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Achievable Rate Optimization for MIMO Systems
with Reconfigurable Intelligent Surfaces
Nemanja Stefan Perovi´
c, Le-Nam Tran, Senior Member, IEEE, Marco Di Renzo, Fellow, IEEE,
and Mark F. Flanagan, Senior Member, IEEE
Abstract—Reconfigurable intelligent surfaces (RISs) represent
a new technology that can shape the radio wave propagation in
wireless networks and offers a great variety of possible perfor-
mance and implementation gains. Motivated by this, we study
the achievable rate optimization for multi-stream multiple-input
multiple-output (MIMO) systems equipped with an RIS, and
formulate a joint optimization problem of the covariance matrix
of the transmitted signal and the RIS elements. To solve this
problem, we propose an iterative optimization algorithm that
is based on the projected gradient method (PGM). We derive
the step size that guarantees the convergence of the proposed
algorithm and we define a backtracking line search to improve
its convergence rate. Furthermore, we introduce the total free
space path loss (FSPL) ratio of the indirect and direct links as a
first-order measure of the applicability of RISs in the considered
communication system. Simulation results show that the proposed
PGM achieves the same achievable rate as a state-of-the-art
benchmark scheme, but with a significantly lower computational
complexity. In addition, we demonstrate that the RIS application
is particularly suitable to increase the achievable rate in indoor
environments, as even a small number of RIS elements can
provide a substantial achievable rate gain.
Index Terms—Achievable rate, gradient projection,
multiple-input multiple-output (MIMO), optimization,
reconfigurable intelligent surface (RIS).
I. INTRODUCTION
In recent years, there has been a tremendous, almost ex-
ponential, increase in the demands for higher data rates. The
main driving forces that constantly increase this demand are
the increasing number of mobile devices and the appearance
of services that require high data rates (e.g., video streaming
and online gaming). Consequently, many technology solu-
tions have been proposed to address this ever-increasing de-
mand, such as massive multiple-input multiple-output (MIMO)
and millimeter-wave (mmWave) communications. In spite of
providing potentially significant achievable rate gains, these
The work of N. S. Perovi´
c and M. F. Flanagan was funded by the Irish
Research Council under grant number IRCLA/2017/209. The work of L. N.
Tran was supported in part by a Grant from Science Foundation Ireland under
Grant number 17/CDA/4786. M. Di Renzo’s work was supported in part by
the European Commission through the H2020 ARIADNE project under grant
agreement number 871464 and through the H2020 RISE-6G project under
grant agreement number 101017011.
N. S. Perovi´
c, L. N. Tran, and M. F. Flanagan are with School of
Electrical and Electronic Engineering, University College Dublin, Belfield,
Dublin 4, Ireland (Email: nemanja.stefan.perovic@ucd.ie, nam.tran@ucd.ie
and mark.flanagan@ieee.org).
M. Di Renzo is with Université Paris-Saclay, CNRS, CentraleSupélec,
Laboratoire des Signaux et Systèmes, 3 Rue Joliot-Curie, 91192 Gif-sur-
Yvette, France (E-mail: marco.di-renzo@universite-paris-saclay.fr).
technologies generally incur additional power and hardware
costs, so that the total benefit of their implementation has to
be independently evaluated for each user scenario. Broadly
speaking, these technologies can be seen as novel transmitter
and receiver features that enable us to achieve higher data
rates. However, they do not have the capability of directly
influencing the propagation channel, the stochastic nature of
which can sometimes limit the efficiency of these proposed
technology solutions.
A possible approach to overcome the aforementioned issue
lies in the use of the recently-developed reconfigurable intelli-
gent surfaces (RISs) [1]. The key component to realize the RIS
function is a software-defined surface that is reconfigurable
in such a way as to adapt itself to changes in the wireless
environment. It consists of a large number of small, low-cost,
and passive elements, each of which can reflect the incident
signal with an adjustable phase shift, thereby modifying the
radio waves. Optimization of the wavefront of the reflected
signals enables us to shape how the radio waves interact with
the surrounding objects, and thus control their scattering and
reflection characteristics [2], [3], [4]. Hence, the introduc-
tion of RISs fundamentally changes the wave propagation in
wireless communication systems and offers a wide variety of
possible implementation gains, thus potentially presenting a
new milestone in wireless communications.
In recent years, researchers have investigated many impor-
tant aspects of RIS-assisted wireless communication systems.
The problem of estimating the required channel state informa-
tion (CSI) was considered in [5] by embedding an active sensor
in the RIS, and in [6] by estimation of a combined transmitter-
RIS-receiver channel. Other emerging body of work studies
an accurate modeling of the interactions (considering reflec-
tion, refraction, diffraction and polarization) of the incident
wave with the RIS, and elucidates the dependence of these
interactions on the size of the RIS elements, the distance
between the adjacent RIS elements, the angle of incidence and
so on [7], [8]. All of these aspects are critical for the practical
implementation of RIS-aided wireless communication systems
to become feasible.
From a theoretical standpoint, the evaluation and opti-
mization of the achievable rate of an RIS-aided wireless
communication system is crucial. This problem is signifi-
cantly more challenging to solve than in the conventional
case without an RIS, since in the case without an RIS the
channel capacity can be completely determined in closed-form
for deterministic (or fixed) channels. A variety of different
optimization methods for enhancing the achievable rate in RIS-
2
aided wireless communication systems have been proposed in
the literature, which attempt to find a near-optimal solution
with a reasonable computational complexity and run time.
The vast majority of these methods are particularly tailored
for downlink communication with single-antenna receive de-
vices. In [9], the authors introduced an optimization method
that increases the receive signal-to-noise ratio (SNR) and
consequently enhances the achievable rate in multiple-input
single-output (MISO) systems. The proposed solution is based
on the alternating optimization (AO) method, which adjusts
the transmit beamformer and the RIS element phase shifts
in an alternating fashion. The AO technique has also been
successfully utilized to increase the data rate for secure com-
munications in environments with multiple RISs and single-
antenna users [10]. In contrast to AO, the spectral efficiency
optimization for a single-user MISO system in [11] was
performed by jointly adjusting the transmit beamformer and
the RIS element phase shifts. In [12], the authors employed
a gradient-based algorithm to enhance the receive signal-to-
interference-plus-noise ratio (SINR), and hence the achievable
rate, for single-antenna users that do not have a direct link with
the base station. The achievable rate optimization for multi-
user downlink communications is specifically considered for
mmWave sparsely scattered channels in [13]. An algorithm
for energy efficiency optimization in a multi-user downlink
communication system was presented in [14]. The sum-rate
optimization for multi-user downlink communications using a
deep reinforcement learning based algorithm was introduced
in [15].
In contrast to the previous papers, the achievable rate opti-
mization in [16] is realized by jointly controlling the phase and
the amplitude adjustment of each RIS element. Furthermore,
in [17] the authors developed a practical phase shift model that
captures the phase-dependent amplitude variation in the RIS
element-wise reflection coefficient and utilized it to enhance
the achievable rate. The achievable rate optimization for an
RIS with discrete phase shifts in multi-user downlink commu-
nications was considered in [18]. A system for serving paired
power-domain non-orthogonal multiple access (NOMA) users
by designing the RIS phase shifts was introduced in [19]. In
[20], the authors studied the joint optimization of the RIS
reflection coefficients and the orthogonal frequency division
multiple access (OFDMA) time-frequency resource block, as
well as power allocations, to maximize the users’ common
(minimum) rate. Energy-efficiency optimization for multi-user
uplink MIMO communications was presented in [21], where
the users’ covariance matrices and the RIS phase shifts are
optimized in an alternating fashion based on partial knowledge
of the CSI.
Although RIS-aided communication systems with single-
antenna devices are well-studied in the literature, there is
only a limited number of papers that consider the design and
analysis of an RIS-aided MIMO communication system. In
particular, the achievable rate optimization in those systems
remains relatively unknown. It was demonstrated in [22] how
an RIS can be implemented and optimized to increase the
rank of the channel matrix, leading to substantial achiev-
able rate gains in multi-stream MIMO communications. The
method proposed in [22] was specifically designed for pure
line-of-sight (LOS) channels, neglecting the presence of any
non-LOS (NLOS) component. Optimization of the achievable
rate for a single-stream MIMO system in an indoor mmWave
environment with a blocked direct link was analyzed in [23].
Since in indoor mmWave communications NLOS channel
components are usually significantly weaker than the LOS
component, all communication links were also modeled as
pure LOS links. Although the proposed optimization schemes
in [23] provide the near-optimal achievable rate, they require
very low computational and hardware complexity. In [24], the
authors utilized the AO method to enhance the achievable rate
of an RIS-aided multi-stream MIMO communication system.
Although this optimization method is simple to implement,
it can require many iterations to converge, especially when
the number of RIS elements is very large, which corresponds
precisely to the case where the RIS is the most useful.
Against this background, the contributions of this paper are
listed as follows:
•To maximize the achievable rate of a multi-stream MIMO
system equipped with an RIS, we formulate a joint
optimization problem of the covariance matrix of the
transmitted signal and the RIS elements (i.e., phase
shifts). We then propose an iterative projected gradient
method (PGM) to solve this nonconvex problem, for
which we present exact gradient and projection expres-
sions in closed form. The proposed method is provably
convergent to a critical point of the considered problem,
which is desirable for a nonconvex program.
•We derive a Lipschitz constant for the proposed PGM,
which is then used to determine an appropriate step
size which guarantees its convergence. Also, to improve
the rate of convergence of the proposed algorithm, we
propose a data scaling step and employ a backtracking
line search, which increases the convergence rate signif-
icantly, and more importantly, outperforms the existing
AO approach in terms of convergence rate.
•As a tool to estimate the applicability of an RIS, we
introduce the concept of the total free space path loss
(total FSPL). Since the computation of the total FSPL
of the indirect link is an intractable problem in a MIMO
system, we instead derive the total FSPLs for a single-
input single-output (SISO) system. We then show that the
ratio of the total FSPL of the indirect and direct links
can be used as an accurate first-order measure of the
applicability of an RIS.
•We show through simulations that the proposed PGM
provides the same achievable rate as the AO, but with
a significantly lower number of iterations. This is partic-
ularly visible in the case where the direct link is blocked,
as in this case the PGM needs just a few iterations to
reach the convergent achievable rate. As a side product,
we demonstrate that the total FSPL of the indirect link
is primarily determined by the RIS position, while the
total length of the indirect link is of relatively minor
importance. Also, we show that scaling the number of
RIS elements with the operational frequency can compen-
3
sate the FSPL increase and ensure communication via the
indirect link at all frequencies. Furthermore, we study the
application of an RIS in an indoor environment and show
that a small number of RIS elements is sufficient to enable
the indirect link to have a higher achievable rate than the
direct link. Last but not least, we demonstrate that the
proposed PGM has a significantly lower computational
complexity compared to the AO method.
The rest of this paper is organized as follows. In Section II,
we introduce the system model and formulate the optimization
problem to maximize the achievable rate of a MIMO system
equipped with an RIS. In Section III, we propose and de-
rive the PGM algorithm to solve the previous optimization
problem. The convergence and the complexity analysis of the
proposed optimization algorithm are presented in Section IV.
The applicability of an RIS in the considered communication
system is discussed in Section V. In Section VI, we illustrate
simulation results of the achievable rate for the proposed PGM
algorithm, and use these to illustrate its advantages. Finally,
Section VII concludes this paper.
Notation: Bold lower and upper case letters represent
vectors and matrices, respectively. Ca×bdenotes the space
of complex matrices of dimensions a×b.(·)T,(·)∗and
(·)Hrepresent transpose, complex conjugate and Hermitian
transpose, respectively. ln(x)denotes the natural logarithm of
x.λmax(X)denotes the largest singular value of matrix X. To
simplify the notation we denote by || · || the Euclidean norm
if the argument is a vector and the Frobenius norm if the
argument is a matrix. diag (x)denotes the square diagonal
matrix which has the elements of xon the main diagonal.
|x|is the absolute value of xand (x)+denotes max(0, x).
arg{x}denotes the argument of x. The l-th entry of vector x
is denoted by xl.Tr(X)is the trace of matrix X, and E{·}
stands for the expectation operator. det(X)is the determinant
of X. The notation A()Bmeans that A−Bis positive
semidefinite (definite). ∇Xf(·)is the gradient of fwith
respect to X∗∈Cm×n, which also lies in Cm×n.vecd(X)
denotes the vector comprised of the diagonal elements of X.
vec(X)denotes the vectorization operator which stacks the
columns of Xto create a single long column vector. A(i, k)
denotes the k-th element of the i-th row of matrix A.
II. SYSTEM MODEL AND PROB LE M FORMULATION
A. System Model
We consider a wireless communication system with Nt
transmit and Nrreceive antennas, whose aerial view is de-
picted in Fig. 1. Both the transmit and receive antennas are
placed in uniform linear arrays (ULAs) on vertical walls that
are parallel to each other. The distance between these walls is
denoted by D. For simplicity, both antenna arrays are parallel
to the ground and are assumed to be at the same height. The
inter-antenna separations of these arrays are denoted by stand
sr, respectively. The direct link is attenuated by an obstacle
(e.g., a building) which is situated between the two antenna
arrays, and for this reason, a rectangular RIS of size a×b
is utilized to improve the system performance. The RIS is
installed on a vertical wall that is perpendicular to the antenna
Nt
.
.
.
1
Nr
.
.
.
1
D
RIS
· · ·
TX
RX
HDIR
H1
H2
dris
Obstacle
lt
lr
Fig. 1. Aerial view of the considered communication system.
arrays and its center is at the same height as the transmit and
the receive antenna arrays1. It consists of reflection elements
placed in an uniform rectangular array (URA) with Naand
Nbelements per dimension respectively (the total number of
reflection elements of the RIS then being Nris =NaNb).
All RIS elements are of size λ
2×λ
2, where λdenotes the
wavelength of operation. The separation between the centers
of adjacent RIS elements in both dimensions is sris =λ
2.
The distance between the midpoint of the RIS and the plane
containing the transmit antenna array is dris. The distance
between the midpoint of the transmit antenna array and the
plane containing the RIS is lt, and the distance between the
midpoint of the receive antenna array and the plane containing
the RIS is lr. We assume that the RIS elements are ideal and
that each of them can independently influence the phase and
the reflection angle of the impinging wave.
The signal vector at the receive antenna array is given by
y=Hx +n,(1)
where H∈CNr×Ntis the channel matrix, x∈CNt×1is
the transmit signal vector and n∈CNr×1is the noise vector
which is distributed according to CN(0, N0I). We assume that
the total average transmit power has a maximum value of Pt,
i.e., E{xHx} ≤ Pt. Let Q0be the covariance matrix of
the transmitted signal, i.e., Q=E{xxH}, then the transmit
power constraint can be equivalently written as
Tr(Q)≤Pt.(2)
B. Channel Model
Since an RIS is present in this system, the channel matrix
can be expressed as
H=HDIR +HINDIR,
where HDIR ∈CNr×Ntrepresents the direct link between the
transmitter and the receiver, and HINDIR ∈CNr×Ntrepresents
the indirect link between the transmitter and the receiver (i.e.,
via the RIS). Adopting the Rician fading channel model, the
direct link channel matrix is given by
HDIR =qβ−1
DIR
√K+ 1(√KHD,LOS +HD,NLOS),(3)
1For ease of exposition, we assume that the RIS, the transmit and the receive
antenna arrays are at the same height. It can be shown by simulations that
introducing different heights has a negligible influence on the achievable rate.
4
where HD,LOS(r, t) = e−j2πdr,t /λ and dr,t is the distance
between the t-th transmit and the r-th receive antenna. The
elements of HD,NLOS are independent and identically dis-
tributed (i.i.d.) according to CN(0,1). The FSPL for the
direct link is given by βDIR = (4π/λ)2dαDIR
0[25], where
d0=pD2+ (lt−lr)2is the distance between the transmit
array midpoint and the receive array midpoint. The path loss
exponent of the direct link, whose value is influenced by the
obstacle present, is denoted by αDIR. The Rician factor Kis
chosen from the interval [0,+∞).
We assume that the far-field model is valid for signal
transmission via the RIS (i.e., for the indirect link), and thus
HINDIR can be written as
HINDIR =qβ−1
INDIRH2F(θ)H1,(4)
where H1∈CNris×Ntrepresents the channel between the
transmitter and the RIS, H2∈CNr×Nris represents the
channel between the RIS and the receiver, and β−1
INDIR repre-
sents the overall FSPL for the indirect link. Signal reflection
from the RIS is modeled by the matrix F(θ) = diag(θ)∈
CNris×Nris , where θ= [θ1, θ2, . . . , θNris ]T∈CNris ×1. In this
paper, similar to related works [9], [24], we assume that the
signal reflection from any RIS element is ideal, i.e., without
any power loss. In other words, we may write θl=ejφlfor
l= 1,2, . . . , Nris, where φlis the phase shift induced by the
l-th RIS element. Equivalently, we may write
|θl|= 1, l = 1,2, . . . , Nris.(5)
Utilizing the Rician fading channel model, the channel
between the transmitter and the RIS H1is given by
H1=1
√K+ 1(√KH1,LOS +H1,NLOS),(6)
where H1,LOS(l, t) = e−j2π dl,t/λ and dl,t is the distance
between the t-th transmit antenna and the l-th RIS element.
The elements of H1,NLOS are i.i.d. according to CN(0,1). It
is worth noting that the channel matrix expression (6) does
not contain any FSPL term.
In a similar way, H2can be expressed as
H2=r1
K+ 1(√KH2,LOS +H2,NLOS)(7)
where H2,LOS(r, l) = e−j2πdr,l /λ and dr,l is the distance
between the l-th RIS element and the r-th receive antenna.
The FSPL for the indirect link can be computed according to
[8], [26], [27, Eqn. (18.13.6)] as
β−1
INDIR =λ4
256π2
(cos γ1+ cos γ2)2
d2
1d2
2
,(8)
where d1=pd2
ris +l2
tis the distance between the transmit ar-
ray midpoint and the RIS center, and d2=p(D−dris)2+l2
r
is the distance between the RIS center and the receive array
midpoint. Also, γ1is the angle between the incident wave
direction from the transmit array midpoint to the RIS center
and the vector normal to the RIS, and γ2is the angle between
the vector normal to the RIS and the reflected wave direction
from the RIS center to the receive array midpoint. Therefore,
we have cos γ1=lt/d1and cos γ2=lr/d2, which finally
gives
β−1
INDIR =λ4
256π2
(lt/d1+lr/d2)2
d2
1d2
2
.(9)
C. Problem Formulation
In this paper, we are interested in maximizing the achievable
rate2of the considered RIS-assisted wireless communication
system. It is well known that for a MIMO channel, Gaussian
signaling provides the maximum achievable rate, and that
for a given input covariance matrix Q, when His known
perfectly at both transmitter and receiver, the following rate is
achievable:
R= log2detI+1
N0
HQHH(bit/s/Hz).(10)
We note that the channel matrix Halso depends on θ. Thus,
for the total power Pt, the problem of the achievable rate
optimization for the considered system can be mathematically
stated as:
maximize
θ,Qf(θ,Q) = ln detI+Z(θ)QZH(θ)(11a)
subject to Tr(Q)≤Pt;Q0;(11b)
θl= 1, l = 1,2, .. . , Nris.(11c)
where
Z(θ) = ¯
HDIR +H2F(θ)¯
H1(12)
¯
HDIR =HDIR/pN0(13)
¯
H1=H1qβ−1
INDIR/N0.(14)
III. SOLUTION APPROACH VIA PROJ EC TE D GRADIENT
METHOD
In contrast to the conventional MIMO channel where the
water-filling algorithm can be used to efficiently find the
maximum achievable rate, problem (11) is nonconvex and thus
difficult to solve. Further, we note that the objective is neither
convex nor concave in the involved variables.
Previously proposed methods for rate optimization in RIS
communication systems were primarily based on the alternat-
ing optimization (AO) technique [9], [24]. The main idea of
this method is that the RIS phase shifts and the covariance
matrix are optimized in an alternating fashion, each indepen-
dently of the other. This method is motivated by the fact that
the optimization over one variable can be performed efficiently
(i.e., in closed form) while others are kept fixed. Although
the AO method is easy to implement, it may require many
iterations to converge, especially when the number of RIS
elements is very large (which corresponds to the case in which
the RIS is the most useful). In other words, the simplicity of
an iteration in the AO method does not necessarily translate
into low actual run time.
2Note that this achievable rate does not correspond to the channel capacity,
as we do not consider the possibility of encoding the transmitted data into
the phase shift values of the RIS. If such encoding is performed, the capacity
of the RIS-aided MIMO system may be achieved [28].
5
Motivated by the above discussion, we propose an opti-
mization method to solve (11), based on the PGM presented
in [29]. Our proposed method is motivated by the fact that the
projection onto the feasible set (albeit nonconvex with respect
to θ) can be performed efficiently.
A. Description of Proposed Algorithm
To describe the proposed algorithm, we define the following
two sets:
Θ = {θ∈CNris×1:θl= 1, l = 1,2, . . . , Nris}(15)
Q={Q∈CNt×Nt: Tr(Q)≤Pt;Q0}(16)
It is clear that the feasible set of (11) is the Cartesian product
of Θand Q. We denote by PU(u)the Euclidean projection
from a point uonto a set U, i.e., PU(u) = arg min
x{||x−u|| :
x∈ U}.
The proposed algorithm is outlined in Algorithm 1 and
follows the projected gradient method in order to solve (11).
The main idea behind Algorithm 1 is as follows. Starting
from an arbitrary point (θ0,Q0), we move in each iteration
in the direction of the gradient of f(θ,Q). The size of this
move is determined by the step size µ > 0(see Section
IV for details regarding the choice of an appropriate step
size). As a result of this step, the resulting updated point
may lie outside of the feasible set. Therefore, before the next
iteration, we project the newly computed points θand Qonto
Θand Q, respectively. As shall be seen shortly, the projection
onto Θor Qcan be determined in closed form. Another
important remark concerning Algorithm 1 is also in order.
Since (11) involves complex variables, we adopt the complex-
valued gradient defined in [30, Eq. (4.37)]. In particular, it
is proved that the directions where f(θ,Q)has maximum
rate of change with respect to θand Qare ∇θf(θ,Q)and
∇Qf(θ,Q), respectively [30, Theorem 3.4].
In our method, all optimization variables are updated simul-
taneously in each iteration. This is in sharp contrast to the AO
method, in which each iteration only updates a single variable.
As a result, the proposed method converges much faster than
the AO method as we demonstrate via extensive numerical
results in Section VI.
B. Complex-valued Gradient of f(θ,Q)
Let K(θ,Q)=(I+Z(θ)QZH(θ))−1. Then we have the
following result.
Lemma 1. The gradient of f(θ,Q)with respect to θ∗and
Q∗is given by
∇θf(θ,Q) = vecdHH
2K(θ,Q)Z(θ)Q¯
HH
1(17a)
∇Qf(θ,Q) = ZH(θ)K(θ,Q)Z(θ).(17b)
Proof: See Appendix A.
C. Projection onto Θand Q
We now show that the projection operations in Algorithm 1
can be carried out very efficiently, and thus Algorithm 1 indeed
Algorithm 1 Proposed projected gradient method (PGM).
1: Input :θ0,Q0, µ > 0.
2: for n= 1,2, . . . do
3: θn+1 =PΘ(θn+µ∇θf(θn,Qn))
4: Qn+1 =PQ(Qn+µ∇Qf(θn,Qn))
5: end for
requires low complexity to implement. Note that the constraint
θl= 1 means that θlshould lie on the unit circle in the
complex plane. Thus, it is straightforward to see that, for a
given point u∈CNris×1,PΘ(u)is the vector ¯
uwhere
¯ul=(ul
|ul|ul6= 0
ejφ , φ ∈[0,2π]ul= 0 , l = 1, . . . , Nris.(18)
Note that ¯ulcan be any point on the unit circle if ul= 0, and
thus the projection onto Θis not unique. Despite this issue,
we are still able to prove the convergence of Algorithm 1,
which is shown in the next section. Next we turn our attention
to the projection onto Q, which general problem has already
been studied previously (e.g., in [31]). For a given Y0,
the projection of Yonto Qis the solution of the following
problem:
minimize
QkQ−Yk2(19a)
subject to Tr(Q)≤Pt;Q0(19b)
Let Y=UΣUHbe the eigenvalue decomposition of Y,
where Σ= diag(σ1, . . . , σNt). Now, we can write Q=
UDUHfor some D0and Tr(Q) = Tr(D). Then, we
obtain kQ−Yk2=kΣ−Dk2. Thus, Dmust be diagonal
to be an optimal solution, i.e., D= diag(d1, . . . , dNt).
Therefore, (19) is equivalent to the following program:
minimize
{di}XNt
i=1(di−σi)2(20a)
subject to XNt
i=1 di≤Pt;di≥0(20b)
The solution to the above problem is achieved by the water-
filling algorithm and is given as
di= (σi−γ)+, i = 1, . . . , Nt,(21)
where γ≥0is the water level.
D. Improved Convergence Rate by Data Scaling
For any first-order method, exploiting the structure of the
optimization problem is key to speed up its convergence. In
this regard, we remark that the effective channel via the RIS is
qβ−1
INDIRH2F(θ)¯
H1which can be some orders of magnitude
weaker or stronger than the direct link ¯
HDIR. This unbalanced
data in (11) makes Algorithm 1 converge slowly.
To increase the convergence speed of Algorithm 1, we
propose a change of variable as follows:
¯
Q=k2Q(22a)
¯
θ=θ/k (22b)
¯
HDIR =HDIR/(kpN0)(22c)
6
for some k > 0. Accordingly, the equivalent optimization
problem with respect to the new variables ¯
θand ¯
Qreads
maximize
¯
θ,¯
Q
f(¯
θ,¯
Q) = ln det I+Z(¯
θ)¯
QZH(¯
θ)(23a)
subject to Tr( ¯
Q)≤¯
Pt;¯
Q0(23b)
¯
θl=1
k, l = 1,2, . . . , Nris (23c)
where ¯
Pt=k2Pt. The above change of variable step is
equivalent to scaling the gradient of the original objective,
which can improve the convergence rate. Now we solve (23)
following the iterative procedure in Algorithm 1, but instead
of Qand θwe use their scaled versions ¯
Qand ¯
θwhich
are defined by the expressions (22a) and (22b), respectively.
Accordingly, the sets that contain all valid ¯
Qand ¯
θare denoted
as ¯
Θand ¯
Q, respectively. The projections of computed ¯
Q
and ¯
θonto ¯
Θand ¯
Qare performed in the same way as the
projections in the previous subsections, and the constraints (2)
and (5) are replaced by (23b) and (23c), respectively. Also, it
should be pointed out that ¯
HDIR is scaled (i.e., divided by k)
in (22c) compared to (13). An appropriate value for kshould
reflect the difference between the direct and indirect links.
When the direct link is absent, kshould take into account the
difference between the feasible sets of Qand θ. From our
extensive numerical experiments, an appropriate value for k,
depending on the presence or absence of the direct link, is
given by3
k=
10 max{1,1
√Pt}r1
β−1/2
INDIR
kHDIRk
kH2H1k,HDIR 6=0
10,HDIR =0.
(24)
Based on the previous expressions, we can see that the PGM
requires only the knowledge of the cascaded channel, and
not the individual channels H1and H2, for the indirect link.
Estimation of this channel can be performed at the receiver or
the transmitter in TDD mode [32], [33].
IV. CONVERGENCE AND COMPLEXITY ANALYSIS
A. Convergence Analysis
In this subsection we prove the convergence of Algorithm 1
for solving (23), following the framework in [29]. To achieve
this, we first show that f(¯
θ,¯
Q)has a Lipschitz continuous
gradient with a Lipschitz constant L, and then assert that
Algorithm 1 is convergent if the step size satisfies µ≤1
L.
For the first part of the proof, recall that a function f(x)is
said to be L-Lipschitz continuous (also known as L-smooth)
over a set Xif for all x,y∈ X we have
||∇f(x)− ∇f(y)|| ≤ L||x−y||.(26)
In our present context, the inequality (26) corresponds to (25),
to prove which we may make use of the following lemma.
3Since the gradients with respect to Qand θare of different sizes, using
two different step sizes for each of them can actually increase the convergence
rate of PGM. In order to preserve the single step size, we introduce the scaling
factor kwhich provides the same effect as if two independent step sizes were
used. The value of the scaling factor kis obtained in a heuristic manner, by
performing numerical experiments.
Lemma 2. The following inequalities hold for ∇¯
θf(¯
θ,¯
Q)and
∇¯
Qf(¯
θ,¯
Q)
||∇¯
θf(¯
θ1,¯
Q1)− ∇¯
θf(¯
θ2,¯
Q2)||
≤(ab +ab3¯
Pt)||¯
Q1−¯
Q2||+ (a2¯
Pt+ 2a2b2¯
P2
t)||¯
θ1−¯
θ2||
(27)
||∇¯
Qf(¯
θ1,¯
Q1)− ∇¯
Qf(¯
θ2,¯
Q2)||
≤b4||¯
Q1−¯
Q2|| + (2ab + 2ab3¯
Pt)||¯
θ1−¯
θ2|| (28)
where
a=λmax(¯
H1)λmax(H2)(29)
b=λmax(¯
HDIR) + k−1λmax (¯
H1)λmax(H2).(30)
Proof: See Appendix B.
With the aid of the above lemma, we assert the smoothness
of f(¯
θ,¯
Q)in the next theorem.
Theorem 1. The objective f(¯
θ,¯
Q)is L-smooth with a
constant Lgiven by
L=qmax(L2
¯
θ, L2
¯
Q),(31)
where
L2
¯
θ=(2ab5+ 4a2b2)+(a3b+ 2ab7+ 8a2b4)¯
Pt
+ (3a3b3+a4+ 4a2b6)¯
P2
t+ (2a3b5+ 4a4b2)¯
P3
t
+ 4a4b4¯
P4
t(32)
L2
¯
Q=(a2b2+b8+ 2ab5) + (2a2b4+a3b+ 2ab7)¯
Pt
+ (a2b6+ 3a3b3)¯
P2
t+ 2a3b5¯
P3
t(33)
Proof: Theorem 1 follows immediately from Lemma 2
and the inequality
2||¯
Q1−¯
Q2|| × ||¯
θ1−¯
θ2|| ≤ ||¯
Q1−¯
Q2||2+||¯
θ1−¯
θ2||2.
Specifically, we have
||∇¯
θf(¯
θ1,¯
Q1)− ∇¯
θf(¯
θ2,¯
Q2)||2
+||∇¯
Qf(¯
θ1,¯
Q1)− ∇¯
Qf(¯
θ2,¯
Q2)||2
≤L2
¯
Q||¯
Q1−¯
Q2||2+L2
¯
θ||¯
θ1−¯
θ2||2
≤max(L2
¯
θ, L2
¯
Q)||¯
Q1−¯
Q2||2+||¯
θ1−¯
θ2||2.
Taking the square root of both sides of the above inequality,
we can see that qmax(L2
¯
θ, L2
¯
Q)is a Lipschitz constant of the
gradient of f(¯
θ,¯
Q); this completes the proof.
The convergence of Algorithm 1 is stated in the following
theorem.
Theorem 2. Assume the step size satisfies µ < 1
L, where
Lis given in (31). Then the iterates (¯
θn,¯
Qn)generated by
Algorithm 1 are bounded. Let (θ∗,Q∗)be any accumulation
point of the set {(¯
θn,¯
Qn)}, then, (θ∗,Q∗)is a critical point
of (11).
Proof: See Appendix C.
Before proceeding further, a subtle point regarding the
convergence of Algorithm 1 is worth mentioning. Specifically,
the proposed method is provably convergent to a critical point
7
∇¯
θf(¯
θ1,¯
Q1)− ∇¯
θf(¯
θ2,¯
Q2)
2+
∇¯
Qf(¯
θ1,¯
Q1)− ∇¯
Qf(¯
θ2,¯
Q2)
21/2
≤L
¯
Q1−¯
Q2
2+
¯
θ1−¯
θ2
21/2(25)
of the considered problem, also known as a stationary solution,
which satisfies the necessary optimality conditions for (11).
However, since (11) is nonconvex, these optimality conditions
may not be sufficient in general, and thus the solution obtained
from Algorithm 1 may not be globally optimal. However, it
is often the case that with a good initialization, a stationary
solution is good enough for practical applications. We note that
the same comments also apply to the AO method proposed in
[24].
B. Complexity Analysis
In this subsection, we analyze the computational complexity
of Algorithm 1 (i.e., the PGM).4To simplify the analysis
while still providing a good approximation to the complexity
of Algorithm 1, we concentrate on the number of complex
multiplications required per iteration. To this end, we first
recall some fundamental results. Specifically, the multiplica-
tion of A∈Cm×nand B∈Cn×pneeds mnp complex
multiplications when Aand Bare dense matrices.5This
complexity reduces to mn for the case of a square diagonal
matrix B∈Cn×n. Calculating vecd(ABC),A∈Cm×n,
B∈Cn×pand C∈Cp×m, needs mp(n+ 1) complex
multiplications, which is justified as follows. Multiplying a
row of Awith Brequires np complex multiplications and
multiplying the resulting row vector with the corresponding
column of Crequires a further pcomplex multiplications.
It is obvious that the complexity of the proposed method
is determined by Steps 3 and 4 in Algorithm 1. The compu-
tation of Z(θ)is dominated by that of the term H2F(θ)¯
H1,
which requires NrNris +NrNtNris complex multiplications.
To compute ∇θf(θ,Q), we also need to compute the term
A=K(θ,Q)Z(θ)∈CNr×Nt. Instead of directly computing
K(θ,Q)=(I+Z(θ)QZH(θ))−1using matrix inversion and
then multiplying K(θ,Q)with Z(θ), we note that Ais in
fact the solution to the linear system I+Z(θ)QZH(θ)X=
Z(θ). To form Z(θ)QZH(θ), we first need NrN2
tmultiplica-
tions to achieve Z(θ)Qand then (N2
r+Nr)Nt/2to multiply
Z(θ)Qwith ZH(θ). Solving the linear system using Cholesky
decomposition, by solving two triangular systems using for-
ward and backward substitution, requires a complexity which
is O(N3
r+N2
rNt). In summary, the computation of Atakes
O(N2
tNr+3
2NtN2
r+N3
r)multiplications. Next, the compu-
tation of AQ requires NrN2
tcomplex multiplications. To cal-
culate vecd(HH
2AQ ¯
HH
1), we need NrisNt(Nr+ 1) complex
multiplications. As Ais also common to (17b), the complexity
of computing ∇Qf(θ,Q)is only NrN2
t. In summary, the
computational complexity of ∇θf(θ,Q)and ∇Qf(θ,Q)is
4Although the PGM is actually implemented in the scaled-variable form
described in Subsection III-D, this scaling does not affect the complexity of
the PGM which is equal to the complexity of Algorithm 1. Therefore, we
analyze the complexity of Algorithm 1 in the sequel.
5Special algorithms can reduce the complexity further, but this is not our
focus in this paper.
TABLE I
COMPARISON OF THE COMPUTATIONAL COMPLEXITY REQUIRED BY THE
PRO POS ED PGM M ETH OD A ND TH E AO METHO D TO RE ACH 9 5 % OF TH E
AVERA GE ACHIE VABLE R ATE AT THE 500T H ITERATI ON .
Direct link Nris IPGM CPGM,IT CPGM IOI CAO
Present
100 19 9436 179284 1 394304
225 6 19311 115866 1 862304
400 4 33136 132544 1 1517504
625 3 50911 152733 1 2359904
Blocked
100 2 9436 18872 1 394304
225 2 19311 38622 1 862304
400 2 33136 66272 1 1517504
625 2 50911 101822 1 2359904
O2NrisNtNr+ 2N2
tNr+3
2NtN2
r+N3
r+NrNris +NtNris.
When Nris is much larger than Ntand Nr, then the complexity
can be approximated by ONrisNtNr).
Next, multiplying µwith ∇θf(θ,Q)and then projecting
the result onto Θrequires 3Nris complex multiplications.
Similarly, we need N2
t/2operations to multiply µwith
∇Qf(θn,Qn). The projection of Qn+µ∇Qf(θn,Qn)onto
Qrequires: O(N3
t)operations for the eigenvalue decompo-
sition, O(N2
t)operations for the water-filling algorithm in
(21) and N2
t+ (N2
t+Nt)Nt/2operations for the matrix
multiplication Q=UDUH. Therefore, the complexity for
the update and projection operations in Step 4 is given by
O(3
2N3
t). Thus, the per-iteration complexity of Algorithm 1
is finally determined as
CPGM,IT =O(2NrisNtNr+ 2N2
tNr+3
2NtN2
r+N3
r
+NrNris +NtNris + 3Nris +3
2N3
t),(34)
while the total complexity CPGM also depends from the num-
ber of required iterations IPGM. The computational complexity
of the proposed PGM and the AO method from [24] are
presented in Table I. The complexity of the AO is expressed
with respect to the number of outer iterations IOI, where one
outer iteration is actually a sequence of Nris + 1 conventional
iterations. Further details and discussion on this complexity
comparison will be presented in Subsection VI-D.
C. Improved Convergence by Backtracking Line Search
It often occurs that the Lipschitz constant given in (31)
is much larger than the best Lipschitz constant for the gra-
dient of the objective. The corresponding step size required
(according to Theorem 2) to guarantee the convergence is
then very small, and adopting this step size can lead to a
very slow convergence. To speed up the convergence of the
proposed PGM, we can employ a backtracking line search
to find a possibly larger step size at each iteration. In the
8
following, we present a line search procedure, based on the
Armijo–Goldstein condition [34], that is numerically shown to
be efficient for our considered problem.
Let L0>0,δ > 0be a small constant, and ρ∈(0,1). In
Steps 3 and 4 of Algorithm 1, we replace the step size µby
Loρknand obtain (35a) and (35b), where knis the smallest
nonnegative integer that satisfies (35c).
θn+1 =PΘ(θn+Loρkn∇θf(θn,Qn)) (35a)
Qn+1 =PQ(Qn+Loρkn∇Qf(θn,Qn)) (35b)
f(θn+1,Qn+1 )≥f(θn,Qn)+
δ||θn+1 −θn||2+||Qn+1 −Qn||2.(35c)
The above backtracking line search can be found through
an iterative procedure, which is guaranteed to terminate after
a finite number of iterations since f(¯
θ,¯
Q)is L-smooth. It
is easy to see that the convergence of Algorithm 1 (i.e.,
Theorem 2) still holds when this procedure is used to find
the step size. We remark that the line search described above
results in increased per-iteration complexity. Suppose that the
line search stops after ILS steps, the additional complexity is
OILS(3Nris + 2N3
t). However, this computational cost turns
out to be immaterial, since the line search can significantly
reduce the required number of iterations and hence the actual
overall run time.
V. TOTAL FPSL RATIO -AMETRIC OF RIS
APPLICABILITY
It can be very useful to have a first-order estimate of the
benefit (if any) provided to a wireless communication system
by adding an RIS. We can achieve this by considering the total
FSPL of the indirect and direct links. Note that when speaking
about the “total FSPL” of the indirect link, we require (in
contrast to (8)) a definition which takes into account also the
RIS phase shift values.
The computation of the total FSPL of the indirect link is an
intractable problem in a MIMO system, since the optimal RIS
element phase shifts are a priori unknown and can only be
obtained by implementing an iterative optimization method.
This problem was approximately tackled only for the single-
stream scenario in [33], and the obtained results were then
used in [35] to quantify the performance of RISs in the far
field regime (which is also the case considered in this paper),
but only for Rayleigh and deterministic LOS channels. To
overcome this issue, we consider the total FSPL of the indirect
link in a SISO system, which is given by
β−1
INDIR,T=β−1
INDIREn|h2F(θ)h1|2o,(36)
where h1models the channel between the transmit antenna
and the RIS, and h2models the channel between the RIS
and the receive antenna. The optimal RIS element phase shift
values in (36) satisfy φi=−arg {h2(i)h1(i)}and as a result
we have h2F(θ)h1=PNris
i=1 |h2(i)h1(i)|. As all of the terms
|h2(i)h1(i)|follow the same distribution, we may write
E{
Nris
X
i=1 |h2(i)h1(i)|} =NrisE{|h2(1)h1(1)|}.(37)
From Jensen’s inequality we obtain
En|h2F(θ)h1|2o=E
Nris
X
i=1 |h2(i)h1(i)|
2
≥
E(Nris
X
i=1 |h2(i)h1(i)|)!2
=N2
ris (E{|h2(1)h1(1)|})2.
(38)
Substituting (38) into (36), we finally obtain
β−1
INDIR,T≥β−1
INDIRN2
ris (E{|h2(1)h1(1)|})2,(39)
which constitutes an upper-bound on the total FSPL. The total
FSPL of the direct link in a SISO system is given by βDIR,T=
βDIR, since the direct link does not alter the average signal
power. Finally, the ratio between the total FSPL of the indirect
and direct links can be expressed as
T=βINDIR,T
βDIR,T
=16
λ2
(d1d2)2
dαDIR
0
1
(lt/d1+lr/d2)2N2
risE,(40)
where E= (E{|h2(1)h1(1)|})2. The obtained Tserves as
a first-order measure of the applicability of an RIS for a
given communication scenario6. For T > 1, the direct link
is expected to be always stronger than the indirect link, even
if the RIS phase shifts are optimally adjusted. Consequently,
the RIS is capable of achieving limited performance gains
with respect to the case when only the direct link is utilized
for communication. For T < 1, the indirect link with the
optimized RIS phase shifts is stronger than the direct link and
the gains of using the RIS are usually more substantial.
VI. SIMULATION RESU LTS
In this section, we evaluate the achievable rate of the
proposed optimization algorithm with the aid of Monte Carlo
simulations. First, the study is conducted for a typical outdoor
propagation environment in two different scenarios: with the
direct link present and with the direct link blocked. For the
case study where the direct link is present, we utilize three
benchmark schemes. The first benchmark scheme is based on
the implementation of the AO method from [24]. The second
and third benchmark schemes are based on the use of the PGM
in the case where only the indirect link is active and where
only the direct link is active, respectively. In the case where the
direct link is blocked, we only consider the AO method as the
benchmark scheme. Additionally, we show the variation of the
achievable rate with the number of RIS elements. Furthermore,
we study the suitability of RIS-aided wireless communications
(with the proposed optimization method) for implementation
in indoor propagation environments. We also present a com-
parison of the proposed and benchmark schemes in terms
of computational complexity and run time. In addition, we
analyze the sensitivity and robustness of the proposed PGM.
Finally, we evaluate the influence of data scaling and the line
search procedure.
6Since (40) is derived for a SISO system, Tis realistically only a rough
measure of the applicability of an RIS in a MIMO system. However, as we
shall see in Section V, this metric is quite useful for MIMO scenarios.
9
In the following simulation setup, the parameters are f=
2 GHz (i.e., λ= 15 cm), st=sr=λ/2=7.5 cm,sris =
λ/2 = 7.5 cm,D= 500 m,Nt= 8,Nr= 4,αDIR = 3,
Nris = 225,K= 1,Pt= 0 dB and N0=−120 dB. The
RIS elements are placed in a 15 ×15 square formation so
that the area of the RIS is slightly larger than 1 m2. The line
search procedure for the proposed gradient algorithms utilizes
the parameters L0= 104,δ= 10−5and ρ= 1/2. Also, the
minimum allowed step size value is the largest step size value
lower than 10−4. Unless otherwise specified, we assume the
initial values θ= [1 1 ··· 1]Tand Q= (Pt/Nt)Ifor all
optimization algorithms. To maintain compatibility with [24],
we set the number of random initializations for the AO to
LAO = 100. All of the achievable rate results, except those
for very large Nris in Figs. 8 and 9, are averaged over 200
independent channel realizations.
A. Achievable Rate in Outdoor Environments
1) Direct link present: In this subsection, we present the
achievable rate simulation results when the considered com-
munication system is located in an outdoor environment. To
obtain a more complete picture, the positions of the transmitter
and the receiver, as well as the position of the RIS, are
varied in simulations. In general, we analyze two cases for
the transmitter and receiver positions: the transmitter and
receiver are at substantially different distances from the plane
containing the RIS (lt6=lr), and the transmitter and receiver
are at the same distance from the plane containing the RIS
(lt=lr). In each of these cases, the position of the RIS is
also varied.
In the first case, we assume lt= 20 m and lr= 100 m.
The variation of the FSPL ratio Tgiven by (40) and the total
indirect link length with the RIS position are shown in Fig. 2.
We observe that the highest FSPL ratio Tis obtained when
the RIS is placed close to the center, and the lowest Tis
obtained when the RIS is placed close to the transmitter or the
receiver. In other words, placing the RIS in the vicinity of the
transmitter or the receiver ensures the lowest signal attenuation
for the indirect communication link. It is interesting to note
from Fig. 2b that in contrast to signal propagation principles
for conventional communication systems, the total length of
the indirect link (d1+d2)does not determine the total FSPL of
that link, and that the relationship between these variables is
not monotonic. For example, the minimum and the maximum
of the FSPL ratio Tin Fig. 2a are obtained for almost the same
total length of the indirect link, as shown in Fig. 2b. Also, the
largest indirect link length in Fig. 2b does not coincide with
the highest FSPL of the indirect link in Fig. 2a. The reason for
this is that the FSPL of the indirect link is determined by the
product of the distances d1and d2rather than by their sum.
Therefore, finding the optimal position for the RIS is not a
straightforward task.
Based on the previous observations, we assume in further
simulations that the RIS is placed in the vicinity of the
transmitter (dris = 40 m) or in the vicinity of the receiver
(dris =D−40 m). The achievable rate results for the proposed
PGM approach and for the benchmark schemes are shown
0 50 100 150 200 250 300 350 400 450 500
0
1
2
3
4
dris
T
(a) The total FSPL ratio (T) versus dris.
0 50 100 150 200 250 300 350 400 450 500
520
540
560
dris
d1+d2
(b) The total indirect link length (d1+d2)versus dris.
Fig. 2. The total FSPL ratio and the indirect link length for lt= 20 m and
lr= 100 m.
in Fig. 3. It can be seen that the proposed gradient-based
optimization method converges relatively fast to the optimum
achievable rate value. On the other hand, the AO requires
significantly more iterations (at least one outer iteration, which
consists of a sequence of Nris + 1 conventional iterations
[24]) to reach its optimum value. It can be also observed that
the initial achievable rate for the AO is higher than for the
PGM. The reason for this is that for the AO, we select from
a large set of randomly generated RIS phase shift realizations
and optimized Qmatrices the ones that provide the highest
achievable rate and use these as a starting point for the AO
(thus, this specifies the initial achievable rate). In contrast to
this, the initial achievable rate for the PGM is obtained for the
aforementioned initial RIS phase shifts and Qmatrix.
In addition, the RIS is capable of providing a significant
enhancement of the achievable rate, which is proportional
to the achievable rate of the indirect link. As expected, this
gain is higher when the RIS is located in the vicinity of the
transmitter, due to the lower FSPL of the indirect link. Finally,
we observe that the FPSL ratio T, which is derived for a
SISO system, is not entirely trustworthy for predicting the
achievable rate in a MIMO system. Although the total FSPL
of the indirect link is lower than the total FSPL of the direct
link when the RIS is placed in the vicinity of the receiver
in Fig. 2a, the direct link will ultimately provide a higher
achievable rate in Fig. 3b.
10
100101102
0
2
4
6
8
10
(a) dris = 40 m.
100101102
0
2
4
6
8
10
(b) dris =D−40 m.
Fig. 3. Average achievable rate of the PGM versus the benchmark schemes.
Here lt= 20 m and lr= 100 m.
0 50 100 150 200 250 300 350 400 450 500
0
1
2
3
4
5
dris
T
Fig. 4. The total FSPL ratio (T) for the case lt=lr= 50 m.
In the second case, we assume lt= 50 m and lr= 50 m.
The variation of the total FSPL ratio Twith the RIS position
is shown in Fig. 4. It can be seen that Tis perfectly symmetric
due to the equal values of the distances ltand lr. The same
is true for the total indirect link length, which is not shown
for brevity reasons. The achievable rate of the proposed PGM
approach versus the benchmark schemes is shown in Fig. 5. As
expected, the PGM has a much higher convergence rate than
the AO. Also, it can be seen that the achievable rate is slightly
higher when the RIS is located in the vicinity of the receiver
than in the vicinity of the transmitter. If the communication
link are used individually, the indirect link has a slightly higher
achievable rate than the direct link.
2) Direct link blocked: If the direct link between the
transmitter and the receiver is blocked, the only means of
100101102
0
2
4
6
8
10
(a) dris = 40 m.
100101102
0
2
4
6
8
10
(b) dris =D−40 m.
Fig. 5. Average achievable rate of the PGM versus the benchmark schemes.
Here lt=lr= 50 m.
signal transmission is via the RIS. It can be easily seen that the
main observations made concerning the optimal RIS position
in the previous subsection are also applicable here. Therefore,
we analyze the achievable rate when the RIS is placed in the
vicinity of the transmitter or in the vicinity of the receiver.
The achievable rate results of the PGM versus the AO are
shown in Fig. 6. In both cases, the optimal achievable7rates
match the achievable rates of the second benchmark scheme
in Fig. 2. The PGM requires only a few iterations to converge
to the optimum value. On the other hand, the AO needs
approximately Nris + 1 iterations (i.e., one outer iteration)
to reach the optimum value. Interestingly, the achievable rate
enhancement during the first outer iteration is higher when the
direct link is blocked. It seems that the absence of the direct
link may have a significant influence on choosing the initial
covariance matrix and RIS phase shifts, before starting the
AO.
B. Scaling with Nris
This subsection consists of three parts. First, we demonstrate
the correctness of the expression (38) in Section V. Then, we
show how increasing the number of RIS elements influences
the achievable rate of the considered system. Finally, we
present the trade-off between the operating frequency and the
number of RIS elements.
7In our simulations, we take the “optimal achievable rate” to be that which
is obtained at the final (i.e., 500th) iteration.
11
100101102
2
4
6
(a) dris = 40 m.
100101102
0
2
4
6
(b) dris =D−40 m.
Fig. 6. Average achievable rate of the PGM versus the AO. The parameter
setup is the same as in Fig. 2.
0 50 100 150 200 250 300 350 400
0
0.5
1
·105
Nris
Fig. 7. Comparison of the left-hand side and right-hand side of (38).
To verify the correctness of the upper-bound expression in
(38), we compare the values of the left and right hand sides of
this expression in Fig. 7. As the expression pertains to single-
antenna systems, we assume Nt=Nr= 1 in this simulation.
For completeness, the presented results are computed and
averaged for different positions of the RIS. The graph shows a
very good match between the two sides of the aforementioned
expression, which means that in practice the total FSPL of the
indirect link in a SISO system is very well approximated by
(39).
In general, it is not easy to assess the expected achievable
rate for some arbitrary value of Nris, when gradient-based
optimization methods are applied. Therefore, to obtain a better
0 200 400 600 800 1,000 1,200 1,400 1,600
6
8
10
12
14
16
18
Nris
Fig. 8. Average achievable rate versus the number of RIS elements Nris.
0 1 2 3 4 5 6 7 8 9 10
0
5
10
15
20
f
Fig. 9. Average achievable rate versus frequency f. The parameter setup is
the same as for Fig. 2.
understanding of the variation of the achievable rate with
Nris, we present a numerical evaluation of the achievable rate
in Fig. 8. The parameter setup is the same as for Fig. 2
and dris = 40 m. In this case, the physical size of the RIS
is actually increasing, while the RIS is always operating in
the far field [8]. As a result, we observe that there is an
increase in the achievable rate when Nris is doubled8and this
increase becomes larger as we increase Nris. Also, the slope
of the achievable rate curve gradually reduces with Nris, as a
consequence of the logarithm function in the achievable rate
expression.
The number of RIS elements that can be placed on an
RIS having a constant physical size, without causing coupling
between the neighboring RIS elements, increases with the
frequency of operation. Therefore, the RIS may consist of
a large number of RIS elements, if it is intended to work
at high frequencies. Motivated by this fact, we analyze the
trade-off between the operating frequency and the number of
RIS elements Nris, and their influence on the achievable rate
in the considered system. We assume that the RIS elements
are placed in an RIS of size 1 m ×1 m and sris =λ/2at
all frequencies. The achievable rate of the PGM versus the
operating frequency fis shown in Fig. 9. In the frequency
8It should be noted that the number of RIS elements is not exactly doubled
in simulations, since we aim to have a square RIS. Therefore, the achievable
rate is computed and plotted for Nris equal to 49, 196 and 784 instead of 50,
200 and 800, respectively.
12
range up to 5 GHz, the achievable rate of the considered
system decreases primarily because of the FSPL increase of
the direct link. At higher frequencies, the achievable rate
remains almost constant regardless of whether the direct link is
present or blocked. In other words, the FSPL of the direct link
is so high in this case that the direct link becomes practically
useless for communicating information. On the other hand,
the indirect link has approximately the same achievable rate
across the entire frequency range. It is because the increase in
the FSPL of the indirect link is compensated by the increased
number of RIS elements in the considered system. Finally, we
conclude that the direct link is only useful at lower frequencies,
while the indirect link can be used at all frequencies if a
sufficient number of RIS elements is provided.
C. Achievable Rate in Indoor Environments
All of the previous simulation results are obtained for
a wireless communication system operating in an outdoor
environment. To further demonstrate the effectiveness of the
proposed gradient-based optimization method, we consider its
implementation in an indoor environment. Since the communi-
cation distances are now much smaller and the communication
bandwidths are usually larger (i.e., typically 20/22 MHz), the
following simulation parameters have the following altered
values: D= 30 m,dris = 5 m,lt= 3 m,lr= 7 m,
Nris = 100,Pt=−30 dB and N0=−100 dB.
The simulation results of the proposed optimization method
versus the benchmark schemes in an indoor environment are
presented in Fig. 10. The PGM again requires a lower number
of iterations than the AO to converge to the optimal achievable
rate. In contrast to the previous simulation results, the achiev-
able rate in indoor environments is almost entirely determined
by the indirect link signal transmission, which can explained
by the following argument. Reducing the distances in the
considered communication system results in the reduction of
the total FSPLs, and the total FSPL of the indirect link is
particularly affected by this, since it is inversely proportional
to the product of distances. Hence, a very small number of
RIS elements is sufficient to enable the indirect link to have a
lower total FSPL than the direct link, and any further increase
of the number of RIS elements will render the direct link
comparatively useless for indoor communications.
D. Computational Complexity and Run Time Results
In reality, it is not practical to the wait for an optimization
algorithm to reach a critical point, but rather some value
that is not too far from it. Hence in this subsection, we
consider the computational complexity required for the PGM
and the AO to reach an achievable rate that is equal to 95 %
of the average achievable rate at the 500th iteration. These
complexities are heavily influenced by the number of iterations
that are needed to achieve this target achievable rate. For the
PGM, the computational complexity per iteration is CPGM,IT
given by (34) and the number of iterations needed to achieve
the optimal achievable rate is denoted as IPGM. Their product
100101102
0
1
2
3
(a) Direct link present.
100101102
0
1
2
3
(b) Direct link blocked.
Fig. 10. Average achievable rate of the PGM versus the benchmark schemes
in an indoor environment.
determines the total computational complexity9CPGM of the
PGM. For the AO, the computational complexity is given by10
CAO and the number of outer iterations needed to achieve the
optimal achievable rate is IOI (see Appendix D). To maintain
compatibility with [24], the number of randomly generated
RIS phase shift realizations at the beginning of the AO is
taken to be LAO = 100.
The computational complexity of the PGM and of the AO
is shown in Table I. The parameter setup is the same as for
Figs. 3b and 6b. In general, the PGM is able to achieve a
significantly lower computational complexity than the AO,
while at the same time it requires a small number of iterations.
If the direct link is present, we observe that IPGM becomes
smaller with an increase of the number of RIS elements, or
in other words, the convergence of the PGM improves. As
a result, the computational complexity of the PGM does not
increase proportionally to the number of RIS elements. On
the other hand, IPGM remains constant when the direct link
is blocked and the computational complexity of the PGM
increases if the number of RIS elements is made larger. The
AO needs one outer iteration to reach the target achievable
9We neglect the number of multiplications needed to compute the achiev-
able rate after every iteration, due to the low number of iterations that PGM
needs to reach the target achievable rate.
10This complexity is derived under the assumption that the achievable rate
is computed at the end of each outer iteration, as was proposed in [24].
Therefore, we are not interested in the number of conventional iterations, but
in the number of outer iterations needed to achieve a certain rate.
13
0 50 100 150 200 250 300 350 400 450 500
4
6
8
10
Fig. 11. Average achievable rate of the PGM and the AO versus the run time.
The parameter setup is the same as for Fig. 3a.
rate, and its computational complexity increases in proportion
to the number of RIS elements.
To make this subsection complete, we also compare in
Fig. 11 the achievable rate of the AO and the PGM with respect
to the run time of the algorithm’s software implementation.
The achievable rate for both methods is computed at the end of
each iteration. It can be seen that the PGM needs an extremely
low run time to converge. Approximately the same time is
needed for the AO just to select the optimal initial point. Since
the first iteration of the AO is executed after the initial point is
chosen, the achievable rate curve for the AO in Fig. 11 starts
from about 10 ms. Even after 500 ms the AO is not entirely
capable of reaching the same achievable rate.
E. Sensitivity of PGM to Initialization
In this subsection, we study the sensitivity of the PGM to
the initial values of θand Q. Hence, we consider four cases,
where the initial value of θis either set to [1 1 ·· · 1]T
(referred to as “fixed θ”) or is randomly generated, and the
initial value of Qis either set to (Pt/Nt)I(referred to as “fixed
Q”) or is randomly generated. The achievable rate results for
the different initial values of θand Qare shown in Fig. 12.
The only visible difference between the considered cases is
in the first few iterations, where the PGM with fixed initial θ
and Qachieves a slightly higher achievable rate than in other
cases. In later iterations, the achievable rates in all four cases
are approximately equal. Hence, the PGM can always reach
the same achievable rate in approximately the same number
of iterations, independently of the initial values of θand Q.
F. Robustness to System Imperfections
In order to better understand the applicability of the pro-
posed PGM, it is necessary to consider the influence of
realistic imperfections in an RIS-aided communication system.
Motivated by this, the achievable rate for the case of discrete
RIS phase shifts and imperfect CSI is shown in Fig. 13.
The achievable rate for the RIS with discrete phase shifts is
obtained by discretizing the continuous RIS phase shifts in the
final iteration of the PGM and then calculating the achievable
rate. The proposed PGM is also directly applicable to discrete
phase shifts, since the projection of a given point onto the set
of discrete phase shifts is equivalent to finding the minimum
100101102
2
4
6
8
10
θQ
θQ
θQ
θQ
Fig. 12. Average achievable rate results for different initial values of θand
Q. The parameter setup is the same as for Fig. 3a.
100101102
4
6
8
10
Fig. 13. Average achievable rate for the case of discrete RIS phase shifts and
imperfect CSI. The parameter setup is the same as for Fig. 3a.
distance between the point and all possible phase shifts. It
can be seen that utilizing 1-bit and 2-bit discrete RIS phase
shifts can reduce the optimal achievable rate by approximately
1.1 bit/s/Hz and 0.2 bit/s/Hz, respectively. Hence, even a very
low resolution of discrete RIS phase shifts is sufficient to
ensure a limited reduction of the optimal achievable rate.
In the case of imperfect CSI, we assume that the estimated
channel matrix can be presented as a sum of the true channel
matrix and an estimation error matrix. The estimation error
matrix consists of i.i.d. elements that are distributed according
to CN(0, σ2), where σ2= 0.2. Also, it is assumed that the
channel matrix FSPLs are not affected by imperfect CSI. From
the results, which are plotted in Fig. 13, we can observe
that the optimal achievable rate decreases by approximately
1 bit/s/Hz, which is an acceptable level of reduction.
G. Influence of Data Scaling and Line Search
In this subsection, we analyze the influence of data scal-
ing and line search on the PGM. Hence, we compare the
proposed PGM with two benchmark schemes. For the first
benchmark scheme (i.e., PGM without line search), the PGM
is implemented without the line search procedure and we
assumed a constant step size equal to 10. For the second
benchmark scheme, the PGM is implemented without data
scaling. The achievable rate results are shown in Fig. 14.
As expected, the proposed PGM has the best achievable rate
results among the considered schemes. PGM without line
search needs significantly more iterations to reach the optimal
14
100101102
4
6
8
Fig. 14. Average achievable rate of the proposed PGM and the two benchmark
schemes (i.e., PGM without line search and PGM without data scaling). The
setup of parameters is the same as for Fig. 3b.
achievable rate, for a step size that is multiple times larger
than the inverse of the Lipschitz constant (see Theorem 2).
Generally, the larger step size enables faster convergence, but
the risk of misconvergence is then higher. Furthermore, PGM
without data scaling has an achievable rate that is not very
significantly worse than the achievable rate of the proposed
PGM.
VII. CONCLUSION
In this paper, we proposed a new PGM algorithm for the
achievable rate optimization in multi-stream MIMO system
equipped with an RIS. Also, we derived a Lipschitz constant
that guarantees the convergence of the PGM. To improve the
rate of convergence of the PGM algorithm, we proposed a
data scaling step and employed a backtracking line search,
which enable the PGM to significantly outperform the existing
AO algorithm. In addition, we defined the new metric of
total FSPL, and showed that the ratio between the total FSPL
of the indirect and direct links can successfully serve as a
first-order measure of the applicability of an RIS. Numerical
results confirm that the PGM requires a significantly lower
number of iterations, and correspondingly a substantially lower
computational complexity, than the AO in order to reach a
target (near-optimal) achievable rate. Furthermore, we showed
that the RIS is particularly convenient for application in an
indoor environment, since a small number of RIS elements is
sufficient to enable the indirect link to have a higher achievable
rate than the direct link.
APPENDIX A
COMPLEX-VALUED GRADIENT OF f(θ,Q)
We first note that (17b) is given in [30, Eq. (6.207)] and
is relatively well known in the related literature. To derive
(17a) we follow the procedure to compute the complex-valued
gradient of a general function detailed in [30, Sect. 3.3.1].
Note that in the following, we adopt the notations introduced
in [30]: df (X)denotes the complex differential of f(X). To
proceed, we recall that the complex differential of f(θ,Q)
with respect to F(θ) = diag(θ)and F∗(θ)is given by
df (θ,Q) = Tr K(θ,Q)dZ(θ)QZH(θ)=
Tr K(θ,Q)d(Z(θ))QZH(θ) + Z(θ)QdZH(θ).(41)
After a few algebraic steps we obtain
df (θ,Q) = vecT¯
H1QZH(θ)K(θ,Q)H2Tvec(dF(θ))
+ vecT¯
H∗
1ZT(θ)QTK(θ,Q)H∗
2Tvec (dF∗(θ)) ,
(42)
where we have used the equality Tr(ATB) =
vecT(A) vec(B). Let Ldbe the matrix used to place
the diagonal elements of a square matrix Aon vec(A), i.e.
vec(A) = Ldvecd(A)[30, Definition 2.12]. Then we can
rewrite df (θ,Q)as
df (θ,Q) = vecT¯
H1QZH(θ)K(θ,Q)H2TLdvec(dθ)
+ vecT¯
H∗
1ZT(θ)QTK(θ,Q)H∗
2TLdvec(dθ∗).(43)
Using [30, Table 3.2] and [30, Eqn. (2.140)] we obtain
∇θf(θ,Q) = LT
dvec HH
2K(θ,Q)Z(θ)Q¯
HH
1
= vecdHH
2K(θ,Q)Z(θ)Q¯
HH
1.(44)
In a similar manner, we can prove the expression for
∇Qf(θ,Q). The details are omitted here due to the page limit.
APPENDIX B
PROO F OF LE MM A 2
To make the proof easy to follow, we first recall the
following inequalities, which are well known or can be proved
easily. For the norm of a matrix product it holds that
||AB|| ≤ λmax(A)||B|| (45a)
||ABC|| ≤ λmax(A)||B||λmax (C)(45b)
where λmax(X)denotes the largest singular value of λmax(X).
Since K(¯
θ,¯
Q) = I+Z(¯
θ)¯
QZ(¯
θ)H−1I, we have
λmax K(¯
θ,¯
Q)≤1.(46)
It is easy to check that
λmax F(¯
θ)=k−1;λmax ¯
Q≤¯
Pt.(47)
A. Proof of (27)
From (17a) we obtain
||∇¯
θf(¯
θ1,¯
Q1)−∇¯
θf(¯
θ2,¯
Q2)|| =||HH
2K(¯
θ1,¯
Q1)Z(¯
θ1)¯
Q1¯
HH
1
−HH
2K(¯
θ2,¯
Q2)Z(¯
θ2)¯
Q2¯
HH
1|| ≤ ||HH
2K(¯
θ1,¯
Q1)Z(¯
θ1)¯
Q1¯
HH
1
−HH
2K(¯
θ1,¯
Q1)Z(¯
θ2)¯
Q2¯
HH
1||+||HH
2K(¯
θ1,¯
Q1)Z(¯
θ2)¯
Q2¯
HH
1
−HH
2K(¯
θ2,¯
Q2)Z(¯
θ2)¯
Q2¯
HH
1||.(48)
The first term on the right-hand side of (48) can be upper-
bounded as
||HH
2K(¯
θ1,¯
Q1)Z(¯
θ1)¯
Q1¯
HH
1−HH
2K(¯
θ1,¯
Q1)Z(¯
θ2)¯
Q2¯
HH
1||
≤aλmax(¯
HDIR)|| ¯
Q1−¯
Q2||
+aλmax(H2)||F(¯
θ1)¯
H1¯
Q1−F(¯
θ2)¯
H1¯
Q2||.(49)
15
Furthermore, we have
||F(¯
θ1)¯
H1¯
Q1−F(¯
θ2)¯
H1¯
Q2|| ≤ k−1λmax(¯
H1)||¯
Q1−¯
Q2||
+λmax(¯
H1)¯
Pt||¯
θ1−¯
θ2||.(50)
Substituting (50) into (49) gives
||HH
2K(¯
θ1,¯
Q1)Z(¯
θ1)¯
Q1¯
HH
1−HH
2K(¯
θ1,¯
Q1)Z(¯
θ2)¯
Q2¯
HH
1||
≤ab||¯
Q1−¯
Q2|| +a2¯
Pt||¯
θ1−¯
θ2||.(51)
Similarly, the second term on the right-hand side (RHS) of
(48) can be upper-bounded as
||HH
2K(¯
θ1,¯
Q1)Z(¯
θ2)¯
Q2¯
HH
1−HH
2K(¯
θ2,¯
Q2)Z(¯
θ2)¯
Q2¯
HH
1||
≤ab ¯
Pt||Z(¯
θ2)¯
Q2Z(¯
θ2)H−Z(¯
θ1)¯
Q1Z(¯
θ1)H||.(52)
Furthermore, we obtain
||Z(¯
θ2)¯
Q2Z(¯
θ2)H−Z(¯
θ1)¯
Q1Z(¯
θ1)H||
≤ ||Z(¯
θ2)¯
Q2(Z(¯
θ2)H−Z(¯
θ1)H)||
+||(Z(¯
θ2)¯
Q2−Z(¯
θ1)¯
Q1)Z(¯
θ1)H||.(53)
The following inequalities hold for the two norms in the RHS
of the above equation:
||Z(¯
θ2)¯
Q2(Z(¯
θ2)H−Z(¯
θ1)H)|| =||(¯
HDIR +H2F(¯
θ2)¯
H1)
ׯ
Q2¯
HH
1(F(¯
θ2)−F(¯
θ1))HHH
2|| ≤ ab ¯
Pt||¯
θ1−¯
θ2|| (54)
and
||(Z(¯
θ2)¯
Q2−Z(¯
θ1)¯
Q1)Z(¯
θ1)H||
≤ ||¯
HDIR(¯
Q2−¯
Q1)Z(¯
θ1)H||
+||H2F(¯
θ2)¯
H1¯
Q2−H2F(¯
θ1)¯
H1¯
Q1Z(¯
θ1)H||.(55)
To upper-bound the two terms on the RHS of (55), we use
¯
HDIR(¯
Q2−¯
Q1)Z(¯
θ1)H
≤bλmax(¯
HDIR)
¯
Q1−¯
Q2
(56)
and
||H2F(¯
θ2)¯
H1¯
Q2−H2F(¯
θ1)¯
H1¯
Q1Z(¯
θ1)H||
≤λmax(H2)||F(¯
θ2)¯
H1¯
Q2−F(¯
θ1)¯
H1¯
Q1||
×λmax(¯
HH
DIR) + λmax (¯
HH
1)λmax(HH
2)
≤k−1ab||¯
Q1−¯
Q2|| +ab ¯
Pt||¯
θ1−¯
θ2||.(57)
Substituting (54), (55), (56) and (57) into (53), we obtain
||Z(¯
θ2)¯
Q2Z(¯
θ2)H−Z(¯
θ1)¯
Q1Z(¯
θ1)H||
≤b2||¯
Q1−¯
Q2|| + 2ab ¯
Pt||¯
θ1−¯
θ2|| (58)
and (52) then implies
||HH
2K(¯
θ1,¯
Q1)Z(¯
θ2)¯
Q2¯
HH
1−HH
2K(¯
θ2,¯
Q2)Z(¯
θ2)¯
Q2¯
HH
1||
≤ab3¯
Pt||¯
Q1−¯
Q2|| + 2a2b2¯
P2
t||¯
θ1−¯
θ2||.(59)
Substituting (51) and (59) into (48), we obtain (27).
B. Proof of (28)
From (17b) immediately have
||∇¯
Qf(¯
θ1,¯
Q1)− ∇¯
Qf(¯
θ2,¯
Q2)||
=||Z(¯
θ1)HK(¯
θ1,¯
Q1)Z(¯
θ1)−Z(¯
θ2)HK(¯
θ2,¯
Q2)Z(¯
θ2)||
≤ ||Z(¯
θ1)HK(¯
θ1,¯
Q1)Z(¯
θ1)−Z(¯
θ1)HK(¯
θ1,¯
Q1)Z(¯
θ2)||
+||Z(¯
θ1)HK(¯
θ1,¯
Q1)Z(¯
θ2)−Z(¯
θ2)HK(¯
θ2,¯
Q2)Z(¯
θ2)||.
(60)
Following the same steps used to prove (27) we can further
upper bound the two norms in the RHS of the above equation
to prove (28). The details are omitted here due to the page
limit.
APPENDIX C
PROO F OF TH EO RE M 2
We recall the following inequality for any function f(x)
which is L-smooth:
f(y)≥f(x) + ∇fx,y−x−L
2||y−x||2.(61)
The projection of ¯
θn+1 onto ¯
Θcan be written as
¯
θn+1 = arg min
¯
θ∈¯
Θ
¯
θ−¯
θn−µ∇¯
θf¯
θn,¯
Qn
2
= arg max
¯
θ∈¯
Θ∇¯
θf¯
θn,¯
Qn,¯
θ−¯
θn−1
2µ||¯
θ−¯
θn||2(62)
where x,y=<(xHy)and we have used the fact that ||a−
b||2=||a||2+||b||2−2<(aHb). Note that when ¯
θ=¯
θn,
the objective in the above problem is equal to 0, and thus we
have
∇¯
θf¯
θn,¯
Qn,¯
θn+1 −¯
θn−1
2µ||¯
θn+1 −¯
θn||2≥0.(63)
An analogous inequality also holds for ¯
Qn+1, i.e.,
∇¯
Qf¯
θn,¯
Qn,¯
Qn+1−¯
Qn−1
2µ||¯
Qn+1−¯
Qn||2≥0.(64)
Applying (61) yields
f(¯
θn+1,¯
Qn+1)≥f¯
θn,¯
Qn+∇¯
θf¯
θn,¯
Qn,¯
θn+1 −¯
θn
+∇¯
Qf¯
θn,¯
Qn,¯
Qn+1 −¯
Qn
−L
2
¯
θn+1 −¯
θn
2−L
2
¯
Qn+1 −¯
Qn
2
≥f¯
θn,¯
Qn+1
2µ−L
2
¯
θn+1 −¯
θn
2
+
¯
Qn+1 −¯
Qn
2.(65)
It is easy to see that f(¯
θn+1,¯
Qn+1)≥f¯
θn,¯
Qnif µ < 1
L.
Since the feasible set of the considered problem is closed and
bounded, the iterate (¯
θn,¯
Qn)is bounded and thus ¯
θn,¯
Qn
has accumulation points. Since, as shown above, f¯
θn,¯
Qn
is nondecreasing, fhas the same value, denoted by f∗, at all
of these accumulation points. From (65) we have
f¯
θn+1,¯
Qn+1−f¯
θn,¯
Qn≥1
2µ−L
2
¯
θn+1 −¯
θn
2
+
¯
Qn+1 −¯
Qn
2,(66)
16
which results in
∞> f∗−f¯
θ1,¯
Q1≥∞
X
n=11
2µ−L
2
¯
θn+1 −¯
θn
2
+
¯
Qn+1 −¯
Qn
2.(67)
Since µ < 1
Lwe can conclude that
¯
θn+1 −¯
θn
→0;
¯
Qn+1 −¯
Qn
→0.(68)
The optimality condition of (62) implies
1
µ¯
θn+1 −¯
θn− ∇¯
θf¯
θn,¯
Qn,¯
θ−¯
θn+1≤0,∀¯
θ∈¯
Θ.
(69)
Similarly we have
1
µ¯
Qn+1 −¯
Qn−∇¯
Qf¯
θn,¯
Qn,¯
Q−¯
Qn+1≤0,∀¯
Q∈¯
Q.
(70)
Let θ∗,Q∗be any accumulation point of ¯
θn,¯
Qn, say
¯
θn,¯
Qn→θ∗,Q∗as n→ ∞. We also note that the gra-
dient of f¯
θn,¯
Qnis continuous and thus ∇¯
θf¯
θn,¯
Qn→
∇¯
θfθ∗,Q∗and ∇¯
Qf¯
θn,¯
Qn→ ∇¯
Qfθ∗,Q∗. By let-
ting n→ ∞ in (69) and (70), we have
−∇¯
θfθ∗,Q∗,¯
θ−θ∗≤0,∀¯
θ∈¯
Θ(71)
−∇¯
Qfθ∗,Q∗,¯
Q−Q∗≤0,∀¯
Q∈¯
Q,(72)
which means that θ∗,Q∗is indeed a critical point of (11).
This completes the proof.
APPENDIX D
COMPUTATIONAL COMPLEXITY FOR ALTERNATIN G
OPTIMIZATION (AO)
The computational complexity for the AO method, intro-
duced in [24], is derived in this appendix. To make the fol-
lowing derivation more accessible, the mathematical notation
in this appendix is the same as in [24].
The channel matrix from the transmitter to the receiver is
given by ˜
H=H+RφT, where H∈CNr×Ntpresents
the direct signal transmission between the transmitter and
the receiver, T∈CNris ×Ntpresents the signal transmission
between the transmitter and the RIS, R∈CNr×Nris presents
the signal transmission between the RIS and the receiver,
and φmodels the RIS response. Let R= [r1,...,rNris ],
T= [t1,...,tNris ]Hand φ= diag[α1, . . . , αNris ], so that
the channel matrix can be written as ˜
H=H+PNris
i=1 αiritH
i.
In the first step of the AO algorithm, LAO independent
realizations of {αm}Nris
m=1 are randomly generated and for each
of these the optimal covariance matrix Qis computed. To
do this, the channel matrix ˜
Hhas to be calculated for every
{αm}Nris
m=1 realization. This calculation starts by computing
all rmtH
mmatrices and for this NrNtNris multiplications are
needed. Further, the computation of all αmrmtH
mmatrices
requires NrNtNris multiplications (per one {αm}Nris
m=1 real-
ization). Hence, the complexity of calculating LAO channel
matrices ˜
His (LAO + 1)NrNtNris.
For each ˜
Hit is required to perform the the truncated
singular value decomposition ˜
H=˜
U˜
Λ˜
VH, where ˜
V∈
CNt×Dand D= min(Nt, Nr). The complexity of this
decomposition is approximately O(D3). Next, Qis computed
as Q=˜
Vdiag{p1, . . . , pD}˜
VH, where {p1, . . . , pD}are ob-
tained using a water-filling algorithm and D= min(Nt, Nr).
The complexity of the water-filling algorithm is O(D2)and
the complexity of the matrix multiplication is NtD+ (N2
t+
Nt)D/2. Therefore, the calculation of LAO covariance matri-
ces Qrequires O(LAO(D3+1
2N2
tD)) multiplications.
In the sequel, the optimal {αm}Nris
m=1 and Qare iteratively
determined. In one conventional iteration, one αmor Qis
adjusted. A set of Nris + 1 successive conventional iterations
constitutes one “outer” iteration, in which all αmand Qare
adjusted. The AO method stops when the convergence criterion
at the end of an outer iteration is fulfilled.
At the beginning of each outer iteration, the eigenvalue
decomposition Q=UQΣQUH
Qis performed, which requires
O(N3
t)multiplications. The calculation of the matrices H0=
HUQΣ
1
2
Q∈CNr×Ntand T0=TUQΣ
1
2
Q∈CNris×Nthas the
complexity O(N3
t+N2
tNris).
To form S=H0+PNris
i=1 αirit0H
i,NrNtNris multiplications
are needed first to obtain all rit0H
iand NrNtNris multiplica-
tions are needed for all αirit0H
i. Hence, the complexity of
computing Sis 2NrNtNris.
The optimization of the m-th RIS element requires the
computation of the following auxiliary matrices
Am=I+1
N0
SmSH
m+1
N0
rmt
0H
mrmt
0H
mH(73)
Bm=1
N0
rmt
0H
mSH
m(74)
where Sm=H0+PNris
i=1,i6=mαirit0H
i=S−αmrmt0H
m. It can
be easily shown that the complexities for calculating Amand
Bmfrom the previous expressions are both O(N2
rNt).
Utilizing the results from Subsection IV-B, the complexity
of computing A−1
mBmis O(2N3
r). The subsequent calculation
of αmand update of S=Sm+αmrmt0H
mrequire a negligible
complexity.
After adjusting all RIS elements, the optimization of Qis
performed according to the aforementioned procedure, which
requires O(D3+1
2N2
tD)multiplications.
If IOI is the number of outer iterations, then the computa-
tional complexity of the AO algorithm is given by
CAO =O((LAO + 1)NrNtNris +LAO(D3+1
2N2
tD)
+IOI[N3
t+N2
tNris + 2NrNtNris
+(2N2
rNt+ 2N3
r)Nris +D3+1
2N2
tD]) (75)
where D= min(Nt, Nr).
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