SER and Outage Probability Analysis of Double RIS Assisted Wireless Communication System

. In this paper, we propose a double Reconfigurable Intelligent Surface (RIS)-assisted wireless communication system with no direct link present be-tween source and destination. Specifically, to get the maximum benefits from double RIS we consider a system model with first RIS (R 1 ) near to the source, second RIS (R 2 ) near to the destination, and both RIS will independently assist the communication between source and destination. In this setup, a moment generating function (MGF) based symbol error rate (SER) analysis is developed over Rayleigh fading channels. Furthermore, a simplified closed form expression for the system’s outage probability in terms of Q-function is derived. The performance of the proposed system is compared with single RIS assisted system with three different placements of RIS. Analytical results are validated using Monte Carlo simulation and simulation results reveal that double RIS assisted Wireless Communication System outperforms the single RIS assisted system in all scenarios with equal number of reflecting elements.


Introduction
Reconfigurable intelligent surface (RIS) is one among the key enabling technology for achieving 6G broadband access at the infrastructure level [1].RIS made up of a large number of metamaterial reflective elements, which are passive in nature, will introduce controllable phase shifts to the electromagnetic (EM) waves incident on it with the help of a controller connected between RIS and source [2,3].Moreover, these phase shifts are intelligent enough to add constructively at the desired user and thereby increasing the end-to-end signal to noise ratio (SNR) of the communication system, from which it is evident that the RIS will convert the random wireless channel into a deterministic entity [4].Henceforth, the RIS has the ability to create a truly smart radio environment by making the wireless radio medium controllable [5].In the literature, extensive studies on single RIS aided systems proven that RIS is a potential energy efficient technology [6]- [13].RIS-assisted single input single output (SISO) system will provide low bit error rate even in low SNR regimes and outperforms the amplify and forward (AF) relay in terms of SER and outage probability [6,7].Moreover, with the help of optimal resource allocation schemes RIS-assisted system is energy efficient compared to regular multi-antenna AF relay scheme in multi-user downlink scenario [8].RIS will helps in expanding coverage, reduce the latency to support ultra-reliable low-latency use cases with less energy requirements [9].The integration of RIS with other systems such as Non Orthogonal Multiple Access (NOMA) enhances throughput with improved energy efficiency [11][12][13].
A recurring theme of the aforementioned works [6]- [13] is that their system models consist of a single RIS.However, in recent times a plethora of research works done in multi RIS-aided wireless systems [14]- [20].The multi RIS system with the direct link between source and destination increases the average achievable rate gain and improves outage probability compared to single RIS system [14].By selecting the RIS with highest instantaneous SNR, multi RIS system outperforms relay assisted system in terms of average sum rate and outage probability [15].The effective design of reflection pattern will increase the achievable rate in multi RIS-aided system [16].Moreover, multi RIS aided system will enhance the physical layer security [17]- [19].Despite of having improved performance, resource allocation and energy efficiency are the prime concerns when using number of RISs' to meet the desired Quality of Service (QoS).The authors [20], looked into the issue of resource allocation in a multi RIS-aided system, proposed an energy efficient optimal solution and concluded that 2 to 4 RISs are enough for small cell network.
Motivated by the above considerations, in this paper, we consider a double RISassisted wireless system in which, R1 placed near to source, R2 placed near to destination and without direct link between source and destination.We consider that each channel experiences independent Rayleigh fading.For the proposed system, MGFbased SER analysis is developed.Furthermore, a simple expression in terms of Q-function is derived to calculate OP.We compare the proposed system with single RIS assisted wireless system by placing the RIS near to source, destination and midway between source and destination.All the analytical results are verified through Monte Carlo simulations.
Next section introduces the system model under consideration.Section 3 presents the analysis of SER and outage probability of proposed system.Section 4 provides theoretical and simulation results.Finally, Section 5 concludes the paper.

System Model
As shown in Fig. 1, we consider a double-RIS-assisted system, consisting of a source with single antenna that communicates with a destination with single antenna, with the help of R1 and R2.The number of reflecting elements in R1 and R2 are L1 and L2 respectively.The reflecting elements of each RIS are programmable with the help of RIS controller and it communicate with the source through a separate backhaul link to coordinate transmission and information exchange, such as channel state information and RIS phase shifts.The direct link between source and destination is blocked due to obstacles.The source and destination are in the far field of R1 and R2 [24].It is assumed that the fading channel is quasi static and flat, with no interference [10].Further, we assume that the cascaded links from source to destination via each RIS are independent.

Fig. 1. Proposed Double RIS-assisted wireless communication system
In the proposed system, source first transmits a signal to R1 and R2, and then each RIS passively reflects and retransmit the signal to the destination.Then the received signal at destination is expressed as follows: where  1, and  2, are the controllable phase shifts induced by i th and k th elements of R1 and R2 respectively,  is the transmitted signal and  ~ (0,  0 ) stands for additive white Gaussian noise.For simplicity, we assume that each RIS have same number of elements i.e., L1=L2=L then (1) modified as: In Eq. ( 2 where   , the average energy transmitted per symbol.By intelligently inducing the phases from each RIS as  , =  , +  , maximized SNR is written as [6]: Here,  =  +  ,  = ∑  stands for average SNR.

Performance Analysis
In this section, we derive the expressions of SER and outage probability for the proposed system.

SER analysis:
The mean and variance of product  , and  , are � ,  , � = variable with one degree of freedom is [22]: From ( 5), the average SER of M-ary phase shift keying (PSK) signaling is calculated as [23]: for binary PSK (BPSK), i.e. for  = 2 , Eq. ( 6) simplifies to By setting  =  2 ⁄ , Eq. ( 7) can be upper bounded as follows: Similarly, we can get the expressions of average SER for M-ary PSK by substituting ( 5) in ( 6)

Outage Probability Analysis:
The probability that the received SNR is less than a specified threshold  ℎ is known as outage [23].As we mentioned, in previous section,  follows a  distribution and hence outage probability can be expressed as [15,16]: Proposition 1: The closed form expression for the outage probability for a threshold  ℎ in terms of Q-function is given as, Proof.Please refer Appendix-A for the proof.

Theoretical and Simulation Results
In this section, we present theoretical and simulation results of SER and outage probability for the single RIS and proposed double RIS system.The simulation setup for the single RIS and proposed system are shown in Fig. 2. and Fig. 3. respectively.In both systems, source and destination are separated by  = 100 metre apart and ℎ = 5 metre.
For the single RIS,  1 is varied as 10, 50, 90 metre and for proposed system,  1 ,  2 are fixed as  1 =  2 = 10 metre.Then using Pythagoras theorem, for single RIS system   = � 1 2 + ℎ 2 metre ,   = �( −  1 ) 2 + ℎ 2 metre and for proposed system, The Path loss exponent set  = 3 and  ℎ = 20  [15].In Fig. 4, we plot average BER of the proposed system using Eq. ( 7) for BPSK signaling scheme with L1 =L2 =64.The results are compared with single RIS system with L=128 by varying the position of RIS.In the single RIS system, placing RIS near to source ( 1 =10 metre) or destination ( 1 =90 metre) achieves better BER performance compared to placing RIS midway between source and destination ( 1 =50 metre).Furthermore, the proposed system provides significant improvement in error performance than single RIS system.For example, when BER requirement set to 10 -4 , single RIS system with RIS near to source or destination achieves a SNR gain of 12 dB compared to RIS placed in the midway.For the same BER requirement, proposed system achieves a SNR gain of 42 and 54dB compared RIS near to source or destination and RIS in the midway cases respectively with same number of reflecting elements per system.Fig. 5, shows the theoretical and simulated average BER performance of the proposed system for BPSK signaling scheme with varying number of reflecting elements using Eq. ( 6) and it shows that the derived theoretical expressions are matching with simulation results.Moreover, we observe significant enhancement in the SNR gain as number of elements increasing from 32 to 256.For instance, to get BER of 10 -6 , a 21 dB SNR gain is observed as number of elements increasing from 32 to 256.In Fig. 6, we plot the average SER performance of proposed system for different Mary PSK signaling schemes with L1=L2=64.The average SER performance is decreasing with increasing modulation order similar to the conventional wireless systems.
In Fig. 7, we compare the outage probability of the proposed system with single RISaided system and found that proposed system outperforms the single RIS system.We observe improved outage performance when RIS placed either near to source or near to destination in single RIS system.The proposed system shown significant improvement than single RIS system and we observe OP decreases sharply as SNR increased.Fig. 8, shows the outage probability performance of proposed system for different number of reflecting element.The outage probability is decreasing with increasing number of elements.We observe SNR gain of 17dB as number of elements varying from 32 to 256.Moreover for fixed L1 and L2, as SNR increases outage probability decreases.For example, for L1=L2=256, as SNR changes from 5 to 7 dB, the outage probability decreasing approximately 100 times.

Conclusion
In this paper, we evaluated the error performance of proposed double RIS-assisted system over Rayleigh fading channels using moment generating function.Assuming both RIS are used to assist communication between source and destination, an exact and simplified analysis for the OP was presented.In addition, the outage probability of the proposed system is compared with single RIS system by placing RIS in three different positions.Our findings revealed that the proposed system outperforms single RIS system in terms of SER and outage probability with same number of reflecting elements per system.

𝜋𝜋 4 and 16 � 16 �.
� ,  , � = 1 −  16 respectively [21].Now, for large number of reflecting elements, according to central limit theorem A and B are converges to Gaussian random variables with equal mean  4 ⁄ and variance  �1 −  .Since A and B independent, their sum, C, is also Gaussian Random variable with mean  2 ⁄ and variance 2 �1 −  Hence the MGF of   , which is non-central chi-square () random

𝑄𝑄1 2 (
, ) = ( − ) + ( + ) ), the channel gains of source to Rm and Rm to destination links are ℎ , =  , and  , denotes the phases of the corresponding channels,    and     denotes the distances from source to center of each Rm and from center of Rm to destination, respectively and  denotes the path loss coefficient.By assuming that the perfect knowledge of  , and  , is available at each RIS then the instantaneous SNR at destination expressed as Eq.(3).