# Joint User Pairing and Power Allocation Scheme Based on Transmission Mode Switching between NOMA-Based Maximum Ratio Transmission and MMSE Beamforming in Downlink MISO Systems

Authors: Zhixin Zhao; Dong Wang; Hongwei Zhang; Haitao Sang
Journal: Mobile Information Systems
 (2021)

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## Abstract

This paper investigates the design of the joint user pairing and power allocation scheme with transmission mode switching (TMS) in downlink multiple-input-single-output (MISO) systems. Firstly, the closed-form expressions of the boundary of achievable rate region of two candidate transmission modes, i.e., non-orthogonal multiple access based maximum ratio transmission (NOMA-MRT) and minimum mean square error beamforming (MMSE-BF), are obtained. By obtaining the outer boundary of the union of the achievable rate regions of the two transmission modes, an adaptive switching method is developed to achieve a larger rate region. Secondly, based on the idea that the solution to the problem of weighted sum rate (WSR) optimization must be on the boundary of the achievable rate region, the optimal solutions to the problem of WSR optimization for NOMA-MRT and MMSE-BF are obtained for the two-user case, respectively. Subsequently, by exploiting the aforementioned optimal solutions for two transmission modes and the high efficiency of TMS, a suboptimal user pairing and power allocation algorithm (JUPA) is proposed to further improve the sum rate performance for the multiuser case. Compared with the exhaustive search-based user pairing and power allocation algorithm (ES-PPA), the proposed JUPA can enjoy a much lower computational complexity and only suffers a slight sum rate performance loss, and it outperforms other traditional schemes. Finally, numerical results are provided to validate the analyses and the proposed algorithms.


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## Body

## 1. Introduction

With the continuous emergence of new application scenarios, one of the challenges faced by the future wireless communication systems is how to provide higher-speed downlink transmission, restricted to the scarce spectrum resources. The traditional downlink transmission schemes used in mobile communication systems are based on orthogonal multiple access (OMA) technology, e.g., frequency division multiple access (FDMA) for the first generation (1G), time division multiple access (TDMA) for 2G, code division multiple access (CDMA) for 3G, and orthogonal frequency division multiple access (OFDMA) [1, 2] for 4G. These conventional multiple access schemes can mitigate or avoid the interuser interference by allocating orthogonal resources (frequency/time/code) to different users, which result in insufficient use of spectrum resources.Recently, non-orthogonal multiple access (NOMA) has been considered as a promising multiple access scheme for 5th Generation (5G) wireless communication systems, owing to its higher spectrum efficiency compared with the conventional orthogonal multiple access schemes [3–5]. Note that Third Generation Partnership Project (3GPP) has considered NOMA as a study item for 5G new radio (NR) in Release 15 and decided to leave it for possible use in Beyond 5G (B5G) [6]. NOMA enables multiple users to share the same time-frequency resource block on the same spatial layer. NOMA is achieved by the combination of superposition encoding and successive interference cancellation (SIC), which is a method to reach the boundary of the capacity region of degraded broadcast channel [7].In order to further enhance system performance, beamforming (BF) was combined with NOMA in multiple-input-single-output (MISO) downlink [8–14]. In [8], the sum rate maximization problem with BF vector being the optimization variable was studied and an one-dimensional iterative algorithm was proposed. However, for each iteration, a second-order cone program convex problem needs to be solved, which results in very high computational complexity. Moreover, as the signals of all users are superposed on one resource block, the algorithm may suffer large process delay and error propagation of SIC for the system with a large number of users. In [9], the sum rate optimization problem with minimum user rate constraint was investigated, and therefore a low-complexity BF scheme and a user clustering scheme were proposed. Since the presented BF scheme and user clustering scheme were designed separately, some sum rate performance loss is suffered. In [10], the problem for maximization of the number of users with an ergodic user rate constraint was considered. A power allocation scheme to satisfy ergodic user rate constraints was proposed and then a user admission algorithm that achieves the maximum number of users was developed to guarantee the minimum user rate requirements. However, as the MRT beamforming is adopted for downlink transmission, the optimization of BF vector is not considered. By the duality between the multiple access channel (MAC) and broadcast channel (BC), the duality scheme for sum rate optimization was developed in [11], which also suffers rather high computational complexity because of needing to solve a quadratic constrained quadratic programs convex problem. Furthermore, since the duality scheme is a quasi-degraded solution to the problem of sum rate optimization actually, it is only feasible in the case, where the channel state information (CSI) of the scheduled users meets the quasi-degraded property. As a result, the application of the duality scheme in practical systems may be heavily restricted. In [12], the robust BF design problem to optimize the worst-case achievable sum rate constrained by the total transmit power was studied. In [13], the optimal BF design problem which minimizes the total transmission power subject to a pair of target interference-level constraints was investigated for two-user MISO-NOMA downlink. Based on the results obtained by Chen et al. [13], Chen et al. [14] further proved that the minimum transmit power of the NOMA transmission scheme was equal to that of dirty-paper coding in two-user case, under the condition of the broadcast channel being quasi-degraded. Furthermore, a hybrid NOMA (H-NOMA) precoding algorithm with low complexity is proposed by combining NOMA with zero-forcing beamforming (ZFBF). The summary of comparison of this paper with the existing works [8–14] is shown in Table 1.Table 1
Comparison of this paper with the existing works [8–14].
Research works[8][9][10][11][12][13][14]This paperScenarioMU MISOMU MISOMU MISOTU MIMOMU MISOTU MISOMU MISOMU MISOMain objectivesSum rateSum rateNumber of usersSum rateWC sum rateTransmit powerTransmit powerSum rateOptimization variableBVBV + CMPowerPrecoding matrixBV + powerBVBV + CMBV + UPM + CPJoint optimization–No––No–YesYesUser number in a clusterMUMUMUTUMUTUTUTUMAC-BC dualityNot usedNot usedNot usedUsedNot usedNot usedNot usedUsedEach BF vector serves single userYesYesYesYesNoYesYesYesComplexityHighMediumMediumHighMediumMediumLowLowNeed for iterationYesYesYesYesYesYesNoNoMain challengesPD and EPPD and EPPD and EPOnly feasible in QDCPD and EPSpectral efficiencySpectral efficiencySpectral efficiency“TU” and “MU” represent two users and multiple users, respectively. “BV” and “WC sum rate” represent BF vector and worst-case sum rate, respectively. “UPM” and “CM” represent user pairing relationship matrix and clustering relationship matrix, respectively. “CP” and “QDC” represent control parameter for TMS and quasi-degraded channel, respectively. “PD” and “EP” represent processing delay and error propagation of SIC, respectively.Dynamic user clustering and user pairing are effective methods to achieve the viable benefits of NOMA in providing high spectral efficiency [15–17]. In [15], the impact of user pairing on the performance of the two-user NOMA systems is characterized. It was shown that the performance gain of NOMA over conventional multiple access can be further enlarged by selecting users whose channel conditions are more distinctive. In [16], the problem of jointly optimizing user association and power control to maximize the overall spectral efficiency was investigated for NOMA-based full-duplex MIOS systems. To be spectrally efficient, the tensor model was introduced to optimize uplink users’ decoding order and downlink users’ clustering. In [17], a novel hybrid user pairing beamforming (HUP) scheme was proposed to maximize the achievable spectral and energy efficiencies for multiuser MISO-NOMA downlink systems with simultaneous wireless information and power transfer (SWIPT). In the proposed HUP scheme, two information users with distinct channel conditions are paired while energy users located near the BS perform energy harvesting.Due to the equivalence between the maximization of the weighted sum rate (WSR) and the acquisition of the maximum WSR point on achievable rate region of the MISO downlink, the problems of characterization of the achievable rate region for MISO broadcast channel [18, 19] and for MISO interference channel [20–22] were investigated, respectively. Specifically, in [18], the set of BF vectors which achieve points on the boundary of achievable rate region of two-user MISO broadcast channel is characterized by a single real valued parameter per user. In [19], the design of adaptive transmission mode switching to derive the larger rate region for the two-user MISO broadcast channel is investigated. An explicit characterization of the boundary of achievable rate region for multiuser MISO interference channel was obtained in [20]. A general framework for finding the maximum sum rate operating points on the boundary of the achievable rate region for the two-user MISO interference channel was proposed in [21]. In [22], the achievable rate region of two-user MISO interference channel for single user detection was obtained.Although the aforementioned schemes can provide efficient solutions with several advantages, they bring few insights about their optimality, compared to the achievable rate regions for downlink MISO systems with transmission mode switching. In this paper, we compare the achievable rate regions obtained by NOMA-MRT and MMSE-BF. Then, we find that when the channel vectors of the two users are greatly correlated, the achievable rate region of NOMA-MRT includes that of MMSE-BF completely. However, when the two channels are almost orthogonal, the opposite is true. Consequently, we focus on the design of adaptive transmission mode switching (TMS) between NOMA-MRT and MMSE-BF to maximize the sum rate for multiuser MISO downlink by utilizing the idea of user pairing.The main contributions of this paper are listed as follows:(1)
By combining MRT beamforming and NOMA, a novel transmission scheme referred to as NOMA-MRT for MISO downlink is proposed and the corresponding closed-form expression of the boundary of the achievable rate region is achieved. Moreover, building on the duality of MAC and BC, the closed-form expression for the boundary of the achievable rate region of MISO broadcast channel with MMSE-BF adopted at BS is given. Subsequently, by obtaining the outer boundary of the union of the achievable rate regions of the two transmission schemes, an adaptive transmission mode switching method is developed to achieve a larger rate region for the two-user case.(2)
Based on the idea that the solution to the problem of WSR optimization must be on the corresponding boundary of the achievable rate region and the closed-form expression of the rate region boundary in any channel condition, the global maximum value of the transformed WSR (the unary function) can be found by selecting the maximum among the function values at the end of the interval and the local maximum values of WSR. Consequently, two optimal power allocation algorithms, i.e., power allocation algorithm for NOMA-MRT mode (NOMA-MRT-PA) and power allocation algorithm for MMSE-BF mode (MMSE-BF-PA), are proposed by focusing on the case with two users.(3)
Building on the NOMA-MRT-PA and MMSE-BF-PA, a novel joint user pairing and power allocation algorithm (JUPA) is proposed for the multiuser case. Subsequently, by the combination of JUPA and the idea of TMS between NOMA-MRT and MMSE-BF, a practical transmission method is developed. By making use of the closed-form expression of the optimal solution by NOMA-MRT-PA (MMSE-BF-PA), JUPA can be performed with a low computational complexity, while the exhaustive search-based user pairing and power allocation algorithm (ES-PPA) requires extremely high complexity, compared with which JUPA only suffers a slight performance loss.The remainder of this paper is organized as follows. Section2 briefly describes the system model and introduces the NOMA-based beamforming scheme. In Section 3, the closed-form expressions of the achievable rate region boundary of MISO broadcast channel by using NOMA-MRT and MMSE-BF are obtained. Consequently, an adaptive switching method is proposed in Section 4. In Section 5, two optimal power allocation algorithms for NOMA-MRT mode and MMSE-BF mode are presented, by which a joint user pairing and power allocation algorithm is also developed, respectively. The numerical results are illustrated in Section 6, and finally conclusions are drawn in Section 7.Before proceeding, we introduce the following notation. Throughout the paper, we denote column vectorsx and matrices X by bold lower-case and upper-case letters, respectively. ⋅H represents the complex conjugate transpose of a vector or matrix. The absolute value of a scalar is denoted by ⋅ and the norm of a vector is denoted by ⋅. x∈ℂM×1 means that x is an M×1 complex vector. CNμ,σ2 denotes a complex Gaussian random variable with mean μ and variance σ2. x,y and ∠x,y denote the inner product and the angle of two complex vectors x and y, respectively.


## 2. Problem Description

### 2.1. System Model

We consider a downlink communication system with oneM-antenna base station (BS) and 2K (assumed to be an even number) single-antenna users, from which K user pairs are selected and the two users in a user pair share the same spectrum. The BS serves one user pair in each time-frequency resource block, and the data transmission for the user pair can adaptively switch between two transmission modes, i.e., NOMA-MRT and MMSE-BF, according to the channel state, as shown in Figure 1. When the channel vectors of the two users in a user pair are greatly correlated, the NOMA-MRT mode is usually selected to achieve larger sum rate in the system. However, when the two channels are almost orthogonal, the MMSE-BF mode is selected normally (see Section 6.1 for details). In the NOMA-MRT mode, MRT beamforming is performed at the BS and SIC is executed at the user with strong channel conditions in the user pair. Specifically, for example, user n with strong channel conditions in user pair 1 first decodes the signal sm,1 of user m in the user pair 1 and subtracts this from its received signal yn,1. As a result, user n can decode its own signal sn,1 without the interference caused by user m. However, user m in user pair 1 simply treats sn,1 as noise and decodes its own signal sm,1. In the MMSE-BF mode, MMSE beamforming is adopted for the user pair at the BS. 2K users are assumed to be uniformly located within a cell with a radius of R, and the BS is deployed at the center of the cell. Here, we consider user pairing, i.e., selecting two users to form a group, in the system model for the following reasons. In the NOMA downlink, users cancel the co-channel interference by performing SIC. However, with the number of users in a group growing, the processing complexity and delay at users dramatically increase [23]. As a result, we consider only grouping two users and adopting user pairing scheme to decrease processing complexity and delay in the NOMA downlink. The observation at user i in the user pair j is given by(1)yi,j=hi,jHx+zi,j,i=m,n;j=1,2,…,K,where hi,j∼CN0,σi,j2IM is the channel vector from BS to user i in the user pair j, zi,j∼CN0,N0) is the additive white Gaussian noise (AWGN) at user i in the user pair j, and N0 is the corresponding noise power. For the channel vector hi,j, the variance of the channel from BS to user i in the user pair j is modeled as [14, 24](2)σi,j2=di,j−α′,ifdi,j>d0,d0−α′,otherwise,where di,j denotes the distance between BS and user i in the user pair j, α′ denotes the path loss exponent, and the parameter d0 avoids the singularity when di,j is small. Furthermore, xj=pm,jwm,jsm,j+pn,jwn,jsn,j is the signal transmitted by the BS for the user pair j, where si,j and wi,j are the scalar signal and normalized BF vector for user i in the user pair j, respectively. pi,j is the transmit power allocated to user ii=n,m in the user pair j. Assume that the BS can obtain the perfect CSI.Figure 1
System model with transmission mode switching between NOMA-MRT and MMSE-BF. (a) NOMA-MRT mode. (b) MMSE-BF mode.
(a)(b)For convenience, we omit the subscriptj of the variables mentioned above in the following. In this paper, we attempt to maximize the sum rate of users by jointly optimizing the pairing relationship matrix U as well as power allocation vector pm,n and control parameter Tm,n for TMS of each user pair. Mathematically, the optimization problem can be formulated as(3a)P1:maxpm,n,U,Tm,n∑m=12K∑n=m+12Kum,nRm+Rn,(3b)s.t.Tm,n∈0,1,1≤m,n≤2K,(3c)um,n∈0,1,1≤m,n≤2K,(3d)um,n=un,m,1≤m,n≤2K,(3e)um,m=0,1≤m,n≤2K,(3f)∑m=12Kum,n=1,1≤n≤2K,(3g)∑n=12Kum,n=1,1≤m≤2K,(3h)0≤um,npm+pn≤P,1≤m,n≤2K,(3i)wm=wn=1,1≤m,n≤2K,where Ri is the date rate of user i in the user pair m,n, i=m,n. U is the user pairing relationship matrix with 2K dimensions, in which the m-th row and n-th column element is denoted by um,n. pm,n=pm,pn and Tm,n are power allocation vector and control parameter for transmission mode switching of the user pair m,n. As shown in (3b), Tm,n=1 denotes that the transmission mode NOMA-MRT is selected for the user pair m,n at BS. Otherwise, MMSE-BF is performed for the user pair m,n. In (3c), um,n=1 represents that users m and n are paired together and constitute a user pair. Otherwise, users m and n are not paired together. (3d) and (3e) imply that U is a symmetric matrix, and the diagonal elements are zero as a user cannot be paired with itself. (3f) and (3g) indicate that a user can be paired with only one user. Furthermore, (3h) represents the total power allocated to a user pair m,n, which is upper bounded to P. (3i) denotes that the BF vectors are normalized in this paper.It is hard to achieve the optimal solution to the problem (P1) due to binary constraints of um,n and the nonconvex property of the achievable rate of user i, which is the data rate of user i with the equality holding in (4) or (7). Therefore, for the treatability of the problem (P1), we select the BF scheme (i.e., control parameter for TMS) with the larger sum rate for a user pair, among MMSE-BF and NOMA-based BF, instead of jointly optimizing BF vector for all users, in the process of jointly optimizing the pairing relationship matrix U, power allocation vector pm,n, and control parameter Tm,n of each user pair.


### 2.2. Achievable Rate Region of the Existing BF Schemes

For conventional linear BF scheme, such as ZFBF, MRT, and MMSE-BF, given a fixed normalized BF vectorwi, the achievable rate region of the MISO broadcast channel (BC) is given by the set of rate tuples Rm,Rn satisfying(4)Ri≤log1+γiB,where(5)γiB=pihiHwi2N0+pφihiHwφi2,is the signal-to-interference-plus-noise ratio (SINR) of user i with pm+pn≤P, i=m,n. The mapping function φi is defined by(6)φi=n,i=m,m,i=n,and P is the total transmit power for each user pair. When the equality holds in (4), the rate tupleRm,Rn achieves on the boundary of the achievable rate region with pm+pn=P .


### 2.3. Achievable Rate Region of NOMA-Based BF Scheme

For two-user MISO downlink with the NOMA-based linear BF scheme adopted at the BS, SIC is implemented at the user with strong channel conditions. Therefore, for a fixed normalized BF vectorwi, when SIC is carried out at user n with strong channel conditions, the achievable rate region of the MISO broadcast channel by using NOMA-based linear BF at the BS can be formulated by the set of rate tuples satisfying(7)Rn≤log1+pnhnHwn2N0,Rm≤minlog1+γm,nB,log1+γmB,where log1+γm,nB denotes the achievable rate for user n to detect user m’s message and γm,nB is the corresponding SINR, i.e.,(8)γm,nB=pmhnHwm2N0+pnhnHwm2.In this paper, the angle betweenhm and hn is denoted by α, where(9)cos2α=hnHhm2hm2hn2.Without loss of generality, we setα∈0,π.


### 2.4. The Equivalence between Maximizing WSR and Enlarging Achievable Rate Region

For a given weight vector, the solution of optimization problem of maximizing WSR must be on the boundary of achievable rate region of the two-user MISO broadcast channel [19]. Thus, maximizing the WSR in MISO downlink systems is equivalent to enlarging the corresponding achievable rate region as much as possible.In this paper, from the perspective of achievable rate region, we investigate the design of the suboptimal algorithm for the solution to the problem (P1) based on transmission mode switching between NOMA-based BF and MMSE-BF.


## 2.1. System Model

We consider a downlink communication system with oneM-antenna base station (BS) and 2K (assumed to be an even number) single-antenna users, from which K user pairs are selected and the two users in a user pair share the same spectrum. The BS serves one user pair in each time-frequency resource block, and the data transmission for the user pair can adaptively switch between two transmission modes, i.e., NOMA-MRT and MMSE-BF, according to the channel state, as shown in Figure 1. When the channel vectors of the two users in a user pair are greatly correlated, the NOMA-MRT mode is usually selected to achieve larger sum rate in the system. However, when the two channels are almost orthogonal, the MMSE-BF mode is selected normally (see Section 6.1 for details). In the NOMA-MRT mode, MRT beamforming is performed at the BS and SIC is executed at the user with strong channel conditions in the user pair. Specifically, for example, user n with strong channel conditions in user pair 1 first decodes the signal sm,1 of user m in the user pair 1 and subtracts this from its received signal yn,1. As a result, user n can decode its own signal sn,1 without the interference caused by user m. However, user m in user pair 1 simply treats sn,1 as noise and decodes its own signal sm,1. In the MMSE-BF mode, MMSE beamforming is adopted for the user pair at the BS. 2K users are assumed to be uniformly located within a cell with a radius of R, and the BS is deployed at the center of the cell. Here, we consider user pairing, i.e., selecting two users to form a group, in the system model for the following reasons. In the NOMA downlink, users cancel the co-channel interference by performing SIC. However, with the number of users in a group growing, the processing complexity and delay at users dramatically increase [23]. As a result, we consider only grouping two users and adopting user pairing scheme to decrease processing complexity and delay in the NOMA downlink. The observation at user i in the user pair j is given by(1)yi,j=hi,jHx+zi,j,i=m,n;j=1,2,…,K,where hi,j∼CN0,σi,j2IM is the channel vector from BS to user i in the user pair j, zi,j∼CN0,N0) is the additive white Gaussian noise (AWGN) at user i in the user pair j, and N0 is the corresponding noise power. For the channel vector hi,j, the variance of the channel from BS to user i in the user pair j is modeled as [14, 24](2)σi,j2=di,j−α′,ifdi,j>d0,d0−α′,otherwise,where di,j denotes the distance between BS and user i in the user pair j, α′ denotes the path loss exponent, and the parameter d0 avoids the singularity when di,j is small. Furthermore, xj=pm,jwm,jsm,j+pn,jwn,jsn,j is the signal transmitted by the BS for the user pair j, where si,j and wi,j are the scalar signal and normalized BF vector for user i in the user pair j, respectively. pi,j is the transmit power allocated to user ii=n,m in the user pair j. Assume that the BS can obtain the perfect CSI.Figure 1
System model with transmission mode switching between NOMA-MRT and MMSE-BF. (a) NOMA-MRT mode. (b) MMSE-BF mode.
(a)(b)For convenience, we omit the subscriptj of the variables mentioned above in the following. In this paper, we attempt to maximize the sum rate of users by jointly optimizing the pairing relationship matrix U as well as power allocation vector pm,n and control parameter Tm,n for TMS of each user pair. Mathematically, the optimization problem can be formulated as(3a)P1:maxpm,n,U,Tm,n∑m=12K∑n=m+12Kum,nRm+Rn,(3b)s.t.Tm,n∈0,1,1≤m,n≤2K,(3c)um,n∈0,1,1≤m,n≤2K,(3d)um,n=un,m,1≤m,n≤2K,(3e)um,m=0,1≤m,n≤2K,(3f)∑m=12Kum,n=1,1≤n≤2K,(3g)∑n=12Kum,n=1,1≤m≤2K,(3h)0≤um,npm+pn≤P,1≤m,n≤2K,(3i)wm=wn=1,1≤m,n≤2K,where Ri is the date rate of user i in the user pair m,n, i=m,n. U is the user pairing relationship matrix with 2K dimensions, in which the m-th row and n-th column element is denoted by um,n. pm,n=pm,pn and Tm,n are power allocation vector and control parameter for transmission mode switching of the user pair m,n. As shown in (3b), Tm,n=1 denotes that the transmission mode NOMA-MRT is selected for the user pair m,n at BS. Otherwise, MMSE-BF is performed for the user pair m,n. In (3c), um,n=1 represents that users m and n are paired together and constitute a user pair. Otherwise, users m and n are not paired together. (3d) and (3e) imply that U is a symmetric matrix, and the diagonal elements are zero as a user cannot be paired with itself. (3f) and (3g) indicate that a user can be paired with only one user. Furthermore, (3h) represents the total power allocated to a user pair m,n, which is upper bounded to P. (3i) denotes that the BF vectors are normalized in this paper.It is hard to achieve the optimal solution to the problem (P1) due to binary constraints of um,n and the nonconvex property of the achievable rate of user i, which is the data rate of user i with the equality holding in (4) or (7). Therefore, for the treatability of the problem (P1), we select the BF scheme (i.e., control parameter for TMS) with the larger sum rate for a user pair, among MMSE-BF and NOMA-based BF, instead of jointly optimizing BF vector for all users, in the process of jointly optimizing the pairing relationship matrix U, power allocation vector pm,n, and control parameter Tm,n of each user pair.


## 2.2. Achievable Rate Region of the Existing BF Schemes

For conventional linear BF scheme, such as ZFBF, MRT, and MMSE-BF, given a fixed normalized BF vectorwi, the achievable rate region of the MISO broadcast channel (BC) is given by the set of rate tuples Rm,Rn satisfying(4)Ri≤log1+γiB,where(5)γiB=pihiHwi2N0+pφihiHwφi2,is the signal-to-interference-plus-noise ratio (SINR) of user i with pm+pn≤P, i=m,n. The mapping function φi is defined by(6)φi=n,i=m,m,i=n,and P is the total transmit power for each user pair. When the equality holds in (4), the rate tupleRm,Rn achieves on the boundary of the achievable rate region with pm+pn=P .


## 2.3. Achievable Rate Region of NOMA-Based BF Scheme

For two-user MISO downlink with the NOMA-based linear BF scheme adopted at the BS, SIC is implemented at the user with strong channel conditions. Therefore, for a fixed normalized BF vectorwi, when SIC is carried out at user n with strong channel conditions, the achievable rate region of the MISO broadcast channel by using NOMA-based linear BF at the BS can be formulated by the set of rate tuples satisfying(7)Rn≤log1+pnhnHwn2N0,Rm≤minlog1+γm,nB,log1+γmB,where log1+γm,nB denotes the achievable rate for user n to detect user m’s message and γm,nB is the corresponding SINR, i.e.,(8)γm,nB=pmhnHwm2N0+pnhnHwm2.In this paper, the angle betweenhm and hn is denoted by α, where(9)cos2α=hnHhm2hm2hn2.Without loss of generality, we setα∈0,π.


## 2.4. The Equivalence between Maximizing WSR and Enlarging Achievable Rate Region

For a given weight vector, the solution of optimization problem of maximizing WSR must be on the boundary of achievable rate region of the two-user MISO broadcast channel [19]. Thus, maximizing the WSR in MISO downlink systems is equivalent to enlarging the corresponding achievable rate region as much as possible.In this paper, from the perspective of achievable rate region, we investigate the design of the suboptimal algorithm for the solution to the problem (P1) based on transmission mode switching between NOMA-based BF and MMSE-BF.


## 3. Analysis on Rate Regions of NOMA-MRT and MMSE-BF

### 3.1. Achievable Rate Region of NOMA-MRT

Geometrically, the BF vectorwi of user i with MRT is aligned with the spatial direction of hi to maximize the length of the projection of wi onto hi Therefore, the maximum of the signal-to-noise ratio for user i can be achieved by using MRT. The normalized BF vector of user i by using MRT is given by(10)wiMRT=hihi,i=m,n.NOMA-MRT is the combination of NOMA and MRT, i.e., performing MRT beamforming at the transmitter and executing SIC at the receiver with strong channel conditions for two-receiver case.Lemma 1.
Whenhm<hn, the achievable rate region boundary of NOMA-MRT for MISO broadcast channel can be expressed as follows:Case (1). θ>0.(11)rn=log1+ρ˜nhn2,rm=log1+ρ˜mhn21−θ1+ρ˜nhn2,ζn,m≤0orζn,m<ρ˜n≤ρ,log1+ρ˜mhm21+ρ˜nhm21−θ,ζn,m≥ρor0≤ρ˜n≤ζn,m,where(12)ζn,m=hn21−θ−hm2hm2hn22θ−θ2,whereθ=sin2α, ρ=P/N0, ρ˜m=ρ−ρ˜n, and ρ˜i=pi/N0,i=m,n.Case (2).θ=0.(13)rn=log1+ρ˜nhn2,rm=log1+ρ˜mhm21+ρ˜nhm2.Proof.
See the Appendix.Lemma 2.
Whenhm=hn=l, the achievable rate region boundary of NOMA-MRT for MISO broadcast channel can be expressed as(14)rn=log1+pnl2N0,rm=log1+pml2N0+pnl2.Proof.
See the Appendix.


### 3.2. Achievable Rate Region of MMSE-BF

The MMSE-BF can optimally trade off fighting interference to other users and the background Gaussian noise, i.e., the MMSE-BF can maximize the output SINR for any value of signal-to-noise ratio (SNR) [25]. Such a beamformer looks like the zero-forcing beamformer when the interuser interference is large and like the MRT beamformer when the interference is small. The geometric description of normalized BF vectors for ZFBF, MRT, and MMSE-BF is shown in Figure 2.Figure 2
The geometric description of BF vectors for ZFBF, MRT, and MMSE-BF. For ZFBF scheme,wiZFBF is orthogonal to hφi, i=m,n. (a) 0≤α≤π/2. (b) π/2≤α≤π.
(a)(b)By MAC-BC duality theory [25], the MMSE beamformer in BC is exactly the MMSE receiver filters in dual MAC. Therefore, the sets of achievable SINRs are the same in both cases with the same total transmit power constraint. Consequently, we have the following lemma.Lemma 3.
By the duality of MAC and BC, the achievable rate region boundary of BC by using MMSE-BF can be written in terms of that of the dual MAC with MMSE receiver filter. The set of rate tupler˜m,r˜n on the achievable rate region boundary of BC satisfies(15)r˜i=log1+γiM,i=m,n,where(16)γiM=ρihi21+ρφihφi2θ2ςi+ρφihφi21−θ,where ςi=ρφi2hφi4θ+2ρφihφi2θ+1, ρi=qi/N0, and qi is the transmit power of user i in the dual MAC, satisfying qi+qφi=P.Proof.
According to the duality between MAC and BC [25], the normalized BF vectors of user i with MMSE-BF in MISO broadcast channel are given by(17)wiMMSE=N0I+qφihφihφiH−1hiN0I+qφihφihφiH−1hi.
By the matrix inversion lemma [26], equation (17) can be written as(18)wiMMSE=I−ρφihφihφiH/1+ρφihφi2hiI−ρφihφihφiH/1+ρφihφi2hi.
The SINRγiM of user i in dual MAC with MMSE receiver filter has the following form:(19)γiM=qihiHwiMMSE2N0+qφihφiHwiMMSE2.
According to the definition of inner product, equation (19) can be rewritten as(20)γiM=qihi2cos2βiN0+qφihφi2cos2α+βi,α∈0,π/2,(21)γiM=qihi2cos2βiN0+qφihφi2cos2α−βi,α∈π/2,π,where βi represents the angle between wiMMSE and hi, α+βi is the angle between wiMMSE and hφi when α∈0,π/2, and α−βi is that when α∈π/2,π, i=n,m, as illustrated in Figure 2.
The square of the cosine value ofβi can be expressed as(22)cos2βi=wiMMSE,hi2hi2.
By substituting (18) into (22), we can derive(23)cos2βi=hi21+ρφihφi2θ2si21+ρφihφi22,where(24)si=I−ρφihφihφiH1+ρφihφi2hi.
According to the definition of inner product,si can be written as(25)si=hi−ρφihφi21+ρφihφi2hi,hφihφihφi2=hi−ρφihφi21+ρφihφi2∏hφihi,where ∏hφihi represents the orthogonal projection of hi onto hφi. The norm of si is given by(26)si=hi1+1−θgi2−2gi,where gi=ρφihφi2/1+ρφihφi2. By substituting (26) into (23), we can obtain(27)cos2βi=1+ρφihφi2θ21+ρφihφi221+1−θgi2−2gi.
Whenα belongs to the interval 0,π/2, the angle between wiMMSE and hφi is α+βi and the square of the cosine value of α+βi can be expressed as(28)cos2α+βi=1−θcos2βi+θsin2βi−2θ1−θsinβicosβi.
Then, we considerβi∈0,π/2. In this case, by substituting (27) into (28), we can derive(29)cos2α+βi=1−θρφi2hφi4θ+2ρφihφi2θ+1=1−θςi.
Whenα belongs to the interval π/2,π, the angle between wiMMSE and hφi is α−βi. Similarly, we have(30)cos2α−βi=1−θςi.
From (29) and (30), we can find that the expression of cos2α+βi with α∈0,π/2 and that of cos2α−βi with α∈π/2,π are the same. Therefore, equations (20) and (21) can be combined into one equation. We substitute (27) into (20) and (21), and then (29) and (30) into (20) and (21), respectively. Subsequently, by combining (20) and (21), we can obtain (15).


## 3.1. Achievable Rate Region of NOMA-MRT

Geometrically, the BF vectorwi of user i with MRT is aligned with the spatial direction of hi to maximize the length of the projection of wi onto hi Therefore, the maximum of the signal-to-noise ratio for user i can be achieved by using MRT. The normalized BF vector of user i by using MRT is given by(10)wiMRT=hihi,i=m,n.NOMA-MRT is the combination of NOMA and MRT, i.e., performing MRT beamforming at the transmitter and executing SIC at the receiver with strong channel conditions for two-receiver case.Lemma 1.
Whenhm<hn, the achievable rate region boundary of NOMA-MRT for MISO broadcast channel can be expressed as follows:Case (1). θ>0.(11)rn=log1+ρ˜nhn2,rm=log1+ρ˜mhn21−θ1+ρ˜nhn2,ζn,m≤0orζn,m<ρ˜n≤ρ,log1+ρ˜mhm21+ρ˜nhm21−θ,ζn,m≥ρor0≤ρ˜n≤ζn,m,where(12)ζn,m=hn21−θ−hm2hm2hn22θ−θ2,whereθ=sin2α, ρ=P/N0, ρ˜m=ρ−ρ˜n, and ρ˜i=pi/N0,i=m,n.Case (2).θ=0.(13)rn=log1+ρ˜nhn2,rm=log1+ρ˜mhm21+ρ˜nhm2.Proof.
See the Appendix.Lemma 2.
Whenhm=hn=l, the achievable rate region boundary of NOMA-MRT for MISO broadcast channel can be expressed as(14)rn=log1+pnl2N0,rm=log1+pml2N0+pnl2.Proof.
See the Appendix.


## 3.2. Achievable Rate Region of MMSE-BF

The MMSE-BF can optimally trade off fighting interference to other users and the background Gaussian noise, i.e., the MMSE-BF can maximize the output SINR for any value of signal-to-noise ratio (SNR) [25]. Such a beamformer looks like the zero-forcing beamformer when the interuser interference is large and like the MRT beamformer when the interference is small. The geometric description of normalized BF vectors for ZFBF, MRT, and MMSE-BF is shown in Figure 2.Figure 2
The geometric description of BF vectors for ZFBF, MRT, and MMSE-BF. For ZFBF scheme,wiZFBF is orthogonal to hφi, i=m,n. (a) 0≤α≤π/2. (b) π/2≤α≤π.
(a)(b)By MAC-BC duality theory [25], the MMSE beamformer in BC is exactly the MMSE receiver filters in dual MAC. Therefore, the sets of achievable SINRs are the same in both cases with the same total transmit power constraint. Consequently, we have the following lemma.Lemma 3.
By the duality of MAC and BC, the achievable rate region boundary of BC by using MMSE-BF can be written in terms of that of the dual MAC with MMSE receiver filter. The set of rate tupler˜m,r˜n on the achievable rate region boundary of BC satisfies(15)r˜i=log1+γiM,i=m,n,where(16)γiM=ρihi21+ρφihφi2θ2ςi+ρφihφi21−θ,where ςi=ρφi2hφi4θ+2ρφihφi2θ+1, ρi=qi/N0, and qi is the transmit power of user i in the dual MAC, satisfying qi+qφi=P.Proof.
According to the duality between MAC and BC [25], the normalized BF vectors of user i with MMSE-BF in MISO broadcast channel are given by(17)wiMMSE=N0I+qφihφihφiH−1hiN0I+qφihφihφiH−1hi.
By the matrix inversion lemma [26], equation (17) can be written as(18)wiMMSE=I−ρφihφihφiH/1+ρφihφi2hiI−ρφihφihφiH/1+ρφihφi2hi.
The SINRγiM of user i in dual MAC with MMSE receiver filter has the following form:(19)γiM=qihiHwiMMSE2N0+qφihφiHwiMMSE2.
According to the definition of inner product, equation (19) can be rewritten as(20)γiM=qihi2cos2βiN0+qφihφi2cos2α+βi,α∈0,π/2,(21)γiM=qihi2cos2βiN0+qφihφi2cos2α−βi,α∈π/2,π,where βi represents the angle between wiMMSE and hi, α+βi is the angle between wiMMSE and hφi when α∈0,π/2, and α−βi is that when α∈π/2,π, i=n,m, as illustrated in Figure 2.
The square of the cosine value ofβi can be expressed as(22)cos2βi=wiMMSE,hi2hi2.
By substituting (18) into (22), we can derive(23)cos2βi=hi21+ρφihφi2θ2si21+ρφihφi22,where(24)si=I−ρφihφihφiH1+ρφihφi2hi.
According to the definition of inner product,si can be written as(25)si=hi−ρφihφi21+ρφihφi2hi,hφihφihφi2=hi−ρφihφi21+ρφihφi2∏hφihi,where ∏hφihi represents the orthogonal projection of hi onto hφi. The norm of si is given by(26)si=hi1+1−θgi2−2gi,where gi=ρφihφi2/1+ρφihφi2. By substituting (26) into (23), we can obtain(27)cos2βi=1+ρφihφi2θ21+ρφihφi221+1−θgi2−2gi.
Whenα belongs to the interval 0,π/2, the angle between wiMMSE and hφi is α+βi and the square of the cosine value of α+βi can be expressed as(28)cos2α+βi=1−θcos2βi+θsin2βi−2θ1−θsinβicosβi.
Then, we considerβi∈0,π/2. In this case, by substituting (27) into (28), we can derive(29)cos2α+βi=1−θρφi2hφi4θ+2ρφihφi2θ+1=1−θςi.
Whenα belongs to the interval π/2,π, the angle between wiMMSE and hφi is α−βi. Similarly, we have(30)cos2α−βi=1−θςi.
From (29) and (30), we can find that the expression of cos2α+βi with α∈0,π/2 and that of cos2α−βi with α∈π/2,π are the same. Therefore, equations (20) and (21) can be combined into one equation. We substitute (27) into (20) and (21), and then (29) and (30) into (20) and (21), respectively. Subsequently, by combining (20) and (21), we can obtain (15).


## 4. Achievable Rate Region of the Adaptive Switching Method

In this section, we first derive the intersection points of achievable rate region boundaries of NOMA-MRT and MMSE-BF. Based on the intersection points, we proposed the adaptive switching method for the case with two users, which obtains the larger achievable rate region than that derived by employing NOMA-MRT or MMSE-BF only.


### 4.1. The Intersection Points of NOMA-MRT’s and MMSE-BF’s Rate Region Boundaries

According to Lemmas1–3, we can compare the achievable rate regions obtained by NOMA-MRT and MMSE-BF. Then, we can find that neither MMSE-BF nor NOMA-MRT is optimal in all channel states. Consequently, an adaptive switching method is preferred, which can achieve a larger rate region than that derived by MMSE-BF or NOMA-MRT only. Combined with the concept of time-sharing [25], the adaptive switching method can achieve a convex hull of the union of MMSE-BF and NOMA-MRT’s achievable rate region.Definition 1.
Ifrmρ˜n,k=r˜mρn,k and rnρ˜n,k=r˜nρn,k, ρ˜n,k∈0,ρ,ρn,k∈0,ρ, we call that the rate region boundaries of NOMA-MRT and MMSE-BF have intersection points at ρn,k , k=1,2,…,κ, where κ is the number of the intersection points.
The intersection points of NOMA-MRT and MMSE-BF’s rate region boundaries are given as follows.Case (1). When hm<hn.By combining (11) with (15), we obtain two equation sets(31)ρ˜nhn2=ρnhn21+ρmhm2θ2ςn+ρmhm21−θ,(32)ρ˜mhn21−θ1+ρ˜nhn2=ρmhm21+ρnhn2θ2ςm+ρnhn21−θ;(33)ρ˜nhn2=ρnhn21+ρmhm2θ2ςn+ρmhm21−θ,(34)ρ˜mhm21+ρ˜nhm21−θ=ρmhm21+ρnhn2θ2ςm+ρnhn21−θ.By substituting (31) and (33) into (32) and (34), respectively, the intersection points can be derived by solving the following two equations:(35)c1,6ρn6+c1,5ρn5+c1,4ρn4+c1,3ρn3+c1,2ρn2+c1,1ρn+c1,0=0,wherec1,6=−θ4ln3lm3, c1,5=θ3ln2lm23ln−3lm−θln+3ρθlnlmc1,4=−θ2lnlm3ρ2θ2ln2lm2−2ρθ2ln2lm+5ρθln2lm−9ρθlnlm2+ρln2lm+θ2lnlm−2θln2−θlnlm+3ln2−6lnlm+3lm2, c1,3=−θ−ρ3θ3ln3lm3+ρ2θ3ln3lm2−ρ2θ2ln3lm2+9ρ2θ2ln2lm3−2ρ2θln3lm2−2ρθ3ln2lm2+3ρθ2ln3lm+ρθ2ln2lm2−3ρθln3lm+12ρθln2lm2−9ρθlnlm3−ρln3lm+ρln2lm2−2θ2ln2lm+θ2lnlm2+θln3+θln2lm−3θlnlm2−ln3+4ln2lm−4lnlm2+lm3, c1,2=−ρ2θ3ln2lm2−ρ2θ2ln3lmvm−3ρ2θ2ln2lm2vm+3ρ2θ2lnlm3+ρ2θln3lmvm−ρ2θln2lm2+ρθ3ln2lm−ρθ2ln3vm+ρθ2lnlm2+ρθln3vm−2ρθln2lmvm−ρθln2lm+6ρθlnlm2vm+2ρθlnlm2−2ρθlm3−ρln2lmvm+θ2ln2vm2−2θ2lnlmvm+4θlnlmvm−θlm2−ln2vm2+lnlmvm−lm2vm, c1,1=2ρθlnlmvmvn−ρθ2lnlmvn+ρθlnlmvn−ρθlnlmvm+ρθ2ln2vmvn+ρ2θlnlm2vm+2ρlnlmvm−ρln2vmvn−ρθlm2vn−θlnvm2−ρlm2vm−lmvmvnlnvm2, and c1,0=ρvmρlm+1lm−ln+θln, vm=ρθlm+1, vn=ρlm+1.(36)c2,6ρn6+c2,5ρn5+c2,4ρn4+c2,3ρn3+c2,2ρn2+c2,1ρn+c2,0=0.wherec2,6=−θ41−θln2lm3;c2,5θ3lnlm21−θ2ln−2lm+3ρθlnlm;c2,4=−θ21−θlm3ρ2θ2ln2lm2+4ρθln2lm−6ρθlnlm2+ρln2lm+2ln2−3lnlm+lm2; c2,3=θ1−θρ3θ3ln2lm3+2ρ2θ2ln2lm2−6ρ2θ2lnlm2+2ρ2θ2ln2lm2+3ρθln2lm−6ρθlnlm2+3ρθlm3+ρln2lm−ρlnlm2+ln2−2lnlm+lm2, c2,2=1−θ2ρ3θ3lnlm3−ρ3θ2ln2lm2−ρ2θ2ln2lm+3ρ2θ2lnlm2−3ρ2θ2lm3−ρ2θln2lm+2ρ2θlnlm2−ρθln2+3ρθlnlm−2ρθlm2+ρlnlm+ln, and c2,1=ρ1−θρθlm+1ρθlm2−ρlnlm−ln,c2,0=0.According to equation (31) (equation (33)) with ρm=ρ−ρn, we can obtain the result(37)ρ˜n=ρn1+ρ−ρnhm2θ2ςn+ρ−ρnhm21−θ.Hereςn=hm4θρ−ρn2+2hm2θρ−ρn+1. If the rate region boundaries of NOMA-MRT and MMSE-BF have intersection point at ρn,k, k=1,2,…,κ, then ρn,k must satisfy equation (37)For equation (35), we consider ρn∈0,ρ,ρ˜n∈0,ρ when ζn,m≤0, or consider ρn∈0,ρ, ρ˜n∈ζn,m,ρ when 0<ζn,m<ρ, as shown in (11). In this case, the intersection point ρn,kk∈1,2,…,κ must satisfy ρn,k∈0,ρ and ρ˜n,k∈0,ρ when ζn,m≤0 or ρn,k∈0,ρ and ρ˜n,k∈ζn,m,ρ when 0<ζn,m<ρ, where(38)ρ˜n,k=ρn,k1+ρ−ρn,khm2θ2ςn,k+ρ−ρn,khm21−θ,(39)ςn,k=hm4θρ−ρn,k2+2hm2θρ−ρn,k+1.Similarly, for equation (36), the intersection point ρn,kk∈1,2,…,κ must satisfy ρn,k∈0,ρ, ρ˜n,k∈0,ρ when ζn,m≥ρ or ρn,k∈0,ρ, ρ˜n,k∈0,ζn,m when 0<ζn,m<ρ.Since deriving the analytic solution of (35) and (36) is not a trivial thing, we use numerical method to derive the solutions.Case (2). When hm=hn=l.By combining (14) with (15), we derive the equation set(40)ρ˜nl2=ρnl21+ρml2θ2ςn+ρml21−θ,(41)ρ˜nl21+ρ˜ml2=ρml21+ρnl2θ2ςm+ρnl21−θ.By substituting (40) into (41), the intersection points can be obtained by solving the following equation:(42)ρn2−ρρn−2ρθl2+2θ−ρl2+1θ2l4ρn−ρρn+1θl2ρn−ρθl2+1θl2ρn=0.By solving equation (42), we can derive the results(43)ρn,1=0,ρn,2=ρ,ρn,3=−1θl2,ρn,4=ρθl2+1θl2,ρn,5,6=ρθl2±ρ2θ2l4+8ρθl2−4ρl2−8θ+42θl2,whereρn,k∈0,ρ, k=1,2,…,6.In the following, we denote byp˜ipi the power allocated to user i in the BC with MMSE-BF (NOMA-MRT) scheme, i=m,n. Since the transmit power qi of user i in dual MAC is not equal to the allocated power p˜i for user i in the BC, we should obtain the allocated power p˜i,k of user i to design the switching method in the BC, which corresponds to the intersection point ρi,k=qi,k/N0. According to the duality between MAC and BC, the SINR at each point on the achievable rate region boundary of BC by performing MMSE-BF is equal to that on the rate region boundary of MAC with MMSE receiver filter at the BS, i.e.,(44)γiB=γiM.We substitute (18) into (5), then substitute (5) and (16) into (44), and replace pi with p˜i by abuse of notation. As a result, p˜i can be expressed as(45)p˜i=qieφiei+ρhi21−θeieφi+1−θρihi2eφi+ρφihφi2ei,whereei=hi4ρi2θ+2hi2ρiθ+1, eφi=ρφi2hφi4θ+2ρφihφi2θ+1, i=m,n.


### 4.2. Adaptive Switching Method to Achieving a Larger Rate Region

Based on the aforementioned intersection points, we develop an adaptive switching method to achieve a larger rate region, i.e., Algorithm1, which outputs some parameters such as the user’s transmit power and BF vector in any channel condition. Combined with time-sharing, the switching method can achieve a convex hull of the union of NOMA-MRT and MMSE-BF’s achievable rate region.Algorithm 1:Adaptive switching method.
Input:hi∈ℂM×1 (assuming hm<hn ),i=m,n.Output:pm,n, wi, i=m,n.(1)
Step 1: obtain the intersection point ρn,k, k = 1, 2, …, κ.(2)
ifhm<hn(3)
Calculateρn,k using (35) and (36)(4)
else ifhm=hn(5)
Obtainρn,k using (43)(6)
end if(7)
Step 2: define as the mode switching point the intersection point excluding the ends of the interval 0,ρ and sort the mode switching points in ascending order, i.e., ρn,1≤⋯≤ρn,j≤⋯≤ρn,Γ, where Γ is the number of mode switching points. Divide rate region boundary into Γ+1 sections by ρn,j.(8)
ifΓ=0(9)
if (15) is larger compared to (11) or (14), in 0,ρ(10)
Go toStep 3(11)
Else(12)
Go toStep 4(13)
end if(14)
else divide region boundary (15) into Γ+1 sections by ρn,j, j=1,2,…,Γ.(15)
for each section between ρn,j and ρn,j+1ρn,0=0,ρn,Γ+1=ρ(16)
if (15) is larger compared to (11) or (14), in this section(17)
Go toStep 3(18)
else(19)
Go toStep 4(20)
end if(21)
end for(22)
end if(23)
Step 3: Applying MMSE-BF(24)
Beamform withwm,wn described in (18) and transmit x=p˜mwmsm+p˜nwnsn, with pm,n=P−p˜n,p˜n, where p˜n is calculated by (45), with qn=ρnN0, ρm=ρ−ρn and ρn∈ρn,j,ρn,j+1.(25)
Step 4: performing NOMA-MRT(26)
Beamform withwm,wn expressed in (10) and transmit x with pm,n=P−pn,pn, where pn=ρ˜nN0, ρ˜n∈ρ˜n,j,ρ˜n,j+1 and ρ˜n,j is calculated by (38) (replacing subscript k with subscript j).


## 4.1. The Intersection Points of NOMA-MRT’s and MMSE-BF’s Rate Region Boundaries

According to Lemmas1–3, we can compare the achievable rate regions obtained by NOMA-MRT and MMSE-BF. Then, we can find that neither MMSE-BF nor NOMA-MRT is optimal in all channel states. Consequently, an adaptive switching method is preferred, which can achieve a larger rate region than that derived by MMSE-BF or NOMA-MRT only. Combined with the concept of time-sharing [25], the adaptive switching method can achieve a convex hull of the union of MMSE-BF and NOMA-MRT’s achievable rate region.Definition 1.
Ifrmρ˜n,k=r˜mρn,k and rnρ˜n,k=r˜nρn,k, ρ˜n,k∈0,ρ,ρn,k∈0,ρ, we call that the rate region boundaries of NOMA-MRT and MMSE-BF have intersection points at ρn,k , k=1,2,…,κ, where κ is the number of the intersection points.
The intersection points of NOMA-MRT and MMSE-BF’s rate region boundaries are given as follows.Case (1). When hm<hn.By combining (11) with (15), we obtain two equation sets(31)ρ˜nhn2=ρnhn21+ρmhm2θ2ςn+ρmhm21−θ,(32)ρ˜mhn21−θ1+ρ˜nhn2=ρmhm21+ρnhn2θ2ςm+ρnhn21−θ;(33)ρ˜nhn2=ρnhn21+ρmhm2θ2ςn+ρmhm21−θ,(34)ρ˜mhm21+ρ˜nhm21−θ=ρmhm21+ρnhn2θ2ςm+ρnhn21−θ.By substituting (31) and (33) into (32) and (34), respectively, the intersection points can be derived by solving the following two equations:(35)c1,6ρn6+c1,5ρn5+c1,4ρn4+c1,3ρn3+c1,2ρn2+c1,1ρn+c1,0=0,wherec1,6=−θ4ln3lm3, c1,5=θ3ln2lm23ln−3lm−θln+3ρθlnlmc1,4=−θ2lnlm3ρ2θ2ln2lm2−2ρθ2ln2lm+5ρθln2lm−9ρθlnlm2+ρln2lm+θ2lnlm−2θln2−θlnlm+3ln2−6lnlm+3lm2, c1,3=−θ−ρ3θ3ln3lm3+ρ2θ3ln3lm2−ρ2θ2ln3lm2+9ρ2θ2ln2lm3−2ρ2θln3lm2−2ρθ3ln2lm2+3ρθ2ln3lm+ρθ2ln2lm2−3ρθln3lm+12ρθln2lm2−9ρθlnlm3−ρln3lm+ρln2lm2−2θ2ln2lm+θ2lnlm2+θln3+θln2lm−3θlnlm2−ln3+4ln2lm−4lnlm2+lm3, c1,2=−ρ2θ3ln2lm2−ρ2θ2ln3lmvm−3ρ2θ2ln2lm2vm+3ρ2θ2lnlm3+ρ2θln3lmvm−ρ2θln2lm2+ρθ3ln2lm−ρθ2ln3vm+ρθ2lnlm2+ρθln3vm−2ρθln2lmvm−ρθln2lm+6ρθlnlm2vm+2ρθlnlm2−2ρθlm3−ρln2lmvm+θ2ln2vm2−2θ2lnlmvm+4θlnlmvm−θlm2−ln2vm2+lnlmvm−lm2vm, c1,1=2ρθlnlmvmvn−ρθ2lnlmvn+ρθlnlmvn−ρθlnlmvm+ρθ2ln2vmvn+ρ2θlnlm2vm+2ρlnlmvm−ρln2vmvn−ρθlm2vn−θlnvm2−ρlm2vm−lmvmvnlnvm2, and c1,0=ρvmρlm+1lm−ln+θln, vm=ρθlm+1, vn=ρlm+1.(36)c2,6ρn6+c2,5ρn5+c2,4ρn4+c2,3ρn3+c2,2ρn2+c2,1ρn+c2,0=0.wherec2,6=−θ41−θln2lm3;c2,5θ3lnlm21−θ2ln−2lm+3ρθlnlm;c2,4=−θ21−θlm3ρ2θ2ln2lm2+4ρθln2lm−6ρθlnlm2+ρln2lm+2ln2−3lnlm+lm2; c2,3=θ1−θρ3θ3ln2lm3+2ρ2θ2ln2lm2−6ρ2θ2lnlm2+2ρ2θ2ln2lm2+3ρθln2lm−6ρθlnlm2+3ρθlm3+ρln2lm−ρlnlm2+ln2−2lnlm+lm2, c2,2=1−θ2ρ3θ3lnlm3−ρ3θ2ln2lm2−ρ2θ2ln2lm+3ρ2θ2lnlm2−3ρ2θ2lm3−ρ2θln2lm+2ρ2θlnlm2−ρθln2+3ρθlnlm−2ρθlm2+ρlnlm+ln, and c2,1=ρ1−θρθlm+1ρθlm2−ρlnlm−ln,c2,0=0.According to equation (31) (equation (33)) with ρm=ρ−ρn, we can obtain the result(37)ρ˜n=ρn1+ρ−ρnhm2θ2ςn+ρ−ρnhm21−θ.Hereςn=hm4θρ−ρn2+2hm2θρ−ρn+1. If the rate region boundaries of NOMA-MRT and MMSE-BF have intersection point at ρn,k, k=1,2,…,κ, then ρn,k must satisfy equation (37)For equation (35), we consider ρn∈0,ρ,ρ˜n∈0,ρ when ζn,m≤0, or consider ρn∈0,ρ, ρ˜n∈ζn,m,ρ when 0<ζn,m<ρ, as shown in (11). In this case, the intersection point ρn,kk∈1,2,…,κ must satisfy ρn,k∈0,ρ and ρ˜n,k∈0,ρ when ζn,m≤0 or ρn,k∈0,ρ and ρ˜n,k∈ζn,m,ρ when 0<ζn,m<ρ, where(38)ρ˜n,k=ρn,k1+ρ−ρn,khm2θ2ςn,k+ρ−ρn,khm21−θ,(39)ςn,k=hm4θρ−ρn,k2+2hm2θρ−ρn,k+1.Similarly, for equation (36), the intersection point ρn,kk∈1,2,…,κ must satisfy ρn,k∈0,ρ, ρ˜n,k∈0,ρ when ζn,m≥ρ or ρn,k∈0,ρ, ρ˜n,k∈0,ζn,m when 0<ζn,m<ρ.Since deriving the analytic solution of (35) and (36) is not a trivial thing, we use numerical method to derive the solutions.Case (2). When hm=hn=l.By combining (14) with (15), we derive the equation set(40)ρ˜nl2=ρnl21+ρml2θ2ςn+ρml21−θ,(41)ρ˜nl21+ρ˜ml2=ρml21+ρnl2θ2ςm+ρnl21−θ.By substituting (40) into (41), the intersection points can be obtained by solving the following equation:(42)ρn2−ρρn−2ρθl2+2θ−ρl2+1θ2l4ρn−ρρn+1θl2ρn−ρθl2+1θl2ρn=0.By solving equation (42), we can derive the results(43)ρn,1=0,ρn,2=ρ,ρn,3=−1θl2,ρn,4=ρθl2+1θl2,ρn,5,6=ρθl2±ρ2θ2l4+8ρθl2−4ρl2−8θ+42θl2,whereρn,k∈0,ρ, k=1,2,…,6.In the following, we denote byp˜ipi the power allocated to user i in the BC with MMSE-BF (NOMA-MRT) scheme, i=m,n. Since the transmit power qi of user i in dual MAC is not equal to the allocated power p˜i for user i in the BC, we should obtain the allocated power p˜i,k of user i to design the switching method in the BC, which corresponds to the intersection point ρi,k=qi,k/N0. According to the duality between MAC and BC, the SINR at each point on the achievable rate region boundary of BC by performing MMSE-BF is equal to that on the rate region boundary of MAC with MMSE receiver filter at the BS, i.e.,(44)γiB=γiM.We substitute (18) into (5), then substitute (5) and (16) into (44), and replace pi with p˜i by abuse of notation. As a result, p˜i can be expressed as(45)p˜i=qieφiei+ρhi21−θeieφi+1−θρihi2eφi+ρφihφi2ei,whereei=hi4ρi2θ+2hi2ρiθ+1, eφi=ρφi2hφi4θ+2ρφihφi2θ+1, i=m,n.


## 4.2. Adaptive Switching Method to Achieving a Larger Rate Region

Based on the aforementioned intersection points, we develop an adaptive switching method to achieve a larger rate region, i.e., Algorithm1, which outputs some parameters such as the user’s transmit power and BF vector in any channel condition. Combined with time-sharing, the switching method can achieve a convex hull of the union of NOMA-MRT and MMSE-BF’s achievable rate region.Algorithm 1:Adaptive switching method.
Input:hi∈ℂM×1 (assuming hm<hn ),i=m,n.Output:pm,n, wi, i=m,n.(1)
Step 1: obtain the intersection point ρn,k, k = 1, 2, …, κ.(2)
ifhm<hn(3)
Calculateρn,k using (35) and (36)(4)
else ifhm=hn(5)
Obtainρn,k using (43)(6)
end if(7)
Step 2: define as the mode switching point the intersection point excluding the ends of the interval 0,ρ and sort the mode switching points in ascending order, i.e., ρn,1≤⋯≤ρn,j≤⋯≤ρn,Γ, where Γ is the number of mode switching points. Divide rate region boundary into Γ+1 sections by ρn,j.(8)
ifΓ=0(9)
if (15) is larger compared to (11) or (14), in 0,ρ(10)
Go toStep 3(11)
Else(12)
Go toStep 4(13)
end if(14)
else divide region boundary (15) into Γ+1 sections by ρn,j, j=1,2,…,Γ.(15)
for each section between ρn,j and ρn,j+1ρn,0=0,ρn,Γ+1=ρ(16)
if (15) is larger compared to (11) or (14), in this section(17)
Go toStep 3(18)
else(19)
Go toStep 4(20)
end if(21)
end for(22)
end if(23)
Step 3: Applying MMSE-BF(24)
Beamform withwm,wn described in (18) and transmit x=p˜mwmsm+p˜nwnsn, with pm,n=P−p˜n,p˜n, where p˜n is calculated by (45), with qn=ρnN0, ρm=ρ−ρn and ρn∈ρn,j,ρn,j+1.(25)
Step 4: performing NOMA-MRT(26)
Beamform withwm,wn expressed in (10) and transmit x with pm,n=P−pn,pn, where pn=ρ˜nN0, ρ˜n∈ρ˜n,j,ρ˜n,j+1 and ρ˜n,j is calculated by (38) (replacing subscript k with subscript j).


## 5. Joint User Pairing and Power Allocation with Transmission Mode Switching

In this section, for the problem of WSR maximization, we first proposed two optimal power allocation algorithms based on the concept of the achievable rate region, i.e., power allocation algorithm for NOMA-MRT mode (NOMA-MRT-PA) and power allocation algorithm for MMSE-BF mode (MMSE-BF-PA), by focusing on two-user case. Then, based on the two aforementioned proposed algorithms, a joint user pairing and power allocation algorithm (JUPA) is then developed for multiuser case. Finally, a practical transmission method is proposed by combining JUPA with transmission mode switching between NOMA-MRT and MMSE-BF.


### 5.1. Achieving Maximum WSR on the Rate Region Boundary for NOMA-MRT Mode

According to [19], the solution to the problem of WSR maximization must be on the boundary of achievable rate region of two-user MISO downlink systems. As a result, when the NOMA-MRT mode is employed at BS, the problem of WSR maximization is formulated as(46)P2:maxpm,nUpm,n≔μmrm+μnrn,s.t.pm,n∈Pm,n,where rii=m,n is the achievable rate of user i, defined in Lemma 1 or Lemma 2, and Pm,n=pm,n|0≤pi≤P,pi+pφi=P,i=m,n is the feasible set of power allocation vector for user m and n, in which the power allocation vector corresponding to rate point on the rate region boundary satisfies full power allocation, i.e., pi+pφi=P.For the simplicity of notation, we letUρ˜n stand for the WSR Upm,n in (46) with pn=ρ˜nN0 and pm=P−pn. The motivations behind proposing NOMA-MRT-PA are as following:(1)
According to Lemmas1 and 2, the expression of the achievable rate region boundary of NOMA-MRT can be given in any channel conditions. Furthermore, the achievable rate ri is a differentiable or piecewise differentiable function of ρ˜n in the interval 0,ρ.(2)
The global maximum value ofUρ˜n in the interval 0,ρ can be found by selecting the maximum among U0,Uρ and those corresponding to which the first-order derivative of Uρ˜n is zeros in the interval 0,ρ. As we know, the local maximum values of Uρ˜n in the interval 0,ρ satisfy that the first-order derivative of Uρ˜n is equal to zero.Theorem 1.
Whenhm<hn, the solution of Problem (P2) is as follows.Case (1). ζn,m≤0.(47)pm,n∗=pm∗,pn∗,wherepn∗=ρ˜n∗N0,pm∗=P−pn∗,ρ˜n∗=argmaxρ˜nU0,Uρ,Uρ˜n,1, and(48)ρ˜n,1=μnhn2ρθ−1+ρhn2μn−μm1−θμnhn2θ,ρ˜n,1∈0,ρ.Case (2). ζn,m≥ρ.(49)pm,n∗=pm∗,pn∗,wherepn∗=ρ˜n∗N0,pm∗=P−pn∗, ρ˜n∗=argmaxρ˜nU0,Uρ,Uρ˜n,2,Uρ˜n,3, and(50)ρ˜n,2,3=−ϵ1±ϵ12−4ϵ2ϵ02ϵ2,whereϵ2=μnhn2hm4θ1−θ, ϵ1=hn2μmϵ3−μnϵ4,ϵ0=μmϵ3−μnhn21+hm2ρ, ϵ3=hm2hm2ρ1−θ+1, ϵ4=hm2hm2ρ1−θ−2θ+1, ρ˜n,2∈0,ρ, and ρ˜n,3∈0,ρ.Case (3). 0<ζn,m<ρ.(51)pm,n∗=pm∗,pn∗,wherepn∗=ρ˜n∗N0,pm∗=P−pn∗, ρ˜n∗=argmaxρ˜nU0,Uρ,Uζn,m,Uρ˜n,1,Uρ˜n,2,Uρ˜n,3, ρ˜n,1∈ζn,m,ρ, ρ˜n,2∈0,ζn,m, and ρ˜n,3∈0,ζn,m.Proof.
See the Appendix.Theorem 2.
Whenhm=hn, the solution of Problem (P2) is pm,n∗=P,0 if μm≥μn. Otherwise, the solution of Problem (P2) is pm,n∗=0,P.Proof.
See the Appendix.
For the simplicity of description of the proposed NOMA-MRT-PA, we assumehm≤hn. By Theorems 1 and 2, NOMA-MRT-PA is proposed, i.e., Algorithm 2.Algorithm 2:NOMA-MRT-PA.
Input:μi, hi∈ℂM×1(assuming hm<hn ), i=m,n.Output:pm,n∗.(1)
ifhm<hn(2)
Calculateζn,m using (12)(3)
ifζn,m≤0(4)
pm,n∗=P−ρ˜n∗N0,ρ˜n∗N0, where ρ˜n∗=argmaxρ˜nU0,Uρ,Uρ˜n,1, Uρ˜n,1 is calculated by (48), ρ˜n,1∈0,ρ.(5)
else ifζn,m≥ρ(6)
pm,n∗=P−ρ˜n∗N0,ρ˜n∗N0, where ρ˜n∗=argmaxρ˜nU0,Uρ,Uρ˜n,2,Uρ˜n,3, Uρ˜n,2 and Uρ˜n,3 are calculated by (50), ρ˜n,2∈0,ρ,ρ˜n,3∈0,ρ.(7)
else(8)
pm,n∗=P−ρ˜n∗N0,ρ˜n∗N0, where ρ˜n∗=argmaxρ˜nU0,Uρ,Uζn,m,Uρ˜n,1,Uρ˜n,2,Uρ˜n,3,ρ˜n,1∈ζn,m,ρ, ρ˜n,2∈0,ζn,m and ρ˜n,3∈0,ζn,m(9)
end if(10)
else(11)
Ifμm≥μn, pm,n∗=P,0. Otherwise, pm,n∗=0,P.(12)
end if


### 5.2. Achieving Maximum WSR on Rate Region Boundary for MMSE-BF Mode

When the MMSE-BF mode is performed at BS, the problem of WSR maximization can be formulated as(52)P3:maxqm,nUqm,n≔μmr˜m+μnr˜n,s.t.qm,n∈Qm,n,where qm,n=qm,qn is transmit power vector for users m and n in the dual MAC, r˜i and qi are defined in Lemma 3, and Qm,n=qm,n|0≤qi≤P,qi+qφi=P,i=m,n is the feasible set of transmit power vector for users m and n.For the simplicity of notation, we letUρm stand for the WSR Uqm,n in (52) with qm=ρmN0 and qn=P−qm. With the similar idea to that, by which Theorem 1 was proposed, we have the following theorem.Theorem 3.
The solution of Problem (P3) is qm,n∗=qm∗,qn∗ in dual two-user MAC, where qm∗=ρm∗N0, qn∗=P−qm∗, ρm∗=argmaxρmU0,Uρ,Uρm,1,Uρm,2,…,Uρm,9, and ρm,i∈0,ρ, which are the roots of the nine-degree equation U′ρm=0, which is given by (54).Proof.
According to Lemma3, the WSR Uρm can be expressed as(53)Uρm=μmlog1+ρmhm21+ρnhn2θ2ςm+ρnhn21−θ+μnlog1+ρnhn21+ρmhm2θ2ςn+ρmhm21−θ,where ρn=ρ−ρm. In this case, Uρm is differentiable function of ρm in the interval 0,ρ. By letting U′ρm=0, we can obtain a nine-degree equation:(54)a9ρm9+a8ρm8+a7ρm7+a6ρm6+a5ρm5+a4ρm4+a3ρm3+a2ρm2+a1ρm+a0=0.
Since the coefficients of equation (54) are very complicated, we omit them here. In this case, the global maximum value of Uρm in the interval 0,ρ can be found by selecting the maximum among U0,Uρ,Uρm,1,Uρm,2,…,Uρm,9. Consequently, the solution of Problem (P3) is qm,n∗=qm∗,qn∗ in dual two-user MAC, where qm∗=ρm∗N0, qn∗=P−qm∗, ρm∗=argmaxρmU0,Uρ,Uρm,1,Uρm,2,…,Uρm,9, and ρm,i∈0,ρ,i=1,2,…,9. If ρm,i∉0,ρ, i=1,2,…,9, we let Uρm,i=0.
By Theorem3, MMSE-BF-PA is proposed, i.e., Algorithm 3.Algorithm 3:MMSE-BF-PA.
Input:μi, hi∈ℂM×1,i=m,n.Output:pm,n∗.(1)
Calculateρm,i using (54), i=1,2,…,9.(2)
ρm∗=argmaxρmU0,Uρ,Uρm,1,Uρm,2,…,Uρm,9, where ρm,i∈0,ρ.(3)
pm,n∗=p˜m∗,P−p˜m∗, where p˜m∗ is calculated by (45), with qm∗=ρm∗N0.


### 5.3. Joint User Pairing and Power Allocation (JUPA) Algorithm

With the aim of maximizing the total sum rate, in the step of forming a user pair in JUPA, we compare the two sum rates obtained by NOMA-MRT-PA and MMSE-BF-PA, select the transmission mode with the larger sum rate, and then choose the corresponding user pair to send data. By focusing on the expression of rate region boundary of NOMA-MRT in (11), it is worthwhile noticing the following properties:(1)
Considering the original user pairing problem which is how to pair useri with a user from the set of other unpaired users to obtain the maximum sum rate, we can divide all of unpaired users besides user i into 3 subsets, i.e., Φ1i, Φ2i, and Φ3i, according to ζi,i′. Particularly, user i′ in Φ1i, Φ2i, and Φ3i, satisfies ζi,i′≤0, ζi,i′≥ρ and 0<ζi,i′<ρ, respectively. The original user pairing problem is equivalent to 3 problems, which are how to pair user i with a user to obtain the local-maximum sum rate from Φ1i, Φ2i, and Φ3i, respectively. Therefore, the solution of the original user pairing problem is the user pair, which corresponds to the maximum among the local maximum sum rates for Φ1i, Φ2i, and Φ3i.(2)
Considering the problem which is how to pair useri with a user from Φ1iΦ2i to achieve local maximum sum rate, user i should be paired with the user, whose channel vector can form the minimum (maximum) angle with that of user i, according to (11).(3)
For the problem which is how to pair useri with a user from Φ3i to achieve local maximum sum rate, we divide Φ3i into 2 subsets, i.e., Φ3,1i and Φ3,2i. The user i′ in Φ3,1i and Φ3,2i satisfies 0<ζi,i′≤ρ/2 and ρ/2<ζi,i′<ρ, respectively. According to (11), the formula ri′=log1+ρ˜i′hi21−θ/1+ρ˜ihi2 provides dominant contribution to rate region boundary formed by user i and i′, when 0<ζi,i′≤ρ/2. In this case, for the treatability of the problem, we use the formula ri′=log1+ρ˜i′hi21−θ/1+ρ˜ihi2 as the expression of ri′ in the whole interval 0,ρ, instead of the piecewise formula as shown in (11). Thus, for the problem which is how to pair user i with a user from Φ3,1i, user i should be paired with the user whose channel vector can form the minimum angle with that of user i. With the similar analysis, user i should be paired with the user in Φ3,2i whose channel vector can form the maximum angle with that of user i.For the simplicity of description of the proposed JUPA (Algorithm4), assume that users are ordered as h1≤⋯≤h2K. Let Φ1ikΦ2ik denote the index of k-th entry in Φ1iΦ2i. Besides, let Φ3,1ikΦ3,2ik denote the index of k-th entry in Φ3,1iΦ3,2i.Algorithm 4:Joint user pairing and power allocation algorithm (JUPA).
Input:hi∈ℂM×1 (assuming h1≤⋯≤h2K ), μi=1,i=1,2,…,2K.Output:Ω and UInitial:Ω=∅ and U=02K×2K.(1)
fori=1:2K−1do(2)
if user i has been paired(3)
Go to line 1(4)
end if(5)
forj=i+1:2Kdo(6)
if user j has been paired(7)
Go to line 5(8)
end if(9)
ifhi and hj is orthogonal(10)
Obtainpi,j by MMSE-BF-PA, Ω=Ω∪​i,j,pi,j,0 and go to line 1(11)
end if(12)
end for(13)
Findk1=argmaxkcos2∠hi,hΦ1ik and obtain the power allocation solution pi,Φ1ik1 for the user pair i,Φ1ik1 by NOMA-MRT-PA(14)
Findk2=argminkcos2∠hi,hΦ2ik and obtain the power allocation solution pi,Φ2ik2 for the user pair i,Φ2ik2 by NOMA-MRT-PA(15)
Findk3,1=argmaxkcos2∠hi,hΦ3,1ik and obtain the power allocation solution pi,Φ3,1ik3,1 for the user pair i,Φ3,1ik3,1 by NOMA-MRT-PA(16)
Findk3,2=argminkcos2∠hi,hΦ3,2ik and obtain the power allocation solution pi,Φ3,2ik3,2 for the user pair i,Φ3,2ik3,2 by NOMA-MRT-PA(17)
Obtain the power allocation vector with the maximum WSR, from the solutions obtained bylines 13–16:pN=argmaxpUpi,Φ1ik1,Upi,Φ2ik2,Upi,Φ3,1ik3,1,Upi,Φ3,2ik3,2.Leti,jN denote the user pair which corresponds to pN.(18)
FindjM=argminjcos∠hi,hj and obtain the power allocation solution pi,jM for the user pair i,jM by MMSE-BF-PA.(19)
ifUpN≥Upi,jM(20)
Ω=Ω∪​i,jN,pN,1 and ui,jN=ujN,i=1.(21)
else(22)
Ω=Ω∪​i,jM,pi,jM,0 and ui,jM=ujM,i=1.(23)
end if(24)
end forIn Algorithm4, line 13 (14, 15, and 16) is the step for selecting a user to be paired with user i and deriving the power allocation solution for the obtained user pair by NOMA-MRT-PA, in the set Φ1i (Φ2i, Φ3,1i, and Φ3,2i). Line 18 is the step for similar operation by MMSE-BF-PA. The output of Algorithm 4, Ω, is termed the pairing and power allocation configuration, in which K entries are included and 4 elements are contained in each entry. The first two elements of i-th entry in Ω denote the indexes of the two users in i-th user pair, and the third and fourth element stand for power allocation vector and transmission mode adopted by the i-th user pair, respectively.When the number of users is odd, the last remained user can be trivially served by OMA. Specifically, if the channel vector of the last remained user ishl, then MRT BF is performed at the BS and the BF vector is given by (10). As a result, the maximum signal-to-noise ratio can be achieved.


#### 5.3.1. Complexity Analysis

The computational complexity of the proposed JUPA mainly comes from the computation of the angle between channels of two users and from solving the nine-degree equation (54), by which the optimal power allocation scheme is obtained for MMSE-BF mode.Note that the computational complexity for computing the angle between channels of two users isOM. In addition, we use upper-case letter “C” to indicate the computational complexity for solving the nine-degree equation (54). As the number of computing the angle is 2K−2i−1−1 in i-th outer “for” loop in Algorithm 4, for the MMSE-BF mode or NOMA-MRT mode, the computational complexity for the user pairing process is O2K/22M. On the other hand, the computational complexity for the power allocation process is O2K/2C. Consequently, the overall computational complexity of Algorithm 4 is O2K/22M+O2K/2C.For the exhaustive search-based scheme, i.e., exhaustive search-based user pairing and power allocation (ES-PPA) described in Section6.2, the number of all possible pairing schemes is 2K−1!!. In each pairing scheme, there are K user pairs. Then, for each user pair, the angle between channels of two users and the allocated power of two users for MMSE-BF mode should be calculated. Therefore, the total computational complexity of exhaustive search-based scheme is O2K−1!!2K/2M+O2K−1!!2K/2C.


### 5.4. JUPA/TMS: A Practical Transmission Method

By combining JUPA with transmission mode switching (TMS) between NOMA-MRT and MMSE-BF, a practical transmission scheme termed JUPA/TMS is proposed. JUPA/TMS is described in Algorithm5, where S is the control parameter for transmission mode switching and Ωij denotes the j-th element in the i-th entry in the set Ω. Noting that TDMA(FDMA) is adopted in Algorithm 5 (line10), it is efficient when M is small. When the scenario with sufficient number of antennas at base station is considered, e.g., M≥2K, the performance can be further improved by combining the proposed JUPA with SDMA, which is beyond the scope of this paper.Algorithm 5:JUPA/TMS.
Input:hi∈ℂM×1si, i=1,2,…,2K.Output:x1,x2,…,xK.(1)
Perform JUPA to obtain pairing and power allocation configurationΩ(2)
fori=1:Kdo(3)
m,n,pm,n,Tm,n←Ωi1,Ωi2,Ωi3,Ωi4(4)
ifTm,n=1(5)
Calculatewm and wn with (10)(6)
else ifTm,n=0(7)
Calculatewm and wn with (18)(8)
end if(9)
xi←pmwmsm+pnwnsn(10)
Transmitxi in i-th frequency/time slot(11)
end for


## 5.1. Achieving Maximum WSR on the Rate Region Boundary for NOMA-MRT Mode

According to [19], the solution to the problem of WSR maximization must be on the boundary of achievable rate region of two-user MISO downlink systems. As a result, when the NOMA-MRT mode is employed at BS, the problem of WSR maximization is formulated as(46)P2:maxpm,nUpm,n≔μmrm+μnrn,s.t.pm,n∈Pm,n,where rii=m,n is the achievable rate of user i, defined in Lemma 1 or Lemma 2, and Pm,n=pm,n|0≤pi≤P,pi+pφi=P,i=m,n is the feasible set of power allocation vector for user m and n, in which the power allocation vector corresponding to rate point on the rate region boundary satisfies full power allocation, i.e., pi+pφi=P.For the simplicity of notation, we letUρ˜n stand for the WSR Upm,n in (46) with pn=ρ˜nN0 and pm=P−pn. The motivations behind proposing NOMA-MRT-PA are as following:(1)
According to Lemmas1 and 2, the expression of the achievable rate region boundary of NOMA-MRT can be given in any channel conditions. Furthermore, the achievable rate ri is a differentiable or piecewise differentiable function of ρ˜n in the interval 0,ρ.(2)
The global maximum value ofUρ˜n in the interval 0,ρ can be found by selecting the maximum among U0,Uρ and those corresponding to which the first-order derivative of Uρ˜n is zeros in the interval 0,ρ. As we know, the local maximum values of Uρ˜n in the interval 0,ρ satisfy that the first-order derivative of Uρ˜n is equal to zero.Theorem 1.
Whenhm<hn, the solution of Problem (P2) is as follows.Case (1). ζn,m≤0.(47)pm,n∗=pm∗,pn∗,wherepn∗=ρ˜n∗N0,pm∗=P−pn∗,ρ˜n∗=argmaxρ˜nU0,Uρ,Uρ˜n,1, and(48)ρ˜n,1=μnhn2ρθ−1+ρhn2μn−μm1−θμnhn2θ,ρ˜n,1∈0,ρ.Case (2). ζn,m≥ρ.(49)pm,n∗=pm∗,pn∗,wherepn∗=ρ˜n∗N0,pm∗=P−pn∗, ρ˜n∗=argmaxρ˜nU0,Uρ,Uρ˜n,2,Uρ˜n,3, and(50)ρ˜n,2,3=−ϵ1±ϵ12−4ϵ2ϵ02ϵ2,whereϵ2=μnhn2hm4θ1−θ, ϵ1=hn2μmϵ3−μnϵ4,ϵ0=μmϵ3−μnhn21+hm2ρ, ϵ3=hm2hm2ρ1−θ+1, ϵ4=hm2hm2ρ1−θ−2θ+1, ρ˜n,2∈0,ρ, and ρ˜n,3∈0,ρ.Case (3). 0<ζn,m<ρ.(51)pm,n∗=pm∗,pn∗,wherepn∗=ρ˜n∗N0,pm∗=P−pn∗, ρ˜n∗=argmaxρ˜nU0,Uρ,Uζn,m,Uρ˜n,1,Uρ˜n,2,Uρ˜n,3, ρ˜n,1∈ζn,m,ρ, ρ˜n,2∈0,ζn,m, and ρ˜n,3∈0,ζn,m.Proof.
See the Appendix.Theorem 2.
Whenhm=hn, the solution of Problem (P2) is pm,n∗=P,0 if μm≥μn. Otherwise, the solution of Problem (P2) is pm,n∗=0,P.Proof.
See the Appendix.
For the simplicity of description of the proposed NOMA-MRT-PA, we assumehm≤hn. By Theorems 1 and 2, NOMA-MRT-PA is proposed, i.e., Algorithm 2.Algorithm 2:NOMA-MRT-PA.
Input:μi, hi∈ℂM×1(assuming hm<hn ), i=m,n.Output:pm,n∗.(1)
ifhm<hn(2)
Calculateζn,m using (12)(3)
ifζn,m≤0(4)
pm,n∗=P−ρ˜n∗N0,ρ˜n∗N0, where ρ˜n∗=argmaxρ˜nU0,Uρ,Uρ˜n,1, Uρ˜n,1 is calculated by (48), ρ˜n,1∈0,ρ.(5)
else ifζn,m≥ρ(6)
pm,n∗=P−ρ˜n∗N0,ρ˜n∗N0, where ρ˜n∗=argmaxρ˜nU0,Uρ,Uρ˜n,2,Uρ˜n,3, Uρ˜n,2 and Uρ˜n,3 are calculated by (50), ρ˜n,2∈0,ρ,ρ˜n,3∈0,ρ.(7)
else(8)
pm,n∗=P−ρ˜n∗N0,ρ˜n∗N0, where ρ˜n∗=argmaxρ˜nU0,Uρ,Uζn,m,Uρ˜n,1,Uρ˜n,2,Uρ˜n,3,ρ˜n,1∈ζn,m,ρ, ρ˜n,2∈0,ζn,m and ρ˜n,3∈0,ζn,m(9)
end if(10)
else(11)
Ifμm≥μn, pm,n∗=P,0. Otherwise, pm,n∗=0,P.(12)
end if


## 5.2. Achieving Maximum WSR on Rate Region Boundary for MMSE-BF Mode

When the MMSE-BF mode is performed at BS, the problem of WSR maximization can be formulated as(52)P3:maxqm,nUqm,n≔μmr˜m+μnr˜n,s.t.qm,n∈Qm,n,where qm,n=qm,qn is transmit power vector for users m and n in the dual MAC, r˜i and qi are defined in Lemma 3, and Qm,n=qm,n|0≤qi≤P,qi+qφi=P,i=m,n is the feasible set of transmit power vector for users m and n.For the simplicity of notation, we letUρm stand for the WSR Uqm,n in (52) with qm=ρmN0 and qn=P−qm. With the similar idea to that, by which Theorem 1 was proposed, we have the following theorem.Theorem 3.
The solution of Problem (P3) is qm,n∗=qm∗,qn∗ in dual two-user MAC, where qm∗=ρm∗N0, qn∗=P−qm∗, ρm∗=argmaxρmU0,Uρ,Uρm,1,Uρm,2,…,Uρm,9, and ρm,i∈0,ρ, which are the roots of the nine-degree equation U′ρm=0, which is given by (54).Proof.
According to Lemma3, the WSR Uρm can be expressed as(53)Uρm=μmlog1+ρmhm21+ρnhn2θ2ςm+ρnhn21−θ+μnlog1+ρnhn21+ρmhm2θ2ςn+ρmhm21−θ,where ρn=ρ−ρm. In this case, Uρm is differentiable function of ρm in the interval 0,ρ. By letting U′ρm=0, we can obtain a nine-degree equation:(54)a9ρm9+a8ρm8+a7ρm7+a6ρm6+a5ρm5+a4ρm4+a3ρm3+a2ρm2+a1ρm+a0=0.
Since the coefficients of equation (54) are very complicated, we omit them here. In this case, the global maximum value of Uρm in the interval 0,ρ can be found by selecting the maximum among U0,Uρ,Uρm,1,Uρm,2,…,Uρm,9. Consequently, the solution of Problem (P3) is qm,n∗=qm∗,qn∗ in dual two-user MAC, where qm∗=ρm∗N0, qn∗=P−qm∗, ρm∗=argmaxρmU0,Uρ,Uρm,1,Uρm,2,…,Uρm,9, and ρm,i∈0,ρ,i=1,2,…,9. If ρm,i∉0,ρ, i=1,2,…,9, we let Uρm,i=0.
By Theorem3, MMSE-BF-PA is proposed, i.e., Algorithm 3.Algorithm 3:MMSE-BF-PA.
Input:μi, hi∈ℂM×1,i=m,n.Output:pm,n∗.(1)
Calculateρm,i using (54), i=1,2,…,9.(2)
ρm∗=argmaxρmU0,Uρ,Uρm,1,Uρm,2,…,Uρm,9, where ρm,i∈0,ρ.(3)
pm,n∗=p˜m∗,P−p˜m∗, where p˜m∗ is calculated by (45), with qm∗=ρm∗N0.


## 5.3. Joint User Pairing and Power Allocation (JUPA) Algorithm

With the aim of maximizing the total sum rate, in the step of forming a user pair in JUPA, we compare the two sum rates obtained by NOMA-MRT-PA and MMSE-BF-PA, select the transmission mode with the larger sum rate, and then choose the corresponding user pair to send data. By focusing on the expression of rate region boundary of NOMA-MRT in (11), it is worthwhile noticing the following properties:(1)
Considering the original user pairing problem which is how to pair useri with a user from the set of other unpaired users to obtain the maximum sum rate, we can divide all of unpaired users besides user i into 3 subsets, i.e., Φ1i, Φ2i, and Φ3i, according to ζi,i′. Particularly, user i′ in Φ1i, Φ2i, and Φ3i, satisfies ζi,i′≤0, ζi,i′≥ρ and 0<ζi,i′<ρ, respectively. The original user pairing problem is equivalent to 3 problems, which are how to pair user i with a user to obtain the local-maximum sum rate from Φ1i, Φ2i, and Φ3i, respectively. Therefore, the solution of the original user pairing problem is the user pair, which corresponds to the maximum among the local maximum sum rates for Φ1i, Φ2i, and Φ3i.(2)
Considering the problem which is how to pair useri with a user from Φ1iΦ2i to achieve local maximum sum rate, user i should be paired with the user, whose channel vector can form the minimum (maximum) angle with that of user i, according to (11).(3)
For the problem which is how to pair useri with a user from Φ3i to achieve local maximum sum rate, we divide Φ3i into 2 subsets, i.e., Φ3,1i and Φ3,2i. The user i′ in Φ3,1i and Φ3,2i satisfies 0<ζi,i′≤ρ/2 and ρ/2<ζi,i′<ρ, respectively. According to (11), the formula ri′=log1+ρ˜i′hi21−θ/1+ρ˜ihi2 provides dominant contribution to rate region boundary formed by user i and i′, when 0<ζi,i′≤ρ/2. In this case, for the treatability of the problem, we use the formula ri′=log1+ρ˜i′hi21−θ/1+ρ˜ihi2 as the expression of ri′ in the whole interval 0,ρ, instead of the piecewise formula as shown in (11). Thus, for the problem which is how to pair user i with a user from Φ3,1i, user i should be paired with the user whose channel vector can form the minimum angle with that of user i. With the similar analysis, user i should be paired with the user in Φ3,2i whose channel vector can form the maximum angle with that of user i.For the simplicity of description of the proposed JUPA (Algorithm4), assume that users are ordered as h1≤⋯≤h2K. Let Φ1ikΦ2ik denote the index of k-th entry in Φ1iΦ2i. Besides, let Φ3,1ikΦ3,2ik denote the index of k-th entry in Φ3,1iΦ3,2i.Algorithm 4:Joint user pairing and power allocation algorithm (JUPA).
Input:hi∈ℂM×1 (assuming h1≤⋯≤h2K ), μi=1,i=1,2,…,2K.Output:Ω and UInitial:Ω=∅ and U=02K×2K.(1)
fori=1:2K−1do(2)
if user i has been paired(3)
Go to line 1(4)
end if(5)
forj=i+1:2Kdo(6)
if user j has been paired(7)
Go to line 5(8)
end if(9)
ifhi and hj is orthogonal(10)
Obtainpi,j by MMSE-BF-PA, Ω=Ω∪​i,j,pi,j,0 and go to line 1(11)
end if(12)
end for(13)
Findk1=argmaxkcos2∠hi,hΦ1ik and obtain the power allocation solution pi,Φ1ik1 for the user pair i,Φ1ik1 by NOMA-MRT-PA(14)
Findk2=argminkcos2∠hi,hΦ2ik and obtain the power allocation solution pi,Φ2ik2 for the user pair i,Φ2ik2 by NOMA-MRT-PA(15)
Findk3,1=argmaxkcos2∠hi,hΦ3,1ik and obtain the power allocation solution pi,Φ3,1ik3,1 for the user pair i,Φ3,1ik3,1 by NOMA-MRT-PA(16)
Findk3,2=argminkcos2∠hi,hΦ3,2ik and obtain the power allocation solution pi,Φ3,2ik3,2 for the user pair i,Φ3,2ik3,2 by NOMA-MRT-PA(17)
Obtain the power allocation vector with the maximum WSR, from the solutions obtained bylines 13–16:pN=argmaxpUpi,Φ1ik1,Upi,Φ2ik2,Upi,Φ3,1ik3,1,Upi,Φ3,2ik3,2.Leti,jN denote the user pair which corresponds to pN.(18)
FindjM=argminjcos∠hi,hj and obtain the power allocation solution pi,jM for the user pair i,jM by MMSE-BF-PA.(19)
ifUpN≥Upi,jM(20)
Ω=Ω∪​i,jN,pN,1 and ui,jN=ujN,i=1.(21)
else(22)
Ω=Ω∪​i,jM,pi,jM,0 and ui,jM=ujM,i=1.(23)
end if(24)
end forIn Algorithm4, line 13 (14, 15, and 16) is the step for selecting a user to be paired with user i and deriving the power allocation solution for the obtained user pair by NOMA-MRT-PA, in the set Φ1i (Φ2i, Φ3,1i, and Φ3,2i). Line 18 is the step for similar operation by MMSE-BF-PA. The output of Algorithm 4, Ω, is termed the pairing and power allocation configuration, in which K entries are included and 4 elements are contained in each entry. The first two elements of i-th entry in Ω denote the indexes of the two users in i-th user pair, and the third and fourth element stand for power allocation vector and transmission mode adopted by the i-th user pair, respectively.When the number of users is odd, the last remained user can be trivially served by OMA. Specifically, if the channel vector of the last remained user ishl, then MRT BF is performed at the BS and the BF vector is given by (10). As a result, the maximum signal-to-noise ratio can be achieved.


### 5.3.1. Complexity Analysis

The computational complexity of the proposed JUPA mainly comes from the computation of the angle between channels of two users and from solving the nine-degree equation (54), by which the optimal power allocation scheme is obtained for MMSE-BF mode.Note that the computational complexity for computing the angle between channels of two users isOM. In addition, we use upper-case letter “C” to indicate the computational complexity for solving the nine-degree equation (54). As the number of computing the angle is 2K−2i−1−1 in i-th outer “for” loop in Algorithm 4, for the MMSE-BF mode or NOMA-MRT mode, the computational complexity for the user pairing process is O2K/22M. On the other hand, the computational complexity for the power allocation process is O2K/2C. Consequently, the overall computational complexity of Algorithm 4 is O2K/22M+O2K/2C.For the exhaustive search-based scheme, i.e., exhaustive search-based user pairing and power allocation (ES-PPA) described in Section6.2, the number of all possible pairing schemes is 2K−1!!. In each pairing scheme, there are K user pairs. Then, for each user pair, the angle between channels of two users and the allocated power of two users for MMSE-BF mode should be calculated. Therefore, the total computational complexity of exhaustive search-based scheme is O2K−1!!2K/2M+O2K−1!!2K/2C.


## 5.3.1. Complexity Analysis

The computational complexity of the proposed JUPA mainly comes from the computation of the angle between channels of two users and from solving the nine-degree equation (54), by which the optimal power allocation scheme is obtained for MMSE-BF mode.Note that the computational complexity for computing the angle between channels of two users isOM. In addition, we use upper-case letter “C” to indicate the computational complexity for solving the nine-degree equation (54). As the number of computing the angle is 2K−2i−1−1 in i-th outer “for” loop in Algorithm 4, for the MMSE-BF mode or NOMA-MRT mode, the computational complexity for the user pairing process is O2K/22M. On the other hand, the computational complexity for the power allocation process is O2K/2C. Consequently, the overall computational complexity of Algorithm 4 is O2K/22M+O2K/2C.For the exhaustive search-based scheme, i.e., exhaustive search-based user pairing and power allocation (ES-PPA) described in Section6.2, the number of all possible pairing schemes is 2K−1!!. In each pairing scheme, there are K user pairs. Then, for each user pair, the angle between channels of two users and the allocated power of two users for MMSE-BF mode should be calculated. Therefore, the total computational complexity of exhaustive search-based scheme is O2K−1!!2K/2M+O2K−1!!2K/2C.


## 5.4. JUPA/TMS: A Practical Transmission Method

By combining JUPA with transmission mode switching (TMS) between NOMA-MRT and MMSE-BF, a practical transmission scheme termed JUPA/TMS is proposed. JUPA/TMS is described in Algorithm5, where S is the control parameter for transmission mode switching and Ωij denotes the j-th element in the i-th entry in the set Ω. Noting that TDMA(FDMA) is adopted in Algorithm 5 (line10), it is efficient when M is small. When the scenario with sufficient number of antennas at base station is considered, e.g., M≥2K, the performance can be further improved by combining the proposed JUPA with SDMA, which is beyond the scope of this paper.Algorithm 5:JUPA/TMS.
Input:hi∈ℂM×1si, i=1,2,…,2K.Output:x1,x2,…,xK.(1)
Perform JUPA to obtain pairing and power allocation configurationΩ(2)
fori=1:Kdo(3)
m,n,pm,n,Tm,n←Ωi1,Ωi2,Ωi3,Ωi4(4)
ifTm,n=1(5)
Calculatewm and wn with (10)(6)
else ifTm,n=0(7)
Calculatewm and wn with (18)(8)
end if(9)
xi←pmwmsm+pnwnsn(10)
Transmitxi in i-th frequency/time slot(11)
end for


## 6. Results and Discussion

In this section, we first provide numerical results on the rate regions of different schemes (ZFBF, MRT, MMSE-BF, and NOMA-MRT) under various channel conditions and validate the performance of the proposed adaptive switching method by comparing with other schemes. Then, we demonstrate the optimality of NOMA-MRT-PA and MMSE-BF-PA. Furthermore, we also verify the performance of the proposed JPUA by comparing with traditional transmission schemes in simulation.


### 6.1. The Adaptive Switching Method

In Figure3, numerical results of performance comparison among NOMA-MRT, MMSE-BF, MRT, and ZFBF are derived in various channel states. When hn=2 and α=π/8 implying that the angle between two users’ channel vectors is smaller in space, NOMA-MRT gets better performance than MMSE-BF absolutely and the rate region of MMSE-BF is completely within that of NOMA-MRT as illustrated in Figure 3(a). In Figure 3(b), when hn=2 and α=π/4, larger Rn can be obtained by implementing NOMA-MRT in high Rm area, while in low Rm area, larger Rn can be derived by using MMSE-BF. When hn=2 and α=3π/8 which implies that the angle between channel vectors of two users becomes larger, MMSE-BF performs better than NOMA-MRT completely as shown in Figure 3(c). In symmetric case where hm=hn=1 and α=π/4, the rate region of MMSE-BF covers that of NOMA-MRT. From Figures 3(a)–3(d), as MMSE-BF can optimally trade off fighting interference to other users and the background Gaussian noise, MMSE-BF gets better performance than ZFBF and MRT absolutely, in any channel state.Figure 3
Rate region comparison of ZFBF, MRT, MMSE-BF, and NOMA-MRT in various channel states.P/N0=5dB,hm=1. (a) hn=2,α=π/8. (b) hn=2,α=π/4. (c) hn=2,α=3π/8. (d) hn=1,α=π/4. Rate region boundaries of the four schemes is without time-sharing.
(a)(b)(c)(d)As demonstrated in Figure4, the adaptive switching method can obtain a larger rate region than the other schemes, which can be illustrated as the convex hull of the union of NOMA-MRT and MMSE-BF’s rate regions by combining with time-sharing.Figure 4
Rate region of the adaptive switching method with time-sharing.P/N0=10dB,hm=1,hn=2,α=π/4. Also plotted for comparison are rate regions of ZFBF, MRT, MMSE-BF, and NOMA-MRT, without time-sharing.


### 6.2. Joint User Pairing and Power Allocation with Transmission Mode Switching

In Figure5, with different weight vectors, the rate points on rate region boundary and the corresponding maximum WSR points are obtained by NOMA-MRT-PA, MMSE-BF-PA, and exhaustive search, respectively. When weight vector u is 0.5,0.5, 0.33,0.67, or 0.25,0.75, rate points obtained by NOMA-MRT-PA or MMSE-BF-PA are superposed together with those derived by exhaustive search, which means that NOMA-MRT-PA and MMSE-BF can derive optimal rate point for different weight vector as shown in Figure 5(a). The rate points obtained by NOMA-MRT-PA for the case u=0.33,0.67 and u=0.25,0.75 are superposed together, since the rate points to achieve the maximum WSR for u=0.33,0.67 and u=0.25,0.75 both occur at the inflection point of the achievable rate region boundary of NOMA-MRT. In Figure 5(b), with u=0.5,0.5, the WSRs obtained by NOMA-MRT (red curve) and MMSE-BF (blue curve) are given, respectively, when the allocated power of user m varies, with P being normalized to 1. The maximum WSR points corresponding to the rate point with u=0.5,0.5 in Figure 5(a) can be obtained by NOMA-MRT-PA and MMSE-BF-PA, respectively, each of which is also superposed with that derived by exhaustive search. Figures 5(c) and 5(d) are the cases for u=0.33,0.67 and u=0.25,0.75, respectively.Figure 5
Rate points and the corresponding maximum WSR points obtained by NOMA-MRT-PA, MMSE-BF-PA, and exhaustive search.P/N0=10dB,hm=2,hn=1,α=π/4. (a) Rate points under different weight vectors. Maximum WSR point with (b) u=0.5,0.5, (c) u=0.33,0.67, and (d) u=0.25,0.75.
(a)(b)(c)(d)To validate the effectiveness of the proposed JUPA scheme, other schemes which combine the existing user pairing algorithms (exhaustive search, greedy algorithm, correlation pairing [27, 28], and random pairing) with NOMA-MRT-PA and MMSE-BF-PA are also considered for comparison in simulation, including exhaustive search-based user pairing and power allocation algorithm (ES-PPA), greedy algorithm-based user pairing and power allocation algorithm (GA-PPA), correlation pairing-based user pairing and power allocation algorithm (COR-PPA), and random pairing-based user pairing and power allocation algorithm (RAN-PPA). In the following, we assume noise power N0=−10dB, the cell radius R=30m, and the path loss exponent α′=3, and TDMA technology is adopted. The bandwidth is normalized to one. Assume that the weights of all users are set to one. The key idea of GA-PPA is to pair the two users with the maximum sum rate obtained by NOMA-MRT-PA or MMSE-BF-PA. Mathematically, for a fixed user i, we pair it with user j∗, if(55)j∗=argmaxjmaxUi,jNOMA−MRT,Ui,jMMSE−BF,j≠i,where Ui,jNOMA−MRT and Ui,jMMSE−BF stand for the sum rate of users i and j, obtained by NOMA-MRT-PA and MMSE-BF-PA, respectively. With the similar architecture to the outer “for” loop in the Algorithm 4, the numbers of computing the angle between channels of two users used by NOMA-MRT-PA and the number of solving the equation (54) used by MMSE-BF-PA are both 2K−2i−1−1 in i-th outer “for” loop for GA-PPA. Therefore, its computational complexity is O2K/22M+O2K/22C in total. In RAN-PPA and COR-PPA, we first obtain user pairing solution by random pairing algorithm and correlation pairing algorithm, respectively, and then derive the sum rate for each user pair by selecting the larger one among the sum rates achieved by NOMA-MRT-PA and MMSE-BF-PA.The key idea of correlation pairing algorithm is only to pair two users with the maximum channel correlation. Specifically, for a fixed useri, we pair it with user j∗ satisfying(56)j∗=argmaxjhjHhi2hi2hj2,j≠i.Since the ideas of forming a user pair in COR-PPA and the proposed JUPA are both based on the angle between channels of the two users, COR-PPA and JUPA have the same computational complexity (considering the complexity of computing the angle between two users’ channels and solving equation (54)), i.e., O2K/22M+O2K/2C. In ES-PPA, we first obtain 2K−1!! possible pairing schemes by exhaustive search and then select the pairing scheme with maximum total sum rate, which is the sum of sum rates of K user pairs.For summary, JUPA and COR-PPA have roughly the same computational complexity. However, GA-PPA and ES-PPA require much higher computational complexity than JUPA, especially for ES-PPA. Due to the random paring method adopted in RAN-PPA, its computational complexity is the lowest, which only results from computing the angle between two users’ channels used by NOMA-MRT-PA and from solving equation (54) utilized by MMSE-BF-PA, for each user pair. The computational complexity of these schemes is summarized in Table 2.Table 2
Computational complexity summary for various schemes.
User pairing and power allocation schemesComputational complexityGA-PPAO2K/22M+O2K/22CCOR-PPAO2K/22M+O2K/2CRAN-PPAO2K/2M+O2K/2CES-PPAO2K−1!!2K/2M+O2K−1!!2K/2CJUPAO2K/22M+O2K/2CIn Figures6 and 7, the sum rates of different schemes, including ES-PPA, RAN-PPA, GA-PPA, COR-PPA, and the proposed JUPA, are compared, when the transmit power P and the number of antennas at the BS vary, respectively. In Figure 6, the sum rates of all schemes grow with the transmit power P increasing. Compared with ES-PPA using exhaustive search to obtain the user pairing solution, the proposed JUPA only suffers very slight performance loss. However, the computational complexity of JUPA is only O2K/22M+O2K/2C, while that of ES-PPA is O2K−1!!2K/2M+O2K−1!!2K/2C. In comparison with other schemes, the JUPA derives better performance and the performance gain boosts with P increasing. Further, JUPA outperforms GA-PPA, since GA does not consider the performance loss caused by the users with poor channels. COR-PPA performs worse than RAN-PPA, since correlation-based user pairing algorithm breaks the channels orthogonality, i.e., the users with orthogonal channel are never paired together.Figure 6
Sum rate versusP for different transmission schemes (M=2, 2K=8).Figure 7
Sum rate versus the number of the antennas for several transmission schemes (P  = 40 dBm, 2K=8).As demonstrated in Figure7, it can be observed that the proposed JUPA results in a slight performance loss compared with ES-PPA. However, the JUPA outperforms other schemes and the performance gain over other schemes decreases as the number of antennas at the BS increases.


## 6.1. The Adaptive Switching Method

In Figure3, numerical results of performance comparison among NOMA-MRT, MMSE-BF, MRT, and ZFBF are derived in various channel states. When hn=2 and α=π/8 implying that the angle between two users’ channel vectors is smaller in space, NOMA-MRT gets better performance than MMSE-BF absolutely and the rate region of MMSE-BF is completely within that of NOMA-MRT as illustrated in Figure 3(a). In Figure 3(b), when hn=2 and α=π/4, larger Rn can be obtained by implementing NOMA-MRT in high Rm area, while in low Rm area, larger Rn can be derived by using MMSE-BF. When hn=2 and α=3π/8 which implies that the angle between channel vectors of two users becomes larger, MMSE-BF performs better than NOMA-MRT completely as shown in Figure 3(c). In symmetric case where hm=hn=1 and α=π/4, the rate region of MMSE-BF covers that of NOMA-MRT. From Figures 3(a)–3(d), as MMSE-BF can optimally trade off fighting interference to other users and the background Gaussian noise, MMSE-BF gets better performance than ZFBF and MRT absolutely, in any channel state.Figure 3
Rate region comparison of ZFBF, MRT, MMSE-BF, and NOMA-MRT in various channel states.P/N0=5dB,hm=1. (a) hn=2,α=π/8. (b) hn=2,α=π/4. (c) hn=2,α=3π/8. (d) hn=1,α=π/4. Rate region boundaries of the four schemes is without time-sharing.
(a)(b)(c)(d)As demonstrated in Figure4, the adaptive switching method can obtain a larger rate region than the other schemes, which can be illustrated as the convex hull of the union of NOMA-MRT and MMSE-BF’s rate regions by combining with time-sharing.Figure 4
Rate region of the adaptive switching method with time-sharing.P/N0=10dB,hm=1,hn=2,α=π/4. Also plotted for comparison are rate regions of ZFBF, MRT, MMSE-BF, and NOMA-MRT, without time-sharing.


## 6.2. Joint User Pairing and Power Allocation with Transmission Mode Switching

In Figure5, with different weight vectors, the rate points on rate region boundary and the corresponding maximum WSR points are obtained by NOMA-MRT-PA, MMSE-BF-PA, and exhaustive search, respectively. When weight vector u is 0.5,0.5, 0.33,0.67, or 0.25,0.75, rate points obtained by NOMA-MRT-PA or MMSE-BF-PA are superposed together with those derived by exhaustive search, which means that NOMA-MRT-PA and MMSE-BF can derive optimal rate point for different weight vector as shown in Figure 5(a). The rate points obtained by NOMA-MRT-PA for the case u=0.33,0.67 and u=0.25,0.75 are superposed together, since the rate points to achieve the maximum WSR for u=0.33,0.67 and u=0.25,0.75 both occur at the inflection point of the achievable rate region boundary of NOMA-MRT. In Figure 5(b), with u=0.5,0.5, the WSRs obtained by NOMA-MRT (red curve) and MMSE-BF (blue curve) are given, respectively, when the allocated power of user m varies, with P being normalized to 1. The maximum WSR points corresponding to the rate point with u=0.5,0.5 in Figure 5(a) can be obtained by NOMA-MRT-PA and MMSE-BF-PA, respectively, each of which is also superposed with that derived by exhaustive search. Figures 5(c) and 5(d) are the cases for u=0.33,0.67 and u=0.25,0.75, respectively.Figure 5
Rate points and the corresponding maximum WSR points obtained by NOMA-MRT-PA, MMSE-BF-PA, and exhaustive search.P/N0=10dB,hm=2,hn=1,α=π/4. (a) Rate points under different weight vectors. Maximum WSR point with (b) u=0.5,0.5, (c) u=0.33,0.67, and (d) u=0.25,0.75.
(a)(b)(c)(d)To validate the effectiveness of the proposed JUPA scheme, other schemes which combine the existing user pairing algorithms (exhaustive search, greedy algorithm, correlation pairing [27, 28], and random pairing) with NOMA-MRT-PA and MMSE-BF-PA are also considered for comparison in simulation, including exhaustive search-based user pairing and power allocation algorithm (ES-PPA), greedy algorithm-based user pairing and power allocation algorithm (GA-PPA), correlation pairing-based user pairing and power allocation algorithm (COR-PPA), and random pairing-based user pairing and power allocation algorithm (RAN-PPA). In the following, we assume noise power N0=−10dB, the cell radius R=30m, and the path loss exponent α′=3, and TDMA technology is adopted. The bandwidth is normalized to one. Assume that the weights of all users are set to one. The key idea of GA-PPA is to pair the two users with the maximum sum rate obtained by NOMA-MRT-PA or MMSE-BF-PA. Mathematically, for a fixed user i, we pair it with user j∗, if(55)j∗=argmaxjmaxUi,jNOMA−MRT,Ui,jMMSE−BF,j≠i,where Ui,jNOMA−MRT and Ui,jMMSE−BF stand for the sum rate of users i and j, obtained by NOMA-MRT-PA and MMSE-BF-PA, respectively. With the similar architecture to the outer “for” loop in the Algorithm 4, the numbers of computing the angle between channels of two users used by NOMA-MRT-PA and the number of solving the equation (54) used by MMSE-BF-PA are both 2K−2i−1−1 in i-th outer “for” loop for GA-PPA. Therefore, its computational complexity is O2K/22M+O2K/22C in total. In RAN-PPA and COR-PPA, we first obtain user pairing solution by random pairing algorithm and correlation pairing algorithm, respectively, and then derive the sum rate for each user pair by selecting the larger one among the sum rates achieved by NOMA-MRT-PA and MMSE-BF-PA.The key idea of correlation pairing algorithm is only to pair two users with the maximum channel correlation. Specifically, for a fixed useri, we pair it with user j∗ satisfying(56)j∗=argmaxjhjHhi2hi2hj2,j≠i.Since the ideas of forming a user pair in COR-PPA and the proposed JUPA are both based on the angle between channels of the two users, COR-PPA and JUPA have the same computational complexity (considering the complexity of computing the angle between two users’ channels and solving equation (54)), i.e., O2K/22M+O2K/2C. In ES-PPA, we first obtain 2K−1!! possible pairing schemes by exhaustive search and then select the pairing scheme with maximum total sum rate, which is the sum of sum rates of K user pairs.For summary, JUPA and COR-PPA have roughly the same computational complexity. However, GA-PPA and ES-PPA require much higher computational complexity than JUPA, especially for ES-PPA. Due to the random paring method adopted in RAN-PPA, its computational complexity is the lowest, which only results from computing the angle between two users’ channels used by NOMA-MRT-PA and from solving equation (54) utilized by MMSE-BF-PA, for each user pair. The computational complexity of these schemes is summarized in Table 2.Table 2
Computational complexity summary for various schemes.
User pairing and power allocation schemesComputational complexityGA-PPAO2K/22M+O2K/22CCOR-PPAO2K/22M+O2K/2CRAN-PPAO2K/2M+O2K/2CES-PPAO2K−1!!2K/2M+O2K−1!!2K/2CJUPAO2K/22M+O2K/2CIn Figures6 and 7, the sum rates of different schemes, including ES-PPA, RAN-PPA, GA-PPA, COR-PPA, and the proposed JUPA, are compared, when the transmit power P and the number of antennas at the BS vary, respectively. In Figure 6, the sum rates of all schemes grow with the transmit power P increasing. Compared with ES-PPA using exhaustive search to obtain the user pairing solution, the proposed JUPA only suffers very slight performance loss. However, the computational complexity of JUPA is only O2K/22M+O2K/2C, while that of ES-PPA is O2K−1!!2K/2M+O2K−1!!2K/2C. In comparison with other schemes, the JUPA derives better performance and the performance gain boosts with P increasing. Further, JUPA outperforms GA-PPA, since GA does not consider the performance loss caused by the users with poor channels. COR-PPA performs worse than RAN-PPA, since correlation-based user pairing algorithm breaks the channels orthogonality, i.e., the users with orthogonal channel are never paired together.Figure 6
Sum rate versusP for different transmission schemes (M=2, 2K=8).Figure 7
Sum rate versus the number of the antennas for several transmission schemes (P  = 40 dBm, 2K=8).As demonstrated in Figure7, it can be observed that the proposed JUPA results in a slight performance loss compared with ES-PPA. However, the JUPA outperforms other schemes and the performance gain over other schemes decreases as the number of antennas at the BS increases.


## 7. Conclusion

In this paper, the design of adaptive TMS between NOMA-MRT and MMSE-BF to maximize the sum rate for downlink MISO systems was investigated. Firstly, the closed-form expressions of the boundary of achievable rate region for NOMA-MRT and MMSE-BF were obtained. It has been shown that when the channel vectors of the two users are greatly correlated, the achievable rate region of NOMA-MRT includes that of MMSE-BF completely. However, when the two channels are almost orthogonal, the opposite conclusion can be drawn. As a result, an adaptive switching method is developed to achieve a larger rate region for the two-user case. Subsequently, the optimal power allocation algorithms to maximize weighted sum rate for both NOMA-MRT mode and MMSE-BF mode were presented. Finally, the low-complexity JUPA is consequently developed to further improve the sum rate performance for the multiuser case. Compared with the exhaustive search-based scheme with the computational complexity ofO2K−1!!2K/2M+O2K−1!!2K/2C, the proposed JUPA can obtain a much lower complexity of O2K/22M+O2K/2C and only suffers a slight sum rate performance loss, and it outperforms other conventional schemes.



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*Source: 6671371-2021-03-09.xml*