Projection-Based Algorithm and Performance Analysis for TDOA Localization in MPR

In addition to unifying the positioning of a source that yields the coordinates if the source is near and the direction of arrival (DOA) if the source is far, the Modified Polar Representation (MPR) of the source position has a distinct advantage for decoupling the estimation of angle and range. This paper explores such characteristics of MPR for source localization and develops two algorithms for separate estimation through the subspace projection techniques. The first algorithm, null-space projection (NSP), estimates the object direction by itself after subspace projection and then uses the estimated direction to obtain the inverse-range. The second algorithm, independent NSP (INSP), determines the object direction and inverse-range independently from each other. While exploiting separated estimation, we show by analysis that NSP is optimum in reaching the Cramér Rao Lower bound (CRLB) performance under Gaussian noise. Indeed, we have proven NSP is equivalent to general trust region subproblem (GTRS) solution of joint estimation in theory, although both come from different approaches and perspectives. INSP has the attractive benefit of independent estimation and is suitable for distributed or parallel processing, albeit its performance is suboptimal. Furthermore, we have evaluated the theoretical estimation bias of NSP that can be subtracted from NSP to yield the NSP bias subtraction (NSPBS) solution, having bias performance better than the Maximum Likelihood Estimator (MLE). Simulation validates the performance and analysis. While NSP is equivalent to GTRS in theory, NSP has a lower complexity and can be a little more resilient to noise in practice.