{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T12:00:36Z","timestamp":1775476836387,"version":"3.50.1"},"reference-count":40,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2023,1,23]],"date-time":"2023-01-23T00:00:00Z","timestamp":1674432000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,1,23]],"date-time":"2023-01-23T00:00:00Z","timestamp":1674432000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Circuits Syst Signal Process"],"published-print":{"date-parts":[[2023,6]]},"DOI":"10.1007\/s00034-023-02292-8","type":"journal-article","created":{"date-parts":[[2023,1,23]],"date-time":"2023-01-23T21:28:56Z","timestamp":1674509336000},"page":"3770-3794","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Optimal Coprime Array: Properties, Optimization, and k-times extension"],"prefix":"10.1007","volume":"42","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1146-3783","authenticated-orcid":false,"given":"Rajen Kumar","family":"Patra","sequence":"first","affiliation":[]},{"given":"Anindya Sundar","family":"Dhar","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,1,23]]},"reference":[{"issue":"6","key":"2292_CR1","doi-asserted-by":"publisher","first-page":"688","DOI":"10.1049\/iet-spr.2017.0252","volume":"12","author":"SA Alawsh","year":"2018","unstructured":"S.A. Alawsh, A.H. Muqaibel, Multi-level prime array for sparse sampling. IET Signal Proc. 12(6), 688\u2013699 (2018)","journal-title":"IET Signal Proc."},{"issue":"6","key":"2292_CR2","doi-asserted-by":"publisher","first-page":"EL567","DOI":"10.1121\/1.5017531","volume":"142","author":"D Bush","year":"2017","unstructured":"D. Bush, N. Xiang, n-tuple coprime sensor arrays. J. Acoust. Soc. Am. 142(6), EL567\u2013EL572 (2017)","journal-title":"J. Acoust. Soc. Am."},{"key":"2292_CR3","volume-title":"Advances in Direction-of-Arrival Estimation","author":"S Chandran","year":"2006","unstructured":"S. Chandran, Advances in Direction-of-Arrival Estimation (Artech House, 2006)"},{"key":"2292_CR4","volume-title":"Introduction to Direction-of-Arrival Estimation","author":"Z Chen","year":"2010","unstructured":"Z. Chen, G. Gokeda, Y. Yu, Introduction to Direction-of-Arrival Estimation (Artech House, 2010)"},{"issue":"3","key":"2292_CR5","doi-asserted-by":"publisher","first-page":"273","DOI":"10.1109\/8.76322","volume":"39","author":"B Friedlander","year":"1991","unstructured":"B. Friedlander, A.J. Weiss, Direction finding in the presence of mutual coupling. IEEE Trans. Antennas Propag. 39(3), 273\u2013284 (1991)","journal-title":"IEEE Trans. Antennas Propag."},{"issue":"8","key":"2292_CR6","doi-asserted-by":"publisher","first-page":"1195","DOI":"10.1109\/5.622504","volume":"85","author":"LC Godara","year":"1997","unstructured":"L.C. Godara, Application of antenna arrays to mobile communications, part II: beam-forming and direction-of-arrival considerations. Proc. IEEE 85(8), 1195\u20131245 (1997)","journal-title":"Proc. IEEE"},{"issue":"5","key":"2292_CR7","doi-asserted-by":"publisher","first-page":"785","DOI":"10.1109\/TAP.1983.1143128","volume":"31","author":"I Gupta","year":"1983","unstructured":"I. Gupta, A. Ksienski, Effect of mutual coupling on the performance of adaptive arrays. IEEE Trans. Antennas Propag. 31(5), 785\u2013791 (1983)","journal-title":"IEEE Trans. Antennas Propag."},{"issue":"1","key":"2292_CR8","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1049\/ip-f-2.1992.0001","volume":"139","author":"S Haykin","year":"1992","unstructured":"S. Haykin, J. Reilly, V. Kezys, E. Vertatschitsch, Some aspects of array signal processing. IEE Proc. F-Radar Signal Process. 139(1), 1\u201326 (1992)","journal-title":"IEE Proc. F-Radar Signal Process."},{"issue":"1","key":"2292_CR9","doi-asserted-by":"publisher","first-page":"110","DOI":"10.1109\/MAP.2011.5773580","volume":"53","author":"E Holzman","year":"2011","unstructured":"E. Holzman, Introduction to direction-of-arrival estimation (Chen, Z.; 2010) [reviews and abstracts]. IEEE Antennas Propag. Mag. 53(1), 110\u2013111 (2011)","journal-title":"IEEE Antennas Propag. Mag."},{"issue":"6","key":"2292_CR10","doi-asserted-by":"publisher","first-page":"1163","DOI":"10.1029\/RS015i006p01163","volume":"15","author":"M Ishiguro","year":"1980","unstructured":"M. Ishiguro, Minimum redundancy linear arrays for a large number of antennas. Radio Sci. 15(6), 1163\u20131170 (1980)","journal-title":"Radio Sci."},{"issue":"4","key":"2292_CR11","doi-asserted-by":"publisher","first-page":"67","DOI":"10.1109\/79.526899","volume":"13","author":"H Krim","year":"1996","unstructured":"H. Krim, M. Viberg, Two decades of array signal processing research: the parametric approach. IEEE Signal Process. Mag. 13(4), 67\u201394 (1996)","journal-title":"IEEE Signal Process. Mag."},{"issue":"9","key":"2292_CR12","doi-asserted-by":"publisher","first-page":"1438","DOI":"10.1109\/LSP.2015.2409153","volume":"22","author":"CL Liu","year":"2015","unstructured":"C.L. Liu, P.P. Vaidyanathan, Remarks on the spatial smoothing step in coarray MUSIC. IEEE Signal Process. Lett. 22(9), 1438\u20131442 (2015)","journal-title":"IEEE Signal Process. Lett."},{"key":"2292_CR13","doi-asserted-by":"crossref","unstructured":"C.L. Liu, P. Vaidyanathan, Super nested arrays: sparse arrays with less mutual coupling than nested arrays, in Proc. IEEE Int. Conf. Acoust. Speech Signal Process. (ICASSP, 2016), pp. 2976\u20132980","DOI":"10.1109\/ICASSP.2016.7472223"},{"issue":"15","key":"2292_CR14","doi-asserted-by":"publisher","first-page":"3997","DOI":"10.1109\/TSP.2016.2558159","volume":"64","author":"CL Liu","year":"2016","unstructured":"C.L. Liu, P.P. Vaidyanathan, Super nested arrays: linear sparse arrays with reduced mutual coupling\u2014part I: fundamentals. IEEE Trans. Signal Process. 64(15), 3997\u20134012 (2016)","journal-title":"IEEE Trans. Signal Process."},{"issue":"16","key":"2292_CR15","doi-asserted-by":"publisher","first-page":"4203","DOI":"10.1109\/TSP.2016.2558167","volume":"64","author":"CL Liu","year":"2016","unstructured":"C.L. Liu, P.P. Vaidyanathan, Super nested arrays: linear sparse arrays with reduced mutual coupling\u2014part II: high-order extensions. IEEE Trans. Signal Process. 64(16), 4203\u20134217 (2016)","journal-title":"IEEE Trans. Signal Process."},{"issue":"13","key":"2292_CR16","doi-asserted-by":"publisher","first-page":"3369","DOI":"10.1109\/TSP.2017.2690390","volume":"65","author":"CL Liu","year":"2017","unstructured":"C.L. Liu, P.P. Vaidyanathan, Hourglass arrays and other novel 2-d sparse arrays with reduced mutual coupling. IEEE Trans. Signal Process. 65(13), 3369\u20133383 (2017)","journal-title":"IEEE Trans. Signal Process."},{"issue":"21","key":"2292_CR17","doi-asserted-by":"publisher","first-page":"5549","DOI":"10.1109\/TSP.2017.2736493","volume":"65","author":"J Liu","year":"2017","unstructured":"J. Liu, Y. Zhang, Y. Lu, S. Ren, S. Cao, Augmented nested arrays with enhanced DOF and reduced mutual coupling. IEEE Trans. Signal Process. 65(21), 5549\u20135563 (2017)","journal-title":"IEEE Trans. Signal Process."},{"issue":"8","key":"2292_CR18","doi-asserted-by":"publisher","first-page":"3010","DOI":"10.1109\/TSP.2005.850882","volume":"53","author":"D Malioutov","year":"2005","unstructured":"D. Malioutov, M. Cetin, A.S. Willsky, A sparse signal reconstruction perspective for source localization with sensor arrays. IEEE Trans. Signal Process. 53(8), 3010\u20133022 (2005)","journal-title":"IEEE Trans. Signal Process."},{"issue":"2","key":"2292_CR19","doi-asserted-by":"publisher","first-page":"172","DOI":"10.1109\/TAP.1968.1139138","volume":"16","author":"AT Moffet","year":"1968","unstructured":"A.T. Moffet, Minimum-redundancy linear arrays. IEEE Trans. Antennas Propag. 16(2), 172\u2013175 (1968)","journal-title":"IEEE Trans. Antennas Propag."},{"issue":"8","key":"2292_CR20","doi-asserted-by":"publisher","first-page":"4167","DOI":"10.1109\/TSP.2010.2049264","volume":"58","author":"P Pal","year":"2010","unstructured":"P. Pal, P.P. Vaidyanathan, Nested arrays: a novel approach to array processing with enhanced degrees of freedom. IEEE Trans. Signal Process. 58(8), 4167\u20134181 (2010)","journal-title":"IEEE Trans. Signal Process."},{"key":"2292_CR21","doi-asserted-by":"crossref","unstructured":"P. Pal, P.P. Vaidyanathan, Coprime sampling and the MUSIC algorithm, in 2011 Digital Signal Processing and Signal Processing Education Meeting (DSP\/SPE) (IEEE, 2011), pp. 289\u2013294","DOI":"10.1109\/DSP-SPE.2011.5739227"},{"key":"2292_CR22","volume-title":"Array Signal Processing","author":"SU Pillai","year":"2012","unstructured":"S.U. Pillai, Array Signal Processing (Springer, 2012)"},{"issue":"6","key":"2292_CR23","doi-asserted-by":"publisher","first-page":"1377","DOI":"10.1109\/TSP.2015.2393838","volume":"63","author":"S Qin","year":"2015","unstructured":"S. Qin, Y.D. Zhang, M.G. Amin, Generalized coprime array configurations for direction-of-arrival estimation. IEEE Trans. Signal Process. 63(6), 1377\u20131390 (2015)","journal-title":"IEEE Trans. Signal Process."},{"issue":"8","key":"2292_CR24","doi-asserted-by":"publisher","first-page":"2052","DOI":"10.1109\/TSP.2019.2901380","volume":"67","author":"A Raza","year":"2019","unstructured":"A. Raza, W. Liu, Q. Shen, Thinned coprime array for second-order difference co-array generation with reduced mutual coupling. IEEE Trans. Signal Process. 67(8), 2052\u20132065 (2019)","journal-title":"IEEE Trans. Signal Process."},{"issue":"7","key":"2292_CR25","doi-asserted-by":"publisher","first-page":"984","DOI":"10.1109\/29.32276","volume":"37","author":"R Roy","year":"1989","unstructured":"R. Roy, T. Kailath, ESPRIT-estimation of signal parameters via rotational invariance techniques. IEEE Trans. Acoust. Speech Signal Process. 37(7), 984\u2013995 (1989)","journal-title":"IEEE Trans. Acoust. Speech Signal Process."},{"issue":"3","key":"2292_CR26","doi-asserted-by":"publisher","first-page":"276","DOI":"10.1109\/TAP.1986.1143830","volume":"34","author":"R Schmidt","year":"1986","unstructured":"R. Schmidt, Multiple emitter location and signal parameter estimation. IEEE Trans. Antennas Propag. 34(3), 276\u2013280 (1986)","journal-title":"IEEE Trans. Antennas Propag."},{"issue":"4","key":"2292_CR27","doi-asserted-by":"publisher","first-page":"806","DOI":"10.1109\/TASSP.1985.1164649","volume":"33","author":"TJ Shan","year":"1985","unstructured":"T.J. Shan, M. Wax, T. Kailath, On spatial smoothing for direction-of-arrival estimation of coherent signals. IEEE Trans. Acoust. Speech Signal Process. 33(4), 806\u2013811 (1985)","journal-title":"IEEE Trans. Acoust. Speech Signal Process."},{"issue":"6","key":"2292_CR28","doi-asserted-by":"publisher","first-page":"1208","DOI":"10.1109\/LCOMM.2018.2821672","volume":"22","author":"J Shi","year":"2018","unstructured":"J. Shi, G. Hu, X. Zhang, H. Zhou, Generalized nested array: optimization for degrees of freedom and mutual coupling. IEEE Commun. Lett. 22(6), 1208\u20131211 (2018)","journal-title":"IEEE Commun. Lett."},{"issue":"2","key":"2292_CR29","doi-asserted-by":"publisher","first-page":"573","DOI":"10.1109\/TSP.2010.2089682","volume":"59","author":"PP Vaidyanathan","year":"2010","unstructured":"P.P. Vaidyanathan, P. Pal, Sparse sensing with co-prime samplers and arrays. IEEE Trans. Signal Process. 59(2), 573\u2013586 (2010)","journal-title":"IEEE Trans. Signal Process."},{"key":"2292_CR30","doi-asserted-by":"crossref","DOI":"10.1002\/0471221104","volume-title":"Detection, Estimation, and Modulation Theory, Optimum Array Processing, Part IV","author":"HL Van Trees","year":"2002","unstructured":"H.L. Van Trees, Detection, Estimation, and Modulation Theory, Optimum Array Processing, Part IV (Wiley, 2002)"},{"key":"2292_CR31","doi-asserted-by":"publisher","first-page":"30","DOI":"10.1016\/j.dsp.2017.11.015","volume":"74","author":"W Wang","year":"2018","unstructured":"W. Wang, S. Ren, Z. Chen, Unified coprime array with multi-period subarrays for direction-of-arrival estimation. Digit. Signal Process. 74, 30\u201342 (2018)","journal-title":"Digit. Signal Process."},{"issue":"10","key":"2292_CR32","doi-asserted-by":"publisher","first-page":"2693","DOI":"10.1109\/TSP.2019.2899292","volume":"67","author":"X Wang","year":"2019","unstructured":"X. Wang, X. Wang, Hole identification and filling in k-times extended co-prime arrays for highly-efficient DOA estimation. IEEE Trans. Signal Process. 67(10), 2693\u20132706 (2019)","journal-title":"IEEE Trans. Signal Process."},{"key":"2292_CR33","doi-asserted-by":"publisher","first-page":"22","DOI":"10.1016\/j.dsp.2017.06.013","volume":"69","author":"X Wang","year":"2017","unstructured":"X. Wang, Z. Chen, S. Ren, S. Cao, DOA estimation based on the difference and sum coarray for coprime arrays. Digit. Signal Process. 69, 22\u201331 (2017)","journal-title":"Digit. Signal Process."},{"issue":"25","key":"2292_CR34","doi-asserted-by":"publisher","first-page":"2068","DOI":"10.1049\/el.2016.3197","volume":"52","author":"M Yang","year":"2016","unstructured":"M. Yang, L. Sun, X. Yuan, B. Chen, Improved nested array with hole-free DCA and more degrees of freedom. Electron. Lett. 52(25), 2068\u20132070 (2016)","journal-title":"Electron. Lett."},{"issue":"1","key":"2292_CR35","doi-asserted-by":"publisher","first-page":"38","DOI":"10.1109\/TSP.2012.2222378","volume":"61","author":"Z Yang","year":"2013","unstructured":"Z. Yang, L. Xie, C. Zhang, Off-grid direction of arrival estimation using sparse Bayesian inference. IEEE Trans. Signal Process. 61(1), 38\u201343 (2013)","journal-title":"IEEE Trans. Signal Process."},{"key":"2292_CR36","doi-asserted-by":"crossref","unstructured":"M. Yangg, A.M. Haimovich, B. Chen, X. Yuan, A new array geometry for DOA estimation with enhanced degrees of freedom, in Proc. IEEE Int. Conf. Acoust. Speech Signal Process. (ICASSP, 2016), pp. 3041\u20133045","DOI":"10.1109\/ICASSP.2016.7472236"},{"issue":"9","key":"2292_CR37","doi-asserted-by":"publisher","first-page":"1784","DOI":"10.1109\/LCOMM.2016.2587674","volume":"20","author":"H Zamani","year":"2016","unstructured":"H. Zamani, H. Zayyani, F. Marvasti, An iterative dictionary learning-based algorithm for DOA estimation. IEEE Commun. Lett. 20(9), 1784\u20131787 (2016)","journal-title":"IEEE Commun. Lett."},{"issue":"12","key":"2292_CR38","doi-asserted-by":"publisher","first-page":"2224","DOI":"10.1109\/LCOMM.2019.2947585","volume":"23","author":"P Zhao","year":"2019","unstructured":"P. Zhao, G. Hu, Z. Qu et al., Enhanced nested array configuration with hole-free co-array and increasing degrees of freedom for DOA estimation. IEEE Commun. Lett. 23(12), 2224\u20132228 (2019)","journal-title":"IEEE Commun. Lett."},{"key":"2292_CR39","doi-asserted-by":"publisher","first-page":"4597","DOI":"10.1109\/TSP.2020.3013389","volume":"68","author":"W Zheng","year":"2020","unstructured":"W. Zheng, X. Zhang, Y. Wang, J. Shen, B. Champagne, Padded coprime arrays for improved DOA estimation: exploiting hole representation and filling strategies. IEEE Trans. Signal Process. 68, 4597\u20134611 (2020)","journal-title":"IEEE Trans. Signal Process."},{"issue":"8","key":"2292_CR40","doi-asserted-by":"publisher","first-page":"7841","DOI":"10.1109\/TVT.2019.2925528","volume":"68","author":"W Zheng","year":"2019","unstructured":"W. Zheng, X. Zhang, Y. Wang, M. Zhou, Q. Wu, Extended coprime array configuration generating large-scale antenna co-array in massive MIMO system. IEEE Trans. Veh. Technol. 68(8), 7841\u20137853 (2019)","journal-title":"IEEE Trans. Veh. Technol."}],"container-title":["Circuits, Systems, and Signal Processing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00034-023-02292-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00034-023-02292-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00034-023-02292-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,12,5]],"date-time":"2023-12-05T07:45:41Z","timestamp":1701762341000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00034-023-02292-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,23]]},"references-count":40,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2023,6]]}},"alternative-id":["2292"],"URL":"https:\/\/doi.org\/10.1007\/s00034-023-02292-8","relation":{},"ISSN":["0278-081X","1531-5878"],"issn-type":[{"value":"0278-081X","type":"print"},{"value":"1531-5878","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,1,23]]},"assertion":[{"value":"18 May 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 January 2023","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 January 2023","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 January 2023","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors have no conflict of interest to declare.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}