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China Communications • November 2020
12
Keywords: non-orthogonal multiple access;
massive connection; active user detection;
channel estimation; multi-user detection and
alternating direction method of multipliers
Massive machine-type communication
(mMTC) plays an essential role for 5G and
beyond (B5G) mobile wireless communica-
tion networks to provide massive connectivity
and lower latency, including the access delay
and the transmission delay, for large-scale
Internet of Things (IoT) devices/users, such
as machines and sensors are connected to the
internet via wireless links [1]. Specically, the
mMTC focuses on the uplink sporadic small
packet communication of a large number of
IoT devices, where only a few of massive
potential devices are active and transmit short
data packet with low transmission rates to the
base station (BS) in any coherence time. To
advocate such massive IoT device connectivi-
ty, the massive multiple-input multiple-output
(MIMO) systems [2,3] or non-orthogonal mul-
tiple access (NOMA) systems [4] are required.
In massive MIMO systems, the BS equipped
with massive antennas serves a large number
of devices. In NOMA system, massive devices
are allowed to access the same resource block
Abstract: This paper proposes some low com-
plexity algorithms for active user detection
(AUD), channel estimation (CE) and multi-us-
er detection (MUD) in uplink non-orthogonal
multiple access (NOMA) systems, including
single-carrier and multi-carrier cases. In par-
ticular, we first propose a novel algorithm to
estimate the active users and the channels for
single-carrier based on complex alternating
direction method of multipliers (ADMM),
where fast decaying feature of non-zero
components in sparse signal is considered.
More importantly, the reliable estimated in-
formation is used for AUD, and the unreliable
information will be further handled based on
estimated symbol energy and total accurate or
approximate number of active users. Then, the
proposed algorithm for AUD in single-carrier
model can be extended to multi-carrier case by
exploiting the block sparse structure. Besides,
we propose a low complexity MUD detection
algorithm based on alternating minimization
to estimate the active users’ data, which avoids
the Hessian matrix inverse. The convergence
and the complexity of proposed algorithms are
analyzed and discussed nally. Simulation re-
sults show that the proposed algorithms have
better performance in terms of AUD, CE and
MUD. Moreover, we can detect active users
perfectly for multi-carrier NOMA system.
Donghong Cai1,2, Jinming Wen1,*, Pingzhi Fan2, Yanqing Xu3, Lisu Yu3
1 The College of Cyber Security, Jinan University, Guangzhou 510632, China
2 The Institute of Mobile Communications, Southwest Jiaotong University, Chengdu 610031, China
3 The State Key Laboratory of Rail Trafc Control and Safety, Beijing Jiaotong University, Beijing 100044, China
4 The School of Information Engineering, Nanchang University, Nanchang 330031, China
* The corresponding author, email: jinming.wen@mail.mcgill.ca
Received: Jun. 14, 2020
Revised: Aug. 30, 2020
Editor: Wei Duan
NEW ADVANCES NON-ORTHOGONAL MULTIPLE ACCESS
China Communications • November 2020 13
Moreover, the AUD and CE algorithm based
on generalized expectation consistent signal
recovery (GEC-SR) is proposed in [14], which
allows the element-wise output mapping being
an arbitrary form. Besides, the closed-form
expressions of block error rates (BLERs) of
dwonlink data with finite block-length cod-
ing under imperfect CSI are derived. Most of
above related works, the results shown that
single antenna can only serve few hundred of
users with short pilot length. Hence, massive
antennas system is considered to improve the
connectivity. For the multiple-antenna BS
case, a joint AUD and CE problem is formulat-
ed as a multiple measurement vector (MMV)
[15]. The scheme utilizes the algorithm based
on approximate message passing (AMP) by
exploiting block sparsity in the MMV called
multiple measurement vector approximate
message passing (MMV-AMP) and analyze
the algorithm performance based on the state
evolution. This AMP based joint estimation is
proposed for joint estimation from linear mea-
surements, where the random Gaussian pilots
are taken into account for benet of analysis.
Besides, the alternating direction method of
multipliers (ADMM) algorithms are used to
estimated the sparse signal and the support set
[16,17]. Interestingly, both ADMM and AMP
algorithms are rst order methods, which are
Hessian-free, i.e., we do not have to compute
the Hessian matrix and its inverse.
However, few of the existing papers inves-
tigate MUD besides AUD and CE, which is
crucial to realize large scale multiple access
systems. In [11,14], the MUD based on a lin-
ear minimum mean-squared error (MMSE)
detector has been investigated. This receiver
exhibits a high-error oor of symbol error rate
(SER). Furthermore, a threshold-aided block
sparsity adaptive subspace pursuit (BSA-
SP) algorithm is proposed to jointly perform
AUD, CE, and MUD, and a superior SER
performance is achieved [18]. These algo-
rithms however require high computational
complexity, which is not tolerated in large-
scale systems. To overcome this challenge,
joint user activity and uplink data detections
simultaneously. For example, multiple users
in the power domain NOMA [5] transmit their
data through the same sub-carrier by allocat-
ing different transmit power; while codewords
of multiple users are multiplexed at the same
orthogonal frequency division multiplexing
(OFDM) sub-carriers in the code domain
NOMA [6]. On the other hand, many grant-
free random access schemes are proposed to
reduce the access delay, comparing to the con-
ventional grant based access scheme. As a re-
sult, the key challenges of such massive grant-
free IoT device connectivity scenario are the
channel estimation (CE), active user detection
(AUD) and data decoding for both massive
MIMO and NOMA techniques [7]. In partic-
ular, a natural question that may arise is how
to design the receiver to efficiently perform
the CE and/or multi-user detection (MUD)
besides the AUD. To this end, a two-phase de-
tection, including the training and the uplink
data transmission phases, is always considered
for uplink grant-free random access systems,
in which the CE or the joint CE and AUD is
performed in the training phase and the MUD
is carried out in the uplink data transmission
phase.
To identify active users and to estimate their
channel coefcients in the training phase, one
way is to formulate the received signal model
in the training phase as a CS problem [8] or a
linear inverse problem [9] taking advantage of
the sparse signal. Then we can solve the sparse
signal recovery problem under the framework
of CS or Bayesian inference. In [12], a joint
AUD and CE scheme is considered for a sin-
gle-antenna system, where the received signal
is formulated as a single measurement vector
(SMV) problem. To solve this problem with
high accurate and low complexity, the authors
combine the message passing algorithm with
block sparse Bayesian learning algorithm. In
[11], the authors propose a CE and AUD al-
gorithm based on the expectation propagation
(EP). In addition, a three-phase transmission
protocol which consists of AUD, CE, uplink
and downlink data transmission for massive
access system [13] is designed and optimized.
This paper proposes
some low complexity
algorithms for active
user detection (AUD),
channel estimation
(CE) and multi-user
detection (MUD) in
uplink non-orthog-
onal multiple access
(NOMA) systems.
文后参考文献不符合我刊规范规定。如题名应放在引号内，切除句首首字母外其余都应为小写。刊名应为斜体。卷号 , 期数 , 年份 , 页码的
顺序不符合我刊规范。
作者简介的照片像素不够，达不到出版要求。
China Communications • November 2020
14
error in the rst stage detection, we considered
two MUD strategies: (1) joint active user and
uplink data detection with estimated CSI; (2)
data detection based on the estimated CSI and
active user set. The aims of these two MUD
strategies are designing high accuracy and low
complexity detection algorithm. In addition,
the impact of the AUD and CE is also investi-
gated.
The main contributions of this paper are
summarized as follows:
• We design some low-complexity two-stage
detection algorithms for AUD, CE and
MUD in uplink grant-free NOMA system,
where the impact of both prior information
and accuracy of the signal detection in the
training phase is considered.
• In particular, we propose two novel algo-
rithms to estimate the active users and the
channel coefficients in the training phase
based on the complex ADMM algorithm.
In order to utilize the fast decaying features
of the nonzero components in sparse sig-
nal, we turn the traditional least absolute
shrinkage and selection operator (LASSO)
problem into iterative weighted problem,
where the estimation of non-zero elements
will be stopped after few times iterations
due to the reliable information of this part.
Moreover, we update the active user set
based on the symbol energy of unreliable
information and the total accurate or ap-
proximate number of active user known at
the BS. In this way, both false alarm rate
and missed detection rate can be reduced,
even if the SMV problem with short pilot
sequence length. In addition, the algorithm
for SMV problem can be extended to the
MMV problem by exploiting the block
sparse structure. We can detect the active
users perfectly with the proposed algorithm
in multiple sub-carriers NOMA system.
• For the data transmission phase, we propose
a low complexity MUD detection algorithm
based on alternating minimization is pro-
posed for the estimated active users’ data
with the estimated channel coefficients.
Specially, the proposed algorithm solves
are investigated in [19,20]. The authors in [19]
proposed a posteriori probability (MAP) based
AUD (MAP-AUD) and the MAP based MUD
(MAP-MUD) scheme, where the extrinsic
information between MAP-AUD and MAP-
MUD is exchanging for each iteration. As a
result, the proposed algorithm improves the
AUD and MUD performance. In [20], a dy-
namic modified orthogonal matching pursuit
(OMP) algorithm is exploited to realize both
AUD and MUD in several continuous time
slots. In particular, the estimated active user
set in the current time slot can be the prior
information to estimate the active users in the
next time slot. In this way, the proposed MUD
scheme achieves much better performance
than that of the conventional CS-based MUD.
However, all these works only consider the
ideal scenario that the BS know the perfect
CSI. The impact of the channel estimation er-
ror is neglected in these algorithms. Whereaf-
ter, a novel AUD technique which can perform
the AUD, CE and uplink MUD all together is
presented in [21]. Specially, this algorithm is
proposed based on the block coordinate decent
(BCD) optimization algorithm, and better per-
formance is achieved at the cost of the com-
plexity.
In this paper, we aim at proposing
low-complexity two-stage detection algorithm
for grant-free large scale code domain NOMA
system. For the training phase in the rst stage,
algorithms based on the ADMM are proposed
for single and/or multiple sub-carriers NOMA
systems, where both the BS known total num-
ber of instantaneous active users and active
user probability are considered. Different from
the sparse signal recovery and support set esti-
mation in [16,17], where the non-zero element
in sparse signal with higher symbol energy
can be easily estimated, we further handle the
inactive user set in the proposed algorithms by
take full advantage of the know information at
the BS. Hence, the AUD performance can be
improved at the cost of a negligible complexi-
ty. Especially for multiple sub-carriers NOMA
systems, the active users can be detected per-
fectly. In order to reveal the impact of AUD
China Communications • November 2020 15
are independently chosen from the set
{
−1,1
}
with equal probability [7]. In grant-free access
scheme, each user access the network without
a grant to transmit [15,22]. It is assumed that
users are synchronized in a frame structure
and active or inactive within the duration of a
complete frame. We consider a block-fading
channel, i.e., the channel coefficients remain
fixed across one frame. Due to only a small
subset of the users are active in a coherence
time, which make the communication sporad-
ic. Without loss of generality, it is assumed
that the BS knows the distribution of each user
activity indicator,
α
n, or the total number of
the active users, K(KN<< ), where follows
Bernoulli distribution, i.e., Pr 1
(
α
n= =∋
)
,
Pr 0 1
(
α
n= = −∋
)
, and KN/≈∋ when N i s
large in a certain frame. The active user trans-
mit a pilot symbol followed by Ld data sym-
bols in a certain frame; while the inactive user
keeps silent in the entire frame.
Let xnp,∈ and xnj,∈ be the pilot sym-
bol and the j-th ( 1, 2, , )jL=d data symbol
for user nn N, {1,..., }∈, respectively, where
is the modulation complex constellation
with size M. If the pilot symbol xnp, is set to 1,
then the spreading sequence sn in the training
phase is the same to the pilot sequence. Each
data symbol is spread by an all one vector
IL, where L is the number of the orthogonal
OFDM sub-carriers. After that, signals from
all active users are superimposed, and then are
transmitted over the sub-carriers.
In training phase, the pilot measurements
yl
p Lp
∈×1 on the l-th (lL=1, 2, ,) sub-carrier
at the BS is
y sz
l p n n nl l
pp
= +
= +S ah z
∑
n
N
=
(
1
diag ,
ςα
( )
h
ll
)
p
(1)
where
ς
p is a normalization factor for the pilot
spreading sequences,S ss s=
ς
pN
[
12
,, ,
]
,is
the pilot observation matrix,
a= ∈
(
αα α
12
, , , 0,1n
)
TN
{ }
is the user activity
indicator vector, hl l l Nl
=
(
hh h
12
, ,,
)
T denotes
the signal detection problem in closed-form
inner, which avoid the Hessian matrix in-
verse, comparing with the AMP-like algo-
rithm [11].
• The convergence and the complexity of the
proposed algorithms are analyzed and dis-
cussed nally. Simulation results show that
the proposed algorithms have better perfor-
mance in terms of AUD, CE and MUD with
low complexity, which can be used to guide
the NOMA system design.
The rest of this paper is organized as fol-
lows. Section II introduces the NOMA system
model. Section III formulates the CE and
AUD problem and describes the low complex-
ity detection algorithm proposed in this paper
for both the single sub-carrier and multiple
sub-carriers models. Section IV proposes two
MUD schemes with the estimated channel
coefcients and the active users, and analyzes
the convergence and the complexity. Finally,
the simulation results and the conclusion are
presented in Section V.
Notations: Boldfaced capital letters and
lower-case letters denote matrix and vector,
respectively.
NN
( )
denotes the complex
(real) N-dimensional vector space. The super-
scripts
(
⋅
)
T and
(
⋅
)
H stand for the transpose and
conjugate transpose, respectively. For a given
vector x∈N, || ||xl, l=0,1 denotes its ll
01
( )
norm.[]⋅ stands for mathematical expecta-
tion, denotes complex Gaussian distribu-
tion and I is the identity matrix.
M
We consider a single-cell uplink grant-free
NOMA system, containing N short-packet
transmission users and a central base station
(BS), where all terminals are equipped with
a single-antenna. Each user is allocated a
special non-orthogonal spreading sequence,
sn∈Lp×1, for the pilot symbol in advance,
where the length of sn is Lp(Lp N<). In ad-
dition, the entries of sn n n Lp n
=
(
ss s
1, 2, ,
, ,,
)
T
China Communications • November 2020
16
user activity and channel estimation algo-
rithm is proposed. Specially, the joint active
user identification and channel estimation
procedure is first formulated as an iterative
weighted LASSO problem. Then a complex
ADMM algorithm is used to solve the iterative
weighted LASSO problem, and a channel es-
timation and user activity preceding algorithm
is derived.
The pilot measurement of yl
p in (1) can be
re-written as
y s z Sh z
l pnnnll ll
p pp
= += +
∑
n
N
=1
ςα
h, (4)
which is a SMV problem. When all the
sub-carriers are considered, problem (4) can be
formulated as a MMV problem. Recall that the
number of potential users is always larger than
the length of pilot symbol spreading sequence,
i.e., Lp N<, especially in large-scale multiple
access of the Internet of things. Hence, (4) is
an underdetermined linear system model.
Mathematically, there are infinitely many
solutions to the problem (4). However, due to
the sparse nature of the signal, the most sparse
solution is more suitable for the actual situ-
ation [9,10]. Therefore, the estimation of the
sparse signal h can be obtained by solving the
following l0-norm minimization problem:
h
min
l∈Nhl0 (5a)
s.t. y Sh
ll
p−≤
2
ε
(5b)
where
ε
is an allowable estimation error. How-
ever, the l0-norm minimization problem in (5)
is an NP-hard problem. Therefore, we consid-
er replacing the convex relaxation of l0-norm
with the l1-norm to obtain the approximate
solution of (5), i.e.,
h
min
l∈Nhl1 (6a)
s.t. y Sh
ll
p−≤
2
ε
. (6b)
It has been shown in literature that channel
hl can be accurately recovered by minimizing
l1-norm if the following conditions can be met
the channel vector with, hnl h
(
0,
σ
2
)
,
σ
h
2
is the variance of the channel distribution, and
zI
p~ (0, )
σ
p Lp
2 is the additive white Gauss-
ian noise (AWGN) vector.
In the uplink data transmission phase, the
received signal ylj
d on sub-carrier l at the BS for
the j-th data symbol is
y x hz
lj n j n nl lj
dd
= +
∑
n
N
=1
,
α
, (2)
where zlj d
d~ 0,
(
σ
2
)
is the AWGN. Then,
the combined received signals over all L
sub-carriers in the uplink data transmission
phase at the BS can be expressed as
y H ax z
dd
j jj
diag
( )
+, (3)
where yd dd d L
j j j Lj
(
yy y
12
, ,,
)
T∈,
H hh h
[
12
,,,
L
]
T∈LN×,
zd dd d L
j j j Lj
(
zz z
12
, ,,
)
T∈ and
zd dd d L
j j j Lj
(
zz z
12
, ,,
)
T∈.
The aim of the BS is estimate the user
activity indicator, the channel coefficients,,
and the data symbols based on the available
measurements in (1) and (3). The proposed
algorithms for user activity, channel and data
symbol estimation consists of two parts: chan-
nel estimation with user activity preprocessing
and low-complexity multiple-user data symbol
detection. In particular, we take advantages
of the features about the true sparse signal,
h ah
l l l N Nl
= =diag , , ,
( ) (
αα α
11 2 2
hh h
)
T, in the
rst part, where the nonzero components have
a fast decaying distribution [16]. Hence, the
information of nonzero entries of estimated
signal is reliable; while the information of zero
entries should be further detected based on the
data symbol detection in the second part. The
following two sections will introduce these
two parts of the proposed detection algorithm,
respectively.
A
A
E
In this section, we focus on the first part of
detection, where a reliable information aided
China Communications • November 2020 17
hz
l
min
,∈Nfg
(
hz
l
)
+
( )
(11a)
s.t. Sh z y
ll
−=p, (11b)
where f
(
h Sh y
l ll
)
= −|| || /2
p2
2,g
(
zz
)
=
λ
|| ||1,w.
The corresponding augmented Lagrangian
function is:
+ −+ −re
ρ
(
(
h zu h z
uhz hz
ll
,,
H
(
ll
)
= +fg
)
)
(
ρ
2
)
( )
2
2
, (12)
where u∈N is a Lagrange multiplier,
ρ
is
a penalty parameter, re
(
⋅
)
represents the real
part. Given h zu
l,,, the ADMM iteration con-
sists of the following three steps:
u u hz
h hzu
z h zu
t tt
ll
t t tt
t tt
++
+ ++
+
11
1 11
1
=
=−−
=
arg min , , ,
arg min , , ,
z
h
∈
l
ρ
∈
N
(
N
l
ρ
ρ
(
(
l
)
.
)
)
(13)
Subproblem of hl-update in ADMM itera-
tion (13) can be expressed as
h Sh y
l ll
kp+1
= − + −−
= −
= −+ −
= −
+−
+ −+ −
+− − −
+− −
h
h
h
h
re
min
min re
min
min
uuuu
ρ
ρ
2
2
l
l
l
l
∈
∈
∈
∈
ρρ ρρ
HH
(
(
(
uhz hz
h
N
N
N
N
hz hz
(
hz hz reuhz
H
l
1
1
1
1
2
2
22
2
ll
)
ll l
(
.
Shy uhz
Sh y
Sh y h z
ll
)
ll l
ll l
ll
H
)
H
(
(
)
)
pH
p
p
2
2
2
2
2
2
2
2
ρ
2
)
)
ρ
ρ
2
(
(
(
2
2
H
(
u
ρ
)
)
2
2
)
)
(14)
Then, the solution of subproblem (14) is
derived by using the following lemma.
Lemma 1: The first and second deriva-
tives of the real function of complex variable,
f
(
x Ax c
)
= −|| ||2
2, are
∂f
∂
(
x
x
)
= −
(
Ax c A
)
H, and ∂2
∂
f
x
(
2
x
)
=AA
H,
k<1 1/+
2
µ
, (7)
where k is the number of nonzero elements
of hl, i.e., the sparsity of signal hl,
µ
is the
mutual correlation coefficient of pilot matrix
S ss s=[, , , ]
12
N, dened as
µ
= ∀=max , , 1,2, ,
in≠ss
ss
in
T
in in N. (8)
when
µ
→1, then, k≤1. This indicates that
the l1-norm minimization is not applicable
to the pilot matrix S with a high correlation
coefcient, especially in a large-scale system
where the number of active users is much
larger than one. Fortunately, the existing result
in [16] have shown that the condition in (7)
can be relaxed if an iterative weighted LASSO
problem is considered. Thus, we derive the
channel estimation and the user activity pro-
cessing algorithm by considered the following
iterative weighted LASSO problem:
h
min
l∈Jhl1,w (9a)
s.t. y Sh
ll
p−≤
2
ε
, (9b)
where || || | |hl n nl1,w=∑n
J
=1wh , hh
nl n nl
α
,
wn=0, n∈Λ, wn=1, n∉Λ. Moreover, the
iterative weighted LASSO problem can be
re-formulated as a non-constrained problem
[23]:
h
min
l∈N
1
2y Sh h
ll l
pp
−+
2
2
λ
1,w . (10)
Therefore, the channel estimation is ob-
tained by solving problem (10) and the user
activity is identified according to the update
of the indicator set Λ. The detail derivation of
proposed algorithm will be shown in the fol-
lowing subsection.
To solve iterative weighted LASSO problem,
a complex ADMM algorithm is introduced.
Note that the iterative weighted LASSO
problem in (10) can be can be written as the
following two separable convex optimization
problems
China Communications • November 2020
18
∂2
∂h
(
l
2
hl
)
= +
1
2
(
SS I
H
ρ
N
)
. (21)
For any hh
ll
∈≠N,0, we have
11
22
h S S I h Sh h
l Nl l l
HH
(
+= +>
ρρ
)
(
22
22
)
0,
thus, SS I
H+
ρ
N a positive definite matrix.
Moreover, ∂2
∂h
(
l
2
hl
)
0,
(
hl
)
is a convex
function. Let ∂
∂h
(
h
l
l
)
0, i.e.,
(
SS I h Sy z
H Hp
+ − − −=
ρρ
Nl l
)
u
ρ
0, (22)
Then, the solution of the -related update
problem in (14) is
h SS I Sy z
l Nl
t H Hp+1= + +−
(
ρρ
)
−1
u
ρ
t
. (23)
Since SS I
H+
ρ
N is a positive definite ma-
trix, according to the Cholesky decomposition,
we get
S S I LL
HH
+=
ρ
N, (24)
where L is a lower triangular matrix. Then
h L L Sy z
ll
t H Hp t+−11
= +−
( )
−1
ρ
u
ρ
t
. (25)
For the minimization of z in subproblem
(13), we have
z z uhz hz
tH+1= + −+ −min re
z∈N
λ
1,w
(
(
ll
)
)
ρ
2
2
2
= + −+min
z∈N
λ
z hz
1,w
ρ
2l
u
ρ
2
2
= + −+min
z∈N
λ
ρρ
z hz
1,w
1
2l
u2
2
= +
hw
l
t+1u
ρρ
t
,
λ
, (26)
where
(
a bj c+,
)
denotes the a soft thresh-
old operator over a complex number eld, it is
dened as:
( , ) sign( ) max(0, )a bj c a a c+= −
+⋅ −j b bcsign( ) max(0,| | ),
re
im
(27)
where c ca a b
re = +
2 22
/
( )
, cim =
cb a b
2 22
/
(
+
)
. If b=0, (27) can be reduced
to the soft threshold operator in real eld:
(
ac a a c, sign max 0,
)
= −
( )
( )
. (28)
respectively, where xc∈∈
mp
,,A∈pm×.
Proof: Let r x Ax c
( )
= −∈p, then we
have
=−+
=−+
f
(
(
(
rx r x rx r x
rx r x rx r x
x Ax c Ax c
re im re m
re im re m
)
(
(
=−−
(
)
)
jj
jj
(
(
)
H
)
)
(
)
)(
(
(
(
)
)
)
i
i
(
(
)
)
)
)
,
(16)
where r xr x
re im
( )
,
( )
∈p are the real and im-
age parts of rx
( )
, respectively. Then, the de-
rivative of f
(
x
)
on xx x= +
re im
j is
∂ ∂∂f ff
∂ ∂∂
(
x xx
x xx
)
= −
1
2
(
re im
)
j
( )
= −
1
2
∂+ ∂+
(
rx r x rx r x
re im re im
22 22
(
∂∂
)
xx
re im
( )
)
j
(
( ) ( )
)
= +
rx r x
re im
( )
∂∂rx r x
∂∂
re im
xx
(
re re
)( ) ( )
−+j
rx r x
re im
( )
∂∂rx r x
∂∂
re im
xx
(
im im
)( ) ( )
. (17)
Since rx Ax Ax Ax
( )
=−+ +( )(
re re re re re re
j
cc
re im
+j), we have
∂f
∂
(
x
x
)
= +
(
r xA r xA
re re i m im
( ) ( )
)
−− +j
(
r xA r xA
re im im re
( ) ( )
)
= ++ −rxA A r xA A
re re im im im re
( )(
jj
) ( ) ( )
=−=r xA r xA r xA
re im
( )
j
( )
H
( )
. (18)
and
∂2
∂
f
x
(
2
x
)
=∂−
(
rx r x
re im
( )
∂x
j
( )
)
A
= +
1
2
j∂−
∂−
(
rx r x
(
re im
rx r x
re im
(
(
∂
)
x
∂
)
re
x
j
re
j
(
(
)
)
)
)
A
= − +− −
1
2
(
A A A AA
re im im re
jj j
( )
)
=AA
H. (19)
Invoking Lemma 1, we have
1
∂
2
∂
h SS I y S z
h
(
ll
H H pH
h
l
l
(
)
=
+− − −
ρρ
)
u
ρ
,
(20)
and
China Communications • November 2020 19
replaced by ∋⋅N.
The proposed complete channel estimation
and user activity preprocessing algorithm is
given in Algorithm 1.
Furthermore, the algorithm for SMV prob-
lem can be extended to the MMV problem as
follows:
• For L sub-carriers, the estimated active set
{Ω }
ˆll
L
=1 and the channel coefficients {}h
ˆll
L
=1
can be obtained based on Algorithm 1;
• The elements in the active user set Ω
ˆl
are arranged in ascending order, and de-
fined as ψl∈K. Then we have a matrix
Φ= Ψ ,Ψ , ,Ψ
[
12
L
]
∈KL×.
• Find the mode from each row of Φ, dened
as
α
kn. The final active user set is {}
α
kn k
K
=1
and the estimated channels for all sub-carri-
ers are H hh h
ˆ ˆˆ ˆ
=[, , , ]
12
L
T.
Therefore, the proposed channel estima-
With the estimated hl
t in the t-th iteration,
the indicator set Λ is updated based on the “rst
signicant jump” rule [16,17,24]:
The components of hl
t are first sorted as
hh h
[1] [2] [ ]l l Nl
< <<, where []n is the order
index of the t-th large component. Then, the
ϑ
t=h[]nl threshold with the smallest n such
that
h hC
[ 1] [ ]
tt
n l nl+− >⋅
h
N
l
t
∞. (29)
Finally, the indicator set Λt is updated by
Λ:
t tt+1= >
{
nh
nl
ϑ
}
.
After T times iterations, the channel coef-
cient is estimated by hh
ˆll
=T. In addition, the
user activity indicator is estimated by
α
ˆn=
1, fo r Λ ,
0, otherwise.
n∉T
(30)
Remark 1: The false alarm (FA) and the
missed detection (MD) of joint channel esti-
mation and active user detection algorithm are
dened, respectively, as
ℑ= = =
FD
{
n: 0, 1
αα
nn
ˆ
}
, (31)
ℑ= = =
MD
{
n: 1, 0
αα
nn
ˆ
}
. (32)
The user activity indicator
α
ˆn is estimat-
ed as one due to it has larger symbol energy.
Hence, there are few false alarm events in the
proposed algorithm. In other words, the esti-
mated ones of user activity indicator vector a
ˆ
is reliable.
Therefore, the estimated active user set,
dened as Ω |1
ln
{
n
α
ˆ=
}
, can be further up-
dated according to the following steps:
• The elements, hn
ˆnl l
,Ω∈c are sorted as
hh h
ˆˆ ˆ
[1] [2 ]ll
22
≤ ≤≤[ Ω]Nl−l
2. (33)
• The added active user set is obtained by
E
[ ] [ 1], , Ω , Ω .
Ω
k NK N n
l
c
= −+ − ∈
{
nh h:,
ˆˆ
[]k l nl
22
=
ll
c
}
(34)
• Ωl is updated by ΩΩ
ˆll
= ∪ EΩl
c.
Remark 2: When only the active probabili-
ty ∋ is known at the BS, the K in above rules is
Algorithm 1. Proposed AUD-CE Algorithm for SMV problem.
Input: ∋ or K and received signal yl
p.
1. while the stopping condition is not met do
2. Channel estimation based on ADMM:
3. for tT=0,1, 2, , do
4. h L L Sy z
ll
t H Hp t+−11
= +−
( )
−1
ρ
u
ρ
t
;
5. zh w
tt++11
= +S
l
u
ρρ
t
,
λ
;
6. u u hz
t t tt+ ++1 11
=−−
ρ
(
l
)
;
7. Support set detection:
8. Update the threshold
ϑ
t=h[]nl with the smallest
[
n
]
such that
h hC
[ 1] [ ]
tt
n l nl+− >⋅
h
N
l
t
∞.
The indicator set Λt is updated by
Λ:
t tt+1= >
{
nh
nl
ϑ
}
.
9. end while
10. Channel estimation and user activity preprocessing:
The channel coefficient is estimated by hh
ˆll
=T, and the n-th user activity
indicator is estimated by
α
ˆn=
1, for Λ ,
0, otherwise,
n∉T
the estimated active user set Ω |1
ln
{
n
α
ˆ=
}
.
11. Update the active user set: ΩΩ
ˆll
= ∪ EΩl
c.
Output: The estimated channel coefcient h
ˆl and the active user set Ω
ˆl.
China Communications • November 2020
20
tive users’ data, which avoids Hessian matrix
inverse. In addition, the convergence and the
complexity of proposed algorithm are ana-
lyzed and discussed.
Different from Algorithm MOMP, where a
joint active user and data symbol detection is
proposed, the proposed data symbol detection
algorithm in this subsection is based on the
estimated active user set and the channel esti-
mation in the training phase. In particular, the
multi-user detection is rst formulated as a al-
ternating minimization problem. The solution
of this problem is then derived in closed-form
expression for each update iteration. The pro-
posed algorithm avoids any matrix inversion
and reduces the complexity.
1) Problem Formulation: With the active
user set Ω
ˆ detected in training phase, the
corresponding channel coefficients of active
users H
ˆΩ
ˆ can be obtained. Then the -th time
slot received signals in (3) for the uplink data
transmission phase can be re-written as
y Hx z
Ω ΩΩ Ω
dj
ˆ ˆˆ ˆ
= +
ˆ. (35)
For the signal detection, the complex-val-
ued system model (35) can be converted into a
corresponding real-valued one as
y Hx z= + , (36)
where yyy= ∈[Re{ },Im{ }]
ΩΩ
d dT
ˆˆ
21Na×,
N KN
aΩ,
ˆ∈∋
{ }
, x xx=[Re{ },Im{ }]T,
z zz=[Re{ }, Im{ }]T, and
H= −
Re Im
Im Re
{
{
HH
HH
}
}
{
{
}
}
22LN×a
. (37)
The task of signal detection is to recover
the transmitted signal x from the received
signal vector y. The equivalent ML detection
problem of the real model (36) can be formu-
lated as
x y Hx
ˆ= −
x
min
∈
χ
2Na
2
2, (38)
where
χη
∈− + − −{ 1, , 1,1, , 1}MM
and
η
is for normalization factor. It is a NP-
tion and active user detection algorithm for
multi-subcarrier (MMV problem) is given in
Algorithm 2. Note that the proposed AUD
algorithm for SMV problem updates the esti-
mated active user set again based on the unre-
liable information and partial active user prior
information. Thus, the performance is im-
proved at the costs of negligible computational
complexity. Moreover, the AUD performance
can be improved signicantly for MMV due to
the block sparse structure.
In the training phase, the channel coeffi-
cients and the active user sets for single-car-
rier and multi-carrier NOMA system can
be estimated based on the Algorithm 1 and
Algorithm 2. Moreover, the obtained channel
coefficients and active user set can be used
to detect the data symbols in the uplink data
transmission phase. However, the error of
channel coefficient and active user set esti-
mation will impacts the performance of data
symbol detection. In addition, the complexity
of multi-user data detection becomes import-
ant in NOMA system. We will introduce two
detection schemes in the following section.
L
With the estimated channel coefficients and
active user set, we propose a low complexity
MUD detection algorithm based on alternating
minimization is proposed for estimated ac-
Algorithm 2. Proposed AUD-CE Algorithm for MMV problem.
Input: ∋orKand received signal {}yll
pL
=1.
1. for lL=1, 2, , do
2. Estimate active set {Ω }
ˆll
L
=1 and the channel coefficients {}h
ˆll
L
=1 based on
Algorithm 1.
3. The elements in the active user set Ω
ˆl are arranged in ascending
order, and dened as a vector Ψl∈K. Then we have a matrix
Φ= Ψ ,Ψ , ,Ψ
[
12
L
]
∈KL×.
4. Find the mode from each row of Φ, dened as
α
kn.
5. end for
Output: The nal active user set {}
α
kn k
K
=1, the estimated channel matrix for all
sub-carriers Hh h
ˆˆ ˆ
=[, , ]
1L
T.
China Communications • November 2020 21
section, we use AltMin to solve the proposed
formulation.
The optimization problem (41) is strict-
ly and jointly convex with respect to x and
{}y
ii
2
=
N
1
a. Moreover, there is no common con-
straint that combines both x
and y
i. Therefore,
in order to efficiently solve this problem, we
rst decompose it into the following two sub-
problems:
Update y
t
i with a xed xi
t:
y yh
t tt
i i ii
+1= −min
y
t
i∑
2
i
N
=1
a
x2
2, (42a)
s.t . , 1, , 2
∑
2
i
N
=1
a
y yl L
i
(
lt l
)
= ∀=
( )
. (42b)
Update xi
t with a xed y
t
i
+1:
xx
i i ii
t tt++11
= −min
xi
t∑
2
i
N
=1
a
yh
2
2, (43a)
s.t. , 1,2, ,2− ≤ ≤ ∀=
δδ
xi N
ia
t. (43b)
Then, an iterative algorithm can be used to
solve the subproblems (42) and (43). Note that
the original problem (41) and the subproblems
(42)-(43) are convex optimization problems
with linear constraints. Therefore, the Karush-
Kuhn-Tucker (K.K.T) conditions are the suf-
cient and necessary for the optimal solutions.
For subproblem (42), the Lagrangian func-
tion can be given by
(
y
t
i,
{
λ
(
l
)
}
l
2
=
L
1
)
=−+ −
∑ ∑∑
22
i li
NN
= = =1 11
aa
yh
tt
i ii i
x yy
2
2
2L
λ
(
l l lt
)
( ) ( )
= −+ −
∑∑ ∑ ∑
22
il l i
NN
= = = =11 1 1
aa
22LL
(
y hx y y
i ii i
(
lt l l l lt
) ( )
t
)
2
λ
( )
( ) ( )
,
(44)
where {}
λ
(
l
)
l
2
=
L
1 are the dual variables. Note
that there are only equality constraints in sub-
problem (42). Then, y
t
i is the optimal solution
if and only if there exists unique {}
λ
(
l
)
l
2
=
L
1 such
that
∇=
y
t
i
(
y
t
i,0
{
λ
(
l
)
}
l
2
=
L
1
)
, (45)
∑
2
i
N
=1
a
y yl L
i
(
lt l
)
= ∀=
( )
, 1,2, ,2. (46)
hard problem to solve the ML detection prob-
lem. Fortunately, it has been proved that the
linear MMSE signal detection algorithm is
near-optimal for uplink multi-user systems,
and the estimated of the transmitted signal
vector x can be obtained by
x HH I Hy
ˆ= +
(
HH
σ
2
2Na
)
−1, (39)
where HH
H is the Gram matrix and
HH I
H+
σ
2
2Na is the MMSE filtering matrix.
Therefore, the computational complexity
of the MMSE filtering matrix inversion is
(
8Na
3
)
, which is still very high for NOMA
systems.
2) Multi-User Data Detection: The re-
ceived signal vector y=
(
yy y
(1) (1) (2 )
, ,,L
)
T
can be decomposed into a linear combination
of 2Na vectors such that yy=∑i
2
=
N
1
ai, where
y
i ii i
= ∈=( , , , ) , 1,2, ,2yy y i L
(1) (2) (2 ) 2 1
LT L×.
The element wise representation of the decom-
posed received signal vector is
y
y
y
(
2
(
(
1
l
L
)
)
)
= + ++
yy
yy
yy
12
(2 ) (2 )
12
12
(1) (1)
() ()ll
LL
y
y
y
2
(2 )
2
2
()
(1)
l
N
N
N
L
a
a
a
, (40)
where yy
(
ll
)
=∑i
2
=
N
1
ai
( )
denotes the contribution
of x to y
(
l
)
.
Now, we relax the non-convexity constraint
on the feasible set
χ
, and approximate the ML
problem (38) based on the above decomposi-
tion as follows:
min
xii
,y
∑
2
i
N
=1
a
yh
i ii
−x2
2 (41a)
s.t . , 1, , 2 ,
∑
2
i
N
=1
a
yyl L
i
(
ll
)
= ∀=
( )
(41b)
− ≤ ≤ ∀=
δδ
xi N
ia
, 1,2, ,2, (41c)
where
δ
= −( 1) / ΓM, x
i is the i-th element
of x and hi is the i-th column of real-valued
channel matrix H. The objective function in
(41a) is a sum of separable terms, each of
which is a function of only one symbol and its
contribution in the received vector. In the next
China Communications • November 2020
22
s.t . , 1, , 2− ≤ ≤ ∀=
δδ
xi L
i
t. (50b)
The corresponding Lagrangian function of
(50) for ∀i is:
(
x y hx
i i ii
tt
,,
µµ
12
(
i i lt lt
) ( )
)
= −
∑
l
2
=
L
1
(
( )(
++11
) ( )( )
)
2
+ −+ +
µδ µδ
12
(
ii
)
(
xx
ii
)
( )
( )
, (51)
where
µ
1
l and
µ
2
l are the dual variables. Then,
we consider the following K.K.T. conditions:
20
∑
l
2
=
L
1
h hx y
i ii i
(
l l lt i i
)
(
( )
t− −+=
( )(
+1
)
)
µµ
12
( ) ( )
, (52)
µδ µδ
12
(
ii
)
(
−= +xx
ii
tt
)
= ≥
0,
0, , 0,
µµ
12
(
(
ii
)
)
(
( )
)
(53)
which are sufcient and necessary conditions
for the optimal solution to the convex optimi-
zation problem in (50). Furthermore,
µµ
12
(
ii
)
,
( )
and xi
t in (52) and (53) can be expressed as
• If
µ
1
(
i
)
=0 and
µ
2
(
i
)
=0, from (52), we have
xi
t=∑
∑
l
l
2
2
=
=
L
L
1
1
(
yh
h
ii
(
i
ll
(
)
k
)
(
)
2
)
. (54)
• If
µ
1
(
i
)
=0 and
µ
2
(
i
)
≠0, then xi
t= −
δ
.
• If
µ
1
(
i
)
≠0 and
µ
2
(
i
)
=0, then xl
i
t=.
• If
µ
1
(
i
)
≠0 and
µ
2
(
i
)
≠0, then xi
t= ±
δ
at the
same time, which is impossible for this
case.
The proposed algorithm is summarized in
Algorithm1.
Remark 3: The proposed MUD based on
alternating minimization converses to the opti-
mal solution, this result can be proved by: (1)
The constraints do not have variables together
and are linear functions; (2) The objective
function is a continuously differentiable con-
vex function, which is also positive definite;
(3) The objective function is uniformly Lip-
schitz continuous function with respect to each
variate; (4) Two separable subproblems (42)-
(43) are convex optimization problems with
linear constraints. Therefore, each subproblem
has a unique optimal solution.
The updates of x and y in alternating minimi-
From (44), the condition in (45) can be fur-
ther expressed as
∂
(
y
∂
i
t,
y
{
i
(
λ
lt
)
(
l
)
}
l
2
=
L
1
)
=
=−−
0
2
(
y hx
i ii
(
lt l l
) ( )
t
)
λ
( )
(47)
Then, we have
y h x il
i ii
(
lt l
)
= +∀
( )
t
λ
2
(
l
)
,,, (48)
By submitting (48) into (46) yields
λ
(
l ll
)
=−∀
N
C
a
y hx l
( )
∑
2
i
N
=1
a
ii
( )
t,, (49)
where C is a scaling factor[25].
For the subproblem (43), the each element
xi
t in the vector x is separable and the con-
straint of each xi
t is independent. Then the
object function min
xi
t∑
2
i
N
=1
a
yh
tt
i ii
+1−x2
2 can be
reduced to 2Na independent object function
min
xi
tyh
tt
i ii
+1−x2
2, i.e., the update of the i-th
element in the vector x can be obtained by
solving the following subproblem:
min
xi
t∑
l
2
=
L
1
(
y hx
i ii
(
lt l
)(
+1
)
−
( )
t
)
2 (50a)
Algorithm 3. Alternating minimization based MUD.
Input: x i x CN
i i ii a
00
=∀= − =0, , V ,
2
∑
i
N
=1
a
yh
t2
2 .
1. while VV
(
tt+1
)
−<
( )
κ
do
2. Update
λ
(
l
)
,∀l:
λ
(
ll
)
= −y hx
( )
∑
2
i
N
=1
a
ii
lt
.
3. Update yi
i
(
l
)
,∀:
y hx
i ii
(
lt l
)
= +
( )
t
λ
2
(
l
)
.
4. Update
xi
i
t,∀:
xi
t=∑
∑
l
l
2
2
=
=
L
L
1
1
(
yh
h
ii
(
i
ll
(
)
k
)
(
)
2
)
.
5. Vx
(
t
)
= −
2
∑
i
N
=1
a
yh
tt
i ii
2
2.
6. end while
Output: The estimated symbol vector is x
ˆ.
China Communications • November 2020 23
cause because the active users are only judged
based symbol energy in sparse signal, i.e., if
the symbol energy in sparse is large than a
certain threshold, the element is judged as one,
otherwise, the element is zero. As a result, the
obtained information of the active user in AD-
MM-ISD is very reliable. However, the active
users whose symbol energy is less than the
threshold value are missed. For example, for
the active user with poor channel, the symbol
energy of this user is smaller than the thresh-
old. In this case, the active user is always
missed in ADMM-ISD. Therefore, in Fug.
1(b), the missed detection rate of ADMM-ISD
algorithm is higher than the proposed AUD
algorithm. For the cases that total number of
active users and the active user probability are
known at the BS, corresponding to the fixed
active users and random active users cases,
the proposed algorithm adjusts the threshold
value adaptively according to the active users.
In particular, for the xed case, the threshold
value is determined based on the certain total
number of active users, then the resulting false
alarm rate and false alarm rate are very low.
In contrast, in the random case, the threshold
value is adjusted according to the approximate
number of active users, resulting in a higher
false alarm rate and false alarm rate than in
the xed case. However, as can be seen from
Figure 1(b), when the pilot sequence length is
150, the error rate of the proposed algorithm
reaches a stationary value, and the perfor-
mance is better than that of ADMM-ISD.
zation based MUD scheme need the computa-
tional complexities are
(
NL
a
)
and
(
Na
)
,
respectively. Therefore, the overall complex-
ity is
(
TN L
a
)
for this detection scheme,
which is notably lower than the complexity of
MMSE computing a Hessian matrix inverse.
In this section, we provide simulation results
to demonstrate the effectiveness of the pro-
posed AUD and MUD algorithms. For simula-
tions, the signal-to-noise ratio (SNR) for AUD
is dened as [26]
SNR = =
E
E
Sh
zl
p
l
2
2
Lpp
K
σ
2. (55)
In addition, the SNR for MUD is 1/
σ
d
2, and
the NN M
b/ dB SNR dB 10 log
0 10
( )
= +
( ) ( )
,
where M is the modulation order. QPSK is
considered for MUD demonstration. The nor-
malized mean squared error (NMSE) for total
user and active user is dened as
NMSE =|| ||
|| ||
hh
hh
ˆ
ˆ
−
−
2
2
2
2
. (56)
The Rademacher matrix is considered as a
pilot matrix to serve users, where the entries
are independently chosen form 1 or -1 with
equal probability. Then the mutual correlations
of pilot matrix with different are shown in the
Table 1 for .
As can be seen from Table 1, the mutual
correlation coefficient increases when the
length of pilot sequence decreases. In addi-
tion, simulation parameters are summarized in
Table 2.
Figure 1 depicts the user detection error
rate of proposed algorithm with regard to the
length of pilot sequence in the single-carrier
NOMA system, where active user detection
error includes of the false alarm rate and the
missed detection rate. As can be seen from
Figure 1(a), the false alarm rate of ADMMISD
algorithm is zero when the length of the pilot
sequence is 100. This can be explained be-
Table I. The mutual correlation coefcient of pilot matrix.
Lp 50 100 150 200 250 300
µ
0.64 0.46 0.36 0.36 0.296 0.26
Table II. Simulation parameters.
Channel Model Independent Rayleigh fading channel
Number of sub-carriers 1~256
Number of potential users 400~1000
Number of active users 40~100
Length of pilot sequence 50~300
SNR or NN
b/0 for MUD -2~15 dB
Modulation QPSK
China Communications • November 2020
24
gle-carrier case. In particular, the false alarm
rate of proposed algorithm is zero when the
pilot sequence length is 100. In addition, with
the increase of the number of sub-carriers,
the user detection performance of proposed
algorithm is improved. When L=10, the false
alarm rate and the missed detection rate of
proposed algorithm with Lp =100
are zeros, i.e., the users are detected per-
fectly. More precisely, false alarm rate is
Figure 2 presents the active user detec-
tion error rate of the proposed algorithm in
multi-carrier NOMA system, where the xed
cases of N=400,K=40 and N=1000,K
=100 are considered. Meanwhile, the ran-
dom case which N=1000 and the probability
of active users set as 0.1 is also considered. In
Figure 2(a), we can see that the user detection
performance of proposed AUD algorithm with
Lp =100 for L=2 is better than that of the sin-
Fig. 1. The false alarm and missed detection rate for single-carrier NOMA.
Fig. 2. The false alarm and missed detection rate for multi-carrier NOMA.
50 100 150 200 250 300
The Length of Pilot Sequence
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
False Alarm Rate
ADMM-ISD
Proposed AUD, Fixed
Proposed AUD, Random
149 150 151 152 153 154
-5
0
5
×10-4
50 100 150 200 250 300
The Length of Pilot Sequence
10-4
10-3
10-2
10-1
Missed Detection Rate
ADMM-ISD
Proposed AUD, Fixed
Proposed AUD, Random
50 100 150 200 250 300
The Length of Pilot Sequence
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Detection Error Rate
False Alarm Rate, L=2
Missed Detection Rate, L=2
False Alarm Rate, L= 10
Missed Detection Rate, L= 10
100 102 104
-5
0
5
10
×10-4
150 152 154
-5
0
5
10
15
×10-4
50 100 150 200 250 300
The Length of Pilot Sequence
0
0.02
0.04
0.06
0.08
0.1
0.12
Detection Error Rate
False Alarm Rate, Fixed
Missed Detection Rate, Fixed
False Alarm Rate, Random
Missed Detection Rate, Random
248 249 250 251
0
5
10
×10-4
198 200 202
-5
0
5
10
15
×10-4
(a) NK= = ∋=400, 40, 0.1,SNR=30dB.
(a) NK= =400, 40,SNR=30dB.
(b) NK= = ∋=400, 40, 0.1,SNR=30dB
(b) NK L= = ∋= =1000, 100, 0.1, 10,SNR=30dB.
China Communications • November 2020 25
optimal MMSE detection algorithm. However,
with the increase of active users, there is a
large gap between AltMin-based and MMSE
detections at high SNR.
zero with very short pilot sequences. Figure
2(b) depicts the user detection performance
of a larger system, we can see that when the
number of sub-carriers is 10, the lengthes of
fixed and random active cases are 200 and
220, respectively. If these parameters are sat-
isfied, false alarm and missed detection rates
are zero, which can achieve perfect user rec-
ognition. This is because we make full use of
the fast fading and block structure features of
coefcient signals in proposed algorithm. This
result is of great help to large-scale IoT users’
identication and data detection.
Figure 3 shows NMSE performance of the
different algorithms, including LS, MMSE
and ADMM-ISD. Only anyone channel of
the systems is considered in this experiment
due to the independence of channels. From
this gure, we can see that the NMSE of AD-
MM-ISD is always worse than that of LS and
MMSE for N=400,Lp = 150, K = 40 and N
= 1000, Lp= 250, K = 20. More importantly,
ADMM-ISD loss about 2 dB performance for
channel estimation. In addition, the NMSEs
of LS and MMSE are the same at high SNR;
while the LS performs better at whole SNR
domain. This result can be used to guide the
channel estimation method choose for the data
detection when the users can be detected per-
fectly.
In Figure 4, we can see the BER perfor-
mance of the proposed algorithm and MMSE
algorithm in a large-scale multi-carrier sys-
tem, where the number of potential users is
1000, the number of sub-carriers is 256, and
the length of pilot sequence is 250. Before
multi-user data detection, we have to do active
user detection and channel estimation. In this
experiment, we use the proposed active user
detection algorithm. After the perfect detec-
tion of active users, we use the LS algorithm
for channel estimation. At last, we used alter-
nating minimization (AltMin) based algorithm
for multi-user data detection. As can be seen
from Figure 5, when the active users are small,
the performance of the AltMin-based detec-
tion algorithm is almost the same as that of the
Fig. 3. The NMSE performance of channel estimation in NOMA System.
Fig. 4. The BER performance of multi-carrier NOMA System, where
N L Lp= = =400, 100, 150.
0 5 10 15 20 25 30 35 40
SNR(dB)
10-3
10-2
10-1
100
NMSE
LS
MMSE
ADMM-ISD
N=1000,L=250,K=20
N=400,L=150,K=40
5 6 7 8 9 10 11 12 13 14 15
Eb/N0 (dB)
10-6
10-5
10-4
10-3
10-2
10-1
100
Bit Error Rate
AltMin, SNR = 20 dB
MMSE, SNR = 20 dB
AltMin, SNR = 35 dB
MMSE, SNR = 35 dB
AltMin, Perfect CSI
MMSE, Perfect, CSI
12 12.005 12.01 12.015
0.5
1
1.5
2
×10-3
China Communications • November 2020
26
analyzed and discussed nally. Simulation re-
sults show that the proposed algorithms have
better performance in terms of AUD, CE and
MUD with low complexity. Moreover, we can
detect active users perfectly for multi-carrier
NOMA system.
A
The work of D. Cai was supported by National
Natural Science Foundation of China (NSFC)
under Grant No. 62001190. The work of J.
Wen was supported by NSFC (Nos. 11871248,
61932010, 61932011), the Guangdong Prov-
ince Universities and Colleges Pearl River
Scholar Funded Scheme (2019), Guangdong
Major Project of Basic and Applied Basic Re-
search (2019B030302008), the Fundamental
Research Funds for the Central Universities
(No. 21618329). The work of P. Fan was
supported by National Key R&D Project
(No.2018YFB1801104) and NSFC Project
(No.6202010600).
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rst propose a novel algorithm to estimate the
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Jinming Wen, received the
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China Communications • November 2020
28
HARQ.
Lisu Yu, received the B.E. de-
gree in Mao Yisheng Honors
College and School of Informa-
tion Science and Technology
from Southwest Jiaotong Uni-
versity, Chengdu, China, in
2014, the Ph.D. degree at Key
Laboratory of Information Cod-
ing and Transmission, Southwest Jiaotong University,
Chengdu, China, in 2019. He was a Visiting Scholar at
University of Arkansas, Fayetteville, AR, USA and Uni-
versity of Houston, Houston, TX, USA from 2017 to
2019. He has served as the student activities chair of
IEEE Communication Society Chengdu Chapter. He is
currently a lecturer in School of Information Engi-
neering, Nanchang University, China. His main re-
search interests include advanced wireless communi-
cations, next generation communication system,
coded modulation, non-orthogonal multiple access
(NOMA), fiber wireless communication, machine
learning, and ultra-dense network (UDN). Award
(2018). His research interests include vehicular com-
munications, wireless networks for big data, signal
design and coding, etc. He served as general chair or
TPC chair of a number of international conferences
including VTC’2016Spring, IWSDA’2017, ITW’2018
etc. He is the founding chair of IEEE VTS BJ Chapter
and IEEE electronic le of a passport type face pho-
tograph should be of more than 300 dpi in resolution
and 25 mm * 30mm in width and height. Biographies
chooses the font of Segoe UI and the font size of 8pt
with names in bold and italics. The thematic title
word “Biographies” is in bold and 10.5p various in-
ternational journals, and 8 books (incl. edited), and is
the inventor of 23 granted patents. He is an IEEE VTS
Distinguished Lecturer (2015-2019), a fellow of IEEE,
IET, CIE and CIC. ComSoc CD Chapter, and IEEE
Chengdu Section. He also served as a board member
of IEEE Region 10, IET(IEE) Council and IET Asia-Pacif-
ic Region. He has over 290 research papers published
in for big data, signal design and coding, etc. He
served as general chair or TPC chair of a number of
international conferences including VTC’2016Spring,
IWSDA’2017, ITW’2018 etc. He is the founding chair
of IEEE VTS BJ Chapter and IEEE ComSoc CD Chapter,
and IEEE Chengdu Section. He also served as a board
Mathematics Institute of Jilin University, Jilin, China,
in 2010, and the Ph.D. degree in applied mathematics
from McGill University, Montreal, QC, Canada, in
2015. He was a Post-Doctoral Research Fellow with
Laboratoire LIP from March 2015 to August 2016, the
University of Alberta from September 2016 to August
2017, and the University of Toronto from September
2017 to August 2018. He has been a Full Professor
with Jinan University, Guangzhou, since September
2018.
Pingzhi Fan (M’93-
SM’99-F’15), received his MSc
degree in computer science
from the Southwest Jiaotong
University, China, in 1987, and
PhD degree in Electronic Engi-
neering from the Hull Universi-
ty, UK, in 1994. He is currently
a distinguished professor and director of the institute
of mobile communications, Southwest Jiaotong Uni-
versity, China, and a visiting professor of Leeds Uni-
versity, UK (1997-), a guest professor Shanghai Ji-
aotong University (1999-). He is a recipient of the UK
ORS Award (1992), the NSFC Outstanding Young Sci-
entist Award (1998), and IEEE VTS Jack Neubauer
Memorial Award (2018). His research interests in-
clude vehicular communications, wireless networks
for big data, signal design and coding, etc. He served
as general chair or TPC chair of a number of interna-
tional conferences including VTC’2016Spring, IWS-
DA’2017, ITW’2018 etc. He is the founding chair of
IEEE VTS BJ Chapter and IEEE ComSoc CD Chapter,
and IEEE Chengdu Section. He also served as a board
member of IEEE Region 10, IET(IEE) Council and IET
Asia-Pacic Region. He has over 290 research papers
published in various international journals, and 8
books (incl. edited), and is the inventor of 23 granted
patents. He is an IEEE VTS Distinguished Lecturer
(2015-2019), a fellow of IEEE, IET, CIE and CIC.
Yanqing Xu, received the B.S.
degree in communication en-
gineering from Hangzhou Di-
anzi University, Hangzhou, Chi-
na, in 2014 and the Ph.D. de-
gree in communication and in-
formation system from the
State Key Laboratory of Rail
Trac Control and Safety, Beijing Jiaotong University,
Beijing, China, in 2019. Since July 2019, he has been
a Senior Engineer with the Department of Standard,
Patent and Pre-Research, Huawei Technologies Co.
Ltd, Beijing, China. From March to September 2017,
he was a Visiting Student at The Chinese University
of Hong Kong, Shenzhen, China. He was also a Visit-
ing Student at Lancaster University, Lancaster, UK,
from October 2017 to October 2018. His current re-
search interests include ultra-reliable and low-latency
communications, nonorthogonal multiple access and
