{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,22]],"date-time":"2025-11-22T11:30:16Z","timestamp":1763811016846,"version":"build-2065373602"},"reference-count":46,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2023,2,15]],"date-time":"2023-02-15T00:00:00Z","timestamp":1676419200000},"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":["62271363"],"award-info":[{"award-number":["62271363"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Remote Sensing"],"abstract":"<jats:p>In a distributed frequency-modulated continuous waveform (FMCW) radar system, the echo data collected are not continuous in the azimuth direction, so the imaging effect of the traditional range-Doppler (RD) algorithm is poor. Sparse Bayesian learning (SBL) is an optimization algorithm based on Bayesian theory that has been successfully applied to high-resolution radar imaging because of its strong robustness and high accuracy. However, SBL is highly computationally complex. Fortunately, with FMCW radar echo data, most of the time-consuming SBL operations involve a Toeplitz-block Toeplitz (TBT) matrix. In this article, based on this advantage, we propose a fast SBL algorithm that can be used to obtain high-angular-resolution images, in which the inverse of the TBT matrix can be transposed as the sum of the products of the block lower triangular Toeplitz matrix and the block circulant matrix by using a new decomposition method, and some of the matrix multiplications can be quickly computed using the fast Fourier transform (FFT), decreasing the computation time by several orders of magnitude. Finally, simulations and experiments were used to ensure the effectiveness of the proposed algorithm.<\/jats:p>","DOI":"10.3390\/rs15041054","type":"journal-article","created":{"date-parts":[[2023,2,15]],"date-time":"2023-02-15T04:47:24Z","timestamp":1676436444000},"page":"1054","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Efficient Implementation for SBL-Based Coherent Distributed mmWave Radar Imaging"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2166-2516","authenticated-orcid":false,"given":"Fengzhou","family":"Dai","sequence":"first","affiliation":[{"name":"National Lab of Radar Signal Processing, Xidian University, Xi\u2019an 710071, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yuhang","family":"Li","sequence":"additional","affiliation":[{"name":"National Lab of Radar Signal Processing, Xidian University, Xi\u2019an 710071, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yuanyuan","family":"Wang","sequence":"additional","affiliation":[{"name":"National Lab of Radar Signal Processing, Xidian University, Xi\u2019an 710071, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hao","family":"Chen","sequence":"additional","affiliation":[{"name":"Infineon Technologies Center of Competence (Shanghai) Co., Ltd., Shanghai 201203, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,2,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Wei, S., Zhou, Z., Wang, M., Wei, J., Liu, S., Shi, J., Zhang, X., and Fan, F. (2021). 3DRIED: A High-Resolution 3-D Millimeter-Wave Radar Dataset Dedicated to Imaging and Evaluation. Remote Sens., 13.","DOI":"10.3390\/rs13173366"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1109\/JMW.2020.3034475","article-title":"Coherent automotive radar networks: The next generation of radar-based im-aging and mapping","volume":"1","author":"Gottinger","year":"2021","journal-title":"IEEE J. Microw."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Wang, Z., Miao, X., Huang, Z., and Luo, H. (2021). Research of Target Detection and Classification Techniques Using Millimeter-Wave Radar and Vision Sensors. Remote Sens., 13.","DOI":"10.3390\/rs13061064"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"24504","DOI":"10.1109\/JSEN.2022.3218454","article-title":"Vehicle Occupancy Detector Based on FMCW mm-Wave Radar at 77 GHz","volume":"22","author":"Munte","year":"2022","journal-title":"IEEE Sensors J."},{"key":"ref_5","first-page":"1","article-title":"Multi-Channel Back-Projection Algorithm for Mmwave Automotive MIMO SAR Imaging with Doppler-Division Multiplexing","volume":"PP","author":"Zhang","year":"2022","journal-title":"IEEE J. Sel. Top. Signal Process."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"637","DOI":"10.1109\/JMW.2022.3196454","article-title":"System Performance of a 79 GHz High-Resolution 4D Imaging MIMO Radar with 1728 Virtual Channels","volume":"2","author":"Schwarz","year":"2022","journal-title":"IEEE J. Microw."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Yang, B., Zhang, H., Chen, Y., Zhou, Y., and Peng, Y. (2022). Urban Traffic Imaging Using Millimeter-Wave Radar. Remote Sens., 14.","DOI":"10.3390\/rs14215416"},{"key":"ref_8","first-page":"5006412","article-title":"Object Classification Based on Enhanced Evidence Theory: Radar\u2013Vision Fusion Approach for Roadside Application","volume":"71","author":"Liu","year":"2022","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Huang, X., Dong, X., Ma, J., Liu, K., Ahmed, S., Lin, J., and Qiu, B. (2021). The Improved A* Obstacle Avoidance Algorithm for the Plant Protection UAV with Millimeter Wave Radar and Monocular Camera Data Fusion. Remote Sens., 13.","DOI":"10.3390\/rs13173364"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1807","DOI":"10.1109\/JSEN.2021.3139124","article-title":"Embedded Sensors, Communication Technologies, Computing Platforms and Machine Learning for UAVs: A Review","volume":"22","author":"Wilson","year":"2021","journal-title":"IEEE Sensors J."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"12173","DOI":"10.1109\/JSEN.2022.3175618","article-title":"Angle-Insensitive Human Motion and Posture Recognition Based on 4D Imaging Radar and Deep Learning Classifiers","volume":"22","author":"Zhao","year":"2022","journal-title":"IEEE Sensors J."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2503","DOI":"10.1109\/JSTARS.2022.3158661","article-title":"Accurate Micro-Doppler Analysis by Doppler and $k$-Space Decomposition for Millimeter Wave Short-Range Radar","volume":"15","author":"Ando","year":"2022","journal-title":"IEEE J. Sel. Top. Appl. Earth Obs. Remote. Sens."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Antolinos, E., Garc\u00eda-Rial, F., Hern\u00e1ndez, C., Montesano, D., Godino-Llorente, J.I., and Grajal, J. (2020). Cardiopulmonary Activity Mon-itoring Using Millimeter Wave Radars. Remote Sens., 12.","DOI":"10.3390\/rs12142265"},{"key":"ref_14","first-page":"8006311","article-title":"Detection of Human Breathing in Non-Line-of-Sight Region by Using mmWave FMCW Radar","volume":"71","author":"Li","year":"2022","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Zhang, Q., Zhang, Y., Zhang, Y., Huang, Y., and Yang, J. (2021). A Sparse Denoising-Based Super-Resolution Method for Scanning Radar Imaging. Remote Sens., 13.","DOI":"10.3390\/rs13142768"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"6534","DOI":"10.1109\/TGRS.2020.2977719","article-title":"TV-Sparse Super-Resolution Method for Radar Forward-Looking Imaging","volume":"58","author":"Zhang","year":"2020","journal-title":"IEEE Trans. Geosci. Remote. Sens."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Cong, J., Wang, X., Lan, X., Huang, M., and Wan, L. (2021). Fast Target Localization Method for FMCW MIMO Radar via VDSR Neural Network. Remote Sens., 13.","DOI":"10.3390\/rs13101956"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"4893","DOI":"10.1109\/TMTT.2021.3092401","article-title":"Distributed Phased Arrays: Challenges and Recent Advances","volume":"69","author":"Nanzer","year":"2021","journal-title":"IEEE Trans. Microw. Theory Tech."},{"key":"ref_19","unstructured":"Coutts, S., Cuomo, K., and McHarg, J. (2006, January 12\u201314). Distributed coherent aperture measurements for next generation BMD radar. Proceedings of the Fourth IEEE Workshop on Sensor Array and Multichannel Processing, Waltham, MA, USA."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"4623","DOI":"10.1109\/JSYST.2020.3031912","article-title":"Bayesian Matching Pursuit-Based Distributed FMCW MIMO Radar Imaging","volume":"15","author":"Seo","year":"2020","journal-title":"IEEE Syst. J."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"537","DOI":"10.1109\/TCI.2018.2875375","article-title":"Sparse Blind Deconvolution for Distributed Radar Autofocus Imaging","volume":"4","author":"Mansour","year":"2018","journal-title":"IEEE Trans. Comput. Imaging"},{"key":"ref_22","first-page":"203","article-title":"Ultra-wideband coherent processing","volume":"10","author":"Cuomo","year":"1997","journal-title":"Linc. Lab. J."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1089","DOI":"10.1109\/TAES.2003.1238761","article-title":"Spectral analysis of periodically gapped data","volume":"39","author":"Larsson","year":"2003","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"4154","DOI":"10.1109\/TSP.2011.2145376","article-title":"Efficient Implementation of Iterative Adaptive Approach Spectral Estimation Techniques","volume":"59","author":"Glentis","year":"2011","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"5043","DOI":"10.1109\/TIP.2017.2728182","article-title":"ISAR imaging of high-speed maneuvering target using gapped stepped-frequency waveform and compressive sensing","volume":"26","author":"Kang","year":"2017","journal-title":"IEEE Trans. Image Process."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"4655","DOI":"10.1109\/TIT.2007.909108","article-title":"Signal Recovery from Random Measurements via Orthogonal Matching Pursuit","volume":"53","author":"Tropp","year":"2007","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1289","DOI":"10.1109\/TIT.2006.871582","article-title":"Compressed sensing","volume":"52","author":"Donoho","year":"2006","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2346","DOI":"10.1109\/TSP.2007.914345","article-title":"Bayesian compressive sensing","volume":"56","author":"Ji","year":"2008","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"6257","DOI":"10.1109\/TSP.2011.2168217","article-title":"Fast Variational Sparse Bayesian Learning with Automatic Relevance Determination for Superimposed Signals","volume":"59","author":"Shutin","year":"2011","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_30","unstructured":"Tipping, M.E., and Faul, A.C. (2003, January 3\u20136). Fast marginal likelihood maximisation for sparse Bayesian models. Proceedings of the International Workshop on Artificial Intelligence and Statistics, Key West, FL, USA."},{"key":"ref_31","first-page":"294","article-title":"A GAMP-Based Low Complexity Sparse Bayesian Learning Algorithm","volume":"66","author":"Schniter","year":"2017","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"2010","DOI":"10.1109\/TSP.2010.2040683","article-title":"Computationally Efficient Sparse Bayesian Learning via Belief Propagation","volume":"58","author":"Tan","year":"2010","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"774","DOI":"10.1109\/LSP.2017.2692217","article-title":"Fast Inverse-Free Sparse Bayesian Learning via Relaxed Evidence Lower Bound Maximization","volume":"24","author":"Duan","year":"2017","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"3065","DOI":"10.1109\/TNNLS.2020.3049056","article-title":"An Efficient Sparse Bayesian Learning Algorithm Based on Gaussian-Scale Mixtures","volume":"33","author":"Zhou","year":"2021","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"368","DOI":"10.1049\/iet-rsn.2013.0150","article-title":"Single transceiver-based time division multiplexing multiple-input\u2013multiple-output digital beamforming radar system: Concepts and experiments","volume":"8","author":"Tang","year":"2014","journal-title":"IET Radar Sonar Navig."},{"key":"ref_36","first-page":"300","article-title":"MIMO radar theory and experimental results","volume":"1","author":"Robey","year":"2004","journal-title":"Conf. Rec. Thirty-Eighth Asilomar Conf. Signals Syst. Comput."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Sun, H., Brigui, F., and Lesturgie, M. (2014, January 13\u201317). Analysis and comparison of MIMO radar waveforms. Proceedings of the 2014 International Radar Conference, Lille, France.","DOI":"10.1109\/RADAR.2014.7060251"},{"key":"ref_38","first-page":"211","article-title":"Sparse Bayesian learning and the relevance vector machine","volume":"1","author":"Tipping","year":"2001","journal-title":"J. Mach. Learn. Res."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"912","DOI":"10.1109\/JSTSP.2011.2159773","article-title":"Sparse Signal Recovery with Temporally Correlated Source Vectors Using Sparse Bayesian Learning","volume":"5","author":"Zhang","year":"2011","journal-title":"IEEE J. Sel. Top. Signal Process."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"2153","DOI":"10.1109\/TSP.2004.831016","article-title":"Sparse Bayesian Learning for Basis Selection","volume":"52","author":"Wipf","year":"2004","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"730","DOI":"10.1007\/BF01200697","article-title":"Circulants, displacements and decompositions of matrices","volume":"15","author":"Gohberg","year":"1992","journal-title":"Integral Equ. Oper. Theory"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"207176","DOI":"10.1155\/2013\/207176","article-title":"The Inverses of Block Toeplitz Matrices","volume":"2013","author":"Lv","year":"2013","journal-title":"J. Math."},{"key":"ref_43","first-page":"1","article-title":"Gohberg-Semencul Factorization-Based Fast Implementation of Sparse Bayesian Learning with a Fourier Dictionary","volume":"60","author":"Dai","year":"2022","journal-title":"IEEE Trans. Geosci. Remote. Sens."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"998","DOI":"10.1109\/JSTARS.2016.2598880","article-title":"Autofocusing for Sparse Aperture ISAR Imaging Based on Joint Constraint of Sparsity and Minimum Entropy","volume":"10","author":"Zhang","year":"2016","journal-title":"IEEE J. Sel. Top. Appl. Earth Obs. Remote. Sens."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1240","DOI":"10.1109\/7.805442","article-title":"Autofocusing of ISAR images based on entropy minimization","volume":"35","author":"Xi","year":"1999","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"2950","DOI":"10.1109\/TAES.2016.150883","article-title":"Efficient ISAR autofocus via minimization of Tsallis Entropy","volume":"52","author":"Kang","year":"2017","journal-title":"IEEE Trans. Aerosp. Electron. Syst."}],"container-title":["Remote Sensing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2072-4292\/15\/4\/1054\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:36:17Z","timestamp":1760121377000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2072-4292\/15\/4\/1054"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,15]]},"references-count":46,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2023,2]]}},"alternative-id":["rs15041054"],"URL":"https:\/\/doi.org\/10.3390\/rs15041054","relation":{},"ISSN":["2072-4292"],"issn-type":[{"type":"electronic","value":"2072-4292"}],"subject":[],"published":{"date-parts":[[2023,2,15]]}}}