Multiple-Antenna Weibull-Fading Wireless Communications Enhanced by Reconfigurable Intelligent Surfaces

The Reconfigurable Intelligent Surface (RIS) has recently gained traction as a promising, enabling technology for future sixth-generation (6G) mobile networks. Thereby, this paper presents a comprehensive statistical, secrecy, and performance analysis of RIS-assisted multiple-antenna wireless communication systems under Weibull fading. The system model encompasses an antenna array at the source transmitting to a typical single-antenna user through both a generally weaker direct link and an enhanced RIS-aided double-Weibull-fading linear cascaded communication channel. Moreover, the RIS is assumed to perform constructive interference towards the user through realistic non-ideal phase cancellation with residual phase errors that follow the von Mises distribution. Then, the statistical distribution of the resultant channel fading is accurately modeled by the analytically derived moment-generated Gamma distribution in conformity with the central limit theorem (CLT). While considering the existence of a malicious eavesdropper directly linked with the source, relevant performance metrics, expressly the secrecy outage probability (SOP), the symbol error rate (SER), and the spectral efficiency of the system, have their expressions determined for the specific Weibull-fading scenario and verified by Monte Carlo simulations. The study highlights the significant performance improvement of the system arising from a larger number of antenna elements of the RIS, as well as through a more accurate phase cancellation response associated with a higher von Mises concentration parameter, $\kappa $ . Furthermore, the effects of diverse fading scenarios and varying line-of-sight (LoS) propagation strength are evaluated through the Weibull shape parameter, $k$ .


I. INTRODUCTION
The intelligent information society of 2030 [1] expects to count on ubiquitous and seamless wireless connectivity throughout space, air, ground, and underwater environments.Indeed, the so-called Internet of Everything (IoE) is a fundamental driver of sixth-generation (6G) mobile networks, as virtually everything is expected to be near-instantly connected, enabling reliable real-time global data access and data-driven optimization.Moreover, information on vision, hearing, smell, taste, touch, and neural activity, captured by smart wearables and transmitted via the Tactile Internet [2], aims at providing a complete sensory experience within a human-centric environment.On top of that, augmented reality (AR) and virtual reality (VR) gadgets powered by extremely high-definition (EHD) video and low-latency communications will usher in a new era for mission-critical remote and autonomous applications [1].
The enhancement of the key performance indicators (KPIs) of current fifth-generation (5G) mobile networks is thus essential, raising the data transmission rate to multi-terabytes per second (Tb/s) through an autonomous network architecture enabled by artificial intelligence (AI).Specifically, the KPIs of 6G networks include peak data rate of at least 1 Tb/s, user-experienced data rate of 1 Gb/s, over-the-air latency inferior to 10 µs, mobility above 1000 km/h, connectivity density of up to 10 7 devices/km 2 , as well as energy and spectrum efficiency 100 and 10 times those of 5G, respectively [3].Thereby, enabling technologies for next-generation mobile networks encompass Terahertz (THz) communication, holographic beamforming (HBF), orbital angular momentum (OAM) multiplexing, visible-light communication (VLC), blockchain-based spectrum sharing, quantum communication and computing, molecular communication, and the Internet of Bio-Nano Things (IoBNT) [1], [3].
The extremely large-scale multiple-input multiple-output (XL-MIMO) antenna configurations of 6G mobile networks are expected to deploy over 10,000 antenna elements in a single array [1].Then, spatial multiplexing and diversity through highly-directional beamforming, while simultaneously serving a multitude of users, will increase reliability and significantly boost system capacity [4].Notably, the Reconfigurable Intelligent Surface (RIS), also referred to as Large Intelligent Surface (LIS), evolved from the promising, advanced antenna technology of metasurfaces.The RIS consists of mostly passive reflectarrays with controllable phase and amplitude response, working as dynamic, spatially continuous reflective surfaces [5], [6].Then, such surfaces could be conveniently spread throughout indoor and outdoor environments, significantly extending the signal coverage area and leading to predominant line-of-sight (LoS) wireless propagation.Furthermore, RIS-assisted wireless communication systems are expected to be very cost-effective at generating holographic beamforming with higher spatial resolution [1], [3].

A. RECONFIGURABLE INTELLIGENT SURFACES
The reconfigurable intelligent surface technology in multiple-antenna wireless communications under Weibullfading propagation is the core subject of the present study and will be further analyzed in the following sections.

1) APPLICATIONS
In the realm of wireless communications, reconfigurable intelligent surfaces have emerged as a promising technology with a multitude of use case scenarios.These scenarios encompass both indoor and outdoor environments, nearfield and far-field propagation, line-of-sight and non-lineof-sight (NLoS) conditions, employment of millimeter-wave (mmWave) frequencies, and smart vehicular networks.
Indeed, there has been extensive research interest in the potential of the RIS to address a wide range of practical challenges in wireless communications.For instance, Perović et al. [7] focused on typical indoor millimeter-wave environments that are sparsely scattered and strongly dependent on line-of-sight propagation, while Garcia et al. [8] explored the near-field electromagnetic response of impedance-controlled RIS, investigating the dependence of the scattered signal power on the RIS dimensions and its distance from the base station (BS).On the other hand, Najafi et al. [9] investigated different transmission modes for path-loss mitigation in RISassisted far-field wireless communication settings.
Beyond indoor environments, the benefits of the RIS technology extend to vehicular networks, once Ai et al. [10] and Makarfi et al. [11] highlighted the secrecy improvement brought by the RIS in vehicle-to-vehicle (V2V), vehicle-toinfrastructure (V2I), and vehicular ad-hoc network (VANET) configurations, considering factors such as eavesdropper distance, secrecy threshold, and the Doppler effect on moving network nodes.Furthermore, on top of the increased bandwidth of millimeter-wave RIS-aided wireless communication systems, the potential of reconfigurable intelligent surfaces also lies in their ability to enhance localization under nonline-of-sight propagation [12].

2) CHALLENGES
However, the practical implementation of the RIS technology in wireless communications is subject to several structural and environment-related constraints, including medium and channel fading variations, diverse scattering patterns, unfavorable surface geometry and topology, distance to the base station, quantization errors, and non-linearity due to electromagnetic coupling and wave reverberation.Rabault et al. [13] raise doubts about the tacit assumption of structural linear RIS parametrization in the traditional linear cascaded channel models widely used in the literature.They present experimental evidence from rich-scattering environments and advocate for the physics-compliant end-to-end channel model PhysFad [14] to address non-linearity in practical RIS-parameterized wireless systems.The non-linearity arises predominantly from the structurally inherent mechanisms of proximity-induced mutual coupling between nearby RIS elements and reverberation-induced long-range coupling between all RIS elements in scenarios of multi-bounced wave trajectories, thus leading to linearity truncation errors after the first term of a Born series [15], [16] and after the second term of a Born-like series, respectively.
Considering the further challenges posed by the particularities of the transmission medium to beyond-fifth-generation (B5G) wireless technologies, given the increased hardware cost and power consumption necessary to cope with generally unfavorable electromagnetic propagation, Boulogeorgos and Alexiou [17] call attention to the fact that the RIS can exploit the implicit randomness of the propagation environment to provide the communication system with amplifyand-forward (AF) capabilities without employing a power amplifier.Garcia et al. [8] use Fresnel zone decomposition and signal characterization to demonstrate that the radiated energy may depend on the fourth, third, or second power of the distance to the RIS, which can be perceived as an object with zero, one, or two dimensions, respectively.Moreover, practical, hardware-limited RIS-aided wireless systems are inevitably affected by quantization errors; hence, Trigui et al. [18] investigated the impact of quantized phase shifts on the mathematical system model.

3) MATHEMATICAL MODEL
Consequently, various mathematical models have been proposed to accommodate different wireless network configurations and fading scenarios.For instance, Basar [19] and Boulogeorgos and Alexiou [17] proposed mathematical frameworks for calculating the symbol error probability (SEP) of RIS-aided wireless communication systems with Rayleigh-fading cascaded channels and non-line-ofsight propagation between source and destination.Similarly, Yang et al. [20] presented a RIS-assisted wireless system model for Rayleigh-fading component channels and derived closed-form expressions for the secrecy outage probability (SOP) under the influence of an unwanted eavesdropper link.In its turn, the work of Trigui et al. [18] extended its analysis of quantized phase shifts and their impact on the probability density function (PDF) and characteristic function (CHF) of the combined signal amplitude to the development of exact Monte-Carlo-verified expressions for the bit error rate (BER) of binary phase shift keying (BPSK) modulation.

4) CHANNEL STATE INFORMATION
Furthermore, the proper operation of reconfigurable intelligent surfaces relies on accurate real-time channel state information (CSI), predominantly obtained through machine learning (ML) techniques.On that account, Taha et al. [21] proposed a novel RIS architecture that yields precise CSI estimation with drastically reduced training overhead through minimal active broadband-connected RIS cells.Lin et al. [22] approach the same channel estimation problem as a constrained minimization of the estimation error, with the estimator being iteratively obtained through Lagrange multipliers and a dual ascent-based estimation scheme following the Karush-Kuhn-Tucker (KKT) conditions.
Notably, the twin deep learning (DL) framework based on convolutional neural networks (CNNs) proposed by Elbir et al. [23] consists of a training stage algorithm that builds a nonlinear relationship between thoroughly transmitted pilot signals from users and the channel state information, yielding reasonably accurate channel estimation with separate data in the subsequent prediction stage.Additionally, practical CSI estimation schemes that partition RIS elements into groups or clusters have also been proposed for wireless communications [9], [24].First, the RIS antenna elements are grouped, and the combined channel for each group is estimated with limited computational overhead; then, the transmit power allocation and RIS reflection coefficients are jointly optimized through iterative numerical computation [24].Alternatively, in the first offline stage, the RIS clusters or tiles are designed to support different transmission modes; then, in the following online optimization stage, the best transmission mode is selected for each fading realization, such that the quality of service (QoS) is maximized [9].

5) ARCHITECTURE
Building upon the benefits of the RIS technology in light of the aforementioned diverse use cases and wireless propagation environments, multiple design techniques have been employed to harness the full potential of reconfigurable intelligent surfaces for next-generation wireless networks.For instance, Wymeersch et al. [12] highlighted the potential of RIS to improve radio-based simultaneous localization and mapping (SLAM), while Hu and Rusek [25] extended the analysis of reconfigurable intelligent surfaces to threedimensional (3D) spherical designs, whose geometrical symmetry favors terminal positioning.Moreover, by coating the signal blockages of cellular networks with malleable reconfigurable intelligent surfaces through a well-planned deployment scenario from a stochastic geometry perspective, multiple alternative indirect paths can be established between users and base stations [26].
Additionally, focusing on the enabling technologies for sixth-generation mobile networks, Mukherjee et al. [27] highlighted the interplay between reconfigurable intelligent surfaces and mobile edge computing (MEC) network architectures.Indeed, RIS and MEC can effectively benefit from each other since the computing, storage, and cache capabilities of edge users can be suitably employed for the MLoptimized high-dimensional operations of the RIS, which, in turn, considerably favors communication among network nodes and cloud servers for the resource allocation of the MEC.Another insightful design strategy consists of leveraging the different possible RIS configurations towards the users by switching between configurations with a suitable periodicity for improved security [28]; however, the throughput performance loss inherent to the switching process reveals an inevitable trade-off between secrecy and data transmission rate following this approach.

6) PERFORMANCE
Comparative studies have indicated that the RIS technology offers significant performance gains over traditional wireless networks [8], [19].Boulogeorgos and Alexiou [17] have shown that RIS-assisted wireless communications outperform standard AF-relaying systems in terms of average signal-to-noise ratio (SNR), outage probability (OP), symbol error rate (SER), and ergodic capacity (EC).Additionally, Yang et al. [24] showcased a superior maximum data transmission rate through RIS assistance, while Ferreira et al. [29] demonstrated a reduced bit error rate for a RIS-aided wireless system model with line-of-sight propagation between source and destination.Approaching the RIS deployment from a stochastic geometry perspective, with BS locations modeled as a Poisson point process (PPP), Kishk and Alouini [26] verified that the RIS considerably increases the line-of-sight probability and the density of visible base stations for a generic user under a predefined path-loss threshold.Accounting for quantization errors, Trigui et al. [18] established that RIS-assisted wireless systems can achieve full diversity order with more than two quantization levels.

7) SECURITY
Moreover, on top of the aforementioned numerous advantages of the smart radio wave propagation environment provided by reconfigurable intelligent surfaces, the security of wireless communications has also shown to be substantially improved [20].Indeed, Ai et al. [10] shed light on the secrecy performance of RIS-assisted vehicular networks in both vehicle-to-vehicle and vehicle-to-infrastructure scenarios; under the presence of an unwanted passive eavesdropper, it is shown that the increase in the number of RIS antenna elements consistently enhances the physical layer security (PLS) through a log-scaled linear relation, whereas the gain from increasing the signal-to-noise ratio reaches a saturation point.Likewise, Makarfi et al. [11] focused on the physical layer security analysis of vehicular networks, showing that a larger number of RIS cells and shorter average sourceto-relay distances significantly improve the average secrecy capacity (ASC) and secrecy outage probability.

8) OVERVIEW
In summary, reconfigurable intelligent surfaces offer significant potential for the enhancement of wireless communications in multiple scenarios.Different aspects of the RIS technology, including its applications, channel modeling and estimation, design strategies, and optimization techniques, have been explored through mathematical, numerical, and experimental analysis.Despite the challenges posed by physical and implementation constraints, the benefits of the RIS are evident in terms of increased capacity, improved security, and enhanced performance.By addressing its main challenges and leveraging these many advantages, the RIS technology shall pave the way for highly sophisticated wireless networks in the future.

B. CONTRIBUTION
This paper presents significant contributions to the field of wireless communications with reconfigurable intelligent surfaces.Building upon previous studies that have extensively investigated RIS-assisted wireless propagation environments through the utilization of various statistical distributions for modeling the propagation fading, such as the Rayleigh [30], Nakagami-m [31], [32], and Rician distributions [33], this work takes a step forward by considering Weibull-fading cascaded channels and extending the error probability analysis of RIS-aided wireless systems to address the spectral efficiency of multi-antenna configurations and the security performance of wireless communication links confronted with line-of-sight propagation from the source to a malicious eavesdropper.A notable insight behind the choice of the Weibull distribution is its justification based on the spatial correlation of non-homogeneous scattering surfaces under nonlinear-environment propagation, as demonstrated by Yacoub [34].From a practical perspective, there are also other sources of non-linearity in the communication link, such as the power amplifier (PA) at the transmitter and the low-noise amplifier (LNA) at the receiver, whose effects on the propagation fading conform to the Weibull distribution.Therefore, this rationale highlights the appropriateness of the Weibull-fading channel model in capturing the complex wireless propagation characteristics encountered in real-world settings.
Indeed, a distinguishing feature of this study is evaluating a wide range of fading scenarios through the variation of the Weibull shape parameter k, which offers increased flexibility in modeling diverse wireless propagation environments.Additionally, the inherent presence of phase errors arising from imperfect channel phase cancellation is suitably modeled using the zero-mean von Mises distribution so that the impact of channel estimation accuracy can be evaluated through the variation of the von Mises concentration parameter κ, reflecting different levels of estimation precision.This aspect is essential for practical systems, where accurate channel estimation strongly impacts the overall performance.On top of that, the incorporation of additional direct links from the source to both the user and the eavesdropper also accounts for a more comprehensive analysis of the system, while the quality of the proposed approximation of the resultant channel fading by the Gamma distribution is validated through the metrics of Hellinger distance and Kullback-Leibler divergence.For the proper structural linearity assumption of a system transfer function free of relevant RIS-related coupling mechanisms [13], [35], this study considers minimal proximity-induced coupling, achieved through optimal topological surface design and vanishing RIS scattering cross-section, with sufficient multiple-wavelength separation between source, destination, and the RIS [13].
This research also aims to cover some gaps in the literature by providing analytical closed-form expressions for the parameters of the resultant channel fading distribution over Weibull-fading components.The previous related work of Badiu et al. [33] follows the central limit theorem to analytically address more basic single-antenna transmissions over Rayleigh-and Rician-fading component channels, while Björnson et al. [36] propose an asymptotic approach that notably accounts for the correlation between RIS elements but is also restricted to the context of a single-antenna transmitter.In fact, the analytical derivation of the resultant channel fading distribution parameters can become quite intricate due to the considerable sums and transformations of distinct random variables.Nevertheless, this study achieves remarkable results for a considerably generalized system model that accommodates direct links with the user and the eavesdropper, multiple antennas at the base station, and imperfect phase correction performed by the RIS.
Furthermore, the additional contributions of this work revolve around the analytical derivation, simulation, and discussion of essential performance metrics, expressly the symbol error rate, secrecy outage probability, and spectral efficiency of the system.These unprecedented evaluations, specially tailored for the scenario of Weibull-fading cascaded channels, provide valuable insights into the behavior and limitations of the proposed RIS-assisted wireless system model, contributing to the advancement of wireless communications.

C. ORGANIZATION
This paper is organized as follows: Section II defines mathematical notation, then Section III presents the system model and initial equations.Section IV follows with analytical expressions for spectral efficiency, average SER upper bound, and SOP.Next, Section V verifies the proposed analytical expressions through Monte Carlo simulation results and further discussions, with Section VI having the final considerations.For presentation clarity, the analytical calculations for the mean and variance of the overall channel fading distribution are left for Appendix A.

II. NOTATION
The mathematical notation adopted in this paper considers E[X ], var(X ), and cov(X ) as the expected value, variance, and covariance of the random variable X , respectively.The function I p (.) is the modified Bessel function of the first kind and order p, and Q(.) is the Gaussian error function.The term X H represents the Hermitian, meaning the transposed conjugate, of the complex matrix X, and z ∈ C is an element of the set of complex numbers.The letter L in subscript, as in h SL , stands for either LIS or RIS.

III. SYSTEM MODEL
The system model covered in this study and represented in Fig. 1 considers a base station at the source equipped with an array of M antennas transmitting the same signal to a single-antenna destination user.Additionally, the transmission is aided by a reconfigurable intelligent surface with N passive reflecting antenna elements.Both channels, from BS to RIS and from RIS to user, have fading modeled by the Weibull distribution.Furthermore, two additional direct links, one complex-Gaussian-distributed between the BS and the user and another Weibull-distributed between the BS and an eavesdropper, are also included in the analysis.Hence, under the assumption of a system transfer function that depends linearly on the structural RIS parameters [13], the signal received at the destination can be determined by where h LD ∈ C N ×1 is the link between the RIS and the destination, h SL ∈ C M ×N is the link between the source and the RIS, and h SD ∈ C M ×1 is the direct link between the source and the destination.The term ∈ C N ×N is a diagonal matrix, whose elements correspond to the phase shifts, e −jφ 1 . . .e −jφ N , performed on the incident electromagnetic waves by the RIS.These phase shift angles, φ i ∀i, are assumed continuous in the interval from 0 to 2π radians.The term = vs represents the precoded signal, where s is the unitary data symbol, and the optimal precoding vector, v, is determined at the BS following the maximum ratio transmission (MRT) criterion, i.e., Finally, the term η ∈ C 1×1 is the additive white Gaussian noise (AWGN), with zero mean and unit variance.
As previously stated, the direct link between the BS and the user is assumed complex-Gaussian-distributed, with zero mean, variance σ 2 SD , and uncorrelated components, effectively representing the Rayleigh envelope distribution.This choice is commonly employed to model propagation fading in far-field urban scenarios, considering the characteristics of the wireless propagation environment.
The magnitudes of component channels, h SL,SD , conform to the Weibull distribution with probability density function given by where k and λ are the shape and scale parameters, respectively.Then, the resultant channel can be represented by whose scalar representation is Perfect phase cancellation occurs when However, such an assumption is entirely unfeasible.Therefore, a residual phase noise, θ ik = φ i − φ SL ik − φ LD i , must be expected.Although perfect phase cancellation is unrealistic, it is possible to estimate an optimal phase adjustment matrix to bring phase errors, on average, to zero.That way, the zero-mean von Mises circular distribution can be suitable for modeling the phase errors associated with each antenna element [37], given its nonzero support in the interval from −π to π and concentration parameter κ associated with the quality of both the accuracy of the channel estimation method and the corresponding phase correction performed by the RIS.Hence, this work models the phase noise as a von Mises random variable with concentration parameter κ.
Furthermore, the moment-generating function (MGF) of the von Mises distribution becomes handy for analytical calculations since complex exponential functions can suitably represent phase shifts.Let X be a von Mises random variable, with an MGF given by ϕ p = E e −jpX = α p + jβ p .Since the zero-mean von Mises distribution is symmetric about zero, the imaginary part of the MGF is β p = E [sin pX ] = 0, while its real component is α p = I p (κ) I 0 (κ) .Hence, the overall channel scalar representation can be rewritten as Considering the precoder to be the normalized hermitian of the overall channel matrix and for unitary noise power, the signal-to-noise ratio of the desired link equals the resultant channel fading as in Then, assuming that γ D can be suitably modeled by the Gamma distribution, with probability density function defined by its shape and rate parameters, α and β, respectively, can be estimated through the statistical moments as follows: where E [γ D ] and var (γ D ) are the expected value and variance of γ D , respectively, as shown throughout Appendix A. Moreover, the Hellinger distance can be useful for assessing how close the resultant channel fading distribution is to the Gamma distribution.According to Beran [38], the Hellinger distance between two arbitrary discrete probability distributions p k and q k can be obtained by where N p is the number of samples available from both distributions to calculate the distance.The Hellinger distance is limited in the interval 0 ≤ D HL ≤ 1 and can be considered an absolute metric.
Likewise, the Kullback-Leibler divergence, whose expression for two arbitrary discrete probability distributions p k and q k is given by can also be employed with the same purpose.In Fig. 2 and 3, Monte Carlo simulations of the overall channel yield histograms and experimental probability mass functions (PMFs) that are compared to the Gamma distribution through both the Hellinger distance and the Kullback-Leibler divergence.The results show that both metrics significantly decrease with the increase in N and M , in accordance with the central limit theorem (CLT).Such convergence motivates the further efforts of this work.

IV. PROBLEM FORMULATION
Having the resultant signal-to-noise ratio of the system been accurately approximated by a Gamma-distributed random variable, the following sections present analytical closed-form expressions for the spectral efficiency, average symbol error rate (SER) upper bound, and secrecy outage probability against an undesired eavesdropper for the specific scenario of Weibull-fading component channels.

A. SPECTRAL EFFICIENCY
The average spectral efficiency of the system can be defined as the expected value of the AWGN channel capacity over the wireless communication bandwidth [39], resulting in given a Gamma-distributed signal-to-noise ratio, γ .The integral (12) has a more generic solution [40] that results in the following expression for the spectral efficiency: where ψ (0) (.) is the digamma function, (.) is the gamma function, (., .) is the incomplete gamma function, and 2 F 2 (., .; ., .; .) is the generalized hypergeometric function.

B. SYMBOL ERROR RATE
The symbol error rate or symbol error probability for the m-ary quadrature amplitude modulation (m-QAM) [41] can be properly approximated by where γ is the signal-to-noise ratio of the system.Then, assuming that γ is Gamma-distributed, the average symbol error probability, PQAM e , can be calculated through where f ∥h∥ 2 (.) refers to the overall channel Gamma distribution PDF, and γ is the quadratic mean of the SNR, γ .Next, the following close upper bound for the symbol error probability of an m-QAM scheme [41], which is further limited by the Chernoff bound, i.e., yields a reasonably tight upper bound for the average symbol error probability as follows: which results in

C. SECRECY OUTAGE PROBABILITY
Considering that an eavesdropper has access to the signal provided by the source, the secrecy capacity associated with the two fading channels involved [42] can be obtained by where γ E represents the signal-to-noise ratio of the link between the source and the eavesdropper, while γ D refers to 107224 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
the legitimate overall channel SNR.Then, the SOP is defined as the probability of the instantaneous secrecy capacity, C, to be less than or equal to a given capacity associated with an SNR threshold, γ th , as follows: where Pr [.] denotes the probability of a random event.
Therefore, considering a Weibull-fading eavesdropper channel, it follows that where k and λ are the parameters of the eavesdropper channel fading distribution, while α and β are the overall channel fading Gamma distribution parameters.
After solving the first integral, the remaining expression can be rearranged as where (α, β(γ th x + x + γ th )) represents the upper incomplete gamma function defined by Then, the term (α) − (α, β(γ th x + x + γ th )) can be rewritten as the lower incomplete gamma function, γ (α, β(γ th x + x + γ th )), defined as which results in the following integral for the SOP: Given the relatively increased generalization of the Weibull distribution, the analytical convergence of the integral (26) for a generic k is curbed by the term e −( x /λ) k .Nonetheless, the Taylor series expansion of the exponential function around the origin, i.e., thus for a sufficiently large λ and through further mathematical development, yields the following expression for the secrecy outage probability: where 1 F1 (a; b; c) is the regularized generalized hypergeometric function.Furthermore, the lower incomplete gamma function in ( 26) can be expressed by its series expansion, i.e., Although the expression in ( 29) is an infinite sum, the approximation given by its first terms can be considerably tight, as demonstrated in the numerical results section.

V. NUMERICAL RESULTS
This section analyzes the accuracy of the proposed model and discusses the performance improvements provided by the RIS.In unspecified cases, this study adopts k = 3 and λ = 1 for the shape and scale parameters of the Weibull distribution, respectively, with variances σ 2 SL = σ 2 LD = 1.Additionally, if not specified, the von Mises concentration parameter is assumed κ = 2, as well as M = 12 antennas at the source, a 16-QAM constellation, and Monte Carlo simulations with 10 6 iterations.Fig. 4 and 5 show the simulated and theoretical SER with varying number of RIS antenna elements and von Mises concentration parameter, respectively.The theoretical SER is obtained under the assumption of a Gamma-distributed overall channel fading.It can be noticed that the larger the number of RIS reflectors, N , the lower the error rate.Likewise, the higher the von Mises concentration parameter, κ, the lower the error probability.
Thus, for a propagation environment with a fixed value of k, the increase in the number of RIS antenna elements, N , as well as the improvement of the κ-related phase adjustment performed by the RIS, can meaningfully reduce the symbol error rate in RIS-aided wireless communication systems.Specifically, by raising the number of RIS antenna elements from 16 to 32, the same symbol error probability can be achieved with a signal-to-noise ratio of 8 dB lower.Moreover, by increasing the von Mises concentration parameter from 2 to 4, the equivalent SER can be obtained with a 2 dB lower SNR; then, the overall error probability performance is less sensitive to further increases in κ.
These results showcase the success rate improvement provided by the RIS and the importance of accurately estimating  its phase shift response through the best optimization method.On the contrary, the worst-case scenario with uniformly distributed phase noise implies a higher symbol error probability, which can be mitigated only by an excessively large number of antennas at the transmitter or reflectors at the RIS.
Reconfigurable intelligent surfaces can boost the line-ofsight strength between the source and destination as a result of beamforming towards the user.Indeed, Fig. 6 shows stronger LoS components with an overall SNR of 2 dB lower for a fixed symbol error rate when the Weibull shape parameter, k, is increased from 2 to 8. Furthermore, the SER upper bound ( 19) is demonstrably tight, especially under a high-SNR regime, with a gap of less than 1dB, as shown in Fig. 7.
For the computation of the spectral efficiency according to the number of RIS antenna elements, N , this study evaluates different scenarios by varying the number of base station antennas at the source, M , the LoS strength through the Weibull shape parameter, k, as well as the von Mises   concentration parameter, κ, as indicated in Fig. 8, 9 and 10, respectively.Notably, the spectral efficiency increases when  the RIS counts on a larger number of reflectors or when there are multiple antennas at the source, thus indicating better spectrum sharing.Likewise, the greater the LoS strength, the higher the spectral efficiency.Furthermore, such efficiency is even more prominent when the phase error distribution has a higher von Mises concentration parameter, κ, reinforcing channel estimation and phase correction accuracy to be pivotal for performance improvement.It is also remarkable that the results from the theoretical spectral efficiency expression perfectly match the ones obtained through Monte Carlo simulations.
The secrecy outage probability over a Weibull-fading eavesdropper link with k = 3 and λ = 1 for the shape and scale parameters of the Weibull distribution, respectively, is shown in Fig. 11.The von Mises concentration parameter is κ = 2, and the base station is equipped with M = 8 antennas.Additionally, Weibull-distributed fading with unit variance is assumed for the channels between the source and  the RIS and between the RIS and the destination.It can be seen that the larger the number of reflectors at the RIS, the lower the SOP.Likewise, if the number of base station antennas, M , is increased while the number of RIS reflectors, N , is kept at 32, a very similar secrecy performance gain is obtained, as shown in Fig. 12.Hence, these results confirm the performance boost brought by multiple-antenna RISaided wireless network configurations also from a security perspective.Moreover, the approximation of the SOP by the first summation terms in the lower incomplete Gamma series expansion can be very close to the simulated results as shown in Fig. 13 and 14.For 70 summation terms, theoretical and simulation results practically coincide.
Finally, to further demonstrate the versatility of the Weibull distribution in representing diverse fading scenarios, given other works in the literature that deal with Nakagami-fading channels [31], [32], [33], for a Weibull shape parameter k = 2, the behavior and performance metrics of the system coincide with the scenario of simulated Nakagami-m fading with shape parameter m = 1, as illustrated in Fig. 15, 16, and 17.

VI. CONCLUSION
In conclusion, this work significantly advanced the field of wireless communications assisted by reconfigurable intelligent surfaces through the in-depth investigation of Weibullfading linear cascaded channels since the Weibull distribution satisfactory captures the intricacies of practical wireless propagation settings.Furthermore, while attesting to the accurate representation of the resultant propagation fading by the Gamma distribution, in conformity with the central limit theorem, this research successfully filled some gaps in the existing literature by deriving analytical closed-form expressions for the parameters of the overall channel fading distribution.Building upon prior related works focused on more straightforward scenarios, restricted environments, or alternative distributions, this study presented a more generalized system model that simultaneously accounts for direct links with both the user and potential eavesdropper, multiple antennas at the base station, and imperfect phase correction performed by the RIS.These results laid the foundations for the analytical derivation of closed-form expressions for the signal-to-noise ratio distribution and performance metrics of the system, specifically tailored for the Weibull-fading scenario.
Moreover, the contributions of this work were not limited to theoretical advancements, as practical, valuable insights were also gained through rigorous numerical Monte Carlo simulations and further discussions of the symbol error rate, secrecy outage probability, and spectral efficiency of the system.Exploring a wide range of fading scenarios through the variation of the Weibull shape parameter showcased an approach with increased flexibility in modeling diverse wireless propagation environments.At the same time, the employment of a standard, versatile m-QAM modulation scheme paved the way for various types of digital modulation through further variable manipulations.On top of that, leveraging the zero-mean von Mises distribution for modeling phase errors allowed for evaluating the crucial impact of channel estimation accuracy on ensuring optimal performance in wireless communication systems.
In general, the numerical evidence indicated that the performance metrics exhibit notable improvement when the RIS configuration counts on a larger number of reflecting antenna elements and also through an extended number of antennas at the base station.Furthermore, the shape parameter k of the Weibull distribution has demonstrated a direct correlation with the line-of-sight propagation strength, while the von Mises concentration parameter, κ, has been verified to be directly proportional to the channel estimation accuracy.Asymptotically, as κ approaches infinity, the probability density function of the zero-mean von Mises distribution becomes highly impulsive at zero, thus indicating a precise phase estimation.Conversely, in the worst-case scenario where κ approaches zero, the phase estimation errors conform to the uniform distribution, demonstrating an impaired beamforming capability that requires a considerably large number of RIS reflectors to be counteracted.
In summary, the analytical rigor and practical relevance of the findings make this work a valuable resource for researchers and engineers seeking to harness the potential of reconfigurable intelligent surfaces to enhance the performance and security of wireless communication systems.In future research, it would be beneficial to extend this analysis to encompass even more generalized fading distributions, such as the α −η −κ −µ distribution [43], [44], which covers other relevant distributions as special cases, as well as fully physics-compliant channel models, such as PhysFad [14], that provide helpful guidance for experimental work and optimal design.The exploration of these broader scenarios can produce deeper performance insights and uncover new opportunities to enhance RIS-assisted multiple-antenna wireless networks.Looking further ahead, it is evident that machine learning schemes hold immense promise in accurately estimating channel state information, potentially leading to optimal RIS response.Therefore, the interplay of ML techniques and reconfigurable intelligent surfaces has attracted significant research interest and may pave the way for further advancements in wireless communication technologies.

APPENDIX A STATISTICS OF THE OVERALL CHANNEL FADING, γ D
The theoretical mean and variance of the overall channel fading distribution are obtained through the following steps.

A. EXPECTED VALUE OF EACH FADING TERM
Since the expected value is a linear operator, it follows that Then, considering that the component channels are independent and identically distributed, and also that E h SD k = 0, and E e jθ ik = α 1 , the expression for the mean results in where are the expected values of each Weibull channel fading.
To obtain the variance of the overall channel fading coefficient, the means of c k = Re{h k } and s k = Im{h k }, i.e., the in-phase and quadrature components of the fading coefficient, need to be calculated.
The in-phase component can be written as while the quadrature component is given by First, the expected value of c k can be expressed by Then, since E Re{h SD k } = 0, and all the summation terms are independent, it follows that which has the same result as In its turn, the expected value of s k can be written as Similarly, E Im{h SD k } = 0, and all the summation terms are independent.Hence, the expression results in given that E [sin θ ik ] = 0.

B. VARIANCE OF THE IN-PHASE AND QUADRATURE COMPONENTS OF EACH FADING TERM
The variance of the in-phase component can be written as Since h SD k is zero-mean, Re{h SD k } is independent of the summation terms, and var Thus, the variance of the term h SL ik h LD i cos θ ik is needed.Considering that the variance of the product of two independent random variables, X and Y , is given by and that the phase noise is independent of the fading magnitudes, it follows that Then, since h LD ik and h SL i are independent, it can be stated that and that Next, as var h LD i = σ 2 LD , and var h SL ik = σ 2 SL , the expression (43) can be rewritten as Then, by using (45) in (42), it follows that which can be simplified into Therefore, the expression for the variance of the in-phase component results in Next, the variance of the quadrature component is given by Then, considering that h SD k is zero-mean, and also that var h SD k = σ 2 SD , and Im{h SD k } is independent of the summation terms, it follows that The Finally, the variance of the quadrature component can be expressed by and the fading terms are independent and identically distributed, then 107230 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Next, by using ( 48) and ( 53), it follows that For the quadrature component, E [s k ] = 0, and, consequently, E s 2 k = var (s k ).Hence, Therefore, the mean value of the overall channel fading can finally be determined by The variance of the sum of the terms Z i is given by var with equally distributed magnitudes.Therefore, var Then, the covariance can be obtained by where The expected value of the product of the in-phase coefficients can be written as Then, by expanding the product, and considering that where the independent terms can be separated as in and in where , and the variance of the Weibull-distributed term, σ 2 LD , is given by For l = m and i = k, it follows that where To obtain the variance according to (62), the term E c where E[c i c k ] can be calculated as follows: Since the moments of c k were previously calculated, ρ c i ,c k can be determined through (74), as shown at the bottom of the next page.
Then, when (74) replaces the correlation coefficient, ρ c i ,c k , in the definition of E[c 2 i c 2 k ] in (71), it follows that The term E[c 2 i s 2 k ] depends on two uncorrelated but not independent random variables c i and s k .Hence, the expression needs to be expanded as in

FIGURE 1 .
FIGURE 1. Wireless communication system model with reconfigurable intelligent surface and eavesdropper link.

FIGURE 2 .
FIGURE 2. Hellinger distance between theoretical and simulated overall channel probability distributions as a function of the number of RIS antenna elements, N, with varying number of base station antennas, M.

FIGURE 3 .
FIGURE 3. Kullback-Leibler divergence between theoretical and simulated overall channel distributions as a function of the number of RIS elements, N, with varying number of base station antennas, M.

FIGURE 4 .
FIGURE 4. Symbol error probability as a function of the SNR with varying number of RIS antenna elements, N.

FIGURE 5 .
FIGURE 5. Symbol error probability as a function of the SNR with varying von Mises concentration parameter, κ.

FIGURE 6 .
FIGURE 6. Symbol error probability as a function of the SNR with varying LoS strength through the Weibull shape parameter, k.

FIGURE 7 .
FIGURE 7. Symbol error rate upper bound as a function of the SNR.

FIGURE 8 .
FIGURE 8. Spectral efficiency as a function of the number of RIS elements, N, with varying numbers of base station antennas, M.

FIGURE 9 .
FIGURE 9. Spectral efficiency as a function of the number of RIS elements, N, with varying Weibull shape parameter, k.

FIGURE 10 .
FIGURE 10.Spectral efficiency as a function of the number of RIS elements, N, with varying von Mises concentration parameter, κ.

FIGURE 11 .
FIGURE 11.Secrecy Outage Probability as a function of the capacity SNR threshold, λ th , with varying number of RIS antenna elements, N.

FIGURE 12 .
FIGURE 12. Secrecy Outage Probability as a function of the capacity SNR threshold, λ th , with varying number of base station antennas, M.

FIGURE 13 .FIGURE 14 .FIGURE 15 .
FIGURE 13.Approximation of SOP for a varying number of summation terms in the lower incomplete Gamma series expansion as a function of the capacity SNR threshold, γ th .

FIGURE 16 .
FIGURE 16.Spectral efficiency as a function of the number of RIS antenna elements, N, with varying number of base station antennas, M, and simulated Nakagami-m fading.

FIGURE 17 .
FIGURE 17. Secrecy outage probability as a function of the capacity SNR threshold, λ th , with varying number of RIS antenna elements, N, and simulated Nakagami-m fading.

2 i c 2 kx 2 yE[c 2 i c 2 k ] = µ 4 c k + 2µ 2 c k 1 +
needs to be computed.Since c 2 i and c 2 k can be assumed to be correlated Gaussian random variables, following the central limit theorem for sufficiently large values of N , then 2 f c i ,c k (x, y)dxdy, (70)where f c i ,c k (x, y) is the joint distribution of two correlated Gaussian random variables c i and c k .After solving the integral (70), it results in 2ρc i ,c k σ 2 c k + 1 + 2ρ 2 c i ,c k σ 4 c k ,(71)whereµ c k = E[c k ], and σ 2 c k = var(c k ).Since var(c i ) = var(c k ) and E[c i ] = E[c k ],the correlation coefficient, ρ c i ,c k , can be obtained by