Secrecy Rate with Friendly Full-Duplex Relay
Saeedeh Parsaeefard and Tho Le-Ngoc
Electrical and Computer Engineering Department,Mcgill University
saeideh.ill.ca and tho.le-ngoc@mcgill.ca
Abstract—This paper considers the use of a friendly full-duplex (FD)relay to increase the secrecy rate over a fading channel between the legitimate source and destination in the presence of a naive or informed eavesdropper.Naive eavesdropper can only decode the received signals either from the source or from the relay,while informed eavesdropper can overhear signals trans-mitted from both the source and relay.Accordingly,we compare the achievable secrecy rates of FD relay with traditional half-duplex(HD)relay in terms of the channel state information(CSI) between nodes,eavesdropper types,and the self-interference (SI)in FD-Relay.We consider the non-convex power allocation problems for the developed FD-relay to maximize the secrecy rate under the power constraints and develop an efficient iterative algorithm based on the d oncave-functions(DC) programming.The analytical and simulation results confirm that FD relay offers significant improvements in the secrecy rate over the HD-Relay.
Index Terms—D oncave-functions(DC)-Programming,full-duplex relay,physical-layer security.
I.I NTRODUCTION
Secrecy rate defined as subtraction of source-destination desired rate and the overheard rate of eavesdroppers,has been drawn a lot of attentions for providing the security of next generation of wireless networks[1].When the source-destination channel gain is less than the source-eavesdropper channel gain,the secrecy rate is equal to zero which is an unfavorable scenario and it means the system is interference limited[2].To increase the chance of a non-zero secrecy rate for single-antenna users,deploying the set of external nodes has been proposed where these nodes can collaboratively send jamming signals to eavesdroppers,cooperatively relay the signals from the source to destination,or jointly interfere with the eavesdropper and relay to help the ,[2]–[5]. In many previous works,the external nodes have been assumed to operate in half-duplex(HD)transmission mode where,infirst time-slot,it receives the data from transmitter, and,in the second time-slot,it sends the data to the receiver. Recently,full-duplex(FD)relay with capability of simulta-neous transmission and reception,has been investigated for different system models of wireless networks[3],[6]where its performance highly depends on the self-interference(SI) from its own transmitter to its receiver antenna.
In the context of secrecy rate,the effect of FD-capable eavesdropper on the secrecy rate is investigated in[7]where multiple-antenna eavesdropper tries to simultaneously over-hear transmit information of legitimate user and send the jamming signals to the corresponding legitimate receiver.[8] studied the secrecy rate of users where its FD-capable receiver simultaneously receives the data from its transmitter and sends the jamming signals to the eavesdropper.In[9],the FD relay is deployed to jam the eavesdropper while sending the information of legitimate user.This paper investigates the advantageous and disadvantageous of applying FD-Relay node in secrecy rate where the FD-Relay assists a single-antenna source in improving the desired rate while eavesdropper is trying to detect the signal of source.In particular,our main question is under which conditions the FD-Relay can assist the legitimate users to increase the secrecy rate. Apparently,using FD transmission(FDT)protocol,the FD-Relay can increase user desired rate.However,the achievable user secrecy rate also depends on the capability of the eaves-droppers to extract the information from overheard signals [2].In this paper,we consider two types of eavesdroppers:1) Naive eavesdropper who does not know about the FD-Relay operation and it can just overhear from the transmitter or relay; and2)Informed eavesdropper who knows about the FD-Relay operation and,hence,can also decode the information from both source and relay.We derive the achievable secrecy rates for FDT based on the capabilities of eavesdroppers.
Our studies show that FDT can considerably increase the probability of non-zero secrecy rate as compared to the HD-Relay.For a naive eavesdropper,FDT can improve the secrecy rate up to twice that offered HD-Relay.For an informed eavesdropper,the performance of FDT much more depends on the SI level,and for low SI,FDT outperforms HD-Relay. We also investigate the FDT power allocation problems to maximize the secrecy rate under transmit power constraints. Based on the d oncave-functions(DC)pro-gramming,we develop an efficient iterative algorithm to solve these non-convex optimization problems.Simulation results confirm the analytical results and are also used to compare the achieved secrecy rates and energy efficiencies of the proposed techniques.
The rest of this paper is organized as follows.Section II describes the system model and problem formulations.Sec-tion III presents a performance comparison analysis of FDT and HD-Relay.Section IV formulates the power allocation problems for secrecy rate maximization under transmit power constraints.Section V provides the simulation results and Section VI concludes the paper.
II.N ETWORK M ODEL AND P ROBLEM F ORMULATION Consider a two-hop transmission where a single-antenna source S communicates with a single-antenna destination D via a friendly FD-Relay R in the presence of eavesdropper E shown in Fig.1.The total bandwidth of B Hz allocated for
2015 IEEE Wireless Communications and Networking Conference (WCNC 2015) - Track 2: MAC and Cross-Layer Design
Fig.1:Operation in one transmit frame(a)HD-Relay,(b)
FDT.
transmission,is divided into K equal-bandwidth sub-carriers forming the set K={1,···,K}.Assume that the sub-carrier bandwidth,B/K,is much less than the channel coherence bandwidth so that the sub-carrier frequency response isflat and represented by the channel power gain h k XY for sub-carrier k over the link from X to Y where X∈{S,R}and Y∈{R,D,E}.Consequently,the overall channel responses over the bandwidth B of the link from X to Y are represented by the channel power gain vector h XY=[h1XY,···,h K XY], indicating the channel state information(CSI).
The transmission time is divided into equal transmission frames and each transmission frame consists of two equal time-slots denoted by t=1and t=2.We assume a block fading environment with h k XY unchanged within a transmission frame,but independently varied from one transmission frame to another.Also,we assume that S-D link is highly attenuated, h k SD≈0,and R uses the decode-and-forward(DF)strategy. In the HD-Relay protocol illustrated in Fig.1(a),in t=1, the source transmits x k[t=1]to R over sub-carrier k where E{|x k[t=1]|2}=P k S,E{.}is the expectation operator, and P k S is transmit power of S on sub-carrier k.The received signals at R and E from S on sub-carrier k are,respectively, r k HD[t=1]=
h k SE x k[t=1]+n k E,
where n k R and n k E are the white Gaussian noise samples at R and E.Without loss of generality,we assume the same noise power ofσat each receiver in all sub-carriers.In thefirst time-slot,the signal-to-noise-power ratio(SNR)at R and E
areγk R(HD)=P k S h k SR
σ,respectively for
all k∈K.In the second time-slot t=2,R forwards f k[t=2] to D,and the received signals at D and at E are,respectively, y k HD[t=2]=
h k RE f k[t=2]+n k E
where E{|f k[t=2]|2}=P k R and P k R is the transmit power of R on sub-carrier k and where n k D is the white Gaussian noise samples at D.At D,the SNR isγk D(HD)=P k R h k RD
σ
.For the naive eavesdropper with only listening capability without knowledge of relay operation,its
effective SNR isγk,1
E
=max{γk SE(HD),γk RE(HD)},for all k∈K,and consequently,the S-D secrecy rate in the case of a naive eavesdropper is SR1= K k=1SR k,1where SR k,1= log2(1+γk HD)−log2(1+γk,1E) +, and[x]+=max{0,x}.However,for an informed eaves-dropper with knowledge of relay operation,and,hence,of
the transmission over two hops,its effective SNR isγk,2
E
= P k S h k SE
σ
,and consequently,the S-D secrecy rate in the case of an informed eavesdropper is SR2= K k=1SR k,2 where
齐白石的人格SR k,2= log2(1+γk HD)−log2(1+γk,2E) +.
The FDT protocol is depicted in Fig.1(b)where the FD-Relay transmits and receives data simultaneously,which cre-ates a self-interference(SI)feedback loop between its transmit and receive antennas.We parameterize this residual SI as self-interference CSI(SI-CSI)between transmit and receive antennas of R,denoted by h SI=[h1SI,···,h K SI],where h k SI is h k SI=ρg k SI,g k SI is SI-channel power gain between transmit and receive antennas[6],and0≤ρ≤1represents the quality in terms of power reduction of SI cancelation technique[8]. Using the FDT protocol,in each time-slot t=1or2,S transmits x k[t]to R,while R simultaneously receives r k FD[t] and forwards f k[t]to D.At the same time,E overhears the signal e k FD[t]on sub-carrier k.The received signals at R,D and E are,respectively,
r k FDT[t]= h k RD f k[t]+n k D,∀t, e k FDT[t]= h k SR x k[t]and
σ+P k
R
h k
SI
,
for all k∈K.Also,the SNR at D isγk D(FDT)=
P k
R
h k
RD
The overheard signal e k FDT[t]contains h k SE x k[t]from S,and h k RE f k[t]from R.For a naive eavesdropper with only a single-user decoding capability,it considers one of these two components as an interference,and extracts eitherγk SE(FDT)=
P k S h k
SE
σ+P k
S
h k
SE
for all k∈K.Based on
the quality ofγk SE andγk RE,the naive eavesdropper decides to listen to either S or R.In other words,in the worst case with respect to S,R,and D(which is the best case to E),the effective overheard SINR of E is
γk,3
E
=max{γk SE(FDT),γk RE(FDT)}.
The S-D secrecy rate with FDT in the case of a naive eavesdropper is SR3= K k=1SR k,3where
SR k,3=2 log2(1+γk FDT)−log2(1+γk,3E) +,∀k∈K. The multiplier2in SR k,3comes from the fact that the FDT protocol allows the similar transmission characteristics in both time-slots t=1and2.
For an informed eavesdropper with knowledge of relay operation in FD mode,and,hence,with capability of correctly detecting signals from both S-E and R-E links,the maximum
effective SINR of E can beγk,4
E =P k S h k SE
σ
,and the
lowest S-D secrecy rate with FDT in the case of an informed eavesdropper can be SR4= K k=1SR k,4,where SR k,4=2 log2(1+γk FDT)−log2(1+γk,4E) +,∀k∈K.
III.P ERFORMANCE C OMPARISONS
In this section,we study and compare the achievable secrecy rates of FDT and HD-Relay protocols.Wefirst focus on the single-carrier transmission scenario.Therefore,the index k can be dropped for simplicity without confusion.We assume no adaptive power allocation applied to S and ,they transmit at their maximum transmit power.Without loss of
generality,consider P max
S =1,andα=P max
S
/P max
R
.Let
assumeσ≪P max
R h SI andσ≪P max
S
h SE.In the following,
wefirst investigate the conditions for FDT and HD-Relay to offer non-zero secrecy rate.
Non-Zero Secrecy Rate Conditions for a Naive Eavesdrop-per:For the naive eavesdropper,by consideringγ1E<γHD and γ3E<γFDT,the conditions to achieve non-zero secrecy rate for the FD and HD-Relay protocols are:
C01:For HD-Relay,SR1>0when h SE<h SR or h SE<αh RD,and h RE<h RD orαh RE<h SR.
C02:For FDT,SR3>0when h SE<h SR h RE/h SI or
h SE<α2h RD h RE/σ,and h RE<h RD h SE/σor h RE<
h SR h SE/α2h SI.
From C01and C02,when h SI<h RE or h SI<h SE,FDT increases the chance of the non-zero secrecy rate compared to that for HD-Relay protocol.
Non-Zero Secrecy Rate Conditions for an Informed Eaves-dropper:For an informed eavesdropper,by consideringγ2E<γHD andγ4E<γFDT,the conditions to achieve non-zero secrecy rate for the FD and HD-Relay protocols are:C03:SR2>0if h SE+αh RE<h SR or h SE+αh RE<αh RD,
C04:SR4>0ifαh SI崇文门新世界店庆
h SR
α αh RE
Proof:See Appendix A.
α h SE+αh RE, C22:h SR≥αh RD+α2h SI h RD/σand h SE<α(h RD−h RE).
Proof:See Appendix B.
expansion as g k(P(l))≈g k(P(l−1))+∇P(l−1)g k(P(l−1))[P(l)−P(l−1)]T for all k∈K where∇P(l−1)g k(P(l−1)) is the gradient vector of g k(P(l−1))with respect to vector P(l−1).Nowθ(P)can be approximated as
θ(P(l))≈
K
k=1f k(P(l))−g k(P(l−1))−(1)
∇P(l−1)g k(P(l−1))[P(l)−P(l−1)]T.
Now the right-hand side of(1)is the convex function since the approximation of g k(P(l))is linear function of P(l).The general DC-based iterative algorithm is given in Table I.In the following,we show that how this iterative algorithm can be applied to each of our specific optimization problems.Wefirst consider the FDT for a naive eavesdropper with the following optimization problem O3
max P≥0
K
k=12(log2(1+min{γk R(FDT),γk D(FDT)})(2)
−log2(1+max{γk SE(FDT),γk RR(FDT)}))
We rewrite(2)as
max
π>0,̟>0,P≥0
K
k=12(πk−̟k)(3)
C33:log2(1+γk R(FDT))≥πk,
C34:log2(1+γk D(FDT))≥πk
C35:log2(1+γk SE(FDT))≤̟k,
C36:log2(1+γk RE(FDT))≤̟k,
whereπ=[π1,···,πK]and̟=[̟1,···,̟K].To use the iterative algorithm in Table I,the convexified version of(3) for iteration l is
max
π>0,̟>0,P(l)≥0
xujieK
k=12(πk−̟k)(4)
C43:πk≤f k R(P(l))−g k R(P(l−1))−
光影交错的
时空∂g k R(P(l−1))
∂P k R
(P k R(l)−P k R(l−1))≤̟k, C46:f k RE(P(l))−g k RE(P(l−1))
−∂g k RE(P(l−1))
σ
),
f k SE(P(l))=log2(σ+P k R(l)h k RE+P k S(l)h k SE),
g k SE(P(l−1))=log2(σ+P k R(l−1)h k RE),
甲基丙烯酸甲脂
f k RE(P(l))=log2(σ+P k R(l)h k RE+P k S(l)h k SE),
g k RE(P(l−1))=log2(σ+P k S(l−1)h k SE).
For an informed eavesdropper,the FDT optimization problem
O4is
max
π>0,̟>0,P≥0
K
k=12(πk−̟k),(5)
<:C31,C32,C33,C34,
C37:log2(1+γk,4
E
)
≤̟k,∀k,
and its DC approximation at iteration l is
max
π≥0,̟≥0,P(l)≥0
K
k=12(πk−̟k),(6)
<:C31,C32,C43,C44,and C37.
The formulations related to the optimization problems O3and
O4can be used for the optimization problems of the HD-Relay
protocol for the naive and informed eavesdropper,respectively,
except that the transmit power of R in t=1and the transmit
power of S in t=2must be set to zero.Now,for single-carrier
transmission,we determine the optimal power allocation for
S and R.Proposition3:For FDT,when the secrecy rate is
non-zero,we have
P∗S=min{P max
S
,
σP max
R
h RD+(P max
R
)
2h RD h SI
2h SI
+ 4h2SI+P max S h SRσ
From Proposition3,the S optimal transmit power,P∗S,is
either P max
S
or proportional to the maximum transmit power of
R,αP max
R
.Therefore,Propositions1-2hold when the transmit
power control for S and D are applied.
Fig.2:FDT and HD-Relay secrecy rates versus d SR
d SE andρ
and for informed eavesdropper.
Fig.4:FDTηS versus P max
S
for naive eavesdropper. SR3
d SE =0.1whenρ=0.03
andρ=0.7.It is because,for largerρ,an increase in P max
S introduces a larger increase inγk FD.
On the other hand,Fig.5highlights that when E is informed,
钻机导管increasing P max
S does not always lead to increase in the secrecy
rate.For example,for d SE
d SE <0.5,th
e S-R transmission is
greater than S-E overheard rate,and,as a result,the secrecy
rate is increased with increasing P max
S ,i.e.,the rate is power-
limited not interference-limited for the FDT in the presence of informed eavesdropper.
VI.CONCLUSION
In this paper,the secure transmission between the source, the relay with full-duplex(FD)capability and the destination
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