Secure Communication in the Low-SNR Regime: A Characterization of the Energy-Secrecy Tradeoff

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by  Mustafa Cenk Gursoy

Secure Communication in the Low-SNR Regime: A
Characterization of the Energy-Secrecy Tradeoff
Mustafa Cenk Gursoy
Department of Electrical Engineering
University of Nebraska-Lincoln, Lincoln, NE 68588
Email: [email protected]
9 Abstract—1 Secrecycapacityofamultiple-antennawiretapchan- to average input power constraints, energy efficiency improves
0nel isstudiedinthelow signal-to-noiseratio (SNR) regime. Expres- as one operates at lower SNR levels, and the minimum bit
0sions for the first and second derivatives of the secrecy capacity energy is achieved as SNR vanishes [11]. Hence, requirements
2withrespecttoSNRatSNR=0arederived.Transmissionstrategies
onenergyefficiencynecessitateoperationinthelow-SNRregime.
requiredto achieve these derivatives are identified.In particular, it
nis shown that it is optimal in the low-SNR regime to transmit in Additionally,operatingatlowSNRlevelshasitsbenefitsinterms
JawthheermeaHximmuamnd-eHigeendvaenluoeteetigheencshpaanceneolfmΦatr=iceHs†masHsomcia−tedNNmweiHth†eHthee ofIlnimtihtiinsgpthapeeirn,teirnferoerndceer itnowaidredlreesssssythsetemtws.o critical issues
0legitimate receiver and eavesdropper, respectively, and Nm and Ne of security and energy-efficiency jointly, we study the secrecy
2arethenoisevariancesatthereceiverandeavesdropper,respectively.
Energy efficiency is analyzed by finding the minimum bit energy capacity in the low-SNRregime. We consider a generalmultiple-

]requiredforsecureandreliablecommunications,andthewideband input and multiple-output (MIMO) channel model and identify
T
slope. Increased bit energy requirements under secrecy constraints the optimal transmission strategies in this regime under secrecy
Iare quantified. Finally, the impact of fading is investigated. constraints. Since secrecy capacity is in general smaller than
.
s
the capacity attained in the absence of confidentiality concerns,
c I. INTRODUCTION
[ energyperbitrequirementsincreaseduetosecrecyconstraints.In
Securetransmissionofconfidentialmessagesisacriticalissue
this work, we quantify these increased energy costs and address
1in communication systems and especially in wireless systems
the energy-secrecy tradeoff.
vdue to the broadcast nature of wireless transmissions. In [1],
0
Wyner addressed the transmission security from an information- II. CHANNEL MODEL
3
theoreticpointofview,andidentifiedtherate-equivocationregion
1 We consider a MIMO channel model and assume that the
3and established the secrecy capacity of the discrete memoryless transmitter, legitimate receiver, and eavesdropper are equipped
.wiretap channel in which the wiretapper receives a degraded
1 with nT,nR, and nE antennas, respectively. We further assume
version of the signal observed by the legitimate receiver. The
0 that the channel input-output relations between the transmitter
9secrecy capacity is defined as the maximum communicationrate and legitimate receiver,and the transmitter and eavesdropperare
0from the transmitter to the legitimate receiver, which can be
given by
:achieved while keeping the eavesdroppercompletely ignorantof
v
ithe transmitted messages. Later, these results are extended to ym =Hmx+nm and ye =Hex+ne, (1)
X
Gaussian wiretap channelin [2]. In [3], Csisza´r and Ko¨rnercon-
respectively.Above,xdenotesthe n ×1–dimensionaltransmit-
rsideredamoregeneralwiretapchannelmodelandestablishedthe T
a ted signal vector. This channel input is subject to the following
secrecycapacitywhenthetransmitterhasacommonmessagefor
average power constraint:
two receivers and a confidential message to only one. Recently,
there has been a flurry of activity in the area of information- E{kxk2}=tr(K )≤P (2)
x
theoretic security, where, for instance, the impact of fading,
where tr denotes the trace operation and K = E{xx†} is
cooperation, and interference on secrecy are studied (see e.g., x
the covariance matrix of the input. In (1), n ×1–dimensional
[4]andthearticlesandreferencestherein).Severalrecentresults R
y and n × 1–dimensional y represent the received signal
also addressed the secrecy capacity when multiple-antennas are m E e
vectors at the legitimate receiver and eavesdropper, respectively.
employed by the transmitter, receiver, and the eavedropper [5]–
Moreover, n with dimension n ×1 and n with dimension
[9]. The secrecy capacity for the most general case in which m R e
n ×1areindependent,zero-meanGaussianrandomvectorswith
arbitrary number of antennas are present at each terminal has E
E{n n† }=N IandE{n n†}=N I,whereIistheidentity
been established in [8] and [9]. m m m e e e
matrix. The signal-to-noise ratio is defined as
In addition to security issues, another pivotal concern in most
wireless systems is energy-efficient operation especially when E{kxk2} P
wireless units are powered by batteries. From an information- SNR= E{kn k2} = n N . (3)
m R m
theoretic perspective, energy efficiency can be measured by the
Finally, in the channelmodels, H is the n ×n –dimensional
energy required to send one information bit reliably. It is well- m R T
channel matrix between the transmitter and legitimate receiver,
known that for unfaded and fading Gaussian channels subject
and H is the n × n –dimensional channel matrix between
e E T
1ThisworkwassupportedinpartbytheNSFCAREERGrantCCF-0546384. thetransmitterandeavesdropper.Whilebeingfixeddeterministic
matrices in unfaded channels, Hm and He in fading channels tive of the secrecy capacity at SNR=0 is given by
are random matrices whose components denote the fading coef-
l
ficients between the corresponding antennas at the transmitting C¨ (0)=−n min α α |u†H† H u |2
s R i j j m m i
and receiving ends. αi∈{[α0i,1}]∀iiX,j=1 (cid:18)
Pli=1αi=1
III. SECRECY IN THELOW-SNRREGIME N2
− m|u†H†H u |2 1{λ (Φ>0)} (8)
Recently, in [8] and [9], it has been shown that when the N2 j e e i max
e (cid:19)
channelmatricesH andH arefixedfortheentiretransmission
m e where l is the multiplicity of λ (Φ) > 0, {u } are the
max i
period and are known to all three terminals, then the secrecy
eigenvectorsthat span the maximum-eigenvalueeigenspace, and
capacity in nats per dimension is given by2 1 if λ (Φ)>0
1{λ (Φ) > 0} = max is the indicator
1 1 max 0 else
Cs =nR tr(KmKxxa(cid:23))x≤0Plogdet(cid:18)I+ NmHmKxH†m(cid:19) function. (cid:26)
Proof: We first note that the input covariance matrix K =
1 x
−logdet I+ N HeKxH†e (4) E{xx†}isbydefinitionapositivesemidefiniteHermitianmatrix.
(cid:18) e (cid:19) AsaHermitianmatrix,K canbewrittenas[13,Theorem4.1.5]
x
where the maximization is over all possible input covariance
matrices Kx (cid:23) 03 subject to a trace constraint. We note that Kx =UΛU† (9)
since logdet I+1/N H K H† is a concave function of
m m x m where U is a unitary matrix and Λ is a real diagonal matrix.
K , the objective function in (4) is in general neither concave
x (cid:0) (cid:1) Using (9), we can also express Kx as
nor convex in K , making the identification the optimal input
x
covariance matrix a difficult task. nT
K = d u u† (10)
In this paper, we concentrate on the low-SNR regime. In this x i i i
i=1
regime, the behavior of the secrecy capacity can be accurately X
where {d } are the diagonalcomponentsof Λ, and {u } are the
predicted by its first and second derivatives with respect to SNR i i
column vectors of U and form an orthonormal set. Assuming
at SNR=0:
that the input uses all the available power, we have tr(K ) =
Cs(SNR)=C˙s(0)SNR+ C¨s(0)SNR2+o(SNR2). (5) ni=T1di =P.NotingthatKx ispositivesemidefiniteandhxence
2 d ≥ 0, we can write d = α P where α ∈ [0,1] ∀i and
i i i i
Moreover,C˙s(0) andC¨s(0)also enableusto analyzethe energy Pni=T1αi = 1. Now, the secrecy rate achieved with a particular
efficiency in the low-SNR regime through [11] covariance matrix Kx can be expressed as
P
NEb0s,min = Cl˙osg(02) and S0 = 2−hCC˙¨ss((00))i2 (6) Is(SNR)= n1R logdet I+nRSNRXin=T1αiHmuiu†iH†m!
n N nT
where NEb0s,min denotes the minimum bit energy required for −logdet I+ RNem SNR αiHeuiu†iH†e!!.
reliablecommunicationundersecrecyconstraints,andS denotes i=1
0 X
(11)
the widebandslope which is the slope of the secrecy capacity in
bits/dimension/(3dB)atthepoint Eb .Thesequantitiespro- whereSNR is definedin (3). Asalso notedin [11], we caneasily
N0s,min
videalinearapproximationofthesecrecycapacityinthelow-SNR show that d
logdet(I+vA)| =tr(A), (12)
regime. While Eb is a performance measure for vanishing dv v=0
lSoNwR,bSut0ntoongzeetrhoeNrS0NwsR,misth.inWNEeb0ns,omteinthcahtatrhaectfeorrizmeutlhaefopretrhfeormmiannimceumat ddv22 logdet(I+vA)|v=0 =−tr(A2). (13)
bit energyis valid if Cs is a concave functionof SNR, which we Now, using (12), we obtain the following expressionfor the first
show later in the paper. derivative of the secrecy rate Is with respect to SNR at SNR=0:
The following result identifies the first and second derivatives nT N
of the secrecy capacity at SNR=0. I˙s(0)= αi tr(Hmuiu†iH†m)− Nmtr(Heuiu†iH†e) (14)
Theorem 1: The first derivative of the secrecy capacity in (4) i=1 (cid:18) e (cid:19)
X
with respect to SNR at SNR=0 is = nT α u†H† H u − Nmu†H†H u (15)
C˙s(0)=[λmax(Φ)]+ = 0λmax(Φ) eiflsλemax(Φ)>0 (7) Xin=T1 i(cid:18) i m m i N Ne i e e i(cid:19)nT
(cid:26) = α u† H† H − mH†H u = α u†Φu
where Φ=H†mHm− NNmeH†eHe. Moreover, the second deriva- Xi=1 i i (cid:18) m m Ne e e(cid:19) i Xi=1 i i i
(16)
2Unlessstatedotherwise,alllogarithms throughoutthepaperaretothebasee. where (15) follows from the property that tr(AB) = tr(BA).
res3p(cid:23)ectaivnedly≻, fdoernoHteerpmoistiiatinvemseamtriicdeesfi.nIifteAand(cid:23)poBsit,ivtehednefiAnit−e pBartiiaslaordpeorsiintigvse, Also, in (16), we have defined Φ=H†mHm−NNmeH†eHe. Since
semidefinite matrix.Similarly, A≻Bimplies thatA−Bispositive definite. Φ is a Hermitian matrix and {ui} are unit vectors,we have [13,
2
Theorem 4.2.2] where (22) is obtainedby using the fact that tr(AB)=tr(BA)
and performing some straightforward manipulations. Note again
u†Φu ≤λ (Φ) ∀i (17)
i i max that{u }aretheeigenvectorsspanningthemaximum-eigenvalue
i
where λ (Φ) denotes the maximum eigenvalue of the matrix eigenspace of Φ. Being necessary to achieve the first derivative,
max
Φ. Recall that α ∈ [0,1] and α = 1. Then, from (17), we thecovariancestructuregivenin(20)isalsonecessarytoachieve
i i i
obtain the second derivative. Therefore, the second derivative of the
I˙ (0)= nT α u†PΦu ≤λ (Φ). (18) secrecy capacity at SNR = 0 is the maximum of the expression
s i i i max in (22) over all possible values of {α }. Hence,
i=1 i
X
Note that this upper bound can be achieved if, for instance, N2
C¨ (0)=−n min α α |u†H† H u |2− m|u†H†H u |2
α1 = 1 and αi = 0 ∀i 6= 1, and u1 is chosen as the s R {αi} i,j i j(cid:18) j m m i Ne2 j e e i (cid:19)
eigenvector that corresponds to the maximum eigenvalue of Φ. αi∈[0,1]∀iX
Heretofore, we have implicitly assumed that λ (Φ) > 0 and Pli=1αi=1
max (23)
all the available power is used to transmit the information in
the direction of the maximum eigenvalue. If λmax(Φ)≤0, then Since C¨s(0) is equal to the expressionin (23) when λmax(Φ)>
all eigenvalues of Φ are less than or equal to zero, and hence 0 and is zero otherwise, the final expression in (8) is obtained
Φ is a negative semidefinite matrix. In this situation, none of by multiplying the formula in (23) with the indicator function
the channels of the legitimate receiver is stronger than those 1{λ (Φ)>0}. (cid:4)
max
corresponding ones of the eavesdropper. In such a case, secrecy
capacityiszero.Therefore,ifλmax(Φ)≤0,wehaveC˙s(0)=0. Remark 1: In the absence of secrecy constraints, the first and
Finally,weconcludefrom(18) andthe abovediscussionthatthe second derivatives of the MIMO capacity at SNR=0 are [11]
first derivative of the secrecy capacity with respect to SNR at n
SNR=0 is given by C˙(0)=λmax(H†mHm) and C¨(0)=− lRλ2max(H†mHm) (24)
C˙s(0)=[λmax(Φ)]+ = 0λmax(Φ) eiflsλemax(Φ)>0 . (19) swehceornedldiesrtihveatmivuesltiaprleicaictyhioefveλdmbaxy(tHra†mnsHmmitt)i.nHgeinncteh,ethmeafixrimstuamnd-
(cid:26)
If λ (Φ) > 0 is distinct, C˙ (0) is achieved when we choose eigenvalue eigenspace of H†mHm, the subspace in which the
max s
K = Pu u† where u is the eigenvector that corresponds transmitter-receiverchannelisthestrongest.Duetotheoptimality
x 1 1 1 of the water-filling power allocation method, power should be
to λ (Φ). Therefore, beamforming in the direction in which
max equally distributed in each orthogonal direction in this subspace
the eigenvalue of Φ is maximized is optimal in the sense of
in order for the second derivative to be achieved.
achieving the first derivative of the secrecy capacity in the low-
SNR regime. More generally, if λmax(Φ)>0 has a multiplicity,
Remark 2: We see from Theorem 1 that when there are se-
any covariance matrix in the following form achieves the first
derivative: crecyconstraints,weshouldatlowSNRstransmitinthedirection
l inwhichthetransmitter-receiverchannelisstrongestwithrespect
Kx =P αiuiu†i (20) to the transmitter-eavesdropper channel normalized by the ratio
Xi=1 of the noise variances. For instance, C˙s(0) can be achieved by
where l is the multiplicity of the maximum eigenvalue, beamforming in the direction in which the eigenvalue of Φ is
{ui}li=1 are the eigenvectorsthat span the maximum-eigenvalue maximized. On the other hand, if λmax(Φ) has a multiplicity,
eigenspace, and {αi}li=1 are constants, taking values in [0,1] theoptimizationproblemin(8) shouldbesolvedtoidentifyhow
andhavingthe sum l α =1.Therefore,transmissionin the thepowershouldbeallocatedtodifferentorthogonaldirectionsin
i=1 i
maximum-eigenvalueeigenspace is necessary to achieve C˙ (0). themaximum-eigenvalueeigenspacesothatthesecond-derivative
s
P C¨ (0) is attained. In general, the optimal power allocation
Next,weconsiderthesecondderivativeofthesecrecycapacity. s
strategy is neither water-filling nor beamforming. For instance,
Again, when λ (Φ) ≤ 0, the secrecy capacity is zero and
therefore C¨ (0)m=ax0. Hence, in the following, we consider the consider parallel Gaussian channels for both transmitter-receiver
s and transmitter-eavesdropper links, and assume that H† H =
case in which λ (Φ) > 0. Suppose that the input covariance m m
max diag(5,4,2) and H†H = diag(2,1,1) where diag() is used to
matrix is chosen as in (20) with a particular set of {α }. Then, e e
i
denote a diagonal matrix with components provided in between
using (13), we can obtain
the parentheses. Assume further that the noise variances are
l 2 equal,i.e.,N =N .Then,itcanbeeasilyseenthatλ (Φ)=
m e max
I¨s(0)=−nR tr  αiHmuiu†iH†m!  3andhasamultiplicityof2.Solvingtheoptimizationproblemin
i=1 (8) provides α = 5/12 and α = 7/12. Hence, approximately,
X 1 2
  2 42% of the power is allocated to the channel for which the
N2 l
+n m tr α H u u†H† (21) transmitter-receiverlinkhasastrengthof5,and58%isallocated
RNe2  i=1 i e i i e!  for the channel with strength 4.
X
 N2
=−nR αiαj |u†jH†mHmui|2− Nm2|u†jH†eHeui|2 Remark 3: When λmax(Φ)>0 is distinct, then beamforming
i,j (cid:18) e (cid:19) in the direction in which λ(Φ) is maximized is optimal in the
X
(22) senseofachievingbothC˙ (0)andC¨ (0).Moreover,inthiscase,
s s
3
we have the considered upper bound is tight and
N2 ′ ′
C¨s(0)=−nR(cid:18)kHmu1k4− Nme2kHeu1k4(cid:19) (25) Cs =mp(axx) p(yr′,mye′i|nx)∈DI(x;yr|ye) (33)
where u is the eigenvector that corresponds to λ (Φ). where D is the set of joint conditional density functions
1 max ′ ′ ′ ′
p(y ,y |x) that satisfy p(y |x) = p(y |x) and p(y |x) =
Remark 4: From [13, Theorem 4.3.1], we know that for two r e r r e
p(y |x). Note thatfor fixed channeldistributions, the mutualin-
HermitianmatricesAandBwiththesamedimensions,wehave e
′ ′
formationI(x;y |y )isaconcavefunctionoftheinputdistribu-
r e
λmax(A+B)≤λmax(A)+λmax(B). (26) tionp(x). Sincethe pointwiseinfimumofa setof concavefunc-
′ ′
Applying this result to our setting yields tions is concave [14], f(p(x)) = minp(yr′,ye′|x)∈DI(x;yr|ye)
is also a concave function of p(x). Concavity of the functional
λ (Φ)≤λ (H† H )−λ NmH†H . (27) f and the fact that maximization is over input distributions
max max m m min(cid:18)Ne e e(cid:19) satisfying E{kxk2} ≤ P lead to the concavity of the secrecy
Therefore, we conclude from Remark 1 that secrecy constraints capacity with respect to SNR.
diminish the first derivative C˙ (0) at least by a factor of WecannowwritethefollowingcorollarytoProposition1and
s
Theorem 1.
λ NmH†H when compared to the case in which there
min Ne e e Corollary 1: The minimum bit energy attained under secrecy
are no such constraints.
(cid:16) (cid:17) constraints is
Remark 5: In the case in which each terminal has a single
E log2
b
antenna, the results of Theorem 1 specialize to = . (34)
N [λ (Φ)]+
0s,min max
N + Remark 6: From Remark 4, we can write
C˙ (0)= |h |2− m|h |2 (28)
s (cid:20) m Ne e (cid:21) Eb = log2 ≥ log2
C¨s(0)=− |hm|4− NNm22|he|4 +. (29) N0s,min [λmax(Φ)]+ λmax(H†mHm)−λmin NNmeH†eHe
(cid:20) e (cid:21) log2 Eb (cid:16) (cid:17)
Inthenextresult,weshowthatthesecrecycapacityisconcave ≥ = (35)
in SNR. λmax(H†mHm) N0min
Proposition 1: The secrecy capacity C achieved under the where Eb in (35) denotes the minimum bit energy in the
averagepowerconstraintE{kxk2}≤P issa concavefunctionof absenceNo0fmisnecrecy constraints. Hence, in general, secrecy re-
SNR. quirements increase the energy expenditure. When secure com-
Proof: Concavity can be easily shown using the time-sharing munication is not possible, [λmax(Φ)]+ =0 and NEb0s,min =∞.
argument.AssumethatatpowerlevelP1andsignal-to-noiseratio The expression for the wideband slope S0 can be readily
SNR1, the optimal input is x1, which satisfies E{kx1k2} ≤ P1, obtained by plugging in the expressions in (7) and (8) into that
and the secrecy capacity is Cs(SNR1). Similarly, for P2 and in (6).
SNR2, the optimal input is x2, which satisfies E{kx2k2} ≤ P2, Remark 7: Energycosts of secrecy can easily be identified in
and the secrecy capacity is Cs(SNR2). Now, we assume that the the single-antenna case. Clearly, the minimum bit energy in the
transmitterperformstime-sharingbytransmittingattwodifferent presence of secrecy is strictly greater than that in the absence of
powerlevels using x and x . More specifically, in θ fractionof such constraints:
1 2
the time, the transmitter uses the input x , transmits at most E log2 log2 E
1 b b
= > = (36)
at P1, and achieves the secrecy rate Cs(SNR1). In the remaining N0s,min |h |2− Nm|h |2 + |hm|2 N0min
(1−θ)fractionofthetime,thetransmitteremploysx2,transmits m Ne e
at most at P2, and achieves the secrecy rate Cs(SNR2). Hence, when Nm|h |2h> 0. Furthermorei, the energy requirement in-
this scheme overall achieves the average secrecy rate of Ne e
creasesmonotonicallyasthevalueof Nm|h |2 increases.Indeed,
Ne e
θCs(SNR1)+(1−θ)Cs(SNR2) (30) when NNme|he|2 = |hm|2, secure communication is not possible
by transmitting at the level θE{kx1k2}+(1−θ)E{kx2k2} ≤ and NEb0s,min =∞.
Pθ =θP1+(1−θ)P2.Theaveragesignal-to-noiseratioisSNRθ = IV. THEIMPACT OF FADING
θSNR1 +(1−θ)SNR2. Therefore, the secrecy rate in (30) is an In this section, we assume that the channel matrices H and
achievablesecrecyrateatSNRθ.Since thesecrecycapacityisthe H are random matrices whose componentsare ergodic ramndom
maximum achievable secrecy rate, the secrecy capacity at SNRθ vaeriables, modeling fading in wireless transmissions. We again
is larger than that in (30), i.e.,
assume that realizations of these matrices are perfectly known
Cs(SNRθ)=Cs(θSNR1+(1−θ)SNR2) (31) by all the terminals. As discussed in [12], fading channel can
be regarded as a set of parallel subchannels each of which
≥θCs(SNR1)+(1−θ)Cs(SNR2), (32)
corresponds to a particular fading realization. Hence, in each
showing the concavity. (cid:4) subchannel, the channel matrices are fixed similarly as in the
We further note that the concavity can also be shown using channelmodelconsideredin the previoussection. In[12], Liang
the following facts. As also discussed in [10], MIMO secrecy et al. have shown that having independent inputs for each
capacity is obtained by proving in the converse argument that subchannel is optimal and the secrecy capacity of the set of
4
parallel subchannels is equal to the sum of the capacities of channels under secrecy constraints is
subchannels. Therefore, the secrecy capacity of fading channels
E log2
b
can be be found by averaging the secrecy capacities attained for = . (42)
N E {[λ (Φ)]+}
different fading realizations. 0s,min Hm,He max
Remark 10: Fading has a potential to improve the low-SNR
Weassumethatthetransmitterissubjecttoashort-termpower performance and hence the energy efficiency. To illustrate this,
constraint.Hence,for eachchannelrealization,the same amount we consider the following example. Consider first the unfaded
ofpowerisusedandwehavetr(K )≤P.Withthisassumption, Gaussian channelin whichthe deterministicchannelcoefficients
x
the transmitter is allowed to perform power adaptation in space are h =h =1. For this case, we have
m e
across the antennas, but notacross time. Under such constraints, +
N E log2
it can easily be seen from the above discussion that the average C˙ (0)= 1− m and b = . (43)
s N N +
secrecy capacity in fading channels is given by (cid:20) e (cid:21) 0s,min 1− Nm
Ne
h i
1 1 Now, consider a Rayleigh fading environment and assume that
C = E max logdet I+ H K H†
s nR Hm,He(tr(KKxx(cid:23))≤0P (cid:18) Nm m x m(cid:19) ahbmlesanwdithhevaarreiainncdeespEen{d|ehnt,|2z}er=o-mEe{a|nh,|G2}au=ssi1a.nTrhanend,omwevcaarin-
m e
1 easily find that
−logdet I+ H K H† (37)
(cid:18) Ne e x e(cid:19)) C˙ (0)=E |h |2− Nm|h |2 + = Ne (44)
where the expectation is with respect to the joint distribution of s hm,he m N e N +N
((cid:20) e (cid:21) ) m e
(H ,H ). Note that the only difference between (4) and (37)
m e
is the presence of expectation in (37). Due to this similarity, the leading to NEb0s,min = NlmoNg+e2Ne. Note that if Ne > 0, NmN+eNe >
following result can be obtained immediately as a corollary to +
1− Nm . Hence, fading strictly improves the low-SNR per-
Theorem 1. Ne
fhormanceiby increasing C˙ (0) and decreasing the minimum bit
Corollary 2: Thefirstderivativeoftheaveragesecrecycapac- s
energyevenwithoutperformingpowercontrolovertime.Further
ity in (37) with respect to SNR at SNR=0 is
gains are possible with power adaptation. Another interesting
C˙ (0)=E {[λ (Φ)]+} (38) observation is the following. In unfaded channels, if N ≥N ,
s Hm,He max m e
the minimum bit energy is infinite and secure communication
where again Φ = H† H − NmH†H . The second derivative
m m Ne e e is not possible. On the other hand, in fading channels, the bit
of the average secrecy capacity at SNR=0 is given by
energy is finite as long as N is finite and N > 0. Clearly,
m e
l even if N ≥ N , favorable fading conditions enable secure
m e
C¨s(0)=−nREHm,He min αiαj |u†jH†mHmui|2 transmission in fading channels.
( αi∈{[α0i,1}]∀iiX,j=1 (cid:18) REFERENCES
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