Communications and Network, 2013, 5, 194-199
http://dx.doi.org/10.4236/cn.2013.53B2037 Published Online September 2013 (http://www.scirp.org/journal/cn)
Artificial Noise Based Security Algorithm for
Multi-User MIMO System
Jian-hua Peng, Kai-zhi Huang, Jiang Ji
National Digital Switching System Engineering & Technological Research Center, Zhengzhou, China
Received June, 2013
ABSTRACT
The existing physical layer security algorithm, which is based on artificial noise, could affect legitimate receivers nega-
tively when the number of users is no less than sending antennas in multi-user MIMO system. In order to improve secu-
rity of multi-user MIMO system under this scenario, we propose a new multi-user MIMO system physical layer security
algorithm based on joint channel state matrix. Firstly, multiple users are processed together, thus a multi-user joint
channel state matrix is established. After achieving Singular Value Decomposition (SVD) of the joint channel state ma-
trix, the minimum singular value is obtained, which can be utilized for precoding to eliminate the interference of artifi-
cial noise to legitimate receivers. Further, we also present an approach to optimize the power allocation. Simulation
results show that the proposed alg orithm can increase secrecy capacity by 0.1 bit/s/HZ averagely.
Keywords: Multi-user MIMO System; Joint Channel State Matrix; Secrecy Capacity; Artificial Noise; Physical Layer
Security
1. Introduction
The multiple-input-multiple-output (MIMO) technology
has become one of the key technologies for the next gen-
eration wireless communication systems. Besides, it has
been introduced in the standards like IEEE 802.11/16 and
3GPP LTE. Due to the broadcast characteristics of the
electromagnetic signal propagation and the openness of
wireless channels, however, transmission of communica-
tion contents in the wireless communication system can
be easily intercepted. Thusly, it has become important
increasingly for protecting the multi-user MIMO system
communication security.
Considering the scenario without null space in the
multi-user MIMO system (number of legitimate users is
more than or equal to the number of antennas at the
transmitter), artificial noise for the physical layer security
will introduce additional noise to the legitimate users.
For this reason, available research about using artificial
noise to achieve multi-user MIMO system physical layer
security is conducted under the scenario of null space
existing in the system. [1,2] proposed a method improv-
ing security of the legitimate users by sending artificial
noise to third-party users: the potential eavesdropper’s
receiving could be inhibited by setting main lobe direct-
ing at the desired users at the multi-antenna transmitting
terminal and transmitting artificial noise in the other
beam. Communication security can be improved via
combining the beam forming and the artificial noise
when the locations of eavesdroppers are unknown [3].
Another way to protect the security of the legitimate us-
ers is achieved by introducing redundancy and transmit-
ting artificial noise in the view of space and frequency
domains combined [4]. An encipherment scheme under
multi-user downlinks situation is proposed using artifi-
cial noise jamming the eavesdroppers [5, 6]. Especially,
[5] discusses the security effectiveness of linear beam
forming with artificial noise integrated respectiv ely in the
MIMO broadcast channel and the MIMO transmission
multicast; besides, the noise power allocation scheme is
also considered with the SINR of desired users un-
changed. [7-9] provide a derivation and prove the secrecy
capacity range when there are multiple legitimate users
and eavesdroppers in Gaussian MIMO and MISO wire-
tap channel. The noise power allocation is specific re-
searched to maximize the secrecy capacity in literature
[10]. In order to ensure the existence of system null space
when the number of antennas in transmitting terminal is
limited, literature [11] studies how to select users from
the downlink multi-users to keep the security of the sys-
tem. However, the application scenarios discussed above
are limited, which restrict the system users’ capacity.
In the paper, a multi-user MIMO system security algo-
rithm based on artificial noise is proposed, aiming at no
null space in multi-user MIMO system scenario, and
translating the problem of eliminating the interference of
artificial noise to design of precoding. Firstly, multiple

J.-H PENG ET AL.
195
legitimate users are processed together at transmitting
terminal, thus a multi-user joint channel state matrix and
its complement matrix are established. Then, after Sin-
gular Value Decomposition of the joint channel state
matrix and the complement matrix respectively, the
minimum singular va lue can b e utilized for precoding , so
as to eliminate the interference of artificial noise to le-
gitimate receivers and multi-user. At last, the paper pre-
sented an approach to optimize the power allocation. The
paper established a joint channel state matrix and suffi-
ciently considered the problems might exist in the
multi-user system, by precoding both the artificial noise
affect to the legitimate users is suppressed and the inter-
ference between the multi-user is eliminated; the artifi-
cial noise is forced to be transmitted via the small-
est-affect sub-channel, while the majority artificial noise
falls on the eavesdropping side. Simulation results show
that the proposed algorithm can increase the secrecy ca-
pacity by 0.1 bit/s/HZ averagely.
2. Security Model of the Multi-user MIMO
System
The encipherment model of the multi-user MIMO system
is shown as Figure 1. Supposing there are T N transmit-
ting antennas at transmitting terminal, the number of re-
ceiving antennas of each K legitimate users is present
as
R
N. There is also a eavesdropper whose number of
receiving antennas is
E
N.
Respectively, the signals received by the -t hk le-
gitimate user and the eavesdropper can be shown as:
K
kkkkkjjkk
jk
yHtbH tbHpzn

 
 (1)
1
K
eejje e
j
yH tzHpzn


 (2)
k
y
e
H
k
H
k
t
k
b
k
w
e
w
e
y
p
z
2
b
1
b
2
t
1
t
2
y
2
H
2
w
1
y
1
H
1
w
Figure 1. The Security model of multi-user MIMO system.
Among them: k b is the transmitting signal of the
-t hk legitimate user, k
H
is the channel state matrix of
the -thk legitimate user, e
H
is the channel state ma-
trix of the eavesdropper, k
t and p, the dimension all
being 1
T
N

, are respectively stand for th e precoding of
eliminating affection to legitimate users caused by
multi-user interference and artificial noise, z is the artifi-
cial noise, k
n and e n separately stand for the additive
white Gaussian noise of the legitimate users and the
eavesdropper.
Then, the signal estimation from the -t h k legitimate
user and the eavesdropper are separately:
''
''
K
kkkkkkk jj
jk
kk kk
y
wHtb wHtb
wHpzwn




 (3)
'''
1
K
eeejjee ee
j
ywHtbwHpzwn



 (4)
Among them: k
w and e w are both 1
T
N

dimen-
sion and stands for beam forming weight for the -thk
legitimate user and the eavesdropper respectively, 'k
w
and 'e
w are the conjugate transposed matrixes of k w
and e
w .
3. A Multi-user MIMO System Security
Algorithm Based on Artificial Noise
When there is more than one legitimate user, security
using artificial noise should meet two requirements: 1) to
prevent mutual interference between multi-user; 2) to
minimize the artificial noise affection to legitimate users.
Based on the two requirements, the main idea of the se-
curity algorithm is the precoding design of p and k
t ,
so that the numerous artificial noise and mutual interfer-
ence between multi-user could be filtered as signal is
transmitted though the legitimate users’ channel and the
eavesdropper could not demodulate correctly due to the
artificial noise affection. From (3) and (4), the legitimate
users are affected by a small part of the artificial noise
interference 'kk wHpz, and the natural additive noise
from the channel 'kk
wn. Meanwhile, the eavesdropper is
effected by the mutual interference between multi-user
'K
ee jj
jk
wH tb

, the most part of the artificial noise inter-
ference 'ee wHpz and the natural additive noise from
the channel 'ee wn. As can be seen from the analysis
above, precoding design of k t and p are directly re-
lated to whether the artificial noise affection and the
multi-user interference to legitimate users could be
eliminated; besides, the precoding design of k t and p
are gotten though constructing complement matrix, joint
channel state matrix and the Singular Value Decomposi-
tion of these two matrixes. Thus, the algorithm could be

J.-H PENG ET AL.
196
divided into two modules: precoding based on the Sin-
gular Value Decomposition of complement matrix and
precoding based on the Singular Value Decomposition of
joint channel state matrix.
3.1. Precoding Based on the Singular Value
Decomposition of Complement Matrix
Taking the modified Zero-Forcing beam forming method,
in the receiving terminal [5], firstly, having process of
beam forming and getting the weight vector l
w, then
defining the complement matrix
111
[ ........]T
lllK
H
hhhh


,
in which the ()
H
T
lll
hwH
 is 1
T
N in dimension ,
so the dimension of l
H
 can be known as (1)T
K
N

 ,
at last, the
H
llll
H
UDV

would be got by the Singular
Value Decomposition of l
H
. At the time of T
K
N

,
the row number of l
H
 is less than the column number,
there would be a null space existing in the Singular
Value Decomposition, then the right singular vector
could be sequentially decomposed to () (0)
[]
s
H
lll
VVV
 ,
while the ()
s
l
V
 is the corresponding right singular vector
of the nonzero singular values and the (0)
l
V
 stands for
the null space of the complement matrix. Therefore, the
precoding (0)
kl
tV based on the Singular Value De-
composition of l
H
 is available.
3.2. Precoding Based on the Singular Value
Decomposition of Joint Channel State
Matrix
To minimize the affection of the artificial noise at the
same time on the K legitimate users, the joint channel
state matrix can be defined as 1
[ ........]T
kK
Hh h , in
which the ()
H
T
lll
hwH
 is a 1
T
N dimension vector
and the k
H
is T
K
N dimension, so
H
kkkk
H
UDV
would be got by the Singular Value Decomposition.
At the time of T
K
N, the row number of k
H
is
less than the column number, there would be a null space
existing in the Singular Value Decomposition, then the
right singular vector could be sequentially decomposed
to () (0) []
s
H
kkk
VVV
 , while the (0)
k
V
 stands for the null
space of the complement matrix. Therefore, the precod-
ing (0)
k
pV based on the Singular Value Decomposi-
tion of k
H
is available, then, the artificial noise inter-
ference to the legitimate users is zero by the precoding at
the moment. At the time of T
K
N, the right singular
vector could be sequentially decomposed to
() (0)
[]
s
H
kkk
VVV
 ,
and there would be no null space existing in the decom-
position of k
H
, that is, there is no null space existing
in the multi-user MIMO system. To make artificial noise
being transmitted via the smallest-affect to the legitimate
users sub-channel and in order to reduce the affection to
the legitimate users, the vector with smallest corre-
sponding singular value in ()
s
k
V
 can be chosen as the
precoding p. The specific calculation steps are as fol-
lows:
1) By the Singular Value Decomposition of k
h, select
the left singular vectors to be the k
w corresponding to
the largest singular value.
2) Constructing the complement matrix
111
[ ........]T
lllK
H
hhhh


,
the precoding k t is available via the Singular Value
Decomposition, which can eliminate the multi-user in-
terference.
3) Constructing the joint channel state matrix
1
[ ........]T
kK
Hh h , the precoding p is available via the
Singular Value Decomposition, to eliminate the artificial
noise affection on the legitimate users.
4. Performance Analysis
4.1. Security Performance Analysis
On the basis of definition of secrecy capacity in literature
[4], the secrecy capacity of the -th
k user in the
multi-user MIMO system in the case of no null space
existing could be derived as:
2
'2
sec 2
'22
2
'2
22
'2'22
log(1)
log(1)
kkku
kkz n
eeku
K
eej ueeze
jk
wHt
CwH p
wHt
wHtwH p











(5)
Wherein, 2222
uzne

 ,,, separately stand for the
signal power, artificial noise power, noise power in the
legitimate user channel and the noise power in the
eavesdropper channel of the -thk user.
2
'2
sec 2
'22
2
'2
22
'2'22
log(1 )
log(1)
kkk u
kkz n
eek u
K
eej ueeze
jk
wHt
CwH p
wHt
wHtwH p











(6)
As can be seen from the formula above, under the total
power constrained conditions, a part of the power is used
to transmit artificial noise, besides, the problem of allo-
cation between signal power and noise power is directly
affect the secrecy capacity. Assuming the power alloca-
tion among the users is uniform and the useful power
allocation coefficient is

, then the power of the -thk

J.-H PENG ET AL.
197
user is kP p
K

 ,the allocated noise power of the -thk
user is

1
n
P
p
K
 
 and the channel capacity of the
-t hk user is:

2
'
2
'2
log(1)
1
kkk
Ak
kk n
wHt P
CwHpPK





(7)
The channel capacity of the eavesdropper is:

2
'
22
'' 2
log(1)
1
eek
Ae K
eej eee
jk
wHt P
C
wHtPwH pPK







(8)
Assuming:
22 2
'' 2'
22
'' 2
,,,,
,,
kkkkkn eek
K
eej eee
jk
A
WHtBWHp C KD WHt
EWHtFWHpGK






Then:
log(1)
(1 )
Ak AP
CBPC




(9)
log(1 )
(1 )
Ae DP
CEPFPG


  (10)
After derivation to the secrecy formula
sec Ak Ae
CCC,
make it equal to zero then get a unary quadratic equation
about

:
 
2
1112 221111112 22222
22
112 20
zabzabzah zbh zahzbh
zh zh

 
  (11)
where in
1212
121221
,,,,
, ,,
aEFDaABbEFb B
hFGhBCzAhzdh


By the discussion and judgment of the formulas above,
the value of

which could maximize the -thk user
secrecy capacity can be available.
4.2. System Performance Analysis
From (7), the -thk user’s channel capacity without arti-
ficial noise, i.e. 1

 , can be acquired.
2
'
2
log(1)
kkk
Ak
n
wHt P
CK

 (12)
Then, the system sum capacity without artificial noise is:
2
'
2
1
log(1)
Kkkk
kn
wHt P
CK



 (13)
Similarly, system sum capacity with artificial noise
added is:

2
'
z2
'2
1log(1 )
1
Kkkk
kkk n
wHt P
CwHpPK






 (14)
So, with artificial noise security loss of system sum
capacity is:

2
'
z2
1
2
'
2
'2
1
=- =log(1)
log(1)
1
Kkkk
kn
Kkkk
kkk n
wHt P
CCC K
wHtP
wH pPK











(15)
Obviously, from (15), the loss of system sum capacity
is closely related with the receiving terminal beam form-
ing vector k
w, the precoding eliminating the multi-user
interference k
t, the precoding eliminating the artificial
noise affection p and the useful power allocation coef-
ficient

. Taking advantages of precoding design prin-
ciples in the previous section, which means the useful
signal is transmitted through the primary channel and the
artificial takes the secondary channel i.e. maximizing the
2
'
kkk
wHt and minimizing the 2
'
kk
wH p , the artificial
noise affection to the legitimate users and the loss of
system sum capacity can be minimized.
5. Simulation and Analysis
5.1. Secrecy Capacity
First of all, according to the method given in [6], a simu-
lation about secrecy capacity and sum capacity is taken
as the null space being existent and security being taken
in the system. Assuming that the numbers of antennas in
the transmitting and receiving terminal are both 4, the
number of legitimate users is 3, while the channel state of
legitimate users is given and normalized and total power
is 500 mW constantly. The simulation results of the
-t h
k user’s secrecy capacity and sum capacity are
shown in Figures 3 and 4.
The abscissa in Figure 2 presents the ratio between
noise power in user channel and in the eavesdropper
channel, the ordinate stands for the secrecy capacity.
Before artificial noise being added, part of the secrecy
capacity is greater than zero due to the affection on the
third-party from the multi-user interference; after artifi-
cial noise added the secrecy capacity is entirely improved
which causes the secrecy capacity is always above zero
in the part of 22
40 100
ne
  and realizes the secure
communication. Figure 3 indicates that with transmitting
power limited there will be some loss of the sum capacity
after encryption; meanwhile, the more noise power dis-
tributed, greater the loss will be.

J.-H PENG ET AL.
198
Secondly, raise the number of legitimate users to 4 and
maintain the other simulation conditions. Now the null
space will be inexistent, and the simulation results of the
-t h k user’s secrecy capacity and sum capacity are
shown in Figures 4 and 5 following the proposed algo-
rithm.
From Figure 4, when the system null space is inexist-
ent, the system secrecy capacity can be also improved via
the artificial noise, nonetheless, as the null space is in-
existent, the secrecy capacity is decreased comparing
with the null space existent condition. In addition, the
system secure communication cannot be achieved when
the useful power allocation coefficient is 0.8 and the ratio
range is 22
70 100
ne
 . With the legitimate user’s
channel conditions deteriorated, the secrecy capacity
when useful power distribution coefficient is 0.6 or 0.4
gets gradatim higher than the condition as useful power
distribution coefficient being 0.8. This indicates that un-
der certain conditions, by more artificial noise transmit-
ted, the secrecy capacity can be improved and secure
Figure 2. Secrecy capacity.
Figure 3. Sum capacity.
communication will be realized.
From Figure 5, when the system null space is inexist-
ent, the artificial noise has partial impact on the legiti-
mate users, so the loss of sum capacity is intensified be-
cause of the artificial noise security; the loss of sum ca-
pacity with useful power allocation coefficients being 0.8
and 0.4 is separately as much as 1/4 and 1/2 of the sum
capacity without encryption. In spite of big loss of sum
capacity, as the users number increasing the sum capacity
after encryption is still higher than sum capacity of the
null space existing system. In summary, the substance of
the proposed algorithm is via a certain loss of sum capac-
ity to improve the secrecy capacity.
5.2. Power Allocation
Assuming the total power to be 500 mW constantly, the
simulation result of the -th k user secrecy capacity in
the system changing with the useful signal power alloca-
tion coefficient is shown as Figure 6. When 22 20
ne


,
known conditions from the simulation are:
Figure 4. Secrecy capacity.
Figure 5. Sum capacity.

J.-H PENG ET AL.
199
Figure 6. The Relationship between secrecy capacity and
useful power allocation coefficient.
4
0.0775, 0.0067, 80,9.78610,
1.0048, 0.0012,4
ABCD
EFG



Substituting results above into (10), the power alloca-
tion coefficient

which maximize the secrecy capacity
(i.e. maximum transmitting speed under security trans-
mission) could be present, similarly when the 22
ne


changes, on ly C need to be changed.
It is shown in Figure 6 that in the multi-user MIMO
system with null space inexistent, the secrecy capacity
will decline rapidly, when the transmission power of the
artificial noise is too small, especially less than the 10
percent of total power, and the secure transmission can-
not be achieved if th e condition g etting even worse. With
legitimate users’ channel conditions deteriorate, the op-
timum allocation coefficien t gradu ally shifts left, n amely,
the allocated artificial noise power will be more and
more if the channel condition of the eavesdropper is bet-
ter than the legitimate us er.
6. Conclusions
In this paper, a multi-user MIMO system security algo-
rithm based on artificial noise is proposed, which em-
phasizes the no null space in multi-user MIMO system
scenario. In the algorithm, the affection of artificial noise
to legitimate user is eliminated by precoding. For this
purpose, the joint channel state matrix is established;
then, after Singular Value Decomposition of the joint
channel state matrix, the precoding is completed based
on the minimum singular value; at last, an optimized
power allocation scheme is proposed. As shown from
simulation result, in the multi-user MIMO system with
null space inexistent, with artificial noise introduced for
physical layer security, the secrecy capacity will be im-
proved efficiently by a certain sacrifice of sum capacity;
especially, with the channel condition of the legitimate
user getting worse, the allocated artificial noise power
will be more and more to maximize the secrecy capacity.
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