I Introduction
Multiple antenna systems exploit spatial diversity not only to increase throughput but also to enhance the reliability of the wireless channel. Alternatively, radio signal propagation via manmade intelligent surfaces has emerged recently as an attractive and smart solution to replace powerhungry active components [1, 2]. Such smart radio environments, that have the ability of transmitting data without generating new radio waves but reusing the same radio waves, can thus be implemented with the aid of reflective surfaces. This novel concept utilizes electromagnetically controllable surfaces that can be integrated into the existing infrastructure, for example, along the walls of buildings. Such a surface is frequently referred to as Reconfigurable Intelligent Surface (RIS), Large Intelligent Surface (LIS) or Intelligent Reflective Surface (IRS). Its tunable and reconfigurable reflectors are made of passive or almost passive electromagnetic devices which exhibit a negligible energy consumption compared to the active elements or nodes. For instance, the RISassisted communication outperforms the conventional relaying techniques in terms of energy efficiency. Since the energy efficiency in turn is a function of data rate, power consumption and frequency/time resource usage, this significant efficiency improvement with RIS can address several major issues arising from future wireless applications such as increasing demand for data rates, spectrum crunch, high energy consumption and environment impact.
This brandnew concept has already been proposed to incorporated into various wireless techniques – multicell multipleinput multipleoutput (MIMO) systems [3], massive MIMO [4], nonorthogonal multiple access (NOMA) [5], energy harvesting [6], optical communications [7] to name a few. The RIS can make the radio environment smart by collaboratively adjusting the phase shifts of reflective elements in real time. This results in the desired signals being constructively interfered at the receiver, whereas and other signals being interfered destructively. Therefore, most existing work on RIS focus on phase optimization of RIS elements [8, 9, 10, 11, 2, 12, 13, 14, 15]. However, there are very limited research efforts explored the communicationtheoretic performance limits [13, 16, 17, 18, 19]. The remainder of this section has an overview of related work, followed by a summary on contributions of this work.
Ia Related Work
An RISenhanced pointtopoint multipleinput singleoutput (MISO) system is considered in [8], which aims to maximize the total received signal power at the user by jointly optimizing the (active) transmit beamforming at the access point and (passive) reflect beamforming at RIS. The authors propose a centralized algorithm based on the technique of semidefinite relaxation (SDR) by assuming the availability of global channel state information (CSI) at the RIS. A similar system model is also considered in [9] where the beamformer at the access point and the RIS phase shifts are jointly optimized to maximize the spectral efficiency. The resultant nonconvex problem is solved with the help of fixed point iteration and manifold optimization techniques. An RISenhanced orthogonal frequency division multiplexing (OFDM) system under frequencyselective channels is considered in [10]
. For the proposed subcarrier grouping method for channel estimation, the achievable rate is maximized by jointly optimizing the transmit power allocation and the RIS passive array reflection coefficients. This nonconvex problem is subsequently solved suboptimally by alternately optimizing the power and array coefficients in an iterative manner. For a phase dependent amplitude in the reflection coefficient, in
[11], the transmit beamforming and the RIS reflect beamforming are jointly optimized based on an alternating optimization technique to achieve a lowcomplex suboptimal solution. For downlink multiuser communication helped by RIS from a multiantenna base station, both the transmit power allocation and the phase shifts of the reflecting elements are designed to maximize the energy efficiency on subject to individual link budget in [2]. There the authors use gradient descent search and sequential fractional programming to solve the resultant nonconvex problem. The weighted sumrate of all users is maximized by joint optimizing the active beamforming at the basestation and the passive beamforming at the RIS for multiuser MISO systems in [12]. The resultant nonconvex problem is first decoupled via Lagrangian dual transform, and then the beamforming vectors are optimized alternatively. Moreover, user fairness is considered for a LISaided downlink of a singlecell multiuser system in
[13], whereas physical layer security issues are considered in [14, 15].As mentioned before, while the optimization of power and/or phase shift have received more attention in recent work, a few have focused on analytical performance evaluation. Therefore, very limited number of results are available so far in this respect. For an LISassisted largescale antenna system, an upper bound on the ergodic capacity is first derived and then a procedure for phase shift design based on the upper bound is discussed in [16]. In [13], an optimal precoding strategy is proposed when the lineofsight (LoS) channel between the base station and and the LIS is of rankone, and some asymptotic results are also derived for the LoS channel of ranktwo and above. An asymptotic analysis of the data rate and channel hardening effect in an LISbased large antennaarray system is presented in [17] where the estimation errors and interference are taken into consideration. For a large RIS system, some theoretical performance limits are also explored in [18] where the symbol error probability is derived by characterizing the receive SNR
using the central limit theorem (CLT). In
[19], the LIS transmission with phase errors is considered and the composite channel is shown to be equivalent to a pointtopoint Nakagami fading channel. Subsequent performance analysis of the system has been conducted based on this equivalent channel model.On the other hand, two–way communications exchange messages of two or more users over the same shared channel [20]. Since this improves the spectral efficiency of the system, two–way techniques will have a significant impact on current and next generation cellular networks applications such as mobile video conferencing, communication between a base station and clients, and devicetodevice communications. While the two–way network provides fullduplex type information exchange for the pointtopoint or D2D communications, it also enables maximum spectral efficiency for relaying network with a fullduplex relay node [21]. The benefits of two–way network are contingent on proper selfinterference cancellation, which is possible with the recent signal processing breakthroughs. Therefore, two–way communications have been recently attracted considerable attention, and have already been thoroughly investigated with respect to most of the novel 4G and 5G wireless technologies such as massive MIMO, fullduplex communications, NOMA, mmWave communications, and cognitive radio, to mention but a few [22, 23, 24]. Thus, the RIS may also serve as a potential candidate for further performance improvement in the two–way Beyond 5G or 6G systems. However, to the best of our knowledge, all these previous work on RIS considered the one–way communications. Motivated by this reason, as the first work, we study the RIS for two–way communications in view of quantifying the performance limits, which is the novelty of this paper.
IB Summary of Contributions
Generally speaking, although the RIS can introduce a delay, it may be negligible compared to the actual data transmission time duration. Therefore, the transmission protocol and analytical model of the RISassisted two–way communication may differ from the traditional relayassisted two–way communications. Fig. 1 summarizes two possible RISassisted transmission schemes which require different number of time slots to achieve the bidirectional data exchange between two users.

Scheme 1 (one timeslot transmission): As shown in Fig. (a)a, two endusers simultaneously transmit their own data to the RIS which reflects received signal with negligible delay. Since the signal is received without delay, each enduser should be implemented with a pair of antennas each for signal transmission and reception, where each user experiences a fullduplex type communication as well.

Scheme 2 (two timeslots transmission): As shown in Fig. (b)b, the user 1 transmits its data to the user 2 in the first time slot, and vice versa in the second time slot. Therefore, each enduser may use a single antenna for signal transmission and reception.
Since Scheme 1 is more exciting and interesting; and also Scheme 2 can be deduced from Scheme 1, we develop our analytical framework based on Scheme 1. In Scheme 1, concurrent transmissions occur from the usertoRIS and the RIStouser. We consider both cases where usertoRIS and RIStouser channels are assumed to be reciprocal and nonreciprocal.
Although the RIS may be implemented with large number of reflective elements for the future wireless networks, fundamental communicationtheoretic foundations for single and moderate number of elements of the RIS have not been wellunderstood under multipath fading. However, such knowledge is very critical for network design. As cellfree massive MIMO is a promising extension to colocated massive MIMO, another further research direction of the RIS will be a distributed RIS system. Such system design is based on the understanding of a simple RIS system where each distributed RIS may have single or very few reflective elements. Further, the endtoend SINR expression is different from the SINR expression of either the conventional amplifyandforward (AF) relaying or the instantaneous relaying because these relay types are implemented with active elements which need proper power control at relaying stage [25]. Therefore, we need new analytical frameworks to evaluate the performance for both singleelement and multipleelement RIS cases.
To support the aforementioned research directions, this paper analyzes a general two–way RIS system where the number of reflective elements can range from one to any arbitrary value, and provides several communication–theoretic properties which have not been wellunderstood yet.
The main contributions of the paper are summarized as follows:

For reciprocal channels with a singleelement RIS, we first derive the exact outage probability and spectral efficiency in closedform for the optimal phase adjustment at the RIS. We then provide asymptotic results for sufficiently large transmit power compared to the noise and interference powers. Our analysis reveals that the outage decreases at rate, whereas the spectral efficiency increases at rate for asymptotically large signaltointerferenceplusnoise ratio (SINR), .

For reciprocal channels with a multipleelement RIS, where the number of elements, , is more than one but not necessarily as large as in LIS. In this respect, the instantaneous SINR turns out to take the form of a sum of product of two Rayleigh random variables (RVs). As well documented in the literature, this does not admit a tractable PDF or CDF expression. To circumvent this problem, we first approximate the product of two Rayleigh RVs with a Gamma RV, and then evaluate the outage probability and spectral efficiency. This seems to be the first paper which uses gamma approximation for a Rayleigh product. Surprisingly, this approximation works well and more accurately than the CLT approximation (which is frequently used in LIS literature), even for a moderate number of elements such as or . Since the tail of the gamma approximation does not follow the exact distribution of the SINR RV, we resort to asymptotic analysis of the exact SINR for singleelement RIS. In this respect. we show that the outage decreases at rate, whereas the spectral efficiency still increases at rate.

For nonreciprocal channels, system performance analysis seems an arduous task, since four different channel phases are involved. In this case, we turn to optimize the phase so as to maximize an important measure: the minimum user SINR, which represents user fairness. With singleelement RIS, closedform solution is derived. However, for multipleelement RIS, the associated problem is nonconvex. To find the solution, through some transformations, we relax the formulated problem to be a semidefinite programming (SDP) problem, the optimal solution of which is achievable and can further render a suboptimal solution for our originally formulated optimization problem.
Overall, this paper attempts to strike the correct balance between the performance analysis and optimization of twoway communications with the RIS.
Before proceeding further, here we introduce a list of symbols that have been used in the manuscript. We use lowercase and uppercase boldface letters to denote vectors and matrices respectively. A complex Gaussian random variable
with zero mean and variance
is denoted by , whereas a real Gaussian random variable is denoted by . The magnitude of a complex number is denoted by and represents the mathematical expectation operator.Ii System Model
A RIS–aided two–way wireless network that consists of two end users (namely, and ) and a reflective surface () where the two–way networks with reciprocal and nonreciprocal channels are shown in Fig. (a)a and Fig. (b)b, respectively. The two users exchange their information symbols concurrently via the passive RIS, which only adjusts the phases of incident signals. Each user is equipped with a pair of antennas for the transmission and reception. The RIS contains reconfigurable reflectors where the th passive element is denoted as . No direct link between two users is assumed, due to transmit power limitation or severe shadowing effect. For simplicity, we assume that both users use the same codebook. The unitenergy information symbols from and , randomly selected from the codebook, are denoted by and , respectively. The power budgets are and for end users and , respectively. We assume that all fading channels are independent. By placing the antennas of users and elements of RIS sufficiently apart, the channel gains between different antenna pairs fade more or less independently and no correlation exist.
Iia Reciprocal Channels
The wireless channel can be assumed to be reciprocal if the overall usertoRIS and RIStouser transmission time falls within a coherence interval of the channel and the pair of antennas are placed sufficiently close distance, see Fig. (a)a.
In this case, we denote the fading coefficients from to the and from to the as and , respectively. The channels are reciprocal such that the channels from the to the two end users are also and , respectively. All channels are assumed to be independent and identically distributed (i.i.d.) complex Gaussian fading with zeromean and variance, i.e., . Therefore, magnitudes of and (i.e., and ) follow the Rayleigh distribution. It is assumed that the two end users know all channel coefficients, and , and the knows its own channels’ phase values and
. This channel information requirement can be satisfied in advance by using compressive sensing or deep learning techniques
[26].Each user receives a superposition of the two signals via the RIS. Thus, the receive signal at at time can be given as
(1) 
where is the adjustable phase induced by the , is the receive residual selfinterference resulting from several stages of cancellation and is the additive white Gaussian noise (AWGN) at which is assumed to be i.i.d. with distribution . Further, the vectors of channel coefficients between the two users and RIS are given as and . The phase shifts introduced by the RIS are given by a diagonal matrix as . Then, we can write (1) as
(2) 
where denotes the self interference term. Since the has the global CSI, it can completely eliminate the selfinterference. Therefore, after the elimination, the received instantaneous SINR at can be written as
(3) 
To avoid loop interference, similar to fullduplex communications, the applies some sophisticated loop interference cancellations, which results in residual interference [27]. Among different models used in the literature for fullduplex communications, in this paper, we adopt the model where is i.i.d. with zeromean, variance, additive and Gaussian, which has similar effect as the AWGN [27]. Further, the variance is modeled as for , where the two constants, and , depend on the cancellation scheme used at the user. Thus, the instantaneous SINR at in (3) can be simplified as
(4) 
Similarly, we can write the instantaneous SINR at as
(5) 
where is the noise variance and is the variance of residual interference at the . It can also be modeled as .
IiB NonReciprocal Channels
Even though the overall usertoRIS and RIStouser transmission time falls within a coherence interval of the channel, the wireless channel can be assumed to be nonreciprocal when the pair of antennas are implemented far apart each other or nonreciprocal hardware for transmission and reception, see Fig. (b)b.
In this case, the fading coefficients from the transmit antenna of to the and from the to the receive antenna of are denoted as and , where , , and denote amplitudes and phases, respectively. Similarly, the respective channels associated with the are denoted as and . All channels are assumed to be independent and identically distributed (i.i.d.) complex Gaussian with zeromean and variance (i.e., ). It is assumed that the two end users have full CSI knowledge, i.e., , , and ; and each element knows its own channels’ phases, i.e., and .
Thus, the receive signal at at time can be written as
(6) 
where denotes the self interference, which can be eliminated due to global CSI,, and thus, subsequently, selfinterference cancellation can be applied. We assume the same statistical properties for selfinterference as in (1) for comparison purposes. Then, the SINR at can be written as
(7) 
By performing the similar signal processing techniques as in , the SINR of can be written as
(8) 
where is the noise variance and is the variance of residual selfinterference at the .
Iii Network With Reciprocal Channels
Iiia Optimum Phase Design at RIS
A careful inspection of the structures of and given in (4) and (5) reveals that the optimal , which maximizes the instantaneous SINR of each user, admits the form
(9) 
This is usually feasible at the RIS as it has the global phase information of the respective channels. Now with the aid of (4) and (5), the maximum SINRs at and can be given as and , respectively, where
(10)  
(11) 
In general, the instantaneous SINR of each user can be written as
(12) 
where
Further, we define as the average SINR, and without loss of generality, we assume and .
IiiB Outage Probability
By definition, the outage probability of each user can be expressed as , where is the SINR threshold. This in turn gives us the important relation
(13) 
where
is the cumulative distribution function (CDF) of
. To evaluate the average spectral efficiency, we need the distributions of the RV . For general case, RV is a summation of independent RVs each of which is a product of two independent Rayleigh RVs. Since the analysis of the general case may give rise to some technical difficulties, we evaluate the average spectral efficiency for and cases separately.IiiB1 When
In this case, the instantaneous SINR of each user is . Since and are identical Rayleigh RVs with parameter , the PDF and CDF expressions can be written as and , respectively. Since the RV is a product of two i.i.d. Rayleigh RVs, its CDF can be derived as , from which we obtain
(14) 
where the last equality results from with denoting the modified Bessel function of the second kind [28, eq. 3.324.1]. For a RV with , we can write its CDF as . By using this fact, the CDF of can be derived as
(15) 
Thus, the outage probability can be written as
(16) 
IiiB2 When
In this case, the instantaneous SINR of each user is given in (12). Let us now focus on deriving the CDF of the RV . However, by using the exact CDF of given in (14), an exact statistical characterization of the CDF seems an arduous task. To circumvent this difficulty, in what follows we first seek an approximation for the PDF and CDF of .
Among different techniques of approximating distributions [29]
, the moment matching technique is a popular one. In the existing literature, the regular Gamma distribution is commonly used to approximate some complicated distributions because it has freedom of tuning two parameters: 1) the shape parameter
; and 2) the scale parameter . The mean and variance of such Gamma distribution are and , respectively. The following Lemma gives the Gamma approximation for the CDF .Lemma 1.
The distribution of the product of two i.i.d. Rayleigh RVs with parameter can be approximated with a Gamma distribution which has the CDF
(17) 
where
Further, is the lower incomplete gamma function [28]. Note that, by definition, the lower and upper incomplete gamma functions satisfy .
Proof:
Since the first and second moments of in (14) are and , the RV has mean and variance. By matching the mean and variance of the RV with the mean and variance of the regular Gamma distribution, we have the above CDF. ∎
Here we assess the accuracy of the approximation using the KullbackLeibler (KL) divergence. In particular, we consider the KL divergence between the exact PDF of and its approximated PDF which is defined as [29]
where the expectation is taken with respect to the exact probability density function (PDF) of
which can be derived as . With the aid of [30, Eq. 2.16.2.2 and 2.16.20.1], this can be calculated as(18)  
(19) 
where is the is Euler’s constant.
With numerical calculation, we plot the KL divergence vs for in Fig. (a)a. We get where this very small value confirms the accuracy of the approximation. Numerical result also clarifies that has a little impact on . Moreover, Fig. (b)b plots the complementary cumulative distribution function (CCDF) of based on the simulation, the exact CDF in (14) and the approximate CDF in (17) for which represent very small, moderate and large variance values. The exact CCDF match tightly with the Gamma approximation for the simulated range for all , confirming the validity of the approximation. The accuracy of the approximation is also shown by the performance curves in Section V.
Recalling that the instantaneous SINR is which admits the alternative decomposition
(20) 
Armed with the above lemma, now we are in a position to derive an approximate average spectral efficiency expression pertaining to the case . It is worth mentioning here that the RV is then a sum of i.i.d. Gamma RVs with the parameters and . Therefore, the RV also follows a Gamma distribution with and parameters. By using the similar variable transformation as in (15), the CDF of can be approximated as
(21) 
Therefore, the outage probability can be written as
(22) 
IiiC Spectral Efficiency
The spectral efficiency can be expressed as [bits/sec/Hz]. Then, the average value can be evaluated as where is the PDF of . By employing integration by parts, can be evaluated as
(23) 
IiiC1 When
IiiC2 When
Thus, with the aid of (21) and (23), the average spectral efficiency can be evaluated as
(25) 
where is the generalized hypergeometric functions [28] and is the logarithmic Gamma function [28]. Here we have represented the Gamma function in terms of hypergeometric functions and subsequently use respective integration in [28, Sec. 7.5].
IiiD Asymptotic Analysis
IiiD1 High Sinr
The behavior of the outage probability at high SINR regime is given in the following theorem.
Theorem 1.
For high SINR, i.e., , the user outage probability of elements RISassisted two–way networks decreases with the rate of over Rayleigh fading channels.
Proof:
See Appendix A. ∎
However, with a traditional multiplerelay network, we observe rate. Since the endtoend effective channel behaves as a product of two Rayleigh channels, we observe rate with a RIS network. This is one of the important observations found through this analysis, and, to the best of our knowledge, this behavior has not been captured in any of the previously published work.
The behavior of the average throughout at high SINR regime is given in the following theorem.
Theorem 2.
For high SINR, i.e., , the user average spectral efficiency of elements RISassisted two–way networks increases with the rate of over Rayleigh fading channels.
Proof:
See Appendix B ∎
Since the residual selfinterference may also be a function of the transmit power, it is worth discussing the behavior of the outage probability and average spectral efficiency when the transmit power is relatively larger than the noise and loop interference powers. For brevity, without loss of generality, we assume . The following lemmas provide important asymptotic results.
Lemma 2.
When the transmit power is relatively larger than the noise and loop interference, i.e., , the outage probabilities for and vary, respectively, as
(26) 
and
(27) 
where is the array gain. While the outage probability decreases with the rate for , there is an outage floor for .
Proof:
In particular, we consider the following two extreme cases:

When , where the interference is independent of the transmit power, we have . Therefore, results can easily be deduced from Theorem 2. Please note that we derive case with upper and lower bounds, it is still unclear the precise expression for the array gain . We thus leave is as a future work.
This completes the proof. ∎
When where , it is not trivial to expand the outage probability expressions with respect to for rational , we omit this case. However, the performance of this case is in between and cases.
Lemma 3.
For , the average spectral efficiency for and vary, respectively, as
(28) 
and
(29) 
While the average spectral efficiency increases with the rate for , there is a spectral efficiency floor for .
Proof:
Since the proof follows the similar steps as Lemma 2, we omit the details. ∎
Lemma 3 also reveals that the average spectral efficiency increases with because is an increasing function. Further, when number of elements increases from to , we have spectral efficiency improvements for any given where
(30) 
On the other hand, we can also save power for any given where
(31) 
Based on the behavior of function, the rates of increment and saving decrease with . Thus, use of a very large number of elements at the RIS may not be effective compared to the required overhead cost for large number of channel estimations and phase adjustments.
IiiD2 For Large (or LIS)
For a sufficiently large number , according to the central limit theorem (CLT), the RV converges to a Gaussian random variable with mean and variance which has the CDF expression
(32) 
where is the Gauss error function [28]. Since the CDF of is given as , the outage probability can be evaluated as
(33)  
(34) 
where is the Marcum’s function and the second equality follows from the results in [31].
However, this CLT approximation may not be helpful to derive the average spectral efficiency in closedform or with inbuilt special functions, which may also be a disadvantage of this approach.
IiiE Discussion on Scheme 2
For Scheme 2, the maximum instantaneous SNR of each user is
(35) 
where is the transmit power. The corresponding optimal phases are for both users with reciprocal channels, for with nonreciprocal channels, and for with nonreciprocal channels. It is important to note that there is no loop interference, i.e., in (4), (5), (7) or (8). Therefore, the SNR of Scheme 2 is always larger than the SINR of Scheme 1, and achieves lower outage probability which can easily be deduced from (16) and (22) replacing as . From Theorem 2, we can conclude that the user outage probability decreases with the rate of over Rayleigh fading channels.
Since only one user communicates in a given frequency or time resource block, we have factor for the average spectral efficiency in (23). It can then be derived by multiplying factor and replacing as of (IIIC1) and (IIIC2).
With respect to the average spectral efficiency, we now discuss which transmission scheme is better for a given transmit power . Since the direct comparison by using the spectral efficiency expressions in (IIIC1) and (IIIC2) for Scheme 1 and Scheme 2 does not yield any tractable analytical expressions for , we compare their asymptotic expressions where the corresponding spectral efficiency expressions for Scheme 2 can be given with the aid of (28) and (29) as
(36) 
Now we seek the condition for which Scheme 1 outperforms Scheme 2.
Lemma 4.
The transmit power boundary where Scheme 1 outperforms Scheme 2 can be approximately given

for as
(37) and

for as
(38)
Iv Network With NonReciprocal Channels
In this case, the SINR at in (7) and the SINR at in (8) can be alternatively given as
(39) 
where , , and .
By looking at the structures of and , finding the optimal , which maximizes the instantaneous SINR of each user, is not straightforward as in the case with reciprocal channels. This stems from the fact that the optimal in this case depends on phases of all channels and , and also the SINR is a function of , and the SINR is a function of . In this section, the optimization problem for maximizing the minimum user SINR, i.e.,, is to be formulated by optimizing the phase of the th element of the RIS, i.e., . We consider and cases separately.
Iva For
In this case, we have
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