I Introduction
Successive relaying (SR) is a well known technique in wireless communication, where two halfduplex (HD) relays simultaneously assist the transmission between a source and destination nodes utilizing the full bandwidth, and thus imitating fullduplex (FD) operation [6]. The main idea is that at a given timeinstant, the source transmits data to one relay, and simultaneously, the other relay forwards the data that it received from the source in the previous transmission timeinstant to the destination. Hence, it becomes quite clear that a main challenge in realizing such a scheme is interrelay interference (IRI). Thus far, authors have adopted different approaches to deal with IRI, such as channel ordering and rate adaptation [3], relay interference cancellation and precoding [16], or by neglecting IRI assuming fixed directional relays or large distance between active relays [8].
However, the recently proposed reconfigurable intelligent surfaces (RISs) can be a game changer for many wireless applications, including SR as we will demonstrate in this paper. Thanks to their ability in tweaking the wireless environment while maintaining low cost and power requirements, RISs have gained much attention as a strong candidate for future wireless networks [14, 13, 4, 10]. In principle, RISs are similar to FD amplifyandforward relaying, with the former providing passive beamforming without the need to include active power amplifiers or radiofrequency chains, while the latter has the ability to provide active amplification at the cost of increased hardware complexity and power consumption.
Instead of dealing with conventional active relaying and RISs as two separate technologies, few works have demonstrated the benefits of integrating RISs in active relaying networks. In particular, the works in [2, 1, 15, 19, 12] have shown that RISs can considerably improve the effective rates of conventional relaybased networks. However, in all of these works the authors considered a conventional relay network with a single relay node aided by an RIS, and no SR or IRI was considered in any of these works. In contrast, in this work we investigate the performance of SR networks with the help of two RISs, where we jointly design the passive beamforming weights for the two RISs to provide a spatial (on the air) suppression of the IRI while maximizing the desired signals gains. Our contributions can be summarized as follows:

We propose a new successive decodeandforward (DF) relaying network with two RISs, and investigate its effective rate performance. The corresponding maxmin optimization problem is formulated and solved via a semidefinite programming (SDP) approach.

In addition, as SDP schemes are known to suffer from a high computational complexity, we propose an efficient evolutionary scheme based on particle swarm optimization (PSO) [9] to tackle the joint phaseshift design for both RISs. Furthermore, modifications of the PSO are proposed exploiting the structure of the problem to guarantee stability and control the speed and accuracy of convergence.

Our results demonstrate that RISs can suppress IRI arising from the SR, and a large gain in achievable rates is obtained compared to the cases where only relays or RISs are utilized. Moreover, the proposed PSO scheme is shown to be highly efficient and can achieve a nearoptimal performance when compared to the SDP solution.
It is worth highlighting that our proposed PSO scheme is novel in the sense that it was designed specifically to tackle the RIS passive beamforming with guaranteed stability and controlled convergence behavior.
The rest of this paper is organized as follows: The adopted system model is introduced in Section II. Section III deals with the SDPbased beamforming design, while the PSObased scheme is introduced and explained in detail in Section IV. Numerical results are presented and discussed in Section V. Finally, conclusions are drawn in Section VI.
Notations
: Matrices and vectors are denoted by uppercase and lowercase boldface letters, respectively.
denote the conjugate, transpose, and Hermitian transpose operators, respectively. In addition, , , and are the th row, th column, and th element of the th column of , respectively, while and both represent the th element of and used interchangeably as appropriate. and are the allzero and identity matrices, respectively. Furthermore, is a diagonal matrix whose diagonal is the elements of , while is a vector whose elements are the diagonal elements of . The trace of is denoted by , while indicates that is a positive semidefinite matrix. Finally, and denote the absolute and expectation operators, respectively.Ii System model
We consider a timedivision duplex scenario where a source node () transmits data to a destination node (). Two HDDF relays and , located between and , assist the communication by adopting SR to exploit the full bandwidth. Furthermore, two RISs denoted by and , located near and , respectively, are utilized to enhance the communication quality and suppress the IRI arising from the SR operation. We assume that each of , , and are equipped with a single omnidirectional antenna, while and each has reflective elements.
At any given transmissiontime slot, one of the two relays operates as a receiver (Rx) to receive the new block of data from , while the other relay operates as a transmitter (Tx) to forward the decoded message received from in the previous time instant to ; see Fig. 1.^{1}^{1}1Note that during the first and last transmission time slots, only a single relay will be active and no IRI will be present. However, in this work we focus on the case where both relays are active. Focusing on the case where operates as an Rx, and assuming that signals reflected from each RIS more than once have very low powers and thus can be neglected, we can write the received signal at as:
(1) 
where and are the transmit powers of and , respectively, and are the diagonal reflection matrices of and , respectively, while , , , , , , , and denote the channels between , , , , , , , and , respectively.^{2}^{2}2The different narrowband fading channels adopted in this work will be explained in detail in Section V. Moreover, and are the information symbols transmitted from and , respectively, with , and is the additive white Gaussian noise (AWGN) at .^{3}^{3}3Throughout this work, we assume perfect channel state information (CSI) is available similar to [18] and [7]. Moreover, we assume a centralized processing such that all channels are known to either or in order to perform the joint phaseshift design.
For more convenience, we express the effective channels between each node (, , and ) and the two RISs in a single vector. Specifically, we define , , , , where . Hence, the signaltointerference plus noise ratio (SINR) at can be expressed as follows:
(2) 
where . On the other hand, we can write the received signal at (assuming no direct link between and exists except through the RISs) as follows:
(3) 
where is a vector containing the channel coefficients between and both RISs, is the channel coefficient between and , and is the AWGN at . Noting that represents interference to the destination in this case, we can write the SINR at as:
(4) 
In the subsequent timeinstant, will transmit a new block of data to which will operate as an Rx, whereas will forward its received signal from in the current timeslot, assuming successful decoding, to . Note that if the distances between and were the same for both , then the system is symmetric, and thus it is sufficient to consider only a single phase to evaluate the achievable rate (in this case is the Rx while is the Tx). Therefore, as we adopt a symmetric system model in this work [20], we can write the effective rate at , in bits/s/Hz, as follows:
(5) 
It is clear from (2) and (4) that the achievable rate depends on the reflection matrix . In the next section, we formulate and solve the joint passive beamforming design problem by utilizing an SDP approach.
Iii Passive Beamforming Design via Semidefinite Programming
In this section, we aim to maximize the achievable rate in (5) via an SDP approach. We start by formulating the corresponding optimization problem as follows:
(6a) 
where . Due to the interference terms in (2) and (4), and the unitmodulus constraint on in (6a), problem (6) is nonconvex and cannot be solved in its current form in polynomial time. Therefore, in the following we adopt change of variables and introduce exponential slack variables to overcome the nonconvexity of (6).
Let , , , , and . Moreover, by defining , , , , and , we can now equivalently represent and as follows:
(7a)  
(7b) 
However, even with relaxing the rankone constraint on , formulating the maxmin optimization problem with the current formulations of and would still result in a nonconvex objective function. Nonetheless, this can now be dealt with by introducing additional slack variables [22]. In particular, let and be exponential slack variables such that
(8a)  
(8b)  
(8c)  
(8d) 
Subsequently, and after dropping the rankone constraint on via semidefinite relaxation [11], we can introduce the following optimization problem^{4}^{4}4It is worth highlighting that is omitted from the objective function without any loss of optimality.
(9a)  
(9b)  
(9c)  
(9d)  
(9e)  
(9f) 
The only nonconvex constraints at this stage are (9b) and (9d). To overcome this drawback, a firstorder Taylor approximation is applied to convert these constraints into convex ones. In particular, we define , where and are the points around which the linearization is made such that . Hence, we can recast problem (9) as
(10a)  
(10b)  
(10c) 
It follows that all constraints and objective function at this stage are convex, and problem (10) can be solved iteratively using software tools such as CVX. However, we firstly initialize and as follows
(11a)  
(11b) 
where is any feasible (i.e. random) solution. Once the algorithm has converged, there is no guarantee that the rank of is equal to . In such case, the value of the objective function corresponding to
represents an upperbound. Nonetheless, one can apply eigenvalue decomposition (EVD) with Gaussian randomization to approximate a rank
near optimal solution [17]. The details are omitted here for brevity, however, it is worth pointing out that such approximation guarantees a minimum accuracy of of the optimal objective value [21]. The steps of the proposed scheme are given in Algorithm 1.Iv Particle Swarm Optimizationbased Beamforming Design
The main drawback for the SDP approach is that it suffers from a complexity order of [5], and hence, it becomes unsuitable when dealing with large optimization problems. Therefore, in this section we propose a lowercomplexity evolutionary method to tackle the phaseshift design, which is based on PSO [9].
We start by generating an initial population of particles with entries (or phaseshifts) drawn randomly in the interval . We evaluate the fitness of the th particle () as follows:
(12) 
where and are the SINR values at and given in (2) and (4), respectively, such that
(13) 
Then, we find the local best and global best solutions. Specifically, the global best solution for the th element of each particle is the th entry of the particle that has the highest fitness value in the population. Thus, the vector of global best solutions is in fact the particle that has the highest fitness in the population, which we denote as .
In contrast, the local best solution for the th element of , denoted by (), is the th entry of the particle with the highest fitness value among only the neighbouring particles of , which are and .^{5}^{5}5We adopt a ring topology, thus, the nighbouring particles for are and , while the neighbours of are and .
Furthermore, the velocities that control the direction of each particle, which we initialize as , are updated at each iteration as follows [9]
(14) 
where and are learning factors, and
are random numbers drawn from a uniform distribution with values between
and , and . To ensure a stable convergence,^{6}^{6}6Note that performing the normalization in (15) is crucial, and without it the values of can become extremely large (or small) after many iterations, preventing the algorithm from finding any good solution. we normalize each column of as(15) 
where is an introduced (auxiliary) parameter. Then, the phaseshifts are updated as
(16) 
It is worth to highlight that the value of in (15) controls the speed and accuracy of convergence, and it represents the highest possible phaseshift difference per reflecting element between any two successive iterations. Finally, we readjust the entries of that are either greater than or less than . Due to the circular symmetry of phaseshifts, the readjustment can be carried out as follows
(17) 
As shown in Algorithm 2, the same procedure (i.e. fitness evaluation, obtaining local best and global best solutions, update of velocities followed by normalization, update of phaseshifts of the population, and readjustment) will be repeated in subsequent iterations until a maximum number of iterations is reached, and the particle that achieves the highest fitness value in all iterations will be selected as the final solution.
When it comes to the periteration complexity of the proposed PSO scheme, we can observe that the (fitness evaluation, update of velocity, normalization, update of population, and adjustment), each has a complexity order of . Thus, and compared to the SDP approach, the PSO becomes particularly attractive when dealing with a large number of reflecting elements, as its complexity scales only linearly with the size of optimization problem.
V Results and Discussion
We start by introducing the different narrowband channels and parameters involved in this paper. All channels from and to each RIS are assumed to follow Rician distribution with both lineofsight (LoS) and nonLoS (NLoS) links, such that , where is the Rician factor, is the deterministic LoS channel vector for the th link, with each element having an absolute value of and a random phase between and , is the Rayleigh distributed NLoS channel vector, where is the distance between the nodes associated with the th link, , while and represent the pathloss exponents for LoS and NLoS links, respectively. In contrast, represents the channel coefficient with Rayleigh distribution for links that do not involve any of the RISs, i.e. . However, for the IRI link, , we consider Rician channel to demonstrate the efficiency of RISs in suppressing the interference even when a strong LoS channel exists.
The locations of different nodes are set as: , , , , , and , all in meter units, see Fig. 2. In addition, we set , , dB, , and . Equal power allocation was utilized and thus
, and we define the transmit signaltonoise ratio (SNR) as
.Our simulations compare between the following schemes:

RISsassisted SR (upperbound): This refers to the results obtained by SDP without performing EVD with Gaussian randomization, hence, the term upperbound. Rician fading is assumed for the IRI channel (i.e. ).

SR without RISs: This case demonstrates the results for SR utilizing only the two relays without the RISs. Rayleigh fading is adopted for all different channels including the IRI link.

RISs only: In this case, we only have two RISs (therefore no SR) and we optimize their phaseshifts to reflect the signal from toward , where the former transmits with a power budget of Watts.

RISsassisted SR (PSO): This case shows the performance of our proposed PSO scheme, assuming Ricianfading for the IRI link.
In Fig. 3, we depict the achievable rate vs. SNR curves. The RISsassisted SR notably outperforms both cases of RISsonly scheme and the SR case without RISs. In particular, to achieve an effective rate of bits/s/Hz, a gain of dB in transmit SNR can be obtained for the proposed RISsassisted SR over the RISsonly approach, given that the number of reflecting elements per RIS is . In contrast, the SR case without RISs shows a very poor performance, which highlights the sever impact of the IRI. Moreover, one can also observe from Fig. 3 that there is only a negligible performance gap between the lowcomplexity PSO scheme and the more complex SDP approach.
In Fig. 4, we show the convergence behavior of the proposed PSO scheme for different number of reflecting elements and different values of the auxiliary parameter . Regardless of the number of reflecting elements, setting to a small value (in this case ) is preferable as it allows the proposed algorithm to converge toward better solutions that are close to the optimal points. In contrast, larger values of enable faster exploration of new and more diverse populations, and thus they are preferable when the number of optimization iterations is small. However, they are more likely to fail in finding near optimal solutions due to the rapid velocity changes of candidate particles, especially when the number of reflecting elements is relatively large.
Vi Conclusion
In this work, a new RISaided relay network architecture was proposed with SR, where two RISs were deployed near the two HDDF relays to provide a spatial suppression of IRI while maximizing the gain of desired signals. An upper bound solution for the achievable rate was obtained via the SDP technique without performing Gaussian randomization and EVD. Subsequently, a lowercomplexity beamforming design based on PSO with controlled convergence speed was proposed, and a new normalization step was introduced which guaranteed the stability of the designed scheme. Numerical results demonstrated that RISs can provide large improvements on the rate performance of SR networks, even when a strong lineofsight link for the IRI existed. Finally, the proposed PSO scheme showed its capability in obtaining near optimal solutions.
Acknowledgment
This work was supported by the Luxembourg National Research Fund (FNR) under the CORE project RISOTTI.
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