We consider the distributed estimation of a Gaussian vector with a linear observation model in an inhomogeneous wireless sensor network, in which a fusion center (FC) reconstructs the unknown vector using a linear estimator. Sensors employ uniform multi-bit quantizers and binary PSK modulation, and they communicate with the FC over orthogonal power-and bandwidth-constrained wireless channels.We study transmit power and quantization rate (measured in bits per sensor) allocation schemes that minimize the mean-square error (MSE). In particular, we derive two closed-form upper bounds on the MSE in terms of the optimization parameters and propose "coupled" and "decoupled" resource allocation schemes that minimize these bounds. We show that the bounds are good approximations of the simulated MSE and that the performance of the proposed schemes approaches the clairvoyant centralized estimation when the total transmit power or bandwidth is very large. We investigate how the power and rate allocations are dependent on the sensors' observation qualities and channel gains and on the total transmit power and bandwidth constraints. Our simulations corroborate our analytical results and demonstrate the superior performance of the proposed algorithms. Index TermsDistributed estimation, Gaussian vector, upper bounds on MSE, power and rate allocation, ellipsoid method, quantization, linear estimator, linear observation model.

We consider a multitype branching random walk on d-dimensional Euclidian space. The uniform convergence, as n goes to infinity, of a scaled version of the Laplace transform of the point process given by the nth generation particles of each type is obtained. Similar results in the one-type case, where the transform gives a martingale, were obtained in Biggins (1992) and Barral (2001). This uniform convergence of transforms is then used to obtain limit results for numbers in the underlying point processes. Supporting results, which are of interest in their own right, are obtained on (i) 'Perron-Frobenius theory' for matrices that are smooth functions of a variable λ ∈ L and are nonnegative when λ ∈ L − ⊂ L, where L is an open set in C d , and (ii) saddlepoint approximations of multivariate distributions. The saddlepoint approximations developed are strong enough to give a refined large deviation theorem of Chaganty and Sethuraman (1993) as a by-product.

A generalization of Biggins' Martingale Convergence Theorem is proved for the multi-type branching random walk. The proof appeals to modern techniques involving the construction of size-biased measures on the space of marked trees generated by the branching process. As a simple consequence we obtain existence and uniqueness of solutions (within a speci ed class) to a system of functional equations .

We consider a multitype branching random walk on d-dimensional Euclidian space. The uniform convergence, as n goes to infinity, of a scaled version of the Laplace transform of the point process given by the nth generation particles of each type is obtained. Similar results in the one-type case, where the transform gives a martingale, were obtained in Biggins (1992) and Barral (2001). This uniform convergence of transforms is then used to obtain limit results for numbers in the underlying point processes. Supporting results, which are of interest in their own right, are obtained on (i) 'Perron-Frobenius theory' for matrices that are smooth functions of a variable λ ∈ L and are nonnegative when λ ∈ L − ⊂ L, where L is an open set in C d , and (ii) saddlepoint approximations of multivariate distributions. The saddlepoint approximations developed are strong enough to give a refined large deviation theorem of Chaganty and Sethuraman (1993) as a by-product.

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We consider distributed estimation of a Gaussian source in a heterogenous bandwidth constrained sensor network, where the source is corrupted by independent multiplicative and additive observation noises, with incomplete statistical knowledge of the multiplicative noise. For multi-bit quantizers, we derive the closed-form mean-square-error (MSE) expression for the linear minimum MSE (LMMSE) estimator at the FC. For both error-free and erroneous communication channels, we propose several rate allocation methods named as longest root to leaf path, greedy and integer relaxation to (i) minimize the MSE given a network bandwidth constraint, and (ii) minimize the required network bandwidth given a target MSE. We also derive the Bayesian Cramér-Rao lower bound (CRLB) and compare the MSE performance of our proposed methods against the CRLB. Our results corroborate that, for low power multiplicative observation noises and adequate network bandwidth, the gaps between the MSE of our proposed methods and the CRLB are negligible, while the performance of other methods like individual rate allocation and uniform is not satisfactory.

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