Source Localization via Doppler Shifts Using Mobile Sensors in ICNets Within Industry 5.0

Source localization plays a significant role in industrial 5.0 applications by availing of the communication networks. For the industrial communication networks (ICNets), Doppler shifts can be measured inexpensively by equipping with some mobile sensors. This paper investigates the localization problem of an unknown source using only Doppler shift (DS) when the signal carrier frequency is unavailable. To deal with the DS-only localization under unknown knowledge of carrier frequency, we first propose a semidefinite programming (SDP) solution by applying the convex relaxation technique. The complexity of the SDP solution is high. We also propose a closed-form solution for estimating both the source position and the carrier frequency. Using the weighted least squares (WLS) method, the closed-form solution is segmented into two stages. A bias-compensated scheme is incorporated to reduce the bias of the estimates in the stage-one WLS solution. Subsequently, the root mean square error (RMSE) performance is improved in the stage-two WLS solution, and we design the bias-compensated two-stage WLS (BCTSWLS) solution. Experiments have demonstrated that, compared to traditional localization methods with known carrier frequency, our approach–utilizing SDP and BCTSWLS–effectively solves the localization problem in high-noise environments. This results in greater robustness and accuracy in practical industrial applications. Specifically, in scenarios with fewer sensors or unknown signal frequency, our method effectively reduces bias, achieving accuracy levels close to the Cramér-Rao Lower Bound (CRLB), thereby demonstrating significant performance advantages.