# Comparing matching prescriptions between pre-equilibrium and hydrodynamic
models in high-energy nuclear collisions

Nicolas Borghini<EMAIL_ADDRESS>Fakultät für Physik,
Universität Bielefeld, D-33615 Bielefeld, Germany  Renata Krupczak
<EMAIL_ADDRESS>Fakultät für Physik, Universität Bielefeld,
D-33615 Bielefeld, Germany  Hendrik Roch<EMAIL_ADDRESS>Department of
Physics and Astronomy, Wayne State University, Detroit, Michigan 48201, USA

###### Abstract

State-of-the-art simulations of high-energy nuclear collisions rely on hybrid
setups, involving in particular a pre-equilibrium stage to let the system
evolve from a far-from-equilibrium initial condition towards a near-
equilibrated state after which fluid dynamics can be applied meaningfully. A
known issue is the mismatch between the equation of state in the fluid-
dynamical evolution and the effective one in the previous stage, which leads
to discontinuities at the interface between the two models. Here we introduce
a new matching prescription at this interface, based on the entropy, and we
compare it with the standard one relying on local energy conservation. We
study the behavior of various quantities at the switching time between the
models and investigate a number of final-state hadronic observables. For the
latter, we show that they are not modified significantly by the choice of
matching prescription, provided an appropriate normalization is chosen for the
initial state. In turn, our approach reduces sizeably the ratio of bulk over
thermodynamic pressure at the beginning of the fluid-dynamical stage.

##  I Introduction

Collisions of heavy nuclei at high energies create a hot and dense strongly-
interacting system [1, 2], whose bulk evolution can satisfactorily be
described by relativistic dissipative fluid dynamics [3] with the equation of
state determined in finite-temperature quantum chromodynamics (QCD)
computations on the lattice [4, 5, 6].

These fluid-dynamical simulations are complemented by further models. First,
initial conditions are needed to describe the direct outcome of the initial
nucleus-nucleus collision, for which there exist parametric models [7, 8] as
well as approaches more grounded on the saturation regime of QCD [9, 10, 11].
Then, since the system thus created is far from thermodynamic equilibrium, so
that (isotropic) fluid dynamics may not be applied meaningfully, pre-
equilibrium models are often employed to describe the hydrodynamization [12],
and in particular to reduce the large gradients present in the initial state
[13]. Eventually, after the end of the hydrodynamic evolution, an
“afterburner” [14] models the interactions of hadrons that become the relevant
degrees of freedom when the system description as a fluid is no longer a good
approximation. For a faithful physical modeling, it is crucial to investigate
the best way to match two successive approaches in such hybrid setups.

In this study, we focus on the interface between the pre-equilibrium and
fluid-dynamical stages. For the early evolution, there are essentially two
classes of models: either free streaming, in which classically the degrees of
freedom propagate at the speed of light for a short amount of time without
interaction [15]. Or one lets the QCD degrees of freedom interact, relying on
an effective kinetic theory description [16, 17]. Both approaches assume
massless degrees of freedom, so that when the system is (almost) equilibrated
the equation of state is the conformal one, $e=3P$. This is at variance with
the equation of state used in the hydrodynamic stage, which realistically
should be non-conformal. As a result, some of the fluid-dynamical quantities
are discontinuous when switching from one model to the other.

An approach to handle this issue is to assume that the free-streaming velocity
$v$ is smaller than the speed of light; $v$ is then treated as a fit parameter
[18, 19]. However, because of the fluctuations in the initial state — not only
event-by-event but also point-to-point in a given event —, such a recipe with
a position independent $v$ can only mitigate the mismatch on average, but
cannot hold equally well at each point in the final state of the pre-
equilibrium evolution.

In the present study, we introduce a new matching prescription between the
pre-equilibrium and fluid-dynamical models, which aims at addressing some of
the issues that arise at this interface. To test this approach, we perform
hybrid numerical simulations, whose setup, together with our proposed matching
recipe, is detailed in Sec. II. We compare simulations with the new approach
to computations with the standard one, focusing first on the switching time
between pre-equilibrium and fluid dynamics, then on a few final-state hadronic
observables (Sec. III). Our findings are summarized and discussed in Sec. IV,
in which we also mention future perspectives.

##  II Model

In this study, we simulate Pb-Pb collisions at 5.02 TeV (without conserved
charges) using a hybrid setup whose successive steps we now detail [20].

###  II.1 Hybrid approach

1. The initial states are generated with a Monte Carlo (MC) Glauber model that determines a transverse profile from the positions of participant nucleons and their binary collisions [21, 22]. The precise parameters of the model are the same as in Ref. [23]: since they are not crucial for the present study, we do not repeat them here, but only mention the Gaussian-smearing radius of 0.4 fm used in the energy deposition, which gives the typical scale of “hot spots” in the initial state. The only difference with the setup of Ref. [23] is that we do not fix the impact parameter value $b$, but sample it randomly with a probability $p(b)\,\differential{b}\propto b\,\differential{b}$. Each MC Glauber transverse profile, scaled by a multiplicative $K$-factor which is set such that it leads to the correct final-state charged multiplicity (see below), is then interpreted as an energy-density distribution $e_{0}(\mathbf{x})$ at midrapidity, multiplied by the starting time $\tau_{0}$ of the dynamical evolution.

2. The profiles from the previous step are used as initial condition, at a fixed proper time $\tau_{0}=0.2$ fm,111Throughout the paper we use $c=\hbar=k_{B}=1$. in KøMPøST [16], which we use to model the dynamics in the pre-equilibrium stage. Since the input consists of an energy density profile, KøMPøST is run in the mode with only energy perturbations, to propagate the inhomogeneities in $e_{0}(\mathbf{x})$. The specific shear viscosity in KøMPøST is taken to be $\eta/s=0.16$ and the effective number of degrees of freedom to be $\nu_{\text{eff}}=40$. The hydrodynamization time $\tau_{\rm hydro}$ at which KøMPøST ends is set equal to 1 fm. KøMPøST then provides at each point of the transverse plane an energy-momentum tensor, which is decomposed using the Landau matching condition into a local energy density $e_{\rm K}(\mathbf{x})$, velocity $u^{\mu}(\mathbf{x})$ and shear stress tensor $\pi^{\mu\nu}(\mathbf{x})$. Since KøMPøST is conformal, the “thermodynamic” pressure is $P_{\rm K}(\mathbf{x})=e_{\rm K}(\mathbf{x})/3$, while the bulk pressure $\Pi_{\rm K}(\mathbf{x})$ vanishes.

3. At $\tau_{\rm hydro}$ the fluid-dynamical stage begins, which we model with the relativistic dissipative hydrodynamics code MUSIC [24, 25, 26] in its boost-invariant mode, and using the lattice QCD (LQCD) equation of state from the HotQCD Collaboration [6], matched at low temperatures with the hadron resonance gas used in SMASH. How the energy density $e_{\rm K}(\mathbf{x})$ from KøMPøST is used for the initial state of MUSIC is precisely the main difference between the two scenarios tested in this paper and will be specified in Sec. II.2. In turn, the velocity and shear stress tensor from KøMPøST are used as is to initialize the energy-momentum tensor in MUSIC. For the subsequent hydrodynamic evolution, we use a constant shear viscosity over entropy density ratio $\eta/s=0.16$, as in KøMPøST. For the bulk viscosity, we choose the temperature-dependent parameterization introduced in Ref. [27] and whose impact on hadronic observables was studied in Ref. [28, 29]. When a MUSIC cell reaches a temperature of 155 MeV, particlization takes place and the fluid cell is converted using the iSS package [30] into a sample of hadrons.

4. The latter are fed into the hadronic transport approach SMASH [14], which models their further rescatterings and possible decays. From the final state of SMASH we can then compute hadronic observables of interest, which we shall detail in Sec. III.2.

Using this overall hybrid setup — with both matching prescriptions between
KøMPøST and MUSIC that will be discussed hereafter —, we generated $2^{15}$
initial states with random impact parameter. Using the semi-empirical formula
for final charged-hadron multiplicity based on the initial energy density
introduced in Ref. [31], with at first a guess for the $K$-factor at the end
of the MC Glauber, we estimated the centrality classes in which the $2^{15}$
events are distributed. The precise value of the $K$-factor was then
determined by fitting the respective centrality-dependent multiplicities to
ALICE data [32].

In each centrality bin, we also computed an average initial state, which we
then let evolve dynamically with KøMPøST+MUSIC+SMASH like one of the MC
Glauber initial states. For the final-state observables presented in Sec.
III.2, we actually computed 5000 random events with the full hybrid setup (for
both matching prescriptions) in each centrality class.

###  II.2 Matching the pre-equilibrium model to fluid dynamics

As stated in the introduction, the mismatch between the conformal relation
$e=3P$ at the end of the pre-equilibrium stage and the LQCD equation of state
unavoidably induces jumps in some fluid-dynamical quantities when one switches
from KøMPøST to MUSIC. In our simulations we compare two matching
prescriptions, with a focus on different thermodynamic quantities.

On the one hand, we use the by now standard recipe, consistent with Landau
matching, in which the energy density $e_{\rm K}(\mathbf{x})$ at a given
transverse position (in practice, in a cell) at the end of the pre-equilibrium
model, here KøMPøST, becomes the energy density $e_{\rm M}(\mathbf{x})$ at the
same point in the hydrodynamic code, namely MUSIC: $e_{\rm
M}(\mathbf{x})=e_{\rm K}(\mathbf{x})$. In the following, this approach will be
referred to as “energy matching”. Since we also feed the final velocity of
KøMPøST as is into MUSIC, this mode ensures local energy conservation.
Eventually, the bulk viscosity in MUSIC is initialized as

$\Pi_{\rm M}(\mathbf{x})\equiv\frac{e_{\rm K}(\mathbf{x})}{3}-P_{\rm
LQCD}\big{(}e_{\rm K}(\mathbf{x})\big{)},$ (1)

which also equals $P_{\rm K}(\mathbf{x})-P_{\rm LQCD}\big{(}e_{\rm
K}(\mathbf{x})\big{)}$, such that the sum $P+\Pi$ is continuous at the model
interface.

On the other hand, our proposed “entropy matching” prescription goes as
follows. From the energy density $e_{\rm K}(\mathbf{x})$ in a cell in KøMPøST
at $\tau_{\rm hydro}$, the effective number of freedom $\nu_{\text{eff}}$, and
the conformal equation of state $e=3P$, we compute the equivalent equilibrium
entropy density $s_{\rm K,eq.}(\mathbf{x})$:

$s_{\rm
K,eq}(\mathbf{x})\equiv\frac{4}{3}\bigg{(}\frac{\nu_{\text{eff}}\pi^{2}}{30}\bigg{)}^{\\!\\!1/4}e_{\rm
K}(\mathbf{x})^{3/4}.$ (2)

The “matching” consists in postulating that this becomes the initial local
equilibrium entropy density $s_{\rm M,eq.}(\mathbf{x})$ in MUSIC: $s_{\rm
M,eq.}(\mathbf{x})=s_{\rm K,eq.}(\mathbf{x})$. We then determine which energy
density $e_{\rm LQCD}$ corresponds to this equilibrium entropy density using
the LQCD equation of state: this defines the energy density $e_{\rm
M}(\mathbf{x})\equiv e_{\rm LQCD}\big{(}s_{\rm M,eq.}(\mathbf{x})\big{)}$ with
which we initialize the components of the energy-momentum tensor in
MUSIC.222That is, we really use an energy density as input in MUSIC, not an
entropy density, even though our $e_{\rm M}(\mathbf{x})$ is obtained via the
corresponding $s_{\rm M,eq.}(\mathbf{x})$.

Obviously, this prescription leads to non-conservation of energy, both locally
and in general globally, at the interface between the pre-equilibrium and
fluid-dynamical descriptions. As we shall discuss below, the recipe $s_{\rm
M,eq.}(\mathbf{x})=s_{\rm K,eq.}(\mathbf{x})$ does not ensure entropy
conservation either, since we only match the (equivalent) equilibrium entropy
densities. However, we can still ensure local momentum conservation, as is
also done in the energy-matching prescription. For that, since we feed the
velocity $u^{\mu}$ and shear-stress tensor $\pi^{\mu\nu}$ from KøMPøST into
MUSIC, it is sufficient to keep the total pressure $P+\Pi$ constant over the
switch. Using $P_{\rm K}=e_{\rm K}/3$ and $\Pi_{\rm K}=0$ in KøMPøST, while
the thermodynamic pressure in MUSIC is related to the energy density via the
LQCD equation of state, we thus determine the initial bulk pressure $\Pi_{\rm
M}(\mathbf{x})$ in MUSIC by

$\Pi_{\rm M}(\mathbf{x})\equiv\frac{e_{\rm K}(\mathbf{x})}{3}-P_{\rm
LQCD}\big{(}e_{\rm M}(\mathbf{x})\big{)}.$ (3)

Since $e_{\rm M}(\mathbf{x})\neq e_{\rm K}(\mathbf{x})$, this differs from Eq.
(1) in the energy-matching prescription.

Before we present the results of our simulations, let us comment on the new
matching prescription we propose. Since the flow velocity
$u^{\mu}(\mathbf{x})$ at the end of KøMPøST is also fed in unchanged into
MUSIC, our recipe ensures the continuity of the product $s_{\rm
eq}(\mathbf{x})u^{\mu}(\mathbf{x})$ at the interface. In the absence of
conserved charge as we assume here, this product coincides in the relativistic
fluid with the entropy current $S^{\mu}(\mathbf{x})$ up to terms of second
order in gradients [33, 34], since $u^{\mu}(\mathbf{x})$ is such that the
associated local rest frame is the Landau frame. Assuming, as is implicitly
done, that the system at the end of the KøMPøST evolution is close to
equilibrium, then $s_{\rm eq,K}(\mathbf{x})u^{\mu}(\mathbf{x})$ is also the
entropy current up to second-order terms. Yet even though the shear stress
tensor is taken to be continuous at the switch between both models, the non-
equilibrium corrections to $S^{\mu}(\mathbf{x})$ they induce do not
automatically match, because they depend on the (equivalent) temperature,
which is not the same in KøMPøST and MUSIC at a given $s_{\rm eq}$ due to the
different equations of state. Also, any correction to the entropy current
involving the bulk pressure is zero in KøMPøST but generally non-zero in
MUSIC. Thus, while we are aware that a better-motivated prescription would be
to ensure the continuity of the total entropy current, yet we were not able to
do it in this exploratory study. In addition, the correct approach would be to
compute $S^{\mu}(\mathbf{x})$ in KøMPøST without assuming that it describes an
almost equilibrated dissipative fluid, but we do not know how this can be
done, unless an underlying kinetic model is assumed. Yet our recipe in the
present paper ensures approximate entropy conservation at the interface
between pre-equilibrium and fluid dynamics, while it is not conserved in the
energy-matching prescription [15].

##  III Results

Figure 1: Time dependence of the energy (left) and (equivalent) equilibrium
entropy (right) per unit rapidity, using the energy-matching (blue) and
entropy-matching (green) prescriptions. The vertical dotted line stands for
the switching time $\tau_{\rm hydro}$ between KøMPøST and MUSIC. The top plots
are for the average central (0–10%) event and the bottom ones are for the
average event in the 40–50% centrality class.

In this section we present our results, starting with the behavior of several
quantities around the transition from KøMPøST to MUSIC (Sec. III.1), before
going to final-state observables (Sec. III.2).

###  III.1 Thermodynamic and fluid-dynamical quantities around the switching
time

Figure 2: Time dependence of the thermodynamic pressure (left) and bulk
pressure (right) per unit rapidity, using the energy-matching (blue) and
entropy-matching (green) prescriptions, for the average central (0–10%, top)
and peripheral (40–50%, bottom) events. The vertical dotted line stands for
the switching time $\tau_{\rm hydro}$ between KøMPøST and MUSIC.

As mentioned in the previous section, the multiplicative factor needed to turn
the output of the MC Glauber into a transverse energy density profile at
$\tau_{0}$ is determined such that the final charged multiplicity fits the
experimental data. Since the two matching prescriptions induce different
discontinuities, which we shall presently look into, at the switch from
KøMPøST to MUSIC, they require different factors, namely $K\simeq 51.43$
GeV/fm2 for energy matching and the somewhat smaller value $K\simeq 47.73$
GeV/fm2 for entropy matching.

In Figs. 1 and 2 we display the time dependence between 0.5 and 2 fm, i.e.
around the hydrodynamization time $\tau_{\rm hydro}$, of several quantities
integrated over the whole transverse plane and within one unit of spatial
rapidity. In both figures, we use the “average events” resulting from the
dynamical evolution of the average initial states in the 0–10% (top panels)
and 40–50% (bottom panels) centrality classes. We checked that these average
events are representative of the findings in individual, “random” events in
the corresponding centrality classes.

Figure 1 shows the evolution of energy (left) and equivalent equilibrium
entropy (right), i.e. the quantities whose continuity at the local level
determines our two matching prescriptions. Accordingly, the energy is
continuous at $\tau_{\rm hydro}$ when using energy matching, while the
integral of $s_{\rm eq.}$ is continuous within the entropy-matching approach.
We recall that $s_{\rm eq.}(\mathbf{x})$ is defined by Eq. (2) in the pre-
equilibrium stage, while it is given by LQCD thermodynamics in MUSIC.

In contrast, the mismatch between the equations of state in KøMPøST and MUSIC
means that there are discontinuities at the switching time. Thus, energy is
discontinuous when applying entropy matching (left, green curves), while
reciprocally the integral of $s_{\rm eq.}$ is discontinuous in the simulation
with energy matching (right, blue curves). Due to the different initial
$K$-factors, the values of the energy — and thus the equivalent equilibrium
entropy — necessarily differ in the two matching prescriptions in the pre-
equilibrium stage. Yet for those quantities with a jump at $\tau_{\rm hydro}$,
the values may become closer in MUSIC, because the final charged multiplicity
should be the same. This trend is not so clear in the peripheral events, but
one should pay attention that the difference between the values in MUSIC after
using the two prescriptions are actually of the same absolute size (about 30
GeV for energy) as in central events. Furthermore, one should keep in mind
that the integral of the equilibrium entropy $s_{\rm eq.}$ is not the total
entropy (per unit rapidity) in the (MUSIC) fluid. In addition to $s_{\rm
eq.}$, one should also consider the non-equilibrium contributions from shear
stresses and bulk pressure [33, 34] — and as we shall discuss next, bulk
pressure in MUSIC does depend on the matching recipe. This also explains why
the slight decrease of the integral of $s_{\rm eq.}$ in the MUSIC stage is not
in contradiction with the second law of thermodynamics. In fact, there are
also parts of the system that have already particlized, and which we do not
take into account in those plots.

In Fig. 2, we show the evolution of thermodynamic (left) and bulk pressure
(right) integrated over the transverse plane. We recall that in KøMPøST
$P=e/3$ and $\Pi=0$, so that the portions of these plots before $\tau_{\rm
hydro}$ are either degenerate with Fig. 1 or trivial. Indeed, the most
interesting is the behavior at $\tau_{\rm hydro}$, where these pressures are
discontinuous. In all cases, the jumps in $P$ or $\Pi$ are smaller with the
entropy-matching prescription than with energy matching. When equating $s_{\rm
eq.}$ in MUSIC and KøMPøST, the energy density at the start of MUSIC is larger
than at the end of KøMPøST: this corresponds to a higher temperature, so that
the LQCD equation of state is closer to being conformal, as it is in KøMPøST,
which explains why the gap in thermodynamic pressure is smaller. In turn,
since the sum $P+\Pi$ is continuous this results of an equally smaller gap in
bulk pressure, such that the initial $\Pi$ in MUSIC is smaller with entropy
matching.

Figure 3: Transverse-plane dependence of the ratio $\Pi/P$ between the bulk
and thermodynamic pressures at the initial time $\tau_{\rm hydro}=1$ fm of the
fluid-dynamical description, for a random central (0–10%, top panels) and
peripheral (40–50%, bottom) event, using energy (left) and entropy matching
(right).

This reduction of the bulk pressure is actually an important positive feature
of the entropy-matching prescription. In Fig. 3 we plot the ratio of bulk over
thermodynamic pressure $\Pi/P$ across the transverse plane in the initial
state of MUSIC at $\tau_{\rm hydro}$, for random central (top panels) and
peripheral (bottom) events with either energy (left) or entropy (right)
matching prescription, where we show only the regions of the system with a
temperature larger than 155 MeV. One sees that $\Pi/P$ is significantly
reduced when using $s_{\rm eq}$-matching: it is at most about 0.5, while the
standard energy-matching recipe leads to the presence of fluid cells with
$\Pi/P>1$, in particular towards the edges of the fireball [13, 19].
Suppressing these regions with $\Pi/P\sim 1$ is important since in that regime
the applicability of dissipative fluid dynamics is questionable.333Since the
bulk pressure $\Pi$ resulting from the matching at $\tau_{\rm hydro}$ is
positive, $|\Pi/P|>1$ does not signal the possible onset of cavitation, as may
happen with a negative $\Pi$ around the QCD crossover region [35, 36, 37, 38].
In that respect, the new proposed prescription makes the fluid-dynamical
evolution more reliable.444Note that in all simulations in the present study,
the normal regulators in MUSIC are used, to avoid the instability in
hydrodynamics. This is one reason why the bulk pressure decreases fast in Fig.
2. Besides, several studies [13, 15] have shown that too large a bulk pressure
affects some final observables like the flow coefficients $v_{n}$ and the
average transverse momentum $\expectationvalue{[p_{T}]}$.

It is possible to investigate the behavior at $\tau_{\rm hydro}$ of further
physical quantities, which are always consequences of the jumps in energy or
equilibrium entropy. For instance, enthalpy is discontinuous with both
prescriptions, being reduced for energy matching and increasing for entropy
matching. If one defines the spatial eccentricities with an energy-density
weight, then there is a small jump in their evolution when the switch from
pre-equilibrium to fluid dynamics is done with entropy matching. However, this
does not affect the (mostly linear) correlation between the initial
eccentricity and the final anisotropic flow coefficient.

###  III.2 Final-state observables

As already mentioned, the final charged-hadron multiplicity was fixed to
determine the initial normalization for each approach, so that $N_{\rm ch.}$
is by construction the same with both matching prescriptions. We also computed
a number of other observables in the final state of the SMASH evolution,555For
that analysis, we used the SPARKX package [39]. to test how much they are
affected by the change in matching recipe. More precisely, we looked at the
average over events of the first five anisotropic-flow harmonics $v_{n}$ and
of the average transverse momentum $[p_{T}]$ of charged hadrons. We also
looked at an observable that is more sensitive to event-by-event fluctuation,
namely the linear Pearson coefficient $\rho([p_{t}],v_{2}^{2})$ for the
correlation between $[p_{T}]$ and $v_{2}^{2}$ [40]. For the flow coefficients
we computed the modulus of the average of ${\rm e}^{{\rm i}n\varphi}$ over the
charged hadrons in an event, where $\varphi$ is the particle azimuthal angle
with respect to the known impact-parameter direction.

Table 1: Final-state observables for 5000 events in the centrality class 0–10%. Observable | Energy matching | Entropy matching
---|---|---
$\expectationvalue{v_{1}}$ (%) | $0.912$ | $0.883$
$\expectationvalue{v_{2}}(\%)$ | $2.652$ | $2.597$
$\expectationvalue{v_{3}}$ (%) | $1.535$ | $1.530$
$\expectationvalue{v_{4}}$ (%) | $0.835$ | $0.830$
$\expectationvalue{v_{5}}$ (%) | $0.506$ | $0.514$
$\expectationvalue{[p_{T}]}$ (MeV) | $702.45$ | $697.57$
$\rho([p_{t}],v_{2}^{2})$ | $0.0911$ | $0.1006$

The values of these observables for central collisions are shown in Tab. 1 —
the trend in peripheral collisions is the same and not shown. One sees that
the values of the observables are very similar with both matching
prescriptions, mostly at the percent level. The only exceptions are $v_{1}$,
which varies by about 3%, and especially $\rho([p_{t}],v_{2}^{2})$, which
changes by almost 10%.

From these few results, we would conclude that the precise matching
prescription used has little impact on “global” observables averaged over
events. More differential observables, in particular those sensitive to event-
by-event fluctuations, may be more impacted. However, for the 5000 events in
the 40–50% centrality class the Pearson coefficient $\rho([p_{t}],v_{2}^{2})$
changes from 0.1708 with energy matching to 0.1773 with entropy matching, i.e.
by only a few percent. Note that our using the regulators in MUSIC may partly
suppress the influence of the different bulk pressures (Fig. 2, right panel)
in the hydrodynamic stage on final-state observables.

##  IV Discussion

In this paper, we have tested a new prescription for the switch to a fluid-
dynamical description (MUSIC) from a pre-equilibrium model (KøMPøST) with a
different equation of state. More precisely, we imposed the continuity in each
fluid cell of the equivalent equilibrium entropy, instead of imposing local
energy conservation as is usually done. We compared two sets of MC Glauber +
KøMPøST + MUSIC + iSS+ SMASH simulations, starting from the same ensemble of
initial conditions with different overall normalizations according to the
matching prescription used. For those simulated events, we found that the
difference in matching recipe only has a minor impact on the observables we
computed, with the possible exception of the most differential one, namely the
correlation between the fluctuations of elliptic flow and average transverse
momentum. Focusing on the switching time between KøMPøST and MUSIC, the new
entropy-matching prescription leads to a smaller bulk pressure, and also a
smaller ratio $\Pi/P$, in the initial state of the fluid-dynamical evolution.

It is clear that the proposed matching recipe has the unpleasant feature that
it does not conserve energy at the interface between pre-equilibrium and fluid
dynamics. This is admittedly inelegant, but one should remember that in the
present state of the art, the absolute energy density at the beginning of
fluid dynamics is determined from the multiplicity in the final state of the
evolution, from which an ad hoc $K$-factor in the initial state (or sometimes
at the beginning of hydrodynamics [15]) is deduced. Since multiplicity is
rather associated with total entropy, a prescription based on the latter would
seem more motivated666See Ref. [41] for a discussion of whether energy or
entropy density should be used in the initial state of fluid dynamics. — at
least as long there is no initial-state model that predicts the value of the
initial energy without any indeterminate $K$-factor. Now, our approach in the
present study does not exactly conserve entropy at the switch between KøMPøST
and MUSIC, but it is already a significant step in that direction, since the
jump in total entropy is only of second order in the gradients, as discussed
at the end of Sec. II.

The other strong point of our prescription is the significant reduction of the
ratio of bulk over thermodynamic pressure in the initial state of fluid
dynamics, which means that the applicability of the latter is better
warranted. This would be even more the case in smaller systems [42], in which
“the edges” constitute a larger fraction of the fireball, or if the switching
time $\tau_{\rm hydro}$ from pre-equilibrium to hydrodynamics is such that the
LQCD equation of state is farther from conformal than what we assumed in this
exploratory study. Accordingly, we plan to further investigate the impact of
the entropy-matching recipe, with a more systematic study of final-state
observables, in particular sensitive to event-by-event fluctuations, and also
across different system sizes, and with various choices for $\tau_{\rm
hydro}$.

Eventually, let us emphasize that here we chose KøMPøST and MUSIC for our
simulations, but the proposed entropy-matching recipe may be used (and tested)
with any other pre-equilibrium or fluid-dynamical model.

###### Acknowledgements.

We thank Travis Dore, Grégoire Pihan, Sören Schlichting, Chun Shen, and Jie
Zhu for helpful discussions and comments. N. B. and R. K. acknowledge support
by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)
through the CRC-TR 211 ’Strong-interaction matter under extreme conditions’ -
project number 315477589 - TRR 211. H. R. was supported by the National
Science Foundation (NSF) within the framework of the JETSCAPE collaboration
(OAC-2004571) and by the DOE (DE-SC0024232). Numerical simulations presented
in this work were performed at the Paderborn Center for Parallel Computing
(PC2) and we gratefully acknowledge their support.

## References

  * [1] W. Busza, K. Rajagopal and W. van der Schee, Heavy ion collisions: The big picture, and the big questions, Ann. Rev. Nucl. Part. Sci. 68 (2018) 339 [arXiv:1802.04801 [hep-ph]].
  * [2] H. Elfner and B. Müller, J. Phys. G 50 (2023) 103001 [arXiv:2210.12056 [nucl-th]].
  * [3] D. A. Teaney, in “Quark Gluon Plasma 4”, (eds. R. C. Hwa and X. N. Wang, World Scientific, Singapore, 2010), p. 207 [arXiv:0905.2433 [nucl-th]].
  * [4] S. Borsanyi, G. Endrődi, Z. Fodor, A. Jakovac, S. D. Katz, S. Krieg, C. Ratti and K. K. Szabo, JHEP 11 (2010) 077 [arXiv:1007.2580 [hep-lat]].
  * [5] S. Borsanyi, Z. Fodor, C. Hoelbling, S. D. Katz, S. Krieg and K. K. Szabo, Phys. Lett. B 730 (2014) 99 [arXiv:1309.5258 [hep-lat]].
  * [6] A. Bazavov et al. [HotQCD Collaboration], Phys. Rev. D 90 (2014) 094503 [arXiv:1407.6387 [hep-lat]].
  * [7] J. S. Moreland, J. E. Bernhard and S. A. Bass, Phys. Rev. C 92 (2015) 011901 [arXiv:1412.4708 [nucl-th]].
  * [8] W. Ke, J. S. Moreland, J. E. Bernhard and S. A. Bass, Phys. Rev. C 96 (2017) 044912 [arXiv:1610.08490 [nucl-th]].
  * [9] B. Schenke, P. Tribedy and R. Venugopalan, Phys. Rev. Lett. 108 (2012) 252301 [arXiv:1202.6646 [nucl-th]].
  * [10] R. Paatelainen, K. J. Eskola, H. Holopainen and K. Tuominen, Phys. Rev. C 87 (2013) 044904 [arXiv:1211.0461 [hep-ph]].
  * [11] O. Garcia-Montero, H. Elfner and S. Schlichting, Phys. Rev. C 109 (2024) 044916 [arXiv:2308.11713 [hep-ph]].
  * [12] S. Schlichting and D. Teaney, Ann. Rev. Nucl. Part. Sci. 69 (2019) 447 [arXiv:1908.02113 [nucl-th]].
  * [13] T. Nunes da Silva, D. Chinellato, M. Hippert, W. Serenone, J. Takahashi, G. S. Denicol, M. Luzum and J. Noronha, Phys. Rev. C 103 (2021) 054906 [arXiv:2006.02324 [nucl-th]].
  * [14] J. Weil et al. [SMASH Collaboration], Phys. Rev. C 94 (2016) 054905 [arXiv:1606.06642 [nucl-th]].
  * [15] J. Liu, C. Shen and U. Heinz, Phys. Rev. C 91 (2015) 064906 [erratum: Phys. Rev. C 92 (2015) 049904] [arXiv:1504.02160 [nucl-th]].
  * [16] A. Kurkela, A. Mazeliauskas, J. F. Paquet, S. Schlichting and D. Teaney, Phys. Rev. C 99 (2019) 034910 [arXiv:1805.00961 [hep-ph]].
  * [17] D. Liyanage, D. Everett, C. Chattopadhyay and U. Heinz, Phys. Rev. C 105 (2022) 064908 [arXiv:2205.00964 [nucl-th]].
  * [18] G. Nijs, W. van der Schee, U. Gürsoy and R. Snellings, Phys. Rev. C 103 (2021) 054909 [arXiv:2010.15134 [nucl-th]].
  * [19] T. Nunes da Silva et al. [ExTrEMe Collaboration], Phys. Rev. C 107 (2023) 044901 [arXiv:2211.10561 [nucl-th]].
  * [20] H. Roch and R. Krupczak, Hendrik1704/EBE-PREEQ-HYDRO-TRANSPORT: v2.1 (2024), https://doi.org/10.5281/zenodo.12694840
  * [21] M. L. Miller, K. Reygers, S. J. Sanders and P. Steinberg, Ann. Rev. Nucl. Part. Sci. 57 (2007) 205 [arXiv:nucl-ex/0701025 [nucl-ex]].
  * [22] D. d’Enterria and C. Loizides, Ann. Rev. Nucl. Part. Sci. 71 (2021) 315 [arXiv:2011.14909 [hep-ph]].
  * [23] N. Borghini, M. Borrell, N. Feld, H. Roch, S. Schlichting and C. Werthmann, Phys. Rev. C 107 (2023) 034905 [arXiv:2209.01176 [hep-ph]].
  * [24] B. Schenke, S. Jeon and C. Gale, Phys. Rev. Lett. 106 (2011) 042301 [arXiv:1009.3244 [hep-ph]].
  * [25] B. Schenke, S. Jeon and C. Gale, Phys. Rev. C 82 (2010) 014903 [arXiv:1004.1408 [hep-ph]].
  * [26] J. F. Paquet, C. Shen, G. S. Denicol, M. Luzum, B. Schenke, S. Jeon and C. Gale, Phys. Rev. C 93 (2016) 044906 [arXiv:1509.06738 [hep-ph]].
  * [27] G. S. Denicol, T. Kodama, T. Koide and P. Mota, Phys. Rev. C 80 (2009) 064901 [arXiv:0903.3595 [hep-ph]].
  * [28] S. Ryu, J. F. Paquet, C. Shen, G. S. Denicol, B. Schenke, S. Jeon and C. Gale, Phys. Rev. Lett. 115 (2015) 132301 [arXiv:1502.01675 [nucl-th]].
  * [29] S. Ryu, J. F. Paquet, C. Shen, G. Denicol, B. Schenke, S. Jeon and C. Gale, Phys. Rev. C 97 (2018) 034910 [arXiv:1704.04216 [nucl-th]].
  * [30] C. Shen, Z. Qiu, H. Song, J. Bernhard, S. Bass and U. Heinz, Comput. Phys. Commun. 199 (2016) 61 [arXiv:1409.8164 [nucl-th]].
  * [31] G. Giacalone, A. Mazeliauskas and S. Schlichting, Phys. Rev. Lett. 123 (2019) 262301 [arXiv:1908.02866 [hep-ph]].
  * [32] J. Adam et al. [ALICE Collaboration], Phys. Rev. Lett. 116 (2016) 222302 [arXiv:1512.06104 [nucl-ex]].
  * [33] W. Israel, Annals Phys. 100 (1976) 310.
  * [34] W. Israel and J. M. Stewart, Annals Phys. 118 (1979) 341.
  * [35] K. Rajagopal and N. Tripuraneni, JHEP 03 (2010) 018 [arXiv:0908.1785 [hep-ph]].
  * [36] G. S. Denicol, C. Gale and S. Jeon, PoS CPOD2014 (2015) 033 [arXiv:1503.00531 [nucl-th]].
  * [37] M. Byres, S. H. Lim, C. McGinn, J. Ouellette and J. L. Nagle, Phys. Rev. C 101 (2020) 044902 [arXiv:1910.12930 [nucl-th]].
  * [38] C. Chiu and C. Shen, Phys. Rev. C 103 (2021) 064901 [arXiv:2103.09848 [nucl-th]].
  * [39] H. Roch, N. Saß and N. Götz, smash-transport/sparkx: v1.2.1-Newton (2024), https://doi.org/10.5281/zenodo.11150134.
  * [40] P. Bożek, Phys. Rev. C 93 (2016) 044908 [arXiv:1601.04513 [nucl-th]].
  * [41] P. Huovinen, in “Quark Gluon Plasma 3”, (eds. R. C. Hwa and X. N. Wang, World Scientific, Singapore, 2004), p. 600 [arXiv:nucl-th/0305064 [nucl-th]].
  * [42] J. F. Grosse-Oetringhaus and U. A. Wiedemann, arXiv:2407.07484 [hep-ex].


power consumption in the network.

####  5.5.4 Energy Efficiency versus Distance

We investigate the average EE versus different reference distances. As can be
observed from figure (5.7), the system EE decreases with increasing the
distance. Consequently, as the channel strength deteriorates by increasing the
distance between transmitter and the receiver due to the effect of the path-
loss, more ID antennas have to be activated, and more power must be allocated
to users to meet the minimum required data-rate. Hence, the average system EE
would decline due to increasing the total power consumption in addition to
decreasing the total data-rate of the network. However, the system EE achieved
by our proposed iterative Algorithm 6 still has superior performance as
compared to the other algorithms. It should be noted that the Methods A-D are
the same as defined earlier in figure (5.5).

Figure 5.7: Energy efficiency versus distance.

####  5.5.5 Average System Throughput versus Maximum Allowed Transmit Power

In figure (5.8), we plot the average throughput or data-rate of the network
versus the maximum allowed transmit power, $p_{max}$, for different schemes.
For $p_{max}\leq 30\leavevmode\nobreak\ \text{dBm}$, it can be perceived that
the average system throughput of the proposed iterative Algorithm 6, that is
the Method A in the figure, raises with the maximum transmit power allowance.
However, the slope of the curve of the system throughput starts to decline and
reach a saturation in the high transmit power regime, i.e., $p_{max}\geq
30\leavevmode\nobreak\ \text{dBm}$. In other words, there is a diminishing
return in the average system throughput when the maximum transmit power is
higher than 30 dBm. In fact, as the maximum transmit power increases, the
interference power level arising from the other SBSs becomes more severe,
which degrades the received users’ signal quality. To compare, we also plot
the curve based on the alternative search method (ASM), i.e., Method B, in
which the original problem is divided into three disjoint subproblems. In this
method, the SBS-subcarrier assignment, power allocation, and antenna selection
are selected based on the values that are determined in the previous round
[122]. Method C is the proposed algorithm based only on the power allocation
when random scheduling of the subcarrier allocation and antenna selection
variables is performed to obtain the resource allocation policy. Method D is
the full power allocation approach with equal power across subcarriers for
each user in which the SBS-subcarrier assignment and antenna selection
variables are chosen based on our proposed algorithm. It can be perceived that
power allocation can significantly increase the performance gain due to
controlling interference term arising from other SBSs.

Figure 5.8: Average system throughput versus maximum allowed transmit power.

####  5.5.6 Average Harvested Power versus Maximum Allowed Transmit Power

Figure (5.9) illustrates the average harvested power versus the maximum
allowed transmitting power, $p_{max}$. As the $p_{max}$ increases, the
harvested energy also increases in all considered Methods A-D. However, it is
noticeable that for a large value of $p_{max}$, the amount of average
harvested power gets saturated. The reason for this inclination is that the
transmitter stops to increase the transmit power for the system EE
maximization. In order to evaluate our performance gain, we also compare our
results from the proposed iterative Algorithm 6, i.e., Method A, with three
baseline Methods B-D. In Method B, we consider the proposed algorithm in [52].
Method C splits the received signal into two power streams for a finite
discrete set of power splitting ratios. Method A works better than Method B
and C due to using a different antenna in each subcarrier, leading to a better
degree of freedom of the network. Finally, Method D is our proposed algorithm
with a different objective. Specifically, we maximize the system throughput
with respect to the same constraints and also further consider minimum energy
harvesting requirements for each user. It can be observed that our proposed
algorithm reaches higher performance gain due to employing the antenna
selection strategy via the multiplexing gain. Moreover, by using the antenna
selection, we enhance the performance gain due to increasing flexibility in
the resource allocation design.

Figure 5.9: Average harvested power versus maximum allowed transmit power.

###  5.6 Summary

In this chapter, we addressed the EE optimization problem for the DL of a
multi-user multi-cell OFDMA network with the generalized AS-based co-located
receivers using SWIPT. Considering a practical linear power model in which the
transmit power consumption, circuit energy consumption, and the harvested
energy (by active receiver EH antennas) are taken into account, our goal was
to maximize the EE whilst satisfying the minimum data-rate requirement for
each user. The EE optimization problem, which involves a joint optimization of
the SBS-subcarrier assignment and power allocation along with an optimal
antenna selection, was non-convex and non-linear. This made the optimization
problem extremely difficult to tackle directly. Hence, to obtain a feasible
solution for this problem, we employed the MM approach by constructing a
sequence of surrogate functions to approximate the non-convex optimization. In
particular, based on the Dinkelbach method, an optimization problem with a
transformed objective function was designed that uses the MM method in its
inner loop. Simulation results revealed the superiority of our proposed method
over existing works. Furthermore, the proposed antenna selection scheme
demonstrated that our algorithm provides a good balance of improving in terms
of the throughput as well as EE.

##  Chapter 6 Conclusion and Future Work

In this chapter, we summarize our overall conclusions. We also propose some of
the future research directions that emerge from this work.

###  6.1 Conclusion

In chapter 3, we introduced a new approach to harvesting ambient energy. In
this approach, a designated portion of the spectrum was used for information
decoding (ID) and the rest for energy harvesting (EH), with two separate
filters used at the receivers. We used neither splitters nor switches, which
significantly simplified the complexity of the receiver. Furthermore, we
formulated an optimization problem to maximize the harvested energy via a
joint subcarrier assignment and power allocation using the simultaneous
wireless information and power transfer (SWIPT) scheme for a downlink (DL) of
a multi-user small single-cell orthogonal frequency division multiple access
(OFDMA) network, fulfilling each user’s minimum data-rate requirement. Our
extensive simulation results indicate that our proposed algorithm outperformed
other algorithms in the literature by complying with the policy designed for
resource allocation.

In chapter 4, we extended the system model for SWIPT-enabled single-cell OFDMA
proposed in the previous chapter to a multi-cell network based on separated
receiver architecture with the goal of maximizing system throughput while
respecting the maximum power transfer allowed, the minimum of energy harvested
for EH receivers, and the minimum amount of data-rate required for ID
receivers. The resulting problem, which jointly optimizes subcarrier
assignment and power allocation, was mixed-integer non-linear programming
(MINLP). This is intractable because of the multiplication of two variables,
the binary subcarrier assignment variable, and the intertwingled interference
term in the data-rate function. We applied the majorization minimization (MM)
approach to manage resource allocation policy in this complex problem. We also
analyzed the design of a low complexity algorithm with an upper bound for the
interference term by imposing a limiting interference threshold in each
subcarrier that can be controlled by the resource allocator to improve system
performance. Simulation results showed the excellent performance gain of our
suggested algorithms.

Finally, in chapter 5, we described a novel harvesting technique at the
receiver that is based on the receiver antenna selection for a multi-user
multi-cell SWIPT OFDMA system with a co-located architecture. This we named a
“generalized antenna switching technique”. We then maximized energy efficiency
(EE) as a performance metric in a two-layered approach to determine a resource
allocation policy that optimizes subcarrier assignment, power allocation, and
antenna selection with few constraints. The underlying problem in this chapter
was neither linear nor convex due to the fractional form of the objective
function, the multiplication of variables, incorporating interference, and
integer variables. We relaxed the integer variable and applied the big-M
formulation to make sure that relaxed variables take binary values. After
that, we used the MM approach based on difference of two convex functions
(D.C.) programming to find a feasible solution for the inner problem,
employing a first-order Taylor approximation to convexify the non-convex
functions. Next we applied the Dinkelback algorithm to transform the objective
function into a non-fractional function. We observed that generalized antenna
switching, also known as “antenna selection strategy”, increases the EE of the
system by providing more degrees of freedom as a result of assigning resources
via different antennas. We concluded the chapter with simulation results that
demonstrate the superiority of our proposed method.

###  6.2 Future Work

The field of SWIPT in energy-constrained communication systems is a remarkably
rich research discipline with great potential. The following fundamental
research directions could be pursued in future work.

NOMA-based SWIPT:

Non-orthogonal multiple access (NOMA) has been suggested as one of the
fundamental techniques for beyond fifth generation (5G) and the forthcoming
sixth generation (6G) in order to enhance spectral efficiency (SE) while
permitting some degree of multiple access interference at receivers by having
users share the same spectrum. NOMA schemes are designed to concurrently serve
two or more users at the same base station (BS) or access point (AP) in a
single orthogonal resource block. They can be categorized into main two
classes – single-carrier NOMA and multi-carrier NOMA. The primary principle of
single-carrier NOMA is allocating the same time/frequency resources to
multiple users by utilizing distinct power levels (“power-domain” NOMA). In
the multi-carrier NOMA scheme, multiple users are multiplexed on different
subcarriers by using different codes (sparse code multiple access) and
patterns (pattern division multiple access) for each subcarrier [131]. The
successive interference cancellation technique is employed at the receiver end
to eliminate expected interference and guarantee improved overall fairness and
throughput. Most importantly, SE can be achieved in NOMA networks. It is worth
noting that the resource allocation design based on SWIPT-enabled NOMA
cellular networks could provide both SE improvement through NOMA and EE
improvement via SWIPT. The very first attempt to maximize EE under multiple
restraints and achieve energy-efficient resource management in SWIPT-enabled
NOMA was conducted in [132]. Research on this topic has just begun to
flourish, with many more potentials awaiting discovery [133].

Massive MIMO- and mmWave-based SWIPT:

Propagation losses of broadcasted radio waves may deteriorate SWIPT
efficiency, which makes SWIPT particularly applicable to short-distance uses.
Massive multiple-input multiple-output (MIMO) and millimeter-wave (mmWave)
technologies would enable a BS or an AP to reliably transfer power to energy-
constrained users through ultra-sharp energy beams (that only concentrate
transmission energy at certain points) as well as provide diversity and
multiplexing gain [134, 135, 136]. Integrating SWIPT with massive MIMO and
mmWave technologies can therefore help overcome SWIPT’s deficiencies and
further enhance performance in terms of achievable data-rate, overall SE, and
EE. Since the beam-width of the massive MIMO and mmWave based SWIPT system is
very small due to the shorter wavelengths (and wider bandwidths), effective
initial beam association, beam selection, and beam alignment algorithms are
desirable. These topics were investigated in [137, 138, 139, 140, 141, 142].
Moreover, considering that the antenna selection technique in multiple antenna
systems would significantly decrease the system’s power consumption, an
exciting research direction would be verifying its applicability to massive
MIMO- and mmWave-based SWIPT-enabled networks.

UAV-based SWIPT:

Unmanned Aerial Vehicles (UAVs) or remotely piloted aircraft have attracted
significant attention. UAVs can improve traditional cellular communication as
relays and BSs. Compared to conventional cellular communication, some of the
advantages of UAVs include increased wireless connectivity, high
maneuverability, extensive coverage, simple implementation, and cost-effective
communication – with increasing affordability. An unobstructed line-of-sight
(LoS) link might be able to help further improve the reliability of wireless
communication systems. UAVs could be effectively integrated with SWIPT by
virtue of the dominant presence of LoS connections – acting as flying BSs or
APs – to provide a variety of services in areas lacking infrastructure.
However, a major drawback to UAV-based applications is that UAV devices are
typically power-hungry devices with limited energy storage performing
operations in flight. Deployment, trajectory design, and resource allocation
must be improved for the efficient utilization of UAVs within a reasonable
range of energy-constrained receivers [143, 144, 145, 146, 147, 148, 149].

MEC-based SWIPT:

Mobile edge computing (MEC) is a new paradigm for facilitating the real-time
implementation of computationally heavy tasks for massive low-power devices
(e.g., sensors) by providing cloud-like computing at the edge of mobile
networks. MEC offers solutions for resource-limited wireless devices with
high-quality wireless services and multi-media applications because it is
capable of offloading some or all the computing tasks of resource-limited
devices to nearby APs or BSs, where integrated MEC servers can remotely handle
their tasks [150]. Although MEC covers the computational aspect of massive
computation-intensive devices via local computing or offloading, one big
challenge remains: providing a sustainable and cost-effective energy supply to
resource-limited energy-constrained wireless devices. Implementing resource
allocation policies for mobile users with high computing tasks via MEC and
SWIPT would be an interesting topic to explore, with MEC-based SWIPT
permitting ubiquitous computing [151, 152, 153, 154, 155, 156, 157].

## References

  * [1] S. C. Dushantha Nalin K.Jayakody, John Thompson and S. Durrani, _Wireless Information and Power Transfer: A New Paradigm for Green Communications_. Springer, 1993.
  * [2] L. R. Varshney, “Transporting information and energy simultaneously,” in _2008 IEEE International Symposium on Information Theory_ , 2008, pp. 1612–1616.
  * [3] M. Salem, P. Ruppel, U. Bareth, and A. Küpper, “X-centric positioning: A combination of device-centric and multi-rat network-centric positioning approaches in ngn,” in _2012 IEEE Globecom Workshops_ , 2012, pp. 1741–1746.
  * [4] P. Grover and A. Sahai, “Shannon meets tesla: Wireless information and power transfer,” in _2010 IEEE International Symposium on Information Theory_ , June 2010, pp. 2363–2367.
  * [5] M. Kamel, W. Hamouda, and A. Youssef, “Ultra-dense networks: A survey,” _IEEE Communications Surveys Tutorials_ , vol. 18, no. 4, pp. 2522–2545, 2016\.
  * [6] H. J. Visser and R. J. M. Vullers, “Rf energy harvesting and transport for wireless sensor network applications: Principles and requirements,” _Proceedings of the IEEE_ , vol. 101, no. 6, pp. 1410–1423, June 2013.
  * [7] H. Nishimoto, Y. Kawahara, and T. Asami, “Prototype implementation of ambient rf energy harvesting wireless sensor networks,” in _SENSORS, 2010 IEEE_ , Nov 2010, pp. 1282–1287.
  * [8] I. Krikidis, S. Timotheou, S. Nikolaou, G. Zheng, D. W. K. Ng, and R. Schober, “Simultaneous wireless information and power transfer in modern communication systems,” _IEEE Communications Magazine_ , vol. 52, no. 11, pp. 104–110, Nov 2014.
  * [9] B. Clerckx, R. Zhang, R. Schober, D. W. K. Ng, D. I. Kim, and H. V. Poor, “Fundamentals of wireless information and power transfer: From rf energy harvester models to signal and system designs,” _IEEE Journal on Selected Areas in Communications_ , vol. 37, no. 1, pp. 4–33, Jan 2019.
  * [10] T. D. Ponnimbaduge Perera, D. N. K. Jayakody, S. K. Sharma, S. Chatzinotas, and J. Li, “Simultaneous wireless information and power transfer (swipt): Recent advances and future challenges,” _IEEE Communications Surveys Tutorials_ , vol. 20, no. 1, pp. 264–302, Firstquarter 2018\.
  * [11] X. Lu, P. Wang, D. Niyato, D. I. Kim, and Z. Han, “Wireless networks with rf energy harvesting: A contemporary survey,” _IEEE Communications Surveys Tutorials_ , vol. 17, no. 2, pp. 757–789, Secondquarter 2015.
  * [12] S. Ulukus, A. Yener, E. Erkip, O. Simeone, M. Zorzi, P. Grover, and K. Huang, “Energy harvesting wireless communications: A review of recent advances,” _IEEE Journal on Selected Areas in Communications_ , vol. 33, no. 3, pp. 360–381, March 2015.
  * [13] Y. Wu, J. Linnartz, H. Gao, M. K. Matters-Kammerer, and P. Baltus, “Modeling of rf energy scavenging for batteryless wireless sensors with low input power,” in _2013 IEEE 24th Annual International Symposium on Personal, Indoor, and Mobile Radio Communications (PIMRC)_ , Sep. 2013, pp. 527–531.
  * [14] J. A. G. Akkermans, M. C. van Beurden, G. J. N. Doodeman, and H. J. Visser, “Analytical models for low-power rectenna design,” _IEEE Antennas and Wireless Propagation Letters_ , vol. 4, pp. 187–190, 2005.
  * [15] M. Ghovanloo and K. Najafi, “Fully integrated wideband high-current rectifiers for inductively powered devices,” _IEEE Journal of Solid-State Circuits_ , vol. 39, no. 11, pp. 1976–1984, Nov 2004.
  * [16] J. A. Hagerty, F. B. Helmbrecht, W. H. McCalpin, R. Zane, and Z. B. Popovic, “Recycling ambient microwave energy with broad-band rectenna arrays,” _IEEE Transactions on Microwave Theory and Techniques_ , vol. 52, no. 3, pp. 1014–1024, March 2004.
  * [17] X. Zhou, R. Zhang, and C. K. Ho, “Wireless information and power transfer: Architecture design and rate-energy tradeoff,” _IEEE Transactions on Communications_ , vol. 61, no. 11, pp. 4754–4767, November 2013\.
  * [18] Q. Wu, G. Y. Li, W. Chen, D. W. K. Ng, and R. Schober, “An overview of sustainable green 5g networks,” _IEEE Wireless Communications_ , vol. 24, no. 4, pp. 72–80, Aug 2017.
  * [19] M. Webb _et al._ , “Smart 2020: Enabling the low carbon economy in the information age,” _The Climate Group. London_ , vol. 1, no. 1, pp. 1–1, 2008\.
  * [20] C. Han, T. Harrold, S. Armour, I. Krikidis, S. Videv, P. M. Grant, H. Haas, J. S. Thompson, I. Ku, C.Wang, T. A. Le, M. R. Nakhai, J. Zhang, and L. Hanzo, “Green radio: radio techniques to enable energy-efficient wireless networks,” _IEEE Communications Magazine_ , vol. 49, no. 6, pp. 46–54, June 2011.
  * [21] R. Mahapatra, Y. Nijsure, G. Kaddoum, N. Ul Hassan, and C. Yuen, “Energy efficiency tradeoff mechanism towards wireless green communication: A survey,” _IEEE Communications Surveys Tutorials_ , vol. 18, no. 1, pp. 686–705, Firstquarter 2016.
  * [22] E. Oh, B. Krishnamachari, X. Liu, and Z. Niu, “Toward dynamic energy-efficient operation of cellular network infrastructure,” _IEEE Communications Magazine_ , vol. 49, no. 6, pp. 56–61, June 2011.
  * [23] A. Fehske, G. Fettweis, J. Malmodin, and G. Biczok, “The global footprint of mobile communications: The ecological and economic perspective,” _IEEE Communications Magazine_ , vol. 49, no. 8, pp. 55–62, August 2011.
  * [24] X. Lu, P. Wang, D. Niyato, and Z. Han, “Resource allocation in wireless networks with rf energy harvesting and transfer,” _IEEE Network_ , vol. 29, no. 6, pp. 68–75, Nov 2015.
  * [25] K. Huang and V. K. N. Lau, “Enabling wireless power transfer in cellular networks: Architecture, modeling and deployment,” _IEEE Transactions on Wireless Communications_ , vol. 13, no. 2, pp. 902–912, February 2014.
  * [26] I.-J. Yoon, “Wireless power transfer in the radiating near-field region,” in _2015 USNC-URSI Radio Science Meeting (Joint with AP-S Symposium)_. IEEE, 2015, pp. 344–344.
  * [27] R. C. Johnson, H. A. Ecker, and J. S. Hollis, “Determination of far-field antenna patterns from near-field measurements,” _Proceedings of the IEEE_ , vol. 61, no. 12, pp. 1668–1694, Dec 1973.
  * [28] J. Guo, S. Durrani, X. Zhou, and H. Yanikomeroglu, “Outage probability of ad hoc networks with wireless information and power transfer,” _IEEE Wireless Communications Letters_ , vol. 4, no. 4, pp. 409–412, Aug 2015.
  * [29] J. Xu and R. Zhang, “Energy beamforming with one-bit feedback,” _IEEE Transactions on Signal Processing_ , vol. 62, no. 20, pp. 5370–5381, Oct 2014.
  * [30] Z. Wang, L. Duan, and R. Zhang, “Adaptively directional wireless power transfer for large-scale sensor networks,” _IEEE Journal on Selected Areas in Communications_ , vol. 34, no. 5, pp. 1785–1800, May 2016.
  * [31] S. Bi, Y. Zeng, and R. Zhang, “Wireless powered communication networks: an overview,” _IEEE Wireless Communications_ , vol. 23, no. 2, pp. 10–18, April 2016.
  * [32] W. Zhang, Y. Qin, W. Zhao, M. Jia, Q. Liu, R. He, and B. Ai, “A green paradigm for internet of things: Ambient backscatter communications,” _China Communications_ , vol. 16, no. 7, pp. 109–119, 2019.
  * [33] H. Ju and R. Zhang, “Throughput maximization in wireless powered communication networks,” _IEEE Transactions on Wireless Communications_ , vol. 13, no. 1, pp. 418–428, January 2014.
  * [34] ——, “Optimal resource allocation in full-duplex wireless-powered communication network,” _IEEE Transactions on Communications_ , vol. 62, no. 10, pp. 3528–3540, Oct 2014.
  * [35] H. Lim and T. Hwang, “User-centric energy efficiency optimization for miso wireless powered communications,” _IEEE Transactions on Wireless Communications_ , vol. 18, no. 2, pp. 864–878, 2019.
  * [36] J. Seo and J. H. Lee, “Energy beamforming for full-duplex wireless powered communication in presence of eavesdropper,” in _2017 IEEE 86th Vehicular Technology Conference (VTC-Fall)_ , Sep. 2017, pp. 1–5.
  * [37] N. Van Huynh, D. T. Hoang, X. Lu, D. Niyato, P. Wang, and D. I. Kim, “Ambient backscatter communications: A contemporary survey,” _IEEE Communications Surveys Tutorials_ , vol. 20, no. 4, pp. 2889–2922, 2018\.
  * [38] P. Ramezani and A. Jamalipour, “Optimal resource allocation in backscatter assisted wpcn with practical energy harvesting model,” _IEEE Transactions on Vehicular Technology_ , vol. 68, no. 12, pp. 12 406–12 410, 2019\.
  * [39] B. Clerckx, Z. Bayani Zawawi, and K. Huang, “Wirelessly powered backscatter communications: Waveform design and snr-energy tradeoff,” _IEEE Communications Letters_ , vol. 21, no. 10, pp. 2234–2237, 2017.
  * [40] K. Han and K. Huang, “Wirelessly powered backscatter communication networks: Modeling, coverage, and capacity,” _IEEE Transactions on Wireless Communications_ , vol. 16, no. 4, pp. 2548–2561, 2017.
  * [41] C. Boyer and S. Roy, “— invited paper — backscatter communication and rfid: Coding, energy, and mimo analysis,” _IEEE Transactions on Communications_ , vol. 62, no. 3, pp. 770–785, 2014.
  * [42] R. Zhang and C. K. Ho, “Mimo broadcasting for simultaneous wireless information and power transfer,” in _2011 IEEE Global Telecommunications Conference - GLOBECOM 2011_ , 2011, pp. 1–5.
  * [43] I. Krikidis, S. Sasaki, S. Timotheou, and Z. Ding, “A low complexity antenna switching for joint wireless information and energy transfer in mimo relay channels,” _IEEE Transactions on Communications_ , vol. 62, no. 5, pp. 1577–1587, 2014.
  * [44] D. W. K. Ng, E. S. Lo, and R. Schober, “Robust beamforming for secure communication in systems with wireless information and power transfer,” _IEEE Transactions on Wireless Communications_ , vol. 13, no. 8, pp. 4599–4615, 2014.
  * [45] W. Lu, Y. Gong, J. Wu, H. Peng, and J. Hua, “Simultaneous wireless information and power transfer based on joint subcarrier and power allocation in ofdm systems,” _IEEEAccess_ , vol. 5, pp. 2763–2770, 2017.
  * [46] W. Lu, S. Fang, S. Hu, X. Liu, B. Li, Z. Na, and Y. Gong, “Energy efficiency optimization for ofdm based 5g wireless networks with simultaneous wireless information and power transfer,” _IEEE Access_ , vol. 6, pp. 75 937–75 946, 2018.
  * [47] D. W. K. Ng, E. S. Lo, and R. Schober, “Energy-efficient resource allocation in multiuser ofdm systems with wireless information and power transfer,” in _2013 IEEE Wireless Communications and Networking Conference (WCNC)_ , April 2013, pp. 3823–3828.
  * [48] K. Huang and E. Larsson, “Simultaneous information and power transfer for broadband wireless systems,” _IEEE Transactions on Signal Processing_ , vol. 61, no. 23, pp. 5972–5986, Dec 2013.
  * [49] X. Zhou, R. Zhang, and C. K. Ho, “Wireless information and power transfer in multiuser ofdm systems,” _IEEE Transactions on Wireless Communications_ , vol. 13, no. 4, pp. 2282–2294, April 2014.
  * [50] D. Xu and H. Zhu, “Outage minimized resource allocation for multiuser ofdm systems with swipt,” _IEEE Access_ , vol. 7, pp. 79 714–79 725, 2019.
  * [51] H. Yu, S. Guo, Y. Yang, L. Ji, and Y. Yang, “Secrecy energy efficiency optimization for downlink two-user ofdma networks with swipt,” _IEEE Systems Journal_ , vol. 13, no. 1, pp. 324–335, 2019.
  * [52] D. W. K. Ng, E. S. Lo, and R. Schober, “Wireless information and power transfer: Energy efficiency optimization in ofdma systems,” _IEEE Transactions on Wireless Communications_ , vol. 12, no. 12, pp. 6352–6370, December 2013.
  * [53] E. Hossain, D. Niyato, and Z. Han, _Dynamic spectrum access and management in cognitive radio networks_. Cambridge university press, 2009.
  * [54] X. Lu, P. Wang, D. Niyato, and E. Hossain, “Dynamic spectrum access in cognitive radio networks with rf energy harvesting,” _IEEE Wireless Communications_ , vol. 21, no. 3, pp. 102–110, June 2014.
  * [55] A. Celik, A. Alsharoa, and A. E. Kamal, “Hybrid energy harvesting cooperative spectrum sensing in heterogeneous crns,” in _2016 IEEE Globecom Workshops (GC Wkshps)_ , 2016, pp. 1–6.
  * [56] K. Zheng, X. Liu, Y. Zhu, K. Chi, and K. Liu, “Total throughput maximization of cooperative cognitive radio networks with energy harvesting,” _IEEE Transactions on Wireless Communications_ , vol. 19, no. 1, pp. 533–546, 2020.
  * [57] L. Ni, X. Da, H. Hu, M. Zhang, and K. Cumanan, “Outage constrained robust secrecy energy efficiency maximization for eh cognitive radio networks,” _IEEE Wireless Communications Letters_ , vol. 9, no. 3, pp. 363–366, 2020.
  * [58] H. Lei, M. Xu, H. Zhang, G. Pan, I. S. Ansari, and K. A. Qaraqe, “Secrecy outage performance for underlay mimo crns with energy harvesting and transmit antenna selection,” in _2016 IEEE Globecom Workshops (GC Wkshps)_ , 2016, pp. 1–6.
  * [59] H. Lei, M. Xu, I. S. Ansari, G. Pan, K. A. Qaraqe, and M. Alouini, “On secure underlay mimo cognitive radio networks with energy harvesting and transmit antenna selection,” _IEEE Transactions on Green Communications and Networking_ , vol. 1, no. 2, pp. 192–203, 2017.
  * [60] Z. Ali, G. A. S. Sidhu, M. Waqas, L. Xing, and F. Gao, “A joint optimization framework for energy harvesting based cooperative cr networks,” _IEEE Transactions on Cognitive Communications and Networking_ , vol. 5, no. 2, pp. 452–462, 2019.
  * [61] C. Xu, M. Zheng, W. Liang, H. Yu, and Y. Liang, “End-to-end throughput maximization for underlay multi-hop cognitive radio networks with rf energy harvesting,” _IEEE Transactions on Wireless Communications_ , vol. 16, no. 6, pp. 3561–3572, 2017.
  * [62] Z. Hu, N. Wei, and Z. Zhang, “Optimal resource allocation for harvested energy maximization in wideband cognitive radio network with swipt,” _IEEE Access_ , vol. 5, pp. 23 383–23 394, 2017.
  * [63] M. Peng, Y. Liu, D. Wei, W. Wang, and H. Chen, “Hierarchical cooperative relay based heterogeneous networks,” _IEEE Wireless Communications_ , vol. 18, no. 3, pp. 48–56, June 2011.
  * [64] M. A. Hossain, R. Md Noor, K. A. Yau, I. Ahmedy, and S. S. Anjum, “A survey on simultaneous wireless information and power transfer with cooperative relay and future challenges,” _IEEE Access_ , vol. 7, pp. 19 166–19 198, 2019.
  * [65] Y. Zeng, H. Chen, and R. Zhang, “Bidirectional wireless information and power transfer with a helping relay,” _IEEE Communications Letters_ , vol. 20, no. 5, pp. 862–865, May 2016.
  * [66] Z. Chen, P. Xu, Z. Ding, and X. Dai, “Cooperative transmission in simultaneous wireless information and power transfer networks,” _IEEE Transactions on Vehicular Technology_ , vol. 65, no. 10, pp. 8710–8715, 2016.
  * [67] Z. Ding, I. Krikidis, B. Sharif, and H. V. Poor, “Wireless information and power transfer in cooperative networks with spatially random relays,” _IEEE Transactions on Wireless Communications_ , vol. 13, no. 8, pp. 4440–4453, 2014.
  * [68] Y. Liu, “Wireless information and power transfer for multirelay-assisted cooperative communication,” _IEEE Communications Letters_ , vol. 20, no. 4, pp. 784–787, 2016.
  * [69] S. Guo, X. Zhou, and X. Zhou, “Energy-efficient resource allocation in swipt cooperative wireless networks,” _IEEE Systems Journal_ , pp. 1–12, 2020.
  * [70] Y. Liu, “Joint resource allocation in swipt-based multiantenna decode-and-forward relay networks,” _IEEE Transactions on Vehicular Technology_ , vol. 66, no. 10, pp. 9192–9200, 2017.
  * [71] S. Gautam, E. Lagunas, S. K. Sharma, S. Chatzinotas, and B. Ottersten, “Relay selection strategies for swipt-enabled cooperative wireless systems,” in _2017 IEEE 28th Annual International Symposium on Personal, Indoor, and Mobile Radio Communications (PIMRC)_ , 2017, pp. 1–7.
  * [72] S. Gautam, E. Lagunas, S. Chatzinotas, and B. Ottersten, “Relay selection and resource allocation for swipt in multi-user ofdma systems,” _IEEE Transactions on Wireless Communications_ , vol. 18, no. 5, pp. 2493–2508, 2019.
  * [73] N. Chiurtu, B. Rimoldi, and E. Telatar, “On the capacity of multi-antenna gaussian channels,” in _Proceedings. 2001 IEEE International Symposium on Information Theory (IEEE Cat. No.01CH37252)_ , 2001, pp. 53–.
  * [74] S. Luo, J. Xu, T. J. Lim, and R. Zhang, “Capacity region of miso broadcast channel for simultaneous wireless information and power transfer,” _IEEE Transactions on Communications_ , vol. 63, no. 10, pp. 3856–3868, Oct 2015.
  * [75] H. Liu, Y. Peng, D. Wang, X. Ma, Y. Yang, Y. Wu, and C. Zhang, “On optimizing power allocation and power splitting for miso swipt systems with heterogeneous users,” in _2017 IEEE Conference on Computer Communications Workshops (INFOCOM WKSHPS)_ , 2017, pp. 577–582.
  * [76] A. A. Okandeji, M. R. A. Khandaker, K. Wong, G. Zheng, Y. Zhang, Z. Zheng, and R. Liu, “Swipt in miso full-duplex systems,” _Journal of Communications and Networks_ , vol. 19, no. 5, pp. 470–480, 2017\.
  * [77] S. Wang, L. Ma, and Y. Xu, “A novel beamforming design for transmission power minimization in swipt system,” in _2018 IEEE International Conference on Communications (ICC)_ , 2018, pp. 1–6.
  * [78] H. Lee, K. Lee, H. Kim, and I. Lee, “Joint transceiver optimization for miso swipt systems with time switching,” _IEEE Transactions on Wireless Communications_ , vol. 17, no. 5, pp. 3298–3312, 2018.
  * [79] Y. Dong, M. J. Hossaini, J. Cheng, and V. C. M. Leung, “Robust energy efficient beamforming in misome-swipt systems with proportional secrecy rate,” _IEEE Journal on Selected Areas in Communications_ , vol. 37, no. 1, pp. 202–215, 2019.
  * [80] R. Zhang and C. K. Ho, “Mimo broadcasting for simultaneous wireless information and power transfer,” _IEEE Transactions on Wireless Communications_ , vol. 12, no. 5, pp. 1989–2001, 2013.
  * [81] K. Xiong, B. Wang, and K. J. R. Liu, “Rate-energy region of swipt for mimo broadcasting under nonlinear energy harvesting model,” _IEEE Transactions on Wireless Communications_ , vol. 16, no. 8, pp. 5147–5161, 2017\.
  * [82] W. Wang, L. Li, Q. Sun, and J. Jin, “Power allocation in multiuser mimo systems for simultaneous wireless information and power transfer,” in _2013 IEEE 78th Vehicular Technology Conference (VTC Fall)_ , 2013, pp. 1–5.
  * [83] S. Zhao, Q. Li, Q. Zhang, and J. Qin, “Antenna selection for simultaneous wireless information and power transfer in mimo systems,” _IEEE Communications Letters_ , vol. 18, no. 5, pp. 789–792, 2014.
  * [84] J. Rubio, A. Pascual-Iserte, D. P. Palomar, and A. Goldsmith, “Joint optimization of power and data transfer in multiuser mimo systems,” _IEEE Transactions on Signal Processing_ , vol. 65, no. 1, pp. 212–227, 2017\.
  * [85] B. Fang, Z. Qian, W. Zhong, and W. Shao, “An-aided secrecy precoding for swipt in cognitive mimo broadcast channels,” _IEEE Communications Letters_ , vol. 19, no. 9, pp. 1632–1635, 2015.
  * [86] C. Song, C. Ling, J. Park, and B. Clerckx, “Mimo broadcasting for simultaneous wireless information and power transfer: Weighted mmse approaches,” in _2014 IEEE Globecom Workshops (GC Wkshps)_ , 2014, pp. 1151–1156.
  * [87] X. Wang, W. Feng, Y. Chen, Z. Chu, and N. Ge, “Energy-efficiency maximization for secure multiuser mimo swipt systems with csi uncertainty,” _IEEE Access_ , vol. 6, pp. 2097–2109, 2018.
  * [88] J. Tang, D. K. C. So, N. Zhao, A. Shojaeifard, and K. Wong, “Energy efficiency optimization with swipt in mimo broadcast channels for internet of things,” _IEEE Internet of Things Journal_ , vol. 5, no. 4, pp. 2605–2619, 2018.
  * [89] S. Boyd and L. Vandenberghe, _Convex optimization_. Cambridge university press, 2004.
  * [90] C. Goh, _Duality in optimization and variational inequalities_. CRC Press, 2002.
  * [91] R. T. Rockafellar, _Conjugate duality and optimization_. Siam, 1974, vol. 16.
  * [92] Y. Sun, P. Babu, and D. P. Palomar, “Majorization-minimization algorithms in signal processing, communications, and machine learning,” _IEEE Transactions on Signal Processing_ , vol. 65, no. 3, pp. 794–816, Feb 2017.
  * [93] L. J. Hong, Y. Yang, and L. Zhang, “Sequential convex approximations to joint chance constrained programs: A monte carlo approach,” _Operations Research_ , vol. 59, no. 3, pp. 617–630, 2011.
  * [94] A. L. Yuille and A. Rangarajan, “The concave-convex procedure,” _Neural computation_ , vol. 15, no. 4, pp. 915–936, 2003.
  * [95] M. Razaviyayn, “Successive convex approximation: analysis and applications,” Ph.D. dissertation, University of Minnesota, 2014.
  * [96] B. R. Marks and G. P. Wright, “A general inner approximation algorithm for nonconvex mathematical programs,” _Operations research_ , vol. 26, no. 4, pp. 681–683, 1978.
  * [97] Q. T. Dinh and M. Diehl, “Local convergence of sequential convex programming for nonconvex optimization,” in _Recent Advances in Optimization and its Applications in Engineering_. Springer, 2010, pp. 93–102.
  * [98] M. Sinaie, P. Lin, A. Zappone, P. Azmi, and E. A. Jorswieck, “Delay-aware resource allocation for 5g wireless networks with wireless power transfer,” _IEEE Transactions on Vehicular Technology_ , vol. 67, no. 7, pp. 5841–5855, July 2018.
  * [99] M. Konstantinos, A. Adamis, and P. Constantinou, “Receiver architectures for ofdma systems with subband carrier allocation,” in _2008 14th European Wireless Conference_ , 2008, pp. 1–7.
  * [100] M. R. Bussieck and S. Vigerske, “MINLP solver software,” _Wiley encyclopedia of operations research and management science_ , 2010.
  * [101] D. T. Ngo, S. Khakurel, and T. Le-Ngoc, “Joint subchannel assignment and power allocation for OFDMA femtocell networks,” _IEEE Transactions on Wireless Communications_ , vol. 13, no. 1, pp. 342–355, Jan. 2014.
  * [102] Y. Shen, X. Huang, B. Yang, and S. Wang, “Alternate convex search method for resource allocation in OFDMA AF relay networks with energy harvesting,” in _Proc. ICCC_ , Shenzhen, China, pp. 1-6, Nov. 2015.
  * [103] M. F. Hanif, Z. Ding, T. Ratnarajah, and G. K. Karagiannidis, “A minorization-maximization method for optimizing sum rate in the downlink of non-orthogonal multiple access systems,” _IEEE Transactions on Signal Processing_ , vol. 64, no. 1, pp. 76–88, Jan 2016.
  * [104] T. Wang and L. Vandendorpe, “Iterative resource allocation for maximizing weighted sum min-rate in downlink cellular ofdma systems,” _IEEE Transactions on Signal Processing_ , vol. 59, no. 1, pp. 223–234, Jan. 2011.
  * [105] J. Lee and S. Leyffer, _Mixed integer nonlinear programming_. Springer Science & Business Media, 2011, vol. 154\.
  * [106] A. Khalili, M. R. Mili, M. Rasti, S. Parsaeefard, and D. W. K. Ng, “Antenna selection strategy for energy efficiency maximization in uplink ofdma networks: A multi-objective approach,” _IEEE Transactions on Wireless Communications_ , pp. 1–1, 2019.
  * [107] E. Che, H. D. Tuan, and H. H. Nguyen, “Joint optimization of cooperative beamforming and relay assignment in multi-user wireless relay networks,” _IEEE Transactions on Wireless Communications_ , vol. 13, no. 10, pp. 5481–5495, 2014.
  * [108] H. H. Kha, H. D. Tuan, and H. H. Nguyen, “Fast global optimal power allocation in wireless networks by local d.c. programming,” _IEEE Transactions on Wireless Communications_ , vol. 11, no. 2, pp. 510–515, February 2012.
  * [109] A. Khalili, S. Zarandi, and M. Rasti, “Joint resource allocation and offloading decision in mobile edge computing,” _IEEE Communications Letters_ , vol. 23, no. 4, pp. 684–687, April 2019.
  * [110] J. B. Andersen, T. S. Rappaport, and S. Yoshida, “Propagation measurements and models for wireless communications channels,” _IEEE Communications Magazine_ , vol. 33, no. 1, pp. 42–49, 1995.
  * [111] 3GPP, “Further advancements for E-UTRA physical layer aspect,” 3rd Generation Partnership Project (3GPP), Technical Specification (TS) 36.814, 2010, v9.0.0.
  * [112] D. W. K. Ng, E. S. Lo, and R. Schober, “Energy-efficient resource allocation in multi-cell ofdma systems with limited backhaul capacity,” in _2012 IEEE Wireless Communications and Networking Conference (WCNC)_ , April 2012, pp. 1146–1151.
  * [113] Y. Kim, B. C. Jung, I. Bang, and Y. Han, “Adaptive proportional fairness scheduling for swipt-enabled multicell downlink networks,” in _2019 IEEE Wireless Communications and Networking Conference (WCNC)_ , April 2019, pp. 1–6.
  * [114] S. Jang, H. Lee, S. Kang, T. Oh, and I. Lee, “Energy efficient beamforming for multi-cell miso swipt systems,” in _2018 IEEE 88th Vehicular Technology Conference (VTC-Fall)_ , Aug 2018.
  * [115] A. A. Nasir, D. T. Ngo, H. D. Tuan, S. Durrani, and D. I. Kim, “Secure beamforming for max-min sinr in multi-cell swipt systems,” in _2016 IEEE Wireless Communications and Networking Conference Workshops (WCNCW)_ , April 2016, pp. 404–409.
  * [116] K. Lange, _MM Optimization Algorithms_. Society for Industrial and Applied Mathematics Philadelphia, 2016\.
  * [117] D. W. K. Ng, E. S. Lo, and R. Schober, “Energy-efficient resource allocation in multi-cell ofdma systems with limited backhaul capacity,” _IEEE Transactions on Wireless Communications_ , vol. 11, no. 10, pp. 3618–3631, 2012.
  * [118] L. Zheng and C. W. Tan, “Maximizing sum rates in cognitive radio networks: Convex relaxation and global optimization algorithms,” _IEEE Journal on Selected Areas in Communications_ , vol. 32, no. 3, pp. 667–680, Mar. 2014.
  * [119] H. Zhang, C. Jiang, N. C. Beaulieu, X. Chu, X. Wen, and M. Tao, “Resource allocation in spectrum-sharing ofdma femtocells with heterogeneous services,” _IEEE Transactions on Communications_ , vol. 62, no. 7, pp. 2366–2377, 2014.
  * [120] D. Tse and P. Viswanath, _Fundamentals of Wireless Communication_. Cambridge university press, 2005.
  * [121] Wei Yu and J. M. Cioffi, “FDMA capacity of gaussian multiple-access channels with ISI,” _IEEE Transactions on Communications_ , vol. 50, no. 1, pp. 102–111, Jan 2002.
  * [122] J. Jalali, A. Khalili, and H. Steendam, “Antenna selection and resource allocation in downlink miso ofdma femtocell networks,” _Accepted for presentation in IEEE VTC Spring_ , 2020.
  * [123] S. Zhao, Q. Li, Q. Zhang, and J. Qin, “Antenna selection for simultaneous wireless information and power transfer in MIMO systems,” _IEEE Communications Letters_ , vol. 18, no. 5, pp. 789–792, May 2014.
  * [124] Z. Ding, C. Zhong, D. Wing Kwan Ng, M. Peng, H. A. Suraweera, R. Schober, and H. V. Poor, “Application of smart antenna technologies in simultaneous wireless information and power transfer,” _IEEE Communications Magazine_ , vol. 53, no. 4, pp. 86–93, April 2015.
  * [125] M. Sandell and J. Coon, “Performance of combined bulk and per-tone antenna selection precoding in coded ofdm systems,” _IEEE Transactions on Communications_ , vol. 60, no. 3, pp. 655–660, 2012.
  * [126] N. P. Le, F. Safaei, and L. C. Tran, “Antenna selection strategies for mimo-ofdm wireless systems: An energy efficiency perspective,” _IEEE Transactions on Vehicular Technology_ , vol. 65, no. 4, pp. 2048–2062, April 2016\.
  * [127] N. B. Mehta, S. Kashyap, and A. F. Molisch, “Antenna selection in lte: from motivation to specification,” _IEEE Communications Magazine_ , vol. 50, no. 10, pp. 144–150, 2012.
  * [128] E. Hossain, M. Rasti, H. Tabassum, and A. Abdelnasser, “Evolution toward 5g multi-tier cellular wireless networks: An interference management perspective,” _IEEE Wireless Communications_ , vol. 21, no. 3, pp. 118–127, 2014.
  * [129] W. Dinkelbach, “On nonlinear fractional programming,” _Management science_ , vol. 13, no. 7, pp. 492–498, 1967.
  * [130] A. Khalili, S. Akhlaghi, H. Tabassum, and D. W. K. Ng, “Joint user association and resource allocation in the uplink of heterogeneous networks,” _IEEE Wireless Communications Letters_ , pp. 1–1, 2020.
  * [131] J. Zeng, T. Lv, R. P. Liu, X. Su, M. Peng, C. Wang, and J. Mei, “Investigation on evolving single-carrier noma into multi-carrier noma in 5g,” _IEEE Access_ , vol. 6, pp. 48 268–48 288, 2018.
  * [132] H. Zhang, M. Feng, K. Long, G. K. Karagiannidis, V. C. M. Leung, and H. V. Poor, “Energy efficient resource management in swipt enabled heterogeneous networks with noma,” _IEEE Transactions on Wireless Communications_ , vol. 19, no. 2, pp. 835–845, 2020.
  * [133] J. Jalali and A. Khalili, “Optimal resource allocation for mc-noma in swipt-enabled networks,” _IEEE Commun. Lett, Early Access_ , June 2020.
  * [134] E. G. Larsson, O. Edfors, F. Tufvesson, and T. L. Marzetta, “Massive mimo for next generation wireless systems,” _IEEE Communications Magazine_ , vol. 52, no. 2, pp. 186–195, 2014.
  * [135] A. L. Swindlehurst, E. Ayanoglu, P. Heydari, and F. Capolino, “Millimeter-wave massive mimo: the next wireless revolution?” _IEEE Communications Magazine_ , vol. 52, no. 9, pp. 56–62, 2014.
  * [136] J. G. Andrews, T. Bai, M. N. Kulkarni, A. Alkhateeb, A. K. Gupta, and R. W. Heath, “Modeling and analyzing millimeter wave cellular systems,” _IEEE Transactions on Communications_ , vol. 65, no. 1, pp. 403–430, 2017\.
  * [137] T. A. Khan, A. Alkhateeb, and R. W. Heath, “Millimeter wave energy harvesting,” _IEEE Transactions on Wireless Communications_ , vol. 15, no. 9, pp. 6048–6062, 2016.
  * [138] W. Hao, G. Sun, Z. Chu, P. Xiao, Z. Zhu, S. Yang, and R. Tafazolli, “Beamforming design in swipt-based joint multicast-unicast mmwave massive mimo with lens-antenna array,” _IEEE Wireless Communications Letters_ , vol. 8, no. 4, pp. 1124–1128, 2019.
  * [139] F. Zhu, F. Gao, Y. C. Eldar, and G. Qian, “Robust simultaneous wireless information and power transfer in beamspace massive mimo,” _IEEE Transactions on Wireless Communications_ , vol. 18, no. 9, pp. 4199–4212, 2019.
  * [140] W. Hao, G. Sun, F. Zhou, D. Mi, J. Shi, P. Xiao, and V. C. M. Leung, “Energy-efficient hybrid precoding design for integrated multicast-unicast millimeter wave communications with swipt,” _IEEE Transactions on Vehicular Technology_ , vol. 68, no. 11, pp. 10 956–10 968, 2019\.
  * [141] L. Dai, B. Wang, M. Peng, and S. Chen, “Hybrid precoding-based millimeter-wave massive mimo-noma with simultaneous wireless information and power transfer,” _IEEE Journal on Selected Areas in Communications_ , vol. 37, no. 1, pp. 131–141, 2019.
  * [142] M. Alageli, A. Ikhlef, F. Alsifiany, M. A. M. Abdullah, G. Chen, and J. Chambers, “Optimal downlink transmission for cell-free swipt massive mimo systems with active eavesdropping,” _IEEE Transactions on Information Forensics and Security_ , vol. 15, pp. 1983–1998, 2020.
  * [143] X. Wang and M. C. Gursoy, “Coverage analysis for energy-harvesting uav-assisted mmwave cellular networks,” _IEEE Journal on Selected Areas in Communications_ , vol. 37, no. 12, pp. 2832–2850, 2019.
  * [144] D. N. K. Jayakody, T. D. P. Perera, A. Ghrayeb, and M. O. Hasna, “Self-energized uav-assisted scheme for cooperative wireless relay networks,” _IEEE Transactions on Vehicular Technology_ , vol. 69, no. 1, pp. 578–592, 2020.
  * [145] S. Yin, Y. Zhao, L. Li, and F. R. Yu, “Uav-assisted cooperative communications with time-sharing information and power transfer,” _IEEE Transactions on Vehicular Technology_ , vol. 69, no. 2, pp. 1554–1567, 2020.
  * [146] X. Sun, W. Yang, and Y. Cai, “Secure communication in noma-assisted millimeter-wave swipt uav networks,” _IEEE Internet of Things Journal_ , vol. 7, no. 3, pp. 1884–1897, 2020.
  * [147] Z. Yang, W. Xu, and M. Shikh-Bahaei, “Energy efficient uav communication with energy harvesting,” _IEEE Transactions on Vehicular Technology_ , vol. 69, no. 2, pp. 1913–1927, 2020.
  * [148] J. Kang and C. Chun, “Joint trajectory design, tx power allocation, and rx power splitting for uav-enabled multicasting swipt systems,” _IEEE Systems Journal_ , pp. 1–4, 2020.
  * [149] X. Wang, M. C. Gursoy, and I. Guvenc, “Simultaneous wireless information and power transfer in uav-assisted cellular iot networks,” in _2020 IEEE 17th Annual Consumer Communications Networking Conference (CCNC)_ , 2020, pp. 1–6.
  * [150] Y. Mao, C. You, J. Zhang, K. Huang, and K. B. Letaief, “A survey on mobile edge computing: The communication perspective,” _IEEE Communications Surveys Tutorials_ , vol. 19, no. 4, pp. 2322–2358, 2017.
  * [151] H. Zheng, K. Xiong, P. Fan, Z. Zhong, and K. B. Letaief, “Fog-assisted multiuser swipt networks: Local computing or offloading,” _IEEE Internet of Things Journal_ , vol. 6, no. 3, pp. 5246–5264, 2019.
  * [152] J. Liu, K. Xiong, P. Fan, Z. Zhong, and K. B. Letaief, “Optimal design of swipt-aware fog computing networks,” in _IEEE INFOCOM 2019 - IEEE Conference on Computer Communications Workshops (INFOCOM WKSHPS)_ , 2019, pp. 13–19.
  * [153] A. I. Akin, H. Mirghasemi, and L. Vandendorpe, “Swipt-based mobile edge computing systems: A stochastic geometry perspective,” in _2019 IEEE 30th Annual International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC)_ , 2019, pp. 1–7.
  * [154] F. Chen, J. Fu, Z. Wang, Y. Zhou, and W. Qiu, “Joint communication and computation resource optimization in fd-mec cellular networks,” _IEEE Access_ , vol. 7, pp. 168 444–168 454, 2019.
  * [155] P. Cai, F. Yang, J. Wang, X. Wu, Y. Yang, and X. Luo, “Jote: Joint offloading of tasks and energy in fog-enabled iot networks,” _IEEE Internet of Things Journal_ , pp. 1–1, 2020.
  * [156] J. Fu, J. Hua, J. Wen, H. Chen, W. Lu, and J. Li, “Optimization of energy consumption in the mec-assisted multi-user fd-swipt system,” _IEEE Access_ , vol. 8, pp. 21 345–21 354, 2020.
  * [157] J. Fu, J. Hua, J. Wen, K. Zhou, J. Li, and B. Sheng, “Optimization of achievable rate in the multi-user satellite iot system with swipt and mec,” _IEEE Transactions on Industrial Informatics_ , pp. 1–1, 2020.