Transmitter Optimization in Slow Fading MISO Wiretap Channel

0.15 MB·

by  Sanjay Vishwakarma

Transmitter Optimization in Slow Fading MISO Wiretap Channel
Sanjay Vishwakarma and A. Chockalingam
Email: [email protected], [email protected]
Department of ECE, Indian Institute of Science, Bangalore 560012, India
Abstract—Inthispaper,weconsiderthetransmitteroptimiza- wiretapcode,denotedby(RD,Rs), we definethenon-outage
tionprobleminslowfadingmultiple-input-single-output(MISO) event as the joint event of the link information rates to K
wiretapchannel.Thesourcetransmitsasecretmessageintended
users be greaterthan or equal to R and the link information
forKusersinthepresenceofJnon-colludingeavesdroppers,and D
6 operates under a total power constraint. The channels between rates to J eavesdroppersbe less than or equal to (RD−Rs).
1 the source and all users and eavesdroppers are assumed to be We minimize the transmit power subject to the total power
0 slow fading, and only statistical channel state information (CSI) constraint and satisfying the probability of the non-outage
2 is known at the source. For a given code rate and secrecy rate eventtobegreaterthanorequaltoadesiredthreshold(1 ǫ).
n pairofthewiretapcode,denotedby(RD,Rs),wedefinethenon- We obtain the achievable (R ,R ) region and the tran−smit
a outageeventasthejointeventofthelinkinformationratestoK beamformingvector.We noteDthatws e differfromthereported
J users be greater than or equal to RD and the link information
1 rates to J eavesdroppers be less than or equal to (RD −Rs). works in [13,14], which also consider multiple eavesdroppers
We minimize the transmit power subject to the total power scenario, in following aspects: i) number of users K can be
3
constraint and satisfyingtheprobabilityof thenon-outageevent morethanone,ii)onlystatisticalCSIoftheuserschannelsare
] to be greater than or equal to a desired threshold (1−ǫ). known, and iii) channel covariance matrices of all users and
T keywords: Physicallayersecurity,MISOwiretapchannel,secrecyrate, eavesdroppers can be arbitrary positive semidefinite matrices.
.I multiple eavesdroppers, slowfading. Notations: A CN1×N2 implies that A is a complex
s matrix of dimension∈N N . A 0 and A 0 imply that
[c I. INTRODUCTION Aisapositivesemidefi1ni×tem2atrixa(cid:23)ndpositive≻definitematrix,

With growing applications on wireless networks, there is respectively. Identity matrix is denoted by I. Transpose and
1
v a need to provide security, along with reliability, from being complex conjugate transpose operations are denoted by [.]T
5 eavesdropped, which can easily happen due to the broadcast and [.]∗, respectively. E[.] denotes the expectation operator,
2 nature of the wireless transmission. Wyner, in his work in and . denotes the 2-norm operator. diag(a) denotes a
2 [1], showed that a message could be transmitted at a rate diagoknalkmatrix with elements of the vector a CN×1 on
0 called secrecy rate, at which the legitimate user could decode its diagonal. Trace of matrix A CN×N is∈denoted by
.0 the message reliably whereas the eavesdropper could be kept Tr(A).h CN×1 (0,H)imp∈liesthath isacircularly
2 ∈ ∼CN
entirely ignorant. The wiretap channel model in [1] was symmetriccomplexGaussianrandomvectorwithmeanvector
0
physicallydegradedand discrete memoryless. Later, the work 0 and covariance matrix H.
6
1 in [1] was extended to more general broadcast channel and
: Gaussian channel in [2] and [3], respectively. Subsequent II. SYSTEMMODEL
v
extension to various multi-antenna wireless wiretap channels Consider a MISO wiretap channel as shown in Fig. 1,
i
X and the corresponding achievable secrecy rates and secrecy which consists of a source S having N transmit antennas, K
r capacitieshave been reportedby manyauthors,e.g., [4]–[11]. users D ,D , ,D each having single antenna, and J
a { 1 2 ··· K}
In [12], secrecy capacity of a quasi-static single-antenna non-colluding eavesdroppers E ,E , ,E each having
1 2 J
{ ··· }
Rayleigh fading channel in terms of outage probability has single antenna. The complex channel gain vector from S to
been characterized. Outage probability characterization of the D is denoted by h C1×N, 1 k K. Likewise,
k k
∈ ≤ ≤
secrecy rate of multiple-input-single-output (MISO) wiretap the complex channel gain vector from S to E is denoted
j
channel with artificial noise has been reported in [13]–[15], by z C1×N, 1 j J. We assume that the channels
j
∈ ≤ ≤
and that of amplify-and-forward relay channel has been re- betweenS toD sandthosebetweenS toE sfadeslowlyand
k j
ported in [16]. Motivated by the need for outage probability independently with h (0,H ), and z (0,Z ).
k k j j
∼CN ∼CN
characterization of secrecy rate in MISO wiretap channel, in These channel gains are assumed to be unknown at S. We
this paper, we consider the transmitter optimization problem assumethatthesourceS operatesundertotalpowerconstraint
in slow fading MISO wiretap channel. The source transmits P . The communication between S and D s happens in n
T k
a secret message intended for K users in the presence of J channeluses.ThesourceS transmitssecretmessageW which
non-colludingeavesdroppers,andoperatesunderatotalpower is equiprobable over 1,2, ,2nRs . For each W drawn
{ ··· }
constraint.The channelsbetweenthe source and all users and equiprobablyfrom the set 1,2, ,2nRs , the source, using
{ ··· }
eavesdroppersare assumed to be slow fading. Only statistical astochasticencoder,mapsW toacodeword x n oflength
channel state information (CSI) is assumed to be known at n, where each x C, i.i.d. (0,1), a{ndi}Ei=[1x 2] = 1.
i i
∈ ∼ CN | |
the source. For a given code rate and secrecy rate pair of the Each codeword {xi}ni=1 belongs to a collection of 2nRD
III. TRANSMITTEROPTIMIZATIONUNDERSECRECY
CONSTRAINT
w h Dk Using (4) and (5), and for a given hk and zj, the informa-
1 2 N k k=1,2,···,K tion rates at D and E are obtained,respectively,as follows:
k j
h w 2
k
R = I(x; y ) = log 1+ | | , (6)
Dk Dk 2 N
(cid:16) 0 (cid:17)
z z w 2
S j REj = I(x; yEj) = log2 1+ | Nj | . (7)
(cid:16) 0 (cid:17)
Further,subject to the constraints in (1) and (3) and using (6)
and (7), the optimization problem to minimize the transmit
power is as follows:
Ej min w 2 (8)
j=1,2,···,J w k k
Fig. 1. System model for MISO wiretap channel with K users and J
s.t. w 2 P , (9)
eavesdroppers. T
k k ≤
h w 2
k
Pr log 1+ | | R , k = 1,2, ,K,
n 2(cid:16) N0 (cid:17) ≥ D ∀ ···
aacnpodpdleitwersaonrthsdmesic(tios.emt.h,pewlewixreewtiagephigtcehodtdwsey)m=wb[howelr1ew,wRh2iDc,h··≥i·s,RwwsNx. ]TTihn∈etsChoeNuri×cth1e RD−Rs ≥ log2(cid:16)1+ |zNjw0|2(cid:17), ∀j = 1,2,··· ,Jo
i (1 ǫ). (10)
channeluse,1 i n.Sincethesourceispowerconstrained, ≥ −
≤ ≤
this implies that Since h s and z s are independent, we rewrite the constraint
k j
(10) in the following equivalent product form:
w 2 P . (1)
T
k k ≤ K
In the following, we will use x to denote the symbols in the Pr hkw 2 2RD 1 N0
codeword {xi}ni=1. Since the channel is slow fading and the kY=1 n| | ≥ (cid:0) − (cid:1) o
CSI is unknown at the source S, we define the non-outage J
bevyent,faonrdaimgipvoesne(tRheDp,Rrosb)abpialiirtyocfotnhsetrwaiinrettoanp codase,fdoellnoowtes:d jY=1Prn|zjw|2 ≤ (cid:0)2(cid:0)RD−Rs(cid:1)−1(cid:1)N0o ≥ (1−ǫ). (11)
E E
We note that solving the optimization problem (8) in its
= R R , k = 1,2, ,K, and
E n Dk ≥ D ∀ ··· original form is hard. So, in order to simplify the analysis,
we replace the product probability constraint in (11) with the
R R R , j = 1,2, ,J , (2)
D− s ≥ Ej ∀ ··· o following K+J individual probability constraints:
Pr( ) (1 ǫ), (3)
E ≥ −
k = 1,2, ,K, and j = 1,2, ,J,
∀ ··· ∀ ···
RwhDekrean(d1R−Eǫj)aries tthheelinnokn-inofuotarmgeatpiornobraabteilsitbyettwhreeesnhoSldt,oaDndk Prn|hkw|2 ≥ (cid:0)2RD −1(cid:1)N0o ≥ (1−ǫ)K+1J, (12)
and S to Ej, respectively. In other words, when the source Pr zjw 2 2 RD−Rs 1 N0 (1 ǫ)K+1J. (13)
selects the target code rate and target secrecy rate pair of the n| | ≤ (cid:0) (cid:0) (cid:1)− (cid:1) o ≥ −
wiretap code as (RD,Rs), the above constraint implies that, We alsonotethatanyw whichsatisfiesallK+J constraints
with probability greater than or equal to (1 ǫ), all Dks will in (12) and (13) will also satisfy the product probability con-
−
be able to successfully decode the transmitted message while straintin(11).However,theconversemaynotalwaysbetrue.
all Ejs will be ignorant about the transmitted message. We Further, since hkw in (12) and zjw in (13) are linear trans-
also note that when the CSI on all the links are known at formationsofcircularlysymmetriccomplexGaussianrandom
S, the achievability of the secrecy rate Rs is shown in [11]. vectors,hkw andzjw arealso circularlysymmetriccomplex
Let yDk and yEj denote the received signals at Dk and Ej, Gaussian random variables, i.e., hkw 0, w∗Hkw ,
respectively. We have and zjw 0, w∗Zjw . This∼fuCrNthe(cid:0)r implies tha(cid:1)t
y = h wx+η , k = 1,2, ,K, (4) |hkw|2 and∼|zjCwN|2(cid:0)are exponen(cid:1)tial random variables, i.e.,
Dk k Dk ∀ ···
yEj = zjwx+ηEj, ∀j = 1,2,··· ,J, (5) |hkw|2 ∼ w∗H1 w exp−w∗Hλkw, λ ≥ 0, (14)
k
wCNhe(r0e,Nηs0).are noise components, assumed to be i.i.d. ∼ |zjw|2 ∼ w∗Z1jw exp−w∗Zλjw, λ ≥ 0. (15)
Using (14) and (15), and by following standard integration the above problemcan be solved using semidefinite program-
steps, we get the following equivalent simplified inequalities ming techniques [17]. But the solution W of the above rank
for the probability constraints in (12) and (13): relaxed optimization problem may not have rank 1. This can
be easily seen from the KKT conditions of the rank relaxed
k = 1,2, ,K, w∗H w a, (16)
∀ ··· k ≥ optimization problem which we discuss in the Appendix. We
∀j = 1,2,··· ,J, w∗Zjw ≤ b, (17) now take the rank-1 approximationas follows. Let w0 be the
where a = (2RD−1)N10 and b = (2(RD−Rs)−1)1N0 . uvanliut-enoorfmWei.gWenedsiurebcsttiiotuntecoWrres=poPnwdingw∗tointhtehelaragbeosvteeriganenk
−ln(1−ǫ)(K+J) −ln(1−(1−ǫ)(K+J)) 0 0
Replacing the constraint in (11) with (16) and (17), we get relaxed optimization problem and solve the resulting linear
the following upper bound for the optimization problem (8): optimization problem for unknown P, i.e.,
min w 2 (18) min P (30)
w k k P
s.t. 0 P P , (31)
s.t. w 2 PT, (19) ≤ ≤ T
k k ≤ k = 1,2, ,K, Pw∗H w a, (32)
k = 1,2, ,K, w∗Hkw a, (20) ∀ ··· 0 k 0 ≥
∀ ··· ≥ j = 1,2, ,J, Pw∗Z w b. (33)
j = 1,2, ,J, w∗Zjw b, (21) ∀ ··· 0 j 0 ≤
∀ ··· ≤
Having obtained the transmit power P from (30), the
We solve the above problem for the following two cases. beamforming vector is √Pw .
0
A. All H s and Z s are diagonal matrices
k j
Remark 1: We note that when the channel CSI h on all
When all H s and Z s are diagonal positive semidefinite k
k j D s are perfectly known at the source S, the constraints (28)
matrices, the optimization problem (18) can be written as the k
and (29) in the optimization problem (26) should be replaced
following equivalent linear optimization problem:
with the following constraints, respectively:
N
Tr(Wh∗h ) (2RD 1)N , (34)
min Pm (22) k k ≥ − 0
P1,P2,···,PN mX=1 Tr(WZ ) (2(RD−Rs)−1)N0. (35)
j ≤ ln(1 (1 ǫ)J1)
N − − −
Remark 2: When the source transmits the symbol x
s.t. m = 1,2, ,N, P 0, P P , (23)
∀ ··· m ≥ m ≤ T from an equiprobable complex finite alphabet set A =
mX=1 a ,a , ,a of size M (e.g., M-ary) with E[x]=0 and
k = 1,2, ,K, N P Hmm a, (24) {E[1x22]=··1·,thMei}nformationratesin(6)and(7)canbewritten
∀ ··· mX=1 m k ≥ in|th|e following forms, respectively:
N h w 2
k
∀j = 1,2,··· ,J, PmZjmm ≤ b, (25) RDk = I(x; yDk) = I(cid:18)| N0| (cid:19), (36)
mX=1 z w 2
j
0w,haenredPZm ==|wdimag|2(,[HZ1k1,=Z2d2i,ag([H,Zk1N1,NH]Tk22),···0,.HTkNheNa]Tb)ov(cid:23)e REj = I(x; yEj) = I(cid:18)| N0| (cid:19), (37)
j j j ··· j (cid:23) where
problem can be easily solved using linear optimization tech-
M
nveiqcutoers.wHiasvi[n√gP1o,b√taPin2e,d··P·1,,√PP2,N·]·T·.,PN, the beamforming I(ρ) =△ M1 Xl=1Z pn(cid:0)y −√ρal(cid:1)
B. Some of Hks or Zjs are not diagonal matrices log pn(y −√ρal) dy, (38)
Here, we considerthe generalcase when H s and Z s are 2 M
Hermitian positive semidefinite matrices and sokme of Hjks or M1 mP=1pn(y −√ρam)
Zjs are not diagonal. Define W =△ ww∗. We rewrite the and p (θ)= 1e−|θ|2. Using the fact that the mutualinforma-
optimizationproblem(18) into the followingequivalentform: n π
tionfunction,I(ρ), isastrictly-increasingconcavefunctionin
min Tr(W) (26) ρ [18,19], K +J constraints in (20) and (21) can be written
W in the following forms, respectively:
s.t. W (cid:23)0, rank(W)=1, Tr(W) ≤ PT, (27) ∀k = 1,2,··· ,K, w∗Hkw ≥ a, (39)
∀k = 1,2,··· ,K, Tr(WHk) ≥ a, (28) ∀j = 1,2,··· ,J, w∗Zjw ≤ b, (40)
j = 1,2, ,J, Tr(WZj) b. (29) where a = I−1(RD)N0 and b = I−1(RD−Rs)N0 . With
∀ ··· ≤ 1 1
−ln(1−ǫ)(K+J) −ln(1−(1−ǫ)(K+J))
Theaboveoptimizationproblemisa non-convexoptimization finite alphabetinput, the optimizationproblem (18) should be
problem. However, by relaxing the rank(W)=1 constraint, solved subject to the constraints in (39) and (40).
IV. RESULTSAND DISCUSSIONS observe that the maximum achievable secrecy rate Rs and
the corresponding minimum transmit power increases with
We haveevaluatedthe secrecyrate throughsimulationwith
increaseinR .Thesecrecyratedropstozerowhentheentire
the following system parameters: N = 3, K = 2, J = D
available power, P =12 dB, is used.
1,2,3, N = 1, ǫ = 0.1, and P = 12 dB. We consider T
0 T ScenarioofSectionIII-A:Here,we take H ,H , Z , Z ,
the scenarios discussed in Section III-A and Section III-B. 1 2 1 2
and Z as the diagonalapproximationof covariance matrices
Scenarioof Section III-B: We have used the followingpos- 3
in (41), (42), (43), (44), and (45), respectively. We solve the
itive definite channel covariance matrices in the simulations:
linear optimization problem (22) using the tools in [20,21],
and we plot the maximum achievable R vs R and the
2.1670, 0.1806+0.0183i, −0.1453−0.3101i s D
H1= 0.1806−0.0183i, 1.9165, 0.0696+0.3374i ≻0 (41) corresponding mimimum transmit power vs RD in Fig 3(a)
 −0.1453+0.3101i, 0.0696−0.3374i, 1.4180  and Fig 3(b), respectively. As in Fig 2(a) and Fig 2(b), we
1.9834, −0.2001+0.0250i, 0.0470−0.3424i
H2= −0.2001−0.0250i, 1.3867, 0.0149−0.2083i ≻0 (42)
 0.0470+0.3424i, 0.0149+0.2083i, 1.4323  1 12
0.0043, 0.0010−0.0003i, 0.0013+0.0009i
Z1= 0.0010+0.0003i, 0.0074, −0.0011−0.0029i ≻0 (43) 0.9 One Eavesdropper 10
 0.0013−0.0009i, −0.0011+0.0029i, 0.0079  8
0.8
0.0069, 0.0004−0.0029i, −0.0014+0.0014i
Z2= 0.0004+0.0029i, 0.0070, −0.0019−0.0002i ≻0 (44) e Two Eavesdroppers 6
 −0.0014−0.0014i, −0.0019+0.0002i, 0.0086  Us0.7 4 One Eavesdropper
Z3= −000..00.0000912016,+−−00.00..0000000926i6,i,+−000..00.0000601463,i,−−000..0.00000111183i−,+000.0..00000051948ii ≻0 (45) Channel00..56 Three Eavesdroppers er (dB) 02 Two Eavesdroppers
For a given (RD,Rs) pair, we solve the semidefinite rank Bits/0.4 Pow−2
relaxed optimization problem (26) using the tools in [20,21]. − −4 Three Eavesdroppers
s0.3
We numerically observe that, for any feasible (RD,Rs) pair, R −6
thesolutionW ofthe rankrelaxedoptimizationproblem(26) 0.2
−8
has rank 1. This implies that for such channel realizations,
0.1
rank-1 approximation is not needed. In Fig 2(a), we plot −10
0 −12
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2
R − Bits/ChannelUse R (Bits/ChannelUse)
1 12 D D
(a) R vs R . (b) Transmit Power vs R .
0.9 One Eavesdropper 10 s D D
8
0.8 Fig. 3. Rs vs RD and Transmit Power vs RD in MISO wiretap channel
Two Eavesdroppers 6 with N = 3, K = 2, J = 1,2,3, N0 = 1, ǫ = 0.1 and PT = 12 dB,
)
e0.7 anddiagonal covariance matrices.
Us 4 One Eavesdropper
nnel0.6 Three Eavesdroppers dB) 2 observe that the maximum achievable secrecy rate Rs and
ha0.5 er ( 0 Two Eavesdroppers the corresponding minimum transmit power increases with
C w increaseinR .Thesecrecyratedropstozerowhentheentire
(Bits/0.4 Po−−42 Three Eavesdroppers available powDer, PT =12 dB, is used.
R s0.3 V. CONCLUSIONS
−6
Weconsideredthetransmitteroptimizationprobleminslow
0.2
−8 fading MISO wiretap channel. Secret message transmitted by
0.1 −10 the source was intended for K users in the presence of J
eavesdroppers. For a given code rate and secrecy rate pair
0 −12
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 of the wiretap code, denoted by (R ,R ), we defined the
R (Bits/ChannelUse) R (Bits/ChannelUse) D s
D D non-outage event and minimized the transmit power subject
(a) R vs R . (b) Transmit Power vs R . to the total power constraint and satisfying the probability of
s D D
the non-outage event to be greater than a desired threshold
(1 ǫ). We obtained the achievable (R ,R ) region and the
Fig. 2. Rs vs RD and Transmit Power vs RD in MISO wiretap channel − D s
with N = 3, K = 2, J = 1,2,3, N0 = 1, ǫ = 0.1 and PT = 12 dB, transmit beamforming vector.
andnon-diagonal covariance matrices.
APPENDIX
the maximum achievable R vs R . In Fig 2(b), we plot Inthisappendix,weanalyzetherankoftheoptimalsolution
s D
the corresponding mimimum transmit power vs R . We W oftherankrelaxedoptimizationproblem(26).Wetakethe
D
Lagrangian[17]oftherankrelaxedoptimizationproblem(26) [8] F.OggierandB.Hassibi, “Thesecrecy capacity oftheMIMOwiretap
as follows: channel,” Proc.IEEEISIT’2008,Jul.2008.
[9] A.KhistiandG.Wornell,“Securetransmissionwithmultipleantennas-
ℓ W, Λ, λ, µ , ν = Tr(W) Tr(ΛW) I:TheMISOMEwiretapchannel,”IEEETrans.Inform.Theory,vol.56,
k j
− no.7,pp.3088-3104,Jul.2010.
(cid:0) (cid:1) K [10] A.KhistiandG.Wornell,“Securetransmissionwithmultipleantennas-
+ λ Tr(W) P + µ a Tr(WH ) II: TheMIMOME wiretap channel,” IEEETrans. Inform. Theory, vol.
T k k
(cid:0) − (cid:1) kX=1 (cid:0) − (cid:1) 56,no.11,pp.5515-5532, Nov.2010.
[11] Y. Liang, G. Kramer, H. V. Poor, and S. Shamai, “Compound wiretap
J channels,” EURASIP Journ. on Wireless Commun. and Net., volume
+ νj Tr(WZj) b , (46) 2009,article ID142374,12pages.doi:10.1155/2009/142374.
−
Xj=1 (cid:0) (cid:1) [12] M. Bloch, J. Barros, M. R. D. Rodrigues, and S. W. McLaughlin,
“Wireless information-theoretic security,” IEEETrans. Inform. Theory,
where Λ 0, λ 0, µ 0, and ν 0 are Lagrangian vol.54,no.6,pp.2515-2534, Jun.2008.
k j
multipliers(cid:23). The K≥KT condi≥tions are as fo≥llows: [13] N.R.Zurita,M.Ghogho,andD.McLernon,“Outageprobabilitybased
power distribution between data and artificial noise for physical layer
security,” IEEE Signal Process. Lett., vol. 19, no. 2, pp. 71-74, Feb.
K1. All the constraints in (27), (28), and (29) excluding the
2012.
constraint rank(W)=1, [14] J Xiong, K. K. Wong, D. Ma, and J. Wei, “A closed-form power
allocation forminimizingsecrecyoutageprobability forMISOwiretap
K2. Tr(ΛW) = 0. Since Λ 0 and W 0, this implies channels via masked beamforming,” IEEECommun. Lett., vol. 16, no.
that ΛW =0, (cid:23) (cid:23) 9,pp.1496-1499, Sep.2012.
[15] X. Zhang, X. Zhou, and M. R. McKay, “On the design of artificial-
K3. λ Tr(W) P =0, noise-aidedsecuremulti-antennatransmissioninslowfadingchannels,”
T
− IEEETrans.Veh.Tech.,vol.62,no.5,pp.2170-2181,Jun.2013.
K4. k(cid:0) =1,2, ,K(cid:1), µ a Tr(WH ) =0, [16] J.Huang,A.Mukherjee, andA.L.Swindlehurst, “Securecommunica-
k k
∀ ··· − tionviaanuntrustednon-regenerative relayinfadingchannels,”IEEE
K5. j =1,2, ,J, ν (cid:0)Tr(WZ ) b =(cid:1) 0, Trans.SignalProcess.,vol.61,no.10,pp.2536-2550, May2013.
j j
∀ ··· − [17] S. Boyd and L. Vandenberghe, Convex optimization, Cambridge Univ.
K6. ∂ℓ = 0 implies that(cid:0)Λ = (1+λ)I (cid:1) K µ H + Press,2004.
∂WJj=1νjZj (cid:23)0, −Pk=1 k k [18] Dm.inGimuuom, Sm.eaSnh-asmquaai(rSehEitrzro),riannGdauSs.siVanercdhua,n“nMelsu,”tuIaElEiEnfTorramnast.ioInnfoarnmd.
ThPe KKT conditions(K2), (K6), (K4), and (K5) imply that Theory,vol.51,no.4,pp.1261-1282, Apr.2005.
(1+λ)Tr(W) K µ a+ J ν b = 0. For W = 0, [19] A. Lozano, A. M. Tulino, and S. Verdu, “Optimum power allocation
− k=1 k j=1 j 6 forparallel Gaussianchannels witharbitrary inputdistributions,” IEEE
thisfurtherimpliesPthatnotallµkPscanbezerosimultaneously. Trans.Inform.Theory,vol.52,no.7,pp.3033-3051, Jul.2006.
With this, we rewrite (K6) in the following form: [20] J. Sturm, “Using SeDuMi 1.03: A MATLABtoolbox for optimization
oversymmetricones,”Opt.MethodsandSoftware,vol.11-12,pp.625-
K J 653,1999.SpecialissueonInteriorPointMethods(CDsupplementwith
Λ+ µ H = (1+λ)I + ν Z 0. (47) software).
k k j j
≻ [21] J. Lofberg, “YALMIP: A toolbox for modeling and optimization in
kX=1 Xj=1 MATLAB,” Proc. CACSD Conf., Taipei, 2004. [Online] Available:
The above equation implies that rank(Λ + http://control.ee.ethz.ch/ joloef/yalmip.php.
K µ H ) = N. This further implies that
k=1 k k
trTPhahanitskrm(aΛen)akn(sW≥th)at≤tNherar−annkk(rPaonfKkkW=(1PµmKkk=aHy1kµn)koHt(abskes)u.omnei(nK.g2)Wim6=pl0ie)s.
For the special case when K = 1 and H is a rank one
1
positivesemidefinitematrix,(47)impliesthatrank(Λ) N
1. Assuming W = 0, (K2) further implies that rank≥(Λ) −=
N 1, and rank(6 W)=1.
−
REFERENCES
[1] A.Wyner,“Thewire-tapchannel,”Bell.SystTech.J,vol.54,no.8,pp.
1355-1387, Jan.1975.
[2] I. Csiszar and J. Korner, “Broadcast channels with confidential mes-
sages,”IEEETrans.Inform.Theory,vol.IT-24,pp.339-348,May1978.
[3] S. K.Leung-Yan-Cheong and M. E.Hellman, “The Gaussian wire-tap
channel,” IEEE Trans. Inform. Theory, vol. IT-24, pp. 451-456, Jul.
1978.
[4] Y. Liang, H. V. Poor, and S. Shamai (Shitz), “Information theoretic
security,” Foundations andTrendsinCommunications andInformation
Theory,NOWPublishers,vol.5,no.4-5,2009.
[5] Z.Li, R.Yates, andW.Trappe, “Secure communication with afading
eavesdropper channel,” Proc.IEEEISIT’2007, Jun.2007.
[6] S.ShafieeandS.Ulukus,“AchievableratesinGaussianMISOchannels
withsecrecy constraint,” Proc.IEEEISIT’2007,Jun.2007.
[7] P. K. Gopala, L. Lai, and H. E. Gamal, “On the secrecy capacity of
fadingchannels,”IEEETrans.Inform.Theory,vol.54,no.10,pp.4687-
4698,Oct.2008.