  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
expectation-maximization (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.
