RANK PROPERTY OF THE MIMO GAUSSIAN WIRETAP ... - IEEE Xplore
RANK PROPERTY OF THE MIMO GAUSSIAN WIRETAP CHANNEL WITH AN AVERAGE POWER CONSTRAINT S. Ali. A. Fakoorian, Jing Huang, A. Lee Swindlehurst Center for Pervasive Communications and Computing University of California Irvine afakoori, jing.huang, [email protected] ABSTRACT This paper considers a multiple-input multiple-output (MIMO) Gaussian wiretap channel, where there exists a transmitter, a legitimate receiver and an eavesdropper, each equipped with multiple antennas. In particular, we study the rank of the optimal input covariance matrix that achieves the secrecy capacity of the multiple antenna MIMO Gaussian wiretap channel under an average power constraint. The rank and other properties of the optimal solution are derived based on certain relationships between the channel matrices for the legitimate receiver and eavesdropper. Such properties are useful steps towards characterizing the general solution to the MIMO wiretap problem with an average power constraint. Index Terms— MIMO Wiretap Channel, Secrecy Capacity, Physical Layer Security 1. INTRODUCTION The broadcast nature of a wireless medium makes it very susceptible to eavesdropping, where the transmitted message is decoded by unintended receiver(s). Recent informationtheoretic research on secure communication has focused on enhancing security at the physical layer. The wiretap channel, first introduced and studied by Wyner [1], is the most basic physical layer model that captures the problem of communication security. Wyner showed that when an eavesdropper’s channel is a degraded version of the main channel, the source and destination can achieve a positive secrecy rate, while ensuring that the eavesdropper receives zero bits of information. The maximum secrecy rate from the source to the destination is defined as the secrecy capacity. The Gaussian wiretap channel, in which the outputs at the legitimate receiver and at the eavesdropper are corrupted by additive white Gaussian noise, was studied in [2]. Determining the secrecy capacity of a Gaussian MIMO wiretap channel is in general a difficult non-convex optimization problem, and has been addressed independently in [3]This work was supported by the U.S. Army Research Office under the Multi-University Research Initiative (MURI) grant W911NF-07-1-0318, and by the National Science Foundation under grant CCF-1117983.
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[9]. Oggier and Hassibi [3] and Khisti and Wornell [4] followed an indirect approach using a Sato-like argument and matrix analysis tools. They considered the problem of finding the secrecy capacity under a constraint on the average total power, and a closed-form expression for the secrecy capacity in the high signal-to-noise-ratio (SNR) regime was obtained in [4]. The rank of the optimal input covariance matrix for the secrecy rate maximization problem is discussed in [5], but the authors were unable to characterize the solution for the general case. For some special cases of the MIMO wiretap channel, where the solution has rank one, the optimal input covariance matrix that achieves the secrecy capacity under the average total power constraint was obtained in [5]-[7]. In [8], Liu and Shamai propose a more informationtheoretic approach using the enhancement concept, originally presented by Weingarten et al. [10], as a tool for the characterization of the MIMO Gaussian broadcast channel capacity. Liu and Shamai have shown that an enhanced degraded version of the channel attains the same secrecy capacity as does a Gaussian input distribution. From the mathematical solution in [8] it was evident that such an enhanced channel exists; however it was not clear how to construct such a channel until the work of [9], which provided a closed-form expression for the secrecy capacity under an input covariance matrix constraint. While this result is interesting since the expression for the secrecy capacity is valid for all SNR scenarios, there still exists no computable secrecy capacity expression for the MIMO Gaussian wiretap channel under an average total power constraint. In this paper, we investigate the rank of the optimal input covariance matrix that achieves the secrecy capacity of the general Gaussian multiple-input multiple-output (MIMO) wiretap channel under the average total power constraint, where the number of antennas is arbitrary for both the transmitter and the two receivers. Throughout this analysis, other interesting properties for the optimal input covariance matrix are revealed as well. Notation: Throughout the paper, boldface uppercase letters are used to denote matrices, and vector-valued random variables are written with non-boldface uppercase letters (e.g.,
Asilomar 2012
X), while the corresponding non-boldface lowercase letter (x) denotes a specific realization of the random variable. Scalar variables are written with non-boldface (lowercase or uppercase) letters. The Hermtian (i.e., conjugate) transpose is denoted by (.)H , the matrix trace by Tr(.), and I indicates an identity matrix. Inequality A B means that A − B is Hermitian positive semi-definite. Mutual information between the random variables A and B is denoted by I(A; B), E is the expectation operator, and CN (0, σ 2 ) represents the complex circularly symmetric Gaussian distribution with zero mean and variance σ 2 . The projection matrix onto the column space of X is denoted by PX = X(XH X)−1 XH , and P⊥ X = I − PX denotes the projection onto the space orthogonal to X. span{X} denotes the space spanned by the column vectors of X, and span{X}⊥ denotes the orthogonal complement space of span{X}.
definite matrices 1
1
(S 2 HH HS 2 + I ,
1
1
S 2 GH GS 2 + I) .
(4)
In particular, there exists an invertible generalized eigenvector matrix C such that [12] h 1 i 1 CH S 2 GH GS 2 + I C = I (5)
h 1 i 1 CH S 2 HH HS 2 + I C = Λ
(6)
where Λ = diag{λ1 , ..., λnt } is a positive definite diagonal matrix and λ1 , ..., λnt represent the generalized eigenvalues. Without loss of generality, we assume the eigenvalues are ordered as
2. SYSTEM MODEL AND PRIOR WORK
λ1 ≥ ... ≥ λb > 1 ≥ λb+1 ≥ ... ≥ λnt > 0
We begin with a multiple-antenna wiretap channel with nt transmit antennas and nr and ne receive antennas at the legitimate recipient and the eavesdropper, respectively:
so that a total of b (0 ≤ b ≤ nt ) are greater than 1. Hence, we can write Λ as Λ1 0 Λ= (7) 0 Λ2
yr = Hx + zr
(1)
ye = Gx + ze
where x is a zero-mean nt × 1 transmitted signal vector, zr ∈ Cnr ×1 and ze ∈ Cne ×1 are additive white Gaussian noise vectors at the receiver and eavesdropper, respectively, with i.i.d. entries distributed as CN (0, 1). The matrices H ∈ Cnr ×nt and G ∈ Cne ×nt represent the channels associated with the receiver and the eavesdropper, respectively. Similar to other papers considering the perfect secrecy rate of the wiretap channel, we assume that the transmitter has perfect channel state information (CSI) for both the legitimate receiver and the eavesdropper. For the Gaussian channel, where Gaussian inputs are an optimal choice, the secrecy capacity is given by [3] Csec = max[I(X; Yr ) − I(X; Ye )] = max R(Q) x
Q0
(2)
where R(Q) = log |HQHH + I| − log |GQGH + I|, and Q = E{xxH } is the input covariance matrix. The maximization problem in (2) is on the channel input, where the channel input is under either an average power constraint or an input covariance matrix constraint, as described below. In [9], the above secret communication problem was analyzed under the input covariance constraint, defined as
where Λ1 = diag{λ1 , ..., λb } and Λ2 = diag{λb+1 , ..., λnt }. We can partition C similarly: C = [C1
C2 ]
(8)
where C1 is the nt ×b submatrix representing the generalized eigenvectors corresponding to {λ1 , ..., λb } and C2 is the nt × (nt − b) submatrix representing the generalized eigenvectors corresponding to {λb+1 , ..., λnt }. Using the above notation, the secrecy capacity of the MIMO wiretap channel under the input covariance constraint (3) can be expressed as [9], [11, Theorem 3]: Csec (S) =
b X
log λi = log |Λ1 |
(9)
i=1
where the optimal input covariance matrix Q∗S that maximizes (2) and attains (9) is given by −1 1 1 (CH 0 1 C1 ) Q∗S = S 2 C CH S 2 . (10) 0 0
(3)
Remark 1 From (5) and (6), one can easily confirm that if HH H GH G, then for any S 0 we have Λ I. In other words, in this case the pencil in (4) has no generalized eigenvalue bigger than 1. Thus, Csec (S) = 0 for any S 0.
where S is a positive semi-definite matrix that defines the input covariance constraint. An explicit expression for the secrecy capacity under (3) was obtained via applying the generalized eigenvalue decomposition to the following two positive
Lemma 1 For the case of HH H GH G, for any nt × nt matrix S 0, all the generalized eigenvalues of the pencil 1 1 1 1 (S 2 HH HS 2 + I , S 2 GH GS 2 + I) are strictly bigger than 1, i.e. Λ I, iff S is full rank, i.e. S 0.
Csec (S) = max log |HQHH + I| − log |GQGH + I| s.t. Q S
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Proof: Subtracting (5) from (6), a straightforward computation yields 1 1 S 2 HH H − GH G S 2 = C−H (Λ − I) C−1 . When HH H GH G, both sides of the above equality are positive semi-definite; i.e., for any S 0 we must have Λ I. If S 0, the left side is full-rank and thus Λ I, and vice versa. ♠ In this paper, we consider the secrecy capacity problem in (2) under the average power constraint: Csec (Pt ) = max log |HQHH + I| − log |GQGH + I| s.t. Tr(E{xxH }) = Tr(Q) ≤ Pt . (11)
b and C b 1 have respectively the same definitions as where C those of C and C1 , given by (5)-(8), but here for the pencil b 12 HH HS b 12 + I , S b 12 GH GS b 21 + I). Note that Q∗ can be (S rewritten as " H # h i b H b −1 b 1 C ( C C ) 0 ∗ 1 1 1 b1 C b2 C b2 b 21 Q =S bH S 0 0 C 2
b 12 C b 1 (C b HC b 1 )−1 C bH S b 12 =S 1 1 1
max S0,Tr(S)≤Pt
Csec (S) ,
(12)
where for any given semidefinite S, Csec (S) should be computed as given by (9). Also note that for any covariance constraint S, the optimal input covariance matrix Q∗S that attains Csec (S) is given by (10), and Q∗S S. In the next section, we investigate the rank of the optimal input covariance matrix Q∗ that attains Csec (Pt ).
(14)
b b H b −1 C b H is the projection matrix where PC b 1 = C1 (C1 C1 ) 1 b 1 . Moreover, let P⊥ = I − P b be onto the space of C b1 C1 C b 1 . We have the projection onto the space orthogonal to C
For this constraint, no computable secrecy capacity expression has been derived to date for the general MIMO case. It was shown in [10, Lemma 1], [11] that, the wiretap channel under the average power constraint (11) is equivalent to the channel under the input covariance constraint (3) through an exhaustive search over the set {S : S 0, Tr(S) ≤ P } : Csec (Pt ) =
1
b 2 Pb S b2 =S C1
1
1
b 2 Pb S b2) Tr(Q∗ ) = Tr(S C1 b Pb ) = Tr(S C1
(15)
b Pb Pb ) = Tr(S C1 C1
(16)
b b ) = Tr(PC b 1 S PC 1
(17)
where (15) comes from the fact that for any A ∈ Ca×b and B ∈ Cb×a , Tr(AB) = Tr(BA), and (16) holds because PC b 1 = PC b 1 PC b 1 . Similarly we have b (P b + P⊥ ) b = Tr (P b + P⊥ ) S Tr(S) b b C1 C1 C C 1
1
b b ) + Tr(P⊥ S b P⊥ ) = Tr(PC b 1 S PC b b C C 1
(18)
b ⊥ ) = Tr(Q∗ ) + Tr(P⊥ b S PC b C
(19)
1
1
3. PROPERTIES OF THE OPTIMAL SOLUTION UNDER AN AVERAGE POWER CONSTRAINT
1
1
where in (18) we used the facts that Tr(AB) = Tr(BA) and ⊥ PC b 1 PC b = 0, and where (19) results from (17). 1
For the following results in the paper, we exclude the special case HH H GH G for which the Csec is trivially 0 for any S 0, and consequently for any Pt , as pointed out in Remark 1. Also we note that problem (11) under Tr(Q) ≤ Pt is equivalent to that under Tr(Q) = Pt , see [3, Eq. (9)], [5]. Considering (11) in the form of (12), and recalling that for any S, Q∗S S, thus Tr(Q∗S ) ≤ Tr(S), the above point implies that we can restrict the set of input covariance constraints to S 0, Tr(S) = Pt instead of Tr(S) ≤ Pt . 3.1. span properties b denote the input covariance constraint that maximizes Let S (12). Consequently, from (10), the optimal input covariance matrix that attains Csec (Pt ) is given by1 b H b −1 0 b (C1 C1 ) b HS b 12 , b 12 C C (13) Q∗ = S 0 0 1 To
be consistent with notation in (10), the optimal input covariance mab should be written as Q∗ , which for trix under the covariance constraint S b S simplicity is denoted just as Q∗ .
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b we have span{C b 1 } = span{S}. b Lemma 2 For the optimal S, Proof: The proof is obtained using (19), and by noting that b we must have Tr(S) b = Tr(Q∗ ) = Pt . for the optimal S b ⊥ ) = 0, or This means that we must have Tr(P⊥ b 1 S PC b1 C b = 0, which consequently shows that equivalently P⊥ S b1 C
b ⊆ span{C b 1 }. To complete the proof we only need span{S} b 1 } ⊆ span{S}. b to show that span{C b 1 , i.e., b b Let c1 denote a column vector of C c1 repreb 12 HH HS b 12 + sents a generalized eigenvector of the pencil (S 1 1 b 2 GH GS b 2 + I) corresponding to a generalized eigenI, S value bigger than 1: 1 b 2 HH HS b 21 + I b b cH S c1 1 1 >1. b 2 GH GS b 12 + I b b cH S c1 1 We can write b c1 as b c1 = PSb b c1 + P⊥ c1 , where PSb = b b S H −1 H b S b S) b b is the projection matrix onto the space of S( S
b and P⊥ = I − Pb is the projection onto the space orS, b S S b Letting a = b b 21 H b 21 c1 and b = thogonal to S. cH 1 S H HS b b 21 H b 12 c1 , we note that a > b, and the only compob cH 1 S G GS b nent of b c1 that affects the values of a and b is PSb b c1 . Now, one can easily show that when a > b, for any x and y ≥ 0, we have a+x+y a+x ≥ b+x b+x+y where equality holds if y = 0. Noting that b cH c1 = 1 b H ⊥b b b b cH P c + c P c = x + y, the above point implies 1 b 1 1 1 b S S that the generalized eigenvalue corresponding to b c1 is maxib we b mized when P⊥ c = 0. Consequently, for the optimal S 1 b S b b b must have P⊥ b C1 = 0, i.e., span{C1 } ⊆ span{S}, which S completes the proof. ♠ b 12 } = Using Lemma 2 in (14), and noting that span{S b = span{C b 1 }, we have span{S} b Q∗ = S.
(20)
The following lemma reveals another property of the optimal input covariance matrix under the average power constraint. b i.e. Q∗ , the pencil (S b 21 HH HS b 12 + Lemma 3 For the optimal S, b 12 + I) has no generalized eigenvalue less than b 12 GH GS I, S one. Proof: Let bb denote the number of generalized eigenvalues b 21 HH HS b 12 + I , S b 12 GH GS b 21 + I) that are of the pencil (S strictly bigger than 1. We show that the rest of the generalized eigenvalues (nt −bb) are equal to one. From Lemma 2, we have b = rank(C b 1 ) = bb, where C b 1 is a nt × bb matrix reprank(S) resenting generalized eigenvectors corresponding to generalized eigenvalues bigger than 1. Also define the nt × nt − bb b t −b b 2 = [b matrix C c2 1 . . . b c2 nt −bb ], where {b c2i }ni=1 represent ⊥ b . From the definition of an orthonormal basis for span{S} the generalized eigenvalue decomposition, we note that b c2i , i = 1, · · · , nt − bb, is a generalized eigenvector corresponding to a generalized eigenvalue equal to 1. Mathematically, we have b 21 HH HS b 12 + I) b b 12 GH GS b 12 + I) b (S c2i = 1 × (S c2i . b all generalized eigenvectors of the Thus for the optimal S, b 21 HH HS b 21 + I , S b 12 GH GS b 21 + I) correspond to pencil (S generalized eigenvalues either bigger than or equal to 1. We have: i h b1 0 Λ H b 12 H b 21 b b C S H HS + I C = 0 I (21) h 1 i I 0 H b2 H b 21 b b C S G GS + I C = 0 I h i b = C b1 C b 2 is the generalized eigenvector matrix, where C b 1 is a bb × bb diagonal matrix with diagonal elements and Λ representing generalized eigenvalues greater that 1. ♠
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3.2. rank property In the following, another property of the optimal Q∗ is revealed. From (21), a straightforward computation yields H 1 b1 − I 0 Λ H b 12 H b b b 2 C S H H−G G S C= 0, 0 0 from which we have 1 b2C b1 = Λ b1 − I 0 . b HS b 12 HH H − GH G S C 1
(22)
Let bb denote bb = rank(Q∗ ). We note from lemma 2 that b 1 ) = rank(Λ b 1 ) = bb. Denote the singurank(Q∗ ) = rank(C b lar value decomposition of C1 as b 1 = Uc Dc VH , C c
(23)
where Uc is a nt × bb matrix whose columns are an orthonorb 1 }, UH Uc = I, Dc is a bb × bb positive mal basis for span{C c definite diagonal matrix, and Vc is a unitary matrix. Recall b 1 } = span{Q∗ }, the eigenvalue from lemma 2 that span{C ∗ decomposition of Q is written as b = Q∗ = U c D q U H , S c
(24)
where Dq 0 is a bb × bb diagonal matrix. Using (23) and (24) in (22), we have 1 1 H H H 2 Vc Dc Dq2 UH c H H − G G Uc Dq Dc Vc b1 − I 0 , =Λ
which results in H H UH c H H − G G Uc 1
1
− H b −1 − 2 0. = Dq 2 D−1 c Vc (Λ1 − I) Vc Dc Dq
(25)
∗ b The above equality for Hshows that the optimal Q with b = ∗ H H rank(Q ), Uc H H − G G Uc is a bb × bb positive definite matrix.
b we have Theorem 1 For the optimal Q∗ , i.e. S, rank(Q∗ ) ≤ m
(26)
where m is the number of positive eigenvalues of the matrix HH H − GH G. Proof: Letting bb = rank(Q∗ ), we want to show that bb ≤ m. To prove that rank(Q∗ ) = bb ≤ m, we only need to b show that there is no Uc ∈ Cnt ×b , UH c Uc = I, such that H H H Uc H H − G G Uc 0 and bb > m. The proof is obtained by contradiction. We assume bb > m and show that under this assumption, there exist specific non-zero vectors ex
H H H for which eH x Uc H H − G G Uc ex = 0, which contradicts the fact that (25) is positive definite. Denote the eigenvalue decomposition of HH H − GH G as HH H − GH G =
m X
nt X
H λ+ i fi fi +
i=1
[4] A. Khisti and G. Wornell, “Secure transmission with multiple antennas II: The MIMOME wiretap channel,” IEEE Trans. Inf. Theory, vol. 56, no. 11, pp. 5515-5532, 2010.
H λ− i fi fi ,
i=m+1
− where λ+ i , λi
respectively represent positive and non-positive eigenvalues, we have m X H H H UH H H − G G U = λ+ c c i ti ti
+
i=1 nt X
H λ− i ti ti 0 ,
i=m+1 b×1 where ti = UH . Assuming bb > m, define T+ = c fi ∈ C [t1 . . . tm ]. Let PT+ denote the projection matrix on to the space of T+ , and let rb = rank(T+ ). We note that any ti corresponding to λ− i can be written as b
ti = PT+ ti +
b b−b r X
υij ej
i = m + 1, ...nt ,
j=1 b−b r where {ej }j= is a set of orthonormal basis vectors for j=1 span{T+ }⊥ , and υij is a complex scalar. Noting that H eH k ej = 1 if j = k and ek ej = 0 when j 6= k, one can easily confirm that b
i=n Xt H 2 H H λ− eH U H H − G G U e = c k k c i |υik | > 0 i=m+1
k = 1, ..., bb − rb .
[3] F. Oggier and B. Hassibi, “The secrecy capacity of the MIMO wiretap channel,” in Proc. IEEE Int. Symp. Information Theory Toronto, ON, Canada, Jul. 2008, pp. 524-528.
(27)
Recalling that λ− i ≤ 0, (27) leads to a contradiction since the right side of (27) is zero (after setting υik = 0 for nonzero λ− i ), while the left side must be positive. This contradiction comes from the guaranteed existence of non-zero vectors b b−b r {ej }j= when bb is assumed to be bigger than m. Thus, j=1 ∗ rank(Q ) = bb ≤ m. ♠ Remark 2 From Theorem 1, one can easily confirm that the optimal Q∗ can be full rank only in the case that m = nt , i.e. HH H GH G. For all other scenarios, the optimal Q∗ will be low rank. 4. REFERENCES [1] A. Wyner, “The wire-tap channel,” Bell. Syst. Tech. J., vol. 54, no. 8, pp. 1355-1387, Jan. 1975. [2] S. K. Leung-Yan-Cheong and M. E. Hellman, “The Gaussian wire-tap channel,” IEEE Trans. Inf. Theory, vol. 24, pp. 451-456, Jul. 1978.
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[5] J. Li and A. P. Petropulu, “Transmitter optimization for achieving secrecy capacity in Gaussian MIMO wiretap channels,” submitted to IEEE Trans. Info. Theory, Available [online]: http://arxiv.org/PS cache/arxiv/pdf/0909/0909.2622v1.pdf. [6] A. Khisti and G. Wornell, “Secure transmission with multiple antennas I: The MISOME wiretap channel,” IEEE Trans. Inf. Theory, vol. 56, no. 7, pp. 3088-3104, 2010. [7] S. Shafiee and S. Ulukus, “Towards the Secrecy Capacity of the Gaussian MIMO Wire-Tap Channel: The 2-2-1 Channel,” IEEE Trans. on Inf. Theory, vol. 55, no. 9, Sep. 2009. [8] T. Liu and S. Shamai (Shitz), “A note on secrecy capacity of the multi-antenna wiretap channel,” IEEE Trans. Inf. Theory, vol. 55, no. 6, pp. 2547-2553, 2009. [9] R. Bustin, R. Liu, H. V. Poor, and S. Shamai (Shitz), “A MMSE approach to the secrecy capacity of the MIMO Gaussian wiretap channel,” EURASIP Journal on Wireless Communications and Networking, vol. 2009, Article ID 370970, 8 pages, 2009. [10] H. Weingarten, Y. Steinberg, and S. Shamai (Shitz), “The capacity region of the Gaussian multiple-input multipleoutput broadcast channel,” IEEE Trans. Inf. Theory, vol. 52, no. 9, pp. 3936-3964, 2006. [11] R. Liu, T. Liu, H. V. Poor, and S. Shamai, “Multipleinput multiple-output Gaussian broadcast channels with confidential messages,” IEEE Trans. Inf. Theory, vol. 56, no. 9, pp. 4215-4227, 2010. [12] R. A. Horn and C. R. Johnson, Matrix Analysis, University Press, Cambridge, UK, 1999.
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[9]. Oggier and Hassibi [3] and Khisti and Wornell [4] followed an indirect approach using a Sato-like argument and matrix analysis tools. They considered the problem of finding the secrecy capacity under a constraint on the average total power, and a closed-form expression for the secrecy capacity in the high signal-to-noise-ratio (SNR) regime was obtained in [4]. The rank of the optimal input covariance matrix for the secrecy rate maximization problem is discussed in [5], but the authors were unable to characterize the solution for the general case. For some special cases of the MIMO wiretap channel, where the solution has rank one, the optimal input covariance matrix that achieves the secrecy capacity under the average total power constraint was obtained in [5]-[7]. In [8], Liu and Shamai propose a more informationtheoretic approach using the enhancement concept, originally presented by Weingarten et al. [10], as a tool for the characterization of the MIMO Gaussian broadcast channel capacity. Liu and Shamai have shown that an enhanced degraded version of the channel attains the same secrecy capacity as does a Gaussian input distribution. From the mathematical solution in [8] it was evident that such an enhanced channel exists; however it was not clear how to construct such a channel until the work of [9], which provided a closed-form expression for the secrecy capacity under an input covariance matrix constraint. While this result is interesting since the expression for the secrecy capacity is valid for all SNR scenarios, there still exists no computable secrecy capacity expression for the MIMO Gaussian wiretap channel under an average total power constraint. In this paper, we investigate the rank of the optimal input covariance matrix that achieves the secrecy capacity of the general Gaussian multiple-input multiple-output (MIMO) wiretap channel under the average total power constraint, where the number of antennas is arbitrary for both the transmitter and the two receivers. Throughout this analysis, other interesting properties for the optimal input covariance matrix are revealed as well. Notation: Throughout the paper, boldface uppercase letters are used to denote matrices, and vector-valued random variables are written with non-boldface uppercase letters (e.g.,
Asilomar 2012
X), while the corresponding non-boldface lowercase letter (x) denotes a specific realization of the random variable. Scalar variables are written with non-boldface (lowercase or uppercase) letters. The Hermtian (i.e., conjugate) transpose is denoted by (.)H , the matrix trace by Tr(.), and I indicates an identity matrix. Inequality A B means that A − B is Hermitian positive semi-definite. Mutual information between the random variables A and B is denoted by I(A; B), E is the expectation operator, and CN (0, σ 2 ) represents the complex circularly symmetric Gaussian distribution with zero mean and variance σ 2 . The projection matrix onto the column space of X is denoted by PX = X(XH X)−1 XH , and P⊥ X = I − PX denotes the projection onto the space orthogonal to X. span{X} denotes the space spanned by the column vectors of X, and span{X}⊥ denotes the orthogonal complement space of span{X}.
definite matrices 1
1
(S 2 HH HS 2 + I ,
1
1
S 2 GH GS 2 + I) .
(4)
In particular, there exists an invertible generalized eigenvector matrix C such that [12] h 1 i 1 CH S 2 GH GS 2 + I C = I (5)
h 1 i 1 CH S 2 HH HS 2 + I C = Λ
(6)
where Λ = diag{λ1 , ..., λnt } is a positive definite diagonal matrix and λ1 , ..., λnt represent the generalized eigenvalues. Without loss of generality, we assume the eigenvalues are ordered as
2. SYSTEM MODEL AND PRIOR WORK
λ1 ≥ ... ≥ λb > 1 ≥ λb+1 ≥ ... ≥ λnt > 0
We begin with a multiple-antenna wiretap channel with nt transmit antennas and nr and ne receive antennas at the legitimate recipient and the eavesdropper, respectively:
so that a total of b (0 ≤ b ≤ nt ) are greater than 1. Hence, we can write Λ as Λ1 0 Λ= (7) 0 Λ2
yr = Hx + zr
(1)
ye = Gx + ze
where x is a zero-mean nt × 1 transmitted signal vector, zr ∈ Cnr ×1 and ze ∈ Cne ×1 are additive white Gaussian noise vectors at the receiver and eavesdropper, respectively, with i.i.d. entries distributed as CN (0, 1). The matrices H ∈ Cnr ×nt and G ∈ Cne ×nt represent the channels associated with the receiver and the eavesdropper, respectively. Similar to other papers considering the perfect secrecy rate of the wiretap channel, we assume that the transmitter has perfect channel state information (CSI) for both the legitimate receiver and the eavesdropper. For the Gaussian channel, where Gaussian inputs are an optimal choice, the secrecy capacity is given by [3] Csec = max[I(X; Yr ) − I(X; Ye )] = max R(Q) x
Q0
(2)
where R(Q) = log |HQHH + I| − log |GQGH + I|, and Q = E{xxH } is the input covariance matrix. The maximization problem in (2) is on the channel input, where the channel input is under either an average power constraint or an input covariance matrix constraint, as described below. In [9], the above secret communication problem was analyzed under the input covariance constraint, defined as
where Λ1 = diag{λ1 , ..., λb } and Λ2 = diag{λb+1 , ..., λnt }. We can partition C similarly: C = [C1
C2 ]
(8)
where C1 is the nt ×b submatrix representing the generalized eigenvectors corresponding to {λ1 , ..., λb } and C2 is the nt × (nt − b) submatrix representing the generalized eigenvectors corresponding to {λb+1 , ..., λnt }. Using the above notation, the secrecy capacity of the MIMO wiretap channel under the input covariance constraint (3) can be expressed as [9], [11, Theorem 3]: Csec (S) =
b X
log λi = log |Λ1 |
(9)
i=1
where the optimal input covariance matrix Q∗S that maximizes (2) and attains (9) is given by −1 1 1 (CH 0 1 C1 ) Q∗S = S 2 C CH S 2 . (10) 0 0
(3)
Remark 1 From (5) and (6), one can easily confirm that if HH H GH G, then for any S 0 we have Λ I. In other words, in this case the pencil in (4) has no generalized eigenvalue bigger than 1. Thus, Csec (S) = 0 for any S 0.
where S is a positive semi-definite matrix that defines the input covariance constraint. An explicit expression for the secrecy capacity under (3) was obtained via applying the generalized eigenvalue decomposition to the following two positive
Lemma 1 For the case of HH H GH G, for any nt × nt matrix S 0, all the generalized eigenvalues of the pencil 1 1 1 1 (S 2 HH HS 2 + I , S 2 GH GS 2 + I) are strictly bigger than 1, i.e. Λ I, iff S is full rank, i.e. S 0.
Csec (S) = max log |HQHH + I| − log |GQGH + I| s.t. Q S
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Proof: Subtracting (5) from (6), a straightforward computation yields 1 1 S 2 HH H − GH G S 2 = C−H (Λ − I) C−1 . When HH H GH G, both sides of the above equality are positive semi-definite; i.e., for any S 0 we must have Λ I. If S 0, the left side is full-rank and thus Λ I, and vice versa. ♠ In this paper, we consider the secrecy capacity problem in (2) under the average power constraint: Csec (Pt ) = max log |HQHH + I| − log |GQGH + I| s.t. Tr(E{xxH }) = Tr(Q) ≤ Pt . (11)
b and C b 1 have respectively the same definitions as where C those of C and C1 , given by (5)-(8), but here for the pencil b 12 HH HS b 12 + I , S b 12 GH GS b 21 + I). Note that Q∗ can be (S rewritten as " H # h i b H b −1 b 1 C ( C C ) 0 ∗ 1 1 1 b1 C b2 C b2 b 21 Q =S bH S 0 0 C 2
b 12 C b 1 (C b HC b 1 )−1 C bH S b 12 =S 1 1 1
max S0,Tr(S)≤Pt
Csec (S) ,
(12)
where for any given semidefinite S, Csec (S) should be computed as given by (9). Also note that for any covariance constraint S, the optimal input covariance matrix Q∗S that attains Csec (S) is given by (10), and Q∗S S. In the next section, we investigate the rank of the optimal input covariance matrix Q∗ that attains Csec (Pt ).
(14)
b b H b −1 C b H is the projection matrix where PC b 1 = C1 (C1 C1 ) 1 b 1 . Moreover, let P⊥ = I − P b be onto the space of C b1 C1 C b 1 . We have the projection onto the space orthogonal to C
For this constraint, no computable secrecy capacity expression has been derived to date for the general MIMO case. It was shown in [10, Lemma 1], [11] that, the wiretap channel under the average power constraint (11) is equivalent to the channel under the input covariance constraint (3) through an exhaustive search over the set {S : S 0, Tr(S) ≤ P } : Csec (Pt ) =
1
b 2 Pb S b2 =S C1
1
1
b 2 Pb S b2) Tr(Q∗ ) = Tr(S C1 b Pb ) = Tr(S C1
(15)
b Pb Pb ) = Tr(S C1 C1
(16)
b b ) = Tr(PC b 1 S PC 1
(17)
where (15) comes from the fact that for any A ∈ Ca×b and B ∈ Cb×a , Tr(AB) = Tr(BA), and (16) holds because PC b 1 = PC b 1 PC b 1 . Similarly we have b (P b + P⊥ ) b = Tr (P b + P⊥ ) S Tr(S) b b C1 C1 C C 1
1
b b ) + Tr(P⊥ S b P⊥ ) = Tr(PC b 1 S PC b b C C 1
(18)
b ⊥ ) = Tr(Q∗ ) + Tr(P⊥ b S PC b C
(19)
1
1
3. PROPERTIES OF THE OPTIMAL SOLUTION UNDER AN AVERAGE POWER CONSTRAINT
1
1
where in (18) we used the facts that Tr(AB) = Tr(BA) and ⊥ PC b 1 PC b = 0, and where (19) results from (17). 1
For the following results in the paper, we exclude the special case HH H GH G for which the Csec is trivially 0 for any S 0, and consequently for any Pt , as pointed out in Remark 1. Also we note that problem (11) under Tr(Q) ≤ Pt is equivalent to that under Tr(Q) = Pt , see [3, Eq. (9)], [5]. Considering (11) in the form of (12), and recalling that for any S, Q∗S S, thus Tr(Q∗S ) ≤ Tr(S), the above point implies that we can restrict the set of input covariance constraints to S 0, Tr(S) = Pt instead of Tr(S) ≤ Pt . 3.1. span properties b denote the input covariance constraint that maximizes Let S (12). Consequently, from (10), the optimal input covariance matrix that attains Csec (Pt ) is given by1 b H b −1 0 b (C1 C1 ) b HS b 12 , b 12 C C (13) Q∗ = S 0 0 1 To
be consistent with notation in (10), the optimal input covariance mab should be written as Q∗ , which for trix under the covariance constraint S b S simplicity is denoted just as Q∗ .
423
b we have span{C b 1 } = span{S}. b Lemma 2 For the optimal S, Proof: The proof is obtained using (19), and by noting that b we must have Tr(S) b = Tr(Q∗ ) = Pt . for the optimal S b ⊥ ) = 0, or This means that we must have Tr(P⊥ b 1 S PC b1 C b = 0, which consequently shows that equivalently P⊥ S b1 C
b ⊆ span{C b 1 }. To complete the proof we only need span{S} b 1 } ⊆ span{S}. b to show that span{C b 1 , i.e., b b Let c1 denote a column vector of C c1 repreb 12 HH HS b 12 + sents a generalized eigenvector of the pencil (S 1 1 b 2 GH GS b 2 + I) corresponding to a generalized eigenI, S value bigger than 1: 1 b 2 HH HS b 21 + I b b cH S c1 1 1 >1. b 2 GH GS b 12 + I b b cH S c1 1 We can write b c1 as b c1 = PSb b c1 + P⊥ c1 , where PSb = b b S H −1 H b S b S) b b is the projection matrix onto the space of S( S
b and P⊥ = I − Pb is the projection onto the space orS, b S S b Letting a = b b 21 H b 21 c1 and b = thogonal to S. cH 1 S H HS b b 21 H b 12 c1 , we note that a > b, and the only compob cH 1 S G GS b nent of b c1 that affects the values of a and b is PSb b c1 . Now, one can easily show that when a > b, for any x and y ≥ 0, we have a+x+y a+x ≥ b+x b+x+y where equality holds if y = 0. Noting that b cH c1 = 1 b H ⊥b b b b cH P c + c P c = x + y, the above point implies 1 b 1 1 1 b S S that the generalized eigenvalue corresponding to b c1 is maxib we b mized when P⊥ c = 0. Consequently, for the optimal S 1 b S b b b must have P⊥ b C1 = 0, i.e., span{C1 } ⊆ span{S}, which S completes the proof. ♠ b 12 } = Using Lemma 2 in (14), and noting that span{S b = span{C b 1 }, we have span{S} b Q∗ = S.
(20)
The following lemma reveals another property of the optimal input covariance matrix under the average power constraint. b i.e. Q∗ , the pencil (S b 21 HH HS b 12 + Lemma 3 For the optimal S, b 12 + I) has no generalized eigenvalue less than b 12 GH GS I, S one. Proof: Let bb denote the number of generalized eigenvalues b 21 HH HS b 12 + I , S b 12 GH GS b 21 + I) that are of the pencil (S strictly bigger than 1. We show that the rest of the generalized eigenvalues (nt −bb) are equal to one. From Lemma 2, we have b = rank(C b 1 ) = bb, where C b 1 is a nt × bb matrix reprank(S) resenting generalized eigenvectors corresponding to generalized eigenvalues bigger than 1. Also define the nt × nt − bb b t −b b 2 = [b matrix C c2 1 . . . b c2 nt −bb ], where {b c2i }ni=1 represent ⊥ b . From the definition of an orthonormal basis for span{S} the generalized eigenvalue decomposition, we note that b c2i , i = 1, · · · , nt − bb, is a generalized eigenvector corresponding to a generalized eigenvalue equal to 1. Mathematically, we have b 21 HH HS b 12 + I) b b 12 GH GS b 12 + I) b (S c2i = 1 × (S c2i . b all generalized eigenvectors of the Thus for the optimal S, b 21 HH HS b 21 + I , S b 12 GH GS b 21 + I) correspond to pencil (S generalized eigenvalues either bigger than or equal to 1. We have: i h b1 0 Λ H b 12 H b 21 b b C S H HS + I C = 0 I (21) h 1 i I 0 H b2 H b 21 b b C S G GS + I C = 0 I h i b = C b1 C b 2 is the generalized eigenvector matrix, where C b 1 is a bb × bb diagonal matrix with diagonal elements and Λ representing generalized eigenvalues greater that 1. ♠
424
3.2. rank property In the following, another property of the optimal Q∗ is revealed. From (21), a straightforward computation yields H 1 b1 − I 0 Λ H b 12 H b b b 2 C S H H−G G S C= 0, 0 0 from which we have 1 b2C b1 = Λ b1 − I 0 . b HS b 12 HH H − GH G S C 1
(22)
Let bb denote bb = rank(Q∗ ). We note from lemma 2 that b 1 ) = rank(Λ b 1 ) = bb. Denote the singurank(Q∗ ) = rank(C b lar value decomposition of C1 as b 1 = Uc Dc VH , C c
(23)
where Uc is a nt × bb matrix whose columns are an orthonorb 1 }, UH Uc = I, Dc is a bb × bb positive mal basis for span{C c definite diagonal matrix, and Vc is a unitary matrix. Recall b 1 } = span{Q∗ }, the eigenvalue from lemma 2 that span{C ∗ decomposition of Q is written as b = Q∗ = U c D q U H , S c
(24)
where Dq 0 is a bb × bb diagonal matrix. Using (23) and (24) in (22), we have 1 1 H H H 2 Vc Dc Dq2 UH c H H − G G Uc Dq Dc Vc b1 − I 0 , =Λ
which results in H H UH c H H − G G Uc 1
1
− H b −1 − 2 0. = Dq 2 D−1 c Vc (Λ1 − I) Vc Dc Dq
(25)
∗ b The above equality for Hshows that the optimal Q with b = ∗ H H rank(Q ), Uc H H − G G Uc is a bb × bb positive definite matrix.
b we have Theorem 1 For the optimal Q∗ , i.e. S, rank(Q∗ ) ≤ m
(26)
where m is the number of positive eigenvalues of the matrix HH H − GH G. Proof: Letting bb = rank(Q∗ ), we want to show that bb ≤ m. To prove that rank(Q∗ ) = bb ≤ m, we only need to b show that there is no Uc ∈ Cnt ×b , UH c Uc = I, such that H H H Uc H H − G G Uc 0 and bb > m. The proof is obtained by contradiction. We assume bb > m and show that under this assumption, there exist specific non-zero vectors ex
H H H for which eH x Uc H H − G G Uc ex = 0, which contradicts the fact that (25) is positive definite. Denote the eigenvalue decomposition of HH H − GH G as HH H − GH G =
m X
nt X
H λ+ i fi fi +
i=1
[4] A. Khisti and G. Wornell, “Secure transmission with multiple antennas II: The MIMOME wiretap channel,” IEEE Trans. Inf. Theory, vol. 56, no. 11, pp. 5515-5532, 2010.
H λ− i fi fi ,
i=m+1
− where λ+ i , λi
respectively represent positive and non-positive eigenvalues, we have m X H H H UH H H − G G U = λ+ c c i ti ti
+
i=1 nt X
H λ− i ti ti 0 ,
i=m+1 b×1 where ti = UH . Assuming bb > m, define T+ = c fi ∈ C [t1 . . . tm ]. Let PT+ denote the projection matrix on to the space of T+ , and let rb = rank(T+ ). We note that any ti corresponding to λ− i can be written as b
ti = PT+ ti +
b b−b r X
υij ej
i = m + 1, ...nt ,
j=1 b−b r where {ej }j= is a set of orthonormal basis vectors for j=1 span{T+ }⊥ , and υij is a complex scalar. Noting that H eH k ej = 1 if j = k and ek ej = 0 when j 6= k, one can easily confirm that b
i=n Xt H 2 H H λ− eH U H H − G G U e = c k k c i |υik | > 0 i=m+1
k = 1, ..., bb − rb .
[3] F. Oggier and B. Hassibi, “The secrecy capacity of the MIMO wiretap channel,” in Proc. IEEE Int. Symp. Information Theory Toronto, ON, Canada, Jul. 2008, pp. 524-528.
(27)
Recalling that λ− i ≤ 0, (27) leads to a contradiction since the right side of (27) is zero (after setting υik = 0 for nonzero λ− i ), while the left side must be positive. This contradiction comes from the guaranteed existence of non-zero vectors b b−b r {ej }j= when bb is assumed to be bigger than m. Thus, j=1 ∗ rank(Q ) = bb ≤ m. ♠ Remark 2 From Theorem 1, one can easily confirm that the optimal Q∗ can be full rank only in the case that m = nt , i.e. HH H GH G. For all other scenarios, the optimal Q∗ will be low rank. 4. REFERENCES [1] A. Wyner, “The wire-tap channel,” Bell. Syst. Tech. J., vol. 54, no. 8, pp. 1355-1387, Jan. 1975. [2] S. K. Leung-Yan-Cheong and M. E. Hellman, “The Gaussian wire-tap channel,” IEEE Trans. Inf. Theory, vol. 24, pp. 451-456, Jul. 1978.
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[5] J. Li and A. P. Petropulu, “Transmitter optimization for achieving secrecy capacity in Gaussian MIMO wiretap channels,” submitted to IEEE Trans. Info. Theory, Available [online]: http://arxiv.org/PS cache/arxiv/pdf/0909/0909.2622v1.pdf. [6] A. Khisti and G. Wornell, “Secure transmission with multiple antennas I: The MISOME wiretap channel,” IEEE Trans. Inf. Theory, vol. 56, no. 7, pp. 3088-3104, 2010. [7] S. Shafiee and S. Ulukus, “Towards the Secrecy Capacity of the Gaussian MIMO Wire-Tap Channel: The 2-2-1 Channel,” IEEE Trans. on Inf. Theory, vol. 55, no. 9, Sep. 2009. [8] T. Liu and S. Shamai (Shitz), “A note on secrecy capacity of the multi-antenna wiretap channel,” IEEE Trans. Inf. Theory, vol. 55, no. 6, pp. 2547-2553, 2009. [9] R. Bustin, R. Liu, H. V. Poor, and S. Shamai (Shitz), “A MMSE approach to the secrecy capacity of the MIMO Gaussian wiretap channel,” EURASIP Journal on Wireless Communications and Networking, vol. 2009, Article ID 370970, 8 pages, 2009. [10] H. Weingarten, Y. Steinberg, and S. Shamai (Shitz), “The capacity region of the Gaussian multiple-input multipleoutput broadcast channel,” IEEE Trans. Inf. Theory, vol. 52, no. 9, pp. 3936-3964, 2006. [11] R. Liu, T. Liu, H. V. Poor, and S. Shamai, “Multipleinput multiple-output Gaussian broadcast channels with confidential messages,” IEEE Trans. Inf. Theory, vol. 56, no. 9, pp. 4215-4227, 2010. [12] R. A. Horn and C. R. Johnson, Matrix Analysis, University Press, Cambridge, UK, 1999.
