Distributed Estimation of a Parametric Field: Algorithms and Performance Analysis

Abstract

This paper presents a distributed estimator for a deterministic parametric physical field sensed by a homogeneous sensor network and develops a new transformed expression for the Cramer-Rao lower bound (CRLB) on the variance of distributed estimates. The proposed transformation reduces a multidimensional integral representation of the CRLB to an expression involving an infinite sum. Stochastic models used in this paper assume additive noise in both the observation and transmission channels. Two cases of data transmission are considered. The first case assumes a linear analog modulation of raw observations prior to their transmission to a fusion center. In the second case, each sensor quantizes its observation to M levels, and the quantized data are communicated to a fusion center. In both cases, parallel additive white Gaussian channels are assumed. The paper develops an iterative expectationmaximization (EM) algorithm to estimate unknown parameters of a parametric field, and its linearized version is adopted for numerical analysis. The performance of the developed numerical solution is compared to the performance of a simple iterative approach based on Newton's approximation. While the developed solution has a higher complexity than Newton's solution, it is more robust with respect to the choice of initial parameters and has a better estimation accuracy. Numerical examples are provided for the case of a field modeled as a Gaussian bell, and illustrate the advantages of using the transformed expression for the CRLB. However, the distributed estimator and the derived CRLB are general and can be applied to any parametric field. The dependence of the mean-square error (MSE) on the number of quantization levels, the number of sensors in the network and the SNR of the observation and transmission channels are analyzed. The variance of the estimates is compared to the derived CRLB.Index Terms-Wireless sensor network, EM algorithm, maximum-likelihood estimation, Cramer-Rao lower bound, distributed parameter estimation.

Distributed Vector Estimation for Power- and Bandwidth-Constrained Wireless Sensor Networks

“…For estimating a deterministic parameter, [6] investigated a bit allocation scheme that minimizes the mean-square error (MSE) when the FC utilizes the best linear unbiased estimator (BLUE) subject to a total bit constraint. Several researchers relaxed the assumption of communication channels being error free [9], [10]. For estimating a vector of deterministic parameters, [9] studied an expectation-maximization algorithm and the CRLB when sensors employ fixed and identical multi-bit quantizers and the communication channel model is additive white Gaussian noise (AWGN).…”

“…Several researchers relaxed the assumption of communication channels being error free [9], [10]. For estimating a vector of deterministic parameters, [9] studied an expectation-maximization algorithm and the CRLB when sensors employ fixed and identical multi-bit quantizers and the communication channel model is additive white Gaussian noise (AWGN). A related problem was studied by [10], in whose work the FC employs a spatial BLUE for field reconstruction and the MSE is compared with a posterior CRLB.…”

Optimal Energy Allocation Scheme in Distributed Estimation for Wireless Sensor Networks over Rayleigh Fading Channels

“…This is not very realistic because the links between the sensors and the FC are affected by attenuation and fading that can degrade estimation performance. In [8][9][10][11], the authors consider an imperfect channel modelled as a Binary Symmetric Channel (BSC) and an Additive White Gaussian Noise (AWGN) channel. However, the channels are not fading in either of the cases.…”

“…(iii) this paper investigates the effect of the sensor noise observation, which has been not considered in previous efforts [6][7][8][9][10][11].…”

Parameter estimation from multiple sensors with mixed resolution of quantization

“…[8], [9]), and with target location estimation with quantized measurements, with different quantization threshold but same quantization resolution [10]. Our work is different mainly for the treatment of sensors with mixed forms of sensing the physical field and the mixed quantization resolution of their measurements.…”