Channel Capacity Optimization Based on Riemannian Trust Region Algorithm in IRS-Aided Communication System

Jinzhi Liu,Dan Wang,Jiamin Liang,Zhiqiang Mei

College of Communication and Information Engineering,Chongqing University of Posts and Telecommunications,Chongqing 400065,China

*The corresponding author,email:81559234@qq.com

Abstract:Intelligent Reflecting Surface(IRS)can offer unprecedented channel capacity gains since it can reconfigure the signal propagation environment.We decide to maximize the channel capacity by jointly optimizing the transmit-power-constrained precoding matrix at the base station and the unit-modulusconstrained phase shift vector at the IRS in IRSassisted multi-user downlink communication.We first convert the resulting non-convex problem into an equivalent problem,then use the alternate optimization algorithm.While fixing the phase shift vector,we can obtain the optimal precoding matrix directly by adopting standard optimization packages.While fixing the precoding matrix,we propose the Riemannian Trust-Region(RTR)algorithm to solve this optimization problem.And the key of the RTR algorithm is the solution of the trust-region sub-problem.We first adopt the accurate solution based on Newton’s(ASNT)method to solve this sub-problem,which can obtain the global solution but cannot guarantee that the solution is optimal since the initial iteration point is difficult to choose.Then,we propose the Improved-Polyline(IPL)method,which can avoid the difficulty of the ASNT method and improve convergence speed and calculation efficiency.The numerical results show that the RTR algorithm has more significant performance gains and faster convergence speed compared with the existing approaches.

Keywords:intelligent reflecting surface(IRS);multiuser;MIMO;channel capacity;unit modulus constraint;Riemannian manifold optimization

I.INTRODUCTION

As the 5-th Generation(5G)wireless communication networks putting into commercialization,individuals are eager for faster data transmission speed.The key technologies of 5G,such as massive multiple-input multiple-output(MIMO)and millimeter wave(mm-Wave)communications,have made practical contributions to realize this function.However,additional high hardware costs,massive power/energy consumption,and the site selection of 5G base stations are the main hindrances to their implementation in practice[1–7].Therefore,to realize green and sustainable development in 5G and beyond wireless networks,research on finding both spectrum-efficient and energy-efficient techniques is vital for sustainable capacity growth[8–12].

To solve the challenges mentioned above,intelligent reflecting surface(IRS)is considered a promising green,high cost-effectiveness,and energy-efficient technique in the 5G and next-generation mobile communication system.IRS is an artificial surface composed of electromagnetic materials[3].Specifically,IRS can smartly tune the wireless propagation environment by integrating numerous low-cost passive reflecting elements on the surface[4].Furthermore,IRS possesses appealing advantages such as low profile and light-weight,which provide high flexibility for actual implementation.For instance,IRS can be easily installed/removed on/from the wall,ceiling,advertising panel,and even clothing[13–17][18].Moreover,IRS can provide additional control signal paths by optimizing its reflection coefficient,which can effectively eliminate unexpected channel interference between users[19].Motivated by the above merits,IRS has a bright prospect in future wireless networks.

Figure 1.IRS-aided multi-user MIMO communication system model.

Recently,there are several works on IRS reflection optimization in IRS-assisted wireless communication systems[7–12].Reference[7]utilized Semi-Definite Relaxation(SDR)method and alternate optimization technology to maximize the received power in IRSassisted multi-user multi-input single-output(MISO)communication system.However,the SDR method only can obtain approximate solutions with high complexity.In[10],the author adopted the gradient descent method and Sequential Fraction Programming(SFP)while designing resource allocation to maximize energy or spectrum efficiency at the same time.However,the unit modulus constraints caused by IRS were solved by a series of approximate sub-problems in the SFP method,which may lead to some errors and performance loss.The author in[11]studied the resource allocation design for secure communication in IRS-assisted communication,and adopted continuous convex approximation to enhance the security of the physical layer.In[12],the author proposed a complex circle manifold(CCM)method based on the gradient descent algorithm to solve the unit modulus constraints.However,the CCM method is a Riemannian first-order algorithm.When the solution is closer to the target value,the step size will be smaller,and the convergence speed will be slower,which causes multiple iterations.In summary,no method can maintain excellent performance while keeping low complexity and fast convergence speed in the IRS-assisted communication system.

Motivated by the above,we consider an IRS-aided multi-user MIMO communication system in a single cell,as shown in Figure 1 in this work.In this system,a multi-antenna base station(BS)serves multiantenna users with the help of IRS with discrete phase shifts[20,21].Therefore,it allows us to formulate the channel capacity maximization problem subject to the maximum transmit power constraint of BS and the unit modulus constraints of IRS.Specifically,we jointly optimize the transmit beamformer at the BS and the phase shifter at the IRS to maximize the channel capacity of all users.However,the formulated optimization problem is hard to address since optimization variables are highly coupled,and the unit modulus constraints are highly non-convex.

The contributions of this work are summarized as follows:

1)For the non-convex optimization problem,we convert the original problem into a more tractable problem by using the equivalent relationship between the channel capacity and the weighted minimum mean square error(WMMSE)[12].Then we use the alternate optimization algorithm to optimize the precoding matrix at the BS and the phase shifts vector of IRS one by one.

2)The resultant optimization problem becomes a convex optimization problem when fixing the phase shift vector.We can regard it as a second-order cone programming(SOCP)problem due to the power constraints.Then the standard optimization package is used to obtain the optimal precoding matrix[22].Given the precoding matrix,the unit modulus constraints of IRS can form a product manifold and embed it into the search space[23],and the constraint problem is reformulated into an unconstrained problem.Finally,we propose the Riemannian trust-region(RTR)algorithm to obtain the local optimal solution.And the solution of the trust-region sub-problem is the key to the RTR algorithm.We first adopt the accurate solution based on Newton(ASNT)method to solve the trust-region sub-problem,which can obtain the global solution.However,this method is difficult to select the initial iteration point and this point must be near the optimal point to ensure convergence.Thus,the ASNT method cannot guarantee that the solution is the global optimal solution.Then,we propose the Improved-Polyline(IPL)method,which can avoid the difficulty of initial value selection in the ASNTmethod and improve the convergence speed and calculation efficiency.

Table 1.Symbol description.

3)Numerical results show that the RTR with the IPL method converges faster than the RTR with the ASNT method.Moreover,it is shown that,compared with the existing benchmark schemes,the RTR algorithm has excellent performance gains while maintaining low complexity and fast convergence speed.Furthermore,numerical results also show that the IRSassisted communication system should make the best tradeoff between the number of reflecting elements of the IRS and the transmit power of the BS.

The symbol description is shown in Table 1 below.

II.SYSTEM MODEL AND PROBLEM FORMULATION

2.1 System Model

As shown in Figure 1,we investigate an IRS-aided multi-user MIMO communication system model in this paper.The IRS,composed of N reflecting elements,assiststhe downlink communication between the BS with M1multi-antennas K and M2users with multi-antennas.We only consider the path reflected by the IRS for the first time,and other paths are ignored owing to the impact of path loss.In addition,we assume that all channels involved are quasi-static flatfading channels and the BS can perfectly know the channel state information(CSI)of them1.Specifically,the IRS configuration in Figure 1 is the semi-passive structure2[24],where additional sensing devices are interlaced with IRS reflecting elements.Thus,the CSI can be obtained by utilizing the deep learning and compressive sensing tools in[25].

Then,the received baseband signalykat the userkcan be written as:

where HAI,k∈CM1×Ndenotes the baseband channel matrix between the BS and IRS;HIU,k∈CM2×Ndenotes the baseband channel matrix between the IRS and user;HAU,k∈CM1×M2denotes the baseband channel matrix between the BS and userk;wk∈CN（0,σ2I）is the additional white Gaussian noise at userk;φ=diag(αφ1,···,αφN)∈CN×Nis a diagonal matrix at the IRS,whereφn=ejθn,n=1,2,···,N,α∈(0,1]is the amplitude coefficient andθnrepresents the phase shift induced by then-th reflecting element.xk=Gksk∈CM1×1denotes the transmitted signal for userkat the BS where Gk∈CM1×qand sk∈Cq×1represents the linear precoding matrix and the information-bearing symbol data at userkrespectively,and q is the data stream of each user.Besides,skis independent,which should satisfyandwhenk/=j.

For ease of implementation,we consider a practical IRS model where we assume that the amplitude coefficientα=1,and the phase shiftθninduced by then-th reflecting element is discrete as in[3,13,26].Specifically,we letddenote the number of bits,then the number of phase shift levelsLps=2d,and assume thatθnis uniformly quantized in the interval[0,2π).Thus,the set of discrete phase-shift values at each reflecting element is given byθ∈F={0,Δθ,...,(Lps-1)Δθ},whereΔθ=2π/Lps[13].

2.2 Problem Formulation

Then,the channel capacity for userkis expressed as:

where Hk=HIU,kφHHAI,k+HHAU,kis the combined channel matrix.We have known that the transmit signal of the userkisxk=Gksk.Then,the sum of the transmitted signal power of all users should be less than or equal to the transmit power threshold:

where Pmax＞0 is the maximum transmit power.

Then,we have known thatφn=ejθnin the IRS diagonal matrixφand discrete phase shiftθnis hard to solve directly.Thus,we denote the collection of diagonal elements ofφbyζ=[ζ1,...,ζN]H,whereζn=ejθn,?n∈N.To make the targeted problem more tractable,the discrete phase shiftθnmay be relaxed to their continuous counterparts.Specifically,we assume that the phase-shift resolution is infinite,i.e.,2d?1,then directly quantize each of the obtained continuous phase shifts to its nearest discrete value inF[1,3].Thus,the discrete phase shift constraintθn∈Fcan be equivalent to the unit modulus constraint[13]as follows:

In summary,the channel capacity maximum problem can be written as:

whereωkdenotes weighted factor at the userk.Constraint(6)ensures that the BS transmit power of all users is kept below the maximum transmit power Pmax,while constraint(7)is the unit modulus constraint at the IRS.

III.CHANNEL CAPACITY MAXIMIZATION

3.1 Problem Reformulation

From the problem(P0),we can easily observe that the precoding matrix and the phase shift vector are coupled,and the constraint(7)is highly non-convex,so the problem is NP-hard.

To solve this problem(P0),we transform it into a more tractable way by exploiting the relationship between the mean square error(MSE)of the optimal decoding matrix and the channel capacity expression[14].For ease of implementation,we use the linear decoding matrix,and then the estimated signal vector?skis expressed as:

where DHk∈Cq×M2denotes the linear decoding matrix for the user.We have known that the transmitted signal and the noise are independent.Thus thek-th user’s MSE matrix can be given by

Then,by introducing a set of auxiliary matrices W=and setting a set of decoding matrices asD={Dk,?k∈K},we define the similar functionrk(W,D,G,ζ)as[15]:

Thus,we can build the relationship between the problem(P0)and this function in(10)by exploiting the relationship between the channel capacity and the meansquare error as[14,15].And the problem(P0)can be reformulated as:

We can observe that the problem(P1)introduces two optimization variables compared with the problem(P0).However,the problem(P1)is more tractable to solve.Since,for any given G orζ,when the auxiliary matrices W and the decoding matrices D are also fixed,rk(W,D,G,ζ)is a concave function for each set of optimization matrices.

In the following,we decide to solve the problem(P1)by adopting the alternate optimization(AO)algorithm to optimize each set of optimization matrices alternately.

We first consider the decoding matrix Dand the auxiliary matrixW by exploiting the AO algorithm since these two optimization variables are only related to the functionrk(W,D,G,ζ).Thus,we can derive the optimal solution for the decoding matrix Dkby fixing W,G,ζ.Specifically,we find the first-order derivative of the functionrk(W,D,G,ζ)with respect to Dkand make it equal to zero.The optimal decoding matrix can be given by:

Similarly,by fixing W,G,D,the optimal auxiliary matrix Wkcan be expressed as:

In the following,we concentrate on optimizing the precoding matrix and the phase shift vector.

3.2 Optimization with Respect to the Precoding MatricesG

In this subsection,we optimize the transmit precoding matrix Gwhile fixingW,D,ζ.Specifically,we substitute Ekinto the problem(P1)and ignore the constant terms.Thus,the precoding matrix optimization problem can be expressed as:

whereAk=ωkHHkDkWkDHkHk.

We can easily observe that the precoding matrix optimization in(14)is a convex optimization problem,and the constraint(15)is a second-order cone constraint.Therefore,this problem can be regarded as a second-order cone programming(SOCP)problem,and the existing standard optimization tool such as CVX[22]can be used to obtain the optimal solution of the transmit precoding matrix.

3.3 Optimization with Respect to the Phase Shift Vectorζ

Similarly,we substitute Ekinto the problem(P1)while fixingW,D,G.Then,by ignoring the constant terms,the phase shift vector optimization problem can be expressed as:

Then,we substitute the combined channel matrix Hk=HIU,kφHHAI,k+HHAU,kinto(18),the optimization problem in(18)can be reformulated as follows:

Note thatζis a collection vector of diagonal elements ofφ,then according to the matrix equation in[16],we have:

Similarly,we denote diagonal elements of the matrix V as,and arrive that Tr.Then,the optimization problem in(20)can be equivalently rewritten as:

where U=B⊙C.

Obviously,the unit modulus constraint in(19)is still the main obstacle to solve this optimization problem.Aiming at this non-convex constraint,we decide to use the Riemannian manifold optimization tool to embed the constraint into the search space[17].Then,this problem can be an unconstrained optimization problem based on the search space.We have known that the CCM method proposed in[12]is a first-order Riemannian manifold optimization method,which mainly uses the Riemannian gradient descent method.This method can obtain the global solution,but it can’t guarantee global optimal solution and convergences slowly.Therefore,we propose a second-order Riemannian manifold optimization method—Riemannian trust-region(RTR)algorithm.Specifically,the RTR algorithm uses second-order geometric shapes in the selected trust-region to avoid saddle points and obtains more accurate result while having low complexity and fast convergence speed.What’s more,the solution of the trust-region sub-problem to obtain search direction is the key of the RTR algorithm.ASNT method is first used to solve this sub-problem[27].However,this method is hard to choose the initial iterative point,and the point must be near the optimal point to guarantee convergence.Therefore,we propose the IPL method based on the accurate solution method in[27]and the piecewise linear interpolation method in[28].

Figure 2.Riemannian manifold geometric interpretation.

3.3.1 RTR Algorithm

First,the unit modulus constraint of the problem(22)can be regarded as the product of N complex circles.Then,we define a complex circle product manifoldMas shown in Figure 2:

Before algorithms are proposed,we first provide background knowledge about manifold optimization[23]:

1)Tangent space

According to Figure 2,we assume thatζ(i)is the current iteration point.For the product manifoldM,the tangent spaceTζ(i)Mis of great significance in determining the search direction of the algorithm.For any pointζ(i)∈M,the tangent space is composed of all tangent vectors passing through the point[23].Then,the tangent spaceTζ(i)Mat the pointζ(i)∈Mis as follows:

where z is the tangent vector at the pointζ(i);0Nis the N-dimensional zero vector.

2)Euclidean gradient and Riemannian gradient

In Euclidean space,the Euclidean gradient represents the direction in which the objective function changes fastest at that point[23].Then,at the pointζ(i)∈M,the Euclidean gradient Gradζ(i)fof the objective function in(22)is as follows:

Similar to the Euclidean gradient,the Riemannian gradient is the tangent direction where the objective function grows fastest.And the Riemannian gradient gradf(ζ(i))of the objective function is defined as a unique element in the tangent space,which satisfies the following conditions[23]:

3)Retraction

As shown in Figure 2,moving along the tangent vector direction of the pointζ(i)∈Mto find the target point on the manifoldMis defined as retraction,which is the mapping from the tangent space to the manifold[23].Riemannian exponential retraction is the most commonly used contraction in Riemannian manifold optimization,but its computational cost is high.Therefore,we decide to use second-order retraction replacing exponential retraction,because it not only has lower computational cost,but also is the approximation of exponential mapping[27].Therefore,the retractionRζ(i)(ηi)at the pointζ(i)∈Mis defined as:

4)Riemannian-Hessian

The Riemannian-Hessian is the covariant derivative of the gradient vector field with respect to the Levi-Civita connectionζ(i).Specifically,The Riemannian-Hessian Hessf（ζ(i)）at the pointζ(i)is a linear mapping from the tangent spaceTζ(i)Mto itself,which is expressed as:

Equation(29)can be then calculated by the directional derivative of the Riemannian gradient and Euclidean gradient,as shown below:

In the following,we proposed the RTR algorithm to solve the unconstraint optimization problem in(22).The RTR algorithm is similar to the trust-region algorithm in Euclidean space[28].The basic idea of the trust-region algorithm is as follows:First,the given trust region radius is as the upper bound of the displacement length.Then,a closed sphere called trust region is determined with the current iteration point as the center and the upper bound as the radius.Next,the key step of the RTR algorithm is to approximate the objective function with a simple function and construct it as a trust-region sub-problem.We decide to address this sub-problem by adopting the ASNT method or the IPL method to obtain the trial step(search direction).Finally,we use an evaluation function to decide whether to accept the trial step and determine the trust-region radius for the next iteration.

Specifically,we first use retraction to promote the objective functionon the manifoldMto the cost functionon the tangent spa ceTζ(i)M.Since the tangent space is Euclidean spa ce,we can define a qua dratic model of(ζi),and use the ASNT method and IPL method on the Euclidean space to calculate the extreme value of the trust-region around the point 0ζi∈TζiM.The choice around the point 0ζiis based on the retraction,we have(0ζi)=f(ζi)[27].Then,the extreme value is retracted from the tangent space to the point on the manifold,and this point is used as a candidate for the next iteration.

Currently,we first need to construct a trust-region sub-problem.

In thei-th iteration,it is sufficient to use the quadratic approximation to define a second-order Taylor expansion of the objective functionf（ζ(i)）on the manifold to define a quadratic model[23].Then,the trust-region sub-problem can be given by the quadratic model at the pointζ(i)∈M:

whereηi∈Tζ(i)Mis the search direction of the current iteration;Δiis the trust-region radius of the current iteration.Then,whether to adjust the trust-region radiusΔiand update the next iteration pointζ(i+1)is determined by the following evaluation function[23]:

We can see that the definition ofρiis the ratio of th（e actual de）cline of the objective functionfRζ(i)(ηi)to the predicted decline of the quadratic model functionmζ(i)(0)-mζ(i)(ηi)in thei-th iteration.Ifρi<1/4,this indicates that the quadratic model is bad and we need to reduce the trust-region radius and recalculate the search directionηi.Ifρi→1,this indicates that the quadratic model is good and the objective function has a good approximation,thus the trust-region radius should be expanded.

The critical steps of the Riemann trust region(RTR)algorithm are summarized in Algorithm 1.

Lemm（a 1.）[29]ζ(*i)is the m（inim）um of f（ζ(i)）,（assu）methat fζ(i)∈C2,gradfζ(i)=0,Hessfζ(i)ispositive defnite,then the sequence generated by theRTR algorithm will converge to ζ*(i),with a secondorder convergence rate.

Algorithm 1.RTR algorithm.Require:maximum trust-re trust-region radiusΔ0 threshold,initial iteratio gorithm iteration stop th gion radiusˉΔ＞0,initial∈（0,ˉΔ）,step acceptance n point ζ(0)∈M and alreshold ε,i=0.1:while■■gradf（ζ(i)）■■2＞ε do 2:Solve the trust-region sub-problem in(32)to obtain the search direction ηi 3:Calculate ρi in(34)4:Adjust the trust-region radius:min（2Δi,ˉΔ）,ρζi＞3 4and‖ηi‖=1 Δi+1=■■■■■■■■ ■■■■■■■4Δi,ρζi<1 4 Δi,others 5:Calculate the next iteration point ζ(i+1)6:if ρi＞ρ′then 7: ζ(i+1)=Rζ(i)(ηi)8:else 9: ζ(i+1)=ζ(i)10:end if 11:i←i+1 12:end while 13:return current iteration point ζ(i)Δi

Thus,we can observe that the RTR algorithm has second-order convergence locally.

Recall that the solution of the trust-region subproblem in step 2 is the crucial difficulty of the RTR algorithm.In the following,we propose two efficient methods to solve it.

3.3.2 ASNT Method

ASNT is the accurate solution method based on Newton’s method.First,we introduce the following lemma proposed in[28]

Lemma 2.η*i is the solution of the trust-region subproblem in(32),if and only if the inequality μ*i≥0exists,which makes:

where Hessf(ζi)+μ*i I is a positive semi-defnite matrix.

According to Lemma 1,we can observe that the main idea of the accurate solution method is to solve the following equations:

Whenμi=0,the solution of trust-region sub-problem isη*i=-Hessf（ζ(i)）-1gradf（ζ(i)）;whenμi＞0,the solution of trust-region sub-problem is obtained by solving the equalityto getμ*iand substituting it into the first equation of the equations in(36).

Therefore,whenμi＞0,we define the function as follows:

We can easily see that the above functions(μi)is a monotonically increasing concave function.Then we decide to use Newton’s method in[28]to find the approximate rootμ*iofs(μi)=0,and then we can get the solution of the trust-region sub-problem as follows:

The key steps of ASNT method are summarized in Algorithm 2.When the algorithm converges,we can obtain the optimal solutionμ*iand substitute it into(38)to obtain the optimal solution of the trust-region subproblemη*i.

Algorithm 2.ASNT method.Require:Hessf（ζ(i)）,initial iteration point μmi=μ0,first-order derivative ψ(μi)=s′(μi),and iteration stop threshold ε.1:calculate η*i 2:for m=0,1,2,···do 3:μm+1）,gradf（ζ(i)i =μmi-s(μmi)ψ(μmi)4:if μm+1i -μmi≤ε then 5: μ*i=μm+1i 6: calculate η*i in(38)7: break;8:else 9: m←m+1 10:end if 11:end for

3.3.3 IPL Method

As the number of IRS reflecting elements N increases,the problem scale becomes bigger.We have known that the ASNT method is not suitable for large-scale problems because the initial iteration point must be controlled near the best point to ensure convergence[28].Therefore,we propose the IPL method to handle the trust-region sub-problem,which can avoid the difficulty of initial iteration point selection in the ASNT method and improve the calculation efficiency and accuracy.

We define the function by the optimal solutionη*iof the ASNT method in(38):

whereμi≥0.

Recall the constraint of the trust-region sub-problem in(33),we decide to solve the equationg(μi)-Δi=0 to get the optimal solution whenμi＞0 and then obtain the optimal solution of this sub-problemη*i.However,g(μi)-Δi=0 is a complex non-linear equation,which is difficult to handle in general.Thus,we adopted the following steps to simplify it.

First,we readily observe thatg(μi)is a continuous monotonic decreasing function.Therefore,for any given trust-region radiusΔi,ifΔi≥,theng(μi)<0 for anyμi≥0,thus we letμi=0 to obtain the optimal solutionη*i;ifΔi≤,we letμi=0 and haveg(0)-Δi＞0,and there must be a real numberμMletg(μM)-Δi<0.According to the zero-point theorem in[30],the rooted interval ofg(μi)-Δi=0 is[0,μM].Then we divide the rooted interval[0,μM]into M equal parts,and adopt the linear interpolation method in[30],which can convert the original complex non-linear equationg(μi)-Δi=0 into a simple linear equation to obtain the root in each cell[μm-1,μm],m=1,...,M,and then getη*i.

The key steps of the IPL method are summarized in Algorithm 3.

3.4 Alternate Algorithm to Solve Problem(P1)

Based on the above optimization analysis of the precoding matrix and the phase shift vector,the key steps to solve the problem(P1)are summarized in Algorithm 4.

3.5 Complexity Analysis

We now analyze the complexity of the AO algorithm.The complexity of calculating the decoding matrix D(i)in step 2 isO（KM32）.The complexity of calculating the auxiliary matrix W(i)in step 3 isO（Kq3）.And in step 4,the complexity of calculating the precoding matrix G(i+1)by using the second-order cone programming in CVX is given byO（KM31）.When utilizing the RTR algorithm to obtain the phase shift vector in step 5,the complexity of calculating projection and retraction are both linear complexity[12],and the complexity mainly depends on the calculation of the Riemannian Hessian while solving the trust-region sub-problem by adopting the ASNT or IPL method,which is given by O（M1N2+N3）.We use Titto represent the total number of iterations of the RTR algorithm,then the complexity of the RTR algorithm isO（Tit（N3+M1N2+N））.

Therefore,the total complexity of the alternate optimization algorithm in Algorithm 4 is:

Algorithm 3.IPL method.Require:Hessf（ζ(i)）,Δi,gradf（ζ(i)）.1:Select appropriate μM randomly,and satisfy g(μM)<Δi 2:Divide the rooted interval[0,μM]into M equal parts,and denote the step size as τ=μm M,m=0,1,...,M.Calculate the node μm=mτ and the function ym at this node,and mark the point(μm,ym)as the interpolation node.3:Utilize the linear interpolation method to construct M straight lines.Then the straight-line equation between two adjacent points μm-1 and μm is as follows:lm(μ)=ym-ym-1 μm-μm-1·μ,m=1,...,M 4:calculate η*i 5:for m=0,1,...,M do 6:if y0≤Δi then 7: η*i=-（Hessf（ζ(i)））-1gradf（ζ(i)）8:else if ym≤Δi≤ym+1 then 9: use the straight line lm+1(μ)to replace g(μi)and solve the equation lm+1(μ)-Δi=0 to get μ*i 10: η*i=-（Hessf（ζ(i)）+μ*i I）-1gradf（ζ(i)）11:else 12: m←m+1 13:end if 14:end for 15:return the solution of the trust-region sub-Problem η*i

In addition,the CCM algorithm proposed in[12],which is compared with the RTR algorithm proposed in this paper,has a computational complexity ofO（TCCMN2+N3）,where TCCMrepresents the total number of iterations of the CCM algorithm.The numerical results in the next section will compare their convergence speed.

IV.NUMERICAL RESULTS

Algorithm 4.AO algorithm.Require:the current number of iterations i=0,the maximum number of iterations imax,the initial precoding matrix G(0)and the initial phase shift vector ζ(0),the error tolerance value ε,the value of the objective function in the problem(P0)1:Given G(i)and ζ(i),calculate the optimal decoding matrix D(i)in(14)2:Given G(i),ζ(i),D(i),calculate the optimal decoding matrix W(i)in(15)3:Given W(i),ζ(i)and D(i),use CVX tool to solve the problem(16)to obtain the optimal precoding matrix G(i+1)4:Given G(i),W(i)and D(i),calculate the optimal phase shift vector ζ(i+1)by solving problem(22)with RTR algorithm of ASNT method or IPL method 5:if|R(G(i+1),ζ(i+1))-R(G(i),ζ(i))|R(G(i),ζ(i)) <ε or i＞imax then 6:break;7:else 8:i←i+1 9:return step1;10:end if

In this section,we provide numerical results to prove that employing IRS can dramatically improve the channel capacity and the proposed algorithm has excellent performance.Refer to[7],we consider an IRSassisted downlink communication system where BS consists of M1=4 antennas,K=4 users consists of M2=2 antennas,the number of data streams is q=2,and the number of IRS reflecting elements is N=20.And we assume that all the involved channels obey the independent Rayleigh distribution and exhibit flat fading.The path loss model in dB is expressed as[12]:

where PL0is the path loss at the reference distanced0=1mand set to-30dB;αis the path loss exponent and set to 3 in our simulation results.Except for some specific parameters specified later,other parameters are summarized in Table 2 below.

According to Table 2,we have known the fixed positions of the BS and IRS,then assume that users are randomly distributed in a circle centered at(xu,0)with radius 10m,as shown in Figure 3.For ease ofimplementation,it is assumed that the user is located near the IRS.In step 2 of the RTR algorithm,if the ASNT method is used,the RTR algorithm is denoted as RTR-ASNT,and if the IPL method is used,the RTR algorithm is denoted as RTR-IPL.

Table 2.Simulation parameters.

Figure 3.IRS-aided MIMO multi-user scenario(top view).

4.1 Convergence of the AO Algorithm

We first investigate the convergence behavior of the AO algorithm.Figure 4 shows the channel capacity versus the number of iterations for the various number of reflecting elements,i.e.,for N=10,30,and 50.We can observe that as N increases,the convergence speed becomes slightly slower since more optimization variables are involved,and then more iterations are required for convergence.However,for different values of N,the proposed algorithms still converge within 150 iterations,which confirms the effectiveness of the proposed algorithms.Furthermore,both the RTR-ASNT and RTR-IPL algorithms in the AO algorithm are tested.We can also observe that the RTRASNT and RTR-IPL algorithm converge to almost the same value.Still,the RTR-IPL algorithm converges faster than the RTR-ASNT algorithm as the number of reflecting elements increases.This is expected because as N increases,the problem-scale becomes bigger,the ASNT method is not suitable for the largescale problem since the initial iteration point must be selected near the best point to ensure convergence,and the IPL method can avoid selecting the initial iteration point in the ASNT method.We can conclude that the optimization performance of the RTR-IPL algorithm is better than that of the RTR-ASNT algorithm.

Figure 4.Convergence behavior of the AO algorithm.

Therefore,in the following numerical results,we always use the IPL method to solve the trust-region subproblem when adopting the RTR algorithm.And we compare the following benchmark schemes with the RTR algorithm:

Figure 5.Comparison of convergence behavior between the RTR and CCM algorithm.

1)CCM algorithm:Use the CCM algorithm in[12]to solve the problem(22)at step 4 of the AO algorithm.

2)Random phase shift(RandPhase):We assumed that the phase shift of each reflecting element is independently and randomly generated from the setθn∈F={0,Δθ,...,(Lps-1)Δθ}.Then in the AO algorithm,we can skip step 5 directly.

3)Without-IRS:Set a system that does not employ IRS.

4.2 Comparison of Convergence Behavior between the RTR and CCM Algorithm

Figure 5 shows the convergence performance of the RTR algorithm at the first iteration of the AO algorithm when N=10 and the comparison of convergence behavior between the RTR and CCM algorithm.From Figure 5,we can see that under the previous settings,both the RTR algorithm and CCM algorithm can achieve convergence within 100 iterations.However,we can obviously observe that the convergence speed of the RTR algorithm enjoys a quadratic local convergence rate and converges significantly faster than that of the CCM algorithm,and the convergence channel capacity value of the RTR algorithms is larger than that of the CCM algorithm.We have known that the complexity of both is lower based on the complexity analysis in the previous section,thus we can conclude that the RTR algorithm possesses better optimization performance compared to the CCM algorithm from Figure 5.

Figure 6.The relationship between the channel capacity and IRS reflecting elements.

Figure 7.The relationship between the channel capacity and IRS’s location.

4.3 Impact of IRS Reflecting Elements

In Figure 6,we can see the relationship between the channel capacity values of all the schemes and the number of reflecting elements at the IRS.We can observe that,as the number of reflecting elements increases,the performance gains of the proposed algorithm become significant compared with the other three benchmark schemes.For example,compared with the Without-IRS case,the performance gain of the RTR algorithm is only 14.2%when N=20,but the performance gain is increased to 43.8% when N=70.We can see that as N increases,all With-IRS schemes have a higher channel capacity value than the Without-IRS scheme since IRS is passive and has no additional radio frequency(RF)chain and power consumption.

Therefore,we can conclude that the deployment of IRS in a wireless communication system is practical and feasible.

Figure 8.The relationship between the channel capacity and user’s location.

4.4 Impact of IRS’s Location

In Figure 7,we compare the influence of the location of the IRS fromxIRS=20m toxIRS=100m on the channel capacity.We can observe that the channel capacity values of all schemes decrease asxIRSincreases when 20

4.5 Impact of User’s Location

In Figure 8,we compare the relationship between the channel capacity and the user’s horizontal distancexu.It can be observed that,compared with the other three benchmark schemes,asxuincreases,when the user is far away from the BS and closer to the IRS,the performance gain of the RTR algorithm is significant.This is because users receive strong reflected signals from the IRS,which can reduce interference among users in the cell.

Figure 9.The relationship between the channel capacity and maximum transmit power.

4.6 Impact of Maximum Transmit Power

The relationship between channel capacity performance and maximum transmit power is plotted in Figure 9.We can see that when the maximum transmit power is too small to be practically realized,i.e.,if the transmit power from the BS to the user is insufficient to meet the user’s rate requirement,then the performance of all schemes corresponds to very low values from Figure 9.Then,as the maximum transmit power value increases,the slope of the curve will increase obviously,and the performance of the RTR algorithm is still the best.Therefore,it can be concluded by combining the analysis of Figure 6 and Figure 8 that the IRS-assisted communication system should make the best tradeoff between the number of reflecting elements of the IRS and the transmit power of the BS.

4.7 Impact of the Number of BS Antennas

Recall to Figure 8,we have already known that the RTR algorithm provides a higher channel capacity value than the benchmark scheme Without IRS by varying the user’s horizontal distancexu.However,we have never considered the scenario where the number of BS antennas exceeds(or equals)that of total data streams.In Figure 10,we compare the relationship between the channel capacity and the number of BS antennas when using an IRS with N=20.We can observe that,as compared to the Without-IRS scheme,the gap in performance gains of the proposed algorithm is getting smaller when the number of BS antennas M1becomes larger.This is because the direct channel between the BS and the user becomes stronger as the antennas become larger.However,additional RF chains and power amplifiers need to be deployed while increasing the number of antenna elements,which may lead to huge energy consumption.Therefore,we suggest that the large-scale passive IRS can be introduced into the future wireless system.

Figure 10.The impact of the number of BS antennas.

Figure 11.The impact of weights.

4.8 Impact of Weights

As mentioned in problem(P0),weights are used to control fairness among users.We assume that the locations of the four users are in the coordinates(30,0),(50,0),(70,0),and(90,0)respectively in Figure 3.Among them,the first user is closer to the BS than the second user,and the fourth user is closer to the IRS than the third user.Two sets of weight values were tested:1)ωk=0.25,?k;2)ω1=0.15,ω2=0.35,ω3=0.3,ω4=0.2.It can be found from Figure 11 that,when the weights are equal,the data rate of user 1 is the highest since user 1 is closest to the BS;the data rate of user 4 is slightly lower than that of user 1and higher than the other two,since user 4 is far from the BS but closest to the IRS,the channel gain of user 4 increases.In addition,the data rate difference between the four users is not significant when the weights are not equal.Therefore,in order to ensure fairness between users,the user with low channel gain should be assigned a higher weight to make the user’s data rate distribution more balanced.

V.CONCLUSION

We aim to maximize the channel capacity in the IRSassisted multi-user communication system in this paper.First,the resulting optimization problem is a nonconvex NP-hard problem since the optimization variables are highly coupled.Therefore,we convert the original problem into an easy-to-solve form.Then we use the alternate optimization algorithm to optimize the precoding matrix and the phase shift vector alternately.Under the unit modulus constraint,we proposed the RTR-ASNT algorithm and RTR-ILP algorithm,where the constrained problem becomes an unconstrained problem.Then the local optimal solution is obtained.Numerical results show that the proposed algorithms achieve significant performance gains compared to the benchmark algorithm while maintaining low complexity and fast convergence speed.What’s more,we can conclude that the deployment of IRS in a wireless communication system is effective and feasible.

ACKNOWLEDGEMENT

This work was supported by the General Program of Natural Science Foudation of Chongqing Province of China(cstc2021jcyj-msxmX0454).

NOTES

1The proposed algorithms in this paper are also applicable in the imperfect CSI case while utilizing the stochastic successive convex approximation technique in[26]

2In general,the acquirement of the CSI is a challenging problem.there are two main approaches for the IRS-involved channel acquisition,depending on whether the IRS elements possess any active RF chains or not,termed as semi-passive IRS and(fully)passive IRS,respectively[24].The first approach with active RF chains at the IRS,conventional channel estimation methods can be adopted for the IRS to estimate the channels of the BS-IRS and IRS-user links,respectively.In contrast,for the second approach without receive RF chains at the IRS,the IRS reflection patterns can be designed together with the uplink pilots to estimate the concatenated BS-IRS-user channels[31][32]