Low-SNR Analysis of Interference Channels under Secrecy Constraints

0.1 MB·English

by  Junwei Zhang

Low-SNR Analysis of Interference Channels under
Secrecy Constraints
Junwei Zhang and Mustafa Cenk Gursoy
Department of Electrical Engineering
University of Nebraska-Lincoln, Lincoln, NE 68588
Email: [e-mail],[e-mail]
Abstract—In this paper, we study the secrecy rates over low-power regime. By comparing the performance of TDMA
9 weak Gaussian interference channels for different transmission and superposition schemes, they concluded that the growth
0 schemes. We focus on the low-SNR regime and obtain the min-
of TDMA-achievable rates with energy per bit is suboptimal
0 imum bit energy Eb values, and the wideband slope regions
2 forbothTDMAanNd0mmiunltiplexedtransmissionschemes.Weshow except in some special cases.
thatsecrecyconstraintsintroduceapenaltyinboththeminimum In this paper, we study secure transmission over Gaussian
n
bitenergy andthesloperegions. Additionally,weidentifyunder weakinterferencechannelsinthelow-powerregime.Theorga-
a
J what conditions TDMA or multiplexed transmission is optimal. nizationoftherestofthepaperisasfollows.InSectionII,we
0 Finally, we show that TDMA is more likely to be optimal in the describe the channel model and obtain the secrecy achievable
presence of secrecy constraints.
2 rate regions for TDMA, multiplexed transmission schemes
and artificial noise schemes, and compare their performances
] I. INTRODUCTION in terms of the achievable rates. In Section III, we compute
T
The open natureof wireless communicationsallows for the the minimum energy per bit and slope at Eb for TDMA
.I signals to be received by all users within the communication and multiplexed transmission schemes. InN0Smeicntion IV, we
s
c range. Thus, wireless communication is vulnerable to eaves- use results in Section III to evaluate how secrecy constraints
[ dropping. This problem was first studied in [1] where Wyner affect the performance in the low-power regime and identify

1 proposed a wiretap channel model. In this model, a single optimaltransmissionschemes.Finally,weprovideconclusions
v source-destination communication link is eavesdropped by a in Section V.
2 wiretapper.Thesecrecylevelismeasuredbythe equivocation
3 rate. Wyner showed that secure communication is possible II. GAUSSIAN INTERFERENCECHANNELS WITH
1 withoutsharingasecretkeyif theeavesdropper’schannelisa CONFIDENTIAL MESSAGES
3
degradedversionofthe main chain.Later,Wyner’sresultwas We consider secure communication over a two-transmitter,
.
1 extendedtotheGaussianchannelin[3]andrecentlytofading two-receiver Gaussian interference channel. The input-output
0
channelsin[4].Inadditiontothesingleantennacase,secrecy relations for this channel model are given by
9
0 in multi-antenna models is addressed in [5] – [8]. Multiple
y =c x +c x +n , and (1)
: accesschannelswithconfidentialmessageswereconsideredin 1 11 1 12 2 1
v [9].Liuetal.[10]presentedinnerandouterboundsonsecrecy y2 =c21x1+c22x2+n2 (2)
i
X capacityregionsforbroadcastandinterferencechannels.They
wherex andx arethechannelinputsofthetransmitters,the
1 2
r also described several transmission schemes for Gaussian coefficients {c } denote the channel gains and are determin-
a ij
interferencechannelsandderivedtheirachievablerateregions
istic scalars, and n and n are independent, circularly sym-
1 2
whileensuringmutualinformation-theoreticsecrecy.Recently,
metric, complex Gaussian random variables with zero mean
Bloch et al. in [11] discussed the theoretical aspects and and common variance σ2. It is assumed that the transmitters
practical schemes for wireless information-theoreticsecurity.
are subject to the following average power constraint:
Another important concern in wireless communications is
theefficientuseoflimitedenergyresources.Hence,theenergy E[|xi|2]6Pi =SNRiσ2, i=1,2. (3)
required to reliably send one bit is a metric that can be
We focus on the weak interference channel i.e., we assume
adopted to measure the performance. Generally, energy-per- that |c12|2 < 1 and |c21|2 < 1. Over this channel, transmitter
bit requirementis minimized,and hence the energyefficiency |c11|2 |c22|2
i for i = 1,2 intends to send an confidential message by
is maximized, if the system operates in the low-SNR regime.
transmitting x to the desired receiver i, which receives y ,
i i
In [12], Verdu has analyzed the tradeoff between the spectral
while ensuring that the other receiver does not obtain any
efficiencyandbitenergyin the low-SNRregimefora general
information by listening the transmission. Following [10], we
class of channels. As argued in [12], two key performance
next consider three transmission schemes and their corre-
measures in the low-power regime are the minimum energy
sponding achievable secrecy rate regions.
per bit Eb required for reliable communication and the
N0min
slope of the spectral efficiency versus Eb curve at Eb . A. Time Division Multiple Access
N0 N0min
Caire et al. in [13] employedthese two measuresto study the In TDMA, the transmission period is divided into two
multiple access, broadcast, and interference channels in the nonoverlappingtimeslots.Transmitters1and2transmitusing
αand1−αfractionsoftime,respectively.Wenotethatunder
0.14
this assumption, the channel in each time slot reduces to a
Gaussian wiretap channel [3], and the following rate region 0.12
can be achieved with perfect secrecy [10]:
0.1
R >0
1
R >0 0.08
2
R2
|c |2SNR |c |2SNR
R 6α log 1+ 11 1 −log 1+ 21 1 0.06
1
(cid:20) (cid:18) α (cid:19) (cid:18) α (cid:19)(cid:21)
|c |2SNR |c |2SNR 0.04
R 6(1−α) log 1+ 22 2 −log 1+ 12 2
2
(cid:20) (cid:18) 1−α (cid:19) (cid:18) 1−α (cid:19)(cid:21)
0.02
(4) TDMA
Multiplexed Tx
0
over all possible transmitting signal-to-noise-ratio pairs 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
R
SNR ∈ [0,P /σ2],SNR ∈ [0,P /σ2] and time allocation 1
1 1 2 2
parameter α.
Fig.1. GaussianInterferenceChannelsecrecyrateachievableRegionP1=
B. Multiplexed Transmission P2=0.1,c11=c22=1,c12=c21=0.2
Inthemultiplexedtransmissionscheme,transmittersare al-
lowedtosharethesamedegreesoffreedom.Bytheconstraint
withartificialnoise providesthe largestachievablerateregion
of information-theoretic security, no partial decoding of the
while TDMA gives the smallest rate region.
other transmitter’s message is allowed at a receiver. Hence,
Ontheotherhand,whenweconsiderthetwoextremecases
the interferenceresults in an increase of the noise floor.Thus,
of high- and low-SNR regimes, the picture changes. In the
thefollowingrateregioncanbeachievedwithperfectsecrecy
high-SNR regime, when we let SNR →∞,SNR →∞ and
[10]: 1 2
limSNR1 =q in (4), (5), and (6), we can see that multiplexed
R1 >0 transSmNRis2sion can not achieve any positive secrecy rate, while
R2 >0 TDMA rates are bounded by R1 < αlog(||cc2111||22), and R2 <
RR12 66lloogg(cid:18)(cid:18)11++ 11++||cc12||12cc||122221SS||22NNSSRRNN12RR21(cid:19)(cid:19)−−lloogg((cid:0)11++||cc1221||22SSNNRR21)(cid:1) (Ra1n1−yiαsse)bcolroeugcn(yd||eccr21da22t||be22y)fo.RFr1oRr<2t.hlTeoghst(ur1s1a+,+teT|||g|cccDc212y12112M|||w|2222λqqAit)h,istbhutehteawrbteieficscatincahlnonoiotciaescewh,irheaevtnee
(5) wewantbothuserstohavesecurecommunicationinthehigh-
over all possible transmitting signal-to-noise-ratio pairs SNR regime.
SNR ∈[0,P /σ2],SNR ∈[0,P /σ2]. In the low-SNR regime (as SNR approaches zero), TDMA
1 1 2 2
andmultiplexedtransmissionachievableregionsbecomeiden-
C. Artificial Noise
tical. They converge to the following rectangular rate region,
This scheme allows one of the transmitters (e.g transmitter as illustrated in Fig.1:
2)togenerateartificialnoise.Thisschemewillsplitthepower
R >0
of transmitter 2 into two parts: λP for generating artificial 1
2
noiseandtheremaining(1−λ)P forencodingtheconfidential R >0
2 2
message. As detailed in [10], the achievable rate region is R 6|c |2SNR −|c |2SNR +o(SNR )
1 11 1 21 1 1
R >0 R 6|c |2SNR −|c |2SNR +o(SNR ) (7)
1 2 22 2 12 2 2
R >0
2 Thus, these schemes have similar performances at vanishing
R 6log 1+ |c11|2SNR1 −log 1+ |c21|2SNR1 SNR levels in terms of the asymptotic rates. However, a finer
1 (cid:18) 1+|c |2SNR (cid:19) (cid:18) 1+|c |2λSNR (cid:19)analysis in the next section will provide more insight. We
12 2 22 2
|c |2(1−λ)SNR note that in the case of transmission with artificial noise,
R2 6log(cid:18)1+ 1+|c 2|22SNR +|c |22λSNR (cid:19) we have R1 6 |c11|2SNR1 − |c21|2SNR1 + o(SNR1) and
21 1 22 2
R 6(1−λ)(|c |2SNR −|c |2SNR )+o(SNR ) which is
|c |2(1−λ)SNR 2 22 2 12 2 2
−log 1+ 12 2 (6) strictly smaller than that in (7). This lets us to conclude that
(cid:18) 1+|c |2λSNR (cid:19)
12 2 introducing artificial noise is not preferable in the low-SNR
over all possible transmitting signal-to-noise-ratio pairs regime.
SNR ∈ [0,P /σ2],SNR ∈ [0,P /σ2] and power splitting
1 1 2 2
parameter λ. We can further enlarge the rate region by III. ENERGY EFFICIENCY IN THELOW-SNRREGIME
reversing the roles of transmitters 1 and 2. The tradeoff of spectral efficiency versus energy per infor-
When the transmitting power is moderate, neither too high mationbitis the key measureof performancein the low-SNR
nor too small, as demonstrated in [10], transmission strategy regime. The two major analysis tools in this regime are the
2
minimum value of the energy per bit Eb , and the slope SNR as
N0min
S of the spectral efficiency versus Eb curve at Eb [12].
These can be obtained from N0 N0min R˙1(0)=|c11|2−|c21|2 (14)
R˙ (0)=|c |2−|c |2. (15)
E log 2 2 22 12
b = e (8)
N0min C˙(0) Using (8), we get the minimum bit energy expressions. (cid:3)
¿From the result of Theorem 1, we see that TDMA and
and multiplexed transmission achieve the same minimum energy
per bit. Next, we consider the wideband slope regions.
2[C˙(0)2]
S = −C¨(0) (9) ThTehne,oforermth2e:GLaeutstshiaenraintetservfaenreisnhcewchhilaenkneeelpwinitghRco1/nRfid2e=ntiθa.l
where C˙(0) and C¨(0) denote the first and second derivatives messages, the slope region achieved by TDMA is
of the channel capacity with respect to SNR at SNR = 0. 06S <2
1
In this section, using these tools, we analyze the perfor-
06S <2
mance in interference channels with confidential messages, 2
S S
following an approach similar to that in [13]. Note that 1 + 2 =1 (16)
2A 2B
in interference channels, we have the achievable rate pairs
(R ,R ).AstheSNRsofbothusersapproachzerointhelow- and the slope region achieved by multiplexed transmission is
1 2
SNR regime, it can be easily seen that R →0 and R →0.
1 2 06S <2
In this regime,we introducethe parameterθ, andassume that 1
theratiooftheratesisR1/R2 =θ asR1 andR2 bothvanish. 06S2 <2
In both TDMA and multiplexed transmissions, we have 2A 2B 4|c |2|c |2|c |2|c |2
11 12 22 21
−1 −1 =
R SNR (|c |2−|c |2) (cid:18)S1 (cid:19)(cid:18)S2 (cid:19) (|c11|4−|c21|4)(|c22|4−|c12|4)
θ = 1 = 1 11 21 . (10) (17)
R SNR (|c |2−|c |2)
2 2 22 12
where
By fixing θ, we can rewrite the achievable rate region of
multiplexed transmission in (5) as A= |c11|2−|c21|2, (18)
|c |2+|c |2
11 21
R >0
1 |c |2−|c |2
R2 >0 B = |c22|2+|c12|2. (19)
22 12
Proof: Note again that for both transmission schemes, we
|c |2SNR
R1 6log1+ 1+|c |2 (1|1c11|2−|c211|2) SNR  have
 12 θ(|c22|2−|c12|2) 1 R˙1(0)=|c11|2−|c21|2, (20)
−log(1+|c |2SNR )
21 1 R˙ (0)=|c |2−|c |2. (21)
2 22 12
|c |2SNR
R2 6log1+ 1+|c |2θ2(|2c22|2−|c212|2)SNR  In TDMA, we also have
−log(1+|c |2SNR2)1. (|c11|2−|c21|2) 2 (11) −R¨1(0)= |c11|4−|c21|4, (22)
12 2 α
|c |4−|c |4
¿From (4) and (11), we can see that when SNR diminishes, −R¨ (0)= 22 12 . (23)
2
the bit energy Eb = SNR for both TDMA and multiplexed (1−α)
N0 R(SNR)
transmission schemes monotonically decreases. Furthermore, Then, using (9), we get
it can be shown that the rates are concave functions of SNR
2α(|c |2−|c |2)
in the low-SNR regime. Thus, the minimum energy per bit is S = 11 21 , (24)
achieved as SNR → 0. The following theorems provide the 1 |c11|2+|c21|2
minimumenergyperbitandtheslopeattheminimumenergy S = 2(1−α)(|c22|2−|c12|2). (25)
per bit. 2 |c |2+|c |2
22 12
Theorem 1: For all θ =R /R , the minimum bit energies
1 2 Considering different values of α leads to the region in (16).
in the Gaussian interference channel with confidential mes-
Similarly, for multiplexed transmission, we can obtain
sages for both TDMA and multiplexed transmissions are
2|c |2|c |2(|c |2−|c |2)
E1 = loge2 , (12) −R¨1(0)=|c11|4−|c21|4+ 11θ(|c12 |2−11|c |2) 21 ,
N |c |2−|c |2 22 12
0min 11 21 (26)
E log 2
2 = e . (13) 2|c |2|c |2θ(|c |2−|c |2)
N |c |2−|c |2 −R¨ (0)=|c |4−|c |4+ 22 21 22 12 .
Proof: From (4) 0amndin(11),22we can12for both cases easily 2 22 12 |c11|2−|c21|2
(27)
computethederivativesoftheachievablerateswith respectto
3
¿From the above expression, we can easily see that
|c |2=0.8, |c |2=0.8
2(|c |2−|c |2) 12 21
S1 = |c |2+|c11|2+ 22|1c11|2|c12|2 , (28) 1.2 ||cc12||22==00..74,, ||cc21||22==00..64
11 21 θ(|c22|2−|c12|2) 1 |c12|2=0.2, |c21|2=0.3
2(|c |2−|c |2) 12 21
S = 22 12 . (29)
2 |c |2+|c |2+ 2|c22|2|c21|2θ 0.8
22 12 |c11|2−|c21|2 S2
0.6
Considering different values of θ leads to the slope region
given in (17). (cid:3) 0.4
IV. THEIMPACT OFSECRECY ON ENERGYEFFICIENCY 0.2
Forcomparison,weprovidebelowtheminimumenergyper
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4
bitandsloperegionwhentherearenosecrecyconstraints[13]. S
1
The minimum bit energies for both TDMA and multiplexed
transmission are Fig.2. SloperegionsintheGaussianinterferencechannelwithconfidential
messagesfortheTDMAschemewith|c11|2=|c22|2=1andvariousvalues
E1 = loge2, (30) of|c12|2,|c21|2 .
N |c |2
0min 11
E log 2
2 = e . (31)
N |c |2
0min 22 |c |2=0.8, |c |2=0.8
12 21
The achievable slope region for TDMA is 1.2 |c12|2=0.7, |c21|2=0.6
|c |2=0.4, |c |2=0.4
06S1 <2 1 |c12|2=0.2, |c21|2=0.3
12 21
06S <2
2 0.8
S1+S2 =2, (32) S2
0.6
while for multiplexed transmission, we have
0.4
06S <2
1
0.2
06S <2
2
2 2 |c |2|c |2 00 0.2 0.4 0.6 0.8 1 1.2 1.4
( −1)( −1)=4 12 21 . (33) S
1
S S |c |2|c |2
1 2 22 11
We can immediately note that the minimum bit energies in Fig.3. SloperegionsintheGaussianinterferencechannelwithconfidential
messagesformultiplexedtransmissionschemewith|c11|2=|c22|2=1and
(30) and (31) are strictly smaller than those given in (12) and various values of|c12|2,|c21|2 .
(13).Thus,thereis anenergypenaltyassociatedwith secrecy.
Moreover, comparing the slope regions in (16) and (17) with
those in (32) and (33), and noting that
achievable rate regions converge to those of multiplexed
A<1 transmissionschemeaspowerdecreases.Furthermore,TDMA
B <1 and multiplexed transmission has the same minimum energy
|c |2|c |2 4|c |2|c |2|c |2|c |2 perbitvalues.Therefore,weshouldconsiderthesloperegions.
4 12 21 < 11 12 22 21 , (34) From Theorem 2, we know that when
|c |2|c |2 (|c |4−|c |4)(|c |4−|c |4)
22 11 11 21 22 12
4|c |2|c |2|c |2|c |2
we can easily verify that the slope region of Gaussian weak 11 12 22 21 <1, (35)
interference channel is strictly larger than the slope region (|c11|4−|c21|4)(|c22|4−|c12|4)
of Gaussian weak interferencechannelwith confidentialmes- the slope region of multiplexed transmission is strictly larger
sages for both TDMA and multiplexedtransmission schemes. thanthesloperegionofTDMA,thusinthiscase, multiplexed
Thus, in addition to the increase in the minimum energy per transmission is preferred. On the other hand, when
bit, secrecy introduces a penalty in terms of the achievable
4|c |2|c |2|c |2|c |2
widebandslope values. In Figs. 2 and 3, we plotthe slope re- 11 12 22 21 >1, (36)
(|c |4−|c |4)(|c |4−|c |4)
gions for TDMA and multiplexed transmissions, respectively, 11 21 22 12
undersecrecyconstraints.Wenotethatregionsbecomesmaller the slope region of TDMA is larger than the slope region of
as |c |2 and |c |2 increase. This is due to the fact that for multiplexed transmission. Hence, TDMA should be used in
12 21
fixed |c |2 and |c |2, the larger values of |c |2 and |c |2 this scenario. Finally, when
11 22 12 21
mean that channel of the unintended receiver gets stronger
4|c |2|c |2|c |2|c |2
and we have to use more energy to achieve the same secrecy 11 12 22 21 =1, (37)
(|c |4−|c |4)(|c |4−|c |4)
transmission rate. 11 21 22 12
We are also interested in determining which transmission thesloperegionsofTDMAandmultiplexedtransmissioncon-
scheme performs better in the low-SNR regime. TDMA vergetothesametriangularregion.Inthiscase,TDMAshould
4
2 2
TDMA
1.8 Multiplexed TX 1.8
secrecy TDMA
1.6 secrecy Multiplexed TX 1.6
1.4 1.4
1.2 1.2
S2 1 S2 1
0.8 0.8
0.6 0.6
0.4 0.4 TDMA
Multiplexed TX
0.2 0.2 secrecy TDMA
secrecy Multiplexed TX
0 0
0 0.5 1 1.5 2 0 0.5 1 1.5 2
S S
1 1
Fig. 4. Slope regions in the Gaussian interference channel. |c11|2 = Fig. 5. Slope regions in the Gaussian interference channel. |c11|2 =
|c22|2=1,|c12|2=0.4,|c21|2=0.5 |c22|2=1,|c12|2=0.1,|c21|2=0.2
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V. CONCLUSION
Workshop,Paris,France, Mar2003.
In this paper, we have studied the achievable secrecy rates
over Gaussian interference channel for TDMA, multiplexed
and artificial noise schemes. Althoughusually TDMA has the
worst performance [10], we have noted that only TDMA can
achieve positive secrecy rates for both users in the high-SNR
regime.In the low-powerregime,we have shownthat TDMA
is optimalwhen 4|c11|2|c12|2|c22|2|c21|2 >1. We have also
(|c11|4−|c21|4)(|c22|4−|c12|4)
shown that secrecy constraints introduce penalty in both the
minimum bit energy and slope. Finally, we have shown that
TDMAismorelikelytobeoptimalinthepresenceofsecrecy
limitations.
5