{"id":"https://openalex.org/W3203592813","doi":"https://doi.org/10.1109/lsp.2021.3116522","title":"On the Size and Redundancy of the Fourth-Order Difference Co-Array","display_name":"On the Size and Redundancy of the Fourth-Order Difference Co-Array","publication_year":2021,"publication_date":"2021-01-01","ids":{"openalex":"https://openalex.org/W3203592813","doi":"https://doi.org/10.1109/lsp.2021.3116522","mag":"3203592813"},"language":"en","primary_location":{"id":"doi:10.1109/lsp.2021.3116522","is_oa":false,"landing_page_url":"https://doi.org/10.1109/lsp.2021.3116522","pdf_url":null,"source":{"id":"https://openalex.org/S120629676","display_name":"IEEE Signal Processing Letters","issn_l":"1070-9908","issn":["1070-9908","1558-2361"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Signal Processing Letters","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5085132596","display_name":"Yuan-Pon Chen","orcid":"https://orcid.org/0000-0002-0850-5009"},"institutions":[{"id":"https://openalex.org/I16733864","display_name":"National Taiwan University","ror":"https://ror.org/05bqach95","country_code":"TW","type":"education","lineage":["https://openalex.org/I16733864"]},{"id":"https://openalex.org/I99613584","display_name":"National Taipei University","ror":"https://ror.org/03e29r284","country_code":"TW","type":"education","lineage":["https://openalex.org/I99613584"]}],"countries":["TW"],"is_corresponding":false,"raw_author_name":"Yuan-Pon Chen","raw_affiliation_strings":["Department of Electrical Engineering, National Taiwan University Graduate Institute of Communication Engineering, 597536 Taipei City, Taiwan, 10617 (e-mail: b05901038@ntu.edu.tw)","Department of Electrical Engineering, Graduate Institute of Communication Engineering, National Taiwan University, Taipei City, Taiwan"],"raw_orcid":"https://orcid.org/0000-0002-0850-5009","affiliations":[{"raw_affiliation_string":"Department of Electrical Engineering, National Taiwan University Graduate Institute of Communication Engineering, 597536 Taipei City, Taiwan, 10617 (e-mail: b05901038@ntu.edu.tw)","institution_ids":["https://openalex.org/I16733864","https://openalex.org/I99613584"]},{"raw_affiliation_string":"Department of Electrical Engineering, Graduate Institute of Communication Engineering, National Taiwan University, Taipei City, Taiwan","institution_ids":["https://openalex.org/I16733864"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5102758279","display_name":"Chun-Lin Liu","orcid":"https://orcid.org/0000-0003-3135-9684"},"institutions":[{"id":"https://openalex.org/I16733864","display_name":"National Taiwan University","ror":"https://ror.org/05bqach95","country_code":"TW","type":"education","lineage":["https://openalex.org/I16733864"]},{"id":"https://openalex.org/I99613584","display_name":"National Taipei University","ror":"https://ror.org/03e29r284","country_code":"TW","type":"education","lineage":["https://openalex.org/I99613584"]}],"countries":["TW"],"is_corresponding":false,"raw_author_name":"Chun-Lin Liu","raw_affiliation_strings":["Department of Electrical Engineering, National Taiwan University, 33561 Taipei, Taiwan, 10617 (e-mail: chunlinliu@ntu.edu.tw)","Department of Electrical Engineering, Graduate Institute of Communication Engineering, National Taiwan University, Taipei City, Taiwan"],"raw_orcid":"https://orcid.org/0000-0003-3135-9684","affiliations":[{"raw_affiliation_string":"Department of Electrical Engineering, National Taiwan University, 33561 Taipei, Taiwan, 10617 (e-mail: chunlinliu@ntu.edu.tw)","institution_ids":["https://openalex.org/I16733864","https://openalex.org/I99613584"]},{"raw_affiliation_string":"Department of Electrical Engineering, Graduate Institute of Communication Engineering, National Taiwan University, Taipei City, Taiwan","institution_ids":["https://openalex.org/I16733864"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.2797,"has_fulltext":false,"cited_by_count":3,"citation_normalized_percentile":{"value":0.55427027,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":94,"max":97},"biblio":{"volume":"28","issue":null,"first_page":"2013","last_page":"2017"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10931","display_name":"Direction-of-Arrival Estimation Techniques","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/1711","display_name":"Signal Processing"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10931","display_name":"Direction-of-Arrival Estimation Techniques","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/1711","display_name":"Signal Processing"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11946","display_name":"Antenna Design and Optimization","score":0.9994999766349792,"subfield":{"id":"https://openalex.org/subfields/2202","display_name":"Aerospace Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10891","display_name":"Radar Systems and Signal Processing","score":0.9986000061035156,"subfield":{"id":"https://openalex.org/subfields/2202","display_name":"Aerospace Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/redundancy","display_name":"Redundancy (engineering)","score":0.7539348602294922},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.5606926679611206},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5427932143211365},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.48750993609428406},{"id":"https://openalex.org/keywords/exponential-function","display_name":"Exponential function","score":0.45595553517341614},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.43629083037376404},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.42433908581733704},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.40990275144577026},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.3968408405780792},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.369326651096344},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.19241541624069214}],"concepts":[{"id":"https://openalex.org/C152124472","wikidata":"https://www.wikidata.org/wiki/Q1204361","display_name":"Redundancy (engineering)","level":2,"score":0.7539348602294922},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5606926679611206},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5427932143211365},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.48750993609428406},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.45595553517341614},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.43629083037376404},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.42433908581733704},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.40990275144577026},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3968408405780792},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.369326651096344},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.19241541624069214},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/lsp.2021.3116522","is_oa":false,"landing_page_url":"https://doi.org/10.1109/lsp.2021.3116522","pdf_url":null,"source":{"id":"https://openalex.org/S120629676","display_name":"IEEE Signal Processing Letters","issn_l":"1070-9908","issn":["1070-9908","1558-2361"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Signal Processing Letters","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G7699267148","display_name":"Low Bitrate Signal Processing and Direction-Of-Arrival Estimation on Sparse Arrays","funder_award_id":"MOST108-2218-E002-043-MY2","funder_id":"https://openalex.org/F4320322795","funder_display_name":"Ministry of Science and Technology, Taiwan"}],"funders":[{"id":"https://openalex.org/F4320322795","display_name":"Ministry of Science and Technology, Taiwan","ror":"https://ror.org/02kv4zf79"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":26,"referenced_works":["https://openalex.org/W653761051","https://openalex.org/W2012227645","https://openalex.org/W2098174516","https://openalex.org/W2125930644","https://openalex.org/W2126700042","https://openalex.org/W2148846143","https://openalex.org/W2148982591","https://openalex.org/W2156985756","https://openalex.org/W2161666352","https://openalex.org/W2162654459","https://openalex.org/W2164390589","https://openalex.org/W2170193181","https://openalex.org/W2172082699","https://openalex.org/W2335128326","https://openalex.org/W2346607232","https://openalex.org/W2586607074","https://openalex.org/W2598194391","https://openalex.org/W2611469656","https://openalex.org/W2625286829","https://openalex.org/W2888963289","https://openalex.org/W2904550993","https://openalex.org/W2918469493","https://openalex.org/W2942508766","https://openalex.org/W3015894443","https://openalex.org/W3157811516","https://openalex.org/W6794341446"],"related_works":["https://openalex.org/W2980963558","https://openalex.org/W1495042958","https://openalex.org/W2494338568","https://openalex.org/W2122678784","https://openalex.org/W2282510344","https://openalex.org/W139987158","https://openalex.org/W2183994405","https://openalex.org/W1875646599","https://openalex.org/W2349023125","https://openalex.org/W1882246857"],"abstract_inverted_index":{"In":[0],"array":[1,31,87],"processing,":[2],"the":[3,24,27,35,41,45,49,52,65,71,74,82,85,89,98,103,106,110,130,133,138],"fourth-order":[4,36,53,75,99,111,115,139],"difference":[5,37,54,76,112,140],"co-array":[6,55],"of":[7,26,34,51,70,73,105,132,137],"sparse":[8,43],"arrays":[9],"with":[10,17],"N":[11],"sensors":[12],"admits":[13],"to":[14,23,101],"resolve$\\mathcal{O}(N^4)$source":[15],"directions":[16],"proper":[18],"assumptions.":[19],"This$\\mathcal{O}(N^4)$property":[20],"is":[21,79],"relevant":[22],"size":[25,72,131],"central":[28,134],"uniform":[29],"linear":[30],"(ULA)":[32],"segment":[33,108,136],"co-array.":[38,77,113,141],"However,":[39],"among":[40],"existing":[42],"arrays,":[44],"fundamental":[46],"limits":[47],"for":[48,59],"sizes":[50],"remain":[56],"a":[57,118],"topic":[58],"further":[60,127],"study.":[61],"This":[62],"paper":[63],"characterizes":[64],"lower":[66,90,120],"and":[67,84,91],"upper":[68,92],"bounds":[69,143],"It":[78],"proved":[80],"that":[81],"ULA":[83,107,135],"exponential":[86],"achieve":[88],"bounds,":[93],"respectively.":[94],"We":[95],"also":[96],"propose":[97],"redundancy":[100,116],"quantify":[102],"efficiency":[104],"in":[109],"The":[114],"owns":[117],"provable":[119],"bound":[121],"depending":[122],"only":[123],"on":[124],"N,":[125],"providing":[126],"insights":[128],"into":[129],"These":[142],"are":[144],"validated":[145],"through":[146],"numerical":[147],"examples.":[148]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2023,"cited_by_count":2}],"updated_date":"2026-08-06T08:24:18.245995","created_date":"2025-10-10T00:00:00"}
