Performance Analysis of Randomly Distributed Reconfigurable Intelligent Surfaces With Different Phase Profiles

Future scenarios foresee the deployment of a large amount of reconfigurable intelligent surfaces (RISs) covering buildings and objects to control the propagation of waves and to realize smart radio environments (SREs). Several works have already investigated the possibility to have a large-scale deployment of RISs indoors and outdoors, providing insightful considerations about the coverage and outage probability, especially in the absence of a direct base station (BS)-user (UE) link. Unfortunately, such works typically consider simplified propagation channels and assume ideal or random phase profiles at the RIS, which do not always fit real scenarios. This paper proposes a communication outage analysis that accounts for realistic RIS selection mechanisms and phase shift profiles. Differently from the literature, we first discuss some RIS association mechanisms and provide a general outage analysis, which is then specialized to the cascaded double Rician fading channel BS-RIS-UE. Finally, we provide extensive numerical evaluation, validated through simulations, to corroborate the proposed model to allow discussing the trade-off in terms of the number of employable RIS antennas and the number of quantization bits for each RIS element, which should be accounted for in the system design.


Performance Analysis of Randomly Distributed
Reconfigurable Intelligent Surfaces With Different Phase Profiles Francesco Guidi , Member, IEEE, Anna Guerra , Member, IEEE, and Alberto Zanella, Senior Member, IEEE Abstract-Future scenarios foresee the deployment of a large amount of reconfigurable intelligent surfaces (RISs) covering buildings and objects to control the propagation of waves and to realize smart radio environments (SREs).Several works have already investigated the possibility to have a large-scale deployment of RISs indoors and outdoors, providing insightful considerations about the coverage and outage probability, especially in the absence of a direct base station (BS)-user (UE) link.Unfortunately, such works typically consider simplified propagation channels and assume ideal or random phase profiles at the RIS, which do not always fit real scenarios.This paper proposes a communication outage analysis that accounts for realistic RIS selection mechanisms and phase shift profiles.Differently from the literature, we first discuss some RIS association mechanisms and provide a general outage analysis, which is then specialized to the cascaded double Rician fading channel BS-RIS-UE.Finally, we provide extensive numerical evaluation, validated through simulations, to corroborate the proposed model to allow discussing the trade-off in terms of the number of employable RIS antennas and the number of quantization bits for each RIS element, which should be accounted for in the system design.
Index Terms-Smart radio environment, reconfigurable intelligent surface, poisson point process, outage probability.

I. INTRODUCTION
T HE next sixth generation (6G) of mobile wireless systems fosters the creation of SREs where the environment becomes an active player in supporting communications and localization, attaining unprecedented performance in terms of data rates, connectivity, latency, and ambient awareness in general [1], [2], [3], [4].Controlling the wireless propagation environment is possible thanks to the advent of new technologies that help the BSs and access points (APs) in transferring and processing the information from and to cellular UEs also in non-line-of-sight (NLOS) situations [5], [6], [7], [8], [9], [10], [11].
Among new technological solutions, large antenna arrays and RISs play a central role in realizing SREs, thanks to their capability to focus the power toward the intended direction in space and to control the electromagnetic response to the incident wave [12], [13], [14].In this direction, recent works have envisioned the possibility of coating objects, walls, and vehicles with RISs, giving rise to the opportunity to optimize the environment following different desiderata (e.g., improve wireless communications and localization), even in challenging scenarios as the urban ones [15], [16], [17], [18].Furthermore, RISs are particularly attractive because their passive nature leads to low power consumption and cost, and their consequent large deployment creates an extended communication coverage and a high spatial diversity helpful for localization purposes without the need of increasing the number of more expensive BSs/APs or non-programmable relays and active repeaters.Indeed, RISs are often fabricated with metasurfaces, composed of layers of metamaterials, or as a discrete array of antennas whose physical properties and phase profiles are programmed to meet the desired behavior.
When adopting such RISs, a typical challenge is properly designing the phases of the RIS elements [19].Indeed, from one side, the possibility to control the phases in a quasi-continuum way allows very precise beam patterns to be formed, adherent to the sensing requirements.But, on the other hand, it is unattainable to design an array in which analog components (e.g., phase shifters) can assume the full spectrum of the phases or where digital components can imprint this phase value antenna by antenna.Usually, hardware imperfections occur in the RF chains, phase shifters, and RIS elements in the form of phase noises and quantization errors [20], [21], [22].These imperfections, including component mismatches and manufacturing defects, should be considered when designing algorithms or analyzing communications and localization performance.In this direction, [23] accounted for the number of bits, employed in massive arrays, in the localization algorithm, whereas [24] accounted for an environment-adaptive mapping scheme in order to mitigate strong non-idealities of antenna arrays which translated into a higher side-lobe level.Regarding RISs, recent works have tackled the problem of adequately designing the optimal phase profile according to the considered application requirements [11], [25].As an example, the beamforming operated by the BS can be optimized together with the RIS phases for the minimization of the transmit power [26], or for the maximization of the sum rate [27].
When dealing with the large deployment of wireless systems, stochastic tools have been often useful for assessing their performance (e.g., through the outage probability) in several applications [28], [29], spanning from vehicular ad-hoc network (VANET) based scenarios [30], device-to-device (D2D) networks [31], relays based networks [32], virtual-multiple-input multiple-output (MIMO) scenarios, as well as for the assessment of the interference in a Poissonian field of nodes [33].Then, it has been also adopted for wireless localization [34], [35], where authors considered the BS locations distributed with a Poisson point process (PPP), and exploited tools from stochastic geometry theory.
Concerning RIS, their large scale deployment has been traditionally considered in a setup where the number and positions of RISs are predefined [44], or without accounting for their random deployment [45].In this sense, the inclusion of the stochastic geometry-based modeling of the RISs positions in the environment is still limited.Among the related works, [37] considers RISs deployed according to a Boolean model and, given such a spatial distribution, they derived the probability to have an indirect path offered by a RIS.In [36], the authors accounted for a simplified channel model under ideal propagation conditions, i.e., without accounting for fading, and using the Poisson model for obstacle distribution.Moreover, the BSs and the RISs are modeled as PPP, and the association strategy is based on the minimum PL: if there exists a line-of-sight (LOS) link with a BS, then the UE exploits this direct link by selecting the BS experiencing the minimum PL; otherwise, it uses a RIS path by adopting the same criterion for the RIS selection.As performance metrics, authors considered the probability of being in a blind spot, that is, the probability that the UE is located in an area not reachable by either the BSs or through the RISs, and the coverage probability, i.e., the probability that the PL is over a predefined threshold.The authors in [38] considered a Rayleigh channel model for both BS-RIS and RIS-UE channels.Even if the channel model includes fading, differently from [36], the authors approximate such fading in a way that does not often fit with actual propagation, and they assume an ideal RIS phase profile without accounting for phase quantization errors.
Recently, [39] considered a scenario with multi-RISs whose locations were deployed according to a PPP.Then, the authors develop a tractable theoretical framework to obtain the outage probability and average rate under the optimum RIS selection policies, assuming a Rayleigh channel model and ideal phase compensation as in [38].The most representative related works are listed in Table I.
Unfortunately, all the previous research on stochastic geometry and multi-RIS has typically considered simplified channel models [36] or, even when more realistic channels are included in the analysis, the problem is simplified such that the overall distribution neglects the statistical information deriving from the adoption of a RIS phase setting which is far from being optimized [38], [39].For example, in [39], the analysis is conducted considering a double Rayleigh random variable (RV), whose summation over the number of RIS elements is easily tractable.In this sense, how to consider also practical RIS phases has yet to be investigated, since its implications in the distribution make the overall analysis more complex.
Motivated by this background, in this paper, we propose an analytical and general framework whose flexibility permits investigating different RIS selection mechanisms, dealing with any cascaded channel condition, and any possible RIS phase design.In a unified framework, we provide the outage analysis under realistic RIS phase profiles and by including a general channel model for both the UE-RIS and RIS-BS links.Then, we specialize our analysis to the case where the UE-RIS and RIS-BS follow a Rician distribution and when different numbers of bits are employed at the RIS.Through numerical results, we show the system performance in outage probability for different path-loss models, number of antennas and phase bits employed at the RISs.Finally, the impact of the RIS selection mechanism is evaluated under different settings.
The main contributions of the paper can be summarized as follows.
• We propose a generalized framework for the random deployment of RISs and blockages in a SRE, distributed according to a homogeneous PPP, and which Starting from the obtained results, we come up with a discussion about the trade-offs between the number of employed antennas and the phase quantization bits.The rest of the paper is organized as follows.Section II describes the scenario, the signalling phases and model, and the RIS selection mechanism.Sec.III presents the outage probability analysis, while Sec.IV provides closed-form expressions for the distribution of the joint channel and RIS gain.Then, Sec.V validate the results and, finally, Sec.VI draws final conclusions.
Notation: Throughout the paper, P{A}, denotes the probability of the event A; µ Z (or E {Z}), σ Z , and σ 2 Z (or var{Z}) represent expectation, standard deviation and variance of the RV Z, respectively.The functions f Z (z), F Z (z), and FZ (z) denote the probability density function (PDF), the cumulative density function (CDF) and the complementary CDF of the RV Z, respectively; ||z|| is the Euclidean norm of the N D -dimensional vector z = (z 1 , z 2 , . . ., z ND ); ℜ{c} and ℑ{c} denote the Real and the Imaginary part of the complex number c. d BR (x) ≜ ||x B − x|| is the distance between a generic RIS located at x = (x 1 , x 2 ) and the BS (with coordinates x B ), d RU (x) ≜ ||x − x U || is the distance between the RIS located at x and the UE (with coordinates x U ) and d BU ≜ ||x U − x B || is the distance between the UE and the BS.

II. SCENARIO
We consider an uplink communication system with a BS, a UE and RISs.The BS and the UE are located at fixed positions, with the BS-UE distance denoted by d BU .Then, we indicate the BS and UE coordinates with (0, 0) and (d BU , 0), respectively.The scenario is populated also by blockages, distributed in an arbitrary 2-dimensional space S, with area A S , according to a homogeneous PPP, with intensity (density) λ 1 .This process can be denoted as Φ(λ 1 ) = {x j }, where x j is the midpoint location of the j th blockage.Each blockage is modeled as a line segment having length ℓ j and is also characterized by an angle θ j , which represents the orientation of the j th blockage with respect to the reference system.In the next, we will consider blockages having the same length and we neglect the use of the index j.

A. RIS Phase Model
The communication between the BS and the UE is assisted by the presence of RISs, which are installed in a subset, Φ R ⊆ Φ, of the blockages, with density λ R = δλ 1 , where 0 < δ ≤ 1; each RIS is characterized by N radiating elements.As in [36], we assume that RISs are deployed on a single side of the blockage.Then, in order to model the behavior of the RIS, we assume that the phase at each radiating element cannot take any analog value between (−π, π], but rather a quantized (digital) value.Let N b indicate the number of quantization bits used to represent this phase value, we have where k = 1, • • • , 2 Nb−1 and, for N b = 0, only one phase shift value (i.e., v) can be assumed at all antennas whereas, for N b > 0, the i th RIS antenna can assume a value belonging to a discretized set.
In this paper, communications between the BS and the UE occur in two ways: i) through direct communication between the BS and UE if there is a LOS condition between the two nodes; ii) when the BS-UE link is obstructed, communications occur using BS-RIS and RIS-UE links, if available.The first case is denoted as direct link (and the correspondent event is defined as D) and the second is denoted as indirect link (and the correspondent event is defined as I).Also, we assume that the blockages equipped by RISs can support the communication between the BS and the UE only if the LOS is guaranteed both for the BS-RIS and RIS-UE links.

B. RIS Selection Mechanism
For indirect communications, the BS must identify the RISs that are supposed to provide reliable quality of communication and select the best RIS to rely upon according to some criteria.Because we suppose that both BS and UE do not have information about their related channels with respect to the RIS, we consider a three-phase mechanism, consisting of two pilot phases and one downlink data transmission phase, as described in the following.
• Pilot-A: During the first phase, the BS broadcasts a set of pilot symbols but only the RISs that are located with a favorable orientation with respect to the BS can reflect the signal.Notably, since the RISs do not know where the UE is located, there is no possibility to adjust the Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
phases in order to facilitate the signal reception at the BS.Hence, no phase is adjusted and, consequently, there is no beamsteering operation toward the UE (i.e., N b = 0).In this phase, to discriminate the signals reflected by different RISs, a possibility is to assign different and orthogonal codewords to the RISs according to a bipolar coding scheme (e.g., ±1). 1 Then, assuming that the BS sends a sequence of pilot signals over a certain temporal window, the RISs, acting as a wall, modulates each backscattered pilot sequence according to its code.Then, thanks to the codes' orthogonality, the UE is capable of discriminating the signal coming from each RIS by performing a de-spreading operation.By making in parallel such operations it is possible to evaluate the signal-to-noise ratio (SNR) for each significant RIS when it is configured as a wall.We consider only the signals arriving to the UE with a SNR larger than a given threshold γ S , sufficiently high to perform basic detection operations like synchronization and channel estimation.
Finally, according to a criterion that we will describe in the next, the UE identifies the RIS that will assist the BS-UE communication.
• Pilot-B: In the second phase, namely Pilot-B, once the UE has identified the RIS that will be used for the BS-UE communication, it broadcasts this information to all the other RISs and, through them, to the BS.• Data-C: In the third phase, which is the real data transmission, namely Data-C, the BS transmits the data, and the selected RIS, which now knows the BS-RIS channel, thanks to phase Pilot-A, and the RIS-UE channel, thanks to phase Pilot-B, can adjust its own phases and reflect properly the signal to the UE.
To describe the RIS selection algorithms considered in the paper we here provide the following definitions.
Definition 1: A RIS is defined to be available if it is in a favorable orientation and there are LOS conditions for the BS-RIS and RIS-UE links, respectively.We define p A (x) as the probability that a generic RIS located in x is available for BS-UE communication.
Definition 2: A RIS is defined to be selectable if: i) it is available, according to Definition 1; ii) the SNR at the UE, after Pilot-B phase, is larger than γ S .We also define N S as the RV representing the number of selectable RISs distributed in an area S.
Based on the previous definitions, we propose in this paper the following RIS selection algorithm: • Random choice among the selectable RISs (RS): The UE randomly selects a RIS among the ones able to provide the UE with a signal larger than γ S during Pilot-A phase.This algorithm is extremely simple and does not require accurate estimation of the received SNR.
The RS algorithm is compared with the following benchmarks that require additional pilot phases: • Random choice among the available RISs (RA): The UE randomly selects a RIS among the set of all the available ones, according to Definition 1.This algorithm assumes that the UE is able to discriminate the signals coming from all the available RISs regardless of the capability of the UE to decode the signal during the pilot phase.This is another well-known reference algorithm for many papers in the literature [36].It is worth noting that the adoption of RA and LA requires that the UE knows the power that all the available RISs can provide it once they perform beamforming.Notably, the implementation of these two mechanisms in realistic scenarios is rather complex, as requires additional steps.On the other way round, LS can be more easily implemented in practice, but it requires the knowledge, at the UE, of all the powers from the selectable RISs.We, therefore, consider algorithms LS, RA, and LA as reference algorithms, and the performance of RS is evaluated with respect to these benchmarks.Fig. 2 depicts the thinning processes from the set of blockages to that of selectable RISs according with RS or LS algorithms.

C. Received Signal and SNR Models
We first provide a general model for the received signal at the UE, which can be specialized for any case of interest considered in this work.Under the condition of slow and flat fading channels, the baseband signal y, received by the UE and reflected from the M RISs located in the surroundings, can be expressed as [38], [46] g BRm,i e jαm,i e −jϕm,i g RUm,i e  link between the BS and the UE, and 0 otherwise, ζ(x m ) is a parameter which takes the deterministic components of the channel gain between the BS and the m th RIS (located in x m ) and between the m th RIS and UE into account.The coefficients g BRm,i and g RUm,i are the amplitudes of fading components between the BS and the i th element of the m th RIS and between the i th element of the m th RIS and the UE, respectively; α m,i and β m,i are the phases of the fading components; ϕ m,i is the m th RIS phase defined in (1), x is the transmitted signal with E |x| 2 = 1 and n is the thermal noise component, modelled as Complex Gaussian RV with E |n| 2 = σ 2 n .Finally, we assume that ν, α m,i and β m,i are uniformly distributed in (−π, π]. According to (2), once the m th RIS in x has been selected for the phase Data-C, assuming 1 D = 0, it holds 2 g BRi e jαi e −jϕi g RUi e jβi x + n (3) and the SNR at the UE can now be written as where g BRi e jφi g RUi Given the aforementioned signal model, in the following we provide a discussion about the outage analysis under different RIS phase settings.

III. OUTAGE PROBABILITY EVALUATION
We now evaluate the outage probability as the probability that the SNR at the UE (indicated with γ) falls below a given threshold, that is P out ≜ P{γ < γ T }.According to 2 From this point, we omit the index m to refer to the selected RIS.
the assumptions of Section II, the outage probability for the considered system can be written as where P{D} is the probability that there is a direct link between UE and BS, P out|D denotes the probability that an outage event occurs in the presence of the direct link, whereas P out| Ĩ accounts for the fact that there is no selectable RIS or, when the indirect link is present, there is an outage event.
To this purpose, P{N S = 0} indicates the probability that there is no selectable RIS, and P out|NS>0 (i.e., P out|I ) represents the outage probability conditioned on the fact that at least one indirect link can be used for BS-UE communications.Note that (6) considers either direct UE-BS communication or, when the UE-BS are in NLOS, communications occur using UE-RIS and RIS-BS links provided that there is at least one selectable RIS (i.e., P{N S > 0}).
In order to characterize P out , in the following we evaluate the probabilities that appear in (6).

A. Evaluation of P{D}
To evaluate P{D} we consider the probability that there is a LOS condition between the BS and the UE, that is, the probability that there are no blockages between the path between the UE and the BS.It is worth noting that the probability that there is a LOS condition between two nodes located at a given distance in the presence of PPP of blocks modelled as lines of length ℓ has been investigated in [47].Such a probability can be specialized for our case of interest as

B. Evaluation of P out|D
When LOS is present between the UE and the BS, outage occurs only if the SNR of the direct link falls below γ T .
In particular where F gBU (•) indicates the CDF of the RV denoted by g BU .

C. Evaluation of the Distribution of N S
To obtain some insights on the distribution of N S we introduce the following Proposition.
Proposition 1: Given the PPP Φ R (λ R ) and Definitions 1 and 2 in Section II-B, it holds: i) The set of RISs defined as selectable, Φ S ⊆ Φ R , is an inhomogeneous PPP with density where x indicates the location of the RIS, Ξ 0 is the SNR during phase Pilot-A, with Ξ 0 obtained from (5) when N b = 0, and p A (x) is the probability that the RIS located at x is available, given by where the terms p LOS-RIS (x) and p E1 (x) are the probability of having a LOS between the BS-RIS and the RIS-UE link, and the probability that the RIS has a favorable orientation with respect to the BS and the UE, respectively.These probabilities are given by and, considering the geometry of Fig. 3, by ii) The mean value of N S is where FΞ0 accounts for the fact that, during the pilot transmission phase, the RIS does not introduce any phase shift (N b = 0).Proof: The proof is given in Appendix A. Remark 1: From the results of Proposition 1 we can infer that N S is a Poisson RV with mean value µ S , therefore P{N S = 0} = e −µS .

D. Evaluation of P out|N S >0
To evaluate this probability we start by considering the algorithm RS, that is we assume that the UE chooses randomly among all the RISs identified as selectable.Under this condition we can define Φ T ⊆ Φ S as the set of RISs that are selectable and, after phase Data-C, with a SNR, namely γ, larger than γ T .We can now introduce the following proposition.
Proposition 2: Let Φ T ⊆ Φ S be the set of RISs such that: a) the RIS is selectable in accordance with Definition 2; and b) the SNR at the UE is larger than γ T , then the following results hold: i) Φ T is an inhomogeneous PPP.
ii) Let N T be the number of RISs satisfying conditions a) and b) above, then the mean value of N T is iii) The last probability in (6) for the case RS is iv) When N b = 0 and γ T ≥ γ S , (15) can be written as Proof: The proof is given in Appendix B. It is worth noting that the evaluation of (14) for N b > 0 is rather cumbersome due to the statistical dependence between Ξ Nb and Ξ 0 .Numerical examples (not presented here for the sake of conciseness) show that, for the Rician fading case and N b > 0, the correlation between Ξ Nb and Ξ 0 is rather weak.The correlation also decreases as the Rician K factor increases.Under these conditions, a simple and robust approximation for (14), when N b > 0, is Note that, by definition of Φ T , RISs belonging to the set Φ T are those not experiencing outage as they are selectable and with γ > γ T .This result has an important impact on the outage probability of the indirect link for LS algorithm.In fact, since the LS selection mechanism identifies the RIS with the largest SNR among all the nodes belonging to Φ S , the outage event occurs only when no RIS satisfies the condition γ > γ T , that, in turn, occurs when the set Φ T is empty.This consideration suggests that the outage probability for the indirect link case with the approach LS is simply given by Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
which represents the probability that no RIS belongs to Φ T .On the other way round, according to (15), the outage probability for the indirect link case with RS can be expressed as Finally, the cases RA and LA can be easily evaluated by using the following Corollary.Corollary 1: Under the hypothesis of Proposition 2, RA and LA become special cases of RS and LS with γ S = 0, so that we can write where Proof: The proof is straightforward.

IV. DISTRIBUTION OF THE JOINT RIS-CHANNEL GAIN
We now aim to characterize the distribution of the gain related to the cascaded channel comprising the presence of the RIS, that is, the BS-RIS-UE link.Typically, with the introduction of large intelligent surfaces, researchers have proposed approximations in order to deal with tractable models for the characterization of the wireless propagation channel.In this direction, [48] assumed the presence of uncorrelated Rayleigh fading whereas, in [46], the UE-RIS and the RIS-BS channels were both considered to have a Rician fading distribution.In this way, assuming a perfect recombination of the phases provided by the RIS (namely, perfect RIS profile), it follows that the sum of the product of two RVs with Rician fading distributions has a Gaussian distribution thanks to the application of the central limit theorem (CLT).Undoubtedly, the assumption of a perfect RIS phase profile leads to a dramatic simplification of the analysis, but it holds only for a very particular case of interest, which represents a benchmark for the performance analysis, but which is unfeasible in practical settings [49].Then, [50] considered a double-Nakagami-m channel, which included a random RIS phase profile such that the overall phase is considered uniformly distributed between 0 and 2π and no impact of phase quantization bits was considered.
Here, as in [46], [49], [51], [52], [53], [54], [55], and [56], we assume that g BU , g BRi and g RUi follow a Rician distribution, with CDF given by where the AB label is chosen in the set {BR i , RU i , BU} and where K AB is the Rician K-factor (or shape parameter), the power of the fading components is normalized to 1 (the scale parameter Ω = 1) and Q 1 (•, •) is the generalized Marcum Q-function of order 1 [57].Under such consideration, ( 22) can be substituted in ( 8) to obtain P out|D , whereas the cascaded channel, i.e., the product given by g RUi g BRi , has a double Rician distribution.Furthermore, under the assumption 1 D = 0 and 1 I = 1, the SNR expressed in (4) reduces to where Ξ Nb is defined in (5) and depends on the number of bits N b .To statistically describe Ξ Nb we operate as follows.1) We first consider a general scenario, where we assume that a finite number of bits N b is employed at each antenna element of the RIS.In this case, if the aim is the minimization of φ i using the resolution given in (1), it is straightforward to show that, if α i and β i are assumed to be uniformly distributed in (−π, π], the distribution of φ i becomes 2 N b +1 being the residual of the quantization error. 2) Secondly, in Sec.IV-B, we consider two special cases, i.e., Ξ 0 and Ξ ∞ .Notably, the first distribution considers N b = 0.In contrast, the second one entails an ideal phase profile at the RIS side, i.e., N b → ∞ and Ξ Nb → Ξ ∞ , where the possible phases at each RIS element are not discretized and can realize any value.In this second case we have ϕ i = α i + β i , ∀i such that φ i = 0, ∀i.In the following, we discuss about the statistical description of Ξ Nb .

A. General Model for Ξ Nb
Since a closed-form expression of Ξ Nb is not easily obtainable, we try to derive a distribution that can approximate Ξ Nb .We start by defining and by considering the Real and Imaginary parts of η separately, so that As a first approximation, we suppose that N is sufficiently high to use the CLT.Under this assumption, a possible approach to characterize Ξ Nb is the one proposed in [58], where the approximation for the CDF of |η| is written as in (27), shown at the bottom of the next page, where ϱ is the correlation between ℜ{η} and ℑ{η}, µ ℜ , σ 2 ℜ and σ 2 ℑ correspond to E {ℜ {η}}, var {ℜ {η}} and var {ℑ {η}}, respectively, 3 the Neumann factor, ϵ n , is equal to 1 for n = 0 and equal to 2 for all n = 1, 2, . .., and where 3 The change of notation is made for the sake of conciseness.Note also that it holds µ ℑ = 0.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
dt is the lower incomplete Gamma function.Using (27), the CDF of Ξ Nb can be easily written as 4 This distribution allows us to attain a very accurate approximation but it has the drawback that both the PDF and the CDF require the evaluation of infinite sums.
To obtain a simpler expression for the CDF we start from the application of the CLT to ℜ{η}) and ℑ{η}.Under this assumption, by defining ξ N b i ≜ g BRi g RUi e jφi , we have Then, we observe that ( 26) is the sum of two squared Gaussian RVs, and we approximate Ξ Nb with the Gamma distribution.This choice is justified by the following observations: 1) The Chi-squared distribution is a special case of the Gamma distribution.
2) The sum of two non-central Chi-squared distributed RVs with different variances can be expressed as a weighted sum of Chi-squared RVs [59].
3) The Gamma distribution can well approximate weighted sums of Chi-squared RVs [60].Based on these considerations, and observing that Ξ Nb is the sum of a non-central Chi-squared distribution (ℜ{η}) with a central chi-squared one (ℑ{η}) 5 and such two RVs have also different variances, we here introduce the following Proposition.
Proposition 3: The distribution of Ξ Nb can be approximated by a Gamma function with CDF where Γ L (a, b) is the lower incomplete Gamma function previously defined, and the shape and scale parameters, k and θ, are obtained as 4 Note that the term F |η| (− √ x) does not appear in (29) since it is equal to 0.
In order to validate Prop.3, we randomly generated data for Ξ Nb by setting χ Nb = π/4.By considering that the PDF of Ξ Nb is given as where k and θ are evaluated as in (32), we obtain the data fitting reported in Fig. 4 for a number of RIS elements equal to N = {4, 10, 100}.Notably, the approximate distribution fits well the generated data also for small values of N (i.e., N = 4 and N = 10), even if the Gaussian approximation with the CLT is not accurate due to the low number of employed antennas.
B. Some Considerations for N b = 0 and N b → ∞ The general model can be specialized for two cases of interest, that is, Ξ 0 and Ξ ∞ .Thus, in the following, we discuss how the channel gain characterization varies in these two cases.
1) Distribution for Ξ 0 : This scenario can be easily derived from the previous analysis.Indeed, in this case, starting from (24), it holds φ i ∼ U [−π, π] and the same considerations used to obtain (31) hold, with the CDF of Ξ Nb evaluated for N b = 0, namely Ξ 0 .
2) Distribution for Ξ ∞ : This scenario implicitly considers φ i = 0, ∀i, so that (5) reduces to where Ξ ∞ refers to the fact that we can use infinite bits for the phase shifts.Recalling the same steps of Prop.3, Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.it is possible to first obtain the mean value and the variance, of ξ ∞ i according to Appendix C.Then, again, to determine the distribution of η = i ξ ∞ i = i g RUi g BRi , it can be assumed that a large number of antennas is employed in the RIS so we can apply the CLT and write Finally, the distribution of Ξ ∞ = η 2 can then be related to that of Ξ∞ = Ξ ∞ /var{η} by considering that Ξ∞ = η 2 /var {η} has a non-central chi-squared distribution with 1-degree of freedom and non-centrality parameter λ NCP = E {η} 2 /var {η}.
Thus we have where z = z/var{η}, Q 1/2 denotes the Marcum Q-function of order 1/2.Indeed, the CLT approximation works well for large values of N , which is in general valid when RIS elements are accounted for.Nevertheless, since we are dealing with RVs that are more tractable than before, in this case it is possible to avoid the use of the CLT, and the CDF F η (x) can be written as that of the sum of N double Rician distributed RV (i.e., of where dt is the upper incomplete Gamma function, and with where κ 1 = E {ξ ∞ i } and κ 2 = var {ξ ∞ i } represent the first and second cumulant, respectively.Finally, by using (39) and considering that z > 0, we obtain which is more general than (38).
In this section we have provided tractable expressions and several considerations concerning the cascaded channel BS-RIS-UE together with the RIS phase profile.In the following, we exploit such results to numerically evaluate the performance of the considered scenario.

V. NUMERICAL RESULTS
In order to assess the communication performance in a SRE where RISs and blockages are randomly deployed, we present some results obtained through numerical evaluation of the proposed analytical framework, which is also compared with Monte Carlo simulations in order to test the validity of the proposed approximations.Regarding the model for the channel gain, we assume where k 0 and β are two propagation coefficients which account for losses among the transmission chain and due to the propagation environment, d BR (x) and d RU (x) are the distances between the base and the RIS (located in x) and between the RIS and the UE, respectively [46]. 6Finally, the coefficient k 0 also encapsulates all the other parameters not depending on the distance.

A. Parameter Settings
For the numerical evaluation of the proposed model, we considered an area with size (2 × 2) km 2 , with the BS located at its center with coordinates x B = (0, 0) m with the UE position varied along the x−axis, that is, x U = (x, 0) m, with x ∈ [10, 800] m.The environment is populated with obstacles (with length ℓ = 10 m) of density λ 1 = 10 −4 obstacles/m 2 , and where the number of RIS is set through λ R = 10 −5 obstacles/m 2 .We assumed pt  impact of the number of phase bits employed at each antenna element, we alternatively considered N b ∈ {0, 1, 2, 4}.This choice comes from the fact that RISs could be either very simple, i.e., with N b = 0, in order to reduce their costs, or, in case N b = 4 are adopted, they are sufficient to achieve performance close to the ideal N b → ∞ case, as detailed in the following.Finally, we considered γ S = 3 dB (as a threshold to be considered as a selectable RIS) and γ T = 10 dB (outage threshold).

B. Results
In the following, we provide a detailed analysis of the outage probability related to the indirect link, i.e., for 1 D = 0 and for 1 I = 1, by including the impact of the number of bits employed at the RIS, as well as on the number of antennas.Finally, we end up with results and considerations regarding the RIS selection mechanism.
1) Indirect Link Analysis: Figure 5-top reports the obtained values of µ A , µ S and µ T , that are respectively the expected number of available, selectable, and non-outage RISs, for k 0 = −68 dB and when N = 4, N = 10 and N = 100 antennas are employed at the RIS.In the case of µ T , we initially considered the adoption of the LS mechanism.Lines refer to the validation of the theoretical approximate analysis, whereas markers reproduce data obtained through Monte Carlo simulations, corresponding with the curves with the same color.Indeed, there is an excellent agreement between theoretical and simulated data, which corroborates the validity of the approximation in the proposed theoretical model.Notably, this also holds when a limited number of antennas are used, e.g., for N = 4.
Then, regarding the behavior of µ A , µ S and µ T , note that while µ A is the same in all three configurations since depends only on the considered scenario, µ S and µ T are strongly affected by the choice of N and of N b .Concerning the impact of N , it is worth mentioning two effects: (i) the larger is N , the closer µ S becomes with µ A ; (ii) the larger is N , the more insensitive the system becomes to the choice of N b .The latter effect can be intuitively explained by the fact that a larger number of antennas allows for improvement of the SNR, guaranteeing good coverage even with a scarce phase resolution at each antenna.Indeed, for lower values of N , the impact of the choice of N b is higher.This is well demonstrated by results for N = 4, where only for N b = 4 it results µ T ≃ µ S (see Fig. 5 top-left).Note also that, according to Fig. 5 for k 0 = −82 dB with N = 4 (bottom-left) and N = 10 (bottom-middle), it results that N b = 4 is sufficient to attain the benchmark performance achieved with N b → ∞. 7On the other way round, in the other two configurations, N b = 1 is sufficient to attain very good performance.Such a result is of particular interest for the design of RISs, since it implies that the number of bits can be preserved low (e.g., N b = 1) provided that a sufficient number of antennas is used.Successively, we evaluated how such variations of µ T affect the outage probability of the indirect link when the LS approach is employed and k 0 = −68 dB.Results are reported in Fig. 6-top, and confirm what was previously highlighted as, in all cases, we have a P out| Ĩ lower than 0.2.
Motivated by these results, we then considered a very challenging scenario from the propagation point of view by setting k 0 = −82 dB.Once again, Fig. 5-bottom shows a very good agreement between the theoretical model (lines) and the simulated data (markers).In addition, results reported in Figs.5-bottom and 6-bottom emphasize the impact of N and of N b .More specifically, now the number of selectable RISs drastically changes with N .Indeed, Fig. 5-bottom highlights a larger difference (with respect to Fig. 5-top) between µ S and µ A .In addition, given the fact that the SNR is reduced, it results that the choice of N b affects also the case for N = 10, where 4 bits are still not enough to achieve the same performance as for N b → ∞.On the contrary, it is very interesting to notice that RIS architectures with N = 100 can even employ 0 bits to achieve outage probability for the indirect link lower than 0.2.
2) Impact of the RIS Selection Mechanism: We now investigate the impact of the choice of the RIS selection mechanism on the outage probability of the indirect link.To this purpose, we considered the proposed RS approach, compared with the three benchmarks, that is, the LA, LS and RA.More specifically, we recall that the LA and LS entail the choice of the RIS with the largest SNR among the available and selectable ones, respectively, whereas the RA and RS entail the choice of a random RIS among the available and selectable ones, respectively.
Figure 7 shows the results obtained for k 0 = −68 dB (top) and k 0 = −82 dB (bottom), and for N = 10 (left) and N = 100 (right).As predictable, the LA mechanism, which is the optimal (ideal) algorithm in terms of received SNR at the user equipment, provides the best performance in all scenarios.Indeed, it might happen that a RIS is initially below γ S (i.e., non-selectable) but then, with a suitable rephasing (i.e. with N b = 1), could be capable to overcome the threshold γ T at the user equipment.Note also that, when RISs with a large number of antennas (i.e., 100) are adopted, the performance of all the RIS selection algorithms is very close in this scenario.

VI. CONCLUSION
In this paper we proposed an analytical analysis for a scenario where RISs, which are randomly deployed like obstacles in the environment, are used to extend the communication coverage when the LOS link between the UE and the BS is obstructed.To this purpose, we have first discussed different RIS selection mechanisms, Successively, through numerical evaluation and Monte Carlo simulations of the considered scenario, we compared the performance of the RIS selection methods presented in the paper.In addition, we also demonstrated that there is the possibility to relax the complexity in terms of phase bits provided that a large number of antennas is used.On the other way round, a larger number of phase bits permits the reduction of the number of antennas to achieve reliable performance.Such outcomes are of particular interest for the design of RIS, which can be relaxed according to the communication requirements.

APPENDIX A
For the proof of point i) of Proposition 1 we recall that the removal of points from the original set Φ R (λ R ) following the rules indicated by Definition 1 represents a spatially dependent (to the location) thinning operation [63]; furthermore, since the operation of thinning occurs independently of each other node, the resulting set of points (and also the complementary set) results in an inhomogeneous PPP, Φ A (λ A (x)), with intensity (density) λ A (x) = λ R p A (x).Similarly, the removal of points from the set Φ R (λ R ) following the rule γ (P) > γ S is still an independent thinning operation over the set Φ R (λ R ).The set Φ S (λ S ), which is obtained as the intersection of the sets originated by the two previous thinning operations, is still an inhomogeneous PPP whose intensity is given by ( 9), therefore N S is a Poisson RV.
To evaluate p A (x), we assume that a RIS is located at the position x = (x 1 , x 2 ).According to the definition of RIS available for the BS-UE communication we can identify three events: (i) E 1 (x): the RIS, located in x, has a favorable orientation with respect to UE and BS; (ii) E 2 (x): there is LOS between BS and RIS located in x; (iii) E 3 (x): there is LOS between UE and RIS located in x.Since the three conditions are mutually independent we can treat them separately.To evaluate the first condition we consider the triangle in Fig. 3 defined by the BS, located in (0, 0), the UE, located in (d BU , 0), and the RIS located in x = (x 1 , x 2 ).From the figure it is straightforward to show that the orientation of the RIS, indicated by the angle θ(x), blocks the UE-RIS or RIS-BS link if θ(x) ∈ (−ψ, ω), where ψ and ω are given by Since we assume that the orientation of the RIS is completely random, that is, the angle of orientation is uniformly distributed between (−π, π], the probability that the RIS has a favorable orientation, p E1 (x), becomes (ψ + ω)/(2π), and, through ( 42) and ( 43), becomes (12).
To evaluate the second condition we use (7), which refers to the same condition, therefore and, consequently, it yields to that, together with ( 12), (44), provides (10).To prove ii) we apply the following result [32], [63] µ S = λ R P{(The RIS is available), (γ (P) > γ S )} A S where f x (x) = 1/A S , ∀x ∈ S, as we have assumed that Φ R (λ R ) is a homogeneous PPP.After some algebra, (47) can be written as (13).

APPENDIX B
For the proof of i) of Proposition 2 it is sufficient to recall that the operation of selection leading to the set Φ T from the original set Φ S has the same properties of the thinning operation that led to Φ S from Φ R , described in Appendix A. Therefore, this thinning operation results in a (inhomogeneous) PPP too.For the proof of ii) we can follow the same methodology adopted for the derivation of µ S in (46) and (47).To prove iii) we start by observing that P{(γ < γ T )|N S > 0} = 1 − P{γ > γ T |(The RIS is available), (γ (P) > γ S )} = 1 − λ R A S P{(γ > γ T ), (The RIS is available), (γ (P) > γ S )} µ S (48) where we have used the Bayesian rule and (46).Moreover, (46) can be also applied to the set Φ T and therefore µ T = λ R A S P{(γ > γ T ), (The RIS is available), (γ (P) > γ S )} which gives (15).Finally, to prove iv), by using ii) and iii) for which leads, under the hypothesis that γ T ≥ γ S , to (16).
Analogously, for computing the variance in (51), we can first write since it is straightforward to show that 1 F 1 (−1; 1; −z) = z+1.Consequently, the variance of ξ ∞ i can be expressed as We now consider the phase term, e jφi with φ i ∼ U [−χ Nb , χ Nb ].Being ξ N b i a complex RV, we can treat its Real and Imaginary parts separately by considering e jφi = cos φ i + j sin φ i , that is We have Then, the expectations of the Real and the Imaginary parts of ξ N b i can be represented as follows Regarding the variance, we can write and it is straightforward to obtain Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

Fig. 2 .
Fig. 2. Thinning processes.From left to right: (a): the scenario is populated by a BS and a UE which are placed in fixed positions.The process Φ(λ 1 ) describes the blockage distribution; (b): The process Φ R (λ R ) ⊆ Φ(λ 1 ) represents the RIS distribution, as a subset of the blockage process; (c): The process Φ A (λ A ) ⊆ Φ R (λ R ) represents the distribution of the available RISs; (d): The process Φ S (λ S ) ⊆ Φ A (λ A ) represents the distribution of the selectable RISs, where the red-circle represents the selected RIS.

2 ( 5 )
and φ i ≜ α i + β i − ϕ i .Quantity Ξ Nb accounts for the fading contributions of the indirect link and the phase shift introduced by the RIS with N b bits of resolution.
loss exponent β = 2 and a 1 m path gain alternatively equal to k 0 = −68 dB and to k 0 = −82 dB.Regarding the Rician distribution of the channel, we considered K BU = K BR = K RU = 5 and, as already mentioned, Ω BU = Ω BR = Ω RU = 1.Then, in order to evaluate the

( 46 )
where P(A, B) = P(A AND B) is the joint probability of events A and B, and A S is the measure of area S;(46) can Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.berewritten asµ S = λ R A S S P{E 1 (x) , E 2 (x) , E 3 (x) , (γ (P) (x) > γ S )} × f x (x)dx = λ R S p A (x) P Ξ 0 > γ S ζ(x)p t /σ 2 N b = 0 we have P{(γ > γ T )|N S > 0} = λ R S p A (x)P Ξ 0 > max{γS,γT} ζ(x)pt/σ 2

TABLE I RELATED
WORK ON RIS SELECTION UNDER SPATIAL RIS RANDOM MODEL.THE NUMBER OF BITS USED FOR REPRESENTING THE RIS PHASE IS INDICATED WITH N b Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
• Largest SNR among the selectable RISs (LS): The UE selects the RIS providing the best SNR during Data-C among the set of the selectable RISs.Note that this algorithm leads to the optimal choice (in terms of SNR) among the set of selectable RISs.For this reason, LS can be taken as a theoretical benchmark for all the RIS selection algorithms that choose the RIS among the set of the selectable ones.•Largest SNR among the available RISs (LA): The UE selects the RIS providing the best SNR during Data-C among the set of the available RISs.Notably, LA outperforms LS since it represents the optimal algorithm in terms of SNR.Indeed, there might be a configuration where a RIS with SNR below γ S in Pilot-A phase might allow overcoming γ T in Pilot-B phase by properly rephasing its elements.
2 ℜ and σ 2 ℑ , let us first define ξ ∞ BRi g RUi .Furthermore, since ξ ∞ and φ i are statistically independent, we can evaluate their means separately.Starting from ξ ∞ we observe that both g BRi and g RUi are Rician distributed and therefore ξ ∞ i follows a double Rician distribution, with mean value E{ξ ∞ i } = E {g BRi } E {g RUi } and variance given by var {ξ ∞ i }= var {g BRi }+E 2 {g BRi } × var {g RUi }+E 2 {g RUi } − E 2 {g BRi } E 2 {g RUi } .
i ≜ g