Outage Performance for Non-Orthogonal Multiple Access With Fixed

This article has been accepted for publicationLETTERS, in a future issue this journal, butXX has2018 not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/LCOMM.2018.2799609, IEEE IEEE COMMUNICATIONS VOL.ofXX, NO. XX, 1 Communications Letters

Outage Performance for Non-Orthogonal Multiple Access With Fixed Power Allocation Over Nakagami-m Fading Channels Tianwei Hou, Xin Sun, Zhengyu Song Abstract—In this letter, we study the outage performance of non-orthogonal multiple access (NOMA) with fixed power allocation over Nakagami-m fading channels, where the fading parameters of NOMA users are different. In this scenario, new closed-form expressions are derived for the outage behavior of individual users and the system, respectively. Next, the diversity orders are obtained under high signal-to-noise ratio condition. The derived analytical expressions are exact and unprecedented in the earlier literature. Finally, simulations are conducted to confirm the validity of the analysis and show the outage performance of NOMA under different fading parameters of Nakagami-m fading channels. Index Terms—Non-orthogonal multiple access, Nakagami-m fading, outage behavior, diversity order.

I. I NTRODUCTION

M

ULTIPLE access technology is a critical topic for wireless communications. As a promising technology for future wireless communication systems, non-orthogonal multiple access (NOMA) has received considerable attention because of its superior spectral efficiency. In NOMA, clustered users multiplex in the power domain, i.e., they share the same frequency and time, but the signal power levels are different. At the receivers, the composite signal of different users is separated by successive interference cancellation (SIC) [1]. Recently, a few different forms of NOMA have been proposed in the literature. The authors in [2] investigated the outage performance of individual users and ergodic sum rate with randomly deployed users. The analytical results show that it is more preferable to group users whose channel gains are more distinctive under Rayleigh fading channels. In [3], the authors estimated the performance of NOMA with fixed power allocation (F-NOMA) and cognitive radio inspired NOMA over standard Rayleigh fading channels, which has demonstrated that only the user with higher channel gain influences the outage performance. However, Rayleigh fading is just a special case of fading channels. Considering the line of sight (LoS) communication, it is more advantageous to investigate the outage performance of NOMA with Nakagami-m fading channels. To our best knowledge, the general case, namely, Nakagami-m fading channels, has not been well considered yet in previous literature. Motivated by such observations, in this letter, we study the outage performance of F-NOMA in downlink scenario over Nakagami-m fading channels. In particular, the individual user and system probabilities that NOMA achieves lower data rate than traditional orthogonal multiple access (OMA) Manuscript received October 23, 2017; revised November 16,2017; accepted January 24, 2018. This work was supported by the China Postdoctoral Science Foundation under Grant 2016M600911. The associate editor coordinating the review of this letter and approving it for publication was Y. Liu. (Corresponding author: Zhengyu Song.) T. Hou, Z. Song and X. Sun are with the School of Electronic and Information Engineering, Beijing Jiaotong University, Beijing 100044, China (email: [email protected], [email protected], [email protected]).

are analyzed, and the impact of fading parameters on the outage performance is specially evaluated to better show the differences between Rayleigh fading channels and Nakagamim fading channels. Interestingly, the analytical results in this letter show that the diversity order of the system depends on both users for Nakagami-m fading channels, which is different from [3], where the diversity order only depends on the user with higher channel gain in Rayleigh fading channels. Besides, the outage probabilities of individual users and the system for F-NOMA over Nakagami-m fading channels are derived, which shows that F-NOMA dramatically increases the data rate of the user with higher channel gain while sacrificing the user with poorer channel gain. II. NOMA W ITH F IXED P OWER A LLOCATION Consider a downlink NOMA system with one base station (BS) and M users, where the channel state information (CSI) of users is perfectly known at the BS [4]. Without loss of generality, it is assumed that the channel gains of users in the system follow the order as 2

2

2

|h1 | ≤ |h2 | ≤ · · · ≤ |hM | .

(1)

Based on the NOMA protocol, theP power allocated to users M should satisfy α1 ≥ · · · ≥ αM , and i=1 αi = 1. In [3], it is indicated that grouping all the users in NOMA is not preferable in practice. Therefore, in this paper, we consider a situation that two users, i.e., user w and user v with v < w, are paired to perform NOMA. In this case, the power allocated to the two users satisfies αv > αw and αv + αw = 1. Accordingly, by applying SIC, the data rates of user w and user v are 2 N Rw = log 1 + αw ρ|hw | , (2) 2

RvN

= log 1 +

αv |hv |

!

2 αw |hv | + 1/ρ

,

(3)

where ρ is the transmit signal-to-noise ratio (SNR). Correspondingly, the data rates of user w and user v in OMA are 1 2 O Rw = log 1 + ρ|hw | , (4) 2 1 2 RvO = log 1 + ρ|hv | . (5) 2 III. O UTAGE B EHAVIOR A. The Outage Probability of Individual Users First, we focus on the probability that the v-th user achieves lower data rate than OMA, which can be expressed as P (RvN < RvO ) = ! ! 2 (6) αv |hv | 1 2 P log 1 + < log(1 + ρ|hv | ) . 2 1 2 αw |hv | + /ρ After some algebraic manipulations, we have

1089-7798 (c) 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.


2 s−s1 −···sm1 −2 s s−s 2 2 P P1 P < =P 1 + ρ|hv | < 1 + ραw |hv | , As,m1 = ··· Denote U = sm1 −1 =0 (7) s,m1 s1=0 s2=0 1 − 2αw 2 s − s1 − · · · sm1 −2 s s − s1 . = P |hv | > · · · , Bs,m1 = 2 ραw s1 s2 sm1 −1 1−2 Note that the marginal probability density function (PDF) of mQ s 2 qr r+1 (qm1−1 )s−s1−···sm1−2 , s = (m1 − 1)(s − s1 ) − (m1 − |hv | is given by [5] r=0 the series expansion to f|hv |2 (y) = $1 f (y)(F (y))v−1 (1 − F (y))M −v , (8) 2)s2 −· · · − sm1−1 , and we also apply ∞ m ys P − Ω1 (−1)t (m1 ys)t 1 the exponential function, i.e., e = , we 2 2 Ωt1 t! and the joint PDF of |hv | and |hw | is given by t=0 can finally attain f|hw |2 ,|hv |2 (x, y) =$2 f (x)f (y)F (x)v−1 (1−F (y))M −w (9) v+i ∞ t t X X (−1) (m1 ys) s s v+i × (F (y) − F (x))w−v−1 , Tm1 y , (16) Λ= (−1)s s Ωt1 t! M! M! s=0 t=0 and $ = . where $1 = (v−1)!(M 2 −v)! (v−1)!(w−v−1)!(M −w)! s Since |hv | and |hw | follow Nakagami-m distribution with where Tm = U As,m1 Bs,m1 . According to binomial coef1 s,m1 fading parameters m1 and m2 , respectively, the channel gain ficients, one can know that also follows a Gamma distribution, with PDF as [6] M −v X 1 M −v m1 m1 y m1 −1 − mΩ1 y = 1. (17) $1 (−1)i e 1 , (10) f1 (y) = m1 i v+i Ω1 Γ(m1 ) i=0 P (RvN

RvO )

and

f2 (y) =

m2 m2 y m2 −1 − mΩ2 y e 2 , Ω2 m2 Γ(m2 )

(11)

where f1 means R ∞ the parameters of PDF are m1 and w1 . Note that Γ(a) = 0 xa−1 e−x dx and Γ(a) = (a − 1)! when a is an integer [7]. Hence, the cumulative distribution function is m1 −1 r m1 y X (ym /Ω ) 1 1 . (12) F1 (y) = 1 − e− Ω1 r! r=0 w Denote I = 1−2α α2w , which is a constant in F-NOMA. By applying the above density functions and binomial expansion, the outage probability in (7) can be expressed as

Z ∞ I P (|hv | > ) = $1 f (y)(F (y))v−1 (1 − F (y))M −v dy I/ ρ ρ Z ∞M −v X M −v = $1 (−1)i f (y)F (y)v+i−1 dy i I/ ρ i=1   M−v v+i $1 X M −v   (−1)i (1 − F (I/ρ) ) . = i v+i i=0 | {z } 2

(13)

1−

Λ=

m1 −1

X (ym1 /Ω1 )r r! r=0

v+i X m ys v+i − 1 = (−1)s e Ω1 s s=0

!v+i

m1 −1

X (ym1 /Ω1 )r r! r=0

(14)

!s .

r

1) Note that qr = (m1 /Ω . Applying successive binomial r! expansion on Λ, we have

m 1 −1 X

(

qr y r )s =

r=0

s s−s X X1 s1 =0 s2 =0

X sm1 −1 =0

s − s1 − · · · sm1 −2 s s − s1 × ··· s1 s2 sm1 −1 m1 −2

×

Y r=0

qrsr+1 y rsr+1 (qm1 −1 y m1 −1 )s−s1 −···sm1 −2 .

for 1 ≤ t < v + i or t ≥ v + i + 1, and v+i X v+i (−1)s (s)t = (−1)v+i (v + i)!, s

(20)

for t = v + i. With these steps and substituting y = I/ρ to the outage probability, (18) can be rewritten as M −v v+i X $1 Tm M −v 1 Pvout =1 − (−1)i i v + i i=0 (21) v+i m1 (v+i) m1 I × . Ω1 ρ The diversity order of the v-th user with high SNR approximation is given by lim −

ρ→∞

s−s1 −···sm1 −2

···

(19)

s=0

In addition, Λ can be rewritten as m y

Recall the following binomial coefficient in [7]: v+i X v+i (−1)s (s)t = 0, s s=0

Λ

− 1 e Ω1

Substituting (16) and (17) into (13), the outage probability of the v-th user in (7) can be rewritten as M −v v+i X $1 s X M − v out Pv = 1− Tm1 (−1)i i i+v s=0 i=0 (18) ∞ t t X (−1) (m ys) s v+i 1 s × (−1) y . s Ωt1 t! t=0

(15)

log Pvout = m1 v. log ρ

(22)

From (22), one can know that the outage probability of the v-th user depends on the fading parameter m1 and v itself. On the other hand, the outage probability of user w is also worth estimating. In NOMA, the w-th user needs to decode the signal intended for user v before decoding its own signal. Therefore, it is assumed that Rv is the target rate for the v-th user, and the probability that the w-th user performs poorer in NOMA than OMA is given by



Pwout

=P

log 1 +

αv |hw |

!

2

!

< Rv

2 αw |hw | + 1/ρ

P1 =

!

2

αv |hw |

> Rv , 2 αw |hw | + 1/ρ 1 2 2 log(1 + αw ρ|hw | ) < log(1 + ρ|hw | ) . 2

+P

log 1 +

(23)

After some algebraic manipulations and denoting C = εv −1 |1−εv αw | , we finally have C C I 2 2 out Pw = P |hw | < +P < |hw | < ρ ρ ρ (24) I 2 . = P |hw | < ρ

m1 Ω1

v+j

m2 Ω2

a+m2 −1

a−1 v+j Tm1 Tm 2 Γ(m2 )b

1−

R2 ρ

b ! ,

(30) where a = w−v +i−j, b = m2 a + m1 (v + j). The diversity order of P1 with high SNR approximation is log P1 = m2 w + (m1 − m2 )v. (31) lim − ρ→∞ log ρ Similar to P1 , P2 can asm1 (v+j) Z ∞be rewritten R1 P2 = Q2 dy, y m2 a−1 R2 ρ ρ

(32)

1 a−1 v+j 1 v+j m2 a+m2 −1 where Q2 = ( m ( Ω2 ) Ω1 ) Γ(m2 ) Tm2 Tm1 . In the high SNR situation, R1 can be considered as a constant. Therefore, the diversity order of P2 is shown as log P2 lim − = m2 w + (m1 − m2 )v. (33) ρ→∞ log ρ

Similar to the argument from (13) to (22), the outage probability of the w-th user can be obtained as From formula (31) and (33), we can know that the diversity w+i m2 (w+i) M−w w+i X M − w m2 I out i $3 Tm2 order is m2 w + (m1 − m2 )v. Besides, the outage probability , Pw = (−1) i w+i Ω ρ 2 of (39) can be transformed into i=0 b ! (25) 1 R2 out M! PSR =1 − Q1 Q2 1− where $3 = (w−1)!(M −w)! , and the diversity order is m2 w. b ρ One can observe that the diversity order of user w only m1 (v+j) ! (34) Z ∞ depends on the fading parameter m2 and w itself, although m2 a−1 R1 − y dy . the w-th user decodes the signal of the v-th user first. R2 ρ ρ

B. The Outage Probability of Sum Rate Next, we consider the probability that the sum rate of NOMA is lower than OMA, which is given by out PSR

=1−

2

P (RvN

+

N Rw

>

RvO

+

O Rw ),

(26)

2

where |hw | ≥ |hv | , which means the w-th user can always decode the v-th user’s signal successfully. By some algebraic handling, we have 2

2

2

2

2

2

out PSR = 1−P (ρ|hv | +ρ|hw | +ρ2 |hv | |hw | > I, |hv | < |hw | ). (27) 2 Note that in this formula, |hv | has an integral range that 2 2 I−ρ|hw |2 < |hv | < |hw | . To guarantee the upper bound is (1+ρ|h |2 )ρ w

2

higher than the lower bound for |hv | , √ an additional constraint 2 2 is imposed on the |hw | , i.e., |hw | > 1+I−1 . Therefore, the ρ outage probability in (27) can be rewritten as R1 2 2 R2 2 out PSR = 1−P < |hv | < |hw | , < |hw | , (28) ρ ρ √ 2 w| where R1 = I−ρ|h , R2 = 1 + I − 1. Again applying the 1+ρ|hw |2 joint PDF in (9), the outage probability is further expanded as Z ∞Z y v−1 out PSR =1− $2 f1 (x)f2 (y)(F1 (x)) R2 R1 (29) ρ ρ × (1 − F2 (y))M −w (F2 (y) − F1 (x))w−v−1 dxdy. Now applying binomial expansion, the outage probability is shown top ofthe following in (39) at the w−v−1 page, where Q1 = M−w P M −w P w−v−1 $2 i (−1) (−1)j v+j . i j i=0 j=0 Substituting (11) and (12) into P1 , and deploying binomial expansion, P1 can be converted as

C. The Outage Probability for the System Finally, the outage probability for the system is given by out N O N O Psystem = 1−P (RvN > RvO , Rw > Rw , RvN +Rw > RvO +Rw ). (35) By some algebraic handling, it is transformed into R1 R1 I R2 I 2 2 out < |hv | < , max , , < |hw | . Psystem = 1−P ρ ρ ρ ρ ρ (36) In addition, I > R1 and I > R2 are always satisfied because 2 of I > 0 and |hv | > 0. Therefore, the outage probability for the system can be rewritten as I I R1 2 2 out < |hv | < , < |hw | . (37) Psystem = 1−P ρ ρ ρ Similar to the argument from (29) to (34), the outage probability for the system is obtained as m1 (v+j) m2 a I 1 I out Psystem = 1−Q1 Q2 1− ρ m2 a ρ (38) m1 (v+j) ! Z ∞ m2 a−1 R1 y dy , − I ρ ρ

where the diversity order is m2 w + (m1 − m2 )v. Unlike [3], where the diversity order is only dependent on the user with better channel gain, (31), (33) and (38) show that the diversity order depends on both users under Nakagami-m fading channels. One can also observe that m2 and w are the dominant part of the diversity order, and v can be omitted in the case of m1 = m2 . IV. N UMERICAL S TUDIES In this section, we demonstrate the outage performance of FNOMA over Nakagami-m fading channels. In the simulations,







Z ∞ v+j  Z ∞   v+j R1 w−v−1−j+i w−v−1−j+i  =1−Q1  f2 (y)F2 (y) F1 (y) dy − f2 (y)F2 (y) F1 dy  . R2 R2 ρ   ρ ρ | {z } | {z } P1

10 0

100

10

Outage probability

Probability

10

Simulation results Asymptotic results The coverage probability of user v, m 1=1

-4

10

1

Simulation results Analytical result, m 1 :m 2 =1:1 Analytical result, m 1 :m 2 =1:2

-2

Analytical result, m 1 :m 2 =2:2

-2

10-4

0.9

0.8 Simulation results Analytical result, m 1 :m 2 =1:1 Analytical result, m 1 :m 2 =1:2

0.7

Analytical result, m 1 :m 2 =2:2

The outage probability of user w, m2=1

Analytical result, α v : α w=0.6:0.4

The coverage probability of user v, m =2 1

10 -6

The outgae probability of user w, m2=2

5

10

15

20

SNR(dB)

(a) Fig. 1.

(39)

P2

Outage probability

out PSR

25

30

10

-6

5

10

15 20 SNR(dB)

(b)

25

30

0.6

Analytical result, α v : α w=0.9:0.1

5

10

15 20 SNR(dB)

25

30

(c)

(a) The probability of individual users. (b) The outage probability of sum rate. (c) The outage probability for the system.

the power allocation factors are set as αv = 0.8 and αw = 0.2. The distances from the BS to user v and user w are fixed to 1 and 0.5, respectively. Then, we can obtain Ω1 = 1 and −γ Ω2 = 12 , where γ is the path loss exponent. Here, we set γ = 3. Without loss of generality, we set the number of users M = 5, w = 3, and v = 2. Fig. 1(a) demonstrates two different but related probabilities for individual users versus transmit SNR in different fading parameters. One is the outage probability of the w-th user, i.e., NOMA user w performs worse than OMA. The other is P (RvN > RvO ), i.e., the probability that the NOMA user with poorer channel gain achieves higher data rate than OMA. In Fig. 1(a), it can be seen that the probabilities decrease with the increase of transmit SNR. In addition, the increase of fading parameters m1 and m2 decreases the above probabilities dramatically. The asymptotic simulations are also provided to confirm the close agreement between analytical results and the simulations. Thus, the correctness of analytical results is verified. In Fig. 1(b), the probability that F-NOMA achieves lower sum rate than OMA is shown as a function of transmit SNR, which decreases monotonically. It is demonstrated that when m2 varies from 1 to 2, the decreasing rate of outage probability is much faster than the case when m1 increases from 1 to 2, which shows that the existence of line of sight (LoS) for the user with higher channel gain is able to dramatically decrease the outage probability. Comparatively, the existence of LoS for the user with poorer channel gain only decreases the outage probability slightly. The probability that the data rate of individual user and the sum rate in F-NOMA are both lower than OMA is illustrated in Fig. 1(c). As can be seen in this figure, the outage probability is not a monotonically decreasing function, and it is always larger than 0.6 no matter what the fading parameters are. In other words, F-NOMA cannot perform better than OMA on both individual user’s data rate and sum rate simultaneously, which

indicates that F-NOMA improves the performance of the user with higher channel gain while sacrificing the user with poorer channel gain. In addition, the dot curve and dash curve are the outage probabilities of αv : αw = 0.6 : 0.4 and αv : αw = 0.9 : 0.1, respectively, when the fading parameters m1 : m2 = 2 : 2. From this result, it is found that the power allocation factors αv and αw affect the system outage probability slightly. V. C ONCLUSIONS In this letter, the outage probability of F-NOMA over Nakagami-m fading channels was studied. Both analytical and numerical results have been provided to demonstrate that FNOMA can offer larger individual rates for the users with higher channel gain, while the data rate of the user with poorer channel gain in F-NOMA is even lower than OMA in high transmit SNR situation. It is also shown that the outage probabilities of the sum rate and the system with F-NOMA over Nakagami-m fading channels depend on both users from the analytical results. R EFERENCES [1] Z. Ding, Y. Liu, J. Choi, Q. Sun, M. Elkashlan, C. L. I, and H. V. Poor, “Application of non-orthogonal multiple access in LTE and 5G networks,” IEEE Commun. Mag., vol. 55, no. 2, pp. 185–191, Feb. 2017. [2] Z. Ding, Z. Yang, P. Fan et al., “On the performance of non-orthogonal multiple access in 5G systems with randomly deployed users,” IEEE Signal Processing Letters, vol. 21, no. 12, pp. 1501–1505, Dec. 2014. [3] Z. Ding, P. Fan, and H. V. Poor, “Impact of user pairing on 5G nonorthogonal multiple-access downlink transmissions,” IEEE Trans. Veh. Technol., vol. 65, no. 8, pp. 6010–6023, Aug. 2016. [4] Y. Liu, Z. Ding, M. Elkashlan et al., “Cooperative non-orthogonal multiple access with simultaneous wirelss information and power transfer,” IEEE J. Sel. Areas Commun., vol. 34, no. 4, pp. 938–953, Apr. 2016. [5] H. N. Nagaraja and H. A. David, Order Statistics. John Wiley, New York, 3rd ed, 2003. [6] X. Yue, Y. Liu, S. Kang, and A. Nallanathan, “Performance analysis of NOMA with fixed gain relaying over Nakagami-m fading channels,” IEEE Access, vol. 5, pp. 5445–5454, 2017. [7] Y. Liu, Z. Qin, M. Elkashlan, and L. Hanzo, “Enhancing the physical layer security of non-orthogonal multiple access in latge-scale networks,” IEEE Trans. Wireless Commun., vol. 16, no. 3, pp. 1656–1672, Mar. 2017.
