RIS-Assisted Full-Duplex Relay Systems

Reconfigurable intelligent surfaces (RISs) have recently been presented as a revolutionary technique and recognized as one of the candidates for beyond fifth-generation wireless networks. The RIS technology becomes especially attractive due to its capability to operate in a full-duplex (FD) mode, which is able to ideally double the half-duplex radio spectrum efficiency. This article analyzes the performance of an RIS-aided multihop FD relaying system, where an intermediate FD relay is employed to defeat the far-field path-loss effect inherent to RIS communication links. To this end, we analytically derive exact and asymptotic closed-form expressions for the outage probability, average spectral efficiency, and average bit error rate. The RIS-aided system demonstrates effective performance improvement for the proposed network. However, from the results, it is noticed that the increase in the number of RIS units within a certain communication link does not provide linear performance improvement due to the RIS channel's far-field path-loss model. Thus, mixing active FD relaying techniques with passive RIS platforms can be an ideal solution to improve further system performance, such as outage probability, average spectral efficiency, and average bit error rate in RIS-enabled networks.

communication rates by offering wider bandwidth [2]. The primary distinctive technical feature of the mmWave communication is that signals suffer from a significant path loss compared to conventional lower frequency bands. This severe path-loss effect in mmWave bands can be overcome using large antenna arrays [3], [4]. For this purpose, multiple-input multiple-output (MIMO) technology is employed to perform beamforming in order to obtain an array gain that overcomes the path-loss impact in an mmWave frequency range [5]. However, in dense urban and indoor areas, the mmWave communications' high directivity antennas become oversensitive to blockages [6]. In addition to this, mmWave frequency range still has restrictions in terms of propagation characteristics, such as diffraction, penetration loss, and scattering [6]. In recent years, reconfigurable intelligent surfaces (RISs) have been proposed to address the blockage issue and improve the coverage in mmWave communications [7]. The RIS is considered to be a key technique in the future beyond fifth-generation(5G) systems, which can be considered as a physical evolution of massive MIMO [8], where multiple reflecting antenna modules can be programmed to control the propagation environment in a constructive manner. Hence, the RIS provides the programmable control of the wireless medium [9]. The RIS can be easily integrated into the interior of buildings and on the surface of large vehicles with minor or no change of equipment and device software, which will significantly reduce the cost of performance improvement, e.g., power efficiency, connectivity, coverage extension, etc., for future wireless networks. Moreover, the RIS can be designed in a lightweight fashion due to its miniature circuitry, which makes it suitable for indoor propagation improvement [10]. The RIS has been studied for different wireless applications, namely, cognitive radio [11], wireless power transfer [12], nonorthogonal multiple access [13], wireless security [14], etc.
The operation principle of the RIS is similar to the full-duplex (FD) MIMO relay from the reflecting characteristic perspective. However, the RIS also refracts the incident signals by passing through its surface, which can effectively reconstruct the channels to blocked users and significantly improve transmission performance. Owing to these RIS characteristics, the RIS is believed to provide better system performance than that of halfduplex (HD)/FD relaying techniques. However, as numerically illustrated in [15], the RIS needs hundreds of reflecting elements and perfect phase-shifting ability to outperform a single-antenna HD decode-and-forward (DF) relay. Furthermore, it has been shown that a sufficiently large number of RIS elements are needed to outperform the FD-DF relay [16]. In addition, the RIS should be located at a short distance between the two nodes to provide maximum performance. The reason is that the source's signal passes through two hops before reaching the destination, which leads to a poor achieved channel gain due to the impact of a far-field path-loss effect [17]. One option to overcome the This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ impact of far-field path loss in RIS communications is to adopt intermediate FD relays that can work as repeaters/boosters of the RIS-assisted signals and enhance spectral efficiency.
The FD transceiver simultaneously transmits and receives in the same frequency band; thus, it has the potential to double the spectrum utilization compared to the traditional HD systems [18]- [20]. The promising spectral efficiency gain in the FD networks also brings challenges in practice as strong self-interference (SI) exists at the transceiver caused by the simultaneous transmission mode [21]. If these interference terms are left without being canceled/removed, it can significantly degrade the FD networks' performance by making FD perform worse than the HD one. Various SI suppression methods have been proposed in the literature, such as passive cancellation [21], analog and digital cancellations [22], etc. The combination of these interference suppression techniques allows suppressing the SI at the FD node down to the background noise floor by making it possible to use FD systems in practice [23]. An appealing advantage of a dual-hop FD relaying has been extensively investigated in different wireless applications, e.g., cognitive radio [24], device-to-device communications [25], massive MIMO [26], and mmWave networks [27]. Hence, a combination of RIS and FD techniques in a future wireless network infrastructure could help achieve a high data rate. In that network, the RIS can be reconfigured to constructively combine the reflected signals at the receiver and reduce the effect of SI at the FD receivers by boosting the desired signal [28].

A. Related Works
RIS-assisted FD systems have been studied in a few works [29]- [32], where all of them considered the FD ability at the terrestrial transmitter/receiver node or both the nodes. These works concluded that the RIS improves the coverage and boosts the signal-to-noise ratio (SNR) level in FD communication systems. Tian et al. [33] proposed an unmanned aerial vehicle as an FD transmitter to deliver signals to uplink and downlink users through the RIS. The results showed that the proposed scheme outperforms the HD-RIS mode. Moreover, the study of RIS with other technologies, such as a device-to-device network [34], physical layer security [35], and nonorthogonal multiple access [36], showed the usefulness of the FD mode to improve the system performance. Despite the considerable FD-RIS literature, only a small number of works concentrated on the RIS-assisted two-hop FD communication systems. For example, Abdullah et al. [37] proposed a hybrid FD-RIS communication network, where the FD relay and the RIS forwarded data to the end user. The results illustrated that the proposed system outperformed the RIS only and RIS-assisted HD relay networks. Ying et al. [38] proposed an architecture consisting of double side-by-side RISs connected through a relay working in an FD mode. The numerical results showed that the proposed model achieved a higher rate than that of the simple RIS and conventional relay techniques, especially at high frequencies.
Another conclusion of the work is that the FD relay-assisted RIS system can provide a certain transmission rate with much fewer elements than a traditional RIS architecture.
The RIS may play a vital role in future wireless networks by providing benefits in terms of reliability and QoS. Specifically, the multihop RIS technique is a good candidate for coverage extension solutions in short-ranged mmWave communication networks. Even though various RIS-aided networks have been studied in recent years, there is a lack of research investigating multihop or cascaded RISs comprehensively. Initial work on the topic of cascaded RIS systems with two relaying RISs has been studied in [39]- [41] to show that the two-hop RIS architecture obtains multiplied beamforming gain compared to the conventional single-hop RIS benchmark. However, these studies do not consider multiple RISs (higher than two) to illustrate the performance limits using multiple RISs within a fixed communication link, e.g., outage probability (OP), bit error rate (BER), spectral efficiency, etc. Hence, whether the increase in the number of RISs within a certain communication link leads to further improve system performance remains unknown. A second question is an interaction between passive RIS and active FD systems to satisfy target system performance metrics, such as the BER or the receiver SNR.

B. Contributions
To address the challenges mentioned above, this article first considers a single RIS-assisted FD relaying network, where the communication is performed through the assistance of both RIS and FD relay. We provide an analytical derivation in terms of the OP, average spectral efficiency, and average BER for the considered system model. Furthermore, we consider two system model cases: 1) communication in a two-hop RIS system and one FD relay and 2) communication in a three-hop RIS network. Here, we numerically show the benefit of the former system model. The main contributions of this article can be summarized as follows.
1) We compare a cascaded multihop RIS system with a multihop RIS-assisted FD relaying. Here, we show that the increase in the number of RIS elements within a communication link does not provide considerable spectral efficiency improvement due to the RIS channel's far-field path-loss model. However, the RIS network can achieve better system performance in terms of spectral efficiency, OP, and BER if it uses an intermediate DF-FD relaying node to combat the path-loss effect. We also show that the network with the FD relay obtains the spectral efficiency of an ideal phase-shift mode at a smaller phase-shift resolution than the cascaded RIS system. 2) The works [28], [32], and [39] offered noncentral chisquare (NCCS) distribution based on a central limit theorem (CLT) to describe the statistical model of RIS-aided communication links. However, as it was demonstrated in [42], the NCCS distribution has limitations such as the significant divergence between theoretical and simulation results and the applicability of this distribution only when the RIS has a large number of reflecting elements. Hence, this study analytically investigates the RIS-assisted FD relaying network, where a channel to/from the RIS is modeled by the Nakagami-m random variable (RV) and the distribution of the cascaded Nakagami-m channels is approximated by the squared generalized-K (K G ) distribution [43], which is a more accurate approximation than the NCCS. The tightness of the K G model is validated over Monte Carlo simulations. 3) We provide exact OP, average spectral efficiency, and average BER performance by deriving the cumulative distribution functions (CDFs) and probability density functions (PDFs) of the proposed two-hop network. Moreover, asymptotic expressions for the analytical performance are provided in simple elementary function forms, which allow us to define the system's diversity order and attain further insights from the analytical studies. It is illustrated that the diversity order depends on the number of RIS elements and the wireless channel parameters. The rest of this article is organized as follows. Section II introduces the system model and transmission protocol of an RIS-assisted FD relaying network. Section IV provides and discusses new closed form and approximated analytical expressions for the OP, average spectral efficiency, and average BER. Section V shows numerical findings to validate the correctness of theoretical analysis. Finally, Section VI concludes this article.

II. SYSTEM MODEL
We consider an FD downlink transmission in an RIS-assisted network, which consists of a source node (S), an FD relay node (R), a destination node (D), and an RIS equipped with N reflecting elements (see Fig. 1). It is assumed that R operates under the DF relaying protocol, 1 and S and D are equipped with a single antenna, while R has one receive and one transmit antennas. Moreover, no direct link exists between S and R due to the presence of environmental obstacles. Thus, S sends a signal to D with the aid of RIS and R.

A. Channel Model
The channel coefficients 2 between the RIS's nth element and S/R are denoted by h sn /h nr , where n ∈ {1, 2, . . . , N}. The RIS's each shifting element reflects the incoming signal with a reflection amplitude of η ∈ [0, 1] and a phase-shift of θ ∈ [0, 2π). Furthermore, h rr denotes the SI channel received directly at the receive antenna of R, while h txn /h nrx denotes the channel between the relay's transmit/receive antennas and the nth element of the RIS. Furthermore, h rd denotes the channel between R and D, while h l ∀l ∈ {1, 2, . . . , L} indicates an ambient interference channel to D. Measurement results in [44] showed that small-scale fading at mmWave is less severe than that in traditional bands, when narrow beam antennas are used.
Therefore, similar to [45], we assume that the channel coefficient h j , j ∈ {sn, nr, rr, rd, l} is modeled by the Nakagami-m distribution with a following PDF:

B. Source-to-Relay Link
By expressing the channel coefficients as h sn = |h sn |e −jψ n and h nr = |h nr |e −jφ n , where |h sn | and |h nr | denote the channel amplitudes, which follow the Nakagami-m distribution, while ψ n and φ n are the channel phases, we can write the signal received at R via the RIS as where P ι , with ∀ι ∈ {S, R}, is the transmit power at node ι; P SI is the SI power at R; d sn indicates the distance of the S-RIS link, while d nr is the distance from the RIS to R; d txn and d nrx denote the distances from the relay's transmit and receive antenna to the RIS, respectively; η n ∈ [0, 1] and θ n ∈ [0, 2π) denote the amplitude reflection coefficient and the effective phase shift introduced by the RIS's nth element, respectively; and τ is the path-loss coefficient. Moreover, signal of interest (SOI) and SI messages are represented by We consider an interference-limited network, where the effect of the additive white Gaussian noise (AWGN) is negligible [47]. Furthermore, considering the principle of the RIS to constructively beamform the signal in desired directions and suppress interference by spatial filtering [48], we assume that there is no interference from the RIS to R [49]. In addition, we also assume that interference at R from interference nodes is neglected compared to the strong SI loop 3 [28]. In light of the above discussions and for the sake of analytical tractability, we rewrite the signal-to-interference ratio (SIR) at R as 3 We justify our assumptions of ignoring the negligible effect of AWGN in the interference-limited network and interference in the FD relay in Section V.
The maximum SIR value can be obtained by setting η n = 1 and the reflected signals need to be cophased 4 by θ n = ψ n + φ n . Thus, the maximum SIR at R can be written as where W =P s Z 2 ; Z = N n=1 X and X = |h sn ||h nr | is double Nakagami-m RV. Moreover, Y = P SI |h rr | 2 is the gammadistributed RV with shape and scale parameters of Gamma(m y ,β y ), whereβ y = P SI β y .
The double Nakagami-m RV X can be presented over the K G distribution [43]. Moreover, as it is stated in [43, eq. (1)], the PDF of the sum of several K G RVs is well approximated by √ U , with U = N n=1 X 2 . Hence, we express the PDF of the RV Z as follows: are the real-valued shaping parameters. However, if these parameters are complex conjugate numbers, they can be set to the magnitude of the estimated complex parameters. Furthermore, parameters a, b, and c are given in [43] and can be calculated using the kth moments of Z as where From (4) and (5), the PDF of the RV W can be expressed as the squared K G distribution 5 as follows: whereΨ = Ψ/P s . Then, the CDF of (8) is derived as [42] where G ·,· ·,· [·] denotes the Meijer G-function [46, eq. (9.301)]. 4 We consider perfect signal cophasing to make the analytical derivations tractable. We also provide a system model with imperfect phase shifting due to the RIS phase shifter's quantization error in numerical results. 5 It has been proven for the RIS channel models that the squared K G distribution is more accurate than the CLT-based NCCS distribution [42].

C. Relay-to-Destination Link
The instantaneous SIR of the R-D link can be written as |h l | 2 is the summation of L independent but not identically distributed Gamma RVs, P l is the transmit power at interference node l, and d l denotes the distance from interfering node l to D. The PDF of U can be closely approximated by [51] where the approximated scale parameterβ u is evaluated by solving a set of equations, m l β l , and the shape parameter is given bym u = μ β u .

III. CASCADED MULTIHOP RIS SYSTEM
This section considers a multihop RIS-assisted cooperative communication system, as shown in Fig. 2, where we assume that, due to the severe blockage by obstacles, there is no direct link and interference leakage at each RIS from the rest RISs. We consider two system models, namely, Cases I and II.
Case I: The S-to-D communication happens through two RISs and one intermediate the FD relay node. In this case, the end-to-end spectral efficiency is a function of SINR received at FD relay and D. The received SINR at the FD relay is identical to (3), while the SINR at D, considering the cascaded RIS channel, can be written as where the channel coefficients between the RIS B's qth element and R/D are denoted by h rq /h qd . Therefore, the spectral efficiency for Case I can be written as Case II: The communication in Case II occurs through a cascaded three RISs and the end-to-end spectral efficiency is a function of received SINR at D, which can be written as in (13), shown at the bottom of the next page.
Then, the spectral efficiency for Case II can be written as To the best of the authors' knowledge, the channel distribution model of cascaded multihop RIS communication links has not been studied and is still a challenging research topic. Therefore, we consider the performance of the multihop RIS system numerically in Section V to show that the intermediate FD relay can improve the spectral efficiency of the cascaded RIS communication link.

IV. PERFORMANCE ANALYSIS
This section provides closed-form solutions for the OP, average spectral efficiency, and average BER for the proposed RIS-aided FD network in Fig. 1. Besides, asymptotic equations at high transmit power are calculated to gain more technical insights into the system performance.

A. Outage Probability
The OP performance can assess the reliability of the communication link. The OP happens when the instantaneous SIR is below a predefined thresholdγ. Hence, the end-to-end OP at D is written as [52] Lemma 1: The end-to-end OP at the destination node can be written as in (16), shown at the bottom of this page.
Proof: We refer interested readers to Appendix A for the derivation steps.
Since (16) expresses the OP in the form of the Meijer Gfunction, it may be limited to provide a deep engineering insight into the outage behavior of an end-to-end system. Therefore, the asymptotic analysis is considered by setting P s = P r → ∞ to indicate diversity gain and attain more useful information regarding the overall system performance. By using [53, eq. (07.34.06.0040.01)] for Meijer G-functions in (16), we can express the asymptotic OP as where t = min(m w , k w ). In addition, (17) indicates that the achievable diversity order is min(m w , k w , m v ). Moreover, as m w and k w are defined from the number of RIS's shifting elements, the diversity order also depends on N .

B. Average Spectral Efficiency
The end-to-end average spectral efficiency of the considered system model can be expressed as [54, eq. (15)] The end-to-end PDF f γ (γ) in (18) can be derived by where f R (γ) and f D (γ) are the respective PDFs of γ R and γ D .

Lemma 2:
The end-to-end average spectral efficiency for D is written in closed form as in (20), shown at the bottom of the next page.
Proof: The detailed derivation of the average spectral efficiency is given in Appendix B.

C. Average BER
The average BER for some modulation schemes can be calculated as [55, eq. (25)] where the values of p and q depend on a modulation scheme chosen. For instance, binary phase-shift keying and differential phase-shift keying (DPSK) are obtained by setting p = 0.5/q = 1 and p = 1/q = 1, respectively. Lemma 3: The end-to-end average BER for D can be written in closed form as in (22), shown at the bottom of this page.
Proof: See Appendix C.
The asymptotic average BER can be found by inserting the CDFs in (17) into (21) and, after some mathematical manipulations, can be derived as It is worth noting that the asymptotic outage (17) and BER (23) are expressed in terms of elementary functions, which do not require advanced software to compute.

V. NUMERICAL RESULTS
This section presents numerical results to investigate the impact of key system parameters on the end-to-end system performance. Monte Carlo simulations based on the interferencelimited system model discussed in Section II are used to verify the correctness of the analytical derivations. We consider the system parameters shown in Table I Fig. 3, we demonstrate some results of the OP, average spectral efficiency, and average BER 7 for the proposed FD-RIS system with a different number of RIS elements, N . It is clearly observed that the analytical results obtained using the K G approximation provide highly tight consistency with the simulation results. In Fig. 3(a), we can observe that the outage performance improves by N . Moreover, it is worth noting that the analytical results are tighter with the simulation ones when the RIS has a higher number of reflecting elements.
Similarly, Fig. 3(b) shows that an increase of N helps obtain better average spectral efficiency, which justifies using RIS in the FD-based networks. However, for the considered FD-RIS model setup, the outage and spectral efficiency results show the advantage of using the RIS only when N ≥ 16. This is due to the far-field channel modeling of RIS communication link, where the RIS with a lower number of reflecting elements is vulnerable to the path-loss impact. However, this impact can be diminished by using the RIS with a larger N , which improves the SNR at the receiving node. Moreover, we also provide simulation results for average spectral efficiency by considering AWGN and interference at both D and R to compare with that of the proposed interference-limited scenario. We consider the noise power spectral of −174 dBm/Hz [48]) and the same number of 6 This is the residual SI power level after several stage of interference suppression at R. For example, considering the transmission power of P r = 30 dBm, R requires −60 dB of SI suppression, which is feasible considering the current achievements in the FD research [23]. 7 Throughout this section, we consider the BER performance of the DPSK modulation.   randomly located interference nodes L = 5 for R and D. First of all, we consider only the effect of AWGN on spectral efficiency by ignoring the impact of interference. It is clear from the curves (blue dot lines) that the effect of AWGN is negligible for all N . Furthermore, we can notice that the joint impact of AWGN and interference (short-dot brown lines) on spectral efficiency is minor. Another observation is that the impact of interference diminishes by increasing the number of RIS elements. These results justify our assumptions on AWGN and interference by showing that the proposed interference-limited system model achieves tight upper-bound results.
Furthermore, the RIS-aided FD network's performance enhancement in terms of the BER is shown in Fig. 3(c). Moreover, the asymptotic results derived in Section IV tightly converge to the exact analysis at high transmission power levels by confirming the accuracy of (17) and (23).
Next, we change the RIS location close to either S or R to identify the best RIS location that provides the maximum system performance for the considered system model. As shown in Fig. 4, the distance of the RIS on the x-axis is changed from 5 to 95 m. From the outage, spectral efficiency, and BER results in the plot, one can notice that the RIS provides better results when it is located closer to S or R. The minimum performance is obtained when the RIS is located in the middle of S and R. This phenomenon can be explained by the far-field channel modeling of the RIS, where the RIS link's path loss is inversely proportional to the multiplication of incoming and outgoing channel distances with power τ . For example, the path loss of the link through the RIS is higher when the RIS is located at an equal distance from S to D than the case when it is closer to one of the nodes. The RIS with N = 4, 8 cannot provide better system performance compared to the no-RIS model in Fig. 3. However, the outage [see Fig. 4(a)] and average spectral efficiency [see Fig. 4(b)] performance of the current system outperform those in Fig. 3 when the RIS is located closer to one of the nodes. Moreover, when N > 16, the RIS can be placed in any position to provide better performance than the no-RIS model. Fig. 5 illustrates the impact of N and small-scale fading parameter m on the average BER performance at different RIS locations on the x-axis. Similar to [45] and [56], we assume LOS and NLOS mmWave links with m = 2 and m = 4, respectively. As expected, the system with N = 24 achieves better BER than that with N = 16. Moreover, it is observed that the increase in the parameter m results in the BER improvement. This happens because the parameter m represents the degree of fading severity of the channel. When m = 1, the channel experiences Rayleigh fading by providing the worse BER performance. Then, when the mmWave NLOS link (m = 2) is considered, the BER performance improves. However, the system achieves the best BER performance when the LOS channel is considered with m = 4. Another observation is that the impact of m is considerable on the BER performance for N = 24. For example, when x = 50 m, the BER enhancements between curves of N = 16 and N = 24 for m = 1 and m = 4 are 1.8 and 14 times, respectively. Fig. 6 illustrates the OP results considering different residual SI levels at R. The plot highlights the importance of SI cancellation accuracy to guarantee the advantages of using the FD relay. For the considered transmit power, SI levels higher than  −50 dBm cause a full outage to the system for any value of N . On the other hand, a certain OP can be achieved at different SI levels for various N . For example, the OP of 10 −4 can be attained at the residual SI of −82 dBm for N = 4. The same OP can be achieved at −60 dBm for N = 32. In other words, the RIS with a higher number of reflecting elements can still improve the performance of the FD relay with considerable residual SI power. Considering these results, it may be less costly to employ the RIS with large N than design an FD relay with a high SI cancellation ability for obtaining certain QoS. Fig. 7 plots the average spectral efficiency results considering a fixed transmission power at R, i.e., P r = 0 dBm. From the results, we can observe that the spectral efficiency curves saturate at 4 bits/s/Hz for all N values. This happens since the end-to-end average spectral efficiency is restricted to min(γ R , γ D ). Moreover, as derived in Section IV, the Nakagami-m parameter (m v ) of the R-D link is one of the parameters that define the diversity order. Hence, the achieved SIR at D becomes a bottleneck for the end-to-end spectral efficiency performance when the SIR at R gets higher. Therefore, it can be asserted that the use of the RIS with N > 32 does not provide spectral efficiency enhancement at higher transmission power for the considered system setup.
In Fig. 8, we compare the proposed FD-RIS model (Model I) with a similar setup where the FD relay is replaced with another RIS model. In the latter model, S transmits the signal to D through the two cascaded RISs (Model II). The transmit power at S for this simulation is 30 dBm. When P r = 0 dBm and N ≤ 16, it is obvious from the plot that Model I obtains a much better average spectral efficiency than Model II. However, when N = 32, the performance of Model I saturates at 4 bit/s/Hz, while the increase in the number of elements provides spectral efficiency improvements for Model II by achieving the maximum average spectral efficiency of 6.8 bit/s/Hz. Furthermore, when the transmit power at R is increased to P r = 10 dBm and N = 32, the spectral efficiency performance of Model I outperforms the result of Model II. Hence, these results show that the FD relay can considerably enhance the system performance with higher transmission power. Besides, Fig. 8 also demonstrates the results when both the models have the same transmit power consumption. 8 Considering a total available transmit power of 30 dBm, Model I uses that power at S (P s = 30 dBm), while, in Model II, S and R use half of that power (P s = P r = 15 dBm). The result shows that Model I achieves 1.57 bits/s/Hz better average spectral efficiency than Model II in their maximum achieved performance. These results can infer that an intermediate FD relay provides better spectral efficiency performance than cascaded RISs due to the latter model's strong channel path loss. Fig. 9 illustrates the average spectral efficiency performance for Cases I and II shown in Fig. 2. Apart from the previous results, herein, we consider an imperfect phase shift at RISs as where b is the bit resolution of the RIS element's phase shifter. The coordinates for nodes are shown in Table II. Moreover, all the RISs have the same reflecting elements N = 32. We observe that the spectral efficiency curves for Case I considerably outperform those for Case II. For example, when P s = 30 dBm and the phase-shift resolution b → ∞, which corresponds to the perfect cophasing of the RIS phase shift, Case I attains 8.5 bit/s/Hz, while Case II achieves 3.1 bit/s/Hz (2.7 times lower). The lower spectral efficiency performance of Case II is due to the far-field path-loss effect of the cascaded four-hop distance of RISs [see (13)]. Moreover, we can observe that, when b = 4, the curve of Case I tightly converges with the perfect phase-shift performance. Furthermore, when the RIS has less phase-shift resolutions b = 1 and b = 2, the respective spectral efficiency performance drops from the ideal phase shift are 4.5% and 18.8%, respectively. Moreover, in the setup when the RIS performs no phase shifting (b = 0), the spectral efficiency degradation from the ideal phase-shift setup is 66%. Regarding Case II performance, we can see that the system setup with b = 4 does not provide tight results to reach the ideal system setup's performance with b → ∞. Another observation is that the use of smaller phase-shift resolution considerably degrades the spectral efficiency performance. For example, compared with the spectral efficiency result of the ideal phase-shift curve at 30 dBm, the performance drop of the curve with b = 1 and b = 2 is 81% and 24%, respectively. If the RIS is set with no phase-shift ability (b = 0), then no QoS is achieved for Case II with the considered system settings. The obtained results from this plot support that using an intermediate FD relay as a repeater/booster in the RIS-based networks can further enhance the system performance by diminishing the path-loss impact of cascaded RIS links.

VI. CONCLUSION
This article studied the performance of RIS-aided dual-hop FD relaying networks. Exact analytical derivations for the OP, average spectral efficiency, and average BER were derived in closed forms. Moreover, asymptotic expressions were derived and the system diversity order was identified as a function of the number of the RIS elements N and channel shape parameter m. An increase in N improved the aforementioned end-to-end system performance. On the other hand, at a large value of N , the considered end-to-end system performance becomes dependent on the R-D link. Moreover, we numerically studied the cascaded multihop RIS system. The results showed that the increase in the number of RISs within the communication link does not provide substantial spectral efficiency improvement due to the RIS channel's far-field path-loss model. However, the RIS network could achieve better system performance when an intermediate DF-FD relaying node was inserted to combat the path-loss effect. The proposed RIS model with active FD relay can be extended in future works considering MIMO and multiuser systems.
The CDF of γ R in (15) can be further written as . Furthermore, applying [58, eq. (11)] into (A.1) and, then, using [58, eq. (21)], (A.1) can be written in closed form as Similarly, we can write the CDF of γ D in (15) as and γ(a, b) Finally, the OP at D is obtained by substituting (A.2) and (A.4) into (15) and written as in (16).