$ are the appropriate Lagrange multipliers and penalty
parameters, accordingly. These are considered hyper-parameters, which balance
the contribution of each loss component to the joint loss function. We start
with a moderate value of $\lambda_{2a}$, $\lambda_{2b}$ and $\lambda_{3}$ and
then iterate for a better value according to the resulting $\mathsf{PAPR}$
loss of each iteration and some predetermined $\mathsf{PAPR}$ threshold value.
Better performance was observed for relatively small $\lambda_{2b}$ values,
with very low $\rho_{2b}$, meaning that $\lambda_{2b}$ was kept almost
constant during training.

The loss function we use for optimizing the signal reconstruction and
detection is the sum of negative log loss function of the predicted output
probability of the real and imaginary parts of each symbol, with
$\text{L}_{2}$ regularization to reduce over-fitting. Denoting by $x$ the
input sample (which is also the output target), $\hat{x}$ as the estimated
signal, $\Theta$ as the model’s weights, and $\lambda_{1}$ as a hyperparameter
for tuning the $\text{L}_{2}$ regularization, the loss function for each
$\mathsf{OFDM}$ $\mathsf{MIMO}$ sample is given by,

$\displaystyle\mathcal{L}_{1}(\mathbb{x},\hat{\mathbb{x}})$
$\displaystyle=-\Bigg{[}\sum_{j=1}^{N_{t}}\sum_{s=1}^{N_{sc}}\sum_{q=1}^{N_{c}}1\Big{\\{}Re\\{x^{j}\\}=l_{q}\Big{\\}}\log
P_{\theta}\Big{(}Re\left\\{\hat{x}^{j}\right\\}=l_{q}\Big{)}$
$\displaystyle+\sum_{j=1}^{N_{t}}\sum_{s=1}^{N_{sc}}\sum_{q=1}^{N_{c}}1\Big{\\{}Im\\{x^{j}\\}=l_{q}\Big{\\}}\log
P_{\theta}\Big{(}Im\left\\{\hat{x}^{j}\right\\}=l_{q}\Big{)}\Bigg{]}+\lambda_{1}\left\|\Theta\right\|_{2}^{2},$
(21)

where $N_{c}=\sqrt{|\mathcal{M}|}$ denotes the number of the real value
possibilities, $l_{q}$, of each of the real and imaginary parts of the
transmitted modulated symbol.

The $\mathsf{PAPR}$ minimization part is handled with two loss components,
where one, $\mathcal{L}_{2a}$, is calculated according to the $\mathsf{BPF}$
input, ${{x}_{n}^{\mathsf{E}}}$, and the other one, $\mathcal{L}_{2b}$,
according to the $\mathsf{BPF}$ output, ${{x}_{n}^{\mathsf{F}}}$ (cf. Fig.
3(a)). These are our equality constraints, defined by

$\displaystyle\mathcal{L}_{2a}(x)$
$\displaystyle=\mathsf{PAPR}\\{{{x}_{n}^{\mathsf{E}}}\\},$ (22)
$\displaystyle\mathcal{L}_{2b}(x)$
$\displaystyle=\mathsf{PAPR}\\{{{x}_{n}^{\mathsf{F}}}\\}.$ (23)

Other than the role of $\mathsf{PAPR}$ minimization handled by either of the
components, $\mathcal{L}_{2a}(x)$, significantly improved the $\mathsf{BER}$
result together with the $\mathsf{ACPR}$, while $\mathcal{L}_{2b}(x)$, enabled
us to control and obtain better spectral performance, meaning lower
$\mathsf{ACPR}$ with lower output back-off ($\mathsf{OBO}$) values. The
$\mathsf{ACPR}$ loss component is given by

$\displaystyle\mathcal{L}_{3}(x)$
$\displaystyle=\mathsf{ACPR}\\{{{x}_{n}^{\mathsf{P}}}\\}-\mathsf{ACPR_{req}},$
(24)

where ${{x}_{n}^{\mathsf{P}}}$ is the PA’s output, and $\mathsf{ACPR_{req}}$
is the required $\mathsf{ACPR}$ value, which is usually dictated by a
standard. $\mathsf{ACPR_{req}}$ was set according to the 5G standard
requirements for high spectral purity:
$\mathsf{ACPR_{req}}\leq{-45}\mathrm{dB}$ [6]; thus, $\mathcal{L}_{3}(x)$
defines our inequality constraint.

We have applied a gradual loss learning technique. In the first stage, the
loss function consisted only of $\mathcal{L}_{1}$ and optimized only the
reconstruction loss. Then, after a predetermined number of epochs, the loss
function defined in (III-D) was used to reduce the $\mathsf{PAPR}$ and improve
the spectral behavior. The gradual loss learning enables better control and
stability in tuning the different criteria’ trade-offs.

##  IV Results and Insights

###  IV-A Data Generation and Experimental Setup

To train and test the proposed data-driven model, the MATLAB® 5G Toolbox™ [46]
was used. This toolbox provides 5G radio-standard-compliant functions to
generate accurate data for $\mathsf{MIMO}$-$\mathsf{OFDM}$ transmission,
according to specified constellation sizes and examined $\mathsf{MIMO}$
setups. $\mathsf{MIMO}$-$\mathsf{OFDM}$ transmissions over fading channels
were simulated, where TDL-D type channels - a 13 delay tap channel with a 30ns
delay spread, as described in the 3gpp specification document [6], were used
for our implemented algorithm.

We consider a $\mathsf{MIMO}$-$\mathsf{OFDM}$ system with $K=$72 subcarriers
over 14 $\mathsf{OFDM}$ symbols per frame. 4375 batches of 32 $\mathsf{MIMO}$
samples each were used for a single training set, where the input and output
of the $\mathsf{CAE}$ sample shape is
$[2/4-antennas,{(72-subcarriers)}\times{(2-complex-
parts)}\times{(4-oversampling)}]$. An oversampling factor $L=4$, and
smoothness factor $p=2$ were considered. We trained three identical
$\mathsf{CAE}$ models on the following setups:

  1. 1.

QPSK modulation scheme with a $2\times 2$ $\mathsf{MIMO}$ setting, with 3GPP
multipath channel.

  2. 2.

16-QAM modulation scheme with a $4\times 4$ $\mathsf{MIMO}$ setting, with 3GPP
multipath channel.

  3. 3.

16-QAM modulation scheme with a $4\times 4$ $\mathsf{MIMO}$ setting, with AWGN
channel.

To provide an unbiased performance evaluation of the final training model, the
training and test data sets were generated independently, i.e. $\mathsf{OFDM}$
symbols, channel realization, and noise were randomized independently. In the
following, we give numerical performance results of our multi-objective
$\mathsf{CAE}$ model compared to a classical $\mathsf{CF}$ algorithm with a
clipping ratio of 4.08 dB, and to $\mathsf{SLM}$ with $U=64$ phase sequences,
with $\mathsf{MLE}$ added for $\mathsf{MIMO}$ detection. The inference part
was performed on 7000 $\mathsf{MIMO}$ samples for each SNR point.

###  IV-B Training Setup

As a part of the experimental analysis, we performed an extensive exploration
of different model structures and hyper-parameters, including the number of
layers, kernel sizes, number of convolutional layer channels, regularization,
dropout, number of decoder iterations, batch-normalization, learning rate,
$\mathsf{AL}$ parameters, trained SNR value, and epoch number. We found that
the best performance versus complexity on both examined $\mathsf{MIMO}$ setups
was achieved for the same model structure, only with different training data
sets. As the constellation, the number of subcarriers per OFDM symbols, and
the number of antennas are higher, the training is longer, and it is harder to
achieve the desired results.

TABLE I: CAE Proposed Structure

| Transmitter | Receiver
---|---|---
Parameter | Value | Kernel | Ch-in | Ch-out | Value | Kernel | Ch-in | Ch-out
Input size | $4\times 720$ | - | - | - | $12\times 144$ | - | - |
Conv (SELU) | - | $1\times 3$ | 1 | 21 | - | $3\times 3$ | 1 | 15
Conv (SELU) | - | $1\times 3$ | 21 | 15 | - | $3\times 3$ | 15 | 21
Conv (SELU) | - | $1\times 3$ | 15 | 21 | - | - | - | -
FC (Linear) output size | $4\times 720$ | - | - | - | $12\times 144$ | - | - | -
Decoder iterations | - | - | - | - | 10 | - | - | -

Conv padding | LR | Epochs num | Grad start | SNR train |  $\lambda_{2a}^{(0)}$, $\lambda_{2b}^{(0)}$, $\lambda_{3}^{(0)}$ |  $\rho_{2a}$, $\rho_{2b}$, $\rho_{3}$
---|---|---|---|---|---|---
2 | 0.001 | 140 | 45 | 40 dB | 0.015, 0.001, 0.005 | 0.0015, 0.00001, 0.001

The proposed $\mathsf{CAE}$ structure for the above $4\times 4$
$\mathsf{MIMO}$ system is described in Table I, where $\lambda_{2a}^{(0)}$,
$\lambda_{2b}^{(0)}$, and $\lambda_{3}^{(0)}$ are the values of the first
iteration when the $\mathsf{AL}$ epochs start, and ’Grad start’ indicates the
number of initial epochs where only the reconstruction loss is counted,
optimizing the unconstrained problem. ’LR’ indicates the learning rate.
Training on any of the data sets with the same best SNR value used for noise
generation, ’SNR train’, showed the top overall inference performance for any
tested SNRs. Adding dropout had no benefit in all examined setups.

###  IV-C BER Analysis

The calculation of $\mathsf{BER}$ versus Peak Signal to Noise Ratio
($\mathsf{P\\_SNR}$) is used here as a key parameter to measure the
reconstruction and detection of the transmitted signal. Considering a
normalized channel, i.e. $\mathbb{E}\left\|\mathbb{H}\right\|^{2}=1$, the
$\mathsf{P\\_SNR}$ is defined as the ratio between the $\mathsf{MIMO}$ system
maximal emitted energy, $P_{T}$, and the noise power, $\sigma_{w}^{2}$, such
that

$\displaystyle\vspace{-0.4cm}\mathsf{P\\_SNR}=\frac{P_{T}}{\sigma_{w}^{2}}.\vspace{-0.4cm}$
(25)

As shown in Fig. 4, the $\mathsf{CAE}$ has competitive $\mathsf{BER}$ vs.
$\mathsf{P\\_SNR}$ performance compared to the other standard examined methods
in most of the $\mathsf{P\\_SNR}$ range, where a visible gain is achieved at
the higher part. As the NN does not assume any specific physical model, it has
better robustness to distortions. That is, the $\mathsf{MIMO}$-$\mathsf{OFDM}$
signal reconstruction and detection of $\mathsf{HPA}$-distorted data as a part
of the multi-objective optimization proposed by our end-to-end $\mathsf{DL}$
scheme has the benefit over the common algorithms.

(a) 16-QAM, 4x4 MIMO with 3GPP multipath channel

(b) 16-QAM, 4x4 MIMO with AWGN channel

(c) 4-QAM, 2x2 MIMO with 3GPP multipath channel

Figure 4: BER vs. $\mathsf{P\\_SNR}$ of the considered methods and setups.

###  IV-D CCDF for PAPR Comparison

To demonstrate the $\mathsf{PAPR}$ performance of different methods,
complementary cumulative distribution function (CCDF) curves are presented in
Fig. 5. The CCDF of the $\mathsf{PAPR}$ denotes the probability that the
$\mathsf{PAPR}$ exceeds a certain threshold, i.e.
$\mathbb{P}(\mathsf{PAPR_{MIMO-OFDM}}>\mathsf{PAPR_{0}})$. The $\mathsf{PAPR}$
is calculated according to the $\mathsf{BPF}$ output, ${x}_{n}^{\mathsf{F}}$.
As can be observed in Fig. 5, the proposed $\mathsf{CAE}$ achieves the better
performance of $\mathsf{PAPR}$ reduction compared to the $\mathsf{CF}$ and
$\mathsf{SLM}$ methods. However, still the $\mathsf{BER}$ and spectral
behavior are more important for performance evaluation.

(a) 16-QAM, 4x4 MIMO with 3GPP multipath channel

(b) 16-QAM, 4x4 MIMO with AWGN channel

(c) 4-QAM, 2x2 MIMO with 3GPP multipath channel

Figure 5: CCDF of PAPR of the considered methods.

###  IV-E Spectrum Analysis

Figure 6 compare the spectral performance in terms of the $\mathsf{PSD}$ of
the transmitted signals for all examined methods. The dashed rectangle shows
perfect spectral behavior for a linear $\mathsf{HPA}$ with no non-linear
components.

The proposed $\mathsf{CAE}$ decreases the out-of-band distortions at the
expense of lower transmitted power efficiency. Observing the spectral behavior
as a part of the experimental analysis showed us that there is a trade-off
between increasing the $\mathsf{IBO}$ and increasing the Lagrange multiplier
associated with the $\mathsf{PAPR}$ loss component, $\lambda_{2b}$. As shown
in Fig. 7, while increasing the $\mathsf{IBO}$ mostly shifts the
$\mathsf{CAE}$ curve downwards, increasing $\lambda_{2b}$ causes the curve to
bend more.

(a) 16-QAM, 4x4 MIMO with 3GPP multipath channel

(b) 16-QAM, 4x4 MIMO with AWGN channel

(c) 4-QAM, 2x2 MIMO with 3GPP multipath channel

Figure 6: PSD for the considered methods.

---

(a)

(b)

(c)

(d)

Figure 7: Spectral mask behavior trade-off between IBO and $\lambda_{2b}$.

The transmitter’s $\mathsf{OBO}$, which evaluates the power efficiency of the
system, is defined as the ratio between the maximal radiated power that is the
maximal power transmitted by all the $\mathsf{HPA}$s in the network, $P_{T}$,
and the mean transmitted power at the $\mathsf{HPA}$s’ input, i.e.

$\displaystyle\vspace{-0.3cm}\mathsf{OBO}=\frac{P_{T}}{\sum_{m=1}^{N_{t}}\mathbb{E}\left(|{{x}_{n,m}^{\mathsf{B}}}|^{2}\right)}.$
(26)

TABLE II: ACPR and OBO | 4QAM 2X2 MIMO | 16QAM 4X4 MIMO
---|---|---
Parameter | CAE | FC-AE | CF | SLM | No-reduction | CAE | FC-AE | CF | SLM | No-reduction
ACPR[dB] | -39.87 | -37.26 | -39.08 | -37.73 | -34.99 | -37.88 | -36.53 | -39.004 | -37.67 | -35.01
OBO[dB] | 5.92 | 6.62 | 6.74 | 6.78 | 6.86 | 6.09 | 6.77 | 6.74 | 6.78 | 6.84
Figure 8: OBO vs. ACPR for the considered methods for 16-QAM with 4x4 MIMO
setup and 3GPP multipath channel.

The maximum radiated power is defined to be $P_{T}=1$. $P_{T}$ is divided
equally between the $\mathsf{HPA}$s. In other words, the maximal transmitted
power of each $\mathsf{HPA}$ is $P_{T}/N_{t}$. As we defined all
$\mathsf{HPA}$s to have the same characteristics, the saturation level of each
$\mathsf{HPA}$ is $A_{0}=\sqrt{P_{T}/N_{T}}$. Table II compares the
$\mathsf{ACPR}$ and the $\mathsf{OBO}$ of the proposed $\mathsf{CAE}$ to the
other methods. As shown, the $\mathsf{ACPR}$ of the $\mathsf{CAE}$ is
comparable with the considered methods.

In Fig. 8 we further compare the $\mathsf{OBO}$ performance for different
$\mathsf{ACPR}$ values. It can be seen that the $\mathsf{CAE}$ system requires
lower $\mathsf{OBO}$s, which is better overall power efficiency, while
maintaining better $\mathsf{BER}$ compared to the other methods.

###  IV-F Autoencoder - FC vs. CNN

We investigated various $\mathsf{NN}$ types for the $\mathsf{AE}$, in
particular, $\mathsf{FC}$ and $\mathsf{CNN}$. Figure 9(a) compares the
$\mathsf{BER}$ performance of two $\mathsf{AE}$ architectures: the proposed
$\mathsf{CAE}$, which contains convolutional layers, and a fully connected
autoencoder (FC-AE), which contains only $\mathsf{FC}$ layers. It can be
observed that the $\mathsf{CAE}$ network has better $\mathsf{BER}$ performance
compared to the FC-AE. As shown in Table II, the $\mathsf{ACPR}$ of the
$\mathsf{CAE}$ is better than that of the FC-AE. Moreover, the $\mathsf{CAE}$
has lower complexity and thus faster training. The three convolutional layers
have a total of $1953$ parameters, while for three $\mathsf{FC}$ layers of
sizes $3500$, $2500$, and $3500$, as were used for the FC-AE in Fig. 9(a) and
Table II, the number of parameters is around $10^{7}$.

(a) FC-AE and CAE

(b) Fixed and gradual loss learning

Figure 9: BER vs. $\mathsf{P\\_SNR}$ learning approaches comparison of 16-QAM,
4x4 MIMO with 3GPP multipath channel.

###  IV-G Fixed vs. Gradual Loss Learning

To show the benefits of using a gradual loss learning procedure, Fig. 9(b)
compares its $\mathsf{BER}$ performance to that of a fixed-loss training
procedure, where the loss function’s weights are fixed for the entire
training. It can be observed that the gradual loss learning procedure
significantly improves the $\mathsf{BER}$. In addition, improving the
$\mathsf{BER}$ while keeping the $\mathsf{PAPR}$ and spectral performance at
the desired levels is easier to control when applying the gradual loss
learning method than manipulating loss function weights in fixed-loss
training. Also, spectral performance and $\mathsf{PAPR}$ reduction were harder
to control and provide similar performance.

##  V Conclusions and Future Work

In this study, we have presented a $\mathsf{CAE}$ model for $\mathsf{PAPR}$
reduction and waveform design in a $\mathsf{MIMO}$-$\mathsf{OFDM}$
communication system. We have applied a gradual loss learning method to
optimize the performance in terms of three objectives: low $\mathsf{BER}$, low
$\mathsf{PAPR}$, and adherence to $\mathsf{ACPR}$ spectral requirements, on
top of the $\mathsf{AL}$ multipliers optimization technique. The presented
$\mathsf{CAE}$ structure trainable parts included a neural $\mathsf{PAPR}$
reduction block, followed by a $\mathsf{BPF}$ filter to optimize the spectral
behavior at the transmitter, and a neural iterative $\mathsf{MIMO}$ detection
block at the receiver, both were simultaneously optimized as a part of the
end-to-end network design. The proposed $\mathsf{CAE}$ was shown to outperform
the $\mathsf{CF}$ and the $\mathsf{SLM}$ algorithms at the examined cases.
Future work can extend the $\mathsf{MIMO}$ scenario to higher modulation
schemes and larger $\mathsf{MIMO}$ setups, aiming to achieve a functional
utility for future wireless communication networks.

## References

  * [1] T. Jiang and G. Zhu, “Complement block coding for reduction in peak-to-average power ratio of ofdm signals,” _IEEE Communications Magazine_ , vol. 43, no. 9, p. S17–S22, 2005.
  * [2] M. Z. Chowdhury, M. Shahjalal, S. Ahmed, and Y. M. Jang, “6g wireless communication systems: Applications, requirements, technologies, challenges, and research directions,” _IEEE Open Journal of the Communications Society_ , vol. 1, pp. 957–975, 2020.
  * [3] A. Bo, Y. Zhi-Xing, P. C.-Y. Z. Tao-Tao, and G. Jian-Hua, “Effects of papr reduction on hpa predistortion,” _IEEE Transactions on Consumer Electronics_ , vol. 51, no. 4, pp. 1143–1147, 2005.
  * [4] C.-X. Wang, M. D. Renzo, S. Stanczak, S. Wang, and E. G. Larsson, “Artificial intelligence enabled wireless networking for 5g and beyond: Recent advances and future challenges,” _IEEE Wireless Communications_ , vol. 27, no. 1, pp. 16–23, 2020.
  * [5] Y. S. Cho, J. Kim, W. Y. Yang, and C. G. Kang, _MIMO-OFDM wireless communications with MATLAB_. John Wiley & Sons, 2010.
  * [6] 3GPP, “3rd generation partnership project (3gpp). study on channel model for frequencies from 0.5 to 100 ghz,” 3rd Generation Partnership Project (3GPP), Technical Specification (TS). [Online]. Available: https://www.3gpp.org/ftp/Specs/archive/38_series/38.901
  * [7] B. Hassibi and H. Vikalo, “On sphere decoding algorithm 1: expected complexity,” _IEEE Transaction on Signal Processing_ , vol. 53, no. 8, pp. 2806–2818, 2005.
  * [8] P. W. Wolniansky, G. J. Foschini, G. D. Golden, and R. A. Valenzuela, “V-blast: An architecture for realizing very high data rates over the rich-scattering wireless channel,” in _1998 URSI international symposium on signals, systems, and electronics. Conference proceedings (Cat. No. 98EX167)_. IEEE, 1998, pp. 295–300.
  * [9] B. Steingrimsson, Z.-Q. Luo, and K. M. Wong, “Soft quasi-maximum-likelihood detection for multiple-antenna wireless channels,” _IEEE Transactions on Signal Processing_ , vol. 51, no. 11, pp. 2710–2719, 2003.
  * [10] N. Samuel, T. Diskin, and A. Wiesel, “Learning to detect,” _IEEE Transactions on Signal Processing_ , vol. 67, no. 10, pp. 2554–2564, 2019.
  * [11] O. Sholev, H. H. Permuter, E. Ben-Dror, and W. Liang, “Neural network MIMO detection for coded wireless communication with impairments,” in _2020 IEEE Wireless Communications and Networking Conference (WCNC)_ , 2020, pp. 1–8.
  * [12] M. Mohammadkarimi, M. Mehrabi, M. Ardakani, and Y. Jing, “Deep learning-based sphere decoding,” _IEEE Transactions on Wireless Communications_ , vol. 18, no. 9, pp. 4368–4378, 2019.
  * [13] K. Pratik, B. D. Rao, and M. Welling, “Re-mimo: Recurrent and permutation equivariant neural mimo detection,” _IEEE Transactions on Signal Processing_ , vol. 69, pp. 459–473, 2020.
  * [14] T. J. O’Shea, T. Erpek, and T. C. Clancy, “Deep learning based mimo communications,” _arXiv preprint arXiv:1707.07980_ , 2017.
  * [15] J. Song, C. Häger, J. Schröder, T. J. O’Shea, E. Agrell, and H. Wymeersch, “Benchmarking and interpreting end-to-end learning of mimo and multi-user communication,” _IEEE Transactions on Wireless Communications_ , 2022.
  * [16] T. O’shea and J. Hoydis, “An introduction to deep learning for the physical layer,” _IEEE Transactions on Cognitive Communications and Networking_ , vol. 3, no. 4, pp. 563–575, 2017.
  * [17] M. A. Albreem, M. Juntti, and S. Shahabuddin, “Massive mimo detection techniques: A survey,” _IEEE Communications Surveys & Tutorials_, vol. 21, no. 4, pp. 3109–3132, 2019.
  * [18] I. Gutman, I. Iofedov, and D. Wulich, “Iterative decoding of iterative clipped and filtered OFDM signal,” _IEEE Transactions on Communications_ , vol. 61, no. 10, pp. 4284–4293, 2013.
  * [19] A. P. More and S. B. Somani, “The reduction of PAPR in OFDM systems using clipping and SLM method,” in _2013 International Conference on Information Communication and Embedded Systems (ICICES)_ , 2013, pp. 593–597.
  * [20] Di-xiao Wu, “Selected mapping and partial transmit sequence schemes to reduce PAPR in OFDM systems,” in _2011 International Conference on Image Analysis and Signal Processing_ , 2011, pp. 1–5.
  * [21] T. Jiang and Y. Wu, “An overview: Peak-to-average power ratio reduction techniques for OFDM signals,” _IEEE Transactions on Broadcasting_ , vol. 54, no. 2, pp. 257–268, 2008.
  * [22] A. Mobasher and A. K. Khandani, “Integer-based constellation-shaping method for PAPR reduction in OFDM systems,” _IEEE Transactions on Communications_ , vol. 54, no. 1, pp. 119–127, 2006.
  * [23] Z. Q. Taha and X. Liu, “An adaptive coding technique for PAPR reduction,” in _IEEE GLOBECOM 2007 - IEEE Global Telecommunications Conference_ , 2007, pp. 376–380.
  * [24] Y. Lee, Y. You, W. Jeon, J. Paik, and H. Song, “Peak-to-average power ratio in mimo-ofdm systems using selective mapping,” _IEEE Communications Letters_ , vol. 7, no. 12, pp. 575–577, 2003.
  * [25] A. Namitha and S. Sameer, “A bandwidth efficient selective mapping technique for the papr reduction in spatial multiplexing mimo-ofdm wireless communication system,” _Physical Communication_ , vol. 25, pp. 128–138, 2017\.
  * [26] A. Zappone, M. Di Renzo, and M. Debbah, “Wireless networks design in the era of deep learning: Model-based, AI-based, or both?” _IEEE Transactions on Communications_ , vol. 67, no. 10, pp. 7331–7376, 2019.
  * [27] I. Sohn, “A low complexity PAPR reduction scheme for OFDM systems via neural networks,” _IEEE Communications Letters_ , vol. 18, no. 2, pp. 225–228, 2014.
  * [28] I. Sohn and S. C. Kim, “Neural network based simplified clipping and filtering technique for PAPR reduction of OFDM signals,” _IEEE Communications Letters_ , vol. 19, no. 8, pp. 1438–1441, 2015.
  * [29] M. Kim, W. Lee, and D. Cho, “A novel papr reduction scheme for OFDM system based on deep learning,” _IEEE Communications Letters_ , vol. 22, no. 3, pp. 510–513, 2018.
  * [30] L. Shi, X. Zhang, W. Wang, Z. Wang, A. Vladimirescu, Y. Zhang, and J. Wang, “PAPR reduction based on deep autoencoder for VLC DCO-OFDM system,” in _2019 IEEE International Symposium on Broadband Multimedia Systems and Broadcasting (BMSB)_ , 2019, pp. 1–4.
  * [31] Y. Huleihel, E. Ben-Dror, and H. H. Permuter, “Low papr waveform design for ofdm systems based on convolutional autoencoder,” in _2020 IEEE International Conference on Advanced Networks and Telecommunications Systems (ANTS)_. IEEE, 2020, pp. 1–6.
  * [32] F. A. Aoudia and J. Hoydis, “Waveform learning for next-generation wireless communication systems,” _IEEE Transactions on Communications_ , 2022.
  * [33] L. Hao, D. Wang, W. Cheng, J. Li, and A. Ma, “Performance enhancement of ACO-OFDM-based VLC systems using a hybrid autoencoder scheme,” _Optics Communications_ , vol. 442, pp. 110–116, 2019.
  * [34] A. Kalinov, R. Bychkov, A. Ivanov, A. Osinsky, and D. Yarotsky, “Machine learning-assisted papr reduction in massive mimo,” _IEEE Wireless Communications Letters_ , vol. 10, no. 3, pp. 537–541, 2020.
  * [35] A. Ivanov and D. Lakontsev, “Selective tone reservation for papr reduction in wireless communication systems,” in _2017 IEEE International Workshop on Signal Processing Systems (SiPS)_. IEEE, 2017, pp. 1–6.
  * [36] A. Ivanov, A. Volokhatyi, D. Lakontsev, and D. Yarotsky, “Unused beam reservation for papr reduction in massive mimo system,” in _2018 IEEE 87th Vehicular Technology Conference (VTC Spring)_. IEEE, 2018, pp. 1–5.
  * [37] P. Kenington, “Methods linearize RF transmitters and power amps,” _Microwaves & RF_, vol. 37, no. 13, pp. 102–116, 1998.
  * [38] S. Santurkar, D. Tsipras, A. Ilyas, and A. Madry, “How does batch normalization help optimization?” in _Advances in Neural Information Processing Systems_ , 2018, pp. 2483–2493.
  * [39] K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image recognition,” in _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 2016, pp. 770–778.
  * [40] G. Klambauer, T. Unterthiner, A. Mayr, and S. Hochreiter, “Self-normalizing neural networks,” in _Advances in neural information processing systems_ , 2017, pp. 971–980.
  * [41] H. E. Rowe, “Memoryless nonlinearities with gaussian inputs: Elementary results,” _The BELL system technical Journal_ , vol. 61, no. 7, pp. 1519–1525, 1982.
  * [42] G. L. Stuber, J. R. Barry, S. W. Mclaughlin, Y. Li, M. A. Ingram, and T. G. Pratt, “Broadband mimo-ofdm wireless communications,” _Proceedings of the IEEE_ , vol. 92, no. 2, pp. 271–294, 2004.
  * [43] I. Loshchilov and F. Hutter, “Decoupled weight decay regularization,” _arXiv preprint arXiv:1711.05101_ , 2017.
  * [44] S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” _Foundations and Trends® in Machine learning_ , vol. 3, no. 1, pp. 1–122, 2011.
  * [45] D. G. Luenberger and Y. Yinyu, _Linear and nonlinear programming_. Springer, 1984, vol. 2.
  * [46] “Matlab 5g toolbox,” 2020, the MathWorks, Natick, MA, USA. [Online]. Available: https://www.mathworks.com/products/5g.html