Secure Transmission with Multiple Antennas: The MIMOME Channel

Transcription

1 1 Secure Transmission with Multiple Antennas: The MIMOME Channel Ashish Khisti and Gregory Wornell. Abstract The Gaussian wiretap channel model is studied when there are multiple antennas at the sender, the receiver and the eavesdropper and the channel matrices are fixed and known to all the terminals. The secrecy capacity is characterized using a Sato type upper bound. A computable characterization of the capacity is provided. The secrecy capacity is studied in several interesting regimes. The high SNR secrecy capacity is attained by simultaneously diagonalizing the channel matrices using the generalized singular value decomposition and independently coding across the resulting parallel channels. An explicit closed form solution in terms of the generalized singular values of the channel matrices is provided in this regime. In addition to the capacity achieving scheme we also study achievable rates from a synthetic noise transmission strategy. Interestingly this rate can also be expressed in terms of the generalized singular values in the high SNR regime. Necessary and sufficient conditions for the secrecy capacity to be zero are provided and further studied when the entries in the channel matrices are sampled i.i.d. CN0, 1, and the dimensions of the matrices go to infinity. In this regime, we show rather interestingly that an asymmetric allocation of dividing the antennas between the sender and receiver in the ratio 2 : 1 maximizes the number of antennas required by the eavesdropper for the secrecy capacity to be zero. I. INTRODUCTION Multiple antennas are a valuable resource in wireless communications. Recently there has been a significant activity in exploring both the theoretical and practical aspects of wireless systems with multiple antennas. In this work we explore the role of multiple antennas for physical layer security, which is an emerging area of interest. The wiretap channel [2] is an information theoretic model for physical layer security. The setup has three terminals one sender, one receiver and one eavesdropper. The goal is to exploit the structure of the underlying broadcast channel to transmit a message reliably to the intended receiver, while leaking asymptotically no information to the eavesdropper. A single letter characterization of the secrecy capacity, when the underlying channel is a discrete memoryless broadcast channel, has been obtained by Csiszár and Körner [3]. An explicit solution for the scalar Gaussian case is obtained in [4], where the optimality of Gausian codebooks is established. In this paper we consider the case where all the three terminals have multiple antennas and naturally refer to it as multiple input, multiple output, multiple eavesdropper MI- MOME channel. In this setup we assume that the channel This work was supported in part by NSF under Grant No. CCF Part of the material in this work appeared in Allerton Conference on Communications, Control and Signal Processing, 2007 [1]. The authors are with the Dept. EECS, MIT, Cambridge, MA, {khisti,gww}@mit.edu matrices are fixed and known to all the three terminals. While the assumption that the eavesdropper s channel is known to both the sender and the receiver is obviously a strong assumption, we remark in advance that our solution provides ultimate limits on secure transmission with multiple antennas and could be a starting point for other formulations. The problem of secure communication with multiple antennas has been extensively studied recently. We summarize some of the literature here. The case when the channel matrices of intended receiver and eavesdropper are square and diagonal follows from the results in [5] [8] that consider secure transmission over fading channels. These works establish that for the special case of independent parallel channels, it suffices to use independent codebooks across the channels. The MIMOME channel is a non-degraded broadcast channel to which the Csiszár and Körner capacity expression [3] applies in principle. Nevertheless, computing the explicit capacity via [3] appears difficult as already noted in e.g., [9] [13]. To our knowledge, the first computable upper bound for the secrecy capacity of the multi-antenna wiretap channel has been reported in our earlier works [14], [15] and has been used to establish the secrecy capacity when the intended receiver has a single antenna MISOME case. This approach involves revealing the output of the eavesdropper s channel to the legitimate receiver to create a fictitious degraded broadcast channel and results in a minimax expression for the upper bound, analogous to the Sato technique used to upper bound the sum capacity of the multi-antenna broadcast channel see e.g., [16]. This minimax upper bound is analytically simplified for the MISOME case and a closed form expression for the secrecy capacity is obtained in [14], [15]. In addition, a number of useful insights are developed into the behavior of the secrecy capacity. In the high-signal-to-noise-ratio SNR regime, a simple masked beamforming scheme, first studied in [10], is shown to be near optimal. Also the scaling behavior of the secrecy capacity in the limit of many antennas is studied. We note that this upper bounding approach has been independently conceived by Ulukus et. al. [17] and further applied to the case [18]. Subsequently, this minimax upper bound has been shown to be tight for the MIMOME case in independent works in [1] and [19] see also [20]. Both works use the minimax upper bound in [15] as a starting point and work with the optimality conditions to establish that the saddle value is achievable with Gaussian inputs in the Csiszár and Körner expression. Subsequently, T. Liu and S. Shamai [21] use a different approach, based on channel enhancement techniques of [22], to also establish the secrecy capacity. In our opinion these two approaches shed complimentary insights into the problem. The minimax

2 upper bounding approach in [1], [19] provides a computable characterization for the capacity expression and identifies a hidden convexity in optimizing the Csiszár and Körner expression with Gaussian inputs, whereas the techniques in [21] do not provide a computable characterization of the capacity. On the other hand [21] establishes the capacity for any covariance constraint on the input distribution, not just the sum power constraint as considered in [1], [19]. Finally, the diversity-multiplexing tradeoff of the multi-antenna wiretap channel has been recently studied in [23]. This paper is organized as follows. We describe the channel model in section II and the main results of the paper are summarized in III. The proof of the secrecy capacity of the MIMOME channel is presented in section IV. We further study the capacity in the high signal-to-noise-ratio SNR regime in Section V. In the high SNR regime, a capacity achieving scheme involves simultaneously diagonalizing both the channel matrices, using the generalized singular value decomposition, and using independent codebooks across the resulting parallel channels. In addition to the capacity achieving scheme, a synthetic noise transmission scheme is also analyzed. This scheme is semi-blind it selects the transmit directions only based on the channel of the legitimate receiver, but needs the knowledge of the eavesdropper s channel for selecting the rate. Interestingly, the high SNR rate of this scheme can also be expressed in terms of the generalized singular values of the channel matrices. In section VI, necessary and sufficient conditions on the channel matrices for the secrecy capacity to be zero are stated. For i.i.d. Rayleigh fading some scaling laws in the limit of many antennas are also developed. The conclusions are provided in section VII II. CHANNEL MODEL We denote the number of antennas at the sender, the receiver and the eavesdropper by n t, n r and n e respectively. y r t = H r xt + z r t y e t = H e xt + z e t, where H r C nr nt and H e C ne nt are channel matrices associated with the receiver and the eavesdropper. The channel matrices are fixed for the entire transmission period and known to all the three terminals. The additive noise z r t and z e t are circularly-symmetric and complexvalued Gaussian random variables. The input satisfies a power constraint E [ 1 n n t=1 xt 2] P. A rate R is achievable if there exists a sequence of length n codes, such that the error probability at the intended receiver and 1 n Iw;yn e both approach zero as n. The secrecy capacity is the supremum of all achievable rates. A. Notation Bold upper and lower case characters are used for matrices and vectors, respectively. Random variables are distinguished from realizations by the use of san-serif fonts for the former and seriffed fonts for the latter. And we generally reserve the symbols I for mutual information, H for entropy, and 1 h for differential entropy. All logarithms are base-2 unless otherwise indicated. The set of all n-dimensional complex-valued vectors is denoted by C n, and the set of m n-dimensional matrices is denoted using C m n. Matrix transposition is denoted using the superscript T, and the Hermitian i.e., conjugate transpose of a matrix is denoted using the superscript, the Moore-Penrose pseudo-inverse is denoted by, while the projection matrix onto the null space is denoted by. Moreover, Null denotes the null space of its matrix argument, and tr and det denote the trace and determinant of a matrix, respectively. The notation A 0 means that A is a positive semidefinite matrix. I denotes the identity matrix and 0 denotes the matrix with all zeros. The dimensions of these matrices will be suppressed and will be clear from the context. III. SUMMARY OF MAIN RESULTS In this section we summarize the main results in this paper. The proof A. MIMOME Secrecy Capacity The secrecy capacity of the MIMOME channel is stated in the theorem below. Theorem 1: The secrecy capacity of the MIMOME wiretap channel is C = min K Φ K Φ max R +,K Φ, 2 where R +,K Φ = Ix;y r y e with x CN0, and { } 0, tr P and where [z r,z e ] CN0,K Φ, with { [ ] } Inr Φ K Φ K Φ K Φ =, K Φ 0 Φ I ne { [ Inr Φ = K Φ K Φ = Φ I ne, 3 ] }, σ max Φ 1. Furthermore, the minimax problem in 2 has a saddle point solution, K Φ and the secrecy capacity can also be expressed as, 4 C = R +, K Φ = log deti + H r H r deti + H e KP H e. 5 1 Connection with Csiszár and Körner Capacity: A characterization of the secrecy capacity for the non-degraded discrete memoryless broadcast channel p yr,y e x is provided by Csiszár and Körner [3], C = max p u,p x u Iu; y r Iu; y e, 6 where u is an auxiliary random variable over a certain alphabet with bounded cardinality that satisfies u x y r, y e. As remarked in [3], the secrecy capacity 6 can be extended in principle to incorporate continuous-valued

3 inputs. However, directly identifying the optimal u for the MIMOME case is not straightforward. Theorem 1 indirectly establishes an optimal choice of u in 6. Suppose that, K Φ is a saddle point solution to the minimax problem in 2. From 5 we have where R +, K Φ = R, 7 R log deti + H r H r deti + H e KP H e is the achievable rate obtained by evaluating 6 for u = x CN0,. This choice of p u, p x u thus maximizes 6. Furthermore note that argmax log deti + H r H r deti + H e H e where the set is defined in 3. Unlike the minimax problem 2 the maximization problem 8 is not a convex optimization problem since the objective function is not a concave function of. Even if one verifies that KP satisfies the optimality conditions associated with 8, this will only establish that KP is a locally optimal solution. The capacity expression 2 provides a convex reformulation of 8 and establishes that is a globally optimal solution in Structure of the optimal solution: The saddle point solution, K Φ satisfies a certain necessary condition that admits an intuitive interpretation. In particular, in the proof of Theorem 1, we show the following: Let S be any matrix that has a full column rank matrix and satisfies = SS and let Φ be the cross-covariance matrix between the noise random variables in 2, c.f. 4, then 8 H e S = Φ H r S. 9 Note that Φ is a contraction matrix i.e., all its singular values are less than or equal to unity. The column space of S is the subspace in which the sender transmits information. So 9 states that no information is transmitted along any direction where the eavesdropper observes a stronger signal than the intended receiver. The effective channel of the eavesdropper, H e S, is a degraded version of the effective channel of the intended receiver, H r S even though the channel matrices may not be ordered a-priori. This condition explains why the genie upper bound, which provides y e to the legitimate receiver c.f. Lemma 1 does not increase the capacity of the fictitious channel. B. Capacity analysis in the High SNR Regime 1 Capacity Achieving Scheme: While the capacity expression in Theorem 1 can be computed numerically, it does not admit a closed form solution. In this section, we develop a closed form expression for the capacity in the high signalto-noise-ratio SNR regime, in terms of the generalized 1 The high SNR case of this problem i.e., max K K log dethrkh r is known as the multiple-discriminantfunction in multivariate statistics and is well-studied; see, e.g., deth ekh e [24]. singular values of the channel matrices H r and H e. The main message here is that in the high SNR regime, an optimal scheme involves simultaneously diagonalizing the channel matrices H r and H e using the generalized singular value decomposition GSVD transform. This creates a set of parallel channels independent channels between the sender and the receivers, and it suffices to use independent Gaussian codebooks across these channels. This architecture for the case of channel is shown in Fig. 1. The reader is referred to section The reader is referred to Appendix VIII for the definition and properties of the GSVD transform. Theorem 2: Let, σ 1 σ 2... σ s, be the generalized singular values of the channel matrices H r and H e. The high SNR secrecy capacity is given as follows. If then else, CP= j:σ j 1 NullH e NullH r = { } 10 lim CP = log σj 2, 11 P j:σ j 1 log σj 2 + log det I + P p H rh e H r o P 1, 12 where p is a constant that depends on H r and H e defined via 138, and o P 1 0 as P, and H e C nt nt is the projection matrix see 147 onto the null space of H e. 2 Synthetic noise transmission strategy: In addition to the capacity achieving scheme, we consider a suboptimal strategy where the choice of transmit vectors only depends on the knowledge of H r. The strategy is only semi-blind the rate allocated does depend on both H r,h e. The performance of a similar strategy was first studied via monte carlo simulations for the MISOME case in by Negi and Goel [10], [11]. Subsequently, several analytical properties and conditions of optimality are reported in [15]. Here we extend that framework for the MIMOME channel. For simplicity, we limit the discussion to the case when rankh r = n r and rankh e = n t. This case normally happens when n r n t and n e n t and the channel matrices have a full row or column rank. The transmission scheme can be described as imposing a particular choice of x, u in the binning scheme 6. Let b 1,...,b nt be independent Gaussian random variables sampled according to CN0, P t, where P t = P n t. Let H r = UΛV r, be the compact SVD of H r. Since rankh r = n r, note that U C nr nr is a unitary matrix and Λ C nr nr is a diagonal matrix. Let V = [v 1,...,v nr ] C nt nr and let {v j } nt j=1 constitute an orthogonal basis in Cnt. Our choice of parameters is, n t x = b j v j, u = b 1,..., b nr. 13 j=1 Here the symbols in u are the information bearing symbols from a corresponding codeword, while the symbols b nr+1,..., b nt are synthetic noise symbols transmitted in

4 Tx H r Rx Tx Ω 1 Σ r Ψ r Rx H e Ev Σ e Ψ e Ev Fig. 1. Simultaneous diagonalization via the GSVD transform. The left figure show the original channel model with 2 2 channel matrices H r and H e. The right figure shows the GSVD transform applied to the channel matrices i.e., H r = Ψ rσ rω 1 and H e = Ψ eσ eω 1, where Ψ r and Ψ e are unitary matrices and Σ r and Σ e are diagonal matrices. the null space of the legitimate receiver s channel in order to confuse a potential eavesdropper. In the high SNR regime the rate expression 83, can be expressed in terms of the generalized singular values of H r,h e. In particular, lim R SNP = P n t j=1 log σ 2 j 14 It is interesting to compare the expression 14 with the high SNR capacity expression 11. While the capacity expression involves summation over only those generalized singular values that exceed unity, the synthetic noise transmission scheme involves summation over all the singular values and hence is sub-optimal. Rather surprisingly, both the capacity achieving scheme and the synthetic noise scheme can be characterized using just the generalized singular values of H r,h e in the high SNR regime. C. Zero-Capacity Condition and Scaling Laws Under what conditions is the secrecy capacity zero? We develop some sharp insights into these conditions in the limit of many antennas. Corollary 1: Suppose that H r and H e have i.i.d. CN0, 1 entries. Suppose that n r, n e, n t, while keeping n r /n e = γ and n t /n e = β fixed. The secrecy capacity 2 CH r,h e converges almost surely to zero if and only if 0 β 1/2, 0 γ 1, and γ 1 2β Figs. 2 and 3 provide further insight into the asymptotic analysis for the capacity achieving scheme. In Fig. 2, we show the values of γ, β where the secrecy rate is zero. If the eavesdropper increases its antennas at a sufficiently high rate so that the point γ, β lies below the solid curve, then secrecy capacity is zero. The MISOME case corresponds to the vertical intercept of this plot. The secrecy capacity is zero, if β 1/2, i.e., the eavesdropper has at least twice the number of antennas as the sender. The 2 We assume that the channels are sampled once, then stay fixed for the entire period of transmission, and are revealed to all the terminals. single transmit antenna SIMOME case corresponds to the horizontal intercept. In this case the secrecy capacity is zero if γ 1, i.e., the eavesdropper has more antennas than the receiver. In Fig. 3, we consider the scenario where a total of T 1 antennas are divided between the sender and the receiver. The horizontal axis plots the ratio n r /n t, while the vertical axis plots the minimum number of antennas at the eavesdropper normalized by T for the secrecy capacity to be zero. We note that the optimal allocation of antennas, that maximizes the number of eavesdropper antennas happens at n r /n t = 1/2. This can be explicitly obtained from the following minimization minimize β + γ subject to, γ 1 2β 2, β 0, γ The optimal solution can be easily verified to be β, γ = 2/9, 1/9. In this case, the eavesdropper needs 3T antennas for the secrecy capacity to be zero. We remark that the objective function in 16 is not sensitive to variations in the optimal solution. If fact even if we allocate equal number of antennas to the sender and the receiver, the eavesdropper needs T T antennas for the secrecy capacity to be zero. IV. MIMOME SECRECY CAPACITY In this section we provide our proof of the secrecy capacity of the MIMOME channel i.e.,theorem 1. Our proof involves two main parts. First we note that the right hand side in 2 is an upper bound on the secrecy capacity. Then we examine the optimality conditions associated with the saddle point solution to establish 7, which completes the proof since C R +, K Φ = R C. We begin with an upper bound on the secrecy capacity of the multi-antenna wiretap channel that was established in [15]. Lemma 1 Upper Bound [15]: An upper bound on the secrecy capacity is given by CP R UB P = min K Φ K Φ max R +,K Φ, 17

5 β= n t /n e C s > 0 n e /n r + n t C s = γ = n /n r e n /n r t Fig. 2. Zero-capacity condition in the γ, β plane. The capacity is zero for any Fig. 3. The minimum number of eavesdropping antennas per sender plus point below the curve, i.e., the eavesdropper has sufficiently many antennas to receiver antenna for the secrecy capacity to be zero, plotted as a function get non-vanishing fraction of the message, even when the sender and receiver of n r/n t. fully exploit the knowledge of H e. K Φ arg min K Φ R +,K Φ Saddle Point:, K Φ arg max R +, K Φ arg max hy r Θy e Φ H rs = H es R +, K Φ = R Fig. 4. Key steps in the Proof of Theorem 1. The existence of a saddle point, K Φ is first established. Thereafter the KKT conditions associated with the minimax expressions are used to simplify the saddle value to show that it matches the lower bound. where R +,K Φ Ix;y r y e 18 is the conditional mutual information expression evaluated with x CN0,, and [z r,z e ] CN0,K Φ, and the domain sets and K Φ are defined via 3 and 4 respectively. It remains to establish that this upper bound expression satisfies 7, which we do in the remainder of this section. We divide the proof into several steps, which are outlined in Fig. 4. A. Convexity of the upper bound We first show that the minimax upper bound is a convexconcave problem with a saddle point solution. Lemma 2 Existence of a saddle point solution: The function R +,K Φ in 18 has the following properties: 1 For each fixed K Φ K Φ, the function R +,K Φ is concave in the variable. 2 For each fixed, the function R +, is convex in the variable K Φ K Φ. 3 There exists a saddle point solution to 17 i.e., and K Φ K Φ, such that R +, K Φ R +, K Φ R +,K Φ 19 holds for each, and each K Φ K Φ. Proof: To establish 1 above, with a slight abuse in notation, let us define R + p x,k Φ = Ix;y r y e, to be the conditional mutual information evaluated when the noise random variables are jointly Gaussian random variables with a covariance K Φ, and with input distribution of p x. As before, R + Q,K Φ denotes the conditional mutual information, evaluated when the noise random variables are jointly Gaussian with covariance K Φ and the input distribution is Gaussian with a covariance Q. Let p 1 x = CN0,Q 1, p 2 x = CN0,Q 2 and p θ x = θp1 x + 1 θp2 x, Qθ = θq θq 2, for some θ [0, 1] and p G x = CN0,Q θ. It suffices to show that R + Q θ,k Φ θr + Q 1,K Φ + 1 θr + Q 2,K Φ, which we do below: R + Q θ,k Φ = R + p G x,k Φ R + p θ x,k Φ 20 θr + p 1 x,k Φ + 1 θr + p 2 x,k Φ 21 = θr + Q 1,K Φ + 1 θr + Q 2,K Φ, where 20 follows from the fact that, as shown in Appendix I, a Gaussian distribution maximizes function R + p θ x,k Φ, among all distributions with a fixed covariance, and 21 from the fact that for each fixed p yr,y e x, the function Ix;y r y e is a concave function in the input distribution see e.g., [7, Appendix I]. To establish the 2, we note that for each x CN0,, the function Ix;y r,y e is convex in the noise covariance

6 K Φ see e.g., [25, Lemma II-3, pg. 3076] for an information theoretic proof. Finally, since the constraint sets and K Φ are convex and compact the existence of a saddle point solution, K Φ as stated in 3 follows from the above properties and the basic theorem in Game theory [26]. B. Saddle Point Properties In this subsection we examine the optimality properties associated with the saddle point solution to establish certain technical conditions which will be used to simplify the saddle value. In the sequel, let, K Φ denote a saddle point solution in 17, and define Φ and Θ via, [ ] Inr Φ K Φ = Φ, 22 I ne Θ = H r KP H e + ΦI + H e KP H e Lemma 3 Properties of saddle-point: The saddle point solution, K Φ to 17 satisfies the following 1 H r ΘH e Φ H r H e = Suppose that S is a full rank square root matrix of, i.e., KP = SS and S has a full column rank. Then provided H r ΘH e 0, the matrix M = H r ΘH e S 25 has a full column rank 3. Proof: The conditions 1 and 2 are established by examining the optimality conditions satisfied by the saddlepoint in 17 i.e., and K Φ arg min K Φ K Φ R +,K Φ 26 argmax R +, K Φ. 27 We first consider the optimality condition in 26 and establish 24. The derivation is most direct when K Φ is nonsingular. The extension to the case when K Φ is singular is provided in Appendix III. The Lagrangian associated with the minimization 26 is L Φ K Φ,Υ = R +,K Φ + trυk Φ, 28 where the dual variable nr ne [ ] n r Υ Υ = 1 0 n e 0 Υ 2 29 is a block diagonal matrix corresponding to the constraint that the noise covariance K Φ must have identity matrices on 3 A matrix M has a full column rank if, for any vector a, Ma = 0 if and only if a = 0. its diagonal. The associated Kuhn-Tucker KKT conditions yield KΦ L Φ K Φ,Υ KΦ = KΦ R +,K Φ 30 KΦ + Υ = 0, where, KΦ R +,K Φ KΦ 31 ] = KΦ [log detk Φ + H t KP H t logdetk Φ KΦ = K Φ + H t KP H t 1 1 K Φ 32 and where we have used H t = [ ] Hr. 33 H e Substituting 32 in 30, and simplifying, we obtain, H t KP H t = K Φ Υ K Φ + H t KP H t, 34 and the relation in 24 follows from 34 through a straightforward computation as shown in Appendix II. To establish 2 above, we use the optimality condition associated with i.e., 27 As in establishing 1, the proof is most direct when K Φ is non-singular. Hence this case is treated first, while the case when K Φ is singular is treated in Appendix VI. arg max R +, K Φ = argmax hy r y e = argmax h y r Θ y e, 35 where Θ = H r H e + ΦH e H e + I 1 is the linear minimum mean squared estimation coefficient of y r given y e. Directly working with the Kuhn-Tucker conditions associated with 35 appears difficult. Nevertheless it turns out that we can replace the objecive function above, with a simpler objective function as described below. First, note that since is an optimum solution to 35, in general arg max h y r Θy e argmax h y r Θ y e 36 holds, since substituting = in the objective function on the left hand side, attains the maximum on the right hand side. Somewhat surprisingly, it turns out that the inequality above is in fact an equality, i.e., the left hand side also attains the maximum when =. This observation is stated formally below, and allows us to replace the objective function in 35 with a simpler objective function on the left hand side in 36. Claim 1: Suppose that K Φ 0 and define H hy r Θy e. 37 Then, arg maxh. 38 The proof involves showing that KP, satisfies the Kuhn- Tucker conditions which we do in Appendix IV.

7 Finally, to establish 2, we note that, arg max H 39 = argmax log deti+j 1 2 Hr ΘH e H r ΘH e J 1 2, where J I + Θ Θ Θ Φ Φ Θ 0 40 is an invertible matrix. We can interpret 40 as stating that is an optimal input covariance for a MIMO channel with white noise and matrix H eff J 1 2H r ΘH e. The fact that H eff S is a full rank matrix, then a consequence of the so called water-filling conditions. The proof is provided in Appendix V. which can be used to establish the second case in 41 as we now do. In particular, we show that R = R +, K Φ R equals zero. Indeed, hy e y r = Ix;y r y e {Ix;y r Ix;y e } = Ix;y e y r = hy e y r hz e z r, = log deti + H e KP H e H e KP H r + Φ H r KP H r + I 1 H r KP H e + Φ = log deti + H e KP H e Φ H r KP H r + I Φ = log deti Φ Φ = hze z r, 47 C. Simplified Saddle Value The conditions in Lemma 3 can be used in turn to establish the tightness of the upper bound in 17. Lemma 4: The saddle value in 17 can be expressed as follows, { 0, H r R UB P = ΘH e = 0, R K 41 P, otherwise, where, R log deti + H r KP H r log deti + H e H e. 42 Proof: The proof is most direct when we assume that the saddle point solution is such that KΦ 0 i.e., when Φ 2 < 1. The extension when K Φ is singular is provided in Appendix VII. First consider the case when H r ΘH e = 0. From 23, it follows that Θ = Φ, using which one can establish the first part in 41: R +, K Φ = Ix;y r y e 43 = hy r y e hz r z e = hy r Θy e hz r Φz e 44 = hz r Θz e hz r Φz e 45 = 0, where 44 follows from the fact that Θ in 23 is the linear minimum mean squared estimation LMMSE coefficient in estimation y r given y e and Φ is the LMMSE coefficient in estimating z r given z e and 45 follows via the relation H r = ΘH e, so that, y r Θy e = z r Θz e. When H r ΘH e 0, combining parts 1 and 2 in Lemma 3, it follows that, Φ H r S = H e S, 46 where we have used the relation 46 in simplifying 47. This establishes the second half of 41. D. Proof of Theorem 1 The proof of Theorem 1 is a direct consequence of Lemma 4. If R +, K Φ = 0, the capacity is zero, otherwise R +, K Φ = R, and the latter expression is an achievable rate as can be seen by setting p u = p x = CN0, in the Csiszár-Körner expression 6. V. CAPACITY ANALYSIS IN THE HIGH SNR REGIME A. High SNR capacity when H e has a column full rank We first prove Theorem 2 when H e has a full column rank. In this case, it is clear that the condition in 10 is satisfied and accordingly we establish Achievability: The achievability part follows by simultaneously diagonalizing the channel matrices H r and H e using the GSVD transform. This reduces the system into a set of parallel independent channels and independent codebooks are used across these channels. More specifically, recall that in the case of interest, the transform is given in 150. Let σ 1 σ 2... σ s be the ordered set of singular values and suppose that σ i > 1 for i ν. We select the following choices for x and u in the Csiszár and Körner expression 6 x = A [ 0nt s u ],u = [0,...,0, u ν, u ν+1,..., u s ], 48 and the random variables u i are sampled i.i.d. according 1 to CN0, αp. Here α = n tσ maxa is selected so that the average power constraint is satisfied. Substituting 48 and 150 into the channel model 1 yields, [ ] 0nt s y r = Ψ r + z D r u r, y e = Ψ e 0 n t s D e u + z e ne n t Since Ψ r and Ψ e are unitary, and D r and D e are diagonal, the system of equations 49 indeed represents a parallel channel model. See Fig. 1 for an illustration of the case.

8 The achievable rate obtained by substituting 49 and 48 into 6, is R = Iu;y r Iu;y e 50 n t = log 1 + αpr2 j 1 + αpe 2 j=ν j = log σj 2 o P 1, 51 j:σ j>1 where o P 1 0 as P. 2 Converse: For the converse we begin with a more convenient upper bound expression to the secrecy capacity 17, R UB = min Φ: Φ 2 1 Θ C nr n t max R ++,Θ,Φ R ++ = log deth eff H eff + I + ΘΘ ΘΦ ΦΘ deti ΦΦ, H eff = H r ΘH e. 52 This expression, as an upper bound, was suggested to us by Y. Eldar and A. Wiesel and was first used in establishing the secrecy capacity of the MISOME channel in [14]. To establish 52, first note that the objective function R +,K Φ in 17 can be upper bounded as follows: R +,K Φ = Ix;y r y e = hy r y e hz r z e = hy r y e log deti ΦΦ = min Θ hy r Θy e log deti ΦΦ = min Θ R ++,Θ,Φ. Thus, we have from 17 that R + P = min K Φ max R +,K Φ 53 = min max min ++,Θ,Φ K Φ Θ 54 min min R ++,Θ,Φ, K Φ Θ 55 as required. To establish the capacity, we show that the upper bound in 52 above, reduces to the capacity expression 11, for a specific choice of Θ and Φ as stated below. Our choice of parameters in the minimization of 52 is as follows Θ = H r H e, where, Φ = Ψ r = diag{δ 1, δ 2,...,δ s }, nt s s ne nt [ n r s s 0 0 ] Ψ e, 56 δ i = min σ i, 1σi, 57 and H e denotes the Moore-Penrose pesudo-inverse of H e c.f Note that with these choice of parameters, H eff = 0. So the maximization over in 52 is not effective. Simplifying 52 with these choice of parameters the upper bound expression reduces to R ++ log deti + D rd 1 e 2 2D r D 1 e deti 2 = log σ j. 2 j:σ j>1 as in 11. B. High SNR capacity when H e is not full column rank When H e is not a full column rank matrix, the capacity result in 12 will now be established. 1 Achievability: To show the achievability, we identify the subspaces S z = NullH e NullH r = span{ψ k p+1,..., ψ k } S s =NullH e NullH r =span{ψ k p s+1,..., ψ k p }. 58 We will use most of the power for transmission in the subspace S z and a small fraction of power for transmissoin in the subspace S s. More specifically, by selecting, we have, x = Ψ t 0 k p s Ω 2 u v 0 nt k, 59 0 nr p s y r = Ψ r D r u + z r, T 32 Ω 1 2 u + Ω 1 3 v y e = Ψ e 0 k p s D e u + z e. 0 ne+p k 60 In 59, we select v = [v 1, v 2,...,v p ] T to be a vector of i.i.d. Gaussian random variables with a distribution CN 0, P P p and u = [0,...,0, u ν,...,u s ] T to be a vector of independent Gausian random variables. Here ν is the smallest integer such that σ j > 1 for all j ν and σ j 1 otherwise. Each u j CN0, α 1 P, where α = n, tσ maxω 2 is chosen to meet the power constraint. An achievable rate for this choice of parameters is R = Iu,v;y r Iu,v;y e 61 = Iu;y r Iu;y e + Iv;y r u, 62 where the last step follows from the fact that v is independent of y e,u c.f. 60. Following 51, we have that Iu;y r Iu;y e = log σj 2 o P1 63 and Iv;y r u = log det = log det = log det j:σ j>1 I + P P p I + P p Ω 1 3 Ω 3 I + P p H rh e H r Ω 1 3 Ω 3 64 o P 1 65 o P 1, 66

9 where 65 follows from the fact that log1+x is a continuous function of x and log deti +X = log1 + λ i X and the last step follows from Converse: To establish the converse, we use the following choices for Θ and Φ in 52. k s p s n e+p k Θ = Ψ r and Φ = Ψ r n r s p 0 s D r D 1 e p F 31 F 32 0 k s p s n e+p k n r s p 0 s p 0 where is defined in 57, and the matrices are selected such that H r ΘH e Ψ e, 67 Ψ e 68 F 32 = T 32 Ω 2 D 1 e F 31 = T 31 F 32 D e T 21 Ω 1 69 = Ψ r [Σ r Ω 1,0 nr n t k] Ψ r ΘΨ e[σ e Ω 1,0 ne n t k]ψ t 70 k p s s p n t k n r s p 0 = Ψ r s 0 Ψ p Ω 1 t The upper bound expression 52 can now be simplified as follows. H eff H eff = H r ΘH e H r ΘH e n r p s s p n r p s 0 = Ψ r s 0 Ψ r, p Ω 1 3 QΩ 3 where Q is related to by, Ψ t Ψ t = k p s s p n t k k p s s p Q n t k and satisfies trq P. From 72, 68 and 67, we have that the numerator in the upper bound expression 52 simplifies as in 74. Using 74 and the Hardamard inequality, we have log deti + H eff ΘH eff + ΘΘ ΘΦ ΦΘ log deti + D r D 1 e 2 2D r D 1 e + log deti + F 31 F 31 + F 32F 32 + Ω 1 3 QΩ 3 75 Substituting this relation in 52, the upper bound reduces to, R + P log deti + D rd 1 e 2 2D r D 1 e deti 2 + max log deti + F 31 F 31 + F 32F 32 + Q 0: Ω 1 3 QΩ 3 trq P 76 Substituting for D r and D e from 139 and for from 57, we have that log deti + D rd 1 e 2 2D r D 1 e deti 2 = log σj 2. j:σ j>1 77 It remains to establish that max log deti + F 31 F 31 + F 32F 32 + Q 0: Ω 1 3 QΩ 3 trq P log det I + P p H rh e H r + o P 1, 78 which we now do. Let γ = σ max F 31 F 31 + F 32F 32, 79 denote the largest singular value of the matrix F 31 F 31 + F 32 F 32. Since log-det is increasing on the cone of positive semidefinite matrices, we have that, max log deti + F 31 F 31 + F 32F 32 + Q 0: Ω 1 3 QΩ 3 trq P max Q 0: trq P = log det = log det = log det log det1 + γi + Ω 1 3 QΩ γi + P p Ω 1 3 Ω 3 I + P p Ω 1 3 Ω 3 I + P p H rh e H r + o P o P 1 + o P 1 82 where 80 follows from the fact that F 31 F 31 +F 32F 32 γi, and 81 follows from the fact that water-filling provides a vanishingly small gain over flat power allocation when the channel matrix has a full rank see e.g., [27] and 82 follows via 149. C. Analysis of synthetic noise transmission scheme We first show, via straightforward computation, that this choice of parameters, results in a rate of R SN P = log det I + ε t Λ 2 + log deth r ε t I + H e H e 1 H r. 83

10 I + H eff ΘH eff + ΘΘ ΘΦ ΦΘ n r s p I = Ψ r s p n r s p s p I + D r D 1 e 2 2D r D 1 e D r D 1 e F 32 F 32 D r D 1 e I + F 31 F 31 + F 32F 32 + Ω 1 3 QΩ 3 Ψ r 74 where ε t = 1 P t. First note that Iu;y e = log deti + P t H r H r = log deti + P t Λ 2 84 In the following, let V n = [v nr+1,...,v nt ] denote the vectors in the null space of H r. Iu;y e = hy e hy e u = log deti + P t H e H e log deti + P t H e V n V nh e = log deti + P t H e H e log deti + P th e I VV H e = log deti + P t H rh r log deti + P t I VV H eh e = logdeti P t I + P t H e H e 1 VV H e H e = logdeti P t V H e H ei + P t H e H e 1 V = logdetv I + P t H eh e 1 V Where we have repeatedly used the fact that deti+ab = deti + BA for any two matrices A and B of compatible dimensions. R SN P = log deti + P t Λ 2 + log detv I + P t H e H e 1 V. Since U and Λ are square and invertible, R SN P = log deti + ε t Λ 2 + log detuλv ε t I + H eh e 1 VΛU = log deti + ε t Λ 2 + log deth r ε t I + H e H e 1 H r, as required. To establish 14, we use the following facts Fact 1 Taylor Series Expansion [28]: Let M be an invertible matrix. Then εi + M 1 = M 1 + Oε, 85 where Oε represents a function that goes to zero as ε 0. Fact 2: Suppose that H r and H e be the channel matrices as in 1, and suppose that rankh r = n r and rankh e = n t and n r n t n e. Let σ 1, σ 2..., σ s denote the generalized singular values of H r,h e c.f Then det H r H eh e 1 H s r = σj 2 86 j=1 Finally, to establish 14, we take the limit ε t 0 in 83 R SN P = log det I + ε t Λ 2 + log deth r ε t I + H eh e 1 H r = log deth r H e H e 1 + Oε t H r + Oε t 87 = log deth r H eh e 1 H r = + log deti + H e H e 1/2 OεH e H e /2 s log σj 2 + Oε t. 88 j=1 where we use Facts 1 and 2 above in 87 and 88 above and the fact that log deti + X = j log1 + λ jx is continuous in the entries of X. VI. ZERO-CAPACITY CONDITION AND SCALING LAWS First we develop a general condition under which the secrecy capacity is zero. Lemma 5: The secrecy capacity of the MIMOME channel is zero if and only if H r v σ max H r,h e sup v C n t H e v Proof: When NullH r NullH e {}, clearly, σ max H r,h e =. Otherwise, it is known see e.g., [29] that σ max is the largest generalized singular value of H r,h e as defined in 140. To establish that the capacity is zero, whenever σ max H r,h e 1, it suffices to consider the high SNR secrecy capacity in 11 in Theorem 2, which is clearly zero whenever σ max 1. If σ max > 1, let v there exists a vector v such that H r v > H e v. Select x = u CN0, Pvv in 6. Clearly CP R P > 0 for all P > 0. By combining Lemma 5 and Fact 3 below which is established in [30, Pg. 642], one can deduce the following condition for the zero-capacity condition in Corollary 1. Fact 3 [30] [31]: Suppose that H r and H e have i.i.d. CN0, 1 entries. Let n r, n e, n t, while keeping n r /n e = γ and n t /n e = β fixed. If β < 1, then the largest generalized singular value of H r,h e converges almost surely to 2 σ max H r,h e a.s β 1 β γ γ 1 β. The proof follows by direct substitution of the GSVD expansion 136 and will be omitted. 90

11 VII. CONCLUSION We establish the secrecy capacity of the MIMOME channel as a saddle point solution to a minimax problem. Our capacity result establishes that a Gaussian input maximizes the secrecy capacity expression by Csiszár and Körner for the MIMOME channel. Our proof uses upper bounding ideas from the MIMO broadcast channel literature and the analysis of optimality conditions provides insight into the structure of the optimal solution. Next, we develop an explicit expression for the secrecy capacity in the high SNR regime in terms of the generalized singular value decomposition GSVD and show that in this case, an optimal scheme involves simultaneous diagonalization of the channel matrices to create a set of independent parallel channel and using independent codebooks across these channels. We also study a synthetic noise transmission scheme that is semi-blind as it selects the transmit directions based on the legitimate receiver s channel only and compare its performance with the capacity achieving scheme. Finally, we study the conditions under which the secrecy capacity is zero and study its scaling laws in the limit of many antennas. ACKNOWLEDGEMENT We thank Ami Wiesel for interesting discussions and help with numerical optimization of the saddle point expression in Theorem 1. APPENDIX I OPTIMALITY OF GAUSSIAN INPUTS We show that a Gaussian input maximizes the conditional mutual information term Ix;y r y e when the noise distribution [z r,z e] CN0,K Φ. Recall that K Φ has the form, [ ] Inr Φ K Φ = 91 Φ and K Φ 0 if and only if Φ 2 < 1. In this case we show that among all distributions p x with a covariance of, a Gaussian distribution maximizes Ix;y r y e. Note that I ne Ix;y r y e = hy r y e hz r z e 92 where = hy r y e log2πe nr deti nr ΦΦ log detλ log deti nr ΦΦ, 93 Λ I + H r H r Φ + H r H e I + H e H e 1 Φ + H e H r 94 is the linear minimum mean squared error in estimating y r given y e and the last inequality is satisfied with equality if p x = CN0,. When K Φ is singular, the expansion 92 is not well defined. Nevertheless, we can circumvent this step by defining an appropriately reduced channel. In particular, let Φ = [ U 1 U 2 ] [ I 0 0 ] [ V 1 V 2 ] 95 be the singular value decomposition of Φ, where σ max < 1 then we have the following Claim 2: Suppose that the singular value decomposition of Φ is given as in 95 and that for the input distribution p x, we have that Ix;y r y e <, then, U 1 z r a.s. = V 1 z e 96a Ix;y r y e = Ix;U 2 y r y e 96b The optimality of Gaussian inputs now follows since the term Ix;U 2 y r y e can be expanded in the same manner as The proof of Claim 2 is provided below. Proof: To establish 96a, we simply note that E[U 1 z rz ev 1 ] = U 1 ΦV 1 = I, i.e., the Gaussian random variables U 1 z r and V 1 z e are perfectly correlated. Next note that R +, K Φ = Ix;y r y e = Ix;U 1 y r,u 2 y r y e 97 = Ix;U 2 y r,u 1 y r V 1 y e y e = Ix;U 2 y r,u 1 H rx V 1 H ex y e. 98 Since by hypothesis, Ix;y r y e <, we have that U 1 H r V 1 H ex = 0, and Ix;y r y e = Ix;U 2 y r y e, establishing 96b. Finally if p x is such that Ix;y r y e =, then from 98, U 1 H r V 1 H e U 1 H r V 1 H e 0 and hence the choice of a Gaussian p x = CN0, also results in Ix;y r y e =. APPENDIX II MATRIX SIMPLIFICATIONS FOR ESTABLISHING 24 FROM 34 Substituting for K Φ and H t in 34 and carrying out the block matrix multiplication gives H r KP H r = Υ 1I + H r KP H r + ΦΥ 2 Φ + H e KP H r H r KP H e = Υ 1 Φ + H r KP H e + ΦΥ 2 I + H e KP H e H e KP H r = Φ Υ 1 I + H r KP H r + Υ 2 Φ + H e KP H r H e KP H e = Φ Υ 1 Φ + H r KP H e + Υ 2I + H e KP H e. 99 Eliminating Υ 1 from the first and third equation above, we have Φ H r H e H r = Φ Φ IΥ2 Φ + H e KP H r. 100 Similarly eliminating Υ 1 from the second and fourth equations in 99 we have Φ H r H e H e = Φ Φ IΥ2 I+H e KP H e. 101 Finally, eliminating Υ 2 from 100 and 101 we obtain Φ H r H e H r = Φ H r H e H e I + H e H e 1 Φ + H e KP H r = Φ H r H e H e Θ 102 which reduces to 24.

12 APPENDIX III DERIVATION OF 24 WHEN THE NOISE COVARIANCE IS SINGULAR Consider the compact singular value decomposition of K Φ : K Φ = W ΩW, 103 where W is a matrix with orthogonal columns, i.e., W W = I and Ω is a non-singular matrix. We first note that it must also be the case that H t = WG, 104 i.e., the column space of H t is a subspace of the column space of W. If this were not the case then clearly Ix;y r,y e = whenever the covariance matrix has a component in the null space of W which implies that, max R +, K Φ =. 105 Since, K Φ is a saddle point, we must have that R +, K Φ R +,I <, and hence 104 must hold. Also note that since R +, K Φ = log deti + H e KP H e + log detg G + Ω detω 106 it follows that Ω in 103 is a solution to the following minimization problem, min Ω K Ω R Ω Ω, R Ω Ω = log detg G + Ω, detω { [ ] } Inr Φ K Ω = Ω WΩW = Φ 0. I ne The Kuhn-Tucker conditions for 107 yield, Ω 1 G G + Ω 1 = W ΥW, G G = ΩW ΥW Ω + G G where Υ has the block diagonal form in 29. Multiplying the left and right and side of 108 with W and W respectively and using 103 and 104 we have that H t KP H t = K Φ Υ K Φ + H t KP H t, 109 establishing 34. Finally note that the derivation in Appendix II does not require the non-singularity assumption on K Φ. APPENDIX IV PROOF OF CLAIM 1 To establish 38 note that since H is a concave function in and differentiable over, the optimality conditions associated with the Lagrangian L Θ, λ,ψ = H + trψ λtr P, 110 are both necessary and sufficient. Thus is an optimal solution to 38 if and only if there exists a λ 0 and Ψ 0 such that H r ΘH e [Γ ] 1 H r ΘH e + Ψ = λi, trψ = 0, λtr P = 0, where Γ is defined via Γ I + Θ Θ Θ Φ Φ Θ H r ΘH e H r ΘH e. 112 To obtain these parameters note that since, K Φ constitutes a saddle point solution, argmax R +, K Φ. 113 Since R +, K Φ is differentiable at each whenever K Φ 0, KP satisfies the associated KKT conditions there exists a λ 0 0 and Ψ 0 0 such that KP R, K Φ +Ψ 0 = λ 0 I KP 114 λ 0 tr P = 0, trψ 0 KP = 0. As we show below, KP R, K Φ =H r ΘH e [Λ ] 1 H r ΘH e, KP where Λ I + H r H r 115 Φ + H r H ei + H e H e 1 Φ + H e H r 116 Λ, satisfies 4 Λ = Γ. Hence the first condition in 114 reduces to H r ΘH e [Γ ] 1 H r ΘH e + Ψ 0 = λ 0 I. 117 Comparing 114 and 117 with 111, we note that, λ 0,Ψ 0 satisfy the conditions in 111, thus establishing 38. It thus remains to establish 115, which we do below. KP R +, K Φ = H th t H t + K Φ 1 H t H ei + H e H e 1 H e. 118 Substituting for H t and K Φ from 33 and 22, K Φ + H t KP H t 1 [ I + Hr KP H r Φ + H r KP H ] 1 e = Φ + H r KP H e I + H e KP H e [ ΛKP 1 Λ 1 ] Θ = Θ Λ 1 I+H e KP H e 1 + Θ Λ 1, Θ 4 To verify this relation, note that Γ is the variance of y r Θy e. When =, note that Θye is the MMSE estimate of y r given y e and Γ is the associated MMSE estimation error.

13 where we have used the matrix inversion lemma e.g., [28], and Λ is defined in 94, and Θ is as defined in 23. Substituting into 118 and simplifying gives KP R +, K Φ KP where the F is of the form [ ν nt ν ν F 0 F 1 F = n t ν F 1 F 2 ]. 128 =H t K Φ + H t KP H t 1 H t H ei + H e KP H e 1 H e as required. = H r ΘH e [Λ ] 1 H r ΘH e APPENDIX V FULL RANK CONDITION FOR OPTIMAL SOLUTION Claim 3: Suppose that KΦ 0 and ˆ be any optimal solution to ˆ arg maxlog deti+j 1 2 Hr ΘH e H r ΘH e J for some J 0 and Θ is defined in 23. Suppose that S P is a matrix with a full column rank such that ˆ = S P S P 120 then H r ΘH e S P has a full column rank. Define H eff J 1 2 Hr ΘH e. It suffices to prove that H eff S P has a full column rank, which we now do. Let rankh eff = ν and let H eff = AΣB 121 be the singular value decomposition of H eff where A and B are unitary matrices, and We now note that ˆF 1 = 0 and ˆF 2 = 0. Indeed if ˆF 2 0, then trˆf 2 > 0. This contradicts the optimality claim in 127, since the objective function only depends on ˆF 0 and one can strictly increase the objective function by increasing the trace of ˆF 0. Finally since ˆF 0 and ˆF 2 = 0, it follows that ˆF 1 = 0. APPENDIX VI FULL RANK CONDITION WHEN K Φ IS SINGULAR In this section we establish 2 in Lemma 3 when K Φ is singular. We map this case to another channel when the saddle point noise covariance is non-singular and apply the results for non-singular noise covariance. When K Φ is singular, we have that Φ has d 1 singular values equal to unity and hence we express its SVD in 95, where σ max < 1. Following Claim 2 in Appendix I we have that U 1 z r a.s. = V 1 z e 129a U 1 H r = V 1 H e, 129b R +, K Φ = Ix;U 2 y r y e,. 129c Thus with Ĥr = U 2 H r, and ẑ r = U 2 z r and Σ = [ ν nt ν ν Σ 0 0 n r ν 0 0 Note that it suffices to show that the matrix has the form Since, ˆF = ]. 122 ˆF B ˆKP B 123 [ ν nt ν ν F 0 0 n t ν 0 0 ˆ arg max log deti + H eff H eff ]. 124 = argmax log deti + AΣB BΣ A = argmax log deti + ΣB BΣ, 125 and if and only if B B, observe that, ˆF arg maxlog deti + ΣFΣ 126 = arg maxlog deti + Σ 0 F 0 Σ 0, 127 we have from 129c, that ŷ r = U 2 y r = Ĥrx + ẑ r, 130 argmaxix;ŷ r y e. 131 Since the associated cross-covariance matrix ˆΦ = E[ẑ r z e ] has all its singular values strictly less than unity, it follows from Claim 1 that where argmax Ĥ 132 Ĥ = hŷ r ˆΘy e, ˆΘ = U 2 H r H e + ΦI + H e KP H e 1. Following the proof of Claim 3 in Appendix V we then have that Ĥr ˆΘH e S = U 2 H r ΘH e S has a full column rank. This in turn implies that H r ΘH e S has a full column rank.

14 APPENDIX VII PROOF OF LEMMA 4 WHEN K Φ IS SINGULAR When K Φ is singular, we assume that the singular value decomposition of Φ is given in 95. First let us consider the case that H r = ΘH e and show that R +, K Φ = 0. Indeed following claim 2 in Appendix I we have that R +, K Φ = Ix;U 2 y r y e and expanding this expression in the same manner as 43-45, we establish the desired result. When H r ΘH e 0, we show that the difference between the upper and lower bounds is zero. R = R +, K Φ R = Ix;y e y r = Ix;V 2 y e y r, 133 where the last step follows from the fact that U 1 z a.s. r = V 1 z e and U 1 H r = V 1 H e c.f. 129a, 129b. Next, note that, hv 2 y e y r = log deti + V 2 H e H e V 2 V 2 H e H r + U 2 I + H r KP H r 1 H r KP H ev 2 + U 2 = log deti + U 2 H r H r U 2 U 2 I + H r H ru 2 = log deti 134 = hv 2 z e U 2 z r = hv 2 z e z r, 135 where we have used c.f. 46 that V 2 Φ H r S = V 2 H es U 2 H rs = V 2 H es, in simplifying 134 and the equality in 135 follows from the fact that U 1 z r is independent of U 2 z r,v 2 z e. APPENDIX VIII GSVD TRANSFORM We begin with a definition of the generalized singular value decomposition [32], [33]. Definition 1 GSVD Transform: Given two matrices H r C nr nt and H e C ne nt, there exist unitary matrices Ψ r C nr nr, Ψ e C ne ne and Ψ t C nt nt, a non-singular, lower triangular matrix Ω C k k, and two matrices Σ r R nr k and Σ e R ne k, such that Ψ r H rψ t = Σ r [ Ω 1,0 k nt k], 136a Ψ eh e Ψ t = Σ e [ Ω 1,0 k nt k], 136b where the matrices Σ r and Σ e have the following structure, Σ r = Σ e = k p s s p n r p s 0 s D r p I k p s s p k p s s D e n e+p k 0 I, 137a, 137b and the constants k = rank [ Hr H e ], p = dim NullH e NullH r, 138 and s depend on the matrices H r and H e. The matrices D r = diag{r 1,...,r s }, D e = diag{e 1,...,e s }, 139 are diagonal matrices with strictly positive entries, and the generalized singular values are given by σ i = r i e i, i = 1, 2,...,s. 140 We provide a few properties of the GSVD-transform that are used in the sequel. 1 The GSVD transform provides a characterization of the null space of H e. Let Ψ t = [ψ 1,..., ψ nt ], 141 where Ψ t is defined via 136. Then S n = NullH e NullH r = span{ψ k+1,...,ψ nt } 142a S z = NullH e NullH r = span{ψ k p+1,...,ψ k } 142b Indeed, it can be readily verified from 136 that H r ψ j = H e ψ j = 0, j = k + 1,...,n t, 143 which establishes 142a. To establish 142b, we will show that for each j such that k p + 1 j k, H e ψ j = 0 and {H r ψ j } are linearly independent. It suffices to show that the last p columns of Σ r Ω 1 are linearly independent and the last p columns of Σ e Ω 1 are zero. Note that since Ω 1 in 136 is a lower triangular matrix, we can express it as Ω 1 = k p s k p s s p Ω 1 1 s T 21 Ω 1 2 p T 31 T 32 Ω By direct block multiplication with 137a and 137b, we have, Σ r Ω 1 = Σ e Ω 1 = k s p s p n r s p 0 s D r T 21 D r Ω 1 2 p T 31 T 32 Ω 1 k p s k s p s p Ω 1 s D e T 21 D e Ω 1 2 n e+p k a 145b Since Ω 3 is invertible, the last p columns of Σ r Ω 1 are linearly independent and clearly the last p columns of Σ e Ω 1 are zero establishing 142b. Furthermore, NullH e = span{ψ k p+1,..., ψ nt }. 146