Coverage Probability of Distributed IRS Systems Under Spatially Correlated Channels
Anastasios Papazafeiropoulos, Cunhua Pan, Ahmet Elbir, Pandelis Kourtessis, Symeon Chatzinotas, John M. Senior
11 Coverage Probability of Distributed IRS SystemsUnder Spatially Correlated Channels
Anastasios Papazafeiropoulos, Cunhua Pan, Ahmet Elbir, Pandelis Kourtessis, Symeon Chatzinotas, John M.Senior
Abstract —This paper suggests the use of multiple distributedintelligent reflecting surfaces (IRSs) towards a smarter control ofthe propagation environment. Notably, we also take into accountthe inevitable correlated Rayleigh fading in IRS-assisted systems.In particular, in a single-input and single-output (SISO) system,we consider and compare two insightful scenarios, namely, a finitenumber of large IRSs and a large number of finite size IRSs toshow which implementation method is more advantageous. Inthis direction, we derive the coverage probability in closed-formfor both cases contingent on statistical channel state information(CSI) by using the deterministic equivalent (DE) analysis. Next,we obtain the optimal coverage probability. Among others,numerical results reveal that the addition of more surfacesoutperforms the design scheme of adding more elements persurface. Moreover, in the case of uncorrelated Rayleigh fading,statistical CSI-based IRS systems do not allow the optimizationof the coverage probability.
Index Terms —Intelligent reflecting surface (IRS), coverageprobability, deterministic equivalents, beyond 5G networks.
I. I
NTRODUCTION
The advancements on metasurfaces have enabled the devel-opment of intelligent reflecting surface (IRS), being a planararray that includes a large number of nearly passive reflectingelements [1], [2]. IRS provides a smart radio environmentby realizing reflecting beamforming through its elements,which can introduce phase adjustments on the impinging wavewith different objectives such as an increase of coverage andavoidance of obstacles. Also, its construction principles allowaffordable and green transmission due to its low-cost hardwareand low energy consumption, respectively.Many works have approached the concept of IRS from thewireless communication point of view due to its appealingadvantages to achieve various tasks by adjusting the phaseshifts of the reflecting surface elements, e.g., see [2]–[9]and references therein. Among them, in [4], we observe amaximization of the sum-rate with a transmit power constraint,in [3], authors achieved maximization of the energy efficiencywith signal-to-interference-plus-noise ratio (SINR) constraints,and in [7], channel estimation, being an interesting researcharea in IRS-assisted systems due to their special characteristics,by using a deep learning approach was proposed.In particular, in the communication-theoretic direction, thestudy of the coverage probability in IRS-assisted systems hasattracted significant attention [8], [10], [11]. However, allprevious works assumed only one IRS while the performance
A. Papazafeiropoulos is with the Communications and Intelligent SystemsResearch Group, University of Hertfordshire, Hatfield AL10 9AB, U. K.,and with SnT at the University of Luxembourg, Luxembourg. C. Panis with the School of Electronic Engineering and Computer Science atQueen Mary University of London, London E1 4NS, U.K. A. Elbir iswith the EE department of Duzce University, Duzce, Turkey. P. Kourtessisand John M. Senior are with the Communications and Intelligent SystemsResearch Group, University of Hertfordshire, Hatfield AL10 9AB, U. K. S.Chatzinotas is with the SnT at the University of Luxembourg, Luxembourg.E-mails: [email protected], [email protected], [email protected]{p.kourtessis,j.m.senior}@herts.ac.uk, [email protected]. of systems aided simultaneously by multiple IRSs, offeringextended advantages such as a more robust avoidance ofobstacles and improved coverage has not been investigatedexcept [12]–[14].In parallel, the assumption of independent Rayleigh fading,which is commonly assumed for tractable performance analysisis unrealistic for IRS-assisted systems [15]. Although severalworks have accounted for the IRS correlation by acknowledgingits importance, they relied on conventional correlation models[6], which are not directly applicable in IRSs as mentioned in[15], where a practical correlation model was suggested.Against the above background, we present the only workproviding the coverage probability in closed-form for single-input and single-output (SISO) systems assisted simultaneouslyby multiple IRSs while accounting for the inevitable correlatedRayleigh fading requiring suitable mathematical manipulations.In particular, we consider two insightful design cornerstones:i) A finite number of large IRSs (the number of elementsper IRS grows large) and ii) a large number of IRSs witheach IRS having finite dimensions. Hence, contrary to [8], weestablish the theoretical framework incorporating correlatedfading into the analysis to identify the realistic potentials ofIRSs before their final implementation and we also study theperformance when the IRSs number becomes large. Comparedto [9], which also assumed distributed IRSs and correlatedfading, we focus on the coverage probability instead of theachievable rate and we rely on a more realistic correlationmodel while we account for the scenario of a large numberof IRSs, which has not been addressed before. Moreover, weprovide a methodology to optimize the reflect beamformingmatrix based on statistical channel state information (CSI)that enables optimization at every several coherence intervalsinstead of frequent optimization at every coherence interval asin works relying on instantaneous CSI.
Notation : Vectors and matrices are denoted by boldfacelower and upper case symbols, respectively. The notations ( · ) T , ( · ) H , and tr( · ) represent the transpose, Hermitian transpose,and trace operators, respectively. The expectation operator isdenoted by E [ · ] while diag ( a ) and diag ( A ) express diagonalmatrices with diagonal elements being the elements of vector a and the diagonal elements of A , respectively. Also, thenotations arg ( · ) and mod ( · , · ) denote the argument functionand the modulus operation while (cid:98)·(cid:99) truncates the argument.Given two infinite sequences a n and b n , the relation a n (cid:16) b n is equivalent to a n − b n a.s. −−−−→ n →∞ . Finally, b ∼ CN ( , Σ ) represents a circularly symmetric complex Gaussian vectorwith zero mean and covariance matrix Σ .II. S YSTEM M ODEL
We consider the smart connectivity between a single-antennatransmitter (TX) and a single-antenna receiver (RX) enabled bymeans of a set of M independent IRSs uniformly distributedin the intermediate space. We assume that each IRS, controlled a r X i v : . [ c s . I T ] F e b hrough a perfect backhaul link by the transmitter, consists ofa two-dimensional rectangular grid of N = N H N V passiveunit elements with N H elements per row and N V elements percolumn that can modify the phase shifts of impinging waves.Severe blockage effects make any direct channel unavailable.To focus on the impact of the multitude of IRSs, we rely on theassumption of perfect CSI, and thus, the results act as upperbounds of practical implementations.Let a block-fading model with independent realizationsacross different coherence blocks for the description of allchannels. In particular, we assume the existence of a directlink and M cascaded channels. The former is described by h d ∼ CN (0 , β d ) with β d expressing the path-loss. Regardingthe IRS-assisted links, h m, = [ h mn, , . . . , h mN, ] T ∈ C N × expresses the channel fading vector between the TX andthe m th IRS while h m, = [ h mn, , . . . , h mN, ] T ∈ C N × corresponds to the link between the m th IRS and the RX.Contrary to existing works, relying on independent Rayleighand Rician fading models, we consider correlated Rayleighfading . Hence, by accounting for both small-scale fading andpath-loss, we have h m, ∼ CN ( , β m, R m, ) , (1) h m, ∼ CN ( , β m, R m, ) , (2)where β m, , β m, describe the path-losses while R m, ∈ C N × N , R m, ∈ C N × N describe the spatial covariancematrices of the respective links . Herein, we account for thecorrelation model proposed in [15] as suitable for IRSs underthe conditions of rectangular IRSs and isotropic Rayleigh fading.Let the size of each IRS element be d H × d V , where d V and d H express its vertical height and its horizontal width, respectively.Then, the ( i, j ) th element of the correlation matrix R m,k ,k ∈ { , } is given by r nm,k = d H d V sinc (2 (cid:107) u n,k − u m,k (cid:107) /λ ) , (3)where u (cid:15),k = [0 , mod ( (cid:15) − , N H ) d H , (cid:98) ( (cid:15) − /N H (cid:99) d V ] T , (cid:15) ∈ { i, j } and λ is the wavelength of the plane wave.Based on a slowly varying flat-fading channel model, thecomplex-valued received signal at the RX through the networkof M IRSs is described by y = (cid:18) M (cid:88) m =1 h H m, Φ m h m, + h d (cid:19) x + n, (4)where n ∼ CN (0 , N ) is the additive white Gaussiannoise (AWGN) sample and x is the transmitted data sym-bol satisfying E [ | x | ] = P , i.e., P denotes the aver-age power of the symbol. The diagonal matrix Φ m =diag ( α m exp ( jθ m ) , . . . , α mN exp ( jθ mN )) ∈ C N × N ex-presses the response of the elements of the m th IRS. Notethat θ mn ∈ [0 , π ] , n = 1 , . . . , N and α mn ∈ (0 , expressthe phase shifts and the fixed amplitude reflection coefficientsintroduced by the corresponding IRS element. The progress onloss-less meta-surfaces allows to set α mn = 1 , which ensuresmaximum reflection [5].III. P ERFORMANCE A NALYSIS
In this section, we present the derivation of the coverageprobability when multiple IRSs are subject to correlated The extension to correlated Rician fading, having a LoS component, is thetopic of future work. The path-losses and the covariance matrices are assumed known by applyingpractical methods, e.g., see [16].
Rayleigh fading by means of the deterministic equivalent (DE)analysis. We focus on two interesting scenarios: a) a finiteset of large IRSs ( N → ∞ ); and b) a large number of IRSs( M → ∞ ) with finite size. A. Main Results
The coverage probability ¯ P c is defined as the probabilitythat the effective received SNR at the RX is larger than a giventhreshold T , i.e., ¯ P c = Pr ( γ > T ) , where γ = γ (cid:12)(cid:12)(cid:12)(cid:12) M (cid:88) m =1 h H m, Φ m h m, + h d (cid:12)(cid:12)(cid:12)(cid:12) (5)is the received SNR in the general case with correlatedfading that is obtained by using (4) and assuming coherentcommunication. Also, γ = P/N is the average transmit SNR.Under independent Rayleigh fading with instantaneous CSI,it is known that the phase configuration φ m,n = arg ( h d ) − arg (cid:0) h ∗ mn, (cid:1) arg ( h mn, ) provides the optimal SNR [2], [5].However, in the practical case of correlated Rayleigh fading,where only statistical CSI is available, we cannot directly obtainthe solution of the phase shifts. Moreover, since correlatedfading renders the exact derivation of the SNR intractable,we resort to the application of the DE analysis to derivethe approximated SNR . In Section IV, we show that thecorresponding P c provides a tight match with Pr ( γ > T ) . B. Finite M and large N analysis In this part, we assume large IRSs, as usually considered inthe existing literature to obtain the coverage probability, e.g.,see [10], [12].
Lemma 1:
The SNR of a SISO transmission, enabled by M large IRSs with correlated Rayleigh fading is approximated by γ (cid:16) γ (cid:0) B M + | h d | (cid:1) , (6)where B M = M (cid:88) m =1 β m tr( R m, Φ m R m, Φ H m ) with β m = β m, β m, . Proof:
See Appendix A.
Proposition 1:
The coverage probability of a SISO trans-mission, enabled by M large IRSs with correlated Rayleighfading for arbitrary phase shifts, is tightly approximated by P c = (cid:40) exp (cid:16) − β d (cid:16) Tγ − B M (cid:17)(cid:17) B M < Tγ B M ≥ Tγ . (7) Proof:
See Appendix B.
Remark 1:
If the aggregate contribution from the IRS-assistedchannels is larger than
T /γ , no outage is detected during thecommunication. Also, the weaker the direct signal ( β d → ,the less severe its impact is on P c and the influence of thecascaded channels becomes more pronounced. Moreover, whenthe path-losses of the cascaded links increase, i.e., β m decreases,the coverage decreases. Remark 2:
From (7), we observe that when the number ofsurfaces M increases, the coverage probability is improved. Inaddition, by increasing the size of each IRS in terms of N , P c is enhanced. Hence, the use of more IRSs or larger IRSs isproved to be beneficial for coverage. Note that the majority of works, deriving the coverage probability inIRS-assisted systems, result in approximations since they are based on CLT. emark 3:
Obviously, the coverage probability depends onthe phase shifts, which could be optimized. However, in thecase of uncorrelated fading, i.e., R m, = R m, = I N , P c becomes independent of the reflect beamforming matrices Φ m .In such case, the phase shifts of the IRS cannot be optimizedto improve the coverage. C. Large M and finite N analysis The previous analysis does not allow to examine the coveragewhen M → ∞ but N is finite. To address this scenario, let g n, = [ h n, , . . . , h Mn, ] T ∈ C M × denote the channel fadingvector between the TX and the n th elements of all IRSs (firstlink). Also, g n, = [ h n, , . . . , h Mn, ] T ∈ C N × expressesthe channel between the n th elements of all IRSs and theRX (second link). Given that IRSs are reasonably far aparteach other, we assume no correlation among them . In otherwords, we have E [ h m, h H l, ] = E [ h m, h H l, ] = N ∀ m (cid:54) = l with m = 1 , . . . , M and l = 1 , . . . , M .Notably, a correlation appears between different channelvectors at each link. Specifically, regarding the first link,let Q np, describe the correlation between the n th and p thelements across all surfaces. It can be written as Q np, = E [ g n, g H p, ]= β diag (cid:0) r np, , . . . , r np,M (cid:1) , (8)where the matrix β = diag ( β , , . . . , β M, ) ∈ C M × M isdiagonal with elements expressing the path-losses between theTX and the M surfaces. Note that β does not depend theindex m but it includes the corresponding path-losses from allIRSs. The matrix Q np, is diagonal due to the independenceamong the IRSs. Also, r np,i with i = 1 , . . . , M expresses the ( n, p ) th element of the correlation matrix of the i th IRS of thefirst link, i.e., R i, . Similarly, for the second link, we have Q np, = β diag (cid:0) r np, , . . . , r np,M (cid:1) , (9)where β = diag ( β , , . . . , β M, ) ∈ C M × M is the diagonalmatrix expressing the path-losses among the IRSs and theRX, and Q np, = diag (cid:0) r np, , . . . , r np,M (cid:1) ∈ C M × M describesthe corresponding spatial correlation. Notably, in the case ofindependent Rayleigh fading, Q np, = Q np, = O for n (cid:54) = p .As a result, the corresponding channel vectors of the first andand second links are formulated as g n, ∼ CN ( , β Q nn, ) , (10) g n, ∼ CN ( , β Q nn, ) . (11)The SNR in (5) can be rewritten in terms of a summationover the number of elements of each IRS as γ = γ (cid:12)(cid:12)(cid:12)(cid:12) N (cid:88) n =1 g H n, Ψ n g n, + h d (cid:12)(cid:12)(cid:12)(cid:12) , (12)where Ψ n = diag (exp ( jθ n ) , . . . , exp ( jθ Mn )) ∈ C M × M . Lemma 2:
The SNR of a SISO transmission, enabled bya large number of finite size IRSs with correlated Rayleighfading is approximated by γ (cid:16) γ (cid:0) B N + | h d | (cid:1) , (13) Note that not only this assumption is quite reasonable but the modeling ofa potential correlation would require the conduct of measurement campaigns,which are currently unavailable. where B N = N (cid:88) n =1 N (cid:88) p =1 tr (cid:0) Q np, Ψ n Q np, Ψ H p (cid:1) . Note that Ψ = diag ( Ψ , . . . , Ψ N ) ∈ C MN × MN , i.e., Ψ is a block diagonalmatrix. Proof:
See Appendix C.
Proposition 2:
The coverage probability of a SISO trans-mission, enabled by a large number of finite size IRSs withcorrelated Rayleigh fading for arbitrary phase shifts, is tightlyapproximated by P c = (cid:40) exp (cid:16) − β d (cid:16) Tγ − B N (cid:17)(cid:17) B N < Tγ B N ≥ Tγ . (14) Proof:
The proof follows similar lines with the proof ofProposition 1.
Remark 4:
We observe that (14) has a similar expressionwith (7). However, the main characteristic of (14) is that itis written in terms of a double summation expressing thecorrelation among the IRS elements instead of one summationin (7). Notably, if the correlation matrix is identical acrossdifferent IRSs, Q np, and Q np, are scaled identity matricesbut the coverage will always be dependent on the phase shiftsdue to the contributions from the off-diagonal terms of theIRSs. Furthermore, we observe a similar dependence from thepath-loss of the direct signal and the number of surfaces andtheir elements, i.e., their increase improves the coverage. Also,under uncorrelated Rayleigh fading conditions, P c does notdepend on the phases, and thus, cannot be optimized. D. Reflecting beamforming optimization
Both Propositions (1) and (2) are described by a similarexpression in terms of a trace that includes the reflectingbeamforming matrices. Hence, their optimization followssimilar steps up to a point. Specifically, to achieve maximum P c , we formulate the optimization problem, relying on thecommon assumption of infinite resolution phase shifters, as ( P
1) max Φ P c s . t | φ mn | = 1 , m = 1 , . . . , M and n = 1 , . . . , N, (15)where P c is given by (7) or (14) and φ mn = exp ( jθ mn ) .The optimization problem ( P is non-convex with respectto the reflect beamforming matrix while having a unit-modulusconstraint regarding φ mn . Use of projected gradient ascent untilconverging to a stationary point can provide a direct solution.In particular, since each surface has a similar solution, wefocus on the m th IRS. At the i th step, we assume the vectors s m,i = [ φ im , . . . , φ imN ] T , which include the phases at thisstep. The next iteration increases P c until its convergence byprojecting the solution onto the closest feasible point based on min | φ mn | =1 ,n =1 ,...,N (cid:107) s m − ˜ s m (cid:107) satisfying the unit-modulusconstraint concerning φ mn with ˜ s m,i +1 = s m,i + µ q m.i , (16) s m,i +1 = exp ( j arg (˜ s m,i +1 )) . (17)Note that µ expresses the step size computed at each iteration bymeans of the backtracking line search [17] while q m,i denotesthe adopted ascent direction at step i with q m,i = ∂P c ∂φ mn ,obtained by Lemma 3 below. Algorithm 1 provides an outlineof the proposed algorithm for Proposition 1 and 2 by setting ˜ Φ = Φ m and ˜ Φ = Ψ n , respectively. lgorithm 1 Projected Gradient Ascent Algorithm for the IRSDesign1.
Initialisation : s m, = exp ( jπ/ N , ˜ Φ = diag ( s m, ) , P = f (cid:16) ˜ Φ (cid:17) given by (15); (cid:15) > Iteration i : for i = 0 , , . . . , do3. [ q m,i ] n = ∂P c ∂ s ∗ m,i , where ∂P c ∂ s ∗ m,i is given by Lemma 3;4. Find µ by backtrack line search ( f (cid:16) ˜ Φ (cid:17) , q m,i , s m,i ) [17];5. ˜ s m,i +1 = s m,i + µ q m,i ;6. s m,i +1 = exp ( j arg (˜ s m,i +1 )) ; ˜ Φ i +1 = diag ( s m,i +1 ) ;7. P i +1c = f (cid:16) ˜ Φ i +1 (cid:17) ;8. Until (cid:107) P i +1c − P i c (cid:107) < (cid:15) ; Obtain ˜ Φ ∗ = ˜ Φ i +1 ;9. end for Lemma 3:
The derivative of the coverage probability withrespect to s ∗ m,i is given by ∂P c ∂ s ∗ m,i = β m β d P c diag ( R m, Φ m R m, ) , Prop . P c N (cid:88) p =1 c p , Prop . (18)when B i < Tγ , i = M, N . Otherwise, it is zero. Note that c p = (cid:2) r p,m r p,m φ m, n , . . . , r Np,m r Np,m φ m, N (cid:3) T . Proof:
See Appendix D.IV. N
UMERICAL R ESULTS
Relying on a Cartesian coordinate system, we consider a cell,where the TX is located at the origin and the RX at (60 , .Also, we assume M = 15 IRSs being uniformly distributedbetween them while each IRS is deployed with N = 225 elements unless otherwise specified. The size of each IRSelement is given by d H = d V = λ/ . The spatial correlationmatrix is given by (1). The large-scale fading coefficientsbetween the TX and the RX are given by β m,i = G t + G r +10 ν e log ( d i / − . , where i ∈ { , } while β d is givensimilarly. Also, the path-loss exponents are ν = ν = 2 and ν d = 3 . . Moreover, we have G t = 3 . dBi and G r = 1 . dBi,the system bandwidth is MHz; the carrier frequency is GHz and the noise variance is − dBm with the noisefigure being dB. Note that the transmitter power is dBm.Monte-Carlo (MC) simulations verify the analytical results andcorroborate that the DE analysis provides tight approximationsas has been already shown in the literature, e.g., see [18].Fig. 1 shows the coverage probability versus the target ratein the scenario described by Proposition 1 . By increasingthe number of elements in each IRS, P c increases. Also, theaddition of more IRSs (an increase of M ) through B M con-tributes to the observation of less outage during communication.Moreover, if no correlation is assumed, P c is lower because itbecomes independent of the reflect beamforming matrix andcannot be optimized. Notably, if the impact from the directsignal through β d becomes weaker, the coverage decreases,and the variation regarding the number of elements as well ascorrelated Rayleigh fading have a greater impact on P c sincethe relevant gaps are larger.In Fig. 2, we depict the coverage probability versus the targetrate by accounting for a large number of surfaces, i.e., we shedlight on the setting referring to Proposition 2. Herein, we notice The theoretical analysis of this proposition relies on finite N but weconsider N ≥ , which is common for practical IRS implementations. Fig. 1. Coverage probability of a SISO system with correlated Rayleigh fadingassisted by M IRSs each having a large number of elements ( N → ∞ ) versusthe target rate T (analytical results and MC simulations). that the coverage is improved as the number of IRSs increases.Similarly, if we increase the number of elements per IRS, weobserve a further improvement. Notably, a comparison betweenFigs. 1 and 2 reveal that an analogous increase concerningthe number of IRSs results in a larger improvement of thecoverage compared to increasing the number of elements perIRS. Specifically, in Fig. 2, it is shown that when M increasesfrom to b/s/Hz ( increase), P c starts decreasingfrom full coverage when T = 1 .
47 dB and . , respectively.On the other hand, in Fig. 1, we observe that for a similarincrease concerning the number of elements per IRS, i.e., when N increases from to , the coverage is much lower. Fig. 2. Coverage probability of a SISO system with correlated Rayleigh fadingassisted by a large number of IRSs ( M → ∞ ) each having N elements versusthe target rate T (analytical results and MC simulations). V. C
ONCLUSION
In this paper, we derived the coverage probability of a SISOsystem assisted with multiple IRSs under the unavoidableconditions of correlated Rayleigh fading. We considered twodistinct scenarios: a finite multitude of large IRSs and a largenumber of finite IRSs. Especially, we managed to derive andptimize the coverage probability with respect to the phaseshifts of the IRS elements in both cases. The results enabledus to show that it is more beneficial to increase the numberof IRSs instead of increasing their elements. Future workson coverage of distributed IRSs should take into account thedesign of multi-user transmission with multiple antennas, andpossibly, under Rician fading conditions.A
PPENDIX AP ROOF OF L EMMA N . Then, we have N γ = γ N (cid:18) M (cid:88) m =1 (cid:12)(cid:12)(cid:12)(cid:12) h H m, Φ m h m, (cid:12)(cid:12)(cid:12)(cid:12) + | h d | + 2Re (cid:32) h ∗ d M (cid:88) m =1 h H m, Φ m h m, (cid:33) + M (cid:88) m =1 M (cid:88) j =1 n (cid:54) = m h H m, Φ m h m, h H n, Φ H n h n, (cid:19) (19) (cid:16) γ N (cid:32) M (cid:88) m =1 β m, β m, tr( R m, Φ m R m, Φ H m )+ | h d | (cid:33) , (20)where, in (20), we have used [18, Lem. 4] for the first, third,and the fourth terms. Especially, the third and fourth terms in(19) vanish as N → ∞ due to the independence among the M IRSs and between the two links, respectively.A
PPENDIX BP ROOF OF P ROPOSITION P c = Pr (cid:18) | h d | > Tγ − B M (cid:19) , (21)where in (21), we have used the SNR from (6). Given that | h d | is exponentially distributed with rate parameter β d , i.e., | h d | ∼ Exp(1 /β d ) , we obtain the first branch in (7), if B M < Tγ . Otherwise, P c = 1 , and we conclude the proof.A PPENDIX CP ROOF OF L EMMA γ with M , we have M γ = γ M (cid:18) N (cid:88) n =1 (cid:12)(cid:12)(cid:12)(cid:12) g H n, Ψ n g n, (cid:12)(cid:12)(cid:12)(cid:12) +2Re (cid:32) h ∗ d N (cid:88) n =1 g H n, Ψ n g n, (cid:33) + | h d | + N (cid:88) n =1 N (cid:88) p =1 p (cid:54) = n g H n, Ψ n g n, g H p, Ψ H p g p, (cid:19) (22) = γ M (cid:18) N (cid:88) n =1 N (cid:88) p =1 tr (cid:0) Q np, Ψ n Q np, Ψ H p (cid:1) + | h d | (cid:19) , (23)where the second term in (22) vanishes as M → ∞ due to theindependence between the two links. Application of [18, Lem.4] at the first and fourth terms in (22) gives (23) after a directcombination of the two resultant traces. A PPENDIX DP ROOF OF L EMMA ∂P c ∂ s ∗ m,i = ∂P c ∂B i ∂B i ∂ s ∗ m,i , (24)where i = M and i = N correspond to Propositions 1 andPropositions 2, respectively. The first-order derivative in (24)becomes ∂P c ∂B i = 1 β d P c . (25)By taking into account the expression of B M , we have ∂B M ∂ s ∗ m,i = β m ∂ ( diag ( R m, Φ m R m, )) T s ∗ m,i ∂ s ∗ m,i (26) = β m diag ( R m, Φ m R m, ) , (27)where we have used the property tr ( A diag ( s m )) =( diag ( A )) T s m . In the case of ∂B N ∂ s ∗ m,i , we obtain ∂B N ∂ s ∗ m,i = N (cid:88) p =1 ∂ ( diag ( Q np, Ψ n Q np, )) T s ∗ m,i ∂ s ∗ m,i (28) = N (cid:88) p =1 diag ( Q np, Ψ n Q np, )= β m N (cid:88) p =1 r p,m r p,m φ m, ... r Np,m r Np,m φ m,N . (29)By substituting (27) or (29) together with (25) into (24), weobtain the desired results.R EFERENCES[1] Q. Wu and R. Zhang, “Towards smart and reconfigurable environment:Intelligent reflecting surface aided wireless network,”
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