Improved Solutions for 3-D Localization Using 1-D Bearing Measurements

Localizing a 3-D source using bearing measurements from linear arrays (LAs) is a recent technological innovation. In contrast to traditional bearing-only localization using planar arrays (PAs), LA-based positioning is simpler and provides a potentially available option. Existing studies on LA-enabled 3-D localization are either confined to specific array deployment (SAD) or computationally heavy. Also, they disregard the influence of bias, which can significantly degrade the localization performance under high noise power or poor geographic distribution. This article focuses on improving the performance and reducing the complexity of LA-based localization, proposing a lower complexity semidefinite programming solution with correction (SDP-C) and a residual successive minimization (RSM) in closed form. SDP-C permits LAs placed arbitrarily. It leverages the semidefinite relaxation (SDR) to obtain a coarse solution, which is subsequently corrected by compensating the approximated deviation. RSM tackles the optimization problem by successively minimizing the residuals, albeit designed for the case of arranging LAs in the same direction. We conduct the mean-square analysis for SDP-C and RSM, as well as the bias analysis for RSM. The analytical covariances of SDP-C and RSM reach the Cramér–Rao lower bound (CRLB) if the noise is mild. Subtracting the theoretical bias from RSM leads to the bias-subtracted RSM (BSRSM) that lowers the bias further. Analytical and numerical comparisons of complexity validate the proposed methods are more computationally efficient. Simulations corroborate the analysis, showing performance superiority in terms of mean-square error (MSE), bias, and computational load.