Semidefinite relaxation method for quantized differential time delay localization in underwater sensor networks

To address the limited node energy and acoustic channel bandwidth in underwater sensor networks, a quantized differential time delay localization method is proposed for the joint estimation of target and source positions. First, an observation model considering quantization intervals and channel bit errors is established, and a quantized observation likelihood function with respect to the target and source positions is constructed. Then, auxiliary variables and matrix lifting are introduced, by which the original localization problem is relaxed into a semidefinite programming problem. Furthermore, unquantized measurements are recovered from the relaxation solution, and nonlinear least squares refinement is performed to improve the localization accuracy. Meanwhile, the quantized differential time delay Cramér-Rao bound for the target position is derived to analyze the quantized localization performance. Simulation results show that the proposed method achieves lower localization errors under different noise powers, bit error rates, quantization bits, and numbers of snapshots, with accuracy close to the Cramér-Rao bound. It also provides a higher localization success rate with fewer receiving nodes or unfavorable geometries. Bellhop simulations under shallow- and deep-water scenarios further verify its stability.