Performance Analysis of RIS-Assisted Full-Duplex Communication Over Correlated Nakagami-$m$ Fading Channel

In this article, we investigate the performance of a reconfigurable intelligent surface (RIS) assisted full-duplex (FD) communication network, where each user is facilitated by a specific RIS in the network. The correlated Nakagami-$m$ fading channel is first considered in a RIS system, which is a general channel model that can capture the spatial correlation effect inherent in the RIS-assisted communication system. Using the two-dimensional Laplace transform and its inverse, the closed form expressions of the mean and variance of the signal power distribution are obtained. Then, the outage probability and average achievable rates of the uplink and downlink users are derived in closed form. Furthermore, the impact of the residual self-interference (SI) on FD communication performance is discussed. It is demonstrated that FD communication outperforms HD communication when the residual SI is below a threshold, and the threshold is derived in closed form. Simulation results are presented to confirm the accuracy of the theoretical analysis and show the negative impact of channel correlation on the system performance. Moreover, it is illustrated that the outage probability and the average achievable rate of the uplink user will converge to a constant when the residual SI is linearly dependent on the transmit power.

Abstract-In this article, we investigate the performance of a reconfigurable intelligent surface (RIS) assisted full-duplex (FD) communication network, where each user is facilitated by a specific RIS in the network.The correlated Nakagami-m fading channel is first considered in a RIS system, which is a general channel model that can capture the spatial correlation effect inherent in the RIS-assisted communication system.Using the two-dimensional Laplace transform and its inverse, the closed form expressions of the mean and variance of the signal power distribution are obtained.Then, the outage probability and average achievable rates of the uplink and downlink users are derived in closed form.Furthermore, the impact of the residual self-interference (SI) on FD communication performance is discussed.It is demonstrated that FD communication outperforms HD communication when the residual SI is below a threshold, and the threshold is derived in closed form.Simulation results are presented to confirm the accuracy of the theoretical analysis and show the negative impact of channel correlation on the system performance.Moreover, it is illustrated that the outage probability and the average achievable rate of the uplink user will converge to a constant when the residual SI is linearly dependent on the transmit power.

I. INTRODUCTION
T HE upcoming wireless communication networks, includ- ing beyond fifth-generation (B5G) and sixth-generation (6G), are expected to provide ubiquitous wireless connectivity for billions of communication devices with high requirements in terms of data rates, latency, spectral and energy efficiency and reliability [1].To meet these significant demands, various innovative physical layer wireless communication technologies have been explored and developed in the recent past, such as massive multiple-input multiple-output (MIMO) [2], millimeterwave (mm-wave) communications [3], ultra-reliable low latency communications (URLLC) [4], and network densification [5].Among these potential technologies, reconfigurable intelligent surface (RIS) assisted wireless communication has gained enormous research interests and has been regarded as a promising physical layer technology in the future wireless communication networks, which can improve the spectral and energy efficiency of wireless communications with low hardware cost [6].Specifically, a RIS is a planar array consisting of a large number of reflecting elements.With the development of metamaterials and metasurfaces, each element is controlled by the RIS controller to perform real-time independent phase shifting of the incident signals [7].With the channel state information (CSI), a RIS is capable of reconfiguring the wireless propagation environment intelligently by adjusting the phase shifts of all the reflecting elements such that the signals can be combined constructively and destructively towards different directions to achieve spatial signal enhancement and nulling, thus enhancing the communication performance [8].In addition, a RIS does not need the active radio frequency (RF) chains, which significantly reduces the hardware cost and energy consumption of RIS [9].Owing to the above advantages, RIS has been investigated in various wireless networks.RIS-empowered orthogonal frequency division multiplexing (OFDM) and RIS-aided non-orthogonal multiple access (NOMA) were investigated in [10] and [11], respectively.The authors in [12] studied the physical layer security in a RIS-assisted wireless network, and the RIS-assisted UAV communication was explored in [13].
As a brand new technology, the performance of RIS-assisted uplink/downlink wireless communications has been studied extensively in various scenarios, where the performance gains brought by deploying RISs in these systems have been illustrated.The authors in [14] analyzed the path loss of a RISassisted wireless system and demonstrated the double fading effect such that the large-scale fading is inversely proportional to the product of the distances of the incident and reflecting channels.Over an independent Rayleigh fading channel model, the authors in [15] analyzed the outage probability, average achievable rate, and symbol error rate of a RIS-assisted downlink communication system with perfect phase alignment.With the same system model, an upper bound on the average achievable rate was given as a concise form in [16] and proved to be asymptotically equivalent in the number of reflecting elements.Considering the phase quantization errors, the authors in [17] analyzed the outage probability and diversity order of a RISassisted communication system with a 1-bit phase adjustment scheme.As a step further, the authors in [18] derived the symbol error rate and diversity order with the general phase errors pattern by using the central limit theorem (CLT) and proved that the composite channel gain is Nakagami-m distributed.More general channel models, e.g., Rician fading and Nakagami-m fading, were considered in other works [19], [20], [21], [22], [23].The authors in [19] derived an approximation and an upper bound of the outage probability and the ergodic capacity of a downlink RIS-assisted communication system over the independent Rician fading channel model, respectively.Under a similar system model, an asymptotic analysis was carried out in [20] to characterize the diversity order of the system.A discrete phase shifting scheme was considered in [21], where the authors derived the required phase shifting levels under a constraint on the capacity degradation.As a step further, the more general Nakagami-m fading channel model was adopted in some recent works.The authors in [22] analyzed the bit error probability of a RIS-assisted communication network over the independent Nakagami-m channel model with and without phase errors, respectively.The outage probability of a RIS-aided system over Nakagami-m channels was given in an integral form with the Gil-Pelaez theorem in [23].It is worth noting that all the above works considered independent channel models.However, channel correlation is inevitable in RIS systems since all the sub-wavelength reflecting elements are closely compacted together in a rectangular array [24].Therefore, the correlated channel model is more reasonable in RIS systems.The authors in [25] and [26] analyzed the outage probability of a RIS-assisted communication scheme over correlated Rayleigh fading channels with optimal and random phase shifting schemes, respectively.With a similar system model, an upper bound on the average achievable rate was proposed in [27].Applying the equal correlation profile, the authors in [28] investigated the outage probability and diversity order of a RIS system with a correlated Rician fading channel model.Moreover, the authors in [29] explored the uplink single-input multiple-output (SIMO) scenario and characterized the optimal signal-to-noise ratio (SNR) over correlated Rician fading channels.
To improve the spectral efficiency of wireless communications, full-duplex (FD) communication technology is regarded as another promising wireless communications technology.Contrary to the conventional half-duplex (HD) system, the FD system enables signal transmission and reception with the same time and frequency resource block, thus boosting spectral efficiency.Therefore, significant research efforts have been devoted to RIS-assisted FD communication to characterize the performance of the combination of these two spectral efficient techniques.The authors in [30] proposed a RIS-assisted FD MIMO communication system and maximized the sum rate by jointly optimizing the beamforming vectors at users and the phase shifting matrix at the RIS.The outage probability, diversity order, and spectral efficiency of a RIS-aided FD communication system over independent Rayleigh fading channels were derived in closed form in [31].The authors in [32] proposed a RIS-assisted FD system, where an FD transceiver simultaneously serves a downlink user and an uplink user over the same frequency band.The total power consumption was minimized by jointly optimizing the transmit power of uplink and downlink transmissions and the reflecting coefficients of the RISs.The authors in [33] analyzed the performance of an FD communication system over independent Nakagami-m fading channel models, where an FD transceiver communicates with a downlink user and an uplink user with the aid of two separate RISs in the FD mode.Considering the external interference from jammers, the authors in [34] derived the outage probability and ergodic capacity of a RIS-assisted FD communication system over Weibull fading channels when the RIS conducts the random phase shifting scheme.The authors in [35] characterized the performance degradation due to channel estimation and phase quantization errors of a RIS-assisted FD system over independent Rayleigh fading channels.The authors in [36] proposed a RIS-assisted FD communication system where two RISs were deployed to serve two end users separately, and the channels were Nakagami-m fading.Outage probability and symbol error probability were then derived in closed form.Following a similar system model, the authors in [37] considered the correlated Rayleigh fading channel model and demonstrated the impact of channel correlation on the system performance.Although the correlated Rayleigh fading channel model has been studied, it can only capture the rich scattering propagation environment without LoS link.The correlated Nakagami-m fading channel model is a more general and accurate channel model for multiple propagation environments, thus it requires more investigation and remains untouched so far.
To fill the research gap, this article presents novel results regarding the RIS-assisted FD communication system with correlated Nakagami-m fading channels.The main contributions of this article are listed below: 1) We study a RIS-assisted FD communication network, where an FD transceiver communicates with a downlink user and an uplink user simultaneously within the same frequency band.Two RISs are set to individually serve the uplink user and the downlink user.The channels associated with the RISs are assumed as correlated Nakagami-m fading.Both the inter-user interference and the residual self-interference (SI) are considered.2) Exact closed form expressions for the mean and variance of the signal power expressions are derived in terms of elementary and special functions by using the definition of Nakagami-m distribution and the two-dimensional Laplace transform, leading to approximations to the distributions of the received signal to interference plus noise ratio (SINR) of different users.Then, the outage probability and the average achievable rates of the uplink and downlink transmissions are obtained in closed form.With the derived expressions, the residual SI threshold below which the FD communication outperforms the HD counterpart is given, and the impact of the residual SI exponent on the performance of the uplink transmission is also investigated.3) Numerical results are presented to confirm the effectiveness of the analysis.Channel correlation will lead to higher outage probability and lower average achievable rates compared to the benchmark of uncorrelated channels.Furthermore, FD communication can perform better than HD communication only when the residual SI is below the given threshold.In the case that the uplink user and the FD transceiver have the same transmit power, and the residual SI is linearly dependent on the transmit power, the uplink performance metrics will converge to constants in the large transmit power regime.The rest of the article is organized as follows.Section II introduced the system model.The outage probability of the uplink and downlink transmissions are analyzed in Section III.Section IV gives the average achievable rates of the uplink and downlink users, and discusses the impact of the residual SI on the FD communication.Section V presents the numerical results to verify the theoretical analysis, followed by Section VI to conclude the paper.
Notations: mod(x, y) and x stand for the remainder of x divided by y and the greatest integer less or equal to x; C N ×1 represents the space of the N × 1 complex vectors; Nakagami(m, Ω) is the Nakagami-m distribution of shape parameter m and scale parameter Ω; E[X], Var(X) and Cov(X, Y ) denote the expectation of a random variable X, variance of X, and the covariance of X and Y ; 2 F 1 (•, •; •; •), Γ(•) and γ(•, •) are the hypergeometric function, gamma function and the lower incomplete gamma function, respectively.CN (μ, σ 2 ) represents the complex Gaussian distribution of mean μ and variance σ 2 ; X T is the transpose of matrix X.

II. SYSTEM MODEL
In this article, we study a RIS-assisted FD wireless communication network in which an uplink (UL) user U A and a downlink (DL) user U B are served by a transceiver (TX) simultaneously.U A and U B are equipped with a single transmit antenna and a single receive antenna respectively, while TX has two antennas, one each for signal transmission and reception, operating in the FD mode. 1 Since both users are assumed to be cell-edge users, two RISs, S A and S B , are deployed in close proximity to U A and U B respectively for improving the wireless propagation environment by tuning the phases of the corresponding channels.As shown in Fig. 1, the direct links between TX and users are assumed to be absent due to obstacles, large path loss and/or shadowing effect.The proposed system model can be generalized to the multi-user scenario with multiple users on both uplink and downlink by using the time division multiple access  (TDMA) transmission scheme.In each time slot, TX serves a downlink user and an uplink user in the FD mode.In different time slots, TX serves different downlink and uplink users.The analysis in this article can be regarded as a RIS-assisted FD TDMA system in a specific time slot.

A. RIS Structure
Two RISs are placed near their corresponding users to enhance the communication performance.S u , u ∈ {A, B} contains M u = M v,u M h,u elements equipped as a rectangular array, where M v,u and M h,u are the number of elements along the vertical and horizontal directions, respectively.Each element of S u is of the dimension of A u = l u × w u with l u and w u denoting the length and width of each element on S u , respectively.As shown in Fig. 2, S u is set on the yOz plane of height v u .Therefore, the coordinate of the mth element of S u is given as where y u (m) and z u (m) denote the y and z indexes of the mth element of S u , respectively.
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Assuming that each element of S u is able to perform independent passive phase shifting without energy dissipation, the passive beamforming matrix of S u can be given as where θ m,u , m ∈ {1, 2, . .., M u } is the phase shift angle of the mth element of S u .

B. Channel Model 1) Small Scale Fading:
represent the normalized small scale fading of the S u -U u , S u -TX and S u -U u channels, respectively, where u ∈ {A, B} and u ∈ {A, B}\u.Both RISs work in the far field of TX and users.In this article, the correlated Nakagami-m fading model is applied.It is worth highlighting that the Nakagami-m fading is a general channel fading model which can model many types of distributions in wireless communications by adjusting the shape parameter, and Rayleigh fading is a special case of Nakagami-m fading with the unit shape parameter [39].Specifically, where m ∈ {1, . .., M u }; a h,u , a g,u and a I,u are the shape parameters of the corresponding channels.|h m,u |, |g m,u | and |I m,u | have the same unit scale parameters.The channel correlation is characterized by the well-known exponential decay power correlation model as introduced in many relative literatures such as [40], [41], [42].Under this model, for the channel S u -U u , we can have the power correlation coefficient of |h m,u | and where 0 < ρ u < 1 is the power correlation coefficient of the channels associated with the nearest neighboring elements on S u ; c m,u − c n,u is the distance between the mth and nth elements on S u ; c 0,u is the smallest element spacing of S u .Under the same framework, it can be learned that (5) The exponential decay power correlated Nakagami-m fading channel model can be reduced to the correlated Rayleigh fading channel model by setting the shape parameter as one.
The normalized small scale fading of the U A -U B channel and the SI channel from the transmit antenna (TA) to the receive antenna (RA) at TX are denoted as I D and I 0 , respectively.I D is circularly symmetric complex Gaussian (CSCG) distributed with unit power such that ).It is assumed that S A and S B are sufficiently far from U B and U A , respectively [37].Furthermore, U A and U B are assumed to be two cell-edge users assisted by RISs which are in sufficiently long distances to TX [6].Therefore, the channel that S A reflects the signal from TX to U B , the channel that S B reflects the signal from U A to TX, and the channel that S B reflects back the signal from TX to itself can be ignored due to the long distances [32].
2) Path Loss: The path loss of the U u -S u -TX, U A -S u -U B and U A -U B channels, where u ∈ {A, B}, can be given respectively as [43] where Λ u , Λ I,u and Λ D are the path loss per unit distance of the U u -S u -TX, U A -S u -U B and U A -U B channels, respectively; d h,u , d g,u , d I,u and d 0 denote respectively the distances of the S u -U u , S u -TX, S u -U u and U A -U B links, where u ∈ {A, B} and u ∈ {A, B}\u; χ stands for the path loss exponent.The position of the RIS affects the path loss of the corresponding channel.The path loss is inversely proportional to the product of the distances of the TX-RIS and the RIS-user channels.The larger the product, the more serious the path loss.

C. Signal Model
The received signal at TX from the uplink transmission and received by U B from the downlink transmission are introduced separately.The received signal at TX can be given as where the first term and the second term refer to the desired signal and the SI, respectively; P A and P T are the transmit power of U A and TX, respectively; s A and s T are the independent transmit constellation symbols of U A and TX with unit power such that is the additive white Gaussian noise (AWGN) with zero mean and variance σ2 T .Similar to [36] and [37], it is assumed that S A only has the channel state information (CSI) of the channels 2 associated with its pairing user U A , which are g A and h A .Therefore, the phase shift of each element on S A is adjusted to co-phase the reflecting links such that (9) After proper phase shifting, the received signal at TX can be rewritten as where Thus, the received SINR of the uplink transmission from U A to TX can be given as In practice, several SI cancellation techniques can be applied to alleviate the SI and lead to the residual SI [44].Among difference frameworks used in existing works to model the residual SI for FD communication, in this article, we adopt the model where the residual SI is subject to a Gaussian random variable with zero mean and variance σ 2 l [45].The variance depends on the transmit power and is modeled as where the two parameters, > 0 and ∈ [0, 1], depend on the SI cancellation method used at TX [46].After several stages of SI cancellation, the received SINR of TX can be rewritten as The received signal at U B can be expressed as where (16) in which the first and second terms denote the interference from U A reflected by S A and S B , respectively; the third term stands for the interference from U A transmitted through the direct link.(17) After phase shifting, the received signal at U B can be rewritten as where Since the channels h u , g u and I u , u ∈ {A, B} are independent with each other, and the phase shifts of S A and S B are manipulated respectively to boost the U A -S A -TX and TX-S B -U B channels, it can be learned that φ m,u , u ∈ {A, B}, m ∈ {1, 2, . .., M u }, are independent and identically distributed (i.i.d.) random variables which follow the uniform distribution over [−π, π).Further, due to the distribution of I D , φ D is also uniformly distributed over [−π, π).Hence, the SINR of U B can be given as where

D. Performance Metrics
In this article, we analyze the outage probability and the ergodic capacity of the uplink and downlink transmissions.
1) Outage Probability: Outage probability is a performance metric to characterize the reliability of wireless communication.It is the probability that the received SINR falls below a threshold, i.e., where γ is the SINR threshold.
2) Ergodic Capacity: Ergodic capacity is a performance metric to evaluate the efficiency of wireless communication, and it is defined as where f γ u (x) denotes the PDF of γ u .

III. OUTAGE PROBABILITY ANALYSIS
In this section, we analyze the outage probability of UL user U A and DL user U B , respectively.The distribution of the composite channel gain H u , u ∈ {A, B} is characterized first, followed by deriving the outage probability expressions of the two users.

A. Channel Distribution Analysis
The random variable H u , u ∈ {A, B} is the sum of products of correlated Nakagami-m random variables.The exact statistical characterization of H u is mathematically intractable.Therefore, in this case, we analyzed the mean and variance of H u first, based on which the distribution of H u is fitted by a gamma random variable.The gamma distribution is often applied to fit some sophisticated distributions due to its advantages in terms of convenient parameter calculations and high accuracy [47].
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Therefore, we derive the mean and variance of H u in the following proposition.
Proposition 1: The mean and variance of H u , u ∈ {A, B} can be given as where each elements of R h,u and R g,u are respectively given as Proof: See Appendix A. Remark 1: When the channels are uncorrelated, the mean and variance of H u , u ∈ {A, B} can be derived directly as Remark 2: Even though the exponential decay power correlation model is adopted in this paper, the proposed method can be generalized to other correlation profiles, such as the equal correlation model [48].
With the mean and variance of H u , u ∈ {A, B}, we can use the moment-matching method to fit the distribution of H u to a gamma distribution, of which the shape parameter κ u and the scale parameter ω u are provided as follows: Hence, the PDF and CDF of H u can be written as

B. Outage Probability of UL and DL Users 1) Outage Probability of U A :
The outage probability of U A can be expressed as Since the CDF of H A has been given in (28b), the outage probability of U A is written as Remark 3: Considering the case of P A = P T , the impact of the residual SI on the outage probability depends on the residual SI exponent .If 0 ≤ < 1, the outage probability will decrease with the increase of P A , and smaller will lead to lower outage probability; If = 1 and P A is large, the outage probability will be independent with P A , such that 2) Outage Probability of U B : As per the expressions of the received SINR of U B and the CDF of H B , the outage probability of U B is computed as Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
To further simplify the outage probability of U B , it is required to figure out the distribution of I, which is summarized in the following proposition.Proposition 2: The aggregate interference power I approximately follows the exponential distribution with the distribution parameter Hence, the PDF and CDF of I can be given as Proof: According to the system model, the aggregate interference power I is written as where Moreover, these two terms are proved to be uncorrelated with each other due to the relation: Hence, with the CLT, ĨA can be approximated as a CSCG random variable with zero mean and variance M A , such that Following the same method, we can obtain that ĨB also follows the CSCG distribution with zero mean and variance M B , i.e., Since I D is also CSCG distributed and applying the property of the sum of CSCG random variables, we can obtain the distribution of Ĩ as Recalling the relationship between exponential and CSCG distributions, we can arrive at the conclusion given in Proposition 2. With the conclusion in Proposition 2, the outage probability of U B can be rewritten as (42) The above integral can be evaluated numerically in efficient ways.On the other hand, we propose to use the Gaussian Chebyshev quadrature (GCQ) method to approximate (42), and derive an accurate closed form expression, which is provided in the following proposition.
Proposition 3: Using the GCQ method, an approximation to the outage probability of U B can be given in closed form as where K is the complexity-accuracy trade-off factor, and the approximation error is negligible for large value of K.
Proof: Based on the integral form of the outage probability of U B , by changing the integration variable of (42) as t = e −ξ I x , we can obtain that where (a) follows from the GCQ method as introduced in [49]; ϕ k and b 1 have been defined in (44a) and (44b), respectively.Remark 4: The GCQ-based approximation is demonstrated to be highly accurate, even for small values of K.The relative error between PB (γ) and P + B (γ) for different values of K is presented numerically in Section V.

IV. AVERAGE ACHIEVABLE RATE ANALYSIS
In this section, we characterize the average achievable rates of the uplink and downlink transmissions, respectively.Firstly, the approximate average achievable rates of U A and U B are derived by using the outage probability expressions of the two users obtained in the last section.Then, the performance of FD and HD communications are compared and discussed, relaying on upper bounds on the average achievable rates, which are in more concise form.

A. Approximate Average Achievable Rate Analysis
In the following lemma, the relationship between the outage probability the average achievable rate is given.
Lemma 1: Assuming the outage probability of U u , u ∈ {A, B} is Pu (γ), where γ is the SINR threshold, the average achievable rate of U u is given as Proof: See Appendix B.

1) Average Achievable Rate of U A :
With Lemma 1 and the outage probability expression given in (30), the average achievable rate of U A can be written as Remark 5: Similar to the outage probability, if P A = P T , the impact of the residual SI on the average achievable rate depends on the residual SI exponent .If 0 ≤ < 1, the average achievable rate will increase with the increase of P A , and smaller will lead to higher average achievable rate; If = 1 and P A is large, the average achievable will be independent with P A , such that The integral form average achievable rate of U A can be evaluated numerically in efficient ways.Further, a closed form approximation to (47) can be obtained as follows.By changing the integration variable of (47) as we can obtain that where (b) is due to the GCQ method; ϕ k and b 1 have been defined in (44a) and (44b), respectively; K is the complexity-accuracy trade-off factor, and the approximation error is negligible for large value of K.
Remark 6: If = 1 and P A is large, a closed form approximation to (48) can be given as 2) Average Achievable Rate of U B : Following the similar method, the average achievable rate of U B can be given as where PB (γ) is the outage probability of U B , derived in (42).Further, a closed form approximation of to R B can be derived by using the approximation to PB (γ), i.e., P + B (γ) given in (43).By changing the integration variable of (52) as the average achievable rate of U B can be transformed as where Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
(c) is due the GCQ method; Q is the complexity-accuracy trade-off factor, and the approximation error is negligible for large value of Q.

B. Discussion on HD Communication
Since a main feature of the FD communication is the residual SI, which affects the performance of U A , we compared the performance of U A with the FD and HD communication schemes here.Firstly, a tight upper bound on the average achievable rate of U A with FD communication is derived with the well-known Jensen's inequality [50], such that for any random variable X and concave function f (x).With Jensen's inequality, an upper bound on the average achievable rate of U A can be given as For comparison, the performance of the HD communication scheme is introduced here.Specifically, TX serves the uplink and downlink users consecutively in two identical time slots.
In the first time slot, TX receives the signal coming from U A .And, TX transmits the signal towards U B in the second time slot.In each time slot, the phases of the corresponding RIS are shifted to maximize the composite gain of the corresponding channels.Therefore, the average achievable rate of U A in the HD communication scheme can be written as where By using Jensen's inequality, an upper bound on the average achievable rate of U A in the HD communication scheme is given by Compared (57) with (60), it can be learned that the received SINR of the uplink transmission is larger with the HD communication scheme due to the lack of SI.However, the average achievable rate of the HD communication scheme is with the penalty of a 1/2 pre-log coefficient.The superiority of FD and HD communication schemes depends on the level of the residual SI after the SI cancellation procedures.It can be derived that As expected, the FD communication scheme performs better as long as the power of the residual SI is lower than the threshold given in (61).

V. NUMERICAL RESULTS
In this section, simulation results are provided to verify the effectiveness of the proposed analysis, and illustrate the impacts of different system parameters on the uplink and downlink communication performance.

A. Simulation Setup
If not otherwise stated, we set the simulation parameters for the Monte Carlo (MC) simulations as follows.For the uplink transmission, the distances of the U A -S A and S A -TX links are given as d h,A = 5 m and d g,A = 20 m, respectively.The path loss per unit distance of the uplink channel is set to be The operating carrier frequency of the system is The path loss exponent is χ = 3.The AWGN noise power at TX and U B are set to be σ 2 T = σ 2 B = −70 dBm.We generate all the correlated Nakagami-m random variables in simulations by using the decomposition method proposed in [51].The proposed analysis can be applied for any number of RIS elements.The number of RIS elements used in simulations is relatively small for clear illustrations.As can be seen from the figures, the derived expressions match well with the simulation results.Increasing the number of elements on S B and reducing the transmit power of U A are both beneficial for the system performance due to the increased desired signal power and the decreased interference power, respectively.For example, when the transmit power of TX is P T = 15 dBm and P A = 30 dBm, the average achievable rates of U B are 3 bits/s/Hz and 4 bits/s/Hz for M B = 50 and M B = 75, respectively.To illustrate the impact of channel correlations on the communication performance, a benchmark Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.that all the channels are assumed to be uncorrelated is given for comparison in Fig. 3.As expected, channel correlation has a negative impact on the system performance.For example, when the transmit power of TX is P T = 10 dBm, the number of elements on S B is M B = 50, and the transmit power of U A is P A = 20 dBm, the outage probability is 10 −3 and 6 × 10 −5 for correlated and uncorrelated channels, respectively.

B. Downlink Transmission Analysis
The relative error between PB (γ) and its GCQ-based approximation P + B (γ) for different values of the complexity-accuracy trade-off factor K is presented in Fig. 5.The relative error is defined as We assume that the number of elements of S A and S B are as follows:  small values of K.For example, the relative error is already less than 10 −3 when K is 30.

C. Uplink Transmission Analysis
Fig. 6 shows the outage probability of the uplink transmission versus the transmit power of U A .RIS elements on S A are deployed in a compact rectangular array with five elements per column, i.e., M v,A = 5.The number of RIS elements per row varies in M h,A ∈ {10, 15, 20}.The Nakagami-m shape parameters are set to be a h,A = a g,A = 2. ρ A = 0.5.Regarding the residual SI, we set = 10 −7 and = 0, representing the case where the residual SI is independent of the transmit power of TX.As illustrated in Fig. 6, increasing the number of RIS elements can improve the communication performance.In other words, the lower outage probability can be achieved with a larger number of RIS elements.For example, to achieve the outage probability of 10 −4 with correlated channels, the required transmit power of U A is P A = 8.5 dBm and P A = 5.5 dBm for N = 75 and N = 100, respectively.Same as the downlink transmission, the channel correlation effect will deteriorate the system performance.When the transmit power of U A is Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.P A = 10 dBm and the number of elements is M A = 50, the outage probability is 4 × 10 −2 and 10 −5 for correlated and uncorrelated channels, respectively.It is worth noting that the proposed gamma approximation is more accurate for the case of uncorrelated channels than the case of correlated channels, especially in the large transmit power regime.And, the approximation is more accurate for larger number of RIS elements.This is due to not only the approximation errors, but also the errors brought by the correlated Nakagami-m random variables generation method [51].However, the proposed method in this article already provided us with expressions in concise form, which can demonstrate the impacts of channel correlations on the system performance.The approximation gap is less than 1 dBm for the same outage probability.Fig. 7 explores the impact of residual SI on the system performance.RIS elements on S A are placed in a rectangular array with five elements per column and fifteen elements per row, such that M v,A = 5 and M h,A = 15.ρ A = 0.3.The Nakagami-m shape parameters are set to be a h,A = a g,A = 2.It is assumed that the transmit power of U A and TX are equal, i.e., P A = P T .Regarding the residual SI, we set = 3 × 10 −5 , and ∈ {0, 0.5, 1}.= 0 and = 1 stand for the cases where the residual SI is independent and linearly dependent of the transmit power, respectively.For different residual SI exponent , it can be observed that the outage probability converges to a constant for = 1, and decreases with the increase of the transmit power for 0 ≤ < 1 in the large transmit power regime.This confirms our analysis in Remark 3. Furthermore, if 0 < < 1, the system performance falls between the cases of = 0 and = 1.
Fig. 8 illustrates the average achievable rate of the uplink transmission versus the transmit power of U A .The element configuration of S A is M v,A = 5, M h,A ∈ {10, 15}, i.e., M A ∈ {50, 75}.The Nakagami-m shape parameters are a h,A = a g,A = 2. ρ A = 0.5.The residual SI coefficient and exponent are given as = 10 −7 and = 0, respectively.In the legend, 'Analytical' and 'Upper Bound' stand for average achievable rate derived from the outage probability expression in (47) and the upper bound derived in (57), respectively.As can be seen from Fig. 8, the analytical results exactly match  with the simulation results, and the upper bounds are also very tight.Similar to the outage probability, increasing the number of elements will improve the average achievable rate.For example, when P A = 15 dBm, the average achievable rates are R A = 2.8 bits/s/Hz and R A = 3.9 bits/s/Hz for M A = 50 and M A = 75, respectively.Fig. 9 demonstrates the impact of the residual SI on the system performance by illustrating the average achievable rate versus the transmit power of U A where the residual SI coefficient and exponent vary in ∈ {10 −5 , 5 × 10 −5 } and ∈ {0, 0.5, 1}.We set a h,A = a g,A = 2; ρ A = 0.5; P A = P T .RIS elements on S A are deployed with M v,A = 5 elements per column and M h,A = 10 elements per column.As shown in the figure, smaller residual SI coefficient will lead to higher average achievable rate.For example, when P A = 45 dBm and = 0, the average achievable rates of U A are 6 bits/s/Hz and 3.7 bits/s/Hz for = 10 −5 and = 5 × 10 −5 , respectively.Further, similar to the case of the outage probability, the average achievable rate converges to a constant value due to the linear dependence between the transmit power and the residual SI.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.The average achievable rate of the HD communication is a constant since the residual SI does not exist in the HD mode.As can be seen from Fig. 10, the FD communication outperforms the HD communication when the residual SI is lower than a threshold value.The boundary given by ( 61) is very precise as proven by the simulations.

VI. CONCLUSION
This article study a RIS-assisted FD communication network, where an uplink user and a downlink user are served by an FD transceiver over the same time and frequency resource block.Due to the size and placement of RIS reflecting elements, the proposed system incorporates a spatially correlated Nakagamim fading channel model.Since the exact performance analysis is mathematically intractable, we apply the moment-matching method to approximate the signal power as a gamma random variable.Then, the outage probability and the average achievable rates of the uplink and downlink users are derived in closed form.The impact of the residual SI is explored.Floors of outage probability and average achievable rate can appear due to the linear relationship between the transmit power and the residual SI.Furthermore, the residual SI threshold below which the FD communication performs better than the HD counterpart is given in the closed form and verified by simulations.Taking the inverse two dimensional Laplace transform [53], the joint distribution of (H m,u , H n,u ) is derived as (A.15) Defining matrices R h,u and R g,u as in (25a) and (25b), we can arrive at the variance of H u in a form as given in (23b).

APPENDIX B
As stated in (22), the average achievable rate of U u can be rewritten as where (a) is obtained by using the relationship between the CDF and PDF of a random variable; (b) is derived by exchanging the order of integration.
n B ∼ CN (0, σ 2 B ) is the additive white Gaussian noise (AWGN) with zero mean and variance σ 2 B .Similar to S A , S B can only obtain the CSI of the channels g B and h B which are the signal transmission channels regarding its pairing user U B , and the phase shifts of elements on S B are controlled to boost the signal transmission through the channel TX-S B -U B , i.e., 3 θ m,B = − arg(g m,B ) − arg(h m,B ), m ∈ {1, 2, . .., M B }.

)
The distribution of ĨA is characterized first.Since φ m,A follows the uniform distribution within (−π, π], the expectation and variance of |I m,A ||h m,A | cos(φ m,A ) and |I m,A ||h m,A | sin(φ m,A ) are derived as E [|I m,A ||h m,A | cos(φ m,A )] = E [|I m,A ||h m,A | sin(φ m,A )] = 0, (37a) For the downlink transmission, the distances of the TX-S B and S B -U B are set as d h,B = 4 m and d g,B = 20 m, respectively.The path loss per unit distance of the downlink channel is Λ B = −20 dB.This reflects the scenario where S A and S B are placed closed to U A and U B , respectively.Regarding the interference channel, the distances of the S A -U B and S B -U A links are set to be d I,A = d I,B = 30 m.The distance of the U A -U B link is d 0 = 25 m.The path loss per unit distance of the interference channels are Λ I,A = Λ I,B = Λ D = −25 dB.

Figs. 3
Figs.3 and 4illustrate the outage probability and average achievable rate of U B versus the transmit power of TX.RIS elements on S B are implemented in a rectangular array with five elements per column, i.e., M v,B = 5.The number of elements per row varies in M h,B ∈ {10, 15}.Hence, the number of elements on S B varies inM B ∈ {50, 75}.The element configuration of S A is: M A = 50, M v,A = 5, M h,A = 10.ρ A = ρ B = 0.5.The transmit power of U A varies in P A ∈ {20 dBm, 30 dBm}.The Nakagami-m shape parameters are set to be a h,B = a g,B = 2.As can be seen from the figures, the derived expressions match well with the simulation results.Increasing the number of elements on S B and reducing the transmit power of U A are both beneficial for the system performance due to the increased desired signal power and the decreased interference power, respectively.For example, when the transmit power of TX is P T = 15 dBm and P A = 30 dBm, the average achievable rates of U B are 3 bits/s/Hz and 4 bits/s/Hz for M B = 50 and M B = 75, respectively.To illustrate the impact of channel correlations on the communication performance, a benchmark

Fig. 3 .
Fig. 3. Outage probability of U B versus the transmit power with different number of elements.The curves of correlated and uncorrelated channels with same M B and P A are circled together.

Fig. 4 .
Fig. 4. Average Achievable rate of U B versus the transmit power with different number of elements.
M h,A = 10; M B = 50, M v,B = 5, M h,B = 10, and ρ A = ρ B = 0.5.The transmit power of U A and TX are P A = 30 dBm and P T = 10 dBm.Our results indicate that the GCQ-based approximation is highly accurate even for

Fig. 7 .
Fig. 7. Outage probability of U A versus the transmit power with different .

Fig. 8 .
Fig. 8. Average achievable rate of U A versus the transmit power with different number of elements.

Fig. 9 .
Fig. 9. Average achievable rate of U A versus the transmit power with different .

Fig. 10
Fig.10is presented to verify the boundary at which the FD communication scheme is better the the HD communication given in (61).The element configuration of S A isM A = 50, M v,A = 5, M h,A = 10.The transmit power of U A varies in P A ∈ {20 dBm, 30 dBm}.The residual SI lies in σ 2 l ∈ (−70 dBm, −20 dBm).The average achievable rate of the HD communication is a constant since the residual SI does not exist in the HD mode.As can be seen from Fig.10, the FD communication outperforms the HD communication when the residual SI is lower than a threshold value.The boundary given by (61) is very precise as proven by the simulations.

1 )
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.