{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,29]],"date-time":"2025-10-29T03:42:50Z","timestamp":1761709370778,"version":"build-2065373602"},"reference-count":27,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2017,7,3]],"date-time":"2017-07-03T00:00:00Z","timestamp":1499040000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61371172"],"award-info":[{"award-number":["61371172"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"International S&amp;T Cooperation Program of China","award":["2015DFR10220"],"award-info":[{"award-number":["2015DFR10220"]}]},{"name":"Ocean 245 Engineering Project of the National Key Laboratory Foundation","award":["1213"],"award-info":[{"award-number":["1213"]}]},{"name":"Fundamental Research Funds 246 for the Central Universities","award":["HEUCF1508"],"award-info":[{"award-number":["HEUCF1508"]}]},{"DOI":"10.13039\/501100005046","name":"Natural Science Foundation of Heilongjiang Province","doi-asserted-by":"publisher","award":["F201337"],"award-info":[{"award-number":["F201337"]}],"id":[{"id":"10.13039\/501100005046","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper considers the underdetermined blind separation of multiple input multiple output (MIMO) radar signals that are insufficiently sparse in both time and frequency domains under noisy conditions, while traditional algorithms are usually applied in the ideal sparse environment. An effective separation method based on single source point (SSP) identification and time-frequency smoothed     l 0     norm (TF-SL0) is proposed. Firstly, a preprocessing step of the moving average filter and a novel argument-based time-frequency SSPs detection are employed to improve the signal-to-noise ratio and signal sparsity of the observed signals, respectively. Then, the mixing matrix is obtained by using clustering algorithms. Secondly, to obtain the optimal solution of underdetermined sparse component analysis, the smoothed     l 0     norm (SL0) is introduced to preliminarily achieve signal separation in the time-frequency domain. Finally, time-frequency ridge estimation is proposed to jointly enhance the reconstruction accuracy of the MIMO radar signals, and the time domain waveforms are recovered by the model of the signals. Simulations illustrate the validity of the method and show that the proposed method outperforms the traditional methods in source separation, especially in the non-cooperative electromagnetic case where the prior information is unknown.<\/jats:p>","DOI":"10.3390\/sym9070104","type":"journal-article","created":{"date-parts":[[2017,7,3]],"date-time":"2017-07-03T10:27:31Z","timestamp":1499077651000},"page":"104","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["A Time-Frequency Domain Underdetermined Blind Source Separation Algorithm for MIMO Radar Signals"],"prefix":"10.3390","volume":"9","author":[{"given":"Qiang","family":"Guo","sequence":"first","affiliation":[{"name":"College of Information and Communication Engineering, Harbin Engineering University, Harbin 150001, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6746-7471","authenticated-orcid":false,"given":"Guoqing","family":"Ruan","sequence":"additional","affiliation":[{"name":"College of Information and Communication Engineering, Harbin Engineering University, Harbin 150001, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yanping","family":"Liao","sequence":"additional","affiliation":[{"name":"College of Information and Communication Engineering, Harbin Engineering University, Harbin 150001, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2017,7,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1016\/j.neucom.2014.02.033","article-title":"A fast blind source separation algorithm based on the temporal structure of signals","volume":"139","author":"Zhang","year":"2014","journal-title":"Neurocomputing"},{"doi-asserted-by":"crossref","unstructured":"Tang, G., Luo, G.G., and Zhang, W.H. (2016). Underdetermined Blind Source Separation with Variational Mode Decomposition for Compound Roller Bearing Fault Signals. Sensors, 16.","key":"ref_2","DOI":"10.3390\/s16060897"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"3367","DOI":"10.1007\/s00034-015-0198-y","article-title":"A Mixing Matrix Estimation Algorithm for Underdetermined Blind Source Separation","volume":"35","author":"Li","year":"2016","journal-title":"Circuits Syst. Signal Process."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2353","DOI":"10.1016\/S0165-1684(01)00120-7","article-title":"Underdetermined blind source separation using sparse representations","volume":"81","author":"Bofill","year":"2001","journal-title":"Signal Process."},{"key":"ref_5","first-page":"2985","article-title":"Blind separation of disjoint orthogonal signals: Demixing N sources from 2 mixtures","volume":"5","author":"Jourjine","year":"2000","journal-title":"Acoust. Speech Signal Process. ICASSP"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1830","DOI":"10.1109\/TSP.2004.828896","article-title":"Blind separation of speech mixture via time-frequency masking","volume":"52","author":"Yilmaz","year":"2004","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1762","DOI":"10.1016\/j.sigpro.2009.03.017","article-title":"An algorithm for mixing matrix estimation in instantaneous blind source separation","volume":"89","author":"Reju","year":"2009","journal-title":"Signal Process."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2604","DOI":"10.1109\/TSP.2009.2017570","article-title":"Underdetermined blind source separation based on subspace representation","volume":"57","author":"Kim","year":"2009","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"623","DOI":"10.1016\/j.neucom.2015.08.008","article-title":"Novel mixing matrix estimation approach in underdetermined blind source separation","volume":"173","author":"Sun","year":"2016","journal-title":"Neurocomputing"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"3397","DOI":"10.1109\/78.258082","article-title":"Matching pursuits with time-frequency dictionaries","volume":"41","author":"Mallat","year":"1993","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1873","DOI":"10.1080\/00207178908953472","article-title":"Orthogonal least squares methods and their application to non-linear system identification","volume":"50","author":"Chen","year":"1989","journal-title":"Int. J. Control"},{"unstructured":"Pati, Y.C., Rezaiifar, R., and Krishnaprasad, P.S. (1993, January 1\u20133). Orthogonal matching pursuit: Recursive function approximation with applications to wavelet decomposition. Proceedings of the Conference Record of the 27th Asilomar Conference on Signals Systems and Compute, Pacific Grove, CA, USA.","key":"ref_12"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1137\/S003614450037906X","article-title":"Atomic decomposition by basis pursuit","volume":"43","author":"Chen","year":"2001","journal-title":"SIAM Rev."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"863","DOI":"10.1162\/089976601300014385","article-title":"Blind source separation by sparse decomposition in a signal dictionary","volume":"13","author":"Zibulevsky","year":"2001","journal-title":"Neural Comput."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2197","DOI":"10.1073\/pnas.0437847100","article-title":"Maximal sparsity representation via l1 minimization","volume":"100","author":"Donoho","year":"2003","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1193","DOI":"10.1162\/089976604773717586","article-title":"Analysis of sparse representation and blind source separation","volume":"16","author":"Li","year":"2004","journal-title":"Neural Comput."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"4378","DOI":"10.1109\/TSP.2009.2025154","article-title":"An iterative Bayesian algorithm for sparse component analysis in presence of noise","volume":"57","author":"Zayyani","year":"2009","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1109\/TSP.2008.2007606","article-title":"A fast approach for overcomplete sparse decomposition based on smoothed l0 norm","volume":"57","author":"Mohimani","year":"2009","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"899","DOI":"10.1109\/TAES.2010.5461664","article-title":"Permutation method for ICA separated source signal blocks in time domain","volume":"46","author":"Amishima","year":"2010","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"3572","DOI":"10.1109\/TMTT.2014.2364220","article-title":"A frequency-division MIMO FMCW radar system based on Delta\u2013Sigma modulated transmitters","volume":"62","author":"Feger","year":"2014","journal-title":"IEEE Trans. Microw. Theory Tech."},{"unstructured":"Ma, J., Huang, G., Zhou, D., Fang, B., and Yu, H. (2012, January 9\u201311). Underdetermined blind sorting of radar signals based on sparse component analysis. Proceedings of the IEEE International Conference on Communication Technology, Chengdu, China.","key":"ref_21"},{"doi-asserted-by":"crossref","unstructured":"Fang, B., Huang, G., and Gao, J. (2013, January 25\u201327). Underdetermined blind source separation for LFM radar signal based on compressive sensing. Proceedings of the 2013 25th Chinese Control and Decision Conference, Guiyang, China.","key":"ref_22","DOI":"10.1109\/CCDC.2013.6561239"},{"key":"ref_23","first-page":"2505","article-title":"Underdetermined blind separation of radar signals based on tensor decomposition","volume":"38","author":"Ai","year":"2016","journal-title":"Syst. Eng. Electron."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"449","DOI":"10.1109\/LSP.2008.924027","article-title":"Comments on discrete frequency-coding waveform design for netted radar systems","volume":"15","author":"Liu","year":"2008","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"645","DOI":"10.1109\/TNN.2005.845141","article-title":"Survey of clustering algorithms","volume":"16","author":"Xu","year":"2005","journal-title":"IEEE Trans. Neural Netw."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1007\/978-3-540-30110-3_34","article-title":"Wavelet de-noising for blind source separation in noisy mixtures","volume":"Volume 3195","author":"Rivet","year":"2004","journal-title":"Independent Component Analysis and Blind Signal Separation"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1650","DOI":"10.1109\/TAES.2009.5310326","article-title":"Orthogonal discrete frequency-coding waveform set design with minimized autocorrelation sidelobes","volume":"45","author":"Liu","year":"2009","journal-title":"IEEE Trans. Aerosp. Electron. 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