On the analysis and comparison between MPR and cartesian for TDOA localization

Modified polar representation (MPR) provides a unified mathematical framework for both point localization and direction finding in near-field and far-field scenarios. Although prior research has shown that MPR alleviates the range thresholding effect, which refers to the sudden degradation in localization accuracy when the source moves beyond a critical distance, a rigorous theoretical explanation and comprehensive comparison with the Cartesian representation (CR) are still lacking. This paper analyzes the advantages and limitations of MPR and CR for time difference of arrival (TDOA)-based localization under both known and unknown signal propagation speeds (SPS). While the Cramér–Rao lower bound (CRLB) and hybrid Bhattacharyya–Barankin bound (HBBB) have been studied previously for known-SPS scenarios in Wang and Ho (2017), we derive and analyze the HBBB under the unknown-SPS setting. HBBB provides a tighter analytical evaluation beyond the CRLB, so it can quantify the thresholding effect when the source is distant or noise is high. Furthermore, an analytical comparison based on the Hessian of the maximum likelihood (ML) cost function is performed to reveal why MPR is more noise-robust in far-field conditions, whereas CR performs better in the near field—findings supported by the HBBB. Additionally, the far-field case is investigated, establishing the equivalence of MPR with the conventional far-field model in estimating both angles and SPS. These results enhance the theoretical understanding of MPR and underscore its practical implications for localization and sensing applications.