Distributed Estimation for Vector Signal in Linear Coherent Sensor Networks

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(1)IEICE TRANS. COMMUN., VOL.E95–B, NO.2 FEBRUARY 2012. 460. PAPER. Distributed Estimation for Vector Signal in Linear Coherent Sensor Networks∗ Chien-Hsien WU†a) , Member and Ching-An LIN† , Nonmember. SUMMARY We introduce the distributed estimation of a random vector signal in wireless sensor networks that follow coherent multiple access channel model. We adopt the linear minimum mean squared error fusion rule. The problem of interest is to design linear coding matrices for those sensors in the network so as to minimize mean squared error of the estimated vector signal under a total power constraint. We show that the problem can be formulated as a convex optimization problem and we obtain closed form expressions of the coding matrices. Numerical results are used to illustrate the performance of the proposed method. key words: distributed estimation, convex optimization, power allocation, wireless sensor network. 1.. Introduction. Low power consumption and efficient bandwidth usage are two critical issues for distributed estimation in wireless sensor networks [1]. In the distributed estimation scenario, observation data is measured by spatially distributed sensors and transmitted to a fusion center (FC) to generate the final estimate. Due to power and bandwidth limitations, many research works (e.g. [2]–[6]) concentrated on parameter estimation by amplify-and-forward scheme and discussed the problems which restrict the amount of power and information sent from each sensor per observation period. Among these works, optimal power allocation schemes, in the form of optimal power gains, that minimize estimation distortion are proposed based on the orthogonal multiple access channel (MAC) model [2]–[4] as well as the coherent MAC model [5], [6]. All the power allocation schemes mentioned above consider single-input single-output (SISO) systems in which the sensor with a scalar measurement/transmitter is regarded as a unit. Recently, multiple-input multiple-output (MIMO) systems have attracted much attention due to their advantages of increased data rates and improved performance [7]. In MIMO wireless sensor networks, each sensor performs vector measurements and has multiple transmitters. Recent works on MIMO sensor networks addressed the dimensionality reduction problem based on the ideal channel assumption [8]–[10] and the optimal design of coding matrices [11]. Manuscript received April 9, 2011. Manuscript revised October 14, 2011. † The authors are with the Department of Electrical and Control Engineering, National Chiao Tung University, 1001 University Road, Hsinchu 300, Taiwan. ∗ Research sponsored by National Science Council under grant NSC 97-2221-E009-046-MY3. a) E-mail: [email protected] DOI: 10.1587/transcom.E95.B.460. In this paper, we consider the distributed estimation of a vector signal using MIMO wireless sensor networks with the coherent MAC model. The measurement for each sensor is encoded by a coding matrix that forms a message for transmission to a FC, where the signal is estimated based on the linear minimum mean-squared error (LMMSE) rule. We study the problem of designing linear coding matrices that minimize the mean-squared error (MSE) under a total power constraint. We show that the problem can be formulated as a convex optimization problem. The solution, which is a water-filling type, gives closed form expressions for the coding matrices. The organization of this paper is as follows. Section 2 is the network model and the problem statement of linear distributed estimation. In Sect. 3, the proposed method is given and closed form expressions for the coding matrices are shown. Section 4 is the simulation results. Finally, Sect. 5 is a brief conclusion. Notations: Throughout this paper, the following notations are used. A lower case letter denotes a scalar, a boldface lower case letter denotes a vector, and a boldface uppercase letter denotes a matrix. In addition, AT and AH denote the transpose of A and the conjugate transpose of A, respectively. The letter In denotes an identity matrix of size n × n. 0 and 0n×m denote, respectively, a zero vector and a zero matrix of size n × m. The operator diag(x1 , · · · , x M ) is a diagonal matrix with its mth diagonal element equal to xm and tr(A) is the trace of A. 2.. System Model and Problem Formulation. We consider a wireless sensor network consisting of L sensors for estimating p random source signals, written in vector form θ = [θ1 , θ2 , · · · , θ p ]T ∈ C p , as shown in Fig. 1.. Fig. 1. Linear distributed estimation with coherent MAC.. c 2012 The Institute of Electronics, Information and Communication Engineers Copyright .

(2) WU and LIN: DISTRIBUTED ESTIMATION FOR VECTOR SIGNAL IN LINEAR COHERENT SENSOR NETWORKS. 461. The lth sensor has kl measurements, which can be expressed in vector form as xl = Fl θ + nl ,. 1≤l≤L. (1). where Fl ∈ Ckl ×p is the observation matrix and nl ∈ Ckl is L the additive noise. Let K = l=1 kl . In vector form, (1) can be written as x = Fθ + n,. Bl . Let D = tr FH BH BR x BH where we use Al = H−1 l −1 +σ2ν IN BF . Thus J = p σ2θ − σ4θ D. Since σθ and p are fixed, minimization of J subject to (5) can be expressed equivalently as max. Bl , 1≤l≤L. D. subject to. (2). L H −H tr H−1 ≤ P. l Bl R xl Bl Hl. (6). l=1. where x = [xT1 xT2 · · · xTL ]T ∈ CK , F = [FT1 FT2 · · · FTL ]T ∈ CK×p , and n = [nT1 nT2 · · · nTL ]T ∈ CK . The measurement of the lth sensor is encoded by a linear coding matrix Al ∈ CN×kl to form the message vector Al xl ∈ CN , where N denotes the number of messages transmitted from the lth sensor. The message vector is then sent to the fusion center (FC) through the channel which is described by the gain matrix Hl ∈ CN×N . From (1), the received signal y at the FC can be expressed as y=. L . Hl Al (Fl θ + nl ) + ν = B(Fθ + n) + ν,. (3). l=1. where ν ∈ C is additive noise at the receiver and B = [B1 B2 · · · BL ] ∈ CN×K with Bl = Hl Al ∈ CN×kl . We assume that i) E[θ] = 0 and E[θθ H ] = σ2θ I p , ii) E[nl ] = 0, E[nl nlH ] = σ2n Ikl , and E[nl nHj ] = 0 for j l, iii) E[ν] = 0 and E[ννH ] = σ2ν IN , iv) E[θnH ] = 0, E[θνH ] = 0, and E[nνH ] = 0, v) Fl and Hl are known at the FC, vi) K ≥ N, K ≥ p, and rank(F) = p, and vii) Hl = diag(hl1 , · · · , hlN ) with hli 0, ∀i, l. Remark 1: Assumption vii) simplifies the derivations to follow. In practice, if the N signals are transmitted using N different frequencies, the assumption is reasonable. Using the received signal y in (3), the LMMSE estimate of θ is [12, p.382] −1 θˆ =E[θyH ] E[yyH ] y −1 =σ2θ FH BH BR x BH + σ2ν IN y, N. where R x = E[xxH ] = σ2θ FFH + σ2n IK , and the corresponding MSE is ˆ − θ) ˆ H ]) J =tr(E[(θ − θ)(θ −1 =tr σ2θ I p − σ4θ FH BH BR x BH + σ2ν IN BF ,. (4). The problem is to minimize J in (4) by designing the coding matrices Al under a total power constraint. The transmitted power for the lth sensor is defined as E[xlH AlH Al xl ] = tr(Al R xl AlH ), where R xl = E[xl xlH ] = σ2θ Fl FlH + σ2n Ikl . If P is the total power that the L sensors can use, then the constraint can be expressed as L H −H tr H−1 ≤ P, l Bl R xl Bl Hl l=1. (5). Remark 2: In general, if E[θθ H ] = Rθ and E[nl nlH ] = Rnl are Hermitian and positive definite, we can write Rθ = 1/2 1/2 −1/2 ˜ R1/2 and Rnl = R1/2 θ, nl Rnl , and introduce θ = Rθ θ Rθ −1/2 1/2 −1/2 1/2 ˜ ˜ n˜ l = Rnl nl , Al = Al Rnl , and Fl = Rnl Fl Rθ . Then ˜ l , F˜ l , θ, ˜ and Al , Fl , θ, and nl in (3) can be replaced by A n˜ l , respectively, and the equivalent model satisfies assumption i) and ii). Hence, the optimization problem can still be formulated to have the same form as (6). 3.. Proposed Approach. In this section, we first consider the objective function of (6) and determine its maximum through singular value decomposition technique. After obtaining the maximum objective function, we consider the power constraint and formulate the problem (6) as a convex optimization problem, which then yields a solution in closed form. Since R x = σ2θ FFH + σ2n IK is positive definite, it can 1/2 1/2 be expressed as R x = R1/2 is Hermitian x R x , where R x and positive definite. Since rank(F) = p by assumption vi), FFH R−1/2 ) = p and we have rank(R−1/2 x x FFH R−1/2 = UC ΛC UCH , R−1/2 x x. (7). where ΛC = diag(c1 , · · · , c p , 0, · · · , 0) with c1 ≥ · · · ≥ c p > 0 and UC ∈ CK×K is unitary. Multiplying (7) by R−1/2 x on the right and R1/2 on the left shows that the nonzero x eigenvalues ci , 1 ≤ i ≤ p, are also the nonzero eigenvalues of FFH (σ2θ FFH + σ2n IK )−1 and it follows that ci ≤ 1/σ2θ , 1 ≤ i ≤ p. Let B¯ = BR1/2 x and use (7), the objective function D in (6) can be expressed as −1 ¯ −1/2 FFH R−1/2 ) D =tr(B¯ H B¯ B¯ H + σ2ν IN BR x x −1 ¯ C ΛC UCH ). (8) =tr(B¯ H B¯ B¯ H + σ2ν IN BU Express B¯ as a singular value decomposition B¯ = UB¯ ΛB¯ VHB¯ ,. (9) √. √. where UB¯ ∈ CN×N is unitary, ΛB¯ = diag( b1 , · · · , bN ), b1 ≥ · · · ≥ bN ≥ 0, and VB¯ ∈ CK×N has orthonormal columns. With (9), D in (8) can be further simplified as −1 (10) D = tr(ΛB¯ Λ2B¯ + σ2ν IN ΛB¯ VHB¯ UC ΛC UCH VB¯ ). To find an upper bound on (10), we need the follow fact..

(3) IEICE TRANS. COMMUN., VOL.E95–B, NO.2 FEBRUARY 2012. 462. Fact. [13, p.326] Let X, Y ∈ Cn×n be positive semidefinite matrices with eigenvalues λ1 ≥ λ2 ≥ · · · ≥ λn ≥ 0 and δ1 ≥ δ2 ≥ · · · ≥ δn ≥ 0 respectively. Then tr(XY) ≤. n . λi δi .. i=1. −1 Applying the fact to (10) with X = ΛB¯ ΛB¯ ΛB¯ +σ2ν IN ΛB¯ and Y = VHB¯ UC ΛC UCH VB¯ , we have D≤. p i=1. ci bi . σ2ν + bi. (11). We note that since X is diagonal, if we choose VB¯ so that VHB¯ UC = [IN 0] ∈ CN×K , where N ≥ p, then equality in (11) holds. Hence the upper bound in (11) is achieved if we choose VB¯ = UC (:, 1 : N),. (12). where UC (:, 1 : N) denotes the first N columns of UC . Since the upper bound is the same for all N > p, we choose N = p. This keeps the number of transmitters for each sensor at a minimum. With the choice VB¯ = UC (:, 1 : p), we obtain D=. p i=1. ci bi σ2ν + bi. (13). where ci > 0 and σ2ν are fixed. Hence, the problem of choosing Bl in√B to maximize D amounts to choosing the singular values bi of B¯ in (9) since the choice of UB¯ is irrelevant ¯ −1/2 . It is clear from (13) that to maximize D, and B = BR x we should choose each bi > 0 as large as possible; however, it can not be chosen arbitrarily large due to the total power constraint. By taking into account the power constraint in (6), we set UB¯ = IN to simplify our analysis and thus we have B = , or equivalently, ΛB¯ VB¯ R−1/2 x ˆ C, l , Bl = ΛB¯ U. 1 ≤ l ≤ L,. (14). ˆ C, l ∈ C p×kl is the lth block matrix in VH¯ R−1/2 = where U B x ˆ C,L ] with VB¯ = UC (:, 1 : p). Substituting (14) ˆ C,1 , · · · , U [U into (6), since Hl are diagonal matrices from assumption vii), the power constraint can be further simplified as ¯ 2¯ = tr RΛ B. p . ri bi ≤ P,. (15). i=1 L ˆ ˆ H −H with diagonal entries ¯ = l=1 H−1 where R l UC, l R xl UC, l Hl ri , 1 ≤ i ≤ p. Note that since R xl = σ2θ Fl FlH + σ2n Ikl , we see ˆ ˆ H −H are positive that the diagonal entries of H−1 l UC, l R xl UC, l Hl and thus ri > 0. From (13) and (15), the problem in (6) with VB¯ = UC (:, 1 : p) and UB¯ = IN can be written as. min −. bi , 1≤i≤p. p i=1. ci bi 2 σν + bi. subject to. p . ri bi ≤ P,. (16). i=1. bi ≥ 0. i = 1, · · · , p. This is a convex optimization problem since the cost function is convex and the constraints are linear. To solve the problem (16), we form the Lagrangian as ⎞ ⎛ p p p ⎟⎟⎟ ⎜⎜⎜ ci bi ⎟⎟⎠ − ⎜ + μ r b − P μi bi , L(bi , μ0 , μi ) = − ⎜ 0 i i ⎝ σ2ν + bi i=1 i=1 i=1 where μ0 ≥ 0 and μi ≥ 0, and the associated KKT conditions [14] are ci σ2ν + μ0 r i − μi = 0 (σ2ν + bi )2 ⎛ p ⎞ ⎟⎟ ⎜⎜⎜⎜ μ0 ⎜⎝ ri bi − P⎟⎟⎟⎠ = 0 −. (17) (18). i=1. μi bi = 0. (19). From (17), we obtain ci 2 bi = σν −1 . σ2ν (μ0 ri + μi ). (20). From (19), if bi > 0, we have μi = 0 and thus (20) can be written as + ci 2 bi = σν −1 (21) σ2ν μ0 ri where (x)+ = max(0, x). From (17) with bi > 0 and μi = 0, we have μ0 > 0 otherwise we have a contradiction μ0 < 0. Hence, from (18), we get p . ri bi = P.. (22). i=1. Let wmi = cmi /rmi and assume wm1 ≥ · · · ≥ wm p , where mi ∈ {1, · · · , p}. We define a function n P + m i=m1 ri σ2ν f (mn ) = wmn × m . n ri ci /σ2ν i=m1 Let 1 ≤ p1 ≤ p be such that f (m p1 ) > 1 and f (m p1 +1 ) ≤ 1. Then we have ⎧ 2 ⎪ σν 2 ⎪ ⎨ μ0 wmi − σν , i ≤ p1 (23) bmi = ⎪ ⎪ ⎩ 0, i > p1 where ⎛ m p ⎞2 1 ⎜⎜⎜ ri ci /σ2ν ⎟⎟⎟⎟ i=m1 ⎜ ⎟⎟⎟ μ0 = ⎜⎜⎜⎝ P m p + i=m1 1 ri ⎠ σ2 ν. is obtained by substituting (23) into √ (22). With the choice ΛB¯ = diag( b1 , · · · , b p ) according.

(4) WU and LIN: DISTRIBUTED ESTIMATION FOR VECTOR SIGNAL IN LINEAR COHERENT SENSOR NETWORKS. 463. to (23), the coding matrix can be written from (14) and Al = H−1 l Bl , and is given by ˆ Al = H−1 l Λ B¯ UC, l ,. 1 ≤ l ≤ L;. (24). moreover, from (4) and (13), the corresponding MSE can be written as J = pσ2θ − σ4θ. p . ci bi . + bi. σ2ν i=1. (25). As P → ∞, from (15), we have bi → ∞ and thus the lower bound of the MSE is Jlow =. p σ2θ. −. σ4θ. p . ci ,. (26). i=1. Note that since ci ≤ 1/σ2θ , we have Jlow ≥ 0. 4.. Numerical Results. In this section, we use numerical simulations to illustrate the analytical results established in the previous section. In all simulations, the random vectors θ, nl , and ν, are complex Gaussian. Specifically, each entry in the vectors is set as a complex Gaussian random variable with zero mean and unit variance. We also take the entries of Fl and the channels hli as complex Gaussian random variables with zero mean and unit variance. The number of source signals is set as p = 5. The average MSE versus N with different power levels P = 5 dB, 10 dB, and 20 dB is shown in Fig. 2, in which we take L = 10 and kl = 8, ∀l. The dash-line denotes the MSE lower bound in (26). We can see that the MSE decreases as N increases and remains a constant as N ≥ p. That is, there is no improvement in network performance as the number of transmitters for each sensor is greater than the number of source signals. The result is consistent with our analysis in Sect. 3 that under a total power constraint, the minimum achieved MSE is (25) for all N > p. From the figure we can also see that the MSE decreases as the power level increases. Fig. 2. MSE versus N with different power levels.. as is expected. Since the performance is the same for all N > p, we set N = p in all the simulations to follow. To see the effect of the number of measurements, we consider L = 10 and assume that kl are equal for all sensors. Figure 3 shows the average MSE versus kl with different power levels P = 5 dB, 10 dB, and 20 dB. We note that as kl increases, the MSEs decreases; moreover, the gap between the lower bound and the power constraint case becomes large as kl increases. This is because although the increase of kl leads to the increase of measurement power, the transmitted power for each sensor is restricted if there is a power constraint. Hence, the performance of the power constraint case has no significant improvement as kl increases compared with that of the unconstraint power case (the lower bound). The comparison of the MSE for the proposed method and the equal power method is plotted in Fig. 4, in which we take L = 10 and kl = 8, ∀l. The equal power method is to set the transmitted powers for all sensors to be equal, that is, we choose the coding matrix Al = αl · [I5 0], where. Fig. 3. MSE versus kl with different power levels.. Fig. 4 MSE comparison between the proposed method and the equal power method..

(5) IEICE TRANS. COMMUN., VOL.E95–B, NO.2 FEBRUARY 2012. 464. αl = P/[Ltr(R xl (1 : 5, 1 : 5))], so that tr(Al R xl AlH ) = P/L, 1 ≤ l ≤ L, here R xl (1 : 5, 1 : 5) denotes the first 5 rows and columns of R xl and [I5 0] is a 5 × 8 matrix with its diagonal entries equal to 1 and other entries equal to 0. As we can see from the figure, the proposed method performs better than the equal power method. Moreover, the MSE of the proposed method goes to the lower bound as P increases while the MSE of the equal power method approaches a constant MSE for P > 25 dB. This is because the proposed method takes into account the effects of the observation and channel matrices in the design of coding matrices, while the equal power method does not use the information of observation matrices and channel matrices and thus the performance improvement is limited as P increases. To estimate a vector signal, [5] proposed an amplifyand-forward scheme based on the SISO sensor network. For comparison, we simulate the scheme in [5], where a scalar measurement xl at the lth sensor is multiplied by an amplified factor al (n) at the nth time instant before it is transmitted to the receiver through a fading channel hl , 1 ≤ l ≤ Q, here Q denotes the total number of sensors in Qthe SISO network. hl al (n)xl +νl (n), At time n, the received signal is y(n) = l=1 where νl (n) is an additive noise. After collecting p received signals {y(1), · · · , y(p)} at the FC, the source vector θ is then estimated based on the LMMSE fusion rule. The proposed MIMO scheme is modelled in (3) with L = 1, a single vector sensor. To compare these two schemes based on the equal condition, we assume the number of measurements in the proposed scheme is equal to the scheme in [5], that is, kl = Q; moreover, the total network powers, consumed by the sensors, are set equal for both schemes. Figure 5 shows that the average MSE of the proposed scheme with kl = 10 is lower than that of the scheme in [5] with Q = 10 when power P is small. This is because in our MIMO sensor network, the antenna diversity are used and thus each received signal comes from individual channel; while in [5], the received signal is a linear combination of transmitted signals without using antenna diversity. From the figure, we can also see that as the power increases, the performance of two. Fig. 5. MSE comparison between the proposed scheme and that in [5].. schemes are very close. Moreover, for the proposed scheme with kl = 15 and the scheme in [5] with Q = 15, the MSEs are almost the same for over all P. In this simulation, we see that as total power less than 20 dB and the number of measurements less than 15, the proposed method has a noticeable improvement compared with the method in [5]. 5.. Conclusion. We study distributed estimation of a random vector signal in power-constrained MIMO sensor networks. We find an upper bound of the objective function and show that the minimum number of transmitters to reach the upper bound is equal to the number of the source signals. By choosing specific singular vectors, we formulate the original problem as a convex optimization problem. The solution then yields closed form expressions for the coding matrices. The proposed MIMO scheme can be viewed as an extension of the SISO scheme proposed in [5]. From simulation results, we see that the improvement in performance of the proposed MIMO scheme over the SISO scheme in [5] is more significant when the transmitted power is not too high and the number of measurements is not too large, while in other cases, they have almost identical performance. Acknowledgement We thank the reviewers for insightful comments that improve the paper. References [1] J.J. Xiao, A. Ribeiro, Z.Q. Luo, and G.B. Giannakis, “Distributed compression-estimation using wireless sensor networks,” IEEE Signal Process. Mag., vol.23, no.4, pp.27–41, July 2006. [2] S. Cui, J.J. Xiao, A.J. Goldsmith, Z.Q. Luo, and H.V. Poor, “Estimation diversity and energy efficiency in distributed sensing,” IEEE Trans. Signal Process., vol.55, no.9, pp.4683–4695, Sept. 2007. [3] I. Bahceci and A.J. Khandani “Linear estimation of correlated data in wireless sensor networks with optimum power allocation and analog modulation,” IEEE Trans. Commun., vol.56, no.7, pp.1146– 1156, July 2008. [4] H. S¸enol and C. Tepedelenlio˘glu, “Performance of distributed estimation over unknown parallel fading channels,” IEEE Trans. Signal Process., vol.56, no.12, pp.6057–6068, Dec. 2008. [5] W. Guo, J.-J. Xiao, and S. Cui, “An efficient water-filling solution for linear coherent joint estimation,” IEEE Trans. Signal Process., vol.56, no.10, pp.5301–5305, Oct. 2008. [6] C.H. Wu and C.A. 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(6) WU and LIN: DISTRIBUTED ESTIMATION FOR VECTOR SIGNAL IN LINEAR COHERENT SENSOR NETWORKS. 465. pp.4339–4353, Aug. 2010. [11] J.-J. Xiao, S. Cui, Z.-Q. Luo, and A.J. Goldsmith, “Linear coherent decentralized estimation,” IEEE Trans. Signal Process., vol.56, no.2, pp.757–770, Feb. 2008. [12] S.M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory, Prentice-Hall PTR, 1993. [13] D.S. Bernstein, Matrix Mathematics: Theory, Facts, and Formulas with Allpication to Linear Systems Theory, Princeton University Press, 2005. [14] S. Boyd and L. Vanderberghe, Convex Optimization, Cambridge Univ. Press, Cambridge, U.K., 2003.. Chien-Hsien Wu received the B.S. degree in electrical engineering from the National Taiwan Ocean University, Keelong, Taiwan, in 2001, and the M.S. degree in electrical and control engineering from the National Chiao Tung University, Hsinchu, Taiwan, in 2003. He is currently a Ph.D. student with the Department of Electrical and Control Engineering, National Chiao Tung University. His current research interests are in signal processing.. Ching-An Lin received the B.S. degree from the National Chiao Tung University, Hsinchu, Taiwan, in 1977, the M.S. degree from the University of New Mexico, Albuquerque, in 1980, and the Ph.D. degree from the University of California, Berkeley, in 1984, all in electrical engineering. He was with the Chung Shan Institute of Science and Technology from 1977 to 1979 and with Integrated Systems Inc. from 1984 to 1986. Since June, 1986, he has been with the Department of Electrical and Control Engineering, National Chiao Tung University, where he is a Professor. His current research interests are in control and signal processing..

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