###### Proof.

. :

2G

    Second Generation
3G

    3${}^{\text{rd}}$ Generation
3GPP

    3${}^{\text{rd}}$ Generation Partnership Project
4G

    4${}^{\text{th}}$ Generation
5G

    5${}^{\text{th}}$ Generation
5GPPP

    5G Infrastructure Public Private Partnership
ACLR

    adjacent channel leakage ratio
ADMM

    alternating direction method of multipliers
ARQ

    automatic repeat request
BADMM

    Bregman alternating direction method of multipliers (ADMM)
BER

    bit error rate
BLER

    block error rate
BPC

    binary power control
BPSK

    binary phase-shift keying
BRA

    balanced random allocation
BS

    base station
BW

    bandwidth
CADMM

     channel state information (CSI)-aware ADMM
CAP

    combinatorial allocation problem
CAPEX

    capital expenditure
CBF

    coordinated beamforming
CCDF

    complementary cumulative distribution function
CFR

    crest-factor reduction
CP

    cyclic prefix
CS

    coordinated scheduling
CPOCS

     CSI-aware projection on convex sets (POCS)
CRUSADE

    convex suppression of unwanted emission and distortion elimination
CRUISE

    convex reduction of unwanted spectral emission
CRUISE-DAYS

    Convex RedUctIon of unwanted Spectral Emission with Davis And Yin Splitting
cruise

    convex reduction of unwanted spectral emission
CRUISE-SSP

     convex reduction of unwanted spectral emission (CRUISE)- semi-analytical spectral precoding (SSP)
CSI

    channel state information
CSIT

    channel state information at the transmitter
D2D

    device-to-device
DCA

    dynamic channel allocation
DCI

    downlink control information
DE

    differential evolution
DFT

    discrete Fourier transform
DIST

    Distance
DL

    downlink
DMA

    double moving average
DMRS

    demodulation reference signal
D2DM

    D2D mode
DMS

    D2D mode selection
DROPE

    Douglas-Rachford-based out-of-band emissions (OOBE) power reduction with error vector magnitude (EVM) constraint
DRS

    Douglas-Rachford splitting
DYS

    Davis-Yin splitting
DPC

    dirty paper coding
DR

    Douglas-Rachford
DRA

    dynamic resource assignment
DSA

    dynamic spectrum access
eMBB

    enhanced mobile broadband
eV2X

    enhanced vehicle-to-everything
EADMM

     EVM-constrained ADMM
EIRP

    equivalent isotropically radiated power
EPOCS

     EVM-constrained POCS
ETSI

    European telecommunications standards institute
EVM

    error vector magnitude
EMSP

    EVM-constrained mask-compliant spectral precoder (MSP)
ENSP

    EVM-constrained notching spectral precoder (NSP)
ESP

    EVM-constrained spectral precoding
ESSP

    EVM-constrained SSP
FD

    fully digital
FDD

    frequency division duplexing
FFT

    (fast) discrete Fourier Transform
GFBS

    generalized forward-backward splitting
HARQ

    hybrid automatic repeat request
HB

    hybrid beamforming
KKT

    Karush-Kuhn-Tucker
KPI

    key performance indicator
IPAPR

    instantaneous peak-to-average power ratio (PAPR)
ICF

    iterative clipping and filtering
IDFT

    Inverse discrete Fourier transform (DFT)
IFFT

    Inverse (fast) discrete Fourier Transform (FFT)
InC

    in-coverage
IoT

    internet of things
ITS

    intelligent transportation systems
L-BFGS

    limited-memory Broyden-Fletcher-Goldfarb-Shanno
LDPC

    low-density parity-check code
LMMSE

    linear minimum mean squared error
LOS

    line-of-sight
LS

    large-scale
LS-MIMO-OFDM

     large-scale (LS)- multiple-input multiple-output (MIMO)- orthogonal frequency division multiplexing (OFDM)
LS-MSP

    large-scale MSP
LTE

    long-term evolution
MAC

    medium access control
MADMM

    modified consensus ADMM
MCS

    modulation and coding scheme
METIS

    Mobile Enablers for the Twenty-Twenty Information Society
MIMO

    multiple-input multiple-output
MIMO-OFDM

     MIMO-OFDM
MISO

    multiple-input single-output
MRC

    maximum ratio combining
MS

    mobile station
MSE

    mean squared error
MSP

    mask-compliant spectral precoder
MMSE

    minimum mean squared error
MTC

    machine type communications
MU-MIMO

    multi-user MIMO
MU-MIMO-OFDM

     multi-user  (MU-MIMO)-OFDM
mMTC

    massive machine type communications
cMTC

    critical machine type communications
NR

    new radio
NSP

    notching spectral precoder
NSPS

    national security and public safety
NWC

    network coding
OFDM

    orthogonal frequency division multiplexing
OOB

    out-of-band
OOBE

    out-of-band emissions
OoC

    out-of-coverage
PA

    power amplifier
PAPR

    peak-to-average power ratio
PDCCH

    physical downlink control channel
PDSCH

    physical downlink shared channel
POCS

    projection on convex sets
PPG

    proximal-proximal-gradient
PRB

    physical resource block
PRBs

    physical resource block
PRG

    precoding resource block group
PSBCH

    physical sidelink broadcast channel
PSFCH

    physical sidelink feedback channel
PSCCH

    physical sidelink control channel
PSP

    proximal splitting based mask compliant spectral precoding
PSSCH

    physical sidelink shared channel
PHY

    physical
PLNC

    physical layer network coding
PRB

    physical resource block
PSD

    power spectral density
QAM

    quadrature amplitude modulation
QCQP

    quadratic constrained quadratic programming
QoS

    quality of service
QPSK

    quadrature-phase shift keying
PaC

    partial coverage
RAISES

    reallocation-based assignment for improved spectral efficiency and satisfaction
RAN

    radio access network
RA

    resource allocation
RAT

    radio access technology
RB

    resource block
RF

    radio frequency
RSRP

    reference signal received power
Rx

    receive
RX

    receive
RxEVM

    receive-EVM
Rx-TxEVM

    received transmit-EVM (TxEVM)
RZF

    regularized zero forcing
SC-FDM

    single carrier frequency division modulation
SEM

    spectral emission mask
SFBC

    space-frequency block coding
SCI

    sidelink control information
SIC

    successive interference cancellation
SINR

    signal-to-interference-plus-noise ratio
SISO

    single-input single-output
SL

    sidelink
SP

    Spectral precoding
SNR

    signal-to-noise ratio
SSP

    semi-analytical spectral precoding
STC

    space-time coding
SU

    single-user
SU-MIMO

    single-user MIMO
SU-MIMO-OFDM

     single-user  (SU-MIMO)-OFDM
SVM

    support vector machine
TDD

    time division duplexing
TDL

    tapped delay line
TOP-ADMM

    three-operator ADMM
Tx

    transmit
TX

    transmit
TxEVM

    transmit-EVM
UE

    user equipment
UL

    uplink
UP

    uniform power
URLLC

    ultra-reliable and low latency communications
VUE

    vehicular user equipment
V2N

    vehicular-to-network
V2X

    vehicle-to-everything
V2V

    vehicle-to-vehicle
V2P

    vehicle-to-pedestrian
WMMSE

    weighted minimum mean squared error
wMMSE

    weighted minimum mean squared error
ZF

    zero forcing
ZMCSCG

    zero mean circularly symmetric complex Gaussian

#  EVM Mitigation with PAPR and ACLR Constraints in Large-Scale MIMO-OFDM
Using TOP-ADMM

Shashi Kant, Mats Bengtsson, Gabor Fodor, Bo Göransson, and Carlo Fischione S.
Kant, B. Göransson, and G. Fodor are with Ericsson AB and KTH Royal Institute
of Technology, Stockholm, Sweden (e-mail: {shashi.v.kant, bo.goransson,
gabor.fodor}@ericsson.com)M. Bengtsson and C. Fischione are with KTH Royal
Institute of Technology, Stockholm, Sweden (e-mail: {matben,
carlofi}@kth.se)The work of S. Kant was supported in part by the Swedish
Foundation for Strategic Research under grant ID17-0114.Parts of this work is
presented at IEEE Asilomar 2021 [1].

###### Abstract

Although signal distortion-based peak-to-average power ratio (PAPR) reduction
is a feasible candidate for orthogonal frequency division multiplexing (OFDM)
to meet standard/regulatory requirements, the error vector magnitude (EVM)
stemming from the PAPR reduction has a deleterious impact on the performance
of high data-rate achieving multiple-input multiple-output (MIMO) systems.
Moreover, these systems must constrain the adjacent channel leakage ratio
(ACLR) to comply with regulatory requirements. Several recent works have
investigated the mitigation of the EVM seen at the receivers by capitalizing
on the excess spatial dimensions inherent in the large-scale MIMO that assume
the availability of perfect channel state information (CSI) with spatially
uncorrelated wireless channels. Unfortunately, practical systems operate with
erroneous CSI and spatially correlated channels. Additionally, most standards
support user-specific/CSI-aware beamformed and cell-specific/non-CSI-aware
broadcasting channels. Hence, we formulate a robust EVM mitigation problem
under channel uncertainty with nonconvex PAPR and ACLR constraints catering to
beamforming/broadcasting. To solve this formidable problem, we develop an
efficient scheme using our recently proposed three-operator alternating
direction method of multipliers (TOP-ADMM) algorithm and benchmark it against
two three-operator algorithms previously presented for machine learning
purposes. Numerical results show the efficacy of the proposed algorithm under
imperfect CSI and spatially correlated channels.

###### Index Terms:

nonconvex PAPR reduction, Three-operator ADMM (TOP-ADMM), EVM, ACLR, MIMO-OFDM

©2022 IEEE. Personal use of this material is permitted. Permission from IEEE
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reprinting/republishing this material for advertising or promotional purposes,
creating new collective works, for resale or redistribution to servers or
lists, or reuse of any copyrighted component of this work in other works. DOI:
10.1109/TWC.2022.3177136

##  I Introduction

Modern mobile and wireless communication systems, such as 5G new radio (NR)
and beyond, adopt orthogonal frequency division multiplexing (OFDM) with a
cyclic prefix [2]. This is because of both its backward compatibility with 4G
long-term evolution (LTE) as well as the fact that OFDM has several appealing
attributes, such as robustness against the adverse effects of time and/or
frequency selective channels, simplicity in terms of single-tap (frequency-
domain) equalization, and resilience in terms of supporting both low and high
symbol rates—thereby supporting various quality of service requirements.

It is well-known that OFDM suffers from high PAPR because of the linear
combination of the subcarriers carrying (pseudo-random) $M$-ary quadrature
amplitude modulation symbols in the inverse discrete Fourier transform (
Inverse  (IDFT)) operation [3].  With the trend of an increasing number of
transmit antennas and support of wider carrier bandwidths with the higher
spectrum and spectral efficiencies, including support of higher frequency
bands, the implementation of radio base station faces many challenges, notably
in terms of power consumption and size—primarily dominated by power
amplifiers. Thus, to reduce the power consumption of radio base stations or
PAs, one must reduce the PAPR of the transmit signals for each transmit
antenna. Additionally, the challenges are to support multiple radio access
technologies, e.g., 5G NR, 4G LTE, and narrowband internet of things (NB-IoT),
by the same radio base station—where NB-IoT carriers can be placed in the
guard bands of NR/LTE. For some classes of base stations, high average
transmit output power of the signals is required for better cell coverage.
Therefore, if PAPR is high and the required average transmit signal power is
high, smaller size and efficient PA can cause non-linear distortions rendering
both inband and OOBE—which may fail the base station to comply with
3${}^{\text{rd}}$ Generation Partnership Project (3GPP) standard and
regulatory requirements. Consequently, if PAPR is kept high, an inefficient
and bigger size PA will be employed such that the signals can operate in a
linear region to avoid non-linear distortions. Hence, it is paramount to
minimize the PAPR of OFDM waveform in large-scale MIMO systems. Furthermore,
low PAPR of the signals with a digital predistorter or a PA linearizer can
improve the PA efficiency by letting PA operate close to its bias point.

Numerous techniques to reduce PAPR, also referred to as crest factor
reduction, have been proposed in the literature over the past decades. These
can be grouped as 1) distortion-based approaches, such as iterative clipping
and filtering [4, 5], peak cancellation [6, 7], and non-linear companding [8];
2) coding-based methods [9]; and 3) multiple signaling-based approaches, such
as selected mapping, partial transmit sequences [10], and tone reservation
[11] (see also review papers [12, 13, 14]). Distortion-based techniques,
particularly clipping and filtering, are the most popular methods in practice
as they are transparent to the standards and to the receiver, see [13].
However, they cause significant inband distortion—quantified in terms of error
vector magnitude (EVM)—and may increase OOBE. Basically, these distortion-
based techniques manage the inherent trade-off between EVM and achievable
PAPR. More precisely, if the desired PAPR threshold is low, then the inband
transmit EVM is high, which consequently penalizes the throughput performance
at the receiver.

##  II Related Works and Contributions

In this section, we first give an overview of some prior works on PAPR
reduction methods with some practical challenges for 5G NR and beyond to
motivate distortion-based PAPR techniques. Most of the principled approaches
for distortion-based PAPR reductions are posed as optimization problems, whose
(near-)optimal solutions cannot be obtained in closed-form and require
efficient iterative optimization techniques to tackle the problem. Thus, we
briefly describe some powerful and efficient first-order optimization
techniques to tackle the PAPR reduction problems for large-scale MIMO-OFDM-
based systems. Finally, we highlight our original contribution with
mathematical notations and paper organization.

###  II-A Practical Research Challenges for PAPR Reduction Techniques in 5G NR
and Beyond

As mentioned above, the base station supports multiple standards from 4G LTE
to current 5G NR releases, including cellular internet of things (IoT),
possibly in the guard bands of LTE/NR. It supports wide bandwidths and
multiple bands with growing demand for energy-efficient and lightweight radio
heads of the base stations equipped with many transmit antennas. Clearly, PAPR
of the multiple carriers of the signals must be reduced, which complements PA
linearization such as digital predistortion to improve PA efficiency—targeting
energy-efficient and lightweight base stations. The challenges for the PAPR
reduction techniques are that 1) one cannot exploit guard bands to deploy many
non-useful subcarriers for PAPR reduction such as in tone reservation [11],
and 2) one cannot send extra non-standard side information to the receiver(s)
because of compliance with the stringent 3GPP standard and regulatory
requirements.

###  II-B Related Works for PAPR Reduction Techniques

We briefly summarize some of the related works for PAPR reduction in OFDM-
based systems, including MIMO, wherever applicable. Moreover, we refer
interested readers to overview papers [12, 13, 14].

####  II-B1 Multiple Signaling and Probabilistic Methods

These techniques reduce PAPR by either generating multiple permutations,
introducing phase shifts or adding peak reducing subcarriers of OFDM signal.

Selective mapping [15, 12, 13, 14] generates multiple statistically
independent input data sequences representing the same useful information,
where the generated signal with the lowest PAPR is selected for the
transmission. However, the receiver must know the chosen mapping at the
transmitter for successful demodulation of the signal via side information.
Hence, unfortunately, selective mapping cannot be employed in 3GPP-compliant
cellular base stations. There are some related methods to selective mapping,
called partial transmit sequences—see [12, 13, 14] and references therein.

Tone reservation [11, 16, 12, 13, 14] is one of the signal distortion-less
techniques for PAPR reduction, which utilizes unused/inactive subcarriers—that
is, adds peak reducing subcarriers—typically in the vicinity of used/active
subcarriers carrying useful information. On the one hand, one of the strengths
of tone reservation is that it does not induce any signal distortions on the
useful data-carrying subcarriers. On the other hand, although tone reservation
offers low complexity, as discussed above, unfortunately, the limited number
of unused/inactive subcarriers can be exploited because of relatively high
spectrum efficiency of 5G NR compared to LTE and nearly no effective guard
bands are available in 5G NR (or even LTE) as NB-IoT carriers can also be
deployed in the guard bands of NR/LTE. Nevertheless, ignoring the guard band
usage restrictions of NR/LTE, in Section V-B, we have evaluated and compared
the performance of two variants of tone reservation schemes against our
proposed method—see particularly Fig. 4.

####  II-B2 Coding-based PAPR Reduction

These coding-based techniques, e.g., linear block codes, Golay complementary
sequences, and turbo coding—see [12, 13, 14] and references therein—reduce the
PAPR by generating encoded data inputs that produce an OFDM symbol with
desired PAPR. However, these techniques typically require an exhaustive search
for a good code, which becomes intractable for an OFDM-based system with more
than a dozen subcarriers. Unfortunately, such techniques are practically
impossible with a 5G NR-like system.

####  II-B3 Distortion-based PAPR Reduction

These distortion-based methods essentially reduce the PAPR by (non-linear)
distortion of the useful transmit signal, where the signal quality degradation
is measured in terms of EVM. Moreover, these distortion-based methods render
OOBE, which must be handled via appropriate filtering in either time or
frequency domain.

Active constellation or active set extension [17] reduces PAPR by modifying or
(pre)distorting the modulation constellation carried on the useful/active
subcarriers with a minimal penalty in terms of EVM. Although no side
information needs to be sent to the receiver, this method has, unfortunately,
limited application to large modulation alphabet, e.g., alphabet size more
than 64QAM [12, 13]. Hence, active constellation extension is not feasible for
a 5G NR-like system where more than 64QAM, e.g., 256/1024QAM, modulation
alphabet can be supported.

Iterative clipping and filtering [4, 5] is one of the simplest approaches,
which clips the high amplitudes of the signal if above a given peak threshold
while retaining the phase of the signals. This peak clipping causes unwanted
distortions inband and out-of-band—i.e., EVM and OOBE. This induced OOBE needs
to be filtered out either in frequency or time domain to maintain desired 3GPP
adjacent channel leakage ratio (ACLR) requirements. Consequently, filtering of
unwanted distortions causes regrowth of the signal peaks. However, after a few
repetitions/iterations of clipping and filtering reduces PAPR but at the cost
of increased signal distortion (EVM) depending on the chosen peak threshold.
If the signal peaks are clipped at a lower threshold, signal
distortion—EVM—increases significantly, penalizing the throughput [18].

Recently, for reciprocity-based or CSI-aware massive/large-scale MIMO,
distortion-based PAPR reduction techniques, see, e.g., [18, 19, 20, 21, 22],
are proposed, which not only reduce the PAPR at the transmitter but also
mitigate the incurred signal distortion (EVM) seen at the receiver by
exploiting downlink channel knowledge available at the base station. Most of
these distortion-based techniques relax the nonconvex PAPR problem by
employing either minimization of peaks, see, e.g., [18], which renders a
suboptimal solution, or semidefinite relaxation, see, e.g., [20], which can
enable a near-optimal solution at the expense of prohibitive computational
complexity. Another drawback is that many of these methods are not robust when
the downlink channel knowledge acquired at the transmitter is either imperfect
or incomplete e.g., [19, 21, 22]. Several of these previously proposed methods
lack the flexibility to adapt the desired PAPR target and OOBE—typically
quantified in terms of ACLR—according to the regulatory requirements.
Furthermore, in realistic systems, the channel estimates at the transmitter
and receiver are imperfect and/or incomplete. Consequently, under imperfect
and/or incomplete channel estimates, these previously proposed non-robust
schemes cannot achieve (near) zero EVM at the receiver (as shown in Section
V). Additionally, these non-robust prior arts do not support the incomplete
channel knowledge case, that is, OFDM symbols carrying broadcast channels
(such as cell-specific control channels), since there is no downlink channel
knowledge as these common channels naturally do not target a specific subset
of the served users. Although [20] offers various robust methods for
reciprocity-based PAPR reduction techniques, in these schemes there is no
constraint on OOBE or ACLR, which are important for PAPR reduction schemes.
Additionally and regrettably, semidefinite programming-based algorithms are
typically solved by interior-point-based methods [23], which have prohibitive
complexity for state-of-the-art base station radio hardware systems.

###  II-C Motivation for Distortion and Optimization Based PAPR Reduction

Considering practical constraints/restrictions in 5G NR and beyond, an
interesting candidate is a distortion-based PAPR reduction scheme, which
utilizes mainly useful/active subcarriers to reduce the PAPR of the OFDM
signal. Concretely, we seek a one-size-fits-all PAPR reduction scheme that can
naturally work for base stations equipped with single or multiple transmit
antennas and adaptively caters to non-CSI-aware, e.g., broadcast channels, and
CSI-aware if possible. In large-scale MIMO, the base station has a large
number of antennas compared to the total number of served/scheduled spatial
layers for the transmission—applicable to both SU-MIMO or MU-MIMO—and
additionally has some good enough quality of downlink channel estimates of all
the served users. Consequently, one could potentially relax the EVM
requirements at the transmitter stipulated by the standards, such as 3GPP NR
[24], by mitigating/reducing the EVM seen at the receivers of the users
accordingly while meeting the desired PAPR and ACLR requirements at the
transmitter of a base station. Moreover, if the base station has no downlink
channel knowledge of the users for instance supporting broadcast or control
channels in some OFDM symbols over all the active subcarriers within a
bandwidth, we propose minimizing the signal distortion energy subject to the
desired PAPR and ACLR constraints. Additionally, we seek a principled approach
to describe the distortion-based PAPR reduction scheme using an optimization
formulation and subsequently develop a (near-)optimal and computationally-
efficient solution. In practice, we do not require highly accurate solution,
especially for the problems of interest in this work. Therefore, the
subsequent section introduces first-order optimization schemes, which offer a
good trade-off between computational complexity and accuracy. Hence, these
first-order methods can tackle complicated composite problems and yield
implementation-friendly algorithms suitable for large-scale MIMO-OFDM systems.

###  II-D Related Works on Operator/Proximal Splitting for Signal Processing
in Communications

In the optimization problems mentioned above, proximal/operator splitting
techniques—typically first-order optimization algorithms—have become popular
thanks to their low computational complexity. This can be seen not only in
wireless communications but also in machine learning applications. Operator
splitting methods employ a divide-and-conquer approach, which essentially
breaks the large problem into smaller, easy-to-solve subproblems.  However,
first-order methods can be very slow to converge to solutions having high
accuracy. Nonetheless, modest accuracy can be sufficient for many practical
applications, such as in this work. There is a plethora of proximal/operator
splitting methods [25, 26, 27, 28, 29, 30, 31, 32, 33, 34] to tackle the
generic composite problems. The operator splitting algorithms can be roughly
grouped into two classes of algorithms [33], namely primal-dual-type—e.g.,
Chambolle-Pock-type [29]—and primal-/dual-type—e.g., ADMM-type [34]
algorithms.

Operator splitting for two operators (or, loosely speaking, optimization
problems in which the objective function is given by the sum of two functions)
has been widely studied [25, 26, 27, 29, 30, 31, 34]. Operator splitting with
more than two composite terms in the objective has been an active research
problem without resorting to problem or product space reformulations [32]
because these techniques are either not straightforward or converge slowly
[32, 33]. In other words, if the optimization problem comprises more than two
functions, then under certain separability and smoothness assumptions, a
typical approach is to reformulate the problem by modeling it as a block-wise
two-operator problem such that one can exploit the separability of grouped
functions within each block. However, as shown in [32, 33, 35], this can lead
to slow convergence or/and exhibit memory inefficiency. Therefore, novel
algorithms are researched and developed to solve composite problems comprising
more than two operators.

Related Works on Three-Operator Splitting: Recently, some authors have
extended both primal-/dual-type (including ADMM-type) algorithms and primal-
dual classes of splitting algorithms from two operators to three operators
[28, 32, 33, 36, 35, 37, 34]. ADMM (primal-/dual-type splitting) algorithms
for three operators have gained interest in the literature.  There are many
varieties of ADMM in the literature, see, e.g., [30]. One of the popular
variants is the so-called proximal ADMM, which essentially regularizes each
subproblem with a quadratic proximal regularization (or weighted norm). A
special case of proximal ADMM is linearized proximal ADMM, which basically
cancels out the inconvenient quadratic terms in the subproblem. However,
“linearization” in the ADMM-type literature can be ambiguous [34]. Besides the
aforementioned linearization of the subproblem, there are also techniques that
linearize the $L$-smooth111See Definition 1 of $L$-smooth function. objective
function within the ADMM-type class of algorithms designed to solve the three-
operator problem [28, 32, 36, 35], which we collectively refer to as three-
operator  (TOP-ADMM)-type algorithms in this work.

###  II-E Contribution

In this paper, we develop scalable, flexible, and computationally efficient
optimization methods to tackle a robust and nonconvex PAPR reduction problem
while catering to both beamforming and broadcasting in standard-compliant
wireless communication systems such as 5G NR. Our contributions are as
follows:

  * •

We formulate a PAPR reduction optimization problem amenable to large-scale
MIMO-OFDM, where the transmit distortion (or EVM) stemming from the PAPR
reduction subproblem is mitigated at the receiver without imposing any
additional signal processing requirement at the receiver.

  * •

Additionally, the proposed problem formulation offers flexibility to adapt the
desired PAPR and ACLR targets, and can seamlessly support both beamforming and
broadcasting carrying resources. Note that user-specific and common resources
may exist in the same OFDM symbol. One could argue that it is not non-trivial
to extend the previously proposed methods supporting both beamforming and
broadcasting, particularly a special case of our proposed formulation.

  * •

We introduce a powerful yet simple optimization solution algorithm based on
our recently proposed TOP-ADMM method [35], which is intended for a class of
three-operator proximal splitting schemes. To benchmark it against state-of-
the-art algorithms, we also provide details on how to apply Bregman  (BADMM)
[1, 28] and Davis-Yin splitting (DYS) [32] to the problem at hand.

  * •

We present simulation results for downlink 5G NR-like systems to substantiate
the capability of the proposed TOP-ADMM-based algorithm. Additionally, we
analyze the performance considering PAPR and EVM metrics under imperfect CSI
available at the transmitter and considering spatially-correlated wireless
channels.

###  II-F Notation

Let the set of complex and real numbers be denoted by $\mathbb{C}$ and
$\mathbb{R}$, respectively. $\Re\\{x\\}$ denotes the real part of a complex
number $x\\!\in\\!\mathbb{C}$. The $i$-th element of a vector
$\boldsymbol{a}\\!\in\\!\mathbb{C}^{m\times 1}$ and $j$-th column vector of a
matrix $\boldsymbol{A}\\!\in\\!\mathbb{C}^{m\times n}$ are denoted by
${a}[i]\\!\coloneqq\left(\boldsymbol{a}\right)_{i}\\!\\!\in\\!\mathbb{C}$ and
$\boldsymbol{A}\left[:,j\right]\\!\in\\!\mathbb{C}^{m\times 1}$, respectively.
An $i$-th element of three-order tensor is denoted in matrix form as
$\boldsymbol{X}[i]\\!\in\\!\mathbb{C}^{m\times n}$. We form a matrix by
stacking the set of higher order vectors
$\left\\{\boldsymbol{a}[n]\\!\in\\!\mathbb{C}^{M\times 1}\right\\}_{n=1}^{N}$
and $\left\\{\boldsymbol{b}[m]\\!\in\\!\mathbb{C}^{1\times
N}\right\\}_{m=1}^{M}$ column-wise and row-wise as
$\boldsymbol{A}\\!=\\!\left[\boldsymbol{a}[1],\ldots,\boldsymbol{a}[N]\right]\\!\in\\!\mathbb{C}^{M\times
N}$ and
$\boldsymbol{B}\\!=\\!\left[\boldsymbol{b}[1];\ldots;\boldsymbol{b}[M]\right]\\!\in\\!\mathbb{C}^{M\times
N}$, respectively. The transpose and conjugate transpose of a vector or matrix
are denoted by $\left(\cdot\right)^{\rm T}$ and $\left(\cdot\right)^{{\rm
H}}$, respectively. The complex conjugate is represented by
$\left(\cdot\right)^{*}$. The $K\times K$ identity matrix is written as
$\boldsymbol{I}_{K}$. The expectation operator is denoted by
$\mathbb{E}\\{\cdot\\}$. An $i$-th iterative update is denoted by
$(\cdot)^{(i)}$; $\boldsymbol{e}_{k}\\!\in\\!\mathbb{R}^{N\times 1}$ is a
standard basis vector. $\boldsymbol{F}$ refers to the DFT matrix;
$\boldsymbol{A}^{\nicefrac{{1}}{{2}}}$ denotes the square root of
$\boldsymbol{A}$.

###  II-G Paper Organization:

The remainder of the paper is organized as follows. The next section briefly
introduces the system and EVM model with useful performance and algorithm
design metrics. Section IV formulates EVM mitigation with PAPR and ACLR
constraints suitable for any practical 5G NR-like system, i.e., supporting
both user-specific and cell-specific signals. Moreover, in Section IV, we
first introduce our novel TOP-ADMM algorithm as a generic splitting method and
thereafter present an algorithm tackling the proposed large-scale problem
using TOP-ADMM. Then, we also present the benchmark algorithms tackling the
same proposed problem using BADMM [1, 28] and DYS [32]. Moreover, we
characterize the computational complexities of these TOP-ADMM-type algorithms
by solving the proposed problem and compare their complexities with two prior
arts.  Section V gives a rich set of numerical results, and Section VI
concludes with a summary.

##  III Preliminaries

This section briefly introduces the downlink MIMO-OFDM system model followed
by a short description of the performance metrics that are useful for
designing distortion-based PAPR reduction algorithms.

###  III-A System Model for Distortion-Based PAPR Reduction in Frequency-
Domain

We consider an OFDM-based MIMO downlink—as depicted in Fig. 1—where the base
station is equipped with $N_{\rm T}$ transmit (Tx) antennas, and the user
equipment (UE) is equipped with $N_{\rm R}$ receive (Rx) antennas. Moreover,
we assume a spatial multiplexing transmission scheme with $N_{\rm
L}\\!\leq\\!\min\left\\{N_{\rm T},N_{\rm R}\right\\}$ spatial layers. For MU-
MIMO, $N_{\rm U}$ represent total number of scheduled users sharing all the
resources for the data transmission, where each $\mu$-th user can have
$N_{{\rm L}_{\mu}}\\!\leq\\!N_{\rm L}$ spatial layers such that $N_{\rm
L}\\!=\\!\sum_{\mu=1}^{N_{\rm U}}N_{{\rm L}_{\mu}}$.

Figure 1: Simplified block diagram of a single-user MIMO-OFDM transceiver with
PAPR reduction scheme.

A (generalized) spatially precoded (or beamformed) symbol matrix in frequency-
domain input to an $\ell$-times oversampled $K$-point IDFT matrix is denoted
by ${\boldsymbol{X}}\\!\in\\!\mathbb{C}^{N_{\rm T}\times\ell K}$ such that at
the $k$-th (useful) subcarrier, ${\mathbb{C}^{N_{\rm T}\times
1}\\!\ni\\!\boldsymbol{X}}[:,k]\,=\mathcal{W}\left(\boldsymbol{S}[:,k]\right)$,
where non-precoded symbol vector, for all the spatial layers of SU-MIMO/MU-
MIMO, $\boldsymbol{S}[:,k]\\!\in\\!\mathcal{S}^{N_{\rm L}\times 1}$ belongs to
a complex-valued finite-alphabet set $\mathcal{S}$, e.g., corresponding to a
$2^{Q}$\- quadrature amplitude modulation (QAM) constellation with
$Q\\!\in\\!\\{2,4,6,8\\}$. The spatial (possibly nonlinear) precoding
$\mathcal{W}\\!:\\!\mathcal{S}^{N_{\rm L}\times
1}\\!\rightarrow\\!\mathbb{C}^{N_{\rm T}\times 1}$ maps the non-precoded
symbol vector $\boldsymbol{S}[:,k]\\!\in\\!\mathcal{S}^{N_{\rm L}\times 1}$ to
spatially precoded symbol vector
$\boldsymbol{X}[:,k]\\!\in\\!\mathbb{C}^{N_{\rm T}\times 1}$ catering either
SU-MIMO or MU-MIMO. For MU-MIMO,
$\boldsymbol{S}[:,k]\\!=\\!\left[\boldsymbol{S}_{1}[:,k];\ldots;\boldsymbol{S}_{N_{\rm
U}}[:,k]\right]$, where
$\boldsymbol{S}_{\mu}[:,k]\\!\in\\!\mathcal{S}^{N_{{\rm L}_{\mu}}\times 1}$
represent $\mu$-th user’s data vector.

####  III-A1 Transmit SU/MU-MIMO Signal Model

Inspired by, e.g., [38, 35], we model a frequency-domain data matrix
$\overline{\boldsymbol{X}}$ having reduced PAPR as $\mathbb{C}^{N_{\rm
T}\\!\times\\!\ell
K}\\!\ni\\!\overline{\boldsymbol{X}}\\!\coloneqq\\!\mathcal{G}\left({\boldsymbol{X}}\right)$,
where a distortion-based PAPR reduction function
${\mathcal{G}}:\\!\boldsymbol{X}\\!\in\\!\mathbb{C}^{N_{\rm T}\\!\times\\!\ell
K}\\!\\!\rightarrow\\!\mathbb{C}^{N_{\rm T}\\!\times\\!\ell
K}\\!\ni\\!\overline{\boldsymbol{X}}$ manipulates the data symbol matrix
$\boldsymbol{X}$. Consequently, this perturbation signal model capable of
reducing PAPR at a given $k$-th subcarrier can be expressed as,

$\overline{\boldsymbol{X}}\left[:,k\right]\\!=\\!\alpha[k]{\boldsymbol{X}}\left[:,k\right]\\!+\\!\underbrace{\bm{\epsilon}[k]}_{\rm
Tx\ distortion}\\!\approx\\!{\boldsymbol{X}}\left[:,k\right]+\bm{\epsilon}[k]$
(1)

where $\alpha[k]\\!\in\\!\mathbb{C}$ is a deterministic scalar and
$\bm{\epsilon}[k]$ is an instantaneous distortion that is uncorrelated with
$\boldsymbol{X}[:,k]$ but statistically dependent. However, appealing to the
works in [39, 40, 41, 42, 35], $\bm{\epsilon}[k]$ stemming from any (possibly
non-linear) source, e.g., clipping, spectral precoding, is correlated across
antennas and consequently beamformed in the similar direction as the signal
$\boldsymbol{X}[:,k]$, for instance, depending on the ratio of transmit
spatial layers and the transmit antenna branches—i.e., rank of the channel,
frequency-granularity of the spatial precoding—i.e., channel coherence
bandwidth. Observe that although the scaling factor $\alpha[k]$ may vary
across subcarriers, on average, the scaling is assumed to be real-valued and
nearly unity—our extensive simulations in Section V show that the approximate
additive EVM model (1) is accurate and ignoring $\alpha[k]$ has nearly
negligible impact on the key performance metrics.

####  III-A2 Received SU/MU-MIMO Signal Model

For MU-MIMO, let $N_{{\rm R}_{\mu}}$ denote the number of receive antennas of
user $\mu$ such that the total number of receive antennas for all the users is
$N_{\rm R}\\!=\\!\sum_{\mu=1}^{N_{\rm U}}N_{{\rm R}_{\mu}}$. Additionally, let
$\boldsymbol{H}_{\mu}\left[k\right]\\!\in\\!\mathbb{C}^{N_{{\rm
R}_{\mu}}\times N_{\rm T}}$ describe the channel matrix between user $\mu$ and
base station such that the composite channel matrix over all the receive
antennas of users is
$\boldsymbol{H}\left[k\right]\\!=\\!\left[\boldsymbol{H}_{1}\left[k\right];\ldots;\boldsymbol{H}_{N_{\rm
U}}\left[k\right]\right]$. Therefore, the received signal model of user $\mu$
can be expressed as

$\displaystyle{\mathbb{C}^{N_{{\rm R}_{\mu}}\times
1}\\!\ni}\boldsymbol{Y}_{\mu}[:,k]=$
$\displaystyle\boldsymbol{H}_{\mu}[k]\overline{\boldsymbol{X}}\left[:,k\right]\\!+\\!\boldsymbol{n}_{\mu}[k]$
(2a) $\displaystyle=$
$\displaystyle\underbrace{\boldsymbol{S}_{\mu}\left[:,k\right]}_{\rm desired}$
$\displaystyle+\\!\underbrace{\left(\boldsymbol{H}_{\mu}\left[k\right]\overline{\boldsymbol{X}}\left[:,k\right]\\!-\\!\boldsymbol{S}_{\mu}\left[:,k\right]\right)}_{\rm
interuser\,interference\,and\,received\,Tx\,distortion}\\!$
$\displaystyle+\boldsymbol{n}_{\mu}[k],$ (2b)
$\displaystyle{\phantom{{\mathbb{C}^{N_{{\rm R}_{\mu}}\times
1}\\!\ni}\boldsymbol{Y}_{\mu}[:,k]}}\approx$
$\displaystyle\boldsymbol{H}_{\mu}\left[k\right]{\boldsymbol{X}}\left[:,k\right]\\!+\\!{\boldsymbol{H}_{\mu}\left[k\right]\bm{\epsilon}\left[k\right]}\\!+\\!\boldsymbol{n}_{\mu}[k]$
(3a) $\displaystyle=$
$\displaystyle\\!\underbrace{\boldsymbol{S}_{\mu}\left[:,k\right]}_{\rm
desired}$
$\displaystyle+\\!\underbrace{\left(\boldsymbol{H}_{\mu}\left[k\right]{\boldsymbol{X}}\left[:,k\right]\\!-\\!\boldsymbol{S}_{\mu}\left[:,k\right]\right)}_{\rm
interuser\ interference}$
$\displaystyle+\\!\underbrace{\boldsymbol{H}_{\mu}\left[k\right]\bm{\epsilon}\left[k\right]}_{{\rm
received\,{Tx}\,{distortion}}}\\!+\boldsymbol{n}_{\mu}[k],$ (3b)

where $\boldsymbol{n}_{\mu}[k]$ is effective noise at user $\mu$, which can be
modeled as a zero-mean complex symmetric white Gaussian noise with
$\mathcal{CN}\left(\boldsymbol{0},N_{0}\boldsymbol{I}_{N_{{\rm
R}_{\mu}}}\right)$, where $N_{0}$ is the noise variance at the receiver. The
transmit distortion on the receiver side of user $\mu$ at subcarrier $k$ can
be expressed by
$\boldsymbol{H}_{\mu}\left[k\right]\bm{\epsilon}\left[k\right]$, which
unfortunately correlates the total noise spatially at the receiver since the
effective noise covariance matrix seen at the user side—ignoring the $k$-th
index for
brevity—$\boldsymbol{R}_{\mu}\\!=\\!\boldsymbol{H}_{\mu}\boldsymbol{R}_{\bm{\epsilon}\bm{\epsilon}}\boldsymbol{H}_{\mu}^{\rm
H}\\!+\\!N_{0}\boldsymbol{I}_{N_{{\rm R}_{\mu}}}$ where
$\boldsymbol{R}_{\bm{\epsilon}\bm{\epsilon}}\\!=\\!\mathbb{E}\left\\{\bm{\epsilon}\bm{\epsilon}^{\rm
H}\right\\}$ is the transmit distortion covariance. Therefore, the impact of
transmit distortion has unfortunately a well-known pernicious effect on the
system-wide throughput, see, e.g., [18, 19, 20, 35] and their references. To
minimize the impact on the system throughput due to transmit distortion, we
focus on reducing the transmitted $\bm{\epsilon}$ or/and received
$\boldsymbol{H}_{\mu}\bm{\epsilon}$ distortion energy. For SU-MIMO received
signal model, all the $N_{\rm L}$ spatial layers target a single user
$\mu\\!=\\!1$ having $N_{\rm R}$ receive antennas. Thus, the received signal
model for both SU-MIMO and over all the receive antennas of $N_{\rm U}$ users
in MU-MIMO is exactly same as $\mathbb{C}^{N_{\rm R}\times
1}\\!\ni\\!\boldsymbol{Y}[:,k]\\!=\\!\left[\boldsymbol{Y}_{1}[:,k];\ldots;\boldsymbol{Y}_{N_{\rm
U}}[:,k]\right]\\!\approx\\!\boldsymbol{H}\left[k\right]{\boldsymbol{X}}\left[:,k\right]\\!+\\!{\boldsymbol{H}\left[k\right]\bm{\epsilon}\left[k\right]}\\!+\\!\boldsymbol{n}[k]$.
Hence, we will use the overall received signal model that is valid for both
SU-MIMO and MU-MIMO.

###  III-B Performance Metrics

We utilize three figures of merit: 1) PAPR and instantaneous  (IPAPR), 2)
inband distortions, and 3) OOBE for the PAPR reduction algorithm design and
its performance.

####  III-B1 PAPR and IPAPR

The PAPR and IPAPR of the MIMO-OFDM signal in the time domain, per the $j$-th
transmit antenna branch, are defined as
$\mathrm{PAPR}_{j}\\!\coloneqq\\!{\left\|\boldsymbol{F}^{\rm
H}\left(\boldsymbol{X}[j,:]\right)^{\rm
T}\right\|_{\infty}^{2}}/{\left\|\boldsymbol{F}^{\rm
H}\left(\boldsymbol{X}[j,:]\right)^{\rm T}\right\|_{2}^{2}}$ and
$\mathrm{IPAPR}_{j}\\!\left[n\right]\\!\coloneqq\\!{\left|\\!\left(\\!\boldsymbol{F}^{\rm
H}\left(\boldsymbol{X}[j,:]\right)^{\rm
T}\\!\right)_{n}\right|^{2}}/{\left\|\boldsymbol{F}^{\rm
H}\left(\boldsymbol{X}[j,:]\right)^{\rm T}\right\|_{2}^{2}}$, respectively,
where $\boldsymbol{F}^{\rm H}\\!\in\\!\mathbb{C}^{\ell K\times\ell K}$ is the
IDFT matrix.

####  III-B2 Inband distortions

The inband distortions are evaluated in terms of three EVM metrics222The EVM
metric evaluation provides insightful and useful information on the link
quality in terms of signal-to-noise ratio (SNR) seen at the receiver due to
the aggregated digital and analogue hardware imperfections [43, 44]. Hence, in
this study, we evaluate the inband performance in terms of proposed EVM
metrics., namely i) (unequalized) EVM at the transmitter, ii) predicted
received EVM, and iii) estimated received EVM. Using (2) and (3), we define
these three instantaneous performance and algorithm design metrics at
subcarrier $k$ for a given MIMO-OFDM symbol.

  1. (i)

Transmit (unequalized) EVM: ${\rm EVM}_{\rm
Tx}\left[k\right]\\!\coloneqq\\!{{\left\|\left(\\!\overline{\boldsymbol{X}}\\!\left[:,k\right]\\!-\\!\boldsymbol{X}\\!\left[:,k\right]\right)\right\|_{2}}/{\left\|\boldsymbol{X}\\!\left[:,k\right]\right\|_{2}}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\approx\\!\left\|\bm{\epsilon}\\!\left[k\right]\right\|_{2}/{\left\|\boldsymbol{X}\\!\left[:,k\right]\right\|_{2}}}$
represents the transmit signal distortion energy. For the performance
evaluation, we assume that the average transmit signal power for each $j$-th
transmit antenna branch
$\mathbb{E}{\left\\{\left\|\boldsymbol{X}[j,:]\right\|_{2}^{2}\right\\}}$ is
fixed, and the total signal energy per MIMO-OFDM symbol is the same before and
after PAPR reduction.

  2. (ii)

Predicted received EVM: ${\rm EVM}_{\rm
pred}\left[k\right]\\!\coloneqq\\!\left\|\boldsymbol{H}\\!\left[k\right]\\!\left(\\!\overline{\boldsymbol{X}}\\!\left[:,k\right]\\!-\\!\boldsymbol{X}\\!\left[:,k\right]\right)\right\|_{2}\\!/\\!\left(\\!\left\|\boldsymbol{H}\\!\left[k\right]\\!\boldsymbol{X}\\!\left[:,k\right]\right\|_{2}\\!\right)$
where $\left\\{\boldsymbol{H}\\!\left[k\right]\right\\}$ is CSI acquired at
the transmitter via the uplink channel for the user-specific transmission.

  3. (iii)

Estimated received EVM per spatial layer after MIMO detection (with channel
equalization): $\allowbreak{\rm EVM}_{\rm
est}\\!\left[l,k\right]\\!\coloneqq\\!{\left|\widehat{\overline{\boldsymbol{S}}}\\!\left[l,k\right]\\!-\\!\boldsymbol{S}\\!\left[l,k\right]\right|}/{\left|\boldsymbol{S}\\!\left[l,k\right]\right|},$
where
$\widehat{\overline{\boldsymbol{S}}}\\!\left[l,k\right]\\!\coloneqq\\!\boldsymbol{G}\\!\left[k\right]\\!\boldsymbol{H}\\!\left[k\right]\\!\overline{\boldsymbol{X}}\\!\left[:,k\right]$
corresponds to the estimated symbol for spatial layer $l$ and subcarrier $k$
with $\boldsymbol{G}\left[k\right]\\!\in\\!\mathbb{C}^{N_{\rm L}\times N_{\rm
T}}$ equalizer at the receiver.

####  III-B3 Out-of-band emissions

In this work, the OOBE requirements are simply characterized by the amount of
distortion energy on the unused subcarriers $k\\!\in\\!\mathcal{T}^{\perp}$
with respect to the useful data-carrying subcarriers $k\\!\in\\!\mathcal{T}$,
which is referred to as the ACLR per $j$-th antenna and can mathematically be
described as
$\mathrm{ACLR}_{j}\\!\coloneqq\\!{\left\|\overline{\boldsymbol{X}}\left[j,\mathcal{T}^{\perp}\right]\right\|_{2}^{2}}/{\left\|\overline{\boldsymbol{X}}\left[j,\mathcal{T}\right]\right\|_{2}^{2}}$.
Note our ACLR definition in the description of optimization problem—posed in
the subsequent section—can be seen as reciprocal to the 3GPP ACLR definition
[24]. However, we interchangeably use both definitions but with the
appropriate sign of the ACLR value.

##  IV Proposed EVM Mitigation with PAPR and ACLR Constraints IN MIMO-OFDM

This section introduces proximal/operator splitting techniques, particularly,
TOP-ADMM proposed recently in [35], that can tackle an appropriate
optimization problem potentially yield computationally efficient algorithm for
the large-scale optimization problem. In the sequel, we formulate a robust
PAPR reduction problem that mitigates the predicted received EVM under channel
estimates’ uncertainty while targeting desired PAPR and ACLR levels.
Subsequently, we formulate a robust problem amenable to the TOP-ADMM-type
algorithms. Finally, we propose a computationally efficient TOP-ADMM-based
algorithm and develop algorithms utilizing two popular BADMM [28] and DYS [32]
techniques.

###  IV-A Efficient Proximal/Operator Splitting Algorithms

As discussed in Section II, we present some proximal/operator splitting
techniques, notably TOP-ADMM, DYS, and BADMM that can render computationally
efficient algorithms, typically using proximal operators, to solve a large-
scale problem in a principled manner. We consider a generic problem,

$\underset{\boldsymbol{x}\in\mathcal{X}}{\operatorname{minimize}}\quad\sum_{m=1}^{M}\\!f_{m}\\!\left({\boldsymbol{x}}\right)\\!+\\!g\left({{\boldsymbol{x}}}\right)\\!+\\!\beta
h\\!\left({{\boldsymbol{x}}}\right),$ (4)

for some $M\\!\geq\\!1$, where $\mathcal{X}$ is a real/complex Euclidean
space, $\left\\{f_{m}(\cdot)\right\\}$, $g(\cdot)$ and $h(\cdot)$ are closed,
convex, and proper functions, and $h$ is $L$-smooth (see Definition 1) with
some scaling $\beta\\!\in\\!\mathbb{R}_{\geq 0}$. Foremost, we layout two
useful definitions in the sequel to introduce proximal/operator splitting
algorithms.

######  Definition 1 ($L$-smooth function [45, 31]).

A differentiable function
$f\\!\left(\\!\boldsymbol{z}\right)\\!\in\\!\mathbb{R}$, where
$\boldsymbol{z}\\!\in\\!\mathbb{C}^{n}$, is $L$-smooth, i.e., has
$L$-Lipschitz continuous gradient (for $L\\!>\\!0$) if
$\left\|\nabla\\!f\\!\left(\boldsymbol{z}_{1}\right)\\!-\\!\nabla\\!f\\!\left(\boldsymbol{z}_{2}\right)\right\|\\!\leq\\!L\\!\left\|\\!\boldsymbol{z}_{1}\\!-\\!\boldsymbol{z}_{2}\\!\right\|\>\forall\boldsymbol{z}_{1},\\!\boldsymbol{z}_{2}\\!\in\\!\mathbb{C}^{n}$.

######  Definition 2 (Proximal mapping [27][31]).

Given a proper closed convex function
$f:\rm{dom}_{f}\mapsto\left(-\infty\right.,\left.+\infty\right]$, then the
proximal mapping of $f$ is the operator given by

$\displaystyle\boldsymbol{\rm{prox}}_{\lambda
f}\left(\boldsymbol{X}\right)\\!=\\!\arg\min_{\boldsymbol{Z}\in\rm{dom}_{f}}\left\\{f\left(\boldsymbol{Z}\right)\\!+\\!\frac{1}{\beta\lambda}\left\|\boldsymbol{X}\\!-\\!\boldsymbol{Z}\right\|_{F}^{2}\right\\}$
(5)

for any $\boldsymbol{X}\in\rm{dom}_{f}$, where $\rm{dom}_{f}$ corresponds to
the domain of a function $f$ and $\lambda>0$. If $\boldsymbol{Z}$ is complex-
valued or real-valued, $\beta=1$ or $\beta=2$, respectively.

######  Definition 3 (Proximal mapping of the indicator function [27][31]).

Let
$f\\!\coloneqq\\!{\delta}_{C}:\rm{dom}_{f}\\!\mapsto\\!\left(-\infty\right.,\left.+\infty\right]$
be an indicator function (also known as characteristic function),

$\displaystyle
f(\boldsymbol{z})={\delta}_{\mathcal{C}}\left(\boldsymbol{z}\right)\coloneqq\left\\{\begin{matrix}0,&\boldsymbol{z}\in\mathcal{C}\\\
+\infty,&\boldsymbol{z}\notin\mathcal{C},\end{matrix}\right.$ (6)

then the proximal mapping of a given set $\mathcal{C}\\!\neq\\!\varnothing$ is
the orthogonal projection operator onto $\mathcal{C}$, i.e.,

$\displaystyle\boldsymbol{\rm{prox}}_{\lambda{\delta}_{\mathcal{C}}}\left(\boldsymbol{z}\right)$
$\displaystyle=\arg\min_{\boldsymbol{y}\in\rm{dom}_{f}}\left\\{{\delta}_{\mathcal{C}}\left(\boldsymbol{y}\right)+\frac{1}{\beta\lambda}\left\|\boldsymbol{z}-\boldsymbol{y}\right\|_{2}^{2}\right\\}$
$\displaystyle\equiv\arg\min_{\boldsymbol{y}\in\mathcal{C}}\left\\{\left\|\boldsymbol{z}-\boldsymbol{y}\right\|_{2}^{2}\right\\}\\!=\\!\boldsymbol{\rm{proj}}_{\mathcal{C}}\left(\boldsymbol{z}\right).$

In the subsequent sections, we present our proposed TOP-ADMM [35] generic
algorithm, and briefly introduce DYS [32] and BADMM [28] algorithms.

####  IV-A1 TOP-ADMM Algorithm [35]

It is one of the generalized algorithms for the classical consensus ADMM—see
details of ADMM and its applications in, e.g., review paper [26]. This TOP-
ADMM can be employed to solve many optimization problems333For instance, see
[35], where we applied TOP-ADMM to tackle a large-scale mask-compliant
spectral precoding problem in the MIMO-OFDM-based wireless communication
systems, particularly 5G NR. that can be expressed as a composite problem such
as (4)—which utilizes the gradient of $L$-smooth function $h$, unlike
classical ADMM. The TOP-ADMM algorithm can be classified as a divide-and-
conquer method, which decomposes a big optimization problem—difficult to solve
in a composite form (IV-A1)—into smaller subproblems that are easy to solve.

Let us reformulate problem (4), without loss of generality, which has at least
one solution, as

$\displaystyle\underset{\left\\{{\boldsymbol{x}}_{m}\in\mathbb{C}^{n}\right\\}\\!,{\boldsymbol{z}}\in\mathbb{C}^{n}}{\operatorname{minimize}}$
$\displaystyle\sum_{m=1}^{M}\\!f_{m}\\!\left({{\boldsymbol{x}}_{m}}\right)\\!+\\!g\left({{\boldsymbol{z}}}\right)\\!+\\!\beta
h\\!\left({{\boldsymbol{z}}}\right)$ $\displaystyle\operatorname{subject\,to}$
$\displaystyle\
{{\boldsymbol{x}}_{m}}-{\boldsymbol{z}}={\boldsymbol{0}},\forall
m\\!=\\!1,\ldots,M.$ (7)

We lay out the general definition of our TOP-ADMM method, which was introduced
in [35].

######  Theorem 1 (TOP-ADMM).

Consider a problem given in (IV-A1) with at least one solution and a suitable
step-size $\tau\\!\in\\!\mathbb{R}_{\geq 0}$. Assume subproblems (8a) and (8)
have solutions, and consider a relaxation/penalty parameter
$\rho\\!\in\\!\mathbb{R}_{>0}$ with some arbitrary initial
$\left(\left\\{\boldsymbol{x}_{m}^{(0)}\right\\},{\boldsymbol{z}^{(0)}},\left\\{\boldsymbol{y}_{m}^{(0)}\right\\}\right)$.
Then

$\displaystyle{\boldsymbol{x}}_{m}^{\left(i+1\right)}\\!=$
$\displaystyle\arg\min_{{\boldsymbol{x}}_{m}}f_{m}\left({\boldsymbol{x}}_{m}\right)\\!+\\!\rho\\!\left\|\\!{{\boldsymbol{x}}_{m}}\\!-\\!{\boldsymbol{z}^{\left(i\right)}}\\!+\\!\frac{{\boldsymbol{y}}_{m}^{\left(i\right)}}{\rho}\right\|_{2}^{2}\;\forall
m$ (8a) $\displaystyle{\boldsymbol{z}}^{\left(i+1\right)}\\!=$
$\displaystyle\arg\min_{{{\boldsymbol{z}}}}g\left({\boldsymbol{z}}\right)$
$\displaystyle+\\!\sum_{m=1}^{M}\rho\left\|{{\boldsymbol{x}}_{m}^{\left(i+1\right)}}\\!-\\!{\boldsymbol{z}}\\!{-\tau\nabla
h\\!\left(\\!{\boldsymbol{z}}^{\left(i\right)}\right)}\\!+\\!\frac{{\boldsymbol{y}}_{m}^{\left(i\right)}}{\rho}\right\|_{2}^{2}$
(8b) $\displaystyle{\boldsymbol{y}}_{m}^{\left(i+1\right)}=$
$\displaystyle{\boldsymbol{y}}_{m}^{\left(i\right)}+\rho\left({{\boldsymbol{x}}_{m}^{\left(i+1\right)}}-{\boldsymbol{z}^{\left(i+1\right)}}\right)\quad\forall
m=1,\ldots,M,$ (8c)

at any limit point, converges to a Karush-Kuhn-Tucker (KKT) stationary point
of (IV-A1).

###### Proof.

We prove dual residual
$\lim_{i\rightarrow+\infty}\\!\left(\\!\boldsymbol{z}^{\left(i+1\right)}\\!-\\!\boldsymbol{z}^{\left(i\right)}\\!\right)\\!=\\!0$
and primal residual
$\lim_{i\rightarrow+\infty}\\!\left(\\!\boldsymbol{x}_{m}^{\left(i+1\right)}\\!-\\!\boldsymbol{z}^{\left(i+1\right)}\\!\right)\\!=\\!0$,
$\forall m\\!=\\!1,\ldots,M$, hold under mild assumptions. Consequently, we
establish the global convergence [46]. ∎

Please observe that the classical consensus ADMM algorithm is a special case
of our proposed TOP-ADMM algorithm when $h\\!=\\!0$ in (IV-A1) or $\nabla
h\\!=\\!0$ in (8). Although classical consensus ADMM can solve many problems
in signal processing for communications, it does not necessarily yield an
implementation-friendly algorithm, particularly, if the proximal operator of
$L$-smooth function $h$ is inefficient to compute.

####  IV-A2 DYS Algorithm [32]

It tackles the subform of (4), i.e.,

$\operatorname{minimize}_{\boldsymbol{x}\in\mathcal{X}}\quad
f_{1}(\boldsymbol{x})\\!+\\!g(\boldsymbol{x})\\!+\\!h(\boldsymbol{x}).$ (9)

######  Theorem 2 (DYS).

Consider a step-size $\tau\\!\coloneqq\\!\left(0,2/L\right)$ with some
arbitrary initial $\boldsymbol{z}^{(0)}\\!\in\\!\mathcal{X}$. Set
$\varrho\\!=\\!2\\!-\\!(\tau L)/2$. Let
$\left(\rho^{(i)}\right)_{i\in\mathbb{N}}$ be a sequence in
$\left[0,\varrho\right]$ such that
$\sum_{i\in\mathbb{N}}\rho^{(i)}\left(\varrho\\!-\\!\rho^{(i)}\right)\\!=\\!+\infty$.
Then, the sequence $\left\\{\boldsymbol{x}^{\left(i+1\right)}\right\\}$
generated by the following iterative scheme

$\displaystyle\boldsymbol{x}^{\left(i+1\right)}\\!=$
$\displaystyle{\boldsymbol{\rm{prox}}}_{\tau
f_{1}}\left(\boldsymbol{z}^{\left(i\right)}\right)$ (10a)
$\displaystyle\boldsymbol{z}^{\left(i+1\right)}\\!=$
$\displaystyle\boldsymbol{z}^{\left(i\right)}\\!+\\!\rho^{(i)}\Bigl{[}\\!{\boldsymbol{\rm{prox}}}_{\tau
g}\\!\bigl{(}2\boldsymbol{x}^{\left(i+1\right)}$ $\displaystyle\hskip
14.22636pt-\\!\boldsymbol{z}^{\left(i\right)}\\!-\\!\tau\nabla
h\\!\left(\boldsymbol{x}^{\left(i+1\right)}\right)\bigr{)}\\!-\\!\boldsymbol{x}^{\left(i+1\right)}\\!\Bigr{]},$
(10b)

converges to a solution of (9).

####  IV-A3 BADMM Algorithm [28]

Although this algorithm may seem to solve a two-operator problem, it utilizes
the Bregman distance to linearize the $L$-smooth function $h$, and
consequently solve the three-operator problem (9). We omit the details, but we
refer the interested reader to [28, Section 2.1].

To this end, we formulate the problem that is amenable to the TOP-ADMM, DYS,
and BADMM.

###  IV-B Proposed Problem Formulations

We propose a principled approach to mitigate (in CSI-aware mode under channel
uncertainty) and/or minimize (in non-CSI-aware mode) signal distortion
according to the appropriate conditions and/or requirements subject to the
PAPR and ACLR constraints. Consequently, we first pose an optimization problem
P0 that essentially minimizes the total transmit EVM over all the allocated
subcarriers and transmit antennas, see (11a), subject to the three performance
metrics introduced in Section III-B, namely nonconvex
$\left\\{\mathrm{PAPR}_{j}\right\\}$, see (11b), frequency-selective predicted
received $\left\\{{\rm EVM}_{\rm pred}\left[k\right]\right\\}$, see (11c), and
nonconvex $\left\\{\mathrm{ACLR}_{j}\right\\}$, see (11d), constraints.
Therefore, the optimization problem can mathematically be described as:

$\displaystyle(\texttt{P0})\quad$
$\displaystyle\underset{\overline{\boldsymbol{X}}\in\mathbb{C}^{N_{\rm
T}\times\ell K}}{\operatorname{minimize}}$
$\displaystyle\left\|\overline{\boldsymbol{X}}-\boldsymbol{X}\right\|_{F}^{2}$
(11a) $\displaystyle\operatorname{subject\,to}$
$\displaystyle\\!\frac{\left\|\boldsymbol{F}^{\rm
H}\overline{\boldsymbol{X}}^{\rm
T}\right\|_{\infty}^{2}}{\left\|\boldsymbol{F}^{\rm
H}\overline{\boldsymbol{X}}^{\rm T}\right\|_{2}^{2}}\preceq\\!\bm{\gamma}_{\rm
par}$ (11b)
$\displaystyle\left\|{\boldsymbol{H}}\\!\left[k\right]\left(\overline{\boldsymbol{X}}\\!-\\!\boldsymbol{X}\right)\boldsymbol{e}_{k}\right\|_{2}\\!\leq\\!\epsilon_{\rm
evm}\left[k\right]\ \forall k\\!\in\\!\mathcal{T}$ (11c)
$\displaystyle{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\frac{\left\|\overline{\boldsymbol{X}}\left[:,\mathcal{T}^{\perp}\right]\right\|_{2}^{2}}{\left\|\overline{\boldsymbol{X}}\left[:,\mathcal{T}\right]\right\|_{2}^{2}}\preceq\bm{\psi}_{\rm
aclr}},$ (11d)

where $\preceq$ denotes element-wise inequality, and $\bm{\gamma}_{\rm
par}\\!\in\\!\mathbb{R}^{N_{\rm T}\\!\times\\!1}$, $\epsilon_{\rm
evm}\\!\left[k\right]\\!\in\\!\mathbb{R}$, and $\bm{\psi}_{\rm
aclr}\\!\in\\!\mathbb{R}^{N_{\rm T}\\!\times\\!1}$ are predefined thresholds
for the desired PAPR, the received EVM per $k$-th useful data-carrying
subcarrier, and ACLR constraint, respectively. These predefined thresholds can
be scalar, i.e., the same values are used for all the antennas. Observe that
this ACLR constraint is not shaping the spectrum such as the methods described
in [38, 35]. However, this constraint could ensure that the minimum 3GPP
requirement of 45 dB [24] is met after some spectrum shaping, e.g.,
${\psi}_{\rm aclr}\\!=\\!-50$ dB, see Section V. Furthermore, although in
(11c) we have used a perfect channel estimate notation, subsequently, we
develop a robust formulation that utilizes imperfect and/or incomplete channel
estimates. Nevertheless, constraint (11c) can essentially exploit the excess
spatial degrees-of-freedom to mitigate the transmit distortion at the
receiver(s)—see Section V-B, particularly, Fig. 11 that shows that the
(predicted/estimated) received EVM decreases with the increasing spatial
degrees-of-freedom or number of base station transmit antennas. Notice that
instead of constraining predicted received EVM in (11c), it is straightforward
to constrain interlayer interference and the received distortion energy—cf.
(2)—that is, replace the left hand of (11c) with
$\left\|\left({\boldsymbol{H}}\\!\left[k\right]\overline{\boldsymbol{X}}\left[:,k\right]\\!-\\!\boldsymbol{S}\left[:,k\right]\right)\right\|_{2}$.
On the contrary, if the base station has no downlink channel knowledge, e.g.,
catering to non-CSI-aware mode for the broadcast/control channel, then
constraint (11c) can be omitted. In the sequel, we will reformulate this
problem, which can easily adapt to CSI-aware and non-CSI-aware modes via
suitable parameter settings.

Clearly, this problem P0 is nonconvex, in particular, due to the PAPR
constraint (11b) and the ACLR constraint (11d), whereas objective (11a) and
constraint (11c) are convex. One could convexify the PAPR constraint by
resorting to peak clipping, i.e., replacing the denominator in (11b), i.e.,
$\left\|\boldsymbol{F}^{\rm H}\overline{\boldsymbol{X}}^{\rm
T}\right\|_{2}^{2}$, with a fixed energy of the OFDM symbol, i.e.,
$\mathbb{E}\left\\{\left\|\boldsymbol{F}^{\rm H}{\boldsymbol{X}}^{\rm
T}\right\|_{2}^{2}\right\\}\\!$, see, e.g., [18]. However, such an approach
would render a suboptimal solution. Therefore, one could employ a semidefinite
relaxation to the PAPR constraint, see, e.g., [20] to improve the accuracy of
the solution at the cost of prohibitive computational complexity
$\text{O}\left(\left(\ell KN_{\rm T}\right)^{4.5}\right)$. Similarly, one
could convexify the ACLR constraint (11d) by replacing the denominator
$\left\|\overline{\boldsymbol{X}}\left[:,\mathcal{T}\right]\right\|_{2}^{2}$
with $\left\|{\boldsymbol{X}}\left[:,\mathcal{T}\right]\right\|_{2}^{2}$.
Thus, even if one convexifies problem P0 by employing fixed values to the
denominators of PAPR and ACLR constraints, computing an optimal solution in
real-time using off-the-shelf convex optimization solvers, typically employs
interior-point methods, such as CVX [47] will result in prohibitively high
computational complexity $\text{O}\left(\left(\ell KN_{\rm
T}\right)^{4.5}\right)$. Thus, an interior-point-based algorithm is infeasible
to deploy on current state-of-the-art radio base stations. Hence, we seek a
computationally efficient algorithm tackling the proposed problem that can be
employed in realistic radio systems.

Toward the goal to develop efficient algorithms, unfortunately proposed
problem P0 is not amenable to first-order optimization methods, notably
operator/proximal splitting—see Section II-D and Section IV-A. Hence, we must
reformulate the problem P0. For the problem reformulation, we now define two
sets:

$\displaystyle\mathbfcal{P}$
$\displaystyle\coloneqq\\!\left\\{\\!\overline{\boldsymbol{X}}\\!\bm{:}\\!\left\|\overline{\boldsymbol{T}}\right\|_{\\!\infty}\\!\preceq\\!\bm{\gamma}_{\rm
par}^{\frac{1}{2}}\\!\left\|\overline{\boldsymbol{T}}\right\|_{2};\overline{\boldsymbol{T}}\\!=\\!\boldsymbol{F}^{\rm
H}\overline{\boldsymbol{X}}^{\rm T}\\!\right\\}$
$\displaystyle\equiv\\!\left\\{\\!\overline{\boldsymbol{X}}\\!\bm{:}\\!\left\|\\!\overline{\boldsymbol{T}}\\!\boldsymbol{e}_{j}\\!\right\|_{\infty}^{2}\\!\leq\\!{\gamma}_{\rm
par}\\!\left\|\\!\overline{\boldsymbol{T}}\\!\boldsymbol{e}_{j}\\!\right\|_{2}^{2};\overline{\boldsymbol{T}}\\!=\\!\boldsymbol{F}^{\rm
H}\\!\overline{\boldsymbol{X}}^{\rm T};j\\!=\\!1,\\!\ldots\\!,N_{\rm
T}\\!\right\\}$ (12) $\displaystyle\mathbfcal{U}$
$\displaystyle\coloneqq\\!\left\\{\\!\overline{\boldsymbol{X}}\\!\bm{:}\\!\left\|\overline{\boldsymbol{X}}\\!\left[:,\mathcal{T}^{\perp}\\!\right]\\!\right\|_{2}\\!\preceq\\!\bm{\psi}_{\rm
aclr}^{\frac{1}{2}}\\!\left\|\overline{\boldsymbol{X}}\\!\left[:,\mathcal{T}\right]\\!\right\|_{2}\right\\}$
$\displaystyle\equiv\\!\left\\{\\!\overline{\boldsymbol{X}}\bm{:}\\!\left\|\\!\left(\overline{\boldsymbol{X}}\\!\left[:,\mathcal{T}^{\perp}\right]\right)\\!\boldsymbol{e}_{j}^{{\rm
T}}\right\|_{2}^{2}\\!\leq\\!{\psi}_{\rm
aclr}\\!\left\|\\!\left(\overline{\boldsymbol{X}}\\!\left[:,\mathcal{T}\right]\right)\\!\boldsymbol{e}_{j}^{{\rm
T}}\right\|_{2}^{2}\\!;\forall j\right\\}$ (13)

corresponding to the nonconvex PAPR (11b) and ACLR (11d) constraints,
respectively. Subsequently, we use an indicator function to these sets
$\mathbfcal{P}$ and $\mathbfcal{U}$, i.e.,
${\delta}_{\mathbfcal{P}}\left(\overline{\boldsymbol{X}}\right)$ and
${\delta}_{\mathbfcal{U}}\left(\overline{\boldsymbol{X}}\right)$, where the
indicator function definition to any set $\mathcal{C}$ is given in (6).
Furthermore, we define $h_{\rm
TxEVM}\left(\overline{\boldsymbol{X}}\right)\\!\coloneqq\\!\left\|\overline{\boldsymbol{X}}\\!-\\!\boldsymbol{X}\\!\right\|_{F}^{2}$.
Consequently, using these definitions, we can reformulate an equivalent
problem to P0, without loss of generality, as

$\displaystyle(\texttt{P1})\quad$
$\displaystyle\underset{\overline{\boldsymbol{X}}\in\mathbb{C}^{N_{\rm
T}\times\ell K}}{\operatorname{minimize}}$ $\displaystyle\quad h_{\rm
TxEVM}\left(\overline{\boldsymbol{X}}\right)\\!+\\!{\delta}_{\mathbfcal{P}}\left(\overline{\boldsymbol{X}}\right)\\!+\\!{\delta}_{\mathbfcal{U}}\left(\overline{\boldsymbol{X}}\right)$
(14a) $\displaystyle\operatorname{subject\,to}$
$\displaystyle\quad\left\|{\boldsymbol{H}}\\!\left[k\right]\left(\overline{\boldsymbol{X}}\\!-\\!\boldsymbol{X}\right)\boldsymbol{e}_{k}\right\|_{2}^{2}\\!\leq\\!\epsilon_{\rm
evm}^{2}\left[k\right]\ \forall k\\!\in\\!\mathcal{T}.$ (14b)

Clearly, problem P1 is equally hard to solve as P0 because of nonconvexity of
indicator functions to nonconvex sets $\mathbfcal{P}$ and $\mathbfcal{U}$.
Question remains whether there are any possible (approximate) techniques to
solve P1. Unfortunately, the answer is still negative unless there is some
reformulation.

Before we progress further on tackling nonconvex problem P0/P1, let us express
convexified problem P0/P1 in an unconstrained form:

$\displaystyle(\texttt{Convexified: P2})$
$\displaystyle\underset{\overline{\boldsymbol{X}}\in\mathbb{C}^{N_{\rm
T}\times\ell K}}{\operatorname{minimize}}\ h_{\rm
TxEVM}\left(\overline{\boldsymbol{X}}\right)\\!+\\!{\delta}_{\mathbfcal{P}^{\left(0\right)}}\left(\overline{\boldsymbol{X}}\right)\\!+\\!{\delta}_{\mathbfcal{U}^{\left(0\right)}}\left(\overline{\boldsymbol{X}}\right)\\!+\\!{\delta}_{\mathbfcal{E}}\left(\overline{\boldsymbol{X}}\right),$
(15)

where convexified PAPR constraint set reads

$\mathbfcal{P}^{(0)}\\!\coloneqq\\!\left\\{\\!\overline{\boldsymbol{X}}\bm{:}\left\|\overline{\boldsymbol{T}}\right\|_{\infty}\\!\preceq\\!\sqrt{\bm{\gamma}_{\rm
par}}\left\|{\boldsymbol{T}}\right\|_{2};\overline{\boldsymbol{T}}\\!=\\!\boldsymbol{F}^{\rm
H}\overline{\boldsymbol{X}}^{\rm T};{\boldsymbol{T}}\\!=\\!\boldsymbol{F}^{\rm
H}\\!{\boldsymbol{X}}^{\rm T}\\!\right\\}$

and convexified ACLR constraint set is

$\mathbfcal{U}^{(0)}\\!\coloneqq\\!\left\\{\overline{\boldsymbol{X}}\bm{:}\left\|\overline{\boldsymbol{X}}\left[:,\mathcal{T}^{\perp}\right]\right\|_{2}\\!\preceq\\!\sqrt{\bm{\psi}_{\rm
aclr}}\left\|{\boldsymbol{X}}\left[:,\mathcal{T}\right]\right\|_{2}\right\\},$

i.e., the respective denominators are fixed as discussed previously. The
convex set corresponding to the predicted received EVM can be deonoted by
$\mathbfcal{E}\\!\coloneqq\\!\left\\{\overline{\boldsymbol{X}}\bm{:}\left\|{\boldsymbol{H}}\\!\left[k\right]\left(\overline{\boldsymbol{X}}\\!-\\!\boldsymbol{X}\right)\boldsymbol{e}_{k}\right\|_{2}^{2}\\!\leq\\!\epsilon_{\rm
evm}^{2}\left[k\right]\ \forall k\\!\in\\!\mathcal{T}\right\\}$.

Although problem P2 is convex, employing off-the-shelf solvers like CVX [47]
is not easible for a realistic radio systems due to prohibitive high
computational complexity. Hence, developing a computationally efficient
algorithm is challenging using any (first-order) proximal splitting algorithm
for problem P1/P2, notably and unfortunately due to the computation of
proximal operator for
${\delta}_{\mathbfcal{E}}\left(\overline{\boldsymbol{X}}\right)$, i.e., an
orthogonal projection operator on an ellipsoid having
$\operatorname{rank}\\!\geq\\!1$ as it involves eigenvalue decomposition—see,
e.g. [35, Theorem 4] and their references. In other words, the orthogonal
projection operator of such an ellipsoid for the predicted received EVM is
computationally expensive. Although the orthogonal projection operators
corresponding to the nonconvex ACLR and PAPR constraints are still notoriously
hard, we will propose later in the subsequent sections how to possibly tackle
these nonconvex constraints. Before proceeding further on attacking predicted
received EVM constraint, let us formulate a robust version of predicted
received EVM against channel uncertainty if channel estimates are available.

Problem P1 Reformulation for Robust Mitigation of Distortion: Dropping
subcarrier index $k$, let the channel estimation error model be described by
$\boldsymbol{H}\\!\coloneqq\\!\widehat{\boldsymbol{H}}\\!+\\!\Delta\boldsymbol{H}$,
where $\boldsymbol{H}$, $\widehat{\boldsymbol{H}}$, and $\Delta\boldsymbol{H}$
denote true, estimated, and error, respectively. Additionally, let the error
at subcarrier $k$ be denoted by
$\Delta{\boldsymbol{X}}\\!\coloneqq\\!\left(\overline{\boldsymbol{X}}\\!\left[:,k\right]\\!-\\!{\boldsymbol{X}}\\!\left[:,k\right]\right)$.
Furthermore, for simplicity, we assume that $\Delta\boldsymbol{H}$ takes
values from the bounded set
$\left\\{\left\|\Delta\boldsymbol{H}\right\|_{F}^{2}\leq\sigma^{2}_{\rm
ce}\right\\}$, where $\sigma^{2}_{\rm ce}\\!>\\!0$ describes the channel
uncertainty that is assumed to be known to the transmitter. Applying the
triangle inequality followed by the Cauchy-Schwarz inequality to (11c)/(14b),
we have

$\displaystyle\Bigl{\|}{\boldsymbol{H}}\Delta\\!{\boldsymbol{X}}\\!\Bigr{\|}_{2}^{2}$
$\displaystyle=\\!\Bigl{\|}\left(\widehat{\boldsymbol{H}}\\!+\\!\Delta\\!\boldsymbol{H}\right)\Delta{\boldsymbol{X}}\Bigr{\|}_{2}^{2}$
$\displaystyle\leq\left\|\widehat{\boldsymbol{H}}\Delta{\boldsymbol{X}}\right\|_{2}^{2}\\!+\\!\left\|\Delta\boldsymbol{H}\Delta{\boldsymbol{X}}\right\|_{2}^{2}$
$\displaystyle\leq\Delta\\!{\boldsymbol{X}}^{\rm
H}\\!\left(\widehat{\boldsymbol{H}}^{\rm
H}\widehat{\boldsymbol{H}}\\!+\\!\sigma^{2}_{\rm
ce}\boldsymbol{I}\\!\right)\Delta\\!{\boldsymbol{X}}.$ (16)

Using the inequality (IV-B) in the constraint (11c)/(14b), the channel
estimation error-aware constraint is guaranteed if
$\left\|\boldsymbol{Q}^{\nicefrac{{1}}{{2}}}\left[k\right]\left(\overline{\boldsymbol{X}}\\!-\\!\boldsymbol{X}\right)\boldsymbol{e}_{k}\right\|_{2}\\!\leq\\!\epsilon_{\rm
evm}[k]$, where matrix $\boldsymbol{Q}$ is defined as

${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbb{C}^{N_{\rm
T}\times N_{\rm
T}}\ni}\boldsymbol{Q}[k]\\!\coloneqq\\!\left(\widehat{\boldsymbol{H}}[k]^{\rm
H}\widehat{\boldsymbol{H}}[k]\\!+\\!{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\nu}\boldsymbol{I}_{N_{\rm
T}}\\!\right),$ (17)

where $\nu$ is a user-defined parameter to regularize the Gram matrix under
channel estimation error. However, the $\nu$ parameter does not have to be the
same as the channel estimation error variance $\sigma^{2}_{\rm ce}$—see Fig.
11, which shows the estimated received EVM metric against channel estimation
error variance for a fixed $\nu$.

To achieve our computationally efficient algorithms for our proposed problem,
let us further reformulate the problem P1 under channel uncertainty by forming
the Lagrangian such that we have

$\displaystyle
L_{\texttt{P1}}\\!\left(\overline{\boldsymbol{X}},\left\\{\lambda_{k}\right\\}\\!\right)\\!\coloneqq$
$\displaystyle\\!h_{\rm
TxEVM}\left(\overline{\boldsymbol{X}}\right)\\!+\\!{\delta}_{\mathbfcal{P}}\left(\overline{\boldsymbol{X}}\right)\\!+\\!{\delta}_{\mathbfcal{U}}\left(\overline{\boldsymbol{X}}\right)$
$\displaystyle\\!+\\!\sum_{k\in\mathcal{T}}\\!\lambda_{k}\\!\left(\\!\left\|{\boldsymbol{Q}}^{\nicefrac{{1}}{{2}}}\\!\left[k\right]\left(\overline{\boldsymbol{X}}\\!-\\!\boldsymbol{X}\\!\right)\\!\boldsymbol{e}_{k}\right\|_{2}^{2}\\!-\\!\epsilon_{\rm
evm}^{2}\left[k\right]\\!\right)\\!.$

As we know from the Lagrange duality [23], the primal problem P1 can
equivalently be expressed as

$\underset{\overline{\boldsymbol{X}}}{\operatorname{minimize}}\
\sup_{\left\\{\lambda_{k}\right\\}}L_{\texttt{P1}}\left(\overline{\boldsymbol{X}},\left\\{\lambda_{k}\right\\}\right).$

It is well-known that the dual function
$d(\left\\{\lambda_{k}\right\\})\\!\coloneqq\\!\inf_{\overline{\boldsymbol{X}}}L_{\texttt{P1}}\left(\overline{\boldsymbol{X}},\left\\{\lambda_{k}\right\\}\right)$
is a concave function due to pointwise infimum of affine functions of
$\lambda_{k}$, even if the primal problem is nonconvex. Moreover, for the
inequality constraint, the Lagrange multipliers are nonnegative
$\lambda_{k}\\!\in\\!\mathbb{R}_{\geq 0}$. Unfortunately, computing a dual
function for problem P1 is not trivial, which yields a lower bound on the
optimal primal value. Therefore, toward the goal to develop implementation-
friendly algorithms, let us assume that we have access to near/sub-optimal
Lagrange multipliers $\lambda_{k}^{\star}\\!\in\\!\mathbb{R}_{\geq 0}$. Hence,
we now reformulate problem P1 that is robust against channel uncertainty and
avoids costly orthogonal projections onto ellipsoids corresponding to
predicted received EVM even though the problem is still nonconvex:

$\displaystyle\underset{\overline{\boldsymbol{X}}\in\mathbb{C}^{N_{\rm
T}\times\ell K}}{\operatorname{minimize}}\\!\Bigl{\\{}\\!L_{\texttt{P1}}$
$\displaystyle\\!\left(\overline{\boldsymbol{X}},\left\\{\lambda_{k}^{\star}\right\\}\right)$
$\displaystyle=$ $\displaystyle h_{\rm
TxEVM}\left(\overline{\boldsymbol{X}}\right)\\!+\\!\sum_{k\in\mathcal{T}}\\!\lambda_{k}^{\star}\\!\left\|{\boldsymbol{Q}}^{\nicefrac{{1}}{{2}}}\\!\left[k\right]\left(\overline{\boldsymbol{X}}\\!-\\!\boldsymbol{X}\right)\boldsymbol{e}_{k}\right\|_{2}^{2}$
$\displaystyle+\\!{\delta}_{\mathbfcal{P}}\left(\overline{\boldsymbol{X}}\right)\\!+\\!{\delta}_{\mathbfcal{U}}\left(\overline{\boldsymbol{X}}\right)\Bigr{\\}}.$
(18)

We now absorb $\lambda_{k}\\!=\\!\lambda_{k}^{\star}$ in
$\boldsymbol{Q}\\!\left[k\right]$ and offer a trade-off between $h_{\rm
TxEVM}$ and the predicted received EVM with a user-defined parameter
$\zeta\\!\in\\!\mathbb{R}_{\geq 0}$. Consequently, we succinctly represent the
above problem as

$(\texttt{P3})\quad\underset{\overline{\boldsymbol{X}}\in\mathbb{C}^{N_{\rm
T}\times\ell K}}{\operatorname{minimize}}\
{\delta}_{\mathbfcal{P}}\left(\overline{\boldsymbol{X}}\right)\\!+\\!{\delta}_{\mathbfcal{U}}\left(\overline{\boldsymbol{X}}\right)\\!+\\!h\left(\overline{\boldsymbol{X}}\right),$
(19)

where

$h\left(\overline{\boldsymbol{X}}\right)\\!\coloneqq\\!\sum_{k\in\mathcal{T}}\left\|{\boldsymbol{Q}}^{\nicefrac{{1}}{{2}}}\\!\left[k\right]\left(\overline{\boldsymbol{X}}\\!-\\!\boldsymbol{X}\right)\boldsymbol{e}_{k}\right\|_{2}^{2}\\!+\\!\zeta
h_{\rm TxEVM}\left(\overline{\boldsymbol{X}}\right)$ (20)

is a differentiable function, the gradient of which is

$\nabla
h\left(\overline{\boldsymbol{X}}\right)=\sum_{k\in\mathcal{T}}\boldsymbol{Q}\\!\left[k\right]\left(\overline{\boldsymbol{X}}\\!-\\!\boldsymbol{X}\\!\right)\\!\boldsymbol{e}_{k}\boldsymbol{e}_{k}^{\rm
T}\\!+\\!\zeta\\!\left(\overline{\boldsymbol{X}}\\!-\\!\boldsymbol{X}\\!\right).$
(21)

If $\zeta$ is a large value, then EVM at the transmitter will be minimized
more than the predicted received EVM. Observe that when the base station has
no downlink channel knowledge, e.g., for broadcast channels, then one
possibility is $\boldsymbol{Q}[k]\\!=\\!\lambda_{k}\boldsymbol{I}_{N_{\rm T}}$
with $\lambda_{k}\\!\in\\!\mathbb{R}_{\geq 0}$ and other possibilities can be
explored in future work. Observe that one could argue that it is not easy to
find optimal dual variables $\lambda_{k}^{\star}$. Therefore, alternatively,
one can envisage to minimize the maximum over all the predicted received EVM,
i.e.,
$\max\left\\{\left\|{\boldsymbol{Q}}^{\nicefrac{{1}}{{2}}}\\!\left[k\right]\left(\overline{\boldsymbol{X}}\\!-\\!\boldsymbol{X}\right)\boldsymbol{e}_{k}\right\|_{2}^{2}\right\\}$—reminiscent
of infinite norm. Unfortunately, this approach is equally implementation
unfriendly as minimizing peaks (or convexified PAPR), i.e., minimizing the
infinite norm. Thus, we do not pursue this approach further in this work.

Before we attack on the nonconvexity of problem P3, we convexify P3 similar to
(IV-B):

$\displaystyle(\texttt{Convexified: P4})$
$\displaystyle\underset{\overline{\boldsymbol{X}}\in\mathbb{C}^{N_{\rm
T}\times\ell
K}}{\operatorname{minimize}}\quad{\delta}_{\mathbfcal{P}^{\left(0\right)}}\left(\overline{\boldsymbol{X}}\right)\\!+\\!{\delta}_{\mathbfcal{U}^{\left(0\right)}}\left(\overline{\boldsymbol{X}}\right)\\!+\\!h\left(\overline{\boldsymbol{X}}\right).$
(22)

Reiterating, although this convex problem P4 can be solved by any off-the-
shelf solvers such as CVX [47], clearly, a general purpose solver is
infeasible in realistic radio systems. Hence, we forage for computationally
efficient algorithms to solve large-scale problem P3 and P4.

###  IV-C Efficient Solution of P3 Utilizing TOP-ADMM-type Algorithms

In this section, we develop three implementation-friendly algorithms utilizing
TOP-ADMM, DYS, and BADMM techniques described in Section IV-A, which solve
problem P3 by invoking our proposed conjecture. Naturally, these designed
algorithms inherently tackle problem P4 as a special case.

In general, the considered proximal/operator splitting techniques, described
in Section IV-A, need to compute the proximal operators of indicator functions
${\delta}_{\mathbfcal{P}}$ and ${\delta}_{\mathbfcal{U}}$ to nonconvex sets,
i.e., orthogonal projections onto the respective nonconvex sets—see Definition
3. Unfortunately, no (efficient) projection operators
${\boldsymbol{\rm{proj}}}_{\mathbfcal{P}}$ and
${\boldsymbol{\rm{proj}}}_{\mathbfcal{U}}$ are known for the nonconvex PAPR
and ACLR sets, i.e., $\mathbfcal{P}$ and $\mathbfcal{U}$, respectively. On the
contrary, for the convexified problem P4, the orthogonal projection operator
${\boldsymbol{\rm{proj}}}_{\mathbfcal{P}^{(0)}}$ for the convexified PAPR
constraint set $\mathbfcal{P}^{(0)}$ is $l_{\infty}$-norm ball for each
transmit antenna, which is provided below in Proposition 1. Furthermore, the
orthogonal projection operator
${\boldsymbol{\rm{proj}}}_{\mathbfcal{U}^{(0)}}$ for the convexified ACLR
constraint set $\mathbfcal{U}^{(0)}$ is $l_{2}$-norm ball for each transmit
antenna, which is given below in Proposition 2.

######  Proposition 1 (Proj onto the $l_{\infty}$-norm ball).

Let $\mathcal{C}\\!=\\!\left\\{\boldsymbol{x}\\!\in\\!\mathbb{C}^{K\times
1}\\!\bm{:}\\!\left\|\boldsymbol{x}\right\|_{\infty}\\!-\\!r\\!\leq\\!0\right\\}$,
where $r\\!\in\\!\mathbb{R}_{\geq 0}$, then the orthogonal projection operator
is
$\boldsymbol{\rm{proj}}_{\mathcal{C}}\left(\boldsymbol{x}\right)\\!=\\!\left(\left(\frac{x\left[k\right]}{\left|x\left[k\right]\right|}\right)\min\Bigl{\\{}\bigl{|}x\left[k\right]\bigl{|},r\Bigr{\\}}\right)_{1\leq
k\leq K}.$

###### Proof.

Invoking the Moreau decomposition theorem [31, Section 6.6] and using the
$l_{1}$-norm proximal operator result [31, Example 6.8], the proof follows
after the rearrangement. ∎

######  Proposition 2 (Proj onto the $l_{2}$-norm ball [31, Lemma 6.26]).

Let $E\subseteq\mathbb{C}^{p\times 1}$ and $E\neq\varnothing$ be given by
$E\\!\coloneqq\\!\mathcal{B}_{\|\cdot\|_{2}}\left[\boldsymbol{c},r\right]\\!=\\!\left\\{\boldsymbol{x}\\!\in\\!\mathbb{C}^{p\times
1}:\left\|\boldsymbol{x}\\!-\\!\boldsymbol{c}\right\|_{2}\\!\leq\\!r\right\\}$,
then the orthogonal projection operator for the $l_{2}$-norm ball, i.e.,
$\mathcal{B}_{\|\cdot\|_{2}}\\!\left[\boldsymbol{c},r\right]$ with a given
center $\boldsymbol{c}$ and the radius $r$, is
$\boldsymbol{\rm{proj}}_{\mathcal{B}_{\|\cdot\|_{2}}\left[\boldsymbol{c},r\right]}\left(\boldsymbol{x}\right)\\!=\\!\boldsymbol{c}\\!+\\!\left(\\!\frac{r}{\max\left\\{\left\|\boldsymbol{x}\\!-\\!\boldsymbol{c}\right\|_{2},r\right\\}}\\!\right)\\!\left(\boldsymbol{x}\\!-\\!\boldsymbol{c}\right).$

Toward the goal to develop efficient algorithms utilizing TOP-ADMM-type
techniques for nonconvex problem P3 (and inherently for P4 as a special case),
we solve the appropriate subproblems at iteration $(i\\!+\\!1)$ by replacing
the nonconvex constraint sets (IV-B) and (IV-B) with the following convex but
employing generated sequence $\overline{\boldsymbol{X}}^{\left(i\right)}$ of
previous iteration $i$ for PAPR and ACLR constraint sets:

$\displaystyle\mathbfcal{P}^{\left(i\right)}$
$\displaystyle\coloneqq\\!\Biggl{\\{}\\!\overline{\boldsymbol{X}}\bm{:}\left\|\overline{\boldsymbol{T}}\right\|_{\infty}\\!\preceq\\!\sqrt{\bm{\gamma}_{\rm
par}}\left\|\overline{\boldsymbol{T}}^{\left(i\right)}\right\|_{2};$
$\displaystyle\hskip
28.45274pt\overline{\boldsymbol{T}}\\!=\\!\boldsymbol{F}^{\rm
H}\overline{\boldsymbol{X}}^{\rm
T};\,\overline{\boldsymbol{T}}^{\left(i\right)}\\!=\\!\boldsymbol{F}^{\rm
H}\left(\overline{\boldsymbol{X}}^{\left(i\right)}\right)^{\\!{\rm
T}}\\!\Biggr{\\}},$ $\displaystyle\mathbfcal{U}^{\left(i\right)}$
$\displaystyle\coloneqq\\!\left\\{\overline{\boldsymbol{X}}\bm{:}\\!\left\|\overline{\boldsymbol{X}}\\!\left[:,\mathcal{T}^{\perp}\\!\right]\\!\right\|_{2}\\!\preceq\\!\bm{\psi}_{\rm
aclr}^{\frac{1}{2}}\\!\left\|\overline{\boldsymbol{X}}^{\left(i\right)}\\!\left[:,\mathcal{T}\right]\\!\right\|_{2}\right\\}.$

Theorem 1 establishes the global convergence of TOP-ADMM for the general
convex problem. However, based on extensive numerical evidence, we conjecture
that the asymptotically vanishing property of the (dual/primal) residual
errors of TOP-ADMM holds for the nonconvex problem P3 with iteration-dependent
sets—see Fig. 6–Fig. 9. Consequently, under this conjecture, the proposed TOP-
ADMM algorithm satisfies the KKT optimality conditions of nonconvex problem
P3—see Appendix A. Additionally, this optimality condition implies that
$\lim_{i\rightarrow\infty}\mathbfcal{P}^{\left(i\right)}\\!=\\!\mathbfcal{P}$
and
$\lim_{i\rightarrow\infty}\mathbfcal{U}^{\left(i\right)}\\!=\\!\mathbfcal{U}$.
Therefore, we argue that a KKT point for nonconvex P3 is not worse than a KKT
point of convex P4 by applying TOP-ADMM-type techniques. Noticeably, in Fig. 5
and Fig. 12, the iteration-dependent set renders a better solution compared to
the fixed set for the considered test cases. However, proving the
conjecture444During the revision process, we came across [48] and its
references, which evidently employs similar techniques within the
“majorization-minimization” framework to approximate the nonconvex constraint
set for a difference of convex functions, i.e., $f(x)\\!-\\!g(x)\\!\leq\\!0$.
is the topic of future research.

Algorithm 1 TOP-ADMM-based PAPR Reduction for P3/P4

1:$\overline{\boldsymbol{X}}^{\left(0\right)},\overline{\boldsymbol{Z}}^{\left(0\right)}\\!,{{\bm{\Lambda}}^{\left(0\right)}}\\!\in\\!\mathbb{C}^{N_{\rm
T}\times\ell K}$, $\tau\\!\in\\!\mathbb{R}$;
$\left\\{\boldsymbol{Q}\left[k\right]\\!\in\\!\mathbb{C}^{N_{\rm T}\times
N_{\rm T}}\right\\}$; $\rho\\!\in\\!\mathbb{R}_{>0}$

2:$\overline{\boldsymbol{X}}^{\left(\cdot\right)}\in\mathbb{C}^{N_{\rm
T}\times\ell K}$ or
$\overline{\boldsymbol{Z}}^{\left(\cdot\right)}\in\mathbb{C}^{N_{\rm
T}\times\ell K}$

3:for $i=0,1,\ldots$ do

4: $\displaystyle\overline{\boldsymbol{X}}^{\left(i+1\right)}\\!$
$\displaystyle\coloneqq\\!\boldsymbol{\rm{proj}}_{\mathbfcal{U}^{\left(i\right)}}\left(\\!{\overline{\boldsymbol{Z}}}^{\left(i\right)}\\!-\\!\frac{{\bm{\Lambda}}^{\left(i\right)}}{\rho}\right)$
(23a) $\displaystyle\overline{\boldsymbol{Z}}^{\left(i+1\right)}\\!$
$\displaystyle\coloneqq\\!\boldsymbol{\rm{proj}}_{\mathbfcal{P}^{\left(i\right)}}\\!\left(\\!\overline{\boldsymbol{X}}^{\left(i+1\right)}\\!-\\!\tau\nabla
h\\!\left(\\!\overline{\boldsymbol{Z}}^{\left(i\right)}\right)\\!+\\!\frac{{\bm{\Lambda}}^{\left(i\right)}}{\rho}\\!\right)$
$\displaystyle\equiv\\!\left[\\!\boldsymbol{F}\boldsymbol{\rm{proj}}_{\mathbfcal{P}_{T}^{\left(i\right)}}\\!\left(\\!\boldsymbol{F}^{\rm
H}\\!\left(\\!\overline{\boldsymbol{X}}^{\left(i+1\right)}\\!-\\!\tau\nabla
h\\!\left(\\!\overline{\boldsymbol{Z}}^{\left(i\right)}\right)\\!+\\!\frac{{\bm{\Lambda}}^{\left(i\right)}}{\rho}\\!\right)^{\\!{\rm
T}}\\!\right)\\!\right]^{\rm T}$ (23b)
$\displaystyle{\bm{\Lambda}}^{\left(i+1\right)}$
$\displaystyle\coloneqq{\bm{\Lambda}}^{\left(i\right)}\\!+\\!\rho\left(\overline{\boldsymbol{X}}^{\left(i+1\right)}\\!-\\!\overline{\boldsymbol{Z}}^{\left(i+1\right)}\right)$
(23c)

5:end for

####  IV-C1 TOP-ADMM-Based Algorithm for P3/P4

For the brevity of the algorithm description, we redefine the PAPR set
${\mathbfcal{P}_{T}}^{\left(i\right)}\\!\coloneqq\\!\left\\{\\!\overline{\boldsymbol{T}}\bm{:}\left\|\overline{\boldsymbol{T}}\right\|_{\infty}\\!\preceq\\!\sqrt{\bm{\gamma}_{\rm
par}}\left\|\overline{\boldsymbol{T}}^{\left(i\right)}\right\|_{2}\\!\right\\}$
and apply TOP-ADMM method (8) to tackle P3/P4 by setting $M\\!=\\!1$,
$f_{1}\\!=\\!{\delta}_{\mathbfcal{U}^{(i)}}$,
$g\\!=\\!{\delta}_{\mathbfcal{P}^{(i)}}$,
$\overline{\boldsymbol{X}}\\!=\\!\boldsymbol{x}_{1}$,
$\overline{\boldsymbol{Z}}\\!=\\!\boldsymbol{z}$,
$\bm{\Lambda}\\!=\\!\boldsymbol{y}_{1}$. The resulting TOP-ADMM-based
algorithm for P3/P4 is given in Algorithm 1. The gradient $\nabla h$ is given
by (21). Observe that both primal updates, i.e., (23a) and (23), are easy to
solve and have closed-form solutions. For the
$\overline{\boldsymbol{X}}^{\left(i+1\right)}$ update in (23a)—corresponding
to $\boldsymbol{\rm{proj}}_{\mathbfcal{U}^{\left(i\right)}}$—we apply
Proposition 2 with center $\boldsymbol{C}\\!=\\!\boldsymbol{0}$ and radius
$r\\!=\\!\sqrt{\bm{\psi}_{\rm
aclr}}\left\|\overline{\boldsymbol{X}}^{\left(i\right)}\left[:,\mathcal{T}\right]\right\|_{2}$.
Subsequently, we invoke Proposition 3 for the
$\overline{\boldsymbol{X}}^{\left(i+1\right)}$ and
$\overline{\boldsymbol{Z}}^{\left(i+1\right)}$ updates in (23a) and (23),
respectively, such that the updates are performed independently for each
transmit antenna. Consequently, the orthogonal projection in (23a), i.e.,
$\boldsymbol{\rm{proj}}_{\mathbfcal{U}^{\left(i\right)}}$, corresponding to
the ACLR constraint for each transmit antenna, is given by Proposition 2. The
orthogonal projection in (23), i.e.,
$\boldsymbol{\rm{proj}}_{\mathbfcal{P}^{\left(i\right)}}$, corresponding to
the PAPR constraint per transmit antenna branch is given by Proposition 1.
Moreover, (23) update exploits the proximal operator property given in
Proposition 4. Notice that to solve P4, the sets for both PAPR and ACLR are
not iteration dependent, i.e., we set
${\boldsymbol{\rm{proj}}}_{\mathbfcal{U}^{(i)}}\\!=\\!{\boldsymbol{\rm{proj}}}_{\mathbfcal{U}^{(0)}}$
and
${\boldsymbol{\rm{proj}}}_{\mathbfcal{P}^{(i)}}\\!=\\!{\boldsymbol{\rm{proj}}}_{\mathbfcal{P}^{(0)}}$
for all $i\\!\geq\\!0$.

######  Proposition 3 (Separation theorem for proximal operators[31, 27]).

If a function
$f\left(\boldsymbol{X}\right)\\!=\\!\sum_{k=1}^{n}f_{i}\left(\boldsymbol{x}_{k}\right)$
is separable across the variables column-wise
$\boldsymbol{X}\\!=\\!\left[\boldsymbol{x}_{1},\ldots,\boldsymbol{x}_{n}\right]$
and row-wise
$\boldsymbol{X}\\!=\\!\left[\boldsymbol{x}_{1};\ldots;\boldsymbol{x}_{N_{\rm
T}}\right]$, then the respective $\boldsymbol{\rm{prox}}$ operators are as
$\boldsymbol{\rm{prox}}_{f}\left(\boldsymbol{X}\right)\\!=\\!\left[\boldsymbol{\rm{prox}}_{f_{1}}\left(\boldsymbol{x}_{1}\right),\ldots,\boldsymbol{\rm{prox}}_{f_{n}}\left(\boldsymbol{x}_{n}\right)\right]$
and
$\boldsymbol{\rm{prox}}_{f}\left(\boldsymbol{X}\right)\\!=\\!\big{[}\boldsymbol{\rm{prox}}_{f_{1}}\left(\boldsymbol{x}_{1}\right);\ldots;\allowbreak\boldsymbol{\rm{prox}}_{f_{N_{\rm
T}}}\left(\boldsymbol{x}_{N_{\rm T}}\right)\big{]}$, respectively.

######  Proposition 4 (Prox map for composition with an affine mapping [31,
Theorem 6.15]).

Let $g:\mathbb{C}^{m\times 1}\\!\mapsto\\!\bigl{(}-\infty,+\infty\bigr{]}$ be
a closed proper convex, and
$f\left(\boldsymbol{x}\right)\\!=\\!g\left(\mathcal{A}\left(\boldsymbol{x}\right)\\!+\\!\boldsymbol{b}\right)$,
where $\boldsymbol{b}\\!\in\\!\mathbb{C}^{m\times 1}$ and
$\mathcal{A}:\mathcal{X}\\!\mapsto\\!\mathbb{C}^{m\times 1}$ satisfying
$\mathcal{A}\circ\mathcal{A}^{{\rm H}}\\!=\\!\nu\boldsymbol{I}$ where
$\nu\\!\in\mathbb{R}_{>0}$, then for any
$\boldsymbol{x}\\!\in\\!\mathbb{C}^{n\times 1}$
$\boldsymbol{\rm{prox}}_{f}\left(\boldsymbol{x}\right)\\!=\\!\boldsymbol{x}\\!+\\!\frac{1}{\lambda}\mathcal{A}^{\rm
H}\\!\left(\boldsymbol{\rm{prox}}_{\lambda
g}\\!\left(\mathcal{A}\\!\left(\boldsymbol{x}\right)\\!+\\!\boldsymbol{b}\right)\\!-\\!\mathcal{A}\\!\left(\boldsymbol{x}\right)\\!-\\!\boldsymbol{b}\\!\right).$

Algorithm 2 Bregman ADMM-based PAPR Reduction for P3/P4

1:$\overline{\boldsymbol{X}}^{\left(0\right)},\overline{\boldsymbol{Z}}^{\left(0\right)}\\!,{{\bm{\Lambda}}^{\left(0\right)}}\\!\in\\!\mathbb{C}^{N_{\rm
T}\times\ell K}$;
$\left\\{\boldsymbol{Q}\left[k\right]\\!\in\\!\mathbb{C}^{N_{\rm T}\times
N_{\rm T}}\right\\}$; $\rho,\rho_{x},\rho_{z},\tau\\!\in\\!\mathbb{R}_{>0}$

2:$\overline{\boldsymbol{X}}^{\left(\cdot\right)}\in\mathbb{C}^{N_{\rm
T}\times\ell K}$ or
$\overline{\boldsymbol{Z}}^{\left(\cdot\right)}\in\mathbb{C}^{N_{\rm
T}\times\ell K}$

3:for $i=0,1,\ldots$ do

4: $\displaystyle\boldsymbol{U}$
$\displaystyle\coloneqq\rho_{x}\overline{\boldsymbol{X}}^{\left(i\right)}\\!-\\!\nabla
h\\!\left(\\!\overline{\boldsymbol{X}}^{\left(i\right)}\right)\\!+\\!\rho\left(\overline{\boldsymbol{Z}}^{\left(i\right)}\\!\\!-\\!\frac{{\bm{\Lambda}}^{\left(i\right)}}{\rho}\right)$
(24a) $\displaystyle\overline{\boldsymbol{X}}^{\left(i+1\right)}\\!$
$\displaystyle\coloneqq\\!\boldsymbol{\rm{proj}}_{\mathbfcal{U}^{\left(i\right)}}\left(\frac{\boldsymbol{U}}{\left(\rho_{x}\\!+\\!\rho_{z}\right)}\\!\right)$
(24b) $\displaystyle\boldsymbol{V}$
$\displaystyle\coloneqq\frac{1}{\left(\rho_{z}\\!+\\!\rho\right)}\\!\left(\\!\rho_{z}\overline{\boldsymbol{Z}}^{\left(i\right)}\\!-\\!\nabla
h\\!\left(\\!\overline{\boldsymbol{Z}}^{\left(i\right)}\right)\\!+\\!\rho\left(\overline{\boldsymbol{X}}^{\left(i+1\right)}\\!+\\!\frac{{\bm{\Lambda}}^{\left(i\right)}}{\rho}\right)\\!\right)$
(24c) $\displaystyle\overline{\boldsymbol{Z}}^{\left(i+1\right)}\\!$
$\displaystyle\coloneqq\\!\left[\\!\boldsymbol{F}\boldsymbol{\rm{proj}}_{\mathbfcal{P}_{T}^{\left(i\right)}}\left(\\!\boldsymbol{F}^{\rm
H}\boldsymbol{V}^{{\rm T}}\right)\\!\right]^{\rm T}$ (24d)
$\displaystyle{\bm{\Lambda}}^{\left(i+1\right)}\coloneqq$
$\displaystyle{\bm{\Lambda}}^{\left(i\right)}\\!+\\!\tau\left(\overline{\boldsymbol{X}}^{\left(i+1\right)}\\!-\\!\overline{\boldsymbol{Z}}^{\left(i+1\right)}\right)$
(24e)

5:end for

To benchmark our proposed Algorithm 1 using TOP-ADMM against state-of-the-art
algorithms, we next present two algorithms applying BADMM and DYS methods to
solve P3/P4. Recall, to solve P4, the sets for both PAPR and ACLR are
iteration independent, i.e.,
${\boldsymbol{\rm{proj}}}_{\mathbfcal{U}^{(i)}}\\!=\\!{\boldsymbol{\rm{proj}}}_{\mathbfcal{U}^{(0)}}$
and
${\boldsymbol{\rm{proj}}}_{\mathbfcal{P}^{(i)}}\\!=\\!{\boldsymbol{\rm{proj}}}_{\mathbfcal{P}^{(0)}}$
for all the iterations $i\\!\geq\\!0$.

###  IV-D Bregman ADMM (BADMM) Algorithm

We develop first reference algorithm using BADMM [28], cf. Section IV-A3, for
benchmarking purposes to solve P3/P4, whose recipe is presented in Algorithm
2—see [1] for additional algorithmic details. The operators
$\boldsymbol{\rm{proj}}_{\mathbfcal{P}^{\left(i\right)}}$,
$\boldsymbol{\rm{proj}}_{\mathbfcal{U}^{\left(i\right)}}$, and $\nabla h$ can
be computed as described for the TOP-ADMM algorithm.

###  IV-E Davis-Yin Splitting (DYS) Algorithm

Lastly, we also designed another reference algorithm for benchmarking purposes
using DYS [32], cf. Section IV-A2 tackling P3/P4—the recipe is given in
Algorithm 3. Similarly, the projection operators
$\boldsymbol{\rm{proj}}_{\mathbfcal{P}^{\left(i\right)}}$ and
$\boldsymbol{\rm{proj}}_{\mathbfcal{U}^{\left(i\right)}}$, and gradient
$\nabla h$ are same as described for the TOP-ADMM algorithm.

###  IV-F Computational Complexity Analysis

We compare the run-time computational complexities of our proposed TOP-ADMM-
type algorithms with two prior arts, namely semidefinite relaxation-based [20]
and Bao et al. [19]. Let $K_{D}\\!\coloneqq\\!\left|\mathcal{T}\right|$ denote
the number of useful data-carrying subcarriers. Consequently, let
$K_{C}\\!\coloneqq\\!\left|\mathcal{T}^{\perp}\right|$ such that
$K\\!=\\!K_{D}\\!+\\!K_{C}$.

Common processing of TOP-ADMM-type algorithms: the major computational
complexity of all the three presented algorithms stem from IDFT and DFT
operations, and gradient $\nabla h$. The cost of IDFT or DFT is typically
$\text{O}\left(\ell K\log\left(\ell K\right)\right)$ per iteration.
Additionally, the worst-case cost in computing the gradient $\nabla h$ (21) is
$\text{O}\left(K_{D}N_{\rm T}^{2}\right)$. Observe that the computation of
$\left\\{\boldsymbol{Q}[k]\right\\}$ matrices is required only once per
subcarrier, which is $\text{O}\left(K_{D}N_{\rm R}N_{\rm T}^{2}\right)$. The
computational cost of orthogonal projection operators is relatively low
compared to IDFT/DFT operation. Hence, the overall complexity of the proposed
TOP-ADMM-type algorithms is $\text{O}\left(\left(N_{\rm T}\ell K\log\left(\ell
K\right)\right)\\!+\\!K_{D}N_{\rm T}^{2}\right)$. Table I summarizes and
compares the complexity of the proposed schemes with three prior arts.
Clearly, the semidefinite relaxation-based method [20] is impractical to
deploy on the current state-of-the-art chipsets due to prohibitive complexity
cost compared to the proposed first-order-based approaches. Moreover, another
drawback of the prior art [19] is the need to resort to iterative numerical
optimization, e.g., a bisection method. This can be used to solve the proximal
operator for the infinite norm—referred to as PROXINF in [19]—which has a
complexity of $\text{O}\left(I^{\prime}N_{\rm T}\ell K\right)$, where
$I^{\prime}$ denotes the number of iterations. Nevertheless, the complexity of
the prior art, Bao et al. [19], is similar to our TOP-ADMM-type schemes.

Algorithm 3 DYS-based PAPR Reduction for P3/P4

1:$\overline{\boldsymbol{X}}^{\left(0\right)},\overline{\boldsymbol{Z}}^{\left(0\right)}\\!,{{\bm{\Lambda}}^{\left(0\right)}}\\!\in\\!\mathbb{C}^{N_{\rm
T}\times\ell K}$;
$\left\\{\boldsymbol{Q}\left[k\right]\\!\in\\!\mathbb{C}^{N_{\rm T}\times
N_{\rm T}}\right\\}$; $\tau,\mu\\!\in\\!\mathbb{R}_{\geq 0}$

2:$\overline{\boldsymbol{X}}^{\left(\cdot\right)}\in\mathbb{C}^{N_{\rm
T}\times\ell K}$ or
$\overline{\boldsymbol{Z}}^{\left(\cdot\right)}\in\mathbb{C}^{N_{\rm
T}\times\ell K}$

3:for $i=0,1,\ldots$ do

4: $\displaystyle\overline{\boldsymbol{X}}^{\left(i+1\right)}$
$\displaystyle\coloneqq\boldsymbol{\rm{proj}}_{\mathbfcal{U}^{\left(i\right)}}\\!\left(\bm{\Lambda}^{\left(i\right)}\right)$
(25a) $\displaystyle\overline{\boldsymbol{Z}}^{\left(i+1\right)}$
$\displaystyle\coloneqq\left[\\!\boldsymbol{F}\boldsymbol{\rm{proj}}_{\mathbfcal{P}_{T}^{\left(i\right)}}\\!\left(\\!\boldsymbol{F}^{{\rm
H}}\\!\left(\\!2\overline{\boldsymbol{X}}^{\left(i+1\right)}\\!-\\!\bm{\Lambda}^{\left(i\right)}\\!-\\!\tau\nabla
h\\!\left(\\!\overline{\boldsymbol{X}}^{\left(i+1\right)}\right)\\!\right)^{\\!{\rm
T}}\right)\\!\right]^{\\!{\rm T}}$ (25b)
$\displaystyle\bm{\Lambda}^{\left(i+1\right)}$
$\displaystyle\coloneqq{\bm{\Lambda}}^{\left(i\right)}+\mu\left(\overline{\boldsymbol{Z}}^{\left(i+1\right)}-\overline{\boldsymbol{X}}^{\left(i+1\right)}\right)$
(25c)

5:end for

TABLE I: Comparison of computational complexities of various PAPR reduction
methods

Method | Complexity per (outer) iteration
---|---
Prior art (non-CSI-aware): Iterative clip & filter [4] | $\text{O}\left(\left(N_{\rm T}\ell K\log\left(\ell K\right)\right)\right)$
Prior art (CSI-aware): Semidefinite relaxation-based [20] | $\text{O}\left(\left({\ell KN_{\rm T}}\right)^{4.5}\right)$
Prior art (CSI-aware): Bao et al. [19] | $\text{O}\left(\left(N_{\rm T}\ell K\left(I^{\prime}\\!+\\!\log\left(\ell K\right)\right)\right)\\!+\\!I_{\rm max}K_{D}N_{\rm T}^{2}\right)$; $I^{\prime}$ and $I_{\rm max}$ iterations for PROXINF and inner loop, resp.
TOP-ADMM-type: TOP-ADMM/BADMM/DYS | $\text{O}\left(\left(N_{\rm T}\ell K\log\left(\ell K\right)\right)\\!+\\!K_{D}N_{\rm T}^{2}\right)$

##  V Simulation Results

We present the performance of the proposed algorithms and benchmark their
performance with the prior art [19] utilizing a 5G NR-like simulator for
physical downlink shared channel (PDSCH). We analyze the PAPR reduction
performance in terms of three figures of merit briefly described in Section
III-B.

###  V-A Simulation Assumptions and Parameters

The general simulation assumptions and parameters are given in Table III, and
3GPP NR wideband (average) EVM requirements (after equalization555The EVM
measurement procedure as described in [24, Annex B for FR1] does not include
any wireless channel, but (realistic) impairments stemming from the
transmitter such as clipping noise and inband distortion due to nonlinear
power amplifier. Therefore, for BS type 1-C and FR1, 3GPP EVM requirements
consider linear equalization of the distortion at the receiver, where the
distortion is solely emanating from the non-ideal transmitter. In these
measurements for 3GPP requirements, the transmitter and receiver are connected
with a wire-like connector. Thus, the 3GPP EVM requirements do not consider
any wireless channel equalization.) for each respective modulation alphabet
are shown in Table III. Furthermore, we assume a base station supporting sub-6
GHz, e.g., frequency ranges between 410 MHz and 7.125 GHz, referred to as FR1
in 5G NR [24, Section 5.1]. Note that we do not model any additional radio
hardware impairments at the transmitter and/or receiver besides the distortion
generated at the transmitter by the considered PAPR reduction methods, if
enabled.

In Fig. 2, we illustrate channel frequency responses, corresponding to the
first receive and transmit antenna branch $|H_{1,1}|$ of one of the
realizations of the tapped delay line (TDL) 3GPP NR channel models [49]: TDL-D
(30 ns, 10 Hz) in Fig. 22(a) and TDL-A (300 ns, 100 Hz) in Fig. 22(b), where
the considered parameters in terms of root mean square delay spread in nano
seconds (ns) and maximum Doppler shift in Hz parameters are shown in the
brackets. Notice that the taps of the TDL-D channel model follow both Ricean
and Rayleigh fading, i.e., line-of-sight (LOS) and non-LOS, whereas the taps
of the TDL-A follow only Rayleigh fading, i.e., non-LOS—cf. [49, Section
B.2.3.2]. Moreover, we have also extended the 3GPP MIMO channel spatial
correlation model for $N_{\rm T}\\!>\\!8$ transmit (with cross-polarized
antennas) and two/four receive antennas—cf. [50, Section B.2.3.2]. The
considered parameters of the MIMO spatial correlation model are given in Table
III.

TABLE II: Simulation Parameters for 5G NR-like PDSCH

Parameters | Test $1$ | Test $2$
---|---|---
Subcarrier spacing | 15 kHz
Carrier bandwidth (PRB allocation) |  $20$ MHz ($100$ PRBs, i.e., active subcarriers $N_{\rm SC}\\!=\\!1200$)
DL SU/MU-MIMO $N_{\rm T},N_{\rm R}$ | $2/4/8/16/32/64/128/256{\rm Tx},2/4{\rm Rx}$; In MU-MIMO, each user with 1 Rx
Spatial layers (${\rm rank}$) | Fixed ${\rm rank}$ 2 when $N_{\rm R}\\!=\\!2$ or 4 when $N_{\rm R}\\!=\\!4$
Spatial precoding | RZF precoding with regularization $\alpha\\!=\\!0.001$
Precoding resource block group (PRG) size | 2 PRBs granularity
Modulation | 256QAM
Channel model [50] | TDL-D ($30$ ns, $10$ Hz, K-factor=$9$) | TDL-A ($300$ ns, $10$ Hz)
MIMO spatial correlation [50] | High-like: $\alpha_{1},\alpha_{2},\beta\\!=\\!0.9$, $\gamma\\!=\\!0.3$ | Low-like: $\alpha_{1},\alpha_{2}\\!=\\!0.09$, $\beta\\!=\\!0.03$, $\gamma\\!=\\!0.0$
DL channel knowledge at the BS |  Noisy channel estimate with SNR=5 dB
DL channel knowledge at the UE | Perfect channel estimate
Other information |  CSI delay is 1 ms, no additional impairments; fixed $\nu\\!=\\!0.001$ for $\left\\{\boldsymbol{Q}[k]\right\\}$—cf. (17)

TABLE III: EVM Req. [24]

Modulation Scheme | EVM Threshold
---|---
QPSK | 17.5%
16QAM | 12.5%
64QAM | 8.0%
256QAM | 3.5%

(a) TDL-D (30 ns, 10 Hz)

(b) TDL-A (300 ns, 10 Hz)

Figure 2: Magnitude of channel frequency responses $\left|H_{1,1}\right|$ of
(a) TDL-D (LOS and non-LOS) and (b) TDL-A (non-LOS) channel models. Figure 3:
PSD of OFDM waveform with and without (raised cosine) transmit windowing.

We depict three channel frequency responses in Fig. 2: one at the UE/receiver
and two at the base station (BS)/transmitter. At the UE/receiver, perfect
channel estimates are utilized for the estimated received EVM. However, at the
BS/transmitter, we have employed a channel estimation error model
$\boldsymbol{H}\\!=\\!\widehat{\boldsymbol{H}}\\!+\\!\Delta\boldsymbol{H}$,
where $\Delta\boldsymbol{H}$ has entries that are independent and identically
distributed and follow zero mean circularly symmetric complex Gaussian
distribution with given variance $\widetilde{\sigma}^{2}_{\rm ce}$. The noisy
channel estimates $\widehat{\boldsymbol{H}}$ with
SNR$\\!=-10\log_{10}\left(\widetilde{\sigma}^{2}_{\rm ce}\right)\\!=\\!5$ dB
are used both for MIMO signal precoding and computation of $\boldsymbol{Q}$
matrices as defined in (17). Both MIMO signal precoding $\boldsymbol{W}$ and
$\boldsymbol{Q}$ matrices have two physical resource blocks (PRBs) for
grouping—see Table III—which implies that the mean channel estimate within
considered precoding resource block group (PRG) is used for the computation of
$\boldsymbol{W}$ and $\boldsymbol{Q}$.

In Fig. 3, we illustrate three plots of power spectral density (PSD) of the
OFDM waveform having a useful signal bandwidth of 20 MHz, see Table III, 1)
without either transmit windowing or any PAPR reduction method, 2) with
transmit windowing but without any PAPR reduction method, and 3) with a naïve
amplitude clipping (PAPR target of 4 dB) but without any transmit windowing.
The transmit windowing is typically employed to shape the spectrum to meet the
minimum 3GPP NR ACLR requirement of 45 dB [24], see, e.g., [51, 38, 35]. In
this study, we perform transmit windowing—using raised cosine windowing, see,
e.g., [51]—after the (proposed) PAPR reduction if enabled. Although windowing
penalizes ~10% of the cyclic prefix, the spectrum-shaped signal has 58.9 dB of
ACLR, which clearly meets the minimum 45 dB ACLR requirement (with a good
implementation margin). However, as we show in the PSD result subsequently,
depending on the ACLR constraint $\bm{\psi}_{\rm aclr}$, the effective ACLR
after the (proposed) PAPR reduction method(s) can be lower than ~59 dB but
must be more than or equal to 45 dB (with some margin) to comply with the
requirement. Clearly, distortion-based PAPR reduction methods, such as
clipping, require filtering or windowing to remove unwanted distortion on the
unused subcarriers.

###  V-B Simulation Results

In this section, we present the simulation results for the test cases given in
Table III. We consider TOP-ADMM-type performance with iteration-dependent sets
targeting problem P3 unless specified explicitly. Additionally, we set
$\zeta\\!=\\!0$ typically for the computation of the gradient of $h$ in (21)
unless stated otherwise. Moreover, the typical MIMO test setup is 64 transmit
antennas, and two receive antennas unless mentioned otherwise. Since the
estimated EVM at the receiver is computed for each spatial layer—see Section
III-B2—we do not differentiate between SU-MIMO and MU-MIMO performance
evaluation.

(a) PSD with 3GPP spectral emission mask

(b) IPAPR distribution

Figure 4: Comparison among 1) distortion-based: ICF and (non-CSI-aware) TOP-
ADMM, and 2) distortion-less: TR1 and TR2. Noticeably, TR1/TR2 has poor PAPR
reduction performance compared to ICF/TOP-ADMM.

We have employed linear regularized zero forcing (RZF)-based MIMO (spatial)
precoding in the downlink, which can be expressed as
$\boldsymbol{X}[:,k]\\!=\\!\mathcal{W}\left(\boldsymbol{S}[:,k]\right)\\!=\\!\boldsymbol{W}[k]\boldsymbol{S}[:,k]$,
where the MIMO precoder reads
$\boldsymbol{W}[k]\\!=\\!\Xi\odot\widetilde{\boldsymbol{W}}[k]$, where $\Xi$
normalizes power discussed later in the following text. The un-normalized RZF
as given by, see, e.g., [52], is $\mathbb{C}^{N_{\rm T}\\!\times\\!N_{\rm
L}}\\!\ni\\!\widetilde{\boldsymbol{W}}[k]\\!=\\!\widehat{\boldsymbol{H}}[k]^{\rm
H}\\!\left(\widehat{\boldsymbol{H}}\\![k]\widehat{\boldsymbol{H}}\\![k]^{\rm
H}\\!+\\!\widetilde{\boldsymbol{R}}\\!+\\!\alpha\\!\boldsymbol{I}_{N_{\rm
R}}\\!\right)^{\\!-1}\\!\coloneqq\\!\left[\widetilde{\boldsymbol{w}}_{1}\\![k],\\!\ldots,\\!\widetilde{\boldsymbol{w}}_{N_{\rm
L}}\\![k]\right]$. The choice of suitable regularizing parameters, namely
Hermitian positive semidefinite $\widetilde{\boldsymbol{R}}$ and
$\alpha\\!\in\\!\mathbb{R}_{\geq 0}$, are typically tunable to maximize the
desired performance metrics of the system, such as the sum-rate, see, e.g.,
[52]. Moreover, the power normalization matrix can be $\mathbb{C}^{N_{\rm
T}\times N_{\rm
L}}\ni\Xi=\\!\left[\frac{\sqrt{\eta}_{1}}{\left\|\widetilde{\boldsymbol{w}}_{1}[k]\right\|_{2}}\boldsymbol{1}_{N_{\rm
T}},\ldots,\frac{\sqrt{\eta}_{N_{\rm
L}}}{\left\|\widetilde{\boldsymbol{w}}_{N_{\rm
L}}[k]\right\|_{2}}\boldsymbol{1}_{N_{\rm T}}\right]$, where
$\left\\{\sqrt{\eta}_{s}\right\\}$ is a set of powers that can be assigned to
respective layers or users. Consequently, the total power allocated to the
$k$-th subcarrier must satisfy
$\left\|\boldsymbol{W}[k]\right\|_{F}^{2}\\!=\\!\sum_{s=1}^{N_{\rm
L}}\eta_{s}$. In our simulations, we have simply set $\Xi$ as all-ones matrix.

Most of all, to motivate the signal distortion-based PAPR reduction schemes
for 5G NR and beyond, in Fig. 4, we compare the distortion-based PAPR
reduction techniques, notably, classical iterative clipping and frequency-
domain filtering ( iterative clipping and filtering (ICF)) [4] and our
proposed TOP-ADMM with two variants of signal distortion-less tone reservation
(TR) schemes under a special case with a single transmit and receive antenna.
In ICF, let $\mathbb{C}^{\ell K\times N_{\rm
T}}\\!\ni\\!{\boldsymbol{T}}\\!=\\!\boldsymbol{F}^{\rm H}{\boldsymbol{X}}^{\rm
T}$ be a time-domain input signal;
$\overline{\boldsymbol{T}}^{\left(I\right)}$ denotes a time-domain output from
ICF after $I$ iterations. Then, within one iteration cycle of ICF, for each
$j$-th transmit antenna and an $i$-th iteration, the amplitude of a discrete
time-domain (oversampled) signal is clipped
$\overline{\boldsymbol{T}}^{\left(i\right)}[n,j]\\!=\\!A\exp\left\\{\iota\phi\left(n\right)\right\\}$
if $\left|\overline{\boldsymbol{T}}^{\left(i-1\right)}[n,j]\right\|\\!>\\!A$,
where $A\\!=\\!\sqrt{\gamma_{\rm par}P_{\rm avg}}$, $P_{\rm avg}$ is average
transmit power of the signal, and $\phi\left(n\right)$ is the phase of
$\overline{\boldsymbol{T}}^{\left(i-1\right)}[n,j]$. In the frequency-domain
of a clipped signal, the unwanted distortions on the unused/inactive
subcarriers are zeroed out. After filtering, the frequency-domain signal is
transformed back to the time-domain signal. We repeat this time-domain and
frequency-domain filtering for $I$ iterations. For tone reservation schemes,
there are at least two popular variants: 1) unconstrained method, referred to
as TR1, i.e.,
$\operatorname{minimize}_{\overline{\boldsymbol{C}}}\,\left\|\boldsymbol{F}^{{\rm
H}}\left(\boldsymbol{X}\\!+\\!\overline{\boldsymbol{C}}\right)\right\|_{\infty}$,
where $\overline{\boldsymbol{C}}$ corresponds to tones/subcarriers in the
guard bands, and 2) peak-constrained method, called TR2, i.e.,
$\left\|\overline{\boldsymbol{C}}\right\|_{F}\\!\leq\\!{\rm threshold}$. In
the unconstrained TR1 scheme, the energy of the subcarriers carrying PAPR
reduction capable information does not have any constraint/restriction on the
energy. Therefore, the unconstrained TR1 method causes peaks, see Fig. 44(a)
for 24 tones (CCs) on each edge of guard bands, on the power spectral density,
which can cause many radio frequency-related implementation issues, including
PA linearization. Hence, the peak-constrained TR2 technique can control the
peaks but penalizes the capability to reduce the PAPR of signal—see Fig. 44(b)
constraining the total energy of 36 CCs for TR2 within 5% of total energy for
useful signal. Additionally, we cannot use many unused tones due to the strict
compliance with the regulatory and 3GPP standard spectral mask requirements.
For high numerical accuracy of solutions of TR1 and TR2 schemes, we employ CVX
since TR1/TR2 does not offer closed-form solutions due to the minimization of
the infinite norm. Noticeably, although TR1 and TR2 schemes offer relatively
low complexity compared to our proposed TOP-ADMM method, TR1 and TR2,
unfortunately, cannot meet desired PAPR targets under practical
constraints—see Fig. 44(b). Moreover, recall that the guard bands of NR/LTE
can be used to deploy NB-IoT carriers, among other practical challenges for
the PAPR reduction in NR/LTE.

Figure 5: Broadcast channel: IPAPR distribution comparison between P3 (with
iteration-dependent sets) and P4 (with fixed sets).

Figure 5 illustrates the complementary cumulative distribution function (CCDF)
of IPAPR considering cell-specific broadcast/control channel or non-CSI-aware,
i.e., the base station has no downlink channel knowledge, under Test 2 (see
Table III) but with a single antenna at the transmitter and the receiver. By
the formulation of our proposed problem P3/P4, it can fall back and support
the case where the base station has no downlink channel knowledge, which can
cover the cell-specific broadcast/control channel case, e.g., by setting
$\left\\{\boldsymbol{Q}[k]\\!=\\!\nu\boldsymbol{I}\right\\}$ with
$\nu\\!\geq\\!0$—see (17)—and $\zeta\\!\geq 0\\!$. For brevity, we have set
the following parameters to evaluate P3/P4 for the non-CSI-aware case:
$\nu\\!=\\!0$, i.e., $\boldsymbol{Q}[k]\\!=\\!\boldsymbol{0}$ and
$\zeta\\!=\\!0.05$. For TOP-ADMM, we have set the step-size $\tau\\!=\\!1.945$
targeting either P3 or P4. As expected by our conjecture, TOP-ADMM targeting
P3, i.e., with iteration-dependent sets, yields a better solution—at least not
worse—than TOP-ADMM targeting P4 in terms of PAPR. Moreover, our proposed
computationally efficient TOP-ADMM-based solution targeting P4 converges to a
similar solution using, high numerical accuracy, CVX wrapper (with SDPT3
solver) [47]. However, CVX is very slow to run on a personal computer even for
single antennas. Therefore, we could not obtain solutions using CVX for P4
with more than one antenna. Under the non-CSI-aware case, the proposed problem
P3 or P4 minimizes the transmit EVM subject to the PAPR and ACLR constraints.
In the sequel, we show numerically that when the base station with many
antennas has (practically even imperfect) downlink channel estimates, the
proposed problem P3 or P4—solved by using our implementation-friendly TOP-
ADMM-based algorithms—mitigates the signal distortion seen at the receiver by
capitalizing on the excess spatial degrees of freedom.

(a) TOP-ADMM ($\tau$): Dual residual vs. Iterations

(b) TOP-ADMM ($\tau$): Primal residual vs. Iterations

Figure 6: Algorithm 1 convergence behaviour

(a) BADMM ($\rho\\!,\rho_{x}\\!,\rho_{z}\\!,\tau$): Dual residual vs.
Iterations

(b) BADMM ($\rho,\\!\rho_{x},\\!\rho_{z},\\!\tau$): Primal residual vs.
Iterations

Figure 7: Algorithm 2 convergence behaviour. Figure 8: DYS ($\tau$), Algorithm
3, convergence behaviour Figure 9: Convergence behaviour of TOP-
ADMM($\tau\\!=\\!1.945$) with $\bm{\psi}_{\rm
aclr}\\!\in\\!\left\\{-50,-100\right\\}{\rm dB}$

For benchmarking purposes, we evaluate ICF [4] as a baseline for distortion-
based PAPR reduction techniques. Additionally, we evaluate the prior art, in
particular the method referred to as PROXINF-ADMM1, which essentially targets
to solve convex programming problem [19, Equation (14)/(15)]. Observe that the
authors in [19] employed double loops, which can be avoided by directly
employing our (proposed) three-operator ADMM-like approaches.

Figures 6–8 exhibit the convergence behaviour of the (proposed) TOP-ADMM-type
algorithms, namely, TOP-ADMM (Algorithm 1), BADMM (Algorithm 2), and DYS
(Algorithm 3). We use a set of tunable parameters to solve the optimization
problem P3 considering Test 1 with four transmit and two receive antennas—see
Table III—for PAPR constraint with $\bm{\gamma}_{\rm par}\\!=\\!4$ dB and ACLR
constraint with $\bm{\psi}_{\rm aclr}\\!=\\!-50$ dB. We do not know an optimal
solution to problem P3 a priori (or through any black-box global optimization
solvers). Thus, as remarked in [26], we characterize the convergence of the
(proposed) TOP-ADMM-type algorithms, see Figure 6, and BADMM, see Figure 7, in
terms of the primal and dual residual errors, i.e.,
$\left\|\Delta\mathbf{r}^{\left(i\right)}\right\|\\!\coloneqq\\!\left\|\overline{\boldsymbol{Y}}^{\left(i\right)}\\!-\\!\overline{\boldsymbol{X}}^{\left(i\right)}\right\|_{F}$
and
$\left\|\Delta\mathbf{x}^{\left(i\right)}\right\|\\!\coloneqq\\!\left\|\overline{\boldsymbol{X}}^{\left(i\right)}\\!-\\!\overline{\boldsymbol{X}}^{\left(i-1\right)}\right\|_{F}$,
respectively. For the convergence analysis of DYS (Algorithm 3), the (fixed-
point) residual error, cf. [32],
$\left\|\Delta\Lambda^{\left(i\right)}\right\|\\!\coloneqq\\!\left\|{\bm{\Lambda}}^{\left(i\right)}\\!-\\!{\bm{\Lambda}}^{\left(i-1\right)}\right\|_{F}$
plays a similar role as the primal and dual residual errors—see Figure 8.

In TOP-ADMM (Algorithm 1), the scaled dual variable
$\nicefrac{{{\bm{\Lambda}}^{\left(i+1\right)}}}{{\rho}}$ could be used such
that Algorithm 1 becomes free from the $\rho$ parameter. Consequently,
Algorithm 1 has only one tunable step-size $\tau$ parameter. In Fig. 6, we
show the residual errors for the step-size
$\tau\\!\in\\!\left\\{0.500,1.000,1.500,1.945,2.000\right\\}$. Similar to TOP-
ADMM with a scaled dual variable, the DYS-based Algorithm 3 also has only one
parameter, where Fig. 8 illustrates the primal and dual residual errors
against iterations for various step-sizes
$\tau\\!\in\\!\left\\{0.500,1.000,1.500,1.945,2.000\right\\}$. In contrast to
TOP-ADMM and DYS, the BADMM algorithm is not blessed with a single parameter
but with four parameters $\left(\rho,\rho_{x},\rho_{z},\tau\right)$. In Fig.
7, the primal and dual residual errors are shown for some choices of
parameters within a bracket. Conspicuously, the primal residual error of the
sequences generated by TOP-ADMM is nearly 10 times smaller than the BADMM.
Although these metrics, i.e., primal and residual errors, are commonly used in
the (numerical) optimization literature, subsequently, we focus directly on
the performance metrics of interest against iteration. Nevertheless, from
these numerical convergence figures, we fix the tunable parameters of the
respective algorithms for the subsequent performance figures. Therefore, for
TOP-ADMM, BADMM, and DYS, we use $\tau\\!=\\!1.945$,
$\left(\rho\\!=\\!0.01,\rho_{x}\\!=\\!1,\rho_{z}\\!=\\!1,\tau\\!=\\!0.01\right)$,
and $\tau\\!=\\!1.945$, respectively, unless mentioned otherwise.
Additionally, in Fig. 9, we show the convergence behaviour of TOP-ADMM for two
different ACLR constraints having $\bm{\psi}_{\rm
aclr}\\!\in\\!\left\\{-50,-100\right\\}$ dB. In terms of primal residual
error, the convergence noticeably improves with the relaxed ACLR constraint
having $\bm{\psi}_{\rm aclr}\\!=\\!-50$ dB. Similar behaviour is observed with
BADMM and DYS. To this end, we fix both ACLR and PAPR constraints with
$\bm{\psi}_{\rm aclr}\\!=\\!-50$ dB and $\bm{\gamma}_{\rm par}\\!=\\!4$ dB,
respectively, unless stated otherwise.

Figure 10: Estimated average received EVM against increasing excess spatial
degrees of freedom for $N_{\rm T}\\!\times\\!2$ and $N_{\rm T}\\!\times\\!4$
DL MU-MIMO/SU-MIMO setup under Test 2.

Figure 11: Estimated average received EVM against increasing UL channel
estimation error variance with a fixed regularizer $\nu\\!=\\!0.001$ for
$\left\\{\boldsymbol{Q}[k]\right\\}$—cf. (17)—under Test 2.

Figure 12: CSI-aware: Comparison between TOP-ADMM (for P3 with iteration-
dependent sets) and TOP-ADMM (for P4 with fixed sets) under Test 1.

Figure 11 shows the inband performance with increasing excess spatial degrees
of freedom, i.e., number of transmit antennas under Test 2 using TOP-ADMM
(targeting P3) with the above discussed fixed parameters. More specifically,
this figure depicts the estimated (averaged) received EVM performance of MU-
MIMO or SU-MIMO with 2/4 users or spatial layers against increasing excess
spatial degrees of freedom. As expected, the proposed problem P3 (or P4)
mitigates the incurred signal distortion when there are good enough excess
spatial degrees of freedom. However, the estimated received EVM may not
necessarily vanish entirely or become zero with very high excess spatial
degrees of freedom due to many practical reasons such as imperfect channel
estimates and channel aging.

Figure 11 illustrates the robustness against the usage of erroneous channel
estimates with a fixed regularizer $\nu\\!=\\!0.001$ in the computation of
distortion mitigation $\left\\{\boldsymbol{Q}[k]\right\\}$—see. (17)—under
Test 2. The channel estimation error variance equal to -100 dB represents
perfect channel estimates at the transmitter. Therefore, as expected, the
received EVM increases if the channel estimation error variance increases.
Hence, if channel estimation error variance is high, then one can instead make
the proposed PAPR reduction without capitalizing on the CSI knowledge if any,
e.g., by simply setting
$\left\\{\boldsymbol{Q}[k]\\!=\\!\boldsymbol{0}\right\\}$ and
$\zeta\\!=\\!0.05$, i.e., the fall back mode, where P3/P4 minimizes the
transmit EVM as required for the broadcast/control channel (or non-CSI-aware
case). Additionally, the practical system employs a link adaptation, where the
modulation and coding scheme for the downlink transmission can be adapted
according to the channel conditions (via CSI report feedback). In Fig. 12, we
compare the TOP-ADMM solution achieved by solving P3 with P4 considering Test
1 ($64\\!\times\\!2$ MIMO setup). As expected, the solution of P3 (utilizing
iteration-dependent sets) renders better performance than the solution of P4
(using fixed constraint sets).

(a) IPAPR vs. Iterations; Test 1 (64Tx)

(b) Predicted EVM vs. Iterations; Test 1 (64Tx)

(c) Transmit EVM vs. Iterations; Test 1 (64Tx)

(d) Estimated EVM vs. Iterations; Test 1 (64Tx)

Figure 13: Performance comparison of proposed TOP-ADMM-type algorithms (at
PAPR target of 4 dB) with two prior arts, namely the non-CSI-aware ICF [4] and
CSI-aware Bao et al. algorithm [19] in Test 1.

(a) IPAPR vs. Iterations

(b) Predicted EVM vs. Iterations

Figure 14: TOP-ADMM:Test 2 (TDL-A; Non-LOS, low spatial corr.) vs. Test 1
(TDL-D; LOS+Non-LOS, high spatial corr.).

(a) Estimated scatter plot of ICF [4]

(b) Estimated scatter plot of Bao et al. [19]

(c) Estimated scatter plot of TOP-ADMM

(d) Estimated received EVM distribution in the subcarrier domain for spatial
layer 1

(e) Estimated received EVM distribution in the subcarrier domain for spatial
layer 2

(f) Throughput versus averaged SNR per receive antenna with link adaptation
enabled

Figure 15: Scatter plot, EVM, ACLR, and Throughput performance comparison of
TOP-ADMM with prior art.

Figure 13 compares the performance metrics against iterations of proposed TOP-
ADMM-type methods with the two considered prior arts. Note that the prior art
by Bao et al. [19] has two inner iterations for each outer iteration, which
are included in these plots for a fair comparison. Figure 1313(a) shows the
achieved IPAPR at 99.99% probability within each OFDM symbol. Clearly, TOP-
ADMM followed by DYS has a relatively faster convergence compared to BADMM.
One of the reasons for the slower convergence of BADMM among the proposed
methods could be due to the curse of the number of tuning parameters. In other
words, a better and finer grid search of tunable parameters of BADMM may be
required to be competitive. Nevertheless, we iterate that the target
optimization problem in [19] minimizes signal peaks subject to equality
constraints on EVM mitigation and ACLR. However, in practice, we need to
adjust at least one of the performance metrics, i.e., PAPR or EVM, according
to the traffic and channel conditions. Therefore, the method in [19] achieves
the ~0.25 dB smaller IPAPR than the proposed methods but at the cost of a
nearly two-fold increase in transmit EVM, see Fig. 1313(c). This two times
increase in the transmit EVM, i.e., ~6 dB increase in the distortion energy,
consequently penalizes the energy of the useful transmit signal by ~6 dB since
the total transmit power is fixed and shared by both the signal and
distortion. It can clearly be argued that this IPAPR gain is worthwhile in
practice. Moreover, for a larger modulation alphabet, such as 1024QAM in the
upcoming NR releases, a lower estimated received EVM would be required
compared to 256QAM. Thus, one might need to lower the target PAPR (or back
off). Hence, the optimization problem (and method) proposed in [19] would be
unfortunately and arguably impractical for realistic radio systems.
Furthermore, Figs. 1313(b),13(d) illustrate the predicted and estimated EVM,
respectively, over iterations. All the methods except ICF meet the desired
estimated EVM target of 5.6% (additionally, as a reference, we have also shown
the 3GPP transmit EVM requirement of 3.5% for 256QAM). However, TOP-ADMM has
slightly better estimated/predicted received EVM performance compared to DYS
and BADMM, and noticeably compared to prior arts.

In Fig. 14, we compare TOP-ADMM performance in terms of IPAPR and predicted
received EVM against the iterations achieved in Test 1, i.e., the TDL-D
channel model with non-LOS and LOS components and high-like spatial
correlation, and in Test 2, i.e., the TDL-A channel model with only non-LOS
components and low-like spatial correlation. We construe the following key
observation from this plot.

######  Observation 1.

The convergence, in terms of the predicted/estimated received EVM over
iterations, slows down for a channel with increasing Rician $K$-factor, i.e.,
LOS component, and spatial correlation.

The slower convergence under such channel conditions could be due to more
active constraints.

For completeness, Fig. 15 presents the scatter plot, PSD, and EVM distribution
after the last considered iteration (simply fixed to 1000 for all the
algorithms) of ICF, the Bao et al. method, and TOP-ADMM. Note that we average
the EVM and PSD metrics over 28 OFDM symbols to analyze instantaneous-like
behaviour. In Figs. 1515(a)–15(c), we illustrate the scatter plot of
estimated/equalized received symbols for ICF, the Bao et al. method, and TOP-
ADMM, respectively. Incontestably, the scatter plot of TOP-ADMM shows
qualitatively the lowest estimated received EVM compared to prior arts
(particularly, the CSI-aware Bao et al. method). We show the distribution of
the average (over time domain OFDM symbols) estimated received EVM for Spatial
Layers 1 and 2 across subcarriers in Fig. 1515(d),15(e), respectively.
Clearly, for the non-robust Bao et al. method, in some sets of subcarriers, we
observe high amounts of estimated received EVM compared to not only TOP-ADMM
but also classical ICF. The estimated received EVM in some resources can be
large due to the composite mismatch effect under channel uncertainty. In Fig.
1515(f), we also depict the (normalized) throughput against the average SNR
per receive antenna after the respective PAPR reduction algorithms with link
adaptation enabled under Test 1. Around 20 dB SNR, we see that prior art [19]
performs poorer than our method. The reason is that the method in [19] renders
high transmit signal distortion energy, see Fig. 1313(c), which consequently
penalizes the useful signal’s transmit energy. However, as expected, at high
enough SNR, our proposed TOP-ADMM and the method in [19] will have a
negligible impact on the throughput performance.

##  VI Conclusion

In this paper, we sought a principled approach to mitigate (CSI-aware catering
to user-specific beamformed channel under channel uncertainty) or minimize
(non-CSI-aware supporting cell-specific non-beamformed broadcast channel)
flexibly signal distortion seen at the receivers subject to nonconvex PAPR and
ACLR constraints for a large-scale MIMO-OFDM-based system. Unfortunately, a
general-purpose solver for such a large-scale (nonconvex) optimization problem
is practically infeasible. Hence, to tackle such a nonconvex problem, we
applied iteration-dependent “convexification” to the nonconvex PAPR and ACLR
constraints. We conjecture that such iteration-dependent “convexification”
approaches the nonconvex constraint set asymptotically. Furthermore, we
capitalized on a divide-and-conquer approach to decompose the large-scale
problem into smaller easy-to-solve subproblems. More concretely, we proposed a
powerful and simple optimization solution using our recently proposed TOP-ADMM
algorithm to tackle such a problem and compared it to two relatively new
algorithms popular within machine learning and (numerical) optimization, the
so-called BADMM and DYS algorithms. Consequently, we developed three
“implementation-friendly” algorithms to tackle the proposed large-scale
(nonconvex) optimization problem. Numerical results showed that the proposed
low-complexity algorithms can meet the estimated (received) EVM target while
meeting the desired PAPR and ACLR constraints. This is arguably the first work
to propose computationally efficient algorithms for a robust EVM mitigation
problem under channel uncertainty with nonconvex PAPR and ACLR constraints.

##  Appendix A Optimality Conditions of TOP-ADMM for P3/P4

We first form the Lagrangian to (19) (considering in (IV-A1) form)—no
convexity assumptions are required:

$\displaystyle\mathcal{L}\\!\left(\overline{\boldsymbol{X}}\\!,\overline{\boldsymbol{Z}}\\!,\bm{\Lambda}\\!\right)\\!\coloneqq$
$\displaystyle\\!{\delta}_{\mathbfcal{U}}\\!\left(\overline{\boldsymbol{X}}\right)\\!+\\!{\delta}_{\mathbfcal{P}}\\!\left(\overline{\boldsymbol{Z}}\right)\\!+\\!h\\!\left(\overline{\boldsymbol{Z}}\right)\\!+\\!2\Re\\!\left\\{\\!{\rm
Tr}\\!\left[\\!\bm{\Lambda}^{\\!{\rm
H}}\\!\left(\overline{\boldsymbol{X}}\\!-\\!\overline{\boldsymbol{Z}}\right)\\!\right]\\!\right\\}\\!.$

It is known that the KKT conditions are first-order optimality conditions
which are necessary conditions for the solution point of (IV-A1) to be optimal
under some suitable constraint qualifications, see, e.g., [23]. Then, we
assume the subdifferential exist for the (nonconvex) indicator function. Let
$\left[\overline{\boldsymbol{X}}^{\star};\overline{\boldsymbol{Z}}^{\star};\bm{\Lambda}^{\star}\right]$
be a KKT point, which fulfils the dual feasibility, i.e.,
$\boldsymbol{0}\\!\in\\!\partial{\delta}_{\mathbfcal{U}}\left(\overline{\boldsymbol{X}}^{\star}\right)\\!+\\!\bm{\Lambda}^{\star}$
and
$\boldsymbol{0}\\!\in\\!\partial{\delta}_{\mathbfcal{P}}\left(\overline{\boldsymbol{Z}}^{\star}\right)\\!+\\!\nabla
h\left(\overline{\boldsymbol{Z}}^{\star}\right)\\!-\\!\bm{\Lambda}^{\star}$
and the primal feasibility, i.e.,
$\overline{\boldsymbol{X}}^{\star}\\!-\\!\overline{\boldsymbol{Z}}^{\star}\\!=\\!\boldsymbol{0}$.
Following [35, Appendix B], we can easily show that the sequences generated by
TOP-ADMM for either P3 (under the conjecture that the residual errors vanish
asymptotically) or P4 (see [46]), at any limit point, converges to a KKT
point.

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