Linear Precoding of Data and Artificial Noise in Secure Massive MIMO Systems
aa r X i v : . [ c s . I T ] A ug Linear Precoding of Data and Artificial Noisein Secure Massive MIMO Systems
Jun Zhu,
Student Member, IEEE , Robert Schober,
Fellow, IEEE , andVijay K. Bhargava,
Life Fellow, IEEE
The University of British Columbia
Abstract
In this paper, we consider secure downlink transmission in a multi-cell massive multiple-inputmultiple-output (MIMO) system where the numbers of base station (BS) antennas, mobile terminals,and eavesdropper antennas are asymptotically large. The channel state information of the eavesdrop-per is assumed to be unavailable at the BS and hence, linear precoding of data and artificial noise(AN) are employed for secrecy enhancement. Four different data precoders (i.e., selfish zero-forcing(ZF)/regularized channel inversion (RCI) and collaborative ZF/RCI precoders) and three different ANprecoders (i.e., random, selfish/collaborative null-space based precoders) are investigated and the cor-responding achievable ergodic secrecy rates are analyzed. Our analysis includes the effects of uplinkchannel estimation, pilot contamination, multi-cell interference, and path-loss. Furthermore, to strike abalance between complexity and performance, linear precoders that are based on matrix polynomials areproposed for both data and AN precoding. The polynomial coefficients of the data and AN precodersare optimized respectively for minimization of the sum mean squared error of and the AN leakage to themobile terminals in the cell of interest using tools from free probability and random matrix theory. Ouranalytical and simulation results provide interesting insights for the design of secure multi-cell massiveMIMO systems and reveal that the proposed polynomial data and AN precoders closely approach theperformance of selfish RCI data and null-space based AN precoders, respectively.
I. I
NTRODUCTION
Massive multiple-input multiple-output (MIMO) systems employing simple linear precodingand combining schemes offer significant performance gains in terms of bandwidth, power, and
This work was presented in part at the European Wireless (EW) Conference, Barcelona, Spain, 2014, and the InternationalSymposium on Communications, Control, and Signal Processing (ISCCSP), Athens, Greece, 2014.
September 2, 2015 DRAFT energy efficiency compared to conventional multiuser MIMO systems as impairments such asfading, noise, and interference are averaged out for very large numbers of base station (BS)antennas [1]–[3]. Furthermore, in time-division duplex (TDD) systems, channel reciprocity canbe exploited to estimate the downlink channels via uplink training so that the training overheadscales only linearly with the number of users and is independent of the number of BS antennas[2]. However, if the pilot sequences employed in different cells are not orthogonal, so-called pilotcontamination occurs and impairs the channel estimates, which ultimately limits the achievableperformance of massive MIMO systems [2], [4].Since secrecy and privacy are critical concerns for the design of future communication systems[5], it is of interest to investigate how the large number of spatial degrees of freedom inmassive MIMO systems can be exploited for secrecy enhancement [6], [7]. If the eavesdropper(Eve) remains passive to hide its existence, neither the transmitter (Alice) nor the legitimatereceiver (Bob) will be able to learn Eve’s channel state information (CSI). In this situation, itis advantageous to inject artificial noise (AN) at the transmitter to degrade Eve’s channel andto use linear precoding to avoid impairment to Bob’s channel as was shown in [8]- [10] and[11], [12] for single user and single-cell multiuser systems, respectively. However, in multi-cellmassive MIMO systems, multi-cell interference and pilot contamination will hamper Alice’sability to degrade Eve’s channel and to protect Bob’s channel. This problem was studied firstin [13] for simple matched-filter (MF) data precoding and null-space (NS) and random ANprecoding. However, it is well known that MF data precoding suffers from a large loss in theachievable information rate compared to other linear data precoders such as zero-forcing (ZF) andregularized channel inversion (RCI) precoders as the number of mobile terminals (MTs) increases[14]. Since it is expected that this loss in information rate also translates into a loss in secrecyrate, studying the secrecy performance of ZF and RCI data precoders in massive MIMO systemsis of interest. Furthermore, while NS AN precoding was shown to achieve a better performancecompared to random AN precoding [13], it also entails a much higher complexity. Similarly, theimproved performance of ZF and RCI data precoding compared to MF data precoding comes atthe expense of a higher complexity. Hence, the design of novel data and AN precoders whichallow a flexible tradeoff between complexity and secrecy performance is desirable.Related work on physical layer security in massive MIMO systems includes [15] where theauthors use the channel between Alice and Bob as secrete key and show that the complexity
September 2, 2015 DRAFT required by Eve to decode Alice’s message is at least of the same order as a worst-case latticeproblem. Physical layer security in a downlink multi-cell MIMO system was considered in [16]-[18]. However, unlike our work, perfect knowledge of Eve’s channel was assumed, AN injectionwas not considered, and pilot contamination was not taken into account. Furthermore, ZF andRCI data precoding were analyzed in the large system limit in [19], [20]. However, neither pilotcontamination nor AN were taken into account and the secrecy rate was not analyzed. Using aconcept that was originally conceived for code division multiple access (CDMA) uplink systemsin [21] and later extended to MIMO systems in [22], reduced complexity linear data precodersthat are based on matrix polynomials were investigated for use in massive MIMO systems in[23]- [25]. However, [23]- [25] did not take into account the effect of AN leakage for precoderdesign and did not study the secrecy performance. Hence, the results presented in [15]- [25] arenot directly applicable to the system studied in this paper.In this paper, we consider secure downlink transmission in a multi-cell massive MIMO systememploying linear data and AN precoding in the presence of a passive multi-antenna eavesdropper.We study the achievable ergodic secrecy rate of such systems for different linear precodingschemes taking into account the effects of uplink channel estimation, pilot contamination, multi-cell interference, and path-loss. The main contributions of this paper are summarized as follows: • We study the performance-complexity tradeoff of selfish and collaborative data and ANprecoders. Selfish precoders require only the CSI of the MTs in the local cell but causeinter-cell interference and inter-cell AN leakage. In contrast, collaborative precoders requirethe CSI between the local BS and the MTs in all cells, but reduce inter-cell interferenceand inter-cell AN leakage. However, since the additional CSI required for the collaborativeprecoders can be estimated directly by the local BS, the additional overhead and complexityincurred compared to selfish precoders is limited. • We derive novel closed-form expressions for the asymptotic ergodic secrecy rate whichfacilitate the performance comparison of different combinations of linear data precoders(i.e., MF, selfish and collaborative ZF/RCI) and AN precoders (i.e., random, selfish andcollaborative NS), and provide significant insight for system design and optimization. • In order to avoid the computational complexity and potential stability issues in fixed pointimplementations entailed by the large-scale matrix inversions required for ZF and RCI dataprecoding and NS AN precoding, we propose polynomial (POLY) data and AN precoders
September 2, 2015 DRAFT and optimize their coefficients. Unlike [24] and [25], which considered polynomial dataprecoders for massive MIMO systems without AN generation, we use free probability theory[23], [26] to obtain the POLY coefficients. This allows us to express the POLY coefficientsas simple functions of the channel and system parameters. Simulation results reveal thatthese precoders are able to closely approach the performance of selfish RCI data and NSAN precoders, respectively.The remainder of this paper is organized as follows. In Section II, we outline the consideredsystem model and review some basic results from [13]. In Sections III and IV, the consideredlinear data and AN precoders are investigated, respectively. In Section V, the ergodic secrecy ratesof different linear precoders are compared analytically for a simple path-loss model. Simulationand numerical results are presented in Section VI, and some conclusions are drawn in SectionVII.
Notation:
Superscripts T and H stand for the transpose and conjugate transpose, respectively. I N is the N -dimensional identity matrix. The expectation operation and the variance of a randomvariable are denoted by E [ · ] and var[ · ] , respectively. diag { x } denotes a diagonal matrix with theelements of vector x on the main diagonal. tr {·} and rank {·} denote trace and rank of a matrix,respectively. C m × n represents the space of all m × n matrices with complex-valued elements. x ∼ CN ( N , Σ ) denotes a circularly symmetric complex Gaussian vector x ∈ C N × with zeromean and covariance matrix Σ . [ A ] kl denotes the element in the k th row and l th column ofmatrix A , and [ x ] + = max { x, } .II. S YSTEM M ODEL AND P RELIMINARIES
In this section, we introduce the considered system model as well as the adopted channelestimation scheme, and review some ergodic secrecy rate results.
A. System Model
We consider the downlink of a multi-cell massive MIMO system with M cells and a frequencyreuse factor of one, i.e., all BSs use the same spectrum. Each cell includes one N T -antenna BS, K ≤ N T single-antenna MTs, and potentially an N E -antenna eavesdropper. The eavesdropperstry to hide their existence and hence remain passive. As a result, the BSs cannot estimate the September 2, 2015 DRAFT eavesdroppers’ CSI. To overcome this limitation, each BS generates AN to mask its information-carrying signal and to prevent eavesdropping [8]. In the following, the k th MT, k = 1 , . . . , K ,in the n th cell, n = 1 , . . . , M , is the MT of interest and we assume that an eavesdropper tries todecode the signal intended for this MT. We note that neither the BSs nor the MTs are assumedto know which MT is targeted by the eavesdropper. The signal vector, x n ∈ C N T × , transmittedby the BS in the n th cell (also referred to as the n th BS in the following) is given by x n = √ p F n s n + √ q A n z n , (1)where s n ∼ CN ( K , I K ) and z n ∼ CN ( N T , I N T ) denote the data and AN vectors for the K MTsin the n th cell, respectively. F n = [ f n , · · · , f nK ] ∈ C N T × K and A n = [ a n , · · · , a nN T ] ∈ C N T × N T are the data and AN precoding matrices, respectively, and the efficient design of these matricesis the main scope of this paper. Thereby, the structure of both types of precoding matrices doesnot depend on which MT is targeted by the eavesdropper. The AN precoding matrix A n hasrank L = rank { A n } ≤ N T , i.e., L dimensions of the N T -dimensional signal space spanned bythe N T BS antennas are exploited for jamming of the eavesdropper. The data and AN precodingmatrices are normalized as tr { F Hn F n } = K and tr { A Hn A n } = L , i.e., their average power perdimension is one. The average powers p and q allocated to the information-carrying signal foreach MT and each AN signal, respectively, can be written as p = φP T K and q = (1 − φ ) P T L , where P T is the total transmit power and φ ∈ (0 , is a power allocation factor which can be optimized.For the sake of clarity, in this paper, we assume that all cells utilize the same value of φ .The vectors collecting the received signals at the K MTs and the N E antennas of theeavesdropper in the n th cell are given by y n = M X m =1 G mn x m + n n and y E = M X m =1 G mE x m + n E , (2)respectively, with Gaussian noise vectors n n ∈ CN ( K , σ n I K ) and n E ∈ CN ( N E , σ E I N E ) ,where σ n and σ E denote the noise variances at one MT and one eavesdropper receive antenna,respectively. Furthermore, G mn = D / mn H mn ∈ C K × N T and G mE = √ β mE H mE ∈ C N E × N T arethe matrices modeling the channels from the m th BS to the K MTs and the eavesdropper inthe n th cell, respectively. Thereby, D mn = diag { β mn , . . . , β Kmn } and β mE represent the path-losses from the m th BS to the K MTs and the eavesdropper in the n th cell, respectively. Matrix H mn ∈ C K × N T , with row vector h kmn ∈ C × N T in the k th row, and matrix H mE ∈ C N E × N T September 2, 2015 DRAFT represent the corresponding small-scale fading components. Their elements are modeled asmutually independent and identically distributed (i.i.d.) complex Gaussian random variables(RVs) with zero mean and unit variance.For the design of the data and noise precoders, we consider two different approaches:
Selfish designs and collaborative designs. For the selfish designs, each BS designs its precoders onlybased on the estimate of the CSI in its own cell, G nn , and without regard for the interferenceand the AN it causes to other cells. In contrast, for the collaborative designs, each BS designsits precoders based on the estimates of the CSI to the MTs in all cells, G mn , m = 1 , . . . , M , inan effort to avoid excessive interference and AN to other cells. Although collaborative designsintroduce more channel estimation overhead at the BS, they may not always outperform selfishdesigns because of the imperfection of the CSI and the limited number of spatial degrees offreedom available for precoder design. B. Channel Estimation and Pilot Contamination
As is customary for massive MIMO systems, we assume that the downlink and uplink channelsare reciprocal and the CSI is estimated in an uplink training phase [1]- [4]. To this end, all MTsemit pilot sequences of length τ ≥ K and with pilot symbol power p τ . We assume that the pilotsequences of the K MTs in a given cell are mutually orthogonal but the same pilot sequencesare used in all cells. This gives rise to so-called pilot contamination [1]- [4]. Furthermore, weassume that the path-loss information changes on a much slower time scale than the small-scalefading. Hence, the path-loss matrices D nm , m = 1 , . . . , M , can be estimated perfectly and areassumed to be known at the BS for minimum mean-square error (MMSE) estimation of thesmall-scale fading gains [4]. At the n th BS, the small-scale fading vector to the k th MT in the m th cell, h knm , can be expressed as h knm = ˆ h knm + ˜ h knm , (3)where the estimate ˆ h knm and the estimation error ˜ h knm are mutually independent and can be statisti-cally characterized as ˆ h knm ∼ CN ( N T , p τ τβ knm p τ τ P Ml =1 β knl I N T ) and ˜ h knm ∼ CN ( N T , p τ τ P Ml = m β knl p τ τ P Ml =1 β knl I N T ) ,respectively, cf. [13]. For future reference, we collect the estimates and the estimation errors at the n th BS corresponding to all K MTs in the m th cell in matrices ˆ H nm = [(ˆ h nm ) T , . . . , (ˆ h Knm ) T ] T ∈ C K × N T and ˜ H nm = [(˜ h nm ) T , . . . , (˜ h Knm ) T ] T ∈ C K × N T , respectively. September 2, 2015 DRAFT
C. Ergodic Secrecy Rate
The performance metric adopted in this paper is the ergodic secrecy rate [7]. In this section, wereview some results for the ergodic secrecy rate in multi-cell massive MIMO systems employinglinear data and AN precoding from [13], as these results will be needed throughout this paper.Combining (1) and (2) we observe that the downlink channel comprising the BS, the k th MT,and the eavesdropper in the n th cell is an instance of a multiple-input, single-output, multi-eavesdropper (MISOME) wiretap channel [6]. Hence, the achievable secrecy rate of the k th MTin the n th cell is bounded by the difference of the capacities of the channel between the BS andthe MT and the channel between the BS and the eavesdropper, see [13, Lemma 1], [17, Lemma2]. Thus, a lower bound on the ergodic secrecy rate of the k th MT in the n th cell is given by[13] R sec nk = [ R nk − C eve nk ] + , k = 1 , . . . , K, (4)where R nk denotes an achievable rate of the k th MT in the n th cell and C eve nk denotes the ergodiccapacity of the channel between the BS and the eavesdropper. In order to obtain a tractablelower bound on the ergodic secrecy rate, we lower bound the achievable rate of the MT as R nk = log (1 + γ nk ) with signal-to-interference-and-noise ratio (SINR) [13, Eq. (10)] γ nk = | E [ p β knn p h knn f nk ] | var[ p β knn p h knn f nk ] + M P m =1 N t P i =1 E [ | p β kmn q h kmn a mi | ] + P { m,l }6 = { n,k } E [ | p β kmn p h kmn f ml | ] + 1 . (5)Furthermore, we make the pessimistic assumption that the eavesdropper is able to cancel thereceived signals of all in-cell and out-of-cell MTs except the signal intended for the MT ofinterest. This leads to an upper bound for the eavesdropper’s capacity, and consequently, to alower bound for the ergodic secrecy rate. Hence, the ergodic capacity of the eavesdropper isgiven by [13, Eq. (7)] C eve nk = E (cid:20) log (cid:0) p f Hnk G HnE X − G nE f nk (cid:1) (cid:21) , (6)where X = q P Mm =1 G mE A m A Hm G HmE ∈ C N T × N T denotes the noise correlation matrix at theeavesdropper under the worst-case assumption that the receiver noise at the eavesdropper is This lower bound is achievable if the eavesdropper has access to the data of all interfering in-cell and out-of-cell MTs, whichmight be the case e.g. if the interfering MTs cooperate with the eavesdropper.
September 2, 2015 DRAFT negligible, i.e., σ E → . Denoting the normalized number of eavesdropper antennas by α = N E /N T , a necessary condition for the invertibility of matrix X is α ≤ M L/N T . Hence, a non-zero secrecy rate can only be achieved if this condition is met. Consequently, a larger L impliesthat the BS is able to tolerate more eavesdropper antennas.If H nE f nk and matrix X are statistically independent, which in turn means for the data andAN precoders that vector f nk and the subspace spanned by the columns of A n are mutuallyorthogonal, a simple and tight upper bound on (6) can be obtained. Since any efficient data/ANprecoder pair has to keep the AN self-interference at the desired MT small, this orthogonalitycondition holds at least approximately in practice. In this case, for α < a L/ ( cN T ) and N T →∞ , where a = 1 + P Mm = n β mE /β nE and c = 1 + P Mm = n ( β mE /β nE ) , a simple and tight upperbound for C eve nk is given by [13, Theorem 1] C eve nk ≤ log (cid:18) αpaqL/N T − cαq/a (cid:19) = log (cid:18) αφβ (1 − φ )( a − cαN T / ( La )) (cid:19) . (7)For M = 1 , we have a /c = M = 1 , i.e., the bound in (7) is applicable in the entirerange of α where C eve nk in (6) is finite. For M > , we have a /c ≤ M , i.e., the bound is notapplicable for La / ( cN T ) ≤ α ≤ M L/N T . However, for strong inter-cell interference, we have β mE ≈ β nE and a /c ≈ M , i.e., the bound is applicable for all α for which C eve nk in (6) isfinite. On the other hand, for weak inter-cell interference, we have β mE ≪ β nE , and matrix X will be ill-conditioned for L/N T ≤ α ≤ M L/N T and C eve nk will become very large. Hence,the bound is again applicable for the values of α (i.e., ≤ α ≤ L/N T ), for which C eve nk in(6) assumes practically relevant values. More generally, [13, Figs. 2-4] and Section VI suggestthat, for N T → ∞ , (7) is applicable and tight for all values of α which permit a non-vanishingsecrecy rate.Combining (4), (5), and (7), we obtain a tight and tractable lower bound on the secrecy rate[13]. It is noteworthy that the upper bound on the capacity of the eavesdropper in (7) is onlyaffected by the dimensionality of the AN precoder, L , but not by the exact structures of A n and F n , as long as f nk and the subspace spanned by the columns of A n are orthogonal. On the otherhand, the achievable rate of the MT in (5) is affected by both the data and the AN precoders.In the following two sections, we analyze the impact of the most important existing data andAN precoder designs on the achievable rate R nk as N T → ∞ , respectively, and propose novellow-complexity data and AN precoders that are based on a polynomial matrix expansion. September 2, 2015 DRAFT
III. L
INEAR D ATA P RECODERS FOR S ECURE M ASSIVE
MIMOIn this section, we analyze the achievable rate of selfish and collaborative ZF/RCI dataprecoding, respectively, and develop a novel POLY data precoder. In contrast to existing analysesand designs of data precoders for massive MIMO, e.g. [19], [20], [23]- [25], the results presentedin this section account for the effect of AN leakage, which is only present if AN is injected atthe BS for secrecy enhancement. We are interested in the asymptotic regime where
K, N T → ∞ but β = K/N T and α = N E /N T are finite. A. Analysis of Existing Data Precoders
For N T → ∞ , analyzing the achievable rate is equivalent to analyzing the SINR in (5).Thereby, the effect of the AN precoder can be captured by the term Q = M X m =1 N t X i =1 E [ | p β kmn h kmn a mi | ] = M X m =1 β kmn E [ h kmn A m A Hm ( h kmn ) H ] (8)in the denominator of (5), which represents the inter-cell and intra-cell AN leakage. This termis assumed to be given in this section and will be analyzed in detail for different AN precodersin Section IV.
1) Selfish ZF/RCI Data Precoding:
The selfish RCI (SRCI) data precoder for the n th cell isgiven by F n = γ L nn ˆ H Hnn , (9)where L nn = ( ˆ H Hnn ˆ H nn + κ I N T ) − , γ is a scalar normalization constant, and κ is a regular-ization constant. In the following proposition, we provide the resulting SINR of the k th MT inthe n th cell. Proposition 1 : For SRCI data precoding, the received SINR at the k th MT in the n th cell isgiven by γ SRCI nk = 1 ˆΓ SRCI +(1+ G ( β,κ )) G ( β,κ ) (cid:18) ˆΓ SRCI + ˆΓSRCI κ β (1+ G ( β,κ )) (cid:19) + P m = n β kmn /β knn , (10)where G ( β, κ ) = 12 (cid:20)s (1 − β ) κ + 2(1 + β ) κ + 1 + 1 − βκ − (cid:21) , (11) September 2, 2015 DRAFT0 and ˆΓ SRCI = Γ SRCI θ nk Γ SRCI ϑ nk +1 with Γ SRCI = β knn K P Mm = n P l = k β kmn + ηQ + KφPT , θ mk = p τ τ ( β kmn ) p τ τ P Ml =1 β kml , ϑ mk = β kmn × p τ τ P Ml = m β kml p τ τ P Ml =1 β kml , and η = q/p . Proof:
Please refer to Appendix A.Regularization constant κ can be optimized for maximization of the lower bound on thesecrecy rate in (4), which is equivalent to maximizing the SINR in (10). Setting the derivativeof γ SRCI nk with respect to κ to zero, the optimal regularization parameter is found as κ , opt = β/ ˆΓ SRCI , and the corresponding maximum SINR is given by γ SRCI nk = 11 / G ( β, κ , opt ) + P m = n β kmn /β knn . (12)On the other hand, for κ → , the SRCI data precoder in (9) reduces to the selfish ZF (SZF)data precoder. The corresponding received SINR is provided in the following corollary. Corollary 1 : Assuming β ≤ , for SZF data precoding, the received SINR at the k th MT inthe n th cell is given by γ SZF nk = 1 β (1 − β )ˆΓ SRCI + P m = n β kmn /β knn . (13) Proof: γ SZF nk in (13) can be obtained from (10) as γ SZF nk = lim κ → γ SRCI nk .
2) Collaborative ZF/RCI Precoding:
The collaborative RCI (CRCI) precoder for the n th cellis given by F n = γ L n ˆ H Hnn , (14)where L n = ( ˆ H Hn ˆ H n + κ I N T ) − with ˆ H n = [ ˆ H Tn . . . ˆ H TnM ] T ∈ C MK × N T , γ is a normalizationconstant, and κ is a regularization constant. The corresponding SINR of the k th MT in the n th cell is provided in the following proposition. Proposition 2 : For CRCI data precoding, the received SINR at the k th MT in the n th cell isgiven by γ CRCI nk = 1 ˆΓ CRCI +(1+ G ( Mβ,κ )) G ( Mβ,κ ) (cid:18) ˆΓ CRCI + ˆΓCRCI κ β (1+ G ( Mβ,κ )) (cid:19) + P m = n β kmn /β knn , (15)where ˆΓ CRCI = Γ CRCI θ nk Γ CRCI ϑ nk +1 with Γ CRCI = β knn KηQ + KφPT . Proof:
The proof is similar to that for the SINR for the SRCI data precoder given inAppendix A and omitted here for brevity.
September 2, 2015 DRAFT1
Furthermore, the optimal regularization constant maximizing the SINR (and thus the secrecyrate) in (15) is obtained as κ , opt = M β/ ˆΓ CRCI , and the corresponding maximum SINR is givenby γ CRCI nk = 11 / G ( M β, κ , opt ) + P m = n β kmn /β knn . (16)On the other hand, for κ → , the CRCI precoder in (14) reduces to the collaborative ZF(CZF) precoder. The corresponding received SINR is provided in the following corollary. Corollary 2 : Assuming β ≤ /M , for CZF data precoding, the received SINR at the k th MTin the n th cell is given by γ CZF nk = 1 Mβ (1 − Mβ )ˆΓ CRCI + P m = n β kmn /β knn . (17) Proof: γ CZF nk in (17) is obtained by letting κ → in (15). Remark 1:
Selfish data precoders require estimation of in-cell CSI, i.e., ˆ H nn , only. In contrast,collaborative data precoders require estimation of both in-cell and inter-cell CSI at the BS, i.e., ˆ H n . Furthermore, since collaborative data precoders attempt to avoid interference not only to in-cell users but also to out-of-cell users, more BS antennas are needed to achieve high performance.This is evident from Corollaries 1 and 2, which reveal that N T > K and N T > M K arenecessary for SZF and CZF data precoding, respectively. On the other hand, if successful, tryingto avoid out-of-cell interference is beneficial for the overall performance. Hence, whether selfishor collaborative precoders are preferable depends on the parameters of the considered system,cf. Sections V and VI. B. Polynomial Data Precoder
The RCI and ZF data precoders introduced in the previous section achieve a higher perfor-mance than simple MF data precoding [13]. However, they require a matrix inversion whichentails a high computational complexity for the large values of K and N T desired in massiveMIMO. Hence, in this section, we propose a low-complexity POLY data precoder which avoidsthe matrix inversion. As the goal is a low-complexity design, we focus on selfish POLY precoders,although the extension to collaborative designs is possible. September 2, 2015 DRAFT2
The proposed POLY precoder, F n , for the n th BS can be expressed as F n = 1 √ N T ˆ H Hnn I X i =0 µ i (cid:18) ˆ H nn ˆ H Hnn (cid:19) i , (18)where ˆ H nn = √ N T ˆ H nn , and µ = [ µ , . . . , µ I ] T are the real-valued coefficients of the precodermatrix polynomial, which have to be optimized. In the following, we show that, for K, N T → ∞ ,the optimum coefficients µ do not depend on the instantaneous channel estimates but are constantand can be determined by exploiting results from free probability [26] and random matrix theory[29]. To this end, we define the asymptotic average mean-square error (MSE) of the users in the n th cell as mse n = lim K →∞ K E [ k e n k ] with error vector e n = ς y n − s n = ς ( G nn ( √ p F n s n + √ q A n z n ) + ˜ n n ) − s n , (19)where ˜ n n = P m = n G mn x m + n n includes Gaussian noise, inter-cell interference, and inter-cellAN leakage. Furthermore, ς is a normalization constant at the receiver, which does not impactdetection performance. The optimal coefficient vector µ minimizes mse n for a given powerbudget φP T for the information-carrying signal, i.e., min µ ,ς mse n s . t . : Tr { F Hn F n } = 1 , (20)where we use the notation Tr {·} = lim K →∞ tr {·} /K . The optimal coefficient vector, µ opt , isprovided in the following theorem. Theorem 1 : For
K, N T → ∞ , the optimal coefficient vector minimizing the asymptotic averageMSE of the users in the n th cell for the POLY precoder in (18) is given by µ opt = γ Π − ψ , (21)where ψ = [ ζ , ζ , . . . , ζ I +1 ] T , [ Π ] i,j = Tr { D nn } ζ i + j + (cid:16) Tr { D nn ∆ n } + Tr { Σ n } + P AN N T p (cid:17) ζ i + j − , Σ n = E [˜ n n ˜ n Hn ] , ∆ n = diag n p τ τ P m = n β nm p τ τ P Mm =1 β nm , · · · , p τ τ P m = n β Knm p τ τ P Mm =1 β Knm o , and P AN = q E (cid:2) Tr (cid:8) G nn A n A Hn G Hnn (cid:9)(cid:3) . Furthermore, ζ l denotes the l th -order moment of the sum of theeigenvalues of ˆ H nn ˆ H Hnn , i.e., ζ l = lim K →∞ K P Kk =1 λ lk , which converges to ζ l = P l − i =0 (cid:0) li (cid:1)(cid:0) li +1 (cid:1) β i l for K → ∞ [23, Theorem 2]. Finally, γ is chosen such that Tr { F Hn F n } = 1 holds. Proof:
Please refer to Appendix B.We note that µ opt does not depend on instantaneous channel estimates, and hence, can becomputed offline. September 2, 2015 DRAFT3
C. Computational Complexity of Data Precoding
We compare the computational complexity of the considered data precoders in terms of thenumber of floating point operations (FLOPs) [28]. Each FLOP represents one scalar complexaddition or multiplication. We assume that the coherence time of the channel is T symbol intervalsof which τ are used for training and T − τ are used for data transmission. Hence, the complexityrequired for precoding in one coherence interval is comprised of the complexity required forgenerating one precoding matrix and T − τ precoded vectors. A similar complexity analysis wasconducted in [23, Section IV] for selfish data precoders without AN injection at the BS. Sincethe AN injection does not affect the structure of the data precoders, we can directly adapt theresults from [23, Section IV] to the case at hand. In particular, the selfish MF, the SZF/SRCI,and the CZF/CRCI precoders require (2 K − N T ( T − τ ) , . K + K )(2 N T −
1) + K + K + K + N T K (2 K −
1) + (2 K − N T ( T − τ ) , and . M K + M K )(2 N T −
1) + M K + M K + M K + N T M K (2 M K −
1) + (2 K − N T ( T − τ ) FLOPs per coherence interval, see [23, SectionIV]. In contrast, for the POLY data precoder, we obtain for the overall computational complexity ( T − τ ) (( I + 1)(2 K − N T + I (2 N T − K ) FLOPs, which assumes implementation of theprecoding operation by Horner’s rule [23, Section IV].The above complexity expressions reveal that the additional complexity introduced by collab-orative data precoders compared to selfish data precoders is at most a factor of M . In addition,the complexity savings achieved with the POLY data precoder compared to the SZF/SRCI dataprecoders increase with increasing K for a given T . We note however that, regardless of theircomplexity, POLY data precoders are attractive as they avoid the stability issues that may arisein fixed point implementation of large matrix inverses.IV. L INEAR
AN P
RECODERS FOR S ECURE M ASSIVE
MIMOIn this section, we investigate the performance of selfish and collaborative NS (S/CNS) andrandom AN precoders. In addition, a novel POLY AN precoder is derived. To the best of theauthors’ knowledge, POLY AN precoding has not been considered in the literature before.
A. Analysis of Existing AN Precoders
For a given dimensionality of the AN precoder, L , the secrecy rate depends on the AN precoderonly via the AN leakage, Q , given in (8), which affects the SINR of the MT. Furthermore, the September 2, 2015 DRAFT4 optimal POLY data precoder coefficients in (21) are affected by the AN precoder via the leakageterm P AN . In this subsection, for N T → ∞ , we will provide closed-form expressions for Q and P AN for the SNS, CNS, and random AN precoders.
1) SNS AN Precoder:
The SNS AN precoder of the n th BS is given by [8] A n = I N T − ˆ H Hnn (cid:16) ˆ H nn ˆ H Hnn (cid:17) − ˆ H nn , (22)which has rank L = N T − K and exists only if β < . We divide the corresponding ANleakage Q SNS into an inter-cell AN leakage Q SNS o and an intra-cell AN leakage Q SNS i , where Q SNS = Q SNS o + Q SNS i . For the SNS AN precoder, Q SNS o is obtained as Q SNS o = X m = n β kmn E (cid:20) h kmn A m A Hm ( h kmn ) H (cid:21) = E (cid:20) tr (cid:8) A m A Hm (cid:9) (cid:21) M X m = n β kmn = ( N T − K ) M X m = n β kmn , (23)where we exploited [24, Lemma 11] and the independence of A m and h kmn . In contrast, theintra-cell AN leakage power is given by Q SNS i = β knn E (cid:20) h knn A n A Hn ( h knn ) H (cid:21) = β knn E (cid:20) ˜ h knn A n A Hn (˜ h knn ) H (cid:21) = ( N T − K ) β knn p τ τ P Mm = n β knm p τ τ P Mm =1 β knm , as the SNS AN precoder matrix lies in the null space of the estimated channels of all K MTsin the n th cell. Similarly, the AN leakage relevant for computation of the POLY data precoderis obtained as P SNSAN = (1 − φ ) P T lim K →∞ K K X k =1 β knn p τ τ P Mm = n β knm p τ τ P Mm =1 β knm . (24)
2) CNS AN Precoder:
For the CNS AN precoder at the n th BS, the AN is designed to lie inthe null space of the estimated channels between all
M K
MTs and the BS, i.e., A n = I N T − ˆ H Hn (cid:16) ˆ H n ˆ H Hn (cid:17) − ˆ H n , (25)which has rank L = N T − M K and exists only if β < /M . The corresponding AN leakage tothe k th MT in the n th cell is given by Q CNS = M X m =1 β kmn E (cid:20) h kmn A m A Hm ( h kmn ) H (cid:21) = ( N T − M K ) M X m =1 β kmn p τ τ P Ml = m β kml p τ τ P Ml =1 β kml . (26)Furthermore, the CNS AN precoder results in the same P AN as the SNS AN precoder, cf. (24). September 2, 2015 DRAFT5
3) Random AN Precoder:
For the random precoder, all elements of A n are i.i.d. randomvariables independent of the channel [13], i.e., A n has rank L = N T . Hence, h kmn and A m , ∀ m ,are mutually independent, and we obtain Q random = M X m =1 β kmn E (cid:20) h kmn A m A Hm ( h kmn ) H (cid:21) = N T M X m =1 β kmn . (27)Furthermore, we obtain P randomAN = (1 − φ ) P T lim K →∞ K P Kk =1 β knn . Remark 2:
If the power and time allocated to channel estimation are very small, i.e., τ p τ → ,the S/CNS AN precoders yield the same qQ and P AN as the random AN precoder. This suggeststhat in this regime all considered AN precoders achieve a similar SINR performance for agiven MT. However, for τ p τ > , the S/CNS AN precoders cause less AN leakage resulting inan improved SINR performance compared to the random precoder at the expense of a highercomplexity. B. POLY AN Precoder
To mitigate the high computational complexity imposed by the matrix inversion required forthe S/CNS AN precoders, while achieving an improved performance compared to the random ANprecoder, we propose a POLY AN precoder. Similar to the POLY data precoder, we concentrateon the selfish design because of the desired low complexity, and hence, set L = N T − K . Theproposed POLY AN precoder is given by A n = I N T − ˆ H Hnn J X i =0 ν j (cid:18) ˆ H nn ˆ H Hnn (cid:19) j ! ˆ H nn , (28)where ν = [ ν , . . . , ν J ] T contains the real-valued coefficients of the AN precoder polynomial,which have to be optimized. In particular, ν is optimized for minimization of the asymptoticaverage AN leakage caused to all MTs in the n th cell P AN . The corresponding optimizationproblem is formulated as min ν P AN = q E (cid:20) Tr { G nn A n A Hn G Hnn } (cid:21) s . t . :Tr { A Hn A n } = 1 /β − . (29)The solution of (29) is provided in the following theorem. Theorem 2 : For
K, N T → ∞ , the optimal coefficient vector minimizing the asymptotic averageAN leakage caused to the users in the n th cell for the AN precoder structure in (28) is given by ν opt = Σ − ω , (30) September 2, 2015 DRAFT6 where [ Σ ] i,j = ζ i + j +1 + ǫζ i + j and ω = [ ζ + ǫζ , . . . , ζ J +2 + ǫζ J +1 ] . Here, ζ l denotes again the l th order moment of the sum of the eigenvalues of matrix ˆ H nn ˆ H Hnn , cf. Theorem 1. ǫ is chosensuch that Tr { A Hn A n } = 1 /β − . Proof:
Please refer to Appendix C.
C. Computational Complexity of AN Precoding
Similarly to the data precoders, the complexity of the AN precoders is evaluated in terms of thenumber of flops required per coherence interval T . For the SNS AN precoder, the computationof A n in (22) requires the computation and inversion of a K × K positive definite matrix,which entails . K + K )(2 N T −
1) + K + K + K FLOPs [28], and the multiplicationof an N T × K , an K × K , and an K × N T matrix, which entails N T ( N T + K )(2 K − FLOPs [28]. Furthermore, the T − τ vector-matrix multiplications required for AN precodingentail a complexity of (2 N T − N T FLOPs [28], respectively. Hence, the overall complexity is . K + K )(2 N T −
1) + K + K + K + N T ( N T + K )(2 K −
1) + (2 N T − N T ( T − τ ) FLOPs.Similarly, for the CNS AN precoder, we obtain a complexity of . M K ) + M K )(2 N T −
1) +(
M K ) + ( M K ) + M K + N T ( N T + M K )(2
M K −
1) + (2 N T − N T ( T − τ ) FLOPs, whereasthe random AN precoder entails a complexity of (2 N T − N T ( T − τ ) FLOPs as only the ANvector-matrix multiplications are required.Similar to the precoded data vector [23, Section IV], the POLY precoded AN vector can begenerated using Horner’s rule. Hence, based on (28), the transmitted AN vector in the n th cellcan be obtained as A n z n = z n − (cid:18) ν ˆ H Hnn ˆ H nn (cid:18) z n + ν ν ˆ H Hnn ˆ H nn ( z n + . . . ) (cid:19)(cid:19) . (31)Hence, A n z n can be computed efficiently by first multiplying ˆ H nn with z n , which requires (2 N T − K FLOPs, then multiplying ˆ H Hnn with the resulting vector, which requires (2 K − N T FLOPs, adding z n to the newly resulting vector, and repeating similar operations ( J + 1) times, see [21], [23] for details of Horner’s rule. Overall, this leads to a complexity of ( J +1) ((2 K − N T + (2 N T − K ) ( T − τ ) FLOPs.V. C
OMPARISON OF L INEAR D ATA AND
AN P
RECODERS
In this subsection, we compare the secrecy performances of the considered data and ANprecoders. Thereby, in order to get tractable results, we focus on the relative performances
September 2, 2015 DRAFT7
TABLE ISINR
OF THE k th MT IN THE n th CELL FOR LINEAR DATA PRECODING AND THE SIMPLIFIED PATH - LOSS MODEL IN (32).F
OR THIS MODEL , ˆΓ SRCI
AND ˆΓ CRCI
SIMPLIFY TO ˆΓ SRCI = Γ SRCI θ Γ SRCI ϑ +1 AND ˆΓ CRCI = Γ CRCI θ Γ CRCI ϑ +1 WHERE Γ SRCI = βφβφρ ( M − − φ ) β ˜ Q + βPT , Γ CRCI = βφ (1 − φ ) β ˜ Q + βPT , θ = p τ τ ap τ τ , AND ϑ = M − ρp τ τ ap τ τ .Data Precoder γ nk SZF θφ (1 − β )(1 − φ ) β ˜ Q + βφ ( a − θ )+( M − ρ θφ (1 − β )+ β/P T SRCI / G ( β,β/ ˆΓ SRCI )+( M − ρ CZF θφ (1 − Mβ )(1 − φ ) β ˜ Q + βφa (1 − θ )+( M − ρ θφ (1 − Mβ )+ β/P T CRCI / G ( aβ,aβ/ ˆΓ CRCI )+( M − ρ MF θφ (1 − φ ) β ˜ Q + βφa +( M − ρ θφ + β/P T of SZF, CZF, and MF [13] data precoders and SNS, CNS, and random AN precoders. Theperformances of SRCI, CRCI, and POLY data precoders and the POLY AN precoder will beinvestigated via numerical and simulation results in Section VI.In order to gain some insight for system design and analysis, we adopt a simplified path-lossmodel. In particular, we assume the path losses are given by β kmn = , m = nρ, otherwise (32)where ρ ∈ [0 , denotes the inter-cell interference factor. For this simplified model, a and c in(7) simplify to a = 1 + ( M − ρ and c = 1 + ( M − ρ . Furthermore, the SINR expressionsof the linear data precoders considered in Section III-A and the MF precoder considered in [13]can be simplified considerably and are provided in Table I, where we use the normalized ANleakage ˜ Q = Q/L . The expressions for the normalized AN leakage ˜ Q , the asymptotic averageAN leakage P AN , and the dimensionality L of the considered linear AN precoders are given inTable II. A. Comparison of SZF, CZF, and MF Data Precoders
In this subsection, we compare the performances achieved with SZF, CZF, and MF dataprecoders for a given AN precoder, i.e., L and ˜ Q are fixed. Since the upper bound on thecapacity of the eavesdropper channel is independent of the adopted data precoder, cf. Section September 2, 2015 DRAFT8
TABLE IIAN
LEAKAGE FOR SIMPLIFIED PATH - LOSS MODEL IN (32). θ AND ϑ ARE DEFINED IN THE CAPTION OF T ABLE
I.AN Precoder ˜ Q P AN L SNS ( a − θ ) (1 − φ ) P T ϑ N T − K CNS a (1 − θ ) (1 − φ ) P T ϑ N T − MK Random a (1 − φ ) P T N T II-C, we compare the considered data precoders based on their SINRs. Exploiting the results inTable I, we obtain the following relations between γ SZF nk , γ CZF nk , and γ MF nk : γ SZF nk γ MF nk = 1 + β ( cγ SZF nk −
1) and γ CZF nk γ SZF nk = 1 − M β − β + a ( a − β − β γ CZF nk . (33)Hence, for γ SZF nk > γ MF nk , we require γ SZF nk > /c = 1 / (1 + ρ ( M − , and for γ CZF nk > γ SZF nk , weneed γ CZF nk > / ( ρa ) = 1 / [ ρ (1 + ρ ( M − . As expected, (33) suggests that for a lightly loadedsystem, i.e., β → , all three precoders have a similar performance, i.e., γ CZF nk ≈ γ SZF nk ≈ γ MF nk .In the following, we investigate the impact of the number of MTs and the pilot power on therelative performances of the considered data precoders.1) Number of MTs : From (33), we find that for γ SZF nk > γ MF nk and γ CZF nk > γ SZF nk to hold, thenumber of MTs has to meet K < K
SZF > MF and K < K
CZF > SZF , where K SZF > MF = θφN T (1 − φ ) ˜ Q + aφ + 1 /P T and K CZF > SZF = ρφθN T (1 − φ ) ˜ Q + [ a (1 − θ ) + ρθM ] φ + 1 /P T , (34)respectively. Interestingly, both the maximum numbers of MTs for which the SZF data precoderis advantageous compared to the MF data precoder, K SZF > MF , and the maximum number of MTsfor which the CZF data precoder is advantageous compared to the SZF data precoder, K CZF > SZF ,decrease with increasing AN leakage, ˜ Q , and increasing number of cells, M , but increase withthe amount of resources dedicated to channel estimation, p τ τ (via θ ), and consequently withthe channel estimation quality. However, while K SZF > MF decreases with increasing inter-cellinterference factor, ρ (via a ), K CZF > SZF increases.2)
Pilot Energy : From (33), we find that for γ SZF nk > γ MF nk and γ CZF nk > γ SZF nk to hold, pilotenergy p τ τ has to fulfill p τ τ > ( p τ τ ) SZF > MF = 1 φ (1 − β ) /β +1 a +1 /P T − a and p τ τ > ( p τ τ ) CZF > SZF = 1 ρφ (1 − β ) /β +1 a +1 /P T − a , (35) September 2, 2015 DRAFT9 where we have assumed that SNS AN precoding is adopted, i.e., ˜ Q = a − θ , to arrive at insightfulexpressions. Similar results can be obtained for other AN precoders. From (35), we observe thatMF, SZF, and CZF data precoding are preferable if < p τ τ < ( p τ τ ) SZF > MF , ( p τ τ ) SZF > MF ≤ p τ τ < ( p τ τ ) CZF > SZF , and p τ τ ≥ ( p τ τ ) CZF > SZF , respectively. In general, the more MTs are inthe system (i.e., the larger β ), the larger the pilot energy has to be to make SZF and CZF dataprecoding beneficial. In fact, from (35) we observe that if β exceeds β MF = φ/ [ a + a/P T + φ − ,MF data precoding is always preferable regardless of the value of p τ τ . Similarly, if β exceeds β SZF = φρ/ [ a + a/P T + φρ − , SZF data precoding is always preferable compared to CZFdata precoding regardless of the value of p τ τ . B. Comparison of SNS, CNS, and MF AN Precoding
In this subsection, we analyze the impact of the AN precoders on the secrecy rate. ANprecoders affect the ergodic capacity of the eavesdropper via L and the achievable rate of theMT via the leakage, ˜ Q . Since the upper bound on the ergodic secrecy rate of the eavesdropperin (7) is a decreasing function in L , we have C eve nk | random ≤ C eve nk | SNS ≤ C eve nk | CNS . (36)On the other hand, from Table II, we observe ˜ Q random ≥ ˜ Q SNS ≥ ˜ Q CNS . Since according to TableI the SINRs for all data precoders are decreasing functions of ˜ Q , for a given data precoder, weobtain for the lower bound on the ergodic rate of the k th MT in the n th cell R nk | random ≤ R nk | SNS ≤ R nk | CNS . (37)Considering (36), (37), and the expression for the ergodic secrecy rate, R sec nk = [ R nk − C eve nk ] + , itis not a priori clear which AN precoder has the best performance. In fact, our numerical resultsin Section VI confirm that it depends on the system parameters (e.g. α , β , M , p τ τ , and ρ ) whichAN precoder is preferable. C. Ergodic Secrecy Rate Analysis
In this subsection, we provide closed-form results for the ergodic secrecy rate for SZF, CZF,and MF data precoding for the simplified path-loss model in (32). Thereby, the simplified path-loss model is extended also to the eavesdropper, i.e., β nE = 1 and β mE = ρ , m = n , is assumed. September 2, 2015 DRAFT0
Combining (4), (7), and the results in Table I, we obtain the following lower bounds for theergodic secrecy rate of the k th MT in the n th cell: R sec nk ≥ (cid:20) log (cid:16) ( ˜ Q +1 /P T ) β +( a − ˜ Q ) βφ + cθφ ( ˜ Q +1 /P T ) β +( a − ˜ Q ) βφ +( c − θφ · − χφ + χ (1 − χ ) φ + χ (cid:17) (cid:21) + for MF , (cid:20) log (cid:16) ( ˜ Q +1 /P T ) β +( a − θ − ˜ Q ) βφ + cθ (1 − β ) φ ( ˜ Q +1 /P T ) β +( a − θ − ˜ Q ) βφ +( c − θ (1 − β ) φ · − χφ + χ (1 − χ ) φ + χ (cid:17) (cid:21) + for SZF , (cid:20) log (cid:16) ( ˜ Q +1 /P T ) β +( a − aθ − ˜ Q ) βφ + cθ (1 − Mβ ) φ ( ˜ Q +1 /P T ) β +( a − aθ − ˜ Q ) βφ +( c − θ (1 − Mβ ) φ · − χφ + χ (1 − χ ) φ + χ (cid:17) (cid:21) + for CZF , (38)where χ = aβα − βcN T aL , and ˜ Q and L are given in Table II for the considered AN precoders.Eq. (38) is easy to evaluate and reveals how the ergodic secrecy rate of the three considered dataprecoders depends on the various system parameters. To gain more insight, we determine themaximum value of α which admits a non-zero secrecy rate. This value is denoted by α s in thefollowing, and can be shown to be a decreasing function of φ for all conidered data precoders.Hence, we find α s by setting R sec nk = 0 in (38) and letting φ → . This leads to α s = a θ ˜ Qa + cθN T /L + a/P T for MF , (1 − β ) a θ ˜ Qa + cθ (1 − β ) N T /L + a/P T for SZF , (1 − Mβ ) a θ ˜ Qa + cθ (1 − Mβ ) N T /L + a/P T for CZF . (39)Eq. (39) reveals that for a given AN precoder, independent of the system parameters, the MFdata precoder can always tolerate a larger number of eavesdropper antennas than the SZF dataprecoder, which in turn can always tolerate a larger number of eavesdropper antennas than theCZF data precoder. This can be explained by the fact that the high AN transmit power requiredto combat a large number of eavesdropper antennas drives the receiver of the desired MT intothe noise-limited regime, where the MF data precoder has a superior performance compared tothe S/CZF data precoders. On the other hand, since α s depends on both ˜ Q and L , it is not apriori clear which AN precoder can tolerate the largest number of eavesdropper antennas. Fora lightly loaded network with small β and small M , according to Table II, we have L ≈ N T for all three AN precoders. Hence, in this case, we expect the CNS AN precoder to outperformthe SNS and random AN precoders as it achieves a smaller ˜ Q . On the other hand, for a heavilyloaded network with large β and M , the value of α s of the CNS AN precoder is compromisedby its small value of L and SNS and even random AN precoders are expected to achieve a larger α s . September 2, 2015 DRAFT1
VI. P
ERFORMANCE E VALUATION
In this section, we evaluate the performance of the considered secure multi-cell massive MIMOsystem. We consider cellular systems with M = 2 and M = 7 hexagonal cells, respectively,and to gain insight for system design, we adopt the simplified path-loss model introduced inSection V, i.e., the severeness of the inter-cell interference is only characterized by the parameter ρ ∈ (0 , . The pilot sequence length is τ = K . The simulation results for the ergodic secrecyrate of the k th MT in the n th cell are based on (4), (6), and the expression for the ergodic rateof the MT [13, Eq. (8)] and are averaged over , random channel realizations. Note that,in this paper, we consider the ergodic secrecy rate of a certain MT, i.e., the k th MT in the n th cell. The cell sum secrecy rate can be obtained by multiplying the secrecy rate of the k th MTby the number of MTs, K , as for the considered channel model, all MTs in the n th cell achievethe same secrecy rate. The values of all relevant system parameters are provided in the captionsof the figures. To enable a fair comparison, throughout this section, we adopted the selfish SNSAN precoder when we compare different data precoders and the selfish ZF data precoder whenwe compare different AN precoders. A. Ergodic Capacity of the Eavesdropper for Conventional AN Precoders
In Fig. 1, we show the ergodic capacity of the eavesdropper for the considered conventionalAN precoders. First, we note that the upper bound in (7) is very tight since the number of BSantennas is large ( N T = 200 ) and α < a L/ ( cN T ) holds for all considered AN precoders andall consider values of α and β . Furthermore, as β increases, the ergodic capacity of all ANprecoders decreases since the power allocated to the information-carrying signal of the user thatthe eavesdropper tries to intercept decreases with increasing β as the total power allocated tothe information-carrying signals of all users is fixed. As expected, the eavesdropper’s capacitybenefits from larger values of α . Furthermore, as predicted in (36), because of their differentvalues of L , the CNS AN precoder yields the largest eavesdropper capacity, while the randomAN precoder yields the lowest. The performance differences between the different AN precodersdiminish for small values of α and β as the dependence of the eavesdropper capacity on L becomes negligible for small α , cf. (7), and L ≈ N T holds for all precoders for small β ,cf. Table II. September 2, 2015 DRAFT2 β C apa c i t y o f E a v e s d r oppe r ( bp s / H z ) CNS Upper boundCNS SimulationSNS Upper boundSNS SimulationRandom Upper boundRandom Simulation α = 0 . α = 0 . α = 0 . Fig. 1. Ergodic capacity of the eavesdropper vs. the normalized number of MTs in the cell, β , for a system with N T = 200 , φ = 0 . , P T = 10 dB, ρ = 0 . , and M = 2 . B. Ergodic Secrecy Rate for Conventional Linear Data Precoders
In Figs. 2 and 3, we show the ergodic secrecy rates of the k th MT in the n th cell vs. thenumber of BS antennas for the MF, SZF, CZF, SRCI, and CRCI data precoders for a lightlyloaded and a dense network, respectively, and a fixed power allocation factor of φ = 0 . . In bothfigures, the analytical results were obtained from (4), (6), and (12) for the SRCI data precoder,(16) for the CRCI data precoder, and (38) for the MF, SZF, and CZF data precoders. For allconsidered precoders, the analytical results provide a tight lower bound for the ergodic secrecyrates obtained by simulations. Furthermore, as expected, the RCI data precoders outperform theZF data precoders for both the selfish and the collaborative strategies, but the performance gapdiminishes with increasing number of BS antennas.For the lightly loaded network in Fig. 2, we assume M = 2 cells, K = 10 users, and a smallinter-cell interference factor of ρ = 0 . . For this scenario, the collaborative designs outperform theselfish designs and C/SZF precoding yields a large performance gain compared to MF precoding.This is expected from our analysis in Section V-A as for the parameters valid for Fig. 2, weobtain from (34), K SZF > MF ≈ and K CZF > SZF ≈ for N T = 400 . Intuitively, as the networkis only lightly loaded, the collaborative data precoder can efficiently reduce interference to theother cell despite the pilot contamination. September 2, 2015 DRAFT3
200 250 300 350 40022.533.544.5
Number of BS antennas N T E r god i c s e c r e cy r a t e ( bp s / H z ) CRCI sim.CRCI ana.CZF sim.CZF ana.SRCI sim.SRCI ana.SZF sim.SZF ana.MF ana.
Fig. 2. Analytical and simulation results for the ergodicsecrecy rate vs. the number of BS antennas, N T , for alightly loaded network with φ = 0 . , P T = 10 dB, p τ = P T /K , α = 0 . , K = 10 , ρ = 0 . , and M = 2 .
200 250 300 350 40000.10.20.30.40.50.60.7
Number of BS antennas N T E r god i c s e c r e cy r a t e ( bp s / H z ) CRCI sim.CRCI ana.CZF sim.CZF ana.SRCI sim.SRCI ana.SZF sim.SZF ana.MF ana.
Fig. 3. Analytical and simulation results for the ergodicsecrecy rate vs. the number of BS antennas, N T , for adense network with φ = 0 . , P T = 10 dB, p τ = P T /K , α = 0 . , K = 20 , ρ = 0 . , and M = 7 . For the dense network in Fig. 3, we assume M = 7 cells, K = 20 users, and a larger inter-cellinterference factor of ρ = 0 . . In this case, for the considered range of N T , the collaborativeprecoder designs are not able to suppress inter-cell interference and AN leakage to other cellssufficiently well to outperform the selfish precoder designs. In fact, for N T = 400 , we obtainfrom (34) K CZF > SZF ≈ , i.e., our analytical results suggest that the SZF precoder outperformsthe CZF precoder for K = 20 which is confirmed by Fig. 3. Nevertheless, for N T > , theergodic secrecy rate for the CZF data precoder will eventually surpass that for the SZF dataprecoder. C. Optimal Power Allocation
In this subsection, we investigate the dependence of the ergodic secrecy rate on the powerallocation factor φ and study the impact of system parameters such as β , M , and ρ on theoptimal φ that maximizes the ergodic secrecy rate. The results in this subsection were generatedbased on the analytical expressions in (4), (6), and (12) for the SRCI data precoder, (16) for theCRCI data precoder, and (38) for the MF, SZF, and CZF data precoders.Fig. 4 depicts the ergodic secrecy rate of the k th MT in the n th cell for the selfish dataprecoders SRCI, SZF, and MF as a function of the power allocation factor φ . All curves are September 2, 2015 DRAFT4 φ E r god i c S e c r e cy R a t e ( bp s / H z ) SRCI sim.SRCI ana.SZF sim.SZF ana.MF sim.MF ana. β = 0 . β = 0 . Fig. 4. Ergodic secrecy rate vs. φ for different selfish dataprecoders for a network with P T = 10 dB, N T = 100 , p τ = P T /K , α = 0 . , ρ = 0 . , and M = 7 . φ E r god i c S e c r e cy R a t e ( bp s / H z ) CRCI sim.CRCI ana.CZF sim.CZF ana.SZF sim.SZF ana. M = 2 M = 7 Fig. 5. Ergodic secrecy rate vs. φ for different dataprecoders for a network with P T = 10 dB, N T = 100 , p τ = P T /K , α = 0 . , β = 0 . , and ρ = 0 . . concave and have a single maximum. For φ = 0 only AN is transmitted, hence R sec nk = 0 resultssince no data can be transmitted. For φ = 1 , no AN is transmitted, hence R sec nk = 0 resultssince the capacity of the eavesdropper becomes unbounded (recall that we make the worst-caseassumption that the eavesdropper can receive noise-free). For < φ < , a positive secrecyrate may result depending on the system parameters and the precoding schemes. Since we keepthe total transmit power fixed, the transmit power per MT decreases with increasing β . Tocompensate for this effect, the portion of the total transmit power allocated to data transmissionshould increase. This is confirmed by Fig. 4 where the optimal value of φ for β = 0 . is largerthan that for β = 0 . . Furthermore, for a given β , the optimal φ is the larger, the better theperformance of the adopted data precoder is, i.e., for a more effective data precoder, transmittingthe data signal with higher power is more beneficial, whereas for a less effective data precoderimpairing the eavesdropper with a higher AN power is more beneficial.In Fig. 5, we show the ergodic secrecy rate vs. φ for the CRCI, CZF, and SZF precoders.Similar to our observations in Fig. 4, for given system parameters, the optimal φ tends to belarger for more effective precoders that achieve a better performance. For the system with M = 7 ,this can be observed by comparing the optimal φ for the SZF and CZF precoders. Furthermore,while for the smaller system with M = 2 cells collaborative precoding is always preferable, for September 2, 2015 DRAFT5 φ E r god i c S e c r e cy R a t e ( bp s / H z ) CNS sim.CNS ana.SNS sim.SNS ana.Random sim.Random ana. β = 0 . β = 0 . Fig. 6. Ergodic secrecy rate vs. φ for different ANprecoders for a network with P T = 10 dB, N T = 100 , p τ = P T /K , M = 2 , ρ = 0 . , and α = 0 . . β α s MFSZFCZF 0 0.5 100.10.20.30.40.50.60.70.80.9 β α s RandomSNSCNS
Fig. 7. α s vs. β for different data and AN precoders fora network with P T = 10 dB, N T = 100 , p τ = P T /K , ρ = 0 . , and M = 2 . M = 7 , SZF precoding outperforms CZF and CRCI precoding for all considered values of φ , asthe collaborative designs are not able to effectively suppress the interference and AN leakage tothe ( M − K = 60 users in the other cells with the available N T = 100 antennas. In particular,from (34), we obtain K CZF > SZF ≤ for M = 2 and K CZF > SZF ≤ for M = 7 , which confirmsthe results shown in Fig. 5. Fig. 6 depicts the ergodic secrecy rate vs. φ for the consideredconventional AN precoder structures. We consider a lightly loaded network with β = 0 . and amoderately loaded network with β = 0 . . For β = 0 . , the CNS AN precoder outperforms theSNS AN precoder since, in this case, for the CNS AN precoder, the negative impact of having(slightly) fewer dimensions available for degrading the eavesdropper’s channel (smaller value of L ) is outweighed by the positive impact of causing less AN leakage (smaller value of ˜ Q ). Onthe other hand, for β = 0 . , the CNS AN precoder has a substantially smaller L than the SNSprecoder which cannot be compensated by its larger ˜ Q . Despite having the largest value of L ,the random AN precoder has the worst performance for both considered cases because of itslarge AN leakage. September 2, 2015 DRAFT6
D. Conditions for Non-Zero Secrecy Rate
In Section V-C, we showed that a positive ergodic secrecy rate is possible only if α < α s . InFig. 7, using (39), we plot α s as a function of β . In the left hand side subfigure, we compareMF, SZF, and CZF data precoding for SNS AN precoding, and in the right hand side subfigure,we compare random, SNS, and CNS AN precoding for SZF data precoding. The comparisonof the data precoders reveals that although SZF and CZF entail a much higher complexity, MFprecoding achieves a larger α s . Therefore, if the eavesdropper has a large number of antennasand small ergodic secrecy rates are targeted, simple MF precoding is always preferable. On theother hand, whether SNS or CNS AN precoder is preferable depends on the system load. Forsmall values of β , CNS AN precoding can tolerate more eavesdropper antennas, whereas forlarge values of β , SNS AN precoding is preferable. Random AN precoding is outperformedby SNS AN precoding for any value of β . A closer examination of (39) reveals that this isalways true if S/CZF data precoders are employed. However, for the MF data precoder, thereare parameter combination for which random AN precoding outperforms SNS and CNS ANprecoding. E. Low-Complexity POLY Data and AN Precoders
In this subsection, we evaluate the ergodic secrecy rates of the proposed low-complexityPOLY data and AN precoders. To this end, we consider again a lightly loaded network withlittle inter-cell interference ( M = 2 , β = 0 . , ρ = 0 . ) and a dense network with more inter-cellinterference ( M = 7 , β = 0 . , ρ = 0 . ). All results shown in this section were obtained bysimulation. For each simulation point, the optimal value of φ was found numerically and applied.In Figs. 8 and 9, we show the ergodic secrecy rate of the k th MT in the n th cell as a functionof the pilot energy, τ p τ . As expected, for all considered schemes, the ergodic secrecy rate ismonotonically increasing in the pilot energy since more accurate channel estimates improve theperformance.In Fig. 8, we depict the ergodic secrecy rates for the proposed POLY data precoder fordifferent values of I and compare them to those of conventional selfish data precoders. Forthe sake of comparison, all data precoders are combined with the SNS AN precoder. As thenumber of terms of the polynomial I increase, the performance of the POLY data precoderquickly improves and approaches that of the SRCI data precoder. The convergence is faster for September 2, 2015 DRAFT7 −10 −5 0 5 10 1500.511.522.533.5
Pilot energy τ p τ (dB) E r god i c s e c r e cy r a t e ( bp s / H z ) M = 2 , ρ = 0 . , β = 0 . −5 0 5 10 150.080.10.120.140.160.180.20.220.240.260.28 Pilot energy τ p τ (dB) E r god i c s e c r e cy r a t e ( bp s / H z ) M = 7 , ρ = 0 . , β = 0 . SRCISZFPOLYMF I = 1 , , I = 1 , , Fig. 8. Ergodic secrecy rate for POLY and conventionalselfish data precoders for a network employing the optimal φ , P T = 10 dB, N T = 200 , and α = 0 . . −10 −5 0 5 10 1500.511.522.533.5 Pilot energy τ p τ (dB) E r god i c s e c r e cy r a t e ( bp s / H z ) M = 2 , ρ = 0 . , β = 0 . SNSPOLYRandom −5 0 5 10 150.060.080.10.120.140.160.180.20.220.240.26
Pilot energy τ p τ (dB) E r god i c s e c r e cy r a t e ( bp s / H z ) M = 7 , ρ = 0 . , β = 0 . J = 1 , , J = 1 , , Fig. 9. Ergodic secrecy rate for POLY, SNS, and randomAN precoders for a network employing the optimal φ , P T = 10 dB, N T = 200 , and α = 0 . . the dense network considered in the right hand side subfigure, where the performance differencebetween all precoders is smaller in general since interference cannot be as efficiently avoidedas for the lightly loaded network.In Fig. 9, we show the ergodic secrecy rates for the proposed POLY AN precoder fordifferent values of J and compare them to those of the random and SNS AN precoders. Forthe sake of comparison, all AN precoders are combined with SZF data precoding. The POLYAN precoder quickly approaches the performance of the SNS AN precoder as the polynomialorder J increases. Similar to the POLY data precoders, the convergence is faster for the densenetwork where the performance differences between different AN precoders are also smaller.For the denser network, even the random AN precoder is a viable option and suffers only froma small loss in performance compared to the SNS AN precoder. F. Complexity-Performance Tradeoff
In this subsection, we investigate the tradeoff between the ergodic secrecy rate performanceand the computational complexity of the proposed data and AN precoders in Figs. 10 and 11,respectively. In particular, Figs. 10 and 11 depict the ergodic secrecy rate on the left hand sideand the computational complexity (in Giga FLOP) on the right hand side, both as a function of
September 2, 2015 DRAFT8
100 200 300 40011.522.533.544.5
Number of MTs in each cell, K E r god i c s e c r e cy r a t e ( bp s / H z ) Ergodic secrecy rate 100 200 300 40000.511.522.533.5
Number of MTs in each cell, K C o m pu t a t i ona l c o m p l e x i t y ( G i ga − F L O P ) Computational complexity
CRCISRCIPOLYMF I = 1 , , I = 1 , , Fig. 10. Ergodic secrecy rate (left hand side) andcomputational complexity (right hand side) of variouslinear data precoders for a network employing P T = 10 dB, N T = 1000 , p τ = P T /K , M = 2 , ρ = 0 . , T − τ = 100 , and an SNS AN precoder.
100 200 3000.511.522.53
Number of MTs in each cell, K E r god i c s e c r e cy r a t e ( bp s / H z ) Ergodic secrecy rate 100 200 30000.511.522.53
Number of MTs in each cell, K C o m pu t a t i ona l c o m p l e x i t y ( G i ga − F L O P ) Computational complexity
CNSSNSPOLYRandom J = 1 , , J = 1 , , Fig. 11. Ergodic secrecy rate (left hand side) andcomputational complexity (right hand side) of variouslinear AN precoders for a network employing P T = 10 dB, N T = 1000 , p τ = P T /K , M = 2 , ρ = 0 . , T − τ = 100 , and an SZF data precoder. the numbers of users in a cell. For the considered setting, the performance gains of collaborativedata and AN precoding compared to selfish strategies are moderate, but the associated increasein complexity is substantial, especially for large K .Fig. 10 illustrates that for the considered setting a POLY data precoder with I = 1 achieves abetter performance than the MF precoder but has substantially lower complexity than the SRCIprecoder. For large I , the POLY data precoder has a lower complexity than the SRCI precoderfor large K . However, even for small K , the POLY precoder may be preferable as it does notincur the stability issues that may arise in the implementation of the large-scale matrix inversionsrequired for the SRCI precoder.Fig. 11 shows that for the considered setting the proposed POLY AN precoder with J = 1 outperforms the random AN precoder. The POLY AN precoder with J = 5 achieves almostthe same performance as the SNS AN precoder but with a substantially lower complexity. Wefurther observe that for small K , because of its efficient implementation via Horner’s scheme,cf. (31), the proposed POLY AN precoder requires an even lower complexity than the randomAN precoder. September 2, 2015 DRAFT9
VII. C
ONCLUSION
In this paper, we considered downlink multi-cell massive MIMO systems employing lineardata and AN precoding for physical layer security provisioning. We analyzed and compared theachievable ergodic secrecy rate of various conventional data and AN precoders in the presence ofpilot contamination. To this end, we also optimized the regularization constants of the selfish andcollaborative RCI precoders in the presence of AN and multi-cell interference. In addition, wederived linear POLY data and AN precoders which offer a good compromise between complexityand performance in massive MIMO systems. Interesting findings of this paper include: 1)Collaborative data precoders outperform selfish designs only in lightly loaded systems wherea sufficient number of degrees of freedom for suppressing inter-cell interference and sufficientresources for training are available. 2) Similarly, CNS AN precoding is preferable over SNS ANprecoding in lightly loaded systems as it causes less AN leakage to the information-carryingsignal, whereas in more heavily loaded systems, CNS AN precoding does not have sufficientdegrees of freedom for effectively degrading the eavesdropper channel and SNS AN precodingis preferable. 3) For a large number of eavesdropper antennas, where only small positive secrecyrates are achievable, MF data precoding is always preferable compared to SZF and CZF dataprecoding. 4) The proposed POLY data and AN precoders approach the performances of theSRCI data and SNS AN precoders with only a few terms in the respective matrix polynomialsand are attractive options for practical implementation.A
PPENDIX
A. Proof of Proposition 1
Considering (3) and (9), the effective signal power, i.e., the numerator in (5), can be expressedas [20] E [ h knn f nk ] = γ E [ h knn L nn (ˆ h knn ) H ] = γ E (cid:20) h knn L n,k (ˆ h knn ) H h knn L n,k (ˆ h knn ) H (cid:21) = γ ( X nk + A nk ) (1 + X nk ) , (40)where L n,k = ( ˆ H nn ˆ H Hnn − (ˆ h knn ) H ˆ h knn + κ I N T ) − , X nk = E [ˆ h knn L n,k (ˆ h knn ) H ] , and A nk = E [˜ h knn L n,k (ˆ h knn ) H ] . On the other hand, the intra-cell interference term in the denominator of(5) can be expressed as E (cid:20) X l = k | h knn f nl | (cid:21) = γ E (cid:20) h knn L n,k ˆ H Hn,k ˆ H n,k L n,k ( h knn ) H (cid:16) h knn L n,k (ˆ h knn ) H (cid:17) (cid:21) = γ ( Y nk + B nk )(1 + X nk ) , (41) September 2, 2015 DRAFT0 where ˆ H n,k is equal to ˆ H nn with the k th row removed, and Y nk = E [ˆ h knn L n,k ˆ H Hn,k ˆ H n,k L n,k (ˆ h knn ) H ] and B nk = E [˜ h knm L n,k ˆ H Hn,k ˆ H n,k L n,k (˜ h knn ) H ] .Due to pilot contamination, the data precoding matrix of the m th BS is a function of the channelvectors between the m th BS and the k th MTs in all cells. Hence, the inter-cell interference fromthe BSs in adjacent cells is obtained as E [ | h kmn f mk | ] = γ ( X nk + A nk ) (1 + X nk ) + 1 + p τ τ P Ml = m β kml p τ τ P Ml =1 β kml . (42)Meanwhile, by exploiting (40), (42), and the definition of the variance, i.e., var[ x ] = E [ x ] − E [ x ] , we obtain for the first term of the denominator of (5), var[ h knn f nk ] = p τ τ P Mm = n β knm p τ τ P Mm =1 β knm .According to [20, Eq. (16)] and [27, Theorem 7], for N T → ∞ and constant β , X nk convergesto G ( β, κ ) defined in (11) and A nk → . Similarly, Y nk and B nk approach Y nk N T →∞ = G ( β, κ ) + κ ∂∂κ G ( β, κ ) (43)and B nk N T →∞ = ϑ nk θ nk (1 + G ( β, κ )) (cid:18) G ( β, κ ) + κ ∂∂κ G ( β, κ ) (cid:19) , (44)respectively, where ∂∂κ G ( β, κ ) = − G ( β,κ )(1+ G ( β,κ )) β + κ (1+ G ( β,κ )) .Moreover, the inter-cell interference from other MTs (i.e., not the k th MTs) is calculated as E (cid:20) h kmn F m,k F Hm,k ( h kmn ) H (cid:21) = E (cid:20) tr (cid:8) F m,k F Hm,k (cid:9) (cid:21) = K − , (45)where F m,k is equal to F m with the k th column removed. The first equality in (45) is due tothe fact that the precoding matrix for the other MTs (i.e., not the k th MTs) in adjacent cells areindependent of h kmn and [24, Lemma 11], while the second equality holds for N T → ∞ .On the other hand, the constant scaling factor γ for SRCI precoding is given by [20, Eq.(22)] γ = 1 G ( β, κ ) + κ ∂∂κ G ( β, κ ) . (46)Hence, employing (40)-(46) in (5), the received SINR in (10) is obtained, which completes theproof of Proposition 1 . September 2, 2015 DRAFT1
B. Proof of Theorem 1
The objective function in (20) can be rewritten as mse n = ς p E (cid:20) Tr (cid:26) I X i =0 µ i (cid:18) ˆ H nn ˆ H Hnn (cid:19) i +1 D nn I X i =0 µ i (cid:18) ˆ H nn ˆ H Hnn (cid:19) i +1 (cid:27)(cid:21) + ς p E (cid:20) Tr (cid:26) I X i =0 µ i (cid:18) ˆ H nn ˆ H Hnn (cid:19) i ˆ H nn ˜ H Hnn D nn ˜ H nn ˆ H Hnn I X i =0 µ i (cid:18) ˆ H nn ˆ H Hnn (cid:19) i (cid:27)(cid:21) − ς √ p E (cid:20) Tr (cid:26) D / nn I X i =0 µ i (cid:18) ˆ H nn ˆ H Hnn (cid:19) i +1 (cid:27)(cid:21) + 1 + ς P AN + ς Tr { Σ n } , (47)where we exploited E [ s n s Hn ] = I K , the definition of P AN given in Theorem 1, the definition of F n in (18), the definition √ N T H nn = ˆ H nn + ˜ H nn , and ˜ H nn = √ N T ˜ H nn .In the following, we simplify the right hand side (RHS) of (47) term by term. To this end,we denote the first three terms on the RHS of (47) by t , t , and t , respectively. Using a resultfrom free probability theory [26], the first term converges to [23, Theorem 1] t = ς p Tr { D nn } E (cid:20) Tr (cid:26)(cid:18) I X i =0 µ i (cid:18) ˆ H nn ˆ H Hnn (cid:19) i +1 (cid:19) (cid:27)(cid:21) , (48)as matrix D nn is free from P I i =0 µ i (cid:18) ˆ H nn ˆ H Hnn (cid:19) i +1 . Similarly, the third term converges to t = − ς √ p Tr (cid:8) D / nn (cid:9) E (cid:20) Tr (cid:26) I X i =0 µ i (cid:18) ˆ H nn ˆ H Hnn (cid:19) i +1 (cid:27)(cid:21) . (49)Furthermore, the second term can be rewritten as t = ς p E (cid:20) Tr (cid:26) ˜ H Hnn D nn ˜ H nn (cid:27) Tr (cid:26) I X i =0 µ i (cid:18) ˆ H nn ˆ H Hnn (cid:19) i ˆ H nn ˆ H Hnn I X i =0 µ i (cid:18) ˆ H nn ˆ H Hnn (cid:19) i (cid:27)(cid:21) (b) = ς pN T Tr { D nn ∆ n } , (50)where (a) follows again from [23, Theorem 1] and (b) results from E [Tr { ˜ H Hnn D nn ˜ H nn } ] =Tr { D nn ∆ n } , where ∆ n is defined in Theorem 1, (18), and the constraint in (20).Exploiting (48)-(50) and the eigen-decomposition of matrix ˆ H nn ˆ H Hnn = TΛT H , where diag-onal matrix Λ = diag ( λ , . . . , λ K ) contains all eigenvalues and unitary matrix T contains thecorresponding eigenvectors, the asymptotic average MSE becomes mse n = E (cid:20) ς p Tr { D nn } Tr (cid:26) Λ (cid:18) I X i =0 µ i Λ i (cid:19) (cid:27) − ς √ p Tr (cid:8) D / nn (cid:9) Tr (cid:26) I X i =0 µ i Λ i +1 (cid:27)(cid:21) +1 + ς P AN + ς Tr { Σ n } + ς pN T Tr { D nn ∆ n } . (51) September 2, 2015 DRAFT2
Next, we introduce the Vandermonde matrix C ∈ R K × ( I +1) , where [ C ] i,j = λ j − i , and λ =[ λ , . . . , λ K ] T , which allows us to rewrite (51) in compact form as mse n = lim K →∞ K E (cid:20) ς p Tr { D nn } µ T C T Λ C µ − ς √ p Tr (cid:8) D / nn (cid:9) µ T C T λ (cid:21) +1 + ς P AN + ς Tr { Σ n } + ς pN T Tr { D nn ∆ n } . (52)Similarly, the constraint in (20) can be expressed as lim K →∞ K E (cid:20) µ T C T ΛC µ (cid:21) = N T . (53)Thus, the Lagrangian function of primal problem (20) can be expressed as L ( µ , ς ) = mse n + ǫ (lim K →∞ K E [ µ T C T ΛC µ ] − N T ) , where ǫ is the Lagrangian multiplier. Taking the gradientof the Lagrangian function with respect to µ , and setting the result to zero, we obtain for theoptimal coefficient vector µ opt : lim K →∞ K E (cid:20) C T Λ (cid:18) Λ + ǫ Tr { D nn } ς p I K (cid:19) C (cid:21) µ = Tr n D / nn o ς √ p Tr { D nn } lim K →∞ K E (cid:2) C T λ (cid:3) . (54)Furthermore, taking the derivative of L ( µ , ς ) with respect to ς and equating it to zero, andmultiplying both sides of (54) by µ T and applying (53), we obtain ǫ ς p = Tr { D nn ∆ n } + P AN + Tr { Σ n } N T p . (55)The expressions involving C , Λ , and λ in (54) can be further simplified. For example, we obtain lim K →∞ E (cid:20) K (cid:20) C T ΛC (cid:21) m,n (cid:21) = lim K →∞ E (cid:20) K P Kk =1 λ m + n − k (cid:21) . Simplifying the other terms in(54) in a similar manner and inserting (55) into (54) we obtain the result in Theorem 1. C. Proof of Theorem 2
Exploiting E [ z n z Hn ] = I N T , the constraint in (29), and a similar approach as was used to arriveat (24), the objective function in (29) can be simplified as P AN = q E (cid:20) Tr (cid:8) G nn A n A Hn G Hnn (cid:9) (cid:21) = q E (cid:20) Tr n D nn ˆ H nn A n A Hn ˆ H Hnn o (cid:21) + (1 − φ ) P T Tr { D nn ∆ n } . (56)Using (28) and a similar approach as in Appendix B, (56) can be rewritten as P AN = (1 − φ ) P T Tr { D nn ∆ n } (57) + qN T Tr { D nn } E (cid:20) − (cid:26) J X j =0 ν j Λ j +2 (cid:27) + Tr { Λ } + Tr (cid:26) Λ (cid:18) J X i =0 ν j Λ j +1 (cid:19) (cid:27)(cid:21) September 2, 2015 DRAFT3
Defining Vandermode matrix C ∈ R K × ( J +1) , where [ C ] i,j = λ j − i , we can rewrite (57) incompact form as P AN = qN T Tr { D nn } lim K →∞ K E (cid:20) − ν T C T Λ λ + T λ + ν T C T Λ C ν (cid:21) + (1 − φ ) P T Tr { D nn ∆ n } , (58)where denotes the all-ones column vector. Taking into account the constraint in (29), we canformulate the Lagrangian as L ( ν ) = P AN + ǫ (lim K →∞ K E [ ν T C T Λ C ν − ν T C T λ ] + 1) with Lagrangian multiplier ǫ . The optimal coefficient vector ν opt is then obtained by taking thegradient of the Lagrangian function with respect to ν and setting it to zero: lim K →∞ E (cid:20) C T Λ ( Λ + ǫ I K ) C (cid:21) ν = lim K →∞ E (cid:20) C T ( Λ + ǫ I K ) λ (cid:21) , (59)where we used ǫ = ǫ qN T Tr { D nn } . Simplifying the terms in (59) by exploiting a similar approachas in Appendix B, we obtain the result in Theorem 2.R EFERENCES [1] F. Rusek, D. Persson, B. K. Lau, E. G. Larsson, T. L. Marzetta, O. Edfors, and F. Tufvesson, “Scaling up MIMO:Opportunities and challenges with very large arrays,”
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