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Blind Decentralized Estimation for Bandwidth Constrained Wireless Sensor Networks Tuncer C. Aysal, Student Member, IEEE, and Kenneth E. Barner, Senior Member, IEEE

Abstract— Recently proposed decentralized, distributed estimation and power scheduling methods for wireless sensor networks (WSNs) do not consider errors occurring during the transmission of binary observations from the sensors to fusion center. In this letter, we extend the decentralized estimation model to the case in which imperfect transmission channels are considered. The proposed estimators, which operate on additive channel noise corrupted versions of quantized noisy sensor observations, are approached from a maximum likelihood (ML) perspective. Complicating this approach is the fact that the noise distribution is rarely fully known to the fusion center. Here we assume the distribution is known but not the defining parameters, e.g., variance. The resulting incomplete data estimation problem is approached from a expectation–maximization (EM) perspective. The critical initialization and convergence aspects of the EM algorithm are investigated. Furthermore, the estimation of the source parameter is extended to the blind case where both the channel and sensor noise parameters are unknown. Finally, numerical experiments are provided to show the effectiveness of the proposed estimators.

(estimator based on unquantized measurements) [11], [12], [16]. Alternatively, when the sensor noise pdf is unknown, pdf–unaware estimators based on quantized sensor data have also been introduced recently [9], [12]. The distributed estimation techniques considered in the previously proposed methods are based on quantized noisy sensor observations. These methods subsequently assume that the transmission of binary observations from sensors to fusion center is perfect. Recently, we have extended the decentralized estimation model to admit additive transmission noise channels and derived optimal and closed-to-optimal suboptimal solutions [17]. However, in [17], we have assumed that all sensing and channel noise parameters are known at the fusion center, an assumption clearly not satisfied in practice.

Index Terms— Sensor networks, imperfect channels, maximum likelihood, expectation–maximization, Gaussian mixtures, binary observations.

In this letter, we consider the case where the quantized noisy sensor observations are corrupted by additive noise during transmission from sensors to fusion center and derive practical and effective algorithms that do not need sensing and channel noise parameters at the fusion center. The considered estimator is hence based on noisy quantized versions of noisy sensor observations. Utilizing this extended WSN model, we derive the maximum likelihood (ML) estimate of a deterministic source signal. This formulation is complicated by the fact that the noise statistics are rarely known entirely in practice. Here we consider the practical case in which the noise pdf is known (e.g., Gaussian) but some parameters of the distribution are unknown. For instance, a case frequently encountered in practice is when the noise pdf is known except for the value of some defining parameters [9]. Moreover, the unlabeled nature of the fusion center observations makes the problem at hand a typical task with incomplete data. We focus on the so-called EM algorithm, which has attracted a great deal of interest over the past few years in a wide range of application involving tasks with incomplete data sets [18], [19]. We integrate the EM algorithm to solve the estimation problem where the channel noise parameter is unknown. The critical initialization and convergence aspects of the EM algorithm are investigated. Furthermore, the estimation of the source parameter is extended to blind case where both the channel and sensor noise parameters are unknown. Finally, numerical experiments are provided to show the effectiveness of the proposed estimators.

I. I NTRODUCTION

T

HE problem of decentralized estimation has been studied in the context of distributed control [1], [2] and tracking [3] and most recently wireless sensor networks [4]–[12]. WSNs comprise a large number of geographically distributed nodes characterized by power constraints and limited computation capability. While a number of earlier works address sensor collaboration for distributed detection [4]–[7], the challenging problem of distributed estimation has only recently received attention [8]–[12]. In distributed estimation for WSNs, each sensor has available a subset of the observations that must be transmitted to a central node, or fusion center. Various WSN implementations and quantizer design issues are considered in [8], [10], [13], [14]. A constraint in many WSNs is that bandwidth is limited, necessitating the use and transmission of quantized binary versions of the original noisy observations. Many recent efforts address the estimation of a deterministic source signal from quantized noisy observations [8], [10]–[12], [15]. When the probability density function (pdf) of the sensor noise is known, transmitting a single bit per sensor leads to minimal loss in estimator variance compared with the clairvoyant estimator

Manuscript received September 12, 2006; revised June 25, 2007 and December 3, 2007; accepted March 3, 2008. The associate editor coordinating the review of this paper and approving it for publication was R. Fantacci. The authors are with the Department of Electrical and Computer Engineering, University of Delaware, Newark, DE 19716 USA (e-mail: {aysal, barner}@mail.eecis.udel.edu). Digital Object Identifier 10.1109/TWC.2008.060687.

A. Summary of Contributions

B. Paper Organization The remainder of this letter is organized as follows. The problem formulation and the extended WSN model admitting

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transmission noise are introduced in Section II. The estimator of a deterministic source signal utilizing the corrupted quantized noisy sensor observations, assuming unknown channel noise parameter, is derived and presented in Section III, along with an analysis of EM initialization and convergence, and an extension of the algorithm to the blind case. Numerical experiments are provided in Section IV. Finally, the conclusions are drawn in Section V.

Consider a set of K distributed sensors, each making observations of a deterministic source signal θ. The observations are corrupted by additive noise and are described by [7]–[10], [12] x(k) = θ + n(k), k = 1, 2, . . . , K. (1) Noise samples {n(k) : k = 1, 2, . . . , K} are assumed zero– mean, spatially uncorrelated and independent. Furthermore, the density function of the sensor noise is denoted by n(k) ∼ fn (u; σn ), where σn denotes the common scale parameter of fn . Suppose a fusion center is to estimate θ based on the noisy sensor observations {x(k) : k = 1, 2, . . . , K}. If the fusion center has knowledge of the sensor noise density function and sensors are capable of sending the observations {x(k) : k = 1, 2, . . . , K} to the fusion center without distortion, then the fusion center can simply perform the ML estimate of θ,  N  ˆ ρ(x(k) − θ) (2) θ = arg min k=1

where ρ(u) = − log fn (u; σn ). This scheme is only applicable in a centralized estimation situation where observations are either centrally located, or can be transmitted to a central location without distortion. Neither of these requirements is realistic in a WSN, where the sensor nodes are bandwidth constrained and the communication links between the fusion center and sensors are noisy. Due to the inherent bandwidth limitations, the {x(k) : k = 1, 2, . . . , K} observations are quantized. To this end, we consider the quantization operation as the construction of a set of indicator variables, which are binary observations [7]–[10], [12], b(k) = 1{x(k) ∈ (τk , +∞)},

k = 1, 2, . . . , K

(3)

where τk ∈ R is a threshold defining b(k), R denotes the set of real numbers, and 1{·} is the indicator function. In addition, due to imperfections in the communication links between the sensor nodes and fusion center, we further extend the model to include channel noise, y(k) = b(k) + w(k),

k = 1, 2, . . . , K

where σ denotes the spread parameter. III. E STIMATION BASED ON N OISY B INARY O BSERVATIONS

II. P ROBLEM F ORMULATION

θ

sensors and the fusion center, are modeled as additive white Gaussian noise (AWGN), the defining pdf of which is given by   u2 1 (5) f (u) = √ exp − 2 2σ σ 2π

(4)

where the {w(k) : k = 1, 2, . . . , K} are assumed to be zero–mean independent channel noise samples, {y(k) : k = 1, 2, . . . , K} are the noisy observations received at the fusion center, and on–off keying is adopted. Moreover, the density function of the link noise is denoted by w(k) ∼ fw (u; σw ), where σw denotes the common scale parameter of fw . The channels between the source signal and the sensors, and the

Consider the most demanding bandwidth constraint case, in which sensors are restricted to transmit one bit per x(k) observation. Furthermore, let every sensor use the same threshold τ to form {b(k) : k = 1, 2, . . . , K}, i.e., b(k) = 1{x(k) ∈ (τ, +∞)}, k = 1, 2, . . . , K. Instrumental to the WSN scheme presented in Section II is the fact that b(k) is a Bernoulli random variable with parameter ψ(θ)  Pr{b(k) = 1} = 1 − Fn (τ − θ)

(6)

where Fn (·) is the cumulative distribution function of n(k). The probability density function of the noisy observations received at the fusion center, i.e., y(k) = b(k) + w(k), for k = 1, 2, . . . , K, is then given by fy (y) = aw (y)[1 − Fn (τ − θ)] + bw (y)

(7)

where aw (y)  [fw (y − 1) − fw (y)] and bw (y)  fw (y). An inspection of the pdf of the observed random variable reveals that y can be modeled as a two–component Gaussian mixture model: fy (u) = Fn (τ − θ)fw (u) + [1 − Fn (τ − θ)]fw (u − 1) (8) where Fn (τ − θ) and [1 − Fn (τ − θ)] are the mixing probabilities and fw (u) and fw (u − 1) are the mixing densities. A realistic approach to the estimation problem in WSNs is to assume that the noise pdf is known (e.g., Gaussian) but that the value of some parameters are unknown [9]. A case frequently encountered in practice is when the noise pdf is known except for its parameter σ (or equivalently its variance). In the following, we consider the estimation problem when the channel noise parameter σw is unknown but the sensor noise parameter σn is known. A. Estimation in Unknown Channel and Known Sensor Noise Given the pdf of the y, the ML estimate of θ is given by θˆML = arg max θ

= arg max θ

K  k=1 K 

fy (y(k)|θ)

(9)

Fn (τ − θ)fw (y(k))

k=1

+ [1 − Fn (τ − θ)]fw (y(k) − 1)

(10)

where fy (·|·) denotes the conditional probability. Let us define ψ  ψ(θ) = 1 − Fn (τ − θ). Note that the ψ is the probability that the binary sensor observation b(k) is unity, i.e., ψ(θ) = Pr{b(k) = 1}, and is restricted to the open interval (0, 1). To simplify the problem, we first derive the estimate for ψ and

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then utilize the invariance of the ML estimate to determine θ using (6). The ML estimate of ψ ∈ (0, 1) thus reduces to ψˆML = arg max Λ(y|ψ) ψ

= arg max ψ

K 

(11)

[1 − ψ]fw (y(k)) + ψfw (y(k) − 1) (12)

k=1

where y = {y(k) : k = 1, 2, . . . , K}. Taking the natural log(·) of the above yields the log–likelihood function, denoted as ΛL (y|ψ), and the ML estimate of ψ is then given by ψˆML = arg max ΛL (y|ψ) ψ

= arg max ψ

K  k=1

(j+1) (j) (j+1) σw,ML − spread parameters. If |ψˆML − ψˆML | > 1 and |ˆ (j) σ ˆw,ML | > 2 , where |·| and i for i = 1, 2, denote the absolute value operator and positive small numbers, respectively, the algorithm returns to the E–step. Otherwise, the iteration is ended. More detail on the EM algorithm and its applications to mixture density parameter estimation–type problems can be found in [18], [19]. Given the estimate of ψ, and utilizing the facts that Fn (·) is a bijection and the ML estimates are invariant, the ML estimate of θ is given by

ψˆML = 1 − Fn (τ − θˆML ) = Fn (θˆML − τ ) ⇒θˆML = F −1 (ψˆML ) + τ

(13)

n

log([1 − ψ]fw (y(k)) + ψfw (y(k) − 1)). (14)

The unknown parameter set for the above estimation is p = {ψ, σw }. Due to the lack of information concerning the labels of {y(k) : k = 1, 2, . . . , K}, the summation formulation and the unknown channel parameter, the typical ML estimation encounters difficulty. The missing label information makes the problem at hand a typical task with an incomplete data set. We focus on addressing this problem utilizing the so-called EM algorithm, which has attracted a great deal of interest over the past few years in a wide range of application involving tasks with incomplete data sets [18]–[20]. Each iteration of the EM algorithm consists of two processes: The E–step and the M–step. In the expectation, or E-step, the missing data are estimated given the observed data and current estimate of the model parameters. This is achieved using the conditional expectation, explaining the choice of terminology. In the M–step, the likelihood function is maximized under the assumption that the missing data are known. The estimate of the missing data from the E– step are used in lieu of the actual missing data [18]–[20]. The followings are the M– and E– steps for the unknown parameter set estimation of finite Gaussian mixture models in the considered WSN application. (E–Step) – Let the parameters estimated at the j–th iteration be marked by a superscript (j). Compute the posterior probabilities (j) ψˆML fw (y(k) − 1|p(j) ) . (j) (j) ψˆML fw (y(k) − 1|p(j) ) + (1 − ψˆML )fw (y(k)|p(j) ) (15) (j+1) (j+1) ˆ (j+1) = {ψˆML , σ (M–step) – The ML estimates, p ˆw,ML } are given by ∆(q) (j+1) (16) ψˆML = K K where ∆(q)  k=1 q(k) and q = {q(1), q(2), . . . , q(K)}, and  1/2 (j+1) σ ˆw,ML = (K)−1/2 ∆(q  (y − 1)2 ) + ∆((1 − q)  y2 ) (17) where  and (·)2 denote the element–wise multiplication and squaring operations, respectively. Note that (17) is a special case of the EM algorithm derived for a two–component Gaussian mixture model with different

q(k) =

3

(18) (19)

where ψˆML is the solution obtained through the EM algorithm and the second equality follows from the fact that the Gaussian distribution is symmetric. 1) EM Initialization: The initialization of the ψ and σw is an important step of the EM algorithm. Consider first the (0) initialization of ψˆML . The ψML is set to: ∆(y) . (20) K This setting is justified as follows. By the law of large numbers, ∆(y) (0) → E{y} (21) ψML = K almost surely, which implies that (0)

ψML =

(0)

ψML → E{b} + E{w} = E{b} = ψ

(22)

where the above follows from the fact that the channel noise is zero–mean. Due to the finite number of fusion center (0) observations, the initial estimate ψML ≈ ψ. Consider next the initialization of the estimate of σw . Note that 2 (23) φ(y) = [1 − Fn (τ − θ)]Fn (τ − θ) + σw since φ(b) = [1 − Fn (τ − θ)]Fn (τ − θ)

(24)

where φ(·) denotes the variance of its argument. Solving for σw yields

σw = φ(y) − [1 − Fn (τ − θ)]Fn (τ − θ). (25) Utilizing the unbiased ML estimator of the variance of y and the loose upper bound [1 − Fn (τ − θ)]Fn (τ − θ) ≤ 1/4

(26)

gives

σ ˆw ≥

2 K  1  1 ∆(y) −  L(σw ). (27) y(k) − K −1 K 4 k=1

Clearly, the upper bound of σw is estimated by 2 K  1  ∆(y) σ ˆw ≤  U(σw ). y(k) − K −1 K

(28)

k=1

Noting that the L(σw ) can take on imaginary numbers if the term inside the square root is negative, the σ ˆw,ML is hence

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initialized as

⎧ ⎪ ⎨ L(σw ) + U(σw ) , Im(L(σw )) = 0 (0) 2 (29) σ ˆw,ML = U(σw ) ⎪ ⎩ , Im(L(σw )) = 0 2 where Im(·) denotes the imaginary part of its argument. 2) Convergence Analysis: Note that the EM algorithm converges to a local (global if there is unique extrema) extrema of the log-likelihood function [18], [19]. It is also shown that it is possible for the EM algorithm to converge to local extrema or saddle points in unusual cases [19]. In the following, we prove that the log–likelihood function is concave in ψ, guaranteeing convergence to the global maximum. The log–likelihood function is rewritten as ΛL (ψ) =

K 

log(aw (y(k))ψ + bw (y(k)))

(30)

k=1

where aw (y(k)) = fw (y(k) − 1) − fw (y(k)), and bw (y(k)) = fw (y(k)). Note that aw (y(k))ψ +bw (y(k)) is a linear concave function in ψ, and that aw (y(k))ψ+bw (y(k)) is monotonically decreasing (increasing) if aw (y(k)) < 0(> 0). Also, recall that log(·) is concave, indicating that log(aw (y(k))ψ + bw (y(k))) is concave in ψ. Finally, noting that the summation preserves concavity concludes the proof. An estimator that requires the least amount of information is one that assumes that the sensor noise parameter σn is, along with the channel noise parameter σw , unknown. This is the case considered in the following. B. Estimation in Unknown Channel and Sensor Noise Consider the estimation of θ in the worst case, where both the sensor noise parameter σn and the channel noise parameter σw are unknown. To estimate θ when σn and σw are unknown, while keeping the bandwidth constraint to one bit per sensor, we divide the sensors in two groups, with each group using a different region (i.e. threshold) to define the binary random observations [9]:  (τ1 , +∞)  S1 , k = 1, . . . , K 2 S(k)  . (31) (τ2 , +∞)  S2 , k = K 2 + 1, . . . , K That is, without loss of generality, the first K/2 sensors quantize their observations using the region S1 , while the remaining K/2 sensors utilize the region S2 . Furthermore, we assume, without loss of generality that τ2 > τ1 . The Bernoulli parameters of the resultant binary observations are expressed in terms of the cdf of the the standard Gaussian random variable, ⎧   τ1 − θ ⎪ ⎪ k = 1, . . . , K  ψ1 , ⎨ 1 − Fs 2 σ n   ψ(k)  . τ − θ ⎪ 2 ⎪ + 1, . . . , K  ψ2 , k = K ⎩ 1 − Fs 2 σn (32) Given the noise independence across sensors, the ML estimates of ψ1 and ψ2 are found, respectively, as the solutions to EM algorithms operating on y1 = {y(1), y(2), . . . , y(K/2)} and y2 = {y(K/2 + 1), y(K/2 + 2), . . . , y(K)}. Mimicking the derivations in Section III-A, we invert Fs (·) and invoke

the invariance property of the ML estimate to obtain the ML estimate of θ [9]: F −1 (1 − ψˆ2,ML )τ1 − Fs−1 (1 − ψˆ1,ML )τ2 θˆML = s −1 Fs (1 − ψˆ2,ML ) − Fs−1 (1 − ψˆ1,ML )

(33)

where ψˆi,ML denotes the ML estimates obtained through the EM algorithm. IV. N UMERICAL E XPERIMENTS This section presents numerical experiments, first analyzing the performance of EM algorithm in WSN problems and second evaluating the performances of EM–based ML estimator with unknown channel parameter (MLU) and the EM– based ML estimator with unknown channel and sensor noise parameters, i.e., the blind ML estimator (MLB). Results are compared to the variance of the clairvoyant estimator (CE), the estimator operating directly on the analog observations, to the variance of binary estimator (BE) [8], [9], [11], the estimator operating directly on the quantized noisy observations, and to the variance of the ML estimator with known channel and sensor parameters (MLK) [17], the estimator operating on the noisy quantized noisy observations. Consider a fusion center operating in a WSN with parameters, τ = 0.5, σn = 1, σw = 0.25 and K = 1000, where the source parameter to be estimated is θ = 0.25. The channel noise parameters are unknown at the fusion center whereas the sensor noise parameters are known. The (normalized) histogram of the fusion center observations are given in Fig. 1 (a). The true pdf of the observed random variable y and the estimated pdf, obtained utilizing the EM procedures, are given in Fig. 1 (b). Note that the estimated pdf closely follows the true pdf. Consider next a fusion center operating in a WSN with parameters τ = 0, σn = 1, σw = 0.5 and K = {250, 500, 1000, 2000, 4000}, where the source parameter to be estimated is θ = 1. The variances of the MLU and MLB are plotted in Fig. 2 (a), along with the variances of the CE, BE and MLK [17]. Note that the MLB performance loss compared to that of the MLU, and the MLU performance loss compared to that of the MLK, are marginal. Also, the estimators exhibits the expected performance order, i.e., φ(θˆCE ) < φ(θˆBE ) < φ(θˆMLK ) < φ(θˆMLU ) < φ(θˆMLB ) (34) where φ(·) denotes the variance of the corresponding random variable. Also considered is the case where the estimators are operating in a WSN with varying SNR, defined as SNR  2 . The MLB, MLU, MLK, BE and CE variances are θ2 /σw plotted in Fig. 2 (b) for K = 1000. Note that in this case, the CE and BE variances appear flat since they consider perfect transmission. Also, the performances of MLU and MLK approach the performance of BE as SNR increases, whereas the performance of the MLB decreases. This is expected since the extended WSN scheme, in this case, reduces to a WSN that disregards the transmission errors occurring between the sensors and the fusion center. Also investigated is the performance of the MLB algorithm with respect to number of iterations carried out in the EM algorithm and with respect to varying threshold; Fig 3. Note

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 5, MAY 2008

1.4

0.04

1.2

0.035 Output Variance

0.8 0.6

0.025 0.02 0.015

0.4

0.01

0.2

0.005

0 −1

CE BE MLK MLU MLB

0.03

1 Histogram

5

−0.5

0

0.5

Value

1

1.5

0 250

2

500

1000 2000 Number of Sensors

(a)

(a) 1.4

Theoretical Estimated

1.2 1 0.8 0.6

CE BE MLK MLU MLB

0.05

Output Variance

Theoretical and Estimated Density Function

4000

0.04 0.03 0.02

0.4 0.01 0.2 0 −1

−0.5

0

0.5 Value

1

1.5

2

(b) Fig. 1. A fusion center is operating on a WSN with parameters: θ = −0.25, τ = 0.5, σn = 1, σw = 0.25 and K = 1000: (a) Histogram of the fusion center observations and (b) Theoretical and Estimated pdf of the observed random variable.

that the EM algorithm converges after approximatively two iterations, showing the effectiveness of both the initialization parameters and the EM algorithm. Moreover, the effect of the thresholding is consistent with the findings in [9]. If thresholds τi , i = 1, 2 are chosen close to θ, relative to the noise standard deviation, the variance decreases and if, |τi − θ| ≈ 0, where i ∈ {1, 2}, the variance increases. It is also noted that the worst case variance, in the case of perfect transmission, is minimized by setting τ1 ≈ θmin and τ1 ≈ θmax , if θ ∈ [θmin , θmax ] [9]. V. C ONCLUDING R EMARKS The decentralized WSN estimation scheme is extended to admit imperfections occurring during the data transmission from sensors to fusion center. Based on the extended decentralized estimation scheme, a maximum likelihood estimator operating on corrupted quantized noisy sensor observations is proposed. A case frequently encountered in practice, when the noise pdf is known except for its parameter (or equivalently its variance), is considered. Due to the lack of information concerning the labels of fusion center observations, the summation formulation of the probability density functions, and

0 −3.5

−2

0

2 SNR (dB)

4

6

(b) Fig. 2. A WSN with the following set of parameters: τ = 0, σn = 1, σw = 0.5, K = 1000 and θ = 1. Variances of MLB, MLU, MLK, BE and CE for (a) varying K and (b) varying SNR (dB).

the missing channel parameter information, the typical ML estimation encounters difficulty. The missing label information makes the addressed problem a typical task with an incomplete data set, which we approach from a expectation–maximization (EM) perspective. The critical initialization and convergence aspects of the EM algorithm are investigated. Furthermore, the estimation of the source parameter is extended to the blind case in which both the channel and sensor noise parameters are unknown. Provided numerical examples show the near– optimal performance of the algorithms and the fact that the variance of the estimators increase slightly compared to the estimator with known channel parameter/known channel and sensor parameter. Finally, it should be noted that although the provided simulations utilize the common Gaussian distribution, represented derivations and algorithm development are general, indicating that they can be applied to any (closed form) distribution cases. R EFERENCES [1] D. Castanon and D. Teneketzis, “Distributed estimation algorithms for nonlinear systems,” IEEE Trans. Autom. Contr., vol. AC–30, no. 5, pp. 418–425, May 1985.

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−3

4.2

x 10

Output Variance

4.1 4 3.9 3.8 3.7 3.6 0

1

2 3 Number of Iterations

4

5

(a) 0.02 0.019

[3] A. S. Willsky, M. Bello, D. Castanon, B. Levy, and G. Verghese, “On the complexity of decentralized decision making and detection problems,” IEEE Trans. Autom. Contr., vol. AC–27, no. 4, pp. 799–813, Aug. 1982. [4] P. K. Varshney, Distributed Detection and Data Fusion. New York: Springer–Verlag, 1997. [5] R. Niu, B. Chen, and P. K. Varshney, “Fusion of decisions transmitted over rayleigh fading channels in wireless sensor networks,” IEEE Trans. Signal Processing, vol. 54, no. 3, pp. 1018–1027, Mar. 2006. [6] V. V. Veeravalli, T. Basar, and V. H. Poor, “Minimax robust decentralized detection,” IEEE Trans. Inform. Theory, vol. 40, no. 1, pp. 35–40, Jan. 1994. [7] J.-J. Xiao and Z.-Q. Luo, “Universal decentralized detection in a bandwidth-constrained sensor network,” IEEE Trans. Signal Processing, vol. 53, no. 8, pp. 2617–2624, Aug. 2005. [8] A. Ribeiro and G. B. Giannakis, “Bandwidth–constrained distributed estimation for wireless sensor networks–part i: Gaussian case,” IEEE Trans. Signal Processing, vol. 54, no. 3, pp. 1131–1143, Mar. 2006. [9] ——, “Bandwidth-constrained distributed estimation for wireless sensor networks-part ii: unknown probability density function,” IEEE Trans. Signal Processing, vol. 54, no. 7, pp. 2784–2796, July 2006. [10] J.-J. Xiao, S. Cui, Z.-Q. Luo, and A. J. Goldsmith, “Power scheduling of universal decentralized estimation in sensor networks,” IEEE Trans. Signal Processing, vol. 54, no. 2, pp. 413–422, Feb. 2006. [11] H. Papadopoulos, G. Wornell, and A. Oppenheim, “Sequential signal encoding from noisy measurements using quantizers with dynamic bias

Output Variance

0.018 0.017

[12]

0.016

[13]

0.015

[14]

0.014 0.013 0.012

[15] 1.4

1.6

τ2

1.8

2

2.2

[16]

(b) Fig. 3. A fusion center is operating on a WSN with parameters: θ = 1, τ1 = 0, τ2 = 2, σn = 1, σw = 0.5 and K = 1000. The performance of the MLB algorithm (a) with respect to number of iteration taken in the EM algorithm and (b) with respect to τ2 ∈ [θ + 0.25, θ + 1.25] and σw = 1.

[17] [18] [19]

[2] J. L. Speyer, “Computation and transmission requirements for a decentralized linear–quadratic–gaussian control problem,” IEEE Trans. Autom. Contr., vol. AC–24, no. 2, pp. 266–269, Apr. 1979.

[20]

control,” IEEE Trans. Information Theory, vol. 47, no. 3, pp. 978–1002, Mar. 2001. Z.-Q. Luo, “An isotropic universal decentralized estimation scheme for a bandwidth constrained ad hoc sensor network,” IEEE J. Select. Areas Commun., vol. 23, no. 4, pp. 735–744, Apr. 2005. J. Gubner, “Distributed estimation and quantizer design,” IEEE Trans. Inform. Theory, vol. 39, no. 4, pp. 1456–1459, July 1993. W. Lam and A. Reibman, “Quantizer design for decentralized systems with communication constraints,” IEEE Trans. Commun., vol. 41, pp. 1602–1605, Aug. 1993. J.-J. Xiao and Z.-Q. Luo, “Decentralized estimation in an inhomogeneous sensing environment,” IEEE Trans. Inform. Theory, vol. 51, no. 10, pp. 3564–3575, Oct. 2005. M. Abdallah and H. Papadopoulos, “Sequential signal encoding and estimation for distributed sensor networks,” in Proc. 2001 IEEE International Conference on Acoustics, Speech, and Signal Processing, Salt Lake City, UT, 2001, pp. 2577–2580. T. C. Aysal and K. E. Barner, “Constrained decentralized estimation over noisy channels for sensor networks,” IEEE Trans. Signal Processing, vol. 56, no. 4, Apr. 2008. A. Dempster, N. Laird, and D. Rubin, “Maximum likelihood from incomplete data via the em algorithm,” J. Royal Statistical Society, Series B, vol. 39, no. 1, pp. 1–38, Nov. 1977. G. McLachlan and T. Krishnan, The EM Algorithm and Extensions. New York: John Wiley & Sons, 1996. R. O. Duda, P. E. Hart, and D. G. Stork, Pattern Classification, 2nd ed. New York: Wiley, 2001.

Blind Decentralized Estimation for Bandwidth ...

Bandwidth Constrained Wireless Sensor Networks. Tuncer C. Aysal ...... 1–38, Nov. 1977. [19] G. McLachlan and T. Krishnan, The EM Algorithm and Extensions.

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