On the Design of Artificial-Noise-Aided Secure Multi-Antenna Transmission in Slow Fading Channels

0.44 MB·English

by  Xi Zhang

1
On the Design of Artificial-Noise-Aided
Secure Multi-Antenna Transmission
in Slow Fading Channels
Xi Zhang, Student Member, IEEE, Xiangyun Zhou, Member, IEEE,
and Matthew R. McKay, Member, IEEE
Abstract—In this paper, we investigate the design of artificial- guarantee secure transmission regardless of the eavesdrop-
noise-aided secure multi-antenna transmission in slow fading per’scomputationalcapability.Assuch,thesetechniqueshave
3
channels. The primary design concerns include the transmit
1 drawn a lot of recent attention from the research community.
power allocation and the rate parameters of the wiretap code.
0
Weconsidertwoscenarioswithdifferentcomplexitylevels:i)the
2
design parameters are chosen to be fixed for all transmissions,
A. Background and Previous Work
n ii) they are adaptively adjusted based on the instantaneous
a channel feedback from the intended receiver. In both scenarios, The notion of perfect secrecy was first introduced by Shan-
J we provide explicit design solutions for achieving the maximal non [1]. Subsequently, pioneering works on physical-layer

0 throughputsubjecttoasecrecyconstraint,givenbyamaximum security [2,3] proved that there exist coding schemes which
2 allowable secrecy outage probability. We then derive accurate
canensuretransmissionreliabilityandperfectsecrecysimulta-
approximationsforthemaximalthroughputinbothscenariosin
] the high signal-to-noise ratio region, and give new insights into neously. Many recent papers have expanded upon these initial
T the additional power cost for achieving a higher security level, contributions, considering different system configurations and
I whilst maintaining a specified target throughput. In the end, assumptions. Particularly, multi-antenna techniques have been
s. the throughput gain of adaptive transmission over non-adaptive extensivelystudiedasameansforachievingsecurityenhance-
c transmission is also quantified and analyzed.
[ ments [4–8]. However, in much of the literature on physical-
IndexTerms—Physical-layersecurity,multi-antennatransmis- layer security, the eavesdropper’s channel state information
3 sion,artificialnoise,powerallocation,secrecyoutageprobability,
(CSI) was assumed to be available at the transmitter, which is
v throughput optimization.
usually impractical. To relax this strong assumption, the au-
0
4 thorsin[9]proposedamulti-antennatransmissionschemethat
6 I. INTRODUCTION insertsartificialnoiseintothetransmittedsignalinacontrolled
3 manner, thus to confuse the malicious eavesdropper. This
2. WITH the rapid development of wireless technology, transmission scheme requires the instantaneous CSI feedback
1 an ever-increasing amount of sensitive data (e.g., pri- from the intended receiver, but not the eavesdropper, which is
2 vate conversation, credit card information) is transmitted over amajoradvancementtowardpracticalsecurecommunications.
1
wireless networks. However, the broadcasting nature of the Building on the ideas from [9], the design and analysis of
:
v wireless medium makes it especially vulnerable to malicious artificial-noise-aided transmission has been further studied for
Xi interception. Currently, cryptographic algorithms, typically both fast and slow fading channels [10–16]. For fast fading
designed without exploiting the physical properties of the channels,thechannelcoherencetimeismuchshorterthanthe
r
a wireless medium, are used to keep broadcasted messages codeword length and the ergodic secrecy rate is often used as
confidential. For such techniques, although the expenditure of theperformancemetricfordesigningbeamformingandpower
interception may be very high, providing robust encryption allocation strategies [10–12]. For slow fading channels, the
algorithms is becoming ever more challenging, due to the channel coherence time is usually longer than the codeword
continuing development of computing devices. By taking the length, and in such scenarios outage-based formulations be-
physicalpropertiesofthewirelesschannelsintoconsideration, come more appropriate. To this end, various secrecy outage
the recently developed physical-layer security techniques can formulations were proposed first in [17,18] and recently
in[19,20].Inparticular,thefirstformulationdevelopedin[17,
Copyright (c) 2012 IEEE. Personal use of this material is permitted. 18] has been used for studying artificial-noise-aided multi-
However, permission to use this material for any other purposes must be
antennatransmissionschemesin[13–16].Thissecrecyoutage
[email protected].
X. Zhang and M. R. McKay are with the Department of Electronic and formulation characterizes the possibility of having a secure
Computer Engineering, Hong Kong University of Science and Technology, andreliabletransmission,withoutdistinguishingsecrecyfrom
HongKong(e-mails:[email protected],[email protected]).
reliability. In other words, a secrecy outage event defined
X. Zhou is with the Research School of Engineering, the Australian
NationalUniversity,Australia(e-mail:[email protected]). therein may occur due to either an insecure link to the eaves-
TheworkofX.ZhangandM.R.McKaywassupportedbytheHongKong dropperoranunreliablelinktotheintendedreceiver.Tobetter
ResearchGrantsCouncil(GrantNo.616312).
assist the secure transmission design, revised secrecy outage
TheworkofX.ZhouwassupportedbytheAustralianResearchCouncil’s
DiscoveryProjectsfundingscheme(ProjectNo.DP110102548). formulations were independently developed in [19] and [20].
2
In these revised outage formulations, a secrecy outage event throughput. Finally, we analyze the throughput gain of
arises solely due to an insecure link to the eavesdropper; adaptive-rate transmission over fixed-rate transmission.
thus, the secrecy and reliability performance can be measured Whilst doing adaptation is always beneficial in terms of
separately. These revised secrecy outage formulations can be increasing the throughput, our analysis shows that in the
utilized to obtain a better understanding and more practically- high SNR region, the throughput gain is most significant
oriented designs for the artificial-noise-aided secure multi- whenthenumberoftransmitantennasissmall.Moreover,
antenna transmission. we show that the throughput gain brought by doing
adaptionisnotverysensitivetotherequiredsecrecylevel.
B. Our Approach and Contributions Wepointoutthat,similartoourwork,averyrecentcontri-
bution [21] also used a secrecy outage formulation along the
Inthispaper,weprovidenewdesignguidelinesforartificial-
lines of [19,20] to study artificial-noise-aided multi-antenna
noise-aided secure multi-antenna transmission in slow fading
transmission. The main focus therein was to minimize the
channels, based on the recently developed secrecy outage
power consumption for given levels of secrecy and quality-
formulation in [20]. This formulation allows us to measure
of-service performance, while the transmission rates were not
the secrecy and reliability performance for any given rate
part of the design consideration. In contrast, we consider a
parameters of the wiretap code. In turn, we are able to set the
transmissionsystemwithafixedtransmitpower,andoptimize
rate parameters to achieve a target secrecy level, given by a
therateparametersofthewiretapcodeaswellasthetransmit
maximumallowablesecrecyoutageprobability.Tothebestof
power allocation to maximize the average throughput, subject
theauthors’knowledge,nopriorworkonartificial-noise-aided
to (s.t.) a secrecy outage constraint.
multi-antenna transmission has considered the rate parameters
of the wiretap code as design parameters.
Our main contributions include explicit design solutions II. SYSTEMMODELANDPERFORMANCEMETRIC
and new performance analysis results for the throughput- We consider the transmission from Alice to Bob in the
maximizing transmission schemes with either fixed-rate or presence of an eavesdropper Eve. Alice is equipped with
adaptive-rate encoder, under a constraint on the level of multiple transmit antennas (N ≥ 2) while Bob and Eve each
secrecy. The design concerns include the rate parameters of has one receive antenna. Thus, the channel from Alice to Bob
the wiretap code, as well as the transmit power allocation ismultiple-inputandsingle-output(MISO).Weassumeanon-
betweentheinformation-bearingsignalandtheartificialnoise. line-of-sight rich scattering environment, and as such, model
Weconsidertwoscenarioswithdifferentsystemcomplexities: allchannelsasuncorrelatedRayleighfading.Itisalsoassumed
• In the first scenario, the design parameters are optimized thatBobcanestimatehischannelaccuratelyanduseaperfect
off-lineandremainfixedforalltransmissions.Wedivide feedback link to inform Alice about his instantaneous CSI.
our design into two steps: The first step minimizes the ThisfeedbacklinkisnotsecureandcanbeinterceptedbyEve.
transmissiondelayforagivendatarate,whilethesecond We further assume that the coherence time is long enough to
step maximizes the average throughput. In the first step, support the wiretap code [2] and the time used for learning
closed-form solutions are derived for the optimal system the channel and feeding back the CSI is negligible. Assuming
parameters and the secrecy-delay tradeoff for fixed-rate Eve is a passive eavesdropper, the instantaneous CSI of Eve
transmission is captured. In the second step, the average is thereby unavailable to Alice.
throughput is maximized by numerically optimizing the The N dimensional symbol vector to be transmitted is
data rate with the optimal designs of all other system defined as x and the received signal at Bob is given by
parameters already given in closed form. To obtain y =hTx+n , (1)
b b
further analytical insights, we derive a high signal-to-
noise ratio (SNR) approximation of the optimal data rate where the N ×1 vector h is the channel fading gain from
to maximize the average throughput. Focusing on the Alice to Bob and n is the receiver noise at Bob. The entries
b
high SNR region, we further investigate the additional of h and n are assumed to be independent and identically
b
power cost incurred by imposing or strengthening the distributed(i.i.d.)zero-meancomplexGaussianvariableswith
secrecy constraint, while guaranteeing a specified target unit variance.
throughput. Similarly, the received signal at Eve is given by
• In the second scenario, the design parameters are dy-
y =gTx+n , (2)
namically adjusted based on the instantaneous channel e e
feedback from the intended receiver. We provide an
wheretheN×1vectorgcapturesthechannelfadinggainfrom
analytical solution to the optimal system parameters that
Alice to Eve and n is the receiver noise at Eve. The entries
e
maximize the achievable data rate for each realization of
of g are assumed to be i.i.d. zero-mean complex Gaussian
the intended channel, under the secrecy constraint, such variables each with variance σ2.
g
thattheaveragethroughputisalsomaximized.Inthehigh
SNR region, we present accurate approximations for the
A. Transmit Beamforming with Artificial Noise Generation
maximal throughput, which enable us to study the addi-
tional power cost incurred by imposing or strengthening The authors in [9] introduced the concept of generating
the secrecy constraint, while achieving a specified target artificial noise to guarantee secure transmission. The key idea
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is outlined as follows. Alice generates an orthonormal basis this paper, we refer to the rates R and R as the rate param-
b s
of CN as W = [w W ], where w = h∗/||h||. Then she etersofthewiretapcode.TherateredundancyR (cid:44)R −R
1 2 1 e b s
mixes some artificial noise with the message symbol u as is intentionally added in order to provide secrecy against
eavesdropping. If the instantaneous channel capacity to Eve
x=w u+W v,
1 2 C = log (1+γ ) is larger than R , perfect secrecy cannot
e 2 e e
whereuiscomplexGaussiandistributedandv istheartificial be achieved and a secrecy outage is deemed to occur.
noise vector. We consider an on-off transmission scheme [20] where
Withthisbeamformingstrategy,by(1)and(2),thereceived Alice decides to transmit or not, based on her knowledge of
signal at Bob becomes Bob’s channel h. To be specific, Alice transmits only if the
effective channel gain ||h||2 exceeds a threshold µ. As will
yb =hTw1u+hTW2v+nb =||h||u+nb , (3) be seen, this on-off scheme is adopted to prevent undesirable
transmissions which incur capacity outages (i.e., R >C ) or
while the received signal at Eve becomes b b
unacceptably high risk of secrecy outage (which occurs when
ye =gTw1u+gTW2v+ne =g1u+g2Tv+ne , (4) Ce >Re). With the on-off transmission strategy, the transmit
probability is
where g =gTw and gT =gTW .
1 1 2 2
ThetotaltransmitpoweravailableatAliceisdenotedasP. ptx(µ)=F¯||h||2(µ)=Γ˜(N,µ) . (6)
The power allocation ratio Φ is defined as the fraction of the
Sincethechannelfadinggainchangesfromtimetotime,Alice
information-bearing signal power to the total transmit power.
would transmit once the effective channel gain exceeds the
Thus, the variance of u is set to PΦ. By (3), it is clear
preselected threshold (i.e., ||h||2 > µ). In this way, the value
that Alice is performing maximal ratio transmission on the
ofthetransmitprobabilityp isdirectlyrelatedtotheaverage
channel to Bob and the instantaneous effective channel gain tx
delay, i.e., the larger p , the shorter the expected delay.
is||h||2.Therefore,theinstantaneouschannelcapacitytoBob tx
(cid:16) (cid:17) Since the rates of the wiretap code may be adaptively
isC =log 1+PΦ||h||2 ,where||h||2 ∼Gamma(N,1).
b 2 adjusted based on Bob’s channel feedback, the rates R and
b
The complementary cumulative distribution function (c.c.d.f.)
R are potentially functions of h. The average throughput is
of ||h||2 is F¯ (r) = Γ˜(N,r), where Γ˜(·,·) is the regu- s
||h||2 then defined as
larized upper incomplete gamma function. All transmit power
not allocated to u is equally assigned to the N −1 entries of η =Eh[Rs(h)] (bits/channeluse), (7)
v, due to the absence of Eve’s CSI. Therefore, the entries of
where R (h)=0 for ||h||2 ≤µ, as a consequence of the on-
v are set to be i.i.d. zero-mean complex Gaussian variables s
each with variance σ2 = P(1−Φ). The power allocation ratio off transmission protocol. Note that this throughput definition
v N−1 ismeaningfulonlywhen:i)theriskofsecrecyoutageisunder
Φ may be dynamically adjusted based on h.
control, ii) the intended receiver can decode the messages
SincethenoisepoweratEveistypicallyunknowntoAlice,
correctly. As will be seen later, our designs can satisfy these
a robust approach, as done in [10], is to design for the worst-
two conditions simultaneously; thus, it is meaningful to apply
case scenario and assume that there is no receiver noise at
this throughput definition.
Eve (i.e., n =0). Therefore, from (4), for a given realization
e
of the channel fading gains (i.e., h and g), the instantaneous
SNR at Eve is given by C. Secrecy Performance Characterization
|g |2σ2 N −1 |g |2 From (5), it is clear that changing the power allocation
γ = 1 u = 1 .
e ||g ||2σ2 Φ−1−1||g ||2 would effectively alter the distribution of the received SNR
2 v 2
at Eve. With on-off transmission, for a given h, if ||h||2 >µ,
Sinceghasi.i.d.complexGaussianentrieseachwithvariance
Alice would specify three parameters: R (h), R (h) and
σ2 and W is unitary, gTW=[g g ] also has i.i.d. complex b s
g 1 2 Φ(h) for transmission; otherwise, she stops transmission.
Gaussian entries each with variance σ2. Using [22, Eq. 19],
g Therefore, the secrecy outage probability is given by
the c.c.d.f. of γ can be characterized as
e
p (R (h),R (h),Φ(h))
F¯γe(γe)=(cid:18)1+γe(cid:18)ΦN−1−−11(cid:19)(cid:19)1−N . (5) =(cid:40)soPr(bCe(Φ(hs))>Rb(h)−Rs(h)) if ||h||2 >µ,
0 other,
As N → ∞, this converges to the c.c.d.f. of an exponential
distribution with mean Φ . Note that if Φ is adaptively where
1−Φ
adjusted based on h, the distribution of the received SNR at
Pr(C (Φ(h))>R (h)−R (h))
Eve would be changed dynamically. e b s
(cid:18) (cid:16) (cid:17)(cid:18)Φ−1(h)−1(cid:19)(cid:19)1−N
= 1+ 2Rb(h)−Rs(h)−1 , (8)
B. Secure Transmission and Throughput N −1
Using the well-known wiretap code [2], data is encoded evaluated from (5).
beforetransmission.Theratesofthetransmittedcodewordand Since we assumed that n = 0, the above-mentioned p
e so
the secret message are denoted as R and R , respectively. In is an upper bound on the actual secrecy outage probability. It
b s
4
depends on the power allocation ratio Φ, but not the transmit constraint. With a minor abuse of notation, the maximal p
tx
power P. for a given R is denoted as pmax(R ), where µ is optimized
s tx s
The overall secrecy outage probability of this transmission already. Then, in the second step, we consider the throughput
system is given by maximization, i.e., maximizing pmax(R )×R w.r.t. R .
tx s s s
p¯ =E(cid:2)p (R (h),R (h),Φ(h))|||h||2 >µ(cid:3) . (9)
so so b s
A. Delay Minimization
In sequel, we consider the optimization of the throughput
From (11), the delay minimization problem is posed as
performance, given a maximum allowable secrecy outage
probability (cid:15). max p (µ)
tx
µ,Rb,Φ
s.t. p¯ (R ,Φ)≤(cid:15),
III. SECURETRANSMISSIONDESIGN so b
R ≤log (1+PΦµ) ,
WITHNON-ADAPTIVEENCODER b 2
R ≤R ,
In this section, we consider the scenario where a single s b
codebook is used by Alice and Bob, thus R and R are 0≤Φ≤1,
b s
carefullychosentohavefixedvalueswhichdonotchangewith 0≤µ. (12)
h. In addition, the value of Φ is also optimized off-line and
To simplify the expressions throughout the paper, we define
remains fixed. We refer to this as the non-adaptive encoding
the quantity:
(NAE) scheme.
Theprimarydesignobjectiveistomeetthesecrecyrequire- λ((cid:15),N):=(N −1)(cid:16)(cid:15)1−1N −1(cid:17) . (13)
ment given on the secrecy outage probability. For the NAE
scheme,thedesignparametersRb,Rs andΦhavefixedvalues The solution to (12) is presented as follows, while the deriva-
for all transmissions (i.e., they do not change with h); thus, tion is relegated to Appendix A.
(8) becomes independent of h. Consequently, the conditional
Proposition 1. With a non-adaptive encoder, for a prescribed
expectation in (9) can be ignored, and the overall secrecy
message rate R and under the secrecy constraint p¯ ≤(cid:15), the
outage probability p¯ in this case is described by (8). In s so
so maximal transmit probability in (12) leading to the minimal
other words, for the NAE scheme, the overall secrecy outage
average delay is given by:
probability is a function of R , R and Φ, and the secrecy
b s
requirement can be written as p¯so(Rb,Rs,Φ)≤(cid:15). (cid:32) 1 (cid:18)(cid:113) (cid:112) (cid:19)2(cid:33)
Meanwhile, as already mentioned, on-off transmission with pmtxax =Γ˜ N,P 2Rsλ((cid:15),N)+ 2Rs −1 . (14)
threshold µ is adopted to prevent undesirable transmissions
which incur capacity outages at Bob. When Alice decides to This is achieved with the following choice of system param-
transmit (i.e., ||h||2 >µ), the channel capacity to Bob is eters:
√
Cb =log2(cid:0)1+PΦ||h||2(cid:1)>log2(1+PΦµ) . Φ∗ = (cid:112) 2Rs −√1 , (15)
2Rsλ((cid:15),N)+ 2Rs −1
Thus, with Rb satisfying Rb ≤log2(1+PΦµ), the transmit- (cid:18)(cid:113) (cid:19)
ted messages can be decoded at Bob with an arbitrarily low Rb∗ =Rs+log2 (1−2−Rs)λ((cid:15),N)+1 , (16)
error probability.
From (6) and (7), the throughput of the NAE scheme is µ∗ = 2Rb∗ −1 .
given by PΦ∗
From (15) and (16), several observations can be made:
η =p (µ)×R . (10)
NAE tx s • Rb∗ and Φ∗ are independent of P. This is explained by
NowweconsiderthroughputoptimizationofthisNAEscheme noting that with the zero-noise assumption of Eve, the
underthesecrecyconstraint.Combiningtheabovediscussions, secrecy outage probability depends only on R and Φ.
b
the throughput optimization problem is posed as follows: Therefore, for a given Φ, whether the secrecy constraint
can be satisfied depends only on R . Since the latter
max p (µ)×R b
µ,Rb,Rs,Φ tx s optimization w.r.t. Φ also does not involve P, the in-
s.t. p¯ (R ,R ,Φ)≤(cid:15), dependence is then observed.
so b s
R ≤log (1+PΦµ) , • Φ∗ increases with N (since λ((cid:15),N) decreases with N,
b 2 as shown in Appendix B), approaching the constant:
0≤R ≤R ,
s b
1
0≤Φ≤1, lim Φ∗ = (cid:113) .
0≤µ. (11) N→∞ 1+ ln(cid:0)1(cid:15)(cid:1)/(1−2−Rs)
This result follows the intuition that when more transmit
We solve the problem in two steps. In the first step, we fix
antennasareinstalled,asmallerfractionofpowercanbe
R and maximize p (µ) with respect to (w.r.t.) µ, R and Φ.
s tx b
usedforgeneratingartificialnoisewhilestillensuringthe
Since R is fixed, by maximizing p , this step can be viewed
s tx
required security level.
as a delay-minimizing design, while satisfying the secrecy
5
byreplacingpmax in(14)withδ andsolvingtheobtained
1 tx
equality w.r.t. P. This comes from the Pareto optimality
N = 4 indicated above and the fact that increasing P would
0.8 effectively expand the feasible region. If extra antennas
ptx
y, N = 3 may be added, the decreasing rate of Pmin is most
bilit 0.6 significant for small N. For the special case δ = 0.5,
ba N = 2 it can be proved that P decreases inverse linearly as
o min
Transmit Pr 0.4 Feasible Region Nt(hhiesrgeoi,nevtsheerlsaerdgoleti,tneeudasrilnibngeehs[2aav5ri,eoErpqrh.oop6lde.2rsl]y.foAsrcsaalsewhdoidwinenvreairnnsegFeliigno.efa2δr,
0.2
curves, shown for reference).
0
0 0.2 0.4 0.6 0.8 1 B. Throughput Optimization and Analysis
SecrecyOutageProbability,p¯so Having maximized p for a given R , from (11) and (14),
tx s
Fig.1. Optimaltradeoffbetweenthesecrecyanddelayperformanceofthe the optimal message rate that maximizes the throughput is
NAE scheme for different numbers of transmit antennas, with P = 10 dB given by
andRs=2bits/channeluse.
R∗ =argmax η =argmax pmax(R )×R , (17)
s NAE tx s s
Rs>0 Rs>0
30 and the corresponding maximal throughput is given by
δ = 0.8
(dB)min25 Iδδn v==e r00s..e52 Linear ηN∗AE =pmtxax(Rs∗)×Rs∗ . (18)
P
er, 20
w
o
mit P 15 4.5 N = 8
Trans 10 4
mal AE3.5 N = 6
ni 5 ηN 3
Mi put, 2.5 N = 4
h
02 4 6 8 10 12 14 16 oug 2
Transmit Antenna Number, N Thr 1.5
Fig.2. MinimalpowerconsumptionoftheNAEschemeunderjointsecrecy 1
anddelayconstraintversusthenumberoftransmitantennasfordifferentdelay 0.5 N = 2
constraints,with(cid:15)=0.01andRs=2bits/channeluse.
1 2 3 4 5 6 7 8
Message Rate, R
s
The points above focus on the optimal system parameters.
Fig.3. ThroughputoftheNAEschemeversusthemessageratefordifferent
Some useful insights regarding the system performance and
numbersoftransmitantennas,withP =20dBand(cid:15)=0.01.
design implications may also be concluded:
• Foragiventransmissionsystem(i.e.,givenN andP)and From(14)and(17),itcanbeshownthatifRs istoosmall,
a prescribed message rate Rs, the optimal tradeoff curve even though ptx may be high (i.e., close to one), the value
betweenthesecrecyanddelayperformanceiscompletely of ηNAE is still very small; whereas, if Rs is too large, the
specifiedby(14).Thisisobservedbynotingthatpmax is value of ptx and therefore ηNAE will also become very small.
tx
achievedwhenp¯so =(cid:15),whilevarying(cid:15)effectivelytraces Moreover, by differentiating ηNAE = pmtxax(Rs)×Rs w.r.t.
the optimal tradeoff. In [23], we considered the mini- Rs, one can verify that the derivative is first positive and then
mization of the overall secrecy outage probability under negativewithincreasingRs;thusRs∗ isunique.Assuch,even
an average delay constraint (i.e., min p¯ s.t.p ≥δ), if the objective function is non-concave, as shown in Fig. 3,
so tx
which is the symmetric problem to (12). Since the we still can solve the derivative to find Rs∗ numerically.
tradeoff curve obtained in [23] coincides with (14), we Though it seems difficult to find a general closed-form
conclude that each point on the optimal tradeoff curve is solution for (17), we can obtain an accurate approximation in
Pareto optimal [24]. As shown in Fig. 1, the area under the high SNR region. Specifically, as derived in Appendix C,
each curve represents the feasible region of (p¯ ,p ), when P →∞, we have
so tx
points of which are achievable.    
• If the system is designed to operate under a joint secrecy R∗ ≈ 1 W  exp(1)N!PN −1 , (19)
and delay constraint (i.e., p¯ ≤ (cid:15) and p ≥ δ), the s Nln(2) 0(cid:16)(cid:112) (cid:17)2N 
so tx λ((cid:15),N)+1
minimumrequiredtransmitpowerP canbecomputed
min
6
where W (·) is the principle branch of the Lambert-W func-
0 14
tion and λ((cid:15),N) is defined in (13). Exact
From (19) and [26, Eq. 65], we know that R∗ increases 12 Approx.
s
withP.Therefore,from(15)and(16),wecanobservethatfor
the throughput-maximizing scheme, as P → ∞, the optimal E10 9.14 dB
A
power allocation ratio Φ∗ and the added rate redundancy *ηN
R∗ =R∗−R∗ converge to ut, 8
e b s hp ε = 1 3.55 dB
g 6
1 u
lim Φ∗ = (cid:112) , (20) hro ε = 0.1
P→∞ λ((cid:15),N)+1 T 4
(cid:16)(cid:112) (cid:17) ε = 0.01
lim R∗ =log λ((cid:15),N)+1 . (21)
e 2 2
P→∞
Furthermore,basedon(19),ahighSNRthroughputapprox-
0
10 15 20 25 30 35 40
imation for the NAE scheme is derived in Appendix D, and
Transmit Power, P (dB)
is given as follows:
Fig. 4. Throughput of the NAE scheme and the high SNR approximation
ηN∗AE((cid:15))≈ηN(cid:15)=A1E−ηNloAssE((cid:15)) , (22) versusthetransmitpowerfordifferentsecrecyconstraints,withN =4.
where
η(cid:15)=1 =log (P)− 1 log (ln(P))+ 1 log ((N−1)!), (23) arrow shows that with (cid:15)1 = 1 and (cid:15)2 = 0.1, the additional
NAE 2 N 2 N 2 power cost is 9.14 dB, while the second arrow suggests that
ηloss ((cid:15))=2log (cid:16)(cid:112)λ((cid:15),N)+1(cid:17). (24) with (cid:15)1 = 0.1 and (cid:15)2 = 0.01, the additional power cost is
NAE 2
3.55dB.Thesetwovaluesareveryclosetotheapproximation
Here η(cid:15)=1 is the high SNR throughput approximation for provided in (25). Note that when N is very small (e.g.,
NAE
the NAE scheme when the system is operating without any N =2), the approximation in (22) becomes inaccurate due to
secrecy constraint (i.e., (cid:15) = 1), while ηloss ((cid:15)) reflects the theapproximationprocedureusedinAppendixDandabetter
NAE
throughputlossforrealizingsecuretransmission.Interestingly, approximation can be obtained by plugging (19) into (18).
the throughput loss in (24) is independent of P and this is a
direct consequence of the zero-noise assumption of Eve. IV. SECURETRANSMISSIONDESIGN
From (22)-(24), several observations can be made: WITHADAPTIVEENCODER
• The benefits of adding extra transmit antennas are two- In this section, we consider the scenario where Alice
fold: i) the system is more capable of achieving a larger dynamically adjusts the system parameters: R , R and Φ
b s
throughput, ii) the throughput loss for securing the trans- for each realization of the channel to Bob h. In other words,
missionwillbereduced.However,thebenefitsintermsof Alice performs adaptive encoding (AE). As before, the target
reducing ηloss ((cid:15)) by adding extra antennas are limited, is to maximize the throughput under the secrecy constraint
NAE
i.e., givenbyamaximumallowablesecrecyoutageprobability(cid:15).In
(cid:32)(cid:115) (cid:18) (cid:19) (cid:33) particular,thevaluesofRb,Rs andΦaredynamicallychosen
1
lim ηloss ((cid:15))=2log ln +1 , to meet the requirement on the secrecy outage probability for
N→∞ NAE 2 (cid:15) each transmission, i.e., p (R (h),R (h),Φ(h)) ≤ (cid:15). This
so b s
design will in turn satisfy the requirement on the overall se-
which depends only on (cid:15), as we may expect.
crecy outage probability. With the secrecy constraint satisfied,
• Even in the high SNR region, the second term in (23)
the problem of maximizing the throughput is equivalent to
is usually insignificant due to the double logarithm.
maximizingR (h)foreachh.Intuitively,theAEschemecan
Therefore, to achieve a specified target throughput, the s
achieve a larger throughput compared with the NAE scheme,
additional power cost for imposing or strengthening the
but demands a higher complexity.
secrecy constraint can be approximated as
∆NAE((cid:15) ,(cid:15) ,N) (dB) (25) A. Message Rate Maximization
P 1 2
20log10(cid:16)(cid:112)λ((cid:15)2,N)+1(cid:17) if (cid:15)2 ≤(cid:15)1 =1, From (8), the problem of maximizing Rs(h) for each h is
≈ (cid:16) (cid:17) given by
N1−01log10 (cid:15)(cid:15)12 if (cid:15)2 ≤(cid:15)1 (cid:28)1, max R (h)
s
whichisindependentofthetargetedthroughput.Itisclear Rb(h),Rs(h),Φ(h)
thatthepowercostinbothcasesin(25)canbeeffectively s.t. pso(Rb(h),Rs(h),Φ(h))≤(cid:15),
reduced by adding extra transmit antennas. R (h)≤log (cid:0)1+PΦ(h)||h||2(cid:1) ,
b 2
In Fig. 4, we show that the throughput approximation 0≤R (h)≤R (h),
s b
in(22)isquiteaccuratebycomparingwiththeexactmaximal
0≤Φ(h)≤1, (26)
throughput,whichisobtainedbynumericallyoptimizing(17).
When the secrecy constraint is strengthened, in order to wherethesecondconstrainteliminatesthecapacityoutagesat
maintain a specified target throughput, for N = 4, the first the intended receiver.
7
The solution to (26) is presented as follows, whilst the
0.31
derivation is relegated to Appendix E.
*Φ 0.3
Proposition 2. With an adaptive encoder, for a given realiza- o,
tionoftheintendedchannelhandunderthesecrecyconstraint ati 0.29
R
ipnso((2R6)bi(shg)i,vRens(bhy):,Φ(h)) ≤ (cid:15), the maximal message rate cation 0.28
o
Rsmax((cid:32)h(cid:112)) Pλ((cid:15),N)||h||2−(cid:112)P||h||2−λ((cid:15),N)+1(cid:33)(27) Power All 00..2267
=2log2 λ((cid:15),N)−1 , mal 0.25 AE ||h||2 = 9
Opti 0.24 NAE
where λ((cid:15),N) is defined in (13). The corresponding optimal AE ||h||2 = 3
system parameters are given by: 0.23
10 15 20 25 30
Transmit Power, P (dB)
µ∗ =λ((cid:15),N)/P , (28)
(cid:114)
(cid:16) (cid:17) Fig.5. OptimalpowerallocationratiosoftheNAEandAEschemesversus
λ((cid:15),N) 1+ 1−λ((cid:15),N) −1 thetransmitpower,withN =8and(cid:15)=0.01.
P||h||2
Φ∗(h)= , (29)
λ((cid:15),N)−1
Rb∗(h)=log2(cid:0)1+PΦ∗(h)||h||2(cid:1) . (30) Fig. 5 illustrates our analysis, showing Φ∗(h) for the AE
scheme for different h. The value of Φ∗ for the NAE scheme
From (27)-(30), several observations can be made:
is also shown for comparison and it is found by numerically
• Intuitively, we may expect that a positive message rate
optimizingR in(17),followedbypluggingitinto(15).Aswe
can always be achieved by properly adjusting the trans- s
mayexpect,theoptimalpowerallocationratiosallconvergeto
mission system; thus, it may not be necessary to in-
the same limit with increasing P. This observation is similar
troduce the on-off transmission protocol. However, the
to the one in [10] where the channel gain was found to be
derivation in Appendix E shows that after guarantee-
irrelevant in finding the optimal power allocation in the high
ing the secrecy performance, a positive R (h) can be
s
SNR region for maximizing the ergodic secrecy rate in fast
achievedonlywhen ||h||2 >λ((cid:15),N)/P.Inotherwords,
fadingchannels.However,besidesthesimilarity,thedifference
when Bob’s channel is not sufficiently good, a positive
is also significant. In the high SNR region, the optimal power
R (h) and the required secrecy performance cannot be
s
allocationratiothatmaximizestheergodicsecrecyrateforfast
achievedsimultaneously,nomatterhowAliceadjuststhe
fadingchannels[10]isaround0.5,whilethelimitingvalueof
powerallocationorencoding.Therefore,tomaximizethe
Φ∗(h) in (31) that maximizes the throughput for slow fading
throughputwhilstguaranteeingtherequiredsecrecylevel,
channelsdependsstronglyontherequiredsecuritylevel(cid:15)and
on-off transmission with µ = λ((cid:15),N)/P is introduced,
the number of transmit antennas N.
as stated in (28).
• The optimal power allocation ratio Φ∗(h) in (29) de-
creases with increasing P, approaching the constant:
B. Throughput Performance Analysis
1
lim Φ∗(h)= (cid:112) , (31) By (7) and (27), the throughput for the optimized AE
P→∞ λ((cid:15),N)+1
scheme is given by
which is independent of h and is identical to that for the
NAE scheme in (20). This convergence suggests that in ηA∗E =Eh[Rsmax(h)] . (33)
the high SNR region, near optimal performance can be
Though it seems difficult to find a closed-form expression,
obtained by employing a non-adaptive power allocation
similartotheNAEcase,progresscanbemadebyappealingto
strategy.
• With optimized power allocation, Rb(h) is set to the the high SNR region. Specifically, as P → ∞, the following
approximation is derived in Appendix F:
instantaneouscapacityofBob’schanneltoguaranteesuc-
cessful decoding, as shown in (30). From (27) and (30), (cid:18) (cid:19)
P ψ(N)
we can observe that as P → ∞, the added rate redun- η∗ ≈log + (34)
AE 2 λ((cid:15),N) ln(2)
dancy R (h)=R (h)−R (h) converges to
e b s (cid:32) (cid:112) (cid:33) (cid:18) (cid:19)
Pl→im∞Re(h)=log2(cid:16)(cid:112)λ((cid:15),N)+1(cid:17) , (32) +2log2 (cid:112)λ(λ(cid:15)(,N(cid:15),N)+)1 Γ˜ N,λ((cid:15)P,N)
which is independent of h and is identical to that for the (λ((cid:15),N)/P)N (cid:18) λ((cid:15),N)(cid:19)
NAE scheme in (21). + N2(N −1)!ln(2)2F2 N,N;N+1,N+1;− P ,
• Furthermore, as shown in Appendix E, Rsmax(h) in (27)
isalwaysachievedwhenp (R (h),R (h),Φ(h))=(cid:15) where λ((cid:15),N) is defined in (13), ψ(N) is the digamma
so b s
andthisimpliesthattheoverallsecrecyoutageprobability function,and F (·,·;·,·;·)isthegeneralizedhypergeometric
2 2
in (9) is also guaranteed to be p¯ =(cid:15). function [27, Eq. 9.14.1].
so
8
To gain more insights, as shown in Appendix F, when
16
P →∞, (34) is further approximated as Exact
14 Approx.
η∗ ((cid:15))≈η(cid:15)=1−ηloss((cid:15)) , (35)
AE AE AE 12 9.13 dB
E
where *ηA10
ψ(N) ut, ε = 1 3.53 dB
η(cid:15)=1 =log (P)+ , (36) hp 8
AE 2 (cid:16)(cid:112) ln(2) (cid:17) hroug 6 ε = 0.1
ηAloEss((cid:15))=2log2 λ((cid:15),N)+1 . (37) T 4 ε = 0.01
Here η(cid:15)=1 is the high SNR throughput approximation for the 2
AE
AE scheme without any secrecy constraint, which coincides
0
with the high SNR approximation of the ergodic capacity 10 15 20 25 30 35 40
of the MISO Rayleigh fading channel [28], while ηloss((cid:15)) Transmit Power, P (dB)
AE
reflects the throughput loss incurred by enforcing the secrecy
Fig. 6. Throughput of the AE scheme and the high SNR approximation
constraint. versusthetransmitpowerfordifferentsecrecyconstraints,withN =4.
From (35)-(37), several observations can be made:
• InthehighSNRregion,thethroughputlossforimposing
C. Throughput Gain of Adaptation: AE over NAE
the secrecy constraint in (37) is independent of P and it
is identical to the throughput loss for the NAE scheme In the following, we analyze the throughput gain brought
in (24). The throughput loss in (37) is actually twice the by doing adaptation. With the throughput approximations for
limiting value of the rate redundancy in (32). This can the NAE and AE schemes in (22) and (35), in the high SNR
be explained by noting that in the high SNR region, the region, the throughput gain can be approximated by
transmitthresholdin(28)isclosetozeroand,inaddition
to the rate redundancy in (32), the transmission system η∗ −η∗ (38)
AE NAE
alsosuffersapowerlosssinceacertainfractionofpower 1 (cid:18)ψ(N) 1 (cid:19)
≈ log (ln(P))+ − log ((N −1)!) .
is used to generate artificial noise, as shown in (31). It is N 2 ln(2) N 2
interesting that this power loss leads to a data rate loss
With this approximation, several observations can be made:
which is identical with the rate redundancy in (32).
• In the high SNR region, to achieve a specified target • In the high SNR region, the throughput gain increases
throughput, the additional power cost for imposing or withP,albeitveryslowly,ascanbeseenfromthedouble
strengtheningthesecrecyconstraintcanbeapproximated logarithm in the first term of (38).
from (35) and the results are identical to those for the • It turns out that as N increases, the second term in (38)
NAEschemein(25).However,althoughtheNAEandAE (i.e., the P-independent term) increases slowly. The first
schemes suffer the same power cost when imposing the term, on the other hand, decreases very fast with N
samesecrecyconstraint,theAEschemestilloutperforms (i.e.,linearly),andmoreover,thistermbecomesdominant
theNAEschemeintermsofthethroughputperformance. when P is large. These observations imply that for high
This is due to the fact that when the secrecy constraint SNR, the throughput gap between the NAE and AE
is removed, the AE scheme is designed to achieve a schemes shrinks when the number of antennas grows.
better throughput performance, compared with the NAE Consequently, for systems operating at high SNRs, if
scheme. Specifically, when (cid:15) = 1, for both schemes, all the number of antennas is not small, it may be more
thetransmitpowerisallocatedtotheinformation-bearing preferable to employ the NAE scheme rather than the
signal and the data will be transmitted without wiretap AE scheme, due to the complexity saving.
coding. In this case, the NAE scheme still chooses a • Thethroughputgainapproximationin(38)isindependent
constant rate to transmit and the transmit threshold is of the secrecy constraint (cid:15). This can be explained by
set to the minimum required effective channel gain to noting that in the high SNR region, the NAE and AE
support the selected rate; while the AE scheme always schemes incur thesame throughput loss for imposingthe
transmits at the instantaneous capacity of Bob’s channel secrecy constraint, as shown in (22) and (35).
and the transmit threshold is therefore set to zero.
Fig. 7 compares the approximation in (38) with the exact
In Fig. 6, we show that the throughput approximation throughput gain. It is clear that the exact throughput gain
in (35) is quite accurate by comparing with the exact max- decreases with increasing N. Note that the variation in the
imal throughput, which is obtained by numerically evaluat- throughput gain is not very significant over a wide range of (cid:15),
ing (33). The additional power cost analysis is confirmed by though the gain does decrease with strengthening the secrecy
the examples therein. Note that when N is very small (e.g., constraint (i.e., reducing (cid:15)). We note that the approximation
N =2), (35) becomes less accurate due to the approximation is not very accurate, since the throughput gain is relatively
procedure used in Appendix F and (34) provides a better small and thus the approximation error in (38) is relatively
approximation. pronounced. What is important is that, as stated above, our
9
possible value. From the second constraint in (12), which
3
eliminates capacity outages at Bob, we know that for a given
Φ, µ should be set to
E2.5
A
*ηN 1 (cid:16) (cid:17)
− E 2 µmin := 2Rbmin −1 ,
*ηA PΦ
ain, 1.5 inordertomaximizethetransmitprobability,whilstsatisfying
G
ut the secrecy constraint.
p
ugh 1 Substituting µmin into (6), the corresponding ptx is given
Thro 0.5 NN == 44 AExpapcrtox. by
N = 8 Approx.
N = 8 Exact p (Φ)=Γ˜(N,ω(Φ)) , (39)
tx
0
0 0.2 0.4 0.6 0.8 1
Secrecy Constraint, ε where
Fig. 7. Throughput gain of the AE scheme over the NAE scheme and the ρ (cid:37) 2Rsλ 2Rs −1
high SNR approximation versus the secrecy constraint for different number ω(Φ)= + , ρ= , (cid:37)= . (40)
1−Φ Φ P P
oftransmitantennas,withP =40dB.
Since the regularized upper incomplete gamma function
decreases with its second parameter, it is clear that p (Φ)
analysis accurately predicts the varying trends of the exact tx
in (39) achieves its maximum when ω(Φ) takes its minimum.
throughput gain.
Note that ω(Φ) is a convex function of Φ; thus, by solving
the derivative of ω(Φ) w.r.t. Φ, the optimal power allocation
V. CONCLUSION
ratio can be found as
This paper investigated the design of artificial-noise-aided √
securemulti-antennatransmissioninslowfadingchannels.We Φ∗ = √ (cid:37)√ ,
provided explicit design solutions to the secrecy-constrained ρ+ (cid:37)
throughput-maximization problem with either non-adaptive
transmission (i.e., the NAE scheme) or adaptive transmission and the corresponding maximal transmit probability pmtxax is
given by
(i.e., the AE scheme). To facilitate practical system design,
we examined the additional power cost for achieving a higher (cid:16) √ √ (cid:17)
secrecy level for both schemes and analyzed the throughput pmtxax :=ptx(Φ∗)=Γ˜ N,( ρ+ (cid:37))2 .
gain of the AE scheme over the NAE scheme in the high
SNRregion.Ouranalysisobtainednewinsightsonthedesign Summarizing the obtained results, we have Proposition 1.
of artificial-noise-aided secure multi-antenna transmission in
slow fading channels.
B. Monotonicity of Quantity in (13)
APPENDIX In this proof, we show that λ in (13) decreases with
A. Proposition 1 increasing N. To this end, we temporarily assume that N is
In this proof, we derive the optimal values of the system a continuous variable. Taking the derivative of λ w.r.t. N, we
parameters R , µ and Φ to maximize the transmit probability have
b
p in(12)(i.e.,minimizetheaveragedelay).Wefirstoptimize (cid:18) (cid:19)
tx dλ ln((cid:15))
Rb and µ for a given Φ, and then optimize Φ. The message dN =(cid:15)1−1N 1+ N −1 −1. (41)
rate R is assumed to be at a prescribed value.
s
AswediscussedatthebeginningofSectionIII,fortheNAE Toshowthattheabovederivativeisnegative,weneedtoshow
scheme, the overall secrecy outage probability in (9) reduces 1
that the expression in the bracket is smaller than (cid:15)N−1. Since
to (8). For a given Φ, from (8), it is clear that the secrecy the secrecy constraint (cid:15) is smaller than one, we know that
constraint in (12) can be satisfied by choosing a large enough ln((cid:15)) < 0. From the inequality 1+x < ex for x < 0, letting
Rb. Substituting (8) into the secrecy constraint in (12) (i.e., xN−=1 ln((cid:15)), we see that the quantity in the bracket in (41)
p¯ (R ,Φ)≤(cid:15)) and solving w.r.t. R , we have N−1
so b b satisfies
(cid:18) (cid:19)
λΦ
Rb ≥Rs+log2 1+ 1−Φ =:Rbmin , 1+ ln((cid:15)) <exp(cid:18) ln((cid:15)) (cid:19)=(cid:15)N1−1 .
N −1 N −1
where λ is defined in (13) and its parameters are omitted for
brevity. Thus, we know that the derivative in (41) is negative. Since
To minimize the average delay, the transmit probability p the discrete N is just a sampled version of the continuous N,
tx
in (6) needs to be maximized. From (6), it is clear that p is the monotonic properties are the same. We then know that λ
tx
maximizedwhenthetransmitthresholdµissettoitssmallest in (13) decreases with increasing N.
10
C. Approximation of Optimal Message Rate in (19) Thus, as P →∞, by ignoring the high order terms, (19) can
be further approximated as
In this proof, we derive an approximation for the optimal
 
message rate Rs∗ in (17). From the discussions after (18), √N!PN
wηNeAkEnoww.r.tth.aRt sonteocfianndsoRlvs∗e. Hthoewdeevreivr,attihvee oefquthaleitythroobutgahinpeudt R˜s∗ = N1 log2ln(cid:18)(ex√pλ(+11)N)2!NPN(cid:19) ,
by setting the derivative to zero is quite complicated and ( λ+1)2N
seemsnotsolvable.Hence,weconsiderfirstapproximatingthe
throughput and then optimizing the approximated throughput. where λ is defined in (13). Note that as P → ∞, 2R˜s∗ →0.
P
The details can be found as follows. Substituting the above approximation into the throughput
Usingtheinequality2Rs >2Rs−1,alowerboundtoηNAE lower bound in (44), we have
can be given by  (cid:32) (cid:33)N
η ≥log (x)×Γ˜(N,ςx)=:ηLB , (42) ηN∗AE ≈R˜s∗1−O 2PR˜s∗  .
NAE 2 NAE
(cid:16)√ (cid:17)2 Taking the leading order term yields
where x=2Rs, ς = 1 λ+1 and λ is defined in (13).
P 1 1
As P → ∞, it is clear that ς → 0. Using the series η∗ ≈log (P)− log (ln(P))+ log (N!)
NAE 2 N 2 N 2
representation of the regularized upper incomplete gamma 1 (cid:18) (cid:18) 1 (cid:19)(cid:19) (cid:16)√ (cid:17)
function [27, Eq. 8.352.4.*], we expand ηLB in (42) as − log N +O −2log λ+1 .
NAE N 2 ln(P) 2
N−1 k (cid:16) (cid:17) (cid:16) (cid:16)√ (cid:17)(cid:17)
ηNLBAE =log2(x)e−ςx (cid:88) (ςkx!) Here O ln(1P) = ln(1P) 1+ln(N!)−2Nln λ+1 ,
which can be ignored for high P, thereby giving (22).
k=0
(cid:32) ∞ k(cid:33)
(cid:88) (ςx)
=log (x)e−ςx eςx−
2 k! E. Proposition 2
k=N
(cid:32) (cid:88)∞ (−ςx)l (cid:88)∞ (ςx)k(cid:33) In this proof, we derive the optimal values of the system
=log2(x) 1− l! k! (43) ipnar(a2m6)etfeorrsaRgbiv,eRnsreaanlidzaΦtiotnoomftahxeiminitzeendtheedcmheasnsnaeglehr,awtehiRlsst
l=0 k=N
(cid:32) N (cid:33) satisfyingthesecrecyconstraint.(NotethatfortheAEscheme,
=log (x) 1− (ςx) +O(cid:0)ςN+1(cid:1) , (44) thedesignparametersR ,R andΦareintrinsicallyfunctions
2 N! b s
of h.) We first optimize R and R for a given Φ, and then
b s
optimize Φ.
where in (43) we used the standard series expansion of the
For a given Φ, to eliminate capacity outages at Bob (i.e.,
exponential function.
the second constraint in (26)), Alice sets R to the capacity
Ignoring the high order terms in the above series represen- b
of Bob’s channel:
tation and taking the derivative w.r.t. x, we have
R (Φ)=C =log (cid:0)1+PΦ||h||2(cid:1) .
dη˜NLBAE = ςN (cid:0)N!ς−N −xN(cid:0)1+ln(cid:0)xN(cid:1)(cid:1)(cid:1) . ForagivenΦb,ifAlicedbecides2totransmitwithR (Φ)above,
dx xln(2)N! b
and with R <R (Φ), the secrecy outage probability for this
s b
Wethensettheabovederivativetozeroandsolvetheobtained transmission is given by (8) as
equality w.r.t. x. After summarizing the obtained results, we
(cid:18) (cid:16) (cid:17)(cid:18)Φ−1−1(cid:19)(cid:19)1−N
have the approximation in (19). pso(Rb(Φ),Rs,Φ)= 1+ 2Rb(Φ)−Rs−1 N−1 ,
whichsimplyincreaseswithincreasingR .Thus,themaximal
s
D. High SNR Throughput Approximation in (22) R will be obtained when the secrecy constraint in (26) is
s
met with equality, i.e., p (R (Φ),R ,Φ) = (cid:15). Solving this
In (19), we have obtained an approximation for the optimal so b s
equality gives
message rate R∗ in (17). Plugging an approximation for (19)
s
(cid:32) (cid:33)
into a lower bound to the throughput in (44) and ignoring the
1+τΦ
high order terms, we can get an elegant approximation for Rs(Φ)=log2 1+ λΦ , (45)
the maximal throughput η∗ . The details can be found as 1−Φ
NAE
follows. whichisafunctionofΦ,τ =P||h||2 andλisdefinedin(13).
In [26, Eq. 65], a series expansion is provided for the From (45), we see that when τ ≤ λ (equivalently,
principle branch of the Lambert-W function W (·), and it ||h||2 ≤ λ), it is impossible to adjust Φ ∈ (0,1) to achieve
0 P
states that as x→∞, a positive R . For the alternative case, when τ > λ
s
(i.e., ||h||2 > λ), the range of Φ which can guarantee that
W0(x)=ln(cid:18) x (cid:19)+O(cid:18)ln(ln(x))(cid:19) . Rs(Φ)>0 isP(0,1− λτ). Henceforth, we focus on the case
ln(x) ln(x) where τ >λ.