Secure Transmission with Artificial Noise over Fading Channels: Achievable Rate and Optimal Power Allocation
aa r X i v : . [ c s . I T ] J un Secure Transmission with Artificial Noise overFading Channels: Achievable Rate and OptimalPower Allocation
Xiangyun Zhou,
Student Member, IEEE , and Matthew R. McKay,
Member, IEEE
Abstract — We consider the problem of secure communicationwith multi-antenna transmission in fading channels. The trans-mitter simultaneously transmits an information bearing signal tothe intended receiver and artificial noise to the eavesdroppers.We obtain an analytical closed-form expression of an achievablesecrecy rate, and use it as the objective function to optimize thetransmit power allocation between the information signal and theartificial noise. Our analytical and numerical results show thatequal power allocation is a simple yet near optimal strategy forthe case of non-colluding eavesdroppers. When the number ofcolluding eavesdroppers increases, more power should be usedto generate the artificial noise. We also provide an upper boundon the signal-to-noise ratio (SNR) above which the achievablesecrecy rate is positive and show that the bound is tight at lowSNR. Furthermore, we consider the impact of imperfect channelstate information (CSI) at both the transmitter and the receiverand find that it is wise to create more artificial noise to confusethe eavesdroppers than to increase the signal strength for theintended receiver if the CSI is not accurately obtained.
Index Terms — Secrecy rate, multi-antenna transmission, arti-ficial noise, power allocation, channel estimation error.
I. I
NTRODUCTION
Security is a fundamental problem in wireless communi-cations due to the broadcast nature of the wireless medium.Traditionally, secure communication is achieved by usingcryptographic technologies such as encryption. On the otherhand, the studies from an information-theoretic viewpoint havefound conditions for reliable secure communication withoutusing secret keys. In the pioneering works on information-theoretic security, Wyner introduced the wiretap channelmodel in which the eavesdropper’s channel is a degradedversion of the receiver’s channel [1]. Csisz´ar and K¨ornerconsidered a general non-degraded channel condition andstudied the transmission of both a common message to tworeceivers and a confidential message to only one of them [2].The results in these early works showed that a positive secrecycapacity can be achieved if the intended receiver has a betterchannel than the eavesdropper.Recently, information-theoretic security with multi-antennatransmission has drawn a lot of attention. Many works have
This paper was presented in part at the Int. Conf. on Signal Processing andCommun. Syst., Omaha, NE, Sept. 2009.Xiangyun Zhou is with the Research School of Information Sciencesand Engineering, the Australian National University, Canberra, ACT 0200,Australia. (Email: [email protected]). His work was supported underAustralian Research Council’s Discovery Projects funding scheme (project no.DP0773898). Matthew R. McKay is with the Department of Electronic andComputer Engineering, Hong Kong University of Science and Technology,Hong Kong. (Email: [email protected]). been devoted to analyzing the secrecy capacity with variousantenna configurations and channel conditions, e.g., [3–6].With multiple antennas at the transmitter, the optimal inputstructure (for Gaussian codes) that maximizes the secrecyrate of Gaussian channels was found to be in the formof beamforming transmission [3, 4]. The secrecy capacity ofGaussian channels with multiple antennas at both the trans-mitter and the receiver was obtained in [5, 6]. One of themain assumptions in the above-mentioned works is that theeavesdropper’s channel is known at the transmitter. Clearlythis assumption is usually impractical, especially for fadingchannels. The ergodic secrecy capacity with and withoutknowing the eavesdropper’s channel was studied for fadingchannels in [7–11]. The authors in [9] studied a fadingbroadcast channel with confidential information intended onlyfor one receiver and derived the optimal power allocationthat minimizes the secrecy outage probability. The authorsin [10] proposed an on-off power transmission with variable-rate allocation scheme for single antenna systems, which wasshown to approach the optimal performance at asymptoticallyhigh signal-to-noise ratio (SNR). The authors in [11] extendedthe ergodic secrecy capacity result to systems with multipleantennas and developed capacity bounds in the large antennalimit.Furthermore, various physical-layer techniques were pro-posed to achieve secure communication even if the receiver’schannel is worse than the eavesdropper’s channel. One ofthe main techniques is the use of interference or artificialnoise to confuse the eavesdropper. With two base stationsconnected by a high capacity backbone, one base station cansimultaneously transmit an interfering signal to secure theuplink communication for the other base station [12, 13]. Inthe scenario where the transmitter has a helping interfereror a relay node, the secrecy level can also be increased byhaving the interferer [14] or relay [15] to send codewordsindependent of the source message at an appropriate rate.When multiple cooperative nodes are available to help thetransmitter, the optimal weights of the signal transmitted fromcooperative nodes, which maximize an achievable secrecy rate,were derived for both decode-and-forward [16] and amplify-and-forward [17] protocols. The use of interference for secrecyis also extended to multiple-access and broadcast channelswith user cooperation [18–20].When multiple antennas are available at the transmitter, itis possible to simultaneously transmit both the informationbearing signal and artificial noise to achieve secrecy in aading environment [21–23]. The artificial noise is radiatedisotropically to mask the transmission of the informationsignal to the intended receiver. In the design of this multi-antenna technique, the transmit power allocation between theinformation signal and the artificial noise is an importantparameter, which has not been investigated in [21, 22]. Asub-optimal power allocation strategy was considered in [23],which aims to meet a target signal to interference and noiseratio at the intended receiver to satisfy a quality of servicerequirement.In this paper, we study the problem of secure commu-nication in fading channels with a multi-antenna transmittercapable of simultaneous transmission of both the informationsignal and the artificial noise. We derive a closed-form ex-pression for an achievable secrecy rate in fading channels.The availability of a closed-form secrecy rate expressiongreatly reduces the complexity of obtaining the optimal powerallocation between transmission of the information signal andthe artificial noise. We also study the critical SNR above whichthe achievable secrecy rate is positive. This is an importantproblem in wideband communications in which a higherthroughput is achieved by reducing the SNR per hertz whileincreasing the bandwidth [24]. Furthermore, perfect channelstate information (CSI) at both the transmitter and the receiveris usually assumed in the existing studies on information-theoretic security. With this assumption, the artificial noiseis accurately transmitted into the null space of the intendedreceiver’s channel. When the CSI is not perfectly known at thetransmitter, the artificial noise leaks into the receiver’s channel.The effects of imperfect CSI on the achievable secrecy rate andthe aforementioned design parameters are investigated in thispaper.The main contributions of this work are summarized asfollows: • In Section III, we derive analytical closed-form lowerbounds on the ergodic secrecy capacity for both non-colluding and colluding eavesdroppers. These closed-form expressions, which give achievable secrecy ratesfor secure communications with artificial noise, greatlyreduce the complexity of system design and analysis, andalso allow analytical insights to be obtained. • In Section IV, we study the optimal power allocationbetween transmission of the information signal and theartificial noise. For the non-colluding eavesdropper case,the equal power allocation is shown to be a simplestrategy that achieves nearly the same secrecy rate as theoptimal power allocation. For the colluding eavesdroppercase, more power should be used to transmit the artificialnoise as the number of eavesdropper increases. Analyticalresults are obtained in the high SNR regime in both cases. • In Section V, we derive an upper bound on the criticalSNR above which the achievable secrecy rate is positive.The bound is shown to be tight at low SNR, hence isuseful in the design and analysis of wideband securecommunications. • In Section VI, we derive an ergodic secrecy capacitylower bound taking into account channel estimation er-rors and investigate the effects of imperfect CSI on the optimal power allocation and the critical SNR for securecommunication. In particular, we find that it is better tocreate more artificial noise for the eavesdroppers than toincrease the signal strength for the intended receiver asthe channel estimation error increases.Throughout the paper, the following notations will be used:Boldface upper and lower cases denote matrices and vectors,respectively. [ · ] T denotes the matrix transpose operation, [ · ] ∗ denotes the complex conjugate operation, and [ · ] † denotes theconjugate transpose operation. The notation E {·} denotes themathematical expectation. k · k denotes the norm of a vectorand | · | denotes the determinant of a matrix.II. S YSTEM M ODEL
We consider secure communication between a transmitter(Alice) and a receiver (Bob) in the presence of eavesdroppers(Eves). Alice has N A antennas ( N A > ) and Bob has asingle antenna. This scenario is representative, for example, ofdownlink transmission in cellular systems and wireless localarea networks. In addition, each Eve is equipped with a singleantenna. We consider two cases, namely non-colluding andcolluding eavesdroppers. In the former case, Eves individuallyoverhear the communication between Alice and Bob withoutany centralized processing. While in the latter case, thereare N E Eves capable of jointly processing their receivedinformation. Therefore, the non-colluding case can be seenas a special colluding case where N E = 1 . We assume that N A > N E for which the reason will become clear in the nextsection. We also assume that Eves are passive, hence theycannot transmit jamming signals. The received symbols at Boband the multiple colluding Eves are given by, respectively, y B = hx + n, (1) y E = Gx + e , (2)where h is a × N A vector denoting the channel between Aliceand Bob and G is an N E × N A matrix denoting the channelbetween Alice and multiple colluding Eves. The elements of h and G are independent zero-mean complex Gaussian randomvariables. n and e are the additive white Gaussian noises atBob and Eves, respectively. Without loss of generality, wenormalize the variance of n to unity. We assume that h isaccurately estimated by Bob and is also known by Aliceusing a noiseless feedback link from Bob . Similar to [21],we assume that the knowledge of both h and G is available atEve, which makes the secrecy of communication independentof the secrecy of channel gains.The key idea of guaranteeing secure communication usingartificial noise proposed in [21] is outlined as follows. Welet an N A × N A matrix W = [ w W ] be an orthonormalbasis of C N A , where w = h † / k h k . The N A × transmittedsymbol vector at Alice is given by x = w u + W v , wherethe variance of u is σ u and the N A − elements of v A reliable feedback link could be achieved by using low rate transmissionwith appropriate quantization schemes. The design of a high-quality feedbacklink and the effect of noisy feedback is beyond the scope of this work.However, we will investigate the effect of imperfect channel knowledge atAlice by considering channel estimation errors at Bob in Section VI. re independent and identically distributed (i.i.d.) complexGaussian random variables each with variance σ v . u representsthe information bearing signal and v represents the artificialnoise. The received symbols at Bob and Eves become y B = hw u + hW v + n = k h k u + n, (3) y E = Gw u + GW v + e = g u + G v + e , (4)where we have defined that g = Gw and G = GW .We consider a total power per transmission denoted by P ,that is, P = σ u + ( N A − σ v . Due to the normalization of thenoise variance at Bob, we also refer to P as the transmit SNR.One important design parameter is the ratio of power allocatedto the information bearing signal and the artificial noise. Wedenote the fraction of total power allocated to the informationsignal as φ . Hence, we have the following relationships: σ u = φP, (5) σ v = (1 − φ ) P/ ( N A − . (6)Since h is known by Alice, she can adaptively change thevalue of φ according to the instantaneous realization of h . Werefer to this strategy as the adaptive power allocation strategy.Alternatively, Alice can choose a fixed value for φ regardlessof the instantaneous channel realization, which we refer toas the non-adaptive power allocation strategy. Note that Alicedoes not know G , and thus equally distributes the transmitpower amongst the artificial noise signal, as given by (6).III. S ECRECY C APACITY L OWER B OUND
The secrecy capacity is the maximum transmission rateat which the intended receiver can decode the data witharbitrarily small error while the mutual information betweenthe transmitted message and the received signal at the eaves-dropper is arbitrarily small. It is bounded from below by thedifference in the capacity of the channel between Alice andBob and that between Alice and Eve [2]. In this section,we derive a closed-form expression for an ergodic secrecycapacity lower bound with transmission of artificial noise.The capacity of the channel between Alice and Bob is givenby C = E h { log (1 + σ u k h k ) } = E h { log (1 + φP k h k ) } . (7)Without loss of generality, we normalize the variance ofeach element of h to unity. It is then easy to see that k h k follows a Gamma distribution with parameters ( N A , .Therefore, for systems with non-adaptive power allocationstrategy, we can rewrite (7) in an integral form as C = 1ln 2 Z ∞ ln(1 + φP x ) x N A − exp( − x )Γ( N A ) d x, = 1ln 2 exp (cid:16) zP (cid:17) N A X k =1 E k (cid:16) zP (cid:17) , (8)where Γ( · ) is the Gamma function, E n ( · ) is the generalizedexponential integral, (8) is obtained using an integral identitygiven in [25], and we have defined z = φ − . Next we study the capacity of the channel between Aliceand multiple colluding Eves. When multiple Eves are present,the noise at each Eve may be different. In addition, the receivernoise levels at Eves may not be known by Alice and Bob.To guarantee secure communication, it is therefore reasonableto consider the worst case scenario where the noises at Evesare arbitrarily small. Note that this approach was also takenin [22]. In this case, we can normalize the distance of eachEve to make the variance of the elements of G equal to unitywithout loss of generality. The noiseless eavesdropper assumption effectively gives anupper bound on the capacity of the channel between Alice andmultiple colluding Eves as C = E h , g , G n log (cid:12)(cid:12)(cid:12) I + σ u g g † ( σ v G G † ) − (cid:12)(cid:12)(cid:12)o = E h , g , G n log (cid:16) N A − z − g † ( G G † ) − g (cid:17)o , (9)where we have again used z = φ − . The expectation over h in (9) is due to the fact that z may be dependent on h (whichhappens when adaptive power allocation strategy is used). It isrequired in (9) that G G † is invertible, which is guaranteedwith the assumption of N A > N E . If the assumption isviolated, the colluding eavesdroppers are able to eliminatethe artificial noise, resulting C = ∞ . Hence, We assume N A > N E for guaranteeing secure communication.Since G has i.i.d. complex Gaussian entries and W isa unitary matrix, GW = [ g G ] also has i.i.d. complexGaussian entries. Therefore, the elements of g and G areindependent. As a result, the quantity g † ( G G † ) − g isequivalent to the signal-to-interference ratio (SIR) of a N E -branch minimum mean square error (MMSE) diversity com-biner with N A − interferers. The complementary cumulativedistribution function of X = g † ( G G † ) − g is givenin [26] as R X ( x ) = P N E − k =0 (cid:0) N A − k (cid:1) x k (1 + x ) N A − . (10)Therefore, we can rewrite (9) in an integral form as C = E h n Z ∞ log (cid:16) N A − z − x (cid:17) f X ( x )d x o = E h n Z ∞ N A − z − (cid:16) N A − z − x (cid:17) − R X ( x )d x o (11) = E h n N E − X k =0 (cid:18) N A − k (cid:19) × Z ∞ (cid:16) z − N A − x (cid:17) − (1 + x ) − N A x k d x o = E h n N E − X k =0 (cid:18) N A − k (cid:19) N A − z − B ( k +1 , N A − − k ) × F (cid:16) , k +1; N A ; z − N A z − (cid:17)o , (12) With the noiseless eavesdropper assumption, the capacity between Aliceand each Eve is determined from the signal-to-artificial-noise ratio. Consid-ering the signal reception at a particular Eve, both the information signal andthe artificial noise are generated from the same source (Alice), and hence theirratio is independent of the large scale fading from Alice to Eve. That is tosay, the signal-to-artificial-noise ratios are i.i.d. random variables for all Eves,regardless of their distances from Alice.
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SNR, dB E r god i c s ec r ec y ca p ac it y l o w e r bound N A = 8, N E = 1N A = 4, N E = 1N A = 4, N E = 2N A = 8, N E = 6 Fig. 1. Ergodic secrecy capacity lower bound C in (14) versus SNR P forsystems with different numbers of antennas. The ratio of power allocation isset to φ = 0 . . where f X ( x ) denotes the probability density function of X ,B ( α, β ) = Γ( α )Γ( β )Γ( α + β ) is the Beta function and F ( · ) is theGauss hypergeometric function. Note that (11) is obtainedusing integration by parts, and (12) is obtained using anintegration identity given in [27].After deriving expressions for C and C , a lower boundon the ergodic secrecy capacity can now be obtained as C =[ C − C ] + , where [ α ] + = max { , α } . This is a data ratethat can be always guaranteed for the secure communication(without knowing the noise level at Eves). For systems withadaptive power allocation, the ergodic secrecy capacity lowerbound is given as C = 1ln 2 h E h n ln (cid:16) Pz k h k (cid:17) − N E − X k =0 (cid:18) N A − k (cid:19) N A − z − × B ( k +1 , N A − − k ) F (cid:16) , k +1; N A ; z − N A z − (cid:17)oi + (13)where z is a function of h . For systems with non-adaptivepower allocation, the ergodic secrecy capacity lower bound isgiven as C = 1ln 2 h exp (cid:16) zP (cid:17) N A X k =1 E k (cid:16) zP (cid:17) − N E − X k =0 (cid:18) N A − k (cid:19) N A − z − × B ( k +1 , N A − − k ) F (cid:16) , k +1; N A ; z − N A z − (cid:17)i + (14)where z is a constant independent of h .Fig. 1 shows the ergodic secrecy capacity lower bound C in (14) for systems with different numbers of antennas. Wesee that the presence of multiple colluding Eves dramaticallyreduces the secrecy rate, compared with the case of non-colluding Eves. Furthermore, the secrecy rate quickly reducesto zero at low to moderate SNR.In the following subsections, we aim to give simplifiedor approximated expressions of the secrecy capacity lowerbound in two special scenarios. These expressions will beused to obtain analytical results and useful insights on the optimal power allocation in Section IV. Note that the derivedapproximation may not be an achievable secrecy rate, althoughit is useful for the design of power allocation. A. Non-colluding Eavesdroppers
In the case where Eves cannot collude, we have N E = 1 .Then C in (12) reduces to C = E h n z − F (cid:16) , N A ; z − N A z − (cid:17)o = E h ( (cid:16) N A − N A − z (cid:17) N A − × ln (cid:16) N A − z − (cid:17) − N A − X l =1 l (cid:16) N A − zN A − (cid:17) l !) , (15)where (15) is obtained using an identity for the Gauss hyper-geometric function derived in Appendix I. This can then besubstituted into C = [ C − C ] + to yield simplified expressionsfor the ergodic secrecy capacity lower bound. B. Large N A Analysis C in (7) can be rewritten as C = E h n log (cid:16) Pz k h k (cid:17)o = log N A + E h n log (cid:16) N A + Pz k h k N A (cid:17)o . (16)The law of large numbers implies that lim N A →∞ k h k /N A =1 . Hence we focus on the non-adaptive power allocationstrategy where z is a constant. In the large N A limit, we have lim N A →∞ ( C − log N A ) = lim N A →∞ E h n log (cid:16) N A + Pz k h k N A (cid:17)o = log Pz . (17)That is to say, the difference between C and log N A ap-proaches log Pz as N A increases. Therefore, in the large N A regime, we have C = log (cid:16) N A Pz (cid:17) + o (1) . (18)From the law of large numbers, we also know that lim N A →∞ G G † / ( N A −
1) = I . Using (9) with the non-adaptive power allocation strategy, we have lim N A →∞ C = lim N A →∞ E g , G n log (cid:16)
1+ 1 z − g † (cid:16) G G † N A − (cid:17) − g (cid:17)o = E g n log (cid:16) z − k g k (cid:17)o = 1ln 2 exp( z − N E X k =1 E k ( z − , (19)where k g k has a Gamma distribution with parameters ( N E , . We see from (19) that altering the number of antennas The notation f ( x ) = o ( g ( x )) implies that lim x →∞ f ( x ) g ( x ) = 0 . This limitis taken w.r.t. N A in (18). Number of antennas at Alice, N A E r god i c s ec r ec y ca p ac it y l o w e r bound N E = 1N E = 2N E = 4N E = 6 large N A approx. Fig. 2. Ergodic secrecy capacity lower bound C in (14) at 10 dB versusthe number of antennas at Alice N A for systems with different numbersof colluding eavesdroppers. The large N A approximations of C in (20) areshown as the dashed lines. The ratio of power allocation is set to φ = 0 . . at Alice does not affect the channel capacity between Aliceand Eves in the large N A limit.The ergodic secrecy capacity lower bound in the large N A regime is then given by C = 1ln 2 h ln (cid:16) N A Pz (cid:17) − exp( z − N E X k =1 E k ( z − o (1) i + . (20)In Section IV, we will use the expression (dropping o (1) )in (20) as an approximation of the secrecy capacity lowerbound for systems with large N A to study the optimal powerallocation.Fig. 2 shows the ergodic secrecy capacity lower bound C in (14) as well as its large N A approximation in (20). We seethat (14) converges to (20) as N A increases. The convergenceis fast for small number of colluding Eves, e.g. N E = 2 , andis slow for large number of colluding Eves, e.g. N E = 6 .IV. O PTIMAL P OWER A LLOCATION
In this section, we study the optimal power allocationbetween the information bearing signal and the artificial noise.As we have discussed, the power allocation strategy canbe either adaptive or non-adaptive. The former depends onevery realization of the channel gain while the latter is fixedfor all channel realizations. The objective function for thisoptimization problem is the ergodic secrecy capacity lowerbound. The closed-form expressions derived in the previoussection greatly reduce the computational complexity of theoptimization process. In the following, we first study the caseof non-colluding eavesdroppers and then look at the case ofcolluding eavesdroppers.
A. Non-colluding Eavesdropper Case
The optimal value of φ or z can be easily found numericallyusing the capacity lower bound expressions derived in Sec-tion III. Moveover, these expressions enable us to analytically obtain useful insights into the optimal z in the high SNRregime as follows.In the high SNR regime, i.e., P ≫ , C in (7) can beapproximated as C ≈ E h n log (cid:16) Pz k h k (cid:17)o = E h { log ( P k h k ) } − E h { log z } . (21)We see in (21) that E h { log ( P k h k ) } is a constant and E h { log z } does not directly depend on h although z maybe a function of h . Therefore, the high SNR approximationof the secrecy capacity lower bound does not have h in itsexpression (except for the expectation over h ). Consequently,for any value of h , the optimal z that maximizes the highSNR approximation of the secrecy capacity lower bound is thesame. In other words, the value of h is irrelevant in findingthe optimal power allocation. Therefore, the adaptive powerallocation strategy does not need to be considered at high SNR.The optimal value of z in the high SNR regime satisfies d C d z = d C d z − d C d z = − z ln 2 − d C d z = 0 , (22)where the derivative of C w.r.t. z can be computed in closed-form using (15).In the special case of N A = 2 , (22) is reduces to − z − z − z −
1) + ln( z − z − = 0 . (23)The solution to the above equation is given by z = 2 . It can beshown that lim z → C d z < . Hence the optimal ratio of powerallocation is given by φ = 0 . , that is to say, equal powerallocation between the information signal and the artificialnoise is the optimal strategy in the high SNR regime for N A = 2 .For large N A , using (19) with N E = 1 , we have d C d z = 1ln 2 (cid:16) exp( z − E ( z − − exp( z − E ( z − (cid:17) = 1ln 2 (cid:16) exp( z − E ( z − − ( z − − (cid:17) . (24)Hence the optimal value of z satisfies − z − e z − E ( z −
1) + 1 z − , (25)which gives z = 1 . . It can be shown that at z = 1 . , d C d z < . Hence the optimal ratio of power allocation is givenby φ = 0 . in the high SNR regime for sufficiently large N A . We see that the difference between the optimal values of φ for the smallest N A (i.e., N A = 2 ) and asymptotically large N A is very small.Fig. 3 shows the optimal values of φ using the non-adaptivepower allocation strategy for systems with different numbersof antennas at Alice N A . The values of φ are shown forSNRs at which the ergodic secrecy capacity lower bound ispositive. The general trend is that more power needs to beallocated to the information signal as SNR or N A increases.In the high SNR regime, we see that the optimal values of φ converge to constant values. For N A = 2 , the optimal valueof φ converges to 0.5, which agrees with our analytical result.
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SNR, dB O p ti m a l r a ti o o f po w e r a ll o ca ti on φ N A = 16N A = 8N A = 4N A = 2 Fig. 3. Optimal ratio of power allocation φ versus SNR P for differentnumbers of antennas at Alice N A . The non-adaptive power allocation strategyis used. The values of φ are shown for SNRs at which the ergodic secrecycapacity lower bound is positive. −10 −5 0 5 10 15 20 25 30024681012 SNR, dB E r god i c s ec r ec y ca p ac it y l o w e r bound N A = 16, optimal φ N A = 8, optimal φ N A = 4, optimal φ N A = 2, optimal φ equal power allocation φ = 0.5 Fig. 4. Ergodic secrecy capacity lower bound C in (14) versus SNR P for different numbers of antennas at Alice N A . The non-adaptive powerallocation strategy is used. The ergodic secrecy capacity lower bound withequal power allocation for each case, indicated by the solid line, is also shownfor comparison. Furthermore, this constant value only increases slightly with N A , and the maximum value is 0.55 which agrees with ourlarge N A analysis. These observations suggest that a near-optimal yet simple power allocation strategy at moderate tohigh SNR values is the equal power allocation between theinformation signal and the artificial noise.Fig. 4 shows the ergodic secrecy capacity lower bound C in (14) with the optimized φ using the non-adaptive powerallocation strategy. For comparison, we also include the ca-pacity lower bound with equal power allocation, i.e., φ = 0 . ,indicated by the solid lines. We see that the equal powerallocation strategy achieves nearly the same secrecy rate asthe optimal non-adaptive power allocation in all cases over awide range of SNR values. This confirms that equal power −10 −5 0 5 10 15 20 25 30024681012 SNR, dB E r god i c s ec r ec y ca p ac it y l o w e r bound N A = 16, adaptive power allocation N A = 8, adaptive power allocation N A = 4, adaptive power allocation N A = 2, adaptive power allocationnon−adaptive power allocation Fig. 5. Ergodic secrecy capacity lower bound C in (13) and (14) versus SNR P for different numbers of antennas at Alice N A . Both the adaptive and non-adaptive power allocation strategies are used, indicated by the markers andthe lines, respectively. allocation is a simple and generic strategy which yields closeto optimal performance in terms of the derived achievablesecrecy rate.Fig. 5 shows the ergodic secrecy capacity lower bound C in (13) and (14) with the optimized φ , using both theadaptive and non-adaptive power allocation strategies. For theadaptive power allocation, we apply a linear search on φ to findthe optimal value that maximizes the secrecy capacity lowerbound for each realization of h . The maximum value of thesecrecy capacity lower bound for each channel realization isrecorded and the ergodic secrecy capacity lower bound is thencomputed using the distribution of h . We see that there is nodifference between the secrecy rate achieved by the adaptiveand non-adaptive strategies over a wide range of SNR values.The adaptive strategy only gives marginal advantage whenthe secrecy rate is close to zero. This result suggests that thenon-adaptive power allocation strategy is sufficient to achievealmost the best possible secrecy rate performance. For thisreason, we will only focus on the non-adaptive scheme in therest of this paper.
B. Colluding Eavesdropper Case
As we have seen in Fig. 1, the presence of multiplecolluding Eves severely degrades the secrecy rate. Therefore,it is essential for Alice to have a relatively large number ofantennas to maintain a good secure communication link. Forany value of N E , the optimal value of φ or z can be easilyfound numerically using the closed-form capacity lower boundexpression given in Section III. As the number of antennas atAlice is desired to be large, we carry out large N A analysisto obtain an asymptotic result on optimal z in the high SNRregime as follows. The same result is found for the colluding Eves case. The numerical resultsare omitted for brevity.
SNR, dB O p ti m a l r a ti o o f po w e r a ll o ca ti on φ N A = 10, N E = 2N A = 10, N E = 4N A = 10, N E = 6N A = 10, N E = 8 Fig. 6. Optimal ratio of power allocation φ versus SNR P for systems withdifferent numbers of colluding Eves N E . The values of φ are shown for SNRsat which the ergodic secrecy capacity lower bound is positive. In the high SNR regime with large N A , C in (20) can beapproximated as C ≈ h ln( N A P ) − ln z − exp( z − N E X k =1 E k ( z − i . (26)By taking the derivative of C w.r.t. z , the optimal z satisfies − z − e z − E N E ( z −
1) + 1 z − . (27)Using e z − E N E ( z − ≈ ( z − N E ) − from [28], whichis accurate when either N E or z is large, (27) reduces to − z − z − N E + 1 z − . (28)Hence the optimal z is given by z ∗ = 1 + p N E . (29)From (29) we see that the optimal value of z only dependson N E in the high SNR and large antenna regime. Moreover,(29) suggests that more power should be used to generateartificial noise when the number of Eves increases.Fig. 6 shows the optimal value of φ for systems withdifferent numbers of colluding Eves N E . Similar to the non-colluding Eves case, we see that more power should be used totransmit the information signal as SNR increases. The optimalvalue of φ stays constant in the high SNR regime. Furthermore,the optimal value of φ for colluding Eves case is usually muchsmaller than 0.5, i.e., equal power allocation, which is nearoptimal for non-colluding Eves case. In particular, the optimal φ reduces as N E grows, which implies that more power shouldbe allocated to generate the artificial noise as the number ofcolluding Eves increases. This observation agrees with ouranalytical insight and intuition.Fig. 7 shows the ergodic secrecy capacity lower bound C in(14) for systems with different N E . Here, we investigate thesensitivity in the secrecy rate to the design of power allocation.Consider a scenario where the total number of Eves that can −5 0 5 10 15 20 250123456789 SNR, dB E r god i c s ec r ec y ca p ac it y l o w e r bound N A = 10, N E = 2 , optimal φ N A = 10, N E = 4 , optimal φ N A = 10, N E = 6 , optimal φ N A = 10, N E = 8 , optimal φ use optimal φ for N A = 10, N E = 8 Fig. 7. Ergodic secrecy capacity lower bound C in (14) versus SNR P forsystems with different numbers of colluding Eves N E . The solid lines withmarkers indicate C achieved with optimal values of φ for the correspondingsystem. The dashed lines indicate C achieved with value of φ optimized for N E = 8 , which represents the case where the power allocation was initiallydesigned for N E = 8 , but the current value of N E reduces from 8 and thepower allocation is not redesigned.
10 15 20 25 30 35 40 45 500.050.10.150.20.250.30.350.40.450.5
Number of antennas at Alice, N A O p ti m a l r a ti o o f po w e r a ll o ca ti on φ N E = 2 , optimal N E = 4 , optimal N E = 6 , optimal N E = 8 , optimallarge antenna approx. Fig. 8. The ratio of power allocation φ at 20 dB versus the number ofantennas at Alice N A for systems with different numbers of colluding Eves N E . The solid lines with markers indicate the optimal values of φ , while thedashed lines indicate the values of φ from the large antenna approximationgiven in (29). collude is 8, and hence Alice has optimized φ for N E = 8 .When N E changes, the power allocation parameter φ does notneed to be optimized again as long as N E stays reasonablyclose to 8, e.g., N E = 6 , since the value of φ optimized for N E = 8 still works well for N E = 6 (with a power lossof 0.2 dB) as shown in Fig. 7. However, redesigning of φ becomes important when N E is considerably different from 8,e.g., N E = 2 to 4. For example, if N E changes from 8 to 4, apower loss of approximately 1 dB will incur if Alice still usesthe value of φ optimized for N E = 8 , as shown in Fig. 7.We also provide numerical verification of the optimal powerallocation obtained from the large antenna approximation C < z N A P N E − k =0 (cid:0) N A − k (cid:1) N A − z − B ( k +1 , N A − − k ) F (cid:16) , k +1; N A ; z − N A z − (cid:17) − N A +12 − . (33) P C < z N A (cid:16) N A − N A − z (cid:17) N A − (cid:18) ln (cid:16) N A − z − (cid:17) − P N A − l =1 1 l (cid:16) N A − zN A − (cid:17) l (cid:19) − N A + 12 − . (34)in the high SNR regime. Fig. 8 shows the ratio of powerallocation φ at 20 dB versus the number of antennas at Alice N A for systems with different numbers of colluding Eves N E .For a fixed N E , we see that the optimal value of φ increaseswith N A and reaches a constant value when N A is sufficientlylarge. This agrees with our analytical insight that the optimalpower allocation depends on N E but not on N A when N A is large. The asymptotic constant value of φ is close to theanalytical value given in (29) obtained from the large antennaapproximation.In the system model, we have assumed fixed power trans-mission over time. When variable power transmission is al-lowed subject to an average power constraint, the achievablesecrecy rate can be increased by having temporal power allo-cation according to the channel gain at each time instant. Fromthe derived secrecy rate expression, we see that the transmitpower only affects the transmission rate between Alice andBob. The existing study on the point-to-point channel capacity,e.g. in [29], showed that the temporal power optimization giveslittle capacity gain provided that the spatial power optimizationis used.In reality, the noise is always present at the eavesdroppersand hence, the designed power allocation strategy is not theoptimal strategy in practice. If the eavesdroppers’ noise levelsare known to the transmitter and hence are taken into accountin the secrecy rate expression, the efficiency of using artificialnoise in degrading the capacity between Alice and Eve isreduced. Therefore, more power should be used to transmitthe information signal.V. C RITICAL
SNR
FOR S ECURE C OMMUNICATIONS
Another important aspect of secure communication is theminimum SNR required for a positive secrecy rate, which isa critical parameter in wideband communications. With theclosed-form expression of the secrecy capacity lower boundderived in Section III, one can numerically find the criticalSNR with low computational complexity. In this section, wederive a closed-form upper bound on the critical SNR whichis useful in the design of wideband communications.Using properties of the exponential integral function in [28],(8) can be bounded from below as C > N A X k =1 zP + k , (30)which is asymptotically tight as the SNR approaches zero,i.e., P → . Using the convexity of (30) in k , we can further Number of antennas at Alice, N A C r iti ca l S N R , d B N E = 6 , exact N E = 4 , exact N E = 2 , exact N E = 1 , exactupper bound Fig. 9. The critical SNR P C versus number of antennas at Alice N A forsystems with different numbers of colluding Eves N E . The ratio of powerallocation is set to φ = 0 . . The solid lines with markers indicate the exactvalue of P C , while the dashed lines indicate the analytical upper bound givenin (33). bound C as C > N AzP + N A +12 , (31)which is also asymptotically tight as the SNR approaches zero.Using the lower bound on C in (31) and C in (12), theergodic secrecy capacity lower bound can be further boundedfrom below as C > N AzP + N A +12 − N E − X k =0 (cid:18) N A − k (cid:19) N A − z − × B ( k +1 , N A − − k ) F (cid:16) , k +1; N A ; z − N A z − (cid:17) . (32)The critical SNR, denoted by P C , is the SNR at which C drops to zero. With the lower bound on C given in (32), anupper bound on P C can be found as (33) on the top of thepage. In the case of non-colluding eavesdroppers, i.e., N E = 1 ,(33) reduces to (34) on the top of the page. The upper boundin (33) or (34) indicates a minimum SNR that guarantees apositive secrecy rate. Since (33) and (34) are asymptoticallytight at low SNR, they can be used to fine tune the powerallocation parameter z to minimize P C .Fig. 9 shows the critical SNR P C versus number of antennasat Alice N A for systems with different numbers of colluding C < − σ h z (cid:16) N A P N E − k =0 (cid:0) N A − k (cid:1) N A − z − B ( k + 1 , N A − − k ) F (cid:16) , k + 1; N A ; z − N A z − (cid:17) − N A + 12 (cid:17) − σ h − , (42)Eves N E . The power allocation is set to φ = 0 . in all cases.The general trend is that P C decreases as N A increases, and ahigher P C is required when N E increases. These observationsagree with intuition. Furthermore, we see that the analyticalupper bound on P C is very accurate for the case of non-colluding Eves. For the case of colluding Eves, the upperbound is reasonably accurate when P C < dB. The differencebetween the exact value of P C and its upper bound graduallyincreases as N E increases, which is mainly due to the increasein P C . When N E is relatively large, e.g., N E = 6 , one shouldallocate more power to generate the artificial noise (i.e., reduce φ ), as suggested in Fig. 6, in order to achieve a lower P C ,which in turn makes the bound tighter.VI. E FFECT OF I MPERFECT C HANNEL S TATE I NFORMATION
So far, we have assumed that the CSI can be perfectlyobtained at Alice and Bob. In this section, we investigate theeffect of imperfect CSI by considering channel estimation er-rors. With imperfect CSI, the beamforming transmission fromAlice to Bob is designed based on the estimated channel gainsrather than the true channel gains. Therefore, the artificialnoise leaks into Bob’s channel.To incorporate imperfect CSI, we consider that Bob per-forms the MMSE channel estimation. Therefore, we have h = ˆ h + ˜ h , (35) σ h = σ h + σ h , (36)where ˆ h denotes the channel estimate and ˜ h denotes theestimation error. σ h denotes the variance of each element in h . σ h and σ h denote the variance of each element in ˆ h and ˜ h ,respectively. As a general property of the MMSE estimator forGaussian signals [30], ˆ h and ˜ h are uncorrelated, each havingi.i.d. complex Gaussian entries.Similar to our system model in Section II, we assume thatthe knowledge of ˆ h is available at Alice and Eves. Therefore,the beamforming vector becomes w = ˆ h † / k ˆ h k , and thereceived symbol at Bob is given by y B = ˆ hx + ˜ hx + n = k ˆ h k u + ˜ hW [ u v T ] T + n. (37)A capacity lower bound for the channel between Aliceand Bob can be obtained by considering ˜ hW [ u v T ] T + n as the worst case Gaussian noise [31]. Note that W is aunitary matrix, hence ˜ hW has the same distribution as ˜ h [32].Therefore, the ergodic capacity lower bound for the channelbetween Alice and Bob is given by ˆ C = E ˆ h n log (cid:16) σ u k ˆ h k σ h P + 1 (cid:17)o . (38)With σ h normalized to unity, we have σ h = 1 − σ h . Sincethe elements of ˆ h is i.i.d. complex Gaussian, k ˆ h k is a sum of i.i.d. exponential distributed random variables, which followsa Gamma distribution with parameter ( N A , − σ h ) . Therefore,we obtain a closed-form expression for ˆ C as ˆ C = 1ln 2 exp (cid:16) z σ h + P − − σ h (cid:17) N A X k =1 E k (cid:16) z σ h + P − − σ h (cid:17) . (39)The presence of channel estimation errors does not affectthe signal reception at Eve given in (4). Therefore, the ergodicsecrecy capacity lower bound can be obtained by subtracting C from ˆ C as C = 1ln 2 h exp (cid:16) z σ h + P − − σ h (cid:17) N A X k =1 E k (cid:16) z σ h + P − − σ h (cid:17) − N E − X k =0 (cid:18) N A − k (cid:19) N A − z − B ( k +1 , N A − − k ) × F (cid:16) , k +1; N A ; z − N A z − (cid:17)i + . (40)Following the steps in Section V, we can also bound C frombelow in order to obtain an upper bound on the critical SNRfor secure communication with channel estimation errors as C > N A z σ h + P − − σ h + N A +12 − N E − X k =0 (cid:18) N A − k (cid:19) N A − z − × B ( k +1 , N A − − k ) F (cid:16) , k +1; N A ; z − N A z − (cid:17) . (41)And the upper bound on the critical SNR is then given in (42)on the top of the page, which is asymptotically tight at lowSNR.We now present numerical results on the optimal powerallocation as well as critical SNR in the presence of thechannel estimation errors. For brevity, we focus on the caseof non-colluding eavesdroppers. The trends on the effect ofchannel estimation errors observed in the following results alsoapply to the case of colluding eavesdroppers.Fig. 10 shows the optimal ratio of power allocation φ withdifferent channel estimation error variances σ h . We see that thechannel estimation error has noticeable impact on the value of φ , especially for small number of antennas at Alice, e.g., N A =2 . The general trend is that less power should be allocatedto information signal as channel estimation error increases.This is mainly due to the fact that the efficiency of improvingBob’s signal reception by boosting the transmit power of theinformation signal reduces as the channel estimation errorincreases. On the other hand, the efficiency of degrading Eve’ssignal reception by boosting the transmit power of the artificialnoise stays the same regardless of the channel estimation error.Hence, it is better to create more noise for Eves than toincrease the signal strength for Bob if the CSI is not accuratelyobtained. SNR, dB O p ti m a l r a ti o o f po w e r a ll o ca ti on φ N A = 4 , error variance = 0 N A = 4 , error variance = 0.1 N A = 4 , error variance = 0.2 N A = 2 , error variance = 0 N A = 2 , error variance = 0.1 N A = 2 , error variance = 0.2 Fig. 10. Optimal ratio of power allocation φ versus SNR P for differentnumbers of antennas at Alice N A and different variances of the channelestimation errors σ h . The values of φ are shown for SNRs at which theergodic secrecy capacity lower bound is positive.TABLE IC RITICAL
SNR (
IN D B) FOR S ECURE C OMMUNICATIONS WITH E QUAL P OWER A LLOCATION
Error variance Number of antennas N A σ h ∞ -0.26 -3.08 -4.76 -5.96 In practical systems, the channel estimation error usuallyreduces as the SNR increases, although their exact relationshipdepends on the training design. From Fig. 10, we can expectthat at low to moderate SNR where the channel estimationerror is usually noticeable, the optimal power allocation isvery different from that in the perfect CSI case. While at highSNR where the channel estimation error is usually small, theoptimal power allocation is expected to be very close to thatof the perfect CSI case. Therefore, in practical systems it isimportant to take channel estimation error into account whendesigning the power allocation at relatively low SNR.Table I lists the exact values of the critical SNR P C as wellas the closed-form upper bound given in (42) with φ = 0 . .The general trend is that the critical SNR increases as thechannel estimation error increases, which agrees with intuition.The upper bound gets tighter as P C reduces (or N A increases),and is accurate for N A ≥ with an error of less than 1 dB.VII. C ONCLUSION
In this paper, we considered the secure communicationin the wireless fading environment in the presence of non-colluding or colluding eavesdroppers. The transmitter isequipped with multiple antennas and is able to simultaneouslytransmit an information signal to the intended receiver andartificial noise to confuse the eavesdroppers. We obtained an closed-form expression for the ergodic secrecy capacitylower bound. We studied the optimal power allocation betweentransmission of the information signal and the artificial noise.In particular, equal power allocation was shown to be a nearoptimal strategy in the case of non-colluding eavesdroppers.When the number of colluding eavesdroppers increases, morepower should be used to generate artificial noise. We alsoderived an upper bound on the critical SNR above whichthe secrecy rate is positive and this bound was shown to betight at low SNR. When imperfect channel state informationwas considered in the form of channel estimation errors,we found that it is wise to create more artificial noise toconfuse the eavesdroppers than to increase the signal strengthfor the intended receiver. The results obtained in this workprovide various insights into the design and analysis of securecommunication with multi-antenna transmission.A
PPENDIX II DENTITY FOR S PECIAL C LASS OF G AUSS H YPERGEOMETRIC F UNCTION
Here we obtain a simplified expression for the Gausshypergeometric function in the form of F (1 , N + 1; x ) or F ( N, N ; N + 1; x ) for integer N ≥ . From [28], we knowthat these two forms of the Gauss hypergeometric function arerelated to each other by F (1 , N + 1; x ) = (1 − x ) N − F ( N, N ; N + 1; x ) . (43)Also, we know from [28] that d N − d x N − F (1 ,
1; 2; x ) = (1) N − (1) N − (2) N − F ( N, N ; N +1; x ) , where ( a ) b is the rising factorial. Therefore, we have F ( N, N ; N + 1; x )= (2) N − (1) N − (1) N − d N − d x N − F (1 ,
1; 2; x )= − N ( N − N − X l =0 (cid:18) N − l (cid:19) d l d x l ln(1 − x ) d N − − l d x N − − l x − , (44)where we have used the identity F (1 ,
1; 2; x ) = − ln(1 − x ) /x from [28]. It is easy to show that d k d x k ln(1 − x ) = − d k − d x k − (1 − x ) − = − ( k − − x ) k , k = 1 , , , ... d k d x k z − = ( − k k ! x k +1 , k = 0 , , , , ... Substituting the above expressions for the derivatives into (44),we obtain an identity expression as F ( N, N ; N + 1; x )= − N ( N − (cid:16) ln(1 − x ) ( − N − ( N − z N − N − X l =1 ( N − l !( N − − l )! ( l − − x ) l ( − N − − l ( N − − l )! x N − l (cid:17) = ( − N Nx N (cid:16) ln(1 − x ) − N − X l =1 l x l ( x − l (cid:17) . (45)sing (43), we also have F (1 , N + 1; x )= ( − N N (1 − x ) N − x N (cid:16) ln(1 − x ) − N − X l =1 l x l ( x − l (cid:17) . (46)A CKNOWLEDGEMENTS
The authors would like to thank Dr. Parastoo Sadeghi foruseful discussions. R
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Xiangyun Zhou (S’08) received the B.E. (hons.) degreein electronics and telecommunications engineering from theAustralian National University, Australia, in 2007. He iscurrently working toward the Ph.D. degree in engineering andinformation technology at the Research School of InformationSciences and Engineering, the Australian National University.His research interests are in signal processing for wirelesscommunications, including MIMO systems, ad hoc networks,relay and cooperative networks, and physical-layer security.