Cooperative User-Scheduling and Resource Allocation Optimization for Intelligent Reflecting Surface Enhanced LEO Satellite Communication

Meng Meng ,Bo Hu,* ,Shanzhi Chen ,Jianyin Zhang

1 State Key Lab of Networking and Switching Technology,Beijing University of Posts and Telecommunications,Beijing 100876,China

2 State Key Lab of Wireless Mobile Communication,China Academy of Telecommunication Technology,Beijing 100191,China

3 Research Institute of China Mobile,Beijing 100053,China

Abstract: Lower Earth Orbit (LEO) satellite becomes an important part of complementing terrestrial communication due to its lower orbital altitude and smaller propagation delay than Geostationary satellite.However,the LEO satellite communication system cannot meet the requirements of users when the satellite-terrestrial link is blocked by obstacles.To solve this problem,we introduce Intelligent reflect surface(IRS)for improving the achievable rate of terrestrial users in LEO satellite communication.We investigated joint IRS scheduling,user scheduling,power and bandwidth allocation (JIRPB) optimization algorithm for improving LEO satellite system throughput.The optimization problem of joint user scheduling and resource allocation is formulated as a non-convex optimization problem.To cope with this problem,the nonconvex optimization problem is divided into resource allocation optimization sub-problem and scheduling optimization sub-problem firstly.Second,we optimize the resource allocation sub-problem via alternating direction multiplier method (ADMM) and scheduling sub-problem via Lagrangian dual method repeatedly.Third,we prove that the proposed resource allocation algorithm based ADMM approaches sublinear convergence theoretically.Finally,we demonstrate that the proposed JIRPB optimization algorithm improves the LEO satellite communication system throughput.

Keywords: convex optimization;intelligent reflecting surface;LEO satellite communication;OFDM

I.INTRODUCTION

With the rapid growth of mobile users and wireless devices,traditional terrestrial networks cannot meet the demands of users.The future wireless communication network is expected to provide extremely high throughput,global and seamless coverage,ultra reliability,low latency and massive connectivity.As an important enabler of this grand target,satellite communication can provide continuous and expansive coverage for areas without terrestrial communication infrastructure[1].Low-earth-orbit(LEO)satellites have been researched extensively among all kinds of satellites running at different orbital altitudes,because of their shorter round-trip delay and lower path loss compared to the geostationary-earth-orbit (GEO) satellite [2].And compared to terrestrial communication,LEO satellite communication achieves wider coverage at a cheaper cost.However,the communication distance between LEO satellites and terrestrial users is still much larger than the communication distance between terrestrial base stations and terrestrial users,which gives give rise to higher path loss.For this reason,the high-power transmitters and high-sensitivity receivers are deployed for mitigating the influence of the path loss of satellite-terrestrial links with higher hardware costs and higher power consumption [3].However,the performance of LEO satellite communication system cannot meet the requirement of terrestrial users when satellite-terrestrial Line-of-Sight(LoS)link cannot meet.We consider IRS for improving the satellite-terrestrial Non-Line-of-Sight (NLoS)link when satellite-terrestrial link is blocked by obstacles.

Intelligent reflecting surface (IRS) has been concerned in the wireless communication field recently because of shaping wireless propagation and establishing a programmable radio environment capability of IRS [4].Therefore,we researched the literature to understand the use of IRS in LEO satellite communications.In order to make up for the lack of HAP-ground channel rank and reduce the power budget in the integrated network,IRS is introduced into the integrated network for suppressing co-channel interference and increase spatial degrees of freedom in[5].In[6],the authors study a multiple intelligent reflecting surface assisted integrated terrestrial satellite network in which IRSs are deployed for low channel gain users cooperatively.The joint beamforming and power allocation has been researched for uplink NOMA transmission in an IRS-assisted cognitive satellite and terrestrial network operating at millimeter wave frequency band in [7].The authors show that an IRS having the size of just two large billboards can improve the link budget of ground to LEO satellite links observably in [8].And a new architecture of IRS-aided LEO satellite communication has been proposed for studying cooperative passive beamforming design over LoS-dominant single-reflection and double-reflection channels in [9],where IRSs are deployed at both sides of the satellite and terrestrial in IRS-aided LEO satellite communication system.The improvement of LoS probability has been analyzed when the large-scale deployment of IRSs[10].However,introducing an IRS into LEO satellite communication system brings both opportunities and challenges for IRS scheduling and resource allocation for multiple users.

For this reason,we proposed joint IRS scheduling,user scheduling,power and bandwidth allocation(JIRPB)optimization algorithm for improving system throughput in IRS enhanced LEO satellite communication in this paper.We find the balance between the user/IRS scheduling and the resource allocation while meeting the minimum data rate requirements of individual terrestrial users.To this end,the IRS scheduling and user scheduling can be obtained through Lagrangian dual method,the bandwidth allocation and power allocation can be obtained via the alternating direction method of multipliers method (ADMM).The main contributions of this paper are summarized as following:

?Problem formulation.We formulate the JIRPB optimization problem for solving the system throughput with the minimum data rate requirements of individual terrestrial users,limited system bandwidth,limited transmit power and spectrum.For this reason,we can obtain the optimization scheme by JIRPB optimization algorithm to improve system throughput in IRS-enhanced LEO satellite communication system.

?Algorithm design.The JIRPB optimization problem is a non-convex optimization problem.We divide this optimization problem into two optimization sub-problem: scheduling optimization sub-problem and resource allocation optimization sub-problem firstly.Second,we optimizing power allocation variable and bandwidth allocation variable with fixed user scheduling variable and IRS-scheduling variable via ADMM.Third,we use the obtained resource allocation variables for optimizing user scheduling variable and IRSscheduling variable via Lagrangian dual method.Finally,we repeatedly optimize scheduling variables and resource allocation variables until meeting the feasible solution.

?Convergence analysis.We prove that the proposed ADMM algorithm approaches sublinear converge with the rate of 1/ktheoretically,kis the number of iterations.We also show that the Lagrangian maximization algorithm based on the ADMM obtains the primal residual convergence.

?Simulation results.Extensive simulations demonstrate the performance gain of the proposed JIRPB algorithm in IRS enhanced satellite communication system.We demonstrate that employing an IRS in LEO satellite communication system can substantially improve the system throughput.

The remainder of this paper is organized as follows.In Section II,we proposed the system model and channel model considered IRS enhanced satellite communication system.In Section III,we design the JIRPB optimization algorithm in IRS enhanced LEO satellite communication system.In Section IV,we evaluate the JIRPB algorithm performance.We conclude this paper in Section V finally.

Notations used in this paper are listed as follows.Throughout the paper,vectors are denoted in bold small letters and boldface capital are reserved for matrices.The operation‖·‖2means the Euclidean norm,(·)Tdenotes transpose,and [x]+means max{x,0}.A?Bdenotes the Kronecker product of two matricesAandB.diag(x)means the diagonal matrix of vectorx.x～CN(μ,σ2)means the circularly symmetric complex Gaussian distribution with meanμand varianceσ2.We summarize the adopted notations of this paper in Table 1.

Table 1.Notations for system parameters.

II.SYSTEM MODEL

We proposed the system model and channel model considered IRS enhanced LEO satellite communication system in this section.The rest of this section,resource allocation variables and IRS/user scheduling variables were introduced in IRS enhanced LEO satellite communication system.

2.1 System Setup

We consider a IRS enhanced satellite system serving multiple terrestrial users providing downlink communication,which consist of a LEO satellite station,a IRS and multiple terrestrial users in this paper,as shown in Figure 1.

Figure 1.The IRS enhanced LEO satellite communication model.

On the other hand,we assume that the satellite station beam has steering and alignment capability in IRS enhanced LEO satellite communication system.Therefore,the satellite station provide downlink service forNterrestrial users via IRS within the satellite coverage.We assume the satellite is a uniform planar array(UPA)withKr×Kctransmit antennas.And IRS is a uniform planar array(UPA)withKr×Kcpassivereflection units(PRUs).Each column/row of the UPA hasKr/KcPRUs with an equal spacing ofdr/dcmeters.And the PRUs can scatter signals with independent reflection coefficient which consists of a phase shift?Kr,Kc ∈[-π,π) and an amplitudea ∈[0,1].?Kr,Kc ∈[-π,π)denotes the phase shift at PRU andameans the fixed reflection loss of IRS.The amplitudeaand phase shift?Kr,Kccan control by IRS controller and we consider infinite resolution phase shifters as a performance upper bound of finite resolution phase shifters[11],[12].

The location of the LEO satellite,IRS and terrestrial users can be denote asLS(t)=[x(t),y(t),HS],LR(t)=[xR,0,HR] andLU(t)=[xn(t),yn(t),0] in time slottfor this IRS enhanced LEO satellite communication system.We assume the distance between the LEO satellite and IRSdSR(t) is invariant when time slottis small enough.In addition,the distance between the IRS and terrestrial usersis fixed in time slottfor this considered system.And we ignore that the signals power which are reflected by IRS two or more times.

Letxs,n(t) andxr,n(t) with E[|xs,n(t)|2]=1 and E[|xr,n(t)|2]=1 denote the signals transmitted by satellite and IRS for terrestrial users.Therefore,the received signals at IRS can be expressed by[13]

wherens,n(t) is the noise of terrestrial usernfrom LEO satellite to IRS in time slott.hSRdenotes the channel matrix from the satellite to the IRS.The received signals at terrestrial users can be written as

wherehRUis the channel matrix from the IRS to terrestrial users andnr,n(t) is the noise from the IRS to terrestrial usernin time slott.fDdenotes the maximum Doppler shift and.fis the carrier frequency,v(t)is the velocity of the satellite to terrestrial receiver,θSRdenotes the angle between the forward velocity of the satellite and the boresight from satellite to terrestrial receiver,andcdenotes the speed of light [13].Utilizing the Doppler property of the LEO satellite channels,we preform Doppler compensation for the received signal at terrestrial users [14].Therefore,the frequencies at terrestrial users are assumed to be perfectly synchronized in the following.

2.2 Channel Model

The channel model of the IRS enhanced LEO satellite communication system is given as following.In the considered system,OFDMA has been adopted to support multiple terrestrial users downlink communication[15].The transmitted wave can be focused on the terrestrial usernvia IRS for improving the received signal strength.The obstruction and reflection disappeared in the satellite-IRS downlink usually when we assume the altitude of IRS isHRand IRS is close to terrestrial users.The orbital altitude of the satelliteHSis fixed usually and it is much larger than the distance between terrestrial users and IRS.Therefore,terrestrial users in the far-field of the IRS and the satellite in the far-field of the IRS.We will use the far-field beamforming scheme for IRS in this paper.

We assume the total system bandwidthBmaxis segmented intoIorthogonal subcarriers with subcarriers spaceBi(t)in time slott.Therefore,the channel vector of the RIS enhanced satellite communication system between satellite and the IRS via subcarrieriin time slottas following[16]

whereγdenotes the channel power gain at distanced=1m.Theis the array response vector and given as following[17]

whereNrandNtdenote the number of receiver antenna and the number of transmitter antenna respectively.ar(θ,?)andat(θ,?)in(5)denote the array response vector of receiver and the array response vector of transmitter,which can be expression as follow

fcdenotes the carrier frequency,θSR(t)means the vertical angles of arrival and?SR(t)denotes the horizontal angles of arrival in time slottfrom the satellite to the RIS.We assume the angles of arrival is equal to the angles of departure from the satellite to the RIS in this paper.Therefore,and cos?SR(t)=The IRS deals with a signal with a carrier frequencyfcandB ?fcholds usually.Therefore,depends on the angle of arrival and is frequency-flat[16].

The distancedSR(t) between the satellite and RIS much larger than the distance of the terrestrial users,then,we assume thatθSR(t)and?SR(t)do not change in time slott.And we assume a far-field model at the RIS becausedSR(t)?(Krdr,Kcdc)in practice.The phase shiftdepends on the subcarrier and all subcarrier are introduce a non-uniform phase.

We introduce the Rician fading channel model for the RIS-user links.The channel vector between the IRS and the terrestrial usernin time slotton subcarrieriis given as following

θRU(t) and?RU(t) mean the vertical and horizontal angles of deviate from the IRS to terrestrial usernin time slott,sinθRU(t)=and cos?RU(t)=

The IRS reflection phase coefficient matrix Ψ(t)in time slottshown as following

The satellite-IRS-user channel for usernvia subcarrieriis given as following

And the LoS component and the scattering component are given as equation(10)and(11),respectively.

We obtain the lower bound of channel matrix by the trace operation in this paper.

Therefore,the channel gain from the satellite to the terrestrial usernvia subcarrieriin time slottcan be given as following

The scattering component of channel gaincan be express as(13).

We focus on the phase control based on the LoS components in this paper.Due to the randomness of fading channels,the proposed LoS-based design may lead to an outage which the data rate is larger than the capacity.The scattering componentcan be given as following by combining the equation (4),(8)and(11).

The phase terms are irrelevant to,so,we have～CN(0,KrKc) [18].Fromwe can get the scattering component follows

2.3 Bandwidth,Power Allocation and IRS Allocation Design

We adopted OFDMA for supporting multiple users concurrently.We introduce a binary variablevn,i(t).If subcarrieriis allocated to the terrestrial usernin time slott,vn,i(t)=1.Otherwise,vn,i(t)=0.Each terrestrial users can be associated with LEO satellite using one subcarrier at most for guaranteeing the orthogonality among terrestrial users.Therefore,the constraints ofvn,i(t)can be described as≤1,?n,?i.The power allocated to the terrestrial usernin time slottvia subcarrieriis denoted aspn,i(t),pn,i(t)≥0 and≤ Pmax.Pmaxdenotes the maximum transmission power.The bandwidth allocation to the terrestrial usernin time slottvia subcarrieriis denoted asBn,i(t),Bn,i(t)≥0 andWe assume the phase of the IRS reflection matrix is aligned for achieving a sizable reflection gain at IRS,which means one terrestrial user can be selected for implementing the phase scheduling in each time slot.The selected terrestrial usernin time slottis called the IRS-enhanced terrestrial user andsn(t)=1.Otherwise,sn(t)=0.We also haveTo maximize the LoS component as shown in(10),the phase shift at PRU(kr,kc)is set as following[17]

The phase control strategy above can maximize the passive beamforming gain of the IRS-enhanced terrestrial user at the IRS.We can see that the phase control strategy in(16)only depends on the locations of satellite and terrestrial users.Therefore,the phase control strategy reduces the signaling overhead of CSI obtainment significantly.It can be see that the phase term in the LoS component in (10) has not affect on the phase control strategy.The phase control at IRS effects all of the subcarrier concurrently,which is a flat frequency response.The upper bound of the terrestrial usernLoS component of channel gain via subcarrieriin time slottcan be rewritten as(17)which follows Minkowski inequality.

The beam pattern response from the IRS to the terrestrial userniswhen the terrestrial usern′is the IRS-enhanced user in time slott,whereWhenn′=n,the terrestrial usernis the IRS enhanced user in time slottandsn′(t)=1,the beamforming gain can be obtained fully.The beamforming depends on the arrive of angle withsn′(t)=1,?n′=n.Therefore,the channel gain of satellite-IRS-user link via subcarrieriin time slottis given as following

The channel model as shown in (18) can be apply to the reverse link directly because the channel model does not depend on the communication direction.

III.PROBLEM FORMULATION

We formulate the throughput maximization problem in IRS enhanced communication system in following.

3.1 Problem Formulation

The achievable rate of the terrestrial usernvia subcarrieriin time slottis given by(19).N0denotes the the noise power spectral density at the terrestrial user.

Therefore,the system throughput in time slottcan be given as(20).

The terrestrial user interference is neglected in throughput because we adopted OFDMA scheme.Then,the optimization problem about the throughput maximization can be formulated by (21) in next page.In (21),constraintC1defines the terrestrial user scheduling.C3defines the power allocation.C5defines the IRS scheduling.C2guarantees that each subcarrier can scheduled at most one terrestrial user in each time slot.C4limits the total transmit power of the satellite in each time slot.C6means the IRSenhanced terrestrial user has at most one in each time slot.C7guarantees that each bandwidth of subcarrier constrained by the subcarrier spacing.C8indicates that the sum of subcarrier is less than the total system bandwidth.The more fair utility metrics need to consider because the imbalance of resource allocation.Therefore,the max-min utility is adopted to maximize the minimum data rate.Max-min utility leads to more fairness by increasing the lower data rate of terrestrial users.Hence,we introducesRmin,nfor denoting the minimum required average data rate of the terrestrial userninC9,where theRn,i,LoS(t)denotes the achievable data rate of the terrestrial usernvia subcarrieriin time slott.

The optimization problemP1is a non-convex optimization problem because the non-convexity of object function.The feasible solution sets for the scheduling variablessn′(t) andvn,i(t) are coupled,which is a challenge for solving the optimization problem via an effective tool in convex optimization theory.In order to solve the non-convex problem,we use the block coordinate descent method to solve it.First,we optimize resource allocation variablespn,i(t) andBn,i(t)with fixed scheduling variablesvn,i(t) andsn′(t) via ADMM.Second,we optimize scheduling variablesvn,i(t) andsn′(t) with fixed resource allocation variablespn,i(t)andBn,i(t)via Lagrangian dual method.Finally,we repeatedly optimize scheduling variables and resource variables until reach the optimal solution.Detailed explanation will be introduced in the rest of section.

3.2 Optimize Resource Allocation Variables with Fixed Scheduling Variables

We aim to find the optimal power allocation and bandwidth allocation by optimizing multiple variables in this section.The function ofRn(t)is increasing withpn,i(t) andBn,i(t).Therefore,the constraintsC4andC8are active when getting the optimal solution,which can be rewritten as

We can transfer the constraintC4into an equality constraint when the variablesBn,i(t) andpn,i(t)achieve optimal.Hence,we can rewritten the optimal problemP1as following

wherex=(Bn,i(t)n∈N,i∈I,pn,i(t)n∈N,i∈I)T.

SinceRn,i,LoS(t) is concave subject toBn,i(t)andpn,i(t).The objective function in the optimal problemP2is concave because it is the sum of the concave functions and the linear functions.So,the optimal problemP2is convex optimization problem.

We can obtain the optimalP2via the primal-dual argument because the concave nature.Hence,we can solve the optimal problemP2by maximizing Lagrangian function and minimizing the dual function.Letm=(m1,m2,···,mN)be the multiplier vector consistent with the minimum data rate constraints inC9.Therefore,the Lagrangian functionL(x,m) of the optimal problemP2in the following form:

And we can maximize the Lagrangian and minimize the dual problem for solving the optimal problemP2as following.

The optimal solution ofP3because the couple of power allocation and bandwidth allocation in the closed expression.For this reason,we obtain the optimal solution ofP2via solvingP3alternatively in the primal-dual arguments.m-denotes the solution of Lagrangian multipliers in the previous iteration.x+is updates by maximizing Lagrangian function in Algorithm 1 and them+is updates by the solution of(27).The optimal solution ofmis obtained via the subgradient method which shown as

whereR(x+)is the vector ofRnand is obtained when the vectorx+all variables.Rmindenotes the vector[R1,min,···,Rn,min]T.κis the step-size.Therefore,the joint power allocation and bandwidth allocation algorithm is performing Algorithm 1 by solving the optimal problemP3.In Algorithm 1,?denotes the convergence tolerance and? ≥0,x-denotes the optimal solution obtained in the previous iteration.

We proposed the joint power allocation and bandwidth allocation algorithm in Algorithm 1.

As we known,the ADMM is an effective method for solving equality-constrained convex optimization problem.We notice that the Lagrangian maximization problem is an equality-constrained convex optimization problem.Therefore,we proposed the Lagrangian maximization algorithm based ADMM for obtaining the optimal solution of in this section.The power allocation and bandwidth allocation via maximizing the augmented Lagrangian about (26) firstly.Then,the multipliers of constrains(22)and(23)are updated with the updated variables.

LetmB,andmpdenote the multipliers of constrains (22) and (23),respectively.We definee=(mB,mp)T.Hence,the augmented Lagrangian of(26)satisfy

where the matrixAdenotes the coefficient matrix of variables in constraints(22)and(23).The vectorbdenotes the coefficient vector in right hand side of constraints(22)and(23).TheηLdenotes the penalty parameter.

The maximization of augmented Lagrangian and the update of multipliers can be achieved alternately according to the principle of ADMM.Therefore,the power allocation and bandwidth allocation variablesm(k+1)can be update via maximizing the augmented Lagrangian at the (k+1)-th iteration.The variables satisfy following

whereekis updated at thekth iteration.The root of(30) can be obtain via Newton method because (30)is a quadratic differentiable function.anddenote that vectors contain all the elements inx(k)exceptBn,i(t)andpn,i(t),respectively.

We update multipliers according to thex(k+1)andekafter updating variables shown in (30).r(k+1)has been introduced for denoting the primal residual in(k+1)-th iteration.

Therefore,the updated multipliere(k+1)in(k+1)iteration has been shown

xk+1andek+1are updating alternatively until that the stop standard is satisfied.The Lagrangian maximization algorithm based on the ADMM is achieved and the optimal solution ofP3is obtained when updating stop.The stop standard is adopted when the primal residual is small enough.

where?r ＞0 denotes the feasibility tolerance of(33).The optimization problemP3is a convex optimization problem,it is guaranteed that the primal solution can be achieved by solving the variables and Lagrange multipliers iteratively when the step size satisfies the infinite travel conditions.We proposed the Lagrangian maximization algorithm based on the ADMM in Algorithm 2.

3.3 Optimize Scheduling Variables Variables with Fixed Resource Allocation Variables

We obtain a non-convex optimization problem via fixed resource allocation variables.

The optimization problemP4is non-convex since the objective function and integer constraintsC1andC3.The feasible solution set ofsn′(t) andvn,i(t) is disjoint which is difficult for solving via convex optimization theory.Therefore,we introduce the auxiliary variableun,n′,i(t),un,n′,i(t)=1 if and only if bothsn′(t)=1 andvn,i(t)=1.The IRS scheduling variablesn′(t) and the subcarrier allocation variablevn,i(t) are relaxed to be a real between zero and one.For the sake of description,we rewrite the optimization as(35).

The transformed optimization problemP5(35)is a convex optimization problem with respect toun,n′,i(t),vn,i(t)andsn′(t).To solve the optimization problem,we use the Lagrangian dual method for solving subcarrier allocation and IRS scheduling.The Lagrangian function ofP5has been given as(36).

Then,the dual problem of the optimization problem(37)can be given by

Therefore,the Lagrangian dual method solving the dual problem iteratively.Specifically,the IRS scheduling and user scheduling optimization problem maximizes the Lagrangian function L via (U,V,S) for given Lagrangian multipliersζ.And the Lagrangian multiplier optimization problem minimizes the Lagrangian function L viaζfor given(U,V,S).

In order to obtain the optimal user scheduling and IRS scheduling,we find the derivative of Lagrangian function with respect toun,n′,i(t),vn,i(t)andsn′(t).

We can see that the Lagrangian function grows linearly withun,n′,i(t),vn,i(t)andsn′(t)for giving the derivative of Lagrangian function.Therefore,we have the root of Lagrangian function to maximize the optimization problem as following

Furthermore,the Lagrange multipliers will update by following:

whereκkdenotes the step size of dual variables ink-th iteration.We notice that Lagrange multipliersandwill remain unchanged becausevn,i(t) andsn′(t)update by (41)～(43).The variables and Lagrange multipliers are updated via (44)～(48) iteratively.The optimization problemP5is a convex optimization problem,it is guaranteed that the primal solution can be achieved by solving the variables and Lagrange multipliers iteratively when the step size satisfies the infinite travel conditions.

Therefore,the joint bandwidth allocation,power allocation,IRS scheduling and user scheduling algorithm is described in Algorithm 3.The convergence of JIRPB has been proved in Appendix A.

IV.SIMULATION RESULT

In this section,we evaluate the algorithm performance and the system throughput when using the proposed algorithm.Referring to the satellite station and satellite-terrestrial link parameter settings in [19],we set the simulation parameters in Table 2.

Table 2.Simulation parameters.

4.1 Algorithm Performance

In this simulation example,we aim to verify the algorithm optimality of the proposed ADMM-based Lagrangian maximization algorithm as illustrated in Figure 2.

Figure 2.The time complexity of the ADMM-based Lagrangian maximization algorithm for different penalty parameter.

The minimum data rate requirements of each terrestrial users set asRmin=5bps/Hz.Figure 2 shows thatrkcloses to zero with the increase of the number of iterations.It indicates that the Lagrangian maximization algorithm based on the ADMM attains the residual convergence,which follows with Theorem 2.Further,we have that the vectoresublinear convergence by (27) sincerkcloses to zero,which shows the ADMM-based Lagrangian maximization algorithm sublinear convergence.And we can see that the rate of convergence follows with Theorem 3.In addition,it can see that the time complexity of the Lagrangian maximization algorithm based on the ADMM is increasing with the decrease of penalty parameter.

Figure 3 shows that the convergence of the proposed JIRPB algorithm with the step-sizeκ=1e-6.It can see that the proposed JIRPB algorithm can converges quickly.We also can be see that the proposed JIRPB algorithm needs fewer iterations with smaller penalty parameter than larger penalty parameter.

Figure 3.The average system throughput of the ADMMbased Lagrangian maximization algorithm for different penalty parameter.

4.2 Performance Comparison

To the best of our knowledge,there is no method that has been proposed for solving the same problem in literatures.We introduce three baseline for comparing our proposed JIRPB algorithm.Baseline 1 is introduced for calculating system performance without IRS and resource allocation,which means that the throughput of terrestrial users in satellite-terrestrial communication system with NLoS link.Baseline 2 is introduced for calculating system performance with resource allocation and without IRS,which means that the throughput of terrestrial users in satellite-terrestrial communication system with NLoS link and resource allocation.Baseline 3 is introduced for calculating system performance with IRS enhanced and without resource allocation,which means that the throughput of IRS enhanced LEO satellite communication system without resource allocation.

We assume that user 1 is selected to be IRS scheduling user during the whole period in Figure 4,i.e.,s1(t)=1,?n.In this case,the location of users are unchanged,the location of IRS and satellite station remains the same.

Figure 4.Average system throughput (bps/Hz) versus the number of time slots.

Figure 4 depicts the average channel power gain versus the number of PRUs at the deployed IRS for the IRS enhanced LEO satellite communication system.We can observe that the channel power gain of IRS enhanced user increase with the increasing number of PRUs due to the passive beamforming gain.Moreover,the channel power gain of IRS assisted user enhance also with the increasing number of PRUs.Compared to the IRS enhanced user,the increase channel power gain of IRS assisted user are smaller.This is because the passive beamforming gain of IRS is magnified via the increasing the number of PRUs.Therefore,the channel power of IRS assisted user increases with the number of PRUs increases.

Figure 5 illustrates the average system throughput versus the number of available time slotsTfor the proposed scheme withPmax=80W,Rmin,n=5bps/Hz,andKr=Kc=150.We can observe that the system throughput is relatively stable for all the three schemes increase with increasingT.The throughput gain of the proposed scheme over baseline 3 is enlarged for a large number of time slots.

Figure 5.Average system throughput (bps/Hz) versus the number of time slots.

Figure 6 shows the average system throughput versus the different transmit powerPmaxwithRmin,n=5bps/Hz,andKr=Kc=150.We can observe that the system throughput of the three methods increase with the total transmit radiated power increase from the satellite.More importantly,the throughput gain of the proposed IRS-assisted optimization algorithm compared to baseline 3 enlarges with increasingPmax.In reality,because of the IRS enhanced passive beamforming gain and the resource allocation gain,the proposed IRS enhanced optimization algorithm can improve the system performance gain by developing the transmit power more efficiently.Additionally,it can be seen that the performance gain of IRS enhanced optimization algorithm forN=8 is higher than that ofN=5 andN=3 which demonstrates the effectiveness of the proposed IRS enhanced optimization algorithm in handling multiple users.

Figure 6.Average system throughput (bps/Hz) versus the transmit power.

Figure 7 shows the average system throughput versus the total transmit bandwidthBmaxwithRmin,n=5bps/Hz,andKr=Kc=150.We can observe that the system throughput of the three methods increase with the total transmit bandwidth increase from the satellite.More importantly,the throughput gain of the proposed IRS enhanced optimization algorithm compared to baseline 3 enlarges with increasingBmax.In reality,because of the IRS enhanced passive beamforming gain and the resource allocation gain,the proposed IRS enhanced optimization algorithm can improve the system performance gain by developing the transmit bandwidth more efficiently.

Figure 7.Average system throughput (bps/Hz) versus the total bandwidth.

Figure 8 shows the average system throughput versus the different number of usersNwithPmax=80W,Rmin,n=5bps/Hz,andKr=Kc=150.We can observe that the system throughput of the three methods increase with the total number of terrestrial users.It can be seen that the performance gain of IRS enhanced optimization algorithm is higher with the number of terrestrial users which demonstrates the effectiveness of the proposed IRS enhanced optimization algorithm in handling multiple users.

Figure 8.Average system throughput (bps/Hz) versus the different number of users.

V.CONCLUSION

This paper proposed the JIRPB optimization algorithm for improving the system throughput in IRS enhanced LEO satellite communication system.First,we formulate the JIRPB optimization problem for optimizating the system throughput under the minimum data rate requirements of terrestrial users,limited transmit bandwidth,and limited transmit power.Therefore,we can obtained the optimization scheme in IRS enhanced LEO satellite communication system through solving the non-convex optimization problem.Second,the non-convex optimization problem is divided into two optimization sub-problem.We optimize allocation variables with fixed scheduling variables via ADMM.Then,we use the obtained resource allocation variables for optimizing scheduling variables via Lagrangian dual method.The IRS-scheduling variable,user scheduling variable,bandwidth allocation variable and power allocation variable are optimized repeatedly until reach the optimal solution.Third,we theoretically prove that the proposed joint bandwidth allocation and power allocation algorithm approaches the global optimalitly with the rate of 1/k.Finally,the simulation results show that employing an IRS in LEO satellite communication systems can substantially improve the system throughput.

ACKNOWLEDGEMENT

This work was supported by the National Key R&D Program of China under Grant 2020YFB1807900,the National Natural Science Foundation of China(NSFC) under Grant 61931005,and Beijing University of Posts and Telecommunications-China Mobile Research Institute Joint Innovation Center.

APPENDIX

Theorem 1.The power and bandwidth allocation algorithm achieved a solution withconvergence rate when k →∞,and the gap of optimal solution in(24)less than. k is the number of iteration,ω=maxx∈B‖R(x)-Rmin‖2and B is the feasible set of x.

Proof.We introducex*(k+1)for the optimal solution of(24)in the(k+1)-th iteration.denotes the average of[x*(0),···,x*(k)].R(x)is concave and continuous withx.Therefore,ωis finite and we have the inequation(.49).

whereF*is the optimal value of(24),m*denotes the optimal solution of(26),m(0)denotes the initial multiplier vector in Algorithm 1 andm(k+1)denotes the obtained multiplier vector in Algorithm 1.

The‖m(k+1)‖2is bounded because of(24)is strict convexity.Therefore,withconvergence rate whenk →∞.It means that a optimal solution of (24) is obtained withconvergence rate whenk →∞.F*is obtained also with less thanof gap ask →∞.we can getx*(k)→→ask →∞fromx*(k)=(k+1)-and→∞ask →∞.For now,the proof of Theorem 1 is complete.

Theorem 2.The power and bandwidth allocation algorithm obtains the primal residual convergence and the optimal solution as k →∞.

Proof.The functionRn,i(t) is continuous and concave.And the functionRn,i(t)respects tox.We note thatL(x,m) is concave,differentiable.Therefore,L(x,m)is closed with(Bn,i(t),pn,i(t))Tbecause the twice-differentiable and strictly concave of the objective function.

Therefore,the saddle pointx*,e*of the augmented Lagrangian(x,e,m)satisfies

wherex*denotes the primal optimal solution of (26)ande*denotes the dual optimal solution of (26).L(x*,m)is the optimal value of(26),and we have

Therefore,the optimal condition inkth iteration can be shown

where?means the differential operator.We can get

via substitutinge(k)=e(k-1)+ηLr(k).Therefore,we have

Add(.54)and(.57),we can get

via substitutingAx*=b.By substitutinge(k)=e(k-1)+ηLr(k),we have

Φ(k)decreases in each iteration,and we have the following inequation by iterating(.60).

Andr(k)→0 whenk →∞,the primal residual convergence.We have

from (.54) and (.57).L(x(k),m)→L(x*,m) andAx(k)→bwhenk →∞.Then,r(k)→0 whenk →∞.For now,the proof of Theorem 2 is completed.

Theorem 3.The power and bandwidth allocation algorithm achieves the optimal solution of(26)with a sublinear convergence rate.

Proof.is decreasing with iterations as shown in (.60).Therefore,x(k)is bounded,and we have

(.60)can be rewritten as

Then,we have

e(∞)is introduced forx(k).We can see that thex(k)sublinear converges toe(∞).x(k)withe(∞)sublinear converges to the optimal solution of (26).Therefore,x(k)sublinear converges to the dual optimal solution for the strict concavity of (26).For now,x(k)converges to the optimal solution of (26) with sublinear rate.The proof of Theorem 3 is completed.