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

10 This paper investigates the design of the joint user pairing and power allocation scheme with transmission mode 11 switching (TMS) in downlink multiple-input-single-output (MISO) systems. Firstly, the closed-form 12 expressions of the boundary of achievable rate region of two candidate transmission modes, i.e., non-orthogonal 13 multiple access based maximum ratio transmission (NOMA-MRT) and minimum mean square error 14 beamforming (MMSE-BF), are obtained. By obtaining the outer boundary of the union of the achievable rate 15 regions of the two transmission modes, an adaptive switching method is developed to achieve a larger rate 16 region. Secondly, based on the idea that the solution to the problem of weighted sum rate (WSR) optimization 17 must be on the boundary of the achievable rate region, the optimal solutions to the problem of WSR 18 optimization for NOMA-MRT and MMSE-BF are obtained for the two-user case, respectively. Subsequently, 19 by exploiting the optimal solutions aforementioned for two transmission modes and the high efficiency of TMS, 20 a suboptimal User pairing and Power Allocation algorithm (JUPA) is proposed to further improve sum-rate 21 performance for the multi-user case. Compared with the Exhaustive Search based user Pairing and Power 22 Allocation algorithm (ES-PPA), the proposed JUPA can enjoy a much lower computational complexity and 23 only suffer a slight sum-rate performance loss, whereas outperforms other traditional schemes. Finally, 24 numerical results are provided to validate the analyses and the proposed algorithms.


Introduction
and orthogonal frequency division multiple access (OFDMA) [1,2] for 4G. These conventional multiple access 32 schemes can mitigate or avoid the inter-user interference by allocating orthogonal resources 33 (frequency/time/code) to different users, which result in insufficient use of spectrum resources. 2 Recently, Non-orthogonal multiple access (NOMA) has been considered as a promising multiple access scheme 35 for 5th Generation (5G) wireless communication systems, owning to its higher spectrum efficiency compared 36 with the conventional orthogonal multiple access schemes [3][4][5]. Note that Third Generation Partnership Project 37 (3GPP) has considered NOMA as a study-item for 5G new radio (NR) in Release 15 and decided to leave it for 38 possible use in Beyond 5G (B5G) [6]. NOMA enables multiple users to share the same time-frequency resource 39 block on the same spatial layer. NOMA is achieved by the combination of superposition encoding and 40 successive interference cancellation (SIC), which is a method to reach the boundary of the capacity region of 41 degraded broadcast channel [7].

42
In order to further enhance system performance, beamforming (BF) was combined with NOMA in multiple-43 input-single-output (MISO) downlink [8][9][10][11][12][13][14]. In [8], the sum rate maximization problem with BF vector being 44 the optimization variable was studied and an one-dimensional iterative algorithm was proposed. However, for 45 each iteration, a second-order cone program convex problem needs to be solve, which results in very high 46 computational complexity. Moreover, as the signals of all users are superposed on one resource block, the 47 algorithm may suffer large process delay and error propagation of SIC for the system with a large number of 48 users. In [9], the sum rate optimization problem with minimum user rate constrain was investigated, and 49 therefore a low complexity BF scheme and a user clustering scheme were proposed. Since the presented BF 50 scheme and user clustering scheme were designed separately, some sum-rate performance loss is suffered. In 51 [10], the problem for maximization of the number of users with an ergodic user rate constrain was considered. A 52 power allocation scheme to satisfy ergodic user rate constrains was proposed and then a user admission 53 algorithm that achieves the maximum number of users was developed to guarantee the minimum user rate 54 requirements. However, as the MRT beamforming is adopted for downlink transmission, the optimization of BF 55 vector is not considered. By the duality between the multiple access channel (MAC) and broadcast channel (BC), 56 the duality scheme for sum rate optimization was developed in [11], which also suffers rather high 57 computational complexity because of needing to solve a quadratic constrained quadratic programs convex 58 problem. Furthermore, since the duality scheme is a quasi-degraded solution to the problem of sum rate 59 optimization actually, it is only feasible in the case, where the channel state information (CSI) of the scheduled 60 users meet the quasi-degrade property. As a result, the application of the duality scheme in practical systems 61 may be heavily restricted. In [12], the robust BF design problem to optimize the worst-case achievable sum rate 62 constrained by the total transmit power was studied. In [13], the optimal BF design problem which minimizes 63 the total transmission power subject to a pair of target interference levels constrains, was investigated for two-64 user MISO-NOMA downlink. Based on the results obtained by [13], [14] further proved that the minimum 65 transmit power of the NOMA transmission scheme was equal to that of dirty-paper coding in two-user case, 66 under the condition of the broadcast channel being quasi-degraded. Furthermore, a Hybrid NOMA (H-NOMA) 67 precoding algorithm with low-complexity, is proposed by combining NOMA with zero-forcing beamforming 68 (ZFBF). The summary for comparison of this paper with the existing works [8][9][10][11][12][13][14], is shown in Table 1. 69 Table 1: Comparison of this paper with the existing works [8][9][10][11][12][13][14]. "TU" and "MU" represent Two-User and 70 Multiple-User, respectively. "BV" and "WC sum rate" represent BF Vector and Worst Case sum rate, 71 respectively. "UPM" and "CM" represent User Pairing relationship Matrix and Clustering relationship Matrix, 72 respectively. CP and QDC represent Control Parameter for TMS and Quasi-Degraded Channel, respectively. 73 "PD" and "EP" represent Processing Delay and Error Propagation of SIC, respectively. providing high spectral efficiency [15][16][17]. In [15], the impact of user pairing on the performance of the two-76 user NOMA systems, is characterized. It was shown that the performance gain of NOMA over conventional 77 multiple access can be further enlarged by electing users whose channel conditions are more distinctive. In [16], 78 the problem of jointly optimizing user association and power control to maximize the overall spectral efficiency,

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Before proceeding, we introduce the following notation. Throughout the paper, we denote column vectors and

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However, user in user-pair 1 simply treat ,1 as noise and decodes its own signal ,1 . In MMSE-BF mode, 153 MMSE beamforming is adopted for the user-pair at the BS. 2K users are assumed to be uniformly located within 154 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., 155 selecting two users to form a group, in the system model for the following reasons. In the NOMA downlink, 156 users cancel the co-channel interference by performing SIC. However, with the number of users in a group 157 growing, the processing complexity and delay at users dramatically increase [23]. As a result, we consider only where 퐡 , 㤵(0, , 2 ) is the channel vector from BS to user in the user-pair , 푧 , 㤵(0, 0 )) is the 161 additive white Gaussian noise (AWGN) at user in the user-pair , and 0 is the corresponding noise power. For 162 the channel vector 퐡 , , the variance of the channel from BS to user in the user-pair is modeled as [14], [24] 163 where is the date rate of user in the user-pair

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It is hard to achieve the optimal solution to the problem (P1) due to binary constraints of is the signal-to-interference-plus-noise ratio (SINR) of user with where log(1 + , ) denotes the achievable rate for user to detect user 's message and In this paper, the angle between 퐡 and 퐡 is denoted by α, where 206 Without loss of generality, we set ∈ [0, ).

Analysis on Rate Regions of NOMA-MRT and MMSE-BF
NOMA-MRT is the combination of NOMA and MRT, i.e., performing MRT beamforming at the transmitter 220 and executing SIC at the receiver with strong channel conditions for two-receiver case.
Proof See the Appendix.

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Lemma 2 When∥ 퐡 ∥=∥ 퐡 ∥= , the achievable rate region boundary of NOMA-MRT for MISO broadcast 229 channel can be expressed as Proof See the Appendix.   249 Proof According to the duality between MAC and BC [25], the normalized BF vectors of user with 250 MMSE-BF in MISO broadcast channel is given by By the matrix inversion Lemma [26], Eq. (13) can be written as The SINR of user in dual MAC with MMSE receiver filter have the following form According to the definition of inner product, Eq. (15) can be rewritten as where represent the angle between MMSE and 퐡 , + is the angle between MMSE and 퐡 ( ) when α ∈ [0, / 255 2) and is that when ∈ /2, , = , , as illustrated in Figure 2.

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Based on the intersection points aforementioned, we develop an adaptive switching method to achieve a larger 327 rate region, i.e., Algorithm 1, described in Table 2, which outputs some parameters such as the user's transmit
Step 2: Define as the mode switching point the intersection point excluding the ends of the interval [0, ], and sort the mode switching points in ascending order, i.e., ,1 ⋯ , ⋯ ,Γ , where Γ is the number of mode switching points. Divide rate region boundary into Γ + 1 sections by , . 8.

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As we know, the local maximum values of ( ) in the interval (0, ) satisfies that the first-order derivative 356 of ( ) equals to zero.

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Proof See the Appendix.
if ,

1.
Calculate , using (45), = 1,2,⋯, .  2) Considering the problem which is how to pair user with a user from Φ 1 ( Φ 2 ) to achieve local-maximum 411 sum rate, user should be paired with the user, whose channel vector can form the minimum (maximum) 412 angle with that of user , according to (7

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The first two elements of -th entry in Ω denote the indexes of the two users in -th user-pair, and the third and 430 fourth element stand for power allocation vector and transmission mode adopted by the -th user-pair, 431 respectively.

432
When the number of users is odd, the last remained user can be trivially served by OMA. Specifically, if the 433 channel vector of the last remained user is 퐡 , then MRT BF is performed at the BS and the BF vector is given 434 by (6). As a result, the maximum signal to noise ratio can be achieved. if user has been paired 3.
if user has been paired 7.
Go to Line 5 8.
if 퐡 and 퐡 is orthogonal 10.
Let , N denote the user-pair which corresponds to N . 18.   Table 6, where is control parameter for transmission mode switching and Ω( ) denotes the -th element in the 456 -th entry in the set Ω. Noting that TDMA(FDMA) is adopted in Algorithm 5 (Line10), it is efficient when is 457 small. When the scenario with sufficient number of antennas at base station is considered, e.g., 2퐾 , the 458 performance can be further improved by combining the proposed JUPA with SDMA, which is beyond the scope 459 of this paper. 460 Table 6: Algorithm 5(JUPA/TMS).

1.
Perform JUPA to obtain pairing and power allocation configuration Ω 2.
end if 9.  Table 7.   As 0, the Eq. (50) has two roots both of which are real numbers and the roots are given by 587 ,1 = ,
where ∈ [0,1] represents the fraction of the time allocated to user . We consider the case in which SIC is 608 executed at user . According to (46)

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Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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The authors declare that there is no conflict of interest regarding the publication of this paper. The geometric description of BF vectors for ZFBF, MRT and MMSE-BF.   Sum rate versus the number of the antennas for several transmission schemes.