{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,21]],"date-time":"2026-07-21T14:44:09Z","timestamp":1784645049950,"version":"3.55.0"},"reference-count":56,"publisher":"MDPI AG","issue":"13","license":[{"start":{"date-parts":[[2022,6,30]],"date-time":"2022-06-30T00:00:00Z","timestamp":1656547200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>In this manuscript, an underwater target tracking problem with passive sensors is considered. The measurements used to track the target trajectories are (i) only bearing angles, and (ii) Doppler-shifted frequencies and bearing angles. Measurement noise is assumed to follow a zero mean Gaussian probability density function with unknown noise covariance. A method is developed which can estimate the position and velocity of the target along with the unknown measurement noise covariance at each time step. The proposed estimator linearises the nonlinear measurement using an orthogonal polynomial of first order, and the coefficients of the polynomial are evaluated using numerical integration. The unknown sensor noise covariance is estimated online from residual measurements. Compared to available adaptive sigma point filters, it is free from the Cholesky decomposition error. The developed method is applied to two underwater tracking scenarios which consider a nearly constant velocity target. The filter\u2019s efficacy is evaluated using (i) root mean square error (RMSE), (ii) percentage of track loss, (iii) normalised (state) estimation error squared (NEES), (iv) bias norm, and (v) floating point operations (flops) count. From the simulation results, it is observed that the proposed method tracks the target in both scenarios, even for the unknown and time-varying measurement noise covariance case. Furthermore, the tracking accuracy increases with the incorporation of Doppler frequency measurements. The performance of the proposed method is comparable to the adaptive deterministic support point filters, with the advantage of a considerably reduced flops requirement.<\/jats:p>","DOI":"10.3390\/s22134970","type":"journal-article","created":{"date-parts":[[2022,7,1]],"date-time":"2022-07-01T01:40:36Z","timestamp":1656639636000},"page":"4970","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Tracking an Underwater Object with Unknown Sensor Noise Covariance Using Orthogonal Polynomial Filters"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2977-8815","authenticated-orcid":false,"given":"Kundan","family":"Kumar","sequence":"first","affiliation":[{"name":"Department of Electrical Engineering, Indian Institute of Technology Patna, Patna 801103, India"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shovan","family":"Bhaumik","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, Indian Institute of Technology Patna, Patna 801103, India"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sanjeev","family":"Arulampalam","sequence":"additional","affiliation":[{"name":"Maritime Division, Defence Science and Technology (DST) Group, Edinburgh, SA 5111, Australia"},{"name":"Faculty of Engineering, Computer & Mathematical Sciences, The University of Adelaide, Adelaide, SA 5005, Australia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,6,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Arulampalam, S., and Ristic, B. (2000, January 24\u201328). Comparison of the particle filter with range-parameterized and modified polar EKFs for angle-only tracking. Proceedings of the AeroSense 2000 Signal and Data Processing of Small Targets 2000, Orlando, FL, USA.","DOI":"10.1117\/12.391985"},{"key":"ref_2","unstructured":"Ristic, B., Arulampalam, S., and Gordon, N. (2003). Beyond the Kalman Filter: Particle Filters For Tracking Applications, Artech House."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"809","DOI":"10.1109\/TSP.2005.861776","article-title":"An asymptotically unbiased estimator for bearings-only and Doppler-bearing target motion analysis","volume":"54","author":"Ho","year":"2006","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1536","DOI":"10.1109\/TAES.2016.140820","article-title":"Passive tracking in heavy clutter with sensor location uncertainty","volume":"52","author":"Guo","year":"2016","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"492","DOI":"10.1109\/JOE.2018.2814218","article-title":"Gaussian sum shifted Rayleigh filter for underwater bearings-only target tracking problems","volume":"44","author":"Radhakrishnan","year":"2018","journal-title":"IEEE J. Ocean. Eng."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Luo, J., Han, Y., and Fan, L. (2018). Underwater acoustic target tracking: A review. Sensors, 18.","DOI":"10.3390\/s18010112"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1161","DOI":"10.1109\/TAES.2013.6494405","article-title":"A Gaussian-sum based cubature Kalman filter for bearings-only tracking","volume":"49","author":"Leong","year":"2013","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"4082","DOI":"10.1109\/TSP.2008.925589","article-title":"Doppler-bearing tracking in the presence of observer location error","volume":"56","author":"Yang","year":"2008","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"69","DOI":"10.4103\/0377-2063.61267","article-title":"Doppler-bearing passive target tracking using a parameterized unscented Kalman filter","volume":"56","author":"Rao","year":"2010","journal-title":"IETE J. Res."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Li, X., Zhao, C., Yu, J., and Wei, W. (2019). Underwater bearing-only and bearing-Doppler target tracking based on square root unscented Kalman filter. Entropy, 21.","DOI":"10.3390\/e21080740"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Borisov, A., Bosov, A., Miller, B., and Miller, G. (2020). Passive underwater target tracking: Conditionally minimax nonlinear filtering with bearing-Doppler observations. Sensors, 20.","DOI":"10.3390\/s20082257"},{"key":"ref_12","unstructured":"Bar-Shalom, Y., Li, X.R., and Kirubarajan, T. (2004). Estimation with Applications to Tracking and Navigation: Theory Algorithms and Software, John Wiley & Sons."},{"key":"ref_13","unstructured":"Anderson, B.D., and Moore, J.B. (2012). Optimal Filtering, Courier Corporation."},{"key":"ref_14","unstructured":"Karaman, S. (1986). Fixed point smoothing algorithm to the torpedo tracking problem. [Master\u2019s Thesis, Naval Postgraduate School Monterey]."},{"key":"ref_15","unstructured":"Karlsson, R. (2002). Various Topics on Angle-Only Tracking Using Particle Filters, Link\u00f6ping University."},{"key":"ref_16","unstructured":"Karlsson, R. (2002). Simulation based methods for target tracking. [Ph.D. Thesis, Link\u00f6pings University]."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"673","DOI":"10.1007\/s11771-019-4038-2","article-title":"A novel estimation algorithm for torpedo tracking in undersea environment","volume":"26","author":"Kumar","year":"2019","journal-title":"J. Cent. South Univ."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"714","DOI":"10.1109\/9.754809","article-title":"Stochastic stability of the discrete-time extended Kalman filter","volume":"44","author":"Reif","year":"1999","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_19","unstructured":"Julier, S.J., and Uhlmann, J.K. (1991, January 21\u201325). New extension of the Kalman filter to nonlinear systems. Proceedings of the AeroSense \u201997, Signal Processing, Sensor Fusion, and Target Recognition VI, Orlando, FL, USA."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1109\/9.847726","article-title":"A new method for the nonlinear transformation of means and covariances in filters and estimators","volume":"45","author":"Julier","year":"2000","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"910","DOI":"10.1109\/9.855552","article-title":"Gaussian filters for nonlinear filtering problems","volume":"45","author":"Ito","year":"2000","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"4441","DOI":"10.1016\/j.apm.2015.11.035","article-title":"Multiple sparse-grid Gauss\u2013Hermite filtering","volume":"40","author":"Radhakrishnan","year":"2016","journal-title":"Appl. Math. Model."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1254","DOI":"10.1109\/TAC.2009.2019800","article-title":"Cubature Kalman filters","volume":"54","author":"Arasaratnam","year":"2009","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"533","DOI":"10.1049\/iet-spr.2012.0085","article-title":"Cubature quadrature Kalman filter","volume":"7","author":"Bhaumik","year":"2013","journal-title":"IET Signal Process."},{"key":"ref_25","unstructured":"Van Der Merwe, R., and Wan, E.A. (2001, January 7\u201311). The square-root unscented Kalman filter for state and parameter-estimation. Proceedings of the 2001 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No. 01CH37221), Salt Lake City, UT, USA."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2589","DOI":"10.1109\/TSP.2007.914964","article-title":"Square-root quadrature Kalman filtering","volume":"56","author":"Arasaratnam","year":"2008","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"617","DOI":"10.1002\/asjc.704","article-title":"Square-root cubature-quadrature Kalman filter","volume":"16","author":"Bhaumik","year":"2014","journal-title":"Asian J. Control"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"59675","DOI":"10.1109\/ACCESS.2021.3073289","article-title":"Extended Kalman filter using orthogonal polynomials","volume":"9","author":"Kumar","year":"2021","journal-title":"IEEE Access"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1333","DOI":"10.1109\/TAES.2003.1261132","article-title":"Survey of maneuvering target tracking. Part I: Dynamic models","volume":"39","author":"Li","year":"2003","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"501","DOI":"10.1049\/rsn2.12198","article-title":"Bearing-only underwater uncooperative target tracking for non-Gaussian environment using fast particle filter","volume":"16","author":"Hou","year":"2022","journal-title":"IET Radar Sonar Navig."},{"key":"ref_31","unstructured":"Abramowitz, M., and Stegun, I.A. (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1016\/0024-3795(95)00595-1","article-title":"The d-variate vector Hermite polynomial of order k","volume":"237","author":"Holmquist","year":"1996","journal-title":"Linear Algebra Its Appl."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1017","DOI":"10.4236\/am.2015.66094","article-title":"A multinomial theorem for Hermite polynomials and financial applications","volume":"6","year":"2015","journal-title":"Appl. Math."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"2910","DOI":"10.1109\/TSP.2006.875389","article-title":"A numerical-integration perspective on Gaussian filters","volume":"54","author":"Wu","year":"2006","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"510","DOI":"10.1016\/j.automatica.2012.11.014","article-title":"High-degree cubature Kalman filter","volume":"49","author":"Jia","year":"2013","journal-title":"Automatica"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1097","DOI":"10.1007\/s12555-014-0228-8","article-title":"Higher degree cubature quadrature Kalman filter","volume":"13","author":"Singh","year":"2015","journal-title":"Int. J. Control Autom. Syst."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"953","DOI":"10.1109\/JPROC.2007.894705","article-title":"Discrete-time nonlinear filtering algorithms using Gauss-Hermite quadrature","volume":"95","author":"Arasaratnam","year":"2007","journal-title":"Proc. IEEE"},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Chalasani, G., and Bhaumik, S. (2012, January 18\u201320). Bearing only tracking using Gauss-Hermite filter. Proceedings of the 2012 7th IEEE Conference on Industrial Electronics and Applications (ICIEA), Singapore.","DOI":"10.1109\/ICIEA.2012.6360970"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1055","DOI":"10.1049\/iet-rsn.2019.0477","article-title":"Parameter estimation of underwater impulsive noise with the class B model","volume":"14","author":"Zhang","year":"2020","journal-title":"IET Radar Sonar Navig."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Mahmood, A., and Chitre, M. (2015, January 18\u201321). Modeling colored impulsive noise by Markov chains and alpha-stable processes. Proceedings of the OCEANS 2015-Genova, Genova, Italy.","DOI":"10.1109\/OCEANS-Genova.2015.7271550"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"154","DOI":"10.1016\/j.aeue.2018.10.004","article-title":"A novel robust filtering strategy for systems with non-Gaussian noises","volume":"97","author":"Zhou","year":"2018","journal-title":"AEU Int. J. Electron. Commun."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1007\/s001900050236","article-title":"Adaptive Kalman filtering for INS\/GPS","volume":"73","author":"Mohamed","year":"1999","journal-title":"J. Geod."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1002\/j.2161-4296.1999.tb02416.x","article-title":"Stochastic modeling for real-time kinematic GPS\/GLONASS positioning","volume":"46","author":"Wang","year":"1999","journal-title":"Navigation"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"33","DOI":"10.5081\/jgps.9.1.33","article-title":"Evaluating the performances of adaptive Kalman filter methods in GPS\/INS integration","volume":"9","author":"Almagbile","year":"2010","journal-title":"J. Glob. Position. Syst."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"04014088","DOI":"10.1061\/(ASCE)AS.1943-5525.0000412","article-title":"Adaptive tuning of the unscented Kalman filter for satellite attitude estimation","volume":"28","author":"Soken","year":"2015","journal-title":"J. Aerosp. Eng."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"1007","DOI":"10.1049\/iet-smt.2015.0020","article-title":"Adaptive Gauss\u2013Hermite filter for non-linear systems with unknown measurement noise covariance","volume":"9","author":"Dey","year":"2015","journal-title":"IET Sci. Meas. Technol."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"594","DOI":"10.1109\/TAC.2017.2730480","article-title":"A novel adaptive Kalman filter with inaccurate process and measurement noise covariance matrices","volume":"63","author":"Huang","year":"2017","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"59362","DOI":"10.1109\/ACCESS.2020.2982407","article-title":"On the identification of noise covariances and adaptive Kalman filtering: A new look at a 50 year-old problem","volume":"8","author":"Zhang","year":"2020","journal-title":"IEEE Access"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"381478","DOI":"10.1155\/2015\/381478","article-title":"A new adaptive square-root unscented Kalman filter for nonlinear systems with additive noise","volume":"2015","author":"Zhou","year":"2015","journal-title":"Int. J. Aerosp. Eng."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"115101","DOI":"10.1088\/1361-6501\/abfef4","article-title":"Correction adaptive square-root cubature Kalman filter with application to autonomous vehicle target tracking","volume":"32","author":"Zhang","year":"2021","journal-title":"Meas. Sci. Technol."},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Ristic, B., Wang, X., and Arulampalam, S. (2017, January 10\u201313). Target motion analysis with unknown measurement noise variance. Proceedings of the 2017 20th International Conference on Information Fusion (Fusion), Xi\u2019an, China.","DOI":"10.23919\/ICIF.2017.8009853"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1016\/j.inffus.2017.11.006","article-title":"Measurement variance ignorant target motion analysis","volume":"43","author":"Ristic","year":"2018","journal-title":"Inf. Fusion"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"4408","DOI":"10.1109\/TSP.2005.857061","article-title":"Complexity analysis of the marginalized particle filter","volume":"53","author":"Karlsson","year":"2005","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_54","unstructured":"Li, X.R., Zhao, Z., and Jilkov, V.P. (2002, January 21\u201326). Estimator\u2019s credibility and its measures. Proceedings of the IFAC 15th World Congress, Barcelona, Spain."},{"key":"ref_55","unstructured":"Li, X.R., Zhao, Z., and Jilkov, V.P. (2001). Practical measures and test for credibility of an estimator. Proceedings Workshop on Estimation, Tracking, and Fusion: A Tribute to Yaakov Bar-Shalom, Citeseer."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"6488","DOI":"10.1109\/TSP.2021.3129599","article-title":"Analysis of propagation delay effects on bearings-only fusion of heterogeneous sensors","volume":"69","author":"Arulampalam","year":"2021","journal-title":"IEEE Trans. Signal Process."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/13\/4970\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:41:28Z","timestamp":1760139688000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/13\/4970"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,6,30]]},"references-count":56,"journal-issue":{"issue":"13","published-online":{"date-parts":[[2022,7]]}},"alternative-id":["s22134970"],"URL":"https:\/\/doi.org\/10.3390\/s22134970","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,6,30]]}}}