Power-Constrained Secrecy Rate Maximization for Joint Relay and Jammer Selection Assisted Wireless Networks

0.75 MB·

by  Haiyan Guo

IEEETRANSACTIONSONCOMMUNICATIONS(ACCEPTEDTOAPPEAR) 1
Power-Constrained Secrecy Rate Maximization for
Joint Relay and Jammer Selection Assisted Wireless
Networks
Haiyan Guo, Zhen Yang, Linghua Zhang, Jia Zhu, and Yulong Zou, Senior Member, IEEE
Abstract—Inthispaper,weexaminethephysicallayersecurity I. INTRODUCTION
7 for cooperative wireless networks with multiple intermediate
1 nodes,wherethedecode-and-forward(DF)protocolisconsidered. DUE to the broadcast nature of wireless medium, the
0 We propose a new joint relay and jammer selection (JRJS)
2 scheme for protecting wireless communications against eaves- confidentialityof wirelesscommunicationsisvulnerable
dropping,whereanintermediatenodeisselectedastherelayfor to an eavesdropping attack [1]. Physical layer security is
n
the sake of forwarding the source signal to the destination and emergingas a promising paradigmagainst eavesdroppingand
a
J meanwhile, the remaining intermediate nodes are employed to has attracted considerable attention recently. Different from
act as friendly jammers which broadcast the artificial noise for
7 the conventional encryption methods [2], it mainly exploits
disturbing the eavesdropper. We further investigate the power
the physical characteristics of wireless channels to enhance
allocation among the source, relay and friendly jammers for
T] maximizing the secrecy rate of proposed JRJS scheme and the security of the signal transmission from its source to
derive a closed-form sub-optimal solution. Specificially, all the the intended destination. Physical layer security work was
I
. intermediate nodes which successfully decode the source signal pioneered by Wyner for a discrete memoryless wiretap
s
areconsideredasrelaycandidates.Foreachcandidate,wederive
c channel [3], which later on, was extended to a Gaussian
[ thesub-optimalclosed-formpowerallocationsolutionandobtain wiretap channel in [4]. The secrecy capacity which is the
thesecrecyrate resultof thecorrespondingJRJS scheme. Then,

1 the candidate which is capable of achieving the highest secrecy maximum of secrecy rate was developed to evaluate the
v rate is selected as the relay. Two assumptions about the channel physical-layersecurity performance.In [4], it showed thatthe
0 state information (CSI), namely the full CSI (FCSI) and partial secrecy capacity can be expressed as the difference between
2 CSI (PCSI), are considered. Simulation results show that the
the channel capacity from source to destination (referred to
8 proposed JRJS scheme outperforms the conventional pure relay
as main channel) and the channel capacity from source to
1 selection,purejammingandGSVDbasedbeamformingschemes
0 in terms of secrecy rate. Additionally, the proposed FCSI based eavesdropper (called wiretap channel) in the presence of an
. power allocation (FCSI-PA) and PCSI based power allocation eavesdropper.
1
(PCSI-PA) schemes both achieve higher secrecy rates than the Multiple-input multiple-output (MIMO) [5]-[11] has been
0
equal power allocation (EPA) scheme.
7 widely considered as an effective way to improve physical
1 Index Terms—Physical layer security, relay selection, jammer layer security by exploiting multiple antennas at both the
v: selection, eavesdropping, secrecy capacity. transmitterandthereceiver.However,duetothebatterypower
i and terminalsize limitations, it may be difficult to implement
X
multiple antennas in some cases e.g. handheld terminals
Manuscript received July 27,2016; revised November07, 2016; accepted
r
a December 31, 2016. The editor coordinating the review of this paper and and wireless sensors. As a consequence, node cooperation is
approving itforpublication wasProf.J.Yuan. considered to be an effective means of enhancing the secrecy
This work was partially supported bythe National Natural Science Foun-
capacity by allowing the network nodes to share each other’s
dation of China (Grant Nos. 61302152, 61302104, 61401223, 61522109
and 61671252), the Major Science Research Project of Jiangsu Provin- antennas [12]-[31]. Generally, there are two efficient ways
cial Education Department (Grant Nos. 13KJA510003, 14KJA510003 and to make use of multiple intermediate nodes: cooperative
15KJA510003), the Natural Science Foundation of Jiangsu Province (Grant
beamforming and cooperative jamming. To be specific,
Nos. BK20140887 and BK20150040), the Priority Academic Program De-
velopment of Jiangsu Higher Education Institutions (PAPD) and Research cooperative bearmforming helps to improve the capacity
FoundationforAdvancedTalents,NanjingUniversityofPostsandTelecom- of main channel, which may include relay selection and
munications (No.NY215020)
beamforming [12]-[18]. By contrast, cooperative jamming
H.GuoandL.ZhangarewiththeKeyLaboratoryofBroadbandWireless
CommunicationandSensorNetworkTechnology(MinistryofEducation)and aims to degrade the capacity of the wiretap channel by
theCollegeofInternetofThings,NanjingUniversityofPostsandTelecommu- sending artificial noise to interfere with the eavesdropper
nications, Nanjing210003,China(e-mail:{guohy,zhanglh}@njupt.edu.cn).
without affecting the legitimate destination [19]-[21]. Both
Z. Yang is with the Key Laboratory of Broadband Wireless Communi-
cation and Sensor Network Technology (Ministry of Education), Nanjing cooperative beamforming and jamming are capable of
UniversityofPostsandTelecommunications,Nanjing210003,China(e-mail: increasing the secrecy capacity of wireless communications
[email protected]).
in the presence of an eavesdropper.
Y. Zou and J. Zhu are with the Key Laboratory of Broadband Wireless
Communication and Sensor Network Technology (Ministry of Education) In [12] and [13], the authors studied the cooperative
andtheSchoolofTelecommunicationsandInformationEngineering,Nanjing beamforming and cooperative jamming separately, where all
UniversityofPostsandTelecommunications,Nanjing210003,China(e-mail:
the intermediate nodes are used either to assist the source
{yulong.zou, jiazhu}@njupt.edu.cn).
Corresponding authorisYulongZou. transmission or to send a jamming signal for interfering with
IEEETRANSACTIONSONCOMMUNICATIONS(ACCEPTEDTOAPPEAR) 2
the eavesdropper. The transmit power at the source and the in [31], where the power allocation between the confidential
weights of intermediate nodes are determined to maximize signal and jamming signals is optimized to maximize the
the secrecy rate subject to a total power constraint. In [14], ergodic secrecy rate.
an optimal relay selection scheme was proposed by taking In this paper, we propose a new JRJS scheme to enhance
into account the channel state information (CSI) of both the physical layer security. Specifically, an intermediate node
the main channel and wiretap channel, which is shown to which can successfully decode the source signal is selected
achieve a better security performance than the conventional to act as the relay for assisting the source transmission, while
relay selection and combining methods. Almost meanwhile, the remaining intermediate nodes are employed as friendly
cooperative jamming was also studied for increasing the jammerstotransmitnull-steeringartificialnoiseforconfusing
secrecy rate. In [19], optimal cooperative jamming was the eavesdropper without affecting the legitimate destination.
studied with an emphasis on how to guarantee a positive We select one intermediate node as the relay to exploit the
secrecy rate. In [20], the use of a jammer was proposed full spatial diversity and the others as jammers for the sake
to send artificial noise along with confidential signal for of reducing the complexity of JRJS without an extra jammer
confusing the wiretap link and enhancing the main link selection. In addition, if we select multiple intermediate
simultaneously. nodes as the relays to help signal transmission in different
The aforementioned studies are based on the ideal channels, there would be more links where the eavesdropper
assumption that the instantaneous CSI of both main and can tap the legitimite transmission. Moreover, if the multiple
wiretap links are known. Considering that obtaining the relays share a common channel to forward their re-encoded
instantaneous CSI of the wiretap channel is challenging in outcomes, the corresponding symbol-level synchronization
many cases, Y. Su et al. [22] investigated the relay selection among the spatially distributed relays is very complex.
andpowerallocationbyassumingthatthecorrelationbetween Consideringthatthesecrecyperformancewouldbeaffected
the real and outdated CSIs is known without requiring the bythepowerallocationamongthesource,relayandjammers,
instantaneous CSI. In [23], X. Gong et al. presented a robust we furtherformulatea powerallocation optimizationproblem
relay beamforming scheme which maximizes the worst-case to maximize the secrecy rate of the proposed JRJS scheme
transmit power of the artificial noise with imperfect CSI. subject to a total power constraint and derive a closed-form
Moreover, in [24], F. S. Al-Qahtani et al. studied the impact sub-optimalpowerallocationsolution.Itisnotedthatthetotal
of CSI feedback delay on the secrecy performance of three power constraint is widely used in literature e.g., [12]-[13],
different diversity combining methods under the imperfect [15]-[17], [21], [25] and [30]-[31]. In our proposed JRJS
CSI case. In addition, C. Wang et al. [25] investigated the scheme, the relay selection and power allocation are jointly
relay selection and power allocation between two sources optimized by taking into account both the transimission
and mulitiple relays for a two-way amplify-and-forward(AF) rate and secrecy rate. Since the instantaneous CSI may be
wireless network based on statistical channel information. unavailable in some practical scenarios, we consider two
In order to take both advantages of cooperative different CSI cases in this paper, which are called the full
beamforming and cooperative jamming, various joint CSI (FCSI) and partial CSI (PCSI), respectively. The FCSI
relay selection and jamming (JRJS) schemes were proposed assumes that the instantaneousCSI of both main channel and
forfurtherenhancingthewirelesssecrecyperformance,where wiretap channel is known, while the PCSI only requires the
some intermediate nodes help the legitimate transmission main channel’s instantaneous CSI along with the statistical
while some other nodes may jam the eavesdropper [26]-[31]. wiretap channel knowledge.
More specifically, the authors of [26] and [27] selected The main contributions of our work are summarized as
one intermediate node as the relay and two other nodes follows.
as jammers in a two-way cooperative network, achieving a Firstly,the relayandjammerselection aswellas the power
better secrecy performance than conventional non-jamming allocation are jointly carried out to maximize the secrecy
schemes. Two different channel knowledge assumptions rate of legitimate transmission. To be specific, for each relay
were considered in [27]: 1) an instantaneous CSI of the candidate, we derive an optimal power allocation solution
eavesdropping channel is assumed to be known, and 2) only and obtain the corresponding secrecy rate result. After that,
an average CSI of the wiretap channelis known. By contrast, the candidate that maximizes the secrecy rate is selected as
in [28] and [29], one intermediate node was chosen as the the relay. Secondly, considering that the secrecy performance
jammer to send artificial noise and the other intermediate would be affected by the power allocation among the source,
nodes were considered to be relays for data transmission. relay and jammers, we formulate a power allocation problem
A second-order convex cone programming algorithm was for maximizing the secrecy rate of proposed JRJS scheme
proposed in [29] to obtain sub-optimal beamforming weights subject to a total power constraint. Moreover, we derive
and power allocation without eavesdropper’s CSI. In [30], closed-form sub-optimal solutions to the formulated power
an intermediate node that achieves the highest instantaneous allocation problem under FCSI and PCSI assumptions,
signal-to-noiseratio (SNR) of the second hop was selected as respectively.
the relay and a node of minimizing the interference imposed The remainder of this paper is organized as follows. In
onthe destinationwas usedas the jammer,wherean outdated Section II, the system model and secrecy rate analysis are
estimated CSI is assumed. Additionally, an opportunistic presented. In Section III, we derive the power allocation
relaying scheme with the artificial jamming was proposed schemes under the FCSI and PCSI assumptions, respectively.
IEEETRANSACTIONSONCOMMUNICATIONS(ACCEPTEDTOAPPEAR) 3
Numerical evaluation results are described in Section IV and Section III.
main conclusions are drawn in Section V. Inthefirstphase,thesourcetransmitsasignals(E(s2)=
| |
II. SYSTEMMODELAND SECRECY RATEANALYSIS
A. System Model
We consider a decode-and-forward (DF) cooperative wire-
less network consisting of one source S, one destination D
andM intermediatenodesin the presenceof aneavesdropper
E as shown in Fig. 1. Each node in the whole network is
only equipped with a single antenna. The solid and dash
lines represent the main and wiretap links, respectively. Both
the main and wiretap links are modeled as Rayeigh fading
channelsandthethermalnoisereceivedatanynodeismodeled
as a complex Gaussian random variable with zero mean and
variance σ2 . We assume that there is a direct link from the
n
source to the eavesdropperand no direct link from the source Fig. 1. A cooperative wireless network consisting of one source, one
tothedestination.M intermediatenodesareexploitedtoassist destination, oneeavesdropper andMintermediate nodes.
the signal transmission from the source to the destination.
In this paper, we propose a new JRJS scheme, where the 1) with power Ps. The intermediate node i and eavesdropper
intermediatenodesaredividedintotwogroups:onerelaynode E thus receive
R and M 1 jammers Ji,i = 1,2,...,M 1, as shown yi = Pshsis+ni (1)
− −
in Fig. 1. More specifically, in the first phase, the source
node broadcasts a signal s with power Ps to M intermediate ye(1) = pPshses+n(e1) (2)
nodes, which all attempt to decode their received signals.
whereh andh represenptafadingcoefficientofthechannel
Due to the broadcasting nature, the signal s is also received si se
fromthesourcetotheintermediatenodeiandeavesdropperE,
by the eavesdropper. For convenience,the intermediate nodes
respectively,n (0,δ2) is additive white Gaussian noise
which succeed in decoding the source signal are denoted (AWGN) at thei ∈intCeNrmediante node i, and n(1) (0,δ2) is
by a decoding set . In the second phase, we select one e ∈CN n
D AWGN at E.
intermediate node from as the relay R to re-encode and
D In the second phase, the relay R is selected from the
transmit its decoded outcome to the destination with power
decodingset forsignal transmissionwith powerP . Mean-
Penra.bMleedantowahciltea,sthferierenmdlayinjianmgmMers−J1,iin=term1,e2d,i.a.t.e,Mnodes1,atroe while, each jaDmmer Ji transmits an artificial noise srignal zi.
i − Itneedsto bepointedoutthatthevectorof theartificialnoise
transmit artificial noise z for confusingthe eavesdropper.We
i z = [z z ... z ]T should be normalized. The total
assumethatthedistancesamongintermediatenodesaremuch 1 2 M−1
transmitpoweratallthejammersisdenotedasP .Therefore,
smallerthanthedistancesbetweenintermediatenodesandthe z
the received signal at the destination is expressed as
source/destination/eavesdropperfollowing [30]. This assump-
tion is reasonable for both WSNs and MANETs associated
y = P h s+ P h z +n (3)
d r rd z jid i d
with a clustered relay configuration. Under this assumption,
i=1,2,...,M−1
thecorrespondingpathlossesamongtherelayandjammersare p X p
approximatelythe same, hence we simplify our work without where hrd represents a fading coefficientof the channel from
considering path loss. the relay R to destination, hjid representsa fading coefficient
In this paper, we assume that the whole wireless network of the channel from the jammer Ji to destination and nd
∈
is to operate under a total power constraint following [12]- CN(0,δn2) represents AWGN at the destination. The received
signal at the eavesdropper is expressed as
[13], [15]-[17], [21], [25] and [30]-[31], which is widely
adopted in both secrecy performance analysis and optimal
y(2) = P h s+ P h z +n(2) (4)
design.Forexample,asindicatedin[17],halfofthemaximum e r re z jie i e
i=1,2,...,M−1
power budget is allocated to the two transceivers and the p X p
remaining half will be shared among all the relay nodes for where h represents a fading coefficient of the channel from
re
symmetricrelayingschemes.In[30],itpointedoutthesecrecy the relay to eavesdropper, h represents a fading coefficient
jie
performanceiseffectedbypowerallocationbetweentherelay of the channel from the jammer J to eavesdropper and
i
and jammers. Futhermore, from a network design of view, n(2) (0,δ2) represents AWGN at the eavesdropper.
such an assumption allows to optimize total power consumed eFo∈r CNsimplnicity, let us define h =
d
in the whole network. In addition, the solved transmit power [h h ... h ]H and h =
j1d j2d jM−1d e
at the relay also provides a guideline for the transmit power [h h ... h ]H. Considering that the artificial
j1e j2e jM−1e
of individual relays. The optimal power allocation among the noise should not interfere with D, we design z as a
source, relay and jammers will be addressed in following normalized signal onto the null space of h so that
d
IEEETRANSACTIONSONCOMMUNICATIONS(ACCEPTEDTOAPPEAR) 4
hHz = h z = 0. As a consequence, we and the instantaneous SNR at eavesdropper as a function of
d i=1,2,...,M−1 jid i
obtain (3) and (4) as R, P , P and P
s r z
P
yd = Prhrds+nd (5) γ (R,P ,P ,P )= |hse|2Ps + |hre|2Pr . (14)
e s r z δ2 δ2 +P hHzzHh
and p n n z e e
y(2) = P h s+ P hHz+n(2) (6) Combining(11)-(14),thesecrecyrateofthesource-destination
e r re z e e
transmissionrelyingontheproposedJRJSschemeisgivenby
which completes thepsignal modpeling of the JRJS scheme.
1 1+γ (R,P ,P )
C =(C C )+ = log ( d s r )+ (15)
s d− e 2 2 1+γ (R,P ,P ,P )
B. Secrecy Rate Analysis e s r z
In[32],avariabletransmissionparameterschemewherethe where (.)+ is a function such that (x)+ = x if x > 0 and
source and relay use different codebooks with different code (x)+ =0 if x 0.
≤
rates is proposed for DF relay networks to further improve
the secrecy performance. In [33], a framework is provided
to determine the wiretap code rates to achieve the locally III. POWERALLOCATIONOF PROPOSED JRJS SCHEME
maximum effective secrecy throughput.In this paper, we still
A. FCSI based Power Allocation (FCSI-PA) Scheme
consider the case where the source and relay transmit signal
In this subsection, we focus on how to select the optimal
usingthe same codewiththe same coderate R in bothmain
d
relay R and allocate the optimal transmit power P , P and
andwiretaplinksandfocusonhowtoallocatetransmitpower s r
P to maximize the instantaneous secrecy rate stated as (15)
among the source, relay and all the jammers to maximize the z
assuming the avaibility of global instantaneous CSI. The as-
secrecyrateoftheproposedJRJSscheme.Althoughwedonot
sumption is reasonable in some cases [34]-[36].For example,
propose a new transmission parameter design scheme in this
we consider the scenario in which a legal user in the wireless
paper, we take into account the effect of R on the secrecy
d
network is captured by Trojan and slaved as an eavesdropper
rate and set R to different values based on the values of
d
P by experimental experience, which will be shown in the to tap the signal transmission. In this case, a broadcasting
δn2 network consisting of multiple legal receivers among which
following experimental section.
some are slaved by Trojan and become eavesdroppers, turns
WhenthesourceandrelaytransmitthesamecodewithR ,
d
into a wiretap system, where the global instantaneous CSI of
the rate of DF transmission from source S via the relay R to
theeavesdroppersmaybeavailable.Foranotherexample,ina
destination D, C , is obtained as
d
scenariowhereasingleactiveeavesdropperisregisteredinthe
Cd =min(Csr,Crd) (7) wirelessnetworkasasubscribeduserandexchangessignaling
messages with the source and intermediate nodes, the global
where C and C representthe rate of the link fromS to R
sr rd
instantaneous CSI of the eavesdropper may be known.
and the rate of the link from R to D, respectively, which are
Under the FCSI assumption, the joint relay selection and
given as
1 h 2P power allocation problem can be formulated as
C = log (1+ | sr| s) (8)
sr 2 2 δ2 1+γ (i,P ,P )
n (R∗,P∗,P∗,P∗) = arg max d s i
and s r z i,Ps,Pr,Pz 1+γe(i,Ps,Pi,Pz)
1 h 2P
C = log (1+ | rd| r). (9) s.t. Ps+Pr +Pz =P
rd 2 2 δn2 Ps 0,Pr 0,Pz 0 (16)
≥ ≥ ≥
It is noted that the relay R is selected from the decoding set log (1+ |hsi|2Ps) R
. That is, it satisfies that 2 δn2 ≥ d
D whereP isthe totalpowerbudgetfortransmittingonesource
h 2P
log (1+ | sr| s) R . (10) symbol, which is also the sum of the transmit power at the
2 δn2 ≥ d source, relay and jammers. Problem (16)can be performedin
Combining (7)-(9), the rate of the link from S to D is given two steps. At the first step, each intermediate node i in the
by decoding set is treated as a candidate of relay. For each
Cd = 12log2(1+ min(|hsr|2δP2s,|hrd|2Pr)). (11) pcaronbdliedmatetoi m∈DaxDim, iwzee tshoelvseectrheecyforlalotewoinfgthpeowcoerrreaslploocnadtiionng
n
JRJS scheme,
Combing (5) and (6), we obtain the rate of the link fromS
to E as (P∗,P∗,P∗) = arg max 1+γd(i,Ps,Pr)
C = 1log (1+ |hse|2Ps + |hre|2Pr ). (12) si ri zi Ps,Pr,Pz 1+γe(i,Ps,Pr,Pz)
e 2 2 δn2 δn2 +PzhHezzHhe s.t. Ps+Pr +Pz =P
For simplicity,let us define the instantaneousSNR at destina- Ps ≥0,Pr ≥0,Pz ≥0 (17)
tion as a function of the relay R, Ps and Pr log2(1+ |hsδi|n22Ps)≥Rd.
min(h 2P , h 2P ) Atthesecondstep,thecandidatewhichleadstothemaximum
γ (R,P ,P )= | sr| s | rd| r (13)
d s r δ2 of the secrecy rate results with the optimal power allocation
n
IEEETRANSACTIONSONCOMMUNICATIONS(ACCEPTEDTOAPPEAR) 5
solution above is selected as the relay R. That is, Then, the solution of problem (23) is
R∗ =argmax 1+γd(i,Ps∗i,Pr∗i) . (18) PsoRpt1 =arg max g1(Ps) (30)
i∈D 1+γe(i,Ps∗i,Pr∗i,Pz∗i) Ps∈{PspR-opt1,Pb1,Pb2}
Then, the optimal power allocation solution of the proposed Ps∈[Pb1,Pb2]
JRJS scheme is the solution of problem (17) when the candi- where x1,x2,...,xn denotes a set consisting of elements
{ }
date is R∗. x1,x2,...,xn.
Itis notedthatproblem(18)canbeeasily solvedbysimple Now we discuss the solution of problem(23) when τ2φ1
−
comparison. Therefore, we focus on how to solve problem τ1φ2 6= 0. Define q1(Pr) = (τ2φ1 − τ1φ2)Ps2 + 2(τ2 −
(17). Without loss of generality, we denote the candidate of τ1)θ1Ps+(φ1 φ2)θ1.Thediscriminantofq1(Pr)isobtained
−
relay as R. Let us define λ = hHzzHh . Then, due to as
e e e
P =P P P , problem (17) can be rewritten as (19).
zDue to− s− r ∆1 =(2(τ2−τ1)θ1)2−4(τ2φ1−τ1φ2)(φ1−φ2)θ1. (31)
min(|hsr|2Ps,|hrd|2Pr)= |hsrδn|22Ps if |hsr|2Ps ≤|hrd|2Pr, When ∆1 >0, the solutions of dgd1P(Pss) =0 are obtained as
δn2 δn2 (|hrdδn2|2Pr if |hsr|2Ps ≥|hrd|2Pr Pp-opt2 = −2(τ2−τ1)θ1+√∆1 (32)
(20) sR 2(τ φ τ φ )
2 1 1 2
−
(19) can be broken down into two sub-problems, which are
and
given by (21) and (22). 2(τ τ )θ √∆
As f1(Ps,Pr) is a decreasingfunctionof Pr, Pr should be PspR-opt3 = − 2(2τ−φ 1 1τ−φ ) 1. (33)
2 1 1 2
minimizedinordertomaximizef (P ,P ).Duetothecondi- −
1 s r
tion h 2P h 2P inproblem(21),wecaneasilyobtain Then, the solution of problem (23) is
sr s rd r
Pr ≥| ||hhrs|rd||22Ps≤. T| hus|, we set Pr = ||hhrsdr||22Ps to obtain a sub- PsoRpt2 =arg max g1(Ps). (34)
aonpdtimPal+soPlutionP,ofwperhoabvleemP(21). Co|hmrdb|2iningPP.rA=ddi||thhiorsdrn||a22lPlys, Ps∈{PsPpR-osp∈t2,[PPbsp1R-o,pPt3b,P2]b1,Pb2}
s r ≤ s ≤ |hsr|2+|hrd|2 When ∆ =0, the solution of dg1(Ps) =0 is
duetolog (1+|hsr|2Ps) R ,wecanobtainP 2Rd−1δ2. 1 dPs
Then, prob2lem (21δ)n2beco≥mesd(23) where s ≥ |hsr|2 n Pp-opt4 = (τ1−τ2)θ1 . (35)
P = 2Rd −1δ2 (24) sR τ2φ1−τ1φ2
b1 h 2 n Then, the solution of problem (23) is
sr
| |
and Popt3 =arg max g (P ). (36)
Pb2 = hsr|2h+rd|2hrd 2P. (25) sR Ps∈P{sP∈spR-[oPptb41,P,Pb1b2,P]b2} 1 s
| | | |
Note that we should use When ∆ < 0, dg1(Ps) = 0 has no solutions. Then, the
R log ( |hsr|2|hrd|2 P +1) (26) solution o1f problemd(P2s3) is
d ≤ 2 h 2+ h 2δ2
| sr| | rd| n PsoRpt4 =arg max g1(Ps). (37)
for signal transmission since Pb1 Pb2. The corresponding Ps∈{Pb1,Pb2}
≤
sub-optimal transmit power at R and all the jammers is Next, we will discuss the solution of sub-problem(22) in a
ProRpt = ||hhrsrd||22PsoRpt and PzoRpt =P −PsoRpt−ProRpt. similar way. As f2(Ps,Pr) is a decreasing functionof Ps, Ps
µ Fo=r sPimλpli+cityδ,2,deθfine=α1µ=δ2,||hhφrsrd||22 =+1µ, βh1 =2 |hs|rαh|2r|dδh|22reλ|2,, stohotuhledcboendmitiionnimhizserd2iPnsordehrrdto2Pmraxinimpirzoeblfe2m(P(s2,2P),r)w.eDsueet
ττ211==α1αλ1eλ|ehe|she|s2r.n|2T,hφe1n2, g=1(Pµ1s1)n|hcsaen|12be−reαw11rδi|n2ttλesern|+as−β1δ1n2 naned (P2s2)=. C||ohhmrsrdb||22inPirngt|oPsob|=tain||hh≥rsard||22|suPbr-|oapntdimPasl+soPlurtio≤nPof, wpreobhlaevme
By taking theg1d(ePrisv)a=tiveθθ11o++f gφφ112(PPPsss)−−wττi12thPPss22re.spect to Ps,(2w7e) lpPorrgo2b≤(l1em|+hsr(||h2|h2s2+srδr)n|2||2h2bPresdc|)2oP≥m.eRsAd(d,3dw8i)teiowcnahanellryoe,btdauine PtorP≥s 2=|hRrdd||−hh|2rs1rdδ||22n2P. Trhaennd,
obtain
2Rd 1
dg (P ) (τ φ τ φ )P2+2(τ τ )θ P +(φ φ )θ P = − δ2 (39)
1 s = 2 1− 1 2 s 2− 1 1 s 1− 2 1. b3 h 2 n
dP (θ +φ P τ P2)2 | rd|
s 1 2 s− 2 s (28) and
As hse 2 is not equal to hsr 2 generally, it always satisfies P = |hsr|2 P. (40)
othfatd|gτd12P(Ps−|s)τ=1 6=0 i0s.eTahsuilsy,owb|htaeinn|eτd2φa1s−τ1φ2 = 0, the solution We can have that (26b4) stil|lhhsor|l2ds+d|uherdto|2Pb3 ≤Pb4. Then, the
φ φ correspondingsub-optimaltransmitpoweratthesourceandall
PspR-opt1 = 2(τ22−−τ11). (29) the jammersis PsoRpt = ||hhrsrd||22ProRpt and PzoRpt =P−PsoRpt−ProRpt.
IEEETRANSACTIONSONCOMMUNICATIONS(ACCEPTEDTOAPPEAR) 6
1+min(|hsr|2Ps,|hrd|2Pr)
(P∗ ,P∗ ) = arg maxf(P ,P )= δn2 δn2
sR rR Ps,Pr s r 1+ |hse|2Ps + |hre|2Pr
δn2 δn2+(P−Ps−Pr)λe
s.t. P +P P
s r
≤
Ps 0,Pr 0 (19)
log≥(1+ |hs≥r|2Ps) R .
2 δn2 ≥ d
1+ |hsr|2Ps
(P∗ ,P∗ ) = arg maxf (P ,P )= δn2
sR rR Ps,Pr 1 s r 1+ |hse|2Ps + |hre|2Pr
δn2 δn2+(P−Ps−Pr)λe
s.t. P +P P
s r
≤
P 0,P 0
s r
log≥(1+ |hs≥r|2Ps) R (21)
2 δn2 ≥ d
h 2P h 2P
sr s rd r
| | ≤| |
1+ |hrd|2Pr
(P∗ ,P∗ ) = arg maxf (P ,P )= δn2
sR rR Ps,Pr 2 s r 1+ |hse|2Ps + |hre|2Pr
δn2 δn2+(P−Ps−Pr)λe
s.t. P +P P
s r
≤
P 0,P 0
s r
log≥(1+ |hs≥r|2Ps) R (22)
2 δn2 ≥ d
h 2P h 2P .
sr s rd r
| | ≥| |
1+ |hsr|2Ps
Popt = arg maxg (P )= δn2
sR Ps 1 s 1+ |hse|2Ps + |hre|2||hhrsrd||22Ps
δn2 δn2+(P−Ps−||hhrsdr||22Ps)λe
s.t. P P P (23)
b1 s b2
≤ ≤
For simplicity, define α = |hrd|2 +1, β = |hrd|2|hse|2, Now we discuss the solution of problem(38) when τ φ
2 |hsr|2 2 |hsr|2 4 3−
φ3 = µ1|hrd|2 −α2δn2λe, τ3 = α2λe|hrd|2, φ4 = µ1β2 − τ3φ4 6= 0. Define q2(Pr) = (τ4φ3 − τ3φ4)Pr2 + 2(τ4 −
α2δn2λe +|hre|2δn2 and τ4 = α2β2λe. Then, g2(Pr) can be τ3)θ1Pr+(φ3−φ4)θ1.Thediscriminantofq2(Pr)isobtained
rewritten as as
g2(Pr)= θθ1++φφ3PPr −ττ3PPr22. (41) ∆2 =(2(τ4−τ3)θ1)2−4(τ4φ3−τ3φ4)(φ3−φ4)θ1. (45)
1 4 r − 4 r When ∆ >0, the solutions of dg2(Pr) =0 are obtained as
By taking the derivative of g (P ) with respect to P , we 2 dPr
2 r r
obtain 2(τ τ )θ +√∆
Pp-opt2 = − 4− 3 1 2 (46)
dg2(Pr) = (τ4φ3−τ3φ4)Pr2+2(τ4−τ3)θ1Pr +(φ3−φ4)θ1. rR 2(τ4φ3−τ3φ4)
dPr (θ1+φ4Pr −τ4Pr2)2 (42) and 2(τ τ )θ √∆
As hse 2 isnotequalto hsr 2 generally,italwayssatisfies PrpR-opt3 = − 2(4τ−φ 3 1τ−φ ) 2. (47)
that τ| τ| = 0. Thus, wh|en τ| φ τ φ = 0, the solution 4 3− 3 4
4 3 4 3 3 4
of dg2(P−r) =60 is obtained as − Then, the solution of problem (38) is
dPr
Popt2 =arg max g (P ). (48)
PrpR-opt1 = 2φ(τ44−−φτ33). (43) rR Pr∈{PrPpR-rop∈t2,[PPbrp3R-o,pPt3b,P4]b3,Pb4} 2 r
Then, the solution of problem (38) is When ∆ =0, the solution of dg2(Pr) =0 is
2 dPr
Popt1 =arg max g (P ). (44)
rR 2 r (τ τ )θ
Pr∈P{rP∈rpR-[oPptb13,P,Pb3b4,P]b4} PrpR-opt4 = τ4φ33− 4τ3φ14. (49)
−
IEEETRANSACTIONSONCOMMUNICATIONS(ACCEPTEDTOAPPEAR) 7
1+ |hrd|2Pr
Popt = argmaxg (P )= δn2
rR Pr 2 r 1+ |hse|2||hhrsrd||22Pr + |hre|2Pr
δn2 δn2+(P−Pr−||hhrsrd||22Pr)λe
s.t. P P P (38)
b3 r b4
≤ ≤
Then, the solution of problem (38) is loss of generality. As log (x) is a concave function of x, due
2
to Jensen’s inequality, we have
Popt3 =arg max g (P ) (50)
rR 2 r
Pr∈{PrpR-opt4,Pb3,Pb4} E[C ]=E[log (1+γ (i,P ,P ,P ))] (53)
Ps∈[Pb3,Pb4] e 2 e s r z
log (1+γ¯(i,P ,P ,P ))
When ∆ < 0, dg2(Pr) = 0 has no solutions. Then, the ≤ 2 e s r z
2 dPr
solution of problem (38) is
where γ¯ (i,P ,P ,P ) is expressed as
e s r z
Popt4 =arg max g (P ). (51)
rR 2 r
Pr∈{Pb3,Pb4} γ¯e(i,Ps,Pr,Pz)=E[γe(i,Ps,Pr,Pz)] (54)
It is noted that problem (30), (34), (36), (37), (44), (48), (50) = Ehse |hse|2 Ps +E |hre|2Pr
and (51) can be easily solved by simple comparison. δ2 δ2 +P hHzzHh
(cid:2) n (cid:3) (cid:20) n z e e(cid:21)
In summary, the sub-optimal solution of problem (19), ǫ P h 2P
1 s re r
(P∗ ,P∗ ),isstatedinTABLEI,andcorrespondingly,P∗ = = +E | | .
P sRP∗rR P∗ . zR δn2 (cid:20)δn2 +Pzλe(cid:21)
− sR− rR
E 1 − 1
B. PCSI based Power Allocation (PCSI-PA) Scheme Defining eE = (cid:20)δn2+Pzλe1(cid:21) δn2+Pzǫ2, we can represent eE
as δn2+Pzǫ2
Consideringthatitisgenerallydifficulttoobtaintheinstan-
taneousCSIofthewiretapchannelinmanypracticalcases,we
E 1 1
propose a PCSI based power allocation (PCSI-PA) scheme in e = δn2+Pzλe − δn2+Pzǫ2 (55)
thissubsection.We assumethatthe fadingcoefficienth and E h 1 i
h are complexGaussian randomswith zero mean valusee and δn2+Pzǫ2
ie
varianceǫ1 andǫ2, respectively,thatis, E[hse]=E[hie]=0, =E ǫ2−λe .
E |hse|2 =δs2e =ǫ1 and E |hie|2 =δi2e =ǫ2. "Pδn2z +λe#
As the CSI of the wiretap channel is unavailable, it is
(cid:2) (cid:3) (cid:2) (cid:3)
infeasible to maximize the instantaneous secrecy rate stated Moreover, we can obtain the mean of λ as
e
in (15). Thus, we aim to maximize the average secrecy rate
of the source-destination transmission E[Cs] = E[Cd Ce]. M−1 M−1
Due to − E[λ ]=E hHzzHh =E h z (h z )H
e e e jie i jie i
" #
i=1 i=1
1 L (cid:2) (cid:3) X X (56)
E[C C ]= lim [C (l) C (l)] (52)
d e dl el
− L→∞L −
l=1 M−1 M−1 M−1
1 XL =E |hjie|2|zi|2+ (hji1ezi1)(hji2ezi2)H
=Ll→im∞L [Cdl(l)−ρ(l)E[Ce]] Xi=1 iX1=1i2=X1,i26=i1
Xl=1 M−1 M−1 
=E h 2 z 2 = E h 2 z 2
where Cdl(l) and Cel(l) denote the instantaneous channel " | jie| | i| # | jie| | i|
capacity of the main link and wiretap link at time l, respec- Xi=1 Xi=1 (cid:2) (cid:3)
M−1
tively, and ρ(l) is a weighting factor satisfying that ρ(l)
L ≥ =ǫ2 |zi|2 =ǫ2.
0, ρ(l) = L. Thus, (52) can be optimized by maximizing i=1
X
l=1
CdPl(l) ρ(l)E[Ce] for each l by finding the optimal ρ(l) and Itcan be observedfrom (55)and (56)that, whenthe variance
−
the optimal transmit power at the source and relay. Since it
ofλ tendsto0,therandomvariableλ convergestoitsmean
e e
is difficult to obtain the optimal value of ρ(l) for each l, we
ǫ , leading to the fact that E 1 approaches 1 .
stuimrnptloyusestetiCngdlρ(l()l)−=E1[Cfoe]raesacahnlo.bAjesctthiveeofputnimctiizoantiionnstoeafdρ(bly) F2ig. 2 illustrates the values ofheδnE2+vPezrλseuisthe varianceδon2f+λPezǫi2n
the cases of P =0 dB, P =10 dB and P =20 dB. In the
is not taken into account, we aim to obtain a lower bound of δn2 δn2 δn2
simulation, P is set to a random value in [0,P].
E[C C ] by using the objective function C (l) E[C ]. z
d e dl e
For−simplicity, we denote Cdl(l) uniformly as C−d without Thus, when the variance of λe is small, E δn2+1Pzλe can
h i
IEEETRANSACTIONSONCOMMUNICATIONS(ACCEPTEDTOAPPEAR) 8
TABLEI
SUB-OPTIMALSOLUTIONOFPROBLEM(19)
condition sub-optimal solution
τ2φ1−τ1φ2 ∆1 τ4φ3−τ3φ4 ∆2 others (Ps∗R,Pr∗R)
0 − 0 − f(PsoRpt1,||hhrsdr||22PsoRpt1)≥f(||hhrsrd||22ProRpt1,ProRpt1)
0 − 6=0 >0 f(PsoRpt1,||hhrsdr||22PsoRpt1)≥f(||hhrsrd||22ProRpt2,ProRpt2) (PsoRpt1,||hhrsrd||22PsoRpt1)
0 − 6=0 0 f(PsoRpt1,||hhrsdr||22PsoRpt1)≥f(||hhrsrd||22ProRpt3,ProRpt3)
0 − 6=0 <0 f(PsoRpt1,||hhrsdr||22PsoRpt1)≥f(||hhrsrd||22ProRpt4,ProRpt4)
6=0 >0 0 − f(PsoRpt2,||hhrsdr||22PsoRpt2)≥f(||hhrsrd||22ProRpt1,ProRpt1)
6=0 >0 6=0 >0 f(PsoRpt2,||hhrsdr||22PsoRpt2)≥f(||hhrsrd||22ProRpt2,ProRpt2) (PsoRpt2,||hhrsrd||22PsoRpt2)
6=0 >0 6=0 0 f(PsoRpt2,||hhrsdr||22PsoRpt2)≥f(||hhrsrd||22ProRpt3,ProRpt3)
6=0 >0 6=0 <0 f(Popt2,|hsr|2Popt2)≥f(|hrd|2Popt4,Popt4)
sR |hrd|2 sR |hsr|2 rR rR
6=0 0 0 − f(PsoRpt3,||hhrsdr||22PsoRpt3)≥f(||hhrsrd||22ProRpt1,ProRpt1)
6=0 0 6=0 >0 f(PsoRpt3,||hhrsdr||22PsoRpt3)≥f(||hhrsrd||22ProRpt2,ProRpt2) (PsoRpt3,||hhrsrd||22PsoRpt3)
6=0 0 6=0 0 f(Popt3,|hsr|2Popt3)≥f(|hrd|2Popt3,Popt3)
sR |hrd|2 sR |hsr|2 rR rR
6=0 0 6=0 <0 f(PsoRpt3,||hhrsdr||22PsoRpt3)≥f(||hhrsrd||22ProRpt4,ProRpt4)
6=0 <0 0 − f(PsoRpt4,||hhrsdr||22PsoRpt4)≥f(||hhrsrd||22ProRpt1,ProRpt1)
6=0 <0 6=0 >0 f(PsoRpt4,||hhrsdr||22PsoRpt4)≥f(||hhrsrd||22ProRpt2,ProRpt2) (PsoRpt4,||hhrsrd||22PsoRpt4)
6=0 <0 6=0 0 f(PsoRpt4,||hhrsdr||22PsoRpt4)≥f(||hhrsrd||22ProRpt3,ProRpt3)
6=0 <0 6=0 <0 f(PsoRpt4,||hhrsdr||22PsoRpt4)≥f(||hhrsrd||22ProRpt4,ProRpt4)
0 − 0 − f(PsoRpt1,||hhrsdr||22PsoRpt1)≤f(||hhrsrd||22ProRpt1,ProRpt1)
6=0 >0 0 − f(PsoRpt2,||hhrsdr||22PsoRpt2)≤f(||hhrsrd||22ProRpt1,ProRpt1) (||hhrsdr||22ProRpt1,ProRpt1)
6=0 0 0 − f(PsoRpt3,||hhrsdr||22PsoRpt3)≤f(||hhrsrd||22ProRpt1,ProRpt1)
6=0 <0 0 − f(PsoRpt4,||hhrsdr||22PsoRpt4)≤f(||hhrsrd||22ProRpt1,ProRpt1)
0 − 6=0 >0 f(PsoRpt1,||hhrsdr||22PsoRpt1)≤f(||hhrsrd||22ProRpt2,ProRpt2)
6=0 >0 6=0 >0 f(PsoRpt2,||hhrsdr||22PsoRpt2)≤f(||hhrsrd||22ProRpt2,ProRpt2) (||hhrsdr||22ProRpt2,ProRpt2)
6=0 0 6=0 >0 f(PsoRpt3,||hhrsdr||22PsoRpt3)≤f(||hhrsrd||22ProRpt2,ProRpt2)
6=0 <0 6=0 >0 f(PsoRpt4,||hhrsdr||22PsoRpt4)≤f(||hhrsrd||22ProRpt2,ProRpt2)
0 − 6=0 0 f(PsoRpt1,||hhrsdr||22PsoRpt1)≤f(||hhrsrd||22ProRpt3,ProRpt3)
6=0 >0 6=0 0 f(PsoRpt2,||hhrsdr||22PsoRpt2)≤f(||hhrsrd||22ProRpt3,ProRpt3) (||hhrsdr||22ProRpt3,ProRpt3)
6=0 0 6=0 0 f(PsoRpt3,||hhrsdr||22PsoRpt3)≤f(||hhrsrd||22ProRpt3,ProRpt3)
6=0 <0 6=0 0 f(PsoRpt4,||hhrsdr||22PsoRpt4)≤f(||hhrsrd||22ProRpt3,ProRpt3)
0 − 6=0 <0 f(Popt1,|hsr|2Popt1)≤f(|hrd|2Popt4,Popt4)
sR |hrd|2 sR |hsr|2 rR rR
6=0 >0 6=0 <0 f(PsoRpt2,||hhrsdr||22PsoRpt2)≤f(||hhrsrd||22ProRpt4,ProRpt4) (||hhrsdr||22ProRpt4,ProRpt4)
6=0 0 6=0 <0 f(PsoRpt3,||hhrsdr||22PsoRpt3)≤f(||hhrsrd||22ProRpt4,ProRpt4)
6=0 <0 6=0 <0 f(PsoRpt4,||hhrsdr||22PsoRpt4)≤f(||hhrsrd||22ProRpt4,ProRpt4)
be approximated by 1 . Then, we have when the variance of λ is small.
δn2+Pzǫ2 e
From (53) and (58), we have
h 2P 1
E | re| r =E h 2P E (57)
δ2 +P λ | re| r δ2 +P λ
(cid:20) n z e(cid:21) (cid:20) n z e(cid:21)
(cid:2) (cid:3) 1
≈E |hre|2Pr δ2 +P ǫ Cd−E[Ce]≥Cd−log2(1+γ˜e(i,Ps,Pr,Pz)) (59)
n z 2 1+γ (i,P ,P )
= (cid:2)ǫ2Pr .(cid:3) =log2(1+γ˜(di,P ,sP ,rP )).
δ2 +P ǫ e s r z
n z 2
Combining (54) and (57), γ¯ (i,P ,P ,P ) can be approxi-
e s r z
mately expressed as
Therefore, we aim to maximize the lower bound of C
d
ǫ P ǫ P E[C ], that is, log ( 1+γd(i,Ps,Pr) ), which is proved to b−e
γ¯ (i,P ,P ,P ) γ˜ = 1 s + 2 r (58) e 2 1+γ˜e(i,Ps,Pr,Pz)
e s r z ≈ e δ2 δ2 +P ǫ effective in our experiments.
n n z 2
IEEETRANSACTIONSONCOMMUNICATIONS(ACCEPTEDTOAPPEAR) 9
φ )θ and g (P )= θ2+φ5Ps−τ5Ps2. Then, we have
6 2 3 s θ2+φ6Ps−τ6Ps2
φ φ
Pp-opt5 = 6− 5 , (64)
sR 2(τ τ )
6 5
−
Popt5 =arg max g (P ), (65)
sR 3 s
Ps∈ Psp-opt5,Pb1,Pb2
{ }
Ps∈[Pb1,Pb2]
2(τ τ )θ +√∆
Pp-opt6 = − 6− 5 2 3, (66)
sR 2(τ φ τ φ )
6 5 5 6
−
2(τ τ )θ √∆
Pp-opt7 = − 6− 5 2− 3, (67)
sR 2(τ φ τ φ )
6 5 5 6
−
Popt6 =arg max g (P ), (68)
sR 3 s
Ps∈{PspR-opt6,PspR-opt7,Pb1,Pb2}
Ps∈[Pb1,Pb2]
cFaigse.s2.of AδPn2n=illu0stdrsBti,onδPn2of=th1e0vadlBueasnodfδePn2E=ver2s0usdBth.evariance ofλe inthe PspR-opt8 = τ(6τφ55−τ6τ)5θφ26, (69)
−
Popt7 =arg max g (P ), (70)
sR 3 s
The PCSI-PA problem can be formulated as Ps∈{PspR-opt8,Pb1,Pb2}
Ps∈[Pb1,Pb2]
(R∗,P∗,P∗,P∗) = arg max 1+γd(i,Ps,Pr) and
s r z i,Ps,Pr,Pz 1+γ˜e(i,Ps,Pr,Pz) PsoRpt8 =arg max g3(Ps). (71)
s.t. P +P +P =P Ps∈{Pb1,Pb2}
s r z
lPosg2≥(10+,P|rhs≥δi|n220P,sP)z≥≥R0d (60) α2Dǫ2e|fihnrde|2β,4φ8==|h|hrβdsr|42|µ2ǫ12,−φ7α2=δn2ǫµ22+|hrǫd2|δ2n2−, τα82δ=n2ǫ2α,2τǫ72β=4,
∆ = 4(τ τ )2θ2 4(τ φ τ φ )(φ φ )θ and
which can be solved in a similar way as problem (16). g 4(P )= θ28+φ−7Pr7−τ7P2r2−. Then8, w7e−hav7e 8 7 − 8 2
For each candidate of relay i , the power allocation 4 r θ2+φ8Pr−τ8Pr2
∈ D
problem is formulated as
φ φ
Pp-opt5 = 8− 7 , (72)
(P∗,P∗,P∗) = arg max 1+γd(i,Ps,Pr) rR 2(τ8−τ7)
si ri zi Ps,Pr,Pz 1+γ˜e(i,Ps,Pr,Pz) Popt5 =arg max g (P ), (73)
s.t. Ps+Pr+Pz =P rR Pr∈ Prp-opt5,Pb3,Pb4 4 r
{ }
Ps 0,Pr 0,Pz 0 (61) Pr∈[Pb3,Pb4]
≥ ≥ ≥
log (1+ |hsi|2Ps) R
2 δn2 ≥ d Pp-opt6 = −2(τ8−τ7)θ2+√∆4, (74)
rR 2(τ φ τ φ )
and the relay is determined as 8 7− 7 8
R∗ =arg mi∈aDx 1+1+γ˜eγ(id,(Pi,s∗Pi,s∗Pi,r∗Pi,r∗Pi)z∗i) (62) PrpR-opt7 = −2(2τ(8τ−8φτ77−)θ2τ7−φ8√)∆4, (75)
which can be also easily solved by simple comparison. ProRpt6 =arg max g4(Pr), (76)
Since P +P +P =P, problem (61) can be rewritten as Pr∈ Prp-opt6,Prp-opt7,Pb3,Pb4
s r z { }
Pr∈[Pb3,Pb4]
1+min(|hsr|2Ps,|hrd|2Pr) (τ τ )θ
[P∗ ,P∗ ]=arg maxf(P ,P )= δn2 δn2 Pp-opt8 = 7− 8 2 , (77)
sR rR Ps,Pr s r 1+ ǫ1δPn2s + δn2+(P−ǫ2PPsr−Pr)ǫ2 rR τ8φ7−τ7φ8
s.t. Pse+Pr ≤P ProRpt7 =arg max g4(Pr) (78)
Plosg≥(10+,P|rhs≥r|20Ps) R . (63) Pr∈P{sP∈rp[-oPptb83,P,Pb3b4,P]b4}
2 δn2 ≥ d and
We can obtain a sub-optimal solution of problem (63) in a Popt8 =arg max g (P ). (79)
rR 4 r
similar way as that in Section III.A. Pr∈{Pb3,Pb4}
Define β3 = |h|hsrrd|2|2ǫ2, µ2 = Pǫ2 +δn2, θ2 = µ2δn2, φ5 = Westatethesub-optimalsolutionofproblem(63)inTABLE
µ h 2 α δ2ǫ , τ = α ǫ h 2, φ = µ ǫ α δ2ǫ + II and then the sub-optimal total transmit power at all the
2| sr| − 1 n 2 5 1 2| sr| 6 2 1− 1 n 2
β δ2,τ =α ǫ ǫ ,∆ =4(τ τ )2θ2 4(τ φ τ φ )(φ jammers P∗ is P P∗ P∗ .
3 n 6 1 1 2 3 6− 5 2− 6 5− 5 6 5− zR − sR− rR
IEEETRANSACTIONSONCOMMUNICATIONS(ACCEPTEDTOAPPEAR) 10
TABLEII
SUB-OPTIMALSOLUTIONOFPROBLEM(63)
condition sub-optimal solution
τ6φ5−τ5φ6 ∆3 τ8φ7−τ7φ8 ∆4 others [Ps∗R,Pr∗R]
0 − 0 − f(PsoRpt5,||hhrsdr||22PsoRpt5)≥f(||hhrsdr||22ProRpt5,ProRpt5)
0 − 6=0 >0 fe(PsoRpt5,||hhrsdr||22PsoRpt5)≥fe(||hhrsdr||22ProRpt6,ProRpt6) hPsoRpt5,||hhrsrd||22PsoRpt5i
0 − 6=0 0 fe(PsoRpt5,||hhrsdr||22PsoRpt5)≥fe(||hhrsdr||22ProRpt7,ProRpt7)
0 − 6=0 <0 fe(PsoRpt5,||hhrsdr||22PsoRpt5)≥fe(||hhrsdr||22ProRpt8,ProRpt8)
6=0 >0 0 − fe(PsoRpt6,||hhrsdr||22PsoRpt6)≥fe(||hhrsdr||22ProRpt5,ProRpt5)
6=0 >0 6=0 >0 fe(PsoRpt6,||hhrsdr||22PsoRpt6)≥fe(||hhrsdr||22ProRpt6,ProRpt6) hPsoRpt6,||hhrsrd||22PsoRpt6i
6=0 >0 6=0 0 fe(PsoRpt6,||hhrsdr||22PsoRpt6)≥fe(||hhrsdr||22ProRpt7,ProRpt7)
6=0 >0 6=0 <0 fe(Popt6,|hsr|2Popt6)≥fe(|hrd|2Popt8,Popt8)
sR |hrd|2 sR |hsr|2 rR rR
6=0 0 0 − fe(PsoRpt7,||hhrsdr||22PsoRpt7)≥fe(||hhrsdr||22ProRpt5,ProRpt5)
6=0 0 6=0 >0 fe(PsoRpt7,||hhrsdr||22PsoRpt7)≥fe(||hhrsdr||22ProRpt6,ProRpt6) hPsoRpt7,||hhrsrd||22PsoRpt7i
6=0 0 6=0 0 fe(Popt7,|hsr|2Popt7)≥fe(|hrd|2Popt7,Popt7)
sR |hrd|2 sR |hsr|2 rR rR
6=0 0 6=0 <0 fe(PsoRpt7,||hhrsdr||22PsoRpt7)≥fe(||hhrsdr||22ProRpt8,ProRpt8)
6=0 <0 0 − fe(PsoRpt8,||hhrsdr||22PsoRpt8)≥fe(||hhrsdr||22ProRpt5,ProRpt5)
6=0 <0 6=0 >0 fe(PsoRpt8,||hhrsdr||22PsoRpt8)≥fe(||hhrsdr||22ProRpt6,ProRpt6) hPsoRpt8,||hhrsrd||22PsoRpt8i
6=0 <0 6=0 0 fe(PsoRpt8,||hhrsdr||22PsoRpt8)≥fe(||hhrsdr||22ProRpt7,ProRpt7)
6=0 <0 6=0 <0 fe(PsoRpt8,||hhrsdr||22PsoRpt8)≥fe(||hhrsdr||22ProRpt8,ProRpt8)
0 − 0 − fe(PsoRpt5,||hhrsdr||22PsoRpt5)≤fe(||hhrsdr||22ProRpt5,ProRpt5)
6=0 >0 0 − fe(PsoRpt6,||hhrsdr||22PsoRpt6)≤fe(||hhrsdr||22ProRpt5,ProRpt5) h||hhrsdr||22ProRpt5,ProRpt5i
6=0 0 0 − fe(PsoRpt7,||hhrsdr||22PsoRpt7)≤fe(||hhrsdr||22ProRpt5,ProRpt5)
6=0 <0 0 − fe(PsoRpt8,||hhrsdr||22PsoRpt8)≤fe(||hhrsdr||22ProRpt5,ProRpt5)
0 − 6=0 >0 fe(PsoRpt5,||hhrsrd||22PsoRpt5)≤fe(||hhrsdr||22ProRpt6,ProRpt6)
6=0 >0 6=0 >0 f(PsoRpt6,||hhrsdr||22PsoRpt6)≤f(||hhrsdr||22ProRpt6,ProRpt6) h||hhrsdr||22ProRpt6,ProRpt6i
6=0 0 6=0 >0 fe(PsoRpt7,||hhrsdr||22PsoRpt7)≤fe(||hhrsdr||22ProRpt6,ProRpt6)
6=0 <0 6=0 >0 fe(PsoRpt8,||hhrsdr||22PsoRpt8)≤fe(||hhrsdr||22ProRpt6,ProRpt6)
0 − 6=0 0 fe(PsoRpt5,||hhrsdr||22PsoRpt5)≤fe(||hhrsdr||22ProRpt7,ProRpt7)
6=0 >0 6=0 0 fe(PsoRpt6,||hhrsdr||22PsoRpt6)≤fe(||hhrsdr||22ProRpt7,ProRpt7) h||hhrsdr||22ProRpt7,ProRpt7i
6=0 0 6=0 0 fe(PsoRpt7,||hhrsdr||22PsoRpt7)≤fe(||hhrsdr||22ProRpt7,ProRpt7)
6=0 <0 6=0 0 fe(PsoRpt8,||hhrsdr||22PsoRpt8)≤fe(||hhrsdr||22ProRpt7,ProRpt7)
0 − 6=0 <0 fe(Popt5,|hsr|2Popt5)≤fe(|hrd|2Popt8,Popt8)
sR |hrd|2 sR |hsr|2 rR rR
6=0 >0 6=0 <0 fe(PsoRpt6,||hhrsdr||22PsoRpt6)≤fe(||hhrsdr||22ProRpt8,ProRpt8) h||hhrsdr||22ProRpt8,ProRpt8i
6=0 0 6=0 <0 fe(PsoRpt7,||hhrsdr||22PsoRpt7)≤fe(||hhrsdr||22ProRpt8,ProRpt8)
6=0 <0 6=0 <0 fe(PsoRpt8,||hhrsdr||22PsoRpt8)≤fe(||hhrsdr||22ProRpt8,ProRpt8)
e e
IV. NUMERICAL RESULTSAND DISCUSSIONS betterthantheEPAstrategyintermsofsecrecyrate.Moreover,
numerical results illustrate that with an increasing number of
This section presents the numerical secrecy rate results
intermediate nodes, the secrecy rates of the proposed JRJS
of our proposed JRJS schemes using FCSI-PA and PCSI-
schemes using FCSI-PA and PCSI-PA strategies increase.
PA strategies. Pure relay selection,pure jamming,generalized
We first discuss the effect of R on the secrecy rate of the
d
singular-value-decomposition (GSVD) based beaforming and
proposed JRJS scheme. As stated in section III, transmission
JRJS schemes using equal power allocation (EPA) are used
rate R plays an important role in the secrecy rate results
d
as benchmark schemes. In our numerical experiments, we
mainly due to two reasons. One reason is that different
have E(hsi 2) = E(hid 2) = 1, E(hse 2) = ǫ1 = 1, transmission rates lead to different decoding sets where the
| | | | | |
E(|hie|2) = ǫ2 = 1 and δn2 = 0 dBm. We show that the relay is selected, which can be seen from (10). The other
proposed JRJS scheme outperforms the pure relay selection,
reason is that secrecy rate results of the proposed FCSI-PA
pure jamming and GSVD based beamformingschemes. Also,
and PCSI-PA JRJS schemes are effected by P and P
b1 b3
our proposed FCSI-PA and PCSI-PA schemes both perform