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RIS-Assisted Cooperative Spectrum Sensing for
Cognitive Radio Networks
Jungang Ge, Ying-Chang Liang, Fellow, IEEE, Shuo Wang, and Chen Sun
Abstract—Cooperative spectrum sensing (CSS) is a key en-
abling technology of cognitive radio networks with multiple
secondary users (SUs). In conventional CSS systems, when
the primary signals are weak, the SUs require long sensing
time to achieve a high detection probability for protecting the
transmission of the primary user (PU), leading to little remaining
time for secondary transmissions. To address this issue, we
propose a reconfigurable intelligent surface (RIS) assisted CSS
system, where multiple RISs are employed to improve the CSS
performance within limited sensing time. Considering that the
dependency of the CSS performance on the received primary
signal strengths at the SUs differs across various CSS schemes,
the RIS configurations could also be optimized differently re-
garding these CSS schemes. Motivated by this, we investigate
the phase shift matrix (PSM) optimization problems to maximize
the cooperative detection probability given a maximum tolerable
false alarm probability, and we consider two typical kinds of
CSS schemes, namely, data fusion and decision fusion. As it is
intractable to directly solve these problems due to the complex
expressions of the cooperative detection probability with respect
to the PSMs, we show that the solutions can be obtained by
transforming these problems into channel gain-related optimiza-
tion problems. Furthermore, we show that the proposed PSM
optimization methods can be extended to the more practical
scenarios where instantaneous channel state information (CSI) is
unavailable. In such cases, we leverage statistical CSI to improve
the CSS performance in the sense of expectation. Subsequently,
we conduct a numerical analysis on the number of reflecting
elements required to achieve a target detection probability in
the statistical CSI case. Finally, simulation results demonstrate
that the proposed PSM optimization methods can significantly
improve the CSS performance within limited sensing time.
Index Terms—Reconfigurable intelligent surface, cooperative
spectrum sensing, phase shift matrix optimization.
I. INTRO DUC TIO N
With the ever-increasing wireless traffics in the modern
digital society, radio spectrum scarcity has become the main
obstacle limiting the development of future wireless commu-
nication systems [2]. To tackle this issue, dynamic spectrum
Part of this work was presented in IEEE GLOBECOM Workshops 2023
[1]. This work was supported in part by the Key Areas of Research and
Development Program of Sichuan Province, China, under Grant 9099911; in
part by the Fundamental Research Funds for the Central Universities under
Grant ZYGX2019Z022; and in part by the Program of Introducing Talents
of Discipline to Universities under Grant B20064. (Corresponding author:
Ying-Chang Liang.)
J. Ge is with the National Key Laboratory of Wireless Communications,
University of Electronic Science and Technology of China (UESTC), Chengdu
611731, China (e-mail: gejungang@std.uestc.edu.cn).
Y.-C. Liang is with the Center for Intelligent Networking and Communi-
cations, University of Electronic Science and Technology of China, Chengdu
611731, China (e-mail: liangyc@ieee.org).
S. Wang and C. Sun are with Research & Development Center
Sony (China) Limited, Beijing, China (e-mail: Shuo.Wang@sony.com,
Chen.Sun@sony.com).
access (DSA) has been identified as an efficient technique to
improve the utilization efficiency of radio spectrum resources
[3]. In DSA systems, the secondary users (SUs) without
licensed spectrum have the opportunity to access the spectrum
licensed to primary users (PUs) while ensuring the protec-
tion of primary transmissions, which can be realized by the
well-known cognitive radio (CR) technique [4]. Particularly,
spectrum sensing is regarded as a critical technique to pro-
tect primary transmissions for its capability of detecting the
occupancy status of primary spectrum bands.
In the existing literature, various spectrum sensing algo-
rithms have been proposed to achieve reliable sensing per-
formance [5]–[7], such as energy detection, cyclostationary
detection, eigenvalue-based detection, and covariance matrix-
based detection. Moreover, for a spectrum sensing system
with multiple SUs, the sensing results of multiple SUs can
be fused to generate a more accurate decision on the PU’s
presence with cooperative spectrum sensing (CSS) techniques
[8]. Despite the improved performance from optimizing the
sensing schemes, it is still quite challenging to detect the PU’s
presence reliably when the primary signal is weak, and the fail-
ure to detect the PU’s presence may cause severe interference
to primary transmission. For conventional spectrum sensing
systems, the only way to improve the sensing performance
is to increase the number of sensing signal samples, which
requires more sensing time. Besides, in opportunistic CR
networks, the SUs generally adopt the Listen-Before-Talk
sensing/transmission structure to access the primary spec-
trum. As a result, long sensing time inevitably leads to little
remaining time for secondary transmissions, hindering the
improvement of spectrum efficiency [9]. This is essentially
because the wireless environment is naturally uncontrollable,
and the signal-to-noise-ratios (SNRs) of the primary signals at
SUs are dependent on the physically uncontrollable wireless
channels. According to the central limit theorem (CLT), we
have to utilize a large number of signal samples to reduce the
uncertainty in identifying the presence of the primary signals.
Recently, the emerging reconfigurable intelligent surface
(RIS) provides a novel way to construct a programmable
wireless environment. Specifically, RIS acts as an intelligent
environmental reflector that can selectively reflect the incident
signals to specific directions by tuning its reflecting coef-
ficient matrix, and therefore it can be deployed in various
wireless communication systems for significant performance
improvement [10]–[13]. In particular, RIS has also been in-
troduced to spectrum sensing systems to improve the sensing
performance for weak primary signals, leading to less required
sensing time and higher spectrum utilization efficiency [14]–
This article has been accepted for publication in IEEE Transactions on Wireless Communications. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/TWC.2024.3393516
© 2024 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information.
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2
[20]. In [14], the authors consider an RIS-assisted spectrum
sensing system with a multi-antenna SU, where the maximum
eigenvalue-based detection method is adopted to identify the
PU’s presence. The asymptotic distribution of the maximum
sample eigenvalue is derived by random matrix theory, and
the number of required reflecting elements (REs) to achieve
a near-1detection probability is quantitatively analyzed. In
[15], the authors investigate the active RIS-assisted spectrum
sensing system compared to the passive RIS-assisted one. The
number of REs to achieve detection probability close to 1is
analyzed for both two kinds of RISs while the direct link is
ignored. Besides, the active RIS-enhanced spectrum sensing
system is also investigated in [16], where the passive and
active RISs are compared in terms of the required power
consumption to achieve a target detection probability. In
[17], an RIS-enhanced energy detection method for spectrum
sensing is investigated, and the expressions of the average
detection probability are derived. In [18], the authors consider
an RIS-assisted spectrum sensing system where the phase shift
matrix (PSM) of RIS is dynamically changed according to a
codebook consisting of some predefined PSM configurations,
and a weighted energy detection method is developed to utilize
the time-variant reflected signal power. In [19], the authors
investigate the impact of RIS configurations on the sensing
performance of SU, where two typical RIS configuration
principles are considered, namely, increasing the SNR at the
primary receiver and enhancing the sensing SNR of the SU.
The analytical average detection probabilities of both these
two configurations are also derived. The authors of [20] study
an RIS-enhanced opportunistic CR system, where the RIS is
employed to enhance the sensing performance and improve
the secondary transmission rate. The overall secondary trans-
mission rate is maximized by jointly optimizing the PSMs in
the sensing and transmission stages.
Most of the above works mainly focus on the single-SU and
single-RIS case, where the RIS can be simply configured by
aligning the PSM to the cascaded channel between the PU and
SU [19]. For the CSS system with multiple SUs, the authors of
[17] propose to improve the CSS performance by partitioning
the RIS into several sub-surfaces and equally allocating the
sub-surfaces to the multiple SUs. Then, the PSM optimization
problem under this setup can be reduced to the single-SU
case, where the sub-surfaces allocated to one SU can be easily
configured to improve the individual sensing performance of
the SU. When the number of the REs is sufficiently large,
this configuration scheme enables each SU to detect the PU’s
presence accurately, and the CSS performance of these SUs
can finally be enhanced. However, for the cases where the
number of REs is limited, the configuration scheme should be
optimized regarding the practical conditions of the wireless
environment to efficiently unleash the potential capability of
the RIS. Besides, for the CSS system with multiple SUs, the
CSS performance depends on not only the individual sensing
performance of each SU but also the adopted fusion scheme.
As a consequence, the dependency of the CSS performance on
the received primary signal strengths at the SUs differs across
various CSS schemes, the RIS configurations should also be
optimized differently for these CSS schemes. On the other
hand, the multiple-RIS system, also known as “distributed
RISs”, can be utilized to further enhance the performance
of RIS-aided wireless communication systems [21]–[23], es-
pecially in scenarios with multiple receivers [22]. Compared
to conventional single-RIS systems, the multiple-RIS system
allows a distributed and flexible deployment of numerous REs,
and the receivers are able to benefit from some REs closer than
the single-RIS structure.
In this paper, we consider a general RIS-assisted CSS
system consisting of multiple SUs and multiple RISs, where
the PSMs of the RISs are jointly optimized to enhance the
CSS performance of the SUs. Our main contributions are
summarized as follows.
•This work expands upon the concept of integrating RIS
to elevate spectrum sensing performance within a more
general spectrum sensing system encompassing multi-
ple SUs, where multiple RISs are deployed to improve
the CSS performance. Particularly, considering that the
dependency of the CSS performance on the received
primary signal strengths at the SUs differs among various
CSS schemes, the PSMs of the RISs are optimized
differently regarding these CSS schemes.
•We delve into the PSM optimization problems focused on
maximizing the cooperative detection probability given
a maximum tolerable false alarm probability, and both
data fusion and decision fusion based CSS schemes are
explored. Given the complexity in directly solving the
cooperative detection probability maximization problems,
we transform these problems to channel gain-related
optimization problems, which allows for easier attainment
of solutions to the PSM optimization problems.
•We demonstrate that the proposed PSM optimization
methods can be extended to more practical situations
where the instantaneous CSI is unavailable. Particularly,
the statistical CSI is utilized to optimize the PSMs for
enhancing CSS performance in the sense of expectation.
Subsequently, we perform a numerical analysis on the
number of REs required to attain a target detection
probability in the statistical CSI case.
•Last but not least, we provide extensive numerical re-
sults to demonstrate the effectiveness of the proposed
PSM optimization methods, as compared to a benchmark
where each RIS is partitioned into several sub-surfaces to
enhance the individual sensing performance of each SU.
Moreover, the characteristics of the PSMs optimized for
different CSS schemes are studied, and the impacts of
the number of signal samples and channel conditions on
the number of required REs to achieve a target detection
probability are investigated.
The rest of this paper is organized as follows. Section II
illustrates the RIS-assisted CSS system encompassing multi-
ple RISs and multiple SUs, and reviews the basics of CSS
techniques. In Section III, the PSM optimization problems
for the considered CSS schemes are formulated based on the
instantaneous CSI, and the corresponding PSM optimization
methods to obtain the solutions are provided. Section IV
extends the PSM optimization methods in Section III to more
This article has been accepted for publication in IEEE Transactions on Wireless Communications. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/TWC.2024.3393516
© 2024 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information.
Authorized licensed use limited to: Sony Corporation. Downloaded on May 05,2024 at 02:06:47 UTC from IEEE Xplore. Restrictions apply.
3
practical scenarios where instantaneous CSI is unavailable,
and conducts a numerical analysis on the number of REs
required to achieve a target detection probability in such cases.
In Section V, we provide extensive numerical simulations
to evaluate the proposed PSM optimization methods. Finally,
Section VI concludes this paper.
Notations: The notations used in this paper are as follows.
The matrices, column vectors, scalars, and sets are denoted by
bold uppercase, bold lowercase, lowercase, and calligraphic
uppercase symbols (e.g., X,x,x, and X), respectively. The
conjugate of xis denoted by x†. The Hermitian of xand
Xare respectively denoted by xHand XH.| · | and k · k
represent the absolute value and Euclidean norm operations.
Tr(·)and blkdiag(·)respectively denote the trace and block
diagonalization operations. x∼ CN(0, σ2)means that xis
a complex Gaussian random variable with zero mean and
variance σ2.x∼ N(0, σ 2)means that xis a real Gaussian
random variable with zero mean and variance σ2.
II. RI S-A S SI STE D COOP ERATI V E SPE CTRU M SE NS I NG
A. System Model
As shown in Fig. 1, we consider a general RIS-assisted
CSS system, where LRISs are deployed to improve the CSS
performance of NSUs. Besides, all the SUs and the PU are
equipped with a single antenna, and the l-th RIS consists of
Ml=Mh
l×Mv
lREs, where Mh
land Mv
lrespectively denote
the numbers of REs along the horizontal and vertical axes.
In such a system, the multiple RISs concurrently reflect the
primary signals to the SUs, and the reflections are controlled
by the PSMs of the RISs. Moreover, the multi-hop links among
the multiple RISs usually suffer from large path loss, therefore
the signals reflected by the RISs twice or more times can be
ignored. Particularly, the channel between the PU and RIS l
is denoted by fl∈CMl,gn,l ∈CMlrepresents the channel
between SU nand RIS l, and the direct channel between SU
nand the PU is denoted by dn. There are two hypotheses
in spectrum sensing problems, namely, the PU-absent case
denoted by H0and the PU-present case denoted by H1. In
each transmission frame, the SUs should first detect the PU’s
presence, and then the secondary transmission can be launched
only when the PU is detected as absent. Specifically, during
the sensing stage, each SU collects Tsignal samples to detect
the PU’s presence, and the t-th signal sample of SU nunder
the two hypotheses can be respectively given by
H0:xn(t) = un(t),(1)
H1:xn(t) = dn+
L
X
l=1
gH
n,lΦlfl!s(t) + un(t),(2)
where Φl= diag([ejφl,1,··· , ejφl,Ml]) ∈CMl×Mlis a
diagonal matrix representing the PSM of RIS lwith φl,i the
phase shift of the i-th RE, s(t)∼ CN(0, σ2
s)denotes the
transmitted symbol of the PU, and un(t)∼ CN(0, σ2
u)is the
additive white Gaussian noise (AWGN) at SU n.
B. Channel Model
For ease of notation, we assume that each RIS has the
same array size, i.e., Mh
l=Mh,Mv
l=Mv,Ml=M,
PU
SU 1
SU nSU N
f1
dn
fl
gn,1
gn,l
RIS 1
RIS l
Fig. 1. The RIS-assisted CSS system.
∀l= 1,···, L. In the RIS-assisted CSS system, it is fair
to assume that the PU-SU channels are of Rayleigh fading
because the line-of-sight (LOS) components are blocked.
Moreover, as the PU and RISs are usually deployed at high
towers and buildings, the PU-RIS channels can be modeled
with LOS channels [24]. Furthermore, the SUs are usually
small base stations surrounded by some scatters, and the
RISs are deployed to provide channels with LOS components.
Therefore, we consider that the RIS-SU channels, namely,
g1,1,···,gN,L, are of Rician fading. Let ηndenote the path
loss of the channel between the PU and SU n, we have
dn∼ CN(0, ηn). Besides, the channel between the PU and
RIS lcan be characterized by
fl=pβlah(θAOA
l, ψAOA
l)⊗av(θAOA
l, ψAOA
l),(3)
where βldenotes the path loss between the PU and RIS l,θAOA
l
and ψAOA
lare respectively the azimuth angle of arrival (AOA)
and elevation AOA, ah(θAOA
l, ψAOA
l)and av(θAOA
l, ψAOA
l)are
the steering vectors defined as
ah(θ, ψ) = [1,··· , e−j2πd
λ(Mh−1) sin(θ) cos(ψ)]T,(4)
av(θ, ψ) = [1,··· , e−j2πd
λ(Mv−1) cos(θ) cos(ψ)]T,(5)
with dthe element space of the RIS that is usually set as λ/2,
and ⊗is the Kronecker product operation. Additionally, the
channel between RIS land SU ncan be denoted by gn,l =
¯
gn,l +˜
gn,l, where ¯
gn,l is the LOS component given by
¯
gn,l =sκn,lζn,l
1 + κn,l
ah(θAOD
n,l , ψAOD
n,l )⊗av(θAOD
n,l , ψAOD
n,l ),(6)
˜
gn,l is the non-line-of-sight (NLOS) component that satisfies
˜
gn,l ∼ CN 0,ζn,l
1 + κn,l
IM.(7)
Specifically, ζn,l denotes the path loss between RIS land SU
n,κn,l is the Rician factor, θAOD
n,l and ψAOD
n,l are the azimuth
angle of departure (AOD) and elevation AOD.
This article has been accepted for publication in IEEE Transactions on Wireless Communications. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/TWC.2024.3393516
© 2024 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information.
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4
TABLE I
DET EC TI ON PR OBA BILI TIE S AN D F AL SE A LARM PRO BAB ILI TI ES O F DI FFE RE NT D ECI SI ON FU SI ON RUL ES
Decision Fusion Rule Cooperative Detection Probability, PdCooperative False Alarm Probability, Pf a
AND QN
n=1 Pd,n QN
n=1 Pf a,n
OR 1−QN
n=1(1 −Pd,n ) 1 −QN
n=1(1 −Pf a,n )
Krank PN
i=KPN
i
j=1 Qn∈KjPd,n Qm /∈Kj(1 −Pd,m)PN
i=KPN
i
j=1 Qn∈KjPf a,n Qm /∈Kj(1 −Pf a,m)
C. Cooperative Spectrum Sensing
In CR networks, energy detection, which allows the SU to
detect the PU’s presence by comparing its collected signal
energy with a given threshold, is one of the most well-
known spectrum sensing methods for its low computational
complexity. In the RIS-assisted CSS system, we consider that
each SU performs individual spectrum sensing with energy
detection. Therefore, the test statistic of SU ncan be formed
with the sensing signal samples as
En=1
T
T
X
t=1 |xn(t)|2.(8)
By comparing Enwith a given threshold γn, SU ncan make
a hard decision Dnon the PU’s presence as
Dn=(0,if En< γn,
1,if En≥γn,(9)
where Dn= 0 means that the PU is detected as absent and
Dn= 1 means that the PU is detected as present. The false
alarm probability and detection probability of SU n, namely,
Pfa,n and Pd,n, are respectively defined as
Pfa,n =P(En> γn|H0), Pd,n =P(En> γn|H1).(10)
In addition, the CSS technique allows multiple SUs to detect
the PU’s presence in a collaborative manner. Specifically, the
sensing results or the hard decisions of the multiple SUs are
reported to a fusion center (FC), and then a fused decision can
be made according the fusion scheme adopted at the FC. In
general, there are two kinds of fusion schemes, namely, data
fusion and decision fusion. In data fusion-based CSS schemes,
each SU reports its individual sensing result Ento the FC, and
the FC combines the collected En’s to form a cooperative test
statistic, which can be written as [9], [25]
EDaF =
N
X
n=1
wnEn,(11)
where wnis the weighing factor for SU n. In practice, equal
gain combining, also known as square law combining [25], is
usually adopted as the data fusion method for its simplicity.
In decision fusion-based schemes, the FC only collects the
hard decisions from the SUs and then makes a fused decision
according to a specified decision fusion rule. As compared to
data fusion based schemes, decision fusion-based schemes can
significantly reduce the overhead of reporting sensing results
for SUs. Particularly, there are three commonly used decision
fusion rules, namely, logic-AND, logic-OR, and Krank, which
can be represented respectively by
DAND =D1&···&DN,(12)
DOR =D1|···|DN,(13)
DKrank =(0,if PN
n=1 Dn< K,
1,if PN
n=1 Dn≥K, (14)
where &and |respectively denote the logic-AND and logic-
OR operations. It is worth noting that the logic-AND and
logic-OR schemes can be regarded as special versions of
the Krank schemes, which can be easily verified by letting
K=Nand K= 1, respectively. The Krank scheme with
K=⌈N/2⌉is also referred to as the well-known “Majority
Rule”. The cooperative detection probabilities and false alarm
probabilities of these decision fusion rules are summarized in
Table I.
III. PHA SE SH IFT MATR IX OPT IMI ZATIO N W IT H
INS TAN TAN EOU S CS I
To improve the CSS performance, we consider maximizing
the cooperative detection probability for a given maximum
tolerable false alarm probability ¯
Pfa. According to Neyman-
Pearson lemma, the detection probability can be maximized
when the actual false alarm probability approaches ¯
Pfa.
Hence, given a maximum tolerable false alarm probability
¯
Pfa,n for SU n, the detection threshold of SU ncan be
determined by letting
P(En> γn|H0) = ¯
Pfa,n.(15)
During the sensing stage, the SUs usually need to collect a
large number of sensing signal samples to detect the PU’s
presence. With the CLT, the distribution of Enunder H0can
be approximated as a Gaussian distribution, i.e.,
En|H0=1
T
T
X
t=1 |un(t)|2∼ N σ2
u,1
Tσ4
u,(16)
and the detection threshold can be determined by
γn=r1
Tσ2
uQ−1(¯
Pfa,n) + σ2
u,(17)
where Q−1(·)is the inverse Q function with Q(x) =
R∞
x1/√2πe−t2/2dt. Besides, the test statistic Enunder H1
can be written as
En|H1=1
T
T
X
t=1 dn+
L
X
l=1
gH
n,lΦlfl!s(t) + un(t)
2
.
(18)
This article has been accepted for publication in IEEE Transactions on Wireless Communications. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/TWC.2024.3393516
© 2024 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information.
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5
0.8 0.9 1 1.1 1.2 1.3 1.4
0
2
4
6
8
10
12
14
Fig. 2. PDFs of En|H1for different ρn’s, while T= 1000,σ2
s=σ2
u= 1.
Given a maximum tolerable false alarm probability, the detection probability
can be enhanced by increasing ρn.
Denoting hn,dn+PL
l=1 gH
n,lΦlfland ρn,|hn|2, the
distribution of Enunder H1can also be approximated with a
Gaussian distribution as
En|H1∼ N σ2
u+ρnσ2
s,(σ2
u+ρnσ2
s)2
T.(19)
From Table I, it can be observed that the performance
of all the CSS schemes can be improved if the individual
sensing performance of each SU is enhanced. To achieve this
goal, one well-known way is increasing the sensing time such
that more sensing signal samples can be collected. This can
be explained with the CLT, i.e., utilizing numerous sensing
signal samples can reduce the variance of the test statistic to
detect the PU’s presence. However, longer sensing time also
means less remaining time for the secondary transmissions,
which will degrade the spectrum efficiency of CR networks.
The proposed RIS-assisted CSS system provides a novel way
to address this issue, where the sensing performance within
limited sensing time can be enhanced by reconstructing the
channels between the PU and SUs. As such, the requirement of
numerous sensing signal samples can be avoided. In Fig. 2, the
probability density functions (PDFs) of the test statistic under
H1are illustrated for different ρn’s. Obviously, for a given
detection threshold achieving the maximum tolerable false
alarm probability ¯
Pfa,n, the detection probability Pd,n can be
improved by enlarging ρn, which is equivalent to maximizing
the sensing SNR [14], [17], [18].
Besides, it should be noted that the CSS performance of
multiple SUs depends on not only the individual sensing
performance of each SU but also the adopted fusion scheme.
For example, if the detection probability of an arbitrary SU
is far lower than 1, the cooperative detection probability of
the logic-AND fusion scheme cannot approach 1, whereas
that of the logic-OR fusion scheme could. Additionally, the
performance of the EGC-based CSS scheme depends on the
combined energy of all SUs instead of the individual sensing
accuracy of each SU. As a result, the dependency of the CSS
performance on the received primary signal strengths at the
SUs differs across various CSS schemes, and the RISs’ con-
figurations could be optimized regarding these CSS schemes.
In the following, we investigate the PSM optimization methods
for different CSS schemes, and we first consider the case
where the instantaneous CSI is available.
A. PSM Optimization for Data Fusion-based CSS
For data fusion-based CSS schemes, we consider the equal
gain combining (EGC) scheme without loss of generality,
where the combining factors are set as w1=··· =wN=
1/N, and the cooperative test statistic can be written as
EEGC =1
N
N
X
n=1
En=1
NT
N
X
n=1
T
X
t=1 |xn(t)|2.(20)
Similar to (16) and (19), the distributions of EEGC under the
two hypotheses can be written as Gaussian distributions with
the CLT, which unfolds as the following proposition.
Proposition 1. Denoting h,[h1··· hN]T∈CN,d,
[d1··· dN]T∈CN, and Gl,[g1,l ··· gN,l ]∈CM×N, we
have h=d+PL
l=1 GH
lΦlfl. Let ρ,khk2, we have
EEGC|H0∼ N σ2
u,1
NT σ4
u,
EEGC|H1∼ N σ2
u+ρ
Nσ2
s,σ4
u
NT +ρ2σ4
s+ 2ρσ2
sσ2
u
N2T.
Proof. The mean and variance can be obtained via its first-
order and second-order moments, and the fact that the variance
of a random variable xis Var[x] = E[x2]−E[x]2.
As EEGC has similar statistical characteristics with En, the
cooperative detection probability of the EGC-based data fusion
scheme can be improved by maximizing ρ. Therefore, the PSM
optimization problem can be formulated as
P-I-EGC: max
Φ1,··· ,ΦL
d+
L
X
l=1
GH
lΦlfl
2
(21a)
s.t. 0≤φl,m ≤2π, ∀l, m. (21b)
Let G= [GT
1··· GT
L]T∈CLM×N,f= [fT
1··· fT
L]T∈
CLM ,Φ= blkdiag(Φ1··· ΦL)∈CLM ×LM ,P-I-EGC can
be equivalently written as
P-I-EGC-1: max
Φkd+GHΦfk2(22a)
s.t. 0≤φl,m ≤2π, ∀l, m. (22b)
Then, let v∈CLM be a vector consisting of the diagonal
elements of Φ,¯
v= [vT1]T, and F= diag(f)∈CLM ×LM ,
denoting U= [GHF d], we have d+GHΦf =U¯
v.
Therefore, with R=UHU, we have kU¯
vk2= Tr(¯
vHR¯
v) =
Tr(R¯
v¯
vH). Besides, we define V=¯
v¯
vH, which should
satisfy V0and rank(V) = 1. By utilizing the semidefinite
relaxation (SDR) technique to relax this rank-1constraint, P-
I-EGC-1 can be reduced to [26]
P-I-EGC-2: max
VTr(RV)(23a)
s.t. Vi,i = 1,∀i= 1,···, LM, (23b)
V0,(23c)
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6
which is a standard convex semidefinite programming (SDP)
problem, and the solution can be obtained via existing opti-
mization solvers, e.g., CVX [27]. Finally, we can recover the
final solution via the Gaussian randomization method [28].
B. PSM Optimization for Decision Fusion-based CSS
For the decision fusion-based schemes, we consider the
PSM optimization problems under the aforementioned fusion
rules in Section II-C, namely, logic-AND, logic-OR, and
Krank. Specifically, we again aim to maximize the cooperative
detection probability given a maximum tolerable false alarm
probability. Furthermore, we assume that all SUs use the same
detection threshold. Besides, as the logic-AND and logic-OR
schemes can be regarded as special versions of the Krank
scheme, we start from these two special cases in the following.
1) Logic-AND: For the logic-AND scheme, the PSM op-
timization problem to maximize the cooperative detection
probability can be formulated as
P-I-AND: max
Φ1,··· ,ΦL
N
Y
n=1
Pd,n(Φ1,···,ΦL)(24a)
s.t. 0≤φl,m ≤2π, ∀l, m. (24b)
As the formulas of Pd,n ’s concerning Φl’s are quite complex,
which makes it almost impossible to obtain the optimal Φl’s
by directly solving problem P-I-AND. To address this issue,
we first obtain a lower bound of (24a) as
min
n{Pd,n(Φ1,··· ,ΦL)}N≤
N
Y
n=1
Pd,n(Φ1,··· ,ΦL).
Then, by maximizing this lower bound, P-I-AND can be
transformed into a max-min problem as
P-I-AND-1: max
Φ1,··· ,ΦL
min
n{Pd,n(Φ1,··· ,ΦL)}(25a)
s.t. 0≤φl,m ≤2π, ∀l, m. (25b)
This transformation can also be explained by the fact that the
fused decision using the logic-AND rule is accurate only when
the local hard decisions of all SUs are accurate. Consequently,
the cooperative detection probability of the logic-AND scheme
is greatly impacted by the minimum value of the detection
probabilities of the SUs. Besides, as the SUs share the same
detection threshold and Pd,n is positively related to |hn|2, the
relationship between the detection probabilities among SUs
can be transformed to that the equivalent channel gains among
SUs. Hence, P-I-AND-1 can be further transformed into a
channel gain-related max-min problem as
P-I-AND-2: max
Φ1,··· ,ΦL
min
n
dn+
L
X
l=1
gH
n,lΦlfl
2
(26)
s.t. 0≤φl,m ≤2π, ∀l, m. (27)
Let v∈CLM be a vector consisting of the diagonal elements
of Φ,F= diag(f)and ¯
v= [vT1]T, we have
dn+
L
X
l=1
gH
n,lΦlfl=dn+gH
nΦf =uH
n¯
v,(28)
where gn= [gT
n,1··· gT
n,L]Tand un= [gH
nFdn]H. Then,
denoting Qn=unuH
n, we have uH
n¯
v2= Tr(¯
vHQn¯
v) =
Tr(Qn¯
v¯
vH). With the definition V=¯
v¯
vHthat satisfies V
0and rank(V) = 1, using the SDR technique, P-I-AND-2 can
be equivalently rewritten as
P-I-AND-3: max
Vmin
nTr(QnV)(29a)
s.t. Vi,i = 1,∀i= 1,···, LM, (29b)
V0.(29c)
By introducing an auxiliary variable t,P-I-AND-3 can be
transformed as
P-I-AND-4: max
Vt(30a)
s.t. Tr(QnV)≥t, ∀n, (30b)
Vi,i = 1,∀i= 1,···, LM, (30c)
V0,(30d)
which can also be directly solved via existing optimization
solvers like CVX [27]. The final solution can be recovered
with the Gaussian randomization method [28].
As the detection probability of each SU satisfies Pd,n ≤1,
we have Pd≤minn{Pd,n}for the logic-AND scheme, i.e.,
the cooperative detection probability is mainly limited by
the minimum detection probability among all SUs, which is
exactly the motivation of the max-min problem in P-I-AND-
1. In the case without RIS, the detection probabilities across
all SUs are uniformly low due to limited sensing time and
weak direct links. Therefore, the solution to P-I-AND-1 lies
in enhancing the detection probability for each SU, which
consequently improves the cooperative detection probability.
It can be envisioned that, with sufficient REs, the solution to
P-I-AND-1 could also work well with other CSS schemes.
However, when the number of REs is limited, this solution
may not be so efficient for the other CSS schemes, such as the
logic-OR and Krank schemes where the fused decision could
be accurate without requiring all SUs’ local hard decisions to
be accurate. Motivated by this, the PSM optimization problems
for the other decision fusion schemes are investigated in the
following.
2) Logic-OR: For the logic-OR scheme, the fused decision
will be D= 1 as long as the local hard decision of any SU is 1.
Hence, a heuristic PSM optimization method for the logic-OR
scheme is utilizing all REs to improve the sensing performance
of the SU that exhibits the best improved sensing performance
when employing all REs to enhance the sensing capability of
a specific SU. Hence, the PSM optimization problem can be
formulated as
P-I-OR: max
Φ1,··· ,ΦL
max
n{Pd,n(Φ1,···,ΦL)}(31a)
s.t. 0≤φl,m ≤2π, ∀l, m. (31b)
With similar transformations from P-I-AND-1 to P-I-AND-3,
P-I-OR can be transformed as
P-I-OR-1: max
Vmax
nTr(QnV)(32a)
s.t. Vi,i = 1,∀i= 1,···, LM, (32b)
V0,(32c)
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7
and the solution to P-I-OR-1 can be obtained as follows. Let
qnand Φnbe the optimized objective value and the solution
of the following sub-problem of P-I-OR-1 given by
P-I-OR-1-Sub: max
VTr(QnV)(33a)
s.t. Vi,i = 1,∀i= 1,···, LM, (33b)
V0,(33c)
whose solution can be obtained using the existing convex
optimization solvers and Gaussian randomization method. The
solution to problem P-I-OR-1 is exactly Φn∗
with n∗=
arg maxn{qn}N
n=1.
3) Krank: The idea behind the PSM optimization method
for the logic-OR scheme can also be extended to the Krank
scheme, where D= 1 as long as the local hard decisions of
any Kof the total NSUs are 1. The fused decision using
the Krank scheme can be a joint decision of KSUs using the
logic-AND rule. Hence, we can utilize all REs to improve the
CSS performance of KSUs that exhibit the best improved
CSS performance when employing all REs to enhance the
sensing capability of Kspecific SUs. Besides, it should also
be noted that there are I=N
Kdifferent subsets of SUs,
and each subset consists of KSUs selected from the total N
SUs. Similar to P-I-AND, the PSM optimization problem to
maximize the cooperative detection probability of the KSUs
in the i-th subset can be written as
P-I-Krank-Sub: max
Φ1,··· ,ΦL
min
n∈Ki{Pd,n(Φ1,···,ΦL)}(34a)
s.t. 0≤φl,m ≤2π, ∀l, m, (34b)
where Kidenotes the i-th subset consisting of KSUs. Then,
it can be transformed as
P-I-Krank-Sub1: max
Φmin
n∈Ki
dn+
L
X
l=1
gH
n,lΦlfl
2
s.t. 0≤φl,m ≤2π, ∀l, m.
P-I-Krank-Sub1 can be solved with the same method for P-
I-AND-2, and the corresponding optimized PSMs for the i-th
subset can be obtained. Then, by exhausting all the subsets,
we can get the optimized PSMs for the Krank scheme. To
be specific, let qiand Φibe the optimized objective value
and PSMs of P-I-Krank-Sub1 for subset Ki, the optimized
PSMs for the Krank scheme can be denoted by Φi∗with i∗=
arg maxi{qi}I
i=1.
It is worth noting that the PSM optimization method for
the Krank scheme is also a general version of those for the
logic-AND and logic-OR schemes, which can be verified by
letting K=Nand K= 1, respectively. When K=N,P-I-
Krank-Sub reduces to P-I-AND-1 as there is only one subset
consisting of NSUs. The procedure to optimize the PSMs for
the Krank scheme with K= 1 is also the same as that for the
logic-OR scheme.
IV. EXT E NS I ON TO STATIS TIC AL C SI CA SE AN D
ANALY SI S O N THE NU MB E R OF RE QUI RE D RES
In Section III, we have investigated the PSM optimization
problems in the case where the instantaneous CSI is available,
and the PSMs are optimized to enhance the instantaneous
cooperative detection probability. However, the instantaneous
CSI is practically difficult to obtain due to the considerable
overhead of channel estimation, especially in the context of CR
[19]. An alternative solution is to optimize the PSMs with the
statistical CSI, which can be easily acquired because it usually
depends on the physical wireless environment, such as the
locations of PU, SUs, and RISs. In the following, we will show
that the proposed PSM optimization methods can be extended
to the statistical CSI case, where the NLOS components of
the channels (dand ˜
gn,l) are unknown and the PSMs can be
optimized with the LOS components.
A. Preliminaries
Due to the unknown NLOS components of the channels,
the equivalent channel gain between PU and SU n, namely,
ρn, is actually a random variable, the detection probability of
SU ncan be evaluated by a double-integral as
Pd,n =Z∞
γn
p(En|H1) dEn
=Z∞
0
p(ρn)Z∞
γn
p(En|ρn) dEn
|{z }
instantaneous detection probability
dρn,(36)
where p(ρn) = dF(ρn)is the PDF with F(ρn)the cumulative
density function (CDF). Let
Pd(ρn) = Z∞
γn
p(En|ρn) dEn,
we have the following proposition.
Proposition 2. Let P′
d,n and Pd,n respectively denote the av-
erage detection probabilities corresponding to the two CDFs,
namely, F′(ρn)and F(ρn), if F′(ρn)≤F(ρn)for arbitrary
ρn, we have P′
d,n ≥Pd,n.
Proof. F′(ρn)≤F(ρn)means the probability of obtaining
larger ρnunder F′is greater than F, and the probability
of obtaining smaller ρnunder F′is smaller than under F.
Moreover, Pd(ρn)is a monotonically increasing function with
respect to ρn. Hence, the probability of obtaining larger
Pd(ρn)under F′is greater than F, and the probability of
obtaining smaller Pd(ρn)under F′is smaller than F, leading
to P′
d,n ≥Pd,n.
Next, we study the impacts of the PSMs on the CDF of ρnas
follows. Let ¯
gn= [¯
gT
n,1··· ¯
gT
n,L]T,˜
gn= [˜
gT
n,1··· ˜
gT
n,L]T,
and ˜gin denotes the i-th element of ˜
gn, we have
hn=dn+
L
X
l=1
gH
n,lΦlfl=¯
gH
nΦf +dn+
LM
X
i=1
fiφi˜gin.
It is worth noting that fiφi˜gin is a complex Gaussian random
variable with zero-mean and variance βξζξ,n /(1 + κξ,n ),
where ξ=⌈i/M⌉. Therefore, ˜
gH
nΦf is a summation of
LM independent complex Gaussian random variables. Noting
that dnis also a complex Gaussian random variable that is
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8
independent of ˜
gH
nΦf,hnis a complex Gaussian random
variable satisfying
hn∼ CN ¯
gH
nΦf, ηn+M
L
X
l=1
ζn,lβl
1 + κn,l !.(37)
Let cn=¯
gH
nΦf, the mean µρnand variance vρnof ρn=|hn|2
are respectively given by
µρn=|cn|2+ηn+bn,
vρn= (ηn+bn)2+ 2|cn|2(ηn+bn),
where
bn=M
L
X
l=1
ζn,lβl
1 + κn,l
.
Then, the distribution of ρncan be approximated by a Gamma
distribution using the moment-matching method, and the PDF
and CDF of ρnare given by
p(ρn) = ρkρn−1
ne−ρn
θρn
θkρn
ρnΓ(kρn),(38)
F(ρn) = γ(kρn,ρn
θρn)
Γ(kρn),(39)
where F(ρn)is exactly the regularized lower incom-
plete gamma function [29], and it can also denoted by
F(ρn;kρn, θρn)parameterized by [30]
kρn=µ2
ρn/vρn, θρn=vρn/µρn.
In the proposed RIS-assisted CSS system, when NLOS
components are unknown, it can be observed that ηnand bn
are independent of Φ, and the PSMs only have impacts on
the value of |cn|2. Then, the monotonicities of kρnand θρn
with respect to |cn|2could be checked as follows. Let xand
adenote |cn|2and ηn+bnrespectively, we have
kρn=(x+a)2
a2+ 2ax = 1 + x2
a2+ 2ax = 1 + 1
a2
x2+2a
x
,
θρn=a2+ 2ax
x+a.
It is obvious that kρnmonotonically increases as |cn|2in-
creases. Moreover, it is also easy to find that θρnmonoton-
ically increases as |cn|2increases by checking its first-order
derivative with respect to |cn|2. According to the properties
of the regularized lower incomplete gamma function [29], the
value of the CDF at arbitrary ρn, namely, F(ρn), decreases
if either kρnor θρnincreases. Consequently, with Proposition
2, one can maximize |cn|2to improve the average detection
probability of SU nfor the statistical CSI case where the
instantaneous NLOS components are unknown. Besides, as
the direct links from PU to SUs are very weak and the gain of
the RISs on the unknown NLOS components is negligible, the
sensing performance gain brought by the RISs mainly comes
from the value of |cn|2, which means that the variations in
ηn+bnamong the SUs could be neglected in comparison
the value of |cn|2. Hence, it is reasonable to assume that
η1+b1≈ ··· ≈ ηN+bNin PSM optimization problems. As
a result, the individual sensing performance of different SUs
0 0.05 0.1 0.15
0
5
10
15
20
25
30
Fig. 3. Comparison of the empirical PDF of khk2with Gamma distribution
and Gaussian distribution approximated by the moment-matching method.
can be represented by the value of |cn|2. Based on the above
conclusions, the optimization methods proposed in Section III
can be extended to the statistical CSI as follows.
B. PSM Optimization with Statistical CSI
1) Data Fusion: Here, we again consider the EGC-based
data fusion scheme without loss of generality. From Section
III-A, we recall that the CSS performance can be improved by
maximizing ρ, which is a random variable due to the unknown
NLOS CSI. Particularly, the statistical characteristics of ρare
given by the following proposition.
Proposition 3. For an arbitrary Φ, denoting c,¯
GHΦf =
[c1··· cN]with cn=¯
gH
nΦf, where ¯
G= [ ¯
GT
1··· ¯
GT
L]T
and ¯
Gl,[¯
g1,l ··· ¯
gN,l], the mean and variance of ρare
respectively given by
µρ=kck2+
N
X
n=1
(ηn+bn),(40)
vρ=
N
X
n=1
(η2
n+b2
n+ 2ηn|cn|2+ 2ηnbn+ 2bn|cn|2).(41)
Proof. See Appendix B.
Besides, the distribution of ρcan also be approximated by a
Gamma distribution using the moment-matching method, and
its PDF and CDF are given by
p(ρ) = ρkρ−1e−ρ
θρ
θkρ
ρΓ(kρ), F (ρ) = γ(kρ,ρ
θρ)
Γ(kρ),(42)
where kρand θρare given by
kρ=µ2
ρ/vρ, θρ=vρ/µρ.
The accuracy of the approximated Gamma distribution is
validated in Fig. 3. It can be observed that the approximated
Gamma distribution fits the empirical PDF of ρwell, whereas
the Gaussian distribution with the same mean and variance
presents a significant difference from the empirical PDF.
Similar to the analysis in Section IV-A, it can also be proved
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9
that both kρand θρmonotonically increase as kck2increases
under the assumption η1+b1≈ ··· ≈ ηN+bN. With
the properties of the regularized lower incomplete gamma
function [29], the value of the CDF at arbitrary ρ, namely,
F(ρ), decreases if either kρor θρincreases. Therefore, when
the instantaneous NLOS components are unknown, the PSM
optimization problem with statistical CSI can be written as
P-S-EGC: max
Φk¯
GHΦfk2(43a)
s.t. 0≤φl,m ≤2π, ∀l, m. (43b)
Note that P-S-EGC is a special version of P-I-EGC, therefore
it can be solved by the same transformations and techniques,
i.e., SDR and Gaussian randomization method.
2) Decision Fusion: As analyzed in Section IV-A, the
detection probability of SU ncan be represented by the value
of |¯
gH
nΦf|2. Thus, the PSM optimization methods in Section
III-B can be extended to the statistical CSI case by replac-
ing the instantaneous equivalent channel gain with |¯
gH
nΦf|2.
Specifically, when the instantaneous NLOS components are
unknown, the PSM optimization problem for the logic-AND
scheme can be formulated as
P-S-AND: max
Φmin
n¯
gH
nΦf2(44a)
s.t. 0≤φl,m ≤2π, ∀l, m. (44b)
Similarly, the PSM optimization problem for the logic-OR
scheme can be rewritten as
P-S-OR: max
Φmax
n¯
gH
nΦf2(45a)
s.t. 0≤φl,m ≤2π, ∀l, m. (45b)
Besides, the PSM optimization problem for the i-th SU subset
in the Krank scheme can be written as
P-S-Krank-Sub: max
Φmin
n∈Ki¯
gH
nΦf2(46a)
s.t. 0≤φl,m ≤2π, ∀l, m. (46b)
It should be noted that the PSM optimization problems under
the statistical CSI case can be regarded as special versions
of those under the instantaneous CSI case, which can be
easily verified by replacing gnwith ¯
gnand setting dn= 0.
Consequently, they can be solved by using the same techniques
for solving the PSM optimization problems under the instan-
taneous CSI case, namely, SDR and Gaussian randomization
method.
C. How Many REs are Required to Achieve a Target Detection
Probability?
In opportunistic CR networks, the SUs are supposed to
achieve a sufficiently high detection probability such that the
primary transmission can be well protected. For example, a
target detection probability of 0.9is required in IEEE 802.22
standard [31]. From the PSM optimization problems in the
statistical CSI case, we can see that employing more REs
can enhance the expectation of the cascaded channel gain
between PU and SUs, thereby improving both the detection
probability of each SU and the cooperative detection prob-
ability. Considering the expenditures of RIS deployment, the
minimum number of REs required to achieve a target detection
probability is also significant in designing the RIS-assisted
CSS system. In the following, we will show how to evaluate
the number of REs required to achieve a target detection
probability under the statistical CSI case.
Firstly, we need to evaluate the impacts of the number of
REs on the cooperative detection probability for the considered
CSS schemes. Due to the unknown NLOS components of the
channels, the expression of the average detection probability
is of a double-integral form as shown in (36), where the inner
integral can be regarded as a Q-function and the outer integral
is for the distribution of the random channel gain. With some
similar approximation methods in [17], i.e., approximating
the Q-function with its alternative series representation and
approximating the outer integral in (36) with the closed-
form expressions from [32], it is feasible to derive a closed-
form expression of the cooperative detection probability with
respect to Φl’s. In the RIS-assisted spectrum sensing system
with single SU [14], [17], the closed-form expression of the
optimized PSM can be easily obtained, making it possible to
derive the closed-form expression of the relationship between
the detection probability and the number of REs. However,
in the RIS-assisted CSS system with multiple SUs, the opti-
mized PSMs are obtained by the convex optimization solvers
and Gaussian randomization method. Consequently, acquiring
closed-form expressions for the PSMs is nearly impossible,
as well as establishing a closed-form relationship between
cooperative detection probability and the number of REs for
straightforward insights. Moreover, using these closed-form
expressions to calculate the detection probability is also very
complex. Therefore, instead of deriving complex formulas of
little significance, in this paper, we choose to utilize the easy-
understanding numerical evaluation method for evaluating the
detection probability.
The cooperative detection probability of the RIS-assisted
CSS system can be evaluated as follows. For the EGC-based
CSS scheme, the cooperative test statistic distribution under
H1is dependent on ρ, where we recall ρ=khk2. Noting that ρ
is also a random variable due to the fading NLOS components
of gn,l and d, the cooperative detection probability of the
EGC-based scheme can be numerically evaluated by
PEGC
d=Z∞
γEGC
p(EEGC|H1) dEEGC
=Z∞
γEGC Z∞
0
p(EEGC|ρ)p(ρ) dρdEEGC .(47)
In addition, we recall that the distribution of the test statistic
and that of the equivalent channel gain could be derived as in
Proposition 1 and Proposition 3. Therefore, the cooperative
detection probability can be efficiently calculated with the
numerical integration methods. For decision fusion-based CSS
schemes, according to (36), the individual detection probability
of SU ncan be numerically evaluated with the distribution of
Enand ρn. As shown in Table I, the cooperative detection
probabilities of the decision fusion-based CSS schemes can be
calculated based on the SUs’ individual detection probabilities.
Then, with the fact that the performance gain of the RISs
is positively related to the number of REs, we can utilize the
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content may change prior to final publication. Citation information: DOI 10.1109/TWC.2024.3393516
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10
bisection method to find the number of REs required to achieve
a target detection probability.
D. Practical Implementation of RIS-Assisted CSS System
Compared to conventional CSS systems without RIS, the
proposed RIS-assisted CSS system can significantly improve
the performance of various CSS schemes. However, the intro-
duction of RISs will also result in additional requirements for
controlling the PSM configurations of the RISs. For the RIS-
assisted CSS system, the RISs could be deployed by the SUs
as they can benefit from RISs’ reflections in terms of increased
secondary transmission time. Moreover, in CSS systems, the
FC plays the role of collecting SUs’ sensing results and
generating the fused decision, so it is actually the manager of
the multiple SUs. Likewise, the FC is also the controller of the
RIS-assisted CSS system for collecting necessary information
to optimize the PSMs and reconfigure the RISs. Besides, on
the deployment of multiple RISs, it should also access the
required number of REs based on the system setup, such
as the locations of the PU and SUs, and the adopted CSS
scheme. On the other hand, the introduction of RISs also leads
to additional communication overhead of collecting necessary
information to optimize the PSM and control signaling to
configure the RISs, especially in the instantaneous CSI case
where the PSMs are re-optimized once the instantaneous CSI
varies. Nevertheless, this overhead is much smaller compared
to the overhead of collecting hundreds of thousands of sensing
signal samples in each sensing period. Further, it can be
reduced by exploiting the statistical CSI. Given the long-term
constancy of statistical CSI, both the information collection
and RIS configuration procedures do not need to be executed
frequently, leading to a substantial reduction of the additional
overhead.
V. SIMU LATIO N RE SU LTS
In this section, we provide extensive simulations to evaluate
the performance of the proposed RIS-assisted CSS system.
Moreover, the effectiveness of the proposed PSM optimization
methods for the considered CSS schemes is demonstrated in
comparison with another heuristic PSM configuration scheme,
which we adopt as a benchmark. Specifically, in the bench-
mark method, each RIS is equally partitioned into Nsub-
surfaces, and the n-th sub-surface is associated with SU
nto enhance its individual sensing performance. For the
statistical CSI case, the phase shifts of the n-th sub-surface
of the l-th RIS, namely, φl,(n−1)M/N+1 ,··· , φl,nM/N (i=
1,···, M/N), can be configured to align the cascaded channel
from PU to SU nvia the n-th sub-surface. Therefore, the phase
shift configuration of the l-th RIS in the benchmark method
is given by (48), shown at the bottom of this page. Obviously,
this benchmark method can improve the individual sensing
performance of each SU since the equivalent channel gain of
each SU can be enhanced. As a result, the performance of all
CSS schemes can be improved via the benchmark.
For the simulation setup, we consider an RIS-assisted CSS
system with L= 2 RISs and N= 4 cooperative SUs.
Specifically, we describe the system with a Cartesian coor-
dinate, where the PU is located at (0m,0m)and the SUs are
randomly located in a circular area with center (400m,0m)and
radius 10 m. Besides, the two RISs respectively are located at
(380m,25m)and (400m,25m). The path loss of the PU-SU,
PU-RIS, and RIS-SU channels are respectively given by [33]
Ld=λ2
(4π)2dα1
1
, Lf=λ2
(4π)2dα2
2
, LG=λ2
(4π)2dα3
3
,
where λis the carrier wavelength, d1,d2, and d3are the
distances between the corresponding nodes, α1,α2, and α3
denote the pathloss exponents. In the simulations, we set
λ= 0.12 m, α1= 3.5, and α2=α3= 2. The transmitting
power of the active PU and the AWGN power at each SU are
30 dBm and −80 dBm, respectively. The Rician factors of the
RIS-SU channels are set as κn,l = 10,∀n, l. Particularly, the
number of signal samples in each sensing period is T= 1000.
Besides, the maximum tolerable false alarm probabilities of
all the considered CSS schemes are set as 0.1. With the
assumption that all SUs use the same detection threshold, the
detection thresholds of these CSS schemes can be calculated
as follows.
•EGC:
γEGC =r1
NT σ2
uQ−1(¯
Pfa) + σ2
u.
•Logic-AND:
γAND
n=r1
Tσ2
uQ−1(¯
P
1
N
f a ) + σ2
u.
•Logic-OR:
γOR
n=r1
Tσ2
uQ−1(1 −(1 −¯
Pfa)1
N) + σ2
u.
•Krank:
γKrank
n=r1
Tσ2
uQ−1(¯
Pfa,n) + σ2
u,
where ¯
Pfa,n is SU n’s maximum tolerable false alarm
probability satisfying
N
X
i=KN
i¯
Pi
fa,n(1 −¯
Pfa,n)N−i=¯
Pfa.
A. Cooperative Detection Probability Comparison
1) Comparison of PSM Optimization Methods: In Fig. 4,
the advantages of introducing the RISs to the CSS system
are demonstrated, and we also show the superiority of the
proposed PSM optimization methods over the benchmark
method. Particularly, the PSMs obtained by the proposed
φl,(n−1)M/N+i= arg[¯gn,(l−1)M+(n−1)M/N+i]−arg[fl,(n−1)M/N+i], i = 1,···, M/N. (48)
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11
0 50 100 150 200
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(a) Data Fusion, EGC
0 50 100 150 200
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(b) Decision Fusion, OR Rule (Krank, K= 1)
0 50 100 150 200
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(c) Decision Fusion, Majority Rule (Krank, K= 2)
0 50 100 150 200
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(d) Decision Fusion, AND Rule (Krank, K= 4)
Fig. 4. The cooperative detection probabilities of the considered CSS schemes for different numbers of REs per RIS.
optimization methods are denoted by “Proposed opt. Φl”,
and those obtained by the benchmark method are denoted by
“Partitioned opt. Φl”. Moreover, the notation “I-CSI” is for
the case where the instantaneous CSI is available to optimize
the PSMs, while the notation “S-CSI” is for the case where
the statistic CSI is utilized to optimize the PSMs. It can be
observed that, in the proposed RIS-assisted CSS system, the
cooperative detection probabilities of all the considered CSS
schemes can be notably improved with the optimized PSMs
via either the proposed PSM optimization methods or the
benchmark approach. Furthermore, with the increase of the
number of REs per RIS, the cooperative detection probability
of each CSS scheme also increases for all the PSM con-
figuration methods, which is consistent with the conclusions
obtained in other RIS-aided wireless communication systems.
Additionally, for the proposed PSM optimization methods, the
PSMs optimized with the instantaneous CSI can outperform
those optimized with the statistical CSI. This stems from the
fact that the NLOS components of the channels cannot be
utilized to optimize the PSMs for enhancing the received
SNRs of the primary signals at the SUs in the statistical
CSI case. Besides, compared with the random PSMs and the
benchmark, the PSMs obtained with the proposed methods
can achieve higher cooperative detection probabilities for the
same number of REs per RIS, demonstrating the effectiveness
of the proposed PSM optimization methods. We can also
see that, for both the data fusion and decision fusion-based
CSS schemes, the performance improvement brought by the
randomly configured PSMs is almost negligible, which means
that the impacts of the RIS reflections intended on some
targeted SUs on the other untargeted SUs can be neglected
in the RIS-assisted CSS system.
In Fig. 5, we further compare the cooperative detection
probabilities of different CSS schemes in the RIS-assisted
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content may change prior to final publication. Citation information: DOI 10.1109/TWC.2024.3393516
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12
0 50 100 150 200
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fig. 5. The cooperative detection probabilities of different CSS schemes
versus the number of REs per RIS.
CSS system, and the results are obtained in the case where
only statistical CSI is available. In the context of decision
fusion-based CSS schemes, we recall that both the logic-
OR and logic-AND fusion rules can considered specialized
variants of the Krank fusion rule. Consequently, an observation
emerges: as the value of Kescalates while maintaining an
identical number of REs per RIS, the cooperative detection
probability of the Krank scheme diminishes. From another
viewpoint, the number of REs per RIS required to achieve
the same cooperative detection probability increases as K
increases, which means that more REs are required when we
try to improve the sensing performance of more SUs at the
same time. Furthermore, we can see that the superiority and
inferiority relationships among these CSS schemes also vary
according to the number of REs involved. When Mis small or
the PSMs are randomly configured, the EGC-based data fusion
scheme can outperform all the considered decision fusion
schemes, and the majority rule-based decision fusion scheme
can slightly outperform the other decision fusion schemes,
which indicates that the optimal Kof the Krank scheme is
2in such cases. When Mincreases to about 40 and the
PSMs are optimized with our proposed methods, it can be
observed that OR rule-based decision fusion scheme achieves
the highest cooperative detection probability, where all the
RISs are configured to enhance the received primary signal
strength at one SU.
As discussed in Section III and Section IV, in the RIS-
assisted CSS system, the PSMs under various CSS schemes
should also be optimized differently to fully utilize the limited
REs. To further illustrate the characteristics of the PSMs
optimized for different CSS schemes, we investigate the
performance gain brought by the PSMs optimized for one
CSS scheme in the other CSS schemes. As shown in Fig.
6, we consider the large Mregime (M= 200), and the
horizontal axis represents the PSMs optimized for different
CSS schemes, where “Opt. ΦlEGC” denotes the PSMs
optimized for the EGC-based CSS scheme and the PSMs
optimized for the other CSS schemes are denoted similarly.
The results suggest that the PSMs optimized for the EGC
Opt. Φl
EGC
Opt. Φl
Logic-OR
Opt. Φl
Krank-2
Opt. Φl
Krank-3
Opt. Φl
Logic-AND
Cooperative Detection Probability, Pd
Fig. 6. Comparison of the cooperative detection probabilities of the considered
CSS schemes achieved by Φl’s optimized for different CSS schemes in the
large Mregime where M= 200.
scheme and those optimized for the logic-OR scheme perform
similarly under all the considered CSS schemes. This can
be explained by the PSM optimization problem for the EGC
scheme, where we aim to maximize the summation of the
received energy at all SUs. As one can imagine, this objective
may be maximized by utilizing all the REs to enhance the
received primary signal strength of the single SU with the
strongest cascaded channel gain. In addition, for the decision
fusion-based CSS schemes, the PSMs optimized for the Krank
scheme with arbitrary Kcan perform well in the cases with
smaller K’s but poorly in the cases with larger K’s. Moreover,
the cooperative detection probability of the logic-AND scheme
is a product of the detection probabilities of all SUs, and it can
approach 1only when the detection probabilities of all SUs are
close to 1. Consequently, in the large Mregime, the PSMs
optimized for the logic-AND scheme can achieve a near-1
cooperative detection probability under all the considered CSS
schemes. In other words, with sufficient REs, the proposed
PSM optimization method for the logic-AND scheme can
provide a general way to configure the RIS-assisted CSS
system regardless of the adopted CSS scheme at the FC.
2) Comparison of Single-RIS and Multiple-RIS Systems: It
should be noted that the proposed PSM optimization methods
are applicable to both multiple-RIS and single-RIS assisted
CSS systems. Here, we would like to provide a comparison
of single-RIS and multiple-RIS systems in CSS scenarios.
Particularly, we keep the total number of REs the same
for the single-RIS and multiple-RIS systems. Besides, we
consider the cases with the number of RISs L= 1,2,4,
and L= 1 stands for the single-RIS system. The RISs are
deployed with a similar setup as in [22], and the RISs are
uniformly located on a circle with center (400m,0m)and
radius 25 m. In Fig. 7, we provide the simulation results under
the EGC-based data fusion and majority rule-based decision
fusion schemes without loss of generality. Under the EGC-
based data fusion scheme, we can see that the multiple-RIS
system with L= 2 achieves almost the same performance
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13
as the single-RIS system, which can be explained as follows.
On the one hand, in comparison with the single-RIS system,
the multiple-RIS system allows the REs to be deployed at
different locations such that the equivalent channel gains of
some SUs can benefit from some closer REs than the single-
RIS system. On the other hand, the deployment of multiple
RISs also introduces multiple cascaded channels, which can
be denoted by a1,···,aLfor simplicity. Considering the
alignment problem of these cascaded channels where we
have kPalk2≤Pkalk2and the equality holds only when
al’s are parallel, the misalignment between these cascaded
channels will result in performance loss. Under the EGC-based
scheme, when L= 2, the benefits and performance loss of
employing multiple RIS systems balance each other in our
simulation setup, thereby leading to a similar performance as
the single-RIS system. When Lincreases to 4, the benefit
from the multiple-RIS system is large enough to compensate
for the performance loss due to the channel misalignment,
therefore the multiple-RIS system with L= 4 can significantly
outperform the single-RIS system. Fig. 7 (b) also shows the
results under the majority rule-based decision fusion scheme,
where we can see that the multiple-RIS system with L= 2
can outperform the single-RIS system. This indicates that, in
the RIS-assisted CSS system, the performance gain of the
multiple-RIS system over the single-RIS system also depends
on the adopted CSS scheme at the FC. To conclude, the
multiple-RIS system can lead to a significant performance gain
in the RIS-assisted CSS system with multiple SUs since it
allows a distributed and flexible deployment of numerous REs.
B. The Number of REs Required to Achieve a Target ¯
Pd
From both the analysis and simulation results, it can be
seen that the performance of all the considered CSS schemes
can be improved by increasing the number of REs for both
the proposed PSM optimization methods and the benchmark
method. Hence, there exists a minimum number of REs for
each CSS scheme to achieve a target cooperative detection
probability, which can be numerically obtained as discussed in
Section IV-C. In Fig. 8, we first investigate the impacts of the
number of sensing signal samples on the minimum number
of REs required to achieve a target cooperative detection
probability of ¯
Pd= 0.9. For all the considered CSS schemes,
with the decrease in the number of sensing signal samples, the
number of REs should be increased to achieve the same coop-
erative detection probability ¯
Pd. This means that the required
sensing time can be significantly reduced by employing more
REs in the RIS-assisted CSS system. Besides, in comparison
to the benchmark, the proposed PSM optimization methods
require fewer REs to achieve the same cooperative detection
probability, suggesting the superiority of the proposed PSM
optimization methods in terms of the utilization efficiency
of the REs. Moreover, for the Krank-based decision fusion
schemes, the number of required REs becomes larger when
Kincreases, and the logic-AND-based decision fusion scheme
requires the most REs to achieve the same ¯
Pd. As a conclusion,
when we try to improve the individual sensing performance
of more SUs simultaneously, we may have to involve a larger
number of REs in the RIS-assisted CSS system.
0 50 100 150 200 250 300 350 400
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(a) Data Fusion, EGC
0 50 100 150 200 250 300 350 400
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(b) Decision Fusion, Majority Rule (Krank, K= 2)
Fig. 7. The cooperative detection probabilities of the considered CSS schemes
for different numbers of REs per RIS.
In the statistical CSI case, for both the proposed methods
and the benchmark, the performance gain brought by the RISs
mainly depends on the strength of the LOS components as the
reflections of the unknown NLOS components are uncontrol-
lable. In Fig. 9, we investigate the impact of the Rician factors
of the RIS-SU channels on the number of REs required to
achieve a target cooperative detection probability. Without loss
of generality, the results are obtained under the logic-AND-
based decision fusion scheme. With the increase of the Rician
factors of the RIS-SU channels, the required number of REs
can be reduced for the same number of sensing signal samples.
This observation is exactly because the mean of the equivalent
channel gain can be enhanced more significantly when the
LOS components of the channels are stronger. Also, it can
be observed that the proposed PSM optimization methods can
notably reduce the number of required REs as compared to the
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content may change prior to final publication. Citation information: DOI 10.1109/TWC.2024.3393516
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14
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
104
0
50
100
150
200
250
Fig. 8. Number of REs per RIS required to achieve a target cooperative
detection probability of 0.9w.r.t. the number of sensing signal samples for
different CSS schemes.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
104
50
100
150
200
250
300
350
Fig. 9. The impacts of the Rician factors of RIS-SU channels on the number
of REs per RIS required to achieve a target cooperative detection probability
of 0.9for logic-AND decision fusion scheme.
benchmark methods for different Rician factors, which further
validates the effectiveness of the proposed PSM optimization
methods.
VI. CO NCL USI ONS
In this paper, we have studied an RIS-assisted CSS system,
where the PSMs of multiple RISs are jointly optimized to
enhance the CSS performance of multiple SUs. Particularly,
noting that the impact of the received primary signal strengths
at the SUs on the CSS performance differs across various CSS
schemes, we have investigated the PSM optimization problems
to optimize the CSS performance for two typical kinds of
schemes, namely, data fusion and decision fusion. Then, we
have shown that the intractable PSM optimization problems
can be transformed into channel gain-related optimization
problems that are easier to deal with. Furthermore, we have
also demonstrated that the proposed PSM optimization meth-
ods can be extended to the more practical scenarios where
instantaneous CSI is unavailable. In these cases, the statistical
CSI is leveraged to improve the CSS performance in the sense
of expectation. Subsequently, we have conducted a numerical
analysis on the number of reflecting elements required to
achieve a target detection probability in the statistical CSI case.
Finally, simulation results have been provided to demonstrate
the effectiveness of the proposed RIS-assisted CSS system and
PSM optimization methods, which can significantly improve
the CSS performance within limited sensing time. Moreover,
the RIS-assisted CSS system can substantially reduce the
required sensing time for achieving a high cooperative detec-
tion probability, thereby providing more remaining time for
secondary access.
APP EN D IX
A. Proof of Proposition 3
Let ¯
Gl,[¯
g1,l ··· ¯
gN,l]and ˜
Gl,[˜
g1,l ··· ˜
gN,l], and
define ¯
G= [ ¯
GT
1··· ¯
GT
L]T,˜
G= [ ˜
GT
1··· ˜
GT
L]T, we have
G=¯
G+˜
G. Denoting c,¯
GHΦf, we have
khk2=kd+˜
GHΦf +¯
GHΦfk2=kd+˜
GHΦf +ck2
=kdk2+k˜
GHΦfk2+kck2+cHd+dHc+dH˜
GHΦf
+fHΦH˜
Gd+cH˜
GHΦf +fHΦH˜
Gc.
As ˜
Gdenotes the NLOS components of the channels between
RISs and SU, we have E[˜
G] = 0. Noting that the expectation
of dis also 0, namely, E[d] = 0, the expectation of ρis thus
E[ρ] = Ekhk2=kck2+Ekdk2+Ek˜
GHΦfk2.(49)
Denoting the i-th row of ˜
Gby ˜
Gi,:, we have
Ek˜
GHΦfk2=E[fHΦH˜
G˜
GHΦf] =
LM
X
i=1 |fi|2Ek˜
Gi,:k2
=M
N
X
n=1
L
X
l=1
βlζn,l
1 + κn,l
=
N
X
n=1
bn,
and the mean of ρis given by
µρ=E[ρ] = kck2+
N
X
n=1
(ηn+bn).(50)
The variance of ρcan be expressed as vρ=Var(khk2) =
Ekhk4−(Ekhk2)2, where Ekhk4is given by
Ekhk4=kck4+Ekdk4+Ek˜
GHΦfk4+2kck2Ekdk2
+2kck2Ek˜
GHΦfk2+2Ekdk2Ek˜
GHΦfk2+2E|cHd|2
+2E|dH˜
GHΦf|2+2E|cH˜
GHΦf|2,(51)
and the variance of ρis therefore
vρ=Ekhk4−(Ekhk2)2
=Ekdk4−(Ekdk2)2+Ek˜
GHΦfk4−(Ek˜
GHΦfk2)2
+ 2E|cHd|2+2E|dH˜
GHΦf|2+2E|cH˜
GHΦf|2.(52)
Specifically, the components of (52) can be derived as follows.
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15
1) Ekdk4−(Ekdk2)2: Using E|dn|4= 2η4
n, we have
Ekdk4= 2
N
X
n=1
η2
n+
N
X
n=1
N
X
m6=n
ηnηm= (Ekdk2)2+
N
X
n=1
η2
n,
which is equivalent to
Ekdk4−(Ekdk2)2=
N
X
n=1
η2
n.(53)
2) Ek˜
GHΦfk4−(Ek˜
GHΦfk2)2: With the fact that
fHΦH˜
G˜
GHΦf =
LM
X
i=1
LM
X
j=1
f†
ifj˜
Gi,:˜
GH
j,:,
we can derive
Ek˜
GHΦfk4=E|fHΦH˜
G˜
GHΦf|2
=
LM
X
i=1
LM
X
j=1
LM
X
p=1
LM
X
q=1
f†
ifjf†
pfqE[˜
Gi,:˜
GH
j,:˜
Gp,:˜
GH
q,:]
=
LM
X
i=1 |fi|4Ek˜
Gi,:k4+
LM
X
i=1
LM
X
j6=i|fi|2|fj|2Ek˜
Gi,:k2Ek˜
Gj,:k2
+
LM
X
i=1
LM
X
j6=i|fi|2|fj|2E|˜
Gi,:˜
GH
j,:|2.
Denoting the n-th element of the i-th row of Gby ˜gin,
with the fact E|˜gin|4= 2(E|˜gin|2)2, we can obtain
Ek˜
Gi,:k4= (Ek˜
Gi,:k2)2+
N
X
n=1
(E|˜gin|2)2.
Hence, we can derive
Ek˜
GHΦfk4−(Ek˜
GHΦfk2)2
=
LM
X
i=1|fi|4
N
X
n=1
[E|˜gin|2]2+
LM
X
i=1
LM
X
j6=i|fi|2|fj|2
N
X
n=1
E|˜gin|2E|˜gjn|2
=
LM
X
i=1
LM
X
j=1 |fi|2|fj|2
N
X
n=1
E|˜gin|2E|˜gjn|2=
N
X
n=1
b2
n.(54)
3) E|cHd|2,E|dH˜
GHΦf|2, and E|cH˜
GHΦf|2: Let cn
denote the n-th element of c, we can obtain
E|cHd|2=E[cHddHc] =
N
X
n=1
ηn|cn|2,(55)
E|dH˜
GHΦf|2=E[fHΦH˜
GddH˜
GHΦf]
=
LM
X
i=1 |fi|2
N
X
n=1
E|˜gin|2ηn=
N
X
n=1
ηnbn,
(56)
E|cH˜
GHΦf|2=E[fHΦH˜
GccH˜
GHΦf]
=
LM
X
i=1 |fi|2
N
X
n=1
E|˜gin|2|cn|2=
N
X
n=1
bn|cn|2.
(57)
By substituting (53), (54), (55), (56) and (57) into (52), we
finally obtain
vρ=
N
X
n=1
(η2
n+b2
n+ 2ηn|cn|2+ 2ηnbn+ 2bn|cn|2).(58)
REF ERE NC E S
[1] J. Ge, S. Wang, C. Sun, and Y.-C. Liang, “RIS-enhanced cooperative
spectrum sensing for opportunistic cognitive radio networks,” in Proc.
IEEE Global Commun. Conf. Workshops, Kuala Lumpur, Malaysia,
2023, pp. 1427–1432.
[2] X. You et al., “Towards 6G wireless communication networks: Vision,
enabling technologies, and new paradigm shifts,” Sci. China Inf. Sci.,
vol. 64, no. 1, pp. 1–74, Nov. 2021.
[3] Y.-C. Liang, Dynamic spectrum management: From cognitive radio to
blockchain and artificial intelligence. Springer Nature, 2020.
[4] Z. Qin, X. Zhou, L. Zhang, Y. Gao, Y.-C. Liang, and G. Y. Li, “20 years
of evolution from cognitive to intelligent communications,” IEEE Trans.
Cogn. Commun. Netw., vol. 6, no. 1, pp. 6–20, Mar. 2020.
[5] Y. Zeng, Y.-C. Liang, A. T. Hoang, and R. Zhang, “A review on spectrum
sensing for cognitive radio: challenges and solutions,” EURASIP J. Adv.
Signal Process., vol. 2010, pp. 1–15, Jan. 2010.
[6] A. Ali and W. Hamouda, “Advances on spectrum sensing for cognitive
radio networks: Theory and applications,” IEEE Commun. Surveys Tuts.,
vol. 19, no. 2, pp. 1277–1304, 2nd quarter 2016.
[7] J. Ge, Y.-C. Liang, Z. Bai, and G. Pan, “Large-dimensional random
matrix theory and its applications in deep learning and wireless commu-
nications,” Random Matrices, Theory Appl., vol. 10, no. 04, p. 2230001,
Jun. 2021.
[8] I. F. Akyildiz, B. F. Lo, and R. Balakrishnan, “Cooperative spectrum
sensing in cognitive radio networks: A survey,” J. Phy. Commun., vol. 4,
no. 1, pp. 40–62, Dec. 2011.
[9] Y.-C. Liang, Y. Zeng, E. C. Peh, and A. T. Hoang, “Sensing-throughput
tradeoff for cognitive radio networks,” IEEE Trans. Wireless Commun.,
vol. 7, no. 4, pp. 1326–1337, Apr. 2008.
[10] Y.-C. Liang, R. Long, Q. Zhang, J. Chen, H. V. Cheng, and H. Guo,
“Large intelligent surface/antennas (LISA): Making reflective radios
smart,” J. Commun. Inf. Netw., vol. 4, no. 2, pp. 40–50, Jun. 2019.
[11] Y. Liu, X. Liu, X. Mu, T. Hou, J. Xu, M. Di Renzo, and N. Al-Dhahir,
“Reconfigurable intelligent surfaces: Principles and opportunities,” IEEE
Commun. Surveys Tuts., vol. 23, no. 3, pp. 1546–1577, 3rd quarter 2021.
[12] M. A. ElMossallamy, H. Zhang, L. Song, K. G. Seddik, Z. Han, and
G. Y. Li, “Reconfigurable intelligent surfaces for wireless communi-
cations: Principles, challenges, and opportunities,” IEEE Trans. Cogn.
Commun. Netw., vol. 6, no. 3, pp. 990–1002, Sep. 2020.
[13] Q. Wu, S. Zhang, B. Zheng, C. You, and R. Zhang, “Intelligent
reflecting surface-aided wireless communications: A tutorial,” IEEE
Trans. Commun., vol. 69, no. 5, pp. 3313–3351, May 2021.
[14] J. Ge, Y.-C. Liang, S. Li, and Z. Bai, “RIS-enhanced spectrum sensing:
How many reflecting elements are required to achieve a detection
probability close to 1?” IEEE Trans. Wireless Commun., vol. 21, no. 10,
pp. 8600–8615, Oct. 2022.
[15] H. Xie, B. Gu, and D. Li, “Enhancing spectrum sensing via recon-
figurable intelligent surfaces: Passive or active sensing and how many
reflecting elements are needed?” arXiv preprint arXiv:2306.13874, 2023.
[16] J. Ge, Y.-C. Liang, S. Sun, and Y. Zeng, “Active RIS enhanced spectrum
sensing for cognitive radio networks,” arXiv preprint arXiv:2311.16568,
2023.
[17] W. Wu, Z. Wang, L. Yuan, F. Zhou, F. Lang, B. Wang, and Q. Wu,
“IRS-enhanced energy detection for spectrum sensing in cognitive radio
networks,” IEEE Wireless Commun. Lett., vol. 10, no. 10, pp. 2254–
2258, Oct. 2021.
[18] S. Lin, B. Zheng, F. Chen, and R. Zhang, “Intelligent reflecting surface-
aided spectrum sensing for cognitive radio,” IEEE Wireless Commun.
Lett., vol. 11, no. 5, pp. 928–932, May 2022.
[19] A. Nasser, H. A. H. Hassan, A. Mansour, K.-C. Yao, and L. Nuaymi,
“Intelligent reflecting surfaces and spectrum sensing for cognitive radio
networks,” IEEE Trans. Cogn. Commun. Netw., vol. 8, no. 3, pp. 1497–
1511, Sep. 2022.
[20] W. Wu, Z. Wang, Y. Wu, F. Zhou, B. Wang, Q. Wu, and D. W. K. Ng,
“Joint sensing and transmission optimization for IRS-assisted cognitive
radio networks,” IEEE Trans. Wireless Commun., vol. 22, no. 9, pp.
5941–5956, Sep. 2023.
This article has been accepted for publication in IEEE Transactions on Wireless Communications. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/TWC.2024.3393516
© 2024 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information.
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16
[21] M. He, W. Xu, H. Shen, G. Xie, C. Zhao, and M. Di Renzo, “Cooper-
ative multi-RIS communications for wideband mmWave MISO-OFDM
systems,” IEEE Wireless Commun. Lett., vol. 10, no. 11, pp. 2360–2364,
Nov. 2021.
[22] Z. Yang, M. Chen, W. Saad, W. Xu, M. Shikh-Bahaei, H. V. Poor,
and S. Cui, “Energy-efficient wireless communications with distributed
reconfigurable intelligent surfaces,” IEEE Trans. Wireless Commun.,
vol. 21, no. 1, pp. 665–679, Jan. 2022.
[23] K. Xu, S. Gong, M. Cui, G. Zhang, and S. Ma, “Statistically robust
transceiver design for multi-RIS assisted multi-user MIMO systems,”
IEEE Commun. Lett., vol. 26, no. 6, pp. 1428–1432, Jun. 2022.
[24] R. Long, Y.-C. Liang, H. Guo, G. Yang, and R. Zhang, “Symbiotic radio:
A new communication paradigm for passive Internet of Things,” IEEE
Internet Things J., vol. 7, no. 2, pp. 1350–1363, Feb. 2019.
[25] D. Teguig, B. Scheers, and V. Le Nir, “Data fusion schemes for
cooperative spectrum sensing in cognitive radio networks,” in Proc.
Military Commun. Inf. Syst. Conf. (MCC), Gdansk, Poland, 2012, pp.
1–7.
[26] Q. Wu and R. Zhang, “Intelligent reflecting surface enhanced wireless
network: Joint active and passive beamforming design,” in Proc. IEEE
Global Commun. Conf. (GLOBECOM), Abu Dhabi, United Arab Emi-
rates, 2018, pp. 1–6.
[27] M. Grant and S. Boyd, “CVX: MATLAB software for disciplined convex
programming, version 2.2,” http://cvxr.com/cvx, Jan. 2020.
[28] Q. Wu and R. Zhang, “Intelligent reflecting surface enhanced wireless
network via joint active and passive beamforming,” IEEE Trans. Wireless
Commun., vol. 18, no. 11, pp. 5394–5409, Nov. 2019.
[29] F. W. Olver, NIST handbook of mathematical functions hardback and
CD-ROM. Cambridge university press, 2010.
[30] T. Van Chien, L. T. Tu, S. Chatzinotas, and B. Ottersten, “Coverage prob-
ability and ergodic capacity of intelligent reflecting surface-enhanced
communication systems,” IEEE Commun. Lett., vol. 25, no. 1, pp. 69–
73, Jan. 2020.
[31] C. R. Stevenson, G. Chouinard, Z. Lei, W. Hu, S. J. Shellhammer, and
W. Caldwell, “IEEE 802.22: The first cognitive radio wireless regional
area network standard,” IEEE Commun. Mag., vol. 47, no. 1, pp. 130–
138, Jan. 2009.
[32] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and
products. Academic press, 2014.
[33] T. S. Rappaport, Wireless Communications: Principles and Practice.
vol. 2. Upper Saddle River, NJ, USA: Prentice-Hall, 1996.
This article has been accepted for publication in IEEE Transactions on Wireless Communications. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/TWC.2024.3393516
© 2024 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information.
Authorized licensed use limited to: Sony Corporation. Downloaded on May 05,2024 at 02:06:47 UTC from IEEE Xplore. Restrictions apply.
