Robust TDoA Localization via Quadratic Weighted Least Squares in Degenerate Geometries

Authors

Abstract

This letter presents a zone-optimized Quadratic Weighted Least Squares (QWLS) algorithm for time-difference-of-arrival (TDoA)-based localization using wireless sensors. QWLS extends previously developed Parametric Three-Stage Weighted Least Squares (P3WLS) method by introducing a generalized formulation that remains robust under a broad range of eigenvalue conditions. Unlike P3WLS, which fails when the smallest eigenvector of the system matrix becomes nearly tangent to the TDoA constraint surface, QWLS introduces a second-order correction in the eigenbasis to enforce tangency and maintain physical consistency. We focus on the specific “worst-case” zones where P3WLS becomes unstable due to near-degenerate geometry and demonstrate that QWLS significantly improves localization accuracy in these regions. Simulation results across a mixed indoor-outdoor sensor deployment show that QWLS consistently outperforms both standard Weighted Least Squares and the original P3WLS method in structured failure zones

Additional Files

Published

2026-10-26

Issue

Vol. 72 No. 4 (2026)

Section

Digital Signal Processing

License

Copyright (c) 2026 International Journal of Electronics and Telecommunication

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

1. License

The non-commercial use of the article will be governed by the Creative Commons Attribution license as currently displayed on https://creativecommons.org/licenses/by/4.0/.

2. Author’s Warranties

The author warrants that the article is original, written by stated author/s, has not been published before, contains no unlawful statements, does not infringe the rights of others, is subject to copyright that is vested exclusively in the author and free of any third party rights, and that any necessary written permissions to quote from other sources have been obtained by the author/s. The undersigned also warrants that the manuscript (or its essential substance) has not been published other than as an abstract or doctorate thesis and has not been submitted for consideration elsewhere, for print, electronic or digital publication.

3. User Rights

Under the Creative Commons Attribution license, the author(s) and users are free to share (copy, distribute and transmit the contribution) under the following conditions: 1. they must attribute the contribution in the manner specified by the author or licensor, 2. they may alter, transform, or build upon this work, 3. they may use this contribution for commercial purposes.

4. Rights of Authors

Authors retain the following rights:

- copyright, and other proprietary rights relating to the article, such as patent rights,

- the right to use the substance of the article in own future works, including lectures and books,

- the right to reproduce the article for own purposes, provided the copies are not offered for sale,

- the right to self-archive the article

- the right to supervision over the integrity of the content of the work and its fair use.

5. Co-Authorship

If the article was prepared jointly with other authors, the signatory of this form warrants that he/she has been authorized by all co-authors to sign this agreement on their behalf, and agrees to inform his/her co-authors of the terms of this agreement.

6. Termination

This agreement can be terminated by the author or the Journal Owner upon two months’ notice where the other party has materially breached this agreement and failed to remedy such breach within a month of being given the terminating party’s notice requesting such breach to be remedied. No breach or violation of this agreement will cause this agreement or any license granted in it to terminate automatically or affect the definition of the Journal Owner. The author and the Journal Owner may agree to terminate this agreement at any time. This agreement or any license granted in it cannot be terminated otherwise than in accordance with this section 6. This License shall remain in effect throughout the term of copyright in the Work and may not be revoked without the express written consent of both parties.

7. Royalties

This agreement entitles the author to no royalties or other fees. To such extent as legally permissible, the author waives his or her right to collect royalties relative to the article in respect of any use of the article by the Journal Owner or its sublicensee.

8. Miscellaneous

The Journal Owner will publish the article (or have it published) in the Journal if the article’s editorial process is successfully completed and the Journal Owner or its sublicensee has become obligated to have the article published. Where such obligation depends on the payment of a fee, it shall not be deemed to exist until such time as that fee is paid. The Journal Owner may conform the article to a style of punctuation, spelling, capitalization and usage that it deems appropriate. The Journal Owner will be allowed to sublicense the rights that are licensed to it under this agreement. This agreement will be governed by the laws of Poland.

By signing this License, Author(s) warrant(s) that they have the full power to enter into this agreement. This License shall remain in effect throughout the term of copyright in the Work and may not be revoked without the express written consent of both parties.