{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,31]],"date-time":"2026-01-31T06:48:39Z","timestamp":1769842119110,"version":"3.49.0"},"reference-count":32,"publisher":"MDPI AG","issue":"20","license":[{"start":{"date-parts":[[2024,10,19]],"date-time":"2024-10-19T00:00:00Z","timestamp":1729296000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the Key Laboratory Fund of the Chinese Academy of Sciences","award":["E32213xxxx"],"award-info":[{"award-number":["E32213xxxx"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Remote Sensing"],"abstract":"<jats:p>The eigen-decomposition of a covariance matrix is a key step in the Direction of Arrival (DOA) estimation algorithms such as subspace classes. Eigen-decomposition using the parallel Jacobi algorithm implemented on FPGA offers excellent parallelism and real-time performance. Addressing the high complexity and resource consumption of the traditional parallel Jacobi algorithm implemented on FPGA, this study proposes an improved FPGA-based parallel Jacobi algorithm for eigen-decomposition. By analyzing the relationship between angle calculation and rotation during the Jacobi algorithm decomposition process, leveraging parallelism in the data processing, and based on the concepts of time-division multiplexing and parallel partition processing, this approach effectively reduces FPGA resource consumption. The improved parallel Jacobi algorithm is then applied to the classic DOA estimation algorithm, the MUSIC algorithm, and implemented on Xilinx\u2019s Zynq FPGA. Experimental results demonstrate that this parallel approach can reduce resource consumption by approximately 75% compared to the traditional method but introduces little additional time consumption. The proposed method in this paper will solve the problem of great hardware consumption of eigen-decomposition based on FPGA in DOA applications.<\/jats:p>","DOI":"10.3390\/rs16203892","type":"journal-article","created":{"date-parts":[[2024,10,21]],"date-time":"2024-10-21T09:58:24Z","timestamp":1729504704000},"page":"3892","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Field Programmable Gate Array (FPGA) Implementation of Parallel Jacobi for Eigen-Decomposition in Direction of Arrival (DOA) Estimation Algorithm"],"prefix":"10.3390","volume":"16","author":[{"given":"Shuang","family":"Zhou","sequence":"first","affiliation":[{"name":"National Space Science Center, Chinese Academy of Sciences, Beijing 101499, China"},{"name":"University of Chinese Academy of Sciences, Beijing 100049, China"}]},{"ORCID":"https:\/\/orcid.org\/0009-0002-3279-8945","authenticated-orcid":false,"given":"Li","family":"Zhou","sequence":"additional","affiliation":[{"name":"National Space Science Center, Chinese Academy of Sciences, Beijing 101499, China"}]}],"member":"1968","published-online":{"date-parts":[[2024,10,19]]},"reference":[{"key":"ref_1","first-page":"1044","article-title":"Hardware-Efficient Beamspace Direction-of-Arrival Estimator for Unequal-Sized Subarrays","volume":"69","author":"Guo","year":"2022","journal-title":"IEEE Trans. Circuits Syst. II"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Chen, Y., Yang, M., Li, J., and Zhang, X. (2023). A Nested\u2013Nested Sparse Array Specially for Monostatic Colocated MIMO Radar with Increased Degree of Freedom. Sensors, 23.","DOI":"10.3390\/s23229230"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Cui, J., Pan, W., and Wang, H. (2024). Direction of Arrival Estimation Method Based on Eigenvalues and Eigenvectors for Coherent Signals in Impulsive Noise. Mathematics, 12.","DOI":"10.3390\/math12060832"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Zhang, J., Shi, Z., Chen, Y., and Liu, M. (2024). Spatial Parameter Identification for MIMO Systems in the Presence of Non-Gaussian Interference. Remote Sens., 16.","DOI":"10.3390\/rs16071243"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1109\/MSP.2023.3258060","article-title":"Twenty-Five Years of Sensor Array and Multichannel Signal Processing: A Review of Progress to Date and Potential Research Directions","volume":"40","author":"Liu","year":"2023","journal-title":"IEEE Signal Process. Mag."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1007\/s11235-023-01000-w","article-title":"Subspace Based DOA Estimation of DS-CDMA Signals","volume":"83","author":"Ghasemian","year":"2023","journal-title":"Telecommun. Syst."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3351","DOI":"10.1109\/TCSI.2021.3083280","article-title":"Towards Low Latency and Resource-Efficient FPGA Implementations of the MUSIC Algorithm for Direction of Arrival Estimation","volume":"68","author":"Butt","year":"2021","journal-title":"IEEE Trans. Circuits Syst. I"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"3500304","DOI":"10.1109\/LGRS.2023.3238334","article-title":"How to Determine an Optimal Noise Subspace?","volume":"20","author":"Xu","year":"2023","journal-title":"IEEE Geosci. Remote Sens. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"491","DOI":"10.1109\/TIE.2018.2823666","article-title":"High-Accuracy Signal Subspace Separation Algorithm Based on Gaussian Kernel Soft Partition","volume":"66","author":"Xu","year":"2019","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Eranti, P.K., and Barkana, B.D. (2022). An Overview of Direction-of-Arrival Estimation Methods Using Adaptive Directional Time-Frequency Distributions. Electronics, 11.","DOI":"10.3390\/electronics11091321"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"172","DOI":"10.1016\/j.jpdc.2016.05.014","article-title":"FPGA, GPU, and CPU Implementations of Jacobi Algorithm for Eigen analysis","volume":"96","author":"Torun","year":"2016","journal-title":"J. Parallel Distrib. Comput."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1573","DOI":"10.1007\/s00034-022-02180-7","article-title":"High-Performance Matrix Eigenvalue Decomposition Using the Parallel Jacobi Algorithm on FPGA","volume":"42","author":"Yan","year":"2023","journal-title":"Circuits Syst. Signal Process."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Yao, B., Li, H., Zhou, T., Chen, B., and Yu, H. (2008, January 6\u20138). Real-Time Implementation of Multiple Sub-Array Beam-Space MUSIC Based on FPGA and DSP Array. Proceedings of the 2008 Fifth IEEE International Symposium on Embedded Computing, Beijing, China.","DOI":"10.1109\/SEC.2008.6"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2941","DOI":"10.1109\/TCSI.2022.3162303","article-title":"Hardware Acceleration of MUSIC Algorithm for Sparse Arrays and Uniform Linear Arrays","volume":"69","author":"Li","year":"2022","journal-title":"IEEE Trans. Circuits Syst. I"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"e870569","DOI":"10.1155\/2015\/870569","article-title":"High Level Synthesis FPGA Implementation of the Jacobi Algorithm to Solve the Eigen Problem","volume":"2015","author":"Bravo","year":"2015","journal-title":"Math. Probl. Eng."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Shiri, A., and Khosroshahi, G.K. (May, January 30). An FPGA Implementation of Singular Value Decomposition. Proceedings of the 2019 27th Iranian Conference on Electrical Engineering (ICEE), Yazd, Iran.","DOI":"10.1109\/IranianCEE.2019.8786719"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1204","DOI":"10.1137\/0613074","article-title":"Jacobi\u2019s Method Is More Accurate than QR","volume":"13","author":"Demmel","year":"1992","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1137\/0906007","article-title":"The Solution of Singular-Value and Symmetric Eigenvalue Problems on Multiprocessor Arrays","volume":"6","author":"Brent","year":"1985","journal-title":"SIAM J. Sci. Stat. Comput."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1722","DOI":"10.1109\/TVLSI.2008.2001939","article-title":"Novel HW Architecture Based on FPGAs Oriented to Solve the Eigen Problem","volume":"16","author":"Bravo","year":"2008","journal-title":"IEEE Trans. VLSI Syst."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Mahale, G.V., and Bartakke, P.P. (2011, January 14\u201315). Eigen Values and Vectors Computations on Virtex-5 FPGA Platform Cyclic Jacobis Algorithm Using Systolic Array Architecture. Proceedings of the 3rd International Conference on Advances in Recent Technologies in Communication and Computing (ARTCom 2011), Bangalore, India.","DOI":"10.1049\/ic.2011.0044"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Sun, C.-C., and Goetze, J. (2013, January 12\u201315). FPGA Implementation of Parallel Unitary-Rotation Jacobi EVD Method Based on Network-on-Chip. Proceedings of the 2013 International Symposium on Intelligent Signal Processing and Communication Systems, Naha-shi, Japan.","DOI":"10.1109\/ISPACS.2013.6704512"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1049\/cje.2016.06.033","article-title":"Fast Implementation for the Singular Value and Eigenvalue Decomposition Based on FPGA","volume":"26","author":"Zhang","year":"2017","journal-title":"Chin. J. Electron."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"6275","DOI":"10.1109\/TVT.2020.2984705","article-title":"Accelerating Parallel Jacobi Method for Matrix Eigenvalue Computation in DOA Estimation Algorithm","volume":"69","author":"Shi","year":"2020","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Ghayoula, R., Amara, W., El Gmati, I., Smida, A., and Fattahi, J. (2022). An Efficient FPGA Implementation of MUSIC Processor Using Cyclic Jacobi Method: LiDAR Applications. Appl. Sci., 12.","DOI":"10.3390\/app12199726"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1020","DOI":"10.1109\/TVLSI.2022.3170526","article-title":"Low-Latency and Reconfigurable VLSI-Architectures for Computing Eigenvalues and Eigenvectors Using CORDIC-Based Parallel Jacobi Method","volume":"30","author":"Sharma","year":"2022","journal-title":"IEEE Trans. VLSI Syst."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Liu, C.-W., Wu, J.-Y., and Huang, K.-C. (2020, January 12\u201314). A Low Latency NN-Based Cyclic Jacobi EVD Processor for DOA Estimation in Radar System. Proceedings of the 2020 IEEE International Symposium on Circuits and Systems (ISCAS), Seville, Spain.","DOI":"10.1109\/ISCAS45731.2020.9180881"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Ibrahim, A., Valle, M., Noli, L., and Chible, H. (2015, January 8\u201310). Assessment of FPGA Implementations of One-Sided Jacobi Algorithm for Singular Value Decomposition. Proceedings of the 2015 IEEE Computer Society Annual Symposium on VLSI, Montpellier, France.","DOI":"10.1109\/ISVLSI.2015.63"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"5782","DOI":"10.1109\/TVT.2022.3221915","article-title":"High-Performance of Eigenvalue Decomposition on FPGA for the DOA Estimation","volume":"72","author":"Zhang","year":"2023","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Rao, T.V.S., Roy, L.P., and Mahapatra, K. (2021, January 13\u201316). Comparative Study on Error in MIMO Radar DOA Estimation. Proceedings of the 2021 IEEE International Conference on Advanced Networks and Telecommunications Systems (ANTS), Hyderabad, India.","DOI":"10.1109\/ANTS52808.2021.9937013"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"e596103","DOI":"10.1155\/2014\/596103","article-title":"Parallel Jacobi EVD Methods on Integrated Circuits","volume":"2014","author":"Sun","year":"2014","journal-title":"VLSI Des."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Cavallaro, J.R., and Luk, F.T. (1987). CORDIC Arithmetic for an SVD Processor, IEEE Computer Society.","DOI":"10.1109\/ARITH.1987.6158686"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"330","DOI":"10.1109\/TEC.1959.5222693","article-title":"The CORDIC Trigonometric Computing Technique","volume":"EC-8","author":"Volder","year":"1959","journal-title":"IRE Trans. Electron. Comput."}],"container-title":["Remote Sensing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2072-4292\/16\/20\/3892\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:16:50Z","timestamp":1760113010000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2072-4292\/16\/20\/3892"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,10,19]]},"references-count":32,"journal-issue":{"issue":"20","published-online":{"date-parts":[[2024,10]]}},"alternative-id":["rs16203892"],"URL":"https:\/\/doi.org\/10.3390\/rs16203892","relation":{},"ISSN":["2072-4292"],"issn-type":[{"value":"2072-4292","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,10,19]]}}}