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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 1
Robust AN-Aided Secure Precoding for an AF
MIMO Untrusted Relay System
Quanzhong Li, Liang Yang, Qi Zhang, and Jiayin Qin
Abstract—Robust artificial noise (AN)-aided secure precod-
ing for an amplify-and-forward multiple-input multiple-output
(MIMO) untrusted relay system is studied, where the relay is
untrusted and willing to help forwarding multiple data streams
from the source to destination. We consider that the available
channel state information (CSI) is imperfect and modeled by
the worst-case model. Our objective is to maximize the worst-
case secrecy rate under the robust transmit power constraints
at the source and relay, by jointly designing the signal and AN
precoding matrices at the source and the precoding matrix at
the relay. The robust secure precoding problem is non-convex
and hard to solve. To overcome this difficulty, we propose the
weighted minimum mean square error (WMMSE) based method,
where the sign-definiteness lemma is used to eliminate the channel
uncertainties and an effective iterative optimization algorithm is
developed. Simulation results are provided to demonstrate the
effectiveness of the proposed scheme.
Index Terms—Robust precoding, untrusted MIMO relay, se-
curity, worst-case.
I. INT ROD UC TI ON
DUE to the openness of wireless transmission medium,
wireless information is susceptible to eavesdropping.
Thus, the secure communication is a critical issue for wireless
systems. Recently, physical layer secure communications for
untrusted multiple-input multiple-output (MIMO) relay sys-
tems are investigated, where the relay is untrusted and cast
as an eavesdropper [1]–[5]. In [1], the authors propose a joint
source and relay secure beamforming scheme to maximize
the achievable secrecy rate and provide asymptotic analysis of
secrecy rate. Considering no direct link between the source and
Copyright (c) 2015 IEEE. Personal use of this material is permitted.
However, permission to use this material for any other purposes must be
obtained from the IEEE by sending a request to pubs-permissions@ieee.org.
This work was supported in part by the National Natural Science Foundation
of China (No. 61672549, No. 61472458), in part by the Guangdong Natu-
ral Science Foundation (No. 2014A030311009, No. 2014A030310374, No.
2014A030311032, No. 2014A030313111, No. 2015A030310264), in part by
the Department of Education of Guangdong Province (No. 2016KZDXM050),
and in part by the Guangzhou Science and Technology Program (No.
201607010098).
Q. Li is with the School of Data and Computer Science, Sun Yat-
sen University, Guangzhou 510006, China, also with the Guangdong Key
Laboratory of Information Security, Guangzhou 510006, China, and also
with the Collaborative Innovation Center of High Performance Computing,
National University of Defense Technology, Changsha 410000, China (e-mail:
liquanzh@mail.sysu.edu.cn).
L. Yang is with the Department of Communication Engineering,
Guangdong University of Technology, Guangzhou 510006, China (e-mail:
liangyang.guangzhou@gmail.com).
Q. Zhang and J. Qin are with the School of Electronics and Information
Technology, Sun Yat-sen University, Guangzhou 510006, China (e-mail:
zhqi26@mail.sysu.edu.cn, issqjy@mail.sysu.edu.cn). J. Qin is also with the
Xinhua College, Sun Yat-sen University, Guangzhou 510520, China.
destination, the authors in [2] propose a joint destination-aided
cooperative jamming and precoding at both the source and
relay scheme. Apart from one-way relaying [1], [2], untrusted
MIMO two-way relay systems are also studied in [3]–[5]. In
[3], jointly optimizing the source and relay beamformers for
maximizing the secrecy sum rate is studied for the two-way
communications. In [4], a physical-layer secret key generation
scheme is proposed, which is based on zero forcing (ZF) and
minimum mean square error (MMSE) channel estimators. In
[5], a technique of the direction rotation alignment is devel-
oped to achieve transmission efficiency as well as security for
MIMO two-way relay channels.
All the above-mentioned works consider perfect channel
state information (CSI). However, in practice, perfect CSI is
hard to obtain due to many reasons, such as channel estimation
and quantization errors [6]. In this paper, we consider the case
of imperfect CSI and study the same system setting as [1], [2],
which consists of one source, one relay and one destination, all
with multi-antennas, where the relay is untrusted and willing
to help forwarding multiple data streams from the source to
destination. We model the imperfect CSI by using the worst-
case model [6]. To enhance the secure communications, we
propose a scheme of joint artificial noise (AN) and signal
precoding at the source and the precoding at the relay, with
the objective of maximizing the worst-case secrecy rate under
robust transmit power constraints at the source and relay.
However, the robust1secure precoding problem is non-convex
and challenging to solve. To deal with this challenge, we
propose the weighted minimum mean square error (WMMSE)
based method, where the sign-definiteness lemma is used to
eliminate the channel uncertainties and an effective iterative
optimization algorithm is developed.
The secure communications over conventional wireless net-
works are considered in [7]–[13], which are different from our
work. The works [7]–[9] focus on point-to-point communica-
tion networks, while the works [10]–[13] study single-antenna
one/two-way relay networks. All the works [7], [8], [10]–
[13] consider perfect CSIs. The robust design with Gaussian
uncertainties is found in [14].
Notations: The AT,A∗,A†,∥A∥,|A|, and tr(A)denote
the transpose, conjugate, conjugate transpose, Frobenius norm,
determinant and trace of the matrix A, respectively.
II. SYSTEM MO DE L
Consider a three node MIMO untrusted relay system as in
[1], [2], where a source, a relay, and a destination is equipped
1The word “robust” refers to reducing the performance degradation by
taking the channel errors into account.
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 2
with M,Nand Lantennas, respectively. Denote the channels
from source to relay, from source to destination, and from relay
to destination as Hsr ∈CN×M,Hsd ∈CL×M, and Hrd ∈
CL×N, respectively. The relay is considered to be untrusted,
which may decode the confidential message of the source,
and thus is regarded as a potential eavesdropper [1], [2]. Note
that the direct signal transmission from source to destination
provides a spatial copy of the source signal, and thus the joint
consideration of the direct link and the relay link transmission
is able to give incremental performance gain, especially when
the direct link is good [1], [3], [6].
The amplify-and-forward (AF) transmission scheme is em-
ployed. Note that the decode-and-forward (DF) relay strategy
is not applicable here since the relay is untrusted and not
expected to decode the received signal from the source. In
the first time slot, the source transmits confidential signal
x∈CM×1and AN z∈CM×1simultaneously. Then the
received signal at the relay and destination is respectively
yr=Hsr(Wx +Vz) + nr,(1)
yd,1=Hsd(Wx +Vz) + nd,1,(2)
where x,z∈ CN (0,I),W,V∈CM×Mis the corre-
sponding precoding matrix for confidential signal and AN,
nr∈ CN (0, σ 2
rI)and nd,1∈ CN (0, σ 2
d,1I)is the additive
Gaussian noise at the relay and destination, respectively.
The transmit power of the source is constrained to PSas
Ps,∥W∥2+∥V∥2≤PS.
In the second time slot, the relay multiplies the received
signal with a precoding matrix F∈CN×Nand forwards it to
the destination. The received signal at the destination is
yd,2=HrdFHsr (Wx +Vz) + Hrd Fnr+nd,2(3)
where nd,2∈ CN (0, σ 2
d,2I)is the additive Gaussian noise at
the destination. The transmit power of the relay is constrained
to PRas Pr,∥FHsrW∥2+∥FHsr V∥2+σ2
r∥F∥2≤PR.
From (1), the achievable rate at the relay is
Rr= 0.5 log |I+W†H†
srZ−1Hsr WW|(4)
where Z=σ2
rI+HsrVV†H†
sr.
Based on (2) and (3), the received signal at the destination
can be rewritten as
yd=yd,1
yd,2=HsdW
HrdFHsr Wx+˜
nd,1
˜
nd,2(5)
where ˜
nd,1=HsdVz +nd,1and ˜
nd,2=HrdFHsr Vz +
HrdFnr+nd,2. Thus, the achievable rate at the destination is
Rd=0.5 log |I+W†H†
sd(σ2
d,1I+HsdVV†H†
sd)−1
×HsdW+ (Hrd FHsr W)†Q−1HrdFHsr W|(6)
where Q=HrdFHsr V(Hrd FHsr V)†+σ2
rHrdF(Hr dF)†+
σ2
d,2I, and the derivation of (6) can be found in Appendix A.
To characterize the imperfect CSI, we adopt the worse-case
CSI model as in [6], i.e., the actual Hij takes the form as
Hij ={Hij |Hij =ˆ
Hij +∆ij ,||∆ij || ≤ ϵij }(7)
where ij ∈ {sr, rd, sd},ˆ
Hij represents the mismatched chan-
nel obtained by means of channel estimation or quantization,
and ∆ij denotes the channel uncertainty lying in a spectral
norm bounded uncertainty region with a given radius ϵij.
III. RO BU ST SE CU RE PR EC OD IN G
Using (4) and (6)-(7), the robust secure precoding problem
for an AF MIMO untrusted relay system is formulated as
max
F,W,Vmin
Hij ∈Hij
[Rd−Rr]+(8a)
s.t. Ps≤PS,(8b)
Pr≤PR,∀Hsr ∈ Hsr.(8c)
The robust precoding problem (8) is hard to solve because
the objective with log |·| functions is of high non-linearity. To
overcome this non-linearity, we employ the WMMSE based
method [15], which can transform the achievable rate into a
WMMSE problem by introducing some auxiliary variables.
To apply the WMMSE based method, we first introduce the
following result.
Lemma 1 [15]: Define the MSE matrix
M,(DHT −I)(DHT −I)†+DRD†(9)
where R≻0. Then we have
−log |M| = max
S≻0log |S| − tr(SM) + tr(I),(10)
log |I+ (HT)†R−1HT|= max
S≻0,Dlog |S| − tr(SM) + tr(I).
Using the linear equalizer Ddfor the destination to recover
the transmit signal x, i.e., ˆ
x=Ddyd, the MSE matrix Mdat
the destination is given by
Md=E[(ˆ
x−x)(ˆ
x−x)†]
= (D1HsdW+D2˜
FW −I)(D1HsdW+D2˜
FW −I)†
+D1HsdV(D1Hsd V)†+σ2
d,1D1D†
1+σ2
d,2D2D†
2
+D2˜
FV(D2˜
FV)†+σ2
rD2HrdF(D2Hr dF)†(11)
where ˜
F=HrdFHsr ,Dd= [D1D2].
Applying Lemma 1, we have
2Rd= max
Sd≻0,Dd
log |Sd| − tr(SdMd) + tr(I).(12)
For Rr, we first rewrite it as
Rr=0.5 log |I+σ−2
rHsr(WW†+VV†)H†
sr|
,Rr,1
−0.5 log |I+σ−2
rHsrVV†H†
sr|
,Rr,2
.(13)
Applying Lemma 1, we can rewrite Rr,1and Rr,2as
−Rr,1= max
Sr≻0−tr(Sr(I+σ−2
rHsr(WW†+VV†)H†
sr))
+ log |Sr|+ tr(I),(14)
Rr,2= max
Ss≻0,Ds
log |Ss| − tr(SsMs) + tr(I)(15)
where Ms= (DsHsrV−I)(DsHsr V−I)†+σ2
rDsD†
s.
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 3
Inserting (12)-(15) into (8), we rewrite problem (8) as
max
F,W,V,Dd,Ds,Si≻0min
Hij ∈Hij
log |Sd|
−tr(SdMd) + log |Ss| − tr(SsMs) + log |Sr|
−tr(Sr(I+σ−2
rHsr(WW†+VV†)H†
sr)) (16a)
s.t.(8b),(8c).(16b)
Introducing the slack variables βi,i∈ {s, r, d}, we can further
rewrite problem (16) as
max
F,W,V,Dd,Ds,Si≻0,βi
log |Sd|+ log |Ss|
+ log |Sr| − βs−βr−βd(17a)
s.t.tr(SdMd)≤βd,∀Hij ∈ Hij , ij ∈ {sr, rd, sd}(17b)
tr(SsMs)≤βs,∀Hsr ∈ Hsr (17c)
tr(Sr(I+σ−2
rHsr(WW†+VV†)H†
sr)) ≤βr,
∀Hsr ∈ Hsr,(17d)
(8b),(8c).(17e)
Although the objective in (17) is convex, the problem (17)
is still intractable due to the semi-infinite constraints. In the
following, we focus on these semi-infinite constraints and try
to eliminate them.
First, we deal with (17b). To proceed, we rewrite the item
tr(SdMd)in (17b) as
tr(SdMd) =
vec(Rd(D1HsdW+D2˜
FW)−Rd)
vec(RdD1HsdV)
vec(σrRdD2HrdF)
vec(RdD2˜
FV)
vec(σd,1RdD1)
vec(σd,2RdD2)
2
.
(18)
where Rd,S
1
2
dand tr(AA†) = ∥vec(A)∥2is used [17].
After inserting (7) into the above equation, and neglect-
ing higher order uncertainty terms, it is possible to recast
tr(SdMd)as tr(SdMd) = ∥ϕd∥2where
ϕd,ˆ
ϕd+Ωrdvec(∆rd ) +
j=r,d
Ωsj vec(∆sj)
,∆d
,(19)
and
ˆ
ϕd=
vec(Rd(D1ˆ
HsdW+D2ˆ
FW)−Rd)
vec(RdD1ˆ
HsdV)
vec(σrRdD2ˆ
HrdF)
vec(RdD2ˆ
FV)
vec(σd,1RdD1)
vec(σd,2RdD2)
(20)
with ˆ
F=ˆ
HrdFˆ
Hsr and subsequently
Ωrd =
(Fˆ
HsrW)T⊗(RdD2)
0
FT⊗(σrRdD2)
(Fˆ
HsrV)T⊗(RdD2)
0
∈Cm×NL ,(21)
Ωsr =
WT⊗(RdD2ˆ
HrdF)
0
VT⊗(RdD2ˆ
HrdF)
0
∈Cm×MN ,(22)
Ωsd =
WT⊗(RdD1)
VT⊗(RdD1)
0
∈Cm×ML ,(23)
where m=N L + 2M L + 3L2and the identity vec(ABC) =
(CT⊗A)vec(B)[17] is used.
Using the Schur complement lemma [17], we can rewrite
the constraint (17b) as
βdˆ
ϕ†
d
ˆ
ϕdI≽ − 0(∆d)†
∆d0.(24)
The constraint (24) still contains the uncertainty. To eliminate
this uncertainty, we need the following sign-definiteness lem-
ma.
Lemma 2 [6]: Given a Hermitian matrix Aand arbitrary ma-
trices {Pi,Qi}N
i=1, the semi-infinite Linear Matrix Inequality
(LMI) of the form
A≽
N
i=1
(P†
iYiQi+Q†
iY†
iPi),∀Yi:∥Yi∥ ≤ ϵi(25)
holds if and only if there exist nonnegative real numbers
λ1, ..., λNsuch that
A−N
i=1 λiQ†
iQi−ϵ1P†
1· · · −ϵNP†
N
−ϵ1P1λ1I· · · 0
.
.
..
.
.....
.
.
−ϵNPN0· · · λNI
≽0.
(26)
Using Lemma 2, and by appropriately choosing its param-
eters as follows
Ad=βdˆ
ϕ†
d
ˆ
ϕdI,Qd1=Qd2=Qd3= [−10](27)
Pd1=0 Ω†
rd,Pd2=0 Ω†
sr,Pd3=0 Ω†
sd,(28)
we can rewrite the constraint (17b) as the following LMI:
βd−3
j=1 λdj ˆ
ϕ†
d
ˆ
ϕdIΘ†
d
Θddiag({λdjI}3
j=1)
≽0,
(29)
where Θd=−[ϵrdPT
d1, ϵsrPT
d2, ϵsdPT
d3]T.
Similarly, we can transform (17c), (17d), and (8c) to the
following corresponding LMIs:
βs−λsˆ
ϕ†
s
ˆ
ϕsIΘ†
s
ΘsλsI
≽0,(30)
βr−λrˆ
ϕ†
r
ˆ
ϕrIΘ†
r
ΘrλrI
≽0,(31)
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 4
PR−λtˆ
ϕ†
t
ˆ
ϕtIΘ†
t
ΘtλtI
≽0,(32)
where
ˆ
ϕs= [vec†(RsDsˆ
HsrV−Rs) vec†(σrRsDs)]†,(33)
ˆ
ϕr= [vec†(σ−1
rRrˆ
HsrW) vec†(σ−1
rRrˆ
HsrV) vec†(Rr)]†,
(34)
ˆ
ϕt= [vec†(Fˆ
HsrW) vec†(Fˆ
HsrV) vec†(σrF)]†,(35)
Ωs= [(VT⊗(RsDs))†0]†,(36)
Ωr= [(WT⊗(σrRr))†(VT⊗(σrRr))†0]†,(37)
Ωt= [(WT⊗F)†(VT⊗F)†0]†,(38)
Θj=−ϵsr[0 Ω†
j],Rj=S
1
2
j, j =s, r, t. (39)
Based on (29)-(32), we rewrite the robust problem (17) as
max
F,W,V,Dd,Ds,Ri≻0,βi,λi≥02 log |Rd|+ 2 log |Rs|
+ 2 log |Rr| − βs−βr−βd(40a)
s.t.(29) −(32),(8b).(40b)
The problem (40) is no longer semi-infinite, but it is still
non-convex. We note that with fixing a subset of the variables,
problem (40) is convex with respective to F[16]. Similar
results are observed for Ri,Di,Wand V. Then we employ
alternating optimization method to solve problem (40), which
is summarized in Algorithm 1.
Algorithm 1 The Proposed WMMSE Based Iterative Algo-
rithm to Solve Robust Secure Precoding Problem (8)
1: Initialization: set l= 0,W=W(0),V=V(0) ,F=
F(0), and an accuracy parameter ε;
2: repeat
3: l:= l+ 1;
4: Solve (40) to update Diwith the other parameters fixed;
5: Solve (40) to update Rifor fixed Difound in the
previous step;
6: Solve (40) to update Ffor fixed Diand Rifound in
the previous step;
7: Solve (40) to update Vand Wfor fixed F,Diand Ri
found in the previous step;
8: until |z(l)−z(l−1)| ≤ εwhere z(l)is the objective value
in the lth iteration.
Remark 1 (Convergence Analysis): Updating the variables
will only increase or maintain the objective value, thus we
obtain a monotonically increasing sequence of the objective
values of problem (40) which has an upper bound since the
transmit power is limited. Therefore, Algorithm 1 converges.
Remark 2 (Computational Complexity): The computational
complexity of Algorithm 1 is mainly from solving problem
(40), which can be solved using the interior-point method
[16] with the complexity of O(n3.5log(1/ε)) where nis the
number of real optimization variables and εis a given solution
accuracy. Thus, the computational complexity of Algorithm 1
5 10 15 20 25
Iterations
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
Average Worst-Case Secrecy Rate (bps/Hz)
dsr=1.1
dsr=0.7
Fig. 1. Average worst-case secrecy rate by the WMMSE based iterative
algorithm versus the number of iterations.
is about
Oliter 4M2+ 6N2+ 6L2+ 53.5log(1/ε)(41)
where liter is the iterative number for the convergence of
Algorithm 1.
IV. SIMULATION RESULTS
In this section, we present the computer simulation results
of our proposed robust secure precoding scheme. We assume
that the elements of the channel responses ˆ
Hsr,ˆ
Hrd, and ˆ
Hsd
are independent and identically distributed (i.i.d.) complex
Gaussian random variables with zero mean and variances d−n
sr ,
d−n
rd , and d−n
sd , respectively, where dsr,drd, and dsd denote the
distance from the source to relay, from relay to destiantion, and
from source to destinatino, respectively, and nis the pathloss
exponent setting to be 4. We also assume that all nodes lies in
a straight line and the distances are normalized by dsd = 1 as
in [1]. For simplicity, the noise powers and channel uncertainty
radius are summed to be the same, i.e., σ2
r=σ2
d,1=σ2
d,2=σ2
and ϵsr =ϵrd =ϵsd =ϵ.
In Fig. 1, we present the average worst-case secrecy rate
achieved by the WMMSE based iterative algorithm versus the
number of iterations, where PS/σ2=PR/σ2=15 dB, ϵ=
0.05,dsr = 0.7,1.1, and M=N=L= 3. The initial
point of the WMMSE based iterative algorithm is generated
randomly to satisfy the constraints in (8). It is observed from
Fig. 1 that it takes about 15 iterations for the WMMSE based
iterative algorithm to converge.
In Fig. 2, we present the average worst-case secrecy rate
versus the distance dsr, where PS/σ2=PR/σ2=15 dB,
ϵ= 0.05, and M=N=L= 3. We compare three schemes
including our proposed robust secure precoding scheme (de-
noted as “With AN”), our proposed robust secure precoding
scheme without AN (denoted as “No AN”), and the proposed
secure precoding scheme with perfect CSI (denoted as “Perfect
CSI”). From Fig. 2, we can see that the average worst-case
secrecy rate increases with the distance dsr increasing, because
that the source-relay link will experience a larger path loss.
We can also observe from Fig. 2 that the “With AN” scheme
performs better than the “No AN” scheme, while the “Perfect
CSI” scheme has the best performance.
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 5
0.2 0.4 0.6 0.8 1 1.2 1.4
dsr
0
0.5
1
1.5
2
2.5
3
3.5
4
Average Worst-Case Secrecy Rate (bps/Hz)
Perfect CSI
With AN
No AN
Fig. 2. Average worst-case secrecy rate versus dsr; comparison of our
proposed “With AN” scheme, “No AN” scheme and “Perfect CSI” scheme.
0.02 0.04 0.06 0.08 0.1 0.12 0.14
ǫ
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Average Worst-Case Secrecy Rate (bps/Hz)
Perfect CSI
With AN
No AN
Fig. 3. Average worst-case secrecy rate versus ϵ; comparison of our proposed
“With AN” scheme, “No AN” scheme and “Perfect CSI” scheme.
In Fig. 3, we present the average worst-case secrecy rate
versus channel error radius ϵ, where PS/σ2=PR/σ2=15 dB,
dsr = 0.7, and M=N=L= 3. From Fig. 3, we can see
that the average worst-case secrecy rates by all the schemes
decrease as ϵincreases. The performance gab between the
“With AN”, “No AN” scheme and the “Perfect CSI” scheme
becomes large when ϵgrows.
V. CONCLUSIONS
In this correspondence, we have proposed the robust AN-
aided secure precoding for an AF MIMO untrusted relay
system, where the relay is untrusted and the worst-case C-
SI model is considered. We have developed the WMMSE
based iterative algorithm to solve the robust secure precoding
problem, during each iteration several convex optimization
problems are solved. Simulation results have shown that the
robust AN-aided secure precoding outperforms the scheme
without AN.
APP EN DI X A
THE DE RI VATI ON OF (6)
From (5), the equivalent channel matrix is
Heq =HsdW
HrdFHsr W,(42)
and the equivalent covariance matrix of noises is given by
Qeq =σ2
d,1I+HsdVV†H†
sd 0
0 Q .(43)
Thus, the achievable rate at the destination in the second
time slot is [15]
Rd=0.5 log |I+H†
eqQ−1
eq Heq|
=0.5 log |I+W†H†
sd(σ2
d,1I+HsdVV†H†
sd)−1
×HsdW+ (Hrd FHsr W)†Q−1HrdFHsr W|.(44)
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