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A Novel Power Minimization Precoding Scheme for MIMO-NOMA Uplink Systems Hong Wang, Rongbin Zhang, Rongfang Song, and Shu-Hung Leung

Abstract—This letter is to investigate the use of non-orthogonal multiple access (NOMA) and new transceiver design for multipleinput multiple-output (MIMO) uplinks. A new NOMA implementation scheme with group interference cancelation is proposed to minimize the total power consumption subject to individual rate requirements. In this scheme, precoders and equalizers for users in the same group are designed jointly, and closed-form design procedures for user precoders are developed. Simulation results show that the proposed NOMA scheme outperforms both orthogonal multiple access transmissions and signal alignment NOMA scheme in terms of total power consumption. Index Terms—NOMA, MIMO, precoding, uplink

I. I NTRODUCTION By using a new dimension, power domain non-orthogonal multiple access (NOMA) is capable of supporting ultrahigh connectivity for foreseeable applications [1]. It is demonstrated that the NOMA is always better than the conventional orthogonal multiple access (OMA) approaches in terms of the system throughput [2], [3]. Different from the well-known water-filling scheme, NOMA tends to allocate more power to the users with weaker channel conditions, and hence the user fairness is maintained [4]. Recently, the combination of multiple-input multiple-output (MIMO) with NOMA has received substantial attention [5]. In the downlink MIMO-NOMA, multiple beams are formed in the spatial domain at the base station (BS), and an interference rejection filter is used to remove inter-beam interference at each receiver [6]. Therefore, the MIMO-NOMA transmission becomes several independent single beam NOMA arrangements. In contrast to the downlink, uplink MIMO-NOMA has a large difference in transmit power allocation policy and the order of successive interference cancelation (SIC) operations. A transmission scheme based on the concept of signal alignment (SA) is proposed for the MIMO-NOMA uplinks [7], in which the received signals of a near user and a far user are aligned at the BS by using a precoding design. In the conventional MIMO-NOMA scheme, users are divided into clusters, each of which consists of a near user and a far user. The SIC is used to separate the intra-cluster users, while This work was supported in part by Natural Science Foundation of Jiangsu Province (BK20170910), by Open Research Foundation of National Mobile Communications Research Laboratory, Southeast University (2018D09), and by Theme Based Research (T42-103/16-N). H. Wang and R. Song are with the School of Communication and Information Engineering, Nanjing University of Posts and Telecommunications, Nanjing 210003, China, and also with the National Mobile Communications Research Laboratory, Southeast University, Nanjing 210096, China (e-mail: [email protected], [email protected]). R. Zhang and S.-H. Leung are with the State Key Laboratory of Millimeter Waves and Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong (e-mail: [email protected], [email protected]).

precoding is employed to suppress inter-cluster interference. Different NOMA schemes have been proposed following this basic principle [2], [6], [7]. The shortcoming of this cluster approach is that the precoding and the detection are not jointly optimized leading to degradation in power efficiency. In this letter, we divide the users into several groups according to effective channel powers, e.g., near user group and far user group. In the uplink, precoders at the transmitters and equalizers at the receiver are optimally designed to suppress the cochannel interference and enhance the signal detection. The contributions of this letter are two-fold. First, a novel MIMO-NOMA scheme is developed from the perspective of the joint precoding designs and group signal cancelations. The users in the different groups are served according to the NOMA principle. The proposed scheme is capable of supporting multiple data stream transmissions for each user. Second, a new precoding method is proposed to minimize the total power consumption while guaranteeing the achievable rate requirement per user. In this method, the user signals per group are decorrelated jointly by user precoders and a minimum mean square error (MMSE) equalizer. By expressing user precoders in a general form, closed-form precoding design algorithms are developed. It is shown that the proposed scheme is superior to the SA-NOMA and OMA methods in terms of overall power consumption by computer simulations. II. S YSTEM M ODEL There are G groups of randomly distributed users in a circular coverage of radius R with one BS located at the center. The BS and each user are equipped with NR and NT antennas, respectively. The users are divided into different groups according to the path loss to the BS with rg,m < rg′ ,m′ for g < g ′ , where rg,m is the distance from user m in the gth group to the BS. There are Mg users in the gth group, in which user m has dg,m data streams to transmit. ∑Mg dg,m ≤ NR and For NOMA transmissions, we assume m=1 ∑G ∑Mg d > N . It is assumed that the uplink transg,m R m=1 g=1 mission signals arrive at the BS synchronously and channel state information is available at the transmitters. The received signal at the BS, y, can be expressed as ∑G ∑Mg (g) (g) y= H(g) (1) m Pm sm + z, g=1

m=1

(g)

where Hm is the channel matrix from user m in the gth (g) (g) group to the BS, Pm and sm are the precoder and transmit signal of user m in the gth group, respectively, and z is the co-channel interference signal plus noise (CIN) vector. The CIN vector can be expressed as z = i + n, where n is a noise vector with zero mean and covariance matrix

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σv2 I. Similar to [7], interference vector i is modeled as a classical shot noise model. Assuming that J interferers are randomly∑ distributed in the annulus of radii R and RI , we √ √ J have i = j=1 βj 1NR ×1 xj / LI,j [7], where 1N ×T is an N × T all-one matrix, βj , LI,j , and xj are the transmit power, path loss, and normalized signal of interferer j, respectively. The channel model is characterized by both path loss and (g) −α/2 (g) small scale fading, i.e., Hm = rg,m Dm , where α is the (g) path loss exponent, and each element in matrix Dm follows a circularly symmetric complex Gaussian distribution with zero mean and unit variance. By stacking the transmit data of all users, the received signal can be written as y=

∑G

(g)T

where s(g) = [s1

(g)

Heff

(g)

g=1

Heff s(g) + z,

(3)

A. Problem Formulation Based on the SIC, the signals of the strongest user group are detected firstly. Before decoding the user messages of the gth group, the recovered signals of 1 ∼ (g − 1)th groups are subtracted from the mixture signals. The equalized user signals of the gth group for g ≤ G − 1 and the Gth group can be respectively expressed as =

(G)

=

ˆ s ˆ s

(g)

Heff = C1/2 g Ug Σg ,

(g) W Heff s(g) + W(g)H zg , (G) W(G)H Heff s(G) + W(G)H z. (g)H

∑G (k) (k) + z is the total CIN in the where zg = k=g+1 Heff s gth group. The MMSE receiver is the optimal linear receiver in terms of signal-to-interference-plus-noise ratio (SINR) per data stream. An MMSE receiver is used at the BS given as ( )−1 (g) (g)H (g) W(g) = Heff Heff + Cg Heff , (4)

where Ug is a unitary matrix, and Σg is a diagonal matrix. The proposed general formulation in (9) can simplify the expressions of received signals and the optimization problem. Based on (3) and (9), the precoder of the mth user in the gth group can be given as (g) † 1/2 P(g) m = (Hm ) Cg Ug,m Σg,m ,

(g+1)

(g+1)H

Heff

CG = E{zzH } =

∑J

+ Cg+1 , g = 1, ..., G − 1,

j=1

(βj /LI,j )1NR ×NR + σv2 I.

(5) (6)

g=1

(g)

s.t. (g)

∑dg,m k=1

(

(11)

where the covariance ( 2 matrix )−2 of the equalized CIN in the gth 2 group, ˜ zg , is Σg Σg + I . The SINR of the kth stream of user m in the gth group can be expressed as −2

(g)

SINRm,k =

[Σg,m ]4n,n ([Σg,m ]2n,n +1) [Σg,m ]2n,n

−2 [Σg,m ]2n,n +1

(

)

= [Σg,m ]2n,n ,

(12)

where [Σg,m ]n,n is the nth diagonal element of Σg,m . Accordingly, the optimization problem Q0 in (7)-(8) can be transformed into problem Q1 given as { } ∑G ∑Mg Q1 : min F = Tr P(g)H P(g) (13) m m {Ug ,Σg }

s.t.

∑dg,m n=1

g=1

m=1

(g)

log2 (1 + [Σg,m ]2n,n ) ≥ Cm,th , (14)

UH g Ug = I.

(15)

Substituting (10) into the trace operation in (13), we have H Tr{P(g)H P(g) m m } = Tr{Σg,m Gg,m Σg,m }

In this letter, we focus on the minimization of the overall power consumption under the constraint of individual quality of service [8]. The optimization problem can be formulated as { } ∑G ∑Mg (g) Q0 : min F = Tr P(g)H P (7) m m {Pm }

(10)

∑m−1 where Ug,m = [Ug ]lm +1:lm+1 with lm = k=1 dg,k , Σg,m = diag{[Σg ]lm +1,lm +1 , ..., [Σg ]lm+1 ,lm+1 }, and (·)† is the pseudo inverse. As a result, the equalized signals can be expressed as

where Cg is the covariance matrix of zg expressed as Cg = Heff

(9)

( )−1 (g) ˆ s(g) = Σ2g Σ2g + I s +˜ zg , g = 1, ..., G,

III. P RECODING D ESIGN P ROCEDURES

(g)

To simultaneously decorrelate the transmit signals and re(g) move inter-user interference, Heff can be designed in a general form given as

(2)

(g)T

, ..., sMg ]T and [ ] (g) (g) (g) (g) = H1 P1 , ..., HMg PMg .

B. Precoding Design

(16)

with Hermitian matrix H/2 †H † 1/2 Gg,m = UH (H(g) (H(g) g,m Cg m ) m ) Cg Ug,m .

(17)

Then, the objective function in (13) can be rewritten as F=

∑G g=1

∑Mg ∑dg,m m=1

n=1

[Σg,m ]2n,n [Gg,m ]n,n .

(18)

m=1

) (g) (g) log2 1 + SINRm,k ≥ Cm,th ,

C. Solutions of Ug and Σg (8)

where SINRm,k denotes the SINR of the kth stream of user (g) m in the gth group, and Cm,th is the rate requirement of user m in the gth group.

The problem Q1 is a non-convex problem. It is more complicated to obtain {Ug } and {Σg } directly. We propose a suboptimal scheme to solve the optimization problem. The proposed scheme is composed of the design of the unitary matrices {Ug } and the diagonal matrices {Σg }, respectively.



1) Design of the Unitary Matrix Ug : It is noted that the objective function in (18) is a monotonically decreasing function of {[Gg,m ]n,n }. Therefore, the unitary matrix Ug can be designed aiming to minimize {[Gg,m ]n,n }. By using the expression of Gg,m in (17), unitary vectors in {Ug [n]} are obtained one by one subject to the orthogonal constraint according to the following sequential algorithm. Without loss of generality, we label the users in the gth group from 1 to Mg according to the descending order of the path loss. Recalling the structure of Ug , ]we have [ the following expression: Ug = Ug,1 , ..., Ug,Mg . In the proposed algorithm, the first unitary vectors of all Ug,m s are computed firstly, then, the second unitary vectors of Ug,m s, and so on. Say when dg,m unitary vectors of user m have been computed, the next round will skip this user. Define [ ] Ξm,n = Ug,1 [1], ..., Ug,Mg [1], ..., Ug,1 [n], ..., Ug,m−1 [n] as the matrix formed by the unitary vectors that have been designed in the previous rounds, i.e., from the 1st unitary vector of user 1 to the nth unitary vector of user m − 1 in the gth group. It is worth mentioning that the matrix Ξm+1,n can be formed by appending a newly computed vector Ug,m [n] to Ξm,n . To satisfy the orthogonal constraint, the new unitary vector Ug,m [n] can be obtained in terms of the orthogonal projection of Ξm,n given by Ug,m [n] =

(I−Ξm,n ΞH m,n )vg,m,n ∥(I−Ξm,n ΞH m,n )vg,m,n ∥

,

with

where Kg,m is number of nonzero diagonal entries in Σg,m . D. Summary of Design Procedures Once {Ug } and {Σg } are obtained, the precoders at the users and the equalizers at the BS can be obtained accordingly. The implementation procedures of the proposed scheme are summarized in Algorithm 1. Algorithm 1 Precoder design procedures For g = G : 1 • Step 1: Compute Cg by (5)-(6) and obtain unitary matrix Ug by the sequential algorithm in Subsection III-C1; • Step 2: Calculate {[Gg,m ]n,n } by (17) for m = 1, ..., Mg , n = 1, ..., dg,m ; 2 • Step 3: Compute {[Σg,m ]n,n } by (24) for m = 1, ..., Mg , n = 1, ..., dg,m ; • Step 4: Compute the precoder for user m in the gth group by (10) and the equalizer at the BS by (4). Endfor

(19)

where vg,m,n aims at minimizing the {[Gg,m ]n,n } in (17). Substituting (19) into (17), vg,m,n can be obtained by vg,m,n = umin {ΠH g,m,n Πg,m,n }

where (t)+ is defined as (t)+ = t for t > 0 and (t)+ = 0 otherwise. λg,m can be obtained to satisfy the equality con∑dg,m (g) straint n=1 log2 (1+[Σg,m ]2n,n ) = Cm,th . Thus, the optimal 2 solution of [Σg,m ]n,n is given by ( C (g) ∏K )+ g,m 2 m,th k=1 [Gg,m ]k,k [Σg,m ]2n,n = − 1 , (24) [Gg,m ]n,n

Remark: In the proposed scheme, the computational complexity is dominated by the computation ∑G ∑Mof eigenvectors. It requires computing eigenvectors g=1 m=1 dg,m times.

(20) IV. S IMULATION R ESULTS

( ) † 1/2 Πg,m,n = (H(g) I − Ξm,n ΞH m ) Cg m,n ,

(21)

where umin {Φ} is the operation for computing the eigenvector associated with the smallest nonzero eigenvalue of Φ. 2) Design of the Diagonal Matrix Σg : For given {[Gg,m ]n,n }, the power allocation parameters can be obtained by the Lagrange multiplier method. By introducing the Lagrange multipliers {λg,m }, the auxiliary objective function is ∑G ∑Mg ∑dg,m L= [Σg,m ]2n,n [Gg,m ]n,n g=1 m=1 n=1   Mg dg,m G ∑ ∑ ∑ (g) − λg,m  log2 (1 + [Σg,m ]2n,n ) − Cm,th  . g=1 m=1

n=1

Differentiating L with respect to {[Σg,n ]2n,n }, we have ∂L ∂[Σg,m ]2n,n

= [Gg,m ]n,n −

λg,m (1+[Σg,m ]2n,n ) ln(2) .

(22)

The optimal solution of [Σg,m ]2n,n can be obtained by setting ∂L/∂[Σg,m ]2n,n = 0. Since [Σg,m ]2n,n > 0, the solution of [Σg,m ]2n,n is expressed in the form of water-filling as )+ ( λ − 1 [Σg,m ]2n,n = [Gg,m ]g,m , (23) ln(2) n,n

In this section, power consumption and performance analysis of the proposed scheme are presented. The performances of three benchmark schemes, namely, SA-NOMA [7], SAOMA, and jointly precoding OMA (JP-OMA), are included for comparison. For SA-OMA, the precoders are designed by the concept of SA in [7]. For JP-OMA, the precoders are jointly designed by using the method proposed in Section III. The simulation parameters are shown in Table I. TABLE I S IMULATION PARAMETERS Parameters Radius of coverage R Number of transmit antennas NT Number of receive antennas NR Number of user groups G Path loss exponent α Number of interferers J Radius of interference zone RI Interference power βj Noise power σv2

Values 500m 8 8 2 3.5 10 800m -30dBm -99dBm

Figure 1 plots the overall transmit power for different user rate budgets where each user has single data stream to transmit. For both SA-OMA and JP-OMA schemes, the near and far user groups are scheduled to transmit on equal



[3] Y. Liu, G. Pan, H. Zhang, and M. Song, “On the capacity comparison between MIMO-NOMA and MIMO-OMA,” IEEE Access, vol. 4, pp. 21232129, May 2016. [4] 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. Technol. Conf., Jun. 2013, pp. 1-5. [5] L. Liu, C. Yuen, Y. L. Guan, and Y. Li, “Capacity-achieving iterative LMMSE detection for MIMO-NOMA systems,” in Proc. IEEE Int. Commun. Conf., May 2016, pp. 1-6. [6] K. Higuchi and Y. Kishiyama, “Non-orthogonal access with random beamforming and intra-beam SIC for cellular MIMO downlink,” in Proc. IEEE Veh. Technol. Conf., Sep. 2013, pp. 1-5. [7] Z. Ding, R. Schober, and H. V. Poor, “A general MIMO framework for NOMA downlink and uplink transmission based on signal alignment,” IEEE Trans. Wireless Commun., vol. 15, no. 6, pp. 4438-4454, Jun. 2016. [8] C. W. Sung and Y. Fu, “A game-theoretic analysis of uplink power control for a non-orthogonal multiple access system with two interfering cells,” in Proc. IEEE Veh. Technol. Conf., May 2016, pp. 1-5.

Total tansmit power (dBm)

50

Proposed NOMA JP−OMA SA−NOMA SA−OMA

40

30

20

10

0 1

1.5

2

2.5

3

3.5

4

Rate requirement per far user (b/s/Hz)

4.5

5

Fig. 1. Total transmit power versus user rate requirements with Mg = 8 and dg,m = 1. The solid, dashed, and dotted lines represent the cases of (1) (2) (1) (2) (1) (2) Cm,th = Cm,th , Cm,th = 1.5Cm,th , and Cm,th = 2Cm,th , respectively. 25

Proposed NOMA JP−OMA SA−NOMA SA−OMA

Total transmit power (dBm)

20 15 10 5 0

−5

−10 −15

1

1.5

2

2.5

3

3.5

4

Rate requirement per far user (b/s/Hz)

4.5

5

Fig. 2. Total transmit power versus user rate requirements with Mg = 4 and dg,m = 2. The solid, dashed, and dotted lines represent the cases of (1) (2) (1) (2) (1) (2) Cm,th = Cm,th , Cm,th = 1.5Cm,th , and Cm,th = 2Cm,th , respectively. 1 0.9 0.8 0.7 0.6

CDF

and orthogonal time resources. It is clearly shown that the power consumption of the proposed scheme is always less than that of SA-NOMA. When the rate requirement per far (2) user is Cm,th = 5b/s/Hz, the performance gap between the proposed NOMA and SA-NOMA is increased to 7dB, 10dB, (1) (2) (1) (2) and 11dB for the cases of Cm,th = Cm,th , Cm,th = 1.5Cm,th , (1) (2) and Cm,th = 2Cm,th , respectively. The reason is that in the proposed scheme, the precoders at the users and equalizer at the BS are jointly designed, whereas in the SA-NOMA method the precoding design focuses on the signal alignment at the BS for users in the same cluster. Besides, the total power consumption of the NOMA schemes is obviously less than that of their corresponding OMA transmissions. It is because in the NOMA schemes both the near and far users have access to more transmission resources. Figure 2 plots the total power consumption when each user transmits multiple data streams. Similar to the case of single stream transmission, the proposed NOMA scheme always outperforms the SA-NOMA and the two OMA schemes in terms of total power consumption. Figures 3 and 4 plot the cumulative distribution function (CDF) of overall transmit power for single and multiple data streams per user, respectively. The statistics are obtained from 106 channel realizations. It is shown that the transmit power of the proposed scheme are concentrated in the small power region for different rate requirements. Correspondingly, the CDF curves of the total transmit power of the proposed NOMA are convergent to 1 rapidly. However, the distribution of the transmit power of the SA-NOMA scheme is spread over a wide range. The spreading becomes large as per user rate requirement increases. It is because for the proposed scheme each user can obtain relatively large channel gains by using the proposed precoding design. However, in the SA-NOMA method, the precoding focuses on received signal alignment, which may lead to some aligned vectors very close in the spatial dimension. Hence, after the received signals pass through the equalizer, the effective channel gains of the users in the SA-NOMA scheme may be very different and some channel gains may be very small. It indicates that the performance of the SA-NOMA scheme highly depends on channel realizations.

0.5

Proposed NOMA SA−NOMA

0.4 0.3

rate requirement per far user C(2) = 5,4,3,2,1b/s/Hz, bottom to top m,th

0.2 0.1 0

0

50

100

150

200

Total transmit power (mW)

(1)

Fig. 3. CDF of total transmit power with Mg = 8, dg,m = 1, and Cm,th = (2) 1.5Cm,th . 1 0.9 0.8 0.7 0.6

CDF

V. C ONCLUSION The novel NOMA scheme based on group signal cancelation for MIMO uplinks is presented. In the scheme, user precoders in the same group are jointly designed, and users in different groups are distinguished by using the NOMA principle. A closed-form design for user precoders is developed to minimize the total power consumption under the constraint of individual rate budget. Simulation results show that the total power consumption of the proposed scheme is less than the OMA transmissions and the SA-NOMA method.

0.5

rate requirement per far user C(2) = 5,4,3,2,1b/s/Hz, bottom to top m,th

0.4 0.3

R EFERENCES

0.2

Proposed NOMA SA−NOMA

0.1

[1] Y. Liu, Z. Qin, M. Elkashlan, Z. Ding, A. Nallanathan, and L. Hanzo, “Nonorthogonal multiple access for 5G and beyond,” Proc. IEEE, vol. 105, no. 12, pp. 2347-2381, Dec. 2017. [2] M. Zeng, A. Yadav, O. A. Dobre, et al., “Capacity comparison between MIMO-NOMA and MIMO-OMA with multiple users in a cluster,” IEEE J. Sel. Areas Commun., vol. 35, no. 10, pp. 2413-2424, Oct. 2017.

0

0

2

4

6

Total transmit power (mW)

8

10

(1)

Fig. 4. CDF of total transmit power with Mg = 4, dg,m = 2, and Cm,th = (2) 1.5Cm,th .
