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A novel distributed algorithm for estimating the maximum of the node initial state values in a network, in the presence of additive communication noise is proposed. Conventionally, the maximum is estimated locally at each node by updating the node state value with the largest received measurements in every iteration. However, due to the additive channel noise, the estimate of the maximum at each node drifts at each iteration and this results in nodes diverging from the true max value. Max-plus algebra is used as a tool to study this ergodic process. The subadditive ergodic theorem is invoked to establish a constant growth rate for the state values due to noise, which is studied by analyzing the max-plus Lyapunov exponent of the product of noise matrices in a max-plus semiring. The growth rate of the state values is upper bounded by a constant which depends on the spectral radius of the network and the noise variance. Upper and lower bounds are derived for both fixed and random graphs. Finally, a two-run algorithm robust to additive noise in the network is proposed and its variance is analyzed using concentration inequalities. Simulation results supporting the theory are also presented.Index Terms-Max consensus, spectral radius, max-plus algebra, wireless sensor networks.

In this paper, localization using narrowband communication signals are considered in the presence of fading channels with time of arrival measurements. When narrowband signals are used for localization, due to existing hardware constraints, fading channels play a crucial role in localization accuracy. In a location estimation formulation, the Cramer-Rao lower bound for localization error is derived under different assumptions on fading coefficients. For the same level of localization accuracy, the loss in performance due to Rayleigh fading with known phase is shown to be about 5dB compared to the case with no fading. Unknown phase causes an additional 1dB loss. The maximum likelihood estimators are also derived.In an alternative distributed detection formulation, each anchor receives a noisy signal from a node with known location if the node is active. Each anchor makes a decision as to whether the node is active or not and transmits a bit to a fusion center once a decision is made. The fusion center combines all the decisions and uses a design parameter to make the final decision. We derive optimal thresholds and calculate the probabilities of false alarm and detection under different assumptions on the knowledge of channel information. Simulations corroborate our analytical results.

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