Double-IRS Aided MIMO Communication under LoS Channels: Capacity Maximization and Scaling
11 Double-IRS Aided MIMO Communicationunder LoS Channels: Capacity Maximizationand Scaling
Yitao Han,
Student Member, IEEE,
Shuowen Zhang,
Member, IEEE,
Lingjie Duan,
Senior Member, IEEE, and Rui Zhang,
Fellow, IEEE
Abstract
Intelligent reflecting surface (IRS) is a promising technology to extend the wireless signal coverageand support the high performance communication. By intelligently adjusting the reflection coefficients ofa large number of passive reflecting elements, the IRS can modify the wireless propagation environmentin favour of signal transmission. Different from most of the prior works which did not consider anycooperation between IRSs, in this work we propose and study a cooperative double-IRS aided multiple-input multiple-output (MIMO) communication system under the line-of-sight (LoS) propagation chan-nels. We investigate the capacity maximization problem by jointly optimizing the transmit covariancematrix and the passive beamforming matrices of the two cooperative IRSs. Although the above problemis non-convex and difficult to solve, we transform and simplify the original problem by exploiting atractable characterization of the LoS channels. Then we develop a novel low-complexity algorithm whosecomplexity is independent of the number of IRS elements. Moreover, we analyze the capacity scalingorders of the double-IRS aided MIMO system with respect to an asymptotically large number of IRSelements or transmit power, which significantly outperform those of the conventional single-IRS aidedMIMO system, thanks to the cooperative passive beamforming gain brought by the double-reflection linkand the spatial multiplexing gain harvested from the two single-reflection links. Extensive numericalresults are provided to show that by exploiting the LoS channel properties, our proposed algorithmcan achieve a desirable performance with low computational time. Also, our capacity scaling analysisis validated, and the double-IRS system is shown to achieve a much higher rate than its single-IRScounterpart as long as the number of IRS elements or the transmit power is not small.
Index Terms
Intelligent reflecting surface (IRS), multiple-input multiple-output (MIMO), double IRSs, alternatingoptimization, capacity scaling order.
Y. Han and L. Duan are with the Engineering Systems and Design Pillar, Singapore University of Technology and Design (e-mail: yitao [email protected], lingjie [email protected]). Y. Han is also with the Department of Electrical and ComputerEngineering, National University of Singapore.S. Zhang is with the Department of Electronic and Information Engineering, The Hong Kong Polytechnic University (e-mail:[email protected]). S. Zhang is the corresponding author.R. Zhang is with the Department of Electrical and Computer Engineering, National University of Singapore (e-mail:[email protected]). a r X i v : . [ c s . I T ] F e b I. I
NTRODUCTION
The ever-growing demand for higher data rate, lower latency and enhanced reliability inwireless communications has driven both industry and academia to advance the communicationtechnologies. Among them, massive multiple-input multiple-output (MIMO), millimeter wave(mmWave), and ultra-dense network (UDN) are the prominent candidates [1]. However, theyincur high costs in energy consumption and hardware investment, by requiring a large numberof antennas and the expensive radio frequency (RF) chains operating at high frequency bands.Moreover, the ever-increasing number of base stations (BSs) and access points (APs) can alsogenerate high interference to each other and deteriorate the overall performance.With the recent research advances in micro electromechanical systems (MEMS) and meta-material [2], it is now feasible to realize an amplitude change and phase shift to the incidentsignal in real time via a programmable surface, which enables an innovative wireless device andnetwork component—intelligent reflecting surface (IRS) [3]. The IRS is a planar reconfigurablemetasurface, which consists of a large number of passive reflecting elements and a smartcontroller. By adaptively configuring the reflection amplitude and phase of each element, the IRScan reshape the electromagnetic environment to fit specific needs, e.g., extending signal coverage,mitigating interference, and supporting massive device-to-device (D2D) communications [4].Different from the conventional AP or relay, the IRS only uses passive reflection and does notrequire high energy consumption or expensive hardware, which also gives it the potential to bedensely deployed to significantly improve the network performance [4].To fully gain the IRS’s benefits, it is crucial to properly design its passive beamforming, whichis a key focus of the IRS research ([5]–[19]). For example, [6] considered the joint optimization ofBS’s active beamforming and IRS’s passive beamforming. [7] studied the capacity maximizationof a single-user MIMO system aided by one IRS, and proposed an alternating optimizationalgorithm for finding a local optimum. [8] tackled the energy efficiency maximization problemusing gradient descent approach and sequential fractional programming. Besides the passivebeamforming design, there were also works focusing on IRS channel estimation (e.g., [20],[21]) and deployment strategy (e.g., [22], [23]).However, the above IRS works only studied the scenario with one IRS or multiple non-cooperative IRSs (each independently serving its associated users). There is a lack of studyfor the inter-IRS reflection channel and the cooperative passive beamforming between IRSs.
In [5], we made the first attempt to study a double-IRS aided wireless communication systemwith single-antenna BS/user. By assuming a rank-one inter-IRS reflection channel, an M -fold cooperative passive beamforming gain can be achieved by the double-reflection link, with M denoting the total number of IRS elements. This leads to a significantly increased achievablerate as compared to the single-IRS aided system with only an M -fold passive beamforminggain [4], especially when M is large.In this paper, we substantially extend this line of research by proposing a cooperative double-IRS aided MIMO system (as illustrated in Fig. 1 later), where one IRS is deployed near amulti-antenna BS and the other IRS is deployed near a multi-antenna user, for assisting theBS-user downlink communication. Note that the IRS has been shown extremely helpful forimproving the MIMO channel capacity by boosting its rank via introducing reflection link. Withone double-reflection link through two IRSs (BS-IRS -IRS -user) and two single-reflection linkseach via one IRS (BS-IRS -user and BS-IRS -user), the double-IRS aided MIMO system isfurther anticipated to achieve a more obvious rank (spatial multiplexing) gain comparing withthe single-IRS aided MIMO system. This motivates our investigation in this paper to not onlyseek the cooperative passive beamforming gain but also the potential spatial multiplexing gainachievable by the double-IRS aided MIMO system. To this end, we consider a line-of-sight(LoS) propagation environment where the involved channels in Fig. 1 are generally of rank-one.Note that this condition can be practically achieved by properly deploying the two IRSs.The main contributions of this paper are summarized as follows. • We are the first to study the capacity maximization problem of the double-IRS aidedMIMO system, by jointly optimizing the transmit covariance matrix and the passive beam-forming matrices of the two IRSs. This new optimization problem is non-convex andmuch more difficult than the single-IRS aided MIMO system (e.g., [7]). Nevertheless, weprovide a tractable characterization of the LoS channels, and successfully derive a passivebeamforming structure that can simultaneously maximize the power gains of the double-reflection link and two single-reflection links. Based on this, we transform and simplify theoriginal problem, and propose a novel low-complexity algorithm by iteratively optimizingthe transmit covariance matrix and the common phase shifts of the two IRSs in closed-form.Our algorithm’s complexity is independent with the total number of IRS elements thanksto the exploitation of the LoS channel characteristics. • Next, we show that the rank of the double-IRS aided MIMO channel can be improved to two (as compared to the single-IRS aided MIMO channel which is always of rank-one),and analytically provide the explicit conditions that correspond to rank-two or rank-oneMIMO channel. Moreover, by considering the case with two antennas at the BS or user, wederive the closed-form channel capacities for rank-two MIMO channel (with asymptoticallylarge transmit power) and for rank-one MIMO channel. Based on the above, we analyzethe capacity scaling orders with respect to an asymptotically large number of IRS elementsor transmit power, which are shown to significantly outperform those of the conventionalsingle-IRS aided MIMO system, thanks to the cooperative passive beamforming gain broughtby the double-reflection link and the spatial multiplexing gain harvested from the two single-reflection links, respectively. • Finally, we present extensive numerical results. We show that by exploiting the LoS channelproperties, our proposed algorithm can achieve a desirable performance with low compu-tational time. We further validate our capacity scaling analysis, and reveal that the double-IRS aided MIMO system significantly outperforms its single-IRS counterpart when the totalnumber of IRS elements or the transmit power is not small. In addition, we draw usefulinsights into the optimal IRS deployment for maximizing the channel capacity.The rest of this paper is organized as follows. In Section II, we introduce the system modeland formulate the capacity maximization problem for the double-IRS aided MIMO system. InSection III, we provide a tractable characterization of the involved LoS channels. In SectionVI, we propose a low-complexity algorithm to solve the considered problem. In Section V, wederive the closed-form channel capacity and conduct the capacity scaling analysis. In SectionVI, we present numerical results for verifying the performance of our proposed algorithm andour considered double-IRS system. In Section VII, we conclude this paper.
Notations: | z | , z ∗ , arg { z } denote the absolute value, conjugate, and angle of a complex number z . max( x, y ) denotes the maximum between two real numbers x and y . C denotes the space ofcomplex numbers, R denotes the space of real numbers, while Z denotes the space of integers.Vectors and matrices are denoted by boldface lower-case letters and boldface upper-case letters,respectively. ( · ) T denotes the transpose operation, while ( · ) H denotes the conjugate transposeoperation. For a vector x , (cid:107) x (cid:107) and ( x ) m denote its l -norm and k th entry, respectively. For anarbitrary-size matrix X , rank( X ) and ( X ) m,n denote its rank and ( m, n ) th entry, respectively. diag { x , · · · , x M } denotes an M × M diagonal matrix with x , · · · , x M being the diagonalelements. I M denotes an M × M identity matrix, and denotes an all-zero matrix with appropriate 𝐮𝐮 𝐮𝐮 𝑡𝑡 𝐮𝐮 𝐮𝐮 𝑟𝑟 𝐓𝐓 𝐒𝐒 𝐓𝐓 𝐯𝐯 ( 𝑎𝑎 ) 𝐯𝐯 ( 𝑏𝑏 ) IRS IRS 𝐯𝐯 ( 𝑎𝑎 ) 𝐯𝐯 ( 𝑏𝑏 ) User … 𝐯𝐯 𝑡𝑡 𝐯𝐯 𝑟𝑟 𝐑𝐑 𝐑𝐑 𝜔𝜔 𝑇𝑇 , 𝑣𝑣 𝑡𝑡 𝜔𝜔 𝑇𝑇 , 𝑣𝑣 𝑡𝑡 … BS 𝜔𝜔 𝑅𝑅 , 𝑣𝑣 𝑟𝑟 𝜔𝜔 𝑅𝑅 , 𝑣𝑣 𝑟𝑟 𝜔𝜔 𝑆𝑆 , 𝑣𝑣 ( 𝑎𝑎 ) 𝜔𝜔 𝑅𝑅 , 𝑣𝑣 ( 𝑎𝑎 ) 𝜔𝜔 𝑇𝑇 , 𝑣𝑣 ( 𝑏𝑏 ) 𝜔𝜔 𝑆𝑆 , 𝑣𝑣 ( 𝑏𝑏 ) 𝜔𝜔 𝑇𝑇 , 𝑣𝑣 𝑡𝑡 𝜔𝜔 𝑇𝑇 , 𝑣𝑣 𝑡𝑡 Fig. 1. A MIMO wireless communication system aided by two cooperative IRSs. dimension. For a square matrix S , det( S ) , tr( S ) , and S − denote its determinant, trace, andinverse, respectively, and S (cid:23) means that S is positive semi-definite. j = √− denotes theimaginary unit. E {·} denotes the statistical expectation. O ( · ) denotes the standard big-O notation.II. S YSTEM M ODEL AND P ROBLEM F ORMULATION
We study a MIMO downlink communication system with N t ≥ antennas at the BS and N r ≥ antennas at the user, with u t ∈ R × and u r ∈ R × denoting the locations of theBS and the user under a three-dimensional (3D) Cartesian coordinate system, respectively. Weconsider the challenging scenario where the direct BS-user link is blocked by obstacles (e.g.,high buildings for urban scenario and hills for suburban scenario), which is thus weak andnegligible. We propose to deploy two cooperative IRSs to assist the communication in sucha heavy-blockage scenario, as shown in Fig. 1. For minimizing the path loss of the reflectionlinks, we place IRS near the BS with its location denoted by u ∈ R × , and IRS nearthe user with its location denoted by u ∈ R × as in [5]. We are given a total number of M ≥ passive reflecting elements as our budget, where IRS i ∈ { , } is equipped with M i elements with M + M = M . For ease of exposition, we further define N t = { , · · · , N t } and N r = { , · · · , N r } as the sets containing all the antennas at the BS and the user, respectively,while M i = { , · · · , M i } as the set containing all the reflecting elements at IRS i ∈ { , } .We denote T i ∈ C M i × N t as the channel matrix from the BS to IRS i , R i ∈ C N r × M i as thechannel matrix from IRS i to the user, for i ∈ { , } , and S ∈ C M × M as the channel matrix All the results can be easily extended to the multi-user setup, where different users take turns to be served in different timeslots by applying our solution independently. from IRS to IRS , respectively. We denote Φ i = diag { φ i, , · · · , φ i,M i } ∈ C M i × M i as thepassive beamforming matrix of IRS i ∈ { , } , where φ i,m i denotes the reflection coefficientof element m i ∈ M i at IRS i . We allow each IRS to be equipped with a smart controller,which can intelligently adjust the reflection coefficients of its elements, i.e., φ i,m i ’s, to alter theeffective BS-user MIMO channel for improving the communication performance. For achievingmaximum reflection, we further set | φ i,m i | = 1 for i ∈ { , } and m i ∈ M i . Under the abovesetup, the effective BS-user MIMO channel aided by two cooperative IRSs is modelled as H = R Φ T + R Φ T + R Φ SΦ T , (1)which is the superposition of two single-reflection links each via an individual IRS (BS-IRS -user and BS-IRS -user) and one double-reflection link through both IRSs (BS-IRS -IRS -user). We further assume that the two IRSs are deployed such that all the involved channels ( T i ’s, R i ’s, and S ) follow the free-space LoS propagation model [24] and can be expressed as ( T i ) m i ,n t = √ αd Ti,mi,nt e − j πλ d Ti,mi,nt , i ∈ { , } , m i ∈ M i , n t ∈ N t ( S ) m ,m = √ αd S,m ,m e − j πλ d S,m ,m , m ∈ M , m ∈ M ( R i ) n r ,m i = √ αd Ri,nr,mi e − j πλ d Ri,nr,mi , i ∈ { , } , n r ∈ N r , m i ∈ M i , (2)where α is the channel power gain at the reference distance of meter (m), λ is the carrierwavelength; d T i ,m i ,n t , d S,m ,m , and d R i ,n r ,m i denote the distance between antenna n t at the BSand element m i at IRS i , the distance between element m at IRS and element m at IRS , and the distance between element m i at IRS i and antenna n r at the user, respectively. Weassume that the above channels are known a priori at the BS, user, and two IRSs, based on theirgeometric relationship. Let x ∈ C N t × denote the transmitted signal vector. The transmit covariance matrix is thusdefined as Q = E { xx H } with Q (cid:23) and tr( Q ) ≤ P , where P denotes the maximum transmitpower of the BS. The received signal vector y ∈ C N r × is given by y = Hx + z , (3)where z ∼ CN (0 , σ I N r ) denotes the independent circularly symmetric complex Gaussian(CSCG) noise vector at the user receiver, with σ denoting the average noise power. According Note that the double-reflection link reflected by IRS and then by IRS is very weak due to the high path loss, so are thelinks reflected by both IRSs more than twice. Thus, they can be neglected. In practice, such LoS channels can also be estimated by mounting each IRS with low-cost sensor arrays and using the IRSsmart controller to send pilot signals. to [24], the MIMO channel capacity is given by C = max Q :tr( Q ) ≤ P, Q (cid:23) log det (cid:18) I N r + 1 σ HQH H (cid:19) . (4)In this paper, we aim to maximize the channel capacity of our considered double-IRS aidedMIMO system, by jointly optimizing the transmit covariance matrix Q and the passive beam-forming matrices of the two IRSs { Φ , Φ } . The optimization problem is formulated as(P1) max Φ , Φ , Q log det (cid:18) I N r + 1 σ HQH H (cid:19) (5)s.t. Φ i = diag { φ i, , · · · , φ i,M i } , i ∈ { , } (6) | φ i,m i | = 1 , i = { , } , m i ∈ M i (7) tr( Q ) ≤ P (8) Q (cid:23) . (9)Note that (P1) is a non-convex optimization problem given the non-concave objective function in(5) and the non-convex unit-modulus constraints in (7). The complexity of exhaustive search forthe optimal solution to (P1) grows exponentially with the total number of IRS elements M . Dueto the coupling between the passive beamforming matrices of the two IRSs in (P1)’s objectivefunction, (P1) is also more challenging than the MIMO channel capacity maximization problemstudied in [7] with a single IRS. One may want to extend the alternating optimization algorithm in[7] to obtain a locally optimal solution to (P1), by iteratively optimizing the transmit covariancematrix and the reflection coefficients at the two IRSs. However, the involved complexity increaseswith M , and this approach does not exploit the unique structure of the LoS channels in thispaper. Alternatively, we will present a more tractable characterization of the LoS channels in(2), based on which we will develop a novel low-complexity algorithm for solving (P1).For ease of reading, Table I summarizes the main symbol notations used in this paper andtheir physical meanings. TABLE IS
YMBOLS AND THEIR PHYSICAL MEANINGS
Symbols Physical meanings Symbols Physical meanings α Channel power gain at reference distance of m P Maximum transmit power of the BS λ Carrier wavelength σ Average noise power n t /N t Antenna n t /number of antennas at the BS K Rank of H n r /N r Antenna n r /number of antennas at the user δ k k th singular value of H m i /M i Element m i /number of elements at IRS i P k Transmit power allocated to δ k M Total number of IRS elements d T i Distance between the BS and IRS il n Antenna spacing d S Distance between IRS and IRS l m IRS element spacing d R i Distance between IRS i and the user u t Location of the BS t iL / t iR Array response of T i at IRS i /the BS u r Location of the user s L / s R Array response of S at IRS /IRS u i Location of IRS i r iL / r iR Array response of R i at the user/IRS i v t Base direction of the BS’s antenna array ρ N t (Θ t ) Correlation between t R and t R v r Base direction of the user’s antenna array ρ N r (Θ r ) Correlation between r L and r L v ( a ) i / v ( b ) i st/ rd base direction of IRS i ω T i ,v t /ω T i ,v ( a ) i /ω T i ,v ( b ) i Angle between T i and v t / v ( a ) i / v ( b ) i m ( a ) i /M ( a ) i Element m ( a ) i /number of elements inthe st base direction of IRS i ω S,v ( a ) i /ω S,v ( b ) i Angle between S and v ( a ) i / v ( b ) i m ( b ) i /M ( b ) i Element m ( b ) i /number of elements inthe rd base direction of IRS i ω R i ,v r /ω R i ,v ( a ) i /ω R i ,v ( b ) i Angle between R i and v r / v ( a ) i / v ( b ) i T i Channel matrix from the BS to IRS i β a Reference phase of BS-IRS -user link S Channel matrix from IRS to IRS β b Reference phase of BS-IRS -user link R i Channel matrix from IRS i to the user β c Reference phase of BS-IRS -IRS -user link Φ i Passive beamforming matrix of IRS i γ i Common phase shift of IRS iφ i,m i Reflection coefficient of element m i at IRS i | a ( Φ ) | Power gain of BS-IRS -user link H Effective channel from the BS to the user | b ( Φ ) | Power gain of BS-IRS -user link Q Transmit covariance matrix | c ( Φ , Φ ) | / | ˜ c ( Φ , Φ ) | Power gain of BS-IRS -IRS -user link III. T
RACTABLE L O S C
HANNEL C HARACTERIZATION
In this section, we provide a tractable characterization of the LoS channels in (2) and thenthat of the effective BS-user MIMO channel H in (1).Without loss of generality, we assume uniform linear array (ULA) for the BS/user and uniformrectangular array (URA) for the two IRSs, as shown in Fig. 1. Specifically, we place the antennasat the BS and the user on lines along base directions v t ∈ R × and v r ∈ R × , respectively,with (cid:107) v t (cid:107) = (cid:107) v r (cid:107) = 1 . We place the reflecting elements at IRS i ∈ { , } on lines along twoorthogonal base directions v ( a ) i ∈ R × and v ( b ) i ∈ R × , respectively, with (cid:107) v ( a ) i (cid:107) = (cid:107) v ( b ) i (cid:107) = 1 .We denote the position of any particular element m i at IRS i as ( m ( a ) i , m ( b ) i ) , which tells theindices in IRS i ’s first and second base directions. Here, we have m ( a ) i ∈ { , · · · , M ( a ) i − } and m ( b ) i ∈ { , · · · , M ( b ) i − } with M i = M ( a ) i M ( b ) i . Thus, we have the unique mapping between m i and ( m ( a ) i , m ( b ) i ) , i.e., m i = 1 + m ( a ) i + m ( b ) i M ( a ) i ∈ { , · · · , M i } . The results of this paper are also applicable to any other antenna/reflecting element configurations by considering theircorresponding array response vectors.
We reasonably consider that the link distances of T i ’s, R i ’s, and S are much larger than thesizes of the BS’s and user’s antenna arrays as well as the sizes of the two IRSs, thus theseLoS channels follow the far-field LoS channel model of rank-one as in [25]. Similar to [5], werewrite T i in (2) as T i = √ αd T i e − j πdTiλ t iL t TiR , i ∈ { , } , (10)which is the product of the path loss √ αd Ti , the reference phase e − j πdTiλ , and the two array responses t iL ∈ C M i × and t iR ∈ C N t × . Specifically, d T i = (cid:107) u i − u t (cid:107) denotes the distance between(antenna at) the BS and (element at) IRS i , and ( t iL ) m i = exp (cid:18) − j πλ (cid:16) m ( a ) i l m cos (cid:0) ω T i ,v ( a ) i (cid:1) + m ( b ) i l m cos (cid:0) ω T i ,v ( b ) i (cid:1)(cid:17)(cid:19) , i ∈ { , } , m i ∈ M i , (11) ( t iR ) n t = exp (cid:18) j πλ ( n t − l n cos (cid:0) ω T i ,v t (cid:1)(cid:19) , i ∈ { , } , n t ∈ N t , (12)with l m denoting the IRS element spacing and l n denoting the antenna spacing. Note that ω T i ,v t denotes the angle between the direction of T i and the base direction v t , i.e., ω T i ,v t =arccos (cid:0) ( u i − u t ) T v t (cid:107) u i − u t (cid:107)(cid:107) v t (cid:107) (cid:1) ∈ [0 , π ] , as shown in Fig. 1. Similar definitions hold for the angle ω T i ,v ( a ) i between T i and v ( a ) i , as well as the angle ω T i ,v ( b ) i between T i and v ( b ) i .Similar to T i in (10), R i in (2) can be rewritten as R i = √ αd R i e − j πdRiλ r iL r TiR , i ∈ { , } , (13)where d R i = (cid:107) u r − u i (cid:107) denotes the distance between IRS i and the user, and the two arrayresponses r iL ∈ C N r × and r iR ∈ C M i × are given by ( r iL ) n r = exp (cid:18) − j πλ ( n r − l n cos (cid:0) ω R i ,v r (cid:1)(cid:19) , i ∈ { , } , n r ∈ N r , (14) ( r iR ) m i = exp (cid:18) j πλ (cid:16) m ( a ) i l m cos (cid:0) ω R i ,v ( a ) i (cid:1) + m ( b ) i l m cos (cid:0) ω R i ,v ( b ) i (cid:1)(cid:17)(cid:19) , i ∈ { , } , m i ∈ M i . (15)Finally, S in (2) is rewritten as S = √ αd S e − j πdSλ s L s TR , (16)where d S = (cid:107) u − u (cid:107) denotes the distance between IRS and IRS , and the two array This is because the link distance is much larger than the antenna/element spacing, and the distance between each pair of BSantenna and IRS element is almost the same and can be well approximated by d T i when calculating the path loss. responses s L ∈ C M × and s R ∈ C M × are given by ( s L ) m = exp (cid:18) − j πλ (cid:16) m ( a )2 l m cos (cid:0) ω S,v ( a )2 (cid:1) + m ( b )2 l m cos (cid:0) ω S,v ( b )2 (cid:1)(cid:17)(cid:19) , m ∈ M , (17) ( s R ) m = exp (cid:18) j πλ (cid:16) m ( a )1 l m cos (cid:0) ω S,v ( a )1 (cid:1) + m ( b )1 l m cos (cid:0) ω S,v ( b )1 (cid:1)(cid:17)(cid:19) , m ∈ M . (18)By combining (10), (13), and (16), we can successfully rewrite the double-IRS aided MIMOchannel in (1) as H = αβ a d R d T r L r T R Φ t L t T R + αβ b d R d T r L r T R Φ t L t T R + α / β c d R d S d T r L r T R Φ s L s TR Φ t L t T R = a ( Φ ) r L t T R + b ( Φ ) r L t T R + c ( Φ , Φ ) r L t T R , (19)where β a = e − j π ( dR dT λ , β b = e − j π ( dR dT λ , and β c = e − j π ( dR dS + dT λ denote the referencephases of the BS-IRS -user link, BS-IRS -user link, and BS-IRS -IRS -user link, repectively.For ease of characterizing H , we further define a ( Φ ) = αβ a d R d T r T R Φ t L b ( Φ ) = αβ b d R d T r T R Φ t L c ( Φ , Φ ) = α / β c d R d S d T ( r T R Φ s L )( s TR Φ t L ) . (20)It is worth noting that | a ( Φ ) | , | b ( Φ ) | , and | c ( Φ , Φ ) | determine the power gains of the BS-IRS -user link, BS-IRS -user link, and BS-IRS -IRS -user link, repectively.IV. P ROPOSED L OW - COMPLEXITY S OLUTION TO (P1)Based on the new tractable LoS channel characteristics in the last section, in this section, weaim to propose a low-complexity algorithm for solving (P1), to fit the practical scenario witha large total number of IRS elements M . Specifically, we first design a passive beamformingstructure that can simultaneously maximize the power gains of the double-reflection link and twosingle-reflection links. Then, we transform the original problem (P1) to a simplified problem (P2)with only three optimization variables, i.e., the transmit covariance matrix and the common phaseshifts of the two IRSs. We further derive the closed-form solutions to the three subproblems of(P2), for optimizing the transmit covariance matrix or one common phase shift with the othertwo variables being fixed. Finally, we propose an alternating optimization algorithm to obtain alocally optimal solution to (P2), by iteratively solving the above three subproblems.Since the BS-IRS distance and the IRS -user distance are much smaller than the IRS -IRS distance and the BS-user distance, S is approximately parallel to R and T , namely, ω R ,v ( a )1 ≈ ω S,v ( a )1 , ω R ,v ( b )1 ≈ ω S,v ( b )1 , ω T ,v ( a )2 ≈ ω S,v ( a )2 , and ω T ,v ( b )2 ≈ ω S,v ( b )2 . Hence, t L in (11)and s L in (17) are almost the same, so are r R in (15) and s R in (18). Therefore, we have c ( Φ , Φ ) ≈ ˇ c ( Φ , Φ ) = α / β c d R d S d T ( r T R Φ t L )( r T R Φ t L ) , and (19) can be well approximatedby H ≈ a ( Φ ) r L t T R + b ( Φ ) r L t T R + ˇ c ( Φ , Φ ) r L t T R . (21)Consequently, | a ( Φ ) | , | b ( Φ ) | , and | ˇ c ( Φ , Φ ) | can be simultaneously maximized by deployingthe following passive beamforming structure: φ ,m = γ ( t L ) ∗ m ( r R ) ∗ m , m ∈ M φ ,m = γ ( t L ) ∗ m ( r R ) ∗ m , m ∈ M , (22)where γ with | γ | = 1 and γ with | γ | = 1 denote the common phase shifts of IRS and IRS , respectively. The maximum power gains of the single-reflection links via IRS and IRS as well as the double-reflection link are thus given by | a ( Φ ) | = αM d R d T , | b ( Φ ) | = αM d R d T , and | ˇ c ( Φ , Φ ) | = α / M M d R d S d T , respectively.Inspired by the above, we propose to adopt the passive beamforming structure in (22), andfurther optimize the common phase shifts γ and γ to maximize the MIMO channel capacity.In this case, the effective MIMO channel in (21) can be rewritten as H = αM β a γ d R d T r L t T R + αM β b γ d R d T r L t T R + α / M M β c γ γ d R d S d T r L t T R , (23)and the optimization problem (P1) is reformulated as(P2) max γ ,γ , Q log det (cid:18) I N r + 1 σ HQH H (cid:19) (24)s.t. | γ i | = 1 , i = { , } (25) tr( Q ) ≤ P (26) Q (cid:23) . (27)Note that our proposed approach transforms the original problem (P1) with M + 1 optimizationvariables { Q } (cid:83) { φ ,m } M m =1 (cid:83) { φ ,m } M m =1 to a simplified problem (P2) with only optimiza-tion variables { Q , γ , γ } . Although (P2) is still a non-convex optimization problem, in thefollowing, we first optimally solve the three subproblems of (P2) in closed-form, which aim tooptimize one variable in { Q , γ , γ } with the other two variables being fixed. We then presentan efficient alternating optimization algorithm for obtaining a locally optimal solution to (P2),by iteratively solving the above three subproblems. A. Optimization of Q with Given γ and γ First, we consider the subproblem of optimizing the transmit covariance matrix Q with given γ and γ . Note that (P2) is a convex optimization problem about Q . The truncated singularvalue decomposition (SVD) of H in (23) is given by H = U∆V H , where V ∈ C N t × K with K = rank( H ) , and the optimal Q (cid:63) is given in the lemma below by following the water-fillingpower allocation [24]. Lemma 1 : The optimal Q (cid:63) with given γ and γ is Q (cid:63) = V diag { P (cid:63) , · · · , P (cid:63)K } V H , (28)where P (cid:63)k is the optimal amount of transmit power allocated to the k th singular value, i.e., P (cid:63)k = max( µ − σ δ k , , k = 1 , · · · , K , with δ k = ( ∆ ) k,k and µ satisfying (cid:80) Kk =1 P (cid:63)k = P . B. Optimization of γ with Given Q and γ Next, we aim to optimize the common phase shift of IRS denoted by γ with given Q and γ . Note that the eigenvalue decomposition (EVD) of Q is given by Q = U Q Σ Q U HQ , where U Q ∈ C N t × N t and Σ Q ∈ C N t × N t . For ease of exposition, we define A = αM β a d R d T r L t T R U Q Σ Q , B = αM β b d R d T r L t T R U Q Σ Q , C = α / M M β c d R d S d T r L t T R U Q Σ Q , and the objective function of (P2) withrespect to γ can be rewritten as f γ = log det (cid:18) I N r + 1 σ HQH H (cid:19) = log det (cid:18) I N r + 1 σ (cid:0) γ A + γ B + γ γ C (cid:1)(cid:0) γ A + γ B + γ γ C (cid:1) H (cid:19) = log det (cid:0) X + γ Y + γ ∗ Y H (cid:1) , (29)where X = I N r + 1 σ (cid:16) AA H + BB H + CC H + γ CA H + γ ∗ AC H (cid:17) , (30)and Y = 1 σ (cid:16) γ ∗ AB H + CB H (cid:17) . (31)Therefore, the subproblem with respect to γ can be formulated as(P2- γ ) max γ log det (cid:0) X + γ Y + γ ∗ Y H (cid:1) s.t. | γ | = 1 . (32) Note that (P2- γ ) has a similar structure as (P1- m ) in [7]. Therefore, the optimal solution to (P2- γ ) can be similarly derived as the optimal solution to (P1- m ) in [7], as shown in the followinglemma. Lemma 2 : The optimal γ (cid:63) with given Q and γ is γ (cid:63) = e − j arg { ν } , if tr (cid:0) X − Y (cid:1) (cid:54) = 01 , otherwise , (33)where ν is the sole non-zero eigenvalue of X − Y . C. Optimization of γ with Given Q and γ Finally, we focus on the subproblem to optimize the common phase shift of IRS denoted by γ with given Q and γ . Similar to the previous subsection for optimizing γ , the subproblemfor optimizing γ is formulated as(P2- γ ) max γ log det (cid:0) X + γ Y + γ ∗ Y H (cid:1) s.t. | γ | = 1 , (34)where X = I N r + 1 σ (cid:16) AA H + BB H + CC H + γ CB H + γ ∗ BC H (cid:17) , (35)and Y = 1 σ (cid:16) γ ∗ BA H + CA H (cid:17) . (36)Therefore, the optimal solution to (P2- γ ) is given by γ (cid:63) = e − j arg { ν } , if tr (cid:0) X − Y (cid:1) (cid:54) = 01 , otherwise , (37)where ν is the sole non-zero eigenvalue of X − Y . D. Overall Algorithm
With all the three subproblems optimally solved in closed-form, we are ready to present theoverall algorithm to solve (P2). Specifically, we initialize with γ = γ = 1 . Then, we iterativelyoptimize the transmit covariance matrix Q or one common phase shift γ i at each time, with theother two variables being fixed. The convergence is reached if the relative increment of (P2)’sobjective function does not exceed a threshold (cid:15) > by optimizing any variable in { Q , γ , γ } .The overall algorithm is summarized in Algorithm 1. Note that each subproblem is optimally Algorithm 1:
Alternating optimization algorithm to (P2)
Input: T , T , R , R , S , P , σ Output: Q (cid:63) , γ (cid:63) , γ (cid:63) Initialize γ = 1 and γ = 1 . Repeat Obtain the optimal Q (cid:63) in (28). Obtain the optimal γ (cid:63) in (33). Obtain the optimal γ (cid:63) in (37). Until
The relative increment of the objective function in (24) does not exceed athreshold (cid:15) > by optimizing any variable in { Q , γ , γ } .solved in closed-form, and the optimization variables are not coupled in (P2)’s constraints.Therefore, Algorithm 1 is guaranteed to converge to at least a locally optimal solution to (P2),since the algorithm yields a non-decreasing objective function over iterations, which is upper-bounded by a finite value. It is worth noting that based on the obtained solution to (P2), asuboptimal solution to (P1) is automatically obtained by substituting γ (cid:63) and γ (cid:63) into (22).As compared to the very recent algorithm for capacity maximization in [7] whose complexityincreases fast with the total number of IRS elements M , the complexity of our proposed algorithmis independent with M , which greatly saves the computational time especially for the practicalscenario with large M .V. C APACITY S CALING A NALYSIS FOR O PTIMIZED D OUBLE -IRS A
IDED
MIMO S
YSTEM
In this section, we derive the capacity scaling orders of the optimized double-IRS aided MIMOsystem with respect to asymptotically large total number of IRS elements M or transmit power P , for drawing more insights about its asymptotic capacity performance. Specifically, we firstanalytically derive the explicit conditions that the effective MIMO channel H in (21) is of rank-two or rank-one. Then, we analyze the MIMO channel capacities as well as the scaling ordersfor rank-two H and rank-one H , respectively. A. Rank Analysis of Double-IRS Aided MIMO Channel
First, we characterize the rank of the double-IRS aided MIMO channel H . As we can seefrom (21), the double-reflection link can be combined with one of the single-reflection links asone rank-one matrix, thus K = rank( H ) is upper-bounded as K ≤ rank (cid:0) a ( Φ ) r L t T R (cid:1) + rank (cid:16) r L (cid:0) b ( Φ ) t T R + ˇ c ( Φ , Φ ) t T R (cid:1)(cid:17) = 2 . (38) Based on the LoS channel characterization, we express the correlation between the two arrayresponses at the BS in the BS-IRS and BS-IRS channels, t R and t R in (12), as ρ N t (Θ t ) = 1 N t (cid:12)(cid:12) t H R t R (cid:12)(cid:12) = 1 N t (cid:12)(cid:12)(cid:12)(cid:12) N t (cid:88) n t =1 e − j π ( n t − t (cid:12)(cid:12)(cid:12)(cid:12) = sin( πN t Θ t ) N t sin( π Θ t ) ∈ [0 , , (39)where Θ t = l n λ (cid:0) cos( ω T ,v t ) − cos( ω T ,v t ) (cid:1) . Similarly, we express the correlation between the twoarray responses at the user in the IRS -user and IRS -user channels, r L and r L in (14), as ρ N r (Θ r ) = 1 N r (cid:12)(cid:12) r H L r L (cid:12)(cid:12) = 1 N r (cid:12)(cid:12)(cid:12)(cid:12) N r (cid:88) n r =1 e − j π ( n r − r (cid:12)(cid:12)(cid:12)(cid:12) = sin( πN r Θ r ) N r sin( π Θ r ) ∈ [0 , , (40)where Θ r = l n λ (cid:0) cos( ω R ,v r ) − cos( ω R ,v r ) (cid:1) . It can be observed from (39) and (40), when ρ N t (Θ t ) = 1 or ρ N r (Θ r ) = 1 , H is of rank-one. This is because ρ N t (Θ t ) = 1 means that t R and t R are the same, and the three rank-one matrices in (21) can be combined as onerank-one matrix. Similar analysis holds for ρ N r (Θ r ) = 1 . While for the case of ρ N t (Θ t ) (cid:54) = 1 and ρ N r (Θ r ) (cid:54) = 1 , t R and t R are linearly independent, so are r L and r L , thus the equality in (38)holds and H is of rank-two. Therefore, we have the following proposition about the rank of H . Proposition 1 : Under the LoS channels, the rank of the double-IRS aided MIMO channel H is given by K = , if ρ N t (Θ t ) (cid:54) = 1 and ρ N r (Θ r ) (cid:54) = 11 , otherwise . (41)It is worth noting that ρ N t (Θ t ) is periodic with period and takes the maximal value at Θ t = 0 ,thus ρ N t (Θ t ) (cid:54) = 1 corresponds to Θ t = l n λ (cid:0) cos( ω T ,v t ) − cos( ω T ,v t ) (cid:1) / ∈ Z . Also, the differencebetween cos( ω T ,v t ) and cos( ω T ,v t ) cannot be more than . Similar analysis holds for ρ N r (Θ r ) .Therefore, we have the following corollary. Corollary 1 : For the case with antenna spacing l n ≤ λ , the explicit conditions in (41) can befurther simplified as K = , if ω T ,v t (cid:54) = ω T ,v t and ω R ,v r (cid:54) = ω R ,v r , otherwise , (42)which means that H is of rank-two if the two angles-of-departure, ω T ,v t and ω T ,v t , are different(as shown in Fig. 1), so are the two angles-of-arrival, ω R ,v r and ω R ,v r .In the following, we analytically derive the MIMO channel capacities and the scaling ordersof the double-IRS aided MIMO system for rank-two H and rank-one H , respectively. For thepurpose of exposition, we consider the setup with N t ≥ antennas at the BS and N r = 2 antennas at the user, without compromising the spatial multiplexing gain, while all the resultsare also applicable to the setup with N t = 2 antennas at the BS and N r ≥ antennas at theuser by leveraging the uplink-downlink duality. Later in Section VI, we will provide numericalresults on the capacity scaling with arbitrary number of antennas at the BS/user. B. Capacity Scaling with Rank-two H We first consider the case that H in (21) is of rank-two, i.e., when ρ N t (Θ t ) (cid:54) = 1 and ρ N r (Θ r ) (cid:54) =1 as in (41). The MIMO channel capacity can be expressed as C = (cid:88) k =1 log (cid:18) P (cid:63)k δ k σ (cid:19) , (43)where P (cid:63)k is the optimal amount of transmit power allocated to the k th singular value of H , δ k > , k = 1 , , as given in (28) with K = 2 . In this case, the capacity scaling order withrespect to an asymptotically large M , i.e., lim M →∞ C log ( M ) , is given in the following proposition. Proposition 2 : If K = 2 , by adopting the passive beamforming structure in (22) with equalnumber of elements at the two IRSs ( M = M = M ), the capacity scaling order of an N t × double-IRS aided MIMO system with respect to an asymptotically large M is maximized to lim M →∞ C log ( M ) = 4 . Proof:
Please refer to Appendix A.Moreover, in the high transmit power regime, where it is asymptotically optimal to evenlyallocate the transmit power to the two eigenchannels ( P (cid:63) = P (cid:63) = P ), (43) can be wellapproximated as C ≈ log (cid:18) P σ ( δ δ ) (cid:19) (a) = log (cid:18) P σ N t (cid:12)(cid:12)(cid:12) a ( Φ ) b ( Φ ) (cid:12)(cid:12)(cid:12) (cid:0) − ρ N t (Θ t ) (cid:1)(cid:0) − ρ (Θ r ) (cid:1)(cid:19) , (44)where (a) = comes from the expression of ( δ δ ) in (53). Surprisingly, the effect of the commonphase shifts { γ , γ } and the double-reflection link ˇ c ( Φ , Φ ) is negligible. This is because thespatial multiplexing gain introduced by the two single-reflection links a ( Φ ) and b ( Φ ) nowtakes the dominant effect on the MIMO channel capacity. In the high transmit power regime,it can be shown that the passive beamforming structure in (22) maximizes the channel capacityin (44) by maximizing (cid:12)(cid:12) a ( Φ ) b ( Φ ) (cid:12)(cid:12) = (cid:12)(cid:12) α M M d R d T d R d T (cid:12)(cid:12) , and it is optimal to further allocate evennumber of elements at the two IRSs ( M = M = M ). Thus, we have the following proposition. 𝐷𝐷 UserIRS 𝑑𝑑 sep IRS BS 𝜔𝜔 𝑇𝑇 , 𝑣𝑣 𝑡𝑡 = 0 𝜔𝜔 𝑇𝑇 , 𝑣𝑣 𝑡𝑡 ≈ 𝜋𝜋 𝜔𝜔 𝑅𝑅 , 𝑣𝑣 𝑟𝑟 = 𝜋𝜋𝜔𝜔 𝑅𝑅 , 𝑣𝑣 𝑟𝑟 ≈ 𝜋𝜋 Fig. 2. Rank-two H with orthogonal t R and t R as well as orthogonal r L and r L , under N t = N r = 2 and l n = λ . Proposition 3 : If K = 2 , by adopting the passive beamforming structure in (22) with equalnumber of elements at the two IRSs ( M = M = M ), the capacity of an N t × double-IRSaided MIMO system with asymptotically large P is maximized as C = log (cid:18) α N t M P d R d T d R d T σ (cid:0) − ρ N t (Θ t ) (cid:1)(cid:0) − ρ (Θ r ) (cid:1)(cid:19) . (45)From (45) we can see that the capacity scaling order with respect to an asymptotically large P is lim P →∞ C log ( P ) = 2 , which means that a -fold spatial multiplexing gain can be achieved inthe high transmit power regime. It is also worth noting that when ρ N t (Θ t ) = 0 (i.e., orthogonal t R and t R ) and ρ (Θ r ) = 0 (orthogonal r L and r L ), the MIMO channel capacity in (45) ismaximized.In Fig. 2, we illustrate a scenario with orthogonal t R and t R as well as orthogonal r L and r L , under N t = N r = 2 and l n = λ . The two IRSs are deployed such that ω T ,v t = 0 and ω T ,v t ≈ π , also ω R ,v r ≈ π and ω R ,v r = π , thus H is of rank-two due to ω T ,v t (cid:54) = ω T ,v t and ω R ,v r (cid:54) = ω R ,v r according to (42). We further have Θ t = l n λ (cid:0) cos( ω T ,v t ) − cos( ω T ,v t ) (cid:1) = − and Θ r = l n λ (cid:0) cos( ω R ,v r ) − cos( ω R ,v r ) (cid:1) = , thus ρ (Θ t ) = 0 and ρ (Θ r ) = 0 , i.e., orthogonal t R and t R as well as orthogonal r L and r L . C. Capacity Scaling with Rank-one H We now consider the case that H in (21) is of rank-one. Here, we assume ρ N t (Θ t ) = 1 , i.e., t R = t R , for the purpose of exposition, H can be thus rewritten as H = (cid:16) a ( Φ ) r L + (cid:0) b ( Φ ) + ˇ c ( Φ , Φ ) (cid:1) r L (cid:17) t T R , (46)which yields only one singular value δ , and the MIMO channel capacity can be expressed as C = log (cid:18) P δ σ (cid:19) , (47) All the analysis is directly applicable to the other case of r L = r L . 𝐷𝐷 UserIRS 𝑑𝑑 sep IRS BS 𝜔𝜔 𝑇𝑇 , 𝑣𝑣 𝑡𝑡 = 𝜔𝜔 𝑇𝑇 , 𝑣𝑣 𝑡𝑡 ≈ 𝜋𝜋 𝜔𝜔 𝑅𝑅 , 𝑣𝑣 𝑟𝑟 = 𝜋𝜋𝜔𝜔 𝑇𝑇 , 𝑣𝑣 𝑡𝑡 = 𝜋𝜋 𝜔𝜔 𝑅𝑅 , 𝑣𝑣 𝑟𝑟 = 𝜔𝜔 𝑅𝑅 , 𝑣𝑣 𝑟𝑟 ≈ 𝜋𝜋 Fig. 3. Rank-one H with t R = t R and r L = r L , under N t = N r = 2 and l n = λ . with its exact expression given in the following proposition. Proposition 4 : If K = 1 , by adopting the passive beamforming structure in (22) and settingthe common phase shifts as γ (cid:63) = β b β c and γ (cid:63) = β b (cid:0) β a β b β c − π Θ r (cid:1) , the capacity of an N t × double-IRS aided MIMO system is maximized as C = log (cid:32)
1+ 2 N t Pσ (cid:18)(cid:16) αM d R d T (cid:17) + (cid:16) αM d R d T + α / M M d R d S d T (cid:17) +2 (cid:16) αM d R d T (cid:17)(cid:16) αM d R d T + α / M M d R d S d T (cid:17) ρ (Θ r ) (cid:19)(cid:33) . (48) Proof:
Please refer to Appendix B.Note that for large M , it is asymptotically optimal to allocate even number of elements at thetwo IRSs ( M = M = M ), since α / M M d R d S d T will be much larger than αM d R d T or αM d R d T , and from(48) we have C = log (cid:32) N t M Pσ (cid:18)(cid:16) α M d R d T (cid:17) + (cid:16) α M d R d T + α / d R d S d T (cid:17) + αM d R d T × (cid:16) α M d R d T + α / d R d S d T (cid:17) ρ (Θ r ) (cid:19)(cid:33) . (49)As we can see from (49), although the spatial multiplexing gain is lost, i.e., lim P →∞ C log ( P ) = 1 ,due to the rank-one H , a capacity scaling order of with respect to an asymptotically large M can still be achieved, i.e., lim M →∞ C log ( M ) = 4 , thanks to the cooperative passive beamforming gainof the double-reflection link. It is worth noting that for the rank-one H with ρ (Θ t ) = 1 (i.e., t R = t R ), the MIMO channel capacity in (49) is maximized with ρ (Θ r ) = 1 ( r L = r L ).In Fig. 3, we visualize a possible scenario with t R = t R as well as r L = r L , under N t = N r = 2 and l n = λ . The two IRSs are deployed such that ω T ,v t = ω T ,v t = ω R ,v r = ω R ,v r ≈ π ,thus we have ρ (Θ t ) = 1 and ρ (Θ r ) = 1 , and consequently H is of rank-one. D. Comparison with Benchmark Single-IRS System
For comparison with our considered double-IRS system, we also present the conventionalsingle-IRS aided MIMO system [7], where only IRS is deployed with all M reflecting elements, TABLE IIC
APACITY SCALING ORDERS UNDER L O S CHANNELS FOR N t × MIMO
SYSTEM
System Capacity scaling order lim P →∞ C log ( P ) lim M →∞ C log ( M ) Single IRS 1 2Double IRSs Rank-one H H H = RΦT = αβ b d R d T r L r TR Φt L t TR , (50)where T ∈ C M × N t and R ∈ C × M are the channels from the BS to the single IRS and fromthe single IRS to the user, respectively, following similar definitions as in (10) and (13), and Φ = diag { φ , · · · , φ M } ∈ C M × M is the passive beamforming matrix of the single IRS.It can be easily shown that by deploying the optimal passive beamformer φ (cid:63)m = ( r R ) ∗ m ( t L ) ∗ m , m ∈ M , the MIMO channel capacity is maximized as C = log (cid:18) α N t M Pd R d T σ (cid:19) . (51)As we can see from (51), there is neither spatial multiplexing gain harvested from the two single-reflection links, nor cooperative passive beamforming gain brought by the double-reflection link,as in the double-IRS aided MIMO system. Thus, the capacity scaling order with respect to M and P are lim M →∞ C log ( M ) = 2 and lim P →∞ C log ( P ) = 1 , respectively, for the single-IRS aided MIMOsystem. Theroem 1 : The capacity scaling orders for the N t × MIMO system aided by double IRSsand single IRS, with respect to asymptotically large M or P , are given in Table II.As we can see from Table II, the capacity scaling order of our considered double-IRS systemwith respect to an asymptotically large M is lim M →∞ C log ( M ) = 4 , regardless of the rank of H ,which substantially outperforms the conventional single-IRS system. While by further deployingthe IRS locations as in (41), we can potentially harvest a -fold spatial multiplexing gain fromthe double-IRS system, i.e., lim P →∞ C log ( P ) = 2 .VI. N UMERICAL R ESULTS
In this section, we provide extensive numerical results to evaluate the performance of ourproposed Algorithm 1 in Section IV and validate our capacity scaling analysis in Section V. We 𝑥𝑥 ( 𝑚𝑚 ) 𝑦𝑦 ( 𝑚𝑚 ) 𝑧𝑧 ( 𝑚𝑚 ) IRS IRS 𝑑𝑑 𝐒𝐒 𝒏𝒏 ( 𝑎𝑎 ) 𝒏𝒏 ( 𝑏𝑏 ) 𝒏𝒏 ( 𝑎𝑎 ) 𝒏𝒏 ( 𝑏𝑏 ) BS 𝜔𝜔 b II , Rx (0,0,0) (0,50,0)(1,0,0) User (1,50,0)
Fig. 4. Illustration of the simulation setup. set the 3D spatial locations of the BS, user, IRS 1, and IRS 2 as u t = [1 , , T , u r = [1 , , T , u = [0 , , T , and u = [0 , , T , respectively, as illustrated in Fig. 4. We set ULA at the BSand user, with the base direction of BS’s antenna array and that of user’s antenna array both setas v t = v r = [ − , , T . We set URA at IRS and IRS , with the base directions of IRS set as v ( a )1 = [ − , √ , T and v ( b )1 = [0 , , T , while those of IRS set as v ( a )2 = [0 , , T and v ( b )2 = [ − , − √ , T . The carrier frequency is set as . GHz, thus the wavelength is λ = 0 . m and the channel power gain at reference distance of m is α = − dB [24]. The antennaspacing is set as l n = λ , while the IRS element spacing is set as l m = λ . Note that under theabove setup, we have ω T ,v t (cid:54) = ω T ,v t as well as ω R ,v r (cid:54) = ω R ,v r , and the effective MIMO channel H is of rank-two according to (42). The noise power level at the user receiver is set as σ = − dBm. All the involved channels are LoS following the definitions in (2). For all the double-IRSaided MIMO systems considered in this section, we set M = M = M since it maximizes thepower gain of the double-reflection link. The convergence threshold of Algorithm 1 in terms ofthe relative increment in (P2)’s objective function is set as (cid:15) = 10 − . A. Comparison of Algorithm 1 with Other Capacity Maximization Algorithms for Double-IRSAided MIMO System
First, we show the advantages of our proposed Algorithm 1, by comparing with two benchmarkalgorithms: • The alternating optimization algorithm based on [7], by iteratively optimizing one variablein Q ∪ { φ ,m } m ∈M ∪ { φ ,m } m ∈M at each time with the other M variables being fixed. • A heuristic design by directly deploying the passive beamformers in (22) with γ = γ = 1 ,i.e., without optimizing the common phase shifts of the two IRSs.
100 200 300 400 500 600 700 800 900 1000Total number of IRS elements, M0123456 A c h i e v ab l e r a t e ( bp s / H z ) Algorithm 1Alternating optimization algorithm in [7]Passive beamformers in (22), with = =11000 1500 2000468 (a) Low transmit power regime with P = − dBm.
200 400 600 800 1000Total number of IRS elements, M68101214161820 A c h i e v ab l e r a t e ( bp s / H z ) Algorithm 1Alternating optimization algorithm in [7]Passive beamformers in (22), with = =1 (b) High transmit power regime with P = 20 dBm.
200 400 600 800 1000Total number of IRS elements, M10 C o m pu t a t i ona l t i m e ( s ) Algorithm 1Alternating optimization algorithm in [7] (c) Computational time versus totalnumber of IRS elements.Fig. 5. Comparison of different capacity maximization algorithms.
We consider the setup with N t = N r = 3 and evaluate the achievable rate in bps/Hz by allthe three algorithms for both the low transmit power regime with P = − dBm in Fig. 5(a)and the high transmit power regime with P = 20 dBm in Fig. 5(b). In Fig. 5(c), we plot thecomputational time in second (s) versus the total number of IRS elements M , using Algorithm1 and the alternating optimization algorithm based on [7], respectively.It can be observed from Fig. 5(a) and Fig. 5(b) that for both power regimes, our proposed low-complexity Algorithm 1 can achieve almost the same performance as the alternating optimizationalgorithm based on [7]. Note that the computational time of the former is independent with M thanks to the exploitation of the LoS channels, while that of the latter increases fast with M .Moreover, in the low transmit power regime (see Fig. 5(a)), our proposed algorithm sig-nificantly outperforms the heuristic design without optimizing the common phase shifts, e.g.,by at M = 1000 . This is because in the low transmit power regime, it is optimal totransmit a single data stream, thus the achievable rate critically depends on the power gain ofthe strongest ( st) eigenchannel, and the maximization of which requires a fine-tuning of thecommon phase shifts such that the three reflection links can be coherently combined. On theother hand, in the high transmit power regime (see Fig. 5(b)), the heuristic design performsclosely to our proposed algorithm as well as the alternating optimization algorithm based on[7]. This is because the achievable rate in the high transmit power regime is dominated by thespatial multiplexing (rank) gain, and all the three algorithms achieve the maximum rank of with the considered IRS deployment. Furthermore, it can be observed that for both the low andhigh transmit power regimes, the achievable rate increases by bps/Hz when doubling M . Thisshows that the capacity scaling order with respect to an asymptotically large M for N t = N r = 3 is still , which is the same as our analytical results for N t × MIMO system in Table II.
100 200 300 400 500 600 700 800 900 1000Total number of IRS elements, M00.511.522.533.544.55 A c h i e v ab l e r a t e ( bp s / H z ) Double IRSs, single- and double-reflection linksSingle IRSDouble IRSs, only single-reflection linksDouble IRSs, only double-reflection link1000 1500 2000456789 (a) Achievable rate versus total number of IRS elements inthe low transmit power regime with P = − dBm.
100 200 300 400 500 600 700 800 900 1000Total number of IRS elements, M024681012 A c h i e v ab l e r a t e ( bp s / H z ) Double IRSs, single- and double-reflection linksSingle IRSDouble IRSs, only single-reflection linksDouble IRSs, only double-reflection link (b) Achievable rate versus total number of IRS elements inthe moderate transmit power regime with P = 5 dBm.
100 200 300 400 500 600 700 800 900 1000Total number of IRS elements, M02468101214161820 A c h i e v ab l e r a t e ( bp s / H z ) Double IRSs, single- and double-reflection linksSingle IRSDouble IRSs, only single-reflection linksDouble IRSs, only double-reflection link (c) Achievable rate versus total number of IRS elements inthe high transmit power regime with P = 20 dBm. -10 -5 0 5 10 15 20Transmit power P in dBm02468101214161820 A c h i e v ab l e r a t e ( bp s / H z ) Double IRSs, single- and double-reflection linksAnalytical expression in (45)Single IRSDouble IRSs, only single-reflection linksDouble IRSs, only double-reflection link (d) Achievable rate versus transmit power with M = 1000 .Fig. 6. Performance comparison between our proposed double-IRS system and the conventional single-IRS system. B. Double-IRS Aided Versus Single-IRS Aided MIMO Systems
Next, we compare in Fig. 6 the performance of the double-IRS aided and single-IRS aidedMIMO systems by considering a × MIMO system, for which the capacity scaling ordersof both systems are derived in Section V. For the double-IRS aided MIMO system, we use ourproposed Algorithm 1 for the capacity maximization. For the single-IRS aided MIMO system, weconsider that all the M elements are equipped at IRS 2 (following a similar setup as in SectionV-D). To identify which of the single-reflection and double-reflection links is performance-dominant in different transmit power regimes, we consider two benchmark scenarios for thedouble-IRS system, i.e., when only the two single-reflection links are present, and when only the double-reflection link is present, respectively.In Figs. 6(a)–(c), we plot the achievable rate versus the total number of IRS elements M fordifferent transmit power regimes. In addition, we show in Fig. 6(d) the achievable rate versusthe transmit power P with M = 1000 . It is observed from Figs. 6(a)–(d) that the double-IRSsystem achieves higher achievable rate over its single-IRS counterpart as long as the number ofIRS elements M or the transmit power P is not small. The capacity scaling orders with respectto asymptotically large M or P for both double-IRS and single-IRS systems derived in Table IIare also verified.Moreover, for the low transmit power regime with P = − dBm in Fig. 6(a), it can beobserved that the double-IRS system with only the double-reflection link performs closely tothat with both single-reflection and double-reflection links. This is because the achievable ratein the low transmit power regime is maximized by transmitting a single data stream over thestrongest eigenchannel, i.e., the beamforming mode , thus the cooperative passive beamforminggain harvested from the double-reflection link plays a dominant role. On the other hand, for thehigh transmit power regime with P = 20 dBm in Fig. 6(c), the double-IRS system with only thesingle-reflection links performs similarly as that with both links, since the MIMO system nowoperates in the spatial multiplexing mode and the rank gain achieved by the two single-reflectionlinks is dominant. While for the moderate transmit power regime with P = 5 dBm in Fig. 6(b),both the single-reflection and double-reflection links contribute to the overall achievable rate ofthe double-IRS system in general.Furthermore, it can be observed from Fig. 6(d) that as the transmit power P increases, the maincontributor of the double-IRS system’s achievable rate changes from the double-reflection link’scooperative passive beamforming gain to the two single-reflection links’ spatial multiplexinggain. We also plot the analytical expression of the MIMO channel capacity with asymptoticallylarge P for rank-two H in (45), which coincides with the achievable rate by employing ourproposed Algorithm 1 in the high transmit power regime. This validates our analytical resultsand also shows that our proposed Algorithm 1 can achieve a near-optimal solution to (P1).In addition, we show in Fig. 7 the singular values of the effective MIMO channel H as wellas the corresponding transmit power allocation for the double-IRS system in different transmitpower regimes. It can be observed from Fig. 7(a) that the singular values of H vary with different P ’s since the IRSs’ common phase shifts are iteratively optimized with the transmit covariancematrix (the power allocation over the different eigenchannels). Particularly, for the small-to-
100 200 300 400 500 600 700 800 900 1000Total number of IRS elements, M10 -4 -3 -2 S i ngu l a r v a l ue Single-IRS system1st singular value (P=20 dBm)2rd singular value (P=20 dBm)1st singular value (P=5 dBm)2rd singular value (P=5 dBm)1st singular value (P=-10 dBm)2rd singular value (P=-10 dBm) 850 900 9503.83.94 10 -3
850 900 9503.43.6 10 -4 (a) Singular values of H (after alternating optimization of { Q , γ , γ } ) versus total number of IRS elements.
100 200 300 400 500 600 700 800 900 1000Total number of IRS elements, M00.10.20.30.40.50.60.70.80.91 P o w e r a ll o c a t i on po r t i on , P k / P (b) Power allocation versus total number of IRS elements.Fig. 7. Singular values of double-IRS aided MIMO channel and corresponding transmit power allocation. moderate transmit power regime, e.g., P = − dBm and P = 5 dBm, it can be observed fromFig. 7(b) that more power needs to be allocated to the strongest eigenchannel, and consequentlythe strongest ( st) singular value is adjusted to be larger comparing with that in the high transmitpower regime, e.g., P = 20 dBm, as illustrated in Fig. 7(a).On the other hand, for the high transmit power regime, e.g., P = 20 dBm, it is asymptoticallyoptimal to evenly allocate the transmit power on the two eigenchannels as shown in Fig. 7(b),thus our proposed algorithm will strike a balance between the two singular values as shownin Fig. 7(a). The above results show that our proposed algorithm for determining the transmitcovariance matrix and IRS common phase shifts is able to adaptively tune the effective MIMOchannel H according to the available transmit power P . For comparison, we also show the solesingular value of the effective MIMO channel under the single-IRS system, which is observed tobe larger than the strongest singular value under the double-IRS system when M is small, andbecomes smaller than the latter as M increases. This is because as M increases, the cooperativepassive beamforming gain of the double-IRS system increases much faster than the passivebeamforming gain of the single-IRS system ( O ( M ) versus O ( M ) ). C. Performance under Different User-IRS Angles
As discussed in Section V, unlike the single-IRS aided MIMO system with only one single-reflection link, the performance of the double-IRS aided MIMO system depends on the anglesamong the antenna/element arrays at the BS, user, and two IRSs, which determine the superpo- 𝐷𝐷 IRS 𝑑𝑑 sep IRS 𝜓𝜓 𝑦𝑦 ( 𝑚𝑚 ) 𝑥𝑥 ( 𝑚𝑚 ) BS User 𝜓𝜓 = 0 𝜓𝜓 = 𝜋𝜋 (a) A top-down view of the location setup. A c h i e v ab l e r a t e ( bp s / H z ) (b) Achievable rate versus angle ψ .Fig. 8. Achievable rate under different user-IRS angles. sition effect of the three reflection links characterized by the angle-determined array responsesin (21). Motivated by this, we aim to evaluate the achievable rate under different user-IRSangles, in order to identify the region that the user can be best served by the double-IRS system.Specifically, as illustrated in Fig. 8(a), we consider the same BS and IRS locations as stated atthe beginning of this section, while the user location is set as u r = [cos( ψ ) , − sin( ψ ) , T with varying ψ ∈ [0 , π/ . Note that the distance between IRS and the user is still d R = 1 m,and ψ determines the angle between the user’s ULA and the IRS -user line. We consider thesetup with N t = N r = 2 , M = 1000 , and a high transmit power P = 20 dBm.In Fig. 8(b), we show the achievable rate versus the angle ψ . It can be observed that theachievable rate decreases as the angle ψ increases, which can be explained as follows. Notethat under the considered setup, the two array responses at the user are given by r L = (cid:2) , exp (cid:0) j πλ l n cos( π ) (cid:1)(cid:3) T = [1 , T and r L = (cid:2) , exp (cid:0) j πλ l n cos( π − ψ ) (cid:1)(cid:3) T = [1 , exp( − jπ cos( ψ ))] T ,respectively. Recall from Section V that the asymptotic capacity of the double-IRS aided MIMOsystem in (45) decreases as the correlation between the array responses at the user, ρ (Θ r ) = (cid:12)(cid:12) r H L r L (cid:12)(cid:12) = (cid:12)(cid:12) − jπ cos( ψ )) (cid:12)(cid:12) , increases, and ρ (Θ r ) is an increasing function of ψ ∈ [0 , π/ . Particularly, for the best case of ψ = 0 as illustrated in Fig. 7 (a), the correlation ρ (Θ r ) is minimized to , thus r L and r L are orthogonal and the achievable rate is maximized (notethat now the array responses at the BS t R and t R are also orthogonal). While for the worstcase of ψ = π , the correlation ρ (Θ r ) is maximized to and the effective MIMO channel H is of rank one, which leads to the lowest achievable rate since the spatial multiplexing gain cannot be harvested. This suggests that in the high transmit power regime, the two IRSs shouldbe carefully placed such that the array responses at the user are orthogonal, so are the arrayresponses at the BS, for maximizing the MIMO channel capacity.VII. C ONCLUSION
In this paper, we investigated a novel double-IRS aided MIMO communication system, wherea multi-antenna user is served by a multi-antenna BS through two single-reflection links andone double-reflection link, under the LoS propagation channels. We formulated the capacitymaximization problem by jointly optimizing the transmit covariance matrix and the passivebeamforming matrices of the two IRSs, which is non-convex and difficult to solve. Nevertheless,by exploiting the unique characteristics of the LoS channels, we proposed a low-complexityalgorithm whose complexity is independent with the total number of IRS elements. Then, weanalytically showed that the rank of the effective MIMO channel can be increased to two byproperly deploying the two IRSs. By considering the case with two antennas at the BS or user,we further analyzed the capacity scaling orders of the double-IRS aided MIMO system withrespect to asymptotically large number of IRS elements or transmit power, which are shown tosignificantly outperform its single-IRS counterpart. Finally, we conducted extensive simulationsto evaluate the performance of our proposed algorithm, validate our analytical results, and revealthe advantages of our considered double-IRS system over the conventional single-IRS system.In future work, it will be interesting to extend the capacity scaling results for the double-IRS aided MIMO system with arbitrary number of transmit/receive antennas, as well as undermulti-path/non-LoS channels. A
PPENDICES
A. Proof of Proposition 2
The MIMO channel capacity for rank-two H can be written as C = log (cid:18) P (cid:63) δ σ + P (cid:63) δ σ + P (cid:63) P (cid:63) δ δ σ (cid:19) . (52)For maximizing the capacity scaling order with respect to an asymptotically large M , equivalentlywe are to maximize the highest exponent of M inside (52)’s logarithm operator. We start with P (cid:63) P (cid:63) δ δ σ , where the expression of ( δ δ ) is given by ( δ δ ) = (cid:12)(cid:12)(cid:12) det (cid:0) HH H (cid:1)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) a ( Φ ) b ( Φ ) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) N t e − j π ( N t − t − (cid:16) N t (cid:88) n t =1 e − j π ( n t − t (cid:17) (cid:19)(cid:18) e − j π Θ r − (cid:16) (cid:88) n r =1 e − j π ( n r − r (cid:17) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) a ( Φ ) b ( Φ ) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) N t e − j π ( N t − t − (cid:16) − e − j πN t Θ t − e − j π Θ t (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e − j π Θ r − (cid:16) − e − j π Θ r − e − j π Θ r (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) a ( Φ ) b ( Φ ) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) N t − (cid:16) e jπN t Θ t − e − jπN t Θ t e jπ Θ t − e − jπ Θ t (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − (cid:16) e j π Θ r − e − j π Θ r e jπ Θ r − e − jπ Θ r (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) = 4 N t (cid:12)(cid:12)(cid:12) a ( Φ ) b ( Φ ) (cid:12)(cid:12)(cid:12) (cid:0) − ρ N t (Θ t ) (cid:1)(cid:0) − ρ (Θ r ) (cid:1) . (53)Clearly, by deploying the passive beamforming structure in (22), we can maximize (cid:12)(cid:12) a ( Φ ) b ( Φ ) (cid:12)(cid:12) in (53) as (cid:0) α M M d R d T d R d T (cid:1) . By further setting M = M = M , ( δ δ ) ’s highest exponent of M is maximized as ; denoting e as δ ’s highest exponent of M and e as δ ’s highest exponentof M , we thus have the following relationship from (53) e + e = 4 . (54)While for P (cid:63) δ σ and P (cid:63) δ σ , we have δ + δ = tr (cid:0) HH H (cid:1) = (cid:88) n r ∈N r ,n t ∈N t (cid:12)(cid:12) ( H ) n r ,n t (cid:12)(cid:12) , (55)where the entries of H are given by ( H ) n r ,n t = a ( Φ ) e − j πlnλ (cid:0) ( n r − ω R ,vr ) − ( n t −
1) cos( ω T ,vt ) (cid:1) + b ( Φ ) e − j πlnλ (cid:0) ( n r − ω R ,vr ) − ( n t − ω T ,vt ) (cid:1) + ˇ c ( Φ , Φ ) e − j πlnλ (cid:0) ( n r −
1) cos( ω R ,vr ) − ( n t −
1) cos( ω T ,vt ) (cid:1) , n r ∈ N r , n t ∈ N t . (56)Hence, by adopting the same passive beamforming structure in (22) together with M = M = M , the highest exponent of M of each (cid:12)(cid:12) ( H ) n r ,n t (cid:12)(cid:12) is maximized as , in turn that of the RHSof (55) is also maximized as , and we have the following relationship from (55): max( e , e ) ≥ . (57)Next, we show that max( e , e ) > cannot hold by contradiction. Specifically, e = e needsto hold for max( e , e ) > , since the terms in δ with M ’s exponent higher than must havethe same amplitudes but opposite signs with those in δ , so that they can be cancelled out onthe LHS of (55) and the highest exponent of M on the RHS of (55) is still . In this case, e + e = 2 e > , which contradicts with (54). Therefore, we must have max( e , e ) = 4 .Particularly, recall that δ is larger than δ , we thus have e = 4 and e = 0 , i.e., δ scales with O ( M ) and δ is a constant for asymptotically large M .Note that given any feasible water-filling power allocation { P (cid:63) , P (cid:63) } , the strongest eigenchan- nel will always be allocated with a strictly positive transmit power, i.e., P (cid:63) > , and we have • If P (cid:63) > and P (cid:63) > , then C = log (cid:18) P (cid:63) δ σ + P (cid:63) δ σ + P (cid:63) P (cid:63) δ δ σ (cid:19) , (58)and the highest exponent of M inside (58)’s logarithm operator is . • If P (cid:63) = P and P (cid:63) = 0 , then C = log (cid:18) P δ σ (cid:19) , (59)and the highest exponent of M inside (59)’s logarithm operator is also .To summarize, the highest capacity scaling order with respect to an asymptotically large M is , i.e., lim M →∞ C log ( M ) = 4 , by deploying the passive beamforming structure in (22). Proposition 2is thus proved. B. Proof of Proposition 4
The MIMO channel capacity for rank-one H can be written as C = log (cid:18) P δ σ (cid:19) . (60)While the expression of δ for the rank-one H in (47) is given by δ = tr (cid:0) HH H (cid:1) = (cid:88) n r ∈N r ,n t ∈N t (cid:12)(cid:12) ( H ) n r ,n t (cid:12)(cid:12) = N t (cid:88) n r =1 (cid:12)(cid:12)(cid:12) a ( Φ ) e − j πλ ( n r − l n cos( ω R ,vr ) + (cid:0) b ( Φ )+ˇ c ( Φ , Φ ) (cid:1) e − j πλ ( n r − l n cos( ω R ,vr ) (cid:12)(cid:12)(cid:12) (a) = N t (cid:88) n r =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a ( Φ ) (cid:12)(cid:12) e j (cid:0) − πλ ( n r − l n cos( ω R ,vr )+ ϕ a (cid:1) + (cid:12)(cid:12) b ( Φ )+ˇ c ( Φ , Φ ) (cid:12)(cid:12) e j (cid:0) − πλ ( n r − l n cos( ω R ,vr )+ ϕ b + c (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) = N t (cid:88) n r =1 (cid:32)(cid:18)(cid:12)(cid:12) a ( Φ ) (cid:12)(cid:12) cos (cid:16) − πλ ( n r − l n cos( ω R ,v r ) + ϕ a (cid:17) + (cid:12)(cid:12) b ( Φ )+ˇ c ( Φ , Φ ) (cid:12)(cid:12) cos (cid:16) − πλ ( n r − l n cos( ω R ,v r ) + ϕ b + c (cid:17)(cid:19) + (cid:18)(cid:12)(cid:12) a ( Φ ) (cid:12)(cid:12) sin (cid:16) − πλ ( n r − l n cos( ω R ,v r ) + ϕ a (cid:17) + (cid:12)(cid:12) b ( Φ )+ˇ c ( Φ , Φ ) (cid:12)(cid:12) sin (cid:16) − πλ ( n r − l n cos( ω R ,v r ) + ϕ b + c (cid:17)(cid:19) (cid:33) = 2 N t (cid:16)(cid:12)(cid:12) a ( Φ ) (cid:12)(cid:12) + (cid:12)(cid:12) b ( Φ )+ˇ c ( Φ , Φ ) (cid:12)(cid:12) (cid:17) +2 N t (cid:12)(cid:12) a ( Φ ) (cid:12)(cid:12)(cid:12)(cid:12) b ( Φ )+ˇ c ( Φ , Φ ) (cid:12)(cid:12) × (cid:88) n r =1 cos (cid:16) π ( n r − r + ( ϕ b + c − ϕ a ) (cid:17) = 2 N t (cid:16)(cid:12)(cid:12) a ( Φ ) (cid:12)(cid:12) + (cid:12)(cid:12) b ( Φ )+ˇ c ( Φ , Φ ) (cid:12)(cid:12) (cid:17) +4 N t (cid:12)(cid:12) a ( Φ ) (cid:12)(cid:12)(cid:12)(cid:12) b ( Φ )+ˇ c ( Φ , Φ ) (cid:12)(cid:12) × ρ (Θ r ) cos (cid:16) π Θ r + (cid:0) ϕ b + c − ϕ a (cid:1)(cid:17) , (61)where (a) = comes from denoting ϕ a = arg (cid:0) a ( Φ ) (cid:1) and ϕ b + c = arg (cid:0) b ( Φ )+ˇ c ( Φ , Φ ) (cid:1) .To maximize δ in (61), we need to maximize | a ( Φ ) | , | b ( Φ ) | , and | ˇ c ( Φ , Φ ) | by employingthe passive beamforming structure in (22), which gives δ = 2 N t (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) αM β a γ d R d T (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) αM β b γ d R d T + α / M M β c γ γ d R d S d T (cid:12)(cid:12)(cid:12)(cid:12) (cid:19) + 4 N t (cid:12)(cid:12)(cid:12)(cid:12) αM β a γ d R d T (cid:12)(cid:12)(cid:12)(cid:12) × (cid:12)(cid:12)(cid:12)(cid:12) αM β b γ d R d T + α / M M β c γ γ d R d S d T (cid:12)(cid:12)(cid:12)(cid:12) ρ (Θ r ) cos (cid:16) π Θ r + (cid:0) ϕ b + c − ϕ a (cid:1)(cid:17) . (62)Clearly, we should optimally set the common phase shift of IRS as γ (cid:63) = β b β c for maximizing (cid:12)(cid:12) αM β b γ d R d T + α / M M β c γ γ d R d S d T (cid:12)(cid:12) , which yields ϕ a = β a β b β c and ϕ b + c = β b γ . We should further optimallyset the common phase shift of IRS as γ (cid:63) = β b (cid:0) β a β b β c − π Θ r (cid:1) for maximizing the cosine termin (62). Therefore, the MIMO channel capacity in (60) is maximized as C = log (cid:32)
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