Joint Reflecting and Precoding Designs for SER Minimization in Reconfigurable Intelligent Surfaces Assisted MIMO Systems
Jia Ye Shuaishuai Guo Mohamed-Slim Alouini
 


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Joint Reflecting and Precoding Designs for SER Minimization in Reconfigurable Intelligent Surfaces Assisted MIMO Systems
    
Jia Ye, Student Member, IEEE, Shuaishuai Guo, Member, IEEE,  
and Mohamed-Slim Alouini,  Fellow, IEEE
J. Ye and M.-S. Alouini are with Electrical Engineering, Computer Electrical and Mathematical Sciences & Engineering (CEMSE) Division, King Abdullah University of Science and Technology (KAUST), Thuwal, Makkah Province, Kingdom of Saudi Arabia, 23955-6900 (email:jia.ye@kaust.edu.sa; slim.alouini@kaust.edu.sa).

S. Guo was with CEMSE Division, KAUST, Thuwal, Makkah Province, Kingdom of Saudi Arabia, 23955-6900. He is now with Shandong Provincial Key Laboratory of Wireless Communication Technologies and and School of Control Science and Engineering, Shandong University, Jinan 250061, China (email: shuaiguosdu@gamil.com).


    December 30, 2023
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This paper investigates the use of a reconfigurable intelligent surface (RIS) to aid point-to-point multi-data-stream multiple-input multiple-output (MIMO) wireless communications. With practical finite alphabet input, the reflecting elements at the RIS and the precoder at the transmitter are alternatively optimized to minimize the symbol error rate (MSER). In the reflecting optimization with a fixed precoder, two reflecting design methods are developed, referred as eMSER-Reflecting and vMSER-Reflecting. In the optimization of the precoding matrix with a fixed reflecting pattern, the matrix optimization is transformed to be a vector optimization problem and two methods are proposed to solve it, which are referred as MSER-Precoding and  MMED-Precoding. The superiority of the proposed designs is investigated by simulations. Simulation results demonstrate that the proposed reflecting and precoding designs can offer a lower SER than existing designs with the assumption of complex Gaussian input. Moreover, we compare RIS with a full-duplex Amplify-and-Forward (AF) relay system in terms of SER to show the advantage of RIS.






Reconfigurable intelligent surface, phase shifts design, precoder design, symbol error rate.







§ INTRODUCTION

Better communication qualities like smaller delay, higher transmission rate, lower symbol error probability (SER), less energy consumption always attract researchers' and users' eyes. In the last decade, wireless networks have been greatly improved thanks to various technological advances, including massive multiple-input multiple-output (Massive MIMO), millimeter wave (mmWave) communications, and ultra-dense deployments of small cells. However, some critical issues such as the hardware complexity and system update cost are still blocking their steps to the practical implementation <cit.>. To satisfy the growing demands with satisfying communication quality and achieve challenge goals in a green and effective way, reconfigurable intelligent surface (RIS) was proposed as a promising solution in the coming 5G or beyond era <cit.>. 

RIS is a planar array comprising of a large number of nearly passive, low-cost, reflecting elements such as positive-intrinsic-negative (PIN) diodes, which are used for altering the phase of the reflected electromagnetic wave with reconfigurable parameters and smart controller. RIS can be implemented by various materials, including reflect arrays <cit.>, liquid crystal metasurfaces <cit.>, ferroelectric films, or even metasurfaces<cit.>.   

In the beginning, reflecting surfaces were not considered in wireless communication systems because these surfaces only had fixed phase shifters, which could not adapt the phase modification in time-varying wireless propagation environments. Recently,  advanced micro-electrical-mechanical systems (MEMS) and metamaterials have been investigated as a solution to this issue, which enables the real-time reconfiguration reflecting surfaces <cit.>. Compared to existing related technologies such as multi-antenna relay <cit.>, backscatter communication<cit.> and active intelligent surface-based massive MIMO <cit.>, passive RIS does not require any dedicated energy source for either decoding, channel estimation, or transmission. It only reflects the ambient radio frequency (RF) signals in a passive way without a transmitter module. Moreover, the reflect-path signal through RIS carries the same useful information as well as the direct-path signal without any information of its own, which will not cause any additional interference. 

RIS stands out among these technologies by smartly adjusting the phase shifts induced by all the elements with advantages like overcoming unfavorable propagation conditions, enriching the channel with more multi-paths, increasing the coverage area, improving the received signal power, avoiding interference, enhancing security/privacy and consuming very low energy. On the other hand, the lightweight and conformal geometry of RIS can enable the installment onto the facades of buildings in outdoor communication environments or the ceilings and walls of rooms in indoor communication environments, which provides high flexibility and superior compatibility for practical implementation <cit.>. Also, integrating RIS into the existing networks can be made transparent to the users without the need for any change in the hardware and software of their devices.



 §.§ Prior Work

Due to the advantages mentioned above, RIS has attracted more and more researchers' attention in the last few years. Many interesting and contributed works appeared in the industry <cit.> and academia field regarding to channel estimation <cit.>, <cit.>, multi-cell networks <cit.>, comparison with relaying systems <cit.> and massive MIMO networks <cit.>, combination with other technologies like SWIPT <cit.>, <cit.>, millimeter wave <cit.>, <cit.>, Terahertz communication <cit.> and so on. A large and growing body of literature has optimized the RIS parameters and systems' structure to improve system performance, such as outage probability <cit.>, signal power <cit.>, signal-to-interference-plus-noise ratio (SINR) <cit.>, data rate <cit.>, <cit.>, spectral efficiency (SE) <cit.>, secrecy achievable rate <cit.> and so on. To describe the status of research, we survey some of them as follows before introducing our work.

In <cit.>, the authors enhanced the mmWave link robustness and optimized the link outage probability by deploying smart reflect-arrays when the direct links are blocked by obstructions. The authors also investigated the optimal beam direction for randomly moving devices without any location information, incorporating the antenna sector selection at the access point (AP) and the mobile user as well as the configuration of the smart reflect-arrays. 

For maximizing the total received signal power, preliminary contributions appeared in <cit.>. Specifically, a centralized algorithm with the global channel state information (CSI) and a low-complexity distributed algorithm for designing the phase shifts were proposed in <cit.>. With the same system setup in <cit.>, the authors of <cit.> considered a more practical case when the RIS only has a finite number of discrete phase shifts in contrast to the continuous phase shifts. It can be seen that the asymptotic squared power gain of RIS-aided multiple-input single-output (MISO) shown in <cit.> with continuous phase shifts still holds with discrete phase shifts with a stable performance loss gap between the two conditions. The authors of <cit.> and <cit.> maximized the energy-efficiency (EE) and SE of a RIS-assisted multi-user MISO system by designing both the transmit power allocation at the BS and the phase elements of the RIS. 

By using the knowledge of only the channel large-scale statistics instead of the global CSI, <cit.> designed an optimal linear precoder and the power allocation at the base station (BS) as well as the RIS phase matrix to maximize the minimum SINR in multi-user RIS-assisted MISO communication systems. It was shown that the RIS-assisted system can achieve power gains with a much fewer number of active antennas at the BS. 

The authors in <cit.> analyzed the approximated uplink ergodic rate of a Rician fading system and derived an optimal size of a RIS unit, where users are mapped to a limited area of the entire RIS. It was shown that RIS can bring improved reliability with a significantly reduced area for antenna deployment compared to massive MIMO. Targeting for practical implementation, <cit.> investigated an optimal phase shift design to maintain an acceptable degradation of the ergodic capacity. The authors in <cit.> proposed a provably convergent, low-complexity method to maximize the system sum-rate. It was shown that a nearly “interference-free” zone can be established in the proximity of the RIS thanks to its spatial interference nulling/cancellation capability. The authors in <cit.> maximized the weighted sum rate in the RIS-assisted multicell MIMO system, where a RIS located at the cell boundary of multiple cells to assist the transmission and alleviate the inter-cell interference. 

Moreover, using RIS to secure wireless communications is also a hot topic in this field. The authors of <cit.> jointly optimized the transmit beamforming with artificial noise jamming to maximize the achievable secrecy rate. The physical-layer security was also studied in <cit.>, which aims to maximizing system secrecy rate under source transmit power constraint on the transmitter and the phase shifts unit modulus constraints on RIS. By jointly designing the transmit beamforming of the AP and the reflect beamforming of the RIS, the authors of <cit.> optimized the secrecy rate of the legitimate communication link considering the presence of an eavesdropper.

In all the studies reviewed here, RIS is a promising technology to satisfy the growing demand for data rates and communication quality. Current research on the joint optimization of beamforming and reflecting in RIS-aided communications paid more attention to maximizing SE, EE, signal-to-noise ratio (SNR) and so on assuming Gaussian-distributed signal input. However, the assumption with Gaussian-distributed input signals is impractical, since the Gaussian-distributed signals are not bounded. In practical, the widely adopted is finite discrete constellation signals, such as phase-shift keying (PSK) or quadrature amplitude modulation (QAM) signals. The authors in <cit.> showed that the probability distribution function of a standard Gaussian-distributed signal is significantly different from the probability mass function of a quadrature phase-shift keying (QPSK) signal.   Based on previous research on MIMO communications without the aid of RIS, the optimization with the Gaussian input assumption is far from optimal in MIMO communications with finite alphabet input, and also the optimization with finite alphabet input is much more complicated <cit.>. In our paper, we are interested in the joint optimization of beamforming and reflecting to minimize SER for RIS-aided MIMO communications assuming finite alphabet input. Although the authors in<cit.> and <cit.> evaluated the error performance of RIS-based communication systems, they do not do the optimization regarding the phase shifts and precoder. To the best of our knowledge, there is no work to jointly optimize the reflecting and precoding for RIS-assisted MIMO communications based on the SER minimization criterion with finite alphabet input. The use of SER as the metric for performance optimization calls for a thorough investigation, which motivates our work.



 §.§ Contributions

As was pointed out in the previous subsection, we are interested in the joint optimization of precoding and reflecting to minimize SER for RIS-aided MIMO communications assuming finite alphabet input. The optimization objective is different from existing works. The challenge of the optimization lies in that the beamforming and reflecting affect all received signal vectors. The optimization complexity increases with the square growth of the number of transmit signal vector candidates, because the mutual Euclidean distances among different noise-free received signal vectors jointly affect the SER. We transform the SER minimization problem into several problems that can be addressed by existing optimization techniques. 
For clearness, we list our contributions as follows:


  * This paper considers a RIS-enhanced point-to-point multiple-data-stream MIMO communication system, where a multiple-antenna transmitter serves a multiple-antenna receiver with the help of a RIS. 
We formulate the SER minimization problem regarding the precoding and reflecting for such a system. Since the precoding and reflecting are coupled in affecting the objective function and isolated physically, we resort to an alternating strategy, which is widely used in literature (e.g., <cit.>) to maximize SE, EE, SNR, etc. 



  * Regarding the reflecting design, we reformulate the reflecting optimization to find the optimal reflecting angle subsequently to minimize the symbol error rate (MSER) and solve it by a coordinate descent method, referred as element-wise MSER (eMSER) reflecting. The eMSER-Reflecting is analyzed to have high computational complexity. To reduce the complexity, we transform a non-convex constraint of the problem to be a convex constraint and solve the optimal reflecting vector jointly through a vector gradient descent method, referred as vMSER reflecting. 


  * Regarding the precoding design, we express the channel matrix, the precoding matrix, and the transmitted symbol vectors in new forms and transform the precoding matrix optimization to be a vector optimization problem. The reformulated problem can be solved by a projected gradient descent method named MSER-Precoding. In the high SNR regime, the minimizing SER criterion is shown to be equivalent to the maximizing the minimum Euclidean distance (MMED) criterion. Based on the fact, the reformulated problem is reduced to be a quadratically constrained quadratic program (QCQP) problem and we propose a corresponding MMED-Precoding method. MMED-Precoding is shown to achieve comparable performance with considerable computational complexity reduction. 


  * Moreover, a thorough investigation is conducted by simulations. The effectiveness of our proposed algorithms is validated by comparing them with the solution gained through an exhaustive search in performance, computation complexity, and central processing unit (CPU) running time. By comparing the proposed designs with existing designs that maximize SNR, we verify that the SER minimization with finite alphabet input could not be replaced by SNR maximization, especially in the high SNR regime. Although the proposed designs are with higher computational complexity than existing designs, they can achieve much better performance in practical communication systems with finite alphabet input. Moreover, numerical comparison with a full-duplex Amplify-and-Forward  (AF) relay system is conducted to show that the RIS-assisted system can achieve the same or even better performance by adopting reflecting elements rather than additional power amplifiers at relays.




 §.§ Organization

The remainder of the paper is organized as follows. Section II describes the system model and formulates the system optimization problem. Section III introduces the proposed reflecting and precoding designs. In Section IV, we analyze the computational complexity. Numerical comparisons are presented in Section V and conclusions are drawn in Section VI.



 §.§ Notations

In this paper, x denotes a scalar; x represents a vector; X stands for a matrix. x_2, x_p and x_∞ represents l_2 norm, l_p norm and l_∞ norm of x respectively. X_F is the Frobenius norm of X.  diag(x) is a diagonal matrix whose diagonal entries are from vector x. x_i denots the ith entry of x, x_i,j is the element in i-th row and j-th column of a matrix X, x_i denots the i-th column of matrix X. tr(X) denotes the trace of X, rank(X) represents the rank of X, vec(X) means the vectorization of matrix X, which is a linear transformation which converts X into a column vector, λ_max(X) is the maximum eigenvalue of matrix X. X≥ 0 means that matrix X is positive semidefinite. ⊙ stands for the Hadamard product, ⊗ denotes Kronecker product. (·)^H is the conjugate transpose; (·)^C represents the conjugate and (·)^T denotes the transpose. ℂ stands for the complex domain while ℝ represents the real domain. 𝒞𝒩(μ, Σ) stands for the circularly symmetric complex Gaussian distribution with mean μ and covariance Σ. E[ ·] represents the expectation operation. I_N denotes an N× N identity matrix. [x]^+ denotes max(x, 0). Re{ x } and Im{ x } represent the real and imaginary part of x, respectively. ∇ denotes the gradient of a function. Q(·) stands for the tail distribution function of the standard normal distribution.






§  SYSTEM MODEL AND PROBLEM FORMULATION 


In this paper, we consider an RIS-assisted MIMO system model as illustrated in Fig. 1. In the model, a transmitter equipped with N_t antennas communicates with a receiver equipped with N_r antennas with the help of a RIS composed of N reflecting units. The RIS acts as a passive relay, which is embedded on a surrounding building. 
The received vector y∈ℂ^N_r × 1 at the receiver can be expressed as 


    y =  √(ρ)(H_2 ΦH_1 +H_d)Fs + n,
where ρ is the SNR; H_2 ∈ℂ^N_r ×N represents the channel between the RIS and the receiver; Φ = diag{ϕ}∈ℂ^N ×N denotes the diagonal matrix accounting for the effective phase shifts applied by the RIS reflecting elements with ϕ =  [ ϕ_1, ϕ_2, ⋯, ϕ_N ]^T, where |ϕ_n| = 1 with ϕ_n∈ℱ≜{exp(j 2 π m/2^b)}_m=0^2^b-1 with b phase resolution in number of bits <cit.>; H_1 ∈ℂ^N ×N_t represents the channel between the transmitter and the RIS, H_d ∈ℂ^N_r ×N_t represents the direct channel between the transmitter and the receiver; F∈ℂ^N_t ×N_s is the precoder to encode N_s data streams; s is the N_s × 1 transmitted data symbol vector with each entry chosen from a M-ary constellation 𝒮_M and there are totally M^N_s legitimate symbol vectors; n∼𝒞𝒩( 0,σ^2I_N_r) is the additive white Gaussian noise (AWGN) vector with each entry obeying a zero-mean variance σ^2 complex Gaussian distribution. In this paper, we assume that the average power of all legitimate symbol vectors {s} is normalized. Let x = Fs denote the transmitted signal vector from the multi-antenna transmitter. It is assumed that x satisfies the maximum average transmit power constraint E[ x_2^2] = E[tr( Fss^HF^H)] = tr( Q) ≤P_max, where Q = ^ΔFF^H is the signal covariance matrix and P_max denotes the maximum average power.

It is considered that the system utilizes all M^N_s feasible transmit vectors, the union bound on SER can thus be written as

    P_S( Φ ,F)    = 1/M^N_s∑_i = 1^M^N_s∑_j = 1,j  i^M^N_s{s_i→s_j},

where {s_i→s_j} denotes the pairwise SER of the vector s_i being erroneously detected as s_j. By using the squared Euclidean distance d_ij^2( Φ ,F) = (H_2ΦH_1+H_d) F( s_i - s_j)^2 between two vectors,  {s_i→s_j} can be computed as {s_i→s_j} = Q( √(ρd_ij^2( Φ ,F)/2σ^2)). 

The objective of our design is to minimize the SER. To make the targeted problem more tractable, we assume that the phase resolution is infinite <cit.> and all involved CSI are known, the optimization can be formulated as problem (P1):

    (P1):    Given:     H_1, H_2, H_d, 𝒮_M 
    Find:     Φ, F
    Minimize:     P_S( Φ ,F)
    Subject to:     tr( Q) ≤P_max
        | ϕ _i| = 1, ∀ i = 1,...,N,

where the first constraint ensures that the BS transmit power is kept below the maximum feasible power P_max and the second constraint means that every reflecting unit only provides phase shift without signal amplification.

The joint optimization problem is non-convex and challenging to obtain the optimal solution due to the coupling effect directly. Thus, we resort to an alternating way to optimize the reflecting and precoding, similarly to <cit.>.



§ JOINT REFLECTING AND PRECODING DESIGN

As stated above, because of the coupling effect between the reflecting elements in Φ and the precoder F, the original problem (P1) is hard to solve. To decouple them, we will first optimize Φ by fixing F and then update F by fixing Φ respectively. Then, we will obtain sub-optimal solutions for both Φ and F by performing the process iteratively until reaching the convergence of the objective function or the solutions to an acceptable level. In the following, we first present the designs for reflecting elements.



 §.§ Reflecting Designs

Based on a given F, the squared Euclidian distance can be re-expressed as d_ij^2( Φ) = H_2ΦH_1x_ij_2^2+ 2ℛ(ϕ^Ta_ij) +H_dx_ij_2^2, where x_ij = F( s_i - s_j) and a_ij is a vector with k-th element a_ij,k = x_ij^Hh_d,2,kh_1,k^Tx_ij, where h_d,2,k is the k-th column of H_d^HH_2 and h_1,k denotes the k-th column of H_1^T. 

Then, we carry out some manipulations to transform H_2ΦH_1x_ij_2^2 as

    H_2ΦH_1x_ij_2^2   =q_ij^HΦ^HH_2^HH_2Φq_ij= tr(Φ^HR_H2ΦΔQ_ij),

where q_ij = H_1x_ij R_H2 = H_2^HH_2 and ΔQ_ij = q_ijq_ij^H. Using the rule that tr( D_x^HAD_yB^T) = x^H( A⊙B)y with D_x = diag{x} and D_y = diag{y} in <cit.>, we can re-express H_2ΦH_1x_ij_2^2 as H_2ΦH_1x_ij_2^2 =ϕ^H(R_H2⊙ΔQ_ij^T)ϕ= ϕ^H C_ijϕ, where C_ij = R_H2⊙ΔQ_ij^T. 

Finally, d_ij^2( Φ) can be finally expressed by the function of ϕ as

    d_ij^2( ϕ) = ϕ^H C_ijϕ + 2ℛ(ϕ^Ta_ij) +H_dx_ij_2^2.




  §.§.§ eMSER-Refelcting Scheme

Expanding the matrix product, we can re-express the minimum Euclidian distance as d_ij^2( ϕ) = ∑_n = 1^N∑_k = 1^Nϕ _nϕ _k^H C_ij,k,n+2ℛ(∑_k = 1^Nϕ _k a_ij,k)+H_dx_ij_2^2,
where C_ij,k,n is the k-th row and n-th column of matrix C_ij. In order to handle the non-convex problem, we use a coordinate descent method to obtain the sub-optimal solution.
 Recall ϕ_n = exp(jθ_n), the objective can be represented as a function of θ_n by fixing remaining phase shifts. In details, we optimize ϕ_n by fixing remaining phase shifts firstly until it meets the stop criterion. Then, we do the process iteratively until we get the sub-optimal solution. Thus, we formulate the gradient of the cost function P_S(θ_n) over θ_n as

    ∂  P_S(θ_n)/∂θ_n = 1/M^N_s∑_i = 1^M^N_s∑_j = 1,j  i^M^N_s∂Q( √(ρd_ij^2( θ_n )/2σ^2))/∂θ_n.


According to Leibniz's integral rule, we can express ∂Q( √(ρd_ij^2( θ_n )/2σ^2))/∂θ_n as

    ∂Q( √(ρd_ij^2( θ_n )/2σ^2))/∂θ_n=-√(ρ/πσ^2 d_ij^2 ( θ_n))exp(  - ρd_ij^2( θ_n)/4σ^2)∂ d_ij^2( θ_n )/∂θ_n,

where ∂ d_ij^2( θ_n )/∂θ_n  = 2ℛ(∑_k = 1,k≠ n^Njexp{j(θ_n-θ_k)}C_ij,k,n)+2ℛ(jexp{jθ_n} a_ij,n).

By using -∂  P_S(θ_n)/∂θ_n as the search direction <cit.> for θ_n and do the process iteratively, we can search the optimized solution as listed in Algorithm 1. Since we optimize θ_n directly, the optimization process will not lead to an infeasible solution. This is because for any real number θ_n, |e^jθ_n|=1 always holds. 







  §.§.§ vMMSR-Reflecting Scheme

To facilitate practical implementation, we investigate low-complexity phase shift designs in this part. Since | ϕ _i| = 1, accordingly we can obtain that tr( ϕϕ^H) = N. In order to handle the non-convex constraint of | ϕ _i| = 1, we relax the problem (P1) into the following optimization  (P2) with a convex ℓ _∞ constraint:

    (P2):    Given:     H_1, H_2, H_d, 𝒮_M, F
    Find:     ϕ
    Minimize:     P_S( ϕ)
    Subject to:    tr( ϕϕ^H) = N
        ϕ_∞≤ 1.


Actually, the original feasible set is a subset of the new feasible set in (P2), i.e.,

    {ϕ∈ℂ^N ×1:| ϕ _i| = 1,  ∀ i = 1,...,N}= {ϕ∈ℂ^N ×1:tr( ϕϕ^H) = N &ϕ_∞≤ 1 },

which is convex due to the convexity of ℓ _∞ norm.

Since the ℓ _∞ constraint is non-differentiable, we exploit the ℓ _p approximation <cit.> with a gradually increased large p, lim_p →∞ϕ_p = ϕ_∞, during the optimization process. To solve (P2), we utilize the barrier method to incorporate the non-negative constraint <cit.> with the logarithmic barrier function I( u ) to approximate the penalty of violating the ℓ _p constraint, i.e., I( u ) = {[ - 1/tln( u ),         u > 0;           ∞ ,        u ≤ 0, ]., where t is used to scale the barrier function's penalty. Thus, we can obtain the following optimization problem:

    min_ϕ∈ℂ^N ×1  g( ϕ, p) =  P_S( ϕ) + I (  1-ϕ_p).

To solve (<ref>) via a gradient method, we formulate the gradient of the cost function g( ϕ, p) over ϕ as follows:

    ∇ _ϕg( ϕ, p)    = ∇ _ϕP_S( ϕ) + ϕ_p^1 - pp_ϕ/2t(  1-ϕ_p),

where p_ϕ∈ℂ^N ×1 is given as
p_ϕ = [ ϕ _1·| ϕ _1|^p - 2,ϕ _2·| ϕ _2|^p - 2,...,ϕ _N·| ϕ _N|^p - 2]^T.

Moreover, the gradient ∇ _ϕP_S( ϕ) can be calculated as

    ∇ _ϕP_S( ϕ) = 1/M^N_s∑_i = 1^M^N_s∑_j = 1,j  i^M^N_s∇ _ϕQ( √(ρd_ij^2( ϕ)/2σ^2)).


According Leibniz's integral rule, ∇ _ϕQ( √(ρd_ij^2( ϕ)/2σ^2))=-√(ρ/πσ^2 d_ij^2 ( ϕ))exp(  - ρd_ij^2( ϕ)/4σ^2)(C_ijϕ+a_ij^C). Thus, ∇ _ϕg( ϕ, p) can be re-expressed as

    ∇ _ϕg( ϕ, p) = -1/M^N_s∑_i = 1^M^N_s∑_j = 1,j  i^M^N_s√(ρ/πσ^2 d_ij^2 ( ϕ))exp(  - ρd_ij^2( ϕ)/4σ^2)(C_ijϕ+a_ij^C)+ϕ_p^1 - pp_ϕ/2t(  1-ϕ_p).


By using -∇ _ϕg( ϕ, p) as the search direction <cit.>, we can search the optimized solution as listed in Algorithm 2.




In each iteration of Algorithm 2, P_S( ϕ_k+1) ≤P_S( ϕ_k).


For any θ̂→ 0, based on (<ref>), the Taylor expansion of P_S( ϕ_k+1) can be derived as

    P_S( ϕ_k+1)    = P_S( ϕ_k)+g_k^H·√(N)w_k^ /w_k^  + 𝒪(θ̂^2)
       ≈P_S( ϕ_k)+g_k^Hw_k^ √(N)/w_k^ .

We can also calculate that g_k^Hw_k^  = (cos^2α-1)w_k ^2≤ 0, where α = arccosϕ_k^Hw_k/ϕ_k^H w_k is the angle between vectors ϕ_k and w_k. Combining it with (<ref>), we have P_S( ϕ_k+1) ≤P_S( ϕ_k).
The proof is completed.


Under this constraint relaxation, the solution obtained from each iteration may cannot guarantee satisfying | ϕ _k,i| = 1. To fix this, we add a element-wise normalization step ϕ̂_k is ϕ̂_k,i = ϕ_k,i/|ϕ_k,i|, conditioned that the objective function is not much sensitive to the normalization. This step ensures the feasibility of the solution and meanwhile keeps good performance.



 §.§ Precoding Designs

Based on the optimized Φ, we can simplify (P1) as

    (P3):    Given:     H_1, H_2, H_d, 𝒮_M, Φ
    Find:     F
    Minimize:     P_S( F)
    Subject to:     tr( Q) ≤P_max.


By introducing  H = H_2ΦH_1+H_d∈ℂ^N_r ×N_t, the received signal can be re-expressed as

    y   = HFs + n=[ h_1,⋯,h_N_t][ [     f_1,1         ⋯   f_1,N_s;         ⋮         ⋱         ⋮;   f_N_t,1         ⋯ f_N_t,N_s ]][ [ s_0;   ⋮; s_1 ]] + n=∑_a = 1^N_t∑_b = 1^N_sf_a,bh_as_b + n.


Considering that (<ref>) has two summations, which are not easy to handle, we rebuild the channel matrix, precoding matrix and the transmitted data stream in a new form <cit.>. We first construct a new channel matrix as H = [ H_1,⋯,H_n_t,⋯,H_N_t] ∈ℂ^N_r ×N_tN_s, in which H_n_t = [ h_n_t,⋯,h_n_t,⋯,h_n_t]_N_s, where each h_n_t repeats N_s times. Meanwhile, the precoding matrix entries in F are collected together as F = diag{vec(F^T)}=diag{[f_1,1, f_1,2,⋯,f_N_t, N_s]^T}=diag{f}. Furthermore,  s = [ s,⋯,s,⋯,s]_N_t^T∈ℂ^N_tN_s×1.

Following the procedure above, we can rewrite (<ref>) to be y = HFs + n. Thus, we can re-express the squared Euclidean distance as d_ij^2( F) = HF( s_i - s_j)^2. Similarly to (<ref>), the squared Euclidean distance can be re-expressed as

    d_ij^2( F)    =(s_i - s_j)^HF^HH^HHF(s_i - s_j)= tr(F^HR_HFΔS_ij)=f^H(R_H⊙ΔS_ij)f = f^H C_ijf,

where R_H = H^HH, ΔS_ij = ( s_i - s_j)(s_i - s_j)^H, f = [f_1,1, f_1,2,⋯,f_N_t, N_s]^T and C_ij = R_H⊙ΔS_ij.

In the following, we will provide several algorithms to solve this problem.



  §.§.§ MMSE-Precoding Scheme

 Based on discussion above, (P3) can be transformed to 

    (P3-a):    Given:     H, 𝒮_M
    Find:     f
    Minimize:     P_S( f)
    Subject to:     tr( ff^H) ≤P_max.


To solve (P3-a), we formulate the Lagrangian function as

    L( f, μ) =  P_S( f) + μ(tr( ff^H) - P_max),

where μ is the Lagrangian multiplier. The optimal solution must satisfy the Karush-Kuhn-Tucker (KKT) conditions as

    {[        ∇ _f L( f, μ) = 0; μ(tr( ff^H) - P_max) = 0;                     μ≥ 0 ]..

Because of its monotonicity with power, L( f, μ) is minimized when the power constraint is met with strict equality. Hence, μ(tr( ff^H) - P_max) = 0. The first equation in (<ref>) can be represented as

    ∇ _f L( f, μ)    = [-1/M^N_sΩ(f)+2μI]f = 0,

where Ω(f) is given by

    Ω(f) = ∑_i = 1^M^N_s∑_j = 1,j  i^M^N_s√(ρ/4πσ^2 d_ij^2( f))exp(  - ρd_ij^2( f)/4σ^2)C_ij.

Clearly, the closed-form solution of (<ref>) is difficult to derive. In this paper, we use the gradient method and project the solution onto the tangent plane of tr( ff^H) = P_max. Specifically, the gradient descent direction of (<ref>) is firstly calculated to be

    r_k = -g_k =Ω(f_k)f_k.

Then, a projection on search direction is conducted by

    r_k^   = r_k - ⟨r_k, f_k⟩f_k/f_k^2.
 
We update the solution by searching along the projected search direction as

    f_k+1 = cosβ̂·f_k+sinβ̂·√(P_max)r_k^ /r_k^ ,

where β̂, (0 ≤β̂≤π/2) can be obtained by 

    β̂  = min_β̂P_S( f_k).

By doing this process iteratively until it reaches the stop criterion r_k^ /r_k ≤ε _4, where ε _4 is the halting criterion, a good solution can be obtained. For clearness, we list the iterative algorithm to search for the solution of (<ref>) in Algorithm 3.
 


In Algorithm 3, P_S( f_k+1) ≤P_S( f_k).


It can be proved similarly to Proposition 1. 


Since (P3-a) is a non-convex problem, (<ref>) is a necessary condition for global optimum and the generated vector f in Algorithm 3 is thus a critical point. This algorithm guarantee the feasibility for all solutions, because

    tr(f_k+1f_k+1^H)   =f_k+1^Hf_k+1=cosβ̂^2f_k^Hf_k+sinβ̂^2P_max+2cosβ̂sinβ̂√(P_max)f_k^H r_k^ /r_k^ 
       = cosβ̂^2P_max+sinβ̂^2P_max=P_max.




  §.§.§ MMED-Precoding Scheme
In the high SNR regime, the SER for a given channel can be simplified as P_S( Φ ,F) ≈λ/M^N_sQ( √(ρd_min^2( Φ ,F)/2σ^2)), where λ is the number of closest symbol pairs, and d_min^2( Φ ,F)= min_∀i,j,i  jd_ij^2( Φ ,F) is the minimum squared Euclidean distance between the noise-free received signal vectors. As 
P_S( Φ ,F) is a monotonically decreasing function of d_min^2( Φ ,F), problem (P1) can be formulated as a maximizing the minimum Euclidian distance (MMED) problem <cit.>.









Based on given Φ and (<ref>), (P1) can be rewritten as 

    (P4):    Given:     H_1, H_2, H_d, 𝒮_M, Φ
    Find:     f
    Maximize:     min f^H C_ijf
    Subject to:     tr( Q) ≤P_max.


By introducing an auxiliary variable r, we have the equivalent epigraph of (P4) as

    (P4-a):    Given:     H, 𝒮_M
    Find:     f
    Maximize:     r
    Subject to:     f^H C_ijf≥ r ∀i,j,i  j
        tr( ff^H) ≤P_max.


However, we can review (P4-a) in another equivalent way <cit.> as

    (P4-b):    Given:    H, 𝒮_M
    Find:     f
    Minimize:     f^2 
    Subject to:     f^H C_ijf≥ d_min ∀i,j,i  j,

where d_min is the desired squared minimum distance. The rationale behind (P4-b) is to guarantee the minimum squared distance, while pursuing the minimum power usage as the objective. It can be seen that (P4-b) is a large-scale non-convex QCQP problem. Since this optimization problem is similar to the problem investigated in <cit.>, we use the same method proposed in <cit.> to solve the problem. The details can be seen in <cit.>. It should be noted that the power constraint is first released in (P4-b). After the optimization through MMED-Precoding, the obtained solution is scaled to satisfy the power constraint to ensure feasibility.






































































































































 §.§ Alternating Optimization

Based on the aforementioned reflecting schemes and precoding schemes, an alternating algorithm can be conducted to minimize SER as listed in Algorithm 4. 




In Algorithm 4, P_S(ϕ_k+1, f_k+1) ≤P_S(ϕ_k, f_k).


From Proposition 1, we can obtain that 
P_S(ϕ_k+1, f_k) ≤P_S(ϕ_k, f_k) and from Proposition 2, we can further obtain that P_S(ϕ_k+1, f_k+1) ≤P_S(ϕ_k+1, f_k). Thus, in Algorithm 4, it yields

    P_S(ϕ_k+1, f_k+1) ≤P_S(ϕ_k+1, f_k) ≤P_S(ϕ_k, f_k).



 The proposition indicates that P_S(ϕ_k, f_k) is a monotone decreasing sequence as k increases. As is well known, the sequence is bounded by a lower limit P_S(ϕ^*, f^*  ) ≥ 0, where (ϕ^*, f^*  ) represents the global optimal solution. According to the monotone convergence theorem, that is, if a sequence of real numbers is decreasing and bounded below, then its infimum is the limit, we can prove that our proposed alternating optimization algorithms are always converged. To observe the convergence speed, we have included a figure showing the number of iterations required for the algorithms in the simulation part. 


In this section, we split the optimization problem (P1) into two separate optimization problem. The approach of iteratively solving the phase shift matrix and precoding matrix can only provide an efficient way to reduce the SER gradually. Due to the non-convexity of the problem, the approach only ensures a sub-optimal solution.  Moreover, the solution obtained through Alternating Optimization is also always feasible since the proposed precoding and reflecting optimization schemes are always feasible. 




§ COMPUTATIONAL COMPLEXITY

In this section, we analyze the computational complexity of the proposed algorithms.



 §.§ Computational Complexity of Reflecting Schemes





  §.§.§ Complexity Order of eMSER-Reflecting
 As can be seen from Algorithm 1, the computational complexity is dominated by the gradient calculation, which involves i) calculating the M^N_s(M^N_s-1) matrix multiplications ϕ^H C_ijϕ for each angle optimization and ii) searching for the best solution for every angle based on other given angle, which is N operations. Firstly, we should obtain the computational complexity of C_ij computation, which is 𝒞_C =  𝒪[ N^2+N^2N_r+N_tN_s+NN_t+N^2 ], where N^2 is the computation complexity for C_ij = R_H2⊙ΔQ_ij^T calculation, N^2N_r is the computation complexity for R_H2 calculation and N_tN_s+NN_t+N^2 is the computation complexity for the calculation of ΔQ_ij^T. It can be further simplified as 𝒞_C = 𝒪[N^2N_r+N_tN_s+NN_t ]. Therefore, the complexity order of Algorithm 1 is :

    𝒞_eM = 𝒪[ NM^2N_s(N^2N_r+N_tN_s+NN_t)].



  §.§.§ Complexity of vMSER-Reflecting
 As can be seen from Algorithm 2, the computational complexity is mainly consumed by the gradient calculation, which involves i) calculating the M^N_s(M^N_s-1) matrix multiplications ϕ^H C_ijϕ, and ii) calculating the ℓ _p norm. Therefore, the complexity order of vMSER-Reflecting Algorithm is :

    𝒞_vM=𝒪[ M^2N_s(N^2N_r+N_tN_s+NN_t) + N∑_n = 1^N_pp^( n )],

where N_p = (p_max-p)/Δ p, p^( n ) = p + (n-1)Δ p, and p, Δ p, p_max are specified in step 1 in Algorithm 2.



 §.§ Computational Complexity of Precoding Schemes




  §.§.§ Complexity of MSER-Precoding
 Similar to Algorithm 2, the complexity of Algorithm 3 is also mainly consumed by the gradient calculation involving matrices f^H C_ijf, while C_ij needs 𝒪[N^2N_r. .+N_rNN_t + N_rN_t^2N_s^2+N_t^2N_s^2] = 𝒪[N^2N_r+N_rNN_t+ N_rN_t^2N_s^2], where N^2N_r+N_rNN_t is the computation complexity of H = H_2ϕH_1, N_rN_t^2N_s^2 is the complexity for computing R_H = H^HH and N_t^2N_s^2 is the complexity of ΔS_ij = ( s_i - s_j)(s_i - s_j)^H. Therefore, the complexity order of Algorithm 3 for each iteration is :

    𝒞_MS = 𝒪[ M^2N_s(N^2N_r+N_rNN_t+ N_rN_t^2N_s^2)].




  §.§.§ Complexity of MMED-Precoding Scheme
 
According to <cit.>, the computational complexity order of MMED-Precoding C_MM is also mainly consumed by the gradient calculation involving matrices f^H C_ijf, which is same as Algorithm 2. 

It should be mentioned that MMED-Precoding does not need SNR information. In other words, MMED-Precoding is applicable for any SNR values, while MSER-Precoding needs to be re-designed as SNR varies. Besides, the number of iterations required for MMED-Precoding scheme is much less owing to the use of the BFGS method <cit.>. Thus, in general, the complexity of the MMED-Precoding scheme is much less than the MMSR-Precoding scheme.







 §.§ Complexity of Alternating Optimization
 
In this subsection, we analyze the overall complexity order of Alternating Optimization by taking different combinations of precoding schemes and reflecting schemes. By process reflecting schemes and three precoding schemes alternatively, the overall complexity order can be expressed as 𝒞=𝒞_i ∈ℝ+𝒞_k∈ℙ, 
where 𝒞_i ∈ℝ and 𝒞_k∈ℙ denote the complexity order of the selected reflecting scheme and precoding scheme with ℝ = {eM, vM} and ℙ = {MS, MM}.





§ NUMERICAL RESULTS

In this section, we first compare the proposed designs with the exhaustive search (ES) solution obtained by an exhaustively searching algorithm and compare their CPU running time numerically. Then, we compare the performance of the proposed reflecting and precoding designs with existing designs which aims to maximize SNR, that is, SDP-Reflecting and Eigen-Precoding in variously configured (N_r, N, N_t, N_s, M, K) systems, where K is the Rician fading parameter. All experiments are performed on a PC with a 3.7GHz W-2145 CPU and 32GB RAM. By adopting Alternating Optimization with different reflecting scheme and precoding scheme combinations, vMSER-MMED, SDP-Eigen, SDP-MMED, and vMSER-Eigen are simulated and compared. Then, we demonstrate the SER performance with a large number of reflecting elements to investigate the impact of the size of the surface on the performance. Furthermore, the system performance using perfect CSI and imperfect CSI is compared to investigate the impact of channel estimation errors. Finally, we make a fair comparison between the RIS-assisted system and a full-duplex Amplify-and-Forward (AF)-relaying system.



 §.§ CPU Running Time Comparison and Superiority of the Proposed Schemes

In Fig. <ref>, we compare the proposed Reflecting-Precoding combinations with ES solution in (1, 2, 2, 1, 2, 5) system under the same channel realization. The ES solution can be obtained by searching each optimal reflecting elements from the feasible set C_ESR and searching every optimal element of the precoding matrix from the feasible set C_ESP. Since the computational complexity of the ES solution exponentially increases with the number of variables 2N_tN_s+N, thus it only can be realized in systems with small N_t, N_s and N. To verify the computation efficiency, we present CPU running time comparisons, which is shown in Table I over the same channel for each algorithm. It is seen that all combinations can achieve almost optimal performance compared to the ES solution, which validates the effectiveness of the proposed algorithms. From Table I, it is observed that eMSER-combinations take much longer time than others since the eMSER-Reflecting needs more time to obtain the sub-optimal solution. However, the vMSER-combinations can achieve close performance to eMSER-combination with much lower computation complexity. Thus, vMSER-combinations would be more suitable for practical implementation. Compared with random design, we conclude that appropriate design for the RIS-assisted system can improve system performance significantly. This results from the fact that more signal power can be centered on the direction of the receiver with the well-designed precoding and reflecting. The number of iterations that the Alternating Optimization converged is one of the key factors dominating the computational complexity, which shown in Fig. <ref>. It is observed  that that Alternating Optimization adopting MMED-Precoding scheme needs more iterations than MSER-Precoding scheme. Moreover, we find that the average number of iterations required for the Alternating Optimization increases as N and N_t increase. This is because  the number of variables and the number of constraints in the optimization problem increase as N and N_t increase.













 §.§ Comparison with Existing Algorithms

To make comparisons with other existing algorithms, we introduce them firstly. 
By using the exponential upper bound of Q-function, another SER upper bound can be given as 

    P_S( Φ ,F)    ≤1/M^N_s∑_i = 1^M^N_s∑_j = 1,j  i^M^N_sexp(-ρd_ij^2( Φ ,F)/4σ^2).

Since the RIS adds new supplementary links to maintain the communication link, the overall system performance can be increased by the in-direct multiple-path without additional interference leading to low required SNR. For this communication situation, the exponential function can be expanded by Taylor expansion e^x=∑_n=0^∞x^n/n !. By only taking the first two terms as exp(-ρd_ij^2( Φ ,F)/4σ^2) ≈ 1-ρd_ij^2( Φ ,F)/4σ^2, (<ref>) becomes P_S( Φ ,F) ≤ -ρ/4M^N_sσ^2∑_i = 1^M^N_s∑_j = 1,j  i^M^N_sd_ij^2( Φ ,F)+M^N_s-1. Based on the given F, the objective function of the optimization problem can be changed for maximizing ∑_i = 1^M^N_s∑_j = 1,j  i^M^N_sd_ij^2( ϕ). Based on (<ref>), the optimization problem can be written as

    (P5):    Given:     H_1, H_2, H_d, 𝒮_M, F
    Find:     ϕ
    Maximize:     ϕ^H Γϕ+γ^Hϕ+ϕ^Hγ
    Subject to:     | ϕ _i| = 1, ∀ i = 1,...,N,

where Γ =  R_H2⊙(∑_i = 1^M^N_s∑_j = 1,j  i^M^N_sΔQ_ij^T) and γ =∑_i = 1^M^N_s∑_j = 1,j  i^M^N_sa_ij^C. This problem can be treated as a homogeneous QCQP, which is similar to the (P3) shown in <cit.> and can be solved in the way shown in <cit.>. We named the solution as SDP-Reflecting, whose computational complexity can be analyzed to be 𝒞_SDP
= 𝒪[N^4+M^2N_sN(N^2N_tN_r+N_tN_s)] <cit.>. 


In the following, we compare the system SER of proposed reflecting schemes with SDP-Reflecting in (3, 2, 4, 2, 4, 3) system. As can be seen from Fig. <ref>, eMSER-Reflecting and vMSER-Reflecting can achieve much better performance than SDP-Reflecting. SDP-Reflecting is about 2-3 dB worse than the other two algorithms in the high SNR regime since the exponential approximation applied in (<ref>) is close to the original Q-function only in the low SNR regime. As a result, the performance gap increases with the increasing SNR in the depicted SNR regime. It is seen that the proposed two reflecting schemes provide apparent favorable SER performance compared to the one with random reflecting. This implies that the phase shift controller can have a significant influence on the system SER. Moreover, it shows that the union bound on SER in (<ref>) is quite tight, which implies that the proposed designs minimizing the upper bound of SER will directly minimize the real SER.




Most RIS-assisted literature optimized precoder based on received signal power, SNR, EE without considering the input vectors. Based on the fact, this problem can be transformed into maximizing the received signal power subject to the maximum transmit power constraint at the transmitter, which can be formulated as

    (P6):    Given:     H, 𝒮_M, Φ
    Find:     F
    Maximize:     HF_F^2
    Subject to:     tr( Q) ≤P_max.


Since F is full-rank matrix, we can easily obtain the solution as F = √(P_max/N_s)W, where the k-th column of W is the k-th eigenvector of H^HH corresponding to k-th eigenvalue with λ_k ≥λ_k+1. We named this solution as Eigen-Precoding in the following comparison. The computational complexity of the Eigen-Precoding scheme is consumed by computing H^HH and its eigenvectors, which is 𝒞_EP=𝒪[N^2N_r+N_rNN_t+N_t^2N_r+N_t^3],
where N^2N_r+N_rNN_t is the computation complexity of H, N_t^2N_r is the computation complexity of H^HH and N_t^3 is the eigenvalue decomposition complexity. In the following, we compare the system SER of proposed precoding schemes with Eigen-Precoding in (3, 20, 3, 2, 4, 1) system. 




As can be seen from Fig. <ref>, all the proposed schemes can provide lower SER than random precoding. We can see that the proposed MSER-Precoding scheme has a slightly better performance than MMED-Precoding scheme. As analyzed in Section V, the MMED-Precoding is of lower complexity than MSER-Precoding. Thus, in realistic systems, MMED-Precoding is a more appealing scheme compared to MSER-Precoding. Eigen-Precoding is much faster than two other schemes but much worse performance. It is obvious that a big performance gap occurs between the Eigen-Precoding and the other two schemes under the given system setup. Although Eigen-Precoding has significant lower computation complexity, it is about 4-5 dB worse than MSER-Precoding and MMER-Precoding. It is validated that the algorithm designed with the Gaussian input assumption should be reconsidered when applied to systems with finite alphabet input. By comparing the proposed schemes with the classical zero-forcing (ZF) precoding, we observe that the proposed schemes can bring significant performance gain in SER. The reason behind the phenomenon is that our proposed precoding schemes aim to minimize the SER directly, while ZF precoding is designed to null the interference. Compared to the performance of random precoder, we can see the importance of well-designed precoding schemes. 



By processing the SDP-Reflecting and Eigen-Precoding alternatively in (2, 5, 3, 2, 4, 2) system, we show the simulation results in Fig. <ref>. Based on the results of Figs. 3-4, there is no doubt that SDP-Eigen would achieve worse performance. Since the SDP-Reflecting maximizing the approximation in high SNR regime and Eigen-Precoding maximizing the SNR is not equivalent to maximizing the Euclidean distances which directly affects the SER. That is why the performance of SDP-Eigen is not comparable to proposed schemes, which aim to minimize SER directly. Even with Alternating Optimization, the system SERs with Eigen-Precoding is not favorable since it does not take inputs into consideration. It proves that the algorithm designed for continues or only finite inputs cannot replace the optimization that needs to be done for limited discrete data. The influence of input symbols taken from finite and discrete constellation should be taken into consideration. 



 §.§ Impact of the Number of Reflecting Elements


In Fig. <ref>, we show the influence of N on the phase shift design, precoding design and joint design in (3, 100, 3, 2, 2, 2) and (3, 120, 3, 2, 2, 2) system. From Fig. <ref>, we can see that phase shift design will play a more important role than precoding design on affecting SER when N is large. There is no obvious SER gain with a larger N if the reflecting is not carefully designed. Obviously, in Fig. <ref>, the joint design can gain optimal performance at the cost of more iteration to gain a better solution. There is 1 dB SER gain with 20 more reflecting elements if both the reflecting and precoding are optimized. In conclusion, the system SER can be significantly decreased by equipping more reflecting elements at the RIS and using the proposed reflecting design. N will not provide any SER gain without phase shift design. 



 §.§ Superiority in Presence of CSI Estimation Errors

Although we were assuming all involved CSI is perfectly known, the channel estimation for the RIS-assisted system is challenging because of its massive number of passive elements without any signal processing capability. The authors in <cit.> proposed a practical transmission protocol based on the received pilot signals from the user to execute channel estimation. The authors in <cit.> presented a two-stage algorithm that includes the sparse matrix factorization stage and the matrix completion stage to do channel estimation, which was shown to be quite accurate. However, it is still an ideal assumption to have perfect and timely channel information in practical application. Based on the fact, the performance of proposed schemes in the presence of channel estimation errors should be probed into.

To show the superiority of the proposed schemes in the presence of CSI estimation errors, we further evaluate the error performance of our schemes with imperfect CSI. SDP-Eigen scheme and vMSER-MMED scheme are chosen as examples. Such imperfection originates from channel estimation and/or feedback errors. The model of imperfect CSI is given by H' = H+H_e, where H_e is the error matrix whose entries follow and i.i.d circularly symmetric complex Gaussian distribution 𝒞𝒩( 0,σ_e^2 ). The error performance of the joint SDP-Eigen scheme with different error levels σ_e^2 = 0 (i.e., perfect CSI), σ_e^2= 0.1, σ_e^2 = 0.3 in (4, 4, 3, 2, 2, 1) system is compared in the Fig. <ref>. As expected, the error performance degrades with imperfect CSI. The SDP-Eigen scheme and vMSER-MMED scheme with imperfect CSI will achieve a floor as SNR grows. It can also be seen that proposed schemes with imperfect CSI still achieve an effective system SER optimization compared to a random joint design with perfect CSI. For example, the SDP-Eigen scheme achieves about 2-3 dB SER gain for σ_e^2 = 0.1 while vMSER-MMED scheme achieves about 10-15 dB SER gain for σ_e^2 = 0.1 and about 5-6 dB SER gain for σ_e^2 = 0.3 compared to random design with perfect CSI, which demonstrates the superiority of our design. Furthermore, vMSER-MMED scheme achieves about 2-3 dB SER gain for σ_e^2 = 0.3 in low SNR regime and about 6-7 dB SER gain for σ_e^2 = 0.1 compared with SDP-Eigen scheme with perfect CSI. This means vMSER-MMED scheme can achieve greatly better performance even with imperfect CSI since the Eigen-Precoding could not achieve a favorable performance with multiple discrete input data streams.



 §.§ Comparison with AF Relay Cooperating System

In this subsection, we compare a   (3, 5, 3, 2, 4, 1) RIS-assisted system with a full-duplex AF relaying system . The reason why using RIS is because of its low complexity and low cost compared to the relay-based system <cit.>. Relay stations equipped with active electronic components to receive and forward with a dedicated power source. The deployment of relays is costly and power-consuming, especially for realizing multiple-antenna designs. 


For fair comparison in performance,  we assume there is a conventional N-antenna AF relay in the place of the RIS structure under same channel realizations and input data streams, and the received signal at the destination can be expressed as y_D =  √(ρ)(H_2 VH_1 +H_d)Fs + H_2 Vn_R +n_D, where n_R ∼𝒞𝒩( 0,σ_R^2I_N_r), n_D ∼𝒞𝒩( 0,σ_D^2I_N_r), V is the diagonal AF matrix and n_R is the additive noise at the AF relay. We jointly optimize V and F alternatively. The optimization for precoder and AF matrix is similar to the optimization for precoder of RIS-assisted system with power constraint. As can be seen from Fig. <ref>, AF relaying system can achieve much lower SER compared to the same N RIS-assisted system when there is no noise at R. The reason is that AF relay consumes additional power to forward received signals, while there is no power gain in RISs. However, we can also see that the performance gap between the RIS-assisted system and AF relaying systems decreases as the variance of noise at relay increasing. As AF-relaying introduces more noise power, the superiority of performance would eliminate. Moreover, the RIS-assisted system outperforms the AF-relaying system by equipping more reflecting elements. Another 10 reflecting elements can offer more 7-8 dB SER gain and achieve almost the same performance to the noise-free AF relaying system. It means the system performance can be greatly improved in a low cost and energy-saving way. 




§ CONCLUSION

In the paper, several joint reflecting and precoding schemes were proposed to minimize the SER in point-to-point RIS-assisted MIMO systems assuming finite alphabet input. The proposed reflecting and precoding schemes can achieve a favorable system performance with different computation complexity. A much lower complex algorithm based on SNR without considering the influence of input data streams can not be used to replace the finite discrete input system. Simulation results demonstrate that our proposed algorithm can significantly improve SER performance with different scheme combinations while reflecting schemes will play a more critical role than precoding schemes. The RIS-assisted system with proper reflecting and precoding design has good robustness with imperfect CSI. By equipping more reflecting elements, it can improve the system performance significantly and surpass the one of relaying systems with consuming additional power.

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