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Active-Passive Cascaded RIS-Aided Receiver
Design for Jamming Nulling and Signal Enhancing
Yifu Sun, Yonggang Zhu, Kang An, Zhi Lin, Cheng Li,
Derrick Wing Kwan Ng, Fellow, IEEE, and Jiangzhou Wang, Fellow, IEEE
Abstract—The utilization of a large-scale antenna array has
led to substantial performance improvements in anti-jamming
communications. However, due to the practical constraints of
hardware cost and power consumption, deploying such a large-
scale antenna array at the user side is impractical. Inspired
by the remarkable advantages of reconfigurable intelligent sur-
faces (RIS), we propose an active-passive cascaded RIS-aided
receiver architecture that facilitates the cost- and energy-efficient
deployment of a large-scale antenna array at the user side,
while also providing additional degrees-of-freedom for effective
beamforming design. Building upon this architectural framework
and taking into account the practical imperfections in the angular
channel state information (CSI), we formulate a worst-case
achievable rate maximization problem for anti-jamming commu-
nications. To address the challenges posed by the intractable non-
convex design problem, we present a low-complexity optimization
framework that obtains semi-closed-form solutions. Specifically,
we first develop a Pareto-dual scheme to handle the general power
constraints in devising the optimal precoder for the base station.
Subsequently, by introducing a novel anti-jamming criterion and
employing the discretization method to transform the imperfect
CSI of jammers into a robust form, we derive two jamming-
nulling feasibility conditions and a unified unit-modulus zero-
forcing scheme to determine the coefficients of the passive RIS.
To strike a satisfactory balance between complexity and perfor-
mance, we further design three computationally-efficient algo-
rithms based on alternating majorization-minimization (AMM)
and conventional/modified cyclic coordinate descent (C/M-CCD)
methods to obtain the coefficients of the active RIS. Finally,
through comprehensive numerical simulations, we validate the
effectiveness of the proposed architecture and optimization
This work is supported by the National NSFC under Grant 61901502,
62201592, 62131005; in part by the Research Plan Project of NUDT under
Grant ZK21-33; in part by the Young Elite Scientist Sponsorship Program of
CAST under Grant 2021-JCJQ-QT-048; in part by the Macau Young Scholars
Program under Grant AM2022011; in part by the National Postdoctoral
Program for Innovative Talents under Grant BX20200101; and in part by the
Graduate Innovation Program of Hunan Province under Grant CX20220008.
The work of D. W. K. Ng is supported by the Australian Research Council’s
Discovery Projects (DP210102169, DP230100603). This work was presented
in part at the IEEE International Conference on Communications (ICC),
Rome, Italy, 2023 [1]. (Corresponding author: Kang An; Yonggang Zhu.)
Y. Sun is with the Sixty-third Research Institute, National University of
Defense Technology, Nanjing 210007, China, and also with the College of
Electronic Science and Technology, National University of Defense Technol-
ogy, Changsha 410005, China (Email: sunyifu.nudt@nudt.edu.cn).
Y. Zhu, K. An, and C. Li are with the Sixty-third Research Institute,
National University of Defense Technology, Nanjing 210007, China (Email:
zhumaka1982@163.com; ankang89@nudt.edu.cn; licheng@nudt.edu.cn).
Z. Lin is with College of Electronic Engineering, National University
of Defense Technology, Hefei 230037, China, and also with the School
of Computer Science and Engineering, Macau University of Science and
Technology, Macau 999078, China (Email: linzhi945@163.com).
D. W. K. Ng is with the School of Electrical Engineering and Telecom-
munications, University of New South Wales, Sydney, NSW 2025, Australia
(e-mail: w.k.ng@unsw.edu.au).
J. Z. Wang is the School of Engineering, University of Kent, Canterbury
CT2 7NZ, U.K. (e-mail: j.z.wang@kent.ac.uk).
framework, demonstrating their capacity to achieve exceptional
performance in a cost-effective manner.
Index Terms—Reconfigurable intelligent surface, anti-jamming
communications, cascaded receiver architecture, low-complexity
robust beamforming design.
I. INTRODUCTION
MALICIOUS jamming attacks have posed a severe threat
to wireless communications due to the broadcast and
superposition characteristics of wireless systems [2], [3].
To cope with this issue, various conventional techniques
have been developed, e.g., frequency hopping (FH) [4] and
direction-sequence spread spectrum (DSSS) [5]. Nevertheless,
these techniques consume extra spectral resources and may be-
come ineffective when the jammers attack multiple frequencies
[6]–[8]. With this focus, benefiting from the ascendancy that
massive multiple-input multiple-output (MIMO) techniques
can simultaneously increase the system capacity and mitigate
jamming by employing a large-scale antenna array at wireless
receivers, numerous MIMO-assisted anti-jamming schemes
have been proposed to suppress jamming in the spatial do-
main, e.g., [9]–[11]. It has been verified that massive MIMO-
aided anti-jamming systems can achieve orders of magnitude
increase in the system capacity by leveraging the additional
degrees-of-freedom (DoFs) to enhance the desired signal
strength and nullify the potential jamming signal. However,
from the scalability and realizability perspective, employing a
large-scale array equipping hundreds of antennas at a receiver
is generally impractical. Hence, a cost-effective receiver archi-
tecture for anti-jamming communications is urgently needed.
Fortunately, the emerging innovative paradigm of reconfig-
urable intelligent surface (RIS) can be adopted to facilitate the
employment of large-scale arrays [12], [13]. To elaborate, RIS
is a large-scale array constituted by massive low-cost meta-
material units, which can constructively increase the desired
signal power or destructively eliminate the interference by
altering the phase and/or amplitude of the incident electro-
magnetic waves [14]. Different from the energy-hungry RF
counterparts in a typical MIMO array, RIS units are composed
of low-cost tunnel diodes and/or active loads, which can serve
as both the phase-shift network and receive (Rx) antennas in an
energy-efficient manner [15]. Thus, RIS serves as a potential
candidate to establish large-scale arrays at the receiver. In the
literature, existing RIS-aided architectures can be roughly di-
vided into two broad categories. The first category applies RIS
as a relay-like reflector to reconfigure the radio propagation
2
environment. Moreover, depending on whether the amplitude
of each RIS unit can be controlled, the RIS-based reflector
can be further divided into a passive and active one. As for
the passive RIS-based reflector whose only the phases can be
adapted, it can provide additional reconfigurable propagation
paths and thus has been widely adopted for increasing the
service coverage [14], [16], [17], maximizing the achievable
rate [18], [19], improving the physical layer security [7], [20]–
[22], and mitigating the interference/jamming [6], [8], [23],
[24]. Nevertheless, due to the potential large-scale fading at-
tenuation in the transmitter-RIS link, the signal strength would
be extremely faint at the users such that directly exploiting
the passive RIS-based reflector architecture at the user may
be ineffective. To address this issue, the active RIS-assisted
reflector has been proposed [25]–[27], where each RIS unit
was connected with a power amplifier that can control both the
phase and amplitude of the reflected signals simultaneously.
Although the active RIS-assisted reflector architecture can
significantly amplify the desired signals, directly deploying
an active RIS at the user in the presence of strong jamming
attacks would also introduce powerful dynamic noise. Besides,
it consumes exceedingly large power at users that results in a
limited performance gain compared to the ones without RIS.
The second category employs RIS as the power-efficient
transmitter for manipulating the radiated electromagnetic
waves. Depending on the number of RIS’s layers, RIS-aided
transmitter architecture can be further categorized into the
single-layer and multi-layer ones. Note that since active RIS
would introduce dynamic noises inevitably, the typical RIS-
aided transmitters usually utilize passive RIS. Although it
has been verified that a single-layer RIS-aided transmitter
architecture can improve the system performance efficiently
[15], [28], [29], employing the single-layer architecture at
the recevier would still suffer from the severe path loss
attenuation. Besides, the power penetrating RIS generally
scatters outside the Rx antennas’s area, as will be speci-
fied in Section V, which leads to significant performance
degradation. To overcome the above shortcomings, the multi-
layer architecture was proposed in [30] and [31], where the
amplitude of incident signals can be partially controlled in
a confined range. However, applying multi-layer RIS at the
receiver leads to bulky hardware cost, and the amplitude of
its units cannot be independently adapted in a wide range.
Thus, the spatial DoFs cannot be fully exploited such that the
pass loss attenuation and power scattering effects can only be
partially overcome. From the above observations, there are no
existing works have investigated the design of RIS-assisted
receiver. More importantly, as stated before, the existing RIS-
assisted architectures cannot be directly applicable to receiver,
which motivates the contributions presented in our work.
On the other hand, jamming nullification and desired signal
enhancement problems have been studied in the past, e.g.,
[7]–[9], [11], [15], [23], [24], [30]. However, due to the
additional unit-modulus constraints and the dynamic noise
term induced by the active RIS, the corresponding problem
with the proposed RIS-aided receiver is non-convex and NP-
hard. Indeed, existing MIMO anti-jamming schemes (e.g., [9]
and [11]) are not applicable to this emerging design problem.
Besides, there are few RIS-related works investigating the
abovementioned problem. For example, two existing RIS-
related works, i.e., [23] and [24], have proposed the alternating
projection and Kronecker decomposition algorithm, respec-
tively, to solve the similar interference eliminating problem.
However, the alternating projection algorithm in [23] can
only address the jamming nulling problem, while the Kro-
necker decomposition algorithm in [24] is only be applicable
to the geometry-based channels. More importantly, applying
their designs to the anti-jamming problem may lead to non-
robust performance since they have not taken the imperfect
channel state information (CSI) into account. Meanwhile,
although several existing works, e.g., [7], [8], [15], and [30]
have proposed various robust algorithms specializing on the
polyblock approximation, successive convex approximation
(SCA), and semidefinite relaxation (SDR) to handle the anti-
jamming problem under various CSI uncertainty models, these
algorithms lift the dimensionality of optimization variables,
resulting in increased computational complexity, especially at
the receiver side. Specifically, the existing robust algorithms
first introduce numerous auxiliary variables to relax the non-
convex problem into its convex counterpart such that an off-
the-shelf optimization toolbox such as CVX can be applied to
obtain a suboptimal solution. However, one of the limitations
in the existing methods is that the optimization toolbox has
extremely high computational complexity. Besides, since the
computation process of the optimization toolbox is not trans-
parent, it is challenging to perform them in real-time FPGA
hardware for practical implementation. Thus, the scalability of
these algorithms remains a bottleneck in the implementation
of large-scale RIS-assisted systems that calls for the design of
scalable and efficient algorithms for achieving better perfor-
mance than the existing ones.
Motivated by the aboves, this paper proposes a novel active-
passive cascaded RIS-aided receiver architecture to circumvent
the curse of dimensionality inherent in deploying a large-
scale array at the user. Building upon this architecture, a low-
complexity optimization framework is established to design an
efficient anti-jamming beamforming scheme. Specifically, the
main contributions of this paper are summarized as follows:
•An active-passive cascaded RIS-aided receiver architec-
ture is proposed to facilitate the employment of a large-
scale array at the user side and overcome the dimension-
ality limit in the beamforming design, where a passive
RIS having NPunits are cascaded with an active RIS
equipping NAunits, and they are vertically stacked in
front of the Rx antennas. To our best knowledge, this
is the first work to exploit active-passive cascaded RIS
at the user side for receiver design. Besides, a rigorous
performance analysis of the proposed architecture is
derived. The theoretical result shows that in contrast to
existing active RIS whose receive power and asymptotic
SINR are proportional to (NP+NA)2and NP+NA, those
of proposed architecture are proportional to N2
AN2
Pand
NANP, respectively, which indicates that the amplitude
of the two RIS layers can be separately controlled in a
larger range, thus providing additional DoFs for design.
3
•Utilizing the proposed receiver to combat the jamming at-
tacks, a worst-case achievable rate maximization problem
is formulated under the imperfect jammers’ angular CSI,
while satisfying the general power constraints at the base
station (BS) and the power budget constraint at the active
RIS. Due to the non-convexity of the formulated problem,
a low-complexity optimization framework is proposed
to optimize the BS’s transmit precoder, the passive and
active RISs’ coefficients, and the Rx digital decoder.
For the design of the BS’s precoder, by exploiting the
Lagrange dual and Pareto optimization theory, the general
power constraints can be transformed into its tractable
counterpart for obtaining the optimal semi-closed-form
precoder at the BS.
•For the optimization of the passive RIS’s coefficient,
after transforming the jammers’ imperfect CSI into a
robust one via the discretization method, we propose
a new anti-jamming criterion where the jamming can
always be nullified at the passive RIS. Based on this
criterion, we further derive two feasibility conditions for
successful jamming nulling and propose a unified unit-
modulus zero-forcing scheme to obtain a semi-closed-
form solution of the passive RIS’s coefficient with low
computational complexity.
•For the design of active RIS’s coefficient, three effi-
cient algorithms, specializing on alternating majorization-
minimization (AMM) and conventional/modified-cyclic
coordinate descent (C/M-CCD) method, are proposed
to obtain the semi-closed-form solutions of the phase
shifts and amplitudes, which can achieve satisfactory
tradeoff between the complexity and performance. Note
that the above three algorithms can also be applicable
to the optimization of passive RIS’s coefficient, which
is viewed as another optimization framework such that
further features our contributions. In addition, we prove
that all the algorithms can converge to a limited KKT
point. Numerical simulations shows that the proposed
architecture and optimization framework can obtain su-
perior performance with lower complexity in comparison
with the existing fully-digital receiver and SDR method.
The remainder of this work is organized as follows. The
system model and the problem formulation are presented in
Section II. In Section III, the low-complexity optimization
framework is proposed to address the formulated beamforming
design problem. Performance analysis of the proposed archi-
tecture is shown in Section IV. Numerical results are provided
in Section V. We conclude this paper in Section VI.
Notation:XH,XT,X∗, and kXkFdenote conjugate
transpose, transpose, conjugate, and Frobenius norm of a
matrix X. The notations
E
{·}, tr{·},<{·},={·} and λ{·}
denote the expectation, trace, real part, imaginary part, and
eigenvalue of a complex number or matrix, respectively. Cm×n
represents the complex space of m×ndimensions. The symbol
Hn×nis the Hermitian matrix of n×ndimensions. [·]n,n
represents the nth diagonal element of a matrix. X0means
that the matrix Xis positive semi-definite. The distribution
of a circularly symmetric complex Gaussian (CSCG) random
Phase-
amplitude
controller
Digital
baseband
processor
Phase
controller
Active
RIS Rx
antennas
DAC DAC
Passive
RIS
DAC
Incident
wave
Signal
transmission
direction
Passive refracting
RIS unit
PA Active refracting
RIS unit
Fig. 1: Active-passive cascaded RIS-aided receiver.
vector with mean vector xand covariance matrix Σis denoted
by CN (x, Σ).
II. SY ST EM MO DE L AN D PROB LE M FOR MU LATI ON
A. Active-Passive Cascaded RIS-Aided Receiver Architecture
To illustrate the considered architecture, Fig. 1 portrays the
proposed active-passive cascaded RIS-aided receiver. Specifi-
cally, the two layers of RISs, namely, the passive and active
RIS, are vertically stacked in front of the Rx antennas to
form a cascaded receiver structure, where the passive RIS
is connected to a phase-shift controller for the jamming
nulling, while the active one is connected to a phase-shift and
amplitude controller for the desired signal amplification. To
elaborate, the passive refracting RIS receives a superposition
of the legitimate and jamming signals from the wireless
channels. It then forwards the signals to the RIS-aided phase
shifters. Subsequently, the phase-shifted signal are conveyed to
the transmit units via the microstrip. Then, the phase delayed
signals can be refracted by the active RIS, where the RIS phase
shifters and power amplifier (PA) further impose the phase
shift and power amplification to them. Naturally, the entire
structure should be contained within an enclosure, which is
surrounded by absorbing materials for reducing the potential
energy loss and protecting the internal channel from external
interference, which can also guarantee that the jammer can
only send the signal to the passive RIS directly while the active
RIS does not receive the jamming signal [27]. Besides, it is
also worth noting that the gap between adjacent vertical layers
is flexible for practical applications, but it is generally compact
for implementation. To clarify the characteristics of the novel
active-passive cascaded RIS-aided receiver architecture, we
summarize its key advantages as follows.
•Low power consumption and hardware cost: The im-
plementation of a fully-digital receiver requires as many
radio-frequency (RF) chains as the number of anten-
nas, which places huge demands on the hardware and
increases the associated power consumption [29]. As a
remedy, the hybrid analog-digital receiver, whose RF
chains are connected to the antennas through an analog
phase-shifter network, has been proposed to reduce the
need for the use of RF combiners and phase shifters
[15]. Nonetheless, deploying practical analog networks
at a receiver still results in bulky circuits cost and
excessive power consumption [15], [29]. Fortunately, the
passive/active RIS only entails low-cost tunnel diodes and
4
active loads, whose power and hardware consumption
are extremely low [29]. Furthermore, as compared to
the single-layer active/passive RIS-aided receiver, the
proposed architecture only divides the total RIS units into
two layers of RIS instead of deploying twice number
of RIS units. Thus, the total number of the proposed
cascaded RIS-receiver is the same as those of single-layer
one, which means that the proposed architecture is still
cost- and energy-efficient as the single-layer ones.
•Miniaturization and high scalability: Benefiting from
the compact axonometric structure and small size of
meta-materials, the size of the proposed receiver is much
smaller than those of the same dimensional single-layer
RIS-aided and typical receivers, which contributes to the
scalability of the proposed architecture. To elaborate,
the typical and single-layer RIS-aided receivers can only
deploy NP+NAantennas/RIS units at the transversal
side, while our proposed receiver can deploy NPand NA
units at two compact layers, respectively. Combining with
the fact that the gap between adjacent layers is generally
very small [29], the miniaturization and scalability of the
proposed architecture is highly desirable.
•Additional DoFs for jamming nulling and signal en-
hancing: Assisted by the separated controllers of each
RIS and the cascaded structure, the proposed receiver can
flexibly amplify the signals in a larger range compared to
these benchmark receivers, which will be stated in The-
orem 6. Furthermore, the receive power of the proposed
architecture is in proportion to the product of the number
of active RIS units and that of passive RIS units NANP,
while the single-layer RIS-aided and typical receivers can
only amplify it in proportional to the total number of
antennas/RIS units NP+NA(please see Theorem 5). The
above findings suggest that additional DoFs are generated
by the proposed architecture for facilitating the design.
•Low power scattering ratio (PSR): Since the transmission
mechanism between the active RIS and the Rx antennas is
air, the power transmitted by active RIS would inevitably
scatter outside the Rx antennas. Thus, we introduce a
new metric of PSR to explicitly quantify the particular
fraction of scattering power to the total transmit power.
Due to the compact size of the proposed receiver and
the DoFs introduced by the active RIS, the PSR of the
proposed receiver is much lower than that of single-layer
RIS-aided one (see Section V).
B. RIS-Receiver-Aided Anti-Jamming Communications
Let us consider the RIS-receiver-aided anti-jamming com-
munications scenario of Fig. 2, where an active-passive cas-
caded RIS-aided receiver is adopted at the user1for enhancing
1Deploying the proposed RIS-aided receiver at the user side is conceived
for improving user’s individual utility, which is independent of the other
users. As such, the key difference between the single-user scenario and multi-
user one is the optimization of BS’s transmit beamforming. Since this paper
focuses on the proposed RIS-aided receiver design, we only consider the
single-user scenario for a better illustration of our design. Furthermore, our
proposed algorithm for optimizing the transmit beamforming in Section III-
A can be easily extended to the multi-user scenario by using some matrix
transformations [10], [32].
BS
Jammer 1
Jammer m
Active-Passive Cascaded
RIS-Aided Receiver
B
G
J,1
g
J,m
g
U
B
U
H
User
Rx antenna
Passive RIS unit
Desired signal Jamming signal
Active RIS unit
Fig. 2: System model.
the desired signal from base station (BS), while simultaneously
nullifying Mjammers’ malicious signals. Furthermore, we
assume that the BS is equipped with NBtransmit antennas, and
the m-th jammer utilizes an omnidirectional single-antenna
to impair the signal reception at the receiver from all an-
gles2. Besides, the proposed cascaded RIS-aided receiver is
composed of a passive RIS having NPunits, an active RIS
having NAunits, and the Rx antennas having NUelements.
For exposition, we denote GB∈
C
NP×NB,gJ,m ∈
C
NP×1,
BU∈
C
NA×NP, and HU∈
C
NU×NAas the channels
spanning from the BS to the passive RIS, from the m-th
jammer to the passive RIS, from the passive RIS to the active
RIS, and from the active RIS to the Rx antennas, respectively.
Denote sUas the desired symbol transmitted by the BS
to the user3, which satisfies
E
n|sU|2o= 1. Prior to trans-
mission, sUis processed by the transmit precoder wB∈
C
NB×1such that the desired signal transmitted by the BS
is wBsU. Meanwhile, the m-th jammer launches the jam-
ming signal sJ,m =pPJ,mbsJ,m to interrupt the legitimate
transmission, where bsJ,m satisfying
E
n|bsJ,m|2o= 1 is the
jamming symbol and PJ,m is the corresponding jamming
power. As such, the proposed cascaded RIS-aided receiver
simultaneously receives both the desired and jamming sig-
nals. Subsequently, the passive and active RIS can impose
the coefficients P∈
C
NP×NPand Ξ∈
C
NA×NAinto
the received signals, respectively. Here, Pand Ξcan be
rewritten as P= diag (p) = diag ej θP,1,· · · , ejθP,NPand
Ξ=e
ΞPΞ= diag (ξ) = diag a1ejθA,1,· · · , aNAejθA,NA,
where θA,θP∈[0,2π)and an∈[0, an,max]represent
the phase and the amplitude, and an,max is predetermined
2The justifications for the single-antenna jammer are two-fold. First, from
the view of flexibility and effective jamming, the single-antenna configuration
for a potential jamming has been regarded as a practical scenario from
the trade-off between cost and performance [33]. Moreover, to disrupt the
legitimate nodes from different angles, the jammer tends to be equipped with
omnidirectional single-antenna in practice. Second, as will be discussed in the
later section, by dividing the multi-antenna jamming channel into multiple
single-antenna ones, the jamming nulling algorithm proposed in Section III-B
can also be extended to the multi-antenna jammers, thus we adopt the single-
antenna jammers here for brevity.
3To guarantee the anti-jamming communications under unfavorable condi-
tion and characterize the lower bound of the system performance, this paper
considers the worst case that the BS only transmits a single-data stream [15].
Moreover, after using some matrix transformations, the proposed algorithms
which are presented in Section III can be also extended to the multi-data
stream case, thus we adopt the single-data stream for simplicity.
5
maximum amplitude of the active load for the n-th unit on the
active RIS. Besides, e
Ξand PΞare the amplitude and phase-
shift matrices of the active RIS’s coefficient Ξ, respectively.
Finally, the user adopts the digital decoder vU∈
C
NU×1
satisfing kvUk= 1 to harness the interference. Therefore, the
received signal is4
yU=e
hH
U GBwBsU+
M
X
m=1
gJ,msJ,m !+hH
Uz+nU,(1)
where e
hU=PHBH
UΞHHH
UvU,hU=ΞHHH
UvU, and nU=
vH
UnU. Here, z∼ CN (0, σ2
zINA)and nU∼ CN (0, σ2
UINU)
are the dynamic noise induced by the active RIS and thermal
noise at the Rx antennas, respectively, where σ2
zand σ2
Uare the
corresponding noise power per-antenna. Note that the dynamic
noise induced by active RIS is related to the its amplitude,
which is consistent with the existing works, e.g., [25]. Then,
the achievable rate at the user can be modeled as
RU(wB,P,Ξ,vU)=log2
1+ e
hH
UGBwB
2
M
P
m=1
PJ,me
hH
UgJ,m
2+eσ2
U
,(2)
where eσ2
U=σ2
z
hU
2+σ2
U.
C. Near/Far-Field Channel Models
In this paper, considering the effects of spherical wave, BU
and HUcan be regarded as the near-field channels, which can
be characterized as the radar illumination model [29], i.e.,
Q=
λ
qbρGD
n,v (θR, ϕR)GR
n,v (θD, ϕD)
4πdn,v
e−j2πdn,v
λ
n,v
,(3)
where Qdenotes the near-field channels between adjacent
vertical layers (i.e., BUand HU), λis the carrier wavelength,
bρdenotes the power efficiency of the RIS, dn,v is the distance
between the n-th RIS unit of the passive RIS and the v-th
unit of the active RIS, and GD
n,v θR, ϕR,GR
n,v θD, ϕDare
the active and passive antenna gains from the n-th and the v-
th unit, respectively. Owing to the short distance between the
adjacent vertical layers, BUand HUare deterministic matrices
and can be precisely measured in advance [15], [29], [30].
Besides, taking into account the effects of plane wave, the
downlink channels GBand gJ,m can be termed as the far-field
channels, which is the superposition of a predominant line-
of-sight (LoS) component and a sparse set of single-bounce
non-LoS (NLoS) components [35]. Thus, GBand gJ,m can
be modeled as in [35], which is omitted here for brevity.
On the other hand, owing to the cooperation between the
BS and the user, the legitimate CSI is available at the BS
and the user by sending pilot signals and exploiting some
efficient estimation methods in [36], [37]. Hence, we assume
that the involved legitimate CSI GBcan be perfectly obtained
during the whole transmission period [7]. However, since the
4When RIS’s reflection is considered, the multi-order-reflection between
adjacent RISs will be introduced into the RIS-receiver. Particularly, the signals
refracted by the passive RIS can be simultaneously refracted and reflected by
the active RIS, then the reflected signals can be reflected by the passive RIS
twice, which leads to the multi-order-reflection effect and the different signal
model. The multi-order-reflection between multiple RISs has been investigated
in [34], but a comprehensive understanding of these effects is still lacking and
shall be addressed in our further works.
jammers are not expected to cooperate with the user for the
channel estimation, the illegitimate CSI gJ,m are challenging
to obtain. Fortunately, the illegitimate CSI depends on the
relative position between the jammer and the user, namely,
the azimuth and elevation angles between them, such that gJ,m
can be estimated by detecting the jamming power transmitted
by the jammers’ RF frontend [10], [15], [38]. Nonetheless,
the estimation of the azimuth and elevation angles may also
be inaccurate. To account for this effect, this paper assumes
that gJ,m belongs to a given continuous angle range, which is
given by5[15], [30], [41]
∆J={gJ,m|θm∈[θm,L, θm,U], ϕm∈[ϕm,L, ϕm,U],
gm∈[gm,L, gm,U],∀m},(4)
where θUand θLdenote the upper and lower bounds of
azimuth angle, ϕUand ϕLare the upper and lower bounds
of elevation angle, respectively, and gUand gLis the upper
and lower bounds of the channel gain amplitude, respectively.
D. Problem Formulation
In this paper, a worst-case achievable rate maximization
problem formulation is considered for providing robustness
against imperfect angular CSI. To elaborate, under the angular
uncertainty ∆J, our goal is to maximize the achievable rate by
jointly optimizing the BS’s transmit precoder wB, the passive
and active RISs’ coefficients Pand Ξ, and the Rx digital
decoder vU, while fulfilling the BS’s power constraint and
the active RIS power budget constraint. In contrast to most
existing RIS-related works subject to only the total power
constraint, e.g., [19], this paper adopts a general formulation:
C1 : wH
BEcwB≤pB,c,∀c= 1,· · · , C, (5)
where Ec0is the weighting matrices for the c-th antenna’s
cluster, pB,c denotes the c-th cluster’s maximum power, and
Cis the total number of the antenna’s cluster. Note that the
general power constraint can be reduced to the conventional
total power constraint by setting C= 1 and Ec=INB.
As such, the corresponding optimization problem can be
formulated as
max
wB,P,Ξ,vU
min
∆J
RU(wB,P,Ξ,vU)(6)
s.t. C1,C2 : max
∆JkΞe
wBk2+σ2
zkΞk2
F≤PR,max,
C3 : [Ξ]n1,n1≤αn1,max,∀n1,C4 : [P]n2,n2= 1,∀n2,
where e
wB=BUPGBwB+PM
m=1 gJ,m,PR,max is the
maximum amplification power budget at active RIS, and
5The reasons that the LoS and NLoS components have the common CSI
accuracy can be two-fold. First, recalling the jamming channel models in
[35], θRx
J,md =θRx
J,m0+ˆ
θd, and ϕRx
J,md =ϕRx
J,m0+ ˆϕd, where ˆ
θdand ˆϕd
denote the random angle deviation [35], we can find that both the LoS and
NLoS components in gJ,m are related to θRx
J,m0and ϕRx
J,m0, which depends
on the location estimation errors of jammers. As such, we can adopt the
imperfect angular uncertainty set ∆Jto account for the location estimation
errors, thereby characterizing the CSI imperfection of both the LoS and NLoS
components. Furthermore, according to [39], ˆ
θdand ˆϕdcan be obtained by
using UPA based phase rotation schemes. Thus, it is resonable to assume
that the LoS and NLoS components share the same accuracy ∆J. Second,
based on the previous findings in [40], the contribution of the RIS-assisted
link is mainly determined by its LoS component. Thus, we assume the same
accuracy ∆Jin the LoS and NLoS components for brevity.
6
αn1,max >1denotes the maximum amplification factor at
n1-th active unit.
Obviously, the optimization problem in (6) is challenging to
solve. To be specific, the unique challenges in addressing (6)
are summarized as follows. First, different from the traditional
hybrid MIMO beamforming schemes whose analog beam-
formers are seperated with the channel matrices, e.g., [42], the
RISs’ coefficients P,Ξand the propagation are tighly coupled
in (6), rendering the optimal solutions challenging to obtain.
Second, due to the cascaded active-passive RIS structure, the
existing passive RIS-aided anti-jamming schemes (e.g., [7],
[8], [15], [30]) are not directly applicable to (6). More impor-
tantly, the abovementioned schemes incur high computational
complexity and suboptimal solutions, thus an efficient low-
complexity algorithm for obtaining the closed-form solution
is needed. Third, the angular uncertainty ∆Jand the existence
of the extra noise term σ2
z
hU
2in RUmake (6) non-convex,
which constitutes another unique challenge for obtaining the
closed-form solution of (6). Thus, in the sequel, we propose
an efficient low-complexity anti-jamming scheme to confront
the foregoing unique challenges.
III. EFFIC IE NT LOW-COMPLEXITY OPTIMIZATION
FRA ME WORK FOR CASCADED RIS-RECEIVER-AIDED
ANT I-JAMMING COMMUNICATIONS
In this section, an efficient low-complexity optimization
framework based on the block coordinate descent (BCD) for
cascaded RIS-receiver-aided anti-jamming communications is
proposed. In particular, (6) is decoupled into four subprob-
lems and then the optimal semi-closed-form solutions can be
derived, thereby significantly reducing the complexity.
A. Pareto-Dual Scheme for wB
Firstly, we focus on optimizing the transmit beamforming
wBunder the general power constraint. By defining e
HU=
GH
Be
hUe
hH
UGBand WB=wBwH
Bwith an implicit constraint
rank (WB)=1, the subproblem for wBis formulated as6
min
WB
f(WB) = −tr e
HUWB(7)
s.t. e
C1 : tr (EcWB)≤pB,c,∀c= 1,· · · , C, C5 :WB0.
Note that the multiple power constraints e
C1 prevent us from
deriving the semi-closed-form solution of (7). Thus, we adopt
Pareto optimization and Lagrangian dual theory [43], [44] to
transform e
C1 into a tractable one via the following proposition.
Proposition 1: Since f(WB)is a matrix-monotone de-
creasing affine function w.r.t WB, subproblem (7) can be
equivalently transformed into
min
WB
f(WB)s.t. e
C1a: tr (EWB)≤PB,C5,(8)
where PB=PC
c=1 pB,c,E=PC
c=1 βcEc, and βc=
ηcPB.PC
i=1 ηipB,i. Note that ηcis the optimal dual variables
6Note that the rank-one constraint is temporality omitted for making (7)
easier to handle, which can be always guaranteed by the derived optimal
semi-closed-form solution later.
associated with e
C1a, which can be obtained by solving the
dual problem of (7):
min
Q0,{ηc}C
c=1≥0XC
c=1 ηcpB,c (9)
s.t. F1=XC
c=1 ηcEc−e
HU−Q0,
where Qis the dual variable associated with WB0.
Benefiting from the single power constraint in (8) and the
dual method in (9), the optimal wBto (7) can be derived in
a closed-form expression with low computational complexity,
which is presented in Proposition 2.
Proof: See Appendix A in the supplemental information.
Based on Proposition 1, we find that given the optimal dual
variable ηcby utilizing the dual method (9), the optimal WB
of (7) can be obtained by solving (8) that provides the way
for derving an optimal closed-form solution. Different from
the widely adopted subgradient method for finding the optimal
ηcin [18], the dual method in (9) solves ηcwithout the need
for iterations. Besides, in contrast to directly solve the dual
problem of (7) whose WBis not constrained, (8) can restrict
WBby introducing tr (EWB) = PBand E=PC
c=1 βcEc.
Therefore, Proposition 1 offers an easy way to address (7),
and theoretically reduces the computational complexity.
Next, armed with Proposition 1, we develop the following
proposition for obtaining the optimal solution of (7).
Proposition 2: The optimal closed-form solution for the
convex optimization problem w.r.t. wBcan be derived by
wB=pPBE−1
2umax E−1
2e
HUE−1
2,(10)
where umax E−1
2e
HUE−1
2is the eigenvector corresponding
to the largest eigenvalue of E−1
2e
HUE−1
2. Note that the
implicit rank-one constraint of (7) can be satisfied by (10).
Proof: Defining WB=E−1
2f
WBE−1
2, problem (8) can be
equivalently rewritten as
max
f
WB
tr e
HUE−1
2f
WBE−1
2
s.t. C1 : tr f
WB≤PB,e
C5 :f
WB0.(11)
It is well known that the optimal f
WBfor the classical MIMO
capacity maximization problem (11) is f
WB=epBumaxuH
max,
where epBis power for e
wB. Then, (11) can be simplified as
max
epBepBtruH
maxE−1
2e
HUE−1
2umaxs.t.0≤epB≤PB.(12)
Obviously, (12) is the positive linear problem, whose optimal
solution is PB. Thus, by aligning the optimal f
WBwith the
largest eigenvalues of E−1
2e
HUE−1
2, we can obtain (10).
B. Unified Unit-Modulus Zero-Forcing Beamforming for P
In this subsection, with a given wB, we investigate the
optimization of the passive RIS’s receive coefficient Pfor
maximazing the achievable rate, which can be formulated as
max
Pmin
∆J
RU(P)s.t. C4 : [P]n2,n2= 1,∀n2.(13)
Although some existing techniques have been widely adopted
for rate maximazation, such as the polyblock approximation
[15], the SCA [7], and the SDR [45], they lead to the
suboptimal or even far away from the optimal solution with
high computational complexity. Thus, we propose a novel anti-
7
jamming criterion7for our considered syetem, which makes
the closed-form solution is much likely available, and thereby
reducing the computational complexity.
Criterion 1: Since the malicious jammers usually transmit
with high power which is extremely detrimental to the user, the
passive RIS at the first layer is applied to nullify the jamming
signals. Besides, the desired signal should be boosted simul-
taneously. As such, the rate maximization of anti-jamming
communications can be regarded as the jamming nulling and
desired signal enhancing:
max
Pmin
∆J
RU(P) :
hH
UBUPgJ,m = 0,∀m,
max
Pe
hH
UGBwB
2
.(14)
However, the jamming-nulling term in (14) is infinite non-
convex due to the angular uncertainty ∆J. According to our
previous works [7], [15], [30], and [41], the equivalent worst-
case CSI can be obtained by the discretization method8, i.e.,
b
gJ,m=XNP1
i=1 XNP2
j=1 (1/NP)g(i,j)
J,m ,(15)
where g(i,j)
J,m is the discrete CSI by discretizing all the angles
in the set of ∆J, which is given by
θ(i)=θL+ (i−1) ∆θ, i = 1,· · · , Q1,
ϕ(j)=ϕL+ (j−1) ∆ϕ, j = 1,· · · , Q2,(16)
where Q1and Q2are the sample number of θand
ϕ, respectively, ∆θ= (θU−θL)/(Q1−1), and ∆ϕ=
(ϕU−ϕL)/(Q2−1). The interested readers can refer to our
previous works [7], [15], [30], [41] for more details, which
is omitted here for brevity. As such, the obstacle introduced
by ∆Jhas been handled. In addition, due to the presence of
active RIS, hUis undetermined such that the jamming nulling
performance is related to the active RIS. In order to nullify
the powerful jamming at the passive RIS, we further need to
ensure that BUPb
gJ,m =0,∀m. To proceed, we divide BU
into multiple vectors bU,i, which is written as
BU= [bH
U,1;bH
U,2;· · · ;bH
U,NA].(17)
Overall, the anti-jamming criterion is given by
bH
U,iPb
gJ,m = 0,∀i, m, max
Pe
hH
UGBwB
2
.(18)
By using Criterion 1, the subproblem w.r.t. p= diag(P)
can be reformulated as
max
pe
bH
Up
2
s.t. C4,C6 :e
gH
J,imp= 0,∀i, m, (19)
7Note that although the proposed criterion provides a subcase of (13), it
can achieve a satisfactory tradeoff between the computational complexity
and system performance. Specifically, as shown in Section V, although the
proposed criterion solves a suboptimal solution, it still obtains superior
performance compared to the existing conventional optimization techniques
and achieves much lower computational complexity than that of algorithms
proposed in Section III-C, which suggests a satisfactory tradeoff has been
achieved by the proposed criterion. On the other hand, the proposed criterion
contributes to the derivation of feasibility condition for jamming nulling,
which offer insightful guidelines for the practical implementation of the
proposed receiver in the anti-jamming communications. In summary, the
proposed criterion provides a satisfactory and useful subcase of (13). However,
finding the one in (14) that yields the optimal performance of (13) is still quite
a challenging task and worth in-depth analysis in our future works.
8The discretization method transforms the orignal problem (6) into a worst-
case one, which can guarantee that the constraint C2 can be satisfied under
any CSI imperfections fulfilling any required conditions [46]. Furthermore,
since there are no quality of service (QoS) constraints, the problem (6) is
always feasible with any time-varying channel fading. As such, the power
budget of the active RIS is always guaranteed under any imperfect CSI and
time-varying channel fading by using the discretization method.
1
J, 1
H
im p
g
2
J, 2
H
im p
g
3
J, 3
H
im p
g
1
J, 1
H
im p
g
2
J, 2
H
im p
g
3
J, 3
H
im p
g
( )
a
( )
b
Fig. 3: An example of NP= 3 with maxl∈[NP][e
gJ,im]l=
[e
gJ,im]3, where (a) satisfies the ZF condition and (b) contra-
dicts the ZF condition.
where e
bH
U=hH
UBUdiag {GBwB}and e
gH
J,im =
bH
U,idiag {b
gJ,m}. However, the non-convex unit-modulus con-
straints C4 prevents us from directly applying the conventional
ZF beamforming. Thus, we propose a unified unit-modulus
zero-forcing (ZF) beamforming to handle (19), which is also
applicable to conventional ZF. Before tackling (19), it is of
importance to investigate the feasibility condition for zero-
forcing, which can be divided into the feasibility condition
w.r.t. the number of passive RIS’s unit NPand w.r.t. the
channels e
gJ,im. In the sequel, we provide the following two
theorems for investigating the zero-forcing feasibility.
Theorem 1 (Feasibility Condition w.r.t. NP): For a rect-
angular passive RIS having NP=NP1 ×NP2 elements, if
min {NP1, NP2 } ≥ 4NAM−13, there exists a feasible
solution to the zero-forcing in problem (19).
Proof: See Appendix B in the supplemental information.
Observing from Theorem 1, to guarantee the feasibility for
ZF, the number of passive RIS units NPshould scale exponen-
tially with the products of the active RIS’s units number and
jammer’s number. However, this is only a sufficient condition.
According to the properties of linear equations C6 and the
simulation results, the feasibility can be ensured if NPis
only slightly larger than NP≥2M(M+ 1) NU, which is
much smaller than that stated in Theorem 1. In particular,
when the number of passive RIS’s units is slightly larger than
the product of jammers’ number and RF chains’ number, the
jamming attacks can be eliminate at the passive RIS9. On the
other hand, note that the equivalent channels e
gJ,im are random,
which may also leads to infeasibility of (19). Thus, in order to
further describe the ZF feasibility condition, we provide the
necessary and sufficient condition w.r.t. e
gJ,im.
Theorem 2 (Feasibility Condition w.r.t. e
gJ,im): The ZF
problem (19) is feasible only if
2 max
l∈[NP][e
gJ,im]l≤XNP
n2=1 [e
gJ,im]n2,∀i, m. (20)
Proof: See Appendix C in the supplemental information.
To further elaborate Theorem 2, as shown in Fig. 3, we
provide an example of NP= 3 with maxl∈[NP][e
gJ,im]l=
[e
gJ,im]3. If the ZF feasibility condition of (19) is satisfied,
we have Fig. 3(a). In particular, a triangle with the length
of three edges are constructed, i.e., [e
gJ,im]1+[e
gJ,im]2≤
9Note that in order to effectively nullify the jamming signals, the RIS-unit
number of the first layer needs to be sufficient when the number of jammer is
large. Thus, deploying the passive RIS at the first layer is appealing to large-
scale deployment as it can obtain satisfactory performance with low power
and hardware consumptions [13]
8
[e
gJ,im]3, which suggests that there always has psatisfing
PNP
n2=1 e
gH
J,imn2pn2= 0. On the contrary, if the condition is
not satisfied, we obtain Fig. 3(b), where a triangle cannot be
constructed, and thus there is no ZF solution. These above
facts suggest that if the equivalent channels e
gJ,im do not
satisfy the general triangle theorem, the adjustment of the
passive RIS’s coefficients cannot guarantee that the sum of
e
gJ,im equals to zero, namely, the jammers’ channels b
gJ,m are
orthogonal to the legitimate channels, thus the ZF feasibility
condition cannot be always ensured.
After checking the feasibility, we turn to solve problem (19).
Specifically, a unified unit-modulus ZF beamforming scheme
is proposed to obtain the semi-closed-form solution of (19)
with low computational complexity. Since an arbitrary phase
rotation of pdoes not alter the value of objective function in
(19), we can transform it into
max
p<ne
bH
Upos.t. C4,C6.(21)
To nullify the jamming signals such that constraint C6 can be
guaranteed, pmust lie in the orthogonal complement of the
subspace span ΩJ[28], namely,
ΩJ= [e
gJ,11,· · · ,e
gJ,NA1,· · · ,e
gJ,1M,· · · ,e
gJ,NAM],(22)
and the orthogonal projector G(ΩJ)can be formulated as [28]
G(ΩJ) = INP−ΩJΩH
JΩJ†ΩH
J.(23)
As such, for psatisfying constraint C6, the optimal pcan be
written as p=G(ΩJ)t, where tis the complex-valued vector
with NPelements to be optimized.
Combining p=G(ΩJ)tand (21), problem (19) can be
equivalently transformed into
max
p,t<ne
bH
Upos.t. C4,e
C6 :p=G(ΩJ)t.(24)
Since pand tare tightly coupled in constraint e
C6, we first
adopt the penalty-based method to handle e
C6 [47], namely,
max
p,t<ne
bH
UG(ΩJ)to−λkp− G (ΩJ)tk2s.t. C4,(25)
where λ > 0is a penalty factor. In the following, we tackle
pand tin an iterative manner, which admits a closed-form
solutions in each iteration.
As for the optimization of pwith given t, problem (25)
w.r.t. pbecomes:
max
p2<pHG(ΩJ)t−tHGH(ΩJ)G(ΩJ)ts.t.C4.(26)
Obviously, the optimal pfor (26) is
[p]opt
n2= [G(ΩJ)t]n2[G(ΩJ)t]n2,∀n2.(27)
Then, as for the optimization of twith given p, problem (25)
w.r.t. tcan be formulated as
max
tg(p,t) = <ne
bH
UG(ΩJ)to−λkp− G (ΩJ)tk2.(28)
Problem (28) is convex such that the optimal tcan be obtained
by setting the gradient of the objective function to zero that
yields
∂tg=GH(ΩJ)e
bU+2λGH(ΩJ) (p−G (ΩJ)t)= 0.(29)
Then, the optimal tis
topt=GH(ΩJ)G(ΩJ)†1
2λGH(ΩJ)e
bU+GH(ΩJ)p
.(30)
Note that under the alternative manner, since the optimal
closed-form solution of subproblem (26) and (28) can be
obtained, the objective value of (25) is monotonically increas-
ing at each iteration. Meanwhile, according to [48], the ZF
problem always has a finite upper bound. Thus, the proposed
algorithm always converges, whose proof is similar to [48].
C. Three Efficient Algorithms for PΞ
After optimizing wBand P, we turn to the design of active
RIS’s coefficients Ξ. After some mathematical manipulations,
we can reformulate the subproblem w.r.t ξ= diag {Ξ}as10
max
ξcHξ2
σ2
zξHR1ξ+σ2
U
(31)
s.t. e
C2 :ξHR2ξ≤PR,max,e
C3 : [ξ]n1≤αn1,max,∀n1,
where
cH=vH
UHUdiag {BUPGBwB},
R1=σ−2
zR3+diag vH
UHUdiag HH
Uv,
R2= diag {e
wB}diag e
wH
B+σ2
zINA.
The above problem (31) are challenging to solve with the con-
ventional RIS-related optimization algorithm and the algorithm
proposed in Section III-B, since the active RIS introduces
additional dynamic noise at the denominator of objective
functions, which leads to a quadratic fractional programming
problem. Although the Dinkelbach method and SDR in [26]
can be also applicable to problem (31) which optimizes the
active RIS’s phase shifts and amplitude jointly, the method
cannot obtain the closed-form solutions and may leads to
suboptimal or even far from the optimal solution, which will
be evalated in Section V. Thus, to obtain the closed-form
solution with low computational complexity, this paper divides
the optimization of Ξinto two subproblems as in [25], namely,
the optimization of active RIS’s phase shifts PΞand the
optimization of its amplitude e
Ξ, which admits the closed-form
solutions of PΞand e
Ξ. In this subsection, we focus on solving
the subproblem w.r.t. PΞ, i.e.,
max
pΞe
cHpΞ2
σ2
zpH
Ξe
R1pΞ+σ2
U
s.t. e
C3a:[pΞ]n1= 1,∀n1,(32)
where PΞ= diag {pΞ},e
cH=cHe
Ξ, and e
R1=e
ΞHR1e
Ξ.
However, problem (32) is still challenging to handle due to
the fractional form of objective function. To address this issue,
we first adopt the Dinkelbach’s method to transform (32) into
an equivalent form [49], i.e.,
min
pΞ,τ h(τ, pΞ) = pH
Ξe
CpΞs.t. e
C3a:[pΞ]n1= 1,∀n1,(33)
where τis the non-negative Dinkelbach’s parameter to be
optimized, and e
C=−e
ce
cH+τσ2
ze
R1. It can be seen that
problem (33) is NP-hard due to the multiplicative variables pΞ
and τ, and the unit-modulus constraints e
C3a. Thus, we propose
three efficient algorithms specializing on the MM and CCD
optimization methods to handle problem (33), which admit the
closed-form solutions for each optimization variables. In the
following, we provide the detials of each algorithm11.
1) AMM Algorithm: We can see that the objective function
of problem (33) is partially convex, namely, given one variable
10Note that after nullifying the jamming signals at passive RIS, the jamming
signals will not consume a lot of power at active RIS, i.e., the constraint
e
C2,
which is one key reason for deploying passive RIS at the first layer. In addition,
after nullifying the jamming signal by the passive RIS at the first layer, the
active RIS at the second layer can further utilize the amplitude’s DoF to boost
the desired signal power and focus the enhanced signal on the Rx antennas
11According to [50], the BSUM optimization framework successively opti-
mizes certain upper bounds or surrogate functions of the original objectives,
possibly in a block-by-block manner, which includes numerous well-known
methods, such as AMM and CCD schemes. Thus, the proposed algorithm
based on AMM and CCD is the variants of BSUM scheme.
9
it is convex in the other. Thus, we can utilize the partial
convexity of the objective and propose alternating minimiza-
tion along with the MM framework, which is called AMM
algorithm. Different from the conventional MM framework
adopted in the existing RIS-aided works (e.g., [19]) where the
cost function is majorized for all the optimization variables and
a gradient projection (GP) method is utilized to tune the step-
size parameter, AMM algorithm only need majorization for
the beamforming vector and the other variable are optimized
in closed-form [51]. As such, the differences lead to efficient
and effective MM implementations.
At the (id+ 1)-th iteration, the variable τis updated by
assuming the previous solution p(id)
Ξfor pΞ. Next, the problem
(33) is addressed for pΞ, using τ(id+1) for variable τ. This
procedure for the abovementioned two subproblems can be
expressed as
τ(id+1) =zτ, p(id)
Ξ,
p(id+1)
Ξ= arg min
pΞ∈A hτ(id+1),pΞ,(34)
where zis the mapping function from τ, p(id)
Ξto closed-
form solution, and A={pn| |pn|= 1,∀n}.
Using the above procedure, the subproblem w.r.t. τadmits
the following solution [49], i.e.,
τ(id+1) =zτ, p(id)
Ξ=e
cHp(id)
Ξ
2
σ2
zp(id),H
Ξe
R1p(id)
Ξ+σ2
U
.(35)
Now, assuming τ(id+1) for variable τ, we turn to solve
the subproblem w.r.t. pΞin (34). However, the minimization
problem is NP-hard. Thus, we first find a majorizing function
for the objective function of the subproblem w.r.t. pΞand
then propose the AMM framework. Note that subproblem
is approximated by using a majorizing function, while the
remaining variable is obtained in closed-form, which is the
key difference between the proposed AMM framework and
the block MM framework. To proceed, we adopt the following
lemma to construct a majorizing function for the subproblem.
Lemma 2 [52]: For the quadratic function pH
ΞSpΞwith S
being a Hermitian matrix, it is majorized by
pH
ΞTpΞ+ 2<pH
Ξ(S−T)p(id)
Ξ+p(id),H
Ξ(T−S)p(id)
Ξ,
at the point p(id)
Ξ, where TSis a Hermitian matrix.
Based on the Lemma 2, we majorize the hτ(id+1),pΞ
and obtain a tight upper bound of it, which is expressed as
hτ(id+1),pΞ≤e
hpΞ;τ(id+1),p(id)
Ξ(36)
=λmax ne
CopH
ΞpΞ
+ 2<pH
Ξe
C−λmax ne
CoINAp(id)
Ξ
+p(id),H
Ξλmax ne
CoINA−e
Cp(id)
Ξ.
Due to the unit-modulus property pH
ΞpΞ=NA, the first
term in e
hpΞ;τ(id+1),p(id)
Ξis independent of pΞ. Hence,
by ignoring the constant term, the majorized subproblem for
pΞin (34) can be formulated as
min
pΞ
<pH
Ξe
C−λmax ne
CoINAp(id)
Ξs.t. e
C3a.(37)
Clearly, it can be seen that (37) admits the following closed-
form solution pΞof (34), i.e.,
pΞ=e−jarg(e
C−λmax{e
C}INA)p(id)
Ξ.(38)
Finally, the overall AMM algorithm is obtained.
2) C-CCD Algorithm: To improve the convergence speed
and the performance of AMM algorithm (see Section V), we
develop C-CCD algorithm, where all variables are concate-
nated into one vector [τ;pΞ]and NA+ 1 scalar subproblems
are addressed upon the block size chosen. Here, we choose the
block size as 1. Note that analog to AMM algorithm, we only
solve the subproblem w.r.t. pΞ, while the remaining variables
are updated block-wise such that the closed-form solutions can
be obtained, which again results in the improvement of CCD
implementation than state-of-art. In the following, we present
the details for the proposed C-CCD algorithm.
At first, the objective function h(τ, pΞ)of problem (33)
can be further expanded as
pH
Ξe
CpΞ=XNA
j=1 XNA
i=1 p∗
Ξ,i e
C(i,j)pΞ,j
=XNA
i=1 p∗
Ξ,i e
C(i,i)pΞ,i +XNA
j6=iXNA
i=1 p∗
Ξ,i e
C(i,j)pΞ,j
=XNA
i=1 e
C(i,i)+<XNA
i=1 p∗
Ξ,iC(i),(39)
where C(i)=Pj<i
j=1 e
C(i,j)pΞ,j +PNA
j>i e
C(i,j)pΞ,j . Note that
the third equation holds due to the unit-modulus property
|pΞ,n1|= 1,∀n1and the fact that e
Cis Hermitian matrix. As
such, we can obtain NA+ 1 scalar subproblems and update
them by the following C-CCD algorithm.
τ(id+1)
1=zτ, p(id)
Ξ,1, p(id)
Ξ,2,· · · , p(id)
Ξ,NA,
p(id+1)
Ξ,1=arg min
pΞ,1∈A hτ(id+1), pΞ,1, p(id)
Ξ,2,· · · , p(id)
Ξ,NA,
.
.
.
p(id+1)
Ξ,i = arg min
pΞ,i∈A
hτ(id+1),· · · , p(id+1)
Ξ,i−1, pΞ,i,· · · , p(id)
Ξ,NA,
.
.
.
p(id+1)
Ξ,NA=arg min
pΞ,NA∈A
hτ(id+1), p(id+1)
Ξ,1, p(id+1)
Ξ,2,· · · , pΞ,NA.
(40)
It is of importance to note that the problem (33) is addressed
for each component pΞ,i. As shown before, given the value for
pΞ, the minimization problem (33) w.r.t. τcan be solved by
the closed-form solution as given in (35). Next, we deal with
the minimization problem w.r.t. the each component pΞ,i of
pΞ. By ignoring the constant terms inside (39), the subproblem
w.r.t. pΞ,i can be expressed as
min
pΞ,i
<p∗
Ξ,i Xj<i
j=1 e
C(i,j)p(id+1)
Ξ,j +XNA
j>i e
C(i,j)p(id)
Ξ,j
s.t. e
C3a:pΞ,i ∈ A.(41)
It is evident that problem (41) admits the closed-form solution
of (33), which is given by
pΞ,i =ejarg−Pj<i
j=1 e
C(i,j)p(id+1)
Ξ,j −PNA
j>i e
C(i,j)p(id)
Ξ,j ,∀i. (42)
3) M-CCD Algorithm: Apart from the proposed C-CCD al-
gorithm, M-CCD algorithm is also proposed to solve problem
(33), where a new update rule is employed so that both the
convergence speed and performance are improved.
Here, we denote that τ(id+1)
ias the i-th inner update of τ
10
at the id-th outer iteration. Then, problem (33) can be solved
by the following subproblems, which is given by
τ(id+1)
1=zτ, p(id)
Ξ,1, p(id)
Ξ,2,· · · , p(id)
Ξ,NA,
p(id+1)
Ξ,1=arg min
pΞ,1∈A hτ(id+1)
1, pΞ,1, p(id)
Ξ,2,· · · , p(id)
Ξ,NA,
.
.
.
τ(id+1)
i=zτ, p(id+1)
Ξ,1,· · · , p(id+1)
Ξ,i−1, p(id)
Ξ,i ,· · · , p(id)
Ξ,NA,
p(id+1)
Ξ,i =arg min
pΞ,i∈A
hτ(id+1)
i,· · · , p(id+1)
Ξ,i−1, pΞ,i,· · · , p(id)
Ξ,NA,
.
.
.
τ(id+1)
NA=zτ, p(id+1)
Ξ,1, p(id+1)
Ξ,2,· · · , p(id)
Ξ,NA,
p(id+1)
Ξ,NA=arg min
pΞ,NA∈A
hτ(id+1)
NA, p(id+1)
Ξ,1, p(id+1)
Ξ,2,· · · , pΞ,NA.
(43)
It can be seen that in the M-CCD procedure, we update the
variable τafter obtaining the solution of pΞ,i. As already
shown, the closed-form solution of each subproblem can be
obtained by the same methods proposed in C-CCD algorithm,
namely, (35) and (42). After the overall iterations, we set
τ(id+1) =τ(id+1)
NA. Based on where we update τ, two different
algorithms are proposed.
It is worth noting that the three efficient algorithms are also
applicable to solve (13) for the closed-form solutions, which
can be regarded as the other algorithm for handling (6), and
it will be evaluated in Section V. In the following, we provide
the convergence guarantees of proposed AMM and C/M-CCD
frameworks, which are given by the following theorems.
Theorem 3 (Convergence and Optimality of AMM): Denote
nτ(id),p(id)
Ξoas the sequence generated by the proposed
AMM algorithm. Then, nτ(id),p(id)
Ξocan converge to the
KKT point of problem (33), which can be regarded as the
coordinatewise optimal solutions [50].
Proof: See Appendix D in the supplemental information.
Theorem 4 (Convergence and Optimality of C/M-CCD):
Let nτ(id), p(id)
Ξ,i obe the sequence generated by the proposed
C/M-CCD algorithm. Then, nτ(id), p(id)
Ξ,i oconverges to the
KKT point of problem (33), which can be regarded as the
coordinatewise optimal solutions [50].
Proof: The proof is similar to that of Theorem 3 and thus
is omitted here for brevity.
D. Effective Countermeasures for e
Ξ
Here, we focus on solving the amplitude matrix of active
RIS e
Ξ. By defining b
c=|c|,R1=PH
ΞR1PΞ, and e
ξ=
diag ne
Ξo, the subproblem w.r.t. e
ξcan be formulated as
max
e
ξb
cTe
ξ
2
σ2
ze
ξTR1e
ξ+σ2
U
(44)
s.t. C2 : e
ξTR2e
ξ≤PR,max,e
C3b: 0 ≤e
ξn1≤αn1,max,∀n1.
Similar to (32), the Dinkelbach’s method is also adopted to
tackle the fractional form of objective function in (44), and
thus proplem (44) can be recast as
max
e
ξ,ρ
rρ, e
ξ=e
ξTb
Ce
ξs.t. C2,e
C3b,(45)
where b
C=b
cb
cT−ρσ2
zR1, and ρis the non-negative
Dinkelbach’s parameter. Obviously, the algorithms proposed
for solving problem (33) can be tailored for problem (45).
Thus, we provide them in the following details.
1) AMM Algorithm: The AMM procedure for problem (45)
can be expressed as
ρ(id+1) =oρ, e
ξ(id),
e
ξ(id+1)= arg min
e
ξ
rρ(id+1),e
ξ.
(46)
According to [49], the closed-form solution of subproblem
w.r.t. ρin (46) can be obtained as
ρ(id+1) =oρ, e
ξ(id)=b
cTe
ξ(id)
2
σ2
ze
ξ(id),T R1e
ξ(id)+σ2
U
.(47)
Next, by utilizing the first-order Taylor series, the subproblem
w.r.t. e
ξcan be rewritten as
max
e
ξe
ξTb
Ce
ξ(id)s.t. C2 : 2e
ξTR2e
ξ(id)≤e
PR,max,(48)
e
C3b: 2e
ξTΛn1e
ξ(id)≤eα2
n1,max,∀n1,
where e
PR,max =PR,max +e
ξ(id),T R2e
ξ(id)and eα2
n1,max =
α2
n1,max +e
ξ(id),T Λn1e
ξ(id). Clearly, problem (48) is convex,
and thus can be directly solved.
2) C-CCD Algorithm: Similar to (41) stated in Section III-C,
the objective function rρ, e
ξin (45) can be expanded as
e
ξTb
Ce
ξ=XNA
i=1 e
ξ2
ib
C(i,i)(49)
+<XNA
i=1 e
ξiXj<i
j=1 b
C(i,j)e
ξj+XNA
j>i b
C(i,j)e
ξj.
Then, problem (45) can be equivalently transformed into
max
e
ξi,ρ e
ξ2
ib
C(i,i)+<e
ξibcamp,i(50)
s.t. b
C2 : e
ξ2
iR2,(i,i)≤e
PR,i,∀i, e
C3b:e
ξ2
i≤α2
i,max,∀i,
where e
PR,i =PR,max −PR1,i,
PR1,i =PNA
j<i e
ξ(id+1),2
jR2,(i,j)+PNA
j>i e
ξ(id),2
jR2,(i,j),
bcamp,i =Pj<i
j=1 b
C(i,j)e
ξ(id+1)
j+PNA
j>i b
C(i,j)e
ξ(id)
j.
As such, the C-CCD procedure can be adopted to solve (50).
Since the C-CCD procedure is similar to (40), we do not
provide the detailed derivation steps for obtaining it for brevity.
As for the update of ρin (49), the closed-form solution can
be obtained by (47), while for the update of e
ξi, by using the
properties of quadratic function, the closed-form solution of
e
ξican be expressed as
e
ξi=di,if b
C(i,i)>0,<(bcamp,i)≥0,
e
ξi= 0,if b
C(i,i)>0,<(bcamp,i)<−2b
C(i,i)di,
e
ξi= arg max
e
ξi={0,di}
re
ξi,
if b
C(i,i)>0,−2b
C(i,i)di≤ < (bcamp,i)<0,
e
ξi=di,if b
C(i,i)<0,<(bcamp,i)≥ −2b
C(i,i)di,
e
ξi=bcamp,i
−2b
C(i,i)
,if b
C(i,i)<0,0<<(bcamp,i)<−2b
C(i,i)di,
e
ξi= 0,if b
C(i,i)<0,<(bcamp,i)≤0,
(51)
11
where di= min re
PR,i.R2,(i,i), ai,max .
3) M-CCD Algorithm: Depending upon where to update ρ,
we can also obtain the M-CCD algorithm for problem (45),
which can be referred from (43). As already shown, each prob-
lem in M-CCD can be solved by the solutions obtained from
C-CCD algorithm, i.e., (47) and (51). Finally, analogous to
Theorem 3, we can also establish the convergence guarantees
to the KKT points for all the abovementioned algorithm.
E. MMSE Decoder for vU
In this subsection, we investigate the design of the digital
decoder vUfor maximizing the receive SINR. As we know, the
linear minimum-mean-square-error (MMSE) detector is the
optimal digital decoder for maximizing the receive SINR [53],
which can balance the interference and noise at the receiver.
Thus, we directly adopt MMSE detector for vU, as given by
vU=wBwH
B+R3+σ2
zHUΞΞHHH
U+σ2
UINU−1
wB
wBwH
B+R3+σ2
zHUΞΞHHH
U+σ2
UINU−1
wB
,(52)
where R3=PM
m=1 PJ,mr3,m rH
3,m,r3,m =HUΞBUPb
gJ,m,
and wB=HUΞBUPGBwB.
F. Convergence and Complexity Analysis
Combining all the proposed algorithms above, the integrated
efficient low-complexity optimization framework is obtained.
In the following, we analyze the convergence of proposed
optimization framework. As already shown, the convergence of
four subproblems have been proven such that a better solution
of each subproblem can be obtained. Thus, the objective func-
tion of (6) RU(wB,P,Ξ,vU)is a monotonically increasing
sequence, which can be expressed as
RUw(n)
B,P(n),Ξ(n),v(n)
U≤RUw(n+1)
B,P(n),Ξ(n),v(n)
U
≤RUw(n+1)
B,P(n+1),Ξ(n+1) ,v(n)
U
≤RUw(n+1)
B,P(n+1),Ξ(n+1) ,v(n+1)
U.(53)
Besides, due to the compact set spanned by constraints C1 and
C2, the optimization framework guarantees to converge.
On the other hand, the computational complexity of the
proposed optimization framework is presented. For the op-
timization of wB, its computational complexity is dominated
by the computation of (9) and (10), which is computed as
O(NB+C)2+NB[47]. As for the design of P, the com-
plexity of unit-modulus ZF beamforming lies in computing
(27) and (30), which can be obtained as O(I1NP(NP+ 1)).
Here, Idenotes the iteration number. Then, we analyze the
computational complexity of AMM and C/M-CCD algorithm.
For the AMM algorithm, the worst-case per iteration com-
plexity lies in updating the auxiliary variables and solving
the desired variables. Therefore, the complexity of optimizing
PΞand e
Ξvia AMM can be obtained as O(2I2NA)and
OI2N2
A+NA, respectively. Note that due to the use
of CVX in solving e
Ξ, the complexity of optimizing e
Ξvia
AMM is much higher than that of PΞ. While for the C-
CCD algorithm, since it updates each element of the opti-
mization variables, the complexities of C-CCD for solving
PΞand e
Ξincrease to both OI2N2
A. Finally, for the M-
CCD algorithm, the computation of each desired variable
is followed by the update of auxiliary variables, thus the
complexities of M-CCD for solving PΞand e
Ξfurther increase
to both OI22N2
A−NA. Overall, the total complexity of
the proposed optimization framework is given by
OAMM =Omax nO(NB+C)2+NB,O(2I2NA),
O(I1NP(NP+ 1)) ,OI2N2
A+NA,
OC−CCD =Omax nO(NB+C)2+NB,O2I2N2
A,
O(I1NP(NP+ 1))}),
OM−CCD =Omax nO(NB+C)2+NB,
O2I22N2
A−NA,O(I1NP(NP+ 1)).(54)
Obviously, we can find that the proposed M-CCD requires the
highest computational complexity, followed by the C-CCD,
while the AMM algorithm achieves the lowest one. Further-
more, it is worth noting that the complexity required by all the
proposed algorithms are significantly lesser than that of the
SDR method requiring ON2
B(C+ 1),ON4
P(NP+ 1),
and ON4
A(NA+ 1)for solving wB,P, and Ξ, respectively
[42]. Particularly, the complexity of SDR for solving (6) is
given by
OSDR =Omax ON2
B(C+ 1),ON4
P(NP+ 1),
ON4
A(NA+ 1).(55)
Comparing (54) with (55), the complexity of the proposed
optimization framework is much lesser than that of SDR, and
thus is beneficial for practical implementation.
To further illustrate the complexity induced by the proposed
architecture, we compare it with those of the other types
of receivers. Here, we assume that all the architectures are
equipped with the same number of antennas or RIS units to
ensure a fair comparison. Specifically, the fully-digital receiver
in [42] can flexibly control both the phase and amplitude due
to the utilization of NA+NPRF chains, resulting in a total
number of 2(NA+NP)optimization variables. Thus, the
complexity of MMSE algorithm induced by the fully-digital
receiver is computed as O(NA+NP)2. In addition,
the single-layer passive RIS-receiver has NA+NPphase
variables that need to be optimized, such that the complexity of
proposed algorithm for it is obtained as O(2I3(NA+NP)).
Clearly, by comparing them with the complexity of proposed
architecture in (54), the fully-digital receiver has lower
complexity since the proposed receiver introduces the need to
design the two layers of RIS’s coefficients with unit-modula
constraint iteratively. Similarly, due to the fact that the
proposed architecture has additional NAamplitude variables
to be optimize and incorporates the extra active RIS’s
power constraints to the formulated problem, the complexity
induced by the proposed architecture is higher than that of
single-layer passive RIS-receiver. On the other hand, the
single-layer active RIS-receiver and fully-connected phase-
array receiver have 2(NA+NP)and NA+NUNAnumbers
of optimization variables to be designed, respectively, thus
complexity induced by them can be given by OSingle−layer =
Omax nO(2I4(NP+NA)) ,OI4(NP+NA)2+NP
12
+NA))})and OPhase−array =
Omax nO2I5(NUNA)2,O(I1NP(NP+ 1))}). Since
the number of optimization variables of the proposed receiver
is lower than those of the single-layer active and phase-array
ones, I4and I5are much higher than I2. Thus, we can obtain
that the complexity induced by proposed architecture is lower
than those of the single-layer active and phase-array ones.
The above facts also confirm the superiority and scalability
of our proposed architecture in the practical implementation.
IV. PERFORMANCE ANA LYSI S
To further analyze the performance gain attained by the pro-
posed cascaded RIS-aided receiver architecture, we consider a
simple SISO scenario, where a single-antenna BS transmits the
desired symbols to a single-antenna user with the assistance of
the proposed RIS-aided receiver. Based on the scenario above,
we have the following theorem characterizing both the power
scaling law and asymptotic SINR of proposed architecture.
Theorem 5: Assuming that gB∼ CN (0, ςgI),BU=
ρ1NA×NP,hU=ρ1NA×1, and the amplification factors of
each active RIS’s units adopt the same value p, if NP→ ∞,
we have the following power scaling law and asymptotic SINR
of the proposed architecture, as given by, respectively,
PProp.→π2−7π+ 16Pmax
Bp2bρ4N2
AN2
Pςg
4,(56)
γProp.→NANPbρ4Pmax
BPR,maxπ2ς2
gNP
16 (PR,max bρ2σ2
zNP+bρ4σ2
UPmax
B).(57)
Proof: See Appendix E in the supplemental information.
Obviously, the receive desired power PProp.is proportional
to p2N2
AN2
P, while the asymptotic SINR γProp.is proportional
to NANPdue to the dynamic noises additionally introduced
by the active components. However, it is worth noting that
although the active components introduce additional dynamic
noises, the proposed cascaded RIS-aided receiver can still
improve the SINR, since the multiple active RIS units can
coherently add up the desired signals at the Rx antenna while
the dynamic noises cannot.
To highlight the superior performance of the proposed cas-
caded RIS-aided architecture, we compare it with the existing
single-layer RIS-aided architecture. According to [25] and
[26], the active RIS can achieve better performance than the
passive one. Thus, we here choose the single-layer active
RIS-aided receiver as the benchmark, which have the same
total number of RIS units with the proposed architecture.
Specifically, similar to Theorem 5, the power scaling law and
asymptotic SINR of the single-layer active RIS-aided receiver
can be expressed as
PSingle →π2−7π+ 16Pmax
Bp2bρ2(NP+NA)2ςg
4,(58)
γSingle→(NP
+NA)bρ2Pmax
BPR,maxπ2ς2
g(NP
+NA)
16 (PR,max bρ2σ2
z(NP
+NA) +bρ2σ2
UPmax
B).
Observing from (58), we can find that the receive desired
power of the single-layer active RIS-aided receiver PSingle is
proportional to (NP+NA)2, whereas its asymptotic SINR is
proportional to NP+NA. Then, comparing (56) with (58), the
ratio PProp./PSingle can be given, respectively, by
PProp.
PSingle
=AProp.
ASingle 2
=bρNANP
NP+NA2
,(59)
where Aidenotes the amplitude of receive signal. Generally,
the RIS’s power efficiency bρis near 0.8 [29], while NA
and NPare always large-scale, such that PProp./PSingle is
significantly larger than 1. In addition, we can also see that the
same conclusion holds for γProp./γSingle. The aforementioned
results suggest that the proposed cascaded active-passive RIS-
aided architecture can flexibly and effectively amplify the
amplitude of the receive signals, which is significantly higher
than that of the single-layer active RIS-aided one.
To further justify the claimed advantages above, we also
provide the following theorem:
Theorem 6: Denoting ynas the signal radiated from the
n-th RIS unit on the second layer, the phase of yncan be
adjusted to any desired angle in [−π, π), while the amplitude
can be adjusted in the range [0, yn,max], namely
y2
n,max =a2
n,max
16π2Zla√NR
2
−la√NR
2
dpxZla√NR
2
−la√NR
2
dpz(60)
l1l2p2
x+l2
1h(px−xn)2+l2
2i
n(p2
x+p2
z+l2
1)h(px−xn)2+ (pz−zn)2+l2
2io2.5,
where lais the side-length of each RIS unit, l1and l2are
the distances between the feed and first layer, the first layer
and second layer, respectively, and (xn, l1+l2, zn)denotes the
position of n-th RIS unit on the second layer.
Proof: See Appendix A in [31].
Obviously, yn,max can be flexibly adapted with a larger
range than those of the typical and single-layer RIS-assisted
receiver whose amplitude is set to a constant one. Combining
with the results in Theorem 5 and Theorem 6, we confirm
that our proposed cascaded RIS-aided receiver generates more
DoFs than the existing RIS-aided receivers, which will be
further validated in the following Section V.
V. SIMULATION RESU LTS
In this section, we present the numerical results to evaluate
the superiority and validity of our proposed algorithms. We
assume that there are M= 2 jammers, the noise power at the
active RIS and the Rx antennas are σ2
z=σ2
U=−70 dBm [19],
and the carrier frequency is 5.8 GHz [30], [54]. Besides, the
total number of the BS’s antenna’s cluster is C= 4 [55], and
the maximum power at BS and active RIS are PB= 30 dBm
and PR,max= 40 dBm [26], respectively. Thus, the maximum
power at each BS’s cluster is pB,c = 0.25 W. Moreover, the
maximum amplification factor is set as αn1,max=10 [26], and
the CSI uncertainty is defined as ∆J=θU−θL= 4◦. The
BS is located at (0 m, 0 m, 60 m), and the two jammers
are located at (15 m, 30 m, 5 m) and (45 m, 5 m, -10 m),
respectively. In addition, the user is located at the direction
{(θ, ϕ)|(45◦,60◦)}with a distance of 110 m w.r.t. the BS.
Here, we compare the following architectures and algorithms:
•Prop. arch.: The proposed cascaded RIS-aided receiver
architecture is adopted.
13
(a) 3D beampattern of prop. arch. (b) 3D beampattern of single-layer arch. (c) 3D beampattern of digital arch.
(d) 2D beampattern of prop. arch. (e) 2D beampattern of single-layer arch. (f) 2D beampattern of digital arch.
Fig. 4: Receive beampattern with different architectures (colorbar on the right is unified for 3 subfigures, unit: dB).
•Digital arch.: The fully-digital receiver architecture in
[15] with N=NA+NPunits is adopted, and the optimal
MMSE decoder in [53] is exploited to design the decoder.
•Phase-array arch.: The fully-connected phase-array re-
ceiver architecture in [15] with N=NARx antennas and
NANUphase shifters is adopted to repalce the active RIS
of proposed receiver.
•Single-layer arch.: The single-layer active RIS-aided
receiver architecture N=NA+NPunits is adopted, and
the M-CCD algorithm is applied to obtain Ξ. To highlight
the superiorities of proposed architecture, we consider
a favorable setting for the single-layer one, where the
maximum amplification is set as 2.5, while ignore active
RIS’s power constraints.
•UM-ZF and AMM(C/M-CCD): Under the proposed
cascaded RIS-aided receiver architecture, the proposed
unit-modulus ZF (UM-ZF) and AMM(C/M-CCD) algo-
rithms are utilized to handle Pand Ξin (6), respectively.
•Double AMM(C/M-CCD): As shown in footnote 5, un-
der the proposed architecture, the proposed AMM(C/M-
CCD) algorithms are adopted to solve both Pand Ξ.
•SDR method: Under the proposed architecture, Dinkel-
bach method and SDR in [26] are used to solve (6).
Fig. 4 shows the normalized receive beampattern of dif-
ferent architectures and evaluates the quality of the beam by
comparing their mainlobes and nulls. Here, we set NP= 8×8,
NA= 4 ×4,NU= 2 ×2, and the signal-to-jamming plus-
noise-ratio (SJNR) is SJNR =hPB.PM=1
m=1 PJ,mi[dB ]=−20
dB. As can be observed, due to the same number of RIS
units constraint, the proposed architecture has a lower lobe
resolution than both the single-layer and fully-digital archi-
tecture. However, the proposed architecture can still accu-
rately generate the nulls towards the jammers’ regions, and
simultaneously align the mainlobes with the desired target,
even under the angular uncertainty. Furthermore, it is worth
noting that the received SINR of proposed architecture at the
BS’s direction is about 0 dB, while those of single-layer and
fully-digital architecture are about -10 dB. This phenomenon
suggests that the proposed active-passive cascaded RIS-aided
architecture can significantly amplify the amplitude of the
desired receive signals, which is consistent with Theorems 5-6
and their corresponding analysis. As such, combining the fact
that the received SINR of all the architectures at the jammer’s
direction are about -50 dB, we can conclude that the mainlobe
to sidelobe ratio of proposed architecture is much higher than
that of benchmark architectures, which implies that additional
DoFs are introduced by the active-passive cascaded RIS-aided
structure for jamming nulling and signal enhancing.
-14
-12
-10
-8
-6
-4
-2
0
Rx antennas' area
(a)
-30
-28
-26
-24
-22
-20
-18
-16
-14
-12
-10
Rx antennas' area
(b)
Fig. 5: Power distribution on Rx antennas for different archi-
tectures (colorbar on the right is unified for each subfigure,
unit: dBW): (a) Proposed architecture with PSR 32.8%; (b)
Single-layer active RIS-aided architecture with PSR 75.9%.
To further reveal the benefits of the active-passive cascaded
RIS-aided structure, the power distribution of the proposed
cascaded and the single-layer active RIS-aided architecture
are presented in Fig. 5. As expected, the Rx antennas’ area is
assigned most of the power in the proposed cascaded architec-
ture, while most of the power scatters outside the Rx antennas’
area in the single-layer active RIS-aided architecture. To
quantify this phenomenon, the PSR metric defined in Section
II-A are adopted here. To elaborate, the proposed cascaded
architecture has a PSR of 32.8%, which is 2.78 times lower
14
than that of single-layer active RIS-aided architecture having
a PSR of 75.9%. The potential reason for the phenomenon is
that the second layer active RIS of the proposed architecture
has fewer units, such that the illumination area of second-layer
active RIS is inherently concentrated on the Rx antennas’ area.
Moreover, benefiting from the amplitude control capability
of the active RIS, the power can be further concentrated on
the targets. On the other hand, similar to Fig. 4, we can
find that the receive power of the proposed architecture is
significantly higher than that of single-layer architecture. This
result means that the amplitude gain caused by the active-
passive cascaded RIS-aided structure can compensate for the
power scattering effects. The abovementioned findings verify
that the proposed active-passive cascaded RIS-aided receiver
can effectively overcome the power scattering effects, such
that obtain the unique performance ascendancy.
Fig. 6: Achievable rate versus NPand NA.
Fig. 6 illustrates the achievable rate versus the number of
passive and active RIS units NPand NA. It can be seen
that when NP<6×6, the proposed architecture obtains
a lower achievable rate than the other three architectures.
This is because the ZF feasibility condition presented in
Theorem 1 cannot be always satisfied when NP<6×6
such that the jamming signals cannot be perfectly eliminated
at the passive RIS and introduce huge dynamic noise at
the active RIS, thereby resulting in significant performance
degradation. Nonetheless, when NPis slightly larger than
6×6, the proposed architecture achieves the highest achievable
rate among the considered schemes. The reason is that after
nullifying the jamming signals at the passive RIS, the desired
signal power can be enhanced in proportion to p2N2
AN2
P,
whose value is significantly higher than those of the other
architectures (see Figs. 4 and 5). The above results suggest
that the ZF feasibility can be ensured if NPis only slightly
larger than NP≥2M(M+ 1) NU, even under ∆J, which
confirms the claims presented in Theorem 1. Furthermore,
due to γProp.∼NANP,γSingle∼(NP+NA)in Theorem 5
and the fact that the amplitude of phase-array architecture
cannot be controlled, we can see that the achievable rate
of all the architectures increase with NPand NA, and the
increasing spped of the proposed architecture is much higher
than that of the other three architectures w.r.t. NPand NA.
The finding indicates that the amplitude of passive RIS can
be also partially altered, which brings about a new DoF for
beamforming design that can be beneficially exploited for
performance enhancement.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
=NA/(NA+NP)
19
20
21
22
Achievable rate (bit/s/Hz)
Prop. arch., corresponding achievable rate
Prop. arch., corresponding energy efficiency
1.1
1.2
1.3
1.4
Energy efficiency (bps/Hz/Joule)
Fig. 7: Achievable rate and energy efficiency versus β.
To find the proper fraction of active RIS’s unit number to
the total unit number which leads to the best performance for
implementation (i.e., β=NA/(NA+NP)) , Fig. 7 shows the
achievable rate and energy efficiency (EE) versus β, and the
EE is calculated by
EE=RU.kwBk2+kΞe
wBk2+σ2
zkΞk2
F+PProp.,(61)
where PProp.= (NP+NA)pris +NAparis +NUpr+Apb,
pris,pr,pb, and paris are the power consumption of each
RIS units, RF chain, baseband processor, and power am-
plifier circuit of active RIS unit, respectively. Here, we set
NP+NA= 100, and the power consumption of each compo-
nent in the proposed architecture is the same as [30]. We can
see that both the achievable rate and the energy efficiency
achieve the maximum at β= 0.36. The results can be
explained through the following reasons. First, as shown in
Theorem 5, the receive desired power PProp.is proportional to
p2N2
AN2
P, thus the desired power increases when NPis closed
to NA, which results in the phenomenon that both achievable
rate and energy efficiency increase with βwhen β < 0.36.
However, as stated in Theorem 1,NPshould be slightly
larger than 2M(M+ 1) NUfor eliminating jamming signal,
otherwise leads to performance degradation. Thus, considering
with the fact that the power consumption increases with β,
both achievable rate and energy efficiency decrease with β
when β>0.36. These findings indicate that β≈0.4should be
chosen for nullifying jamming and enhancing signal.
Next, the convergence performance of the proposed op-
timization framework is illustrated in Fig. 8, where the
convergence behaviors of two major algorithms, including
the UM-ZF scheme in Section III-B and AMM (C/M-CCD)
algorithms in Section III-C, are also illustrated. Here, SJNR
is SJNR = −20 dB, and NU= 2 ×2. It can be seen that
the abovementioned algorithms can monotonically converge
to a stationary point for all settings of NPand NA. Besides,
comparing Fig. 8 (a) and (b), we can observe that the iteration
number of UM-ZF algorithm in per iteration is significantly
smaller than that of AMM (C/M-CCD) algorithms. Thus,
combining with the complexity analysis in Section III-F and
the fact shown in Fig. 8 (c) and (d) that all the algorithms
using UM-ZF converge faster than the remaining algorithms,
we confirm that the proposed UM-ZF algorithm achieves the
lowest complexity, which further supports the footnote 5. On
the other hand, as shown in Fig. 8 (b)-(d), the C/M-CCD
algorithms not only converge faster than AMM algorithm in
the per iteration, but also have smaller number of iterations in
15
(a) (b)
(c) (d)
Fig. 8: Convergence of UM-ZF, AMM (C/M-CCD), and the
overall algorithms under different settings of NPand NA.
the overall algorithm, which further highlights the superiority
of C/M-CCD algorithm. It is also important to highlight
the fact that due to the modified update for τ, the M-CCD
algorithm can achieve better convergence performance than
the C-CCD algorithm.
Fig. 9 presents the achievable rate versus SJNR, where
NP= 8 ×8,NA= 4 ×4, and NP= 2 ×2. It is important
to highlight the fact that the double C/M-CCD algorithms
significantly outperform the algorithms using UM-ZF scheme,
while the double AMM algorithm only slightly outperforms
them. One of the potential reasons is that, C/M-CCD update
each piand admit a closed-form solution, while both the AMM
and UM-ZF algorithms directly optimize overall p. Combining
with the complexity analysis in Fig. 8, we can find that in
the double AMM( C/M-CCD) algorithms, the computational
complexity is sacrificed for the performance enhancement.
Nevertheless, it can be observed that the algorithms using
UM-ZF scheme can still obtain the superior performance in
comparison to the single-layer architecture as well as the SDR
method, and its achievable rate is also close to that of digital
architecture. Therefore, the algorithms using UM-ZF scheme
obtains a good trade-off between the computational complexity
and the performance, thereby contributing to the scalability
of the UM-ZF algorithm. Furthermore, the UM-ZF scheme
can provide the essential ZF feasibility conditions, which
guides us for practical implementation. Besides, in the M-
CCD algorithm, the modified update for the auxiliary variable
improves the achievable rate in comparison to the C-CCD
algorithm. Finally, as expected, the achievable rate of all the
algorithms increases with SJNR, and all the algorithms related
to the proposed architecture increase faster than the remaining,
especially the AMM algorithm. This is because the unit
number at the first-layer passive RIS is smaller than those of
single-layer and digital architecture, such that its beam-pattern
resolution is also lower, thereby resulting in the fact that it
-30 -25 -20 -15 -10 -5
SJNR (dB)
16
18
20
22
24
26
28
Achievable rate (bit/s/Hz)
UM-ZF + AMM
Double AMM
UM-ZF + C-CCD
Double C-CCD
UM-ZF + M-CCD
Double M-CCD+PR= 40 dBm
SDR method
Digital arch.
Single-layer arch.
Phase-array arch.
Double M-CCD+PR= 30 dBm
Fig. 9: Achievable rate versus SJNR.
Fig. 10: Total power consumption versus NPand NA.
is more sensitive to jamming power. Overall, our proposed
algorithms can obtain satisfactory performance in terms of
both the achievable rate and complexity, and thus provide a
flexible choice of algorithm for practical applications.
On the other hand, we can observe that if the active RIS
is equipped with a larger amplification power budget, the
proposed RIS-receiver can achieve a better performance. In
addition, as higher amplification power budget can introduce a
larger amplification gain for the desired signals, the achievable
rate gap between the schemes with 40 dBm and that with 30
dBm decreases with the increasing SJNR. Furthermore, even
under PR,max= 30 dBm, the proposed RIS-receiver can still
obtain the higher achievable rate than the other benchmarks,
which further confirms the superiority of our proposed archi-
tecture.
To further illustrate the low-cost feature of proposed ar-
chitecture, the total power consumption versus NPand NA
for different architectures is presented in Fig, 10. Here, the
total power consumptions of different architectures are calcu-
late based on the parameters in [42], [56], [57]. It can be
seen that the total power consumptions of the digital and
phase-array architecture are much higher than that of the
proposed cascaded RIS-receiver, due to the excessive power
consumption of the RF chains and analog network for large
numbers of antennas in the former. To elaborate, the analog
network is comprised of power splitters, phase shifters, and
line connections, which lead to high power consumption and
hence reduce energy efficiency. Furthermore, we can also
observe that the proposed cascaded RIS-receiver have almost
the same power consumption as the single-layer one due to
the utilization of the same number of RIS units. This finding
confirms the proposed cascaded RIS-receiver is inherently
energy- and cost-efficient compared to the benchmarks.
16
VI. CONCLUSIONS
This paper investigated an active-passive cascaded RIS-
aided receiver architecture for anti-jamming communications,
which facilitates the deployment of a large-scale antenna array
at the user side and provides additional DoFs for practical
beamforming design. Utilizing this architecture and taking the
imperfect angular CSI into account, a worst-case achievable
rate maximization problem was formulated for simultaneously
nullifying the jamming and boosting the desired signal. To
handle the formulated intractable problem, a low-complexity
optimization framework was developed by leveraging La-
grange dual theory, Pareto optimization theory, unified unit-
modulus zero-forcing scheme, and AMM (C/M-CCD) method,
which admits the semi-closed-form solutions of all optimiza-
tion variables. Furthermore, the performance analysis of the
proposed architecture was provided, and two jamming-nulling
feasibility conditions were derived. The theoretical and simu-
lation results showed that the receive power and the asymptotic
SINR of proposed architecture are proportional to N2
AN2
Pand
NANP, respectively, and the proposed algorithm can attain
the excellent performance with low complexity if the channels
satisfy the general triangle theorem and the number of passive
units is slightly larger than 2M(M+ 1) NU, which further
confirmed the superiority of the proposed architecture and
optimization framework in comparison with the existing ones.
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Yifu Sun received the B.Eng. in Communications
Engineering from National University of Defense
Technology (NUDT), Changsha, China, in 2019,
where he is currently pursuing the Ph.D. degree
in Information and Communications Engineering
with College of Electronic Science and Technology.
His current research interests are in anti-jamming
communications, reconfigurable intelligent surface,
physical layer security, cooperative and cognitive
communications, massive MIMO systems, and sig-
nal processing for wireless communications.
Yonggang Zhu received the B.S. degree in elec-
tronical engineering and Ph.D. degree in science of
military equipment from PLA University of Science
and Technology, Nanjing, China, in 2004, and 2009,
respectively. Currently, he is a research associate
professor with the Sixty-third Research Institute, Na-
tional University of Defense Technology, Nanjing,
China. His research interests include compressive
sensing, statistical signal processing, reconfigurable
intelligent surface, and anti-jamming communica-
tion.
Kang An received the B.E. degree in electronic
engineering from Nanjing University of Aeronautics
and Astronautics, Nanjing, China, in 2011, and PhD
degree in communication engineering from Army
Engineering University, Nanjing, China, in 2017.
He is currently an associate professor with the
Sixty-third Research Institute, National University
of Defense Technology, Nanjing, China. His cur-
rent research interests include reconfigurable intel-
ligent surface, anti-jamming communications, satel-
lite/aerial communications, physical-layer security,
signal processing and machine learning for wireless communications. He has
published more than 100 peer-reviewed research papers in leading journals
and flagship conferences and many of them are ESI highly cited papers.
He was listed in the Worlds Top 2%Scientists identified by Stanford
University in 2022 and 2023. He was a recipient of exemplary Reviewer for
IEEE Transactions on Communications and IEEE Communications Letters
in 2022. He was the recipient of the Outstanding Ph.D. Thesis Award of
Chinese Institute of Command and Control in 2019. He is also serving as an
Editor for Frontiers in Communications and Networks and Frontiers in Space
Technologies. He was the corecipient of the Best Paper Awards at the IEEE
IWCMC 2023 and the IEEE ICCT 2023.
18
Zhi Lin received the B.E. and M.E. degrees in
information and communication engineering from
the PLA University of Science and Technology and
the Ph.D. degree in electronic science and tech-
nology from the Army Engineering University of
PLA, Nanjing, China, in 2013, 2016, and 2020,
respectively. From March 2019 to June 2020, he was
a visiting Ph.D. student with the Department of Elec-
trical and Computer Engineering, McGill University,
Montr´
eal, Canada. Since February 2023, he is a
Postdoctoral Fellow with the School of Computer
Science and Engineering, Macau University of Science and Technology,
Macau, China. Since January 2021, he has been with the College of Electronic
Engineering, National University of Defense Technology, Hefei, China, where
he is currently an Associate Professor.
Dr. Lin’s research interests include array signal processing, physical layer
security, reconfigurable intelligent surface and satellite-aerial-terrestrial inte-
grated networks. He was the Symposium Co-Chair of IEEE WCSP’22 and
TPC members of IEEE flagship conferences, including IEEE ICC, Globecom,
Infocom, VTC, and so on. He was listed in the World’s Top 2% Scientists
identified by Stanford University in 2023. He was the recipient of the
Outstanding Ph.D. Thesis Award of Chinese Institute of Electronics in 2022,
the Macao Young Scholars Fellowship in 2022 and the Best Paper Award
from IEEE IWCMC 2023 conference and the IEEE ICCT 2023. He has
been serving as an Academic Editor for the Wireless Communications and
Mobile Computing since 2023. He was also a Lead Guest Editor of the IET
Communications Special Issues on Reconfigurable Intelligent Surfaces Aided
Physical Layer Security in 6G Wireless Networks.
Cheng Li received the B.S. degree in information
engineering, the M.S. and Ph.D. degree in infor-
mation and communication engineering from Na-
tional University of Defense Technology, Changsha,
in 2007, 2009 and 2015, respectively. He is cur-
rently an assistant research fellow of the Sixty-third
Research Institute, National University of Defense
Technology, Nanjing. His current research interests
include signal processing, wireless communication
and electromagnetic countermeasure.
Derrick Wing Kwan Ng (Fellow, IEEE) received
a bachelor’s degree with first-class honors and a
Master of Philosophy (M.Phil.) degree in electronic
engineering from the Hong Kong University of Sci-
ence and Technology (HKUST) in 2006 and 2008,
respectively. He received his Ph.D. degree from the
University of British Columbia (UBC) in Nov. 2012.
He was a senior postdoctoral fellow at the Institute
for Digital Communications, Friedrich-Alexander-
University Erlangen-N¨
urnberg (FAU), Germany. He
is now working as a Scientia Associate Professor at
the University of New South Wales, Sydney, Australia. His research interests
include global optimization, physical layer security, IRS-assisted communica-
tion, UAV-assisted communication, wireless information and power transfer,
and green (energy-efficient) wireless communications.
Dr. Ng has been listed as a Highly Cited Researcher by Clarivate Analytics
(Web of Science) since 2018. He received the Australian Research Council
(ARC) Discovery Early Career Researcher Award 2017, the IEEE Communi-
cations Society Leonard G. Abraham Prize 2023, the IEEE Communications
Society Stephen O. Rice Prize 2022, the Best Paper Awards at the WCSP
2020, 2021, IEEE TCGCC Best Journal Paper Award 2018, INISCOM 2018,
IEEE International Conference on Communications (ICC) 2018, 2021, IEEE
International Conference on Computing, Networking and Communications
(ICNC) 2016, IEEE Wireless Communications and Networking Conference
(WCNC) 2012, the IEEE Global Telecommunication Conference (Globecom)
2011, 2021 and the IEEE Third International Conference on Communications
and Networking in China 2008. He served as an editorial assistant to the
Editor-in-Chief of the IEEE Transactions on Communications from Jan. 2012
to Dec. 2019. He is now serving as an editor for the IEEE Transactions on
Communications and an Associate Editor-in-Chief for the IEEE Open Journal
of the Communications Society.
Jiangzhou Wang (Fellow, IEEE) is a Professor with
the University of Kent, U.K. He has published more
than 400 papers and four books. His research focuses
on mobile communications. He was a recipient of
the 2022 IEEE Communications Society Leonard
G. Abraham Prize and IEEE Globecom2012 Best
Paper Award. He was the Technical Program Chair
of the 2019 IEEE International Conference on Com-
munications (ICC2019), Shanghai, Executive Chair
of the IEEE ICC2015, London, and Technical Pro-
gram Chair of the IEEE WCNC2013. He is/was the
editor of a number of international journals, including IEEE Transactions on
Communications from 1998 to 2013. Professor Wang is a Fellow of the Royal
Academy of Engineering, U.K., Fellow of the IEEE, and Fellow of the IET.
1
Active-Passive Cascaded RIS-Aided Receiver
Design for Jamming Nulling and Signal Enhancing
(Supplemental Information)
Yifu Sun, Yonggang Zhu, Kang An, Zhi Lin, Cheng Li,
Derrick Wing Kwan Ng, Fellow, IEEE, and Jiangzhou Wang, Fellow, IEEE
APPENDIX A
PROO F OF PROPOSITION 1
Here, we first establish the equivalence between problems
(7) and (8), and then provide the proof for the dual method
in (9). Now, we define the optimal solution of (7) as Wopt
B,
and subsequently, it can be proved to be an optimal Pareto
solution of following problem (A.1) by using contradiction:
min
WB{tr (EcWB)}C
c=1 s.t.f (WB) = fWopt
B.(A.1)
Specifically, if Wopt
Bis not the optimal Pareto solution of
(A.1), there must exist a solution WB,1satisfing f(WB,1) =
fWopt
Band tr (EcWB)≤tr (EcWB,1). In other words,
WB,1can achieve the same performance as Wopt
B, but
consumes less transmit power. Obviously, this conclusion
contradicts the assumed optimality of Wopt
B, thus the optimal
solution of the original problem (7) Wopt
Bis an optimal Pareto
solution of (A.1). To effectively handle the multi-objective
optimizatipn problem (A.1), we can apply the well-known
scalarization method to convert it into
min
WBXC
c=1 βctr (EcWB)s.t. f (WB) = fWopt
B,(A.2)
where βc≥0is the non-negative dual variable which enables
tr EcWopt
B≤pB,c. Then, based on the Lagrangian dual
theory, problem (A.2) can be transformed into (8). Obviously,
recalling the matrix-monotone decreasing property of objective
function f(WB), we can readily obtain that the optimal
solution of (A.2) is the same as that of (8).
In the sequel, we drive an accurate mapping between ηc
and βc. To be specific, the Lagrangian function of the original
optimization problem (7) is written as
L1WB,Q,{ηc}C
c=1=f(WB)+XC
c=1
ηc(tr (EcWB)−pB,c)
=f(WB) + µ(tr (EWB)−PB)−tr (QWB).(A.3)
Clearly, we obtain βc=ηcPB.PC
i=1 ηipB,i and µ=
PC
i=1 ηipB,i.PB. Then, the related KKT conditions are
∂WBf(WB)+XC
c=1
ηcEc−Q=∂WBf(WB)+µE−Q=0,
ηc(tr (EcWB)−pB,c)=0,QWB=0,∀c, Q0.(A.3)
Since µcan be regarded as the positive dual variable associated
with the unified single power constraint tr (EWB)≤PB, the
first equality in (A.3) and tr (EWB)−PB= 0 constitute
the KKT optimality of (8). As such, utilizing the mapping
βc=ηcPB.PC
i=1 ηipB,i, the equivalence between problems
(7) and (8) are proved by the dual theory.
Finally, we provide the proof for (9). Recalling the La-
grangian function of (7),
L1WB,Q,{ηc}C
c=1= tr (F1WB)−XA
c=1 ηcpB,c,(A.4)
and the dual objective is expressed as [28]
min
WB0max
Q0,{ηc}C
c=1≥0LWB,Q,{ηc}C
c=1.(A.5)
Note that there are no constraints on WB, if F1are not
positive semidefinite, (A.5) will approach negative infinity.
Besides, Q, ηcshould be chosen so that (A.5) is finite. Thus,
we can obtain the dual problem in (9).
APPENDIX B
PROO F OF TH EO RE M 1
At first, we consider the simplified case that NP1 = 1 with
the line-of-sight (LoS) channel model and ∆J= 0. In this
case, the b
gJ,m and bU,i can be rewritten as
b
gJ,m =hgJ,m0,·· · , gJ,m0ej2πd2(N2−1)
λcos θRx,
J,md iT
,
bU,i = [bU,1, bU,2,·· · , bU,NP2]T.(B.1)
Thus, we have
e
gH
J,im=gJ,m0bH
U,1, bH
U,2zm,·· ·, bH
U,NP2 zNP2−1
m,(B.2)
where zm=ej2πd2
λcos θRx
J,m0. Then, we define f2(zm)as a
polynomial of degree NP2 with unit modulus coefficients, i.e.,
f2(zm)=gJ,m0bH
U,1p1+· ·· +bH
U,NP2 zNP2−1
mpNP2 ,(B.3)
where |pn2|= 1,∀n2. Thus, to nullify the jamming signals,
the ZF constraint C6 becomes
f2(zm)=0,∀m= 1,2,· ·· , M. (B.4)
Clearly, the original of jamming nulling problem is transm-
formed into the polynomial problem in (B.4), so that zm’s
are the roots. By using Lemma 1 as follows, we have if
NP2 =4NAM−13, the polynomial f2exists.
Lemma 1 [23]: Assuming p1,··· , pnare on the complex
unit circle C={x∈
C
:|x|= 1}, there exists a polynomial
function f2of degree Pn
i=1 4i−1with unit modulus coeffi-
cients, which makes pn’s are the only zeros of f2on C.
To extend the theorem to the rectangular array with NP1 >
1, we can divide the rectangular uniform array into NP1
columns of uniformly linear array with NP2 ×1units.
Obviously, increasing NP1 can lead to the more DoF for
jamming nulling, thus the ZF condition can be also achieved
if NP2 =4NAM−13. Finally, if we exchange the roles of
NP2 and NP1, we can have the same results.
2
APPENDIX C
PROO F OF TH EO RE M 2
Based on the ZF constraints C6, we obtain
e
gH
J,imlpl=−PNP
n26=le
gH
J,imn2pn2,∀i, m. Then,
we take the absolute value at both sides yields:
e
gH
J,imlpl=PNP
n6=le
gH
J,imn2pn2.Due to |pn2|= 1,∀n2
and the triangle inequality, we have
e
gH
J,iml=XNP
n26=le
gH
J,imn2pn2(C.1)
≤XNP
n26=le
gH
J,imn2pn2=XNP
n26=le
gH
J,imn2.
Thus, by adding e
gH
J,imlinto both sides, (C.1) becomes:
2e
gH
J,iml≤XNP
n2=1 e
gH
J,imn2.(C.2)
Clearly, the inequality (C.2) should be satisfied for any
l∈[NP], and thus (20) can be obtained by adding maxl∈[NP]
into the left side of (C.2).
APPENDIX D
PROO F OF TH EO RE M 3
First, we show the descent of the objective function
h(τ, pΞ), which is expressed as
hτ(id),p(id)
Ξ(a)
≥hτ(id+1),p(id)
Ξ(b)
=e
hp(id)
Ξ;τ(id+1),p(id)
Ξ
(c)
≥hp(id+1)
Ξ;τ(id+1),p(id)
Ξ(d)
≥hp(id+1)
Ξ;τ(id+1),p(id+1)
Ξ.
(D.1)
Inequality (a) holds from the update of τ[49], equation (b)
follows because e
hp(id)
Ξ;τ(id+1),p(id)
Ξis a valid majorizer
over pΞ, inequality (c) holds from (34), and inequality (d)
follows from the MM’s descent property. Thus, the sequence
nhτ(id),p(id)
Ξois monotonically decreasing.
Next, we turn to provide that the sequence generated by
AMM algorithm converges to a KKT point of (33). As for
the convergence for τ, we have the following inequalities to
demonstrate it, namely,
hτ(im),p(im)
Ξ(a1)
≥hτ(im+1),p(im)
Ξ
(b1)
≥hτ(im+1),p(im+1)
Ξ(c1)
≥hτ(im+1),p(im+1 )
Ξ.(D.2)
These inequalities follow from the decreasing descent
in (D.1). Then, when m→ ∞, based on the con-
vergence of nτ(im),p(im)
Ξo and the continuity of
h(τ, pΞ)and e
hpΞ;τ(id+1),p(id)
Ξ, we have hτ, p(∞)
Ξ≥
hτ(∞),p(∞)
Ξ. The above suggests that τ(∞)is a block-wise
minimizer of h(τ, pΞ). Thus, the partial KKT conditions w.r.t.
τare satisfied, i.e., ∂τhτ(∞),p(∞)
Ξ= 0. Then, we focus
on pΞ, and begin with the following claim.
Claim 1:τ(im), τ (im+1) →τ(∞).
To prove the above claim, the following equivalent condi-
tions should be first proven:
τ(im+1) ≥τ(im),lim
im→∞ τ(im)=τ(+) > τ(∞).(D.3)
According to [49], (D.3) can be always guaranteed such that
Claim 1 is proved. Now, we adopt the claim in the following
steps.
Considering the KKT condition w.r.t. pΞ, we have
e
hpΞ;τ(im+1),p(im)
Ξ(a2)
≥e
hp(im+1)
Ξ;τ(im+1),p(im)
Ξ
(b2)
≥hτ(im+1),p(im+1)
Ξ(c2)
≥hτ(im+1),p(im+1 )
Ξ.(D.4)
The above inequalities follow from the earlier (D.1) and the
upper bound property of the majorizing function. Thus, when
m→ ∞, combining with the convergence of τ, we can have
e
hpΞ;τ(∞),p(∞)
Ξ≥e
hp(∞)
Ξ;τ(∞),p(∞)
Ξ.(D.5)
Clearly, p(∞)
Ξis a minimizer w.r.t. pΞfor e
h, as given by
∂pΞe
hp(∞)
Ξ;τ(∞),p(∞)
Ξ+ 2αp(∞)
Ξ= 0,
∂pΞhτ(∞),p(∞)
Ξ+ 2αp(∞)
Ξ= 0,(D.6)
where αis the dual-variable associated with the unit-modulus
constraints. Note that by using the gradient consistency con-
dition, (D.6) can be obtained. Thus, the KKT conditions w.r.t.
pΞis satisfied. Finally, combining the partial KKT conditions
and (D.6) w.r.t. τand pΞ, we have
∂τhτ(∞),p(∞)
Ξ
∂pΞhτ(∞),p(∞)
Ξ
+ 2 0
ατ
pΞ=0.(D.7)
Thus, based on (D.1) and (D.7), nhτ(id),p(id)
Ξo mono-
tonically decreases to the KKT point. Besides, according to
Theorem 1 in [50], the KKT solutions of AMM are equivalent
to the coordinatewise optimal solutions. Thus, the proposed
AMM converge to the coordinatewise optimal solutions.
APPENDIX E
PROO F OF TH EO RE M 5
First, we prove (56). Defining GB=gB,hH
U=HU, and
wB=ω, the power received at the usr can be expressed as
PProp.=phH
UPΞBUPgBω2, which can be expanded as
hH
UPΞBUPgB=
NA
X
i2=1
hH
U,i2ejθΞ,i2
NP
X
i1=1
BU,(i2,i1)ejθP,i1gB,i1
BU=ρ1NA×NP
hU=ρ1NA×1
≤ρ2
NA
X
i2=1
ejθΞ,i2
NP
X
i1=1
ejθP,i1gB,i1.(E.1)
Obviously, the optimal phase shifts are θP,i1=−∠gB,i1and
θΞ,i2= 0. Thus, PProp.can be further expressed as
PProp.=p2ρ4Pmax
B
NA
X
i2=1
NP
X
i1=1 |gB,i1|
2
.(E.2)
Since gB∼ CN (0, ςgI),|gB,i1|follows the
Rayleigh distribution with mean √πςg2and
variance (4 −π)ςg/2. By using central limit theorem,
PNP
i1=1 |gB,i1| ∼ CN NP√πςg2, NP(4 −π)ςg/2can be
obtained. As a result, we can transform (E.2) into (56).
Next, we provide the proof for (57). According to (1), the
γProp.maximization problem is
max
p,ω,P,PΞ
γProp.=phH
UPΞBUPgBω2
p2
hH
UPΞ
2σ2
z+σ2
U
(E.3)
s.t. C1 :|ω|2≤Pmax
B,C2 −C4.
3
The optimal solutions to problem (E.3) can be obtained by
using the Lagrange multiplier method:
ωopt =pPmax
B, θP,i1=−∠gB,i1, θΞ,i2= 0,
p=v
u
u
u
u
t
PR,max
Pmax
BNAρ2NP
P
i1=1 |gB,i1|
2
+σ2
zNA
.(E.4)
By substituting (E.4) into (E.3), the maximum γProp.of
proposed architecture is
γProp.=(E.5)
Pmax
BPR,maxρ2NA
NP
P
i1=1 |gB,i1|
2
PR,maxρ2σ2
zNA+σ2
U Pmax
BNAρ2NP
P
i1=1 |gB,i1|
2
+σ2
zNA!.
By letting NP→ ∞ for (E.5), we can obtain (57) via the
central limit theorem.
