arXiv:2004.13318v4 [cs.IT] 15 May 2021 EETASCIN NWRLS COMMUNICATIONS WIRELESS ON TRANSACTIONS IEEE nelgn eetn ufc:Sse oeigand Modeling System Surface: Reflecting Intelligent sgnrlysbpia ntrso hogptprui cost unit per throughput of while terms in costs, suboptimal individual densit generally IRS/BS their zero is (i.e., given IRS without cost network conventional deployment total a an exists ratio there that unveiled is it Moreover, h EEb edn eus [email protected] to b request must a purposes other sending any by for IEEE material the this use to permission niern,Ntoa nvriyo igpr,Singapore Singapore, 3610 of China [email protected]). University Electri University, National of Department Xiamen Information the Engineering, with is Education), Marine Zhang R. of [email protected]). and (Ministry Communication ogy Acoustic Underwater Nation R-26 and the R-261-518-005-720 and Grant Resea Research 20720190008), under Fundamental (No. Singapore the Universities Scienc 2019J05002), Central Natural (No. the the Province 61771017), No. Fujian and of 61801408 (No. China of eut nonly in results rwho ieesdt aebe otnal prigen- communi- spurring wireless innovative continually and been new inventing have in data thusiasm wireless of growth iepsiewrls ewr,sailtruhu,stoch analysis. throughput, performance spatial ometry, network, wireless tive/passive yrdntoktruhu gis h ovninlnetwor conventional which the IRS, against enhanci demonstr in without throughput and IRSs network analysis distributed deploying hybrid our of validate effectiveness numeric to Extensive the presented large. becomes are elements effects results IRS hardening of channel spatially number exhibit under the network which the channels, interf in and related need IRSs signal analysis distributed the performance by our both reflected of only, the distributions BSs power characterizing active the derive with on networks performance works wireless on key of prior based pertinent therein to BSs/IRSs other locati Compared random deployed as and the fading hybrid well of channel both new as over thro averaged a downlink metrics spatial paper achievable the its (BSs this in characterize stations and in base future. solutio IRSs, active the study cost-effective passive both in comprising we a network growth problem, be wireless capacity this can sustainable networks address their wireless achieve in to deplo IRSs large-scale the whether of problem syst open wireless an passiv remains IRS-aided still link-le various low-cost the for in optimization massive advances formance recent tuning Despite elements. via fav a reflecting constructing channels spectral by networks, the communication wireless improve of substantially efficiency energy to paradigm promising .Lui ihteSho fIfrais n e aoaoyo Laboratory Key and Informatics, of School the with is Lyu J. oyih 22 EE esnlueo hsmtra spermi is material this of use Personal IEEE. ©2021 Copyright hswr a upre npr yteNtoa aua Scien Natural National the by part in supported was work This yrdAtv/asv ieesNtokAddby Aided Network Wireless Active/Passive Hybrid h rlfrto fmbl plctosadexplosive and applications mobile of proliferation The Terms Index Abstract htmxmzstehbi ewr hogptsbetto subject throughput network hybrid the maximizes that Itlietrflcigsrae(R)i e and new a is (IRS) surface reflecting —Intelligent agnlyicesdinterference increased marginally Itlietrflcigsrae yrdac- hybrid surface, reflecting —Intelligent infiatybot h inlpower signal the boosts significantly .I I. NTRODUCTION inbnLyu, Jiangbin efrac Analysis Performance pia R/Sdensity IRS/BS optimal tcatcgeometry stochastic ebr IEEE Member, ntenetwork. the in a n Computer and cal lUiest of University al bandfrom obtained e . 3-000-E86-112. 153(email: 117583 td However, tted. c ud for Funds rch eFoundation ce Foundation e 5(e-mail: 05 si ge- astic e per- vel Technol- ratio) y ughput m,it ems, gthe ng erence orable yment and ) when cor- to s ons . but the ate nd To al k n e f . n u Zhang, Rui and , civ t utial aaiygot ntefuture. t the solution in cost-effective growth a capacity be sustainable its would achieve network nodes/components active wireless more the and in exis more mmWave the adding whether by fre- at doubtful paradigm is operating radio it when Therefore, costly bands. especially requiring frequency chains, antennas (RF) active mounti quency more as well with as base network more the them of in deployment relays the and/or (BSs) to stations due consumptio cost, energy hardware more higher increasingly and incurred also Although net they wireless SE, [1]. network the enhanced significantly communication ultra-dense technologies (mmWave) these include wave an decade (MIMO), millimeter multiple-output last technologies multiple-input massive wireless the (UDN), prominent in some proposed others, Among deployment/operati affordable cost. an spectral/en at higher yet (SE/EE) achieve efficiency would that technologies cation o-rhgnlmlil ces(OA 1][7,physica [14], [15]–[17], [13] (NOMA) access system multiple (OFDM) as non-orthogonal orthogona multiplexing such IRS-aided division setups, [6]–[12], frequency different design under beamforming and/or passive aspects s various wireless IRS-aided from investigating in recently and interest flexibility of high with as networks well wireless as in cost scalable deployed compatibility. and densely low be a to can at need IRS the such, standar operating without As and systems infrastructure seamless current WiFi and their or modify ceilings), cellular and object into walls existing integrated the (e.g., from environment removed the or in to attached easily Moreove be [2]. can nee (SIC) the cancellation without self-interference (FD) costly full-duplex in operat- efficiently t yet spectral consumption leading energy ing and thus cost reflection, hardware wave lower transmission/recept much passive signal leverages for simply chain but RF active any does IRS require relaying/beamforming, active conventional Co the cancellation. to beamform and/or nulling passive collaboratively interference (3D) spatial thus ing, three-dimensional as them, des such various and/or by achieve functions to amplitude reflected propagation the signal signal elements the alter reconfiguring the reflecting to of of dynamically passive tuned consisting phase low-cost be surface of can planar that number a uncontrollabl massive is and a random IRS be Specifically, to [2]–[5]. deemed traditionally environm is propagation new that wireless the on the based reconfiguring solution of promising concept a as emerged recently has otcl hscalne nelgn eetn ufc (I surface reflecting intelligent challenge, this tackle To h peln datgso R aeatatdagetdeal great a attracted have IRS of advantages appealing The elw IEEE Fellow, mpared ystems s[2]. ds ,IRS r, work ergy onal of d ting RS) ired not ion ent ng ly o o d n e s 1 - l l IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 2 layer security [18]–[21], wireless information and power trans- fading effect. To our best knowledge, the modeling of a general fer [22]–[24], and so on. The above works on IRS-aided wire- multi-cell hybrid wireless network aided by randomly located less systems mainly aim to optimize the system performance IRSs and the characterization of the distribution of the users’ at the link level with one or more IRSs at fixed locations, achievable signal-to-interference-plus-noise ratio (SINR) as which show that the IRS-aided system can achieve significant well as the spatial throughput of the hybrid network have not energy efficiency [12] and/or spectral efficiency improvement been investigated yet in the literature. over the traditional system without IRS, with optimized IRS Motivated by the above, in this paper, we model a hybrid reflection coefficients. In [25], the authors investigate a multi- active/passive wireless network under the general multi-cell user system aided by multiple intelligent surfaces (equivalent setup and derive the distributions of the signal power, inter- to a single large IRS) co-located at a random location in ference power, and thereby the users’ achievable SINR in the the network. However, since IRS typically serves users in its network, with the ultimate goal of characterizing the spatial proximity, distributed IRSs should be deployed in the network throughput of the network, defined as the achievable rate per to serve distant groups of users and thereby boost the network user equipment (UE) averaged over both the wireless channel throughput. Motivated by this, in our prior work [26], the fading and the random BS/IRS locations. Thus, this work is a spatial throughput of a single-cell multi-user system aided by substantial extension of our prior work [26] under the single- distributed IRSs located at random locations is characterized, cell setup to the more general multi-cell setup. We focus on which is compared favorably with the conventional system the downlink communication from the BSs to the UEs while aided by distributed relays but with significantly reduced active the proposed analytical framework can be similarly extended antennas, under their respectively optimized deployment. to the uplink communication, which is left for our future Furthermore, for large-scale deployment of IRSs in future work. Compared to other prior works on characterizing the wireless systems, one critical issue is the modeling, design performance of wireless networks with active BSs only (see, and performance characterization of the IRS-aided multi-cell e.g., [33] and references therein), our analysis needs to derive hybrid wireless network comprising both distributed active the power distributions of both the signal and interference BSs and passive IRSs subjected to the inter-cell interference. reflected by distributed IRSs in the network under spatially There have been some recent works (e.g., [27]–[32]) along correlated channels, which exhibit channel hardening effects this line. Considering a finite number of co-channel/interfering when the number of IRS elements becomes large. Our main BSs, joint active/passive beamforming design with a cell-edge contributions are summarized as follows. IRS is investigated for the users’ weighted-sum-rate maximiza- tion [27] or minimum-rate maximization [28], respectively. In • First, we model the random BS/IRS locations by indepen- [29], the authors consider the quasi-static phase-shift design dent homogeneous Poisson point processes (HPPPs) and of one IRS in the presence of one interfering BS, based propose a practical UE-to-IRS association rule when they on the statistical channel state information (CSI) assuming are in close proximity. Then, for a typical BS-IRS-UE link given BS/IRS locations. In [30], the sum rate of multiple with their given locations, we derive its channel power transmit-receive (Tx-Rx) pairs aided by multiple distributed distribution in terms of the number of reflecting elements IRSs at given locations is maximized. However, the above per IRS, denoted by N, based on which the mean channel works only consider a given number of BSs and IRSs at fixed power is shown to scale with N in the order of O(N 2) locations, but do not investigate the impact of their spatial and O(N) for the cases with reflect beamforming by the random locations on the performance of large-scale hybrid associated IRS and random scattering by non-associated active/passive wireless networks. IRSs, respectively. Furthermore, we define the network Due to practical space constraints and heterogeneous/dense coverage probability and spatial throughput in terms of BS deployment, modern cellular networks typically exhibit an key system parameters including the BS/IRS densities and increasing degree of spatial irregularity, for which the con- network loading factor. ventional grid-based BS deployment models become no more • Next, we propose an analytical framework for the IRS- suitable. As field trials are costly and system-level simulations aided hybrid network based on stochastic geometry, and are time-consuming, stochastic geometry has been extensively address its new challenges. In particular, for a typical UE applied as a tractable analytical tool to model the spatial 0 with randomly distributed IRSs nearby, its nearest IRS 0 distribution of heterogeneously/densely/irregularly deployed (and hence the IRS 0-UE 0 distance d0) typically has the wireless nodes, which provides meaningful performance lower dominant impact on the mean signal or interference power bounds and scaling laws for practical wireless networks [33] compared to other (farther) IRSs, under practical values of [34]. Based on stochastic geometry, the authors in [31] model the IRS density. Moreover, with reflect beamforming by the IRSs by boolean line segments in a large-scale network the associated IRS, the signal link exhibits channel hard- and derive the probability that a given IRS is capable of ening when N becomes large, while the extent of channel providing an indirect path for a given Tx-Rx pair (i.e., the hardening varies with the distance d0, rendering it difficult reflection probability). The authors in [32] further exploit to characterize the signal power distribution and thus the the deployment of IRSs for providing indirect line-of-sight SINR distribution. To overcome this difficulty, we propose (LoS) paths for blocked links, thus improving the coverage to approximate the conditional signal power distribution probability in a large-scale network. However, these two works by the Gamma distribution, whose shape parameter kS do not consider the inter-cell interference and the small-scale specifies the extent of channel hardening conditioned on IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 3
d0. The conditional SINR distribution is then obtained in BS 0 terms of the interference power Laplace transform and its signal derivatives up to integer-order kS. Moreover, we propose an interpolation method for non-integer kS, and apply the IRS j reflect normal approximation of the signal power in the case with scatteringg beamforming large kS in order to reduce the computational complexity. IRS 0 These new analytical methods jointly yield accurate and UE 0 efficient characterization of the network SINR distribution and hence its spatial throughput. • Finally, extensive numerical results are provided to validate our analytical results. It is found that increasing IRS density interference in a hybrid wireless network can significantly enhance the signal power but with only marginally increased interfer- BS m ence, thus greatly improving its throughput as compared Fig. 1: IRS-aided multi-cell wireless network in the downlink. to the traditional wireless network with active BSs only, especially when the BS density, network loading factor or N is large. Moreover, it is unveiled that there exists an division multiple access (OFDMA) scheme and assume that optimal IRS/BS density ratio ζ∗ for maximizing the spatial the transmission bandwidth and each time slot are equally throughput of the new hybrid network under a given total divided into orthogonal resource blocks (RBs), each randomly deployment cost, where ζ∗ is shown to increase with the assigned by a BS to one of its served UEs, over which the BS/IRS cost ratio and the network loading factor, while the channel is assumed to be frequency-flat and constant, while the conventional network without IRS (i.e., zero IRS/BS density channels may vary over different frequency bands or different ratio), or the hybrid network with excessively large ζ (where time slots. We assume that the network has a homogeneous the BS density is too low to provide enough signal power for traffic load, where all BSs have a common loading factor p effective IRS passive beamforming), is generally suboptimal (0 < p ≤ 1), i.e., each of the RBs is active with probability in terms of throughput per unit cost. Furthermore, it is p independently. Consider one typical RB used by the typical shown that the maximum spatial throughput of the hybrid UE 0 located at the origin, which is associated with its nearest network with the optimal IRS/BS density ratio ζ∗ grows BS 0 with distance l0, as shown in Fig. 1. As a result, the BSs ′ almost linearly with the total cost, thus providing a new that transmit on the same RB form a thinned HPPP ΛB with ′ , and cost-effective approach to achieve sustainable capacity density λB pλB. growth for future wireless networks. We consider that distributed IRSs are deployed to assist the BS-UE communications in the network. Assume that all The rest of this paper is organized as follows. The new IRSs are of the same height equal to m,2 while the model of the proposed hybrid wireless network is presented HI IRSs’ horizontal locations are modeled by a 2D HPPP in Section II. The distributions/mean values of the signal ΛI (independent of ) on the ground plane with given density and interference powers are then characterized in Section III ΛB IRSs/m2. Denote the set of IRS horizontal locations as and Section IV, respectively. Next, the SINR distribution and λI , w R2 , where w is the 2D coordinate the network spatial throughput are obtained in Section V. W { j ∈ |j ∈ ΛI} j of an IRS . Denote as the horizontal distance Numerical results are provided in Section VI. Finally, we j ∈ ΛI dj between UE 0 and IRS , as shown in Fig. 1. Since the IRS conclude the paper in Section VII. j typically provides signal enhancement via reflect beamforming in a local region [26], we consider the practical scenario II. SYSTEM MODEL where UE 0 is associated with its nearest IRS 0 for dedicated In this paper, we consider an IRS-aided multi-cell wire- reflect beamforming, if IRS 0 is within a certain threshold 3 less network shown in Fig. 1, and focus on the downlink distance D1, i.e., d0 ≤ D1. On the other hand, if there communication from the BSs to UEs. Assume that the BSs is no IRS within D1, then UE 0 is served by BS 0 only 4 are of the same height equal to HB meters (m), while the without any associated IRS. Moreover, for the purpose of BSs’ horizontal locations are modeled by a 2-dimensional exposition, we assume that each IRS is always on and reflects (2D) HPPP ΛB on the ground plane with given density 2 2 λB BSs/m . Assume that the UE locations follow another Our analytical framework is applicable to any given height of BSs/IRSs, independent HPPP on the ground plane,1 such that we can which can also be extended to account for their random heights by employing the 3-dimensional point process where the height of each BS/IRS is randomly focus on one typical UE 0 in this HPPP to analyze the average set within a certain range. 3 UE performance without changing the location distribution The threshold D1 is practically set such that each IRS serves a finite of other UEs according to the Slivnyak’s theorem [33]. To number of UEs in its neighborhood only. 4To implement this, each IRS controller can sense the nearby UEs and facilitate our analysis, we consider the orthogonal frequency decide to associate with them or not based on their signal strengths, then send the associated UEs’ identification back to the BS (via a separate control 1The HPPP assumption implies the uniformly random UE distribution, link) for RB allocations. Note that such IRS-UE associations change only which is more accurate for UEs of homogeneous distribution and/or higher when the UEs move in/out from the coverage of each IRS, which usually mobility, but in general can serve as a good baseline to evaluate the network happens not so frequently for typical IRS-aided scenarios (e.g., hotspot with performance with heterogeneous user distribution/mobility in practice. local users). IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 4 the received wave at all time, regardless of whether there is any framework by treating it as equivalent random perturbation in UE associated with it.5 As a result, UE 0 receives the reflected the locations of the BSs/IRSs (see Sections III.G and VI.A in signal/interference by all IRSs (including its associated IRS 0 [35] and the references therein), which is ignored in this work if any). To model the randomly reflected signal/interference for simplicity. The channel power gain from BS m to UE 0 by non-serving IRSs accurately while maintaining analytical is thus given by Approximation 1 tractability, we adopt the that only the IRSs 2 , 2 2 −α/2 |hd,m| gd,mξd,m = β(lm + HB) ξd,m, (2) within a sufficiently large threshold distance D2 (D2 > D1) , from UE 0, denoted by the set J {j ∈ ΛI|dj ≤ D2}, will where gd,m is the average channel power gain, ξd,m accounts contribute the signal/interference to it. Finally, to maximize for channel fading, lm denotes the BS-UE horizontal distance, the passive beamforming gain of the IRS to each served UE, 4πfc −2 and β = ( c ) denotes the average channel power gain at we assume that its served UEs are assigned in orthogonal- a reference distance of 1 m based on the free-space path-loss time RBs, i.e., time division multiple access (TDMA) or time model, with fc denoting the carrier frequency, and c denoting sharing is adopted for the UEs served by the same IRS.6 the speed of light. Similarly, the channel power gains from BS m to the n-th element of IRS j, and from the latter to UE 0 A. Channel Model are given by
Assume for simplicity that the BSs and UEs are each (j) 2 (j) (j) 2 2 −α/2 (j) |h | , g ξ = β r + (HB − HI) ξ , equipped with a single antenna, while each IRS has N i,m,n i,m i,m,n m,j i,m,n (3) reflecting elements. The baseband equivalent channels from and −α/2 BS m to IRS j, from IRS j to UE 0, and from BS m to UE 0 |h(j)|2 , g(j)ξ(j) = β d2 + H2 ξ(j), (4) are denoted by h(j) , [h(j) , ··· ,h(j) ]T ∈ CN×1, h(j) , r,n r r,n j I r,n i,m i,m,1 i,m,N r (j) (j) T CN×1 C 7 where g(j) and g(j) denote the average channel power gains, [hr,1 , ··· ,hr,N ] ∈ , and hd,m ∈ , respectively, i,m r C T (j) (j) where denotes the set of complex numbers and [·] denotes ξi,m,n and ξr,n account for channel fading, rm,j and dj denote (j) , (j) (j) 9 the matrix transpose. Let φ [φ1 , ··· , φN ] and further the BS-IRS and IRS-UE horizontal distances, respectively. (j) iφ(j) iφ(j) denote Φ , diag{[e 1 , ··· ,e N ]} (with i denoting the imaginary unit) as the phase-shifting matrix of IRS j, (j) B. BS-IRS-UE Channel Power Statistics where φn ∈ [0, 2π) is the phase shift by element n of the IRS on the incident signal,8 and diag{x} denotes a diagonal In this subsection, we derive the BS-IRS-UE cascaded matrix with each diagonal element being the corresponding channel power statistics, which is new for the IRS-aided hybrid element in x. Each element of the IRS receives the superposed network and essential to our subsequent performance analysis multi-path signals from the BS, and scatters the combined (j) (j) for it. Assume that the channels hd,m, hi,m,n and hr,m,n, signal with adjustable phase as if from a single point source. m ∈ ΛB, j ∈ ΛI, n = 1, ··· ,N are independent. For the Therefore, the cascaded BS-IRS-UE channel can be modeled cascaded BS m-IRS j-UE 0 link in (1), the channel reflected as a concatenation of three components, namely, BS-IRS link, through each element n is given by IRS reflecting with phase shifts, and IRS-UE link, given by (j) (j) i (j) , (j) φn [6] hir,m,n hi,m,nhr,ne i (j) ∠ (j) ∠ (j) N (j) (j) φn + hi,m,n+ hr,n (j) (j) (j) i (j) = |hi,m,n||hr,n|e , (5) , h T Φ(j)h(j) (j) φn hir,m [ i,m] r = hi,m,nhr,ne ,m ∈ ΛB, nX=1 where the channel amplitude |h(j) | , |h(j) ||h(j)| resem- (1) ir,m,n i,m,n r,n (j) (j) (j) i (j) ∠ (j) , , φn bles a doubly-faded RV while the channel phase h where hir,m,n hi,m,nhr,ne denotes the BS m-IRS j-UE ir,m,n (j) ∠ (j) ∠ (j) 0 channel reflected by element n, n =1, ··· ,N. φn + hi,m,n + hr,n is adjustable via controlling the phase (j) For the BS-IRS, IRS-UE and BS-UE links, we assume a shift φn exerted by IRS j. simplified fading channel model without shadowing, which dist. For simplicity, assume that ξd,m = ξ ∼ Exp(1) is an consists of distance-dependent path loss with path-loss expo- exponential random variable (RV) with unit mean accounting nent α ≥ 2 and an additional random term ξ accounting for for the small-scale Rayleigh fading. Therefore, the ampli- small-scale fading. Note that the shadowing effect can also tude |hd | follows the Rayleigh distribution [36] with scale be incorporated into the stochastic geometry based analytical ,m parameter gd,m/2, denoted by R gd,m/2 , while hd,m 5The results in this paper can be extended to the general case where each follows thep circularly symmetric complex p Gaussian (CSCG) IRS is independently on or off with a certain probability. Nevertheless, we distribution [36] with mean zero and covariance gd,m, denoted consider that all IRSs are on to characterize the worst-case interference. by . We also assume Rayleigh faded channel for 6 CN (0,gd,m) It is shown in [17] that for IRS-aided multiple access, the TDMA scheme (j) (j) dist. is in general superior over the FDMA scheme due to the hardware limitation of the BS-IRS and IRS-UE links, i.e., ξi,m,n, ξr,n = ξ ∼ Exp(1), IRS passive reflection, which can be made time-selective, but not frequency- in order to investigate the worse-case propagation condition selective [2]. for the IRS and hence characterize the achievable performance 7The subscripts “i”, “r” and “d” represent the BS-to-IRS channel, IRS- reflected channel (i.e., the IRS-to-UE channel), and direct BS-to-UE channel, respectively. 9For the purpose of exposition, we consider far-field propagation for all 8 In this paper, we assume (maximum) unit amplitude for each reflection links, and accordingly assume HB ≥ 1 m and HI ≥ 1 m, which also avoid coefficient to maximize the IRS beamforming gain to its served UE [6]. unbounded power gain when the horizontal distance lm or dj becomes zero. IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 5 lower bound for the IRS-aided hybrid network.10 Therefore, The average BS 0-IRS 0-UE 0 signal power is the second (j) (j) (j) (j) (0) moment of |hir | and thus given by we have |hi,m,n| ∼ R gi,m/2 and |hr,n| ∼ R gr /2 . ,0 q q As a result, the channel amplitude (j) in (5) is a (0) , E (0) 2 E (0) 2 (0) |hir,m,n| gir,0 {|hir,0| } = {|hir,0|} + var{|hir,0|} double-Rayleigh RV. Note that for a double-Rayleigh dis- 2 2 π π (0) tributed RV Y = X X with independent X ∼ R(δ ) and = N 2 + 1 − N g g(0), (12) 1 2 1 1 16 16 i,0 r X2 ∼ R(δ2), its mean and variance are respectively given by [37] which is proportional to the average channel power product E , (0) (0) , π2 2 {Y } πδ1δ2/2, (6) gi,0 gr , with the beamforming gain coefficient Gbf 16 N + π2 2 , 2 2 2 1 − 16 N that grows with N in the order of O(N ). var{Y } 4δ1δ2 (1 − π /16). (7) On the other hand, for any IRS j that does not provide Therefore, the mean and variance of the channel amplitude reflect beamforming for UE 0 (including IRS 0 if the dis- (j) |hir,m,n| are respectively given by tance d0 > D1), it scatters the incoming signal from BS m without passive beamforming, thus resulting in uniformly E (j) , π (j) (j) (j) hir gi gr , (8) random channel phase ∠h due to the uniformly random ,m,n 4 q ,m ir,m,n phases ∠h(j) and ∠h(j). In this case, we have the following ( j) , 2 (j) (j) i,m,n r,n var hir,m,n (1 − π /16)gi,mgr . (9) proposition.
In the case where IRS 0 provides reflect beamforming ser- Proposition 2. With random scattering by IRS j, the BS m- vice for the desired signal from BS 0 to UE 0 (i.e., d0 ≤ D1), IRS j-UE 0 channel in (1) for practically large N can be ∠ (0) (0) we assume that the cascaded channel phase hi,0,nhr,n for approximated by the CSCG distribution, i.e., each reflected path n = 1, ··· ,N can be obtained via IRS- N customized channel estimation methods (please refer to [13] approx. h(j) = h(j) ∼ CN 0,Ng(j) g(j) . (13) [14] for more details). As a result, IRS 0 can then adjust ir,m ir,m,n i,m r nX=1 the phase shift φ(0) such that the N reflected paths of the desired signal are of the same phase at UE 0’s receiver by (j) Proof: The channel hir,m,n reflected by each element n (0) ∠ (0) (0) 11 setting φn = − hi,0,nhr,n ,n = 1, ··· ,N. Therefore, has zero mean and independent in-phase and quadrature- the amplitude of the BS 0-IRS 0-UE 0 channel is given by 1 (j) (j) phase components each with variance 2 gi,mgr , respec- N tively, with the detailed derivations given in Appendix A. (0) h(0) T h(0) (0) (j) |hir,0| = | i,0 | | r | = hir,0,n . (10) Moreover, since the channels hir,m,n, n = 1, ··· ,N are nX=1 i.i.d., based on Approximation 2, the independent in-phase
Furthermore, we adopt the Approximation 2 that by the and quadrature-phase components of the combined channel (j) N (j) central limit theorem (CLT), given N i.i.d. RVs X1, ··· ,XN hir,m = n=1 hir,m,n can be each approximated by an inde- each with mean and variance 2, the sum N P 1 (j) (j) µ ω Y = n=1 Xn pendent normal distribution N (0, 2 Ngi,mgr ) for practically can be approximated by the normal distribution N (Nµ,NωP 2) large N. As a result, the combined BS m-IRS j-UE 0 channel for sufficiently large N. As a result, we have can be approximated by the CSCG distribution given by (13). Proposition 2 is thus proved. Proposition 1. With reflect beamforming by IRS 0, the BS 0- IRS 0-UE 0 channel amplitude in (10) for practically large Therefore, the average BS m-IRS j-UE 0 channel power is N can be approximated by the normal/Gaussian distribution, given by (j) , E (j) 2 (j) (j) i.e., gir,m {|hir,m| } = Ngi,mgr , (14) which is proportional to the average channel power product (0) approx. E (0) (0) |hir,0| ∼ N N hir,0,n ,N var hir,0,n (j) (j) , gi gr , with the scattering gain coefficient Gsc N that ,m 2 grows linearly with N. π (0) (0) π (0) (0) = N N g gr ,N 1 − g g . (11) 4 i,0 16 i,0 r q Proof: Based on Approximation 2 with large N,12 the BS 0-IRS 0-UE 0 channel amplitude in (10) is the sum of N i.i.d. C. SINR, Coverage Probability, and Spatial Throughput double-Rayleigh RVs h(0) , n =1, ··· ,N, each with mean ir,0,n Denote the downlink transmit power on each RB as P0. and variance given by (8) and (9), respectively. Therefore, Then, the overall signal power (normalized by ) from BS 0 P0 Proposition 1 is proved. to UE 0 is given by 2 10 The proposed analytical method in this paper can be extended to other S , h + h(j) , (15) fading channel models such as Rician fading. d,0 ir,0 11 jX∈J For the ease of practical implementation, we consider reflect beamforming for enhancing the desired signal power only, instead of nulling any co-channel which accounts for the direct path and reflected paths via all interference. 12According to our simulations, when N > 25, this approximation is IRSs j ∈ J . Further denote the total received interference already quite accurate for our considered setup. power (normalized by P0 as well) from all co-channel BSs IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 6
′ m ∈ ΛB \{0} by and IV, respectively, and then deriving the SINR distribution 2 and hence the spatial throughput in Section V. In particular, I , I = h + h(j) . (16) for the signal (or interference) power characterization, we first m d,m ir,m m∈XΛ′ \{0} m∈XΛ′ \{0} jX∈J derive its conditional distribution conditioned on the distances B B l0 and d0, based on which we are then able to obtain the SINR The received SINR at UE 0 is thus given by distribution. Moreover, we also characterize the unconditional S mean signal/interference power averaged over the channel γ , , (17) I + W fading and random BS/IRS locations, in order to reveal the impact of IRS on them, which helps illustrating their respective where W , σ2/P , and the receiver noise is assumed to be 0 effects to the SINR distribution. additive white Gaussian noise (AWGN) with power σ2. The corresponding achievable rate in bits/second/Hz III. SIGNAL POWER DISTRIBUTION (bps/Hz) is given by In this section, we characterize the conditional signal power , distribution as well as the (unconditional) mean signal power. R log2(1 + γ). (18) Since in our considered IRS-aided hybrid network, the exact Note that the signal power S, interference power I, and signal power distribution entails a more complicated form (as thus SINR γ and achievable rate R are all RVs depending will be shown in Section III-A) as compared to the conven- on the random channel fading as well as random BS/IRS tional case without IRS, the well-known analytical method locations. An outage event occurs when the rate R is lower proposed in [33] for deriving the SINR distribution directly than a minimum required target R¯. The coverage probability cannot be applied in our context. Therefore, we propose of the typical UE 0 is then defined as the average non- to approximate the conditional signal power distribution by outage probability over the random channel fading and random the Gamma distribution (which belongs to the exponential BS/IRS locations, i.e., distribution family [36]) based on its first and second moments conditioned on l0 and d0, under three different cases based on Pcov , P{R ≥ R¯} = P{γ ≥ γ¯} , 1 − Fγ (¯γ), (19) d0, i.e., with IRS reflect beamforming (d0 ≤ D1), with IRS , R¯ where γ¯ 2 − 1 denotes the corresponding minimum re- scattering only (D1 < d0 ≤ D2), and without any nearby quired SINR, and Fγ (·) is the cumulative distribution function IRS (d0 >D2). Such Gamma approximation incorporates the (cdf)13 of γ. Accordingly, we can define the spatial throughput BS-IRS-UE channel power statistics derived in Section II-B of the network in bps/Hz/m2 as as well as the impact of randomly distributed IRSs, which is tailored to the new IRS-aided hybrid network and thus differs ν , P Rλ¯ ′ = P Rpλ¯ . (20) cov B cov B from that for the conventional network without IRS. Moreover, In order to obtain the spatial throughput ν, we need to for the case with IRS reflect beamforming, we characterize the characterize the cdf of the SINR γ that depends on the impact of d0 on the mean signal power and the d0-dependent distributions of the signal power S and interference power channel hardening effect to draw useful insights. Finally, the I. Note that the instantaneous S and I are independent due mean signal power is obtained by integrating the conditional to their independent small-scale fading, while their large-scale mean signal power over the distributions of l0 and d0. statistics averaged over fading are dependent in general due Note that the first and second moments of the signal power to the common BS and IRS locations. Specifically, under the S depend on the average channel power gains gd,0 (BS 0-UE 0 (j) (j) distance-based association rule, the distribution of the BS 0- link), gi,0 (BS 0-IRS j link) and gr (IRS j-UE 0 link), which UE 0 link distance l0 affects not only the mean signal power, further depend on the corresponding horizontal link distance 2 but also the mean interference power since the interfering BSs l0, r0,j and dj that are related by the cosine law, i.e., r0,j = 2 2 are located at distances more than l0 from UE 0. Moreover, l0 + dj − 2l0dj cos ϕ, where ϕ is the BS 0-UE 0-IRS j angle for the IRS-aided downlink communication, both the signal projected on the ground plane. The exact expressions for the and interference are reflected by the same set (J ) of IRSs moments of S conditioned on l0 and d0, though not expressible within D2 of UE 0, and hence the path-losses of the BS-IRS in closed-form, can be obtained by numerical integrals over and/or IRS-UE links are random but correlated in general. ϕ and/or dj (as illustrated in Appendix B). Nevertheless, for Such correlation introduced by randomly distributed IRSs near simplicity in this section, we apply the Approximation 3 that (j) UE 0 imposes new difficulty to the system-level performance r0,j ≈ l0 and hence gi,0 ≈ gd,0, j ∈ J , in order to obtain analysis, which is challenging to deal with. In addition, the closed-form approximations for the moments of S|l0,d0 . Such IRS 0-UE 0 distance d0 determines whether there is reflect approximation is reasonable since the IRSs considered here are beamforming provided by IRS 0 and thus the IRS 0-UE 0 in the local region of UE 0 (and hence with small dj ), which channel power gain, which also has a significant impact on is also verified by our numerical examples later in Section VI the system performance. where the analytical results are shown to match well with the To tackle the above challenges, we decompose the perfor- Monte Carlo (MC) simulation results. mance analysis into three parts, by first characterizing the sig- nal power and interference power distributions in Sections III A. The Case with IRS Reflect Beamforming 1) Gamma Approximation with Moment Matching: First, 13 The cdf of an RV X is defined as FX (x) , P{X 2 2 −α/2 consider the case with D1 < d0 ≤ D2, where IRS 0 is outside gd(l) , β(l + H ) , (24) B the association range D1 of UE 0 and hence just randomly , 2 2 −α/2 scatters its received signal, like all the other IRSs in J . In gr(d) β(d + HI ) , (25) this case, we derive the first and second moments of the signal and represents the expectation of the sum of IRS- N · EI1(d0) power S conditioned on l0 only, by averaging over the random UE channel powers from all IRSs within horizontal distance locations of all IRSs outside D1 of UE 0. Specifically, for all (d0,D2] from UE 0, with EI1(d0) given by IRSs j ∈ J , the BS 0-IRS j-UE 0 channel follows (13) and D2 is given by (j) approx. (j) EI1(d ) , E gr(d ) =2πλI gr(d)d dd (j) 0 j hir,0 ∼ CN 0,Ngi,0 gr . (28) Zd=d0 d0 dist. where ξ0 = ξ ∼ Exp(1) and S¯ is the mean signal power IRS reflect beamforming, with IRS scattering only, and with- under given BS/IRS locations. Based on Approximation 3 out nearby IRS), we have respectively approximated S|l0,d0 (j) that gi,0 ≈ gd,0 = gd(l0), we hence have by the Gamma distribution with the same first and second moments. Therefore, Proposition 3 follows. S¯ ≈ gd(l ) 1+ N gr(d ) . (30) Note that the above signal power distribution as a function 0 j jX∈J of d0 is generally not continuous at the boundary point D1 As a result, the conditional mean signal power is given by (or D2), whereas the discontinuity gap is smaller when D1 (or D2) is larger. E E E ¯ (31) {S}|l0 = {ξ0} {S}|l0 ≈ gd(l0) 1+ NEI1(D1) , 2) Impact of the Nearest IRS 0 versus Other IRSs: To see the respective impact of the nearest IRS 0 and other IRSs in where EI1(D1)= EI1(d0) based on (26). Similarly, the d0=D1 conditional second moment of S is given by J , for the case with IRS scattering only, we first express the mean signal power in (30) under given BS/IRS locations as E 2 E 2 E ¯2 {S }|l0 = {ξ0 } {S }|l0 ¯ 2 2 S ≈ gd(l0) 1+ Ngr(d0)+ N gr(dj ) . (36) ≈ 2[gd(l0)] 1+2NEI1(D1)+ N EI3(D1) , (32) j∈JX \{0} Then, the mean signal power conditioned on l and d can be where EI3(D1)= EI3(d0) and EI3(d0) is given by 0 0 d0=D1 obtained as 2 E 2 {S}|l0,d0 ≈ gd(l0)κsc(d0), (37) EI3(d ) , E gr(d ) = EI1(d ) +EI2(d ), 0 j 0 0 d0 , (j) 2 where EB0 represents the expectation of the BS 0-UE 0 direct composite interference power Im hd,m + j∈J hir,m channel power, given by follows the exponential distribution and is given by P ∞ (j) (j) EB0 , E gd(l0) = gd(l0)fl (l0) dl0 I , I¯ ξ = gd + N g g ξ , (45) Z 0 m m m ,m i,m r m l0=0 jX∈J 2−α λ πH2 2 = βλ πH e B B E α (λ πH ), (41) B B 2 B B dist. where ξm = ξ ∼ Exp(1) and I¯m is the average interference 2 , −λBπl0 where fl0 (l0) 2πλBl0e is the pdf of the BS 0-UE power. Therefore, the total interference power I under given 0 distance l , and E α (·) is the exponential integral function 0 2 BS/IRS locations is the sum of independent but not identically α ′ [38] with parameter 2 , which is available in MATLAB. distributed exponential RVs Im,m ∈ ΛB \{0}, and thus Similarly, for the case with IRS scattering only (i.e., D1 < follows the generalized Erlang distribution [36]. d0 ≤ D2), we have Note that the mean interference power under given BS/IRS E locations is given by {S}|d0 ≈ κsc(d0)EB0, (42) I¯ , I¯m with the occurrence probability given by ′ m∈XΛB\{0} 2 2 , −λIπD1 −λIπD2 Psc e − e . (43) (j) (j) = gd + N g g . (46) ,m r i,m Moreover, for the case without nearby IRS (i.e., d0 >D2), we m∈XΛ′ \{0} jX∈J m∈XΛ′ \{0} E , B B have {S}|d0 ≈ EB0, with the occurrence probability Pwo 2 −λIπD Based on the definitions of g (l) and g (d) in (24) and (25), e 2 . Finally, we integrate E{S}|d over d0 to obtain the d r E 0 (j) unconditional mean signal power {S}, i.e., as well as Approximation 4 that gi,m ≈ gd,m = gd(lm) for ∞ j ∈ J , we have I¯m ≈ ηgd(lm) and hence E E S = S d0 fd0 (d0)dd0 { } Zd =0 { }| 0 I¯ ≈ η gd(lm), (47) D1 D2 m∈XΛ′ \{0} EB0 κbf(d0)fd (d0)dd0 + κsc(d0)fd (d0)dd0 + Pwo B ≈ Z 0 Z 0 0 D1 , D1 where η 1+N j∈J gr(dj ) is the relative power gain of all = EB0 κbf(d0)fd0 (d0)dd0 + PscEI1(D1)+ Pwo . (44) paths (including the scattering paths from all IRSs in J ) over Z P d0=0 the BS-UE direct path. For each IRS j ∈ J , its contributed term Ngr(dj ) decreases with the IRS-UE distance dj in the −α order of O(dj ), which decays quickly and becomes negligi- IV. INTERFERENCE POWER DISTRIBUTION ble beyond a certain distance, thus justifying Approximation 1. In this section, we first characterize the interference power distribution due to random channel fading, under given BS/IRS B. Conditional Laplace Transform and Cdf locations. Based on this result, we then characterize the inter- For complete characterization of the interference power ference power distribution conditioned on the given distances distribution conditioned on l0 and d0, we derive its Laplace l0 and d0, by deriving its Laplace transform [38] and hence transform in the following. From (16), (45) and (47), we have cdf. Finally, we derive the mean interference power in the network. I ≈ η gd(lm)ξm. (48) ′ Note that for analytical tractability in this section, we apply m∈XΛB\{0} (j) the Approximation 4 that rm,j ≈ lm and hence gi,m ≈ gd,m, For simplicity, to derive the Laplace transform in the sequel, j ∈ J , which can be similarly justified as in the second we adopt the Approximation 5 that η is approximately paragraph of Section III. Moreover, the distance lm is from replaced with its mean value for the three cases conditioned the non-serving BSs m 6= 0 in other cells, which is at least on d0 (i.e., with IRS reflect beamforming, with IRS scattering larger than the distance l0 from the serving BS 0 according only, and without nearby IRS), respectively,14 which is given to our assumed distance-based user-BS association. Therefore, by the approximation of rm,j ≈ lm is more accurate in this case κsc(d0), if d0 ≤ D1; since dj is much smaller than lm. , η¯ 1+ NEI1(D1), if D1 < d0 ≤ D2; (49) 1, otherwise. A. Interference Power Distribution Given BS/IRS Locations Note that η¯ as a function of d0 is in general not continuous Based on (13), the BS m-IRS j-UE 0 channel can be ap- at the boundary point D1 (or D2), similar to the case in (35). (j) (j) As a result, we have the following proposition. proximated by the CSCG distribution CN 0,Ngi,mgr .Asa (j) 14 result, the composite interference channel hd,m + j∈J hir,m As discussed in Section III-B2, the nearest IRS 0 (and hence the distance d from BS m ∈ Λ′ \{0} is the sum of independent CSCG 0) typically has the dominant impact on the randomly scattered signal or B P interference compared to other (farther) IRSs, whereas such impact also RVs, and hence is CSCG distributed with mean zero and decays quickly as d0 increases. Therefore, Approximation 5 is practically E 2 E (j) 2 covariance {|hd,m| } + j∈J {|hir,m| }. Therefore, the reasonable, which is also verified by simulation results in Section VI. P IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 10 Proposition 4. Based on Approximations 1, 2, 4 and 5, the Note that the scattering gain coefficient Gsc , N is generally Laplace transform of the conditional interference power I|l0,d0 not large enough to compensate the pathloss of the IRS-UE is given by link. As a result, κsc(d0) is dominated by the preceding term of , E −sI ′ 1 that accounts for the direct BS-UE link, and the IRS-reflected LI| (s) {e }|l0,d0 ≈ exp − 2πλBU(¯ηs) , (50) l0,d0 interference is non-negligible only when d0 is sufficiently where the function U(·) is defined as small. Moreover, similar to the analysis in Section III-B2, the nearest IRS 0 (i.e., with the smallest distance ) has the π 2 d0 U(x) , (βx) α dominant impact on the mean interference power compared to α sin( 2π ) α other IRSs. 2 2 l0 + HB 2 2 1 − · 2F1 1, , 1+ , − , (51) Finally, based on (55) and the definition of η, the uncondi- 2 α α gd(l0)x tional mean interference is given by with gd(l0) = gd(l)|l=l0 given by (24), and 2F1 denoting the Gauss hypergeometric function [38]. E{I}≈E{η}E gd(lm) = 1+ NEI1(0) EB2, m∈XΛ′ \{0} B Proof: Please refer to Appendix C. (57) Finally, the cdf of the conditional interference power I|l0,d0 can be obtained by taking the inverse Laplace transform of where N · EI1(0) = N · EI1(d0) based on (26), which d0=0 (50), i.e., represents the expectation of the sum of IRS-UE channel powers from all IRSs j within horizontal distance D2 from −1 1 FI| (x)= L LI| (s) (x), (52) UE 0. l0,d0 s l0,d0 which can be computed directly in MATLAB. V. SPATIAL THROUGHPUT CHARACTERIZATION C. Mean Interference Power In this section, we first derive the conditional SINR dis- Based on (48), (49) and i.i.d. ξm ∼ ξ ∼ Exp(1), ∀m, the tribution and non-outage probability based on the conditional mean interference power conditioned on l0 and d0 is given by distributions of the signal power S and interference power I obtained in Sections III and IV, respectively. Then the E{I}| ≈ E{ξ}E{η}| E gd(l ) l0,d0 d0 m coverage probability and hence spatial throughput are further m∈XΛ′ \{0} l0 B obtained by integrating over the distributions of the distances E = {η}|d0 EB1(l0), (53) l0 and d0. In particular, the conditional non-outage probability is ex- where E (l ) represents the expectation of the sum of direct B1 0 pressed in terms of the conditional interference power Laplace channel powers from interfering BSs m ∈ Λ′ \{0} conditioned B transform and its derivatives. Note that although we can also on l , which is given by 0 approximate the conditional interference power distribution by ∞ (b) ′ the Gamma distribution and obtain closed-form expressions for EB1(l0) , E gd(lm) = 2πλ gd(l)l dl B Z the conditional non-outage probability based on the approach m∈XΛ′ \{0} l0 l=l0 B in [39], it is found by simulations that the resulted SINR ′ 2πλBβ distribution is not accurate in our considered setup and thus = 2 2 α −1 . (54) (α − 2)(l0 + HB) 2 is not adequate to be used to further obtain the coverage Note that (b) is due to the HPPP-distributed BS locations, probability that requires integration over the distributions of which is essential to simplify the infinite interference summa- l0 and d0. tion as a spatial integral over the 2D plane. Furthermore, we can integrate over the distribution of l0 to obtain the conditional mean interference A. Conditional SINR Distribution and Non-Outage Probabil- ity E E {I}|d0 ≈ {η}|d0 EB2, (55) The non-outage probability conditioned on l0 and d0 is with EB2 defined as defined as ∞ , E , P P EB2 EB1(l0) = EB1(l0)fl0 (l0) dl0 Pno|l0,d0 {γ > γ¯}|l0,d0 = {S > γ¯(I + W )}|l0,d0 , (58) Zl0=0 ′ 2πλBβ 4−α λ πH2 2 which is related to the conditional SINR distribution via = λ πH e B B E α (λ πH ), (56) B B 2 −1 B B (α − 2) Pno|l0,d0 = 1 − Fγ (¯γ)|l0,d0 , with Fγ (·)|l0,d0 denoting the conditional SINR cdf. where fl (l0) is the pdf of l0, and E α −1(·) is the exponential 0 2 In Section III, we have approximated the conditional signal integral function with parameter α − 1. 2 power distribution by the Gamma distribution Γ[k ,θ ] given In the case with d ≤ D , we have E{η}| = κ (d ) and S S 0 2 d0 sc 0 in (35). With similar derivations in [40], we have the following hence E{I}| in (55) is proportional to the IRS-dependent d0 lemma. scattering gain κsc(d0)=1+ Ngr(d0)+ NEI1(d0) in (38). IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 11 Lemma 1. For a Gamma-distributed RV S ∼ Γ[kS,θS] with k˜S, we have integer kS and an independent RV X, we have P P (64) − no|l0,d0 = {I Pno|l0,d0 , we can then obtain the coverage probability, i.e., As a result, for integer kS, the conditional non-outage average non-outage probability in the network, by integrating probability in (58) is given by over the distributions of l0 and d0, given by γ¯(I+W ) ∞ ∞ Γ kS, E θS Pno|l0,d0 ≈ I Pcov = Pno|l0,d0 fd0 (d0)fl0 (l0) dd0 dl0 Γ(kS ) Z Z l0,d0 l0=0 d0=0 ∞ D kS −1 i i 1 (−1) ∂ P = L (s) , (60) = no|l0,d0 fd0 (d0)fl0 (l0) dd0 dl0 i Y |l0,d0 s=1 i! ∂s Zl0=0 Zd0=0 Xi=0 ∞ + P P | f (l ) dl where Y , γ¯(I+W ) . Based on (50), the Laplace transform of sc no l0,D1 1 1 0.62 0.8 0.8 0.6 0.58 0.6 0.6 -95 -94.5 -94 0.4 0.4 0.2 0.2 0 -105 -100 -95 -90 -85 -80 -75 -70 -65 -60 0 -102 -100 -98 -96 -94 -92 -90 -88 -86 -84 -82 Fig. 2: Cdf of the conditional signal power S l0,d0 with reflect beam- Fig. 3: Cdf of the conditional interference power I l ,d , under λB = | | 0 0 forming from the associated IRS, under λB = 10λ0, λI = 100λ0, 10λ0, λI = 100λ0, l0 = 50 m and different d0, N and p. l0 = 50 m and different d0 and N. 1 A. Performance with Given BS Density 0.8 1) Conditional Signal/Interference Power Distribution: The cdf of the conditional signal/interference power in (35)/(52) 0.6 0.94 under λB = 10λ0, λI = 100λ0, l0 = 50 m and different 0.9 0.4 d0 and N is plotted in Fig. 2 and Fig. 3, respectively. It is observed from Fig. 2 that, with reflect beamforming from the 0.86 0.2 0.82 associated IRS 0, the signal power is significantly enhanced -7 -6 -5 and exhibits channel hardening when N increases and/or the 0 -20 -10 0 10 20 30 IRS 0-UE 0 horizontal distance d0 decreases. For the interference power distribution shown in Fig. 3, the Fig. 4: Coverage probability with different Q and N, under λ = IRSs within range D2 of the target UE 0 randomly scatter the B signal from interfering BSs, which only slightly increase the 10λ0 and p = 0.5. interference power even when d0 is very small (e.g., d0 = 1 m) and N is very large (e.g., N = 8000), compared to the under the same Q, a smaller N leads to higher IRS density λI, case without IRS. Moreover, as d0 increases (e.g., d0 ≥ 10 which helps cover more UEs with low-SINR requirement. In m), it can be seen that there is negligible difference on the contrast, in the case with large Q (e.g., Q = 10/m2), choosing conditional interference power distribution for cases without higher N under the same Q yields higher coverage probability versus with IRS (even when N = 8000 and under larger in both low and high SINR threshold regions. This is because D2, e.g., D2 = 100 m). These observations validate our in this case λI is already sufficiently large to cover most UEs analysis in Section III-B2 that under practical values of the IRS and thus the IRS passive beamforming gain that grows with N 2 density, the nearest IRS 0 (and hence the distance d0) typically in O(N ) is more effective in enhancing UEs’ signal power has the dominant impact on the randomly scattered signal and hence the coverage probability. or interference compared to other (farther) IRSs, whereas 3) Mean Signal/Interference Power and Spatial Through- such impact also decays quickly as d0 increases. This could put: The mean signal/interference power in (44)/(57) under be attributed to the severe product-distance/double-pathloss different BS/IRS densities and loading factor p is plotted attenuation of the signal/interference randomly reflected by in Fig. 5 for comparison. First, it is observed that for the IRS. In contrast, the interference power significantly increases case with BS only (i.e., λI = 0), increasing λB from 20λ0 when the network loading factor p increases (corresponding to to 40λ0 brings 2.1 dB (4.0 dB) gain to the mean signal more UEs per RB) regardless of with or without IRSs in the (interference) power. In other words, when the BS density network, since more co-channel/interfering BSs become active is large, the mean interference power increases faster than on the same RB. the mean signal power by adding more BSs. In contrast, for 2) Impact of λI and N on Coverage Probability: Next, the hybrid BS/IRS network, as the IRS density increases, we investigate the impact of the IRS density λI and element the mean signal power increases significantly while the mean number N on the network coverage probability Pcov given by interference power increases only marginally. This is mainly (65). Specifically, denote Q , NλI as the total number of IRS due to the different scaling laws of the mean BS-IRS-UE elements per m2. The coverage probability under different Q channel power for the cases with and without IRS reflect and N is plotted in Fig. 4 for comparison. beamforming as discussed in Section II-B, which are also First, it is observed that increasing Q helps improve the consistent with the earlier observations made on Figs. 2 and coverage probability. Second, in the case with small Q (e.g., 3. Q = 1/m2), choosing smaller N under the same Q leads to Next, the spatial throughput ν under different BS/IRS den- slightly higher coverage probability at the low-SINR threshold sities and loading factor p is plotted in Fig. 6. It is observed region (see the zoomed plot inside Fig. 4). The reason is that, that, under given BS density, increasing IRS density always IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 13 Fig. 5: Mean signal (interference) power under different BS/IRS densities and loading factor p, with N = 2000. 120 100 Fig. 7: Spatial throughput ν versus IRS/BS density ratio ζ with total cost C = 80λ0c0, under different BS/IRS cost ratio KN (with N = 80 2000) and network loading factor p. 60 that given the total cost C, there exists an optimal IRS/BS ∗ 40 density ratio ζ that attains the maximum spatial throughput ν∗, which is significantly higher than that of the BS-only 20 0 50 100 150 200 250 300 network (i.e., ζ = 0) as well as the hybrid network with excessively large ζ (e.g., ζ = 10), where the BS density is Fig. 6: Spatial throughput ν under different BS/IRS densities and too low to provide enough signal power for effective IRS loading factor p, with N = 2000. reflection and passive beamforming. Second, it is observed that the optimal ratio ζ∗ is roughly proportional to the BS/IRS enhances the spatial throughput. Moreover, the speed of such cost ratio KN and the network loading factor p, where a larger K suggests that more IRSs should be deployed each with increase is faster under a higher BS density λB and/or higher N network loading factor p. The reasons are two-fold. First, relatively lower cost as compared to BS, while a higher p increasing IRS density helps enhance the signal power with corresponds to higher traffic demand (or more UEs per RB) only marginally increased interference power, as shown in Fig. which thus requires deploying more IRSs. 5. Second, a higher BS density shortens the BS-IRS distance Next, the spatial throughput ν versus total cost C under on average and thus helps enhance the IRS reflected signal different IRS/BS density ratio ζ is plotted in Fig. 8, for the power for its served UEs, while a higher network loading case with p = 1 and KN = 5. First, it is observed that for factor implies more UEs per RB and thus more UEs served by the BS-only network (i.e., ζ =0), the spatial throughput first each IRS on average, both leading to more substantial spatial increases and then decreases as the BS density increases, due throughput improvement with increasing IRS density. to the more severe interference as compared to the improved signal power. Second, when the BS density is larger than a ∗ certain value (say, 40λ0), the optimal IRS/BS density ratio ζ B. Spatial Throughput Subject to Total BS/IRS Cost is approximately 2.5, while the maximum ν∗ increases almost In this subsection, we investigate the network spatial linearly with the total cost C, which significantly outperforms throughput subject to a given total cost of BSs and IRSs the BS-only network. deployed. Denote c0 as the cost of each BS, and assume that Finally, in Fig. 9, we plot the spatial throughput ν versus , the cost of each IRS with N elements is cIRS,N c0/KN , total cost C for the cases with different p and KN , under where KN > 0 is the BS/IRS cost ratio which could be their corresponding optimal IRS/BS density ratio ζ∗. First, any positive value according to practical BS and IRS costs. it is observed that when the BS density is larger than a Nevertheless, the cost of one IRS is expected to be lower than certain value, the optimal IRS/BS density ratio ζ∗ is roughly one BS, thanks to the IRS’s passive signal reflection without proportional to the BS/IRS cost ratio KN and the network the need of any signal processing/regeneration [2] [3]. Denote loading factor p. Second, given the optimal ratio ζ∗, the 2 ζ , λI/λB as the IRS/BS density ratio. The total cost per m spatial throughput always increases with the total cost, where in the IRS-aided hybrid wireless network is then given by the speed of such increase is faster under a larger BS/IRS cost ratio or network loading factor , due to similar C , λ c + λ c = λ c + λ c /K KN p B 0 I IRS,N B 0 I 0 N reasons provided for Fig. 7. The above results demonstrate = λBc0 + ζλBc0/KN = λBc0(1 + ζ/KN ). (66) that the IRS-aided hybrid (active/passive) wireless network The spatial throughput ν versus the IRS/BS density ratio ζ can significantly enhance the network throughput as compared under given total cost C is shown in Fig. 7. First, it is observed to the conventional network with active BSs only, when the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 14 which thus provides an appealing alternative architecture for 150 wireless networks, especially for their migration to higher frequency bands in the future. 100 ACKNOWLEDGEMENT The authors would like to thank Dr. Qingqing Wu and Dr. Weidong Mei for their helps and the anonymous reviewers for 50 their suggestions. APPENDIX A 0 (j) 20 40 60 80 100 120 MEAN AND VARIANCE OF hir,m,n FORTHE IRS RANDOM SCATTERING CASE Fig. 8: Spatial throughput ν versus total cost C under different For the cascaded BS m-IRS j-UE 0 link in (1), the channel (j) IRS/BS density ratio ζ, with p = 1 and given KN = 5 (with hir,m,n of the reflected path via each element n is given N = 2000). , (j) (j) by (5), with the amplitude A |hi,m,n||hr,n| and phase 200 150 , (j) ∠ (j) ∠ (j) ψ φn + hi,m,n + hr,n. For the case without passive beamforming, the phase ψ is uniformly random in [0, 2π) 150 100 while the amplitude A follows the double-Rayleigh distribu- tion with mean and variance given by (8) and (9), respectively. 100 Note that A and ψ are independent. Denote X , A cos ψ and 50 Y , A sin ψ as the in-phase and quadrature-phase components 50 (j) of hir,m,n, respectively. In the following, we derive the mean and variance of X, while those of Y can be obtained similarly. 0 0 20 40 60 80 100 120 20 40 60 80 100 120 Due to the uniformly random ψ in [0, 2π), the first two moments of X are given by E{X} = E{A cos ψ} = 0 and E 2 E 2 2 E 2 E 1+cos 2ψ 1 E 2 {X } = {A cos ψ} = {A } { 2 } = 2 {A }, Fig. 9: Spatial throughput ν versus total cost C for (a) p = 1 and (b) respectively. As a result, the mean of X is 0 and its variance p = 0.5, under different KN (with N = 2000) and the corresponding is given by optimal IRS/BS density ratio ζ∗. 1 var A +(E A )2 var X = E X2 (E X )2 = E A2 = { } { } densities of BSs and IRSs are optimally set based on their total { } { }− { } 2 { } 2 1 π2 π2 1 cost constraint as well as other relevant network parameters. = 1 g(j) g(j) + g(j) g(j) = g(j) g(j). (67) 2 − 16 i,m r 16 · i,m r 2 i,m r VII. CONCLUSIONS This paper investigates a new hybrid active/passive wireless APPENDIX B network with large-scale deployment of BSs and IRSs, and 2 2 FIRST AND SECOND MOMENTS OF h1, h2, |h1| AND |h2| proposes a new analytical framework based on stochastic For , under given and , we have geometry and probability theory to characterize its spatial h1 l0 d0 (0) i∠ throughput as well as other key performance metrics averaged E E hd,0 {h1}|l0,d0 = (|hd,0| + |hir,0|)e over both random channel fading and BS/IRS locations. Ex- l0,d0 (0) i∠hd,0 = E |hd,0| + |h | E e =0, (68) tensive numerical results are provided to validate our analysis ir,0 l0,d0 l0,d0 and show the effectiveness of deploying IRSs to significantly i∠hd,0 due to E e = 0. For h2, it is the sum of enhance the signal power but with only marginally increased l0,d0 independent CSCG RVs and hence is CSCG distributed with interference, thus greatly improving the network throughput zero mean, i.e., E{h2} = 0. Based on similar derivations in as compared to the traditional wireless network with active E 2 E 2 (68), we have {h1}|l0,d0 =0 and {h2}|l0,d0 =0. BSs only, especially when the BS density and network loading 2 For |h1| , its first moment conditioned on l0 and d0 is given factor are large. Furthermore, it is shown that the new hybrid by network with optimal IRS/BS density ratio can achieve a 2 (0) 2 E h1 l ,d = E ( hd,0 + h ) l ,d linear capacity growth with the network deployment cost, {| | }| 0 0 { | | | ir,0| }| 0 0 2 (0) (0) 2 2 thus providing a fundamentally new approach to achieve = E hd,0 + 2 hd,0 h + h l ,d = E hd,0 l ,d + {| | | || ir,0| | ir,0| }| 0 0 {| | }| 0 0 sustainable capacity growth for future wireless networks. Fi- (0) (0) 2 2E hd,0 l ,d E h l ,d + E h l ,d nally, incorporating multi-antenna BSs in the IRS-aided hybrid {| |}| 0 0 {| ir,0|}| 0 0 {| ir,0| }| 0 0 gd,0 π π (0) (0) (0) (0) network can further enhance the network performance, which = gd,0 + 2 Eϕ N g gr + Eϕ Gbfg g · r 2 r 2 · 4 i,0 i,0 r is worth investigating in future work. Nevertheless, this paper q π (0) (0) (0) (0) unveils that the new hybrid wireless network consisting of = gd,0 + N πgr gd,0Eϕ g + Gbfgr Eϕ g , (69) 4 q i,0 i,0 active BSs with single antenna only (thus much less costly q (0) than the conventional massive MIMO BSs) and passive IRSs where gi,0 is the average channel power gain of the can already achieve network capacity scaling cost-effectively, BS 0-IRS 0 link with horizontal link distance r0,0 = IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 15 2 2 l0 + d0 − 2l0d0 cos ϕ and ϕ being the BS 0-UE 0-IRS APPENDIX C p0 angle projected on the ground plane. The expectation of DERIVATION OF INTERFERENCE POWER LAPLACE (0) RANSFORM functions of gi,0 can thus be obtained by simple integrals T over ϕ in [0, 2π). Similarly, we can obtain exact expressions The Laplace transform of the interference power condi- (though not expressible in closed-form) for the moments of tioned on and is given by 2 2 l0 d0 |h1| , |h2| and hence the signal power S conditioned on l0 and , by simple integrals over and/or . L (s) , E{e−sI }| d0 ϕ dj I|l0,d0 l0,d0 ≈ E exp − sη¯ g (l )ξ d m m m∈XΛ′ \{0} l0,d0 B (c) Nevertheless, for simplicity, we apply the approximation of ′ = EΛ Eξ exp − sηg¯ d(lm)ξ (j) B ′ gi,0 ≈ gd,0, j ∈ J (discussed in the second paragraph of m∈ΛY\{0} l0,d0 B Section III) in the above expressions to obtain closed-form ∞ (d) ′ E approximations for the moments of S|l0,d0 . Specifically, by = exp − 2πλB 1 − ξ exp − sηg¯ d(l)ξ l dl (j) (j) Zl0 applying gi,0 ≈ gd,0 = gd(l0) and gr = gr(dj ) given by (24) ∞ E 2 (e) ′ 1 and (25), respectively, {|h1| }|l0,d0 in (69) is approximated = exp − 2πλB 1 − l dl 1+ sηg¯ d(l) by Zl0 ′ = exp − 2πλBU(sη¯) , (74) 2 π E{|h | }| ≈ gd(l ) 1+ N πgr(d )+ Gbfgr(d ) . 1 l0,d0 0 4 0 0 dist. p where (c) follows from i.i.d. ξm = ξ ∼ Exp(1), ∀m and (70) ′ independent ΛB; (d) is based on the probability generating Similarly, we have −sξ functional of HPPP [33]; (e) is due to the fact that Eξ{e } , 4 (0) 4 4 1 E h1 l ,d = E ( hd,0 + h ) l ,d = E hd,0 + for ξ ∼ Exp(1); and U(·) is defined in (51). {| | }| 0 0 { | | | ir,0| }| 0 0 | | 1+s 3 (0) 2 (0) 2 (0) 3 (0) 4 4 hd,0 h + 6 hd,0 h + 4 hd,0 h + h l ,d | | | ir,0| | | | ir,0| | || ir,0| | ir,0| | 0 0 2 3 3 APPENDIX D [gd(l0)] 2+ π 2 N gr(d0) + 6Gbfgr(d0)+ ≈ 4 PROOF OF LEMMA 1 p 3 3 2 π2 π N 3πN (1 16 ) 3 For a Gamma-distributed RV S ∼ Γ[kS,θS], its comple- 2√π + − [gr(d )] 2 + 64 4 0 mentary cdf (ccdf) is given by P{S > x} =Γ k , x /Γ(k ). S θS S 2 4 4 2 3 π 2 As a result, for an independent RV X, we have π N 3π N (1 16 ) 2 π 2 2 + − + 3N 1 [gr(d0)] . (71) 256 8 − 16 Γ k , X S θS P{S>X} = EX . (75) Γ(kS) Let Y = X/θS. Using the fact that Γ kS,y /Γ(kS) = kS −1 i −y i=0 y e /i! for integer kS [38], we have Finally, under given BS/IRS locations, h2 is CSCG dis- P kS −1 (j) (j) P E i −Y tributed with zero mean and covariance j∈J \{0} Ngi,0 gr , {S>X} = Y Y e /i! 2 and thus |h2| follows the exponential distributionP with mean Xi=0 (j) (j) 2 kS −1 i Ngi gr . Therefore, the two moments of |h2| (−1) j∈J \{0} ,0 = E (−1)iY ie−Y . (76) conditionedP on l0 and d0 are respectively approximated by i! Y Xi=0 2 (j) ( ) i E E j ∂ −sY i −sY {|h2| }|l0,d0 = Ngi,0 gr Based on the fact that as well as the ∂si e = (−Y ) e j∈JX \{0} l0,d0 interchangeable operators of expectation and differentiation, we thus have ≈ E Ngd(l0) gr(dj ) = Ngd(l0)EI1(d0), kS −1 j∈JX \{0} l0,d0 (−1)i ∂i P{S>X} = E e−sY (72) i! Y ∂si s=1 Xi=0 k −1 2 S i i (−1) ∂ −sY E 4 E (j) (j) = EY e . (77) {|h2| }|l0,d0 = 2 Ngi,0 gr i! ∂si Xi=0 s=1 j∈JX \{0} l0,d0 2 Therefore, Lemma 1 follows. 2 2 ≈ 2N [gd(l )] E gr(d ) 0 j X d0 j∈J \{0} APPENDIX E 2 2 =2N [gd(l0)] EI3(d0), (73) DERIVATIVES OF THE COMPOSITE FUNCTION exp V (s) where EI1(d0) and EI3(d0) are given in (26) and (33), respec- The high-order derivatives of the composite function tively. exp V (s) can be evaluated efficiently using the Fa`a di IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 16 Bruno’s formula [40]. Since the outer function is an expo- [9] Q. Wu and R. Zhang, “Beamforming optimization for wireless network nential function, we have aided by intelligent reflecting surface with discrete phase shifts,” IEEE Trans. Commun., vol. 68, no. 3, pp. 1838–1851, Mar. 2020. ∂i ∂1V (s) ∂iV (s) [10] S. Zhang and R. Zhang, “Capacity characterization for intelligent exp V (s) = exp V (s) B , ··· , , reflecting surface aided MIMO communication,” IEEE J. Sel. Areas ∂si i ∂s1 ∂si Commun., vol. 38, no. 8, pp. 1823–1838, 2020. (78) [11] Z. Zhang, Y. Cui, F. Yang, and L. Ding, “Analysis and optimization where Bi(x1, ··· , xi) is the i-th complete Bell polynomial of outage probability in multi-intelligent reflecting surface-assisted with fixed and known coefficients. It remains to compute the systems,” 2019. [Online]. Available: https://arxiv.org/abs/1909.02193 [12] C. Huang, A. Zappone, G. C. Alexandropoulos, M. Debbah, and derivatives of the inner function V (s) up to order kS −1, which C. Yuen, “Reconfigurable intelligent surfaces for energy efficiency in γ¯η¯ resorts to the derivatives of U(x) in (51). Define x˜ , . The wireless communication,” IEEE Trans. Wireless Commun., vol. 18, no. 8, θS 1st-order derivative of V (s) is given by pp. 4157–4170, Aug. 2019. [13] Y. Yang, B. Zheng, S. Zhang, and R. Zhang, “Intelligent reflecting 1 1 surface meets OFDM: Protocol design and rate maximization,” IEEE ∂ V (s) γW¯ ′ ∂ U(x) = − − 2πλ x˜ . (79) Trans. Commun., vol. 68, no. 7, pp. 4522–4535, 2020. ∂s1 θ B ∂x1 S x=sx˜ [14] B. Zheng and R. Zhang, “Intelligent reflecting surface-enhanced OFDM: For , we have Channel estimation and reflection optimization,” IEEE Wireless Com- i> 1 mun. Lett., vol. 9, no. 4, pp. 518–522, 2020. ∂iV (s) ∂iU(x) [15] G. Yang, X. Xu, and Y. Liang, “Intelligent reflecting surface assisted = −2πλ′ x˜i . (80) non-orthogonal multiple access,” in Proc. IEEE Wireless Commun. Netw. ∂si B ∂xi x=sx˜ Conf. (WCNC), 2020, pp. 1–6. [16] Z. Ding and H. Vincent Poor, “A simple design of IRS-NOMA trans- Finally, the high-order derivatives of U(x) are given by mission,” IEEE Commun. Lett., vol. 24, no. 5, pp. 1119–1123, 2020. i [17] B. Zheng, Q. Wu, and R. Zhang, “Intelligent reflecting surface-assisted ∂ U(x) δ−i (δ)iH(x) multiple access with user pairing: NOMA or OMA?” IEEE Commun. = (δ)ib1x −b2 +Li(x) ,i ≥ 1, (81) ∂xi xi Lett., vol. 24, no. 4, pp. 753–757, 2020. [18] M. Cui, G. Zhang, and R. Zhang, “Secure wireless communication via 2 2 2 , 2 , πβ α , l0+HB , intelligent reflecting surface,” IEEE Wireless Commun. Lett., vol. 8, where δ , b1 2π , b2 , b3 gd(l0), α α sin( α ) 2 no. 5, pp. 1410–1414, Oct. 2019. H(x) , 2F1 1, δ, 1+ δ, −1/(b3x) , (δ)i , δ(δ − 1) ··· (δ − [19] J. Chen, Y. Liang, Y. Pei, and H. Guo, “Intelligent reflecting surface: A programmable wireless environment for physical layer security,” IEEE i + 1) is the falling factorial, and Li(x) are polynomial Access, vol. 7, pp. 82 599–82 612, 2019. fractions that can be readily obtained via symbolic tools like [20] D. Xu, X. Yu, Y. Sun, D. W. K. Ng, and R. Schober, “Resource Mathematica. For example, we have allocation for secure IRS-assisted multiuser MISO systems,” in IEEE Proc. GLOBECOM Workshops, 2019, pp. 1–6. δb [21] X. Guan, Q. Wu, and R. Zhang, “Intelligent reflecting surface assisted L (x) , − 3 , (82) 1 1+ b x secrecy communication: Is artificial noise helpful or not?” IEEE Wireless 3 Commun. Lett., vol. 9, no. 6, pp. 778–782, 2020. δb3 δ + b3(δ − 1)x [22] Q. Wu and R. Zhang, “Weighted sum power maximization for intelligent , (83) reflecting surface aided SWIPT,” IEEE Wireless Commun. Lett., vol. 9, L2(x) − 2 , x(1 + b3x) no. 5, pp. 586–590, 2020. 2 2 2 2 [23] C. Pan, H. Ren, K. Wang, M. Elkashlan, A. Nallanathan, J. Wang, and δb3 2b x + δ (1 + b3x) δ(1 + b3x)(2+3b3x) L x , 3 − . L. Hanzo, “Intelligent reflecting surface aided MIMO broadcasting for 3( ) 2 3 − x (1 + b3x) simultaneous wireless information and power transfer,” IEEE J. Sel. (84) Areas Commun., vol. 38, no. 8, pp. 1719–1734, 2020. [24] Q. Wu and R. Zhang, “Joint active and passive beamforming opti- mization for intelligent reflecting surface assisted SWIPT Under QoS REFERENCES constraints,” IEEE J. Sel. Areas Commun., vol. 38, no. 8, pp. 1735– [1] J. G. Andrews et al., “What will 5G be?” IEEE J. Sel. Areas Commun., 1748, 2020. vol. 32, no. 6, pp. 1065–1082, June 2014. [25] T. Hou, Y. Liu, Z. Song, X. Sun, Y. Chen, and L. Hanzo, “MIMO assisted [2] Q. Wu and R. Zhang, “Towards smart and reconfigurable environment: networks relying on large intelligent surfaces: A stochastic geometry Intelligent reflecting surface aided wireless network,” IEEE Commun. model,” 2019. [Online]. Available: https://arxiv.org/abs/1910.00959 Mag., vol. 58, no. 1, pp. 106–112, Jan. 2020. [26] J. Lyu and R. Zhang, “Spatial throughput characterization for intelligent [3] E. Basar, M. Di Renzo, J. De Rosny, M. Debbah, M. Alouini, and reflecting surface aided multiuser system,” IEEE Wireless Commun. R. Zhang, “Wireless communications through reconfigurable intelligent Lett., vol. 9, no. 6, pp. 834–838, 2020. surfaces,” IEEE Access, vol. 7, pp. 116 753–116 773, Sept. 2019. [27] C. Pan, H. Ren, K. Wang, W. Xu, M. Elkashlan, A. Nallanathan, [4] M. Di Renzo, M. Debbah, D.-T. Phan-Huy, A. Zappone, M.-S. and L. Hanzo, “Multicell MIMO communications relying on intelligent Alouini, C. Yuen, V. Sciancalepore, G. C. Alexandropoulos, J. Hoydis, reflecting surfaces,” IEEE Trans. Wireless Commun., vol. 19, no. 8, pp. H. Gacanin, J. De Rosny, A. Bounceur, G. Lerosey, and M. Fink, “Smart 5218–5233, 2020. radio environments empowered by reconfigurable AI meta-surfaces: An [28] H. Xie, J. Xu, and Y. Liu, “Max-min fairness in IRS-aided multi-cell idea whose time has come,” EURASIP J. Wireless Commun. and Netw., MISO systems with joint transmit and reflective beamforming,” in Proc. vol. 2019, no. 1, pp. 1–20, 2019. IEEE Int. Conf. Commun. (ICC), 2020, pp. 1–6. [5] C. Huang, S. Hu, G. C. Alexandropoulos, A. Zappone, C. Yuen, [29] Y. Jia, C. Ye, and Y. Cui, “Analysis and optimization of an intelligent R. Zhang, M. Di Renzo, and M. Debbah, “Holographic MIMO surfaces reflecting surface-assisted system with interference,” in Proc. IEEE Int. for 6G wireless networks: Opportunities, challenges, and trends,” IEEE Conf. Commun. (ICC), 2020, pp. 1–6. Wireless Commun., vol. 27, no. 5, pp. 118–125, 2020. [30] J. He, K. Yu, and Y. Shi, “Coordinated passive beamforming for [6] Q. Wu and R. Zhang, “Intelligent reflecting surface enhanced wireless distributed intelligent reflecting surfaces network,” in Proc. IEEE Veh. network via joint active and passive beamforming,” IEEE Trans. Wireless Technol. Conf., 2020, pp. 1–5. Commun., vol. 18, no. 11, pp. 5394–5409, Nov. 2019. [31] M. Di Renzo and J. Song, “Reflection probability in wireless networks [7] Y. Han, W. Tang, S. Jin, C. Wen, and X. Ma, “Large intelligent surface- with metasurface-coated environmental objects: An approach based on assisted wireless communication exploiting statistical CSI,” IEEE Trans. random spatial processes,” EURASIP J. Wireless Commun. Netw., vol. Veh. Technol., vol. 68, no. 8, pp. 8238–8242, Aug. 2019. 2019, no. 1, p. 99, 2019. [8] X. Yu, D. Xu, and R. Schober, “MISO wireless communication sys- [32] M. A. Kishk and M.-S. Alouini, “Exploiting randomly located blockages tems via intelligent reflecting surfaces,” in Proc. IEEE/CIC Int. Conf. for large-scale deployment of intelligent surfaces,” IEEE J. Sel. Areas Commun. China (ICCC), Aug. 2019, pp. 735–740. Commun., vol. 39, no. 4, pp. 1043–1056, 2021. IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 17 [33] J. G. Andrews, F. Baccelli, and R. K. Ganti, “A tractable approach to Rui Zhang (S’00-M’07-SM’15-F’17) received the coverage and rate in cellular networks,” IEEE Trans. Commun., vol. 59, B.Eng. (first-class Hons.) and M.Eng. degrees from no. 11, pp. 3122–3134, Nov. 2011. the National University of Singapore, Singapore, [34] W. Lu and M. Di Renzo, “Stochastic geometry modeling of cellular and the Ph.D. degree from the Stanford University, networks: Analysis, simulation and experimental validation,” in Proc. Stanford, CA, USA, all in electrical engineering. ACM Int. Conf. MSWiM, 2015, p. 179–188. From 2007 to 2010, he worked at the Institute [35] J. G. Andrews, A. K. Gupta, and H. S. Dhillon, “A primer on cellular for Infocomm Research, ASTAR, Singapore. Since network analysis using stochastic geometry,” 2016. [Online]. Available: 2010, he has been working with the National Uni- https://arxiv.org/abs/1604.03183 versity of Singapore, where he is now a Provost’s [36] C. Forbes, M. Evans, N. Hastings, and B. Peacock, Statistical distribu- Chair Professor in the Department of Electrical and tions. John Wiley & Sons, 2011. Computer Engineering. He has published over 250 [37] J. Salo, H. M. El-Sallabi, and P. Vainikainen, “The distribution of journal papers and over 190 conference papers. He has been listed as a the product of independent Rayleigh random variables,” IEEE Trans. Highly Cited Researcher by Thomson Reuters/Clarivate Analytics since 2015. Antennas Propag., vol. 54, no. 2, pp. 639–643, 2006. His current research interests include UAV/satellite communications, wireless [38] F. W. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST power transfer, reconfigurable MIMO, and optimization methods. handbook of mathematical functions. Cambridge university press, 2010. He was the recipient of the 6th IEEE Communications Society Asia- [39] R. W. Heath, M. Kountouris, and T. Bai, “Modeling heterogeneous Pacific Region Best Young Researcher Award in 2011, the Young Researcher network interference using Poisson point processes,” IEEE Trans. Signal Award of National University of Singapore in 2015, the Wireless Commu- Process., vol. 61, no. 16, pp. 4114–4126, Aug. 2013. nications Technical Committee Recognition Award in 2020, and the IEEE [40] R. Tanbourgi, H. S. Dhillon, J. G. Andrews, and F. K. Jondral, “Dual- Signal Processing and Computing for Communications (SPCC) Technical branch MRC receivers under spatial interference correlation and Nak- Recognition Award in 2020. He was the co-recipient of the IEEE Marconi agami fading,” IEEE Trans. Commun., vol. 62, no. 6, pp. 1830–1844, Prize Paper Award in Wireless Communications in 2015 and 2020, the IEEE June 2014. Communications Society Asia-Pacific Region Best Paper Award in 2016, the IEEE Signal Processing Society Best Paper Award in 2016, the IEEE Communications Society Heinrich Hertz Prize Paper Award in 2017 and 2020, the IEEE Signal Processing Society Donald G. Fink Overview Paper Award in 2017, and the IEEE Communications Society Stephen O. Rice Prize in 2021. His co-authored paper received the IEEE Signal Processing Society Young Author Best Paper Award in 2017, and the IEEE Communications Society Young Author Best Paper Award in 2021. He served for over 30 international conferences as the TPC co-chair or an organizing committee member, and as the guest editor for 3 special issues in the IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING and the IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS. He was an elected member of the IEEE Signal Processing Society SPCOM Technical Committee from 2012 to 2017 and SAM Technical Committee from 2013 to 2015, and served as the Vice Chair of the IEEE Communications Society Asia-Pacific Board Technical Affairs Committee from 2014 to 2015. He served as an Editor for the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS from 2012 to 2016, the IEEE JOURNAL ON SELECTED AREAS IN COMMU- NICATIONS: Green Communications and Networking Series from 2015 to 2016, the IEEE TRANSACTIONS ON SIGNAL PROCESSING from 2013 to 2017, and the IEEE TRANSACTIONS ON GREEN COMMUNICATIONS AND NETWORKING from 2016 to 2020. He is now an Editor for the IEEE TRANSACTIONS ON COMMUNICATIONS. He serves as a member of the Steering Committee of the IEEE Wireless Communications Letters. He served as a Distinguished Lecturer of IEEE Signal Processing Society and IEEE Communications Society from 2019 to 2020. Jiangbin Lyu (S’12, M’16) received his B. Eng. degree (Hons.) in control science and engineering (under the Chu Kochen Honors Program) from Zhe- jiang University, Hangzhou, China, in 2011, and the Ph.D. degree from NUS Graduate School for Inte- grative Sciences and Engineering (NGS) (under the NGS scholarship), National University of Singapore (NUS), Singapore, in 2015. He was a Post-Doctoral Research Fellow with the Department of Electrical and Computer Engineering, NUS, from 2015 to 2017. He is currently an assistant professor in the School of Informatics, Xiamen University, China, with research interests in unmanned aerial vehicle communications, intelligent reflecting surface, cross-layer network optimization, etc. Dr. Lyu was a recipient of the IEEE Communications Society Heinrich Hertz Prize Paper Award in 2020, and also the Best Paper Award at Singapore- Japan International Workshop on Smart Wireless Communications in 2014. He served as the Invited Track Co-Chair at the 2021 IEEE/CIC ICCC conference, a TPC member for IEEE GLOBECOM and ICC, and a reviewer for various IEEE journals. X = E S = (s) , X − i Y s=1 where , and is the conditional cdf of { } Γ(kS ) i! ∂s L z µ/γ¯ − W FI|l ,d (·) Xi=0 0 0 (59) I given by (52). where Γ(·, ·) is the upper incomplete Gamma function [38], and . Y = X/θS B. Coverage Probability and Spatial Throughput Proof: Please refer to Appendix D. After obtaining the conditional non-outage probability