Alleviating Sensor Position Error in Source Localization Using Calibration Emitters at Inaccurate Locations

A previous study shows that the use of a calibration emitter whose position is known exactly can significantly reduce the loss in time differences of arrival (TDOA) based source localization accuracy when the available sensor positions have random errors. This paper extends the previous work to a more practical scenario where the exact position of a calibration emitter is not known. By modeling the calibration position error as additive Gaussian noise, the amount of reduction in localization accuracy due to calibration position error is derived through Cramer-Rao lower bound (CRLB) analysis. In addition, the analysis also affirms the previous studies on Bayesian sensor network localization that it remains possible to improve the localization accuracy even if the calibration position is completely unknown. Next, a performance analysis illustrates that the penalty could be very high if one simply pretends the calibration position is accurate and ignores its error. A closed-form solution is then developed by accounting for the calibration position error and it is proved analytically to reach the CRLB accuracy when the sensor and calibration position errors are small relative to the distance between the calibration emitter and the sensor. Finally, the results are generalized to the case where multiple calibration emitters are available. When deploying multiple calibration emitters, although their positions may not be known exactly, we show that it is possible to completely eliminate the sensor position error and recover the best localization accuracy that is limited by the measurement noise in TDOAs only. All the theoretical developments are corroborated by simulations.