Cooperative Non-Orthogonal Multiple Access in

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS

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Cooperative Non-Orthogonal Multiple Access in Multiple-Input-Multiple-Output Channels Yiqing Li, Miao Jiang, Qi Zhang, Member, IEEE, Quanzhong Li, and Jiayin Qin

Abstract—Cooperative non-orthogonal multiple access (NOMA) systems inherit advantages of the NOMA protocol and the cooperative relay. In this paper, we propose cooperative NOMA systems in multiple-input-multiple-output channels. The whole transmission is divided into two phases. In the first phase, the base station broadcasts signals using the NOMA protocol to a central user and a cell-edge user. In the second phase, the central user helps the base station cooperatively relay signals intended for the cell-edge user. Our objective is to maximize achievable rate from the base station to the cell-edge user under transmit power constraints and achievable rate constraint from the base station to the central user. The difficulty of this problem is the joint beamforming of the base station and the central user in the second phase. We propose a constrained convex-concave procedure (CCCP)-based algorithm. To reduce computational complexity, we also propose a closed-form search based suboptimal algorithm. Simulation results demonstrate that our proposed cooperative NOMA system with CCCP-based algorithm outperforms the conventional NOMA scheme. When achievable rate constraint to the central user is low, our proposed cooperative NOMA system with the closed-form search based suboptimal algorithm outperforms the NOMA scheme. Index Terms—Constrained convex-concave procedure (CCCP), cooperative relay, multiple-input-multiple-output (MIMO), nonorthogonal multiple access (NOMA).

I.

INTRODUCTION

Non-orthogonal multiple access (NOMA) has been considered as a breakthrough technology for fifth generation (5G) wireless networks [1]–[9]. The main principle of NOMA is that the same frequency and time resources can be shared by multiple users and the users which have better channel conditions are able to employ successive interference cancellation (SIC) to detect and remove signals intended for the users which have poorer channel conditions. Early works on NOMA systems mainly focus on those in single-input-single-output (SISO) channels [3] or multiple-input-single-output (MISO) channels [6]. In [7], the NOMA technique was applied to multiple-input-multiple-output (MIMO) systems. In [8], the This work was supported in part by the National Natural Science Foundation of China under Grant 61672549 and Grant 61472458, in part by the Guangdong Natural Science Foundation under Grant 2015A030310264, and in part by the Guangzhou Science and Technology Program under Grant 201607010098. Y. Li, M. Jiang, Q. Zhang and J. Qin are with the School of Electronics and Information Technology, Sun Yat-sen University, Guangzhou 510006, Guangdong, China (e-mail: [email protected], [email protected], [email protected], [email protected]). J. Qin is also with Xinhua College of Sun Yat-sen University, Guangzhou 510520, Guangdong, China. Q. Li is with the School of Data and Computer Science, Sun Yatsen University, Guangzhou 510006, Guangdong, China, and also with the Guangdong Province Key Laboratory of Big Data Analysis and Processing, Guangzhou 510006, Guangdong, China (e-mail: [email protected]).

ergodic sum rate optimization problem in an MIMO NOMA system was investigated. To further improve the signal transmission rate from a base station to the users which have poorer channel conditions, Ding et al. proposed a novel cooperative NOMA transmission scheme where the user which has better channel condition behaves like a cooperative relay [10]. The whole transmission is divided into two phases. In the first phase, the base station broadcasts signals using the NOMA protocol to all the users, and in the second phase, the user which has better channel condition helps the base station cooperatively relay signals to the users which have poorer channel conditions. The conventional cooperative relay schemes were extensively investigated in the literature [11]–[14]. It is noted that in [15], [16], turbo codes and low-density-parity-check codes are employed in full-duplex and half-duplex cooperative relay systems where non-orthogonal channels are exploited. In [10], [15] and [16], the proposed cooperative relay transmission schemes are designed for SISO channels. In [17], Xu et al. proposed a beamforming scheme for MISO cooperative NOMA systems. In this paper, we investigate the cooperative NOMA systems in MIMO channels. The whole transmission is divided into two phases. In the first phase, the multi-antenna base station broadcasts signals using the NOMA protocol to a multiantenna central user and a multi-antenna cell-edge user. In the second phase, the central user helps the base station cooperatively relay signals intended for the cell-edge user. Our objective is to maximize achievable rate from the base station to the cell-edge user in two phases under transmit power constraints at the base station and the central user as well as achievable rate constraint from the base station to the central user in the first phase. The difficulty of the aforementioned optimization problem is the joint beamforming of the base station and the central user in the second phase. Furthermore, from the obtained joint beamforming matrix in the second phase, recovery of the individual beamforming matrices at the base station and the central user is also a problem. In this paper, we propose a constrained convex-concave procedure (CCCP)-based algorithm. The CCCP-based algorithm involves iteratively solving semidefinite programs (SDPs). To reduce the computational complexity, we also propose a closed-form search based suboptimal algorithm. The rest of this paper is organized as follows. In Section II, we describe the system model and problem formulation. In Section III, we describe the proposed CCCP-based algorithm. In Section IV, we give the details of the closed-form search based suboptimal algorithm. Simulation results are provided


first phase

... Central user ...

Hbc

2

Hce

Base station

second phase

respectively. Since the duration of the first phase is τ1 , the transmit energy of the base station in the first phase is ( ) ( ) τ1 tr V1 V1† + τ1 tr U1 U†1 . (2)

...

The received signal vectors at the central and cell-edge users in the first phase are given by

Cell-edge user

yc = Hbc x + nc , ye,1 = Hbe x + ne,1 ,

Hbe Fig. 1. System model of a cooperative NOMA transmission system in MIMO channels.

in Section V. We conclude our paper in Section VI. Notations: Boldface lowercase and uppercase letters denote vectors and matrices, respectively. The conjugate transpose, determinant, Frobenius norm and trace of the matrix A are denoted as A† , |A|, ∥A∥ and tr(A), respectively. By A ≽ 0, we mean that A is positive semidefinite. CN (0, σ 2 ) denotes the distribution of a circularly symmetric complex Gaussian random variable with zero mean and variance σ 2 . ei denotes the vector with the ith entry to be one and other entries to be zero. II. S YSTEM M ODEL A ND P ROBLEM F ORMULATION Consider a half-duplex cooperative NOMA transmission system, which consists of a base station, a central user and a cell-edge user, as shown in Fig. 1. The whole transmission is divided into two phases. The first and second phases are with durations of τ1 and 1 − τ1 , respectively. In the first phase, the base station broadcasts signals using the NOMA protocol to the central and cell-edge users. In the second phase, the central user helps the base station cooperatively relay the signals intended for the cell-edge user using decode-and-forward (DF) relaying. The base station, central user and cell-edge user are equipped with Nb , Nc and Ne antennas, respectively, where Nb ≥ Nc and Nb ≥ Ne are assumed. These assumptions are reasonable because the number of antennas at the base station is generally more than or equal to that at the central or cell-edge user. Denote the channel response matrices from the base station to the central user, from the base station to the cell-edge user and from the central user to the cell-edge user as Hbc ∈ CNc ×Nb , Hbe ∈ CNe ×Nb and Hce ∈ CNe ×Nc , respectively. A. The First Phase In the first phase, the base station broadcasts signals using the NOMA protocol to the central and cell-edge users. Denote the symbol vectors intended for the central and cell-edge users as sc ∈ CNb ×1 and se,1 ∈ CNb ×1 , respectively, with E[sc s†c ] = I and E[se,1 s†e,1 ] = I. The broadcast signal from the base station is written as x = V1 sc + U1 se,1

(1)

where V1 ∈ CNb ×Nb and U1 ∈ CNb ×Nb denote the precoding matrices for the central and cell-edge users in the first phase,

(3) (4)

respectively, where nc ∼ CN (0, σc2 I) and ne,1 ∼ CN (0, σe2 I) denote the additive Gaussian noise vectors at the central user and the cell-edge user, respectively. At the cell-edge user, the achievable rate to detect se,1 considering sc as interference is given by ζ1 = τ1 log2 I + Hbe U1 U†1 H†be ( )−1 † † 2 . σe I + Hbe V1 V1 Hbe (5) At the central user, SIC is implemented according to the NOMA protocol. The central user detects se,1 first considering sc as interference and then detects sc by cancelling se,1 . The achievable rate to detect se,1 considering sc as interference is ξe = τ1 log2 I + Hbc U1 U†1 H†bc )−1 ( . (6) σc2 I + Hbc V1 V1† H†bc The achievable rate to detect sc by cancelling se,1 is ξc = τ1 log2 I + σc−2 Hbc V1 V1† H†bc .

(7)

B. The Second Phase If the transmission rate from the base station to the celledge user in the first phase is less than ξe , the central user is able to decode se,1 and re-encode it as se,2 ∈ CNe ×1 [18]. In the second phase, both the base station and central user transmit se,2 to the cell-edge user. The received signal vector at the cell-edge user is given by ye,2 = Hbe U2 se,2 + Hce V2 se,2 + ne,2

(8)

where U2 ∈ CNb ×Ne and V2 ∈ CNc ×Ne denote the precoding matrices at the base station and central user in the second phase, respectively; ne,2 ∼ CN (0, σe2 I) denotes the additive Gaussian noise vector at the cell-edge user in the second phase. The achievable rate for the cell-edge user to detect se,2 is ζ2 = (1 − τ1 ) log2 I + σe−2 (Hbe U2 + Hce V2 ) † (Hbe U2 + Hce V2 ) . (9) According to [18], the achievable rate from the base station to the cell-edge user in two phases is expressed as follows ζe = min (ξe , ζ1 + ζ2 ) .

(10)

Remark 1: Considering that the central user is able to successfully detect symbol vectors intended for the cell-edge user in the first phase, sum achievable rate for the cell-edge user is ζ1 +ζ2 . In the first phase, achievable rate for the central


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user to detect symbol vectors intended for the cell-edge user is ξe . Thus, we have (10). Take an example similar to [15], [16]. If ξe = 1 bps/Hz, ζ1 = 0.5 bps/Hz, ζ2 = 0.5 bps/Hz and τ1 = 0.5, the base station transmits uncoded symbol vectors intended for the cell-edge user at actual transmission rate of 1 bps/Hz in the first phase. The central user can successfully detect symbol vectors whereas the cell-edge user cannot. In the second phase, both the base station and central user encode symbol vectors using the same LDPC codes with rate 0.5. After encoding, only the parity check bits are transmitted [15], [16]. After receiving both uncoded symbol vectors and parity check bits in two phases, the cell-edge user is able to successfully detect the symbol vectors intended for itself.

The constraints in (11d) and (11e) are rewritten as ( ) tr D1 FD†1 ≤ (Pb − P )/(1 − τ1 ), ( ) tr D2 FD†2 ≤ Pc /(1 − τ1 ),

max

[ ] D1 = INb ×Nb 0Nb ×Nc , [ ] D2 = 0Nc ×Nb INc ×Nc .

(17)

Wc = V1 V1† ,

(19)

U1 U†1 .

(20)

U1 , U2 , V1 , V2 , τ1 , P, ζe

s.t.

(11a)

ξc ≥ ξ¯c , ξe ≥ ζe , ζ1 + ζ2 ≥ ζe , ) ( ) ( tr V1 V1† + tr U1 U†1 ≤ P/τ1 , ) ( tr U2 U†2 ≤ (Pb − P )/(1 − τ1 ), ( ) tr V2 V2† ≤ Pc /(1 − τ1 ),

(11b)

0 ≤ P ≤ Pb , 0 ≤ τ1 ≤ 1

(11f)

(18)

Furthermore, let

We =

ζe

(16)

respectively, where

C. Problem Formulation In this paper, our objective is to maximize achievable rate from the base station to the cell-edge user in two phases under transmit power constraints at the base station and the central user as well as achievable rate constraint from the base station to the central user in the first phase, which is formulated as follows

(15)

The constraints in (11b) can be rewritten as τ1 log2 I + σc−2 Hbc Wc H†bc ≥ ξ¯c , τ1 ln σc2 I + Hbc (Wc + We ) H†bc − τ1 ln σc2 I + Hbc Wc H†bc ≥ ζe ln 2, τ1 ln σe2 I + Hbe (Wc + We ) H†be − τ1 ln σe2 I + Hbe Wc H†be + ζ2 ln 2 ≥ ζe ln 2.

(21)

(22)

(23)

(11c) Thus, the problem in (11) is recast as follows (11d) (11e)

where ξ¯c denotes the achievable rate constraint from the base station to the central user in the first phase, Pb denotes the transmit power constraint at the base station, Pc denotes the transmit power constraint at the central user, P denotes the transmit power allocated for transmission in the first phase at the base station. The problem in (11) is non-convex because of the non-convexity of the constraints in (11b), (11c), (11d) and (11e).

max Wc , We , F, τ1 , P, ζe

s.t.

ζe

(24a)

(21), (22), (23), (15), (16), (11f),

(24b)

tr (Wc ) + tr (We ) ≤ P/τ1 , F ≽ 0, Wc ≽ 0, We ≽ 0, rank (F) ≤ Ne

(24c) (24d) (24e)

where the constraint in (24e) is included such that from the obtained F, we can recover (V2 , U2 ). To proceed, we have the following theorem. Theorem 1: The problem in (24) is equivalent to the following rank-relaxation problem

III. CCCP-BASED A LGORITHM In this section, using variable substitution, we rewrite the non-convex functions in the constraints of (11) as perspective functions and their combinations [19]. In doing so, the problem in (11) is transformed into a difference of convex (DC) program [20] whose locally optimal solution can be found by employing CCCP-based algorithm [21]–[23]. By letting [ ] G = σe−1 Hbe σe−1 Hce , (12) [ ] ] U2 [ † F= (13) U2 V2† , V2 the achievable rate ζ2 in (9) is reformulated as follows ζ2 = (1 − τ1 ) log2 I + GFG† . (14)

max Wc , We , F, τ1 , P, ζe

s.t.

ζe (21), (22), (23), (15), (16), (11f), (24c), (24d).

(25)

Proof : See Appendix A. The problem in (25) is still non-convex because of the nonconvex constraints in (21), (22), (23), (15), (16) and (24c). Introducing a slack variable τ2 = 1 − τ1 and letting ¯ c = τ1 Wc , W ¯ e = τ1 We , W ¯ = τ2 F, F

(26) (27) (28)


4

the constraints in (21), (22) and (23) are reexpressed as ¯ c H† ≥ ξ¯c , τ1 log2 I + σc−2 τ1−1 Hbc W (29) bc ( ) ¯ c+W ¯ e H† τ1 ln σc2 I + τ1−1 Hbc W bc 2 † −1 ¯ c H ≥ ζe ln 2, − τ1 ln σc I + τ1 Hbc W (30) bc ( ) ¯ c+W ¯ e H† τ1 ln σe2 I + τ1−1 Hbe W be 2 ¯ c H† − τ1 ln σe I + τ1−1 Hbe W be ¯ † ≥ ζe ln 2. + τ2 ln I + τ2−1 GFG (31) It is noted that the functions of the type such as τ ln |I + τ −1 HWH† |, where H is a constant matrix, τ > 0 and W ≽ 0, are perspective functions [19]. Perspective functions are convex functions. Although the constraints in (30) and (31) are still non-convex since the left-hand sides of (30) and (31) contain both positive and negative perspective functions, the problem in (25) can be transformed into a DC program [20] as follows max

¯ c ≽ 0, W ¯ e ≽ 0, W ¯ ≽ 0, τ1 , τ2 , P, ζe F

s.t.

ζe (29), (30), (31), (11f), ) ( ¯ † ≤ (Pb − P ), tr D1 FD 1 ( ) ¯ † ≤ Pc , tr D2 FD 2 ( ) ( ) ¯ c + tr W ¯ e ≤ P, tr W τ1 + τ2 = 1.

(32a)

into convex constraints as follows ( ) ¯ c+W ¯ e H† τ1 ln σc2 I + τ1−1 Hbc W bc ( ) (l) (l) ¯ ¯ − f1 Wc , τ1 ; Wc , τ1 ≥ ζe ln 2, ( ) ¯ c+W ¯ e H† τ1 ln σe2 I + τ1−1 Hbe W be ( ) (l) ¯ c , τ1 ; W ¯ (l) , τ − f2 W c 1 −1 ¯ † ≥ ζe ln 2 + τ2 ln I + τ2 GFG

In (39) and (40), ( (l) Υ1

(32c)

(l)

(32e) (32f)

(l)

Υ2 = (l)

ψ2 =

)−1 † ¯ (l) H Hbc , σc2 I + (l) Hbc W c bc τ1 [ ( ] ) (l) (l) Nb ∑ λ1,i λ1,i ln 1 + (l) − (l) , (l) τ1 τ1 + λ1,i i=1 ( )−1 1 † 2 (l) † ¯ Hbe σe I + (l) Hbe Wc Hbe Hbe , τ1 [ ( ] ) (l) (l) Nb ∑ λ2,i λ2,i ln 1 + (l) − (l) (l) τ1 τ1 + λ2,i i=1

(41) (42)

(43) (44)

(l) (l) ¯ c(l) H† where λ1,i and λ2,i are the ith eigenvalues of Hbc W bc ¯ c(l) H† , respectively. In the (l +1)th iteration of the and Hbe W be proposed CCCP-based algorithm, we solve

max

¯ c ≽ 0, W ¯ e ≽ 0, W ¯ ≽ 0, τ1 , τ2 , P, ζe F

ζe (29), (37), (38), (11f), (32c), (32d), (32e), (32f).

(34)

(35)

1

H†bc

s.t.

where )−1 ( ˜ † H, Υ = H† I + τ˜−1 HWH ) ] N [ ( ∑ λi λi − ψ= ln 1 + τ˜ τ˜ + λi i=1

=

ψ1 =

(32d)

( ) [ ( )] ˜ τ˜ + tr Υ W − W ˜ f (W, τ ) ≤f W, + ψ (τ − τ˜)

(38)

where ( ) 1 (l) (l) † ¯ c , τ1 ; W ¯ (l) , τ ¯ (l) H f1 W = τ1 ln σc2 I + (l) Hbc W c c 1 bc τ1 [ ( )] ( ) (l) ¯ c−W ¯ (l) + ψ (l) τ1 − τ (l) , + tr Υ1 W (39) c 1 1 ( ) 1 (l) (l) † (l) (l) 2 ¯ ¯ c , τ1 ; W ¯ ,τ I + H W H f2 W = τ σ ln be c c 1 1 be (l) e τ1 [ )] ( ( ) (l) ¯ c−W ¯ (l) + ψ (l) τ1 − τ (l) . + tr Υ2 W (40) c 2 1

(32b)

We propose a CCCP-based algorithm to find a locally optimal solution to the problem in (32) [21]. To continue, we have the following lemma. Lemma 1: The first order Taylor expansion of f (W, τ ) = τ ln I + τ −1 HWH† (33) ( ) ˜ τ˜ is around the point W,

(37)

(45)

The problem in (45) is a convex SDP, which can be solved effectively using the interior point method [19]. We summarize the proposed CCCP-based algorithm in Algorithm 1. Algorithm 1 Proposed CCCP-Based Algorithm.

(36)

˜ † , N is the in which λi is the ith eigenvalue of HWH ˜ †. dimension of HWH Proof : See Appendix B. In the (l + 1)th iteration of the proposed CCCP-based ¯ c(l) , W ¯ e(l) , τ (l) ) which is optimal in the algorithm, given (W 1 lth iteration, the constraints in (30) and (31) are transformed

1: 2:

3:

(0)

(0)

(0)

Initialize: l = 0, Wc , We and τ1 ; Repeat: l := l + 1; Solve the problem in (45) to obtain the solution ¯ c(l) , W ¯ e(l) , F ¯ (l) , τ (l) , τ (l) , P (l) , ζe(l) ); (W 1 2 Until: Convergence.

Complexity Analysis: For Algorithm 1, the computational complexity is mainly from solving the problem in (45). The


5

problem in (45) is an SDP. From [24], the complexity of solving an SDP is ( ) 2 2.5 3 0.5 O msdp n3.5 (46) sdp + msdp nsdp + msdp nsdp where msdp is the number of constraints and nsdp denotes the dimension of the semidefinite cone. For the problem in (45), we have msdp = 9 and nsdp = Nb + Nc . Thus, the computational complexity of Algorithm 1 is ( ) L · O (Nb + Nc )3.5 (47) where L denotes the iteration number of the proposed CCCPbased algorithm. Remark 2: Using Lemma 1, we only achieve linear approximations of the negative perspective functions contained in (30) and (31). However, it is proved in [21] that our proposed CCCP-based algorithm converges to a local optimum of the problem in (32). IV. C LOSED -F ORM S EARCH BASED S UBOPTIMAL A LGORITHM The proposed CCCP-based algorithm involves iteratively solving SDPs. To reduce the computational complexity, we propose a closed-form search based suboptimal algorithm in this section. Given τ1 and P , the optimization of F in (24) is decoupled with the optimization of Wc and We . If only the optimization of F in (24) is considered, the closed-form optimal solution of F is provided in Appendix A, i.e., F∗ = B−1/2 CΛC† B−1/2

(48)

where B, C and Λ are defined in (86), (87) and (90), respectively. Given τ1 and P , the optimization of Wc and We in (24) is coupled because of the constraints in (22), (23) and (24c). We propose to minimize tr(Wc ) under the constraint in (21) by neglecting the functions of Wc in (22) and (23). Thus, the proposed algorithm is suboptimal. The optimization of Wc is formulated as min

tr (Wc ) s.t. log2 I + σc−2 Hbc Wc H†bc ≥ ξ¯c /τ1 .

Wc ≽0

ˆ† ˆ bc Γ ˆ V Hbc = U bc bc

Nc ∑

(50)

( ) log2 1 + σc−2 pi γbc,i = ξ¯c /τ1 .

(53)

i=1

Given τ1 and P , if only the optimization of We is considered, the problem in (24) is reduced to max κ

We ≽0,κ

s.t.

ˆ bc We H ˆ † ≥ κ, log2 I + H bc ˆ be We H ˆ † ≥ κ − ζ2 /τ1 , log2 I + H be

tr (We ) ≤ Pˆ

(54)

where ( )− 12 ˆ bc = σ 2 I + Hbc Wc H† Hbc , H c bc ( )− 12 ˆ be = σ 2 I + Hbe Wc H† H Hbe , e be κ = ζe /τ1 , Pˆ = P/τ1 − tr (Wc ) .

(55) (56) (57) (58)

Note in (57), we assume that τ1 > 0. This is because if τ1 = 0, i.e., the duration of the first phase is zero, there is no need to optimize Wc and We which determine the beamforming matrices in the first phase. For the problem in (54), we propose a generalized singular value decomposition (GSVD) based suboptimal solution. ˆ bc and H ˆ be be Let the GSVD of H ˆ bc = U ˘ bc Λbc [R, 0ϕ×(N −ϕ) ]V ˘ †, H b ˆ be = U ˘ be Λbe [R, 0ϕ×(N −ϕ) ]V ˘† H b

(59) (60)

˘ bc ∈ CNc ×Nc , U ˘ be ∈ CNe ×Ne and V ˘ ∈ CNb ×Nb where U ϕ×ϕ are unitary matrices, R ∈ C is a non-singular uppertriangular matrix, Λbc ∈ CNc ×ϕ and Λbe ∈ CNe ×ϕ are non-negative diagonal matrices with Λ†bc Λbc + Λ†be Λbe = I, ˆ† ,H ˆ † ]† }. The diagonal entries of Λ† Λbc , and ϕ = rank{[H bc be bc denoted as λbc,1 , λbc,2 , · · · , λbc,ϕ , are ordered such that

(49)

Denote the reduced singular value decomposition (SVD) of Hbc as 1 2

where ν denotes the constant water-level which satisfies

0 ≤ λbc,1 ≤ λbc,2 ≤ · · · ≤ λbc,ϕ .

(61)

The diagonal entries of Λ†be Λbe , denoted as λbe,1 , λbe,2 , · · · , λbe,ϕ , are ordered such that λbe,1 ≥ λbe,2 ≥ · · · ≥ λbe,ϕ ≥ 0.

(62)

ˆ bc ∈ CNc ×Nc , V ˆ bc ∈ CNb ×Nc and Γ ˆ bc = where U diag (γbc,1 , γbc,2 , · · · , γbc,Nc ) with γbc,1 ≥ γbc,2 ≥ · · · ≥ γbc,Nc . The closed-form solution to the problem in (49) is [25]

Furthermore, λbc,i + λbe,i = 1 for i = 1, 2, · · · , ϕ. The suboptimal beamforming matrix We∗ is constructed by

ˆ bc ΣV ˆ† Wc∗ = V bc

We∗ = ΘQΘ†

(51)

where Σ = diag (p1 , p2 , · · · , pNc ). In the matrix Σ, diagonal entries are obtained from the water-filling power allocation scheme )+ ( 1 , i = 1, 2, · · · , Nc (52) pi = ν − γbc,i

where

[ ˘ Θ=V

R−1

0(Nb −ϕ)×ϕ

(63) ] (64)

and Q = diag (q1 , q2 , · · · , qϕ ) denotes a ϕ×ϕ diagonal matrix with power allocation factors qi , i = 1, 2, · · · , ϕ, being its


6

diagonal entries. Substituting (63) into the problem in (54), we have max κ

(65a)

{qi ≥0},κ

s.t.

ϕ ∑ i=1 ϕ ∑ i=1 ϕ ∑

log2 (1 + qi λbc,i ) ≥ κ,

(65b)

log2 (1 + qi λbe,i ) ≥ κ − ζ2 /τ1 ,

(65c)

qi θi ≤ Pˆ

(65d)

i=1

where θi denotes the ith diagonal entry of the matrix Θ† Θ. We have the following lemma. Lemma 2: Denote the ith diagonal entry of the matrix Θ† Θ as θi . We have θi > 0. Proof : We prove Lemma 2 by contradiction. Suppose θi = 0. From (64), we have Θ† Θ = (R−1 )† R−1 .

(66)

For the matrix R−1 , if θi = 0, all the entries in the ith row of R−1 should be zero. This contradicts the inversion property of non-singular upper-triangular matrix R. With Lemma 2, the optimal solution to the problem in (65) is given by the following lemma. Lemma 3: If λbc,i = 1, λbe,i = 0, the optimal solution to the problem in (65) is qi∗ =

ω1∗ − ω3∗ θi ln 2 . ω3∗ θi ln 2

(67)

If λbc,i = 0, λbe,i = 1, we have qi∗ =

ω2∗

ω3∗ θi

ln 2 − . ω3∗ θi ln 2

(68)

If λbc,i > 0, λbe,i > 0, we have qi∗ =

−(ω3∗ θi

ln 2 − λbc,i λbe,i ) 2λbc,i λbe,i ω3∗ θi ln 2

(69) √ ∗ (ω3 θi ln 2 − λbc,i λbe,i )2 − 4λbc,i λbe,i ω3∗ θi ϑi ln 2 + . 2λbc,i λbe,i ω3∗ θi ln 2 ω1∗ ,

ω2∗

ω3∗

and are as follow. In (67), (68) and (69), Case I: (ω1∗ = 0, ω2∗ = 1) or (ω1∗ = 1, ω2∗ = 0). In this case, we obtain ω3∗ by solving Pˆ −

ϕ ∑

qi θi = 0.

(70)

i=1

Case II: ω1∗ > 0 and ω2∗ > 0. we can obtain ω1∗ , ω2∗ and ω3∗ by solving the following equations ϕ ∑ i=1

log2 (1 + qi λbc,i ) =

ϕ ∑

log2 (1 + qi λbe,i ) + ζ2 /τ1 , (71)

i=1

ω1∗ + ω2∗ = 1

(72)

and (70). Proof : See Appendix C.

Given τ1 and P , using the aforementioned methods, we obtain the closed-form optimal solution of F and closed-form suboptimal solutions of Wc and We to the problem in (24). Thus, the suboptimal solution to the problem in (24) can be obtained by two-dimensional search over τ1 and P . We summarize the proposed closed-form search based suboptimal algorithm in Algorithm 2, where α1 and α2 denote the search stepping sizes for τ1 and P , respectively. Algorithm 2 Proposed Closed-Form Search Based Suboptimal Algorithm. ˆ 1 = ⌈1/α1 ⌉ + 1, L ˆ2 = 1: Initialize: τ1 = 0, P = 0, L ˆ ˆ ⌈Pb /α2 ⌉ + 1, l1 = 1, l2 = 1; ˆ1 2: While ˆ l1 ≤ L ˆ ˆ2 While l2 ≤ L ∗ Obtain F by (48); Obtain Wc∗ by (51); Obtain We∗ by (63) and Lemma 3; P := P + α2 ; ˆl2 := ˆl2 + 1; End While τ1 := τ1 + α1 ; ˆl1 := ˆl1 + 1; 3: End While Complexity Analysis: For Algorithm 2, the computational complexity to obtain F∗ by (48) is mainly from the SVD of m×n matrix GB−1/2 ∈ CNe ×(Nb +Nc ) . For a matrix , ( 2 )A ∈ C the computation complexity of SVD is O mn [26]. Thus, the computation complexity to obtain F∗ is ( ) 2 O Ne (Nb + Nc ) . (73) Similarly, the computational complexity to obtain Wc∗ by (51), which is mainly from the SVD of matrix Hbc ∈ CNc ×Nb , is ) ( (74) O Nc Nb2 . The computational complexity to obtain We∗ , which is mainly ˆ bc ∈ CNc ×Nb and H ˆ be ∈ from the SVD of matrices H Ne ×Nb C , is ( ) O (Nc + Ne )Nb2 . (75) Thus, the computational complexity of Algorithm 2 is ( ) ˆ1 · L ˆ 2 · O Ne (Nb + Nc )2 + (2Nc + Ne )Nb2 L

(76)

ˆ 1 and L ˆ 2 are defined in Algorithm 2. where L V. S IMULATION R ESULTS In this section, we evaluate our proposed cooperative NOMA transmission schemes through computer simulations. The cooperative NOMA transmission system consists of a base station, a central user and a cell-edge user. The base station, the central user and the cell-edge user are equipped with Nb = 6, Nc = 2 and Ne = 2 antennas, respectively, if not specified. The entries of channel response matrices from the base station to the central user, from the base station to the cell-edge user and from the central user to the cell-edge user, i.e., Hbc , Hbe and Hce , are independent and identically distributed (i.i.d.)


2.5 C−NOMA, CCCP C−NOMA, CFS NOMA OMA

ζe (bps/Hz)

2

1.5

1

0.5

0

0

1

2

3 ξ¯c (bps/Hz)

4

5

6

Fig. 2. Average achievable rate of the cell-edge user, ζe , versus the achievable rate constraint from the base station to the central user, ξ¯c ; performance comparisons of our proposed cooperative NOMA transmission scheme with the CCCP-based algorithm and closed-form search based suboptimal algorithm, the conventional NOMA and OMA schemes without cooperation where Pb = 30 dBm and Pc = 20 dBm.

1.6 1.4 1.2

C−NOMA, CCCP C−NOMA, CFS NOMA OMA

1 ζe (bps/Hz)

complex Gaussian random variables with zero mean. Their variances, assigned by adopting a path loss model, are d−2 bc , −2 d−2 and d , respectively, where d , d and d denote bc be ce ce be the distances from the base station to the central user, from the base station to the cell-edge user and from the central user to the cell-edge user, respectively. In the simulations, we assume that dbc = 500 m, dbe = 1000 m and dce = 500 m. Furthermore, we assume that σc2 = σe2 = −80 dBm. These simulation scenarios and parameters model a typical urbane cellular transmission environment [5]. In Fig. 2, we present the average achievable rate region (ζe , ξ¯c ) of different schemes, i.e., the achievable rate of celledge user, ζe , versus the achievable rate constraint from the base station to the central user, ξ¯c , where Pb is 30 dBm and Pc is 20 dBm. In the legend, “C-NOMA, CCCP” denotes our proposed cooperative NOMA transmission scheme with CCCP-based algorithm. “C-NOMA, CFS” denotes our proposed cooperative NOMA transmission scheme with the closed-form search based suboptimal algorithm. “NOMA” denotes the conventional NOMA scheme without cooperation, i.e., τ1 = 1. “OMA” denotes the conventional orthogonal multiple access (OMA) scheme. From Fig. 2, it is observed that while maintaining the achievable rate from the base station to the central user, ξ¯c , at 3.1 bps/Hz, our proposed “C-NOMA, CCCP” scheme provides the average achievable rate of the cell-edge user at about 1.55 bps/Hz. When ξ¯c is 3.1 bps/Hz, the “NOMA” and “OMA” schemes achieve the average achievable rates of about 1.15 bps/Hz and 0.75 bps/Hz, respectively. From Fig. 2, it is also found that when ξ¯c is below 2.6 bps/Hz, our proposed “C-NOMA, CFS” scheme outperforms the “NOMA” scheme. The price paid for this performance improvement is the cooperative transmission requirement of the central user and slight computational complexity increasing. This is because the “NOMA” scheme is a special case of our proposed “C-NOMA, CFS” scheme where τ1 = 1. In the “NOMA” scheme, to obtain precoding matrices for the central and celledge users, i.e., V1 and U1 , the CCCP-based algorithm should ˘ · O(N 3.5 ) be employed whose computational complexity is L b ˘ where L denotes the iteration number of the CCCP-based algorithm. Remark 3: From Fig. 2, ξ¯c should be chosen such that 0 ≤ ¯ ξc ≤ ξ¯cmax (Pb ), where ξ¯cmax (Pb ) = max log2 I + σc−2 Hbc V1 V1† H†bc . tr(V1 V1† )≤Pb (77) In Fig. 3, we present the average achievable rate comparisons of the cell-edge user for different transmit power constraints at the base station, Pb , where Pc = 20 dBm and the achievable rate constraint from the base station to the central user is ξ¯c = 0.5ξ¯cmax (Pb ). From Fig. 3, it is observed that the average achievable rate of the cell-edge user increases with the increase of Pb . From Fig. 3, it is also found that the our proposed “C-NOMA, CCCP” scheme outperforms the “NOMA” and “OMA” schemes. When Pb is between 20 dBm and 29 dBm, our proposed “C-NOMA, CFS” scheme outperforms the “NOMA” scheme. When Pb is 30 dBm, our proposed “C-NOMA, CFS” scheme is not superior to the

7

0.8 0.6 0.4 0.2 0 20

22

24

26

28

30

Pb (dBm)

Fig. 3. Average achievable rate of the cell-edge user, ζe , versus Pb ; performance comparisons of our proposed cooperative NOMA transmission scheme with the CCCP-based algorithm and closed-form search based suboptimal algorithm, the conventional NOMA and OMA schemes without cooperation where Pc = 20 dBm and ξ¯c = 0.5ξ¯cmax (Pb ).

“NOMA” scheme. This is because in our proposed “C-NOMA, CFS” scheme, we minimize tr(Wc ) under the constraint in (21) by neglecting the functions of Wc in (22) and (23). Thus, our proposed “C-NOMA, CFS” scheme is suboptimal. Note that at high Pb or other configurations, the “NOMA” scheme cannot outperform our proposed “C-NOMA, CCCP” scheme. The reason is that the “NOMA” scheme is a special case of our proposed “C-NOMA, CCCP” scheme where τ1 = 1. In Fig. 4, we present the average achievable rate comparisons of the central user for different transmit power constraints at the base station, Pb , where Pc = 20 dBm and the achievable rate constraint from the base station to the cell-


8

5.5 5 4.5

2.2 C−NOMA, CCCP C−NOMA, CFS NOMA OMA


2 1.8

4 ζe (bps/Hz)

ζc (bps/Hz)

1.6 3.5 3

1.4 1.2

2.5 1

2

0.8

1.5 1 20

22

24

26

28

30

10

15

20 Pc (dBm)

Pb (dBm)

edge user, denoted as ζ¯e . Here, we choose ζ¯e = 0.5ζ¯emax (Pb ) where ζ¯emax (Pb ) is defined as log2 I + σe−2 Hbe U1 U†1 H†be . ζ¯emax (Pb ) = max † tr(U1 U1 )≤Pb (78) From Fig. 4, it is shown that the our proposed “C-NOMA, CCCP” scheme outperforms the “NOMA” and “OMA” schemes. When Pb is between 20 dBm and 28 dBm, our proposed “CNOMA, CFS” scheme outperforms the “NOMA” scheme. In Fig. 5, we present the average achievable rate comparisons of the cell-edge user for different transmit power constraints at the central user, Pc , where Pb = 30 dBm and the achievable rate constraint from the base station to the central user is 3.1 bps/Hz. From Fig. 5, it is observed that without cooperation of the central user, the “NOMA” and “OMA” schemes obtain the constant average achievable rates of the cell-edge user. From Fig. 5, it is also found that when Pc is higher than 22 dBm, our proposed “C-NOMA, CFS” scheme outperforms the “NOMA” scheme. In Fig. 6, we present the average achievable rate comparisons of the cell-edge user for different number of transmit antennas at the base station, Nb , where Pb = 20 dBm, Pc = 20 dBm, and the achievable rate constraint from the base station to the central user is ξ¯c = 0.5ξ¯cmax (Pb ). From Fig. 6, it is observed that the average achievable rate increases with the increase of Nb for all the schemes. In Fig. 7, we show the obtained duration of the first phase, τ1 , versus the achievable rate constraint from the base station to the central user, ξ¯c . For the “OMA” scheme, we assume that the first phase is for signal transmission from the base station to the central user and the second phase is for signal transmission to the cell-edge user. From Fig. 7, it is observed that the duration of the first phase, τ1 , increases with the increase of ξ¯c . When ξ¯c = 0 bps/Hz, our proposed cooperative NOMA transmission schemes reduce to the cooperative relay

30

Fig. 5. Average achievable rate of the cell-edge user, ζe , versus Pc ; performance comparisons of our proposed cooperative NOMA transmission scheme with the CCCP-based algorithm and closed-form search based suboptimal algorithm, the conventional NOMA and OMA schemes without cooperation where Pb = 30 dBm and ξ¯c = 3.1 bps/Hz.

0.9 0.8 0.7 0.6 ζe (bps/Hz)

Fig. 4. Average achievable rate of the central user, ζc , versus Pb ; performance comparisons of our proposed cooperative NOMA transmission scheme with the CCCP-based algorithm and closed-form search based suboptimal algorithm, the conventional NOMA and OMA schemes without cooperation where Pc = 20 dBm and ξ¯e = 0.5ξ¯emax (Pb ).

25


0.5 0.4 0.3 0.2 0.1 0

4

5

6

7

8 Nb

9

10

11

12

Fig. 6. Average achievable rate of the cell-edge user, ζe , versus Nb ; performance comparisons of our proposed cooperative NOMA transmission scheme with the CCCP-based algorithm and closed-form search based suboptimal algorithm, the conventional NOMA and OMA schemes without cooperation where Pb = 20 dBm, Pc = 20 dBm and ξ¯c = 0.5ξ¯cmax (Pb ).

transmission scheme. On the other hand, with the increase of ξ¯c , the achievable rate constraint from the base station to the central user becomes stringent, which causes more transmission time to be allocated to the first phase, i.e., larger τ1 . When τ1 = 1, ζe is reduced to 0 bps/Hz. Thus, our proposed “C-NOMA, CCCP” scheme performs the same as the “OMA” scheme. VI. C ONCLUSION In this paper, we have proposed a CCCP-based algorithm and a closed-form search based suboptimal algorithm for cooperative NOMA systems in MIMO channels. It is shown


9

where µ1 ≥ 0 and µ2 ≥ 0 are the Lagrange dual multipliers, ( ( )) L (F, µ1 , µ2 ) = ln I + GFG† + µ1 η1 − tr D1 FD†1 ( )) ( + µ2 η2 − tr D2 FD†2 . (84)

1 0.9 0.8 0.7

With fixed µ1 and µ2 , the Lagrangian dual function is equivalent to g (µ1 , µ2 ) = max ln I + GFG† − tr (BF) (85)

τ1

0.6 0.5 0.4

F≽0

0.3

where

0.2

C−NOMA, CCCP C−NOMA, CFS OMA

0.1 0

0

1

2

3 ξ¯c (bps/Hz)

4

5

6

Fig. 7. Obtained duration of the first phase, τ1 , versus ξ¯c ; comparisons of our proposed cooperative NOMA transmission scheme with the CCCPbased algorithm and closed-form search based suboptimal algorithm, and the conventional OMA scheme without cooperation where Pb = 30 dBm and Pc = 20 dBm.

through simulation results that our proposed cooperative NOMA system with CCCP-based algorithm outperforms the conventional NOMA and OMA schemes. When achievable rate constraint from the base station to the central user is low, our proposed cooperative NOMA system with the closed-form search based suboptimal algorithm outperforms the conventional NOMA scheme.

In (24), if only the optimization of F is considered, the problem in (24) is reduced to max ln I + GFG† F≽0 ( ) ( ) s.t. tr D1 FD†1 ≤ η1 , tr D2 FD†2 ≤ η2 ,

To continue, we have the following lemma. Lemma 4: The necessary condition that the problem in (85) has a bounded optimal value is µ1 > 0 and µ2 > 0. Proof : See Appendix D. From (17) and (18), we have B ≻ 0. Thus, B−1 exists. Let the reduced singular value decomposition (SVD) of the matrix GB−1/2 be given by ˇΓ ˇ 1/2 C† GB−1/2 = U

(79)

ˇ = diag (γ1 , · · · , γN ) Γ e

F∗ = B−1/2 CΛC† B−1/2

(89)

( ) ˇ1, · · · , λ ˇN , Λ = diag λ e ˇ i = (1 − 1/γi )+ , i ∈ {1, · · · , Ne }. λ

(90) (91)

From the structure of F∗ , we have rank (F∗ ) ≤ rank (C) = Ne . Thus, the optimal solution to the problem in (82) is also the optimal solution to the problem in (79). A PPENDIX B P ROOF OF L EMMA 1 The first order Taylor expansion of f (W, τ ) is given by

(81)

f (W, τ ) [ ] ( ) ( ) ∂f (W, τ ˜ ) ˜ τ˜ + tr ˜ ≤ f W, W−W ∂W ˜ W=W ( ) ˜ ∂f W, τ + (τ − τ˜) . (92) ∂τ τ =˜ τ

The computation of

The Lagrangian dual function of (82) is defined as F≽0

(88)

with γ1 ≥ γ2 ≥ · · · ≥ γNe ≥ 0. It has been shown in [27] that the optimal solution to the problem in (85) with arbitrary B ≻ 0 has the following form

(80)

in which P ∗ and τ1∗ denote the optimal solutions to the problem in (24). The rank-relaxation problem of (79) is max ln I + GFG† F≽0 ( ) ( ) s.t. tr D1 FD†1 ≤ η1 , tr D2 FD†2 ≤ η2 . (82)

g (µ1 , µ2 ) = max L (F, µ1 , µ2 )

(87)

ˇ ∈ CNe ×Ne , C ∈ C(Nb +Nc )×Ne and where U

where Pb − P ∗ , η1 = 1 − τ1∗ Pc η2 = 1 − τ1∗

(86)

where

A PPENDIX A P ROOF OF T HEOREM 1

rank (F) ≤ Ne

B = µ1 D†1 D1 + µ2 D†2 D2 .

(83)

∂f (W,˜ τ) ∂W

is

( )−1 ∂f (W, τ˜) = H† I + τ˜−1 HWH† H ∂W

(93)


where in the derivation ˜ ) ∂f (W,τ tation of is ∂τ (

˜ τ ∂f W,

)

= W−1 is used. The compu-

∂ ln|A| ∂W

[

N ( ∏

∂ τ ln

∂τ

∂

∂τ ( τ ln 1 +

N ∑ i=1

= =

1+

i=1

=

10

λi τ

)

]

λbc,i λbe,i ω3∗ θi qi2 ln 2 + (ω3∗ θi ln 2 − (ω1∗ + ω2∗ )λbc,i λbe,i ) qi + ϑi = 0 (101) where

) λi

ϑi = ω3∗ θi ln 2 − ω1∗ λbc,i − ω2∗ λbe,i < 0.

τ

∂τ ( ) ] λi λi ln 1 + − τ τ + λi

N [ ∑ i=1

(94)

˜ † and N is the in which λi is the ith eigenvalue of HWH † ˜ dimension of HWH . A PPENDIX C P ROOF OF L EMMA 3 The Lagrangian function of the problem in (65) is ( ϕ ) ∑ ˆ L =κ + ω1 log2 (1 + qi λbc,i ) − κ i=1

+ ω2

( ϕ ∑ (

+ ω3

Pˆ −

{qi ≥0},κ

) qi θi

where ω1 ≥ 0, ω2 ≥ 0 and ω3 ≥ 0 are the Lagrange dual multipliers. Define ˆ gˆ (ω1 , ω2 , ω3 ) = max L. {qi ≥0},κ

(96)

Since the problem in (65) is convex with strong duality, we can solve it by ω1 ≥0,ω2 ≥0,ω3 ≥0

gˆ (ω1 , ω2 , ω3 ) .

(97)

Denote the optimal solution to the problem in (97) as (ω1∗ , ω2∗ , ω3∗ ). For i = 1, 2, · · · , ϕ, qi needs to maximize Lî which is defined as follows [28] Lî = ω1∗ log2 (1 + qi λbc,i ) + ω2∗ log2 (1 + qi λbe,i ) − ω3∗ qi θi ∫ qi 1 = fi (x) dx + Ci (98) ln 2 0 where Ci is a constant and fi (x) is defined as fi (x) =

ω1∗ λbc,i ω2∗ λbe,i + − ω3∗ θi ln 2. 1 + xλbc,i 1 + xλbe,i

(99)

To obtain the optimal qi that maximize Lî in (98), we consider the following two cases: Case I: ω1∗ λbc,i +ω2∗ λbe,i −ω3∗ θi ln 2 ≤ 0. In this case, fi (x) is a decreasing function for x ≥ 0 and fi (0) = ω1∗ λbc,i + ω2∗ λbe,i − ω3∗ θi ln 2 ≤ 0. Hence, (99) is always non-positive and the maximum of Li is achieved when qi∗ = 0. Case II: ω1∗ λbc,i +ω2∗ λbe,i −ω3∗ θi ln 2 > 0. In this case, fi (x) is a decreasing function for x ≥ 0. We also have fi (0) = ω1∗ λbc,i + ω2∗ λbe,i − ω3∗ θi ln 2 > 0 and lim fi (x) = −ω3∗ θi ln 2 < 0.

x→∞

(100)

i=1

+

(95)

i=1

min

In the derivation, λbc,i + λbe,i = 1 is employed. To continue, we have the following lemma. Lemma 5: For the optimal solution to the problem in (97), i.e., (ω1∗ , ω2∗ , ω3∗ ), we have ω3∗ > 0. Proof : We prove this lemma by contradiction. Suppose ω3∗ = 0. From the Karush-Kuhn-Tucker (KKT) conditions, we have ∂ Lˆ = 1 − (ω1∗ + ω2∗ ) = 0. (103) ∂κ ω1 = ω1∗ , ω2 = ω2∗ , The power allocation factors {qi } that maximizes Lˆ is obtained by solving ( ϕ ) ∑ ∗ max κ + ω1 log2 (1 + qi λbc,i ) − κ (104)

log2 (1 + qi λbe,i ) − κ + ζ2 /τ1 ϕ ∑

(102)

ω3 = ω3∗ ,

)

i=1

Therefore, there exits a unique solution qi such that fi (qi ) = 0. Let fi (qi ) = 0, we have

ω2∗

( ϕ ∑

)

log2 (1 + qi λbe,i ) − κ + ζ2 /τ1

.

ω1∗ + ω2∗ = 1.

(105)

i=1

From (103), we have Because ω1∗ ≥ 0 and ω2∗ ≥ 0, ω1∗ and ω2∗ cannot be equal to zero simultaneously. By letting qi → +∞, the optimal value of (104) will be unbounded, which contradicts the optimality of (ω1∗ , ω2∗ , ω3∗ ). From Lemma 5, ω3∗ > 0. From Lemma 2, θi > 0. We also have λbc,i + λbe,i = 1. Based on the different values of λbc,i and λbe,i , we consider the following three conditions: 1) λbc,i = 1, λbe,i = 0. Under this condition, from (101), the maximum of Lî is achieved when ω ∗ − ω ∗ θi ln 2 . (106) qi∗ = 1 ∗ 3 ω3 θi ln 2 2) λbc,i = 0, λbe,i = 1. Under this condition, the maximum of Lî is achieved when ω ∗ − ω ∗ θi ln 2 qi∗ = 2 ∗ 3 . (107) ω3 θi ln 2 3) λbc,i > 0, λbe,i > 0. Under this condition, the equation (101) have two roots. One root is larger than zero and the other one is smaller than zero. This is because ϑi < 0. (108) λbc,i λbe,i ω3∗ θi ln 2 Since qi∗ ≥ 0, the maximum of Lî is achieved when −(ω3∗ θi ln 2 − λbc,i λbe,i ) (109) 2λbc,i λbe,i ω3∗ θi ln 2 √ ∗ (ω3 θi ln 2 − λbc,i λbe,i )2 − 4λbc,i λbe,i ω3∗ θi ϑi ln 2 + . 2λbc,i λbe,i ω3∗ θi ln 2

qi∗ =


11

In (106), (107) and (109), ω1∗ , ω2∗ and ω3∗ are still unknown. According to the complementary slackness condition, we have ( ) ϕ ∑ ∗ ˆ ω3 P − qi θi = 0. (110) i=1

ω3∗

From Lemma 5, > 0, we have (70). From (105), i.e., ω1∗ + ω2∗ = 1, we consider the following two cases: Case I: (ω1∗ = 0, ω2∗ = 1) or (ω1∗ = 1, ω2∗ = 0). In this case, substituting the optimal closed-form expression of qi∗ into (70), we can obtain ω3∗ . Case II: ω1∗ > 0 and ω2∗ > 0. Since the ω1∗ and ω2∗ are the Lagrange dual multipliers for the constraints in (65b) and (65c), respectively, the constraints in (65b) and (65c) are active. Thus, we have ϕ ∑ i=1 ϕ ∑

log2 (1 + qi λbc,i ) = κ,

(111)

log2 (1 + qi λbe,i ) = κ − ζ2 /τ1 .

(112)

i=1

Combining (111) and (112), we obtain (71). Solving (71), (72) and (70) simultaneously, we obtain ω1∗ , ω2∗ and ω3∗ . A PPENDIX D P ROOF OF L EMMA 4 We prove Lemma 4 by contradiction. Suppose that µ1 ≤ 0 or µ2 ≤ 0. If µ1 ≤ 0, let F = β1 e1 e†1 (113) with β1 > 0 being any positive scalar. Substituting (113) into (85) yields ( ) max ln 1 + β1 ∥Ge1 ∥2 − β1 µ1 . (114) β1 >0

Since the probability ∥Ge1 ∥ > 0 is close to 1, the objective in (114) becomes unbounded when β1 → ∞. Thus, the presumption that µ1 ≤ 0 cannot be true. If µ2 ≤ 0, let F = β2 eNb +Nc e†Nb +Nc

(115)

with β2 > 0 being any positive scalar. Substituting (115) into (85) yields ( ) max ln 1 + β2 ∥GeNb +Nc ∥2 − β2 µ2 . (116) β2 >0

Since the probability ∥GeNb +Nc ∥ > 0 is close to 1, the objective in (116) becomes unbounded when β2 → ∞. Thus, the presumption that µ2 ≤ 0 cannot be true. R EFERENCES [1] Y. Saito, Y. Kishiyama, A. Benjebbour, T. Nakamura, A. Li, and K. Higuchi, “Non-orthogonal multiple access (NOMA) for cellular future radio access,” in Proc. IEEE Veh. Tech. Conf. 2013, pp. 1-5. [2] Z. Ding, X. Lei, G. K. Karagiannidis, R. Schober, J. Yuan, and V. K. Bhargava, “A survey on non-orthogonal multiple access for 5G networks: Research challenges and future trends,” IEEE J. Sel. Areas Commun., vol. 35, no. 10, pp. 2181-2195, Oct. 2017.

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Yiqing Li received the B.Eng. degree in software engineering from Sun Yat-sen University (SYSU), Guangzhou, China, in 2015. She is currently pursuing the Ph.D. degree with the School of Electronics and Information Technology, SYSU, Guangzhou, China. Her research interests include non-orthogonal multiple access, cooperative communications, and multiple-input-multiple-output communications.

Miao Jiang received the B.Eng. degree in software engineering from Sun Yat-sen University (SYSU), Guangzhou, China, in 2015. He is currently pursuing the Ph.D. degree with the School of Electronics and Information Technology, SYSU, Guangzhou, China. His research interests include non-orthogonal multiple access, cooperative communications, and multiple-input-multiple-output communications.

Qi Zhang (S’04-M’11) received the B.Eng. (Hons.) and M.S. degrees from the University of Electronic Science and Technology of China, Chengdu, China, in 1999 and 2002, respectively, and the Ph.D. degree in electrical and computer engineering from the National University of Singapore (NUS), Singapore, in 2007. He is currently an Associate Professor with the School of Electronics and Information Technology, Sun Yat-sen University, Guangzhou, China. From 2007 to 2008, he was a Research Fellow with the Communications Lab, Department of Electrical and Computer Engineering, NUS. From 2008 to 2011, he was with the Center for Integrated Electronics, Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, Shenzhen, China. His research interests include non-orthogonal multiple access, wireless communications powered by energy harvesting, cooperative communications, ultra-wideband communications.

Quanzhong Li received the B.S. and Ph.D. degrees from Sun Yat-sen University (SYSU), Guangzhou, China, both in information and communications engineering, in 2009 and 2014, respectively. He is currently a Lecturer in the School of Data and Computer Science, SYSU. He is also with the Guangdong Province Key Laboratory of Big Data Analysis and Processing, Guangzhou, China. His research interests include non-orthogonal multiple access, wireless communications powered by energy harvesting, cognitive radio, cooperative communications, and multiple-input-multiple-output communications.

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Jiayin Qin received the M.S. degree in radio physics from the Huazhong Normal University, Wuhan, China, in 1992 and the Ph.D. degree in electronics from Sun Yat-sen University (SYSU), Guangzhou, China, in 1997. He is currently a Professor with the School of Electronics and Information Technology, SYSU. From 2002 to 2004, he was the Head of the Department of Electronics and Communication Engineering, SYSU. From 2003 to 2008, he was the Vice Dean of the School of Information Science and Technology, SYSU. His research interests include wireless communications and submillimeter wave technology. He was the recipient of the IEEE Communications Society Heinrich Hertz Award for Best Communications Letter in 2014, the Second Young Teacher Award of Higher Education Institutions, Ministry of Education (MOE), China in 2001, the Seventh Science and Technology Award for Chinese Youth in 2001, and the New Century Excellent Talent, MOE, China in 1999.

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