Joint Transmit Precoding and Reflect Beamforming Design for IRS-Assisted MIMO Cognitive Radio Systems
11 Joint Transmit Precoding and ReflectBeamforming Design for IRS-Assisted MIMOCognitive Radio Systems
Weiheng Jiang,
Member, IEEE,
Yu Zhang
Abstract
Cognitive radio (CR) is an effective solution to improve the spectral efficiency (SE) of wirelesscommunications by allowing the secondary users (SUs) to share spectrum with primary users (PUs).Meanwhile, intelligent reflecting surface (IRS) has been recently proposed as a promising approachto enhance SE and energy efficiency (EE) of wireless communication systems through intelligentlyreconfiguring the channel environment. In this paper, we consider an IRS-assisted downlink CR system,in which a secondary access point (SAP) communicates with multiple SUs without affecting multiplePUs in the primary network and all nodes are equipped with multiple antennas. Our design objectiveis to maximize the achievable weighted sum rate (WSR) of SUs subject to the total transmit powerconstraint at the SAP and the interference constraints at PUs, by jointly optimizing the transmit precodingat the SAP and the reflecting coefficients at the IRS. To deal with the complex objective function,the problem are reformulated by employing the well-known weighted minimum mean-square error(WMMSE) method and an alternating optimization (AO)-based algorithm is proposed. Furthermore, aspecial scenario with only one PU is considered and an AO-based algorithm with lower complexityis proposed. Finally, some numerical simulations have been done to demonstrate that the proposedalgorithm outperforms other benchmark schemes.
Index Terms
Intelligent Reflecting Surface (IRS), Multiple-Input Multiple-Output (MIMO), Cognitive radio (CR),Resource Allocation, Alternating Optimization (AO).
Weiheng Jiang and Yu Zhang are with the School of Microelectronics and Communication Engineering, Chongqing University,Chongqing 400044, China, email: [email protected], [email protected] a r X i v : . [ c s . I T ] F e b I. I
NTRODUCTION
It is known that the spectral efficiency (SE) and energy efficiency (EE) are the two essentialcriteria for designing future wireless networks [1]. While cognitive radio (CR) has been proposedas one effective way to enhance the radio SE and EE [2]. Moreover, it has great potentialin reducing the cost and the complexity and as well as energy consumption of the future5G technologies such as massive MIMO with excessive antennas, and also can support thedevelopment of the sustainable and green wireless networks in the coming years [3]. Meanwhile,recently, a new technology following the development of the Micro-Electro-Mechanical Systems(MEMS) named as intelligent reflecting surfaces (IRS) has been proposed, which can reconfigureand achieve a smart wireless propagation environment via software-controlled reflection [4]–[6].One the one hand, for the IRS assisted wireless communications, in [7], the problem ofjointly optimizing the access point (AP) active beamforming and IRS passive beamformingwith AP transmission power constraint to maximize the received signal power for one pairof transceivers was discussed. Based on the semidefinite relaxation (SDR) and the alternateoptimization (AO), both the centralized algorithm and distributed algorithm were proposedtherein. The work [8] extended the previous work to the multi-users scenario but with theindividual signal-to-noise ratio (SNR) constraints, where the joint optimization of the AP activebeamforming and IRS passive beamforming was discussed to minimize the total AP transmissionpower, and two suboptimal algorithms with different performance-complexity tradeoff werepresented. Huang et al. considered the IRS-based multiple-input single-output (MISO) downlinkmulti-user communications for an outdoor environment, where [9] studied optimizing the basestation (BS) transmission power and IRS phase shift with BS transmission power constraint anduser signal-to-interference-and-noise-ratio (SINR) constraint to maximize sum system rate. Sincethe formulated resource allocation problem is non-convex, Majorization-Minimization (MM) andAO was jointly used, and the convergence of the proposed algorithm was analyzed. Differentfrom the continuous phase shift assumption of the IRS reflecting elements in existing studies,[10] considered that each IRS reflecting element can only achieve discrete phase shift andthe joint optimization of the multi-antenna AP beamforming and IRS discrete phase shift wasdiscussed under the same scenario as [7]. Then the performance loss caused by the IRS discretephase shift was quantitatively analyzed via comparing with the IRS continuous phase shift. It issurprised that, the results have shown that as the number of IRS reflecting elements approaches infinity, the system can obtain the same square power gain as IRS with continuous phase shift,even based on 1-bit discrete phase shift. Furthermore, [11] and [12] discussed the joint APpower allocation and IRS phase-shift optimization to maximize system energy and spectrumefficiency, where the user has a minimum transmission rate constraint and the AP has a totaltransmit power constraint. Due to the presented problem is non-convex, the gradient descent(GD) based AP power allocation algorithm and fractional programming (FP) based IRS phaseshift algorithm were proposed therein. For the IRS assisted wireless communication system,Han and Tang et al. [13] analyzed and obtained a compact approximation of system ergodiccapacity and then, based on statistical channel information and approximate traversal capacity,the optimal IRS phase shift was proved. The authors also derived the required quantized bitsof the IRS discrete phase shift system to obtain an acceptable ergodic capacity degradation. In[14], a new IRS hardware architecture was presented and then, based on compressed sensingand deep learning, two reflection beamforming methods were proposed with different algorithmcomplexity and channel estimation training overhead. Similar to [14], Huang and Debbah etal. [15] proposed a deep learning based algorithm to maximize the received signal strengthfor IRS-assisted indoor wireless communication environment. Some recently studies about theIRS assisted wireless communications could be found in [16]–[19], and they were focused onthe IRS assisted millimeter band or non-orthogonal multiple access (NOMA) based wirelesscommunications.On the other hand, for the IRS assisted CR networks, considering that all nodes were equippedwith single antenna, in [20], an IRS was deployed to assist in the spectrum sharing between aprimary user (PU) link and an secondary user (SU) link. For which, authors aimed to maximizethe achievable SU rate subject to a given SINR target for the PU link, by jointly optimizing theSU transmit power and IRS reflect beamforming. Different from [20], the CR system consistedof multiple SUs and the secondary access point (SAP) with multiple antennas was introducedin [21]. Specifically, based on both bounded channel state information (CSI) error model andstatistical CSI error model for PU-related channels, robust beamforming design was investigatedand the transmit precoding matrix at the SAP and phase shifts at the IRS were jointly optimized tominimize the total transmit power at the SAP subject to the quality of service of SUs, the limitedinterference imposed on the PU and unit-modulus of the reflective beamforming. In addition,in [22] and [23], Jie Yuan considered a CR system including multiple PUs and single SU, andmaximized the transmission rate of SUs under the constraints of the maximum transmitting power at the SAP and the interference leakage at the PUs. Moreover, the research result was extendedto the case with imperfect CSI. It is noted that the system mentioned in [22] contained only oneIRS, while multiple IRSs were introduced in [23]. Furthermore, in [24], the beamforming vectorsat the base station and the phase shift matrix at the IRS are jointly optimized for maximizationof the sum rate of the secondary system based on the network with multiple PUs and SUs, andthe suboptimal solution was obtained by the AO-based algorithm. Considered the same scenario,[25] minimized the transmit power at the SAP equipped with multiple antennas and the AO-based algorithm was proposed as well. Furthermore, in [26], a IRS-assisted secondary networkemployed an full-duplex base station for serving multiple half-duplex downlink and uplink userssimultaneously. The downlink transmit beamforming vectors and the uplink receive beamformingvectors at the full-duplex base station, the transmit power of the uplink users, and the phase shiftmatrix at the IRS are jointly optimized for maximization of the total sum rate of the secondarysystem. The design task is formulated as a non-convex optimization problem taking into accountthe imperfect CSI of the PUs and their maximum interference tolerance, for which, an iterativeblock coordinate descent (BCD)-based algorithm was developed. Combining IRS with physicallayer security, [27] aimed to solve the security issue of CR networks. Specifically, an IRS-assistedMISO CR wiretap channel was studied. To maximize the secrecy rate of SUs subject to a totalpower constraint for the transmitter and interference power constraint for a single antenna PU,an AO algorithm is proposed to jointly optimize the transmit covariance at transmitter and phaseshift coefficient at IRS by fixing the other as constant.Note that, all the existing research about the IRS-assisted CR network only consider that PUsand SUs are equipped with single antenna [20]–[27]. However, in order to further improve theperformance of the wireless systems, multi-antenna enabled technologies are adopted by manycommercial standards, such as the IEEE 802.11ax, LTE and the fifth generation (5G) mobilenetworks. Therefore, it is necessary to study the IRS assisted CR system with multiple-transmitand multiple-receive antennas, and this is the focus of this work. After finishing the researchof this work, we note that similar work has been done in [28] but with single PU. In specific,the author proposed an IRS assisted CR system which includes a SAP, a PU and multiple SUs.However, it should be noted that compared with [28], our work in this paper is more universaland the scenario studied in [28] can be seen as a special case of our work. Specifically, thecontributions of this paper are summarized as follows. ‚ In this paper, an IRS-assisted downlink multiple-input multiple-output (MIMO) CR system is discussed, that is, a SAP communicates with multiple SUs with the assistance of anIRS but without affecting multiple PUs in the primary network. We aim at maximizing theachievable WSR of SUs by jointly optimizing the transmit precoding matrix at the SAPand the reflecting coefficients at the IRS, subject to a total transmit power constraint at theSAP and interference temperature constraints at PUs. To deal with the complex objectivefunction, the problem are reformulated by employing the well-known weighted minimummean-square error (WMMSE) method and an AO-based algorithm is proposed. That is, forthe auxiliary matrix, decoding matrices, SAP precoding matrix and IRS reflection coefficientmatrix, one of them is iteratively obtained while keeping the others fixed, and the processcontinues until convergence. ‚ In addition, for the scenario with single PU, i.e., the same scenario as [28], a lowercomplexity algorithm is proposed. Since the algorithm proposed in [28] involves the inverseoperation of the matrix in the iterative process for SAP precodind matrix optimization undergiven auxiliary matrix, decoding matrix and IRS reflection coefficients, whose complexity isvery high. In this paper, the special structure of the matrices are used to remove the inverseoperation. Further, without loss the performance, the algorithm with lower complexity ispresented by the Lagrangian dual decomposition and successive convex approximation(SCA) method, and the optimal solution of the subproblem is obtained. ‚ Finally, some numerical simulations have been done to demonstrate that the proposedalgorithm outperforms other benchmark schemes. Note that, simulation results include twopart which are corresponding to the general scenario with multiple PUs and the specialscenario with only one PU.The rest of this paper is organized as follows. In Section II, the system model and the con-sidered optimization problem are presented. In Section III, the considered problem is discussedand solved, and an AO-based algorithm is proposed. Then the discussion is extended to a specialscenario with only one PU in Section IV and an AO-based algorithm with lower complexity isproposed therein. The simulation results are presented in Section V and we conclude at last.
Notation:
We use uppercase boldface letters for matrices and lowercase boldface letters forvectors. E t‚u stands for the statistical expectation for random variables, and | ‚ | , arg p‚q , (cid:60) t‚u and p‚q ˚ denote the absolute value, the argument, the real part and the conjugate of a complexnumber, respectively. det p‚q and T r p‚q indicate the determinant and trace of a matrix, whereas p‚q T , p‚q H , p‚q ´ and p‚q : represent the transpose, conjugate transpose, inverse and pseudo- SAPIRS
SU-1PU-1
PAP
PU-2 PU-K
SU-2 SU-L H pk H rk H sk H pl H rl H sl H sr H pr Fig. 1 System Model.inverse of a matrix, respectively. In addition, I denotes the identity matrix with appropriate size,and diag p‚q represents a diagonal matrix whose diagonal elements are from a vector. A ě and A ą indicate that A are positive semi-definite and positive definite matrix.II. S YSTEM M ODEL AND T HE P ROBLEM
In this section, firstly, we present the system model of the intelligent reflecting surface (IRS)-assisted downlink multiple-input multiple-output (MIMO) cognitive radio (CR) system, referringto it as the IRS-MIMO-CR system. Then, we illustrate the signal model for our consideredsystem, which includes the channel model and IRS reflecting model. Note that, as [7] and[8], the signals that are reflected by the IRS multi-times are ignored due to significant pathloss. Moreover, to characterize the performance limit of the considered IRS-assisted securecommunication system, the quasi-static flat-fading channel model is adopted herein and all theCSI are perfectly known at the SAP [29]. Finally, we formulate the discussed optimizationproblem.
A. System Model
Consider the IRS-MIMO-CR system, as shown in Fig. 1, where a secondary access point(SAP) serves multiple secondary users (SUs) without affecting the communications betweenprimary access point (PAP) and multiple primary users (PUs) in the primary network. All nodes are equipped with multiple antennas and the number of antennas at the PAP, PUs, SAP andSUs are N P A , N P U , N SA and N SU , respectively. Denote the sets of PUs and SUs as K “t , , ..., K u and L “ t , , ..., L u , respectively. In addition, an IRS composed of M passiveelements, which are denoted as the set M “ t , , ..., M u , is installed on a surrounding wall toassist the communications between the SAP and SUs. The IRS has a smart controller, who hasthe capability of dynamically adjusting the phase shift of each reflecting element based on thepropagation environment learned through periodic sensing [7].The number of data streams for each SU is assumed to be d , satisfying ď d ď min t N SA , N SU u .The transmit signal from the BS is given by x “ L ÿ l “ F l s l , (1)where s l P C d ˆ , @ l P L is the d ˆ data symbol vector designated for the l th SU satisfying E “ s l s Hl ‰ “ I and E “ s i s Hj ‰ “ , @ i, j P L , i ‰ j . In addition, F l P C N SA ˆ d , @ l P L is the linearprecoding matrix used by the SAP for the l th SU. As a result, the received signals, which aretransmitted from SAP, at the k th PU and l th SU are given by y k “ r k ` H sk x ` H rk ΘH sr x ` n k “ r k ` p H sk ` H rk ΘH sr q x ` n k “ r k ` G sk p Θ q x ` n k , (2) y l “ H sl x ` H rl ΘH sr x ` n l “ p H sl ` H rl ΘH sr q x ` n l “ G sl p Θ q x ` n l , (3)where r k P C N PU ˆ , @ k P K stands for the signal which is transmitted from PAP and receivedby the k th PU. Moreover, the baseband equivalent channels from the SAP to the k th PU, fromthe SAP to the l th SU, from the SAP to the IRS, from the IRS to the k th PU, from the IRSto the l th SU are modelled by matrices H sk P C N PU ˆ N SA , H sl P C N SU ˆ N SA , H sr P C M ˆ N SA , H rk P C N PU ˆ M and H rl P C N SU ˆ M , respectively. Let Θ “ diag p θ q P C M ˆ M denote the diagonalreflection matrix of the IRS, with θ “ r θ , θ , ..., θ M s T P C M ˆ and | θ m | “ , @ m P M .Therefore, the effective MIMO channel matrix from the SAP to the k th PU and l th SU receiveris given by G sk p Θ q “ H sk ` H rk ΘH sr and G sl p Θ q “ H sl ` H rl ΘH sr , respectively. n k „ CN p , σ k I q represents the additive white Gaussian noise at the k th PU, and n l „ CN p , σ l I q is the equivalent noise at the l th SU, which captures the joint effect of the received interferencefrom the primary network and thermal noise. σ k and σ l denote the corresponding average noisepower at the k th PU and the l th SU. By substituting (1) into (3), the received signals of the l thSU are reformulated as y l “ G sl p Θ q F l s l ` L ÿ i “ ,i ‰ l G sl p Θ q F i s i ` n l . (4)Denotes the collection of precoding matrixs used by the SAP as F “ t F , F , ..., F L u . Hence,the transmit data rate (bit/s/Hz) of the l th SU is written as R l p F , Θ q “ log det ` I ` G sl p Θ q F l F Hl G Hsl p Θ q J ´ l ˘ , (5)where J l “ L ř i “ ,i ‰ l G sl p Θ q F i F Hi G Hsl p Θ q ` σ l I represents the interference-plus-noise covariancematrix of the l th SU. Moreover, the interference signal power imposed on the k th PU is denotedas IT k “ L ÿ l “ T r ` G sk p Θ q F l F Hl G Hsk p Θ q ˘ . (6) B. Problem Formulation
As mentioned earlier, we discuss the joint optimization of the transmit precoding matrix atthe SAP and the reflection coefficients at the IRS to maximize the achievable WSR of SUssubjected to the total transmit power constraint at the SAP, interference constraints at PUs andthe reflection coefficient constraint at the IRS. Thus we have the following OP1, max F , Θ L ÿ l “ ω l R l p F , Θ q s.t. C L ÿ l “ T r ` F Hl F l ˘ ď P max C L ÿ l “ T r ` G sk p Θ q F l F Hl G Hsk p Θ q ˘ ď Γ k , @ k P K C | Θ mm | “ , @ m P M . (7)In which, C1 characterizes the total transmit power constraint at the SAP, C2 defines theinterference constraints at PUs, and C3 represents the IRS reflecting coefficient constraint, Γ k denotes the maximum received interference power at the PU k . It is obvious that the OP1 is anon-convex nonlinear programming with coupled variables F and Θ and also the uni-modularconstraint on each reflection coefficient Θ mm , which makes it difficult to solve. Therefore, inthe sequel, we pursue the suboptimal approach to handle the OP1. III. A
LTERNATING O PTIMIZATION BASED J OINT O PTIMIZATION A LGORITHM
In this section, a suboptimal algorithm is proposed to solve the OP1. As aforementioned that,our formulated OP1 is a non-convex nonlinear programming, therefore, we first transform theOP1 into a more tractable one, which allows the decoupling of precoding matrices and thereflection coefficient matrix. Then, alternating optimization (AO) algorithm is proposed to solvethe transformed problem.
A. Reformulation of the Original Problem
To deal with the complex objective function, we reformulate the OP1 by employing the well-known WMMSE [30] method. Specifically, the linear decoding matrix is applied to estimate thesignal vector for each SU, which is denoted by ˆ s l “ U Hl y l , @ l P L (8)where U l P C N SU ˆ d is the decoding matrix for the l th SU. Then, the MSE matrix for the l thSU is given by E l “ E s , n ” p ˆ s l ´ s l q p ˆ s l ´ s l q H ı “ ` U Hl G sl p Θ q F l ´ I ˘ ` U Hl G sl p Θ q F l ´ I ˘ H ` L ÿ i “ ,i ‰ l U Hl G sl p Θ q F i F Hi G Hsl p Θ q U l ` σ l U Hl U l , @ l P L (9)where s and n denote the collections of data symbols and noise vectors of all SUs, respectively.Denoting the sets of decoding matrices as U “ t U l , @ l P L u and introducing a set of auxiliarymatrices W “ t W l ě , @ l P L u , the OP1 can be reformulated as the following OP2 max W , U , F , Θ L ÿ l “ ω l h l p W , U , F , Θ q s.t. C , C and C . (10)In which, h l p W , U , F , Θ q is given by h l p W , U , F , Θ q “ log det p W l q ´ T r p W l E l q ` d . (11)Note that, by iteratively obtaining one set of variables while keeping the others fixed, the objectivefunction in OP2 is much easier to handle with AO. Since the decoding matrices U and auxiliary matrices W only appear in h l p W , U , F , Θ q , the optimal solution of U and W can be obtainedby setting the first-order derivative of h l p W , U , F , Θ q with respect to U l and W l to zero whilekeeping the other matrices fixed. Hence, the optimal solutions are denoted as ˆU l “ ` J l ` G sl F l F Hl G Hsl ˘ ´ G sl F l , (12) ˆW l “ ´ ˆE l ¯ ´ . (13)Herein, given the reflection coefficient matrix Θ at the IRS, G sl p Θ q is simplistically denotedas G sl . Moreover, ˆE l is obtained by inserting ˆU l into the MSE matrix of the l th SU, yielding ˆE l “ I ´ F Hl G Hsl ` J l ` G sl F l F Hl G Hsl ˘ ´ G sl F l (14)In the following, the precoding matrices F and reflection coefficient matrix Θ are optimizedwith given U and W . B. Optimization of the precoding matrices
In this subsection, we discuss the precoding matrices optimization at the SAP for given thereflecting coefficients at the IRS, decoding matrices and auxiliary matrices. Hence, substituting E into the objective function of OP2 and discarding the constant terms, the precoding matricesoptimization problem can be transformed as the following OP3, min F L ÿ l “ T r ` F Hl X F l ˘ ´ L ÿ l “ Re (cid:32) T r ` Y Hl F Hl ˘( s.t. L ÿ l “ T r ` F Hl F l ˘ ď P max L ÿ l “ T r ` F Hl X k F l ˘ ď Γ k , @ k P K . (15)In which, G sk is the abbreviated form of G sk p Θ q with given the reflection coefficient matrix Θ at the IRS. In addition, X “ L ř m “ ω m G Hsm U m W m U Hm G sm ě , X k “ G Hsk G sk ě , @ k P K and Y l “ ω l W l U Hl G sl , @ l P L . It is obvious that OP3 is quadratically constrained quadraticprogramming (QCQP) problem [31]. Since X ě and X k ě , OP3 is convex and can beeasily solved with standard interior-point methods. C. Optimize the IRS reflecting coefficients
In this subsection, given the precoding matrices at the SAP, decoding matrices and auxiliarymatrices, the optimization of the reflecting coefficient matrix Θ at IRS is discussed. Particularly,we have the following OP4. min Θ L ÿ l “ T r ` ω l W l U Hl G sl p Θ q Q s G Hsl p Θ q U l ˘ ´ L ÿ l “ Re (cid:32) T r ` ω l W l U Hl G sl p Θ q F l ˘( s.t. T r ` G sk p Θ q Q s G Hsk p Θ q ˘ ď Γ k , @ k P K | Θ mm | “ , @ m P M (16)where Q s “ L ř m “ F m F Hm . By using G sl p Θ q “ H sl ` H rl ΘH sr , we have ω l W l U Hl G sl p Θ q Q s G Hsl p Θ q U l “ ω l W l U Hl H rl ΘH sr Q s H Hsr Θ H H Hrl U l ` ω l W l U Hl H rl ΘH sr Q s H Hsl U l ` ω l W l U Hl H sl Q s H Hsr Θ H H Hrl U l ` ω l W l U Hl H sl Q s H Hsl U l (17) ω l W l U Hl G sl p Θ q F l “ ω l W l U Hl H rl ΘH sr F l ` ω l W l U Hl H sl F l (18) G sk p Θ q Q s G Hsk p Θ q “ H rk ΘH sr Q s H Hsr Θ H H Hrk ` H rk ΘH sr Q s H Hsk ` H sk Q s H Hsr Θ H H Hrk ` H sk Q s H Hsk (19)Based on the above conversions, discarding the constant terms, the OP4 can be transformed intothe following formulation, min Θ T r ` B ΘCΘ H ˘ ` Re (cid:32) T r ` D H Θ H ˘( s.t. T r ` B k ΘCΘ H ˘ ` Re (cid:32) T r ` D Hk Θ H ˘( ď ˜Γ k , @ k P K | Θ mm | “ , @ m P M . (20)Herein, B “ L ř l “ ω l H Hrl U l W l U Hl H rl ě , B k “ H Hrk H rk ě , C “ H sr Q s H Hsr ě , D “ L ř l “ ω l H sr Q s H Hsl U l W l U Hl H rl ´ L ř l “ ω l H sr F l W l U Hl H rl , D k “ H sr Q s H Hsk H rk and ˜Γ k “ Γ k ´ Tr ` H sk Q s H Hsk ˘ . Since Θ “ diag p θ q is a diagonal matrix, adopting the matrix identity in [32],it follows that T r ` B ΘCΘ H ˘ “ θ H p B d C q θ T r ` B k ΘCΘ H ˘ “ θ H p B k d C q θ . (21)Further, denote d “ ” r D s , , r D s , , ..., r D s M,M ı T and d k “ ” r D k s , , r D k s , , ..., r D k s M,M ı T as the collections of diagonal elements of D and D k . We thus have T r ` D H Θ H ˘ “ θ H d ˚ , T r ` D Hk Θ H ˘ “ θ H d ˚ k . (22)Therefore, the problem (20) can be simplified as min θ θ H Υ θ ` Re (cid:32) θ H d ˚ ( s.t. θ H Υ k θ ` Re (cid:32) θ H d ˚ k ( ď ˜Γ k , @ k P K | θ m | “ , @ m P M . (23)Herein, since B ě , B k ě and C ě , there are Υ “ B d C T ě and Υ k “ B k d C T ě . However, due to the non-convexity of the uni-modulus constraint on each reflectioncoefficient θ m , the problem (23) is non-convex. Hence, penalty function (PF) and successiveconvex approximation (SCA) are adopted. Specifically, introducing the slack factor λ ď , theproblem (23) can be reformulated as min θ θ H Υ θ ` Re (cid:32) θ H d ˚ ( ´ λ θ H θ s.t. θ H Υ k θ ` Re (cid:32) θ H d ˚ k ( ď ˜Γ k , @ k P K | θ m | ď , @ m P M . (24)The term λ θ H θ could ensure that the uni-modulus constraint | θ m | “ , @ m P M is hold atthe optimal solution when λ Ñ `8 . Note that, the objective function of the problem (24) isthe sum of a convex function and a concave function which means (24) is non-convex. Hence,the SCA-based algorithm [33], [34] is used to handle (24) and the convex part of the objectivefunction is approximated by its first order Taylor expansion. In specific, given the initial point θ p n q and discarding the constant terms, the sub-problem can be denoted as min θ θ H Υ θ ` Re (cid:32) θ H d ˚ ( ´ λRe (cid:32) θ H θ p n q ( s.t. θ H Υ k θ ` Re (cid:32) θ H d ˚ k ( ď ˜Γ k , @ k P K | θ m | ď , @ m P M . (25) Now, (25) is convex and can be solved by standard interior-point methods [31]. Therefore, (24)can be handled by solving a series of convex problems iteratively and the details are summarizedin the Algorithm 1 as below.
Algorithm 1:
SCA-based Algorithm to Solve (24)S1: Initialize θ p q , ε ą , n “ , calculate the objective value of(24) as z ´ θ p q ¯ ;S2: Given θ p n q , obtain ˆ θ p n q by solving problem (25) with CVX;S3: If ˇˇˇ z ´ ˆ θ p n q ¯ ´ z ´ θ p n q ¯ˇˇˇ ą ε , set z ´ θ p n ` q ¯ “ z ´ ˆ θ p n q ¯ , θ p n ` q “ ˆ θ p n q , n “ n ` , go back to S2; else set ˆ θ “ ˆ θ p n q ;S4: Output ˆ θ ; Based on the above discussion, we formulate the PF-based algorithm to solve the OP4 as thefollowing Algorithm 2.
Algorithm 2:
PF-based Algorithm to Solve OP4S1: Initialize θ p q , ε ą , λ p q , n “ ;S2: Given λ p n q , obtain ˆ θ p n q by solving problem (24) using algorithm 1;S3: If M ř m “ ´ˇˇˇ ˆ θ p n q m ˇˇˇ ´ ¯ ą ε , set θ p n ` q “ ˆ θ p n q , n “ n ` ,go back to S2; else set ˆ θ “ ˆ θ p n q ;S4: Output ˆ θ ; Note that, we adopt the stopping criterion M ř m “ ´ˇˇˇ ˆ θ p n q m ˇˇˇ ´ ¯ ą ε to ensure the uni-modulusconstraint is hold at the optimal solution for the Algorithm 2. D. Overall Algorithm
In this subsection, the overall algorithm for the OP1 is provided. As mentioned, the algorithmis based on alternating optimization, which optimizes the objective function with respect todifferent subsets of optimization variables in each iteration while the other subsets are fixed.Therefore, it is summarized as the following Algorithm 3. Algorithm 3:
AO-based Algorithm to Solve OP1S1: Initialize F p q , θ p q , ε ą , n “ , calculate the WSR of all SUs as R ´ F p q , Θ p q ¯ ;S2: Given F p n q and θ p n q , obtain ˆU p n q and ˆW p n q according to (12) and(13);S3: Given ˆU p n q , ˆW p n q and θ p n q , obtain ˆF p n q by solving the OP3 withCVX;S4: Given ˆU p n q , ˆW p n q and ˆF p n q , obtain ˆ θ p n q by solving the OP4 usingalgorithm 2;S5: If ˇˇˇ R ´ ˆF p n q , ˆΘ p n q ¯ ´ R ´ F p n q , Θ p n q ¯ˇˇˇ ą ε , set F p n ` q “ ˆF p n ` q θ p n ` q “ ˆ θ p n q , R ´ F p n ` q , Θ p n ` q ¯ “ R ´ ˆF p n q , ˆΘ p n q ¯ , n “ n ` , go back to S2; else set ˆF “ ˆF p n q , ˆ θ “ ˆ θ p n q ;S6: Output ˆF and ˆ θ ; where ˆU p n q , ˆW p n q , ˆF p n q and ˆ θ p n q represent the stable solutions obtained by solving the subprob-lems in the n th iteration, and R p F , Θ q denotes the WSR of all SUs. Since the original problemis bounded and the progress of the alternative optimization is monotonically non-decreasing, thusthe above algorithm is surely convergent. Furthermore, we analyze the computational complexityof the proposed Algorithm 3. The complexity of the algorithm mainly depends on the Step 3and the Step 4, for which have the complexities of O p Ld N SA q and O p T T M q [31]. In which, T , T and T denote the iteration numbers of the Algorithm 1, 2 and 3, respectively. Hence,the complexity of the overall algorithm is denoted as O p T p Ld N SA ` T T M qq .IV. T HE S PECIAL S CENARIO W ITH O NLY O NE PUIn this section, the IRS-assisted CR network with single PU is discussed and in which, an AO-based algorithm with the lower complexity is proposed. Specifically, let Γ p denote the maximumreceived interference power at the unique PU. H sp P C N PU ˆ N SA and H rp P C N PU ˆ M represent thebaseband equivalent channels from the SAP and the IRS to the PU, respectively. Therefore, theeffective MIMO channel matrix from the SAP to the PU is given by G sp p Θ q “ H sp ` H rp ΘH sr . So, the OP1 can be simplified as the following OP5, max F , Θ L ÿ l “ ω l R l p F , Θ q s.t. C L ÿ l “ T r ` F Hl F l ˘ ď P max C L ÿ l “ T r ` G sp p Θ q F l F Hl G Hsp p Θ q ˘ ď Γ p C | Θ mm | “ , @ m P M . (26)Note that, the model adopted in this section differs from that in the previous section only inthe number of PUs, which only affects the number of interference constraints at PUs. Similarly,the OP5 is transformed to the following OP6 with the WMMSE method, max W , U , F , Θ L ÿ l “ ω l h l p W , U , F , Θ q s.t. C , C and C . (27)Based on the above, the AO algorithm is adopted again and the optimal solution of U and W can be calculated by (12) and (13). In the following, the precoding matrices F and reflectioncoefficient matrix Θ are optimized with given U and W . A. Optimization of the precoding matrices
In this subsection, we discuss the precoding matrices optimization at the SAP for given thereflecting coefficients at the IRS, decoding matrices and auxiliary matrices. Hence, substituting E into the objective function of OP6 and discarding the constant terms, the precoding matricesoptimization problem can be transformed as following OP7, min F L ÿ l “ T r ` F Hl X F l ˘ ´ L ÿ l “ Re (cid:32) T r ` Y Hl F Hl ˘( s.t. C L ÿ l “ T r ` F Hl F l ˘ ď P max C L ÿ l “ T r ` F Hl X p F l ˘ ď Γ p . (28)Herein, G sp is the abbreviated form of G sp p Θ q with given the reflection coefficient matrix Θ at the IRS. In addition, X “ L ř m “ ω m G Hsm U m W m U Hm G sm ě , X p “ G Hsp G sp ě and Y l “ ω l W l U Hl G sl , @ l P L . Same as the OP3, since X ě and X p ě , the OP7 is QCQPconvex optimization problem [31], which can be solved by the standard convex solver packagessuch as CVX. However, the computational complexity is high. In the following, we provide alow-complexity SCA-based algorithm which can obtained the optimal solution of the OP7 bysolving a series of simple convex problem with Lagrangian dual decomposition method. Forwhich, we introduce the following proposition. Proposition 1:
Let f p p F q “ L ř l “ T r ` F Hl X p F l ˘ , X p ě , Z p “ λ p I and λ p denotes themaximum eigenvalue of the X p . Then for @ F and given F p n q , there exists ˜ f p ` F | F p n q ˘ “ L ÿ l “ T r ` F Hl Z p F l ˘ ` L ÿ l “ T r ´ F p n q Hl p Z p ´ X p q F p n q l ¯ ´ L ÿ l “ Re ! T r ´ F p n q Hl p Z p ´ X p q F l ¯) (29)which satisfies the following three conditions:1) ˜ f p ` F p n q | F p n q ˘ “ f p ` F p n q ˘ ,2) ∇ F ˚ ˜ f p ` F | F p n q ˘ˇˇˇ F “ F p n q “ ∇ F ˚ f p p F q| F “ F p n q ,3) ˜ f p ` F | F p n q ˘ ě f p p F q . Proof:
Please see the Appendix A. (cid:4)
Based on Proposition 1, the interference constraint C2 in OP7 can be replaced by the followinginequality, L ÿ l “ T r ` F Hl Z p F l ˘ ´ L ÿ l “ Re ! T r ´ F p n q Hl p Z p ´ X p q F l ¯) ď ˜Γ p . (30)Herein, ˜Γ p “ Γ p ´ L ř l “ T r ´ F p n q Hl p Z p ´ X p q F p n q l ¯ . Therefore, we have the following problem min F L ÿ l “ T r ` F Hl X F l ˘ ´ L ÿ l “ Re (cid:32) T r ` Y Hl F Hl ˘( s.t. p C q , p q (31)We note that (31) is convex and thus traditional interior-point methods (IPM) can be used tohandle it. However, to avoid high computational complexity of the IPM, herein, a low-complexityalgorithm based on Lagrange duality decomposition is presented.Specifically, assuming the optimal solution of (31) is ˆF , herein, according to whether thepower constraint (28 C1) is an active constraint at ˆF , two cases are discussed in the following. Case 1 : Assuming the power constraint (28 C1) is an inactive constraint at ˆF , problem (31)can be transformed into the following problem, min F L ÿ l “ T r ` F Hl X F l ˘ ´ L ÿ l “ Re (cid:32) T r ` Y Hl F Hl ˘( s.t. p q (32)Introducing the Lagrange multiplier µ associated with the interference constraint (30), the La-grangian function for problem (32) can be derived as follows L p F , µ q “ L ÿ l “ T r ` F Hl p X ` µ Z p q F l ˘ ´ L ÿ l “ Re (cid:32) T r ` Y Hl F Hl ˘( ´ µ ˜Γ p ´ L ÿ l “ µRe ! T r ´ F p n q Hl p Z p ´ X p q F l ¯) . (33)The dual function can be obtained by solving the following problem g p µ q ∆ “ min F L p F , µ q , (34)and the dual problem is given by max µ g p µ q s.t. µ ě . (35)By setting the first-order derivative of L p F , µ q w.r.t. F to zero matrix, we can obtain the optimalsolution ˆF p µ q “ ! ˆF l p µ q , @ l P L ) as follows: ˆF l p µ q “ p X ` µ Z p q : ´ Y Hl ` µ p Z p ´ X p q F p n q l ¯ (36)where pseudo inverse is adopted due to the fact that the matrix X ` µ Z p is not full rank when X is not full rank and µ “ . The value of µ should be chosen such that the complementaryslackness condition for constraint (30) is satisfied, namely, µ ˜ L ÿ l “ T r ` F Hl p µ q Z p F l p µ q ˘ ´ L ÿ l “ Re ! T r ´ F p n q Hl p Z p ´ X p q F l p µ q ¯) ´ ˜Γ p ¸ “ . (37)Hence, if the following condition holds L ÿ l “ T r ` F Hl p q Z p F l p q ˘ ´ L ÿ l “ Re ! T r ´ F p n q Hl p Z p ´ X p q F l p q ¯) ď ˜Γ p , (38)the optimal solution to problem (32) is given by ˆF p µ q| µ “ . Otherwise, we need to find µ whichsatisfies the following equation: J p µ q “ L ÿ l “ T r ` F Hl p µ q Z p F l p µ q ˘ ´ L ÿ l “ Re ! T r ´ F p n q Hl p Z p ´ X p q F l p µ q ¯) “ ˜Γ p . (39) For which, the following proposition is introduced.
Proposition 2: J p µ q is a monotonically non-increasing function of µ . Proof:
Please see the Appendix B. (cid:4)
Based on Proposition 2, the bisection search method can be used to find the solution ofequation (39) and the algorithm is shown as below.
Algorithm 4:
Bisection Search Method for (39)S1: Initialize ε ą , n “ and the bounds µ p q l and µ p q u of µ ;S2: Let µ p n q “ p µ p n q l ` µ p n q u q{ , calculate ˆF ´ µ p n q u ¯ and J ´ µ p n q ¯ according to (36) and (39);S3: If J ´ µ p n q ¯ ě ˜Γ p , set µ p n ` q l “ µ p n q and µ p n ` q u “ µ p n q u ;otherwise set µ p n ` q l “ µ p n q l and µ p n ` q u “ µ p n q ;Let n “ n ` ;S4: If ˇˇˇ µ p n q u ´ µ p n q l ˇˇˇ ą ε , go back to S2; else, set ˆ µ “ µ p n q ;S5: Output ˆ µ ; In each iteration of Algorithm 4, we need to calculate ˆF p µ q in (36), which involves thecalculation of p X ` µ Z p q : with a complexity of O p N SA q . If the total number of iterations is T ,the total complexity to calculate p X ` µ Z p q : is O p T N SA q , which may be excessive. Here, weprovide one method to reduce the computational complexity. Specifically, as X is a semi-definitepositive matrix, it can be decomposed as X “ QΛQ H by using the eigenvalue decomposition,where QQ H “ Q H Q “ I and Λ is a diagonal matrix with non-negative diagonal elements. Then,we have p X ` µ Z p q : “ Q p Λ ` µλ p I q : Q H since Z p “ λ p I . Hence, in each iteration, we onlyneed to calculate the product of matrices, which has much lower complexity than calculatingthe inverse of matrices with the same dimension. Case 2 : Assuming the power constraint (28 C1) is an active constraint at ˆF , problem (31) canbe transformed into the following problem, min F L ÿ l “ T r ` F Hl X F l ˘ ´ L ÿ l “ Re (cid:32) T r ` Y Hl F Hl ˘( s.t. C L ÿ l “ T r ` F Hl F l ˘ ď P max C L ÿ l “ Re ! T r ´ F p n q Hl p Z p ´ X p q F l ¯) ě " Γ p . (40) In which, " Γ p “ λ p P max ´ ˜Γ p . Using lagrangian dual decomposition method and introducing theLagrange multiplier λ associated with the power constraint, the partial Lagrangian function forproblem (40) can be derived as follows L p F , λ q “ L ÿ l “ Tr ` F Hl X F l ˘ ´ L ÿ l “ (cid:32) Tr ` Y Hl F Hl ˘( ` λ L ÿ l “ T r ` F Hl F l ˘ ´ λP max . (41)The dual function can be obtained by solving the following problem g p λ q ∆ “ min F L p F , λ q s.t. p C q , (42)and the dual problem is given by max λ g p λ q s.t. λ ě . (43)In order to solve the dual problem (43), we need to derive the expression of dual function g p λ q by solving problem (42) with given λ . By introducing dual variable µ ě associated with theinterference constraint (30), the Lagrangian function for problem (42) is given by L p F , µ q “ L ÿ l “ T r ` F Hl p X ` λ I q F l ˘ ´ L ÿ l “ Re (cid:32) T r ` Y Hl F Hl ˘( ´ λP max ` µ " Γ p ´ L ÿ l “ µRe ! T r ´ F p n q Hl p Z p ´ X p q F l ¯) (44)By setting the first-order derivative of L p F , µ q w.r.t. F to zero matrix, we can obtain the optimalsolution ˆF p µ q “ ! ˆF l p µ q , @ l P L ) as follows ˆF l p µ q “ p X ` λ I q : ´ Y Hl ` µ p Z p ´ X p q F p n q l ¯ . (45)In which, the value of µ should be chosen such that the complementary slackness condition forconstraint (40 C2) is satisfied µ ˜ L ÿ l “ Re ! T r ´ F p n q Hl p Z p ´ X p q ˆF l p µ q ¯) ´ " Γ p ¸ “ . (46)Hence, if the following condition holds L ÿ l “ Re ! T r ´ F p n q Hl p Z p ´ X p q ˆF l p µ q| µ “ ¯) ě " Γ p , (47) the optimal solution to (42) is given by ˆF p µ q| µ “ . Otherwise, the optimal µ is given by µ “ " Γ p ´ L ř l “ Re ! T r ´ F p n q Hl p Z p ´ X p q p X ` λ I q : Y Hl ¯) L ř l “ T r ´ F p n q Hl p Z p ´ X p q p X ` λ I q : p Z p ´ X p q F p n q l ¯ . (48)With dual function, we start to solve the dual problem (43) to find the optimal λ . Given λ ,denote the optimal solution of the problem (42) as ˆF p λ q . The value of λ should be chosen suchthat the complementary slackness condition for power constraint is satisfied λ ˜ L ÿ l “ T r ´ ˆF Hl p λ q ˆF l p λ q ¯ ´ P max ¸ “ . (49)If the following condition holds L ÿ l “ T r ´ ˆF Hl p λ q| λ “ ˆF l p λ q| λ “ ¯ ď P max , (50)the optimal solution is given by ˆF p λ q | λ “ . Otherwise, we need to find λ such that the followingequation holds: P p λ q “ L ÿ l “ T r ´ ˆF Hl p λ q ˆF l p λ q ¯ “ P max (51)For which, the following proposition is introduced. Proposition 3: P p λ q is a monotonically non-increasing function of λ . Proof:
The proof is similar to than for Proposition 2 and thus it is omitted herein. (cid:4)
Based on Proposition 3, the bisection search method can be used to find the solution ofequation (51) and we formulate the algorithm shown below, i.e., the Algorithm 5.
Algorithm 5:
Bisection Search Method for (51)S1: Initialize ε ą , n “ and the bounds λ p q l and λ p q u of λ ;S2: Let λ p n q “ p λ p n q l ` λ p n q u q{ , if the condition (47) is satisfied, µ “ ; otherwise, calculate µ according to (48);S3: Calculate ˆF ´ λ p n q u ¯ and P ´ λ p n q ¯ according to (45) and (51);S3: If P ´ λ p n q ¯ ě P max , set λ p n ` q l “ λ p n q and λ p n ` q u “ λ p n q u ;otherwise set λ p n ` q l “ λ p n q l and λ p n ` q u “ λ p n q ;Let n “ n ` ;S4: If ˇˇˇ λ p n q u ´ λ p n q l ˇˇˇ ą ε , go back to S2; else, set ˆ λ “ λ p n q ;S5: Output ˆ λ ; Similarly, in each iteration of Algorithm 5, we need to calculate ˆF p λ q in (45), which involvesthe calculation of p X ` λ I q : . Herein, the same approach as Algorithm 4 can be adopted toreduce the complexity.Based on the above discussion, the details of the SCA algorithm to solve OP7 is summarizedas the Algorithm 6 shown below. Algorithm 6:
SCA-based Algorithm to Solve OP7S1: Initialize ε ą , F p q , n “ , calculate the objective value of the OP7as z ´ F p q ¯ ;S2: Given F p q , obtain ˆF p n q by solving (32) using algorithm 4;S3: if ˆF p n q satisfies the constraint (28 C1), go to S5;S4: Given F p q , obtain ˆF p n q by solving (40) using algorithm 5;S5: If ˇˇˇ z ´ ˆF p n q ¯ ´ z ´ F p n q ¯ˇˇˇ ą ε , set z ´ F p n ` q ¯ “ z ´ ˆF p n q ¯ , F p n ` q “ ˆF p n q , n “ n ` , go back to S2; else set ˆF “ ˆF p n q ;S6: Output ˆF ; Also, we have the convergence conclusion for the Algorithm 6 as below.
Proposition 4:
The sequence generated by Algorithm 6, i.e., ! ˆF p n q , n “ , , , ... ) convergesto the KKT optimum point of OP7. Proof:
Please see the Appendix C. (cid:4)
Now, we briefly analyze the complexity of Algorithm 6. Firstly, assuming the number ofiterations for Algorithm 6 is T , the number of iterations for Algorithm 4 and Algorithm 5 toconverge are given by log ` µ u ´ µ l ε ˘ and log ` λ u ´ λ l ε ˘ , respectively. Note that, the main complexitylies in calculating F in each iteration of Algorithm 6. Taking advantage of the structures of X ` µ Z p and X ` λ I , the computation of p X ` µ Z p q : and p X ` λ I q : can be simplifiedas the product of matrices by the eigenvalue decomposition of X before entering algorithm,whose complexities is O p N SA q . Assuming the complexities of calculating F in each iterationof Algorithm 4 and Algorithm 5 are denoted by n and n , therefore, the total complexity ofAlgorithm 6 is given by O ` N SA ` T L ` n log ` µ u ´ µ l ε ˘ ` n log ` λ u ´ λ l ε ˘˘˘ . B. Optimize the IRS reflecting coefficients
In this subsection, we focus on optimizing the reflecting coefficients at the IRS while fixingthe other parameters. Based on the problem (20), the reflecting coefficients optimization problem in the simplified model is given by the following OP8, min Θ T r ` B ΘCΘ H ˘ ` Re (cid:32) T r ` D H Θ H ˘( s.t. T r ` B p ΘCΘ H ˘ ` Re (cid:32) T r ` D Hp Θ H ˘( ď ˜Γ p | Θ mm | “ , @ m P M . (52)In which, B p “ H Hrp H rp ě , D p “ H sr Q s H Hsp H rp and ˜Γ p “ Γ p ´ T r ` H sp Q s H Hsp ˘ . Beingsame as the last section, the OP8 can be further transformed to min θ θ H Υ θ ` Re (cid:32) θ H d ˚ ( s.t. θ H Υ p θ ` Re (cid:32) θ H d ˚ p ( ď ˜Γ p | θ m | “ , @ m P M . (53)Herein, Υ p “ B p d C T ě and d p “ ” r D p s , , r D p s , , ..., r D p s M,M ı T . Note that, the problem(53) could not be solved directly since the non-convexity of the uni-modular constraint, hence,SCA approach is adopted again. For which, we introduce the following proposition. Proposition 5:
Let f p θ q “ θ H Υ θ , Υ ě , X “ λ , max I M ˆ M and λ , max denotes themaximum eigenvalue of the Υ . Hence, for @ θ , given θ p n q , there exists ˜ f ` θ | θ p n q ˘ “ θ H X θ ´ Re (cid:32) θ H p X ´ Υ q θ p n q ( ` ` θ p n q ˘ H p X ´ Υ q θ p n q (54)which satisfies the following three conditions:1) ˜ f ` θ p n q | θ p n q ˘ “ f ` θ p n q ˘ ,2) ∇ θ ˚ ˜ f ` θ | θ p n q ˘ˇˇˇ θ “ θ p n q “ ∇ θ ˚ f p θ q| θ “ θ p n q ,3) ˜ f ` θ | θ p n q ˘ ě f p θ q . Proof:
The proof is similar to that for Proposition 1 and thus it is omitted herein. (cid:4)
Same as ˜ f ` θ | θ p n q ˘ in Proposition 5, let ˜ f p ` θ | θ p n q ˘ “ θ H X p θ ´ Re (cid:32) θ H p X p ´ Υ p q θ p n q ( ` ` θ p n q ˘ H p X p ´ Υ p q θ p n q . (55)Given θ p n q , we have max θ ˜ f ` θ | θ p n q ˘ ` Re (cid:32) θ H d ˚ ( s.t. ˜ f p ` θ | θ p n q ˘ ` Re (cid:32) θ H d ˚ p ( ď ˜Γ p | θ m | “ , @ m P M (56) Since θ H θ “ M , we have θ H X θ “ M λ , max and θ H X p θ “ M λ p, max , which is a constant. Byremoving all constant terms, the problem (56) can be rewritten as follows: max θ Re ! θ H q p n q ) s.t. C Re (cid:32) θ H q p n q p ( ě ˜Γ p n q p C | θ m | “ , @ m P M , (57)where ˜Γ p n q p “ ” M λ p, max ` ` θ p n q ˘ H p X p ´ Υ p q θ p n q ´ ˜Γ p ı M , q p n q “ p λ , max I M ˆ M ´ Υ q θ p n q ´ d ˚ and q p n q p “ p λ p, max I M ˆ M ´ Υ p q θ p n q ´ d ˚ p . The problem (57) could not be solved directly sincethe non-convexity of the uni-modular constraint. Therefore, a price mechanism is introduced tosolve the problem (57) that can obtain the globally optimal solution. In specific, we considerthe following problem by introducing α ě : max θ Re ! θ H q p n q ) ` αRe (cid:32) θ H q p n q p ( | θ m | “ , @ m P M (58)For given α , the globally optimal solution is given by θ p α q “ e j arg p q p n q ` αq p n q p q (59)Our objective is to find a α such that the following complementary slackness condition is satisfied: α ´ g p α q ´ ˜Γ p n q p ¯ “ . (60)Herein, g p α q “ Re ! θ p α q H q p n q p ) . To solve this equation, we consider two cases: 1) α “ ; 2) α ą ;. Case 1 : Consider α “ , θ p q “ e j arg p q p n q q needs to satisfy constraint (57 C1). Otherwise, α ą . Case 2 : Consider α ą , equation (60) holds only when g p α q “ ˜Γ p n q p . To solve this equation,we first provide the following proposition. Proposition 6: g p α q is a monotonically non-decreasing function of α . Proof:
The proof is similar to that for Proposition 2 and thus it is omitted herein. (cid:4)
Based on Proposition 6, the bisection search method can be adopted to find the solution ofthe equation (60) and the algorithm is provided in the following. Algorithm 7:
Bisection Search Method for (60)S1: Calculate g p q . If g p q ď ˜Γ p n q p , set ˆ α “ and terminate.Otherwise, go to S2.S2: Initialize ε ą , n “ and the bounds α p q l and α p q u of α ;S3: Set α p n q “ p α p n q l ` α p n q u q{ and calculate g ´ α p n q ¯ ;S4: If g ´ α p n q ¯ ě ˜Γ p n q p , set α p n ` q l “ α p n q and α p n ` q u “ α p n q u ;otherwise set α p n ` q l “ α p n q l and α p n ` q u “ α p n q ;Let n “ n ` ;S5: If ˇˇˇ α p n q u ´ α p n q l ˇˇˇ ą ε , go back to S3; else, set ˆ α “ α p n q ;S6: Output ˆ α ; Although the problem (57) is a non-convex problem, in the following theorem, we prove thatAlgorithm 7 can obtain the globally optimal solution.
Theorem 7:
Algorithm 7 can obtain the globally optimal solution of problem (57).
Proof:
The proof is similar to the proof of Theorem 2 in [35] and omitted for simplicity.Based on the above, we now provide the details of solving OP8 in Algorithm 8.
Algorithm 8:
SCA-based Algorithm To Solve The OP8S1: Initialize θ p q , ε ą , n “ , calculate the objective value of the OP8as z ´ θ p q ¯ ;S2: Given θ p q , obtain ˆ θ p n q by solving (57) using algorithm 7;S3: If ˇˇˇ z ´ ˆ θ p n q ¯ ´ z ´ θ p n q ¯ˇˇˇ ą ε , set z ´ θ p n ` q ¯ “ z ´ ˆ θ p n q ¯ , θ p n ` q “ ˆ θ p n q , n “ n ` , go back to S2; else set ˆ θ “ ˆ θ p n q ;S4: Output ˆ θ ; In the following theorem, we prove that the sequence of ! ˆ θ p n q , n “ , , , ... ) generated byAlgorithm 8 converges to the KKT optimal point of OP8. Theorem 8:
The sequences of the objective value produced by Algorithm 8 is guaranteed toconverge, and the final solution satisfies the KKT point of the OP8.
Proof:
The proof is similar to the proof of Theorem 3 in [35] and omitted for simplicity.Now, we further analyze the complexity of Algorithm 8. Firstly, we assume that the numberof iterations for Algorithm 8 is T . Note that, the main complexity of the Algorithm 8 lies incalculating the maximum eigenvalue of Υ and Υ and the bisection search in the Algorithm 7.Herein, the maximum eigenvalue of Υ and Υ , whose complexities are O p M q , only need tobe calculated once before entering algorithm. Moreover, the number of iterations for Algorithm7 to converge is characterized by log ` α u ´ α l ε ˘ . Assuming the complexities of each iteration of the bisection search in the Algorithm 7 is denoted by n , the total complexity of Algorithm 8is given by O ` M ` T n log ` α u ´ α l ε ˘˘ . C. Overall Algorithm
In this subsection, the overall algorithm based on alternating optimization for OP5 is providedand summarized as the Algorithm 9 as follows.
Algorithm 9:
AO-based Algorithm to Solve OP5S1: Initialize F p q , θ p q , ε ą , n “ , calculate the WSR of all SUs as R ´ F p q , Θ p q ¯ ;S2: Given F p n q and θ p n q , obtain ˆU p n q and ˆW p n q according to (12) and(13);S3: Given ˆU p n q , ˆW p n q and θ p n q , obtain ˆF p n q by solving the OP7 usingalgorithm 6;S4: Given ˆU p n q , ˆW p n q and ˆF p n q , obtain ˆ θ p n q by solving the OP8 usingalgorithm 8;S5: If ˇˇˇ R ´ ˆF p n q , ˆΘ p n q ¯ ´ R ´ F p n q , Θ p n q ¯ˇˇˇ ą ε , set F p n ` q “ ˆF p n ` q θ p n ` q “ ˆ θ p n q , R ´ F p n ` q , Θ p n ` q ¯ “ R ´ ˆF p n q , ˆΘ p n q ¯ , n “ n ` , go back to S2; else set ˆF “ ˆF p n q , ˆ θ “ ˆ θ p n q ;S6: Output ˆF and ˆ θ ; where ˆU p n q , ˆW p n q , ˆF p n q and ˆ θ p n q represent the stable solutions obtained by solving the subprob-lems in the n th iteration, and R p F , Θ q denotes the WSR of all SUs. Since the original problemis bounded and the progress of the alternative optimization is monotonically non-decreasing,thus the above algorithm is surely convergent. Furthermore, the complexity of Algorithm 9 ismainly concentrated in Algorithm 7 in step 3 and Algorithm 8 in step 4, whose complexitieshave been explained in the above, hence, it will not be analyzed here.V. S IMULATION A NALYSIS
In this section, the performances of the proposed algorithms are evaluated by numericalsimulations. Corresponding to the general scenario with multiple PUs in the section III andthe special scenario with only one PU in the section IV, simulation results are given by twosubsections. A. General Scenario With Multiple PUs
In this subsection, simulation results about the general scenario are provided. In the consideredIRS-MIMO-CR system, the SAP and the IRS are located at p , q and p , q in meter ( m ) in atwo-dimensional plane, respectively. In addition, there are three PUs and three SUs and they areuniformly and randomly scattered in a circle with radius 2 m and centered at p , q and p , q ,respectively. The other system parameters used in the simulations are following [35] and [36],that is, we set the antenna numbers of the SAP, PUs and SUs as N SA “ and N P U “ N SU “ ,respectively. The number of data streams for each SU is set as d “ . The noise power at all PUsand SUs is set as σ R “ σ E “ ´ dBm, @ k P K , l P L and the maximum total transmitted poweris set as P max “ W . The maximum interference at all PUs is set as IT k “ ˆ ´ W, @ k P K .Without loss of generality, all the channels are modeled as [8] H “ a β { κ ` ` ? κ H LoS ` H NLoS ˘ , where κ is the Rician factor, while H LOS and H NLOS represent the deterministic line-of-sight(LoS) and Rayleigh fading/non-LoS (NLoS) components, respectively. β represents the path loss,and is given by β “ β ´ α log p d { d q . Herein, β denotes the path loss at the referencedistance d “ m , α and d represent the path loss exponent and the distance between thecorresponding nodes. Assuming that the location of IRS can be carefully selected, the channelsfrom IRS to PUs and SUs have LoS component and experience Rayleigh fading, simultaneously.However, the channels from SAP to PUs, SUs and IRS, only experience Rayleigh fading. Hence,the Rician factors are set as κ rk “ κ rl “ and κ sk “ κ sl “ κ sr “ . In addition, path lossexponents of all channels are set as α sk “ α sl “ α sr “ α rk “ α rl “ .Furthermore, in order to better understand the positive effects of the IRS in improving theperformance for the IRS-MIMO-CR system and the performance gain of the proposed algo-rithms, some benchmark schemes are introduced in simulations for performance comparisonand analysis. Thus, following three algorithms are evaluated in the simulations, i.e., ‘no-IRS’,‘fixed-IRS’ and ‘AO’. no-IRS : That is, no IRS is used in the system and the WSR is obtained by directly optimizingthe OP1 under the conditions G sk p Θ q “ H sk and G sl p Θ q “ H sl . fixed-IRS : That is, all reflecting coefficients of the IRS have same phases, which are set tozeros, namely, arg p θ m q “ , m “ , .., M . AO : It is our proposed AO-based algorithm, i.e., the Algorithm 3. The maximum transmit power W S R ( bp s / H z ) no-IRSfixed-IRSAO with M=20AO with M=40AO with M=80 Fig. 2 WSR VS the maximum transmit powerAt first, the WSR of different algorithms is evaluated by varying the available transmissionpower at the SAP, i.e., the P max P r , s W , and the result is shown in Fig. 2. The ‘AO with M “ ’, ‘AO with M “ ’, and ‘AO with M “ ’ shown in the figure are used toidentify Algorithm 3 when the number of IRS reflection elements is 20, 40, and 80, respectively.Apparently, with the increase of the available transmission power at the SAP, the WSR for allthese algorithms is increased. Moreover, algorithms ‘no-IRS’ and ‘fixed-IRS’ obtain the worstperformance and they are exceedingly close. Introducing the IRS optimization, the performanceof ‘AO with M “ ’, ‘AO with M “ ’ and ‘AO with M “ ’ is improved obviouslywith the increase of the number of IRS reflection elements and better than that of ‘no-IRS’ and‘fixed-IRS’. It is worth mentioning that the performance gaps between the various mechanismsbecome larger with the increase of the transmission power at the SAP, which means that theintroduction of the IRS reflection coefficient optimization brings more significant performancegains at the higher transmission power.Obviously, the performance of Algorithm 3 improves significantly as the number of IRSreflection units increases in Fig. 2. Then, to more intuitively verify the impact of the number ofIRS reflection elements, Fig. 3 shows the trend of the WSR with the number of IRS reflectionelements under three mechanisms. The maximum transmission power of the SAP is set as P max “ W , and the number of IRS reflection elements varies from 20 to 100. The ‘AO with P max “ W ’ shown in the figure is used to identify Algorithm 3 with P max “ W . Note
20 30 40 50 60 70 80 90 100
The number of IRS. W S R ( bp s / H z ) no-IRSfixed-IRSAO with Pmax=5W Fig. 3 WSR VS the number of reflecting elements
The number of iterations W S R ( bp s / H z ) AO with M=20AO with M=40AO with M=80
Fig. 4 WSR VS the number of iterationsthat, the curve corresponding to ‘AO with P max “ W ’ shows a significant upward trend, andis significantly improved compared to the other two mechanisms, while the performance of‘no-IRS’ and ‘fixed-IRS’ mechanisms is not affected by the number of the IRS elements inthe system. The fact means that the wireless communication environment can be improved byincreasing the number of IRS reflection elements appropriately.In order to evaluate the convergence of the proposed algorithm, the variation of WSR withthe number of iterations is given in Fig. 4. Considering algorithms ‘AO with M “ ’, ‘AO with M “ ’ and ‘AO with M “ ’, the performance corresponding to 30 iterations is calculated in the figure. Note that, the proposed algorithm can converge quickly, i.e., no more than 10 outeriterations can surely promise the convergence of the AO algorithm. Whereafter, the performanceof the algorithm is still improved as the iteration progresses but very small. In addition, with thenumber of IRS reflection elements increases, the performance of the proposed algorithm is alsoimproved, which further proves that the wireless environment can be improved by appropriatelyincreasing the number of IRS reflection elements. B. Special Scenario With Only One PU
In this subsection, the simulation results about the special scenario are given. Note that, thelocation of the unique PU is set to p , q , whereas the other system parameters are sameas those in the general scenario. Furthermore, to prove the performance gain of the proposedalgorithms, two benchmark algorithms used in last subsection are retained but the algorithm ‘AO’is used to indicate the proposed Algorithm 9. Meanwhile, with same evaluated parameters, thesimulation results shown below are the same as those obtained in the last section, which meansthat the Algorithm 9 has a much lower computational complexity than Algorithm 3 without theperformance loss. The maximum transmit power W S R ( bp s / H z ) no-IRSfixed-IRSAO with M=20AO with M=40AO with M=80 Fig. 5 WSR VS the maximum transmit powerSimilarly, the WSR of different algorithms is evaluated by varying the available transmissionpower at the SAP, i.e., the P max P r , s W , and the result is shown in Fig. 5. The ‘AO with M “ ’, ‘AO with M “ ’, and ‘AO with M “ ’ shown in the figure are used to identify algorithm 9 when the number of IRS reflection elements is 20, 40, and 80, respectively.Obviously, the WSR for all these algorithms is increased with the increase of the availabletransmission power at the SAP. Moreover, algorithms ‘no-IRS’ and ‘fixed-IRS’ obtain the worstperformance and they are exceedingly close. Introducing the IRS optimization, the performanceof ‘AO with M “ ’, ‘AO with M “ ’ and ‘AO with M “ ’ is improved distinctlywith the increase of the number of IRS reflection elements and better than that of ‘no-IRS’ and‘fixed-IRS’. It is worth mentioning that the performance gaps between the various mechanismsbecome larger with the increase of the transmission power at the SAP, which means that theintroduction of the IRS reflection coefficient optimization brings more significant performancegains at the higher transmission power.
20 30 40 50 60 70 80 90 100
The number of IRS. W S R ( bp s / H z ) no-IRSfixed-IRSAO with Pmax=5W Fig. 6 WSR VS the number of reflecting elementsThen, Fig. 6 shows the trend of the WSR with the number of IRS reflection elements underthree mechanisms to prove the impact of the number of IRS reflection elements visually. Themaximum transmission power of the SAP is set as P max “ W , and the number of IRS reflectionelements varies from 20 to 100. The ‘AO with P max “ W ’ shown in the figure is used to identifyalgorithm 9 with P max “ W . Note that, compared to other two mechanisms, the performanceof ‘AO with P max “ W ’ is significantly improved, while the performance of ‘no-IRS’ and‘fixed-IRS’ mechanisms is not affected by the number of the IRS elements in the system. Thefact means that the wireless communication environment can be improved by increasing thenumber of IRS reflection elements appropriately. The number of iterations W S R ( bp s / H z ) AO with M=20AO with M=40AO with M=80
Fig. 7 WSR VS the number of iterationsFinally, the convergence behavior of the proposed algorithm 9 is evaluated and the result isshown in Fig. 7. Considering algorithms ‘AO with M “ ’, ‘AO with M “ ’ and ‘AO with M “ ’, the performance corresponding to 30 iterations is calculated in the figure. Note that,the proposed algorithm can converge quickly, i.e., no more than 6. Whereafter, the performanceof the algorithm is still improved as the iteration progresses but very small. In addition, with thenumber of IRS reflection elements increases, the performance of the proposed algorithm is alsoimproved, which further proves that the wireless environment can be improved by appropriatelyincreasing the number of IRS reflection elements. It is worth mentioning that algorithm 9 has afaster convergence speed than algorithm 3.VI. C ONCLUSION
In this paper, the joint transmit precoding and reflect beamfroming for the IRS-MIMO-CRsystem is discussed and solved. Our design objective is to maximize the achievable WSRof SUs by jointly optimizing the transmit precoding matrices at the SAP and the reflectingcoefficients at the IRS, subject to a total transmit power constraint at the SAP and interferenceconstraints at PUs. Since the formulated problem is non-convex with coupled variables, thusthe WMMSE is adopted to transfer it to a tractable one and then an AO-based algorithm isproposed. Furthermore, a special scenario with only one PU is considered and an AO-basedalgorithm with lower complexity is presented. Numerical simulation results confirm that theproposed algorithms can obtain significantly performance gain over the benchmark schemes. In addition, for our considered scenario, the beamforming optimization at IRS can bring much moreperformance improvement as higher transmission power is allowed at the SAP. It is important tonote that, the work in this paper based on the hypothesis of perfect CSI, however, the channelestimation is bound to have a certain degree of error in practice. Therefore, the problem withimperfect CSI is more interesting and this is one of our feature work.A PPENDIX AP ROOF OF P ROPOSITION T “ Z p ´ X p and f p F q “ L ř l “ T r ` F Hl TF l ˘ . Given F p n q , the first orderTaylor expansion of the function f p F q is ˜ f ` F | F p n q ˘ ∆ “ L ÿ l “ Re ! T r ´ F p n q Hl TF l ¯) ´ L ÿ l “ T r ´ F p n q Hl TF p n q l ¯ . Hence, we have ˜ f ` F p n q | F p n q ˘ “ f ` F p n q ˘ and ∇ F ˚ ˜ f ` F | F p n q ˘ˇˇˇ F “ F p n q “ ∇ F ˚ f p F q| F “ F p n q . Sub-stitute T “ Z p ´ X p into the previous equation, the conditions 1) and 2) are proved.Moreover, since Z p “ λ p I and λ p is the maximum eigenvalue of the X p , T ě . which meansthat f p F q is the convex function with respect to F . Hence, f p F q ě ˜ f ` F | F p n q ˘ . Similarly, substitute T into the above equation and transfer the term, we obtain the condition3). That is, we have the proposition. (cid:4) A PPENDIX BP ROOF OF P ROPOSITION µ and µ where µ ą µ . Let ˆF p µ q and ˆF p µ q be theoptimal solutions of problem (34) with µ and µ , respectively. Since ˆF p µ q is the optimalsolution of (34) with µ “ µ , we have L ´ ˆF p µ q , µ ¯ ď L ´ ˆF p µ q , µ ¯ . Meanwhile, we also have L ´ ˆF p µ q , µ ¯ ď L ´ ˆF p µ q , µ ¯ . By adding these two inequalities and simplifying them, we have p µ ´ µ q J p µ q ď p µ ´ µ q J p µ q .Since µ ą µ , we have J p µ q ď J p µ q . Therefore, we have this proposition. (cid:4) A PPENDIX CP ROOF OF P ROPOSITION ˆF . Moreover, let f p F q , f p F q and f p F q denote the objective function, left parts of the power constraint and the interferencepower constraint of the OP7, respectively.Given the initial point ˆF , we construct the problem (31) and let ˜ f ´ F | ˆF ¯ denote the leftpart of the approximate interference power constraint. The optimal solution of the problem is ˆF since it is the convergent solution obtained by algorithm 6. Now, there exist λ ˚ ě and µ ˚ ě which satisfy the KKT conditions of the problem (31) as follows $’’’’’’’’’&’’’’’’’’’% f ´ ˆF ¯ ď P max ˜ f ´ ˆF | ˆF ¯ ď ˜Γ p λ ˚ ´ f ´ ˆF ¯ ´ P max ¯ “ µ ˚ ´ ˜ f ´ ˆF | ˆF ¯ ´ ˜Γ p ¯ “ ∇ f ´ ˆF ¯ ` λ ˚ ∇ f ´ ˆF ¯ ` µ ˚ ∇ ˜ f ´ ˆF | ˆF ¯ “ . In addition, given ˆF , there are f ´ ˆF ¯ “ ˜ f ´ ˆF | ˆF ¯ ` L ř l “ T r ´ F p n q Hl p Z p ´ X p q F p n q l ¯ and ∇ f ´ ˆF ¯ “ ∇ ˜ f ´ ˆF | ˆF ¯ , so the KKT conditions of the OP7 is satisfied with λ ˚ and µ ˚ asfollows, $’’’’’’’’’&’’’’’’’’’% f ´ ˆF ¯ ď P max f ´ ˆF ¯ ď Γ p λ ˚ ´ f ´ ˆF ¯ ´ P max ¯ “ µ ˚ ´ f ´ ˆF ¯ ´ Γ p ¯ “ ∇ f ´ ˆF ¯ ` λ ˚ ∇ f ´ ˆF ¯ ` µ ˚ ∇ f ´ ˆF ¯ “ . Meanwhile, the OP7 is a convex optimization problem. Hence, ˆF obtained by algorithm 6 isthe optimal solution of the OP7 [31]. Therefore, we have this proposition. (cid:4) R EFERENCES [1] J. Joung, C. K. Ho, and S. Sun, “Spectral efficiency and energy efficiency of OFDM systems: Impact of poweramplifiers and countermeasures,”
IEEE J. Sel. Areas Commun. , vol. 32, no. 2, pp. 208–220, 2014. [Online]. Available:https://doi.org/10.1109/JSAC.2014.141203 [2] J. M. III and G. Q. M. Jr., “Cognitive radio: making software radios more personal,” IEEE Wirel. Commun. , vol. 6, no. 4,pp. 13–18, 1999. [Online]. Available: https://doi.org/10.1109/98.788210[3] C. Liaskos, S. Nie, A. Tsioliaridou, A. Pitsillides, S. Ioannidis, and I. F. Akyildiz, “A new wireless communicationparadigm through software-controlled metasurfaces,”
IEEE Commun. Mag. , vol. 56, no. 9, pp. 162–169, 2018. [Online].Available: https://doi.org/10.1109/MCOM.2018.1700659[4] H. Yang, X. Cao, F. Yang, J. Gao, S. Xu, M. Li, X. Chen, Y. Zhao, Y. Zheng, and S. Li, “A programmable metasurfacewith dynamic polarization, scattering and focusing control,”
Rep , vol. 6, p. 35692, 2016.[5] M. D. Renzo, M. Debbah, D. T. P. Huy, A. Zappone, M. Alouini, C. Yuen, V. Sciancalepore, G. C. Alexandropoulos,J. Hoydis, H. Gacanin, J. de Rosny, A. Bounceur, G. Lerosey, and M. Fink, “Smart radio environments empowered byreconfigurable AI meta-surfaces: an idea whose time has come,”
EURASIP J. Wirel. Commun. Netw. , vol. 2019, p. 129,2019. [Online]. Available: https://doi.org/10.1186/s13638-019-1438-9[6] Q. Wu and R. Zhang, “Towards smart and reconfigurable environment: Intelligent reflecting surface aided wireless network,”
IEEE Communications Magazine , vol. 58, no. 1, pp. 106–112, 2020.[7] ——, “Intelligent reflecting surface enhanced wireless network: Joint active and passive beamforming design,” in
IEEEGlobal Communications Conference, GLOBECOM 2018, Abu Dhabi, United Arab Emirates, December 9-13, 2018 , 2018,pp. 1–6.[8] ——, “Intelligent reflecting surface enhanced wireless network via joint active and passive beamforming,”
IEEE Trans.Wireless Communications , vol. 18, no. 11, pp. 5394–5409, 2019.[9] C. Huang, A. Zappone, M. Debbah, and C. Yuen, “Achievable rate maximization by passive intelligent mirrors,” in , 2018, pp. 3714–3718.[10] Q. Wu and R. Zhang, “Beamforming optimization for wireless network aided by intelligent reflecting surface with discretephase shifts,”
IEEE Trans. Communications , vol. 68, no. 3, pp. 1838–1851, 2020.[11] C. Huang, A. Zappone, G. C. Alexandropoulos, M. Debbah, and C. Yuen, “Large intelligent surfaces for energy efficiencyin wireless communication,”
CoRR , vol. abs/1810.06934, 2018. [Online]. Available: http://arxiv.org/abs/1810.06934[12] C. Huang, G. C. Alexandropoulos, A. Zappone, M. Debbah, and C. Yuen, “Energy efficient multi-user MISO communicationusing low resolution large intelligent surfaces,” in
IEEE Globecom Workshops, GC Wkshps 2018, Abu Dhabi, United ArabEmirates, December 9-13, 2018 , 2018, pp. 1–6.[13] Y. Han, W. Tang, S. Jin, C. Wen, and X. Ma, “Large intelligent surface-assisted wireless communication exploiting statisticalCSI,”
IEEE Trans. Vehicular Technology , vol. 68, no. 8, pp. 8238–8242, 2019.[14] A. Taha, M. Alrabeiah, and A. Alkhateeb, “Enabling large intelligent surfaces with compressive sensing and deeplearning,”
CoRR , vol. abs/1904.10136, 2019. [Online]. Available: http://arxiv.org/abs/1904.10136[15] C. Huang, G. C. Alexandropoulos, C. Yuen, and M. Debbah, “Indoor signal focusing with deep learning designedreconfigurable intelligent surfaces,” in , 2019, pp. 1–5.[16] X. Tan, Z. Sun, J. M. Jornet, and D. Pados, “Increasing indoor spectrum sharing capacity using smart reflect-array,” in , 2016,pp. 1–6.[17] X. Tan, Z. Sun, D. Koutsonikolas, and J. M. Jornet, “Enabling indoor mobile millimeter-wave networks based on smartreflect-arrays,” in , 2018, pp. 270–278. [18] G. Yang, X. Xu, and Y. Liang, “Intelligent reflecting surface assisted non-orthogonal multiple access,” CoRR , vol.abs/1907.03133, 2019. [Online]. Available: http://arxiv.org/abs/1907.03133[19] S. Abeywickrama, R. Zhang, and C. Yuen, “Intelligent reflecting surface: Practical phase shift model and beamformingoptimization,”
CoRR , vol. abs/1907.06002, 2019. [Online]. Available: http://arxiv.org/abs/1907.06002[20] X. Guan, Q. Wu, and R. Zhang, “Joint power control and passive beamforming in irs-assisted spectrum sharing,”
IEEECommun. Lett. , vol. 24, no. 7, pp. 1553–1557, 2020. [Online]. Available: https://doi.org/10.1109/LCOMM.2020.2979709[21] L. Zhang, C. Pan, Y. Wang, H. Ren, K. Wang, and A. Nallanathan, “Robust beamforming design for intelligent reflectingsurface aided cognitive radio systems with imperfect cascaded csi,” 2020.[22] J. Yuan, Y. Liang, J. Joung, G. Feng, and E. G. Larsson, “Intelligent reflecting surface (irs)-enhanced cognitive radiosystem,” in .IEEE, 2020, pp. 1–6. [Online]. Available: https://doi.org/10.1109/ICC40277.2020.9148890[23] ——, “Intelligent reflecting surface-assisted cognitive radio system,”
IEEE Trans. Commun. , vol. 69, no. 1, pp. 675–687,2021. [Online]. Available: https://doi.org/10.1109/TCOMM.2020.3033006[24] D. Xu, X. Yu, and R. Schober, “Resource allocation for intelligent reflecting surface-assisted cognitive radio networks,” in . IEEE, 2020, pp. 1–5. [Online]. Available: https://doi.org/10.1109/SPAWC48557.2020.9154252[25] J. He, K. Yu, Y. Zhou, and Y. Shi, “Reconfigurable intelligent surface enhanced cognitive radio networks,” arXiv e-prints ,2020.[26] D. Xu, X. Yu, Y. Sun, D. W. K. Ng, and R. Schober, “Resource allocation for irs-assisted full-duplexcognitive radio systems,”
IEEE Trans. Commun. , vol. 68, no. 12, pp. 7376–7394, 2020. [Online]. Available:https://doi.org/10.1109/TCOMM.2020.3020838[27] H. Xiao, L. Dong, and W. Wang, “Intelligent reflecting surface-assisted secure multi-input single-output cognitive radiotransmission,”
Sensors , vol. 20, no. 12, p. 3480, 2020. [Online]. Available: https://doi.org/10.3390/s20123480[28] L. Zhang, Y. Wang, W. Tao, Z. Jia, T. Song, and C. Pan, “Intelligent reflecting surface aided MIMO cognitiveradio systems,”
IEEE Trans. Veh. Technol. , vol. 69, no. 10, pp. 11 445–11 457, 2020. [Online]. Available:https://doi.org/10.1109/TVT.2020.3011308[29] S. Zhang and R. Zhang, “Capacity characterization for intelligent reflecting surface aided MIMO communication,”
IEEE J.Sel. Areas Commun. , vol. 38, no. 8, pp. 1823–1838, 2020. [Online]. Available: https://doi.org/10.1109/JSAC.2020.3000814[30] Q. Shi, M. Razaviyayn, Z. Luo, and C. He, “An iteratively weighted MMSE approach to distributed sum-utilitymaximization for a MIMO interfering broadcast channel,”
IEEE Trans. Signal Process. , vol. 59, no. 9, pp. 4331–4340,2011. [Online]. Available: https://doi.org/10.1109/TSP.2011.2147784[31] S. P. Boyd and L. Vandenberghe,
Convex Optimization . Cambridge University Press, 2014. [Online]. Available:https://web.stanford.edu/%7Eboyd/cvxbook/[32] X. Zhang, “Matrix analysis and applications,”
Int.j.inf.syst , vol. 309, no. 1, p. i, 2017.[33] G. Scutari, F. Facchinei, and L. Lampariello, “Parallel and distributed methods for constrained nonconvex optimization- part I: theory,”
IEEE Trans. Signal Process. , vol. 65, no. 8, pp. 1929–1944, 2017. [Online]. Available:https://doi.org/10.1109/TSP.2016.2637317[34] G. Scutari, F. Facchinei, L. Lampariello, S. Sardellitti, and P. Song, “Parallel and distributed methods for constrainednonconvex optimization-part II: applications in communications and machine learning,”
IEEE Trans. Signal Process. ,vol. 65, no. 8, pp. 1945–1960, 2017. [Online]. Available: https://doi.org/10.1109/TSP.2016.2637314[35] C. Pan, H. Ren, K. Wang, M. Elkashlan, A. Nallanathan, J. Wang, and L. Hanzo, “Intelligent reflecting surface aided MIMO broadcasting for simultaneous wireless information and power transfer,”
IEEE J. Sel. Areas Commun. , vol. 38,no. 8, pp. 1719–1734, 2020. [Online]. Available: https://doi.org/10.1109/JSAC.2020.3000802[36] W. Jiang, Y. Zhang, J. Wu, W. Feng, and Y. Jin, “Intelligent reflecting surface assisted secure wireless communicationswith multiple- transmit and multiple-receive antennas,”