Adaptive transmission mode switching in downlink MISO-NOMA systems: sum rate maximization and the achievable rate region

In this paper, the design of adaptive transmission mode switching (TMS) to maximize the sum rate for the downlink multiple-input-single-output based non-orthogonal multiple access (MISO-NOMA) systems is investigated. Firstly, the colsed-form expressions of the boundary of achievable rate region of two candidate transmission mode, i.e., NOMA-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 mode, 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 achievable rate region, the optimal solutions to the problem of WSR optimization for NOMA-MRT and MMSE-BF are obtained, respectively. Subsequently, by exploiting the optimal solutions aforementioned for two transmission modes and the high eﬃciency of TMS, a low-complexity Joint User pairing and Power Allocation algorithm (JUPA) is proposed to further improve sum-rate performance for the multi-user case. Compared with the Exhaustive Search based user Pairing and Power Allocation algorithm (ES-PPA), the proposed JUPA can enjoy a much lower computation complexity and only suﬀer a slight sum-rate performance loss, whereas outperforms other traditional schemes. Finally, numerical results are provided to validate the analyses and the proposed algorithms. beamforming; WSR:weighted sum rate; MAC:multiple-access channel; CSI:the channel state information; BC:broadcast channel; SIC:successive interference cancellation; NOMA-MRT-PA:Power Allocation algorithm for NOMA-MRT mode; MMSE-BF-PA:Power Allocation algorithm for MMSE-BF mode; BS:base station; SINR:signal-to-interference-plus-noise ratio; JUPA:Joint User pairing and Power Allocation algorithm; ES-PPA:Exhaustive Search based user Pairing and Power Allocation algorithm; GA-PPA:Greedy Algorithm based user Pairing and Power Allocation algorithm; COR-PPA:CORrelation-based-pairing based user Pairing and Power Allocation algorithm; RAN-PPA:RANdom pairing based user Pairing and Power Allocation algorithm; AWGN:additive white Gaussian noise.


Introduction
With the continuous emergence of new application scenarios, one of 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 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 inter-user interference by allocating orthogonal resources (frequency/time/code) to different users, which result in sufficient 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, owning to its higher spectrum efficiency compared with the conventional orthogonal multiple access schemes [3][4][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][9][10][11][12][13][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 solve, 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 constrain 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 constrain was considered. A power allocation scheme to satisfy ergodic user rate constrains 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 computation 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 meet the quasi-degrade 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 levels constrains, was investigated for two-user MISO-NOMA downlink. Based on the results obtained by [13], in [14], it was 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).
Due to the equivalence between the maximization of the weighted sum rate and the acquisition of the maximum weighted sum rate (WSR) point on achievable rate region for the MISO downlink, the problems for characterization of the achievable rate region for MISO broadcast channel [15,16] and for MISO interference channel [17][18][19], were investigated, respectively. Specifically, in [15], the set of BF vectors which achieve points on the boundary of achievable rate region of two-user MISO broadcast channel, are characterized by a single real valued parameter per user. In [16], 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 [17]. A general framework for finding the maximum sumrate operating points on the boundary of the achievable rate region for the two-user MISO interference channel, was proposed in [18]. In [19], 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-NOMA systems. In this paper, from the perspective of the achievable rate region, we focus on the design of adaptive transmission mode switching (TMS) to maximize the sum rate for multi-user MISO downlink, by utilizing the idea of user pairing.
The main contributions for this paper is 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 colsed-form expression of the boundary of 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 equivalence between the maximization of the weighted sum rate and the acquisition of the maximum weighted sum rate point on achievable rate region for the MISO downlink, 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 multi-user case. Subsequently, by the combination of JUPA and the idea of transmission mode switching between NOMA-MRT and MMSE-BF, a practical transmission method is developed. By making use of the closed-expression of the optimal solution by NOMA-MRT-PA(MMSE-BF-PA), JUPA can be performed with a low computation complexity O(K 2 M ), where 2K is the number of users and M stands for the number of antennas at the BS, while the computation complexity of the Exhaustive Search based user Pairing and Power Allocation algorithm (ES-PPA) is O((2K − 1)!!KM ), compared with which JUPA only suffers a slight performance loss.
The remainder of this paper is organized as follows. Section 2 briefly describes the system model and introduces NOMA-based MISO downlink transmission schemes. In Section 3, the closed-form expressions of the achievable rate region boundary of MISO broadcast channel by using NOMA-MRT and MMSE-BF is 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. The numerical results is 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 vectors x 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 by · .
x ∈ C M ×1 means that x is an M × 1 complex vector. CN (µ, σ 2 ) denotes a complex Gaussian random variable with mean µ and variance σ 2 . x, y and ∠(x, y) denotes the inner product and the angle of two complex vectors x and y, respectively.

System Model
We consider a downlink communication system with one M-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. 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 perform SIC to cancel the co-channel interference. However, with the number of users in a group growing, the processing complexity and delay at uses dramatically increase [20]. As a result, we consider only grouping two users and adopting user pairing scheme to decrease processing complexity and delay in the NOMA downlink. Assume that user m and n are paired over the shared spectrum. The observation at user i is given by where is the channel vector from BS to user i, z i ∼ CN (0, N 0 ) is the additive white Gaussian noise (AWGN) at user i and N 0 is the corresponding noise power. For the channel vector h i , the variance of the channel from BS to user i is modeled as [14], [21] where d i denotes the distance between BS and user i, α denotes the path loss exponent and the parameter d 0 avoids the singularity when d i is small. Furthermore, x = √ p m w m s n + √ p n w n s n is the signal transmitted by the BS, where s i and w i are the scalar signal and normalized BF vector for user i, respectively. p i is the transmit power allocated to user i. Assume that the BS can obtain perfect CSI.
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 p and BF matrix W of each user-pair. Mathematically, the optimization problem can be formulated as where R i is the rate of user i, i = m, n. U is the user pairing relationship matrix with 2K-dimension, in which the m-th row and n-th column element is denoted by u m,n . p m,n = [p m , p n ] and W m,n = [w m , w n ] are power allocation vector and BF matrix of the user-pair{m, n}. As shown in (2b), u m,n = 1 represents that user m and n are paired together and constitute a user-pair. Otherwise, user m and n are not paired together. (2c) and (2d) imply that U is a symmetric matrix, and the diagonal elements are zero as a user can't be paired with itself. (2e) and (2f) indicate that a user can be paired with only one user. Furthermore, (2g) represents the total power allocated to a user-pair {m, n}, which is upper bounded to P . (2h) 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 u m,n and the nonconvex property of the achievable rate of user i, which is the rate R i of user i with the equality holding in (3) or (5). Therefore, for the treatability of the problem (P1), we select the BF scheme 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 , and power allocation vector p and BF matrix W of each user-pair.

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 vector w i , the achievable rate region of the MISO broadcast channel (BC) is given by the set of rate tuples (R m ,R n ) satisfying where is the signal-to-interference-plus-noise ratio (SINR) of user i with p m + p n ≤ P , i = m, n. The mapping function ϕ(i) is defined by and P is the total transmit power for each user-pair. When the equality holds in (3), the rate tuple (R m , R n ) achieves on the boundary of rate region with p m + p n = P .

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. Specifically, for a fixed pair of users m and n, the user n with strong channel conditions, first decodes s m and subtracts this from its received signal y n . As a result, user n can decode s n without the interference caused by user m. However, user m simply treat s n as noise and decodes its own signal s m . Therefore, for a fixed normalized BF vector w i , when SIC is carried out at user n, 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 where log(1+γ B m,n ) denotes the achievable rate for user n to detect user m's message and γ B m,n is the corresponding SINR, i.e., In this paper, the angle between h m and h n is denoted by α as, where Without loss of generality, we set α ∈ [0, π).

2.4
The equivalence between maximizing weighted sum rate and enlarging achievable rate region For a given weight vector, the solution of optimization problem of maximizing the weighted sum rate (WSR) must be on the boundary of achievable rate region of the two-user MISO broadcast channel [16]. Thus, maximizing the weighted sum rate 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 vector w i of user i with MRT is aligned with the spatial direction of h i to maximize the length of the projection of w i onto h i . 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 NOMA-MRT is the combination of NOMA and MRT.
Lemma 1 When h m < h n , the achievable rate region boundary of NOMA-MRT for MISO broadcast channel can be expressed as Case 1) θ > 0 r n = log(1 +ρ n h n 2 ), where Proof See the Appendix.

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) [22]. Such a beamformer looks like the Zero-Forcing beamformer when the inter-user 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 Fig. 1.
By MAC-BC duality theory [22], the MMSE beamformer in BC, is exactly the MMSE receive filters in dual MAC. Therefore, the sets of achievable SINRs are the same in both cases with the same total transmit power constrain. Consequently, we have the following Theorem.
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 receive filter. The set of rate tuple (r m ,r n ) on the achievable rate region boundary of BC satisfies where Proof According to the duality between MAC and BC [22], the normalized BF vectors of user i with MMSE-BF in MISO broadcast channel is given by By the matrix inversion Lemma [23], Eq.(13) can be written as The SINR γ M i of user i in dual MAC with MMSE receiver filter have the following form 1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 According to the definition of inner product, Eq.(15) can be rewritten as where β i represent the angle between w MMSE i and h i , α + β i is the angle between w MMSE i and h ϕ(i) when α ∈ [0, π/2) and α − β i is that when α ∈ [π/2, π), as illustrated in Fig. 1.

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.

The intersection points of NOMA-MRT's and MMSE-BF's rate region boundaries
According to Lemma 1, Lemma 2 and Lemma 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 perferred, which can achieve a larger rate region than that derived by MMSE-BF or NOMA-MRT only. Combined with the concept of time-sharing [22], the adaptive switching method can achieve a convex hull of the union of MMSE-BF and NOMA-MRT's achievable rate region.
The intersection points of NOMA-MRT and MMSE-BF's rate region boundaries, are given as following.
Case 2) When h m = h n = l By combining (10) with (11), we derive the equation set .
By substituting (34) into (35), the intersection points can be obtained by solving the following equation where ρ n,k ∈ [0, ρ], k = 1, 2, · · · , 6. We denote byp i (i = m, n) the power allocated to user i in the BC with MMSE-BF scheme. Since the transmit power q i of user i in dual MAC is not equal to the allocated powerp i for user i in the BC, we should obtain the allocated powerp i,k of user i to design the switching method in the BC, which corresponds to the mode switching point ρ i,k = q i,k /N 0 . 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 receive filter at the BS, i.e., We substitute (14) into (4), substitute (4) and (12) into (38) and then replace p i withp i by abuse of notation. As a result,p i can be expressed as where 1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63 64 65

Adaptive Switching Method to Achieving a Larger Rate Region
Based on the intersection points aforementioned, we develops an adaptive switching method to achieve a larger rate region, described in Algorithm 1, which outputs some parameters such as the user's transmit power and BF vector in any channel condition. Combining with time-sharing, the switching method can achieve a convex hull of the union of NOMA-MRT and MMSE-BF's achievable rate region.

Joint User Pairing and Power Allocation with Transmission Mode Switching
In this section, for the problem of weighted sum rate (WSR) maximization, we first proposed two optimal power allocation algorithms based on the concept of achievable rate region, i.e.,

Achieving maximum WSR on rate region boundary for NOMA-MRT mode
According to [16], the solution to the problem of maximizing the WSR must be on the boundary of achievable rate region of two-user MISO downlink systems. As a result, when NOMA-MRT mode is employed at BS, the WSR maximization Problem is formulated as where r i (i = m, n) is the achievable rate of user i, defined in Lemma 1 or Lemma 2, and P m,n = {p m,n |0 ≤ p i ≤ P, p i + p ϕ(i) = 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., p i + p ϕ(i) = P . For the simplicity of notation, we let U (ρ n ) stand for the the WSR U (p m,n ) in (40) with p n =ρ n N 0 and p m = P − p n . The motivations behind proposing NOMA-MRT-PA are as following: 1) According to Lemma 1 and 2, the expression of achievable rate region boundary of NOMA-MRT can be given in any channel conditions. Furthermore, the achievable rate r i is a differentiable or piecewise differentiable function ofρ m in the interval [0, ρ].
2) The global maximum value of U (ρ n ) in the interval [0, ρ] can be found by selecting the maximum among U (0), U (ρ) and those, corresponding to which the first-order derivative of U (ρ n ) is zeros in the interval (0, ρ). As we know, the local maximum values of U (ρ n ) in the interval (0, ρ) satisfies that the first-order derivative of U (ρ n ) equals to zero.
If h m < h n Calculate ρ n,k using (30) and (31) Else h m = h n Calculate ρ n,k using (37) Step 2: Define as the mode switching point the intersection point excluding the end of the interval [0, ρ], and sort the mode switching points in ascending order, i.e., ρ n,1 ≤ · · · ≤ ρ n,j ≤ · · · ≤ ρ n,Γ , where Γ is the number of mode switching points. Divide rate region boundary into Γ + 1 sections by ρ n,j .
Proof See the Appendix. Proof See the Appendix. For the simplicity of description of the proposed NOMA-MRT-PA, we assume h m ≤ h n . By Theorem 1 and Theorem 2, NOMA-MRT-PA is proposed, which is described in Algorithm 2.

Achieving maximum WSR on rate region boundary for MMSE-BF mode
When MMSE-BF mode is performed at BS, the WSR maximization Problem can be formulated as where q m,n = (q m , q n ) is transmit power vector for user m and n in the dual MAC, r i and q i are defined in Lemma 3 and Q m,n = {q m,n |0 ≤ q i ≤ P, q i + q ϕ(i) = P, i = m, n} is the feasible set of transmit power vector for user m and n.
For the simplicity of notation, we let U (ρ m ) stand for the WSR U (q m,n ) in (43) with q m = ρ m N 0 and q n = P − q m .

Joint User Pairing and Power Allocation (JUPA) Algorithm
In order to obtain a maximum 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 (7), it is worthwhile noticing the following properties: 1) Considering the original problem which is how to pair user i with a user from other unpaired users to obtain the maximum sum rate, we can divide all of unpaired users besides user i into 3 subsets, i.e., Φ i 1 , Φ i 2 and Φ i 3 , according to ζ i,i . Particularly, user i in Φ i 1 , Φ i 2 and Φ i 3 , satisfies ζ i,i ≤ 0, ζ i,i ≥ ρ and 0 < ζ i,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 Φ i 1 , Φ i 2 and Φ i 3 , respectively. Therefore, the solution of the original user pairing problem is the one corresponding to the maximum among the local-maximum sum rates for Φ i 1 , Φ i 2 and Φ i 3 . 2) Considering the problem which is how to pair user i with a user from Φ i 1 (Φ i 2 ) 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 (7).
3) For the problem which is how to pair user i with a user from Φ i 3 to achieve local-maximum sum rate, we divide Φ i 3 into 2 subsets, i.e., Φ i 3,1 and Φ i 3,2 . The user i in Φ i 3,1 and Φ i 3,2 satisfies 0 < ζ i,i ≤ ρ/2 and ρ/2 < ζ i,i < ρ, respectively. According to (7), the formula r i = log(1 +ρ i hi 2 (1−θ) 1+ρi hi 2 ) provides dominant contribution to rate region boundary formed by user i and i , when 0 < ζ i,i ≤ ρ/2. In this case, for the treatability of the problem, we use the formula r i = log(1 +ρ i hi 2 (1−θ) 1+ρi hi 2 ) as the expression of r i in the whole interval [0, ρ], instead of the piecewise formula as shown in (7). Thus, for the problem which is how to pair user i with a user from Φ i 3,1 , 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 Φ i 3,2 whose channel vector can form the maximum angle with that of user i.

17:
Select the power allocation vector with the maximum WSR, from the solutions obtained by Line 13-16: Let {i, j N } denote the user-pair which corresponds to p N .
Note that the computation complexity for obtaining the angle between channels of two users is O(M ), and the number of computing the angle is 2K −2(i−1)−1 in i-th outer 'for' loop, for MMSE-BF mode and NOMA-MRT mode. Therefore, the overall computation complexity of Algorithm 4 is O(( 2K 2 ) 2 M ). For exhaustive research based scheme, i.e., Exhaustive Search based user Pairing and Power Allocation (ES-PPA) described in Section VI, the number of all possible pairing schemes is (2K − 1)!!. In each pairing scheme, there are K user-pairs and the angle between channels of two users in each user-pair should be calculated. Therefore, the computation complexity of exhaustive research based scheme, is O((2K − 1)!!KM ).

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 as JUPA/TMS is proposed. JUPA/TMS is described in algorithm 5, where S is control bit for transmission mode switching and Ω(i) j denotes the j-th element in the i-th entry in the set Ω. Noting that TDMA(FDMA) is adopted in Algorithm 5 (Line 10), 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.

The Adaptive Switching Method
In Fig.2, numerical results of performance comparison among NOMA-MRT, MMSE-BF, MRT and ZFBF, are derived in various channel state. When h n = 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 MMSE-BF as illustrated in (a). In (b), when h n = 2 and α = π/4, larger R n can be obtained by implementing NOMA-MRT in high R m area, while in low R m area larger R n can be derived by using MMSE-BF.When h n = 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 (c). In symmetric case where h m = h n = 1 and α = π/4, the rate region of MMSE-BF covers that of NOMA-MRT. From (a) to (d), as MMSE-BF is the optimal linear beamformor in MISO broadcast channel, MMSE-BF get better performance than ZFBF and MRT absolutely, in any channel state. As demonstrated in Fig.3, the adaptive switching method can obtain a larger rate region than the other schemes, which can be illustrated as the convex hull the union of NOMA-MRT and MMSE-BF's rate regions by combining with time-sharing. In the following, we assume noise power N 0 = −10dBm, the cell radius R = 30m, 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 Similarly, in ES-PPA, we first obtain (2K − 1)!! possible pairing schemes by exhaustive research and then select the pairing scheme with maximum total sum rate, which is the sum of the sum rates of K user-pairs. In Fig.6 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 antenna at the S varies, respectively. In Fig.6, the sum rates of all schemes grow with the transmit power P increasing. Compared with optimal ES-PPA using exhaustive search to obtain the user pairing solution, the proposed JUPA only suffers very slight performance loss. However, the computation complexity of proposed JUPA is only In comparison with other schemes, the JUPA derives better performance and the performance gain boosts with P/N 0 increasing. Further, JUPA outperforms GA-PPA, since GA dose 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.
As demonstrated in Fig.7, it is can be observed that the proposed JUPA results in a slight performance loss compared with optimal ES-PPA. However, the JUPA outperforms other schemes and the performance gain over other schemes decreases as the number of antenna at the BS increases.

Conclusion
In this paper, the design of adaptive TMS between NOMA-MRT and MMSE-BF to maximize the sum rate for MISO-NOMA systems, was investigated. Firstly, the colsed-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 include 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, by exploiting the optimal solutions by the aforementioned power allocation algorithms for two transmission modes and the high efficiency of TMS, the low-complexity JUPA is consequently developed to further improve sum-rate performance for the multi-user case. Compared with the exhaustive research based scheme with the computation complexity of O((2K − 1)!!KM ), the proposed JUPA can obtain a much lower complexity of O(K 2 M ) and only suffers a slight sum-rate performance loss, whereas outperforms other conventional schemes .  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 As θ > 0, we solve Eq. (50) with quadratic formula. The discriminant of the Eq. (50) can be expressed as As ≥ 0, the Eq. (50) has two roots both of which are real numbers and the roots are given bỹ In this case, due to h m < h n , f (ρ n ) > 0. As a result, we can derive (9).