Resource Allocation for Secure OFDMA Networks with Imperfect CSIT

Transcription

1 Resoure Alloation for Seure OFDMA Networs with Imperfet CSIT Derri Wing Kwan Ng, Ernest S. Lo, and Robert Shober Department of Eletrial and Computer Engineering The University of British Columbia Centre Tenològi de Teleomuniaions de Catalunya CTTC Abstrat In this paper, we formulate an optimization problem for resoure alloation and sheduling in orthogonal frequeny division multiple aess OFDMA networs. Our problem formulation taes into aount artifiial noise generation to ombat a passive multiple antenna eavesdropper and the effets of imperfet hannel state information at the transmitter CSIT in slow fading. The optimization problem is solved by dual deomposition whih results in an iterative resoure alloation algorithm with a fast speed of onvergene. The paet data rate, serey data rate, power, and subarrier alloation poliies are optimized to maximize the average serey outage apaity bit/s/hz seurely and suessfully delivered to the users. Simulation results illustrate that our proposed iterative algorithm onverges to the optimal solution in a small number of iterations and guarantees a nonzero serey data rate for given target serey outage and hannel outage probability requirements. I. INTRODUCTION Orthogonal frequeny division multiple aess OFDMA is a promising andidate for high speed wireless ommuniation networs inluding IEEE 80.6 Worldwide Interoperability for Mirowave Aess WiMAX, Long Term Evolution Advaned LTE-A, and IEEE 80. Wireless Regional Area Networs WRAN, not only beause of its flexibility in resoure alloation but also its robustness against multipath fading. Reently, a large amount of wor has been devoted to information-theoreti physial layer PHY seurity -6, as a omplement to the traditional ryptographi enryption adopted in the appliation layer. The pioneering wor on PHY seurity by Wyner 7 showed that a soure and a destination an exhange perfetly seure messages with a non-zero rate if the desired reeiver enjoys better hannel onditions than the passive eavesdroppers. In and, resoure alloation in multi-arrier systems with PHY seurity onsiderations were studied for the ase of single-user and two-user systems, respetively. In these wors, the hannel state information CSI of the eavesdroppers is assumed to be nown at the base station BS suh that seure ommuniation an be guaranteed. Yet, eavesdroppers are usually passive and silent in order to hide their existene. Thus, the CSI of the eavesdroppers annot be measured at the BS by estimating handshaing signals or be obtained via feedba from the eavesdroppers. On the other hand, seure ommuniation systems employing multiple antennas have been proposed in the absene of the eavesdropper s CSI. By exploiting the extra degrees of freedom in a multiple antenna system, artifiial noise or interferene is generated in the nullspae of the desired users for degrading the hannels of the eavesdroppers. In 3 and 4, the authors studied the power alloation problem for maximizing the ergodi serey apaity in single-user single-arrier systems with artifiial noise generation, assuming perfet hannel state information at the transmitters CSIT of the desired users is available. However, the ergodi hannel assumption annot be justified for delay sensitive appliations in pratie sine the transmitted paets of these appliations experiene quasi-stati slow fading. Hene, a serey outage ours whenever the sheduled serey data rate exeeds the serey apaity between the BS and the eavesdroppers, whih introdues a quality of servie QoS onern for serey. In 6, the authors studied a resoure alloation algorithm whih taes into aount the artifiial noise generation and serey outage under the assumption of perfet CSIT of the desired users. Yet, in pratie, the CSI of the desired users may be outdated at the transmitter even if the users are moving with pedestrian speeds. Imperfet CSIT introdues two inds of performane degradations whih have been overlooed in the literature -6. First, in quasi-stati fading with imperfet CSIT, a transmitted paet is orrupted whenever the transmit data rate exeeds the hannel apaity between the ative legitimate transeivers, despite the use of hannel apaity ahieving odes for error protetion. Seond, for imperfet CSIT, the artifiial noise not only interferes the eavesdropper, but also the desired users sine their nullspae is not exatly nown. As a result, the optimization problem formulation hanges fundamentally and the studies in the literature are not appliable. Therefore, for pratial implementation, a resoure alloation algorithm whih taes into aount serey outage, hannel outage, and the potentially detrimental effet of artifiial noise generation is needed. II. OFDMA DOWNLINK NETWORK MODEL A. Channel Model We onsider an OFDMA downlin networ whih onsists of a BS with antennas, an eavesdropper with N E antennas, and K mobile users equipped with a single antenna,.f. Figure. The BS adopts multiple-input multiple-output beamforming MIMO-BF to enhane the system performane. We assume that > N E to enable seure ommuniation. The eavesdropper is passive and its goal is to deode the information transmitted by the BS without ausing interferene to the ommuniation hannels. The impulse responses of all hannels are assumed to be time-invariant slow fading. We onsider an OFDMA system with subarriers. The reeived symbols at user and the eavesdropper on subarrier i {,..., } are given by, respetively, y B, i = h B, ix i + n i and y B,E i = G B,E ix i + ei, where x i C denotes the transmitted symbol vetor and C N M is the spae of all N M matries with omplex entries. h B, i C denotes the hannel matrix between the BS and user on subarrier i and G B,E i C N E is the hannel matrix between the BS and the eavesdropper on subarrier i. Both variables, h B, i and G B,E i, inlude the effets of path loss and multipath fading of the assoiated hannels. n i C and ei C NE are the additive white Gaussian noise AWGN in subarrier i at user and An eavesdropper with N E antennas is equivalent to multiple eavesdroppers with a total of N E antennas whih are onneted to a ommon proessing unit.

2 Passive BS eavesdropper Fig.. Illustration of an OFDMA downlin networ. There are one BS with = 4 antennas, K = 9 desired users equipped with a single antenna, and one eavesdropper with N E = antennas. For an effetive eavesdropping, the eavesdropper hooses a loation loser to the BS ompared to all the desired users. the eavesdropper, respetively. Eah entry in both vetors has distribution CN 0, N 0, where N 0 is the noise power spetral density. Here, CN ν, σ denotes a omplex Gaussian random variable with mean ν and variane σ. For the sae of notational simpliity and without loss of generality, a normalized noise variane of N 0 = is assumed for all reeivers in the following. B. Channel State Information In the following, sine path loss is a slowly varying random proess whih hanges in the order of seonds, we assume that the path loss an be estimated perfetly. Although we also assume that for signal detetion purposes the users an obtain perfet estimates of the BS-to-user fading gains h B, i, i {,..., }, {,..., K}, the orresponding CSI may be outdated at the BS due to the users s CSI feedba delay and the mobility of the users. To apture this effet, we model the multipath fading CSIT of the lin between the BS and user on subarrier i as h B, i = ĥ B, i + h B, i, 3 where ĥ B, i and h B, i denote, respetively, the estimated CSI vetor and the CSIT error vetor. ĥ B, i and h B, i are Gaussian random vetors and eah vetor has independent and identially distributed i.i.d. elements. Besides, the elements of vetors h B, i, ĥ B, i, and h B, i have zero means and variane σ h B,, σ h B, σ e, and σ e, respetively. Assuming a minimum mean square error MMSE estimator. On the other hand, sine the eavesdropper is assumed to be passive and the orresponding CSI is unavailable at the BS, in order to seure the desired wireless ommuniation lins, artifiial noise signals are generated at the BS to degrade the hannels between the BS and the eavesdropper. C. Artifiial Noise Generation The BS hooses x i as the linear ombination of the information bearing signal and an artifiial noise signal, i.e., x i = ˆb iu i P B, iα B, i Desired Signal + V B, ivi, 4 Artifiial Noise where u i C is the information bearing signal, vi C is the artifiial noise vetor whose elements are i.i.d. omplex Gaussian random variables with variane σ vi. Sine ĥ B, i is nown at the BS, MIMO-BF an be used to maximize the reeived signal-to-noise SNR ratio at the desired reeivers. The beamforming vetor adopted at the BS, i.e, ˆb i C, is hosen to be the eigenvetor orresponding to the maximum eigenvalue of ĥ B, iĥb,i. Here, denotes the onjugate transpose operation. Furthermore, we define an orthogonal basis, V B, i C, by using the remaining eigenvetors of ĥ B, iĥ B, i. P B, i is the total transmitted power on subarrier i for user and 0 < α B, i represents the fration of power devoted to the information bearing signal on subarrier i for user. The remaining power on subarrier i is equally distributed into dimensions for generating the artifiial noise at the BS, i.e., σvi = α B,iP B, i. Hene, the reeived signal at user and the eavesdropper in and an be rewritten as y B, i = h B, i P B, iα B, iˆb iu i + V B, ivi +n i and y B,E i = G B,E i ˆb iu i P B, iα B, i respetively. +V B, ivi + ei, 5 III. RESOURCE ALLOCATION AND SCHEDULING A. Instantaneous Channel Capaity and Outages Given perfet CSI at the reeiver CSIR, i.e., h B, iˆb i, the instantaneous hannel apaity based on 5 between the BS and user on subarrier i is given by C B, i = log + α B,iP B, iˆb ih B, ih B,iˆb i + α B, P B, iσe. 6 On the other hand, sine the BS does not have any CSI of the eavesdropper, we follow the approah in 3, 4 and onsider a apaity upper bound for the eavesdropper for resoure alloation purposes assuming the absene of thermal noise at the eavesdropper. Therefore, the apaity of the eavesdropper is upper bounded by C B,E i log + Γ B,E i, Γ B,E i = α B,i g α B, i ig ig i g i, 7 where g i = G B,E iˆb i and G i = G B,E iv B, i. Hene, the maximum ahievable serey data rate R se i for perfetly seure ommuniation between the BS and user on subarrier i with outage onsideration an be expressed as R se i i < C B, i i C B,E i > R se i, 8 where denotes an indiator funtion whih is when the event is true and 0 otherwise. R data i is the atual paet data rate transmitted from the BS to user. There are two types of outage measures in the onsidered system. The first one is nown as hannel outage whih orresponds to the first inditor funtion in 8. A hannel outage ours whenever the transmit data rate exeeds the instantaneous hannel apaity between two desired transeivers, i.e., R data i > C B, i. If i > C B, i, any transmitted paet between two legitimate ative transeivers is orrupted even if a hannel apaity ahieving ode is applied for error protetion. Indeed, hannel outage an be avoided by paet data rate adaptation when the CSIT of the desired hannel is perfet. Yet, CSIT with high auray is diffiult to obtain if the users are not stati. The seond type of outage measure is serey outage whih orresponds to the

3 seond indiator funtion in 8. If the CSI of the eavesdropper is available at the BS, the resoure alloator an set the target serey data rate R se i to math the hannel onditions 3, i.e., R se i < Rdata i C B,E i and R data i > C B,E i, suh that a paet with serey rate R se i and paet data rate R data i an be seurely delivered and suessfully deoded by the desired user. However, here the eavesdropper is assumed to be passive and its CSI is not available at the BS, i.e., C B,E i is a random variable for the BS. Hene, a serey outage ours whenever the target serey data rate R se i exeeds the serey apaity, i.e., R data i C B,E i. In order to model the unreliability and the inseurity due to both hannel outage and serey outage, respetively, we onsider the performane in terms of the average serey outage apaity, whih is defined as the total average bits/s/hz seurely and suessfully delivered to the K mobile users averaged over multiple sheduling slots and is given by U se P, R, S = K w = Pr i= s i R se i Pr i < C B, i ĥb,i i C B,E i > R se i ĥb,i, 9 where P and S are the power and subarrier alloation poliies, respetively. Poliy R inludes the alloation of serey data rate R se i and paet data rate Rdata i. s i {0, } is the ombinatorial subarrier alloation indiator. w is a positive onstant provided by the upper layers, whih allows the resoure alloator to give different priorities to different users and to enfore ertain notions of fairness suh as proportional fairness and max-min fairness. B. Optimization Problem Formulation The optimal power alloation poliy, P, data rate serey rate and data rate alloation poliy, R, and subarrier alloation poliy, S, an be obtained from arg max U sep, R, S P,R,S,α B, i s.t. C: Pr R data i C B, i ĥb,i ε,, i, C: Pr i i C B,E i ĥb,i δ,, i, C3: C4: K R se = i= P B, is i, K s i, i; C5: s i = {0, }, i,, = C6:P B, i 0, i, ; C7: 0 < α B, i, i,. 0 Here, C represents the required paet outage probability due to the imperfet CSI of the BS to user hannel on subarrier i. In C, δ denotes the required serey outage probability in the system, i.e., C is a QoS metri for ommuniation seurity. C3 is the maximum instantaneous transmit power onstraint for the BS. Constraints C4 and C5 are the subarrier alloation onstraints whih guarantee that eah subarrier will serve only one user. In other words, intra-user interferene is ompletely avoided. C6 and C7 are the boundary onstraints for the power alloation variables. IV. SOLUTION OF THE OPTIMIZATION PROBLEM A. Transformation of the Optimization Problem For derivation of an effiient resoure alloation algorithm, we replae the -signs in C and C by = -signs and the resulting optimization problem may be viewed as a restrited version of the original problem 0 sine replaing inequality signs by equality signs redues the feasible set. We are now ready to introdue the following Lemma. Lemma Equivalent Objetive Funtion: For a given hannel outage probability ε and a given serey outage probability δ in C and C, respetively, the equivalent serey data rate in high SNR on subarrier i for user is given by R se i > with i log + α B, iλ Ei αb, i + i = log + Γ B, i, Γ B, i = α B, ip B,iFχ ε, i + αb, P B,iσe Λ E i = F δ, i, α B,i =, ΛE i, where x + = max{0, x}, F, i denotes the inverse funtion NE n=0 n z n +z of F z z, i =, and Fχ, i represents the inverse umulative distribution funtion df of a non-entral hi-square random variable with degrees of freedom and nonentrality parameter ĥ B, iĥ B, i. Proof: Please refer to the Appendix. There are two important observations from the above lemma. First, the asymptotially optimal αb, i in Lemma reveals that in high SNR, the optimal fration of power devoted to the artifiial noise only depends on the statisti of the eavesdropper hannel. Seond, the signal-to-interferene-plus-noise α ratio SINR of the eavesdropper, B, i α iλ Ei, approahes a B, onstant value at high SNR. In other words, the SINR of the eavesdropper on eah subarrier is independent of the transmit power variables at the BS. This important observation will be verified in Setion V via simulation. By substituting into 9, a modified objetive funtion is obtained and the onsidered problem beomes an NP-hard mixed ombinatorial and onvex optimization problem, where the ombinatorial nature omes from the binary onstraints in the subarrier assignment. Therefore, we follow the timesharing approah in 8 and relax onstraint C5 in 0 suh that s i is allowed to be any real value between zero and one. To failitate the time sharing on eah subarrier, we introdue a new variable and define it as P B, i = P B, is i. This variable is the atual transmit power of the BS on subarrier i for user under the time-sharing assumption. Then, problem 0 an be re-written as Problem Transformed Optimization Problem: se R max K P,R,S = i= s.t. C4, C6 K C3: P B, i, = i= C5: 0 s i, i, se w s i R i 3 where i = Rse i is the ahievable PB, i= P B, i/s i serey data rate for user on subarrier i. Mathematially, the + operators in the objetive funtion in 3 destroy the onavity of the problem. Nevertheless, as will be seen in

4 the Karush-Kuhn-Tuer KKT onditions in 8, users with negative serey data rate will not be onsidered in the subarrier seletion proess, sine, seure ommuniation annot be guaranteed for those users. Therefore, we an remove the + se operators from R i in, while preserving the onavity of the transformed problem. On the other hand, the onstant term is removed from the transformed objetive funtion for simpliity as it does not affet the values of the arguments whih maximize the objetive funtion. Besides, C7 is also removed from the optimization problem as the asymptotially optimal αb, i in Lemma always satisfied the onstraint. Now, the transformed problem is jointly onave with respet to all optimization variables and under some mild onditions 9, it an be shown that solving the dual problem is equivalent to solving the primal problem. B. Dual Problem Formulation In this subsetion, we solve the onsidered problem by solving its dual. For this purpose, we first need the Lagrangian funtion of the primal problem. Upon rearranging terms, the Lagrangian an be written as Lλ, β, P, R, S = λ K = i= P i K w s i = i= K = i= se R i βis i + λ + i= βi,4 where λ 0 is the Lagrange multiplier orresponding to the power onstraint at the BS. β is the Lagrange multiplier vetor onneted to the subarrier usage onstraints with elements βi 0, i {,..., }. The boundary onstraints C5 and C6 will be absorbed into the KKT onditions when deriving the optimal solution in Setion IV-C. Thus, the dual problem is given by min max λ,β 0 P,R,S Lλ, β, P, R, S. 5 In the following setions, we solve the above dual problem iteratively by deomposing it into two parts nested loops: the first part inner loop onsists of sub-problems with idential struture; the seond part outer loop is the master dual problem to be solved with the gradient method. C. Dual Deomposition and Solution By dual deomposition, the BS first solves the subproblem max P,R,S Lλ, β, P, R, S 6 for a fixed set of Lagrange multipliers. Note that the above subproblem is the inner loop optimization in 5. Using standard optimization tehniques, the optimal power alloation for user on subarrier i is given by P B,i = ΩB, iω B, iλ log + 4Ξ B, iw Ξ B, i λ log + α B,iσe + Ω B, i, 7 Ξ B, i where Ξ B, i = α B, iσeω B, i + σe α B, i and Ω B, i = Fχ ε, iα B, i. The optimal power alloation in 7 has the form of multi-level water-filling. The water level of eah user depends not only on his/her priority via w, but also on the hannel statisti of the desired hannel, i.e., F χ ε, iα B, i. In order to obtain the optimal subarrier alloation, we tae the derivative of the subproblem with respet to s i, whih yields Lλ, β, P, R, S s i = A i βi, 8 where A i 0 is the marginal benefit 0 for alloating subarrier i to user and is given by A i = w log + Γ B, i log + α B, iλ Ei αb, i Γ B, i ln + Γ B, i + σ epb, i α B, i, 9 where Γ B, i = Γ B,i. A i 0 suggests PB, i=pb, i that if user has a negative serey data rate on subarrier i, he/she will not be seleted as he/she an only provide a negative marginal benefit to the system. On the ontrary, if a user has good hannel onditions with positive serey data rate on subarrier i, he/she an provide a higher marginal benefit to the system. Thus, the optimal subarrier seletion determined by the BS on subarrier i is given by { if A s i = max A b i 0 & A b i βi i = b. 0 0 otherwise The dual variable βi ats as the ost in using subarrier i in the system. Only the user who an provide a large marginal benefit to the system has a hane to be seleted by the resoure alloator. Note that eah subarrier will be used for serving only one user eventually. Finally, the optimal transmitted paet data rate R data i and serey data rate R se i are obtained by substituting 7 into the equivalent paet data rate and serey data rate in Lemma for the subarrier with s i =. D. Solution of the Master Problem Sine the dual funtion is differentiable, the gradient method an be used to solve the master problem outer loop in 5 whih leads to a Lagrange multiplier update equation: λt + = λt ξt K = i= P B, is i +, where t 0 is the iteration index and ξt is the positive step size. Then, the updated Lagrange multiplier in is used for solving the subproblems in Setion IV-C. Sine the transformed problem is jointly onave with respet to all optimization variables, it is guaranteed that the iterative algorithm onverges to the optimal solution if the hosen step sizes satisfy the general onditions stated in, Chapter.. In summary, the master problem adjusts the water-levels of 7 through the gradient update equation until the power onstraint of the BS is satisfied. On the other hand, updating βi is not neessary as it has the same value for eah user. Therefore, setting βi = 0 in eah iteration does not affet the subarrier alloation in 0. V. RESULTS In this setion, we evaluate the system performane using simulations. A single ell with a radius of m is onsidered. The number of subarriers is = 64 with arrier enter frequeny.5 GHz, bandwidth B = 5 MHz, and w =,. Eah subarrier has a bandwidth of 78 Hz and a noise variane N 0 = 5 dbm. The eavesdropper is loated 35 m referene

5 8 6 = 5, σ /σhb, e = 0.0, 0 iterations 7 = 5, σ /σhb, e = 5, σ /σhb, e = 0.0, 5 iterations = 0.05, 0 iterations Average serey outage apaity 5 Average serey outage apaity bit/s/hz = 5, σ /σhb, e = 5, σ /σhb, e = 5, σ /σhb, e = 0.05, 5 iterations = 0.0, eavesdropper s average throughput = 0.05, eavesdropper s average throughput Average serey outage apaity bit/s/hz 4 3 = 44 dbm, δ = 0.5 = 44 dbm, δ = 0. Average eavesdropper throughput = 44 dbm, δ = 0.05 = 44 dbm, δ = dbm Fig.. Average serey outage apaity versus transmit power for different values of hannel estimation error-to-signal ratio ESR. The eavesdropper is equipped with N E = antennas and is loated 35 m away from the BS Number of antennas at the eavesdropper N E Fig. 3. Average serey outage apaity versus the number of eavesdropper antennas N E for different serey outage requirements and = 7 antennas at the BS. distane away from the BS for an effetive eavesdropping. Desired users are uniformly distributed between the referene distane and the ell boundary. The 3GPP path loss model is adopted. The small sale fading oeffiients of the BS-to-user and BS-to-eavesdropper lins are modeled as i.i.d. Rayleigh fading. The target serey outage probability and hannel outage probability are set to δ, ε = The hannel estimation errorto-signal ratio ESR is set to e σ = 0.05, unless further σ h B, speified. The average serey outage apaity is obtained by ounting the number of paets seurely delivered and deoded by the users averaged over both the marosopi and mirosopi fading. A. Average Serey Outage Capaity versus Transmit Power Figure illustrates the average serey outage apaity and the throughput of the eavesdropper versus the total transmit power for K = 5 users for different values of ESRs. The eavesdropper is equipped with N E = antennas. The number of iterations for the proposed iterative resoure alloation algorithm is 5 and 0. It an be seen that the performane for 5 and 0 iterations is virtually the same. In other words, the algorithm onverges to the optimal solution in a few iterations. On the other hand, we observe that the system performane dereases as the ESR inreases. This is beause the nullspae information of the desired users at the transmitter beomes less aurate as the ESR inreases whih inreases the artifiial noise leaage. For a better illustration of the effetiveness of the artifiial noise generation, Figure also inludes the performane of the eavesdropper in terms of average throughput. The average throughput between the BS and the eavesdropper does not sale with the transmit power in the high transmit power regime due to the artifiial noise introdued by the BS whih is in good agreement with Lemma. Besides, it an be observed that the artifiial noise generation maintains the same effetiveness for different hannel estimation error varianes sine an inreasing ESR does not enhane the average throughput of the eavesdropper. B. Average Serey Outage Capaity versus N E Figure 3 depits the average serey outage apaity versus the number of reeive antennas N E employed at the eavesdropper for different serey outage requirements δ with K = 5 users. There are = 7 transmit antennas at the BS. The number of iterations for the proposed algorithm is 5. It an be observed that the system performane dereases as N E inreases, sine more of the transmitted power has to be devoted Average serey outage apaity bit/s/hz = 7, N E =, = 4 dbm = 5, N E =, = 4 dbm = 3, N E =, = 4 dbm Performane gain Number of users Fig. 4. Average serey outage apaity versus the number of desired users for different transmit antennas at the BS with a total transmit power = 4 dbm. The eavesdropper is equipped with N E = antennas and is loated 35 m away from the BS. The double arrows demonstrate the performane gain ahieved by an inreasing number of transmit antennas. to the artifiial noise generation for ombatting the eavesdropper, whih results in less transmit power for information transmission. Yet, a non-zero serey outage apaity an still be guaranteed as long as > N E due to the artifiial noise. Moreover, we observe that a more stringent serey outage probability requirement does not neessarily lead to a higher average serey outage apaity. This is beause a larger fration of power has to be alloated to the artifiial noise for degrading the hannel of the eavesdropper and less power is available for information transmission. C. Average Serey Outage Capaity versus Number of Users Figure 4 depits the average serey outage apaity versus the number of users for different numbers of transmit antennas. The number of iterations is 5. The eavesdropper is equipped with N E = antennas. It an be observed that the average serey outage apaity grows with the number of users as the proposed resoure alloation and sheduling algorithm is able to exploit multi-user diversity, despite the existene of the eavesdropper. Besides, it an be observed that an inreasing number of transmit antennas provides a substantial performane gain to the legitimate users in terms of average serey outage apaity. However, there is a diminishing return when is large due to the hannel hardening effet in the desired hannels.

6 VI. CONCLUSION In this paper, we formulated the resoure alloation and sheduling design for OFDMA systems as a non-onvex and ombinatorial optimization problem, where a multiple antenna eavesdropper, artifiial noise generation for seure ommuniation, and the negative effet of imperfet CSIT were taen into onsideration. An effiient iterative resoure alloation algorithm with losed-form power, serey data rate, paet data rate, and subarrier alloation was derived by dual deomposition. Simulation results not only showed that the performane of the proposed algorithm onverges to the optimal performane within a small number of iterations, but also demonstrated the ahievable non-zero serey outage apaity for a required hannel outage probability and serey outage probability. APPENDIX-PROOF OF LEMMA The proof of Lemma involves three steps. We first derive the hannel outage data rate between the BS and user on subarrier i. By onsidering the hannel outage probability requirement C in 0, we obtain i + α B, ip B, iσ Pr e α B, ip B, i ˆb ih B, ih B,iˆb i ĥb,i i + α B, ip B, iσ = F e χ α B, ip B, i i = log + α B,iP B, ifχ ε, i + α B, P B, iσe, i = ε, where F χ, i denotes the df of a non-entral hi-square random variable with degrees of freedom and non-entrality parameter ĥ B, iĥ B, i. Fχ, i represents the inverse funtion of F χ, i. Then, we derive the serey outage data rate by alulating the serey outage probability in C. Without loss of generality, we define the serey data rate as R sei = log r se i and the data rate as Rdata i = log r data i. Then, the serey outage probability an be expressed as Pr R data i C B,E i R se i ĥb,i = δ r data i α B, i = Pr r sei α B, i Θ i g ig ig i g i ĥb,i = δ, 3 Z i where Z i is an unnown random variable for the BS, g i = G B,E iˆb i, and G i = G B,E iv B, i. Sine the supermatrix B i = ˆb i V B, i is a unitary matrix, B ig B,E i has i.i.d. omplex Gaussian entries. As a result, Z i is equivalent to the signal-to-interferene ratio SIR of a N E -branh minimum mean square error MMSE diversity ombiner for interferers. Hene, the orresponding omplementary umulative distribution funtion df is given by 4, 3 F z z, i = N E n=0 NT n z n / + z. 4 Therefore, for a target serey outage probability of δ, Θ i defined in 3 an be expressed as Θ i = Fz δ, i. Thus, solving Θ i = F δ, i for R se i yields Rse i =, 5 i log + α B,i α B, i F δ, i where Fz δ, i is the inverse df of random variable Z i. Note that both inverse funtions Fz δ, i and Fχ ε, i, an be omputed effiiently by numerial solvers or implemented as a loo-up table for pratial implementation. The final step in deriving the lemma is to alulate the optimal αb, i in high SNR P B,i. Under suh onditions and using and 5, the serey data rate is lower bounded by R se i > log + P B,iFχ ε, iαb, i σep B, i log + α B, if δ, i αb, i. 6 In fat, the term P B,iF χ ε,iα B, i σe P an be interpreted as a B,i signal-to-interferene ratio SINR under a virtual interferer with interferene power σep B, i for P B, i. By standard optimization tehniques, the optimal αb, i whih maximizes the lower bound of the serey data rate on subarrier i for user in 6 is given by αb,i = Φ i + Φ iλ E iφ i Λ E i + Φ iλ E i a ΛE i Λ E i 7 ΛE i where Φ i = Fχ ε, i/σe and Λ E i = Fz δ, i. Note that a is due to Φ i Λ E i whih is always valid for reasonably small hannel estimation error variane σe e.g. σe σh B, and serey outage requirement δ e.g. δ. REFERENCES E. A. Jorswie and A. Wolf, Resoure Alloation for the Wire- Tap Multi-Carrier Broadast Channel, in Pro. International Conf. on Teleommun., June 008, pp. 6. Z. Li, R. Yates, and W. 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