On the Performance of RIS-Assisted Communications with Direct Link in κ-μ Shadowed Fading Channels

Reconfigurable intelligent surfaces (RISs) is a new technology that can be used to create a virtual line-of-sight (LOS) link when the direct link is blocked. The signal-to-noise ratio (SNR) can be significantly improved by optimizing the reflecting elements’ phases to make the reflected signals add coherently at the receiver. Nevertheless, the RIS cannot control the phase of the direct link if it exists. In such scenarios, the RIS phase can still be controlled such that the direct and reflected signals add coherently, however, the gain obtained by using the RIS might be hindered. Therefore, this paper considers the performance analysis of such scenarios where a novel analytical framework is developed to evaluate the SNR, outage probability and bit error rate (BER). To capture a broad range of fading conditions, the channels are modeled as independent but not identically distributed generalized κ-μ shadowed fading channels. The Laguerre expansion is used to derive the probability density function (PDF) and cumulative distribution function (CDF) of the instantaneous channel fading, which are used to derive the PDF and CDF of the instantaneous SNR. The paper also considers deriving the asymptotic PDF, CDF, moment generating function (MGF) of the SNR, as well as the outage probability and BER. The derived expressions are used to evaluate the system performance in various fading environments such as Rayleigh, One-Sided Gaussian, Nakagamim̄, Rician, and Rician-shadowed distributions since they are special cases of the κ-μ shadowed distribution. The obtained analytical results corroborated by Monte Carlo simulation show that a strong direct link can generally eliminate the gain obtained using the RIS. Therefore, the overhead required for the RIS operation becomes a burden on the network and may cause severe throughput degradation. In such scenarios, the RIS involvement should dynamically controlled to avoid unnecessary complexity and throughput reduction.


I. INTRODUCTION
The demand for mobile data access has been growing rapidly in the past few years, driven by the large number of new subscribers, high data rate applications, and emerging Internet of Things (IoT) applications, which require massive connectivity. Therefore, the limited capacity problem became prominent [1], [2]. According to Ericsson mobility report [3], mobile network data traffic grew 44% between 2020 and 2021, and reached 72 Exabyte per month generated by about 8 billion subscribers. As per Cisco report [4], 50% of all IoT networked devices in 2023 will be connected to cellular networks. Therefore, the wireless network is expected to support about 15 billion IoT devices in addition to 8 billion mobile users. Although the advancements achieved by fifth generation (5G) wireless networks are considered substantial as compared to the fourth generation (4G), 5G networks capacity improvement is still way below the 1000× capacity increase specified by the International Mobile Telecommunications-2020 (IMT-2020) vision.
To resolve the limited capacity problem, extensive research efforts are being dedicated to design efficient solutions. For example, spectral capacity solutions aim at introducing more spectrum by utilizing new frequency bands, particularly high frequencies such as millimeter waves (mmW). However, mmW poses several challenges [5] such as the poor propagation characteristics of mmW as compared to classical sub-6 GHz spectrum. Moreover, in highly-dense urban areas, the high-rise buildings cause shadowing and mostly block lineof-sight (LOS) connectivity, which makes it more challenging for high frequency transmission. Consequently, reconfigurable intelligent surfaces (RISs) were introduced to control the wireless environment [6], [7]. An RIS panel consists of twodimensional meta-surfaces arrays in which each meta-surface element can independently change the phase-shift of the incident electromagnetic signal to create a preferable wireless environment [8]. RIS panels do not require high hardware complexity or cost overhead during their integration, while they can provide substantial gain in terms of reliability and network capacity [9].

A. Related Work
RIS has recently attracted extensive research activities, particularly performance analysis. The performance analysis was obtained for various system and channel models using several performance metrics such as signal-to-noise ratio (SNR), bit error rate (BER), symbol error rate (SER), outage probability and capacity. The system model can be generally classified into two main categories, RIS without direct link [6], [8], [10]- [12], [12]- [16], which is more common, and with direct link [17]- [21]. Table I shows the classification of selected recent articles based on their consideration of a direct link, system model, channel model, and performance metric. More specifically, In [6], [8], [10]- [16], the authors provide closedform approximations for several performance metrics through modeling an end-to-end RIS channel amplitude by Gaussian [6] . K G [12], Gamma [8], and Nakagami-m [10] distribution. Moreover, the channel in [11] are based on Central Limit Theorem (CLT) which does not accurate for small number of reflecting elements whereas, in [15] a unified analytical frameworks based on Fox-H function are developed which are shown to be highly accurate even for small number of reflecting elements, particularly for continuous phase shifts.

B. Motivation and Contribution
Although the above mentioned works provide good results about the performance of the wireless system in existences of RIS, they are neglecting the direct link between base stations (BS) and user equipment (UE). In practice, it is not necessary that the direct link is completely blocked since the system suffers from an intrinsic product path loss [27]. i.e; the direct link cannot be ignored as generally it has relatively a smaller path loss [16]. Motivated by this, the direct link is considered in References [18]- [21], [24]- [26], as summarized in Table I. In these works (except [24] that considered K-fading model where the approximation of the performance of the system based on the tight approximation of end-to-end SNR was derived), the authors considered Nakagami-m [25] and Rayleigh [18]- [20], [26] models for the direct link. It is worth mentioned that these fading models do not accurately fit since there is a high probability that all links experience composite fading/shadowing. To this end, κ − µ shadowed distribution is widely used in literature for modeling composite fading/shadowing. It has a clear physical interpretation, good analytical properties and unifies popular fading models such as One-Side Gaussian, Rayleigh, Nakagami-m, Rician, κ-µ, and η-μ fading channels as special cases [28], and [29, Table. I].
To the best of our knowledge, no prior work in literature has derived an analytical framework for studying RIS-assisted wireless communication systems in the more generalized κ-µ shadowed fading channel. All scenarios introduced in literature considered Rice [24], Nakagmai-m [25] and Rayleigh [26], which are derived as special cases in our framework. To be more specific, the main contribution of this study can be summarized as follows: • A new and generalized analytical framework model to characterize the performance of the RIS-assisted wireless communication systems has been developed. • Generalized and exact closed-from expressions for the probability density function (PDF), moment generating function (MGF), and cumulative distribution function (CDF) of the end-to-end SNR have been derived. • Generalized and closed-form expression for the Outage Probability (Pout), and BER have also been derived. • To get more insights, at high SNR, new and accurate closed-form expression for the asymptotic of Pout and BER have been obtained, and the system diversity order and coding gain are also provided.

C. Paper Organization and Notations
The remainder of this work is organized as follows: Section II System Model. Next, Section III Channel Model. Performance Evaluation presented in Section IV while, Section V reports respective numerical results and discussions. Finally, conclusions are provided in Section VI. The notations used throughout the paper are listed in Table II.

A. System and Channel Models
The simplified system model is given in Fig.1, where an RIS-assisted wireless communications network that includes a BS, RIS panel equipped with N reflecting elements, and K UEs. Each UE can access the RIS through an orthogonal multiple access protocol such as time division multiple access (TDMA). In this work we consider that the UE can receive the transmitted signal through two different links, an indirect link through the RIS and a direct link from the BS. Because the distance between the RIS panel, BS and UE is much larger than the distance between the reflecting elements within the RIS panel, then all RIS elements are assumed to be at the same distance from the BS and UE. Therefore, the distances from BS to UE, BS to RIS, and RIS to UE are respectively denoted as d 0 , d h , and d g . Unlike the distances, the channel gains between the BS and each RIS element are typically assumed to be independent and identically distributed (i.i.d.) because the elements are spaced no less than half the wavelength of the carrier frequency. Therefore, the channel gains between the BS and RIS, RIS and UE are denoted as h i , and g i , respectively, i = 1, 2, . . . , N . The channel gain between the BS and UE is denoted as h 0 . The channel state information (CSI) of each UE is communicated to the BS by means of channel reciprocity algorithms, or via a feedback channel. The BS controls the phase of each RIS element through programmable controller

Head Head E[·]
Statistical expectation Moment generating function f X (·) Probability density function attached to the RIS panel through a feedback channel [6], [33], [34]. For mathematical tractability, we consider that the BS and UE are equipped with single antenna [35]- [37]. Therefore, the received signal at any UE can be expressed as where Y D is the signal received through the direct link, Y R is the signal received through the RIS, and n ∼ CN (0, N 0 ) is the additive white Gaussian noise (AWGN). The direct signal can be written as [25] h where P is the transmitted power, ζ = c 4πfc 2 denotes the average channel gain at a reference distance of 1 m based on the free space path-loss model, c is the speed of light, f c is the carrier frequency, s is the transmitted information symbol, |h 0 | and θ 0 are the magnitude and phase of the channel from the BS to UE, and α D is the pathloss exponent at direct link.
Similarity, Y R denotes the received signal at UE through the RIS link, which can be written as, where |h i | and θ i denote the magnitude and phase of h i , respectively, while, |g i | and ϕ i are the magnitude and phase of g i the UE, respectively. The ith RIS element gain and phase are v i and Ψ i . Without loss of generality, we consider that v i = 1 [8], [35]- [38]. Therefore, we can write Consequently, the overall channel gain can be expressed as

B. Channel Model
Due to the wide range of use cases and applications of 5G and sixth generation (6G) wireless networks, conventional channel models that consider large scale shadowing will be mostly insufficient to capture the complex propagation effects of such applications [42]. In the literature, several composite channel models such as the κ-µ/inverse and η-μ/inverse gamma fading are used to capture the statistical behavior of the combined small-scale and large-scale fading in various propagation environments [43]. However, such models assume that the shadowing affects the scattered and dominant waves equally, and thus, such models are referred to as multiplicative shadow fading models [44], [45]. However, in practice the shadowing often affects only the dominant components. Therefore, such model is referred to as LOS shadow fading model [46], [47].
The κ-µ shadowed model, denoted as κ-µ, is a generalized distribution that plays an essential role in modeling fading for LOS scenarios. The κ-µ shadowed model has an additional parameter ω, as compared to κ-µ model, which is related to shadowing. Consequently, it has a clear physical interpretation, flexible analytical properties and can represent common fading models such as One-Side Gaussian, Rayleigh, Nakagami-m, Rician, κ-µ, and η-μ fading channels as special cases [28], [29, Table. I]. In addition, it can be used in several real-world applications [46], [48]- [50]. By limiting the fading parameters µ and ω to integer values, minor differences are obtained in practice when fitting field measurements to the κ-µ, while it will make the mathematical analysis more tractable and may lead to efficient closed-form expressions [51]. Consequently, in this work we consider that all channel gains follow the κ-µ with integer µ and ω parameters. Therefore, the PDF and CDF are respectively given by [51] f and where M , C i , m i , and Ω i are given in Table III.

C. PDF of the Received Signal Power
As can be noted from (5) and (6), the received signal power depends on V , and thus, its PDF should be derived. Towards this goal, define |h i ||g i | ≜ Z i , and for notational simplicity we drop the reflector index i unless it is necessary to include. Therefore, the PDF of Z can be evaluated as [52], By substituting the PDFs h ∼ κµ(M 1 , C i , m i , Ω i ), and g ∼ κµ(M 2 , C j , m j , Ω j ) expressed in (7), into (9), and using the identity [30, eq. (3.471.9)] we obtain, Moreover, the mean and variance of Z are respectively given by, and In the special case where all the links are identical, i.e., M 1 = M 2 = M, Ω j = Ω i = Ω l , and C j = C i = C l ), then According to [53], [54], the PDF and CDF of random variable (RV) Z = N i=1 Z i can be tightly approximated by the first term of a Laguerre expansion as follows, and where the parameters a and b are related to the mean and variance of Z as where Zi . In case of i.i.d. channel gains, µ Z = N µ Zi and σ 2 Z = N σ 2 Zi . By applying the transformation method between two random variables in (3), the PDF and CDF of H R can be given by and where Similarly, the PDF and CDF of H D can be expressed as, and  [51].
and Ω q are given in Table III.
By noting the superposition of H D and H R in (6), then the PDF of V can be obtained by applying the convolution theorem to (21) and (23). Thus,

D. Moment Generating Function of V
By applying the Laplace transform to (25), we obtain the MGF of V , where L v a+mq 1 F 1 m q ; a + m q + 1; Therefore, (26) becomes

E. The CDF of V
By noting that the MGF in (28) and f V (v) has the relation then F V (v) we can be obtained as By substituting (28) in (30), we obtain Using [55, eq. (4.24.3)], the Laplace inverse can be evaluated as, Substituting (32) into (31) gives, where the function Φ 2 can be efficiently computed by means of an inverse Laplace transform [56].

F. End-to-End SNR
The instantaneous end-to-end SNR at the UE can be expressed as where P s is the transmitted symbol power, N 0 is thermal noise power, andγ is the average SNR. Therefore, the PDF, CDF and MGF of γ Y are given by, To obtain more insights about the system performance, we derive the asymptotic PDF, CDF and MGF in the high SNR region, i.e.,γ → ∞. For the PDF, by noting that [57] lim γ→∞ 1 F 1 m q , a + m q + 1; then we obtain from (35), Similarly, and

III. PERFORMANCE EVALUATION A. Outage Probability
The derived distributions can be used to compute the outage probability, which is defined as where γ out is a predetermined SNR threshold. From (41), it can be concluded that P out can be approximated as where G o and G d denote the coding gain and diversity order which are respectively given by where w = 2 a+mq+1 , and

B. Bit Error Rate (BER)
Based on the MGF expressed in (37), the BER can be expressed as, where a m and b m are parameters that depend on the modulation scheme. By substituting (37) in (47), the BER becomes By performing the change of variable t = cos 2 (ϑ), and after some algebraic manipulations, we obtain where the values of w 2 , u 1 , and u 2 are respectively given by and For the asymptotic case whereγ → ∞, we obtain where the asymptotic BER can also be expressed in terms of coding G p and diversity gain G d , i.e., eq. (1)] and Rayleigh

IV. NUMERICAL RESULTS
This section presents the theoretical and Monte Carlo simulation results for various cases of interest. The main simulation parameters are given in Table IV, unless is specified otherwise. Fig. 2 demonstrates the effectiveness and validity of the derived PDFs by comparing the generalized theoretical expression of the PDF in (17) to Monte Carlo simulation results. The values of µ, κ and ω of the individual PDFs are selected to correspond to some of the widely used PDFs, while N is fixed at 10. As the figure shows, the theoretical expression perfectly matches the simulation results for all considered scenarios. Moreover, the generalized distribution model can be used to represent the widely known fading distributions such as the the Rayleigh, Nakagami-m, Rician, and Rician-Shadowed. In Fig. 3, PDF f Z (z) is evaluated for fixed and equal fading parameters where N ∈ {1, 3, 5, 7, 10}. As the figure shows, the analytical and simulation results match perfectly, which confirms the accuracy of the derived expressions. Moreover, it can be noted that increasing N can significantly improve the fading conditions. Fig. 4 compares the analytical, asymptotic and simulated outage probability versus SNR using various number of reflecting elements N . As can be noted from the figure, the analytical and simulation results match very well for all the considered scenarios. It can be also observed that at high SNRs, the asymptotic and exact results converge. The improvement gained by increasing the number of reflectors increases non-linearly versus N . For example, increasing the number of reflectors from 5 to 10 reduced the SNR required to achieve P out of 10 −5 from 8.25 dB to −1.45 dB, i.e., the gain is about 9.7 dB. However, increasing N from 10 to 20 reflectors offers only ∼ 6 dB gain.    Fig. 6 and Table. V evaluate the impact of the various system and channel conditions on the outage and BER performance. As can be noted from Fig. 5, outage probability (OP) is significantly dominated by the strength and quality of the direct path that is indicated by µ D . For µ D = 1, the fading of the direct path is severe, and thus P out > 0.1 forγ > 0 dB. Increasing the value of N can reduce P out , but the impact  appears generally forγ > 0 dB. At low SNRs, all P out values approach that of the direct path regardless the value of N . Table V shows the variation of the outage probability with respect to the shadowing parameter of the direct path ω D between BS and UE for different values of κ D and N . The results are obtained for µ R = µ D = 1. The shadowing parameters between BS-RIS and RIS-UE links is ω R = 30. As can be noted from the table, P out is generally dominated by ω D . For example, when ω D = 1, the value of P out remains fixed regardless the vale of κ D . However, when ω D improves, then P out improves by improving κ D . Moreover, it can be noted that ω D also limits the impact of N , where increasing  Bit Error Rate (BER) Direct Link Only RIS-aided with direct Link the value of N from 1 to 10 reduced P out only by a factor of 10, while the improvement for the same scenario is much more significant for ω D = 15. For high values of ω D , it can be observed that the impact of κ D becomes more tangible where P out decreases by increasing κ D . Fig. 6 shows the effect of changing the number of reflecting elements N on the BER. The exact (51) and asymptotic  Diverist (55) BERs are included and compared to the Monte Carlo simulation results. The figure shows that the analytical and simulation results match very well, even for the case of very small number of reflectors such as N = 1 and 3. The BER improvement gained by using RIS seems to be significant, even when for small values of N . The asymptotic BER approaches the exact when the exact BER becomes roughly linear. For the case of no RIS and N = 1, the curves converge forγ ≳ 30 dB, while for the case of N = 3 they converge forγ ≳ 28 dB. Fig. 7 evaluates the impact of the path loss exponent at RIS link (α R ) on the BER using (51). As can be noted from the figure, the BER achieved by the RIS increases by increasing α R , and ultimately approaches the BER of the direct link regardless the value of N . Such performance is obtained because increasing α R causes severe signal attenuation, and hence, the signal reflected by the RIS becomes extremely weak. However, when α R is small, a significant performance gain can be achieved by increasing N . Fig. 8 demonstrates the advantage of using RIS-aided transmission over the direct link. Therefore, the figure illustrates the effect of increasing the distance between the RIS and UE (d g ) on Pout. In particular, the graph provides a comparative analysis of three transmission schemes, which are the direct transmission only, RIS without direct transmission, and RIS transmission with a direct link. It can be noted that if the direct link is available, it is beneficial to combine both signals, notably when N increases. Moreover, it is generally difficult for the direct path alone to achieved reliable P out when the outage threshold P out is high. Therefore, the location of the RIS should be optimized by taking into consideration the possibility of exploiting the direct and reflected links to minimize the outage and BER. Fig. 9 presents the diversity order of the system versus the number of reflecting elements N for various values of ω D . Results indicate that ω D has a significant impact on the diversity order for high values of N . When ω D ≫ 1, all the reflected links have generally strong signals, and thus the diversity gained by using the RIS is limited. For low values of ω D , the channel variations increases significantly, which increases the diversity order achieved by using the RIS.
V. CONCLUSIONS In this paper, we developed a new analytical framework model to investigate the end-to-end SNR, outage probability and BER performance of RIS assisted wireless communication systems over more generalized κ-µ shadowed distribution channels. Novel accurate closed-form expressions for the PDF, CDF of end-to end SNR have been derived. Moreover, closedfrom expressions have been derived for the outage and BER using the Lagueree expansion, which provided near-exact results. The asymptotic outage and BER are derived to provide more insights on the achievable diversity order. The derived expressions were used to holistically study the performance of RIS systems in the presence of a direct link between the BS and UE. The obtained results show that the direct path can significantly affect the RIS system performance, and hence, should be taken into consideration during the planning and design stages of the system. Monte Carlo simulation was used to validate all the derived expressions.