I Introduction
Conventional active relaying, such as decodeandforward (DF), is a well known technology that is used to extend the coverage, and/or to enhance the qualityofservice (QoS) between a pair of transceiving nodes [10]. Ideally, the locations and/or number of relays should be optimized based on a certain cost function, such as to maximize the rate or to minimize the transmit power while satisfying a given QoS constraint [13, 11].
In contrast, the reconfigurable intelligent surface (RIS) technology is a new concept in wireless communications, where a large number of lowcost, nearlypassive reflecting elements are utilized to direct the impinging signal toward a desired destination, such that the multiple signal paths are constructively combined at the receiver [17]. One of the most attractive aspects about RISs is that they do not require powerdemanding active radiofrequency chains. Another benefit compared to traditional active relaying is that RISs work onthefly, i.e. they do not introduce additional delays due to internal signal processing. Thanks to their lowcost and low powerconsumption, it is highly anticipated that RISs will have a key role in future wireless networks [14, 4, 9].
However, due to the large degradation of the signal power with distance, which is caused by the absence of active amplification at the RIS and the double pathloss, few studies have shown that considerably large surfaces are required to outperform a conventional singleantenna relay [3, 5]. Motivated by this fact, many researchers started adopting classical relays, such as DF or amplifyandforward (AF), to enhance the performance of RISassisted networks [2, 1, 18, 20, 19, 12, 7, 8]. However, in all these works, only a single relay was utilized to enhance the RISassisted transmission. Moreover, even though the work in [8] considers three RISs, only one of them was deployed near the single relay, whereas the other two were located within short distances of the source and destination.
However, in many realworld scenarios, the signal might go through multiple hops before reaching the destination. Therefore, we aim to find the optimal way of combining relays with RISs when there is more than a single RIS between the two transceiving nodes. In particular, we consider the doubleRIS reflection case where the signal is subject to reflections from two RISs before reaching the destination, and we propose three different halfduplex (HD) relayaided network architectures, and compare their effective rates under a total power constraint. The findings of this work can help understand how to perform optimal route optimization, relay placement, and RISrelay pairing for future multihop RISrelay assisted networks.
Notations
: Matrices and vectors are denoted by boldface uppercase and lowercase letters, respectively.
, , and are the transpose, conjugate, and Euclidean norm of a vector , respectively, and is the th element of . and are the absolute value and the phase of a complex number , respectively. is the expected value of , while is the identity matrix. is a diagonal matrix whose diagonal contains the elements of , while is a vector whose elements are the diagonal of . Finally, denotes the real part of a complex number .Ii System Model and Phase Optimization
We consider a time division duplex scenario where there is a singleantenna source () aiming to transmit a signal to a singleantenna destination () with the help of two RISs ( and ). Due to large distances, obstacles and pathloss, we assume that has a direct link with only , and similarly has a direct link with only .
To enhance the performance of the doubleRIS channel, we deploy a singleantenna HDDF relay(s) between the two ends. In particular, we investigate three different relayaided scenarios. The first one corresponds to the case of a single relay (R) present between and , and the communication takes place over two timeslots. In the second scenario, we assume that there are two relays and , placed near and , respectively, and only a single node in the network can transmit at any given timeinstant. In the last scenario, we aim to enhance the second scenario by allowing the second relay to transmit to while transmits its data to
. Note that throughout this work, we assume perfect channel estimation for all links.
^{1}^{1}1Note that the authors in [21, 22] have proposed channel estimation schemes for the doubleRIS channels with satisfactory estimation accuracy. We also assume centralized processing for the different doubleRIS communication schemes. Fig. 1 shows the different relayaided doubleRIS network configurations, where we compare with the norelay scenario as a benchmark scheme. We next start formulating the received signals and corresponding achievable rates for each scenario.Iia Transmission through only the RISs
In this scenario, we assume that the transmission is realized through the two RISs only. Therefore, the received signal at the destination can be written as
(1) 
where the superscript in indicates that this is a singlehop transmission, is the time index, is the total transmit power at any given timeinstant , and are the channels between , and , respectively,^{2}^{2}2The different narrowband fading channels adopted in this work will be explained in detail in Section III. and is the number of reflecting elements at each RIS. is the channel between the two RISs; while and are the reflection matrices for and , respectively. is the information symbol transmitted from with , and is the additive white Gaussian noise (AWGN) at the destination. Therefore, the received signaltonoise ratio (SNR) at the destination is given as
(2) 
where . The achievable rate in this case is
(3) 
Clearly, the achievable rate depends on both and . However, to optimize the achievable rate, we first need to reformulate the cascaded channel as follows:
(4)  
Now we can present the following optimization problem
(5a) 
where is the set of all reflecting elements at each RIS. The optimization problem in (5) is nonconvex, due to the unitmodulus constraint and the coupled optimization variables. Therefore, we adopt an alternating approach where we fix to solve for , and viceversa.
IiA1 Optimizing for a given
Let , now we can formulate the following optimization problem
(6a) 
The solution to the above optimization problem can be given in a closedform as follows:
(7) 
IiA2 Optimizing for a given
Similar to the previous approach, and after defining , we can write . Therefore, the solution to is
(8) 
We alternate the optimization process between and until the increment in the achievable rate between two successive iterations falls below a certain threshold, or we reach a maximum number of optimization iterations.
IiB Communication through RISs and a single relay
In this case, we assume that there is one HDDF relay (), placed in the middle between and ,^{3}^{3}3The choice of placing the single relay between and comes intuitively to balance the SNRs of first and second hops. and the transmission is carried out through two timeslots.
IiB1 FirstHop
In the first timeslot, transmits its data to through the direct link and the reflected signal from . Therefore, the received signal at is given as
(9) 
where the superscript in indicates that there are two hops in this case, is the channel vector between and , is the channel between and , and is the AWGN at . To maximize the received SNR at ,^{4}^{4}4Note that there exists another path from , however, this path is neglected here as the received signal through will be dominant due to shorter travel distance and less reflections. the phaseshifts of should be selected as follows
(10) 
. Then, the received SNR at with optimal phaseshifts can be expressed as
(11) 
IiB2 SecondHop
During the second timeslot, the relay retransmits the signal, with power , to the destination through the direct link and . Assuming successful decoding of at the relay, the received signal at can be given as
(12) 
where and are the channels between and , respectively, and is the AWGN at . Assuming perfect phaseshifts at for , the received SNR at is
(13) 
and the corresponding achievable rate is
(14) 
and the () prelog factor is due to the twohop transmission.
IiC Communication through RISs and two relays
In this case, we assume that there are two HDDF relays, and , to assist the communication between and . In particular, we assume that is placed in close proximity to , while is placed near . Furthermore, the transmission takes place over three timeslots, since we assume that only one node can transmit at any given timeinstant.
IiC1 FirstHop
In the firsthop, transmits its signal to through the direct link and . Therefore, the received signal at is
(15) 
where the superscript in indicates that there are three hops and transmits a new block of data every three transmission timeslots, is the channel vector between and , and is the AWGN at . Assuming perfect phaseoptimization for at , the received SNR at is
(16) 
IiC2 SecondHop
During the secondhop, retransmits the signal, assuming successful decoding, to through the direct link, direct reflection links from both and , as well as doublereflection link. Therefore, and assuming perfect decoding of at , the received signal at can be given as shown in (17) at the top of the next page, where we have , , and are the channels between , , and , respectively. and are the reflection matrices for and , respectively, during the secondhop, and is the AWGN at .
(17) 
Let , , , , and . Then, the received SNR at can be written as follows:
(18) 
Clearly the SNR depends on both and , therefore, we can formulate the following optimization problem
(19a) 
the above optimization problem is also nonconvex. Accordingly, we adopt an alternating optimization scheme where we fix one of the optimization variables and solve for the other one. In particular, and for a given , we have , where , and . Therefore, it is straightforward to see that the optimal phaseshift for the th element of is
(20) 
Similarly, to optimize the phaseshifts of , we can fix to obtain , where , and . It follows that the optimal phaseshift for the th element of is
(21) 
IiC3 ThirdHop
After receiving the information, will decode the message and then will retransmit it to through the direct link and . Assuming successful decoding at , the received signal at the destination can be written as
(22) 
and the corresponding received SNR at , assuming perfect phaseoptimization at for , can be expressed as
(23) 
and the corresponding achievable rate can be written as
(24) 
where the spectral efficiency is reduced by a factor of since a new block of data is transmitted every three timeslots.
(25) 
(26) 
IiD Enhanced transmission with RISs and tworelays
The main setback for the previous scenario is the prelog factor, which can be costly at high SNRs. Therefore, we further present an enhanced transmission scheme such that while is transmitting its signal to , transmits a new data packet to . Note that for the first two timeslots (), all equations in the previous subsection hold in terms of SNRs and phaseoptimization; as for the subsequent frames (i.e. when ), the received signals and transmit powers will change as will be thoroughly explained here.
IiD1 Received signal at
At a given oddtime instant
(), both and transmit data to and , respectively. Our focus here is on the received signal at .Clearly, will receive an interfering signal from in addition to the desired signal from , as shown in (IIC3) at the top of the next page, where the superscript in denotes that there are hops, and transmits a new data packet every two timeslots, and are the transmit powers at and , respectively, with to maintain the total transmit power budget, and and are the reflection matrices at and , respectively. The nd term in (IIC3) represents the interference from , which can be canceled in different ways. For example, if a global channel state information (CSI) is available, then can cancel this interference perfectly (assuming perfect channel estimation), since the signal transmitted from , i.e. , can be viewed as the signal that transmitted in the previous timeslot. Otherwise, can estimate the overall effective channel between itself and . This can be performed according to the maximumlikelihood estimation by multiplying the received signal at with the conjugate of the transmitted signal from at time (which is ), as shown in (26).^{5}^{5}5Here we assume that this interference signal is suppressed by one of the two methods explained above. Note that another way of suppressing the interference at is through the passive beamforming at and . However, we will leave this approach for investigation in our future work. After performing interference cancellation at , we can rewrite the received signal in (IIC3) as follows:
(27) 
where is the residual interference cancellation error at
, which is usually assumed to follow normal distribution such that
. Assuming perfect phase optimization for at to maximize the power of received signal at from S, we can formulate the received signaltointerference plus noise ratio (SINR) at as follows:(28) 
Next we focus on the received signal at the destination.^{6}^{6}6Note that the received signal at in the next timeslot (i.e. at time ()) will not be affected by this enhanced transmission scheme, since only will be transmitting data to with a power budget of while the source will be silent. Therefore, we have
IiD2 Received signal at D
While is transmitting its data to , transmits the decoded signal from in the previous timeslot to . The received signal at can be expressed as
(29) 
Note that is intended for , and therefore it represents interference to . As a result, the SINR at is
(30) 
where , , , and . Now we can formulate the following optimization problem:
(31a) 
This problem belongs to fractional programming [6]. As such, we formulate the following parametric program:
(32a) 
where is an introduced parameter. Although problem (32) is nonconvex, it can be solved using the iterative majorizationminimization (MM) method. In particular, we can introduce the following upperbound of (32) [15, 16]:
(33) 
where ,
is the maximum eigenvalue of
, , , , and is the solution to in the previous iteration of the MM scheme. Accordingly, minimizing the upperbound of (32) can be simplified as:(34a) 
For a given value of , the term is a constant. As such, the optimal phase for the th element at any given iteration of the MM scheme can be given as follows:
(35) 
and the value of is updated after each iteration as follows:
(36) 
However, before the MM algorithm starts, we initialize based on any feasible solution for in (36), and then utilize both and to find .
Proposition: The value of in (31) is monotonically nonincreasing with , where is the number of iterations for the MM scheme.
Proof: see Appendix A.
For each iteration of the MM scheme, we find the corresponding values of and , then we optimize the phaseshifts based on (35). The same procedure will be repeated until convergence or reaching a maximum number of iterations. It follows that the achievable rate utilizing this enhanced transmission scheme can be expressed as follows:
(37) 
Iii Results and Discussion
We start by introducing the wireless channels adopted in our work. All links from and to the RISs were assumed to experience Rician fading with both lineofsight (LoS) and nonLoS (NLoS) channels. In particular, , where is the Rician factor, contains the LoS channels, with each link having a deterministic absolute value of , where is the distance between two RISs and is the pathloss exponent for LoS links. In contrast,
is the complex Gaussian Rayleigh fading channel, where each link has a zero mean and a variance of
, where is the pathloss exponent for NLoS channels. Similarly, we have , where each element of has a fixed absolute value of ; while , . In contrast, we assume pure Rayleigh fading between nodes that do not include any of the two RISs, such that , . Moreover, was located at the origin of a plane such that , while , , , , , and , all in meters (see Fig.1). In addition, we set , , , ; while the maximum number of iterations to optimize any phaseshift vector was (which was shown to be enough for convergence), and the optimization convergence threshold was set to . Furthermore, for the enhanced transmission scheme, we have . We define the interferencetonoise ratio (INR) as , while the transmit SNR was defined as .As demonstrated in Fig. 2, the doubleRIS communication without including relaying suffers from a notably low rate performance. This is due to the large loss in signal power due to the lack of active amplification. In contrast, utilizing one or two relays can provide a significant performance gain. To be more specific, and regardless of the value of SNR, adopting the enhanced tworelays transmission is always better than the singlerelay case as long as the interrelay interference is suffiently suppressed, i.e. . Otherwise, the choice between a single relay and two relays depends purely on the value of SNR. At low SNRs, deploying two relays such that no two nodes can transmit in the network at the same time, can still achieve higher rates than the single relay case despite the prelog penalty; while at high SNRs, the single relay case leads to higher rates.
Finally, Fig. 3 shows the performance of different transmission schemes for a wide range of number of reflecting elements. Once again, our results indicate that adopting two relays to assist the transmission is highly desirable for the doubleRIS assisted communication even when the considered RISs are sufficiently large with hundreds of reflecting elements. For example, to achieve , the enhanced tworelay transmission requires reflecting elements per RIS, compared to and elements for the singlerelay and norelay cases, respectively, given that the interrelay interference is suppressed to the noise level.
Iv Conclusion and Future work
We investigated the DF relayaided doubleRIS reflection channels for coverage extension. Three different relayaided network architectures were proposed for effective rate maximization under a total power constraint. Our results demonstrated that deploying two relays for the doubleRIS channel achieves higher rates at low and medium SNRs; while at high SNRs, deploying a single relay to assist the two RISs is better only if the interrelay interference was high. The generalization to multihop with arbitrary numbers of RISs and relays is subject to future investigations.
Acknowledgment
This work was supported by the Luxembourg National Research Fund (FNR) under the CORE project RISOTTI.
Appendix A
Let us denote by , by , and the right hand side of (IID2) by . Let and be the phaseshift values of before and after running a single iteration of the MM scheme, and let denote the value of that corresponds to . Then, from the left hand side of (IID2), we have , where (a) holds from (IID2), (b) holds since minimizes , and (c) holds from the definition of in (36). Therefore, we have .
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