X03bc; Random Variates

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Effective Rate Evaluation of RIS-Assisted Communications Using the Sums of Cascaded U-` Random Variates

LONG KONG1, YUN AI2, (Member, IEEE), SYMEON CHATZINOTAS1, (SENIOR MEMBER, IEEE), and BJÖRN OTTERSTEN1, (FELLOW, IEEE) 1The Interdisciplinary Centre for Security, Reliability and Trust (SnT), University of Luxembourg, Luxembourg City 1855, Luxembourg (e-mail: {long.kong, symeon.chatzinotas, björn.ottersten}@uni.lu.) 2Faculty of Engineering, Norwegian University of Science and Technology, 2815 Gjøvik, Norway (e-mail: [email protected]) Corresponding author: Long Kong (e-mail: [email protected]). This work has been supported by the Luxembourg National Research Fund (FNR) projects, titled Exploiting Interference for Physical Layer Security in 5G Networks (CIPHY, FNR11607830), and Reconﬁgurable Intelligent Surfaces for Smart Cities (RISOTTI).

ABSTRACT This paper begins with the characterization of the sum of " versions of cascaded U- ` (equivalently, # ∗ (U-`)) distributed random variates (RVs), and later on, explores the feasibility and applicability of the obtained results in the recent proposed reconfigurable intelligent surface (RIS)-assisted communications. More specifically, the sum of cascaded U-` distributed RVs is exactly characterized by the probability density function (PDF) expressed in terms of the programmable multivariate Fox’s - function, and approximated with the mixture of Gaussian (MoG) distribution. The approximation accuracy is further validated by the Kolmogorov-Smirnov (KS) goodness of fit test. Based on the obtained statistical results, a promising application, i.e., RIS-assisted communication, is considered and evaluated from the effective rate perspective. The exact, asymptotic, and approximated effective rate are derived with closed- form expressions. Finally, our analytical results are further validated using the Monte-Carlo simulation, their accuracy is confirmed with the analytical error indicator. One can conclude that (i) the exact effective rate expression is complex in the manner of mathematical representation; (ii) the MoG-based effective rate expression is simple and provides a highly tight approximation to the exact expression, besides it indeed acts as a highly tight upper bound for the exact effective rate expression; and (iii) the MoG-based solution outperforms the multivariate Fox’s -function based approach with reducing complexity.

INDEX TERMS Effective rate, quality-of-service (QoS), cascaded U-`, reconﬁgurable intelligent surfaces (RIS), Kolmogorov-Smirnov (KS) goodness of ﬁt test, Fox’s -function, mixture of Gaussian (MoG).

I. INTRODUCTION physical layer and link layer, and is used to explore the UTURE generation of communication systems are performance of various wireless networks under certain delay F bound to change the landscape of industry thanks to constraint [3]. The concept of effective rate has been widely the Industrial Internet of Things (IIoT) enabled by extreme applied over the last few years to investigate the trade-off low latency and massive number of connections. Due to the among latency, reliability, energy efficiency, and security. highly time-varying characteristics of the wireless channel, Examples can be found in [3], especially in some tradi- deterministic delay constraints are normally difficult to meet. tional and future promising applications, including Cellular Instead, the statistical quality-of-service (QoS) provisioning communication, device-to-device (D2D) communications, turns out to be a useful instrument to evaluate the delay bound peer-to-peer video streaming, visible light communication, QoS guarantees for realistic wireless real-time traffic. full-duplex communications, non-orthogonal multiple access (NOMA) [4]–[7], vehicle communications [8], and ultra- The effective rate serves a statistical QoS metric defined as reliable low latency communications (URLLC). the maximum constant arrival rate that a time-varying service process can support with statistical latency guarantees [1]– [3]. It acts as a joint mathematical framework, connecting

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L. Kong et al.: Effective Rate Evaluation of RIS-Assisted Communications Using the Sums of Cascaded U − ` RV

A. BACKGROUND AND RELATED WORKS key fading parameters, i.e., non-linearity of the propagation The effective rate investigation of these applications are medium U and the clustering of the multipath waves `. The conducted under different fading channel models. The effec- advantage of these two factors is regarded as a useful tool tive rate metric of various small-scale (or multipath), large- to vividly depict the inhomogeneous surroundings compared scale (or shadowing), and composite fading channels was with other existing fading models, such as Rayleigh and widely examined, including Weibull [9], U-` [10] (equiv- Nakagami-<. Most of these are based on the assumption of alently, generalized gamma), ^-` [11], generalized-K [12], a rarely-encountered scenario of homogeneous (scattering) [13], Fisher-Snedecor F [14], U-[-`/Gamma [15], two-wave environment. Fortunately, the U-` fading model was later on with diffuse-power (TWDP) [16], double shadowed Rician found valid and feasible in many realistic situations [34]– [17], U-[-^-`1 [18], [19], ^-`/inverse gamma and [-`/inverse [39], including the vehicle-to-vehicle (V2V) communication gamma [20], etc. In [21], the authors investigated the effec- networks and wireless body area networks (WBAN). tive rate of multiple-input multiple-output (MIMO) keyhole Moreover, the sum of U-` distributed random variates channels (i.e., rank one MIMO channel). Rare efforts have (RVs) has been investigated in [40], where a new U-` dis- ever been seen about the investigation for the analytical tributed RV is used to highly approximate the sum. Similarly, analysis of effective rate over cascaded fading channels. the sums of other general fading models, e.g., [-`, ^-`, Fisher-Snedecor F , fluctuating two-ray (FTR), and Fox’s B. MOTIVATIONS -function distributed RVs are also characterized in [41]– To this end, the focus of this paper lies in the investigation [47], where their application to the maximal-ratio-combining of the effective rate analysis over cascaded fading models. (MRC) diversity receiver are correspondingly investigated It is worth mentioning that the cascaded fading models can with the assistance of the obtained statistical results [42]– capably represent a mobile-to-mobile (M2M) scenario both [47]. through analysis and measurement [22]. The cascaded fading To this end, the core of interest in this paper is to models can be physically interpreted as follows: the received characterize the sum of cascaded U-` RVs. Different from signals are generated by the product of a large number of the MRC diversity receivers, in this paper we consider the rays reflected via # statistically independent scatters [23]– newly proposed RIS-assisted communications as a possible [25]. Those models are applicable for the multi-hop non- and promising application based on the obtained statistical regenerative amplify-and-forward (AF) relaying system with results. In continuation with the aforementioned discussion fixed gain [26], the propagation in the presence of keyholes, about the effective rate analysis, the usefulness of our find- the keyhole phenomena in MIMO systems [21], and recon- ings is further evaluated from the effective rate perspective. figurable intelligent surfaces (RIS)-assisted wireless system C. CONTRIBUTIONS AND ORGANIZATION [27]. An RIS, in other words, is also equivalent to digitally controllable scatters and software-defined surface [28]. The The main contributions of this paper are summarized as large number of small, low-cost, and passive artificial ‘meta- follows: atoms’ embedded in an RIS are able to smartly change the 1) Characterizing the probability density function (PDF) reflection direction towards any desired users by tuning a for the sum of " version of cascaded U − ` distributed series of phase shifters [29], [30]. More specifically, the RVs, which is expressed in terms of the multivariate authors of [29], [30] explored the feasibility of applying Fox’s -function. For the purpose of providing a re- the RIS technology to the future sixth-generation (6G) and ducing complexity solution, the sum of cascaded U- beyond fifth-generation (5G) networks from the viewpoints ` RVs is also modeled in the manner of the mixture of modulation and multiple antenna, respectively. of Gaussian (MoG) distribution, the approximation In [24], the authors proposed a flexible and general ‘multi- accuracy is further confirmed using the Kolmogorov- plicative’ stochastic model, namely, the cascaded U-` (#∗(U- Smirnov (KS) goodness of fit test. `)) fading channel. It is flexible and mathematical tractable, 2) Initially applying the sum of cascaded distributed RVs since it can elaborate the cascaded Rayleigh, Nakagami-< to the novel RIS-assisted communication, and explor- [23], exponential, Gamma, Weibull, and lognormal (U → ing the feasibility and advantages of using the obtained ∞, ` → ∞) fading by simply attributing the fading pa- statistical result to the RIS-assisted communications rameters U and ` to selected values. For instance, setting from the effective rate perspective. U = 2 and ` = 1, it will reduce to cascaded Rayleigh fading, 3) Deriving the exact, asymptotic, and approximated ef- whereas choosing U = 2 and ` = < will make it correspond fective rate expressions of the RIS-aided communi- to cascaded Nakagami-< fading. More importantly, the U- cations over U-` fading channels. Especially when ` distribution was proposed by Yacoub in [31] to model " = 1, the exact effective rate analysis fills the gap the small-scale variation of fading signal under line-of-sight of lacking effective rate analysis over cascaded fading conditions [32], [33]. It is physically described with two channels. 4) Validating the accuracy and tightness of the exact 1As shown in [18], the U-[-^-` distribution includes the U-`, ^-`, and and asymptotic behavior of the analytical results with [-` distributions as its special cases. Monte-Carlo simulations.

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L. Kong et al.: Effective Rate Evaluation of RIS-Assisted Communications Using the Sums of Cascaded U − ` RV

5) Providing the asymptotic effective rate slope at high Apparently, the cascaded U-` distribution is a special case signal-to-noise ratio (SNR) regime. of the Fox’s -function distribution. To this end, applying 6) One can surely conclude that (i) the exact, asymptotic, the results obtained in [47, Eq. (7)], the PDF of the sum of and approximated effective rate expressions are an- independent but not identically distributed (i.n.i.d.) cascaded " alytically correct; (ii) the RIS-aided communications Í U-` distributed RVs, i.e., Z = Z9 , is given in terms of the is a suitable application of the sum of cascaded U- 9=1 ` distributed RVs, and the increase of " leads to multivariate Fox’s -function2, i.e., an enhancement of the overall effective rate; and (iii) the MoG-based approach is quite advantageous since 0, 0 − it provides a simple and easy solution to analyze the [ © 0, 1 (1; 1 ) ª " " 5Z (I) = C# , 9 I 9=1:"® effective rate over various wireless fading channels I # 9 , 1 (1, 1) ® ® with reducing complexity. 1, # 9 ℬ « 9=1:" 9=1:" ¬ The rest of this paper is organized as follows: Section (1) [ ¹ ¹ = " ··· Φ(B , ··· ,B ) II provides the preliminary results regarding the sum of " 1 " (2cF) I L1 L" cascaded U-` distributed RVs. Based on the obtained results, " Ö B9 we present a system conﬁguration of the RIS-assisted com- × q(B 9 )(C9 I) 3B1, ··· , 3B" , (3) munications in Section III. Next, the exact, asymptotic, and 9=1 approximated effective rate expressions of our proposed system model are subsequently analyzed in SectionIV. Section " Î K# , 9 1 V demonstrates the numerical results and some interesting where [" = C , i8 = `8, U , ℬ = 9=1 # , 9 8 discussions. Finally, SectionVI concludes this paper. √ (i1, ··· , i# ), and L 9 is the integral path from X 9 − F∞ Mathematical Functions and Notations: F = −1, 5- (G) Í" to X 9 + F∞, Φ(B1, ··· ,B" ) = Γ( 9=1 B 9 ), and q(B 9 ) = represents the probability density function (PDF) of RV -, # <,= Î Γ − B9 Γ( ) ( ) ?,@ [·] is the univariate Fox’s H-function [48, Eq. (1.2)], `8 U B 9 . Step 1 is developed for the sake of 8=1 8 (·) <,=;<1,=1;··· ;

`8 U ` −1 U8 ! etc. The intrinsic parameters used to represent the MoG U8 ` G 8 8 G ( ) = 8 − distribution are implemented via the completely unsuper- 5I8 G U8 `8 exp `8 Ω Γ(` ) Ω8 8 8 (1) vised expectation-maximization (EM) learning algorithm. In (0) − [52], the MoG distribution is proved extremely advantageous = ^ 1,0 _ G , 8 0,1 8 (` − 1 , 1 ) to provide an easy and simple solution when analyzing 8 U8 U8 the physical layer security performance over various fading 1 1 ` U8 ` U8 channels. Motivated by the promising outcomes obtained in where ^ = 8 , _ = 8 , Ω = Γ(`8 ) , 8 = 1, ··· ,#. 8 Ω8 Γ(`8 ) 8 Ω8 8 1 Γ `8 + [51], [52], the following subsection is subjected to apply the U8 Step (0) holds by using [48, Eq. (1.125)]. The PDF of Z is MoG distribution for remodeling the PDF of Z. thereafter given by [24] B. MIXTURE OF GAUSSIAN (MOG) DISTRIBUTION − ( ) K # ,0 C 5Z I = # 0,# # I , (2) For the purpose of differentiating the Fox-based result, 5 ˆ (I) Φ1, ··· , Φ# Z and Zˆ (I) are utilized herein to denote the MoG-based PDF # # where K = Î ^ , C = Î _ , Φ = ` − 1 , 1 . For # 8 # 8 8 8 U8 U8 8=1 8=1 2Due to the limited space and for brevity of notations, the multivariate simplicity, let B = (Φ , ··· , Φ ). As a result, Z is also 1 # Fox’s -function is presented in the manner of the one given in [47]. 1# called a cascaded U-` distributed RV. denotes the # -dimensional all-one vector.

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L. Kong et al.: Effective Rate Evaluation of RIS-Assisted Communications Using the Sums of Cascaded U − ` RV

TABLE 1: MOG PARAMETERS FOR Z OVERCASCADED U- ` FADING CHANNELS WHEN U = 4.5, ` = 3.5, " = 3, 2.5 1 # = 2, AND = 2 0.9

2 0.8 ; F; e; [; 1 0.224375 3.025526 0.299281 0.7 2 0.775625 2.770252 0.289974 1.5 0.6 0.5

1 0.4 and CDF, given by 0.3 2 ! Õ F; (I − e;) 0.5 0.2 5Zˆ (I) = √ exp − , (4a) 2 0.1 ;=1 2c[; 2[; 0 0 !! Õ F I − e 0 2 4 6 -5 0 5 10 (I) = ; 1 + erf √ ; , (4b) Zˆ 2 ;=1 2[; FIGURE 1: The PDF and CDF of Z over cascaded U-` (i.e., where represents the number of Gaussian components3. U = 4.5, ` = 3) fading channels for selected values of ", F; > 0, e; and [; are respectively the ;th weight, mean, and where # = 2. Í variance with the constraint of F; = 1. ;=1 As shown in Fig. 1, we have plotted the PDF of Z for selected value of ", where U and ` is chosen as an illustrative hm g example based on the experimental and analytical results ob- m RIS tained in [34]–[39], the corresponding estimated parameters, i.e., , F;, e;, [; when " = 3, are provided in Table 1. S D Interestingly, one can observe that there exists a highly close agreement between the simulated and approximated PDFs. FIGURE 2: A wireless communication system configuration Besides, the MoG distribution not only can model both the with one source (S) and one destination (D) assisted with an composite and non-composite fading channels, herein we RIS. prove that the MoG distribution is also beneficial to offer a simple but effective approach to model the sum of " versions of cascaded U-` distributed RVs. accuracy. The KS test statistic obtained from Fig. 1 when " = 1, ··· , 5 are respectively given as 2.2 × 10−3, 2.3 × 10−3, −3 −3 −3 C. APPROXIMATION ACCURACY 1.6 × 10 , 2.8 × 10 , and 1.7 × 10 . Herein, in our KS 5 In order to evaluate the accuracy of the MoG distribution, the goodness of fit test, let U = 5%, and E = 10 . Thus, we have −3 KS goodness of fit test is accordingly conducted [23], [53]. <0G = 0.0043 = 4.3 × 10 . The hypothesis H0 is surely The KS test statistic is to maximize the difference between accepted for the cases " = 1, ··· , 5 due to < <0G. the simulated and approximated CDFs, i.e., III. SYSTEM MODEL FOR RIS-AIDED | ( ) − ( )|| , max Z I Zˆ I , (5) COMMUNICATIONS As discussed in Section I, the application of cascaded fading where Z (I) is the empirical CDF of RV Z, and Zˆ (I) is the approximated CDF given in (4b). Let the hypothesis channels is widely used in many communication scenarios, H0 denotes that Z follows the MoG distribution. Such an e.g., M2M communications, multihop AF relaying commu- hypothesis is accepted only when is smaller than the nication, MIMO keyhole/pinhole systems, and also the newly critical value <0G, i.e., < <0G, which is given by proposed RIS-enabled communication systems. To this end, p in the following subsection, we have introduced the RIS- <0G ≈ −(1/2E) ln(g/2), aided wireless communication systems as an illustrative application of the preliminary results. where g is the significance level, and E is the number of random samples of Z. Obviously, Fig.1 presents that the MoG-based CDF of Z A. SYSTEM MODEL approximated the Monte-Carlo simulation results with good Suppose a source node (S) intends to transmit messages to a destination (D) with the assistance of an RIS, as shown in 3The number of components is selected automatically using the Fig. 2 [54], which is composed of " passive and low-cost Bayesian information criterion (BIC) method given in [51, Section III- C], whereas the corresponding parameters for the MoG distribution are reflectors. It is assumed that both S and D are equipped with evaluated using the EM algorithm. a single antenna. The instantaneous received signal at D is

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L. Kong et al.: Effective Rate Evaluation of RIS-Assisted Communications Using the Sums of Cascaded U − ` RV

\) given by where = ln(2) , where \ > 0, ), and represent the " Õ delay exponent, the block length, and the system bandwidth, p + H = B ℎ

<=1 outage probability, which is mathematically formulated as where B is the transmit power, ℎ< = h< exp(−Fq<) and ln (%A(& ≥ &<0G)) 6< = g< exp(−Fk<) are independent and identically dis- \ = − lim , &<0G →∞ & tributed (i.i.d.) U-` RVs4, which correspond to the channel <0G coefficients from S to the <-th reflector element and the <- where %A(0 ≥ 1) stands for the probability of 0 being larger th reflector element to D, respectively. h<, g<, q<, and k< than or equal to 1. Herein, & denotes the queue length at denote the amplitudes and phases of the corresponding fading the transmitter buffer side, and &<0G is the predetermined channel gains. D< = w< (\<) exp(F\<) is the reflection threshold of the queue length. Practically speaking, \ → 0 coefficient produced by the <-th element of the RIS, herein implies the delay-tolerant communication, while \ → ∞ w< (\<) = 1 for the ideal phase shifts, < = 1, ··· ,". B is the implies the delay-limited communication [56], e.g., voice transmitted signal with unit energy. n is the additive white calls. Gaussian noise (AWGN) with zero mean and #0 variance. The instantaneous received effective SNR at D, W, is B. EXACT EFFECTIVE RATE ANALYSIS expressed as Theorem 1. The effective rate of the RIS-assisted communications over U-` fading channel is given by " 2 B Õ W = h

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L. Kong et al.: Effective Rate Evaluation of RIS-Assisted Communications Using the Sums of Cascaded U − ` RV the effective rate of the RIS-assisted communications over Remark 3. The asymptotic behavior of effective rate at high " Nakagami-< (` = <) and Rayleigh (` = < = 1) fading Í SNR regime is given by under the condition that 2 > b 9 , channel when " = 1 are expressed as follows, in terms of the 9=1 <,= 5 Meijer’s -function, i.e., ?,@ (·) , b  " Γ Í" 9 1 K# 1  [ Õ b 9 9=1 2 R = − Δ R∞ ≈ −  " © − ª ( log2 , where " log2 Γ C# Γ() 2Γ() 2 ® Í"  9=1 Γ 9=1 b 9 2  « ¬  # +1,1 C# 1   , Nakagami-<, (16)  1,# +1 W¯ <1 , " # b9  Δ =  # Ö Ö b 9 C# , 9  2 × lim (B − b ) Γ ` − Γ(b ) √ , # +1,1 C 1 9 9 8 9   # B→b9 U8 W¯  1,# +1 W¯ , Rayleigh. 9=1 8=1,8≠2  1# ,    where b 9 = min U 9,8 ` 9,8, 2 = arg min U 9,8 ` 9,8 . (20) Proof. When U = 2, then deploying the elementary property 8=1,··· ,# 8=1,··· ,# of Fox’s -function [58, Eqs. (2.1.5), (2.1.4), and (2.9.1)], the proof is obtained. Especially when # = 1,(11) is then expanded at b 9 = U 9 ` 9 Í" with constraint 2 − 9=1 b 9 > 0. C. ASYMPTOTIC EFFECTIVE RATE ANALYSIS

The asymptotic behavior of effective rate over - b9 b9 Remark 2. U  Γ − Í" Í" 1  [ 9=1 2 9=1 2 ` fading channel at high SNR regime when " = 1 is given ∞  " R" ≈ − log2 by 2Γ() Í"  Γ 9=1 b 9  (21) 1 ^#  ∞ ∞ " b R ≈ − log2 √ Δ , Fℎ4A4 C 9  ( 2 W¯Γ() Ö # , 9  × U 9 Γ b 9 √  . # 2−1 = W¯   Î 2 C# 9 1  2Γ() Γ `8 − √ , 2 < g,  U8 W¯ (17)  ∞  8=1 Δ = g g g # U2 Γ( − )Γ( )Γ `8 − Proof. See AppendixB.  Î 2 2 U8 , 2 > g.  C 1−g  8=1,8≠2 √#  W¯  Remark 4. Observing from (17) and (20), the high SNR Proof. Assuming all the residues of the Fox’s -function are C slope of effective rate is deﬁned as simple, when W¯ goes to ∞, we have √# → 0, the asymptotic W¯ behavior of (12) is evaluated at b = 1 − min (g, 2) where R∞ (∞ = , (22) log2 W¯ g = min U8 `8, 2 = arg min U8 `8, (18) 8=1,··· ,# 8=1,··· ,# subsequently, substituting R∞ and R∞ into (22), we have subsequently making expansion of (12) at B = b [59], and ( " using the residue theorem [60] under the constraint < g , 2 ∞ 1, 2 < g, we have (( = g (23a) , 2 > g. " # ∞ Ö 1 B 1 B Δ ≈ Res Γ `8 − + Γ − + U8 U8 2 2 " " 8=1 ∞ 1 Õ b 9 Õ − ( = , 2 > b , (23b) 1 B C B " 2 9 ×Γ − √# , 1 − 2 9=1 9=1 2 2 W¯ # −B where g and b 9 are given in (18) and (20), respectively. Ö 1 B 1 B C# = lim 2Γ `8 − + Γ − √ B→1−2 U U 2 2 8=1 8 8 W¯ As discussed in this Section, the exact and asymptotic # − behaviors of the effective rate for the RIS-assisted system Ö 2 C 2 1 = 2Γ() Γ ` − √# . (19) setup are characterized. It is noted that the exact effective rate 8 U 8=1 8 W¯ is derived with closed-form expression, given in terms of the multivariate Fox’s -function. In the following subsection, The case 2 < g is similarly accomplished. the highly tight effective rate with the assistance of the MoG The exact effective rate is expressed in terms of the distribution is derived. multivariate Fox’s -function, the asymptotic behavior of effective rate is further provided to gain more insights at the D. AN ALTERNATIVE APPROXIMATION METHOD high SNR regime. Remark 5. Motivated by (9b), one can observe that W can 5Note that the univariate Meijer’s -function is available, i.e., MATLAB be modeled in terms of the normally distributed random meijerG(0, 1, 2, 3, G), Maple MeijerG([0B , 1B ], [2B , 3B ],I), and Math- Í ematica MeijerG[{0 } , {0 } , {1 } , {1 } ,I]. variable, i.e., W = W;, and W; v N(e;,[;), consequently, 8 8=1:= 8 8==+1:? 8 8=1:< 8 8=<+1:@ ;=1

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L. Kong et al.: Effective Rate Evaluation of RIS-Assisted Communications Using the Sums of Cascaded U − ` RV the effective rate of our proposed system conﬁguration can 11 be highly approximated as follows 10 8 " 1 Õ 2 R "> ≈ − log F 9 7 2 ; ( + 2) ;=1 3 1 We¯ ; 8 !# 6 1 1 7 + √ + √ . 10 12 14 2 2 6 6(1 + W¯(e; + 3[;) ) 6(1 + W¯(e; − 3[;) ) (24) 5 4 Proof. Performing a revisit to U given in (10), then substituting (9b) into U, yields 3 2 ∞ − 1 p 2 ! Õ F ¹ W 2 ( W/W¯ − e;) U = ; exp − 3W 1 p (1 + W) 2 0 5 10 15 20 ;=1 8cW[¯ ; 0 2[; ∞ ! (2) Õ F ¹ 1 (H − e )2 (a) # = 2 = √ ; exp − ; 3H, (1 + WH¯ 2) 2 ;=1 2c[; 0 2[; (25) 10 where step (2) is developed by interchanging of variables 9 7 q W H = W¯ , then applying the result given in [61, Eq. (4)], i.e., 8 6.5 F( ) = 1 H (1+WH¯ 2) is a real function of H, herein H follows 7 6 the normal distribution with mean e; and variance [;, the 12 13 14 15 6 expectation of F(H) can√ be highly approximated√ at three points, namely, e;, e; + 3[;, and e; − 3[; [62], 5 √ √ E[F(H)] ≈ F (e;) + F (e; + 3[;) + F (e; − 3[;). (26) 4

Finally, substituting (26) into (10), the proof is accomplished. 3 2 0 5 10 15 20 V. NUMERICAL RESULTS AND DISCUSSION For the simplicity of notifications, assuming that all " ver- (b) " = 3 sion of cascaded U-` distributed RVs have the same fading FIGURE 3: Effective rate over U-` fading channels against W¯ ··· ··· parameters, i.e., U1 = = U# = U, and `1 = = `# = ` for selected values of " and #, where = 4, and U = ` = 3. in the simulation. The U-` distributed RVs are implemented using the WAFO toolbox [63]. It is noteworthy that in this paper when " = 1, # is an integer, which corresponds to the multi-hop AF non-regenerative relaying system [22] or and 3) in addition, the effective rate expression derived in MIMO keyhole systems [21]; when # = 1, " is an integer, Remark. 5 provides a highly tight approximation to the exact which might probably be single-hop multiple-input single- result. This fact can be further confirmed in the following output (MISO) systems [10]; and when # = 2, and " is an subsection. Besides, the MoG-based effective rate expression integer, it is the RIS-aided system under consideration. becomes gradually tight as # decreases and " increases, Currently, no existing mathematical packages are avail- respectively. able at MATLAB, Python, or Mathematica to compute the The asymptotic behavior of effective rate at high SNR multivariate Fox’s -function, but it is still numerically regime is depicted in Fig. 4, the analytical behavior of R programmable and computable, see existing examples using given in (20) becomes gradually tight when W¯ increases. Python code [50] and MATLAB code [64]. The multivariate One can observe that when # = 1, U = 2, ` = 1, and Fox’s -function is implemented herein by using the method = 4, the values of effective rate at 40 dB and 30 dB when given in [50]. " = 1, 2, 3 are 3.718, 7.686, 11.59 and 2.888, 6.011, 9.081, respectively. As a result, the slopes of the asymptotic effective ∞ A. NUMERICAL RESULTS rate R" when " = 1, 2, 3 are correspondingly given by g U` As shown in Fig.3 (a) and (b), the effective rate of the RIS- (3.718 − 2.888) × log10 (2) = 0.2499 ≈ 0.25 = 2 = 2 , g U` assisted communications over U-` fading channels given in (7.686 − 6.011) × log10 (2) = 0.5042 ≈ 0.5 = 2 = , and 3g 3U` (11) and Remark. 5 is presented for selected values of " (11.59 − 9.081) × log10 (2) = 0.7553 ≈ 0.75 = 2 = 2 . and #. One can conclude that 1) the effective rate certainly Overall, the asymptotic slope of effective rate given in (23b) increases as " increases; 2) R decreases as # increases; is verified, and the asymptotic analysis at high SNR regime

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L. Kong et al.: Effective Rate Evaluation of RIS-Assisted Communications Using the Sums of Cascaded U − ` RV

12 2

10 1.5 8

6 1

4 0.5 Rayleigh 2

0 0 -10 0 10 20 30 40 10-2 10-1 100 101 102

FIGURE 4: Effective rate over U-` (i.e., U = 2, ` = 1) fading FIGURE 5: Effective rate over Rayleigh (` = 1) and channels against W¯, where # = 1 and = 4. Nakagami-< (` = < = 3.5) fading channels against \ for \) selected values of #, where = ln(2) , ) = 1, and " = 1. is proved to be efficient with reducing complexity compared to the exact effective rate expression. One can obtain that the 18 high SNR slope is linearly associated with U and `. More 16 specifically, large U and ` lead to a steep high SNR slope 14 curve. In other words, large U and ` contributes to higher effective rate performance. Such a conclusion is also verified 12 in Figs.5 and8, which is because the increase of U and ` 10 physically accounts for the less severe fading environment. 8 In Fig.5, the effective rate over Rayleigh and Nakagami- < (< = 3.5) fading channels, given in (16), are compared 6 versus the delay QoS exponent \. We observe that the gain 4 in effective rate degrades as the QoS limitation becomes 2 stricter. As discussed in [22], the cascaded Rayleigh distribu- 0 tion was found feasible to model the M2M communication 0 10 20 30 40 50 scenario, an increase of # herein means the increase of hops, resultantly leading to a worse effective rate performance. FIGURE 6: Effective rate over U-` fading channels against The impact of # and fading parameter ` are also presented. W¯, where " = 1, = 3, U = 2, and ` = 5. One can observe that the effective rate is improved with the increase of fading parameter `, i.e., multipath fading parameter. Barnes integrals as " becomes larger, e.g., 200. Conclu- Fig.6 presents the asymptotic effective rate at high SNR sively, the MoG-based solution alternatively showcases its regime, together with the exact and simulated results. One importance and high priority when doing performance calcu- can observe that the analytical asymptotic effective rate, lation compared to the multivariate Fox’s -function based as expected, gradually behaves accurately as W¯ increases. solution. Besides, inspired by [18], we also verified the asymptotic Fig.8 plots the effective rate of the RIS-aided system effective rate slope at high SNR regime, theoretically given against \ over Weibull (U is the fading parameter, and ` = 1) ∞ in (23a), for # = 2, the R( is 15.02 and 11.7 at 50 dB and fading channels with consideration of the impacts from the 40 dB, respectively. The high SNR slope is then computed as fading parameter U. One can observe that (i) the increase of (15.02 − 11.7) × log10 (2) = 0.9994 ≈ 1. Consequently, as W¯ U is beneficial for R. More importantly, the gain in effective ∞ increases, then (( will surely converge to 1. capacity benefiting from the increase of U becomes obvious In Fig.7, we investigate the effective rate of the proposed as the QoS limitation \ tends to high, which is quite different RIS-assisted system, where the number of reflecting elements from Fig.5; (ii) as expected, the effective rate demonstrates of the RIS ranges from 100, 150, 200, 300, 400, and 500. A an improvement as decreases, in other words, a smaller de- perfect match between the Monte-Carlo simulation and the lay constraint results in a higher effective rate performance; MoG-based analytical results is validated. It is noteworthy and (iii) as shown in Fig. 5, the multipath effect has a negative that the exact effective rate expression becomes difficult impact on the effective rate performance, i.e., lower values of and time-consuming to implement the multivariate Mellin- U and `, which indicate the serious nonlinearity and sparse

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results are highly accurate; and (ii) compared with error>G, 23 error"> is always negative. Such a fact indicates that the 22 MoG-based expression acts as a highly tight upper bound for 21 the exact effective rate. 20

19 VI. CONCLUSION

18 In this paper, we have characterized the sum of cascaded U-` RVs, and studied its application in the RIS-assisted 17 communications from the effective rate perspective. The 16 exact, asymptotic, and approximated behaviors of the effec- 15 tive rate performance for the RIS-aided communications are 14 explored and further derived with closed-form expressions. 13 The asymptotic effective rate slope at high SNR regime was 0 2 4 6 8 10 12 14 also provided. The accuracy of our analytical results were supported by the simulated results, and tightly approached FIGURE 7: Effective rate over U-` fading channels against by its asymptotic behavior at high SNR regime. W¯, where = 4, # = 2, and U = ` = 3. Interesting and useful insights can be thereafter obtained that (i) the obtained analytical results are not only applicable for the RIS-aided systems, when " = 1 or # = 1, it corre- 4.5 sponds to the multi-hop AF non-regenerative relaying system

4 [22]/MIMO keyhole systems [21] or the single-hop MISO systems [10]; (ii) the effective rate can take advantage of the 3.5 increase of " and the decrease of the number of statistically independent scatters #; and (iii) fading parameters, U and `, 3 i.e., nonlinearity and clustering of the wireless propagation 2.5 medium, are advantageous to enhance the effective rate performance. In addition, compared with the multivariate Fox’s 2 -function based effective rate expression, the MoG-based solution is highly accurate and serves as a tight and simple 1.5 upper bound for the exact effective rate solution with re- 1 ducing complexity. Overall, this paper mainly considered the 10-2 10-1 100 101 102 ideal phase-shifting, the correlation of signals in RIS-aided systems is a worthy subject and left for future investigation. FIGURE 8: Effective rate over Weibull (U is the fading . \) parameter, ` = 1) fading channels against \, where = ln(2) , # = 2, " = 5, and W¯ = 0 dB. APPENDIX A PROOF FOR THEOREM 1 Revisiting the multivariate Fox’s -function in (9a) in terms of its deﬁnition from (3), and then Plugging the result and clustering of propagation medium, lead to the poor effective (14) into U, we have U given in (28), shown at the top of rate performance. next page. It is noted that step (3) in (28) is developed by interchanging the order of two integrals, subsequently, U1 B. ACCURACY ANALYSIS is further evaluated with the help of the deﬁnition of Mellin For the purposes of illustrating the tightness of our analytical transform for Fox’s -function [57, Eq.(2.25.2.1)], results, a useful measure, namely analytical error, is used as " " Õ B 9 Õ B 9 the accuracy indicator [52], [61], U = Γ © ª Γ © − ª . (29) 1 2 ® 2 ® analytical results 9=1 9=1 analytical error = 1 − × 100%. (27) « ¬ « ¬ simulation results After some algebraic manipulations and deploying the def- For the brevity of notations, let error>G denote the analytical inition of multivariate Fox’s -function, the effective rate error between the simulated results and the Fox’s -function of the RIS-assisted system setup over U-` fading channel is based results, similarly, error"> represents the analytical obtained. error between the simulated results and the MoG-based results. APPENDIX B PROOF FOR REMARK 3 Based on Fig. 3. (a), as shown in Table. 2, we have two in- Since the exact effective rate expression is given in terms teresting observations: (i) both error>G and error"> ranges of the multivariate Fox’s -function, then re-expressing the from −1% to 1%, such a fact can be interpreted that both multivariate Fox’s -function in the manner of the multiple

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TABLE 2: ANALYTICAL ERROR AGAINST W¯(3) WHEN = 4, " = 4, # = 2, U = ` = 3.

W¯ 0 4 8 12 16 20 R −3 −2 −2 −4 −2 −3 error>G -1.92 ∗ 10 % 3.05 ∗ 10 % 2.17 ∗ 10 % 5.34 ∗ 10 % 1.76 ∗ 10 % -5.19 ∗ 10 % −2 −2 error"> -0.13% -0.11% -0.11% -0.12% -8.74 ∗ 10 % -9.86 ∗ 10 %

B9 ¹ ∞ ¹ ¹ " C q(B ) r B9 −1 [" 1,1 (1 − , 1) 1 Ö 9 9 W U = W ··· Φ(B1, ··· ,B" ) √ 3B1, ··· , 3B" 3W 2Γ() 1,1 (0, 1) (2cF)" W¯ 0 L1 L" 9=1 WW¯ " B −" " B9 ∞ Í 9 − (28) (3) [ (2cF) ¹ ¹ C ¹ 2 1 ( − ) " Ö 9 9=1 1,1 1 , 1 = ··· Φ(B1, ··· ,B" ) q(B 9 ) √ W W 3W 3B1, ··· , 3B" , 2Γ() 1,1 (0, 1) L1 L" 9=1 W¯ 0 | {z } U1

Mellin-Barnes type integral according to its definition. When Using the residue theorem, Res [ K ( B 1 ,B 2,B3),B1 = b1 = U1 `1] W¯ → ∞, we have can be developed as follows 1 U8, 9 1 Res[K(B ,B ,B ), b ] = lim (B − b )K(b ,B ,B ) # ` Γ `8, 9 + 1 2 3 1 1 1 1 2 3 C# , 9 1 Ö 8, 9 U8, 9 B1→b1 √ = √ → 0. + + Γ(` ) C U1 `1 Γ − U1 `1 B2 B3 W¯ W¯ 8=1 8, 9 1,1 2 = −U1Γ (U1 `1) √ (32) Applying the result given in [59], each single Mellin-Barnes W¯ Γ(U1 `1 + B2 + B3) B2 B3 integral is then further simplified and expanded at the single U1 `1 + B2 + B3 C1,2 C1,3 × Γ q(B2)q(B3) √ √ . residue b 9 of the corresponding integrands at the pole closest 2 W¯ W¯ to the contour [47], namely, the minimum pole on the right of Next, substituting (32) into (31), we have the contour for small Fox’s arguments, and subsequently + + applying the residue theorem, we can arrive at (20). U1 `1 ¹ ¹ U1 `1 B2 B3 U1Γ (U1 `1) C1,1 Γ − To be specific, we herein take the case " = 3 and Δ ≈ √ 2 (2cF)2 W¯ Γ(U ` + B + B ) # = 1 as an example to illustrate the steps for arriving at L2 L3 1 1 2 3 B2 B3 the asymptotic effective rate behavior. As such, the effective U1 `1 + B2 + B3 C1,2 C1,3 × Γ q(B2)q(B3) √ √ 3B23B3. rate is re-expressed as follows in terms of the definition of 2 W¯ W¯ multivariate Fox’s -function, (33) 1 [" In continuation with the asymptotic analysis, the integrals R = − log2 Δ , where 2Γ() over L2 and L3 are similarly evaluated at the residue points 1 ¹ ¹ ¹ b = U ` and b = U ` , respectively. Next applying the Δ = K(B ,B ,B )3B 3B 3B , 2 2 2 3 3 3 3 1 2 3 1 2 3 residue theorem, the proof for (20) is finished. (2cF) L1 L2 L3 When # = 1, then b = U ` , each single Mellin-Barnes  3 C B9 9 9 9  K(B ,B ,B ) = Φ(B ,B ,B ) Î q(B ) √1, 9  1 2 3 1 2 3 9 W¯ integral is expanded at b 9 , i.e.,  9=1  + + + +  − B1 B2 B3 B1 B2 B3 B9 B9 and Γ 2 Γ 2 C1, 9 C1, 9 Φ(B ,B ,B ) = Res q(B ) √ , b = lim (B − b )q(B ) √ ,  1 2 3 Γ(B1+B2+B3) 9 9 9 9 9  W¯ B9 →b9 W¯  1  q(B 9 ) = Γ ` 9 − B 9 Γ(B 9 ), 9 = 1, 2, 3.  U9 subsequently, following the steps for (20), the proof for (21)  (30) is obtained. C When W¯ goes to ∞, we have √1, 9 → 0. According to W¯ [58, Theorem 1.7], the Mellin-Barnes integral over L1 is REFERENCES approximated by evaluating the residue at the closest pole [1] Dapeng Wu and R. Negi, “Effective capacity: a wireless link model for support of quality of service,” IEEE Trans. Wireless Commun., vol. 2, located to the right-hand side, i.e., b1 = U1 `1. As such, the no. 4, pp. 630–643, Jul. 2003. contour L1 is clockwise and −1 is needed to be multiplied, [2] M. C. Gursoy, “MIMO wireless communications under statistical queue- [47] ing constraints,” IEEE Trans. Inf. Theory, vol. 57, no. 9, pp. 5897–5917, ¹ ¹ ¹ Sep. 2011. 1 [3] M. Amjad, L. Musavian, and M. H. Rehmani, “Effective capacity in Δ = K(B1,B2,B3)3B13B23B3 3 wireless networks: A comprehensive survey,” IEEE Commun. Surveys (2cF) L1 L2 L3 −1 ¹ ¹ Tuts., vol. 21, no. 4, pp. 3007–3038, Fourthquarter 2019. = Res[K(B ,B ,B ),B = b ]3B 3B . [4] W. Yu, A. Chorti, L. Musavian, H. Vincent Poor, and Q. Ni, “Effective 2 1 2 3 1 1 2 3 (2cF) L2 L3 secrecy rate for a downlink NOMA network,” IEEE Trans. Wireless (31) Commun., vol. 18, no. 12, pp. 5673–5690, Dec. 2019.

10 VOLUME 4, 2016

L. Kong et al.: Effective Rate Evaluation of RIS-Assisted Communications Using the Sums of Cascaded U − ` RV

[5] V. Kumar, B. Cardiff, S. Prakriya, and M. F. Flanagan, “Delay violation [28] M. D. Renzo, A. Zappone, M. Debbah, M. Alouini, C. Yuen, J. D. Rosny, probability and effective rate of downlink NOMA over U-` fading chan- and S. Tretyakov, “Smart radio environments empowered by reconfig- nels,” IEEE Trans. Veh. Technol., vol. 69, no. 10, pp. 11 241–11 252, Oct. urable intelligent surfaces: How it works, state of research, and road 2020. ahead,” IEEE J. Sel. Areas Commun., vol. 38, no. 11, pp. 2450–2525, Nov. [6] M. Amjad, L. Musavian, and S. Aïssa, “Effective capacity of NOMA 2020. with finite blocklength for low-latency communications,” arXiv preprint [29] E. Basar, “Reconfigurable intelligent surface-based index modulation: A arXiv:2002.07098, 2020. new beyond MIMO paradigm for 6G,” IEEE Trans. Commun., vol. 68, [7] V. Kumar, B. Cardiff, S. Prakriya, and M. F. Flanagan, “Effective rate of no. 5, pp. 3187–3196, May 2020. downlink NOMA over ^-` shadowed fading with integer fading parame- [30] J. Zhang, E. Björnson, M. Matthaiou, D. W. K. Ng, H. Yang, and D. J. ters,” in Proc. IEEE Int. Conf. Commun. (ICC), Dublin, Ireland, Ireland, Love, “Prospective multiple antenna technologies for beyond 5G,” IEEE 2020, pp. 1–7. J. Sel. Areas Commun., vol. 38, no. 8, pp. 1637–1660, Aug. 2020. [8] C. Guo, L. Liang, and G. Y. Li, “Resource allocation for low-latency [31] M. D. Yacoub, “The U-` distribution: A physical fading model for the vehicular communications: An effective capacity perspective,” IEEE J. Sel. stacy distribution,” IEEE Trans. Veh. Technol., vol. 56, no. 1, pp. 27–34, Areas Commun., vol. 37, no. 4, pp. 905–917, Apr. 2019. Jan. 2007. [9] M. You, H. Sun, J. Jiang, and J. Zhang, “Effective rate analysis in Weibull [32] A. K. Papazafeiropoulos and S. A. Kotsopoulos, “Generalized phase- fading channels,” IEEE Wireless Commun. Lett., vol. 5, no. 4, pp. 340–343, crossing rate and random FM noise for U-` fading channels,” IEEE Trans. Aug. 2016. Veh. Technol., vol. 59, no. 1, pp. 494–499, Jan. 2010. [10] J. Zhang, L. Dai, Z. Wang, D. W. K. Ng, and W. H. Gerstacker, “Effective [33] E. J. Leonardo and M. D. Yacoub, “Product of U-` variates,” IEEE rate analysis of MISO systems over U-` fading channels,” in IEEE Wireless Commun. Lett., vol. 4, no. 6, pp. 637–640, Dec. 2015. GLOBECOM, Dec 2015, pp. 1–6. [34] U. S. Dias and M. D. Yacoub, “On the U-` autocorrelation and power [11] X. Li, J. Li, L. Li, J. Jin, J. Zhang, and D. Zhang, “Effective rate of MISO spectrum functions: Field trials and validation,” in IEEE GLOBECOM, systems over ^-` shadowed fading channels,” IEEE Access, vol. 5, pp. Honolulu, HI, USA, Nov. 2009, pp. 1–6. 10 605–10 611, 2017. [35] Q. Wu, D. W. Matolak, and I. Sen, “5-GHz-Band vehicle-to-vehicle [12] Z. Ji, C. Dong, Y. Wang, and J. Lu, “On the analysis of effective capacity channels: Models for multiple values of channel bandwidth,” IEEE Trans. over generalized fading channels,” in Proc. IEEE Int. Conf. Commun. Veh. Technol., vol. 59, no. 5, pp. 2620–2625, Jun. 2010. (ICC), June 2014, pp. 1977–1983. [36] P. Karadimas, E. D. Vagenas, and S. A. Kotsopoulos, “On the scatterers’ [13] M. You, H. Sun, J. Jiang, and J. Zhang, “Unified framework for the mobility and second order statistics of narrowband fixed outdoor wireless effective rate analysis of wireless communication systems over MISO channels,” IEEE Trans. Wireless Commun., vol. 9, no. 7, pp. 2119–2124, fading channels,” IEEE Trans. Commun., vol. 65, no. 4, pp. 1775–1785, Jul. 2010. Apr. 2017. [37] P. K. Chong, S. E. Yoo, S. H. Kim, and D. Kim, “Wind-blown foliage and [14] F. S. Almehmadi and O. S. Badarneh, “On the effective capacity of human-induced fading in ground-surface narrowband communications at F Electron. Lett. Fisher–Snedecor fading channels,” , vol. 54, no. 18, pp. 400 MHz,” IEEE Trans. Veh. Technol., vol. 60, no. 4, pp. 1326–1336, May 1068–1070, 2018. 2011. [15] H. Al-Hmood and H. S. Al-Raweshidy, “Unified approaches based ef- [38] A. Michalopoulou, A. A. Alexandridis, K. Peppas, T. Zervos, F. Lazarakis, fective capacity analysis over composite U-[-`/gamma fading channels,” K. Dangakis, and D. I. Kaklamani, “Statistical analysis for On-Body Electron. Lett., vol. 54, no. 13, pp. 852–853, 2018. spatial diversity communications at 2.45 GHz,” IEEE Trans. Antennas [16] H. Kaur, V. Kansal, and S. Singh, “Effective rate analysis of two-wave with Propag., vol. 60, no. 8, pp. 4014–4019, Aug. 2012. diffuse-power fading channels,” IET Commun., vol. 13, no. 3, pp. 339–344, [39] J. Reig and L. Rubio, “Estimation of the composite fast fading and shad- 2019. owing distribution using the log-moments in wireless communications,” [17] R. Singh, M. Rawat, and P. M. Pradhan, “Effective capacity of wireless net- IEEE Trans. Wireless Commun., vol. 12, no. 8, pp. 3672–3681, Aug. 2013. works over double shadowed Rician fading channels,” Wireless Networks, [40] D. B. da Costa, M. D. Yacoub, and J. C. S. Santos Filho, “Highly accurate vol. 26, no. 2, pp. 1347–1355, 2020. closed-form approximations to the sum of U-` variates and applications,” [18] Y. Ai, A. Mathur, L. Kong, and M. Cheffena, “Effective throughput IEEE Trans. Wireless Commun., vol. 7, no. 9, pp. 3301–3306, Sep. 2008. analysis of U-[-^-` fading channels,” IEEE Access, vol. 8, pp. 57 363– 57 371, 2020. [41] D. B. Da Costa and M. D. Yacoub, “Accurate approximations to the sum of generalized random variables and applications in the performance analysis [19] H. Al-Hmood and H. Al-Raweshidy, “On the effective rate and energy of diversity systems,” IEEE Trans. Commun., vol. 57, no. 5, pp. 1271– detection based spectrum sensing over U-[-^-` fading channels,” IEEE 1274, May 2009. Trans. Veh. Technol., vol. 69, no. 8, pp. 9112–9116, Aug. 2020. [20] S. K. Yoo, S. L. Cotton, P. C. Sofotasios, S. Muhaidat, and G. K. Kara- [42] K. P. Peppas, “Sum of nonidentical squared ^-` variates and applications giannidis, “Effective capacity analysis over generalized composite fading in the performance analysis of diversity receivers,” IEEE Trans. Veh. channels,” IEEE Access, vol. 8, pp. 123 756–123 764, 2020. Technol., vol. 61, no. 1, pp. 413–419, Jan. 2012. [21] C. Zhong, T. Ratnarajah, K. . Wong, and M. . Alouini, “Effective capacity [43] O. S. Badarneh, D. B. da Costa, P. C. Sofotasios, S. Muhaidat, and S. L. of multiple antenna channels: correlation and keyhole,” IET Commun., Cotton, “On the sum of Fisher–Snedecor F variates and its application to vol. 6, no. 12, pp. 1757–1768, Aug. 2012. maximal-ratio combining,” IEEE Wireless Commun. Lett., vol. 7, no. 6, pp. [22] I. Z. Kovacs, P. C. F. Eggers, K. Olesen, and L. G. Petersen, “Investigations 966–969, Dec. 2018. of outdoor-to-indoor mobile-to-mobile radio communication channels,” in [44] H. Du, J. Zhang, J. Cheng, and B. Ai, “Sum of Fisher-Snedecor F random Proceedings IEEE 56th VTC, vol. 1, Vancouver, BC, Canada, 2002, pp. variables and its applications,” IEEE Open J. Commun. Soc., vol. 1, pp. 430–434 vol.1. 342–356, 2020. [23] G. K. Karagiannidis, N. C. Sagias, and P. T. Mathiopoulos, “# ∗Nakagami: [45] P. Zhang, J. Zhang, K. P. Peppas, D. W. K. Ng, and B. Ai, “Dual-hop A novel stochastic model for cascaded fading channels,” IEEE Trans. relaying communications over Fisher-Snedecor F-fading channels,” IEEE Commun., vol. 55, no. 8, pp. 1453–1458, Aug. 2007. Trans. Commun., vol. 68, no. 5, pp. 2695–2710, May 2020. [24] L. Kong, G. Kaddoum, and D. B. da Costa, “Cascaded U − ` fading [46] J. Zheng, J. Zhang, G. Pan, J. Cheng, and B. Ai, “Sum of squared channels: Reliability and security analysis,” IEEE Access, vol. 6, pp. fluctuating two-ray random variables with wireless applications,” IEEE 41 978–41 992, 2018. Trans. Veh. Technol., vol. 68, no. 8, pp. 8173–8177, Aug. 2019. [25] O. S. Badarneh, S. Muhaidat, P. C. Sofotasios, S. L. Cotton, K. Rabie, and [47] Y. Abo Rahama, M. H. Ismail, and M. S. Hassan, “On the sum of D. B. da Costa, “The # ∗Fisher-Snedecor F cascaded fading model,” in independent Fox’s -function variates with applications,” IEEE Trans. IEEE WiMob, Limassol, Cyprus, 2018, pp. 1–7. Veh. Technol., vol. 67, no. 8, pp. 6752–6760, Aug. 2018. [26] Z. Hadzi-Velkov, N. Zlatanov, and G. K. Karagiannidis, “Level crossing [48] A. M. Mathai, R. K. Saxena, and H. J. Haubold, The -function: theory rate and average fade duration of the multihop Rayleigh fading channel,” and applications. Springer Science & Business Media, 2009. in IEEE ICC, 2008, pp. 4451–4455. [49] L. Kong, G. Kaddoum, and H. Chergui, “On physical layer security over [27] L. Yang, F. Meng, Q. Wu, D. B. da Costa, and M. Alouini, “Accurate Fox’s -function wiretap fading channels,” IEEE Trans. Veh. Technol., closed-form approximations to channel distributions of RIS-aided wireless vol. 68, no. 7, pp. 6608–6621, Jul. 2019. systems,” IEEE Wireless Commun. Lett., vol. 9, no. 11, pp. 1985–1989, [50] H. R. Alhennawi, M. M. H. El Ayadi, M. H. Ismail, and H. M. Mourad, Nov. 2020. “Closed-form exact and asymptotic expressions for the symbol error

VOLUME 4, 2016 11

L. Kong et al.: Effective Rate Evaluation of RIS-Assisted Communications Using the Sums of Cascaded U − ` RV

rate and capacity of the -function fading channel,” IEEE Trans. Veh. YUN AI (Member, IEEE) received the M.Sc. de- Technol., vol. 65, no. 4, pp. 1957–1974, Apr. 2016. gree in electrical engineering from the Chalmers [51] B. Selim, O. Alhussein, S. Muhaidat, G. K. Karagiannidis, and J. Liang, University of Technology, Göteborg, Sweden, in “Modeling and analysis of wireless channels via the mixture of Gaussian 2012, and the Ph.D. degree from the University distribution,” IEEE Trans. Veh. Technol., vol. 65, no. 10, pp. 8309–8321, of Oslo, Oslo, Norway, in 2018. In 2016, he was Oct. 2016. a Visiting Researcher with the Department of In- [52] L. Kong, S. Chatzinotas, and B. Ottersten, “Unified framework for secrecy formation and Computer Science, Keio University, characteristics with mixture of Gaussian (MoG) distribution,” IEEE Wire- Tokyo, Japan. In 2016, funded by a stipend from less Commun. Lett., vol. 9, no. 10, pp. 1625–1628, Oct. 2020. the Norwegian University Center, Saint Peters- [53] J. Zhang, M. Matthaiou, G. K. Karagiannidis, and L. Dai, “On the multivariate Gamma–Gamma distribution with arbitrary correlation and burg, Russia, he was a Visiting Researcher with applications in wireless communications,” IEEE Trans. Veh. Technol., the Faculty of Applied Mathematics and Control Processes, Saint Petersburg vol. 65, no. 5, pp. 3834–3840, May 2016. State University, Saint Petersburg, Russia. He is currently a Researcher with [54] L. Yang, Y. Yang, M. O. Hasna, and M. Alouini, “Coverage, probability the Norwegian University of Science and Technology (NTNU), Norway. He of SNR gain, and DOR analysis of RIS-aided communication systems,” has also been with the Signal Processing Group, Technische Universität IEEE Wireless Commun. Lett., vol. 9, no. 8, pp. 1268–1272, Aug. 2020. Darmstadt, Darmstadt, Germany, as a Research Student and also with [55] E. Basar and I. Yildirim, “Indoor and outdoor physical channel modeling Ericsson as an Engineer. His current research interests include the broad and efficient positioning for reconfigurable intelligent surfaces in mmwave areas of wireless communications with focuses on communication theory, bands,” arXiv preprint arXiv:2006.02240, 2020. wireless sensor networks, and smart grids. [56] S. W. H. Shah, M. M. U. Rahman, A. N. Mian, A. Imran, S. Mumtaz, and O. A. Dobre, “On the impact of mode selection on effective capacity of device-to-device communication,” IEEE Wireless Commun. Lett., vol. 8, no. 3, pp. 945–948, Jun. 2019. [57] A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series: More special functions. Gordon and Breach Science Publishers, 1990, vol. 3. [58] A. A. Kilbas, H-Transforms: Theory and Applications. Analytical Methods and Special Functions. CRC Press, 2004. [59] H. Chergui, M. Benjillali, and S. Saoudi, “Performance analysis of project- and-forward relaying in mixed MIMO-pinhole and Rayleigh dual-hop channel,” IEEE Commun. Lett., vol. 20, no. 3, pp. 610–613, Mar. 2016. [60] L. Kong, G. Kaddoum, and Z. Rezki, “Highly accurate and asymptotic analysis on the SOP over SIMO U–` fading channels,” IEEE Commun. Lett., vol. 22, no. 10, pp. 2088–2091, Oct. 2018. [61] G. Pan, C. Tang, X. Zhang, T. Li, Y. Weng, and Y. Chen, “Physical-layer security over non-small-scale fading channels,” IEEE Trans. Veh. Technol., vol. 65, no. 3, pp. 1326–1339, Mar. 2016. [62] J. M. Holtzman, “A simple, accurate method to calculate spread-spectrum multiple-access error probabilities,” IEEE Trans. Commun., vol. 40, no. 3, pp. 461–464, Mar. 1992. [63] P. Brodtkorb, P. Johannesson, G. Lindgren, I. Rychlik, J. Rydén, and E. Sjö, “WAFO - a Matlab toolbox for the analysis of random waves and loads,” in Proc. 10’th Int. Offshore and Polar Eng. Conf., ISOPE, Seattle, USA, vol. 3, 2000, pp. 343–350. [64] H. Chergui, M. Benjillali, and M. Alouini, “Rician K-factor-based analysis of XLOS service probability in 5G outdoor ultra-dense networks,” IEEE Wireless Commun. Lett., vol. 8, no. 8, pp. 428–431, Apr. 2019.

SYMEON CHATZINOTAS (Senior Member, IEEE) received the M.Eng. degree in telecommu- LONG KONG received the B.S. degree in nications from the Aristotle University of Thes- telecommunication engineering from Hainan Uni- saloniki, Thessaloniki, Greece, in 2003, and the versity, Haikou, Hainan, China, in 2010, the M.S. M.Sc. and Ph.D. degrees in electronic engineering degree in communication engineering from Xi- from the University of Surrey, Surrey, U.K., in amen University, Xiamen, China, in 2013, and 2006 and 2009, respectively. the Ph.D. degree from the École de Technologie He is currently a Full Professor/Chief Scientist Supérieure (ÉTS), Université du Québec, Mon- I and Co-Head of the SIGCOM Research Group tréal, QC, Canada. Since September 2019, he has with the SnT, University of Luxembourg. He was been a research associate at the Interdisciplinary a Visiting Professor with the University of Parma, Italy, and he was involved Centre for Security, Reliability and Trust (SnT), in numerous research and development projects with the National Center University of Luxembourg, Luxembourg. From January 2019 to April 2019, for Scientiﬁc Research Demokritos, the Center of Research and Technology he was an exchange student with the center of wireless communications Hellas, and the Center of Communication Systems Research, University of (CWC), University of Oulu, Oulu, Finland, supported by the Mitacs Glob- Surrey. He has authored or coauthored more than 400 technical papers in alink program. From August 2013 to November 2014, he was a Research refereed international journals, conferences, and scientiﬁc books. Assistant in electronic and information engineering (EIE) with The Hong Prof. Chatzinotas was a co-recipient of the 2014 IEEE Distinguished Kong Polytechnic University, Hong Kong. His research interests include Contributions to Satellite Communications Award, the CROWNCOM 2015 physical layer security, cooperative communications, and chaos-based com- Best Paper Award, and the 2018 EURASIC JWCN Best Paper Award. He munication systems. is currently on the Editorial Board of the IEEE OPEN JOURNAL OF VEHICULAR TECHNOLOGY and the International Journal of Satellite Communications and Networking.

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L. Kong et al.: Effective Rate Evaluation of RIS-Assisted Communications Using the Sums of Cascaded U − ` RV

BJÖRN OTTERSTEN (Fellow, IEEE) was born in Stockholm, Sweden, in 1961. He received the M.S. degree in electrical engineering and applied physics from Linköping University, Linköping, Sweden, in 1986, and the Ph.D. degree in electrical engineering from Stanford University, Stanford, CA, USA, in 1990. He has held research positions with the Depart- ment of Electrical Engineering, Linköping Univer- sity, the Information Systems Laboratory, Stanford University, the Katholieke Universiteit Leuven, Leuven, Belgium, and the University of Luxembourg, Luxembourg. From 1996 to 1997, he was the Di- rector of Research with ArrayComm, Inc., a start-up in San Jose, CA, USA, based on his patented technology. In 1991, he was appointed a Professor of signal processing with the Royal Institute of Technology (KTH), Stockholm, Sweden. From 1992 to 2004, he was the Head of the Department for Signals, Sensors, and Systems, KTH, and from 2004 to 2008, he was the Dean of the School of Electrical Engineering, KTH. He is currently the Director for the Interdisciplinary Center for Security, Reliability and Trust, University of Luxembourg. As Digital Champion of Luxembourg, he acts as an Adviser to the European Commission. He was a recipient of the IEEE Signal Processing Society Technical Achievement Award in 2011 and the European Research Council advanced research grant twice, in 2009–2013 and in 2017–2022. He has co-authored journal papers that received the IEEE Signal Processing Society Best Paper Award in 1993, 2001, 2006, and 2013, respectively, and seven other IEEE conference papers Best Paper Awards. He has served as an Associate Editor for the IEEE TRANSACTIONS ON SIGNAL PROCESSING and the Editorial Board for the IEEE Signal Processing Magazine. He is currently a Member of the Editorial Boards of the EURASIP Signal Processing Journal, the EURASIP Journal of Advances in Signal Processing, and the Foundations and Trends of Signal Processing. He is a Fellow of EURASIP.

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