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We consider a distributed sensor network in which the single antenna sensornodes observe a deterministic unknown parameter and after encoding the observedsignal with a phase parameter, the sensor nodes transmit it simultaneously to amulti-antenna fusion center (FC). The FC optimizes the phase encoding parameterand feeds it back to the sensor nodes such that the variance of estimationerror can be minimized. We relax the phase optimization problem to asemidefinite programming problem and the numerical results show that theperformance of the proposed method is close to the theoretical bound. Also,asymptotic results show that when the number of sensors is very large and thevariance of the distance between the sensor nodes and FC is small, multipleantennas do not provide a benefit compared with a single antenna system; whenthe number of antennas $M$ is large and the measurement noise at the sensornodes is small compared with the additive noise at the FC, the estimation errorvariance can be reduced by a factor of $M$.

Phase-Only Analog Encoding for a Multi-Antenna Fusion Center