IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. X, NO. X, MONTH YEAR 1 Disrupting MIMO Communications with Optimal Jamming Signal Design Qian Liu, Member, IEEE, Ming Li, Member, IEEE, Xiangwei Kong, Member, IEEE, and Nan Zhao, Member, IEEE Abstract—This paper considers the problem of intelligent jamming attack on a MIMO wireless communication link with a transmitter, a receiver, and an adversarial jammer, each equipped with multiple antennas. We present an optimal jamming signal design which can maximally disrupt the MIMO transmission when the transceiver adopts an anti-jamming mechanism. In particular, signal-to-jamming-plus-noise ratio (SJNR) at the receiver is used as the anti-jamming reliability metric of the legitimate MIMO transmission. The jamming signal design is developed under the most crucial scenario for the jammer where the legitimate transceiver adopt jointly designed maximum-SJNR transmit beamforming and receive filter to suppress/mitigate the disturbance from the jammer. Under this best anti-jamming scheme, we aim to optimize the jamming signal to minimize the receiver’s maximum-SJNR under a given jamming power budget. The optimal jamming signal designs are developed in different cases with accordance to the availability of channel state information (CSI) at the jammer. The analytical approximations of the jamming performance in terms of average maximumSJNR are also provided. Extensive simulation studies confirm our analytical predictions and illustrate the efficiency of the designed optimal jamming signal on disrupting MIMO communications. Index Terms—Artificial interference, beamforming, jamming, multiple-input multiple-output (MIMO), power allocation, signalto-jamming-plus-noise ratio (SJNR). I. I NTRODUCTION HE past decade witnesses the rapid worldwide deployment of wireless multiple-input multiple-output (MIMO) systems. Consequently, there is an urgent request of MIMO jamming technology because of its vital usage in military networks for adversary defense and proactive attack to hostile forces. The optimization designs of jamming signals and strategies to maximally impair legitimate MIMO communication links have been becoming an important research topic and received significant attention in recent years. The goal of a jammer is to intentionally disturb legitimate wireless transmissions by degrading the receiving performance T Manuscript received October 16, 2014; revised March 27, 2015; accepted May 27, 2013. This work was supported in part by the Fundamental Research Funds for the Central Universities (Grant No. DUT14RC(3)103 and Grant No. DUT14QY44) and the National Natural Science Foundation of China (NSFC) under Grant 61201224, and the Foundation for Innovative Research Groups of the NSFC (Grant No. 71421001). The associate editor coordinating the review of this paper and approving it for publication was R. Zhang. The corresponding author is Nan Zhao Qian Liu is with the Department of Computer Science and Engineering, State University of New York at Buffalo, Buffalo, NY, 14260, USA (E-mail: [email protected]). Ming Li, Xiangwei Kong, and Nan Zhao are with the School of Information and Communication Engineering, Dalian University of Technology, Dalian, Liaoning, 116024, P. R. China (E-mail: {mli,kongxw,zhaonan}@dlut.edu.cn). Digital Object Identifier XXXXX in order to trigger reliability outage and service disruption [1]. The jammer’s ability of disrupting the legitimate communications can be substantially enhanced if certain priori knowledge (for example transmit signal, channel state information (CSI), etc.) is available at the jammer side. When a jammer can fully or partially acquire the transmit signal (by eavesdropping for example), it can efficiently disrupt the legitimate communication by deliberately generating jamming signal correlated to the transmit signal. Such “correlated jamming” approaches have been investigated under various assumptions [2]-[9]. In [2], [3], a so-called Gaussian test channel was considered with the jamming signals correlated with the output of the encoder [2] and the input to the encoder [3]. A more general class of jammers was studied in [4] where the jammer chooses the transition probability from a set of allowed channels to minimize the capacity. Another popular approach is to model the legitimate transmitter and the adversary jammer as players in a game-theoretic formulation to identify the optimal transmit strategies for both parties. In [5], the best transmitter/jammer strategies were found for an additive white Gaussian noise (AWGN) channel with one user and one jammer who participate in a zero-sum mutual information game. In [6], Kashyap et al. pursued related strategies for a single-user MIMO fading channel where the jammer has full knowledge of the source signal. The studies of mutual information games with correlated jamming were then extended to relay channels [7], multiple access channels [8], and multiuser channels [9]. In [10], Diggavi and Cover showed that for high signal power, the maximum entropy noise is the worst additive noise with a banded covariance constraint. On the other hand, in [11]-[14] several “uncorrelated jamming” designs are developed where the jammer attempts to launch more effective jamming attacks by disrupting the channel estimation procedure (period) of MIMO communications. However, such approaches require protocol knowledge and accurate system synchronization to target channel estimation and CSI feedback procedure. In [15]-[17], uncorrelated jamming on MIMO channels is investigated in information-theoretic context which aims to minimize the capacity (or mutual informatin) of the legitimate MIMO channels by injecting a spatially-correlated Gaussian interference signal. In these studies, full channel state information (CSI) of both legitimate and jamming channels is assumed, but the jammer has no knowledge of the legitimate users’ signals. In [18], instead of capacity, Jorswieck et al. used the expectation of the trace of some arbitrary matrix-monotone function as the average performance metric for the jamming design problems. Two 2 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. X, NO. X, MONTH YEAR common performance metrics (mutual information and meansquare-error (MSE)) can be obtained by selecting different matrix-monotone functions. The optimal jamming strategies were proposed for the cases of no CSI, statistical knowledge, and perfect CSI. In addition to the usage in electronic warfare area, jamming signal can also be utilized to improve the physical layer security of wireless transmission in presence of eavesdroppers [19]-[31]. In these works, the legitimate users transmit secret messages along with the sophisticatedly designed jamming signal (artificial interference) which can disturb the eavesdropper’s reception but has no harm to the legitimate receivers. In this paper, we study that how an intelligent adversary jammer can maximally disrupt MIMO communications by generating optimal jamming signal. We consider a MIMO communication link in the presence of an adversary jammer equipped with multiple antennas. We are interested in an uncorrelated jamming problem in which the jammer does not have any knowledge of signals transmitted by the legitimate user. But the jammer may have full or partial CSI about jamming channel and legitimate channel. Under such assumption, we seek an optimal jamming approach for the adversary jammer to maximally disrupt the legitimate MIMO transmission. Previous uncorrelated jamming works [15]-[18] utilize information-theoretic metric. However, the channel capacity (or mutual information) can only provide an upper bound of the achievable throughput of the legitimate transmission link. With the attack of a malicious jammer, we need to evaluate the maximum reliable transmission rate of the legitimate link equipped with a practical anti-jamming mechanism. Furthermore, in order to evaluate the channel capacity, we need the full CSI knowledge of both legitimate and jamming channels, which may not be always available in practice. Therefore, another practical reliability metric needs to be considered for the jamming signal design. In this paper, we consider a more practical signal processing context and use legitimate receiver’s signal-to-jamming-plusnoise ratio (SJNR) as our metric of the reliability of the legitimate MIMO transmission. Particularly, we explore the jammer design under the most crucial scenario for the jammer where the legitimate transmitter and receiver adopt jointly designed maximum-SJNR transmit beamforming and receiver filter to suppress/mitigate the disturbance from the jammer. With such best anti-jamming scheme employed by the legitimate transceiver, we aim to optimize the jamming signal for the jammer to minimize the legitimate receiver’s maximumSJNR under a given jamming power budget constraint. The optimal jamming signal designs are developed in different cases with accordance to the availability of CSI for the jammer. Comparing to the information-theoretic metric [15]-[18], maximum-SJNR is a practical measure to evaluate the legitimate transmission link under a jamming attack. It can reflect the achievable quality of service (QoS) of the legitimate transmission link with practical transmit and receive implementation (e.g. beamforming, simple coding, etc.). Therefore, using SJNR as the jamming design metric, the performance of the proposed jammer is practically achievable in wireless MIMO networks. It is also worth pointing out that in [18] the objective metric can be set as MSE or sum-SJNR by selecting an appropriate matrix-monotone function. However, these metrics can only reflect the average performance of the legitimate link, while maximum-SJNR indicates the best achievable QoS at the legitimate receiver. To evaluate and optimize the jamming attack, we are more interested in how to maximally degrade the best legitimate QoS. Therefore, minimizing the maximumSJNR is more meaningful in practice. In addition, unlike [11][14], the proposed jamming scheme aims at disrupting the data transmission procedure of MIMO communications rather than the channel estimation period. Since the data transmission procedure is the dominant component of the entire communications, delicate synchronization is not a necessity. As a result, the proposed jamming approach is easier to implement than exiting approaches [11]-[14]. The rest of the paper is organized as follows. The jamming signal optimization problem is formulated in Section II. Optimal jamming signal designs are developed in Section III under various assumptions of CSI availability. In Section IV, simulation results illustrate the jamming performance of our developments and, finally, a few conclusions are drawn in Section V. The following notation is used throughout the paper. Boldface lower-case letters indicate column vectors and boldface upper-case letters indicate matrices; C is the set of all complex numbers; (·)H denotes transpose-conjugate operation; IL is the L × L identity matrix; and E{·} represents statistical expectation. X ≽ 0 states that X is positive semidefinite; Tr{X} is the trace of X; X† denotes the Moore-Penrose pseudo inverse of X. II. S YSTEM M ODEL We consider a wireless communication system from a legitimate transmitter to a legitimate receiver in the presence of an adversarial jammer who attempts to disrupt the transmission. For convenience, we follow the common language in the field and name the transmitter, receiver, and jammer, Alice, Bob, and Jeff, respectively. A simple diagram is shown in Fig. 1 to illustrate this basic communication scenario. Alice, Bob, and Jeff are equipped with Na , Nb , and Nj antennas, respectively. The channel between Alice and Bob is assumed to be flat Rayleigh fading and is represented by Hab ∈ CNb ×Na where Hab (i, j) is the channel coefficient from the jth transmit antenna at Alice to the ith receive antenna at Bob. Similarly, let Hjb ∈ CNb ×Nj be the flat Rayleigh fading channel matrix between Jeff and Bob. Alice transmits a symbol multiplied by the corresponding transmit beamforming vector (also referred to as spatial signature or linear precoder in the literature). The transmitted signal has a form of √ (1) x = Pa s t where Pa > 0 denotes Alice’s transmit power, s ∈ C, E{s} = 0, E{|s|2 } = 1, denotes the unit energy information symbol, and t ∈ CNa , ∥t∥ = 1, denotes the beamforming vector, respectively. LI et al.: DISRUPTING MIMO COMMUNICATIONS WITH OPTIMAL JAMMING SIGNAL DESIGN 3 where λab,1 is the largest eigenvalue of matrix Hab HH ab and 2 λab,1 = σab,1 . The statistics of λab,1 of a Wishart matrix Hab HH ab has been intensively investigated [32], [33]. We also define the average maximum-SNR as SNRmax , EHab {SNRmax } = Pa λab,1 σn2 (8) where λab,1 , EHab {λab,1 }. In the rest of this paper, we will use SNRmax and SNRmax as the performance benchmark. B. Jamming Signal Present Fig. 1. Now we consider the case in which the jammer attempts to disturb Alice’s signal by emitting jamming signal z ∈ CNj via Nj antennas. Let Rz , E{zzH } be the autocorrelation matrix of jamming signal z and the power of the emitted jamming signal is Pj = Tr{Rz }. When the jamming signal is present, the signal vector y ∈ CNb received by Bob can be expressed as √ y = Pa sHab t + Hjb z + n. (9) MIMO channel in presence of an adversary jammer. A. Jamming Signal Absent We first investigate the case in which no jamming is presented and consider the performance in such a case as a benchmark. Without any jamming signal from jammer, the signal vector y ∈ CNb received by Bob can be expressed as √ y = Pa sHab t + n (2) where n ∼ CN (0, σn2 I) represents circularly symmetric complex AWGN vector with power σn2 . Bob utilizes a normalized receive beamforming vector (filter) w ∈ CNb , ∥w∥ = 1, to retrieve information symbol s. The signal after filtering is √ sb = wH y = Pa swH Hab t + wH n. (3) The two terms in (3) are desired signal and noise, respectively. The output signal-to-noise ratio (SNR) of filtering is {√ } E | Pa swH Hab t|2 SNR = E {|wH n|2 } Pa |wH Hab t|2 = . (4) σn2 Let Hab , be the singular value decomposition (SVD) of Hab where Uab ∈ CNb ×Nb and Vab ∈ CNa ×Na are two unitary matrices, Σab is an Nb × Na diagonal matrix with nonnegative real numbers σab,1 ≥ . . . ≥ σab,m on the diagonal, m = min{Na , Nb }. In order to maximize the output SNR in (4), Alice and Bob adopt the optimal transmit beamformer tmaxSNR = Vab (1) (5) H Uab Σab Vab and receive filter wmaxSNR = Uab (1), (6) where Vab (1) and Uab (1) are the first (leftmost) vector of Vab and Uab corresponding to the largest singular value σab,1 . With those optimal transmit beamformer (5) and receive filter (6), the maximum-SNR of Bob is SNRmax = 2 Pa σab,1 Pa λab,1 = 2 σn σn2 (7) The retrieved signal after filtering by w is √ sb = wH y = Pa swH Hab t + wH Hjb z + wH n. (10) The three terms in (10) are desired signal, interference from jammer, and noise, respectively. The output SJNR of filtering is {√ } E | Pa swH Hab t|2 SJNR = E {|wH Hjb z + wH n|2 } Pa wH Hab ttH HH ab w . = (11) H H 2 w (Hjb Rz Hjb + σn INb )w In order to mitigate/suppress the disturbance from jammer Alice and Bob take the best anti-jamming effort and adopt joint transmit beamformer and receiver filter design to maximize the output SJNR. It is well known that, for any given transmit beamformer t, the maximum-SJNR filter wmaxSJNR ∈ CNb can be obtained by [34] 2 −1 wmaxSJNR = c (Hjb Rz HH Hab t jb + σn INb ) (12) where a scale c > 0 guarantees the normalization of the filter [34]. We need to point out that Hjb and Rz may be not available for the legitimate users. But the disturbance autocor2 relation matrix Hjb Rz HH jb + σn INb can be estimated at Bob’s e = Hjb z + n when by observing interference and noise only y signal of interest is absent and then averaging the observed ∑N 1 2 e samples as Hjb Rz HH + σ I ≈ y (n)e y(n)H . N n b jb n=1 N With the SJNR-optimum filter wmaxSJN R , the maximized output SJNR is H 2 −1 SJNRmax = Pa tH HH Hab t (13) ab (Hjb Rz Hjb + σn INb ) which is a direct function of the transmit beamforming vector t. The SJNR-maximization beamforming vector t, which maximizes output SJNR in (13), can be obtained by ( ) H 2 −1 tmaxSJNR = Umax HH Hab (14) ab (Hjb Rz Hjb + σn INb ) where Umax (A) denotes the right eigenvector of matrix A corresponding to the largest eigenvalue λmax (A), i.e. 4 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. X, NO. X, MONTH YEAR AUmax (A) = λmax (A)Umax (A). The attained maximum-SJNR by such joint optimal transmit beamformer (12) and receive filter (14) design is ( ) H 2 −1 SJNRmax = Pa λmax HH Hab . ab (Hjb Rz Hjb + σn INb ) (15) While the anti-jamming technologies have been intensively studied for the legitimate transmissions, in this work we stand at jammer’s side and consider the problem from jammer’s design perspective. We focus our attention on the problem of jamming signal design to maximize jammer’s impairment on the Alice-to-Bob transmission even when the maximumSJNR transmit beamforming and receive filter are adopted by Alice and Bob. In particular, our technical goal is to adaptively design the correlation matrix Rz of the jamming signal z to minimize Bob’s maximum-SJNR under a given jamming power budget. The optimal jamming signal design will be developed in the next section in different scenarios according to the availability of Hjb and Hab for the jammer. III. O PTIMAL JAMMING S IGNAL D ESIGNS We first consider the case that the jammer has the knowledge of both Hjb and Hab . The knowledge of Hab may also be obtained by the jammer if the locations of Alice and Bob are known or projected/anticipated. Another scenario, in which such assumption will be valid, is that the jammer can also obtain the knowledge of Hab by eavesdropping the feedback channel in which Alice and Bob share Hab . Regarding to Hjb , jammer can estimate it by analyzing the observed signal transmitted by Bob when Bob is also a transmitter for other transmission tasks. For example, Alice-Bob may be a timedivision duplex (TDD) communication link and Bob will send data to Alice in other time slots. Our objective is to design the autocorrelation matrix Rz of jamming signal z to minimize Bob’s maximum-SJNR with a jamming power budget constraint Pj . This optimization problem can be formulated in the following form = s. t. arg min Rz ∈CNj ×Nj where λmax (A, B) is the maximum generalized eigenvalue of matrices A and B. To successfully disturb the intended signal, the jamming signal should be very strong and overwhelm the noise signal. Therefore, for development convenience, we neglect the noise component σn2 INb in (20) and approximate the output SJNR as1 ( ) H SJNRmax ≈ Pa λmax Hab HH (21) ab , Hjb Rz Hjb . Plugging (21) into our original optimization problem (16)(18) and ignoring the constant power Pa , we can (approximately) re-formulate the objective as ( ) H Ropt = arg min λmax Hab HH z ab , Hjb Rz Hjb (22) Rz ∈CNj ×Nj A. Known both Hjb and Hab Ropt z By Theorems 1 and 2, we can re-formulate SJNRmax in (15) as ( ) H 2 −1 SJNRmax = Pa λmax HH Hab ab (Hjb Rz Hjb + σn INb ) ( ) 2 −1 = Pa λmax (Hjb Rz HH Hab HH jb + σn INb ) ab ( ) H 2 = Pa λmax Hab HH ab , Hjb Rz Hjb + σn INb (20) SJNRmax Tr{Rz } ≤ Pj , Rz = RH z , Rz ≽ 0. (16) (17) (18) Note that constraint (18) assures that the optimized matrix Rz is an autocorrelation matrix which should be Hermitian and positive semidefinite. Before developing the jamming signal design, we first recall following two theorems [35]. Theorem 1: Suppose that A ∈ Cm×n and B ∈ Cn×m . Then AB has the same non-zero eigenvalues as BA. Theorem 2: Suppose that A ∈ Cm×m and B ∈ Cm×m , B is invertible. Let λi and qi , i = 1, . . . , m, be the generalized eigenvalues and corresponding eigenvectors of matrices A and B, respectively. Then, we have Aqi = λi Bqi ⇒ B−1 Aqi = λi qi . (19) −1 Therefore, the eigenvalues and eigenvectors of matrix B A are the same as the generalized eigenvalues and eigenvectors of matrices A and B. s. t. Tr{Rz } ≤ Pj , Rz = RH z , Rz (23) ≽ 0. (24) Note that the true SJNRmax in (20) contains the noise component and is always less than the noise-excluded approximated form in (21) with σn > 0. Then, the approximated optimization problem (22)-(24) is essentially aiming to minimize the upper bound of SJNRmax . To solve the optimization problem in (22)-(24), we first consider the case Nj ≥ Nb and transform it into an equivalent form as described in Proposition 1 whose proof is offered in Appendix A. Proposition 1: When Nj ≥ Nb , the optimization problem in (22)-(24) is equivalent to ( ) † H Ropt = arg min λmax H†jb Hab HH z ab (Hjb ) , Rz (25) Rz ∈CNj ×Nj s. t. Tr{Rz } ≤ Pj , Rz = RH z , Rz ≽ 0. (26) (27) The autocorrelation matrix Rz can be constructed by eigendecomposition Rz := Uz Λz UH (28) z where Uz ∈ CNj ×Nj is a unitary matrix and Λz = diag{λz,1 , . . . , λz,Nj } ≽ 0. After obtaining the equivalent form of the optimization problem, the optimal correlation matrix Rz of jamming signal can be designed by following proposition whose proof is offered in Appendix B. Proposition 2: Consider the optimization problem in (25)(27). Let q1 , q2 , . . . , qNj be the eigenvectors of matrix Rh , † H H†jb Hab HH with corresponding positive eigenvalues ab (Hjb ) λh,1 ≥ λh,2 ≥ . . . ≥ λh,Nj . When Nj ≥ Nb , the optimal 1 The noise component will be considered and accounted for SJNR evaluation in our simulation studies. LI et al.: DISRUPTING MIMO COMMUNICATIONS WITH OPTIMAL JAMMING SIGNAL DESIGN correlation matrix Rz of jamming signal that minimizes Bob’s SJNRmax can be constructed by opt opt Ropt := Uopt z z Λz Uz H (29) and the optimal Uopt and Λopt are z z Uopt z = [q1 , . . . , qNj ], (30) Λopt z = opt diag{λopt z,1 , . . . , λz,Nj }, (31) where λopt z,i = Pj λh,i , i = 1, . . . , Nj . ∑Nj d=1 λh,d (32) 5 the degrees-of-freedom of the legitimate transmission from Nb to Nb − Nj , the reliability of the legitimate transmission can still be degraded by the jamming signal to a certain level. Therefore, in the case of Nj < Nb , we still suggest to use the jamming signal design in Proposition 2 which can evenly disturb the jamming subspace and can force Alice and Bob to use the non-jamming subspace only. The jamming e ab ). These findings are performance SJNRmax ≈ SNRmax (H also available for unknown Hab (and unknown Hjb ) cases as long as Nj < Nb and would not be further discussed in the next two subsections. With Ropt optimally designed in (29), the maximum generz ∑Nj λ d=1 h,d alized eigenvalue λmax (Rh , Ropt z ) is minimized at Pj and the maximum output SJNR of Bob is minimized at ∑Nj Pa d=1 λh,d SJNRmax = . (33) Pj The closed-form expression of the average maximum-SJNR, i.e. SJNRmax , EHab ,Hjb {SJNRmax }, is very difficult to be derived for all possible Pj , Nj , Na , and Nb . Therefore, we turn to find out a practical approximated expression which is summarized in Proposition 3. The derivation of Proposition 3 can be found in Appendix C and the accuracy of the approximation expression is verified by simulation studies shown in Section IV. Proposition 3: Consider jamming with known Hjb and Hab , Nj > Nb . When the optimal jamming signal design in Proposition 2 is used, the jamming performance in terms of SJNRmax can be approximated as SJNRmax ≈ Pa Na Nb . Pj (Nj − Nb ) (34) The above results illustrate that SJNRmax is inversely proportional to Nj and the larger Nj can enhance the jamming performance. On the other side, SJNRmax is directly proportional to Na and Nb . The larger Na and/or Nb can mitigate the disturbance due to jamming signal. These findings also coincide with our intuition. While the jamming design for the case Nj ≥ Nb is developed and summarized in Propositions 1-3, now we attempt to investigate the jamming design when Nj < Nb . Let Hjb , H Ujb Σjb Vjb be the SVD of Hjb . When Nj < Nb , we can e jb , 0N ×(N −N ) ]T , Σ e jb is a Nj × Nj diagonal form Σjb , [Σ j j b e jb , Ujb ], U e jb ∈ CNb ×Nj is the jamming matrix, and Ujb , [U Nb ×(Nb −Nj ) subspace, Ujb ∈ C is the non-jamming subspace. If Bob uses a jamming zero forcing filter w ∈ span(Ujb ), then the jamming signal can be totally suppressed. When jamming power is sufficient and jamming signal is sophisticatedly designed to evenly disturb all basis of the jamming subspace, such jamming zero forcing filter is the best choice for Alice and Bob. Then, in such case, the transmission over a jamming channel is statistically equivalent to the transmission e ab ∈ C(Nb −Nj )×Na with a over a jamming free channel H maximum-SNR transmitter and receiver designs as discussed in Section II-A. Since this effort by the jammer can reduce B. Known Hjb but unknown Hab Now we consider the scenario that the jammer has only the knowledge of the channel Hjb between himself and Bob, but the channel Hab between Alice and Bob is unknown. The jamming signal design solution developed in the previous section cannot be adopted due to the lack of access to Bob’s SJNR. Therefore, we turn to evaluate the power of post-filtered jamming signal which can be expressed as σj2 H H H = E{wmaxSJN R Hjb zz Hjb wmaxSJN R } H H = wmaxSJN R Hjb Rz Hjb wmaxSJN R (35) where Hjb Rz HH jb is the autocorrelation matrix of Bob’s received jamming signal Hjb z. Let n = min{Nj , Nb } denote the rank of the received jamming signal autocorrelation matrix Hjb Rz HH jb and let ui , i = 1, . . . , n, denote orthnormal spatial basis of the received jamming signal Hjb z with corresponding power αi . Then, the autocorrelation matrix Hjb Rz HH jb can be expressed in a ∑n H α u u form of Hjb Rz HH = jb i=1 i i i . To suppress the postfiltered jamming signal power σj2 , the maximum-SJNR receive filter wmaxSJN R is adaptively designed to avoid the spatial subspace ui with large αi . However, the filter wmaxSJN R calculated by (12) is unkown by the jammer when Hab is not available. Therefore, by common intuition, the best effort by the jammer is to isotropically/equally maximize the power of each spatial basis ui of Bob’s received jamming signal in such a way to statistically maximize the post-filtered jamming signal power. Specifically, we aim to design Rz to maximize the received jamming powers α1 , . . . , αn among different spatial basis subject to α1 = . . . = αn . The optimal design problem can be formulated as follows Ropt z = s. t. arg max Rz ∈CNj ×Nj Hjb Rz HH jb α =α (36) n ∑ ui uH i , (37) i=1 Tr{Rz } ≤ Pj , Rz = RH z , Rz (38) ≽ 0. (39) The optimal design of Rz for the problem (36)-(39) is based on the waterfilling principle and is described in Proposition 4. The proof is provided in Appendix D. Proposition 4: Consider the optimization problem (36)H (39). Let Hjb , Ujb Σjb Vjb be the SVD of Hjb . Let Nj ×n e Vjb ∈ C , n = min{Nj , Nb }, be a matrix containing the 6 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. X, NO. X, MONTH YEAR n most-left vectors in Vjb . The optimal correlation matrix Rz of jamming signal to the optimization problem (36)-(39) can opt opt H be constructed by Ropt = Uopt and the optimal z z Λz Uz Uopt and Λopt are z z Uopt z Λopt z where λopt z,i e jb , = V opt = diag{λopt z,1 , . . . , λz,n }, = Pj ∑n (40) (41) 1 2 σjb,i 1 2 p=1 σjb,p , i = 1, . . . , n. (42) With∑ the optimal Ropt z , α is maximized to a value αmax = n Pj /( p=1 σ21 ). jb,p When Nj ≥ Nb , constraint (37) can be simplified as Hjb Rz HH jb = αINb . In such case, the optimal jamming signal received by Bob behaves like AWGN and the power of filtered jamming signal ∑n can be always maintained at a maximum level σJ2 = Pj /( p=1 σ21 ) with any receive filter. Therefore, with jb,p a transmit beamformer t and receive filter w, the output SJNR of Bob can be expressed as SJNR = Pa |wH Hab t|2 Pa |wH Hab t|2 ∑n = . 1 σJ2 + σn2 Pj /( p=1 σ2 ) + σn2 (43) jb,p While the power of filtered jamming signal is independent of transmit beamformer t and receive filter w, in an effort to maximize the SJNR, Alice and Bob adopt the SNR-optimum transmit beamformer tmaxSN R = Vab (1) and receive filter wmaxSJR = Uab (1) (See Section II-A). The jammingdegraded maximum-SJNR of Bob is SJNRmax = Pa λab,1 . σJ2 + σn2 (44) With the maximum-SINR result in (44), the average jamming performance SJNRmax is evaluated and shown in Proposition 5 whose proof is provided in Appendix E. Proposition 5: Consider jamming case with known Hjb , but unknown Hab , Nj > Nb . When the optimal jamming design in Propostion 4 is used, then the jamming performance SJNRmax has an expression as SJNRmax = Pa λab,1 Nj −Nb Pj Nb + σn2 . (45) C. Unknown Hjb and unknown Hab If both Hjb is unknown, Bob’s received jamming signal Hjb z cannot be predicted and amended by appropriately adjusting the jammer’s emitted signal z. Therefore, no matter Hab is known or not, the best strategy for jammer is to inject individual jamming signal via each antenna with equal power. The autocorrelation matrix Rz of the jamming signal generated by such approach is Pj IN . Rz = Nj j (46) The jamming performance with this jamming design is evaluated in the following Proposition whose detailed derivation is provided in Appendix F. Proposition 6: Consider jamming case with unknown Hjb , unknown Hab , and Nj > Nb . When the optimal jamming design in (46) is used and the jamming power Pj is sufficient, then the jamming performance SJNRmax has an approximated expression as SJNRmax ≈ Pa λab,1 Nj . (Pj + σn2 )(Nj − Nb ) (47) It is also worth emphasizing that this interference injection strategy will spread the jamming signal over all spatial basis of Bob’s received signal and can cause more difficulty for Alice and Bob to avoid/suppress those multiple-spatial interference. It can provide better jamming performance than using only single jamming antenna or multiple jamming antennas with the same jamming signal. In those jamming approaches, Bob’s received jamming signal has only one spatial subspace and is easy to be avoided and suppressed. The average maximumSJNR of a MIMO transmission in presence of jamming signal from single antenna is shown in the following Proposition whose proof is offered in Appendix G. Proposition 7: When only single antenna is used at the jamming side, the average jamming performance has a lower bound as ( ) 1 Pa λab,1 1− SJNRmax ≥ (48) Pj Nb ( ) 1 = SNRmax 1 − (49) . Nb Compare the findings in Propositions 3, 5, 6, and 7, we can conclude that: i With single-antenna jamming, jamming performance is independent of Pj and increasing jamming power would not enhance the jamming performance. ii With multi-antenna jamming, for all three cases, jamming performance is strictly proportional to Pj and Nj , and inversely proportional to Nb (and Na ). iii Multi-antenna jamming will provide much better jamming performance than single-antenna jamming. These discussions will be verified by intensive simulation results in Section IV. D. Jamming Signal Generation The optimal jamming signal designs developed in the previous three subsections focused on the second-order statistics optimization and provided the optimal jamming signal autoopt correlation matrix Ropt , the jamming signal z z . With Rz can be simply generated by z= Nj ∑ zi λz,i uz,i (50) i=1 where λz,i and uz,i are eigenvalue and eigenvector of Ropt z , zi , i = 1, . . . , Nj , are independent random variables with zeromean and unit-variance. The jammer can emit the jamming LI et al.: DISRUPTING MIMO COMMUNICATIONS WITH OPTIMAL JAMMING SIGNAL DESIGN IV. S IMULATION S TUDIES In the following, we present extensive simulation studies that we obtained from the implementation of the optimal jamming signal designs. In all simulations, the channels Hab and Hjb are modeled as Rayleigh fading with the channel matrix comprising independent and identically distributed samples of a complex Gaussian random variable with zero mean and unit variance. The noise power is fixed at σn2 = 1. We first set the number of antennas for each node as Na = 4, Nb = 4, Nj = 6, respectively. Alice’s transmit power is fixed at Pa = 10dB. The average SJNRmax of Bob over 106 channel realizations is plotted in Fig. 2 as a function of jammer’s power budget Pj which varies from 10dB to 30dB. Three multi-antenna jamming strategies are examined under various assumptions about the knowledge of CSI: i) known both Hjb and Hab ; ii) known Hjb but unknown Hab ; and iii) unknown Hjb and unknown Hab . As a reference, average SJNRmax of Bob in presence of a single-antenna jammer using the same jamming power budget is also included. In addition to the simulation results (blue solid curves), the analytical approximation SJNRmax of each jamming case are plotted (red dash curves). Finally, as the jamming performance benchmark, average SNRmax with no jamming signal is also illustrated in the same figure. It can be observed from Fig. 2 that single-antenna jamming can just merely degrade Alice-to-Bob transmission and increasing the jamming power would not further affect the quality of the legitimate link. Multiple-antenna jamming, all three cases, can offer significant better jamming performance than single-antenna jamming and the jamming performance is proportional to the jamming power. Particularly, the optimal jamming signal design with known Hjb and Hab can keep the SJNRmax of Bob at the lowest values and provides best jamming performance. In addition, the developed analytical jamming performance approximation SJNRmax can almost perfectly predict the jamming performance for the known Hjb cases while the accuracy becomes a little worse but is still satisfactory for the unknown Hjb and Hab case. 20 Known Hjb, known Hab Average SINR (dB) 15 Known Hjb, unknown Hab Unknown Hjb, unknown Hab 10 Single−antenna jamming No jamming 5 0 −5 Known Hjb, known Hab, theo Known Hjb, unknown Hab, theo −10 Unknown Hjb, unknown Hab, theo Single−antenna jamming, theo −15 10 15 20 Jamming power Pj (dB) 25 30 Fig. 2. Average SINRmax of Bob versus jamming power Pj (Na = 4, Nb = 4, Nj = 6, Pa = 10dB). 1 0.9 0.8 Probability of outage signal z via its antennas to maximally interrupt the MIMO communication between Alice and Bob. While SJNR is the second-order statistics metric of the signal, any type of distribution of jamming signal zi can be adopted to pursue optimal jamming performance in terms of SJNR. Gaussian jamming distribution may be the most common choice since it can minimize the mutual information of an additive noise/interference channel with Gaussian inputs [36]. However, the Gaussian distribution is not necessarily optimal on minimizing the mutual information when the inputs are not Gaussian. It is shown in [37] that the best jamming distribution for additive noise channels with binary inputs is a mixture of two lattice probability mass functions. This suggests to further impair the reliable transmission rate by optimizing the jamming signal distribution in conjugation with the autocorrelation matrix Rz (i.e. the second-order statistics). The jamming signal distribution optimization is beyond the scope of this paper and we will consider this problem in future studies. 7 0.7 0.6 0.5 0.4 Known H , known H jb 0.3 ab Known H , unknown H jb ab 0.2 Unknown Hjb, unknown Hab 0.1 Single−antenna jamming 0 10 15 20 Jamming power P (dB) 25 30 j Fig. 3. Probability of outage versus jamming power Pj (Na = 4, Nb = 4, Nj = 6, Pa = 10dB, γ = 3). To further illustrate the impact on Bob’s reception due to the jamming signal, we also investigate the outage probability of Bob, which is defined as the probability that Bob’s instantaneous SJNRmax is less than a certain outage threshold γ = 3. The outage probabilities under different jamming signal designs are plotted in Fig. 3 as a function of jammer’s power budget Pj . The single-antenna jamming always has zero outage probability no matter how large the jamming power is. This means that the single-antenna jamming cannot destroy the MIMO transmission if the anti-jamming scheme is employed. Therefore, multi-antenna jamming is always suggested if the legitimate transmission is over an MIMO channel. Moreover, the multi-antenna jamming design with known Hjb always provide higher outage probability of Bob (i.e. better jamming performance) than unknown Hjb case. In Figs. 4 and 5, we repeat the same study with a larger number of jamming antennas Nj = 8. Comparing to Figs. 2 and 3, we notice the jamming performance can be improved 8 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. X, NO. X, MONTH YEAR 20 20 Known Hjb, known Hab ab Unknown Hjb, unknown Hab 10 5 0 −5 −10 jb ab Known Hjb, unknown Hab Unknown H jb 10 Single−antenna jamming max Single−antenna jamming No jamming Average SJNR Average SINR (dB) jb Known H , known H 15 Known H , unknown H of Bob (dB) 15 Known Hjb, known Hab, theo 5 0 Known Hjb, unknown Hab, theo −5 Unknown Hjb, unknown Hab, theo Single−antenna jamming, theo −15 10 15 20 Jamming power Pj (dB) 25 Fig. 4. Average SINRmax of Bob versus jamming power Pj (Na = 4, Nb = 4, Nj = 8, Pa = 10dB). 4 6 8 10 12 14 Number of jamming antennas Nj 16 18 20 Fig. 6. Average SINRmax of Bob versus number of jamming antennas Nj (Na = 4, Nb = 4, Pa = 10dB, Pj = 16dB). 1 1 0.9 0.9 Known H , known H 0.8 0.8 Known H , unknown H 0.7 0.7 Unknown Hjb, unknown Hab jb Probability of outage Probability of outage 2 30 0.6 0.5 0.4 Known H , known H jb 0.3 ab Known Hjb, unknown Hab 0.2 ab Single−antenna jamming 0.6 0.5 0.4 0.3 0.2 Unknown H , unknown H jb 0.1 ab 0.1 Single−antenna jamming 0 0 10 ab jb 15 20 Jamming power P (dB) 25 30 2 4 6 8 10 12 14 Number of jamming antennas N Fig. 5. Probability of outage versus jamming power Pj (Na = 4, Nb = 4, Nj = 8, Pa = 10dB, γ = 3). by using a larger number of jamming antennas. Then, we turn to evaluate the effect of the number of jamming antennas on the jamming performance. Jammer’s power and Alice’s power are fixed at Pj = 16dB and Pa = 10dB, respectively. The numbers of Alice’s antennas and Bob’s antennas are Na = 4 and Nb = 4, respectively. The jamming performances in terms of average SJNRmax and outage probability are illustrated in Figs. 6 and 7, respectively, as a function of the number of jamming antennas which varies from 1 to 20. Clearly, when Nj < Nb , the jamming signal can be drastically suppressed by Alice and Bob’s optimal anti-jamming beamforming and filtering. When Nj ≥ Nb , the impact of the adversary jammer on the quality of the MIMO communication link becomes notable and significant. The disruption due to the jamming signal will be more severe when the number of jamming antennas increases. Finally, we investigate the jamming performance with different number of Bob’s antennas. Jammer’s power and Alice’s 16 18 20 j j Fig. 7. Probability of outage versus number of jamming antennas Nj (Na = 4, Nb = 4, Pa = 10dB, Pj = 16dB, γ = 3). power are fixed at Pj = 30dB and Pa = 10dB, respectively. The numbers of Alice’s antennas and jammer’s antennas are Na = 4 and Nj = 8, respectively. The jamming performances in terms of average SJNRmax and outage probability are illustrated in Figs. 8 and 9, respectively, as a function the number of Bob’s antennas which varies from 2 to 16. Jammer with optimal signal design can successfully disrupt the MIMO communication link as long as Nj ≥ Nb . When Nj < Nb , all three multi-antenna jamming strategies have the same performance. This simulation result verifies our discussion in the end of Section III-A. V. C ONCLUSIONS We investigated the problem of optimal jamming signal design to intelligently attack a MIMO communication system. With a given jamming power budget constraint, our goal was to optimize the jamming signal for the jammer to minimize receiver’s maximum-SJNR. The optimal jamming signal designs LI et al.: DISRUPTING MIMO COMMUNICATIONS WITH OPTIMAL JAMMING SIGNAL DESIGN When Nj = Nb , Hjb is an invertible square matrix and † H−1 jb = Hjb . Then, the generalized eigen decomposition (51) can be re-formulated as 25 Average SJNRmax of Bob (dB) 20 † H H†jb Hab HH ab (Hjb ) ui = λi Rz ui , i = 1, . . . , Nb , 15 (52) where ui , HH jb qi . The equivalence is proven for the case Nj = Nb . Now, we consider the case that Nj > Nb . Let Hjb , H Ujb Σjb Vjb be the SVD of Hjb . Then, we can rewrite (51) as 10 5 0 −5 Known Hjb, known Hab Known Hjb, unknown Hab −10 Unknown H jb −15 Single−antenna jamming 2 4 6 8 10 12 Number of Bob’s antennas Nb 14 16 Fig. 8. Average SINRmax of Bob versus number of Bob’s antennas Nb (Na = 4, Nj = 8, Pa = 10dB, Pj = 30dB). 1 Known H , known H jb ab Known Hjb, unknown Hab 0.9 Unknown Hjb, unknown Hab 0.8 Probability of outage 9 Single−antenna jamming 0.7 0.6 0.5 H T H Hab HH ab qi = λi Ujb Σjb Vjb Rz Vjb Σjb Ujb qi , (53) H H T ⇒ UH jb Hab Hab Ujb pi = λi Σjb Vjb Rz Vjb Σjb pi , (54) where pi , UH = 1, . . . , Nb . Form Σjb , jb qi , i e e [Σjb , 0Nb ×(Nj −Nb ) ], Σjb is an Nb × Nb diagonal matrix, and e jb , Vjb ], V e jb ∈ CNj ×Nb , Vjb ∈ CNj ×(Nj −Nb ) . Vjb , [V Then, we have H Σjb Vjb Rz Vjb ΣTjb ][ ] [ ][ e H Rz V e jb e jb V 0Nb ×(Nj −Nb ) Σ jb e = Σjb , 0 . 0 0(Nj −Nb )×(Nj −Nb ) 0(Nj −Nb )×(Nb ) (55) Note that the component of Rz in subspace Vjb always results 0 in (55) and will not affect the eigenvalues. Thus, H = the optimal design Ropt must have property that Vjb Ropt z z 0(Nj −Nb )×Nb . After plugging (55) into (54), we obtain 0.4 H UH jb Hab Hab Ujb pi = [ ][ eH e jb Vjb Rz V e λi Σjb , 0 0 0.3 0.2 0.1 ][ 0 0 e jb Σ 0 ] pi . (56) Left multiply Σ†jb on both sides of (56), we obtain 0 2 4 6 8 10 12 Number of Bob’s antennas N 14 16 b Fig. 9. Probability of outage versus number of Bob’s antennas Nb (Na = 4, Nj = 8, Pa = 10dB, Pj = 30dB, γ = 3). were developed in different cases according to the availability of CSI for the jammer. The analytical approximations of the jamming performance in terms of the average maximumSJNR were also provided. Extensive simulation studies illustrated the importance of using multiple jamming antennas and demonstrated the benefits of the designed jamming signal on degrading the performances of MIMO transmission in terms of maximum-SJNR as well as outage probability. As a natural next step in future work, we will investigate the case of imperfect CSI of the jamming channel which is more practical in realistic systems. We plan to study the performance degradation due to the presence of imperfect CSI and develop robust jamming signal designs. A PPENDIX A - P ROOF OF P ROPOSITION 1 Let λi and qi , i = 1, . . . , Nb , be the generalized eigenvalue H and eigenvector pairs of matrices Hab HH ab and Hjb Rz Hjb : H Hab HH ab qi = λi Hjb Rz Hjb qi , i = 1, . . . , Nb . (51) H Σ†jb UH jb Hab Hab Ujb pi = [ H e Rz V e jb V jb λi 0 ⇒ † H H Σ†jb UH jb Hab Hab Ujb (Σjb ) gi 0 0 ][ [ = λi e jb Σ 0 ] pi e H Rz V e jb V jb 0 (57) 0 0 ] gi (58) H where gi , ΣTjb pi . Plugging the properties Vjb Rz Vjb = 0, H e jb = 0, and V e H Rz Vjb = 0 for the optimal Rz into Vjb Rz V jb (58), we have † H H Σ†jb UH jb Hab Hab Ujb (Σjb ) gi = [ ] e jb V e H Rz Vjb e H Rz V V jb jb λi gi (59) H e jb VH Rz Vjb V Rz V jb jb † H H H ⇒ Σ†jb UH jb Hab Hab Ujb (Σjb ) gi = λi Vjb Rz Vjb gi (60) † H H H ⇒ Vjb Σ†jb UH jb Hab Hab Ujb (Σjb ) Vjb vi = λi Rz vi (61) H † where vi , Vjb gi . Since H†jb = Vjb Σjb Ujb , we finally obtain † H H†jb Hab HH (62) ab (Hjb ) vi = λi Rz vi . 10 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. X, NO. X, MONTH YEAR A PPENDIX B -P ROOF OF P ROPOSITION 2 The min-max optimization problem in (25) is equivalent to H hs minimize the maximum value of Rayleigh ratio ssH R for Rz s H hs any s ∈ CNj , i.e. minimize max{ ssH R , s ∈ CNj }. Let Rz s A PPENDIX D - P ROOF OF P ROPOSITION 4 ∑Nb When Nj ≥ Nb , α i=1 ui uH i = αINb . Then, with Hjb , H Ujb Σjb Vjb , the constrain (37) can be further rewritten as H H Ujb Σjb Vjb Rz Vjb ΣH jb Ujb = αINb Rz ∈CNj ×Nj si ∈ CNj , i = 1, . . . , Nj , be orthonormal vectors and s1 is H hs the vector which has maximum value of ssH R for a given Rz s H Rz . For convenience, we define αi , si Rh si , βi , sH i Rz si , sH R s and siH Rhz sii = αβii , i = 1, . . . , Nj . Then, we can obtain αβ11 ≥ i ∑Nj ∑Nj αNj ∑Nj α2 i=1 αi = i=1 λh,i , and i=1 βi = β2 ≥ . . . ≥ βNj , ∑Nj i=1 λz,i = Tr{Rz }. If the design in Proposition 2 is not optimal, then there exist a Rz and corresponding s1 such that ∑Nj d=1 λh,d Pj > sH 1 Rh s1 , sH 1 Rz s1 (63) † H ⇒ Rz = αVjb Σ†jb (ΣH jb ) Vjb . (68) 1 † Note that Σ†jb (ΣH , . . . , σ2 1 , 0, . . . , 0} and 2 jb ) = Diag{ σjb,1 jb,Nb then rewrite (68) as e jb Diag{ 1 , . . . , 1 }V eH Rz = α V (69) jb 2 2 σjb,1 σjb,N b e jb ∈ CNj ×Nb is a matrix containing the Nb most-left where V vectors in Vjb . Therefore, the design in Proposition 4 satisfies this constrain and is also optimal based on the waterfilling principle. The proof for the case Nj < Nb is similar and omitted. A PPENDIX E - P ROOF OF P ROPOSITION 5 ∑Nj ⇒ i=1 Pj λh,i > αNj α1 α2 ≥ ≥ ... ≥ . β1 β2 βNj (64) ∑Nj ∑Nj To satisfy (64) with i=1 αi = i=1 λh,i , we need to have ∑Nj β = Tr{R } > P which is a contradiction with the i z j i=1 ∑Nj = λh,d must be the constrain = Tr{Rz } ≤ Pj . Therefore, Pj optimal result of the optimization problem and the design in Proposition 2 is optimal. d=1 With maximum-SJNR (44), the jamming performance can be expressed as } { Pa λ2ab,1 SJNRmax = EHab ,Hjb σJ2 + σn2 A PPENDIX C - D ERIVATION OF P ROPOSITION 3 Recall that λab,1 , EHab {λ { ab,1 } (see Section } II-A) and ∑Nb 1 H −1 b EHjb { p=1 σ2 } = EHjb Tr{(Hjb Hjb ) } = NjN−N b jb,p (by random matrix theory [32]). Then, we apply these results into (70) and obtain SJNRmax = Pa † H EHab ,Hjb {Tr{H†jb Hab HH ab (Hjb ) }}.(65) Pj Loosely speaking, if we can consider H†jb as a random matrix, then H†jb has elements with variance Var(H†jb ) = † Nb Nb 1 (Nj −Nb )Nj Nb = (Nj −Nb )Nj and Var(Hjb Hab ) = (Nj −Nb )Nj . With these results, we have { } † H EHab ,Hjb Tr{H†jb Hab HH ab (Hjb ) } = Nb Na Nb Na Nj = . (Nj − Nb )Nj (Nj − Nb ) (66) Applying (66) into (65), we finally have SJNRmax = Pa Na Nb . Pj (Nj − Nb ) Pa λab,1 Nj −Nb Pj Nb + σn2 . (71) { } H By random matrix theory [32], E Tr{(Hjb HH ) } = jb Nb Nj −Nb . (70) jb,p With the maximum-SJNR in (33), the SJNRmax is } { ∑Nj λh,d Pa d=1 SJNRmax = EHab ,Hjb Pj = Pa EHab {λ2ab,1 } . ∑Nb 1 Pj /EHjb { p=1 } + σn2 σ2 A PPENDIX F - D ERIVATION OF P ROPOSITION 6 When jamming channel is Hjb and the jamming signal P has Rz = Njj INj , the maximized output SJNR of the filter wmaxSJNR is Pj 2 −1 SJNRmax = Pa tH HH Hjb HH Hab t. (72) ab ( jb + σn INb ) Nj If the jamming power Pj is sufficient large and Pj ≫ σn2 , the maximum-SJNR in (72) can be approximated in a form of SJNRmax = (73) If Alice and Bob adopt the transmit beamformer t = Vab (1) which is optimal for the jamming-absent case but not for jamming-present case, SJNRmax has a lower bound: Pa Nj H −1 Vab (1)H HH Hab Vab (1) ab (Hjb Hjb ) Pj H Uab Σab Vab Vab (1) Pa λab,1 Nj −1 = Uab (1)H (Hjb HH Uab (1).(74) jb ) Pj SJNRmax ≥ (67) Pa Nj H H −1 t Hab (Hjb HH Hab t. jb ) Pj LI et al.: DISRUPTING MIMO COMMUNICATIONS WITH OPTIMAL JAMMING SIGNAL DESIGN Recall that Uab (1) and Hjb are independent and select a transmit beamformer t = Vab (1) which is optimal for no jamming case but not for jamming case, we can obtain SJNR performance lower bound as −1 (Hjb HH Uab (1} jb ) EHab ,Hjb {Uab (1) 1 = E{Tr{Hjb HH jb }} Nb Nb 1 1 = = . Nb Nj − Nb Nj − Nb H Pa λab,1 Nj . Pj (Nj − Nb ) (75) (76) When Nj → ∞, we have Hjb HH jb ≈ Nj INb and SJNRmax Pa λab,1 Pa H (86) − 2 Vab (1)H HH ab vv Hab Vab (1) σn2 σn ) Pa λab,1 ( = 1 − |Uab (1)H v|2 . (87) 2 σn SJNRmax ≥ Applying (75) into (74), we obtain an approximation SJNRmax ≥ 11 2 −1 = Pa λmax (HH Hab ) ab (Pj INb + σn INb ) Pa λab,1 = (77) Pj + σn2 Now we turn to evaluate the statistics of random variable |Uab (1)H v|2 . Recall that Uab (1) is the basis of channel matrix Hab and v , hjb /∥hjb ∥. Since Hab and hjb are two independent Rayleigh fading channels, Uab (1) and v are two independent normalized random vectors of length Nb and have statistic property: EHab ,hjb {|Uab (1)H v|2 } = and consequently SJNRmax (Nj → ∞) = Pa λab,1 . Pj + σn2 (78) With the asymptotic performance bound (78), we reformulate the approximation expression in (76) in order to compensate the noise term which is ignored during the derivation: Pa λab,1 Nj SJNRmax ≈ . (79) (Pj + σn2 )(Nj − Nb ) Appendix G - Proof of Proposition 7 When jamming signal is present but only single antenna is used at the jamming side, the signal vector y ∈ CNb received by Bob can be expressed as √ y = Pa sHab t + hjb z + n (80) where hjb ∈ CNb is the channel vector between jammer and Bob, E{|z|2 } = Pj . Similarly, the maximized output SJNR of the filter wmaxSJNR is −1 2 H Hab t. SJNRmax = Pa tH HH ab (Pj hjb hjb + σn INb ) (81) H e Rewrite Pj hjb hH where Pej = Pj ∥hjb ∥2 and v = jb as Pj vv hjb /∥hjb ∥. By matrix inversion lemma, we have −1 2 (Pj hjb hH = (Pej vvH + σn2 INb )−1 (82) jb + σn INb ) e 1 Pj = 2 INb − vvH . (83) 2 2 σn σn (σn + Pej ) Applying (83) into (81), we can obtain Pa H H Pa Pej H tH HH t Hab Hab t− ab vv Hab t. 2 σn σn2 (σn2 + Pej ) (84) e P In jamming scenario, Pej ≫ σn2 and (σ2 +jPe ) ≈ 1. Then, (84) j n can be rewritten as Pa H H H Pa SJNRmax ≈ 2 tH HH ab Hab t − 2 t Hab vv Hab t. (85) σn σn SJNRmax = Clearly, the optimal transmit beamformer t is the largest H 2 −1 eigenvector of matrix HH Hab . If we ab (Pj hjb hjb + σn INb ) 1 . Nb Finally, we apply the results (88) into (87) and obtain ( ) Pa λab,1 1 SJNRmax ≥ 1 − . σn2 Nb (88) (89) R EFERENCES [1] R. A. Poisel, Modern Communications Jamming Principles and Techniques. Boston, MA: Artech House Publishers, 2006. [2] T. Basar, “The gaussian test channel with an intelligent jammer,” IEEE Trans. Inf. Theory, vol. 29, no. 1, pp. 152-157, Jan. 1983. [3] T. Basar and Y.-W. Wu, “A complete characterization of minimax and maximin encoder-decoder policies for communication channels with incomplete statistical description,” IEEE Trans. Inf. Theory, vol. 31, no. 4, pp. 482-489, July 1985. [4] R. J. McEliece, Communication in the Presence of Jamming - An information theoretic approach, pp. 127C165, Springer-Verlag, Wien, 1983. [5] M. Medard, “Capacity of correlated jamming channels,” in Proc. 35th Allerton Conf., Monticello, IL, Oct. 1997, pp. 1043-1052. [6] A. Kashyap, T. Basar, and R. Srikant, “Correlated jamming on MIMO Gaussian fading channels,” IEEE Trans. Inf. Theory, vol. 50, no. 9, pp. 2119-2123, Sept. 2004. [7] T. Wang and G. B. Giannakis, “Mutual information jammer-relay games,” IEEE Trans. Inf. Forens. Security, vol. 3, no. 2, pp. 290-303, June 2008. [8] S. Shafiee and S. Ulukus, “Capacity of multiple access channels with correlated jamming,” in Proc. IEEE Military Commun. Conf. (MILCOM), Atlantic city, NJ, Oct. 2005, vol. 1, pp. 218-224. [9] S. Shafiee and S. Ulukus, “Mutual information games in multi-user channels with correlated jamming,” IEEE Trans. Inf. Theory, vol. 55, no. 10, pp. 4598-4607, Oct. 2009. [10] S. N. Diggavi and T. M. Cover, “The worst additive noise under a covariance constraint,” IEEE Trans. Inf. Theory, vol. 47, no. 7, pp. 30723081, Nov. 2001. [11] R. Miller and W. Trappe, “On the vulnerabilities of CSI in MIMO wireless communication systems,” IEEE Trans. Mobile Computing, vol. 11, no. 8, pp. 1386-1393, Aug. 2012. [12] R. Miller and W. Trappe, “Subverting MIMO wireless systems by jamming the channel estimation procedure,” in Proc. ACM Conf. Wireless Network Security, Hoboken, NJ, March 2010, pp. 19-24. [13] X. Zhou, D. Niyato, and A. Hjørungnes, “Optimizing training-based transmission against smart jamming, IEEE Trans. Veh. Technol., vol. 60, no. 6, pp. 2644-655, Jul. 2011. [14] H. Pezeshki, X. Zhou, and B. Maham, “Jamming energy allocation in training-based multiple access systems,” IEEE Commun. Lett., vol. 17, no. 6, pp. 1140-1143, June 2013. [15] A. Bayesteh, M. Ansari, and A. K. Khandani, “Effect of jamming on the capacity of MIMO channels,” in Proc. 42nd Allerton Conf., Monticello, IL, Oct. 2004, pp. 401-410. 12 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. X, NO. X, MONTH YEAR [16] M. H. Brady, M. Mohseni, and J. M. Cioffi, “Spatially-correlated jamming in Gaussian multiple access and broadcast channels,” in Proc. Conf. Inf., Science, and Systems (CISS), Princeton, NY, March 2006, pp. 1635-1639. [17] M. R. D. Rodrigues and G. Ramos, “On multiple-input multiple-output Gaussian channels with arbitrary inputs subject to jamming,” in Proc. IEEE Intern. Symp. Inf. Theory (ISIT), Seoul, Korea, June 2009, pp. 2512-2516. [18] E. A. Jorswieck, H. Boche, and M. Weckerle, “Optimal transmitter and jamming strategies in Gaussian MIMO channels,” in Proc. Veh. Technol. Conf. (VTC), Stockholm, Sweden, May 2005, vol. 2, pp. 978-982. [19] S. Goel and R. Negi, “Guaranteeing secrecy using artificial noise,” IEEE Trans. Wireless Commun., vol. 7, no. 6, pp. 2180-2189, June 2008. [20] A. Mukherjee and A. L. Swindlehurst, “Robust beamforming for secrecy in MIMO wiretap channels with imperfect CSI,” IEEE Trans. Signal Process., vol. 59, no. 1, pp. 351-361, Jan. 2011. [21] R. Zhang, L. Song, Z. Han, and B. Jiao, “Physical layer security for two-way untrusted relaying with friendly jammers,” IEEE Trans. Veh. Technol., vol. 61, no. 8, pp. 3693-3704, Oct. 2012. [22] I. Stanojev and A. Yener “Improving secrecy rate via spectrum leasing for friendly jamming,” IEEE Trans. Wireless Commun., vol. 12, no. 1, pp. 134-145, Jan. 2013 [23] G. Zheng, L.-C. Choo, and K.-K. Wong, “Optimal cooperative jamming to enhance physical layer security using relays,” IEEE Trans. Signal Process., vol. 59, no. 3, pp.1317-1322, Mar. 2011. [24] H.-M. Wang, Q. Yin, and X.-G. Xia, “Distributed Beamforming for Physical-Layer Security of Two-Way Relay Networks,” IEEE Trans. Signal Process., vol. 61. no. 5. pp 3532-3545. Jul. 2012. [25] H.-M. Wang, M. Luo, Q. Yin, and X.-G. Xia, “Hybrid cooperative beamforming and jamming for physical-layer security of two-way relay networks,” IEEE Trans. Inf. Forens. Security, vol. 8, no. 12, pp. 20072020, Dec. 2013. [26] H.-M. Wang, M. Luo, X.-G. Xia, and Q. Yin, “Joint cooperative beamforming and jamming to secure AF relay systems with individual power constraint and no eavesdropper’s CSI,” IEEE Signal Process. Lett., vol. 20, no. 1, pp. 39-42, Jan. 2013. [27] J. Zhu, R. Schober, and V. K. Bhargava, “Secure transmission in multicell massive MIMO systems”, Trans. Wireless Commun., vol. 13, no. 9, pp. 4766-4781, Sept. 2014. [28] J. Zhu, R. Schober, and V. K. Bhargava, “Linear precoding of data and artificial noise in secure massive MIMO systems”, submitted to Trans. Wireless Commun. Available: http://arxiv.org/abs/1505.00330. [29] F. Zhu, F. Gao, M. Yao, and H. Zou, “Joint information- and jammingbeamforming for physical layer security with full duplex base station,” IEEE Transa. Signal Process.vol. 62, no. 24, pp. 6391-6401, Dec. 2014. [30] M. Li, S. Kundu, D. A. Pados, and S. N. Batalama, “Secure waveforms for SISO channels,” in Proc. IEEE Int. Conf. Acoustics, Speech, and Signal Process. (ICASSP), Vancouver, Canada, May 2013, pp. 47134717. [31] M. Li, S. Kundu, D. A. Pados, and S. N. Batalama, “Waveform design for secure SISO transmissions and multicasting,” IEEE J. Sel. Areas Commun., vol. 31, no. 9, pp. 1864-1874, Sept. 2013. [32] A. M. Tulino and S, Verdu, “Random matrix theory and wireless communications,” Journal Communications and Information Theory, vol. 1, no 1, pp. 1-182, June 2004. [33] A. Edelman, “Eigenvalues and condition numbers of random matrices” SIAM. J. Matrix Anal. & Appl., vol. 9, no. 4, pp. 543-560, Oct. 1988. [34] D. G. Manolakis, V. K. Ingle, and S. M. Kogon. Statistical and adaptive signal processing: Spectral estimation, signal modeling, adaptive filtering and array processing. Boston, MA: McGraw-Hill, 2000. [35] R. A. Horn and C. R. Johnson. Matrix Analysis. NY, USA: Cambridge University Press, 1985. [36] T. M. Cover and J. A. Thomas. Elements of Information Theory, 2nd ed. Hoboken, NJ: John Wiley, 2006. [37] S. Shamai and S. Verd¨u, “Worst-case power-constrained noise for binary input channels,” IEEE Trans. Inf. Theory, vol. 38, no. 5, pp. 1494-1511, Sept. 1992. Qian Liu (S’09, M’14) received B.S. and M.S. degrees from Dalian University of Technology, China, in 2006 and 2009, and Ph.D. degree from State University of New York at Buffalo (SUNYBuffalo) in 2013. She is now a postdoctoral fellow in Ubiquitous Multimedia Lab at SUNY-Buffalo. She received the Best Paper Runnerup award in ICME 2012, and the Best Student Paper Finalist in ISCAS 2011. Her current research interests include multimedia transmission over MIMO systems, IEEE 802.11 wireless networks and LTE networks. Ming Li (S’05, M’11) received the M.S. and Ph.D. degrees in electrical engineering from the State University of New York at Buffalo, Buffalo, in 2005 and 2010, respectively. From Jan. 2011 to Aug. 2013, he was a Post-Doctoral Research Associate with the Signals, Communications, and Networking Research Group, Department of Electrical Engineering, State University of New York at Buffalo. From Aug. 2013 to June 2014, Dr. Li joined Qualcomm Technologies Inc. as a Senior Engineer. Since June 2014, he has been with the School of Information and Communication Engineering, Dalian University of Technology, Dalian, China, where he is presently an Associate Professor. His research interests are in the general areas of communication theory and signal processing with applications to interference channels and signal waveform design, secure wireless communications, cognitive radios and networks, data hiding and steganography, and compressed sensing. Xiangwei Kong (M’06) received the B.E. and M.S. degrees from Harbin Shipbuilding Engineering Institute, Harbin, China, in 1985 and 1988, respectively, and the Ph.D. degree from Dalian University of Technology, Dalian, China, in 2003. She is a Professor with the School of Information and Communication Engineering and the vice director of the Information Security Research Center, Dalian University of Technology, Dalian, China. She is also the vice director of the Multimedia Security Session of the Chinese Institute of Electronics. Her research interests include multimedia security and forensics, digital image processing, and pattern recognition. Nan Zhao (S’08-M’11) is currently an associate professor in the School of Information and Communication Engineering at Dalian University of Technology, China. He received the B.S. degree in electronics and information engineering in 2005, the M.E. degree in signal and information processing in 2007, and the Ph.D. degree in information and communication engineering in 2011, from Harbin Institute of Technology, Harbin, China. From Jun. 2011 to Jun. 2013, Nan Zhao did postdoctoral research in Dalian University of Technology, Dalian, China. His recent research interests include Interference Alignment, Physical Layer Security, Cognitive Radio, Wireless Power Transfer, and Optical Communications. He has published nearly 50 papers in refereed journals and international conferences. Dr. Zhao is a member of the IEEE. He serves as an Area Editor of AEUInternational Journal of Electronics and Communications, and an Editor of Ad Hoc & Sensor Wireless Networks. Additionally, he served as a technical program committee (TPC) member for many interferences, e.g., Globecom, VTC, WCSP. He is also a peer-reviewer for a number of international journals, as IEEE Trans. Commun., IEEE Trans. Wireless Commun., IEEE Trans. Veh. Tech., etc.

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