Linear estimation of correlated data in wireless sensor networks with optimum power allocation and analog modulation

In this paper, we study the energy-efficient distributed estimation problem for a wireless sensor network where a physical phenomena that produces correlated data is sensed by a set of spatially distributed sensor nodes and the resulting noisy observations are transmitted to a fusion center via noise- corrupted channels. We assume a Gaussian network model where (i) the data samples being sensed at different sensors have a correlated Gaussian distribution and the correlation matrix is known at the fusion center, (ii) the links between the local sensors and the fusion center are subject to fading and additive white Gaussian noise (AWGN), and the fading gains are known at the fusion center, and (iii) the central node uses the squared error distortion metric. We consider two different distortion criteria: (i) individual distortion constraints at each node, and (ii) average mean square error distortion constraint across the network. We determine the achievable power-distortion regions under each distortion constraint. Taking the delay constraint into account, we investigate the performance of an uncoded transmission strategy where the noisy observations are only scaled and transmitted to the fusion center. At the fusion center, two different estimators are considered: (i) the best linear unbiased estimator (BLUE) that does not require knowledge of the correlation matrix, and (ii) the minimum mean- square error (MMSE) estimator that exploits the correlations. For each estimation method, we determine the optimal power allocation that results in a minimum total transmission power while satisfying some distortion level for the estimate (under both distortion criteria). The numerical comparisons between the two schemes indicate that the MMSE estimator requires less power to attain the same distortion provided by the BLUE and this performance gap becomes more dramatic as correlations between the observations increase. Furthermore, comparisons between power-distortion region achieved by the theoretically optimum system and that achieved by the uncoded system indicate that the performance gap between the two systems becomes small for low levels of correlation between the sensor observations. If observations at all sensor nodes are uncorrelated, the uncoded system with MMSE estimator attains the theoretically optimum system performance.