Optimal Power Allocation for Secure Communications in Large-Scale MIMO Relaying Systems

0.18 MB·English

by  Jian Chen

Optimal Power Allocation for Secure
Communications in Large-Scale MIMO Relaying
Systems
Jian Chen†, Xiaoming Chen†, Xiumin Wang‡, and Lei Lei†
†College of Electronic and Information Engineering, Nanjing University of Aeronautics and Astronautics, China.
‡School of Computer and Information, Hefei University of Technology, Hefei, China.
Email: [e-mail]
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Abstract—In this paper, we address the problem of optimal and thus the secrecy performance is degraded. To solve it,

n power allocation at the relay in two-hop secure communica- a joint jamming and beamforming scheme at the relay in
a tions. In order to solve the challenging issue of short-distance the case without eavesdropper CSI was proposed in [9]. The
J interception in secure communications, the benefit of large-scale
relay transmits the artificial noise signal in the null space
9 MIMO (LS-MIMO) relaying techniques is exploited to improve
1 the secrecy performance significantly, even in the case without of the legitimate channel together with the forward signal,
eavesdropper channel state information (CSI). The focus of this so the quality of the interception signal is weakened. This
] paper is on the analysis and design of optimal power allocation schemeimprovesthesecrecyperformanceatthecostofpower
T for the relay, so as to maximize the secrecy outage capacity. We
efficiency.
I reveal the condition that the secrecy outage capacity is positive,
s. prove that thereisoneandonly oneoptimal power, and present Recently,LS-MIMOrelayingtechniquesareintroducedinto
c an optimal power allocation scheme. Moreover, the asymptotic secure communications to improve the secrecy performance
[ characteristics of the secrecy outage capacity is carried out to [10]. It is found that even without eavesdropper CSI, LS-

provide some clear insights for secrecy performance optimiza- MIMOtechniquescanproduceahigh-resolutionspatialbeam,
1
tion. Finally, simulation results validate the effectiveness of the
v thentheinformationleakagetotheeavesdropperisquitesmall.
proposed scheme.
6 More importantly, the secrecy performance can be enhanced
7 by simply adding the antennas. Thus, the challenging issue
3 I. INTRODUCTION
of short-distance interception in secure communications can
4
Wireless security is always a critical issue due to the be well solved. Note that in two-hop secure systems, the
0
. open nature of the wireless channel. Traditionally, high-layer transmit power at the relay has a great impact on the secrecy
1
encryptiontechniquesareadoptedtoguaranteesecurecommu- performance, since the power will affect the signal quality
0
5 nications.However,information-theoreticstudyshowsthatthe at the destination and the eavesdropper simultaneously. An
1 originallyharmfulfactorsofwirelesschannels,suchasfading, optimal power allocation scheme for a multi-carrier two-hop
: noiseandinterference,canbeusedtorealizewirelesssecurity, single-antennarelayingnetworkwasgivenbymaximizingthe
v
i namely physical layer security [1] [2], then the complicated sum secrecy rate in [11]. However, the power allocation for
X encryption can be partially replaced, especially in mobile a multi-antenna relay, especially an LS-MIMO relay, is still
r communications. an open issue. In this paper, we focus on power allocation
a
It has been proved repeatedly that the secrecy performance for secure two-hop LS-MIMO relaying systems under very
is determined by the rate difference between the legitimate practicalassumptions,i.e.,noeavesdropperCSIandimperfect
channel and the eavesdropper channel [3] [4]. To improve legitimate CSI. The contributionsof this paper are three-fold:
the secrecy performance, multi-antenna relaying techniques
1) We reveal the relation between the secrecy outage ca-
are commonlyused in physicallayer security [5]. On the one
pacity and the defined relative distance-dependent path
hand,theuseoftherelayshortenstheaccessdistance,andthus
loss, andthengivethe conditionthatthe secrecyoutage
increasesthelegitimatechannelrate.Ontheotherhand,multi-
capacity is positive.
antenna techniques can be applied to impair the interception
2) We prove that there is one and only one optimal power
signal. The beamforming schemes at the MIMO relay based
at the relay, and propose an optimal power allocation
on global channel state information (CSI) for amplify-and-
scheme.
forward (AF) and decode-and-forward (DF) relaying systems
3) We present several clear insights for secrecy perfor-
werepresentedin[6]and[7],respectively.Notethatthebeam
mance optimization through asymptotic analysis.
design in secure communications requires both legitimate
and eavesdropper CSI [8]. However, it is usually difficult to The rest of this paper is organized as follows. We first
obtain eavesdropper CSI due to the well hidden property of give an overview of the secure LS-MIMO relaying system
the eavesdropper. In this context, the beam is not optimal, in Section II, and then analyze and design an optimal power
allocationschemefortherelayinSectionIII.InSectionIV,we wheresisthenormalizedGaussiandistributedtransmitsignal,
presentsomesimulationresultstovalidatetheeffectivenessof P is the transmit power at the source, n is the additive
S R
the proposed scheme. Finally, we conclude the whole paper Gaussian white noise with zero mean and unit variance at the
in Section V. relay.
Then, the relay adopts an amplify-and-forward(AF) relay-
II. SYSTEMMODEL ing protocol to forward the received signal. Due to the low
complexity and good performance in LS-MIMO systems, we
combine maximum ratio combination (MRC) and maximum
ratio transmission (MRT) at the relay to process the received
signal. We further assume that the relay has perfect CSI
Eavesdropper
hR,E about h by channel estimation and gets partial CSI about
a a R,E S,R
S,RhS,R hbeRt,wDeednuethtoecehsatinmnealterdecCipSroIchˆityinTanDdDthsyestreemals.CTShIehrelatioins
R,D R,D
Source given by
h =√ρhˆ + 1 ρe, (2)
R,D R,D
−
a where e is the error noise vector wpith i.i.d. zero mean and
R,Dh unit variance complex Gaussian entries, and is independent
Relay R,D
of hˆ . ρ, scaling from 0 to 1, is the correlation coefficient
R,D
between hˆ and h . Then, the normalized signal to be
R,D R,D
transmitted at the relay can be expressed as
Destination
Fig.1. Anoverview ofasecureLS-MIMOrelaying system. rAF =FyR, (3)
where F is the processing matrix, which is given by
Consideratimedivisionduplex(TDD)two-hopLS-MIMO
relaying system, as shown in Fig.1. It consists of one source, hˆ 1 hH
F= R,D S,R . (4)
aonseindgelsetinanattieonnnaanedacohn,eapnadssoivneeeraevlaeysdrwoipthpeNr, equaniptepnendaws.itIht khˆR,Dk PSαS,RkhS,Rk2+1khS,Rk
R
is worth pointing out that N is quite large in this LS- Thus, the received spignals at the destination and the eaves-
R
MIMO relaying system, i.e. N =100 or larger. In addition, dropper are given by
R
it is assumed that the distance between the source and the
y = P α hH rAF +n , (5)
destination is so long that it is impossible to transmit the D R R,D R,D D
information from the source to the destination directly. The and p
wholesystem worksin a half-duplexmode,whichmeansthat
y = P α hH rAF +n , (6)
a complete transmission requires two time slots. Specifically, E R R,E R,E E
in the first time slot, the source sends the signal to the relay, respectively,where Pp is the transmit power of the relay, n
R D
and then the relay forwards the post-processing signal to the
andn aretheadditiveGaussianwhitenoiseswithzeromean
E
destination during the second time slot. We assume that the
and unit variance at the destination and the eavesdropper.
eavesdropperisfarawayfromthesourceandclosetotherelay,
Since thereis no knowledgeof the eavesdropperchannelat
since it thought the signal comes from the relay. Then, the
the source and the relay, it is impossible to provide a steady
eavesdropper only monitors the transmission from the relay
secrecy rate over all realizations of the fading channels. In
to the destination. Note that this is a common assumption
this paper, we take the secrecy outage capacity C as
SOC
in previous related literatures, because it is difficult for the
the performance metric, which is defined as the maximum
eavesdropper to monitor both the source and the relay.
available rate under the condition that the outage probability
We use √αS,RhS,R, √αR,DhR,D and √αR,EhR,E to rep- that the real transmission rate surpasses the secrecy rate is
resent the channels from the source to the relay, the relay to
equal to a given value ε, namely
thedestination,andthe relayto theeavesdropperrespectively,
where αS,R, αR,D and αR,E are the distance-dependentpath Pr(CSOC >CD CE)=ε, (7)
−
losses and h , h , and h are channel small scale
S,R R,D R,E
fading vectors with independent and identically distributed where CD and CE are the legitimate and the eavesdropper
(i.i.d.)zeromean andunitvariancecomplexGaussian entries. channel rates, respectively.
It is assumed that the channels remain constant during a time Note that CSOC is not an decreasing function of PR, since
slot and fade independently over slots. Thus, the received both CD and CE increase as PR adds. Then, it makes sense
signal at the relay in the first time slot can be expressed as to select an optimal PR. The focus of this paper is on the
optimal power allocation at the relay, so as to maximize the
y = P α h s+n , (1) secrecy outage capacity for a given outage probability.
R S S,R S,R R
p
III. OPTIMAL POWERALLOCATION Remarks:It is knownthatfromTheorem1, 0<rl <1 is a
preconditionforpowerallocationinsuchanLS-MIMOrelay-
In this section, we first analyze the condition that the
secrecy outage capacity is positive, prove the existence of ing system. Given channel conditions and outage probability,
one and only one optimal power, and then design an optimal there is a constraint on the minimum number of antennas at
power allocation scheme for the relay. Finally, we present the the relay in order to fulfill 0 < rl < 1. Then, we have the
asymptotic characteristics of the secrecy outage capacity. following proposition:
Note that accurate performance analysis is the basis of Proposition1:ThenumberofantennasNR attherelaymust
powerallocation.Priorto designingtheoptimalpoweralloca- be greater than −αR,Elnε.
ραR,D
tion scheme, we first reveal the relation between the secrecy Note that even with a stringent requirement on the outage
outagecapacityandthetransmitpower.Basedonthereceived probability, −αR,Elnε canbealwaysmetbyaddingtheanten-
ραR,D
signals in (3) and (4), the signal-to-noise ratio (SNR) at the nas, which is an advantage of an LS-MIMO relaying system.
destination and the eavesdropper can be expressed as In what follows, we only consider the case of 0<rl <1.
γ = PSPRαS,RαR,D|hHR,DhˆR,D|2khS,Rk2 , B. Existence and Uniqueness
D P α hH hˆ 2+ hˆ 2(P α h 2+1)
R R,D| R,D R,D| k R,Dk S S,Rk S,Rk As shown in (10), the secrecy outage capacity is not an
(8)
increasing function of P . Then, there may be an optimal
R
and
power for the relay in the sense of maximizing the secrecy
γ = PSPRαS,RαR,E|hHR,EhˆR,D|2khS,Rk2 . outage capacity. In this subsection, we aim to prove that the
E P α hH hˆ 2+ hˆ 2(P α h 2+1) optimal power exists and is unique.
R R,E| R,E R,D| k R,Dk S S,Rk S,Rk (9) Prior to seeking the optimal power, we first check two
Then, the legitimate and the eavesdropper channel rates are extremecases ofPR. On the onehand,if PR is largeenough,
given by C = W log (1+γ ) and C = W log (1+γ ) the terms B +1 in (10) is negligible, so the secrecy outage
D 2 D E 2 E
respectively, where W is a half of the spectral bandwidth, capacity is reduced as CSOC = W log2 1+ PPRRAAB −
since a complete transmission requires two time slots. Thus,
for the secrecy outage capacity, we have the follow lemma: W log2 1+ PPRRAABrrll = 0. In other word(cid:16)s, when PR(cid:17) is
very large, the SNRs at the destination and the eavesdropper
Lemma 1: For a given outage probability by ε, the (cid:16) (cid:17)
asymptotically approach the same value. Thus, the secrecy
secrecy outage capacity of an LS-MIMO relaying
outage capacity becomes zero. On the other hand, when
system with imperfect CSI can be expressed as
CSOC = W log2 1+ PRαPRS,DPRρNαSR,+RPαSRα,DSρ,RNNR2R+1 − PCR ten=dsWtolozgero,1t+he s0ecrecyWoultoagge c1ap+acit0y is=eq0u.alUnto-
W log2 1+ PPRSαPRR,EαSln,Rεα−(cid:16)RP,SEαNSR,RlnNεR −1 . (cid:17) deSrOthCissituation2,(cid:16)boththBe+r1a(cid:17)te−soflegit2im(cid:16)ateanBd+e1a(cid:17)vesdropper
Pro(cid:16)of:Thesecrecyoutagecapacity(cid:17)canbeobtainedbased channelstend to zero, and thus the secrecy outage capacity is
on (7) by making use of the propertyof channelhardeningin also zero.
LS-MIMOsystems[12].Weomittheproof,andthedetailcan According to Theorem 1, the secrecy outage probability is
be referred to our previous work [10]. positive when 0 < r < 1, so the maximum secrecy outage
l
capacitymustappearatmediumP regime.Then,we getthe
A. Positiveness R
following theorem:
Itisworthpointingoutthatthesecrecyoutagecapacitymay
Theorem2:Fromtheperspectiveofmaximizingthesecrecy
benegativeorzerofromapuremathematicalview.Therefore,
outage capacity, the optimal power at the relay in an LS-
it makes sense to find the condition that the positive secrecy
MIMO relayingsystem exists and is unique, once the relative
outage capacity exists.
distance-dependentpath loss r is less than 1.
Let ρα N = A, α lnε = A r , P α N = l
B, whereRr,D=R−αR,Eln−ε isR,dEefined as th·egrlelatSiveSd,RistaRnce- Proof: Please refer to Appendix II.
l ραR,DNR
dependentpathloss. Then,thesecrecyoutagecapacitycan be
C. Optimal Power Allocation
rewritten as
From Theorem 2, it is known that as long as 0 < r < 1,
P AB l
C = W log 1+ R there is always a unique optimal power. In other words, if
SOC 2 P A+B+1
(cid:18) R (cid:19) the relay applies the optimal power, the LS-MIMO relaying
W log 1+ PRABrl . (10) system gets the maximum secrecy outage capacity. Then, we
− 2(cid:18) PRArl+B+1(cid:19) have the following theorem:
Observing the secrecy outage capacity in (10), we get the Theorem 3: When the relay uses the power PR⋆ =
following theorem: PSαS,RNR+1 , the LS-MIMO relaying system gets
−αR,EραR,DNRlnε
Theorem 1: If and only if 0 < rl < 1, the secrecy outage tqhe maximum secrecy outage capacity, which is given
capacity in an LS-MIMO relaying system in presence of
imperfect CSI is positive. by Cmax = Wlog 1+ PSαS,RNR
Proof: Please refer to Appendix I. SOC 2 1+r−ρααRR,,DENlnRε(1+PSαS,RNR) −
 
x 104
W log 1+ PSαS,RNR . 14
P2roof: Su1b+srtit−uαtρiαRnR,gE,DltnNhεRe(o1+ptPiSmαaSl,RpNoRw)er PR in (14) into 12 PPSS==3200ddBB
CSOC in (10), we can derive the maximum secrecy outage b/s)10 PS=10dB
capRaecmitayr.ks: The optimal power at the relay PR⋆ is an in- Capacity( 8
creasing function of source transmit power P , source-relay e
S g
a
path loss αS,R and outage probability ε, and is a decreasing Out 6
afunndctrieolnayo-feaCvSesIdarcocpuprearcypaρth, relolasys-dαeRs,tDin.atIinonadpdatihtiolno,ssdαueR,tDo Secrecy 4
r = −αR,Elnε <1, the maximum secrecy outage capacity is
l ραR,DNR 2
an increasing function of P , α , α , ε, N and ρ, and
S S,R R,D R
is a decreasing function of αR,E. 00 0.2 0.4 0.6 0.8 1 1.2 1.4
r
l
D. Asymptotic Characteristic
As analyzed above, the optimal power at the relay P⋆ is Fig. 2. Secrecy outage capacity with different relative distance-dependent
R pathlosses.
an increasing function of the power at the source P . Next,
S
we carry out asymptotic analysis to P and get the following
S
theorem: x 104
4.5
andThtheoermemax4im: uAmt tsheecrleocwyoPuStagreegcimapea,ctihtyeCopmtaimx atlenpdowtoerzeProR⋆. 4 POSp t=im −a1l0 PdoBwer
SOC P = 0dB
In the high P region, the maximum secrecy outage capacity 3.5 S
will bPerosoaft:urPalSteeadseanrdefeisritnodAeppepnednednitxoIfIIP.S. pacity (b/s) 3 OPOSpp tt=iimm 10aalld PPBoowweerr
As P approaches zero, the source does not transmit any Ca2.5
S e
g
information to the relay in the first slot, so the maximum uta 2
O
secrecyoutagecapacitytendstozero.WhilePS issufficiently ecy 1.5
large,theforwardnoiseattherelayisalsoamplified,andthus ecr
S
the secrecy outage capacity is saturated and is independentof 1
PS and PR. 0.5
0
IV. SIMULATION RESULTS −40 −30 −20 −10 0 10 20 30 40
SNR (dB)
R
Toexaminetheeffectivenessoftheproposedoptimalpower
allocation scheme for the AF LS-MIMO relaying system, we Fig.3. Secrecycapacity withdifferent SNRR.
present several simulation results in the following scenarios:
we set N = 100, W = 10KHz, ρ = 0.9 and ε = 0.01. We
R
assume that the relay is in the middle of the source and the capacity approaches zero both when P tends to zero and
S
destination. For convenience, we normalize the pass loss as infinity,andthe uniqueoptimalpowerassociated to the maxi-
α = α = 1 and use α to denote the relative path mumsecrecyoutagecapacityappearsinthemediumregionof
S,R R,D S,E
loss. Specifically, α >1 means the eavesdropperis closer P .Furthermore,itisfoundthatbothP⋆ andCmax improves
R,E S R SOC
to therelay thanthedestination.We use SNR =10log P as P increases, which confirms our theoretical claims again.
S 10 S S
and SNR = 10log P to represent the transmit signal- Then, we testify the accuracy of the theoretical expression
R 10 R
to-noise ratio (SNR) in dB at the source and the relay, ofthemaximumsecrecyoutagecapacitywithSNR =10dB.
S
respectively. As seen in Fig.4, the theorem results are well consistent with
First, we show the impact of r on the secrecy outage thesimulationsinthewholeα regionwithdifferentoutage
l R,E
capacity with SNR = 20dB. As seen in Fig.2, the positive probability requirements, which proves the high accuracy of
R
secrecy outage capacity exists only when 0 < r < 1, which the derived performanceexpression. As claimed above, given
l
confirms the claims in Theorem 1. Given a r , the secrecy an outage probability bound by ε, as α increases, the
l R,E
outage capacity increases gradually as P adds. However, maximum outage secrecy capacity decreases. This is because
S
the performance loss by reducing P from 30dB to 20dB is theinterceptioncapabilityoftheeavesdropperenhanceswhen
S
smaller than that by reducing P from 20dB to 10dB. This the interception distance becomes small. What’s more, given
S
is because in the largeP region,the secrecy outage capacity a α , the maximum secrecy outage capacity increases with
S R,E
tends to be saturated. the increase of ε.
Second, we validate the existence and uniqueness of the Next, we show the performance gain of the proposed
optimal power P⋆. As showed in Fig.3, the secrecy outage optimalpowerallocationschemecomparedwithafixedpower
R
x 104 x 104
6.5 5
Theoretical(e=0.10)
6 Simulation(e=0.10) 4.5
Theoretical(e=0.05)
4
acity (b/s)5.55 SSSiiimmmuuulllaaatttiiiooonnn(((eee===000...000511))) acity (b/s)3.35
p p
a a
C C
e 4.5 e 2.5
g g
a a
Secrecy Out3.45 Secrecy Out1.25 aaa RREE===012...500
1 RE
3
0.5
2.5 0
0.2 0.6 1 1.4 1.8 −40 −30 −20 −10 0 10 20 30 40
a SNR (dB)
R,E S
Fig.4. Comparisonoftheoretical andsimulation results. Fig.6. Maximumsecrecycapacity withdifferent SNRS.
x 104
7 We present the condition that the secrecy outage capacity is
positive, prove the existence and uniqueness of the optimal
6 Optimal Power Allocation
power at the relay, and propose an optimal power allocation
Fixed Power Allocation
scheme.Moreover,werevealtheasymptoticcharacteristicsof
s) 5
y (b/ themaximumsecrecyoutagecapacityincasesoflowandhigh
cit 4 source transmit powers.
a
p
a
C
e 3 APPENDIX A
g
a
ut PROOF OF THEOREM1
O
y 2
ec To get the condition that the secrecy outage capacity is
cr
Se 1 positive, we first rewrite (10) as
P AB
0 C = Wlog 1+ R
SOC 2 P A+B+1
(cid:18) R (cid:19)
−1
0 0.5 1 1.5 2 2.5 3 3.5 4 P AB
a Wlog 1+ R . (11)
R,E − 2 PRA+ Br+l1!
Fig.5. Performance gainwithdifferent αR,E. Examining(11), it is foundthat if and only if 0<rl <1, the
secrecyoutagecapacityispositive.Accordingtothedefinition
of the relative distance-dependent path loss r = −αR,Elnε,
allocationschemewithSNRS =10dB.Itisworthpointingout 0<r <1 is equivalent to the following condlition:ραR,DNR
l
the fixed scheme uses a fixed power P = 20dB regardless
R
α lnε
of channel conditions and system parameters. As seen in N > − R,E . (12)
R
Fig.5, the optimal power allocation scheme performs better ραR,D
than the fixed scheme. Even with a large αR,E, such as In other words, only when N > −αR,Elnε, the secrecy
αR,E = 4, namely short-distance interception, the optimal outage capacity is positive. TherRefore, wρeαgRe,Dt Theorem 1 and
scheme can still achieve a high performance gain, which
Proposition 1.
proves the effectiveness of the proposed scheme.
Finally, we show the effectof PS on the maximumsecrecy APPENDIX B
outage capacity. As seen in Fig.6, when PS tends to zero, PROOF OF THEOREM2
the maximum secrecy outage capacity with different αR,E Atfirst,wetakederivativeof(10)withrespecttoPR,which
approacheszero.InthelargePS region,themaximumsecrecy is given by (13) at the top of the nextpage. Let C′ =0, we
soc
outage capacity will be saturated for a given ε, which proves get two solutions
the Theorem3again.Consistentwithourtheoreticalanalysis,
1
the performance ceiling is an decreasing function of α . P = r (B+1), (14)
R,E R l
Ar
l
V. CONCLUSION and p
This paper focus on the optimal power allocation for a 1
P = r (B+1). (15)
secure AF LS-MIMO relaying system with imperfect CSI. R −Ar l
l
p
C′ = W B(1+B) A Arl .
soc ln2 (P A+B+1)2+P AB(P A+B+1) − (P Ar +B+1)2+P ABr (P Ar +B+1)
(cid:18) R R R R l R l R l (cid:19)
(13)
Considering P > 0, (14) is the unique optimal solution B
R W log 1+
hinavtehiCs ′case>. 0W.hOatth’sermwoisree,,iwf Phen>PR1< Ar1rl(Br+l(1B),+w1e)h,awvee ≈ 2 rl(B+1)!
C′ <s0oc.Specifically,C imRprovAerslasPl pincreasesinthe p B
rPeRsgoicoinncfrreoamses0intothAe1rlrpegriSol(nOBCfr+om1),A1wrlhilperlC(BSRO+C1)detcoreiansfiensitays. −W log21+ qBr+l1 (17)
OnlywhenP = 1 r (B+1),thesecrecyoutagecapacity B B
asochluietivoensetxhiestRsmaanxdAimrislupmunilvqauleu.eH. eInnceo,thpweer gweotrtdhse,Tthheeoorepmtim2a.l ≈ W log2(cid:18)1+ √rlB(cid:19)−W log21+ rBl(18)
 q 
B
= W log 1+ W log 1+ r B ,
2 rrl!− 2 l
(cid:16) p (cid:17)
APPENDIX C B
PROOF OF THEOREM4 = W log rl ,
2√qr B
l
1 
According to Theorem 3, the maximum secrecy outage = W log . (19)
capacity can be expressed as 2(cid:18)rl(cid:19)
ρα N
= W log R,D R , (20)
2 α lnε
r (B+1)B (cid:18)− R,E (cid:19)
Cmax = W log 1+ l where(17) and(18) holdtruebecause whenB is big enough,
SOC 2
rl(Bp+1)+rl(B+1)! theconstantterm“1”isnegligible.Hence,wegettheTheorem
p rl(B+1)B 3.
Wlog 1+ ,
− 2 rl(pB+1)+(B+1)! REFERENCES
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p 1
Tech.J.,pp.656-715,Oct.1949.
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