arXiv:2105.05618v4 [cs.IT] 11 Oct 2021 1 Joint Optimization for RIS-Assisted Wireless Communications: From Physical and Electromagnetic Perspectives Xin Cheng, Yan Lin, Weiping Shi, Jiayu Li, Cunhua Pan, Feng Shu, Yongpeng Wu, and Jiangzhou Wang, Fellow, IEEE Abstract—Reconfigurable intelligent surfaces (RISs) are envi- sioned to be a disruptive wireless communication technique that is capable of reconfiguring the wireless propagation environment. In this paper, we study a free-space RIS-assisted multiple- input single-output (MISO) communication system in far-field operation. To maximize the received power from the physical and electromagnetic nature point of view, a comprehensive optimization, including beamforming of the transmitter, phase shifts of the RIS, orientation and position of the RIS is formulated and addressed. After exploiting the property of line-of-sight (LoS) links, we derive closed-form solutions of beamforming and phase shifts. For the non-trivial RIS position optimization problem in arbitrary three-dimensional space, a dimensional-reducing theory is proved. The simulation results show that the proposed closed- form beamforming and phase shifts approach the upper bound of the received power. The robustness of our proposed solutions in terms of the perturbation is also verified. Moreover, the RIS significantly enhances the performance of the mmWave/THz communication system. Index Terms—Reconfigurable intelligent surface, intelligent reflecting surface, far-field, closed-form beamforming and phase shifts, position optimization, millimeter wave communication. I. I NTRODUCTION I N recent years, wireless communication has witnessed great success in various aspects such as rate, stability and security. However, most of the existing techniques mainly rely on the transceiver design at both the transmitter and the receiver. The wireless propagation environment is left untouched. Unfortunately, the propagation loss and multi-path fading deteriorate the communication performance. Due to the This work was supported in part by the National Natural Science Foundation of China (Nos. 62071234 and 61771244), and the Scientific Research Fund Project of Hainan University under Grant KYQD(ZR)-21008 and KYQD(ZR)- 21007. Xin Cheng, Yan Lin, Weiping Shi, and Jiayu Li are with the School of Electronic and Optical Engineering, Nanjing University of Science and Technology, Nanjing, 210094, China. (e-mail: [email protected]). Cunhua Pan is with the School of Electronic Engineering and Computer Science , Queen Mary University of London, Mile End Road London E1 4NS, U.K. (e-mail: [email protected]). Feng Shu is with the School of Information and Communication Engi- neering, Hainan University, Haikou 570228, China. and also with the School of Electronic and Optical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China. (e-mail: [email protected]). Yongpeng Wu is with the Shanghai Key Laboratory of Navigation and Location Based Services, Shanghai Jiao Tong University, Minhang 200240, China. (e-mail: [email protected]). Jiangzhou Wang is with the School of Engineering and Digital Arts, University of Kent, Canterbury CT2 7NT, U.K. (e-mail: [email protected]). rapid development of radio frequency (RF), micro electrome- chanical systems (MEMS) and metamaterial, a metasurface called reconfigurable intelligent surface (RIS) [1], [2] has at- tracted a lot of attention. In [3], a new wireless communication paradigm based on the concept of RIS was proposed, which can adaptively tune the propagation environment. The benefits and challenges were discussed in [4]. A RIS is composed of an array of low-cost passive reflective elements, each of which can be controlled by a control loop to re-engineer the electromagnetic wave (EM) including steering towards any desired direction full absorption, polarization manipulation. The EM programmed by many reflective ele- ments can be integrated constructively to induce remarkable effect. Unlike relay which requires active radio frequency (RF) chains, the RIS is passive because it dose not adopt any active transmit module (e.g., power amplifier) [5]. Hence, the RIS is more energy efficient than the relay scheme. Due to the above appealing features, RIS-assisted commu- nication systems have been studied extensively. For exam- ple, owing to low-cost and passive reflecting elements, RIS can achieve high spectrum and energy efficiency for future wireless networks [6]. The joint design of beamforming and phase shifts was also investigated in various communication scenarios. The contributions in [7]–[10] showed that the RIS offers performance improvement and coverage enhancement in the single-user multiple-input single-output (MISO) system. In downlink multi-user MISO case, the advantages of introducing RISs in enhancing the cell-edge user performance were con- firmed in [11]. Employing RISs to wireless information and power transfer (SWIPT) system in multi-user MIMO scenarios was shown to beneficially enhance the system performance in terms of both the link quantity and the harvested power [12]. The advantages of introducing RIS were demonstrated in a secure multigroup multicast MISO communication system in [13]. To minimize the symbol error rate (MSER) of an RIS-assisted point-to-point multi-data-stream MIMO wireless communication system, the reflective elements at the RIS and the precoder at the transmitter were alternately optimized in [14]. With the assistance of RISs, secrecy communication rate i.e., physical layer security can be significantly improved [15], [16]. The work of [17] examined the performance gain achieved by deploying an RIS in covert communica- tions. RIS was proposed to create friendly multipaths for directional modulation (DM) such that two confidential bit streams (CBSs) can be transmitted from Alice to Bob in [18].
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Abstract—Reconfigurable intelligent surfaces (RISs) are envi-sioned to be a disruptive wireless communication technique thatis capable of reconfiguring the wireless propagation environment.In this paper, we study a free-space RIS-assisted multiple-input single-output (MISO) communication system in far-fieldoperation. To maximize the received power from the physicaland electromagnetic nature point of view, a comprehensiveoptimization, including beamforming of the transmitter, phaseshifts of the RIS, orientation and position of the RIS is formulatedand addressed. After exploiting the property of line-of-sight (LoS)links, we derive closed-form solutions of beamforming and phaseshifts. For the non-trivial RIS position optimization problem inarbitrary three-dimensional space, a dimensional-reducing theoryis proved. The simulation results show that the proposed closed-form beamforming and phase shifts approach the upper boundof the received power. The robustness of our proposed solutionsin terms of the perturbation is also verified. Moreover, theRIS significantly enhances the performance of the mmWave/THzcommunication system.
Index Terms—Reconfigurable intelligent surface, intelligentreflecting surface, far-field, closed-form beamforming and phaseshifts, position optimization, millimeter wave communication.
I. INTRODUCTION
IN recent years, wireless communication has witnessed
great success in various aspects such as rate, stability and
security. However, most of the existing techniques mainly
rely on the transceiver design at both the transmitter and
the receiver. The wireless propagation environment is left
untouched. Unfortunately, the propagation loss and multi-path
fading deteriorate the communication performance. Due to the
This work was supported in part by the National Natural Science Foundationof China (Nos. 62071234 and 61771244), and the Scientific Research FundProject of Hainan University under Grant KYQD(ZR)-21008 and KYQD(ZR)-21007.
Xin Cheng, Yan Lin, Weiping Shi, and Jiayu Li are with the Schoolof Electronic and Optical Engineering, Nanjing University of Science andTechnology, Nanjing, 210094, China. (e-mail: [email protected]).
Cunhua Pan is with the School of Electronic Engineering and ComputerScience , Queen Mary University of London, Mile End Road London E1 4NS,U.K. (e-mail: [email protected]).
Feng Shu is with the School of Information and Communication Engi-neering, Hainan University, Haikou 570228, China. and also with the Schoolof Electronic and Optical Engineering, Nanjing University of Science andTechnology, Nanjing 210094, China. (e-mail: [email protected]).
Yongpeng Wu is with the Shanghai Key Laboratory of Navigation andLocation Based Services, Shanghai Jiao Tong University, Minhang 200240,China. (e-mail: [email protected]).
Jiangzhou Wang is with the School of Engineering and Digital Arts,University of Kent, Canterbury CT2 7NT, U.K. (e-mail: [email protected]).
rapid development of radio frequency (RF), micro electrome-
chanical systems (MEMS) and metamaterial, a metasurface
called reconfigurable intelligent surface (RIS) [1], [2] has at-
tracted a lot of attention. In [3], a new wireless communication
paradigm based on the concept of RIS was proposed, which
can adaptively tune the propagation environment. The benefits
and challenges were discussed in [4].
A RIS is composed of an array of low-cost passive reflective
elements, each of which can be controlled by a control loop to
re-engineer the electromagnetic wave (EM) including steering
towards any desired direction full absorption, polarization
manipulation. The EM programmed by many reflective ele-
ments can be integrated constructively to induce remarkable
effect. Unlike relay which requires active radio frequency (RF)
chains, the RIS is passive because it dose not adopt any active
transmit module (e.g., power amplifier) [5]. Hence, the RIS is
more energy efficient than the relay scheme.
Due to the above appealing features, RIS-assisted commu-
nication systems have been studied extensively. For exam-
ple, owing to low-cost and passive reflecting elements, RIS
can achieve high spectrum and energy efficiency for future
wireless networks [6]. The joint design of beamforming and
phase shifts was also investigated in various communication
scenarios. The contributions in [7]–[10] showed that the RIS
offers performance improvement and coverage enhancement in
the single-user multiple-input single-output (MISO) system. In
downlink multi-user MISO case, the advantages of introducing
RISs in enhancing the cell-edge user performance were con-
firmed in [11]. Employing RISs to wireless information and
power transfer (SWIPT) system in multi-user MIMO scenarios
was shown to beneficially enhance the system performance
in terms of both the link quantity and the harvested power
[12]. The advantages of introducing RIS were demonstrated in
a secure multigroup multicast MISO communication system
in [13]. To minimize the symbol error rate (MSER) of an
RIS-assisted point-to-point multi-data-stream MIMO wireless
communication system, the reflective elements at the RIS and
the precoder at the transmitter were alternately optimized in
[14]. With the assistance of RISs, secrecy communication
rate i.e., physical layer security can be significantly improved
[15], [16]. The work of [17] examined the performance
gain achieved by deploying an RIS in covert communica-
tions. RIS was proposed to create friendly multipaths for
directional modulation (DM) such that two confidential bit
streams (CBSs) can be transmitted from Alice to Bob in [18].
The useful power of the received signal can be expressed
by
Pr = |hHIRΘH
HTIv|2. (4)
B. Physical and Electromagnetic Perspectives
In this subsection, we shall express the far-field amplitude
gain and phase change of the RIS link from physical and
electromagnetic perspectives. The parameters mentioned in
this subsection are explained in Table I.
In far-field operation, the length/width of the RIS is much
smaller than the distances of communication parties. Accord-
ing to the propagation principle of electromagnetic wave, the
amplitude gains from different antenna elements to different
RIS elements can be assumed the same.
aTI,p,q ≈ aTI , aIR,q ≈ aIR, p = 1 · · ·N, q = 1 · · ·L. (5)
This kind of approximation is named far-field amplitude ap-
proximation. Since it’s irrational to divide the effectiveness of
RIS by aTI or aIR, the joint term aTIaIR , aTIR is defined
to represent the final amplitude gain from the transmitter to
the receiver through individual reflective element of the RIS1.
Let us concretize aTIR. Referring to the the electric fields
model described in [26], we have
aTIR =
√
|Er|22Z0
Ar
=
√
Z0GtGdxdyF (θt, ϕt)F (θr , ϕr)Γ2d−2TId
−2IR
16π2Z0
Grλ2
4π
=
√
GtGrGdxdyλ2F (θt, ϕt)F (θr, ϕr)Γ2
64π3d−1TId
−1IR
, δTIRd−1TId
−1IR. (6)
1The amplitude gain is actually a combination of channel gain and antennagain.
4
(a)
T
(b)
Fig. 2. Geometrical model of the RIS-assisted free-space wireless communication system. (a) Angular relationships between
the incident/reflected wave and the orientation of RIS. (b) Positional relationships between the ULA of the transmitter and
reflective elements of the RIS.
TABLE I
NOTATIONS ABOUT PHYSICAL AND ELECTROMAGNETIC FACTORS
Symbol Definition
Z0 The characteristic impedance of the air
Gt Antenna gain of the transmitter
G Gain of the RIS
F Normalized power radiation pattern
µTI The complement angle of the DOA of the RIS at the transmitter
θt The elevation angle from the reference point of the RIS to the transmitter
ϕt The azimuth angle from the reference point of the RIS to the transmitter
θr The elevation angle from the reference point of the RIS to the receiver
ϕr The azimuth angle from the reference point of the RIS to the receiver
Γ The reflection coefficient of the RIS
At The aperture of the transmit antenna
Ar The aperture of the receive antenna
Er The electric field intensity of the received signal
Note that normalized power radiation pattern of the RIS is
denoted as F (θt, ϕt)F (θr, ϕr). The general normalized power
radiation pattern of a single reflective element is in the form
of
F (θt, ϕt) =
cosk θt θt ∈ [0,π
2], ϕt ∈ [0, 2π]
0 θt ∈ (π
2, π], ϕt ∈ [0, 2π],
(7a)
F (θr, ϕr) =
cosk θr θr ∈ [0,π
2], ϕr ∈ [0, 2π]
0 θr ∈ (π
2, π], ϕr ∈ [0, 2π],
(7b)
where k ≥ 02. Thus, turning a RIS at a fixed position impacts
the amplitude gain aTIR. Fig. 2(a) illustrates this problem.
From above, the amplitude gain of the RIS is essentially a
function of the position and orientation of the RIS.
We now turn the attention to the phase changes of the EM
in the RIS link. The phase changes in the channel HHTI are
2The form cosk can be used to match the normalized power radiationpattern of different unit cell and antenna designs with an appropriate k [28].
equivalently written as
θTI,q,p = 2πdTI,p,q
λ, 2π
dTI +∆dTI,p,q
λ, (8a)
θIR,q = 2πdIR,p
λ, 2π
dIR +∆dIR,q
λ, (8b)
where λ is the carrier wavelength. Because the size of the
reflective element and the antenna element separation is the
same order or sub-order of the carrier wavelength, unlike
amplitude gains, different phase changes can’t be assumed to
be the same. However, in far-field operation, a tight distance
approximation is suitable to apply, which is shown at the
top of next page. In the formula (9), θt,p/ϕt,p represents the
elevation/azimuth angle at the RIS from the p-th transmitting
antenna to the RIS. mq is the index number of columns of
the q-th element and nq is the index number of rows of the
q-th element. It needs to be mentioned that there are sin(·)functions instead of cos(·) functions in existing works [33],
[34] due to the use of the supplementary angle. This kind of
approximation is named far-field phase approximation.
5
∆dTI,p,q = dTI,p,q − dTI,p,0 + dTI,p,0 − dTI
a≈ dTI,p,q − dTI,p,0 +∆dT,p cosµTI
b≈ − sin θt,p cosϕt,p(mq −MI + 1
2)dx − sin θt,p sinϕt,p(nq −
NI + 1
2)dy + (
N + 1
2− p)∆dT cosµTI
c≈ − sin θt cosϕt(mq −MI + 1
2)dx − sin θt sinϕt(nq −
NI + 1
2)dy
︸ ︷︷ ︸
∆dTI,q
+(N + 1
2− p)∆dT cosµTI
︸ ︷︷ ︸
∆dIT,p
, (9a)
∆dIR,q = dIR,q − dIR ≈ − sin θr cosϕr(mq −MI + 1
2)dx − sin θr sinϕr(nq −
NI + 1
2)dy
︸ ︷︷ ︸
∆dRI,q
. (9b)
In more details, approximation (a) requires dTI ≫ ∆dT,p,
similar to the traditional MIMO model [35]. Approxima-
tion (b) is based on the formula eq. 32 in [26], which
requires dTI ≫√
m2qd
2x + n2
qd2y . We assume that the el-
evation/azimuth angles at the RIS from different antennas
of the transmitter are identical, deriving approximation (c).This assumption is reasonable when dTI ≫ ∆dT,p. As
a conclusion, The conditions of far-field approximationare
summarized as follows
dTI ≫ N∆dT , dTI ≫ L√
d2x + d2y, dIR ≫ L√
d2x + d2y.
(10)
C. Problem Formulation
From above, we can see that the channel gain and phase
changes are both tightly related to the position and the
orientation of the RIS. Some existing works, considering
a fixed RIS, have designed the beamforming vector v and
phase-shift matrix Θ jointly to improve the received power,
namely the information achievable rate. In this paper, the
position of the RIS rI and the orientation of the RIS ξ are
also under-determined variables, thus they can be utilized
to further enhance the received power. Considering practical
limitations, the joint optimization problem of RIS-assisted
wireless communication is given by
(P1) : maxv,Θ,rI ,ξ
Pr
s. t. vHv ≤ Pt
Θ ∈ B
{rI , ξ} ∈ S0
rI ∈ S1 ∩ S2, (11)
where S0 is a set that guarantees the EM to propagate from T
to R through the RIS directly, which is specified by application
environment. In more details, it is determined by the relative
positional relationship of communication parties, obstacles and
the orientation of RIS. S1 can be expressed as (dTI ≥ r0) ∩(dIR ≥ r1). r0 and r1 are the minimum distance to guarantee
the far-field condition, respectively. S2 is the feasible area that
the RIS can be fixed at. Herein, we denote the feasible set of
Θ as B here. Without loss of generality, we assume that the
antennas in T and R are omni-directional3.
The problem (P1) is a joint optimization of four variables,
and it is challenging to solve directly. In the following,
we devote to solving it by two phases. In the first phase,
we assume rI is fixed, and the closed-form global optimal
solutions of Θ and v as well as the optimal ξ are proposed. In
the second phase, based on the optimal solutions of the other
three variables for a fixed rI , the problem (P1) become an
unadulterated position optimizing problem (P2). Substituting
the optimal position of (P2) back to the optimal Θ and v
and ξ, which can be treated as functions of rI , the global
optimal solutions of (P1) are all concretized4. Note that, the
sub-optimization problems in each phase can also be viewed
as meaningful independent works.
III. OPTIMAL STRATEGY WITH A FIXED RIS
To decouple the solutions of Θ, v, ξ with rI in problem
(P1), we consider a fixed RIS in this section. We find that
the Θ and v are coupled with each other, but irrelevant to
ξ. In addition to derive optimal solutions of Θ, v and ξ, we
propose an anti-decay designing principle of manufacturing
RIS for free-space THz communication.
A. Optimal Θ and v
The solutions of Θ and v for the fixed RIS can be obtained
using the iterative method proposed in [7]. However, it’s
challenging to solve the joint optimization problem (P1) based
on it. Herein, by exploiting the tightly coupled property of
channel elements in the free space, we provide the analytic
and optimal solutions of Θ and v.
3If we consider the directional antenna, only placing the RIS in the mainlobe is meaningful. Let S3 denotes the area in the main lobe. Therefore, anew restriction rI ∈ S3 should be added to (P1).
4However, since we focus on far-field communications, the channels havebeen already approximated according to the far-field approximations in (P1).Therefore, the global optimal solutions of (P1) are near-optimal to the primalproblem without approximations.
6
For any given Θ, it is widely known that maximum-ratio
transmission (MRT) is the optimal transmit beamforming to
maximize the received power [7], i.e.,
v⋆ =
√
Pt
(hHIRΘH
HTI)
∗
∥∥hH
IRΘHHTI
∥∥. (12)
Applying the optimal v⋆, the received power can be expressed
by
Pr =∥∥h
HIRΘH
HTI
∥∥2Pt. (13)
According to the far-field gain approximation and far-field
phase approximation, the channel matrix HHTI can be rewritten
as
HHTI = aTIe
j 2πλ
dTIabT , (14a)
a =[
ej2πλ
∆dTI,1 , ej
2πλ
∆dTI,2 , · · · , ej 2π
λ∆dT
I,L ,]
, (14b)
b =[
ej2πλ
∆dIT,1 , ej
2πλ
∆dIT,2 , · · · , ej 2π
λ∆dI
T,N ,]
. (14c)
To maintain the uniform, we rewrite the hHIR as
hHIR = aIRe
j 2πλ
dIRcT , (15a)
c =[
ej2πλ
∆dRI,1 , ej
2πλ
∆dRI,2 , · · · , ej 2π
λ∆dR
I,L ,]
. (15b)
After substituting them into (13), the new form of the
received power is given by
Pr = aTIaIR∥∥c
TΘab
T∥∥2Pt
= aTIR
∥∥θ
Tdiag(cT )abT∥∥2Pt
= aTIR
∥∥θ
Tdb
T∥∥2Pt = aTIRN(θT
d)2Pt, (16)
with
d =[
ej2πλ
∆dTI,1+∆dR
I,1 , · · · , ej 2πλ
∆dTI,L+∆dR
I,L ,]
. (17)
Hence, the optimal θ to maximize the received power is
θ⋆ = d
∗ (18)
More clearly, considering the expressions of ∆d terms in (9),
the optimal phase shift of reflective element q is designed as
ϕ⋆q ,2π
1
λ
((sin θt cosϕt + sin θr cosϕr)(mq −
MI + 1
2)dx
+ (sin θt sinϕt + sin θr sinϕr)(nq −NI + 1
2)dy
),
(19a)
θ⋆q = ejϕ⋆q . (19b)
With the closed-form θ⋆, the decoupled analytic solution of
v⋆ is given by
v⋆ =
√Pt√N
b∗. (20)
More clearly, considering the expressions of ∆d terms in (9),
the p-th element of v is designed as
v⋆p =
√Pt√N
ej−2πλ
(N+12 −p)∆dT cosµTI . (21)
Under the proposed closed-form beamforming and phase
shifts, the optimal received power becomes
Pr = NL2a2TIRPt. (22)
As seen, the growth of the received power follows a N scaling
with the antenna elements and L2 scaling with the reflective
elements, which has been recognized in several recent works
[4], [27].
The optimal received power can be expended as
Pr = (Lλ)2GtGrGdxdyF (θt, ϕt)F (θr, ϕr)Γ
2
64π3d−2TId
−2IRNPt
=λ2
dxdy
S2RISGtGrGF (θt, ϕt)F (θr , ϕr)Γ
2
64π3d−2TId
−2IRNPt,
(23)
where SIRS denotes the total area of the RIS. The formula
(23) indicates an anti-decay designing principle of manufac-
turing RIS for mmWave/THz communication in free space.
To maintain the the received power as the frequency of carrier
wave increases, the RIS must manufactured according to the
wavelength. In more details, there are two alternatives: One
is when the area of total RIS is limited, the size of the
reflective element should be in a fix proportional to the carrier
wavelength. The other is when the area of reflective element
is fixed, the number of the reflective elements is in a fixed
inverse ratio to the carrier wavelength.
B. Optimal Orientation of the RIS
As mentioned before, the orientation of the RIS can be
adjusted to improve the received power via maximizing
F (θt, ϕt)F (θr, ϕr). The optimal value is denoted as F ⋆
hereinafter. Due to large distances between the RIS and
communication parties in the far-field case, it’s reasonable to
treat all reflective elements as the reference point of the RIS
when calculating F ⋆. As illustrated in Fig. 2(a), the problem
of turning a RIS to reap the max gain is organized as
maxθt,θr
F (θt, ϕt)F (θr, ϕr)
s. t. θt ∈ (0,π
2)
θr ∈ (0,π
2)
θ0 ≤ θt + θr ≤ 2π − θ0. (24)
It is found that when θt = θr = θ02 , the objective function in
(24) reaches the max value, denoted as F ⋆, which is given by
F ⋆ = (cos2θ02)k = (
1
2cos θ0 +
1
2)k
= (d2TI + d2IR − d2TR
4dTIdIR+
1
2)k. (25)
From the above, the optimal orientation ξ is just to make the
RIS perform specular reflection, that is, the incident signal
is mainly reflected towards the mirror direction (θr = θt).It is worth mentioned that this result in the MISO system is
consistent with the counterpart in the SISO system [26].
7
IV. OPTIMAL POSITION OF THE RIS
In this section, jointly with the optimal solutions of Θ, v,
ξ, we aim to study the optimal position of the RIS in problem
(P1).As the objective function to optimize the position of the
RIS, the maximum received power (23) is simplified as
Pr = NL2δ2TIRd−2TId
−2IR = NL2 δ
2TIR
F ⋆F ⋆d−2
TId−2IR
= NL2 δ2TIR
F ⋆(d2TI + d2IR − d2TR
4dTIdIR+
1
2)kd−2
TId−2IR
︸ ︷︷ ︸
Fobject
. (26)
According to (6),δ2TIR
F⋆ is a constant with no relationship to
the position of the RIS. The position change of the RIS only
impacts the value of dTI and dIR in this formula.
The position optimizing question is formulated as
(P2) :maxrI
Fobject
s. t. rI ∈ S, (27)
where S = S0 ∩ S1 ∩ S2 represents the feasible space
to place RIS. In this section, we’ll study the property of
optimal position in a two-dimensional plane, then extend it
to arbitrarily three-dimensional space.
It is known that a three-dimensional feasible space can be
fully split into parallel two-dimensional subspaces. Then, the
problem (P2) is replaced by many parallel subproblems, where
the position of the RIS is an two-dimensional variable. It is
noted that the principle of splitting is flexible and has no
influence to the final result.
We start from not considering the constraints of placing
RIS,. Then the parallel subproblems is to optimize the position
of RIS on an infinite large plane, named S. Adequately, we
consider the 12 plane of S due to the symmetry. Hereinafter,
the plane S represents the half plane. The scenario of finding
optimal position on plane S is illustrated in Fig. 3. The points
T ′ and R′ represent the projection of T and R on plane S
respectively. The line l denotes the line containing T ′ and R′
on plane S. The term lR′T ′ denotes the line segment whose
endpoints are T′
and R′
. The term l−−−→T ′R′
accounts for the
half-line staring from T′
and containing R′
. By contrary, the
half-line l−−−→T ′R′
stars from R′
and contains T′
. The term νTI
is the intersection angle of lTR and lTT ′ .
It is obvious that the values of dIR and νTI can uniquely
determine a position of the RIS on plane S. The optimal
problem (P2) is reformulated as:
maxdIR,νTI
Fobject
s. t. (dIR, νTI) ∈ C
dTI =h1
cos νTI
, (28)
where, C represents possible unions of dIR and νTI . For a
given νTI , the feasible set of the RIS on plane S is a circle
marked as CνTI. After excavating the quasi-convex property
of Fobject, the following theorems are obtained.
Fig. 3. Diagram of placing the reference point of the RIS on
a plane S.
Fig. 4. Illustration of area D and corollary 1. The yellow
point denotes the possible optimal position of the RIS on the
intersection of A and C(νTI).
Theorem 1: If k = 0 in (7), on plane S, the optimal position
of the RIS is must on the line segment lR′T ′ . If k > 0 in (7),
on the area S-D, the optimal position is must on line l.The
area D is a set of (dTI ≤ dTR) ∩ (dTI ≤ dTR), as shown in
Fig. 4.
Proof 1: See Appendix A and Appendix B.
As illustrated in Fig. 4, we put forward a useful corollary
for placing the RIS on an arbitrary two-dimensional area A.
Corollary 1: For an arbitrary closed feasible area A on the
whole plane S when k = 0 or S−D when k > 0, the optimal
position of the RIS is must on A’s boundary and the feasible
part on line l.
Eventually, based on the fact that a three-dimensional space
can be split to the parallel planes fully, we extend above corol-
laries to the feasible space S. Similar to the two-dimensional
case, a special space is defined as the forbidden space, whose
cross section created by plane S are the area D. The following
corollary is derived.
Corollary 2: For an arbitrary closed feasible space S in the
whole three-dimensional space when k = 0 or in the three-
8
dimensional space except the special space when k > 0, the
optimal position of the RIS is must on the surface of S 5.
Remark 1: Note that the area D diminishes as the h1 and
h2 becomes either larger or smaller. For most cases, the area
D can be neglected or does not exist. Accordingly, for three-
dimensional cases, the special space can be neglected or does
not exist in most cases. Based on our proposed corollaries,
the dimension of the area of interest can be reduced. Thus the
computational complexity of any numeral algorithms can be
reduced.
V. EXTENSIONS AND DISCUSSIONS
A. Extensions to the Existing of Direct Link
We consider adding a RIS to enhance the wireless commu-
nication in free space. Now, besides the RIS link, the direct
link also exists. The direct channel can be expressed as
hTR = aTR
[
ej2πdTR,1
λ , ej2πdTR,2
λ , · · · , ej2πdTR,N
λ
]
, (29)
with
aTR =
√GtGrλ2
4πd−1TR, (30)
where dTR,p is the distance from antenna p to the receiver,
p = 1, ..., N . In this scenario, the received signal is a sum
from two pathes, which is given by
Pr = |(hHIRΘH
HTI + h
HTR)v|2. (31)
The joint optimal Problem P1 is unchanged except replacing
the objective function with P′
r . For this scenario, we also
propose a closed-form phase shifts as follows.
ϕ⋆q ,
π
2(O
|O| − 1)− 2πdTI + dIR − dTR +∆dTI,q +∆dRI,q
λ
=π
2(O
|O| − 1)− 2π1
λ(dTI + dIR − dTR)
+ 2π1
λ
((sin θt cosϕt + sin θr cosϕr)(mq −
MI + 1
2)dx
+ (sin θt sinϕt + sin θr sinϕr)(nq −NI + 1
2)dy
),
(32a)
θ⋆q = ejϕ⋆q , (32b)
where
O =sinc(N∆dT (cosµTI−cosµTR)π
λ)
sinc(∆dT (cosµTI−cosµTR)πλ
). (33)
Proof 2: See Appendix C.
Interestingly, compared to (19a), only the term π2 (
O|O| −
1) − 2π 1λ(dTI + dIR − dTR) is polymeric in (32a). It’ not
hard to obtain the corresponding bemforming by using MRT,
thus omitted here. With the solutions, the received power is
expressed as
Pr = NL2a2TIRPt +Na2TRPt + 2NLaTRaTIROPt. (34)
5Note that, due to the plane S is an arbitrary plane, the feasible part online l can be avoided by selecting the plane S suitablely.
The derived results about the orientation and position of
the RIS can also be extended to this scenario naturally. It is
obvious that the optimal orientation of RIS is in accordance
with the counterpart in Section III-B. The aforementioned
conclusions of the optimal position of RIS has to be adjusted
slightly. The analytic process is similar to that shown in
Section IV. Differently, the plane S fitting in this scenario can’t
be arbitrary anymore, and becomes a special plane where the
ULA of the transmitter is perpendicular to it. While analysing
in the same manner as the derivation in Appendix B, it is
found that only the results under fixed dTI is available due
to the the existing of O. Because when dTI is fixed, µTI
is fixed, resulting in a certain value of O. By this way, the
Fobject is consistent with the objective function of Problem
P2. However, the similar results don’t exist anymore when
dIR is fixed. Therefore, the theorems and corollaries in Section
IV also hold by replacing area D (dTI ≤ dTR)∩(dIR ≤ dTR)with area D1 (dTI ≤ dTR). Besides, for three-dimensional
cases, a now space needs to be added where the optimal
position may lies in. That is the feasible part on the plane
consisting of the ULA and line l.
B. Extension to UPA Cases
We have assumed the transmit antenna to be a ULA in
system model. Actually, our work can be transplanted to
uniform planar array (UPA) case seamlessly as long as the
far-field condition is satisfied. The corresponding adjustment
is to reformulate the distance approximations at the transmitter.
In more details, we reformulate ∆dIT,p in (9) as
∆dIT,p = − sin θT cosϕT (mTp − MT + 1
2)dTx
− sin θT sinϕT (nTp − NT + 1
2)dTy , (35)
where MT denotes the columns of the UPA and mTp denotes
the index number of columns of antenna p. NT denotes the
rows of the UPA and nTp denotes the index number of rows of
antenna p. dTx × dTy denotes the unit two-dimensional interval
in the UPA. θT and ϕT account for the elevation angle and
the azimuth angle from the transmit antenna to the RIS at
the transmitter side, respectively. Note that it is similar to the
approximations at the RIS since they are both planar array.
With the adjustment, our analysis is unchanged for UPA cases.
C. General Beamforming and Phase Shifts
A main limitation of the aforementioned results for the joint
optimization is that they are derived under far-field operation.
In far-field operation, the channel gain is obtained by far-field
amplitude approximation and far-field phase approximation.
These approximations may be unreliable under near-field
operation, when RIS is close to the transmitter or to the
receiver or RIS has a large size. In order to extend application
scenarios, we develop the proposed beamforming and phase
shifts to both near-field and far-field cases and propose SVD-
9
TABLE II
THE PHYSICAL AND ELECTROMAGNETIC PARAMETERS OF RIS-ASSISTED WIRELESS COMMUNICATION
Fig. 6. The received power versus the distances of communication parties. (a) dx = dy = 0.01 m. (b) dx = dy = 0.06 m.
-70
-60
200
Pow
er (
dBm
)
300
-50
y (m)
2000
x (m)
100
-40
0-100-200 -200
(a) .
-70
-60
200
Pow
er (
dBm
) -50
200y (m) x (m
)0
-40
0-200 -200
(b) .
Fig. 7. The received power versus the position of the RIS on plane S with only RIS link. (a) h = 80 m. (b) h = 40 m.
Moreover, we set dTR = 200 m and the direction of ULA is
perpendicular to plane S (for convenience of simulation but
not necessary). The coordinate origins of plane S is selected
as T′
. The positive direction of X-axis is selected as l−−−→T ′R′
and
the Y-axis is determined correspondingly on plane S. Without
loss of generality, we let h1 = h2 = h.
Fig. 7 reveals the received power versus the position of RIS
on plane S via traversal grid when the direct link is blocked.
As seen the optimal solution is near T′
or R′
and on line l,
which is consistent with our analysis. Similarly, we investigate
the received power for existing the direct link in Fig. 8. As
seen, there are many ripples due to the balance of the RIS
link and direct link to maximize the received power (34). In
more details, they are determined by the antennas structure and
DOAs of the RIS/receiver at the transmitter, which is given
by (33). We also find that the amount of the ripples is half of
the number of antennas at the transmitter. It is observed that
the optimal solution is close to R′
, and on line l, which is
consistent with with our analysis. Moreover, the power at R′
is obvious larger than that at T′
due to the existence of direct
link.
Fig. 9 illustrates the optimal received power versus the
carrier wavelength. Moreover, we let h = 80m and fix the RIS
at R′
on plane S. Note that the RIS in this figure satisfies the
proposed anti-decay designing principle where dx
λ=
dy
λ= 1
3 .
It is observed that, as the wavelength decreases, the optimal
received power from direct link decays largely. But with the
help of RIS link, the performance degradation is alleviated.
11
-45
-44.5
200
-44
Pow
er (
dBm
)
200
-43.5
y (m) x (m)
0
-43
0-200 -200
(a) .
-46
-44
200
Pow
er (
dBm
)
-42
200y (m) x (m
)0
-40
0-200 -200
(b) .
Fig. 8. The received power versus the position of the RIS on plane S with both RIS link and direct link. (a) h = 80 m. (b)
h = 40 m.
Therefore, the RIS has absolute advantages to assist in free-
space mmWave/THz communication.
C. The Robustness of Our Proposed Beamforming and Phase
shifts
In practice, the obtained position of RIS may be obtained
imperfectly due to the error of measurement or the mobility
of the RIS. Therefore we demonstrates the robustness of our
proposed solutions of beamforming and phase shifts in Fig. 10.
The geometric setup is same to the last subsection, and the
direct link is blocked. Assuming the known position of the
RIS is T′
(origin). According to it, the closed-form beam-
forming and phase shifts are derived, named the estimated
solutions. Fig. 10 illustrates the normalized power deviation
using estimated solutions versus the practical position of the
RIS on plane S. The normalized power deviation is used to
quantity the deviation of the received power, which is given
by
Pr =‖Pr − Pr‖
max{Pr, Pr}, (39)
where Pr denotes the received power using estimated solutions
and Pr denotes the ideal optimal received power. It is found
that the area for Pr < 0.1 is large than 5 m×5 m. Therefore,
our proposed solutions are strongly robust to the position
perturbation of the RIS. Note that the performance of the
proposed SVD-based solutions is consistent with the proposed
closed-form solutions in the far-field operation, thus omitted
here.
VII. CONCLUSION
In this paper, comprehensive optimization of incorporating
a RIS to MISO wireless communication in free space has been
0.01 0.02 0.03 0.04 0.05Wavelength (m)
-55
-50
-45
-40
-35
Rec
eive
d po
wer
(dB
m)
with RIS (S=1 m2)without RIS
with RIS (S=2 m2)
Fig. 9. The optimal received power versus the carrier wave
length.
considered from electromagnetic and physical perspectives.
The closed-form solutions of transmitter’s beamforming and
phase shifts have been proposed and extended. Considering
the general power radiation pattern, we have proved that
the optimal orientation of the RIS is just to satisfy specular
reflection. United with the above contributions, the position
optimizing problem of placing a RIS has been studied. For
most three-dimensional space to place the RIS, a substantial
dimensionality reduction theory was provided. In simulation
part, the proposed closed-form solutions of beamforming and
phase shifts approach the power upper bound. Besides, the
robustness in terms of position perturbation is verified. The
simulation results indicate that adding a RIS is remarkable in
12
0
0.2
20
0.4
Nor
mal
ized
pow
er d
evia
tion
0.6
0.8
y (m)
0 2010
x (m)0-10-20 -20
(a) .
0
0.2
20
0.4
0.6
Nor
mal
ized
pow
er d
evia
tion
0.8
1
y (m)
0 2010
x (m)0-10-20 -20
(b) .
Fig. 10. The normalized received power deviation versus the position perturbation of RIS on plane S. (a) h = 80 m. (b)
h = 40 m.
mmWave/THz communication.
APPENDIX A
PROOF OF THEOREM. 1 (k = 0)
For any fixed νTI , the orbit of feasible position is a circle
C(νTI) on plane S. If νTI is fixed, the value of dTI is
also certain. Fobject(dIR) decreases from p1(νTI) to p2(µTI)along C(νTI). Fobject(dIR) is a decreasing function to dIR,
the max value of Fobject arrives at point p1(νTI). Therefore,
for an arbitrarily position on the plane S, there must exists a
position on the half-line l−−−→T ′R′
with the same νTI , at which
the value of the objective function is equal or bigger. As a
conclusion, the optimal position is must on the half-line l−−−→T ′R′
.
With the same manner, when starting from fixing νRI , we
obtain a conclusion that the optimal position is must on the
half of the half-line l−−−→R′T ′
. Therefore, on plane S, the optimal
position of RIS is must on line segment lR′T ′ .
APPENDIX B
PROOF OF THEOREM. 1 (k > 0)
Necessarily, we’ll exploit the quasi-convex property of
Fobject. For convenient expression, we simplify dIR as x and
denote√Fobject(x) as F . To maximize the value of Fobject is
equal to maximize the value of F . Then the objective function
F (x) can be written as
F (x) = x−1(ax−1 + bx+1
2︸ ︷︷ ︸
f(x)
)k2 x > 0. (40)
Wherein, the constant a =d2TI−d2
TR
4dTI, b = 1
4dTI> 0, k > 0 and
0 < f(x) < 1, deducing from (25).
The derivation of F (x), denoted as F′
(x) is given by
F′
(x) = −f(x)k2 x−2 +
k
2x−1f(x)(
k2−1)f
′
(x), (41)
where,
f′
(x) = (−ax−2 + b). (42)
proof (0 < k ≤ 2):
F′
(x) = [f(x)(k2−1)x−2] (−f(x) +
k
2xf
′
(x))︸ ︷︷ ︸
g(x)
. (43)
Because 0 < f(x), g(x) determines whether F′
(x) is positive
or negative.
g(x) = −(ax−1 + bx+1
2) +
k
2(−ax−1 + bx)
= −(k
2+ 1)ax−1 + (
k
2− 1)bx− 1
2< 0. (44)
So, F (x) is a quasi-convex (quasilinear) function for a >0, 0 < k ≤ 2.
proof (k ≥ 2): The condition for F′
(x) = 0 is
f(x) =k
2xf
′
(x). (45)
If the solution doesn’t exist, then F (x) is a quasi-convex
(quasi-linear) function. If it exists, since f(x) > 0, (45)implies f
′
(x) > 0. The second derivative of F (x) is given
by
F′′
(x) = (x−3f(x)k2−2)
(−k
2xf(x)f
′
(x) + 2f(x)2 − k
2xf(x)f
′
(x)
+k
2(k
2− 1)x2f
′
(x)2 +k
2x2f(x) 2ax−3
︸ ︷︷ ︸
f′′ (x)
)
, (x−3f(x)k2−2)h(x), (46)
13
Due to f(x) > 0, whether F′′
(x) is negative or positive
is determined by h(x). The expression of h(x) is further
expressed as
h(x) = f(x) [−k
2xf
′
(x) + f(x)]︸ ︷︷ ︸
Q
+f(x)2
− k
2xf
′
(x) [f(x) − k
2xf
′
(x)]︸ ︷︷ ︸
Q
− k
2x2f
′
(x)2 +k
2x2f(x)f
′′
(x)
a= f(x)2 − k
2x2f
′
(x)2 +k
2x2f(x)f
′′
(x)
b= (
k2
4− k
2)x2f
′
(x)2 + kax−1f(x) > 0, (47)
wherein, (a) is due to Q = 0 when (45) holds. (b) results
from the substitution of (45) and the expression of f′′
(x).Therefore, F (x) is a quasi-convex function for a > 0, k > 2.
As a conclusion, when dTI > dTR, Fobject(dIR) is a quasi-
convex function for a fixed dTI . Due to the symmetry of dTI
and dIR in the function Fobject, it also holds that when dIR >dTR, Fobject(dTI) is a quasi-convex function for a fixed dIR.
Let D1 account for the area where dTI ≤ dTR and D2 account
for the area where dIR ≤ dTR. The union set of D1 and D2
is denoted as D.
Eventually, based on the basic property of quasi-convex
function (Section 3.4.2 in [36]), extending Appendix A, we
derive a conclusion for k > 0. For k > 0, on the area S-D,
the optimal position is must on the line l. Note that it is the
line l, not the line segment lR′T ′ due to the discrepancy of
monotonicity and quasi-convex property.
APPENDIX C
OPTIMAL PHASE SHIFTS FOR EXISTING DIRECT LINK
We already know the MRT is the optimal method to design
beamforming for point to point communication. After applying
MRT, the received power can be expressed as
Pr =∥∥∥aTIRe
j2πdTI+dIR
λ θTdb
T + aTRej2π
dTRλ e
T∥∥∥
2
Pt
=
∥∥∥∥∥∥
aTIR ej2πdTI+dIR−dTR
λ θTd
︸ ︷︷ ︸
A′
bH + aTRe
T
∥∥∥∥∥∥
2
Pt, (48)
where,
eT =
[
ej2π
∆dRT,1λ , ej2π
∆dRT,2λ , · · · , ej2π
∆dRT,Nλ ,
]
, (49a)
∆dRT,p = ((N + 1)
2− p) cosµTR∆dT . (49b)
We equivalently represent A′ as Mejx, which is given by
Mejx =L∑
q=1
ej(ϕq+2π(dIR+dTI−dTR+∆dT
I,q+∆dR
I,q)
λ)
︸ ︷︷ ︸
A′
q
. (50)
It indicates that, via changing ϕq , the amplitude M can be an
arbitrary value in the feasible set [0, L] and the amplitude xcan be an arbitrary value in the feasible set [0, 2π].
Substituting (50) into (48) with new formulation of A’, we
obtain
Pr =N∑
p=1
|aTIRME1(p) + aTRE2(p)|2Pt
= NM2a2TIRPt +Na2TRPt
+ 2MaTIIRaTR
N∑
p=1
cos (∠E1(p)− ∠E2(p))
︸ ︷︷ ︸
E3
Pt, (51)
where,
E1(p) = ejx+j2π((N+1)
2 −p) cosµTI∆dT
λ , (52a)
E2(p) = ej2π((N+1)
2 −p) cosµTR∆dT
λ , (52b)
E3 =
N∑
p=1
cos (x−Kp),
Kp , 2π(p− (N + 1)
2)(cosµTI − cosµTR)
∆dTλ
.
(52c)
We deduce the expression of E3 further as follows.
E3 = cos(x)N∑
p=1
cos(Kp) + sin(x)N∑
p=1
sin(Kp)
a= cos(x)
N∑
p=1
cos(Kp) + j cos(x)
N∑
p=1
sin(Kp)
= cos(x)
N∑
p=1
ejKp
b= N cos(x)
sinc(N∆dT (cosµTI−cosµTR)πλ
)
sinc(∆dT (cosµTI−cosµTR)πλ
)︸ ︷︷ ︸
O
, (53)
where (a) is due to∑N
p=1 sin(Kp) = 0 and (b) results from
ejKp(p = 1 : N) is a geometric progression. It is verified that
the conditions of maximizing Pr are M = L and cos(x) =O|O| . The variable A′ can achieve this requirement if and only
if
ϕ⋆q ,
π
2(O
|O| − 1)− 2πdTI + dIR − dTR +∆dTI,q +∆dRI,q
λ
=π
2(O
|O| − 1)− 2π1
λ(dTI + dIR − dTR)
+ 2π1
λ
((sin θt cosϕt + sin θr cosϕr)(mq −
MI + 1
2)dx
+ (sin θt sinϕt + sin θr sinϕr)(nq −NI + 1
2)dy
).
(54)
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