Performance Analysis of Millimeter Wave Wireless Power Transfer With Imperfect Beam Alignment
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Performance Analysis of Millimeter Wave Wireless
Power Transfer With Imperfect Beam Alignment
Man Wang, Chao Zhang, Xiaoming Chen, Senior Member, IEEE, Suhua Tang, Member, IEEE
Abstract—In this paper, the impact of imperfect beam align- networks [3], [4]. At present, RF-WPT has been successfully
ment (IBA) on millimeter wave (mmWave) wireless power trans- applied in various wireless networks [1]–[4].
fer (WPT) is investigated. We consider a mmWave WPT network,
arXiv:2003.10290v2 [cs.IT] 21 Feb 2021
For addressing the demand of ever-increasing data trans-
where the location of the energy transmitters follows a Poisson
point process. Instead of the mostly used flat-top antenna model, mission rates, millimeter wave (mmWave) frequencies are
we adopt the Gaussian antenna model suggested by the 3rd leveraged to fulfill the multi-Gigabit information transmission
Generation Partnership Project (3GPP) for better accuracy. Two requirement [5]. Due to the small wavelength and blockage
beam alignment error (BAE) models, i.e., truncated Gaussian sensitivity of mmWave signals, strong directional antenna and
and uniform models, are introduced to represent different BAE small cell structure are suggested to enhance the energy and
sources. We derive the probability density functions (PDFs) of
the cascaded antenna gain with both BAE models and then spectrum efficiency of mmWave communications [6]. As WPT
provide the approximated PDFs for tractability. With the help of also suffers from the severe power propagation loss and needs
Fox’s H function, the analytic expression for the energy coverage to avoid interference to existing wireless information networks,
probability with nonlinear energy harvesting model is derived. mmWave also benefits WPT. It has been proven by [7], [8] that
Besides, we deduce a closed-form expression of the average mmWave WPT outperforms WPT with lower frequencies. In
harvested radio frequency (RF) energy. The simulation results
verify our theoretical results and demonstrate the performance [9], lens array based mmWave WPT was proposed to charge
degradation incurred by BAE. It also shows that in the imperfect multiple energy receivers. In [10], simultaneous wireless in-
beam alignment scenario, the Gaussian antenna model can formation and power transfer (SWIPT) was applied in the
accurately represent the performance of mmWave WPT networks hybrid precoding mmWave system. Besides, the effect of rain
with actual beam pattern, while the flat-top antenna model cannot attenuation on mmWave WPT was investigated by [11]. All
always provide accurate performance evaluation.
that literature shows mmWave WPT is feasible.
Index Terms—Beam alignment error, energy coverage proba- To evaluate the system-level performance of mmWave WPT,
bility, millimeter wave, wireless power transfer, stochastic geom- stochastic geometry has been utilized to capture the effects of
etry.
propagation loss and blockage, which dominate the received
signal power [12], [13]. The energy coverage probability and
I. I NTRODUCTION average harvested energy of mmWave WPT were studied by
[7], where the location of energy transmitters follows the
I N the foreseeable future, there would be huge amounts of
low-power devices in wireless networks to perform infor-
mation forwarding, data collecting, situation sensing [1]. These
Poisson point process (PPP). Furthermore, SWIPT was also
introduced into stochastic mmWave networks [8], [14]. Both
devices could be the nodes in Internet of Things (IoT), wireless works verified that mmWave could improve the performance
sensor networks (WSNs) or Device-to-Device (D2D) systems. of SWIPT compared to lower frequency solutions. In [15],
These low-power devices are usually powered by batteries and the energy coverage probability in the presence of human
deployed in a broad area [2]. In order to prolong the network blockage was derived. Discretizing harvested energy into a
lifetime and maintain the sustainability of these nodes, far- finite number of power levels, the total coverage probability
field wireless power transfer (WPT) via radio frequency (RF) integrating information and energy transmission was derived
has been considered as a promising technology to energize by [16]. A beam-training based mmWave WPT scheme was
massive battery-powered devices. The reason is that RF-WPT proposed in [17], where the energy transmitter steers the
can provide a flexible and long-distance charging service, energy beam along the direction in which it receives the
while being compatible with the existing wireless information strongest training signal. Considering the nonlinear behavior
of energy harvesting, the coverage probability and average
Copyright (c) 2015 IEEE. Personal use of this material is permitted. harvested energy were studied by [18]. In [19], the location of
However, permission to use this material for any other purposes must be mmWave powered users was modeled as the Poisson cluster
obtained from the IEEE by sending a request to pubs-permissions@ieee.org.
M. Wang, C. Zhang and X. Chen are with School of Information and Com- process and the energy and information coverage probabilities
munications Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, were derived.
China. (e-mail: manwang@stu.xjtu.edu.cn, chaozhang@mail.xjtu.edu.cn, xi- Most of the aforementioned works, such as [7], [8], [14]–
aoming.chen@mail.xjtu.edu.cn)
S. Tang is with Department of Computer and Network Engineering, The [19], employ the flat-top antenna pattern for mathematical
University of Electro-Communications, 1-5-1 Chofugaoka, Chofu, Tokyo 182- tractability. Nevertheless, it may incur great inaccuracy for
8585, Japan. (e-mail:shtang@uec.ac.jp) evaluating system-level performance [20], [21]. Therefore,
The work is supported in part by National Key Research and Develop-
ment Program of China (No.2020YFB1807000), and the Key Research and some literature considers more realistic antenna patterns in the
Development Program of Shaanxi Province (No. 2020GY-023). performance analysis of mmWave WPT. In [22], the microstrip2
patch antenna and end-fire antenna models were used to the non-associated Txs to the typical Rx are simultaneously
analyze the coverage performance of SWIPT. [23] adopted the taken into consideration. Besides, the nonlinear energy har-
Fejér kernel model to represent the actual array antenna gain of vesting model in the typical receiver is also assumed. With
energy transmitters. Both [22] and [23] assumed each energy these models, we derive the analytic expression of the system-
transmitter is equipped with a directional antenna, while the level performance of the mmWave WPT in the IBA scenario.
antenna of energy receiver was assumed to be isotropic. In For clarity, we summarize our main contribution as follows:
[24], both Tx and Rx were assumed to be equipped with • We adopt the Gaussian antenna gain model suggested by
directional antennas and the average harvested energy was the 3GPP to represent the realistic directional antenna
derived. pattern and also assume all transmitters and receivers are
Most of the existing works on mmWave communica- equipped with the same directional antenna. Considering
tions/WPT have assumed that the beam direction of the the truncated Gaussian and uniform BAE distributions,
associated Tx-Rx pair is perfectly aligned. However, due we derive the probability density functions (PDFs) of
to direction estimation error and hardware imperfection, the the cascaded antenna gains of a Tx-Rx pair. Taking into
beam alignment error (BAE) is always inevitable, and affects account of the distribution characteristics of BAE and the
the received signal strength greatly [25]. Therefore, it is strong directivity of mmWave antenna, we also provide
necessary to investigate the performance of the mmWave the approximated PDFs for tractability.
transmission systems with imperfect beam alignment (IBA). • With the help of the Fox’s H function and its series
By now, much effort has been dedicated to evaluating the expansion, we provide a novel solution to derive the
performance of the mmWave wireless information networks analytic expression of the energy coverage probability of
in the IBA scenario, such as [21], [26]–[30]. Taking into the mmWave WPT in the presence of BAE. Besides, the
account of the BAEs in the associated Tx-Rx pair and the closed-form expression of average harvested RF energy
Interfering Txs-Rx pairs, the information coverage probability is derived. We define the relative energy loss (REL) in
with the flat-top antenna gain model was derived by [26]. order to further investigate the performance degradation
With the same BAE setup, while, [27] adopted the cosine incurred by BAE.
antenna and isotropic antenna patterns to model the antenna • Through Monte-Carlo simulations, we verify the derived
gains of the transmitters and receivers, respectively. In [28], energy coverage probability and average harvested RF
the approximated ergodic capacity loss of the mmWave ad energy. It is found that the widely used flat-top antenna
hoc network with both flat-top and Gaussian antenna gain gain model cannot always exactly evaluate the perfor-
models was derived. In [21], the authors employed the sinc mance of the mmWave WPT system in the presence
and cosine antenna patterns to derive the information coverage of BAE. We also conclude that BAE indeed lowers the
probability, just considering the BAEs between the typical energy coverage probability and the average harvested RF
receiver and interfering transmitters. Alternatively, using the energy.
antenna gain model suggested by the 3rd Generation Part- Notation: Z, R, and C represent the sets of all integers,
nership Project (3GPP), [29] investigated the impact of beam real numbers and complex numbers, respectively. E{x} is
misalignment from interfering transmitters, and resorted to the the expectation of random variable (r.v.) x and Pr(A) is the
curve-fitting method to derive the probability density function probability of event A. For r.v. x, fx (·) and Fx (·) stand
of the antenna gain with BAE. Furthermore, [30] employed for the probability density function (PDF) and cumulative
both the 2D and 3D directional antenna gains and derived the distribution function (CDF) of x, respectively. For x ∈ C,
information coverage probability with BAE incurred by the |x| is the modulus of x. Given x ∈ R2 , ||x|| means the
strongest interfering link. Euclidean norm of x. For a complex vector x, xT and xH
To the best of our knowledge, the impact of BAE on stand for transpose and conjugate transpose of x, respectively.
mmWave WPT system has not been investigated and well x ∼ Γ(k, θ) means the r.v. x follows Gamma distribution with
understood to date. Additionally, in this paper, we consider x
xk−1 e− θ
PDF fx (x) = Γ(k)θk ∀x > 0, k, θ > 0, where Γ(k) is
the comprehensive IBA scenarios and the realistic 3GPP direc- Rx 2
tional antenna model, which has not been well studied from the the Gamma function. erf (x) = √2π 0 e−t dt is the Gaussian
perspective of system-level performance evaluation in either error function and δ(x) is the Dirac delta function. ln x is
mmWave communications or mmWave WPT. Specifically, the natural logarithm of x. For k ≤ K and k, K ∈ Z, K k
K!
there are two aspects of our system model to be highlighted. represents the binomial coefficient and equals to k!(K−k)! . For
First, unlike [25], [26], we employ the 3GPP Gaussian gain r.v. x, Lx (a) = E{e−ax} is the Laplace transform of fx (x).
model, which can approximate the realistic directional antenna
pattern exactly, to explore the impact of BAE. The reason is II. S YSTEM M ODEL
that the flat-top model lacks the capability of depicting the
roll-off effect of the mainlobe and thus is not suitable for A. Network and Node Models
accurately evaluating the impact of BAE on mmWave systems In this paper, we consider a mmWave wireless-powered
[21], [27]–[29]. Second, different from [21], [27]–[30], we network which consists of two types of nodes, i.e., Energy
consider a more realistic IBA scenario, where all Txs and Rxs Transmitter (ETx) and Energy Receiver (ERx) (See the Fig.1
are equipped with directional antennas. Moreover, not only the on the next page). The ETxs are connected to stable power
BAE between the associated Tx-Rx pair but also the BAEs of sources, thus having the capability of emitting energy signals.3
accuracy, we employ the random shape theory model, by
which the analytical LOS probability is coincident with the
ࢥ[ empirical expression of the 3GPP blockage model [6]. Hence,
given the distance r, the LOS and the NLOS probabilities
[ ij[ can be separately expressed as PL (r) = e−βr and PN (r) =
1 − e−βr , where the blockage parameter β is determined by
U the average number and average perimeter of buildings in the
R interested area. Similar to most literature, e.g., [12]−[30], the
correlation of LOS probabilities for different mmWave links
is ignored. Consequently, the ETxs in Φt can be divided into
two independent sets, namely, LOS ETx set and NLOS ETx
set. Both of which follow the inhomogeneous PPP (IPPP)
with intensity PL (r)λt and PN (r)λt , denoted as ΦL and ΦN ,
(QHUJ\7[ 7\SLFDO(QHUJ\5[ respectively. As the ETx at x0 has the distance r0 to the typical
ERx, the performance analysis in the LOS scenario is very
(QHUJ\%HDP %ORFNDJHV similar to that in the NLOS scenario. Furthermore, considering
the harvested energy plays a crucial role in maintaining the
Fig. 1. System model lifetime of wireless-powered nodes, we assume the ETx only
selects the ERx experiencing LOS as its target receiver like
While, the ERxs have to harvest energy to maintain their [21], [32], [33].
routine operation and information transmission. The location
of all ETxs follows the homogeneous Poisson point process C. Antenna Pattern
(HPPP) Φ with intensity λt on the two-dimensional Euclidean We assume that all ETxs and ERxs are equipped with the
plane. Suppose a saturated service scenario, where each ETx same directional antenna to perform mmWave beamforming.
is assumed to serve one dedicated ERx in a WPT block [31], It is also assumed that all ETxs have the same transmit power
[32]. We study the performance of the typical ERx located Pt . For the typical ERx, the received RF energy transmitted
at the origin. Let the ETx associated by the typical ERx be by the ETx at x can be given by
located at x0 ∈ R2 and ||x0 || = r0 .1 Then, due to Slivnyak’s
Theorem [31], except the ETx at x0 , the location of other εx,i = Pt ℓi (rx )hx,i G(φx )G(ϕx ), i ∈ {L, N } (1)
ETxs still forms a PPP with the same intensity λt , denoted by
Φt = Φ \ {x0 }. Without loss of generality, the WPT duration where G(φx ) and G(ϕx ) stand for the antenna gains of ETx
is assumed as unit time. It means in following context, the and ERx, respectively. φx and ϕx are the orientation angles
harvested energy and the harvested power have an equivalence. relative to the boresights of ETx and ERx, respectively, and
belong to the interval [−π, π). Note that we ignore the energy
harvested from the noise, as it is trivial compared with the
B. Channel Model received RF energy [18].
For mmWave links, line-of-sight (LOS) and non-line-of- In order to theoretically evaluate the system-level perfor-
sight (NLOS) channels have sharply different propagation mance, it is inevitable to depict the PDF of the cascaded
characteristics. Given a propagation distance r, the path loss antenna gain Ωx = G(φx )G(ϕx ). For the uniform linear array
of a LOS mmWave link can be modeled as ℓL (r) = CL r−αL , (ULA), the Fejér kernel based sinc and cosine antenna patterns
while in the NLOS case, the path loss is ℓN (r) = CN r−αN . are employed to represent the directional antenna gain [21],
Here αL ( αN ) is the LOS (NLOS) path loss exponent and CL [23], [27], [34]. Unfortunately, it is difficult to directly derive
(CN ) represents the intercept of LOS (NLOS) link. In general, the exact analytic expression of the PDF of Ωx using the sinc
we have αN > αL > 0 and CL ≥ CN [6], [13]. Moreover, or cosine antenna gain model. In [21], [23], [27], the receiver
we model the small-scale fading of each mmWave link as was assumed to be equipped with omni-directional antenna to
independent Nakagami fading with parameters mL and mN avoid the cascaded directional antenna gain. Although [34]
for LOS and NLOS scenarios, respectively. Thus, we denote considered a cascaded antenna gain with the sinc antenna
ρL and ρN as the small-scale channel gains for LOS and pattern, the nodes were not stochastically deployed.
NLOS links, respectively, and then express the LOS power Alternatively, the flat-top antenna gain model is mostly used
gain as hL = |ρL |2 ∼ Γ(mL , 1/mL ) and NLOS power gain in stochastic geometry coverage analysis for its mathematical
as hN = |ρN |2 ∼ Γ(mN , 1/mN ). tractability, e.g., [7], [8], [14]–[19], [25], [26], [32], [33]. In
The probability of a mmWave link experiencing LOS is the perfect beam alignment or slight BAE scenarios, the flat-
modeled as a function with respect to the distance from Tx top antenna gain model can provide the tractable theoretical
to Rx [5]. Several empirical and analytical blockage models expression of the system-level performance with acceptable
had been reported in [5] and [6]. Considering tractability and accuracy. However, the flat-top antenna model is lack of the
1 Herein r can be seen as the maximum allowable WPT distance for an
capability of depicting the roll-off effect of the mainlobe and
0
ETx-ERx pair. Accordingly, in this work we investigate the worst performance incurs significantly inaccuracy for evaluating the system-level
of the considered mmWave WPT system [31], [32]. performance in the IBA scenario [21], [28], [29], [35], [36].4
It is worth mentioning that in [20] a generalized flat-top For the associated ETx-ERx pair, as the BAE appears after
model was proposed to approximate the practical antenna gain. the beam aligning procedure, φx0 and ϕx0 are usually modeled
Nevertheless, in the IBA case, it could incur relatively high as independent and identically distributed truncated Gaussian
complexity to derive the PDF of the cascaded antenna gain. random variables with zero mean and standard deviation σ ≥ 0
In [28], [35], a Gaussian antenna pattern, i.e., [26]–[28]. Such that, the PDF of φx0 can be expressed as
2 ψ2
G(θ) = (Gm − Gs )e−ηθ + Gs , e− 2σ2
fφx0 (ψ) = √ , ψ ∈ [−π, π) (4)
where Gm is the maximum mainlobe gain, Gs is the sidelobe 2πσ 2 erf √π2σ
gain and η is determined by the 3dB beamwidth, was used to
represent the mmWave directional antenna gain. Nevertheless, which is named as the Gaussian BAE model. The standard
to obtain the analytic PDF of the cascaded antenna gain Ωx , deviation σ is usually used to indicate the variability of
the side lobe Gs was ignored in the analysis of [28], [35]. BAE. The larger σ is, the stronger statistical dispersion the
Moreover, the loss in ergodic capacity derived by [28], [35] BAE exhibits, which means the BAE becomes more severe.
only involves the truncated Gaussian BAE. Differently, in the Observing (4), if σ → 0, fφx0 (ψ) gradually converges to δ(ψ).
non-stochastic mmWave networks [29], [36], [37] adopted the It is corresponding to the fact that if the beam is perfectly
3GPP Gaussian antenna model, which can depict the roll-off aligned, the antenna gain equals to Gm with probability 1.
characteristics and match the measurement well. In this paper, For the non-associated ETx-ERx pairs, an ETx in ΦL or
therefore, we also employ the 3GPP Gaussian antenna model ΦN just aligns its beam with the boresight of its paired ERx.
(refer to [36], and the references therein), i.e., To facilitate understanding and representation, for the typical
( ERx, given the ETx at x ∈ Φi , i ∈ {L, N }, we can also treat
2
Gm e−ηθ |θ| ≤ θ0 , φx and ϕx as BAEs, which are usually modeled as independent
G(θ) = (2) uniform distribution over [−π, π) [21], [26], [27], i.e.,
Gs θ0 < |θ| ≤ π,
1
where fφx (ψ) = , ψ ∈ [−π, π), x ∈ Φi , i ∈ {N, L}. (5)
0.3
θ02 2π
θ2
π10 3dB It is named as the uniform BAE model. Note that it is assumed
Gm = ,
Θ(θ0 , θ3dB ) + π − θ0 that BAEs at ETxs and ERxs are independently distributed
π [25], [26], [28].
Gs = ,
Θ(θ0 , θ3dB ) + π − θ0
2 2
Z θ0 θ −x E. Energy Harvesting Model
0.3 02
θ
Θ(θ0 , θ3dB ) = 10 3dB dx, As the small-scale gains of different mmWave links are
0
independently distributed, the harvested RF power of the
η = 0.3θ2ln 10 , 2θ0 is the mainlobe (20dB) beamwidth, and typical ERx can be written by
3dB
2θ3dB is the half-power (3dB) beamwidth. Since Gs =
2
Gm e−ηθ0 , the continuity is ensured. According to the practical εRF = ε0 + εL + εN , (6)
π P
measurement reported by [36], [38], when 24 ≤ θ0 ≤ π6 , θ0 where εL = εx,L is the harvested power from the
x∈Φ PL
is approximately equal to 2.6θ3dB . Then, with this empirical ETxs in ΦL , εN = x∈ΦN εx,N is the harvested power from
approximation, we further obtain Θ(θ0 , θ3dB ) = 42.6443θ0, the ETxs in ΦN , and ε0 = εx0 ,L is the harvested power from
π102.028
Gm = 42.6443θ 0 +π
, and Gs = 10−2.028 Gm . For convenience, the ETx located at x0 . Then, the harvested direct current (DC)
we introduce the normalized antenna gain G(θ)e = G(θ)/Gm , power at the typical ERx is
i.e., ( 2 εDC = ζ(εRF ). (7)
e e−ηθ |θ| ≤ θ0 ,
G(θ) = (3)
g θ0 < |θ| ≤ π, Note that ζ(εRF ) is the RF-DC power conversion function.
In practice, ζ(εRF ) is a nonlinear function with respect to
in which g = 10−2.028 and η = 2.028θ2ln 10 . Obviously, with the input RF power εRF [18], [23], [39]. Using the practical
0
the empirical expression θ0 = 2.6θ3dB , the Gaussian antenna nonlinear energy harvesting model proposed in [39], we can
pattern is only determined by θ0 . We herein employ (3) to write the harvested DC power as
reduce the parameter number of the antenna radiation pattern.
pm (1 − exp(−pa εRF ))
Besides, the cascaded normalized antenna gain is denoted by εDC = , (8)
e x = G(φ
e x )G(ϕ
e x ) = Ωx /G2m . 1 + exp(−pa (εRF − pb ))
Ω
where pm is the maximum DC power that can be harvested
by the ERx and pa and pb are the constants determined by the
D. Imperfect Beam Alignment Models rectifier circuit [39].
According to (3), if we intend to maximize the harvested
energy, we can let φx = 0 and ϕx = 0. In practice, however, III. T HE PDF S OF A NTENNA G AINS
φx and ϕx are not necessarily equal to zero due to the direction In this section, we derive the PDFs of the normalized an-
estimation error and hardware imperfection [25]–[27]. tenna gains with the BAE following the truncated Gaussian or5
uniform distributions. Then, the PDFs of the cascaded antenna C. The Approximated PDFs of the Cascaded Antenna Gains
gains with two BAE models are derived. For tractability, we Although Theorem 1 and 2 show the PDFs of Ω e with the
also provide the approximated PDFs of the cascaded antenna Gaussian and uniform BAE models respectively, the arctan
gains. functions in (12) and (13) make further analysis less tractable.
Therefore, we provide two approximated PDFs, by considering
A. The PDFs of the Normalized Antenna Gains the distribution characteristics of BAE models and the strong
directivity of mmWave antenna.
Lemma 1. If the PDF of the stochastic BAE ψ is fψ (x), the 1) The Approximated PDF With the Gaussian BAE model:
PDF of the normalized Gaussian antenna gain G e is given by In the mmWave WPT system, the associated ETx-ERx pair
s ! is expected to employ elaborately designed beam alignment
1 − ln y
fGe (y) = √ fψ + (1 − P0 )δ(y − g), (9) algorithms, such as [40], to minimize |φx0 | and |ϕx0 | as much
y −η ln y η as possible. It is therefore reasonable to infer that |φx0 | and
Rθ |ϕx0 | are far less than θ0 in most cases. For instance, it is
where y ∈ [g, 1] and P0 = Pr(|ψ| ≤ θ0 ) = −θ0 0 fψ (x)dx.
straightforwardly assumed by [41] that φx0 and ϕx0 lie in
Proof: See Appendix A. [−θ0 , θ0 ]. The authors of [28] and [35] ignored the cascaded
Based on the Lemma 1, we have following two corollaries antenna gain involving the sidelobe gain, considering it has the
according to (4) and (5). relatively small value and happens in a very low probability in
the Gaussian BAE scenario. Following these works, we also
Corollary 1. If the BAE ψ follows the truncated Gaussian ignore the sidelobe gain in the cascaded antenna gain, i.e., the
distribution with zero mean and variance σ 2 over [−π, π), component of (12) over [g 2 , g). As a result, fΩ
e x (Ω) with the
the PDF of G e can be expressed as 0
Gaussian BAE model can be approximately presented as
1
−1 1
y 2ησ2 Ω 2ησ2
−1
fGe (y) = p √ + (1 − P0G )δ(y − g), fΩ ,
π e x (Ω) ∼ Ω ∈ [g, 1]. (14)
2πησ 2 erf √2σ 2
− ln y 0
2ησ 2 erf 2 √2σ
π
2
R θ0 (10)
where y ∈ [g, 1] and P0G = f φ (ψ)dψ = When σ = 0, we have fΩ e x (Ω) = δ(Ω − 1).
−θ0 x 0 0
erf √θ0
/erf √ π
. 2) The Approximated PDF With the Uniform BAE model:
2σ2 2σ2 Recalling the expression of Gm , the strong directional antenna
Thus, for the associated ETx-ERx pair, the PDF of the means θ0 is far less than π. Hence, in the uniform BAE case,
normalized antenna gain of ERx or ETx is equal to (10). the event that Ωe x equals to the product of two mainlobes
occurs in an extremely small probability, i.e., θ02 /π 2 ≪ 1.
Corollary 2. If the BAE ψ follows the uniform distribution
e can be written by Furthermore, by (B.2) in the Appendix B, it can be inferred
over [−π, π), the PDF of G
that the arctan term in (13) is generated by the product
1 of two mainlobes. Therefore, it is reasonable to neglect the
fGe (y) = √ + (1 − P0U )δ(y − g), (11)
2πy −η ln y arctan term of (13) in the uniform BAE case. Then, we can
Rθ approximate (13) by fΩ e x (Ω) ∼
where y ∈ [g, 1] and P0U = −θ0 0 fφx (ψ)dψ = θπ0 .
1−P0U
For the non-associated ETx-ERx pairs, the PDF of involved √ q + δ(Ω − g 2 )(1 − P0U )2 , Ω ∈ [g 2 , g)
πΩ η − ln Ω
g (15)
antenna gains is illustrated by the Corollary 2. 1 , Ω ∈ [g, 1]
4πηΩ
3) Verification of the Approximated PDFs: To verify our
B. The PDFs of the Cascaded Antenna Gains approximations, we draw fΩ e (Ω) with Gaussian and uniform
BAE models in Fig. 2 and Fig. 3, respectively. For the
By (1), the cascaded antenna gain Ωx plays a crucial role e π
Gaussian BAE model, we draw the PDF of G(θ) with θ0 = 12
in the system performance. Herein, we intend to derive the 2
e x with both BAE models respectively. To this end, as an example to verify our approximation. From the left
PDFs of Ω
subfigure of Fig. 2, we can see that with σ = θ0 /7, θ0 /6, θ0 /5,
we have following two theorems.
and θ0 /4, the approximated PDFs match the theoretical PDFs
Theorem 1. The PDF of the cascaded normalized antenna closely. As σ decreases, the PDF of Ω e x0 tends to be a pulse-
e x = G(φ
gain Ω e x )G(ϕ
e x ) with truncated Gaussian BAE model like function. We can infer that it asymptotically converges to
can be written as (12) at the top of the next page. δ(Ω − 1) as σ → 0. While, in the right subfigure of Fig.
2, for σ = θ0 /3, there is a slight difference between the
Proof: See Appendix B
results of (12) and (14). It is because the probability that
Theorem 2. The PDF of the cascaded normalized antenna the sidelobe appears in the cascaded antenna gain becomes
gain Ωe x = G(φ
e x )G(ϕ
e x ) with uniform BAE model can be larger when σ grows. Thus, with σ = θ0 /2, the difference
written as (13) at the top of the next page. generated by our approximation seems extremely apparent.
Proof: The proof of Theorem 2 is similar to that of e x shows the similar accuracy
2 ∀θ ∈ [ π , π ], the approximated PDF of Ω
0 24 6 0
Theorem 1, therefore, we omit the proof for clarity. with various σ.6
1 −1
Ω 2ησ2 Ω ∈ [g, 1]
2ησ2 erf 2 √π
fΩ
2σ2 !
e x (Ω) = 1 −1 1 1 (12)
2
ln
−
2(1−P0G )g 2ησ2 2ησ2
−1
Ω 2ησ arctan qΩ−2 ln g +√ Ω
q + δ(Ω − g 2 )(1 − P0G )2 Ω ∈ [g 2 , g)
πησ2 erf 2 √π 2 ln g ln Ω
g 2πησ erf √
2 π − ln Ω
g
2σ2 2σ2
1
4πηΩ Ω ∈ [g, 1]
!
fΩ
e x (Ω) = ln 1−P0U (13)
1
2π2 ηΩ arctan qΩ−2 ln g + √ q 1 + (1 − P0U )2 δ(Ω − g 2 ) Ω ∈ [g 2 , g)
2 ln g ln Ω Ωπ η − ln Ω
g g
Approximation in (14) Approximation in (14)
of (12). 3 Moreover, from Fig. 3 we can see that the curves
4.5 25
Theoretical Result in (12) Theoretical Result in (12), = 0
/2 of (15) approach those of (13) extremely closely. Summarily,
Theoretical Result in (12), = /3
4 0
it is verified that both approximated PDFs can be used to
3.5
20
7
analyze the system-level performance under our considered
= 0/7
circumstance.
3 6.5
15
= 0/6 6
2.5
IV. E NERGY C OVERAGE A NALYSIS
PDF
PDF
= 0/5
-0.04 0 0.04 0.08
2
10
In this section, we focus on analyzing the energy coverage
2
1.5 probability of the typical ERx. Energy coverage probability
1
1.5 is defined as the probability that the harvested DC energy
5
= 0/4 1
is larger than a pre-defined threshold, which is always the
0.5 -0.04 0 0.04
minimum required energy for information transmission or
0
0 0.2 0.4 0.6 0.8 1
0
0 0.2 0.4 0.6 0.8 1
other operations.
In [42], the Meijer G-function was used to derive the
analytic expression of information coverage probability of
e x with truncated Gaussian error, θ0 =
Fig. 2. The PDF of Ω π
. mmWave transmission. Theoretically speaking, we can also
0 12
adopt this successful approach in our analysis. By [43],
however, to obtain the analytic expression in the form of the
Meijer G-function, the path loss exponents should be positive
integers, which limits the application of the Meijer G-function
based method. Alternatively, in this paper, without path loss
10 3 exponent limitation, we provide an analytic expression of en-
Approximation in (15)
ergy coverage probability with the help of Fox’s H function.4
Theoretical Result in (13)
10 2 Letting εth be the DC energy threshold, we can write the
1
10
energy coverage probability of the typical ERx as
10 1
-0.02 0 0.02 0.04 Pec = Pr(εDC > εth ) = Pr(εRF > ε̃th ), (16)
10 0
where ε̃th = − p1a ln pm +εpth
m −εth
PDF
= /6 exp(pa pb ) is the equivalent RF
10 -1 0
0
= /12
= /24
energy threshold. By the expression of ε̃th , If εth ≥ pm ,
0
-2
10
3 Given σ = θ30 , there is always P0G (θ0 ) ≈ 0.9973 for θ0 ∈ [ 24 π π
, 6 ].
θ0 G π π
While, given σ = 2 , we have P0 (θ0 ) ≈ 0.9545 for θ0 ∈ [ 24 , 6 ].
10 -3
Consequently, the probabilities of two mainlobes cascading with σ = θ30 and
10 -4
σ = θ20 , i.e., P0G (θ0 ) · P0G (θ0 ), are about 0.9946 and 0.9111, respectively.
Apparently, for σ = θ30 , ∀θ0 ∈ [ 24 π π
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
, 6 ], there is P0G (θ0 ) · P0G (θ0 ) ≈ 1.
That is why we here choose σ ≤ θ0 /3 as the approximation condition. Note
that it is weaker than the assumption that φx0 , ψx0 ∈ [−θ0 , θ0 ] adopted by
e x with uniform error.
Fig. 3. The PDF of Ω [41].
4 The Fox’s H function is a general function which can encompass almost
all commonly used functions, e.g., Meijer G-function. Although the Fox’s H
function is defined by an integral in a nonanalytic form, like the widely used
Gamma function, Q-function, Hypergeometric function, Meijer G-function,
etc., a look-up-table (LUP) storing the values of Fox’s H function can be
generated via numerical methods. A Matlab program for evaluating the Fox’s
H function was provided in [44]. More details about the Fox’s H function can
Accordingly, if σ ≤ θ0 /3, (14) is an appropriate approximation be found in [45].7
TABLE I following equation [46],
D EFINITIONS OF VARIABLES IN S ECTION IV Z 1
b(1 + zt)−a tb−1 dt =2 F1 (a, b; 1 + b; −z),
pm −εth
ε̃th = − p1a ln pm +εth exp(pa pb ) 0
∀z ≥ −1, a > 0, b > 0, z, a, b ∈ R.
A = K(K!)−1/K
Ak
When σ = 0, i.e., the perfect beam alignment scenario, we
ak = ε̃th can easily obtain
F(x) = x̟ 2 F1 (mL , ̟; 1 + ̟; −γk x) −mL
ak Pt G2m CL r0−αL
1 Lε0 (ak ) = 1 + . (19)
̟= 2ησ2 mL
ak Pt G2m CL r0 L
−α
γk = mL
e x hx,L , x ∈ ΦL
ωx,L = Ω
z
χL,z = E{ωL e z }E{hz }, z ∈ R+
} = E{Ω L
z
χN,z = E{ωN e z }E{hz }, z ∈ R+
} = E{Ω N
B. The Analytic Expression of LεL (ak )
there is Pec = 0. Then, we let εth < pm to investigate the e x hx,L , x ∈ ΦL . As ΦL follows the
Define ωx,L , Ω
performance of Pec . Note that ε0 , εL and εN are independent
IPPP with intensity PL (rx )λt , the 2-tuple {ωx,L } × ΦL forms
of each other. Following the widely adopted Gamma r.v.
1 a marked IPPP (MIPPP). Due to the probability generating
approximation [7], [18], i.e., using µK ∼ Γ(K, K ), K ∈ Z+ ,
functional (PGFL) of MIPPP [31], we have
instead of 1, we can rewrite Pec as ( )
Y
ε0 + εL + εN LεL (ak ) = EΦL ,ωx,L
2 −αL
e−ak Pt Gm CL rx ωx,L
Pec ≈ Eε0 ,εL ,εN Pr µK <
ε̃th x∈ΦL
( K ) R ∞ n o
(a) A(ε0 + εL + εN ) −2πλt 1−E ω e−ak Pt G2
m CL r
−αL ω
L e−βr rdr
(20)
≈ Eε0 ,εL ,εN 1 − exp − , =e 0 L
ε̃th nR 2 −αL ω o
(c) −2πλt 1
−EωL ∞
e−βr−ak Pt Gm CL r L rdr
XK =e β2 0
.
K
= (−1)k Lε0 (ak ) LεL (ak ) LεN (ak ) , k ∈ Z+ In (c), due to the Fubini’s theorem [47], we exchange the order
k
k=0
of integral and expectation operations. Due to [48, (1.9.5)], we
(17)
have
Z ∞
in which ak = ε̃Ak and A = K(K!)−1/K . (a) is based on 2 −αL
th
Lemma 5 in [7]. Next, we derive the analytic expressions of e−βr−ak Pt Gm CL r ωL
rdr
0 " #
Lε0 (ak ), LεL (ak ), and LεN (ak ), respectively. (21)
1 2,0 2
1
= H β ak Pt Gm CL ωL αL
.
αL β 2 0,2 (2,1)(0, α1 )
L
A. The Analytic Expression of Lε0 (ak ) Note that H2,0
0,2 [·] is the Fox’s H function and is defined by
[45], [48]. Consequently, there is LεL (ak ) =
e x0 are independent random variables,
As for ε0 , hx0 ,L and Ω
1
we have
−2πλt β12 − α 1
2 EωL H2,0
2
β(ak Pt Gm CL ωL ) αL
n o Lβ 0,2
2 e (2,1)(0, 1 )
Lε0 (ak ) = EΩe x ,hx ,L e−ak Pt Gm ℓL (r0 )Ωx0 hx0 ,L e αL
.
0 0
( !−mL ) (22)
ak Pt G2m CL r0−αL Ωe x0
= EΩe x 1+
0 mL Before solving the expectation of H2,0
0,2 [·] with respect to ωL ,
Z 1 !−mL 1 −1
2
we introduce the Lemma 2 as follows.
ak Pt G2m CL r0−αL Ωe x0 Ωe x2ησ
≈ 1+ 0
e x0
dΩ
g mL 2ησ 2 erf 2 √2σ
π
Lemma 2. For t ∈ Z+ , there is
2
(b) F(1) − F(g) (a1 ,A1 )···(ap ,Ap )
= m,n
Hp,q xy =
erf 2 √2σ
π
2
(b1 ,B1 )···(bq ,Bq )
1
t
(18) ∞ 1 − x B1
b1 X (a1 ,A1 )···(ap ,Ap )
m,n
where F(x) , x̟ 2 F1 (mL , ̟; 1 + ̟; −γk x), ̟ = 1 x B1
Hp,q y .
2ησ2 , t! (t+b1 ,B1 )(b2 ,B2 )···(bq ,Bq )
t=0
ak Pt G2m CL r0 L
−α
γk = mL , and 2 F1 (a, b; c; z) is the Gauss hy-
pergeometric function [43]. Note that in (b) we resort to the Proof: See [45, (1.88)].8
By Lemma 2, we can further attain Next, we derive the expressions of L1 (ωN ) and L2 (ωN )
separately.
α1
H2,0 2
0,2 β ak Pt Gm CL ωL
L Firstly, after some manipulations, it is easy to know
(2,1)(0, α1 )
t
L 1 2
2 2
1
L1 (ak ) = (ak Pt Gm CN ) χN, α2 Γ 1 −
αN
, (27)
1 − ωL
αL 2 N αN
2 X
∞
1
H2,0 2
α
= ωL L
0,2 β(ak Pt Gm CL )
αL z e z }E{hz }, z ∈ R+ , which can
t! (t+2,1)(0, α1 )
where χN,z , E{ωN } = E{Ω N
t=0
be also written in the analytic form due to Proposition 1 and
L
q+2
(d)
t
∞ X
X (−1)q qt
α
ωL L 1 2. Following the derivations from (21) to (23), similarly, we
= H2,0 2
0,2 β(ak Pt Gm CL )
αL . can obtain
t=0 q=0
t! (t+2,1)(0, α1 )
L q t
t (−1) χN, q+2
1 XX
∞
1 q αN
In (d), we apply Binomial theorem. Hence, there is L2 (ak ) ≈ 2 − ×
β αN β 2 t=0 q=0 t!
q t
X∞ X t (−1) χL, q+2 (28)
1 1 q αL
LεL (ak ) ≈ exp −2πλt 2 − 2,0
1
β(ak Pt G2m CN ) αN .
β αL β 2 t=0 q=0 t! H0,2
(t+2,1)(0, α1 )
N
1
H2,0 2
0,2 β(ak Pt Gm CL )
αL , Consequently, the analytic expression of LεN (ak ) is also
(t+2,1)(0, 1 )
derived. Therefore, we can get the analytic expression of Pec
αL
(23) with Lε0 (ak ), LεL (ak ), and LεN (ak ).
z
where χL,z , E{ωL e z }E{hz }, z ∈ R+ .
} = E{Ω It is worth noting that for applying Proposition 1 and 2,
L
we need to let 1/αL ∈ R+ , 1/αN ∈ R+ . Clearly, it always
Proposition 1. If the PDF of Ωe follows (15), ∀z ∈ R+ , there holds for the practical condition αL > αN > 0. So, we put no
is limitation on the path loss exponents. In addition, for clarity,
√ √
1 − P0U g z πerf −z ln g 1 − gz we summarized all newly-defined variables in Section IV in
e z
E{Ω } = √ √ + Table I.
π η z 4πηz (24)
U 2 2z
+ (1 − P0 ) g ,
Proof: By (15), it is straightforward to obtain Proposition V. AVERAGE H ARVESTED E NERGY
1. The detailed proof is omitted to save space.
Proposition 2. If h ∼ Γ(m, 1/m), m ∈ Z+ , ∀z ∈ R+ , there Although the derived Pec can be used to evaluate the energy
is coverage performance of mmWave WPT, it may not provide
Γ(m + z) explicit and direct insight into the effect of BAE. Thus, in this
E{hz } = , (25)
Γ(m)mz section, the average harvested energy is addressed to further
Proof: By the PDF of h, it is conveniently to prove investigate the effect of BAE.
Proposition 2. By [7], [18], the average harvested DC energy can be
Therefore, substituting Proposition 1 and 2 into (23), we expressed as
Z ∞
can obtain the analytic expression of LεL (ak ).
εavg = εmin Pec (εmin ) + Pec (ε)dε, (29)
εmin
C. The Analytic Expression of LεN (ak ) where εmin is the minimum energy threshold. Apparently, it
e x hx,N , we have
Similarly to (20), defining ωx,N = Ω is extremely difficult to give an analytic expression of εavg
because of the complicated expression of energy coverage
Y probability.
−ak Pt G2
−αN
LεN (ak ) = EΦN ,ωx,N e m CN r x ωx,N
To achieve the closed-form result, we herein consider the
x∈ΦN
linear energy harvesting (EH) model, i.e., ζ(εRF ) = ζεRF
R∞
2 −αN ω
like [8], [16], [17], [19], [24]. Specifically, we set ζ = 1 as
−2πλt 1−EωN e−ak Pt Gm CN r N (1−e−βr )rdr
=e 0 [49], which means we investigate the performance of average
harvested RF energy at the typical ERx. With this linear EH
Z ∞ n o model, we have
2 −αN
= exp −2πλt 1 − EωN e−ak Pt Gm CN r ωN
rdr
0 εavg = E{ε0 } + E{εL } + E{εN }. (30)
| {z }
L1 (ak )
Obviously, there is
Z e x0 }E{hx0 ,L }
∞ n 2
−ak Pt Gm CN r −αN
ωN
o
−βr
E{ε0 } = Pt G2m CL r0−αL E{Ω
− 1 − EωN e e rdr . 1
+1
|0 {z } 1 − g 2ησ2 . (31)
L2 (ak )
≈ Pt G2m CL r0−αL
2 √π
(26) (2ησ 2 + 1)erf 2σ29
As for εL , due to Campell’s Theorem [31], we have TABLE II
( ) PARAMETERS IN S IMULATIONS
X
2 −αL
E{εL } = EΦL ,ωx,L Pt Gm ωx,L CL rx,L
Symbol Definition Default Value
x∈ΦL
Z ∞ −αL Pt ETx transmit power 40 dBm
= 2πλt EωL Pt G2m ωL CL rL e−βr rdr λt Density of ETxs 5 × 10−4 /m2
0 , (32)
Z ∞ r0 Distance between Typical ETx-ERx 50 m
(e)
≈ 2πλt Pt G2m CL χL,1 r−αL e−βr rdr κ Spacing distance/wavelength (d/ν) 0.25
1
αL −1 β αL Path loss exponent of LOS 2.1
= 2πλt Pt G2m CL χL,1 β 2 −1
e− 2 W− αL −1 , 2−αL (β) αN Path loss exponent of NLOS 2.92
2 2
61.4
CL Path loss intercept of LOS 10− 10
where Wa,b (x) is the Whittaker W function [50, (3.381.6)] 72
and can be efficiently calculated by Matlab. To avoid the CN Path loss intercept of NLOS 10− 10
singularity incurred by the simplified path loss model [31], ML Gamma fading parameter of LOS 3
in (e) we only consider the far field energy signals. In the MN Gamma fading parameter of NLOS 2
same way, we obtain β Blockage parameter 0.0071
( ) pm Maximum harvested power 10 mW
X pa Circuit parameter 1500
2 −αN
E{εN } = EΦN ,ωx,N Pt Gm ωx,N CN rx,N
pb Circuit parameter 0.0022
x∈ΦN
Z ∞ K Gamma approximation parameter 5
−αN
≈ 2πλt E ωN Pt G2m ωN CN rN (1 − e−βr )rdr .
1
= 2πλt Pt G2m CN χN,1 × II. These values are based on the simulation parameters in [13],
1 αN −1 β [18]. The antenna pattern parameters in our simulations are
− β 2 −1 e− 2 W− αN −1 , 2−αN (β)
αN − 2 2 2 shown in Table III. In the figure legends, ‘Theory’ means the
(33) theoretical results obtained by our derived analytic expressions
and others are the results from the Monte Carlo simula-
Then, substituting (31)−(33) into (30), the average harvested
tions. For comparison, we also simulated the performance of
RF energy εavg is obtained.
mmWave WPT system with the flat-top antenna model and the
To investigate the difference between the average harvested
actual beam pattern of ULA model. To let the flat-top model
RF energy with BAE and without BAE, we need to give the
have the same maximum mainlobe gain and 3dB mainlobe
average harvested RF energy without BAE. As the beam angle
beamwidth as the Gaussian antenna model for fair comparison,
differences from the non-associated ETxs in ΦL or ΦN are
we define the flat-top antenna model as
inevitable [28], the difference of average harvested RF energy (
only happens in E{ε0 }. Apparently, if there is no BAE, the Gm , |θ| ≤ θ3dB ,
e x0 = 1 equals to 1. Therefore, by (31), E{ε0 }
probability of Ω GF (θ) = (36)
Gs , |θ3dB | < |θ| ≤ π.
on the condition of Ω e x0 = 1 is
While, by [21], [51], the actual beam gain of ULA model can
e x0 = 1} = Pt G2 CL r−αL .
E{ε0 |Ω (34)
m 0 be written as
!
Then, we define the relative energy loss (REL) of average sin2 ( N2a θ )
harvested RF energy in the IBA scenario as GU (θ) = Na , 0 ≤ |θ| ≤ π, (37)
Na2 sin2 ( 2θ )
e x0 = 1} − E{ε0 }
E{ε0 |Ω
∆ε = where Na is the antenna number of ULA and must be an
E{ε0 |Ωe x0 = 1} integer. Since it is hard to let the ULA model have the
1
+1 (35) same maximum mainlobe gain and mainlobe beamwidth as
1 − g 2ησ2
=1 − , the Gaussian antenna model, we herein force both models to
(2ησ 2 + 1)erf 2 √π
2σ2 achieve the same mainlobe beamwidth 2θ0 . According to [51],
we let Na be approximately equal to [5.64/θ0 ]Z , where [x]Z
where ∆ε ∈ [0, 1]. If ∆ε = 0, it shows the BAE incurs no
is the closest integer to the real number x. For θ0 = π/6,
energy loss compared to the ideal case, i.e., perfect beam
π/12, π/24, therefore, Na shall be 11, 22, 43, respectively.
alignment. While, if ∆ε = 1, it means no energy can be
In addition, by comparing (3) with (37), the roll-off factor of
harvested by the typical ERx with BAE. Observe (35), if
ULA antenna model is slightly different from that of Gaussian
σ 2 → 0, there is ∆ε → 0. That is to say the derived εavg
model.
can cover the perfect beam alignment case.
In Fig. 4, we show the energy coverage performance of
π
the mmWave WPT systems with θ0 = 12 in various IBA
VI. S IMULATION R ESULTS scenarios. First, we can see that in the perfect beam alignment
In this section, we verify our theoretical results by Monte case, i.e., σ = 0, the flat-top and Gaussian models achieve the
Carlo simulations. The carrier frequency is 28 GHz. Unless same energy coverage probabilities and the theoretical curve
otherwise specified, the system parameters are listed in Table matches the simulation curve very well. As the ULA model10
TABLE III
PARAMETERS OF G AUSSIAN A NTENNA G AIN 1
Theory, 0 = /6
0.9 Simulation, 0 = /6, =0
2.028
θ0 0.3 ln 10 π10
θ0 θ3dB = 2.6 η= 2
θ3dB
Gm = 42.6443θ0 +π 0.8
Simulation, 0 = /6, = 0 /4
Simulation, = /6, = /2
Energy Coverage Probability
0 0
π/24 0.0503 272.5250 38.4103 0.7 Theory,
0
= /12
Simulation, = /12, =0
0
π/12 0.1007 68.1313 23.4227 0.6
Simulation, 0 = /12, = 0 /4
0.5 Simulation, 0 = /12, = 0 /2
π/6 0.2014 17.0328 13.1559 Theory, 0 = /24
0.4 Simulation, 0 = /24, =0
0.3 Simulation, 0 = /24, = 0 /4
Simulation, 0 = /24, = 0 /2
Theory, =0
Gaussian, =0
0.2
1
Flat-top, =0
ULA, =0 0.1
0.9
Theory, = /40
0
0.8 Gaussian, = 0/4
-60 -50 -40 -30 -20 -10 0
Energy Coverage Probability
Flat-top, = /4 DC Energy Threshold (dBm)
0
0.7 ULA, = 0/4
Theory, /3
0.6 0 Fig. 5. Energy coverage probability with various mainlobe beamwidths versus
Gaussian, = 0/3
DC energy threshold.
0.5 Flat-top, = /3
0
ULA, = 0/3
0.4
Theory, = 0/2
0.3 Gaussian, = 0/2
Flat-top, = 0/2 Theory
0.2 Gaussian-Linear
ULA, = 0/2
Gaussian-Nonlinear
Average Harvested Energy(dBm)
0.1 Theory, = 0 Flat-top-Linear
Gaussian, = Flat-top-Nonlinear
0 ULA-Linear
0 Flat-top, = 0
-60 -50 -40 -30 -20 -10 0 ULA-Nonlinear
DC Energy Threshold(dBm) ULA, = 0
π
Fig. 4. Energy coverage probability versus DC energy threshold, θ0 = 12
,
Na = 22.
has a different maximum mainlobe gain from the flat-top and
Gaussian antenna models, it achieves the slightly different
performance when σ = 0. When σ = θ40 , θ30 , the theoretical
results approach the simulation results of Gaussian antenna
model closely. While, for σ = θ20 and θ0 , the gap between the
theoretical and simulation curves appears at the low threshold Fig. 6. Average harvested energy versus BAE standard deviation, θ0 =
π
regime and it gets larger when σ grows. This phenomenon 12
, Na = 22, λt = 10−4 /m2 .
is consistent with the observations in Fig. 2. The reason is
that we ignore the sidelobe gain of the Gaussian BAE model.
Therefore, it can be concluded that if P0G (θ0 )·P0G (θ0 ) ≈ 1, the can also see that the larger σ is the lower Pec appears.
derived analytic expression of the energy coverage probability To investigate the effect of mainlobe beam width θ0 , we
can accurately evaluate the performance of the considered exhibit the energy coverage performance of the Gaussian
π π π
mmWave WPT systems. antenna model with θ0 = 24 , 12 , 6 in Fig. 5. Firstly, for
θ0
On the other hand, for σ 6= 0, the energy coverage σ = 0 and 4 , all theoretical results generated by our derived
performance of the flat-top model is drastically different expression match the simulation results closely. So, these
from that of the 3GPP Gaussian model we used. It reveals curves verify the theoretical results. For σ = θ20 , the theoretical
that the flat-top antenna model is not suitable for evaluating results in all three θ0 cases generate nearly the same gap
the performance of the mmWave WPT systems in the IBA compared with the corresponding simulation results. Secondly,
scenario. Moreover, for σ = θ0 /4, θ0 /3, θ0 /2, θ0 , the Gaussian as the threshold εth increases, the energy coverage probability
antenna model achieves similar energy coverage probability to decreases in all cases. Comparing the curves with the same
the ULA model, especially in small σ cases. The performance mainlobe width, we can also see that the BAE indeed degrades
difference between Gaussian antenna model and ULA model the performance of the mmWave WPT system. For example,
π
in the imperfect beam alignment case mainly results from in the scenario of θ0 = 24 , the energy coverage probability
θ0
the slight divergence on antenna gain and roll-off factor. with σ = 4 reduces from 0.78 to 0.51 when εth = −40
Therefore, the Gaussian antenna model can be regarded as a dBm. It means that minimizing BAE is one of the most
useful and tractable mathematical tool to analyze the system- crucial issues for mmWave WPT systems. Besides, using the
level performance of mmWave WPT networks in the presence analytic expression of Pec , we can choose the proper mainlobe
of BAE, while guaranteeing a certain degree of accuracy. We beamwidth θ0 , if Pec and σ are given.11
-15
Theory, =0
Simulation, =0
Theory, = /4
0
Average Harvested Energy(dBm)
-20 Simulation, = /4 = /24
0
0
Theory, = /2
0
Simulation, = 0 /2
-25
REL
-30
Theory, 0= /6 Flat-top, 0= /12
Gaussian, 0= /6 ULA, 0= /12
Flat-top, = /6 Theory, = /24
0 0
-35
ULA, 0= /6 Gaussian, 0= /24
= /6 Theory, 0= /12 Flat-top, 0= /24
0
Gaussian, 0= /12 ULA, 0= /24
-40
-4 -3 -2
10 10 10
t
(m -2 )
Fig. 7. Average harvested energy versus density of ETxs. Fig. 8. Relative energy loss versus BAE standard deviation, λt = 10−4 /m2 .
Fig. 6 illustrates the effect of the BAE standard devia- we stated earlier, when σ = 0, all RELs equal to zero, which
tion σ on the average harvested energy. We consider six means no energy loss incurred by BAE in those cases. As
mmWave WPT systems, i.e., Gaussian antenna model with σ increases, REL increases in all scenarios. With the same
linear EH, Gaussian antenna model with nonlinear EH, Flat- σ, we can see the mmWave WPT system with θ0 = π/24
top antenna model with linear EH, Flat-top antenna model produces the largest REL among all three cases. Therefore,
with nonlinear EH, ULA model with linear EH, and ULA we can deem that the BAE leads to larger performance degra-
model with nonlinear EH. Apparently, as σ increases, the dation for mmWave WPT system with stronger directional
average harvested energy decreases. Observing the purple antenna. Besides, given θ0 , the flat-top model incurs lower
dashed line, the theoretical average energy fits the simulation REL than the Gaussian model in the small σ regime, e.g.,
results very well. It verifies our derived close-form expression for θ0 = 12π π
and σ < 20 . Differently, the REL of the ULA
of the average harvested RF energy. Furthermore, regardless model approaches that of Gaussian antenna model closely
of linear or nonlinear EH model, the flat-top antenna model when σ < 60 π
. Moreover, the REL of ULA model is always
has quite different results from the Gaussian antenna model less than those of Gaussian and flat-top antenna models. The
and the ULA model in small σ cases, i.e., σ < 10 16 θ0 . When performance gap mainly results from the difference in the
10
0 < σ < 16 θ0 , the flat-top antenna model gains more energy maximum mainlobe gain, the sidelobe gain, and roll-off factor
than the other two antenna models. Nevertheless, as σ grows, between the ULA model and the other two antenna models.
8
such as σ > 16 θ0 , the ULA model gains more energy than It is worth mentioning that for θ0 = 24 π
, even when σ = π6 ,
the other two models. Given σ, the linear EH model always i.e., σ = 4θ0 , the REL is still less than 1. This is because
harvests more energy than the nonlinear EH model. This is the average harvested energy can not be zero no matter how
because we set the RF-DC conversion efficiency of linear EH severe the BAE is.
model as 1. From this point, we can also conclude that the
Gaussian antenna model is an accurate approximation of the
ULA model for the small σ situations, and the performance
gap between both models is acceptable for the sake of analysis VII. C ONCLUSIONS
tractability.
Fig. 7 illustrates the performance of the average harvested The impact of imperfect beam alignment on wireless power
energy with different mainlobe widths. To demonstrate these transfer at millimeter wave frequencies has been investigated
π π
curves distinctively, we herein take θ0 = 24 , 6 as represen- in this paper. The beam alignment error (BAE) from the
−4
tatives. When λt ≤ 2 × 10 , all theoretical results match associated ETx-ERx transmission is modeled as the truncated
the simulation results exactly. As λt increases, the average Gaussian distribution, while, the BAE from the non-associated
harvested energy begins to grow. The reason is that the average ETx-ERx transmission follows the uniform distribution. Then,
distance between ETx and ERx gets closer. Additionally, in we derive the probability density functions of the cascaded
this case the sidelobe gain plays a more and more significant antenna gains with both mentioned stochastic BAE models
role in the average harvested energy and the theoretical results and their approximated expressions with more tractability are
based on the approximated PDFs start to be less than the also provided. The analytic expression of energy coverage
simulation results. Certainly, enlarging λt can compensate the probability has been derived. Moreover, we also give the
performance degradation caused by BAE. closed-form expression of average harvested energy under
In Fig. 8, we show the performance of REL versus σ. First, linear energy harvesting model. Finally, the simulation results
the theoretical curves match the simulation curves exactly. As verify our theoretical expressions.12
A PPENDIX A we have
P ROOF OF L EMMA 1 1
−1 Z Ω
Ω 2ησ2 1 g 1 1
F1 (Ω) = √ q dy
e
π
2πησ 2 erf √2σ g − ln y − ln Ω y
Observe (3), when |ψ| ≤ θ0 , G(ψ) is a continuous r.v. with 2 y
e .
respect to ψ. While if θ0 < |ψ| ≤ π, G(ψ) is a discrete r.v. 1
−1
with probability mass function (PMF) Pr (G e = g) = Pr(θ0 < Ω 2ησ2 ln Ω − 2 ln g
= arctan q
e
|ψ| ≤ π). Therefore, G(ψ) is the mixed r.v. over [g, 1]. 2
πησ erf √ π
2 ln Ω ln g Ω
2σ2
Firstly, we derive the PDF of the continuous component. (B.2)
Denote the normalized mainlobe gain as G e m = e−ηψ2 . For an
arbitrary y ∈ [g, 1], the CDF of Gem is given by If Ω ∈ [g, 1], there is
1
−1
2
FGem (y) = Pr(e−ηψ ≤ y, |ψ| ≤ θ0 ) = Ω 2ησ2
F1 (Ω) = . (B.3)
s ! s ! π
2ησ 2 erf √2σ
− ln y − ln y 2
Pr ≤ |ψ| ≤ θ0 = F|ψ| (θ0 ) − F|ψ|
η η And then we have
Ry Z 1
where F|ψ| (y) = −y fψ (y)dy is the CDF of |ψ|. Then, the G II Ω 1
F2 (Ω) = (1 − P0 )δ(y − g)fG dy
e m can be written as g y y
PDF of G 1 1
s ! − −1
(1 − P0G )g 2ησ2 Ω 2ησ2
dFGem (y) 1 − ln y = p q , Ω ∈ [g 2 , g)
fGem (y) = = √ fψ . (A.1) π
2πησ 2 erf √2σ − ln Ω
dy y −η ln y η 2 g
(B.4)
e s , the general-
Denoting the normalized sidelobe gain as G Furthermore, there is
ized PDF of Ge s can be expressed as [52], Z
2 1
Ω 1
e = g)δ(y − g) = (1 − P0 )δ(y − g),
fGes (y) = Pr(G (A.2) F3 (Ω) = 1 − P0G δ(y − g)δ dy
g y y (B.5)
in which P0 = F|ψ| (θ0 ). So, we have fGe (y) = fGem (y) + = (1 − P0G )2 δ(Ω −g )2
fGes (y). Then, Lemma 1 is proven.
According to the domains of F1 (Ω), F2 (Ω) and F3 (Ω), we
can achieve the overall PDF of Ω as (12). .
A PPENDIX B ACKNOWLEDGMENT
P ROOF T HEOREM 1 The authors would like to thank the anonymous reviewers
for their valuable comments and suggestions to improve the
e x ) and G(ϕ
As G(φ e x ) independently follow the PDF f e (y) expression and quality of this paper. This work has also
G
in (10), the PDF of Ω e x can be given by benefited from suggestions by Dr. Zhengdao Wang.
Z 1
Ω 1 R EFERENCES
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