Novel Channel Estimation for Non-orthogonal Multiple Access Systems

Abstract

Non-orthogonal multiple access (NOMA) is a promising technology in future mobile communications. In this letter, we study the channel estimation and power allocation problem for the two-user NOMA downlink system with one strong user and one weak user. Firstly, we introduce a new type of linear estimator that aims at maximizing the average effective signal-to-interference-and-noise ratio (SINR) of the strong user with bounded average effective SINR guaranteed for the weak user. We propose a constrained concave convex procedure (CCCP)-based iterative algorithm to solve the estimation problem. Secondly, we also derive the maximum average effective SINR of the strong user under the traditional maximum-likelihood (ML)-based estimator and linear minimum-mean-square-error (LMMSE)-based estimator, respectively. Simulation results have shown that the proposed estimator outperforms the traditional ML and LMMSE estimators, indicating a new way of channel estimation and power allocation for the NOMA downlink systems.

Full text

IEEE SIGNAL PROCESSING LETTERS, VOL. 23, NO. 12, DECEMBER 2016 1781
Novel Channel Estimation for Non-orthogonal
Multiple Access Systems
Yizhi Tan, Jingrong Zhou, and Jiayin Qin
Abstract—Non-orthogonal multiple access (NOMA) is a promis-
ing technology in future mobile communications. In this letter, we
study the channel estimation and power allocation problem for the
two-user NOMA downlink system with one strong user and one
weak user. Firstly, we introduce a new type of linear estimator that
aims at maximizing the average effective signal-to-interference-
and-noise ratio (SINR) of the strong user with bounded average
effective SINR guaranteed for the weak user. We propose a con-
strained concave convex procedure (CCCP)-based iterative algo-
rithm to solve the estimation problem. Secondly, we also derive
the maximum average effective SINR of the strong user under the
traditional maximum-likelihood (ML)-based estimator and linear
minimum-mean-square-error (LMMSE)-based estimator, respec-
tively. Simulation results have shown that the proposed estimator
outperforms the traditional ML and LMMSE estimators, indicat-
ing a new way of channel estimation and power allocation for the
NOMA downlink systems.
Index Terms—Channel estimation, linear minimum-mean-
square-error (LMMSE), maximum-likelihood (ML), non-
orthogonal multiple access (NOMA), power allocation.
I. INTRODUCTION
NON-ORTHOGONAL multiple access (NOMA) with suc-
cessive interference cancellation (SIC) receiver is consid-
ered as a promising technology in 5G networks [1], [2]. The
key idea of NOMA is to have the communication resources
shared by multiple users with superposition coding, which is
fundamentally different from conventional orthogonal multiple
access (OMA) technologies [3]. NOMA allocates more power
to the users with poorer channel conditions, with the aim to
Manuscript received July 21, 2016; revised September 11, 2016; accepted
October 7, 2016. Date of publication October 21, 2016; date of current version
November 4, 2016. This work was supported in part by the National Natural Sci-
ence Foundation of China under Grant 61472458 and Grant 61672549, in part
by the Guangdong Natural Science Foundation under Grant 2014A030311032,
Grant 2014A030313111, and Grant 2014A030310374, in part by the Guangzhou
Science and Technology Program under Grant 201607010098, and in part by
the Fundamental Research Funds for the Central Universities under Grant
15lgzd10 and Grant 15lgpy15. The associate editor coordinating the review
of this manuscript and approving it for publication was Dr. Feifei Gao. (Corre-
sponding author: Y. Tan.)
Y. Tan is with the School of Electronics and Information Technology, Sun
Yat-Sen University, Guangzhou 510006, Guangdong, China and also with
the Guangdong University of Technology, Guangzhou 510006, Guangdong,
China (e-mail: 5718331@qq.com).
J. Zhou is with the School of Electronics and Information Technology,
Sun Yat-Sen University, Guangzhou 510006, Guangdong, China (e-mail:
zhjrong@mail2.sysu.edu.cn).
J. Qin is with the School of Electronics and Information Technology, Sun
Yat-Sen University, Guangzhou 510006, Guangdong, China and also with
the Xinhua College, Sun Yat-Sen University, Guangzhou 510520, Guangdong,
China (e-mail: issqjy@mail.sysu.edu.cn).
Digital Object Identifier 10.1109/LSP.2016.2617897
facilitate a balance between system throughput and user
fairness [4].
In [5], for a single-input single-output (SISO) NOMA down-
link system, it is shown that, with carefully chosen user rates
and power coefficients, NOMA can achieve superior ergodic
sum rate and outage performance comparing to OMA. In [6],
the capacity of cooperative relaying systems with NOMA is
analyzed, and by appropriate power allocation the sum rate
performance advantage over cooperative relaying systems with
OMA is also revealed. [7] further applies the NOMA principle
to a multiple-input single-output (MISO) system, and solves the
downlink sum rate maximization problem. To consider a more
practical situation, in [8] Zhang et al. assume imperfect channel
state information as a result of channel estimation errors, and
solves the worst-case achievable sum rate problem for NOMA
systems in MISO channels. Different from [8] which assumes
knowledge of channel estimation errors in advance irrespective
of practical estimators, we take a step further by considering
practical channel estimators. In particular, for a typical two-
user NOMA downlink system, as practical channel estimators
introduce estimation errors, and the SIC receiver of the strong
user uses channel estimates to perform SIC, it will subsequently
decrease the signal-to-inference-and-noise ratio (SINR) of the
strong user in data transmission phase. On the other hand, the
SINR of the strong userdepends also on the power allocation
strategy. Therefore, it is necessary to jointly consider the chan-
nel estimation and power allocation for the NOMA downlink
systems.
In this letter, we focus on the two-user SISO NOMA downlink
system. Firstly, a novel scheme for joint channel estimation
and power allocation is provided. In particular, a non-convex
optimization problem is formulated in terms of maximizing the
average effective SINR of the strong user while with the bounded
constraints on the average effective SINR of the weak user.
Secondly, a constrained concave convex procedure (CCCP)-
based iterative algorithm is proposed to solve the non-convex
problem [9]. Finally, in order to compare the performance of
the proposed estimation method, we also provide the maximum
average effective SINR under both maximum-likelihood (ML)
estimation and linear minimum-mean-square-error (LMMSE)
estimation in Section IV.
Notations: Boldface lowercase and uppercase letters denote
vectors and matrices, respectively. The A∗,A†,AT, and (A)−1
denote the conjugate, conjugate transpose, transpose, and in-
verse of the matrix A, respectively. adenotes the 2-norm of
the vector a.Re(·)and Im(·)denote the real and imaginary part
of the complex argument inside, respectively. Iis the identity
matrix. E{·} denotes the statistical expectation, and j=√−1
is the imaginary unit.
1070-9908 © 2016 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.

1782 IEEE SIGNAL PROCESSING LETTERS, VOL. 23, NO. 12, DECEMBER 2016
II. SYSTEM MODEL
Consider a SISO NOMA downlink system with a base sta-
tion (BS) and two users UEi,i∈M={1,2}. The channel
response from BS to UEi, denoted as hi,isassumedtobea
zero-mean circular symmetric complex Gaussian random vari-
able with variance δ2
i=E{|hi|2}. Without loss of generality,
we assume UE1is the strong user, and UE2is the weak user.
Thus, we have δ2
1≥δ2
2.
The signal sent out by the BS is denoted as s=
i∈M √αiPs
i, where siis the signal intended for the ith user
satisfying E{|si|2}=1.Pis the transmit power of the BS, and
0≤αi≤1represents the power allocation ratio of the ith user
with i∈M αi=1. The received signal at the ith user is given
by
yi=hi
m∈M αmPs
m+ni(1)
where ni∼CN(0,σ
2)denotes the additive white Gaussian
noise (AWGN) at the ith user.
In the NOMA downlink system, each user should estimate the
channel response before signal detection. During the channel
estimation phase, BS sends the training sequence, denoted as t
with length N, to both users. Thus, the received training signal
at the ith user, i∈M,is
zi=hit+˜
ni(2)
where ˜
ni∼CN(0,σ
2I)denotes the AWGN vector at the ith
user during the channel estimation phase. The ith user should
employ the channel estimation methods to estimate hibased on
zi.
III. PROPOSED NOVEL CHANNEL ESTIMATION
In this section, we propose a novel channel estimation scheme
for both strong and weak users in terms of maximizing the
average effective SINR of the strong user, while maintaining
certain average effective SINR of the weak one. It is different
from traditional methods with separating channel estimation
and power allocation, which will also be presented in the next
section.
A. Problem Formulation
Consider a linear estimator of hiwith a general form as
ˆ
hi=v†
izi(3)
where viis the unknown vector to be designed.
For the strong user UE1, it performs SIC before detecting its
own signal. At first, UE1will use the estimated channel ˆ
h1to
detect the weak user UE2’s signal based on the following signal
model:
y2
1=ˆ
h1α2Ps
2+(h1−ˆ
h1)
2

i=1αiPs
i+ˆ
h1α1Ps
1+n1
(4)
where the second and third terms are interferences caused by a
channel estimation error and the transmitted signal of the strong
user, respectively. Then, upon successful decoding s2, which is
removed from y2
1in (4) in a successive manner [10], we have
y1
1=ˆ
h1α1Ps
1+(h1−ˆ
h1)
2

i=1 αiPs
i+n1(5)
upon which the detection of the s1is carried out.
Based on (4), we define the average effective SINR for the
strong user to detect the UE2’s signal as [11]
¯γ2
1=E{|ˆ
h1|2}α2P
E{|h1−ˆ
h1|2}P+E{|ˆ
h1|2}α1P+E{|n1|2}.(6)
Substituting (3) into (6), and after some mathematical
manipulations, we obtain
¯γ2
1=(1 −α1)v†
1R1v1
(1 + α1)v†
1R1v1−δ2
1(v†
1t+t†v1)+η1
(7)
where R1δ2
1tt†+σ2Iis the covariance matrix of z1,η1
δ2
1+σ2/P .
Similarly, based on (5), the average effective SINR of the
strong user detecting its own signal s1is obtained as follows:
¯γ1
1=α1v†
1R1v1
v†
1R1v1−δ2
1(v†
1t+t†v1)+η1
.(8)
Meanwhile, for the weak user UE2with channel estimate ˆ
h2,
it detects its own signal s2treating the strong user UE1’s signal
as noise, then we have the signal model as
y2
2=ˆ
h2α2Ps
2+(h2−ˆ
h2)
2

i=1αiPs
i+ˆ
h2α1Ps
1+n2.
(9)
Based on (6) and (7), we can accordingly obtain the average
effective SINR for UE2detecting its own signal as
¯γ2
2=E{|ˆ
h2|2}α2P
E{|h2−ˆ
h2|2}P+E{|ˆ
h2|2}α1P+E{|n2|2}
=(1 −α1)v†
2R2v2
(1 + α1)v†
2R2v2−δ2
2(v†
2t+t†v2)+η2
(10)
where R2δ2
2tt†+σ2Iis the covariance matrix of z2,η2
δ2
2+σ2/P .
The objective of the proposed scheme is to maximize the
average effective SINR ¯γ1
1of the strong user UE1while assuring
that the minimum average effective SINR of the weak user UE2
is not less than a predefined threshold, denoted as γ0. Thus, the
optimization problem is formulated as
max
{0≤α1≤1},{vi}¯γ1
1s.t. min
i∈M{¯γ2
i}≥γ0(11)
The optimization problem (11) is non-convex because of the
non-convexities of the objective and the average effective SINR
constraint. In the following, we propose a CCCP-based iterative
algorithm to solve problem (11).

TAN et al.: NOVEL CHANNEL ESTIMATION FOR NON-ORTHOGONAL MULTIPLE ACCESS SYSTEMS 1783
B. CCCP-Based Iterative Algorithm
The formulated optimization problem (11) is recast as
max
{0≤α1≤1},{vi}¯γ1
1s.t. ¯γ2
i≥γ0∀i∈M.(12)
By letting μ1=1/α1and introducing the slack variable τ, prob-
lem (12) is equivalently rewritten as
max
{μ1≥1},{vi},τ τ/μ1(13a)
s.t. v†
1R1v1−δ2
1(v†
1t+t†v1)+η1−v†
1R1v1
τ≤0
(13b)
v†
iRivi+1+ 1
γ0v†
iRivi
μ1−δ2
i(v†
it+t†vi)
+ηi−1
γ0
v†
iRivi≤0∀i∈M.(13c)
To simplify, problem (13) is recast as
min
{μ1≥1},{vi},τ −ln τ+lnμ1s.t. (13b),(13c).(14)
It is noted that the functions −ln μ1,v†
iRivi, and v†
iRivi
μ1,
where μ1>0,Ri0, are convex [12]. The problem (14)
is a difference of convex (dc) programming [13], therefore,
it can be solved using the CCCP algorithm in [9]. In par-
ticular, let ξ(v1,τ)=v†
1R1v1
τ,ζi(vi)=v†
iRivi,i∈M, and
ρ(μ1)=−ln μ1. The first-order Taylor expansions of above
equations around the point (˜
vi,˜τ, ˜μ1)are computed as [14]
ξ(v1,τ,˜
v1,˜τ)=2Re{˜
v†
1R1v1}
˜τ−˜
v†
1R1˜
v1
˜τ2τ(15)
ζi(vi,˜
vi)=2Re{˜
v†
iRivi}−˜
v†
iRi˜
vi,i∈M (16)
ρ(μ1,˜μ1)=−ln˜μ1−(μ1−˜μ1)/˜μ1.(17)
In the (l+1)th iteration of the proposed CCCP-based iterative
algorithm, we solve the following convex optimization problem
min
{μ1≥1},{vi},τ −ln τ−ρμ1,˜μ1(l)
s.t. v†
1R1v1−δ2
1(v†
1t+t†v1)+η1
−ξv1,τ,˜
v(l)
1,˜τ(l)≤0,
v†
iRivi+1+ 1
γ0v†
iRivi
μ1−δ2
i(v†
it+t†vi)
+ηi−1
γ0
ζi(vi,˜
v(l)
i)≤0∀i∈M (18)
where the point (˜
v(l)
i,˜τ(l),˜μ(l)
1)denotes the solution to problem
(18) at the lth iteration. The problem (18) is convex that can
be effectively solved by using the interior-point method [12].
The proposed CCCP-based iterative algorithm for channel esti-
mation is summarized in Algorithm 1, where the feasible initial
point (˜
v(0)
i,˜τ(0),˜μ(0)
1)is found by the iterative feasibility search
algorithm in [15].
Algorithm 1: The Proposed CCCP-Based Iterative Algo-
rithm.
1: Initialization: l=0,˜
v(0)
i,˜τ(0),˜μ(0)
1,i∈M;
2: Repeat:
Solve problem (18) to obtain ˜
v(l+1)
i,˜τ(l+1),˜μ(l+1)
1,
i∈M;
l:= l+1;
3: Until: Convergence.
IV. MAXIMUM AVERAGE EFFECTIVE SINR UNDER ML AND
LMMSE CHANNEL ESTIMATION
Based on (2) and (3), the ML and LMMSE channel estimators
have the linear forms as v(k)
i,k∈{ML,LMMSE}, which are
given as [16] follows
v(k)
i=⎧
⎪
⎪
⎨
⎪
⎪
⎩
t
t2,k=ML
δ2
it
σ2+δ2
it2,k=LMMSE.
(19)
Putting v(k)
iinto (7), (8), and (10), we obtain the average
effective SINR of the strong user and the weak users under ML
and LMMSE channel estimation as follows:
¯γ(k)
1=(k)
1α(k)
1
b(k)
1
(20)
min
i∈M{¯γ2,(k)
i}=¯γ2,(k)
2
=(k)
2(1 −α(k)
1)
(k)
2α(k)
1+b(k)
2
(21)
where
¯γ2,(k)
i=(k)
i(1 −α(k)
1)
(k)
iα(k)
1+b(k)
i
(22)
(k)
i=⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
δ2
i+σ2
t2,k=ML
δ4
it2
σ2+δ2
it2,k=LMMSE
(23)
b(k)
i=⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
σ2
t2+σ2
P,k=ML
δ2
i+σ2
P−δ4
it2
σ2+δ2
it2,k=LMMSE.
(24)
For k=ML, obviously ¯γ2,(k)
iis an increasing function of δ2
i,
thus ¯γ2,(k)
1≥¯γ2,(k)
2. First line of (21) under ML estimation is
obtained. For k=LMMSE, substituting (k)
iand b(k)
iinto (22)
and after some mathematical manipulations, we have ¯γ2,(k)
i=
1−α(k)
1
α(k)
1−1+1+ σ2
δ2
iP1+ σ2
δ2
it2. It can be seen that ¯γ2,(k)
iis also an
increasing function of δ2
i. We obtain ¯γ2,(k)
1≥¯γ2,(k)
2, and under
LMMSE estimation first line of (21) can also be derived.

1784 IEEE SIGNAL PROCESSING LETTERS, VOL. 23, NO. 12, DECEMBER 2016
We formulate the average effective SINR maximization prob-
lem similar to (11) as follows:
max
{0≤α(k)
1≤1}
¯γ(k)
1s.t. ¯γ2,(k)
2≥γ0(25)
where k∈{ML,LMMSE}. Based on (20) and (21), we know
that ¯γ(k)
1is a monotonic increasing function of α(k)
1, while ¯γ2,(k)
2
is a monotonic decreasing function of α(k)
1. Hence, the maxi-
mization of ¯γ(k)
1is achieved when ¯γ2,(k)
2equals γ0, and α(k)
1
achieves the maximum which is
α(k)
1=(k)
2−γ0b(k)
2
(1 + γ0)(k)
2
.(26)
Since γ0>0, it can be seen that (26) satisfies the constraint of
α(k)
1≤1. To guarantee α(k)
1≥0,γ0should be γ0≤(k)
2
b(k)
2
.
Finally, the maximum average effective SINR ¯γ(k)
1of the
strong user, k∈{ML,LMMSE}, is obtained as follows
¯γ(k)
1=(k)
2−γ0b(k)
2(k)
1
(1 + γ0)b(k)
1(k)
2
.(27)
V. SIMULATION RESULTS
In this section, we numerically investigate the performance
of our proposed channel estimation method and compare it
with that of the conventional ML and LMMSE estimations.
The channels h1,h
2, and the noise are assumed as circularly
symmetric complex Gaussian random variables, with variances
δ2
1=1,δ
2
2=0.1, and σ2=1, respectively. The length of the
training sequence tis set to N=8, where each symbol is
randomly generated as a circularly symmetric complex Gaus-
sian random variable with zero mean and unit variance. The
maximum iteration number of the proposed CCCP algorithm
is 20.
Fig. 1 evaluates the maximum average effective SINR ¯γ1of
the strong user versus transmission power Punder the pro-
posed, ML and LMMSE channel estimation methods. It can
be observed from the figure that ¯γ1increases with Pin small
Pregime under all three channel estimation methods, and be-
comes smooth in high Pregime. For the proposed method,
consistent improvement over the LMMSE approach is observed
at all settings of P. The ML approach has nearly the same aver-
age effective SINR of the strong user as the proposed method in
high Pregime, however, significant rate degradation of the ML
approach is seen in low Pregime compared with the proposed
method.
Fig. 2 shows how the power allocation ratio of the strong
user behaves under the proposed, ML and LMMSE chan-
nel estimation methods. As can be observed, more power is
allocated to the strong user in the proposed method, com-
pared with the LMMSE approach in all settings of Pand the
ML approach in low Pregime. This means that, with joint
consideration of channel estimation and power allocation, the
proposed method can have more efficient use of the transmission
γ
γ
γ
γ
γ
γ
Fig. 1. Averageeffective SINR of the strong user versus Punder the proposed,
ML and LMMSE estimation methods.
α
γ
γ
γ
γ
γ
γ
Fig. 2. Power allocation coefficient α1under proposed, ML and LMMSE
estimation methods.
power compared with the traditional approaches with separating
channel estimation and power allocation.
VI. CONCLUSION
In this letter, we study channel estimation and power allo-
cation for the two-user SISO NOMA downlink systems. We
propose a novel estimation scheme to maximize the average
effective SINR of the strong user with bounded average effec-
tive SINR guaranteed for the weak user. We also consider the
maximum average effective SINR of the strong user under the
ML and LMMSE estimators. From the simulation, it shows
that the proposed estimator outperforms the traditional ML and
LMMSE estimators.

TAN et al.: NOVEL CHANNEL ESTIMATION FOR NON-ORTHOGONAL MULTIPLE ACCESS SYSTEMS 1785
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