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In this paper, we consider distributed estimation of an unknown random vector by using wireless sensors and a fusion center (FC). We adopt a linear model for distributed estimation of a vector source where both observation models and sensor operations are linear and the multiple access channel (MAC) is coherent. Two cases are considered: Noiseless fusion center and Noisy fusion center. In the case of a Noiseless fusion center (where there is no noise at the fusion center), the sensor precoders are designed to minimize the mean square error (MSE) at the fusion center. A closed form solution is found and it is shown that the system performance approaches the benchmark as long as the number of messages transmitted by each sensor is equal to the length of the vector source. Subsequently, if there is noise at the fusion center and one is interested in a closed form solution, a filter is designed to cancel out the effect of noise at the fusion center. This separate design will come at the expense of losing performance. An alternative iterative solution can be found when considering noise at the fusion center where sensor precoders are designed to minimize the MSE at the fusion center subject to transmit power constraints at each sensor. It is shown that the proposed scheme always converges. Finally, simulations are provided to verify the analysis and present the performance of the proposed schemes.