{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,28]],"date-time":"2026-01-28T13:54:59Z","timestamp":1769608499160,"version":"3.49.0"},"reference-count":41,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2022,5,31]],"date-time":"2022-05-31T00:00:00Z","timestamp":1653955200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"China NSF Grant","award":["61971217"],"award-info":[{"award-number":["61971217"]}]},{"name":"China NSF Grant","award":["61971218"],"award-info":[{"award-number":["61971218"]}]},{"name":"China NSF Grant","award":["61631020"],"award-info":[{"award-number":["61631020"]}]},{"name":"Jiangsu NSF Grant","award":["BK20200444"],"award-info":[{"award-number":["BK20200444"]}]},{"name":"National Key Research and Development Project Grant","award":["2020YFB1807602"],"award-info":[{"award-number":["2020YFB1807602"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>The features of quasi-stationary signals (QSS) are considered to be in a direct position determination (DPD) framework, and a real-valued DPD algorithm of QSS for nested arrays is proposed. By stacking the vectorization form of the signal\u2019s covariance for different frames and further eliminating noise, a new noise-eliminated received signal matrix is obtained first. Then, the combination of the Khatri\u2013Rao subspace method and subspace data fusion method was performed to form the cost function. High complexity can be reduced by matrix reconstruction, including the modification of the dimension-reduced matrix and unitary transformation. Ultimately, the advantage of lower complexity, compared with the previous algorithm, is verified by complexity analysis, and the superiority over the existing algorithms, in terms of the maximum number of identifiable sources, estimation accuracy, and resolution, are corroborated by some simulation results.<\/jats:p>","DOI":"10.3390\/s22114209","type":"journal-article","created":{"date-parts":[[2022,6,1]],"date-time":"2022-06-01T21:43:42Z","timestamp":1654119822000},"page":"4209","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Real-Valued Direct Position Determination of Quasi-Stationary Signals for Nested Arrays: Khatri\u2013Rao Subspace and Unitary Transformation"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1832-5527","authenticated-orcid":false,"given":"Haowei","family":"Zeng","sequence":"first","affiliation":[{"name":"College of Electronic and Information Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China"},{"name":"Key Laboratory of Dynamic Cognitive System of Electromagnetic Spectrum Space, Ministry of Industry and Information Technology, Nanjing 211106, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Heng","family":"Yue","sequence":"additional","affiliation":[{"name":"College of Electronic and Information Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China"},{"name":"Key Laboratory of Dynamic Cognitive System of Electromagnetic Spectrum Space, Ministry of Industry and Information Technology, Nanjing 211106, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jinke","family":"Cao","sequence":"additional","affiliation":[{"name":"College of Electronic and Information Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China"},{"name":"Key Laboratory of Dynamic Cognitive System of Electromagnetic Spectrum Space, Ministry of Industry and Information Technology, Nanjing 211106, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaofei","family":"Zhang","sequence":"additional","affiliation":[{"name":"College of Electronic and Information Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China"},{"name":"Key Laboratory of Dynamic Cognitive System of Electromagnetic Spectrum Space, Ministry of Industry and Information Technology, Nanjing 211106, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,5,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"3444","DOI":"10.1109\/JSEN.2015.2477443","article-title":"Sound Localization Sensors for Search and Rescue Biobots","volume":"16","author":"Latif","year":"2016","journal-title":"IEEE Sens. J."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Zhao, K., Zhao, T., Zheng, Z., Yu, C., Ma, D., Rabie, K., and Kharel, R. (2020). Optimization of Time Synchronization and Algorithms with TDOA Based Indoor Positioning Technique for Internet of Things. Sensors, 20.","DOI":"10.3390\/s20226513"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"L\u00f3pez-Sastre, R.J., Herranz-Perdiguero, C., Guerrero-G\u00f3mez-Olmedo, R., O\u00f1oro-Rubio, D., and Maldonado-Basc\u00f3n, S. (2019). Boosting Multi-Vehicle Tracking with a Joint Object Detection and Viewpoint Estimation Sensor. Sensors, 19.","DOI":"10.3390\/s19194062"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"13033","DOI":"10.1109\/TVT.2020.3018489","article-title":"A Direct Position Determination Approach for Underwater Acoustic Sensor Networks","volume":"69","author":"Wang","year":"2020","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Qi, J., and Liu, G.-P. (2017). A Robust High-Accuracy Ultrasound Indoor Positioning System Based on a Wireless Sensor Network. Sensors, 17.","DOI":"10.3390\/s17112554"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"8841","DOI":"10.1109\/TVT.2016.2517151","article-title":"A Novel TOA-Based Mobile Localization Technique under Mixed LOS\/NLOS Conditions for Cellular Networks","volume":"65","author":"Zhou","year":"2016","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2880","DOI":"10.1109\/TITS.2017.2769488","article-title":"Angle of Arrival-Based Cooperative Positioning for Smart Vehicles","volume":"19","author":"Fascista","year":"2018","journal-title":"IEEE Trans. Intell. Transp. Syst."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"De-La-Llana-Calvo, \u00c1., L\u00e1zaro-Galilea, J.L., Alc\u00e1zar-Fern\u00e1ndez, A., Gardel-Vicente, A., Bravo-Mu\u00f1oz, I., and Iamnitchi, A. (2022). Accuracy and Precision of Agents Orientation in an Indoor Positioning System Using Multiple Infrastructure Lighting Spotlights and a PSD Sensor. Sensors, 22.","DOI":"10.3390\/s22082882"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1109\/TWC.2011.102611.110728","article-title":"An Efficient Constrained Weighted Least Squares Algorithm for Moving Source Location Using TDOA and FDOA Measurements","volume":"11","author":"Yu","year":"2012","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"853","DOI":"10.1109\/TVT.2012.2225074","article-title":"A Semidefinite Relaxation Method for Source Localization Using TDOA and FDOA Measurements","volume":"62","author":"Wang","year":"2013","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"513","DOI":"10.1109\/LSP.2004.826501","article-title":"Direct position determination of narrowband radio frequency transmitters","volume":"11","author":"Weiss","year":"2004","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2348","DOI":"10.1016\/j.sigpro.2007.03.008","article-title":"A decoupled algorithm for geolocation of multiple emitters","volume":"87","author":"Amar","year":"2007","journal-title":"Signal Process."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"3056","DOI":"10.1016\/j.sigpro.2010.05.010","article-title":"Direct detection and position determination of multiple sources with intermittent emission","volume":"90","author":"Oispuu","year":"2010","journal-title":"Signal Process."},{"key":"ref_14","unstructured":"Demissie, B., Oispuu, M., and Ruthotto, E. (July, January 30). Localization of multiple sources with a moving array using subspace data fusion. Proceedings of the 2008 11th International Conference on Information Fusion, Cologne, Germany."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"276","DOI":"10.1109\/TAP.1986.1143830","article-title":"Multiple emitter location and signal parameter estimation","volume":"34","author":"Schmidt","year":"1986","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1016\/j.cja.2017.11.007","article-title":"A direct position determination method with combined TDOA and FDOA based on particle filter","volume":"31","author":"Lu","year":"2018","journal-title":"Chin. J. Aeronaut."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"7859","DOI":"10.1109\/JSEN.2020.2980012","article-title":"Direct Position Determination of Moving Sources Based on Delay and Doppler","volume":"20","author":"Ma","year":"2020","journal-title":"IEEE Sens. J."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Bar-Shalom, O., and Weiss, A.J. (2007, January 17\u201320). Direct position determination of OFDM signals. Proceedings of the 2007 IEEE 8th Workshop on Signal Processing Advances in Wireless Communications, Helsinki, Finland.","DOI":"10.1109\/SPAWC.2007.4401287"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1049\/iet-rsn:20080097","article-title":"Efficient direct position determination of orthogonal frequency division multiplexing signals","volume":"3","author":"Weiss","year":"2009","journal-title":"IET Radar Sonar Navig."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Yin, J., Wang, D., and Wu, Y. (2018). An Efficient Direct Position Determination Method for Multiple Strictly Noncircular Sources. Sensors, 18.","DOI":"10.3390\/s18020324"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"4167","DOI":"10.1109\/TSP.2010.2049264","article-title":"Nested Arrays: A Novel Approach to Array Processing with Enhanced Degrees of Freedom","volume":"58","author":"Pal","year":"2010","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"573","DOI":"10.1109\/TSP.2010.2089682","article-title":"Sparse Sensing with Co-Prime Samplers and Arrays","volume":"59","author":"Vaidyanathan","year":"2011","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Zhang, Y., Ba, B., Wang, D., Geng, W., and Xu, H. (2018). Direct Position Determination of Multiple Non-Circular Sources with a Moving Coprime Array. Sensors, 18.","DOI":"10.3390\/s18051479"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"101106","DOI":"10.1109\/ACCESS.2019.2929805","article-title":"Direct Localization of Multiple Noncircular Sources with a Moving Nested Array","volume":"7","author":"Kumar","year":"2019","journal-title":"IEEE Access"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"9950518","DOI":"10.1155\/2021\/9950518","article-title":"Direct Position Determination of Noncircular Sources with Multiple Nested Arrays: Reduced Dimension Subspace Data Fusion","volume":"2021","author":"Qian","year":"2021","journal-title":"Wirel. Commun. Mob. Comput."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2448","DOI":"10.1016\/j.sigpro.2009.04.009","article-title":"Direct position determination of cyclostationary signals","volume":"89","author":"Reuven","year":"2009","journal-title":"Signal Process."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2168","DOI":"10.1109\/TSP.2009.2034935","article-title":"DOA Estimation of Quasi-Stationary Signals with Less Sensors Than Sources and Unknown Spatial Noise Covariance: A Khatri\u2013Rao Subspace Approach","volume":"58","author":"Ma","year":"2010","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"6538","DOI":"10.1109\/ACCESS.2017.2695581","article-title":"Direction of Arrival Estimation of Quasi-Stationary Signals Using Unfolded Coprime Array","volume":"5","author":"Li","year":"2017","journal-title":"IEEE Access"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/j.sigpro.2017.09.029","article-title":"Sparsity-aware DOA estimation of quasi-stationary signals using nested arrays","volume":"144","author":"Wang","year":"2018","journal-title":"Signal Process."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"975","DOI":"10.1109\/78.80927","article-title":"A unitary transformation method for angle-of-arrival estimation","volume":"39","year":"1991","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1548","DOI":"10.1109\/TSP.2014.2298384","article-title":"Real-Valued MUSIC for Efficient Direction Estimation with Arbitrary Array Geometries","volume":"62","author":"Yan","year":"2014","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"716","DOI":"10.1109\/TAC.1974.1100705","article-title":"A new look at the statistical model identification","volume":"19","author":"Akaike","year":"1974","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1109\/TASSP.1985.1164557","article-title":"Detection of signals by information theoretic criteria","volume":"33","author":"Wax","year":"1985","journal-title":"IEEE Trans. Acoust. Speech Signal Process."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1744","DOI":"10.1109\/78.510621","article-title":"A model selection rule for sinusoids in white Gaussian noise","volume":"44","author":"Djuric","year":"1996","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"5658","DOI":"10.1109\/TSP.2007.899344","article-title":"Reduced-Rank MDL Method for Source Enumeration in High-Resolution Array Processing","volume":"55","author":"Huang","year":"2007","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"320","DOI":"10.1109\/89.841214","article-title":"Convolutive blind separation of non-stationary sources","volume":"8","author":"Parra","year":"2000","journal-title":"IEEE Trans. Speech Audio Process."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1837","DOI":"10.1109\/78.942614","article-title":"Blind separation of instantaneous mixtures of nonstationary sources","volume":"49","author":"Cardoso","year":"2001","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"406","DOI":"10.1109\/TSP.2003.820988","article-title":"Blind identification of MIMO FIR systems driven by quasistationary sources using second-order statistics: A frequency domain approach","volume":"52","author":"Rahbar","year":"2004","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"2569","DOI":"10.1109\/LCOMM.2021.3078877","article-title":"Difference Coarray-Based Direction Finding via Covariance Fitting","volume":"25","author":"Huang","year":"2021","journal-title":"IEEE Commun. Lett."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"3529","DOI":"10.1109\/TSP.2020.2996155","article-title":"Beamspace Direct Localization for Large-Scale Antenna Array Systems","volume":"68","author":"Zhao","year":"2020","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"2335","DOI":"10.1109\/LWC.2021.3091989","article-title":"Pseudo Noise Subspace Based DOA Estimation for Unfolded Coprime Linear Arrays","volume":"10","author":"Huang","year":"2021","journal-title":"IEEE Wirel. Commun. Lett."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/11\/4209\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:23:00Z","timestamp":1760138580000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/11\/4209"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,31]]},"references-count":41,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2022,6]]}},"alternative-id":["s22114209"],"URL":"https:\/\/doi.org\/10.3390\/s22114209","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,5,31]]}}}