{"id":"https://openalex.org/W1979005107","doi":"https://doi.org/10.1137/1031127","title":"Kronecker Products, Unitary Matrices and Signal Processing Applications","display_name":"Kronecker Products, Unitary Matrices and Signal Processing Applications","publication_year":1989,"publication_date":"1989-12-01","ids":{"openalex":"https://openalex.org/W1979005107","doi":"https://doi.org/10.1137/1031127","mag":"1979005107"},"language":"en","primary_location":{"id":"doi:10.1137/1031127","is_oa":false,"landing_page_url":"https://doi.org/10.1137/1031127","pdf_url":null,"source":{"id":"https://openalex.org/S160107561","display_name":"SIAM Review","issn_l":"0036-1445","issn":["0036-1445","1095-7200"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Review","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/A5110213788","display_name":"Phillip A. Regalia","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Phillip A. Regalia","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5097004120","display_name":"Mitra K. Sanjit","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Mitra K. Sanjit","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":2.9899,"has_fulltext":false,"cited_by_count":183,"citation_normalized_percentile":{"value":0.89569764,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":96,"max":100},"biblio":{"volume":"31","issue":"4","first_page":"586","last_page":"613"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11797","display_name":"graph theory and CDMA systems","score":0.9944000244140625,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11797","display_name":"graph theory and CDMA systems","score":0.9944000244140625,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic 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/T11034","display_name":"Digital Filter Design and Implementation","score":0.9937000274658203,"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/T10767","display_name":"Advanced Photonic Communication Systems","score":0.9926999807357788,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic 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/kronecker-product","display_name":"Kronecker product","score":0.8933430910110474},{"id":"https://openalex.org/keywords/kronecker-delta","display_name":"Kronecker delta","score":0.8066909909248352},{"id":"https://openalex.org/keywords/notation","display_name":"Notation","score":0.6746718287467957},{"id":"https://openalex.org/keywords/matrix-multiplication","display_name":"Matrix multiplication","score":0.6087349057197571},{"id":"https://openalex.org/keywords/unitary-state","display_name":"Unitary state","score":0.5863736867904663},{"id":"https://openalex.org/keywords/unitary-matrix","display_name":"Unitary matrix","score":0.5594611763954163},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.5403795838356018},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.5295986533164978},{"id":"https://openalex.org/keywords/hadamard-transform","display_name":"Hadamard transform","score":0.5157440304756165},{"id":"https://openalex.org/keywords/signal-processing","display_name":"Signal processing","score":0.4859853684902191},{"id":"https://openalex.org/keywords/product","display_name":"Product (mathematics)","score":0.4705629348754883},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.4523353576660156},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.4033306837081909},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.39704903960227966},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.26316511631011963},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.25353890657424927},{"id":"https://openalex.org/keywords/digital-signal-processing","display_name":"Digital signal processing","score":0.23969295620918274},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.06911727786064148}],"concepts":[{"id":"https://openalex.org/C46030957","wikidata":"https://www.wikidata.org/wiki/Q1238125","display_name":"Kronecker product","level":3,"score":0.8933430910110474},{"id":"https://openalex.org/C39482219","wikidata":"https://www.wikidata.org/wiki/Q192826","display_name":"Kronecker delta","level":2,"score":0.8066909909248352},{"id":"https://openalex.org/C45357846","wikidata":"https://www.wikidata.org/wiki/Q2001982","display_name":"Notation","level":2,"score":0.6746718287467957},{"id":"https://openalex.org/C17349429","wikidata":"https://www.wikidata.org/wiki/Q1049914","display_name":"Matrix multiplication","level":3,"score":0.6087349057197571},{"id":"https://openalex.org/C67820243","wikidata":"https://www.wikidata.org/wiki/Q179164","display_name":"Unitary state","level":2,"score":0.5863736867904663},{"id":"https://openalex.org/C96214315","wikidata":"https://www.wikidata.org/wiki/Q727103","display_name":"Unitary matrix","level":3,"score":0.5594611763954163},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.5403795838356018},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.5295986533164978},{"id":"https://openalex.org/C60292330","wikidata":"https://www.wikidata.org/wiki/Q1014065","display_name":"Hadamard transform","level":2,"score":0.5157440304756165},{"id":"https://openalex.org/C104267543","wikidata":"https://www.wikidata.org/wiki/Q208163","display_name":"Signal processing","level":3,"score":0.4859853684902191},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.4705629348754883},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4523353576660156},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4033306837081909},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.39704903960227966},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.26316511631011963},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.25353890657424927},{"id":"https://openalex.org/C84462506","wikidata":"https://www.wikidata.org/wiki/Q173142","display_name":"Digital signal processing","level":2,"score":0.23969295620918274},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.06911727786064148},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.0},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C9390403","wikidata":"https://www.wikidata.org/wiki/Q3966","display_name":"Computer hardware","level":1,"score":0.0},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C84114770","wikidata":"https://www.wikidata.org/wiki/Q46344","display_name":"Quantum","level":2,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/1031127","is_oa":false,"landing_page_url":"https://doi.org/10.1137/1031127","pdf_url":null,"source":{"id":"https://openalex.org/S160107561","display_name":"SIAM Review","issn_l":"0036-1445","issn":["0036-1445","1095-7200"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Review","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":38,"referenced_works":["https://openalex.org/W1581693007","https://openalex.org/W1966264494","https://openalex.org/W1967795974","https://openalex.org/W1967809501","https://openalex.org/W1982251867","https://openalex.org/W1991153863","https://openalex.org/W2014727541","https://openalex.org/W2016960691","https://openalex.org/W2017911681","https://openalex.org/W2023368870","https://openalex.org/W2026517766","https://openalex.org/W2034310980","https://openalex.org/W2039544376","https://openalex.org/W2048916503","https://openalex.org/W2061171222","https://openalex.org/W2062483276","https://openalex.org/W2063678701","https://openalex.org/W2069285223","https://openalex.org/W2086393337","https://openalex.org/W2092726343","https://openalex.org/W2095535355","https://openalex.org/W2095590702","https://openalex.org/W2105097640","https://openalex.org/W2119386911","https://openalex.org/W2141612050","https://openalex.org/W2142930126","https://openalex.org/W2154543331","https://openalex.org/W2162463358","https://openalex.org/W2167430018","https://openalex.org/W2170317615","https://openalex.org/W2186435531","https://openalex.org/W2296520333","https://openalex.org/W2323936516","https://openalex.org/W2610857016","https://openalex.org/W2796941408","https://openalex.org/W4206194712","https://openalex.org/W4239894305","https://openalex.org/W4285719527"],"related_works":["https://openalex.org/W123417539","https://openalex.org/W2079407403","https://openalex.org/W2060299328","https://openalex.org/W4375957393","https://openalex.org/W4233239985","https://openalex.org/W2117336295","https://openalex.org/W2073634779","https://openalex.org/W3102722572","https://openalex.org/W2090103374","https://openalex.org/W577870507"],"abstract_inverted_index":{"Discrete":[0],"unitary":[1,71],"transforms":[2,72,114],"are":[3,95,118],"extensively":[4],"used":[5],"in":[6,12,45,104,120],"many":[7],"signal":[8,105],"processing":[9,106],"applications,":[10],"and":[11,35,64,93,115],"the":[13,60,98,101,121],"development":[14],"of":[15,32,38,69,100,112,123],"fast":[16],"algorithms":[17],"Kronecker":[18,33,62,124],"products":[19,34],"have":[20],"proved":[21],"quite":[22],"useful.":[23],"In":[24],"this":[25],"semitutorial":[26],"paper,":[27],"we":[28],"briefly":[29],"review":[30],"properties":[31,58],"direct":[36],"sums":[37],"matrices,":[39],"which":[40,53],"provide":[41],"a":[42,66,77],"compact":[43],"notation":[44,85,103],"treating":[46],"patterned":[47],"matrices.":[48],"A":[49],"generalized":[50],"matrix":[51,91],"product,":[52],"inherits":[54],"some":[55,109],"useful":[56],"algebraic":[57],"from":[59,76],"standard":[61],"product":[63],"allows":[65],"large":[67],"class":[68],"discrete":[70],"to":[73,89],"be":[74],"generated":[75],"single":[78],"recursion":[79],"formula,":[80],"is":[81,86],"then":[82],"introduced.":[83],"The":[84],"intimately":[87],"related":[88],"sparse":[90],"factorizations,":[92],"examples":[94],"included":[96],"illustrating":[97],"utility":[99],"new":[102],"applications.":[107],"Finally,":[108],"novel":[110],"characteristics":[111],"Hadamard":[113],"polyadic":[116],"permutations":[117],"derived":[119],"framework":[122],"products.":[125]},"counts_by_year":[{"year":2026,"cited_by_count":3},{"year":2025,"cited_by_count":6},{"year":2024,"cited_by_count":5},{"year":2023,"cited_by_count":7},{"year":2022,"cited_by_count":9},{"year":2021,"cited_by_count":15},{"year":2020,"cited_by_count":8},{"year":2019,"cited_by_count":14},{"year":2018,"cited_by_count":6},{"year":2017,"cited_by_count":6},{"year":2016,"cited_by_count":6},{"year":2015,"cited_by_count":4},{"year":2014,"cited_by_count":4},{"year":2013,"cited_by_count":3},{"year":2012,"cited_by_count":8}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}