Skip to main content

Main menu

  • Home
  • Current Issue
  • Archive
  • About Us
    • About NAVIGATION
    • Editorial Board
    • Peer Review Statement
    • Open Access
  • More
    • Email Alerts
    • Info for Authors
    • Info for Subscribers
  • Other Publications
    • ion

User menu

  • My alerts

Search

  • Advanced search
NAVIGATION: Journal of the Institute of Navigation
  • Other Publications
    • ion
  • My alerts
NAVIGATION: Journal of the Institute of Navigation

Advanced Search

  • Home
  • Current Issue
  • Archive
  • About Us
    • About NAVIGATION
    • Editorial Board
    • Peer Review Statement
    • Open Access
  • More
    • Email Alerts
    • Info for Authors
    • Info for Subscribers
  • Follow ion on Twitter
  • Visit ion on Facebook
  • Follow ion on Instagram
  • Visit ion on YouTube
Research ArticleOriginal Article
Open Access

Weiss–Weinstein Bound of Frequency Estimation Error for Very Weak GNSS Signals

Xin Zhang, Xingqun Zhan, Jihong Huang, Jiahui Liu, and Yingchao Xiao
NAVIGATION: Journal of the Institute of Navigation September 2024, 71 (3) navi.654; DOI: https://doi.org/10.33012/navi.654
Xin Zhang
School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai, China
  • Find this author on Google Scholar
  • Find this author on PubMed
  • Search for this author on this site
Xingqun Zhan
School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai, China
  • Find this author on Google Scholar
  • Find this author on PubMed
  • Search for this author on this site
Jihong Huang
School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai, China
  • Find this author on Google Scholar
  • Find this author on PubMed
  • Search for this author on this site
Jiahui Liu,
School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai, China
  • Find this author on Google Scholar
  • Find this author on PubMed
  • Search for this author on this site
Yingchao Xiao
School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai, China
  • Find this author on Google Scholar
  • Find this author on PubMed
  • Search for this author on this site
  • Article
  • Figures & Data
  • Supplemental
  • References
  • Info & Metrics
  • PDF
Loading

REFERENCES

  1. ↵
    1. Bacharach, L.,
    2. Renaux, A.,
    3. Korso, M. N. E., &
    4. Chaumette, É.
    (2017). Weiss–Weinstein bound on multiple change-points estimation. IEEE Transactions on Signal Processing, 65(10), 2686–2700. https://doi.org/10.1109/TSP.2017.2673804
  2. ↵
    1. Barankin, E. W.
    (1949). Locally best unbiased estimates. The Annals of Mathematical Statistics, 20(4), 477–501. https://doi.org/10.1214/aoms/1177729943
  3. ↵
    1. Bell, K. L.,
    2. Steinberg, Y.,
    3. Ephraim, Y., &
    4. Van Trees, H. L.
    (1997). Extended Ziv-Zakaï lower bound for vector parameter estimation. IEEE Transactions on Information Theory, 43(2), 624–637. https://doi.org/10.1109/18.556118
  4. ↵
    1. Bellini, S., &
    2. Tartara, G.
    (1974). Bounds on error in signal parameter estimation. IEEE Transactions on Communications, 22(3), 340–342. https://doi.org/10.1109/TCOM.1974.1092192
  5. ↵
    1. Bellini, S., &
    2. Tartara, G.
    (1975). Correction to ‘Bounds on error in signal parameter estimation’. IEEE Transactions on Communications, 23(4), 486. https://doi.org/10.1109/TCOM.1975.1092812
  6. ↵
    1. Berens, P.
    (2009). CircStat: A MATLAB toolbox for circular statistics. Journal of Statistical Software, 31 (10), 1–21. https://doi.org/10.18637/jss.v031.i10
    CrossRefPubMed
  7. ↵
    1. Betz, J.
    (2016). Engineering satellite-based navigation and timing: Global navigation satellite systems, signals, and receivers. John Wiley & Sons, Inc. http://doi.org/10.1002/9781119141167
  8. ↵
    1. Bhattacharyya, A.
    (1946). On some analogues of the amount of information and their use in statistical estimation. Sankhyā: The Indian Journal of Statistics, 8(1), 1–14.
  9. ↵
    1. Bobrovsky, B., &
    2. Zakaï, M.
    (1975). A lower bound on the estimation error for Markov processes. IEEE Transactions on Automatic Control, 20(6), 785–788. https://doi.org/10.1109/TAC.1975.1101088
  10. ↵
    1. Bobrovsky, B., &
    2. Zakaï, M.
    (1976). A lower bound on the estimation error for certain diffusion processes. IEEE Transactions on Information Theory, 22(1), 45–52. https://doi.org/10.1109/TIT.1976.1055513
  11. ↵
    1. Brown, L. D., &
    2. Liu, R. C.
    (1993). Bounds on the Bayes and minimax risk for signal parameter estimation. IEEE Transactions on Information Theory, 39(4), 1386–1394. https://doi.org/10.1109/9780470544198.ch28
  12. ↵
    1. Chaumette, E., &
    2. Fritsche, C.
    (2018). A general class of Bayesian lower bounds tighter than the Weiss-Weinstein family. Proc. of the 21st International Conference on Information Fusion (FUSION 2018) Cambridge, UK, 159–165. https://doi.org/10.23919/ICIF.2018.8455577
  13. ↵
    1. Chaumette, E.,
    2. Renaux, A., &
    3. Korso, M. N. E.
    (2017). A class of Weiss-Weinstein bounds and its relationship with the Bobrovsky-Mayer-Wolf-Zakaï bounds. IEEE Transactions on Information Theory, 63(4), 2226–2240. https://doi.org/10.1109/TIT.2017.2671883
  14. ↵
    1. Chazan, D.,
    2. Zakaï, M., &
    3. Ziv, J.
    (1975). Improved lower bounds on signal parameter estimation. IEEE Transactions on Information Theory, 21 (1), 90–93. https://doi.org/10.1109/TIT.1975.1055325
  15. ↵
    1. Closas, P.
    (2009). Bayesian signal processing techniques for GNSS receivers: From multipath mitigation to positioning [Ph.D. thesis, Department of Signal Theory and Communications, Universitat Politecnica de Catalunya]. https://core.ac.uk/download/pdf/81575184.pdf
  16. ↵
    1. Cramér, H.
    (1946). Mathematical methods of statistics. Princeton University Press. https://doi.org/10.1515/9781400883868
  17. ↵
    1. DeLong, D. F.
    (1993). Use of the Weiss-Weinstein bound to compare the direction-finding performance of sparse arrays. Technical Report No. 982, Lincoln Lab., Massachusetts Institute of Technology, Lexington, MA.
  18. ↵
    1. Denis, F.
    (2009). GNSS modulation: A unified statistical description with application to tracking bounds [M.Eng. thesis, Department of Electrical & Computer Engineering, McGill University]. https://escholarship.mcgill.ca/concern/theses/736664959
  19. ↵
    1. Emmanuele, A.
    (2012). Signal design and theoretical bounds for time-of-arrival estimation in GNSS applications [Ph.D. thesis, DSP laboratory of the Dipartimento di Ingegneria dell’Informazione, University of Pisa]. https://etd.adm.unipi.it/t/etd-04202012-125157/
  20. ↵
    1. Fisher, N. I.
    (1993). Statistical analysis of circular data. Cambridge University Press.
  21. ↵
    1. Fritsche, C.,
    2. Orguner, U., &
    3. Gustafsson, F.
    (2018). Bobrovsky-Zakaï bound for filtering, prediction and smoothing of nonlinear dynamic systems. Proc. of the 21st International Conference on Information Fusion (FUSION 2018) Cambridge, UK, 171–178. https://doi.org/10.23919/ICIF.2018.8455541
  22. ↵
    1. Gifford, W. M.,
    2. Dardari, D., &
    3. Win, M. Z.
    (2022). The impact of multipath information on time-of-arrival estimation. IEEE Transactions on Signal Processing, 70, 31–46. https://doi.org/10.1109/TSP.2020.3038254
  23. ↵
    1. Graff, A. M.,
    2. Blount, W. N.,
    3. Iannucci, P. A.,
    4. Andrews, J. G., &
    5. Humphreys, T. E.
    (2021). Analysis of OFDM signals for ranging and communications. Proc. of the 34th International Technical Meeting of the Satellite Division of the Institute of Navigation (ION GNSS+ 2021) St. Louis, MO, 2910–2924. https://doi.org/10.33012/2021.17991
  24. ↵
    1. Gusi-Amigó, A.,
    2. Closas, P.,
    3. Mallat, A., &
    4. Vandendorpe, L.
    (2018). Ziv-Zakaï bound for direct position estimation. NAVIGATION, 65(3), 463–475. https://doi.org/10.1002/navi.259
  25. ↵
    1. Ibragimov, I. A., &
    2. Hasminskii, R. Z.
    (1981). Statistical estimation: Asymptotic theory. Springer-Verlag. https://doi.org/10.1007/978-1-4899-0027-2
  26. ↵
    1. IS-GPS-200K
    . (2019). Accessed June 27, 2023 at https://www.gps.gov/technical/icwg/IS-GPS-200K.pdf.
  27. ↵
    1. Joseph, A.
    (2010). GNSS solutions: Measuring GNSS signal strength. InsideGNSS, November/December, 20–25. https://insidegnss.com/measuring-gnss-signal-strength/
  28. ↵
    1. Kaplan, E. D., &
    2. Hegarty, C.
    (2006). Understanding GPS: Principles and applications (2nd Ed.). Artech House. https://books.google.com/books/about/Understanding_GPS.html?id=V_5OAAAAMAAJ
  29. ↵
    1. Lehmann, E. L.
    (1983). Theory of point estimation. Wiley. https://doi.org/10.1007/b98854
  30. ↵
    1. Lubeigt, C.,
    2. Ortega, L.,
    3. Vilà-Valls, J.,
    4. Lestarquit, L., &
    5. Chaumette, E.
    (2020). Joint delay-Doppler estimation performance in a dual source context. Remote Sensing, 12(23), 3894. http://dx.doi.org/10.3390/rs12233894
  31. ↵
    1. Lubeigt, C.,
    2. Ortega, L.,
    3. Vilà-Valls, J.,
    4. Lestarquit, L., &
    5. Chaumette, E.
    (2023). Untangling first and second order statistics contributions in multipath scenarios. Signal Processing, 205, 108868. https://doi.org/10.1016/j.sigpro.2022.108868
  32. ↵
    1. Manlaney, R. A.
    (2020). Reduced power use in mobile GPS-based technologies (U.S. patent USRE48206E1). U.S. Patent and Trade-mark Office. https://patents.google.com/patent/USRE48206E1/en
  33. ↵
    1. Manzano-Jurado, M.,
    2. Alegre-Rubio, J.,
    3. Pellacani, A.,
    4. Seco-Granados, G.,
    5. López-Salcedo, J. A.,
    6. Guerrero, E., &
    7. García-Rodríguez, A.
    (2014). Use of weak GNSS signals in a mission to the moon. In Proc. of 7 th ESA Workshop on Satellite Navigation Technologies and European Workshop on GNSS Signals and Signal Processing (NAVITEC), Noordwijk, Netherlands, 1–8 https://doi.org/10.1109/NAVITEC.2014.7045151
  34. ↵
    1. Mardia, K. V.
    (1972). Statistics of directional data. Academic Press. https://doi.org/10.1016/C2013-0-07425-7
  35. ↵
    1. Mardia, K. V.
    (1999). Directional statistics. John Wiley & Sons. https://doi.org/10.1002/9780470316979
  36. ↵
    1. McAulay, R., &
    2. Hofstetter, E.
    (1971). Barankin bounds on parameter estimation. IEEE Transactions on Information Theory, 17(6), 669–676. https://doi.org/10.1109/TIT.1971.1054719
  37. ↵
    1. Mcphee, H.,
    2. Ortega, L.,
    3. Vilà-Valls, J., &
    4. Chaumette, E.
    (2023a). Accounting for acceleration – Signal parameters estimation performance limits in high dynamic applications. IEEE Transactions on Aerospace and Electronic Systems, 59(1), 610–622. https://doi.org/10.1109/TAES.2022.3189611.
  38. ↵
    1. Mcphee, H.,
    2. Ortega, L.,
    3. Vilà-Valls, J., &
    4. Chaumette, E.
    (2023b). On the accuracy limits of misspecified delay-Doppler estimation. Signal Processing, 205, 108872. https://doi.org/10.1016/j.sigpro.2022.108872.
  39. ↵
    1. Nitzan, E.,
    2. Routtenberg, T., &
    3. Tabrikian, J.
    (2016). A new class of Bayesian cyclic bounds for periodic parameter estimation. IEEE Transactions on Signal Processing, 64(1), 229–243. https://doi.org/10.1109/TSP.2015.2478758
  40. ↵
    1. Ortega, L.,
    2. Lubeigt, C.,
    3. Vilà-Valls, J., &
    4. Chaumette, E.
    (2023). On GNSS synchronization performance degradation under interference scenarios: Bias and misspecified Cramér-Rao bounds. NAVIGATION, 70(4), 691–712. https://doi.org/10.33012/navi.606
  41. ↵
    1. Pany, T.
    (2010). Navigation signal processing for GNSS software receivers. Artech House.
  42. ↵
    1. Pany, T., &
    2. Eissfeller, B.
    (2006). Use of a vector delay lock loop receiver for GNSS signal power analysis in bad signal conditions. Proc. of the 2006 IEEE/ION Position, Location, and Navigation Symposium (PLANS 2006), Coronado, CA, 893–903. https://doi.org/10.1109/PLANS.2006.1650689
  43. ↵
    1. Rapoport, I., &
    2. Oshman, Y.
    (2004). Recursive Weiss–Weinstein lower bounds for discrete-time nonlinear filtering. Proc. of the 43rd IEEE Conference on Decision and Control (CDC 2004), Nassau, Bahamas, 2662–2667. https://doi.org/10.1109/CDC.2004.1428862
  44. ↵
    1. Rapoport, I., &
    2. Oshman, Y.
    (2007a). Weiss-Weinstein lower bounds for Markovian systems. Part 1: Theory. IEEE Transactions on Signal Processing, 55(5), 2016–2030. https://doi.org/10.1109/TSP.2007.893208
  45. ↵
    1. Rapoport I., &
    2. Oshman Y.
    (2007b). Weiss–Weinstein lower bounds for Markovian systems. Part 2: Applications to fault-tolerant filtering. IEEE Transactions on Signal Processing, 55(5), 2031–2042. https://doi.org/10.1109/TSP.2007.893209
  46. ↵
    1. Reece, S., &
    2. Nicholson, D.
    (2005). Tighter alternatives to the Cramér-Rao lower bound for discrete-time filtering. Proc. of the International Conference on Information Fusion Philadelphia, PA, 101–106). https://doi.org/10.1109/ICIF.2005.1591842
  47. ↵
    1. Ren, T., &
    2. Petovello, M. G.
    (2017). A stand-alone approach for high-sensitivity GNSS receivers in signal-challenged environment. IEEE Transactions on Aerospace and Electronic Systems, 53(5), 2438–2448. https://doi.org/10.1109/TAES.2017.2699539
  48. ↵
    1. Renaux, A.,
    2. Forster, P.,
    3. Larzabal, P.,
    4. Richmond, C., &
    5. Nehorai, A.
    (2008). A fresh look at the Bayesian bounds of the Weiss-Weinstein family. IEEE Transactions on Signal Processing, 56(11), 5334–5352. https://doi.org/10.1109/TSP.2008.927075
  49. ↵
    1. Richmond, C. D., &
    2. Horowitz, L. L.
    (2015). Parameter bounds on estimation accuracy under model misspecification. IEEE Transactions on Signal Processing, 63(9), 2263–2278. https://doi.org/10.1109/TSP.2015.2411222
  50. ↵
    1. Seidman, L. P.
    (1970). Performance limitations and error calculations for parameter estimation. Proceedings of the IEEE, 58(5), 644–652. https://doi.org/10.1109/PROC.1970.7720
    CrossRef
  51. ↵
    1. Soloviev, A.,
    2. Van Grass, F., &
    3. Gunawardena, S.
    (2009). Decoding navigation data messages from weak GPS signals. IEEE Transactions on Aerospace and Electronic Systems, 45(2), 660–666. https://doi.org/10.1109/TAES.2009.5089548
  52. ↵
    1. U-blox
    . (2018). Product summary of ZED-F9P: u-bloc F9 high precision GNSS module. u-blox AG. Accessed on January 26, 2018 at www.u-blox.com.
  53. ↵
    1. Van Trees, H.
    (1968) Detection, estimation and modulation theory (Vol. 1). John Wiley & Sons. https://doi.org/10.1002/0471221082
  54. ↵
    1. Van Trees, H. L., &
    2. Bell, K. L.
    (2007). Bayesian bounds for parameter estimation and nonlinear filtering/tracking. IEEE Press.
  55. ↵
    1. Villares, J.
    (2005). Sample covariance based parameter estimation for digital communications [Ph.D. thesis, Department of Signal Theory and Communications, Universitat Politecnica de Catalunya]. https://upcommons.upc.edu/handle/2117/94206
  56. ↵
    1. Watson, W. D.,
    2. Basu, P.,
    3. Beaudeau, J. P., &
    4. Couto, D. J.
    (2020). Techniques for phase modulated signals having poor autocorrelation (U.S. patent US10666475B2). U.S. Patent and Trade-mark Office. https://patents.justia.com/patent/20200136870
  57. ↵
    1. Weinstein, E.
    (1988). Relations between Belini-Tartara, Chazan-Zakaï-Ziv, and Wax-Ziv lower bounds. IEEE Transactions on Information Theory, 34(2), 342–343. https://doi.org/10.1109/18.2648
  58. ↵
    1. Weinstein, E., &
    2. Weiss, A.
    (1988). A general class of lower bounds in parameter estimation. IEEE Transactions on Information Theory, 34(2), 338–342. https://doi.org/10.1109/18.2647
  59. ↵
    1. Weiss, A., &
    2. Weinstein, E.
    (1983). Fundamental limitations in passive time delay estimation-Part I: Narrow-band systems. IEEE Transactions on Acoustics, Speech, and Signal Processing, 31(2), 472–486. https://doi.org/10.1109/TASSP.1983.1164061
  60. ↵
    1. Weiss, A., &
    2. Weinstein, E.
    (1985). A lower bound on the mean-square error in random parameter estimation. IEEE Transformation Information Theory, 31 (5), 680–682. https://doi.org/10.1109/TIT.1985.1057094
  61. ↵
    1. Wu, Y.,
    2. Hu, D.,
    3. Wu, M., &
    4. Hu, X.
    (2006). A numerical-integration perspective on Gaussian filters. IEEE Transactions on Signal Processing, 54(8), 2910–2921. https://doi.org/10.1109/TSP.2006.875389
  62. ↵
    1. Xu, W.
    (2001). Performance bounds on matched-field methods for source localization and estimation of ocean environmental parameters [Ph.D. thesis, Department of Ocean Engineering, Massachusetts Institute of Technology]. https://dspace.mit.edu/handle/1721.1/91333
  63. ↵
    1. Xu, W.,
    2. Baggeroer, A. B., &
    3. Richmond, C. D.
    (2004). Bayesian bounds for matched-field parameter estimation. IEEE Transactions on Signal Processing, 52(12), 3293–3305. https://doi.org/10.1109/TSP.2004.837437
  64. ↵
    1. Ziv, J., &
    2. Zakaï, M.
    (1969). Some lower bounds on signal parameter estimation. IEEE Transactions on Information Theory, 15(3), 386–391. https://doi.org/10.1109/TIT.1969.1054301
PreviousNext
Back to top

In this issue

NAVIGATION: Journal of the Institute of Navigation: 71 (3)
NAVIGATION: Journal of the Institute of Navigation
Vol. 71, Issue 3
Fall 2024
  • Table of Contents
  • Index by author
Print
Download PDF
Article Alerts
Sign In to Email Alerts with your Email Address
Email Article

Thank you for your interest in spreading the word on NAVIGATION: Journal of the Institute of Navigation.

NOTE: We only request your email address so that the person you are recommending the page to knows that you wanted them to see it, and that it is not junk mail. We do not capture any email address.

Enter multiple addresses on separate lines or separate them with commas.
Weiss–Weinstein Bound of Frequency Estimation Error for Very Weak GNSS Signals
(Your Name) has sent you a message from NAVIGATION: Journal of the Institute of Navigation
(Your Name) thought you would like to see the NAVIGATION: Journal of the Institute of Navigation web site.
Citation Tools
Weiss–Weinstein Bound of Frequency Estimation Error for Very Weak GNSS Signals
Xin Zhang, Xingqun Zhan, Jihong Huang, Jiahui Liu,, Yingchao Xiao
NAVIGATION: Journal of the Institute of Navigation Sep 2024, 71 (3) navi.654; DOI: 10.33012/navi.654

Citation Manager Formats

  • BibTeX
  • Bookends
  • EasyBib
  • EndNote (tagged)
  • EndNote 8 (xml)
  • Medlars
  • Mendeley
  • Papers
  • RefWorks Tagged
  • Ref Manager
  • RIS
  • Zotero
Share
Weiss–Weinstein Bound of Frequency Estimation Error for Very Weak GNSS Signals
Xin Zhang, Xingqun Zhan, Jihong Huang, Jiahui Liu,, Yingchao Xiao
NAVIGATION: Journal of the Institute of Navigation Sep 2024, 71 (3) navi.654; DOI: 10.33012/navi.654
Twitter logo Facebook logo Mendeley logo
  • Tweet Widget
  • Facebook Like
  • Google Plus One
Bookmark this article

Jump to section

  • Article
    • Abstract
    • 1 INTRODUCTION
    • 2 PROBLEM FORMULATION
    • 3 WWB FOR GNSS FREQUENCY ESTIMATION
    • 4 TWO-STEP OPTIMIZATION OF THE WWB
    • 5 SIMULATIONS
    • 6 CONCLUSIONS
    • HOW TO CITE THIS ARTICLE
    • CONFLICT OF INTEREST
    • ACKNOWLEDGEMENTS
    • APPENDIX A WWB
    • APPENDIX B BCRB
    • REFERENCES
  • Figures & Data
  • Supplemental
  • References
  • Info & Metrics
  • PDF

Related Articles

  • Google Scholar

Cited By...

  • No citing articles found.
  • Google Scholar

More in this TOC Section

  • Thirty Years of Maintaining WGS 84 with GPS
  • Doppler Positioning Using Multi-Constellation LEO Satellite Broadband Signals as Signals of Opportunity
  • Federated Learning of Jamming Classifiers: From Global to Personalized Models
Show more Original Article

Similar Articles

Keywords

  • Bayesian bound
  • FLL
  • lower bound
  • MAP
  • Von Mises
  • Weiss Weinstein
  • Ziv Zakaï

Unless otherwise noted, NAVIGATION content is licensed under a Creative Commons CC BY 4.0 License.

© 2025 The Institute of Navigation, Inc.

Powered by HighWire