# Maximizing the Rate of IRS-Assisted Downlink NOMA Systems

Authors: Xinying Li; Haixia Wei; Haiyan Huang; Cuiran Li; Lijun Zhang
Journal: International Journal of Antennas and Propagation
 (2022)

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## Abstract

Nonorthogonal multiple access (NOMA) techniques and intelligent reflecting surface (IRS) are being explored as potential essential technologies for future wireless communications. Accordingly, this paper provides a network framework for IRS-aided downlink NOMA transmission, in which IRS is employed to improve the NOMA system’s transmission performance, and optimization problems are raised to maximize the achievable rate. Given the fractional structure of multivariate coupling as presented in this study, the fractional problem first converts to a linear form; then, the semidefinite relaxation (SDR) algorithm is proposed to address nonconvex issues for a single-user scenario. As for a multiuser scenario, the alternating optimization (AO) algorithm is raised based on transmit beamforming and reflection phase shift matrix to settle relevant issues and mitigate computational complexity. The simulation results suggest that the algorithm described in this paper can significantly increase the signal’s achievable rate compared to the nondeployed IRS and IRS random phase-shifting schemes.


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## Body

## 1. Introduction

In recent years, upgrading mobile communication systems have raised a higher demand for multiple access techniques [1]. Traditional cellular mobile communication systems mostly employ orthogonal multiple access (OMA) methods, which readily segregate the information conveyed by multiple user signals with little complexity. Nonetheless, the user amount supported by OMA is limited by the number of orthogonal resources available. Meanwhile, the orthogonality of the time-frequency code resources adopted in the system is frequently lost due to various impacts, such as time delay, frequency deviation, and Doppler Effect [2]. The nonorthogonal multiple access technique of new radio (NR) has been publicly discussed since the 3GPP RAN1#84bis meeting in April 2016, followed by various proposed schemes regarding NOMA techniques in recent years. Li A. Lan Y et al., for instance, offered a downlink NOMA scheme using successive interference cancellation (SIC) receivers and demonstrated in the paper that the scheme should enhance the capacity and cell edge user throughput, whose performance improvement is irrelevant to the availability of the frequency selective channel quality indicator (CQI) on the base station side [3, 4]. In order to improve large-scale connectivity, Mounchili and Hamouda investigated the minimal pairing distance to distinguish between far and near users; an analytical expression of the distance threshold on pairing was presented in the case of fixed power allocation [5]. During this time, researchers also proposed a slew of other advanced communication technologies, such as Terahertz (THz) communication, large-scale Multiple Input Multiple Output (MIMO) [6, 7], Ultradense Network (UDN) [8, 9], and so on, all of which greatly support the spectral efficiency of wireless communication systems. However, widespread adoption of these technologies necessitates high hardware costs and significant energy consumption, which has driven the wireless communication domain to pursue better data transmission speeds and spectral efficiency while imposing higher demands on the system’s energy efficiency [10]. Besides, the intelligent reflecting surface is perceived as a possible technical solution for further lowering wireless network costs with increasing energy efficiency [11–14].As communication technology is evolving, researchers have begun to integrate IRS with NOMA techniques to upgrade the performance of the communication system and are committed to relevant studies regarding the optimization of IRS-aided NOMA communications systems. Long et al. provided an overview of IRS, including the signal model, hardware architecture, and challenges of IRS application and deployment in 6G networks [15]. Wu et al. explain reflection and channel models, hardware architecture, and practical limitations of IRS-assisted wireless communication, as well as a variety of attractive applications in wireless networks. In addition, they highlight important directions worthy of further research in future work [13]. de Sena et al. summarized the main roles and key challenges of IRS in MIMO-NOMA systems and gave a comprehensive discussion of the main performance gains that can be achieved in IRS-assisted large-scale MIMO-NOMA networks and outlined the future use scenarios of IRS-NOMA systems. The main challenges and future research directions are also revealed [16]. Fang et al. studied an IRS-assisted downlink MISO-NOMA system to optimize system energy efficiency by jointly optimizing the transmitting beam shaping of BS and the reflected beam shaping of IRS [17]. Zhu et al. developed an IRS-aided MISO-NOMA system that effectively adjusts the wireless channel; it further minimizes the transmission power by optimizing the beamforming vector and the IRS phase shift matrix. Moreover, an enhanced quasi-degradation is proposed using IRS, ensuring that NOMA reaches the capacity region with a higher probability [18]. Cheng et al. studied a downlink NOMA system that expands base station coverage by increasing the number of IRS deployments and derives the diversity order pursuant to the asymptotic approximation in the range of high signal-to-noise ratio [19]. In response to dual-cell IRS-aided NOMA system with joint detection, Wang et al. put forward an uplink power minimization issue to realize joint optimization of the system’s power allocation and phase shift [20]. Ding and Vincent Poor highlighted the IRS-NOMA transmission scheme for a multiuser NOMA downlink scenario, assuming no direct link exists between the base station and cell-edge users. This scheme permits to serve more users in each direction of orthogonal space than the space division multiple access schemes. Also, the impact of hardware damage on IRS-NOMA is incorporated in their studies [21]. They also investigated the effect of two phase shift designs, coherent phase shift, and random discrete phase shift, on the performance of IRS-assisted NOMA [22]. In addition, Mu et al. studied ideal and nonideal RIS-assisted multiple input and multiple output scenarios and further proposed a new NOMA decoding order [23].IRS can be divided into active IRS and passive IRS. In the scenario where the direct link between the transmitter and receiver is strong, since the active IRS is equipped with an additional integrated active reflection-type amplifier, the reflective element has the function of amplifying the signal, which will strengthen the reflective link. However, with active components, the IRS needs to consume extra power to amplify the signal, and the thermal noise introduced by the active IRS cannot be ignored like the passive IRS. Passive IRS consume virtually zero direct-current power and can be used to send pilot signals and process baseband signals, whereas active IRS do not have this capability and can only reflect and amplify the signal to strengthen the reflected link [24]. So, this paper uses passive IRS auxiliary signal transmission.An antenna array has been formed by transmitting antennas due to the adoption of NOMA technology; in this case, regarding the devices used to achieve beamforming, such as equipping antennas with digital-to-analog converters, the cost of the system will increase. The intelligent reflecting surface (IRS) comprises numerous low-cost passive reflecting units, and it actively modifies the wireless communications environment to enhance network performance. In summary, associating IRS with NOMA techniques has become a principal research focus since both are expected to play active roles as potential essential technologies in future wireless communications. Yet, the rate optimization issue of the IRS-aided NOMA system has received little attention in academic research. In the case of bounded channel’s uncertainty, research on the algorithm of IRS-aided NOMA system’s rate maximization comes with theoretical significance. Therefore, this paper makes contributions to propose a network framework for IRS-aided NOMA downlink transmission. Multiple users can share the same resource block simultaneously; drawing upon IRS, it is viable to enhance the transmission performance of the NOMA system by adjusting the wireless communication channel. The problem of maximizing the achievable signal rate is proposed; meanwhile, the relationship between the power distribution coefficient and the reflection phase shift matrix is derived while meeting the limitations of maximum transmission power and reflection unit phase shift. Since the problem is involved in a fractional structure of multivariate coupling, the solving process is quite challenging. This study first reduces the complex fractional problem to a linear problem and proposes a semidefinite relaxation (SDR) technique based on maximum ratio transmission (MRT) to solve nonconvex problems for single-user scenarios. Then, in respect of multiuser scenarios, a minimum mean squared error (MMSE) estimator is applied to cope with multiuser interference between multiantenna access points. Besides, the alternating optimization algorithm is put forward based on transmit beamforming and the reflection phase shift matrix to reduce computational complexity. Furthermore, the IRS can only reflect the signal, while signal transmission suffers from significant path loss due to the sophisticated communications environment. On the other hand, given the simple installation of IRS, the free-space model is exploited to deploy the IRS in a position with a direct path to the base station. It also keeps a relatively close distance to the region to be enhanced, adjusts the phase of each reflecting unit on the reflective display panel, and reduces the path loss via the convergence function. The simulation findings show that the SDR and AO schemes presented in the paper can dramatically enhance the achievable rate of the IRS-aided NOMA downlink system, along with the rate value growing as the number of IRS reflection units increases.


## 2. System Model

As presented in Figure1, the proposed IRS aims to aid the NOMA system in completing downlink transmission, in which a base station (BS) with N number of antennas, an IRS with L number of passive reflection units, and M number of users (User, UE) are deployed. The channel response parameters of the link from the base station to the intelligent reflecting surface, from the intelligent reflecting surface to the user i, and from the base station to the user i are represented by H∈CL×N, gr,iH∈C1×L and hr,iH∈C1×N, respectively, where i∈1,2,...,M.Figure 1
Block diagram of the system model.The user set is described asM¯=1,2,...,M while the channel set is N¯=1,2,...,N. The total system bandwidth B is evenly allocated to a set of channels, and the bandwidth of each channel is presented as W=B/N. According to the power multiplexing techniques of NOMA, multiple users can be allocated on a subchannel to achieve frequency sharing. Supposing each user is assigned to one channel, tn denotes the number of users distributed on the n-th channel while Tn signifies the maximum number of users possible to be allocated on each channel. When Tn=1, the IRS-NOMA system is transformed into the IRS-OMA system. The IRS reflection coefficient matrix is indicated as Θ, Θ∈CL×L while Θ=diagA1ejθ1,A2ejθ2,...,ALejθL. In the formula, Alejθl represents the reflection coefficient of the l-th reflection unit, θl reflects the phase coefficient of the l-th reflection unit while θl∈0,2π, and Al represents the reflection coefficient of the l-th reflection unit while Al∈0,1. The following scenarios can be considered for IRS phase optimization: modulating the phase, modulating the amplitude, or modulating both simultaneously. A simple circuit features the choice of merely modulating the amplitude while its performance seems poor. Yet, modulating both phase and amplitude contributes to improved performance, while the hardware design can be highly complicated. Therefore, the available research can only modulate the stage, thereby achieving an ideal balance between performance and complexity [25]. It is assumed that each reflection unit on the IRS performs maximum reflection; that is, the amplitude coefficient of each reflection unit conforms to A1=A2=...=AL=1.For passive IRS, the reflection unit has no sensing element installed, so the IRS cannot receive pilot signals directly from BS and UE. The system adopts a full duplex (FD) mode, as shown in Figure2, each transmission frame (TL) is divided into two transmission stages. In the first transmission phase (TL1≪TL), UE sends pilot signals to BS through the uplink channel, and IRS changes the reflection coefficient in real time according to the preset reflection mode to reflect the pilot signals sent by UE to BS. BS receives a dual-link pilot signal that contains both a direct link and a reflected link. Based on these signals, BS estimates the direct link UE-BS and the reflected link UE-IRS-BS. Then, BS uses the estimated channel state information (CSI) to transmit the passive beam configuration parameters to IRS through the feedback link. The transmitted beam configuration of BS and the reflection phase shift matrix of IRS are jointly designed to realize data transmission. Assuming that all channels are quasi-static flat fading channels, due to the reciprocity of channels, CSI obtained by uplink pilot training can also be used for downlink data transmission. In the second transmission phase (TL2≜TL−TL1), IRS sets the reflection phase shift matrix according to the passive beam configuration parameters fed back by BS to assist in downlink data transmission between BS and UE.Figure 2
Transmission mode.The signal will have a large path loss in the process of multiple reflection transmission, assuming that the signal reflected by IRS two or more times is ignored. In addition, in order to facilitate the theoretical performance gain brought by IRS, it is assumed that BS knows all channel state information, which can be obtained by the method in reference [26].The signal received by the userUm is as follows:(1)ym=gr,mHΘH+hs,mH∑i=1Mwisi+em,∀m∈M¯,where wi represents the beamforming vector of the user i, si represents the signal transmitted by BS to Ui, and em denotes additive white Gaussian noise with a mean value of 0 and a variance of σ2 [27].As presented in Figure3, NOMA adopts the serial interference cancellation technique at the receiving end [28]. Specifically, the user U1 directly decodes the signal si after receiving the signals and treats the remaining signals as noise. When m≥2, the user detects the received signals individually; later, it restores the signal amplitude and detects the remaining signals after excluding the detected signals of other users from the received signal. This operation continues to repeat until the demanded signals are decoded.Figure 3
SIC schematic diagram.The sequence ofM number of users is hs,12≤hs,22≤...hs,M2, where hs,m2 signifies the channel quality between user Um and BS. Therefore, the signal of the user Um before decoding the signal can be denoted as follows:(2)y=m′gr,mHΘH+hs,mH∑i=mMwisi+em,∀m∈M¯.The signal-to-noise ratio of the userUm’s decoded signal sm is indicated as follows:(3)SINRm=gr,mHΘH+hs,mH⋅wm2∑i=m+1Mgr,mHΘH+hs,mH⋅wi2+σ2,∀m∈M¯.According to Shannon’s theorem [29], the achievable rate of the user Um receiving signal sm is presented as follows:(4)Rm=log21+SINRm=log21+gr,mHΘH+hs,mH⋅wm2∑i=m+1Mgr,mHΘH+hs,mH⋅wi2+σ2,∀m∈M¯.Therefore, the throughput of the system is expressed as follows:(5)Rsys=∑m=1Mlog21+SINRm=∑m=1Mlog21+gr,mHΘH+hs,mH⋅wm2∑i=m+1Mgr,mHΘH+hs,mH⋅wi2+σ2.


## 3. Scheme Design for Maximizing the Transmission Rate

When the transmit power meets the constraints, alternating optimization is conducted on the BS’s active transmit beamforming and the IRS’s passive reflection phase shift, allowing the NOMA system to reach the maximum rate during downlink transmission.


### 3.1. Problem Modeling

The optimization problem can be stated as follows:(6)P1:maxw,Θlog21+gr,mHΘH+hs,mH⋅wm2∑i=m+1Mgr,mHΘH+hs,mH⋅wi2+σ2s.t.0≤θl≤2π,l=1,2,...,L,∑m=1Mwm2≤Pmax,where Pmax refers to the maximum transmit power offered by BS.The solving process can be challenging since the objective function and constraints are nonconvex [30]. To simplify the calculation, the single-user scenario is initially considered, followed by a semidefinite relaxation technique to approximate the solution; afterward, extending this process to the multiuser scenario with an AO algorithm for problem solving.


### 3.2. Single-User System

We consider a single-user system, i.e.,M=1, in this case, the system is comparable to IRS-OMA. Besides, there is no mutual interference while only one user is in the system; hence, problem P1 can be reduced to the following:(7)P2:maxw,Θlog21+grHΘH+hsH⋅w2σ2s.t.0≤θl≤2π,l=1,2,...,L,w2≤Pmax.Regarding the analysis of the problemP2, when the signal-to-noise ratio takes the maximum value, the system’s achievable rate reaches the maximum accordingly; thus, the problem P2 is equivalent to the following:(8)P3:maxw,ΘgrHΘH+hsH⋅w2σ2s.t.0≤θl≤2π,l=1,2,...,L,w2≤Pmax.As variable coupling exists in the objective function ofP3, the problem P3 is regarded as nonconvex. The transmit beam at BS and the phase shift matrix at IRS are alternately optimized until convergence. In addition, w remains constant before solving Θ. The problem P3 can be simplified to as follows:(9)P4:maxθgrHΘH+hsH⋅w2σ2s.t.0≤θl≤2π,l=1,2,...,L.Letv˜=v1,v2,...,vLH, where vl=ejθl,∀l. Thus, the constraints in P4 can be equivalent to the unit modulus constraint: vl2=1,∀l. Using variables to replace: grH⋅Θ⋅H=v˜⋅Φ, Φ=diaggrH⋅H∈CL×N and ξ=P/σ2, then problem P4 is equivalent to the following:(10)P5:maxv˜ξ⋅v˜H⋅Φ⋅ΦH⋅v˜+ξ⋅v˜H⋅Φ⋅hsH+ξ⋅hsH⋅ΦH⋅v˜+ξ⋅hsH2s.t.vl2=1,l=1,2,...,L.ProblemP5 is a nonconvex quadratically constrained quadratic programming (QCQP) [31], which can be reformulated as homogeneous QCQP. Specifically, the auxiliary variable t is introduced, allowing the problem P5 to be equivalently rewritten as follows:(11)P6:maxv¯v¯H⋅R⋅v¯+ξ⋅hsH2s.t.vl2=1,l=1,2,...,L.In the formula,R˜=Φ⋅ΦHΦ⋅hsHhsH0, v¯=v˜t, R=ξ⋅R˜. The problem P6 remains nonconvex here; the applied formula is v¯⋅R˜⋅v¯H=trR˜⋅v¯⋅v¯H, let V=v¯⋅v¯H, and V needs to meet V≥0 while rankV=1. Since the rank one constraint is non-convex, SDR is used to relax this constraint. Question P6 is simplified to the following:(12)P7:maxVtrR⋅V+ξ⋅hsH2s.t.Vl,l=1,l=1,2,...,L,V≥0.Since problemP7 is convex semidefinite programming (SDP), the optimal solution can be obtained through CVX [32].The beamforming of the base station is ready to be optimized once the reflection coefficient matrixΘ of the IRS has been optimized. For any given θ, the optimal transmission beamforming is solved by maximum ratio transmission (MRT) [33], let w˜=P⋅grH⋅Θ⋅H+hsHH/grH⋅Θ⋅H+hsH, where P represents the base station’s transmit power. Problem P7 is equivalent to the following:(13)P8:maxPδ⋅Ps.t.P≤Pmax.In the formula,δ=grH⋅Θ⋅H+hsH2/σ2. At this point, we have transformed the optimization goal in passive beamforming into the standard positive semidefinite programming in (12), which can be optimized through CVX toolbox.Typically, the relaxed problemP8 may not necessarily obtain a rank one solution, i.e., rankV≠1. This means that the optimal target value of problem P8 only describes the upper limit of P7. Hence, an additional solution step is required to construct an effective suboptimal solution of problem P8 from the objective function of problem P8. Decompose V by randomization, let V=QAQH, where Q=e1,e2,⋯,eL is a unitary matrix and A=λ1,λ2,⋯,λL is a diagonal matrix, both of size L×L, and then obtain a suboptimal solution of problem P6. Let v¯=QA1/2x, where x∈CL×1 is a random vector generated by CN0,IL, and CN0,IL is a circularly symmetric complex Gaussian(CSCG) distribution with a mean of 0 and a variance of IL. Since the Gaussian random vector x is independent, the target value of problem P6 is approximately the maximum reached by the best v¯ of all x. Given enough random numbers, the SDR algorithm can ensure that π/4 is used to approximate the optimal target value of problem P7. SDR is an approximate solution algorithm that eliminates the rank-one constraint. Compared with Lagrange algorithm, although the computational complexity is higher, but its solution accuracy is higher, so the SDR algorithm is often used in convex optimization solution.


### 3.3. Multiuser Scenario

Multiuser (M>1) scenario is considered on the basis of the single-user scenario. Unlike the maximum ratio transmission criterion adopted in the single-user condition without interference, the minimum mean square error (MMSE) [34] estimator is used to describe the transmit beamforming at BS in the multiantenna access point scenario. In addition, because the receiver in NOMA system uses SIC technology, the BS allocates stronger transmitting power to users with poor channel quality. The receiver decoding is also demodulated preferentially. After demodulation is completed, the modulated signals are deleted from the superimposed signals, and so on until all users have completed signal demodulation. With the maximum signal-to-noise ratio of the signal, the achievable rate is maximized accordingly. Thus, when the BS’s transmit beamforming is determined, the problem P1 is simplified to the following:(14)P9:maxθgr,mH⋅Θ⋅H+hs,mHwm2∑i=m+1Mgr,mH⋅Θ⋅H+hs,mHwi2+σ2s.t.0≤θl≤2π,l=1,2,...,L.Letv⟶=v1,v2,...,vL, where vl=ejθl,∀l. The constraints in problem P9 are equivalent to vl2=1,∀l. The applied substitution variables are presented as ϕm,j=diaggr,mH⋅H⋅wj∈CL×N, then gr,mH⋅Θ⋅H⋅wj=v⟶H⋅ϕm,j, hs.mH⋅wj=εm,j, and gr,mH⋅Θ⋅H+hs,mH⋅wm2=v⟶⋅ϕm,m+εm,m2, the problem P9 hence is simplified to as follows:(15)P10:maxθv⟶H⋅ϕm,m⋅ϕm,mH⋅v⟶+v⟶H⋅ϕm,m⋅εm,mH+εm,m⋅ϕm,mH⋅v⟶+εm,mH2∑i=m+1Mv⟶H⋅ϕm,i⋅ϕm,iH⋅v⟶+v⟶H⋅ϕm,i⋅εm,iH+εm,i⋅ϕm,iH⋅v⟶+εm,iH2+σ2s.t.vl2=1,l=1,2,...,L.Yet, problemP10 remains a nonconvex quadratically constrained quadratic programming problem; thus, the SDR technique will be used to approximate this problem. Specifically, the auxiliary variable t is introduced, let v¯=v⟶t, Rm,j=ϕm,j⋅ϕm,jHϕm,j⋅εm,jHϕm,jH⋅εm,j0, and the problem P10 is rewritten as follows:(16)P11:maxv¯v¯H⋅Rm,m⋅v¯+εm,m2∑i=m+1Mv¯H⋅Rm,i⋅v¯+εm,i2+σ2s.t.vl2=1,l=1,2,...,L.The formulav¯⋅Rm,j⋅v¯H=trRm,j⋅v¯⋅v¯H is applied with V=v¯⋅v¯H defined, and V is required to meet V≥0 while rankV=1; relaxing the constraints, and the problem P11 is further turned into the following:(17)P12:maxVtrRm,m⋅V+εm,m2∑i=m+1MtrRm,i⋅V+εm,i2+σ2s.t.Vl,l=1,l=1,2,...,L,V≥0.It is noticed that the problemP12 is a common positive semidefinite programming problem whose optimal solution can be obtained through the existing CVX toolbox.According to the above optimization of the achievable rate of the user’s signal, the maximum throughput of the system can be expressed as follows:(18)P13:maxV∑m=1MtrRm,m⋅V+εm,m2∑i=m+1MtrRm,i⋅V+εm,i2+σ2s.t.Vl,l=1,l=1,2,...,L,V≥0.


## 3.1. Problem Modeling

The optimization problem can be stated as follows:(6)P1:maxw,Θlog21+gr,mHΘH+hs,mH⋅wm2∑i=m+1Mgr,mHΘH+hs,mH⋅wi2+σ2s.t.0≤θl≤2π,l=1,2,...,L,∑m=1Mwm2≤Pmax,where Pmax refers to the maximum transmit power offered by BS.The solving process can be challenging since the objective function and constraints are nonconvex [30]. To simplify the calculation, the single-user scenario is initially considered, followed by a semidefinite relaxation technique to approximate the solution; afterward, extending this process to the multiuser scenario with an AO algorithm for problem solving.


## 3.2. Single-User System

We consider a single-user system, i.e.,M=1, in this case, the system is comparable to IRS-OMA. Besides, there is no mutual interference while only one user is in the system; hence, problem P1 can be reduced to the following:(7)P2:maxw,Θlog21+grHΘH+hsH⋅w2σ2s.t.0≤θl≤2π,l=1,2,...,L,w2≤Pmax.Regarding the analysis of the problemP2, when the signal-to-noise ratio takes the maximum value, the system’s achievable rate reaches the maximum accordingly; thus, the problem P2 is equivalent to the following:(8)P3:maxw,ΘgrHΘH+hsH⋅w2σ2s.t.0≤θl≤2π,l=1,2,...,L,w2≤Pmax.As variable coupling exists in the objective function ofP3, the problem P3 is regarded as nonconvex. The transmit beam at BS and the phase shift matrix at IRS are alternately optimized until convergence. In addition, w remains constant before solving Θ. The problem P3 can be simplified to as follows:(9)P4:maxθgrHΘH+hsH⋅w2σ2s.t.0≤θl≤2π,l=1,2,...,L.Letv˜=v1,v2,...,vLH, where vl=ejθl,∀l. Thus, the constraints in P4 can be equivalent to the unit modulus constraint: vl2=1,∀l. Using variables to replace: grH⋅Θ⋅H=v˜⋅Φ, Φ=diaggrH⋅H∈CL×N and ξ=P/σ2, then problem P4 is equivalent to the following:(10)P5:maxv˜ξ⋅v˜H⋅Φ⋅ΦH⋅v˜+ξ⋅v˜H⋅Φ⋅hsH+ξ⋅hsH⋅ΦH⋅v˜+ξ⋅hsH2s.t.vl2=1,l=1,2,...,L.ProblemP5 is a nonconvex quadratically constrained quadratic programming (QCQP) [31], which can be reformulated as homogeneous QCQP. Specifically, the auxiliary variable t is introduced, allowing the problem P5 to be equivalently rewritten as follows:(11)P6:maxv¯v¯H⋅R⋅v¯+ξ⋅hsH2s.t.vl2=1,l=1,2,...,L.In the formula,R˜=Φ⋅ΦHΦ⋅hsHhsH0, v¯=v˜t, R=ξ⋅R˜. The problem P6 remains nonconvex here; the applied formula is v¯⋅R˜⋅v¯H=trR˜⋅v¯⋅v¯H, let V=v¯⋅v¯H, and V needs to meet V≥0 while rankV=1. Since the rank one constraint is non-convex, SDR is used to relax this constraint. Question P6 is simplified to the following:(12)P7:maxVtrR⋅V+ξ⋅hsH2s.t.Vl,l=1,l=1,2,...,L,V≥0.Since problemP7 is convex semidefinite programming (SDP), the optimal solution can be obtained through CVX [32].The beamforming of the base station is ready to be optimized once the reflection coefficient matrixΘ of the IRS has been optimized. For any given θ, the optimal transmission beamforming is solved by maximum ratio transmission (MRT) [33], let w˜=P⋅grH⋅Θ⋅H+hsHH/grH⋅Θ⋅H+hsH, where P represents the base station’s transmit power. Problem P7 is equivalent to the following:(13)P8:maxPδ⋅Ps.t.P≤Pmax.In the formula,δ=grH⋅Θ⋅H+hsH2/σ2. At this point, we have transformed the optimization goal in passive beamforming into the standard positive semidefinite programming in (12), which can be optimized through CVX toolbox.Typically, the relaxed problemP8 may not necessarily obtain a rank one solution, i.e., rankV≠1. This means that the optimal target value of problem P8 only describes the upper limit of P7. Hence, an additional solution step is required to construct an effective suboptimal solution of problem P8 from the objective function of problem P8. Decompose V by randomization, let V=QAQH, where Q=e1,e2,⋯,eL is a unitary matrix and A=λ1,λ2,⋯,λL is a diagonal matrix, both of size L×L, and then obtain a suboptimal solution of problem P6. Let v¯=QA1/2x, where x∈CL×1 is a random vector generated by CN0,IL, and CN0,IL is a circularly symmetric complex Gaussian(CSCG) distribution with a mean of 0 and a variance of IL. Since the Gaussian random vector x is independent, the target value of problem P6 is approximately the maximum reached by the best v¯ of all x. Given enough random numbers, the SDR algorithm can ensure that π/4 is used to approximate the optimal target value of problem P7. SDR is an approximate solution algorithm that eliminates the rank-one constraint. Compared with Lagrange algorithm, although the computational complexity is higher, but its solution accuracy is higher, so the SDR algorithm is often used in convex optimization solution.


## 3.3. Multiuser Scenario

Multiuser (M>1) scenario is considered on the basis of the single-user scenario. Unlike the maximum ratio transmission criterion adopted in the single-user condition without interference, the minimum mean square error (MMSE) [34] estimator is used to describe the transmit beamforming at BS in the multiantenna access point scenario. In addition, because the receiver in NOMA system uses SIC technology, the BS allocates stronger transmitting power to users with poor channel quality. The receiver decoding is also demodulated preferentially. After demodulation is completed, the modulated signals are deleted from the superimposed signals, and so on until all users have completed signal demodulation. With the maximum signal-to-noise ratio of the signal, the achievable rate is maximized accordingly. Thus, when the BS’s transmit beamforming is determined, the problem P1 is simplified to the following:(14)P9:maxθgr,mH⋅Θ⋅H+hs,mHwm2∑i=m+1Mgr,mH⋅Θ⋅H+hs,mHwi2+σ2s.t.0≤θl≤2π,l=1,2,...,L.Letv⟶=v1,v2,...,vL, where vl=ejθl,∀l. The constraints in problem P9 are equivalent to vl2=1,∀l. The applied substitution variables are presented as ϕm,j=diaggr,mH⋅H⋅wj∈CL×N, then gr,mH⋅Θ⋅H⋅wj=v⟶H⋅ϕm,j, hs.mH⋅wj=εm,j, and gr,mH⋅Θ⋅H+hs,mH⋅wm2=v⟶⋅ϕm,m+εm,m2, the problem P9 hence is simplified to as follows:(15)P10:maxθv⟶H⋅ϕm,m⋅ϕm,mH⋅v⟶+v⟶H⋅ϕm,m⋅εm,mH+εm,m⋅ϕm,mH⋅v⟶+εm,mH2∑i=m+1Mv⟶H⋅ϕm,i⋅ϕm,iH⋅v⟶+v⟶H⋅ϕm,i⋅εm,iH+εm,i⋅ϕm,iH⋅v⟶+εm,iH2+σ2s.t.vl2=1,l=1,2,...,L.Yet, problemP10 remains a nonconvex quadratically constrained quadratic programming problem; thus, the SDR technique will be used to approximate this problem. Specifically, the auxiliary variable t is introduced, let v¯=v⟶t, Rm,j=ϕm,j⋅ϕm,jHϕm,j⋅εm,jHϕm,jH⋅εm,j0, and the problem P10 is rewritten as follows:(16)P11:maxv¯v¯H⋅Rm,m⋅v¯+εm,m2∑i=m+1Mv¯H⋅Rm,i⋅v¯+εm,i2+σ2s.t.vl2=1,l=1,2,...,L.The formulav¯⋅Rm,j⋅v¯H=trRm,j⋅v¯⋅v¯H is applied with V=v¯⋅v¯H defined, and V is required to meet V≥0 while rankV=1; relaxing the constraints, and the problem P11 is further turned into the following:(17)P12:maxVtrRm,m⋅V+εm,m2∑i=m+1MtrRm,i⋅V+εm,i2+σ2s.t.Vl,l=1,l=1,2,...,L,V≥0.It is noticed that the problemP12 is a common positive semidefinite programming problem whose optimal solution can be obtained through the existing CVX toolbox.According to the above optimization of the achievable rate of the user’s signal, the maximum throughput of the system can be expressed as follows:(18)P13:maxV∑m=1MtrRm,m⋅V+εm,m2∑i=m+1MtrRm,i⋅V+εm,i2+σ2s.t.Vl,l=1,l=1,2,...,L,V≥0.


## 4. Simulation Result Analysis

In this section, the influence of the SDR algorithm and AO algorithm proposed in the article on the signal’s achievable rate is verified. It is compared with the following two schemes: (1) IRS random phase-assisted NOMA downlink transmission system: the phase shift of channelH and gR,m is a random value in the range of 0,2π. Then, MRT is performed at the BS according to the combined channel to maximize the signal achievable rate; (2) Non-IRS-aided NOMA downlink transmission system: Set the model as IRS is absent, based on w=Phs/hs. Then, the BS optimizes transmit beamforming according to the CSI of the channel BS-UE to maximize the signal achievable rate.


### 4.1. Simulation Parameter Settings

The reference center antennas at BS and IRS are, respectively, set to (0,0) and (60m,60 m) in the two-dimensional coordinate system. The system’s path loss model is expressed as follows:(19)Ld=C0dD0−α,where C0 denotes the path loss at a distance D0=1, d reflects the distance, and α signifies the path loss index. With the ignorance of their heights, the distances between BS and IRS, IRS and UEs, and BS and UEs are represented by d1,d2,d3 respectively. The amplitude attenuation coefficients of channels H, gR,m, and hs,m are described as L1=C0d1/D0−α1, L2=C0d2/D0−α2, and L3=C0d3/D0−α3 respectively; where C0=−30dB, the path loss index of the channel between BS and IRS is set as α1=2, i.e., the free space fading model. The path loss index of the channel between IRS and UEs is set as α2=2.8, and the path loss index of the channel directly connected between BS and UEs is set as α3=3.5.Assuming that the channels involved in the simulation process all conform to the Rician fading channel model, the channelH between BS-IRS is as follows:(20)H=β11+β1HLoS+1β1+1HNLoS,where β1 denotes the fading coefficient, HLoS signifies a certain visible path, and HNLoS refers to the Rayleigh fading channel. When β1 tends to infinity, the channel model is simplified into a LoS channel; by contrast, when β1 tends to 0, the channel model is simplified into a Rayleigh channel. Then, multiply H by the square root of the distance-dependent path loss in (19). β2 and β3 are, respectively, applied to reflect the channel fading coefficients of the two links known as IRS-UEs and BS-UEs. β1=∞ is set, which means the channel between BS-IRS is the LoS path; β2=β3=0, so there is a Rayleigh scenario between IRS-UEs and BS-UEs. Other simulation parameters are shown in Table 1.Table 1
Simulation parameters.
ParameterNumericalNumber of UEs(M)10Number of antennas of the BS(N)8Number of elements at IRS(L)100Number of antennas per UE(U)2Noise power (σ2)−80 dBmDistance between BS-IRS602 mCentral location of user group(60 m,10 m)BS transmit power30 dBm


### 4.2. Simulation Analysis

Figure4 displays the condition when ten users are randomly placed in a circle with the center d,10 and a radius of 5m, it further reflects the signals’ achievable rate required by the users at various d values. In addition, three outcomes are noticed in the figure. First and foremost, it can be discovered that the signal achievable rate of the NOMA downlink system with IRS aid is higher than the signal achievable rate of the NOMA downlink system without IRS support. The IRS function is to enhance desired signals while suppressing interference signals. Second, it is obvious that compared to the scheme with absent IRS, the two optimization algorithms adopted present a remarkable peak when d=60m, showing that in the case of adopting these two optimization schemes, when the distance between the user and the IRS is the smallest, the achievable signal rate will be the highest; because at this time, the reflected signal received by the user is the strongest, and a peak will not appear in the scheme without IRS. Finally, it can be seen that the achievable signal rates of the IRS random phase system and the system without IRS aid are sufficiently proximate; this is due to the IRS with missing passive beamforming, which cannot accurately reflect the signal received by the BS to a specific user.Figure 4
Change curve of achievable rate with distance.Figure5 depicts the relationship between the achievable signal rate and the distance between the user and the IRS using the AO scheme. At the same time, different numbers of transmitting antennas are set in BS. Based on the simulation results, it is summarized that the signal’s achievable rate becomes higher as the number of transmitting antennas of BS extends, revealing that increasing the number of antennas of the base station contributes to boosting the signal’s achievable rate required by the user.Figure 5
Change curve of achievable rate with distance.The link between the signal’s achievable rate and the BS’s maximum transmit power threshold is depicted in Figure6. According to the simulation results, the signal’s attainable rate will increase monotonically when the BS maximum transmit power threshold increases in all circumstances. Specifically, as the BS transmit power threshold rises, the amount of power allocated to users expands, leading to an increasing attainable rate of the signal required by the user. Furthermore, it can be observed that the signal with the suggested algorithm can attain the maximum rate under the same conditions since the SDR algorithm and the AO algorithm can modulate the reflection phase shift of each reflection unit on the IRS to direct the signal point to a specific user. Therefore, optimizing the reflected phase shift while increasing the BS maximum transmit power threshold might raise the attainable rate of the signal required by the user to a higher extent.Figure 6
Change curve of the achievable rate with the BS transmit power.Figure7 depicts the relationship between the weighted sum rate and the number of users. The results show that as the number of users increases, the weighted sum rate is almost constant, which also means that the average speed of signal transmission to the user decreases.Figure 7
Change curve of the achievable rate with the number of users.Figure8 shows that the weighted sum rate can converge to a stable value when using four transmission schemes for the system. Due to the without IRS and IRS phase-shift randomness, pilot training of IRS is not required before data transmission, so the convergence speed of these two schemes is faster. When using the two schemes of SDR and AO algorithm, in order to achieve more efficient data transmission, it is necessary to conduct pilot training for IRS before data transmission to ensure that IRS will send signals to users in a more concentrated manner. Therefore, compared with the benchmark scheme, the convergence rate is slower.Figure 8
Change curve of the achievable rate with the number of iteration.Figure9 describes the link between the number of IRS’s reflection units and the signal’s achievable rate under the premise of a given maximum transmit power threshold in BS. It can be noticed in the figure that the achievable signal rate of the IRS-assisted system increases with the growing number of IRS reflection units as a result of an enhanced signal strength required by the user, which is due to the setting of IRS in the system. At the same time, the attainable rate of the system signal using the improved algorithm suggested in this study is much higher than that of the IRS random phase. It shows that as the number of IRS reflection units increases, when the reflection phase shift matrix of the IRS has an optimal value, the signal’s rate performance required by the user in the assisted system is optimized, thereby achieving a higher gain. In addition, the algorithm proposed in the paper acquires a significant rate gain when the reflection unit is larger since more IRS reflection units will accordingly reflect the greater power of the signal received from the base station, resulting in more significant power gain. Besides, it can be drawn from the figure that in the NOMA downlink transmission system without IRS assistance, the signal’s achievable rate required by the user does not change with the increasing number of IRS reflection units.Figure 9
Change curve of the achievable rate with the number of reflecting units.


## 4.1. Simulation Parameter Settings

The reference center antennas at BS and IRS are, respectively, set to (0,0) and (60m,60 m) in the two-dimensional coordinate system. The system’s path loss model is expressed as follows:(19)Ld=C0dD0−α,where C0 denotes the path loss at a distance D0=1, d reflects the distance, and α signifies the path loss index. With the ignorance of their heights, the distances between BS and IRS, IRS and UEs, and BS and UEs are represented by d1,d2,d3 respectively. The amplitude attenuation coefficients of channels H, gR,m, and hs,m are described as L1=C0d1/D0−α1, L2=C0d2/D0−α2, and L3=C0d3/D0−α3 respectively; where C0=−30dB, the path loss index of the channel between BS and IRS is set as α1=2, i.e., the free space fading model. The path loss index of the channel between IRS and UEs is set as α2=2.8, and the path loss index of the channel directly connected between BS and UEs is set as α3=3.5.Assuming that the channels involved in the simulation process all conform to the Rician fading channel model, the channelH between BS-IRS is as follows:(20)H=β11+β1HLoS+1β1+1HNLoS,where β1 denotes the fading coefficient, HLoS signifies a certain visible path, and HNLoS refers to the Rayleigh fading channel. When β1 tends to infinity, the channel model is simplified into a LoS channel; by contrast, when β1 tends to 0, the channel model is simplified into a Rayleigh channel. Then, multiply H by the square root of the distance-dependent path loss in (19). β2 and β3 are, respectively, applied to reflect the channel fading coefficients of the two links known as IRS-UEs and BS-UEs. β1=∞ is set, which means the channel between BS-IRS is the LoS path; β2=β3=0, so there is a Rayleigh scenario between IRS-UEs and BS-UEs. Other simulation parameters are shown in Table 1.Table 1
Simulation parameters.
ParameterNumericalNumber of UEs(M)10Number of antennas of the BS(N)8Number of elements at IRS(L)100Number of antennas per UE(U)2Noise power (σ2)−80 dBmDistance between BS-IRS602 mCentral location of user group(60 m,10 m)BS transmit power30 dBm


## 4.2. Simulation Analysis

Figure4 displays the condition when ten users are randomly placed in a circle with the center d,10 and a radius of 5m, it further reflects the signals’ achievable rate required by the users at various d values. In addition, three outcomes are noticed in the figure. First and foremost, it can be discovered that the signal achievable rate of the NOMA downlink system with IRS aid is higher than the signal achievable rate of the NOMA downlink system without IRS support. The IRS function is to enhance desired signals while suppressing interference signals. Second, it is obvious that compared to the scheme with absent IRS, the two optimization algorithms adopted present a remarkable peak when d=60m, showing that in the case of adopting these two optimization schemes, when the distance between the user and the IRS is the smallest, the achievable signal rate will be the highest; because at this time, the reflected signal received by the user is the strongest, and a peak will not appear in the scheme without IRS. Finally, it can be seen that the achievable signal rates of the IRS random phase system and the system without IRS aid are sufficiently proximate; this is due to the IRS with missing passive beamforming, which cannot accurately reflect the signal received by the BS to a specific user.Figure 4
Change curve of achievable rate with distance.Figure5 depicts the relationship between the achievable signal rate and the distance between the user and the IRS using the AO scheme. At the same time, different numbers of transmitting antennas are set in BS. Based on the simulation results, it is summarized that the signal’s achievable rate becomes higher as the number of transmitting antennas of BS extends, revealing that increasing the number of antennas of the base station contributes to boosting the signal’s achievable rate required by the user.Figure 5
Change curve of achievable rate with distance.The link between the signal’s achievable rate and the BS’s maximum transmit power threshold is depicted in Figure6. According to the simulation results, the signal’s attainable rate will increase monotonically when the BS maximum transmit power threshold increases in all circumstances. Specifically, as the BS transmit power threshold rises, the amount of power allocated to users expands, leading to an increasing attainable rate of the signal required by the user. Furthermore, it can be observed that the signal with the suggested algorithm can attain the maximum rate under the same conditions since the SDR algorithm and the AO algorithm can modulate the reflection phase shift of each reflection unit on the IRS to direct the signal point to a specific user. Therefore, optimizing the reflected phase shift while increasing the BS maximum transmit power threshold might raise the attainable rate of the signal required by the user to a higher extent.Figure 6
Change curve of the achievable rate with the BS transmit power.Figure7 depicts the relationship between the weighted sum rate and the number of users. The results show that as the number of users increases, the weighted sum rate is almost constant, which also means that the average speed of signal transmission to the user decreases.Figure 7
Change curve of the achievable rate with the number of users.Figure8 shows that the weighted sum rate can converge to a stable value when using four transmission schemes for the system. Due to the without IRS and IRS phase-shift randomness, pilot training of IRS is not required before data transmission, so the convergence speed of these two schemes is faster. When using the two schemes of SDR and AO algorithm, in order to achieve more efficient data transmission, it is necessary to conduct pilot training for IRS before data transmission to ensure that IRS will send signals to users in a more concentrated manner. Therefore, compared with the benchmark scheme, the convergence rate is slower.Figure 8
Change curve of the achievable rate with the number of iteration.Figure9 describes the link between the number of IRS’s reflection units and the signal’s achievable rate under the premise of a given maximum transmit power threshold in BS. It can be noticed in the figure that the achievable signal rate of the IRS-assisted system increases with the growing number of IRS reflection units as a result of an enhanced signal strength required by the user, which is due to the setting of IRS in the system. At the same time, the attainable rate of the system signal using the improved algorithm suggested in this study is much higher than that of the IRS random phase. It shows that as the number of IRS reflection units increases, when the reflection phase shift matrix of the IRS has an optimal value, the signal’s rate performance required by the user in the assisted system is optimized, thereby achieving a higher gain. In addition, the algorithm proposed in the paper acquires a significant rate gain when the reflection unit is larger since more IRS reflection units will accordingly reflect the greater power of the signal received from the base station, resulting in more significant power gain. Besides, it can be drawn from the figure that in the NOMA downlink transmission system without IRS assistance, the signal’s achievable rate required by the user does not change with the increasing number of IRS reflection units.Figure 9
Change curve of the achievable rate with the number of reflecting units.


## 5. Conclusion

In response to the IRS-aided NOAM downlink transmission system, this paper adopts two schemes of semidefinite relaxation and alternating optimization, thereby maximizing the achievable rate of the signal required by the user. Given the multivariate coupling of the original problem and the nonconvex constraints, the solving process is quite complex. Therefore, this paper exploits the characteristics of the optimization problem to deduce the relationship between the BS transmit beamforming and the IRS reflection phase shift matrix, further converting the situation into a linear programming problem expecting to reduce the difficulties in the solution process. Furthermore, the SDR algorithm and AO algorithm are used to optimize the transmit beamforming, as well as the reflection phase shift matrix. The simulation results show that compared with the IRS random phase shift system and the system without IRS, the two optimization methods proposed in this paper can significantly enhance the achievable rate of the signal required by the user, indicating that using a large number of reflection units at the IRS while optimizing the phase shift is effective.



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*Source: 8901752-2022-05-17.xml*