{"id":"https://openalex.org/W2249626609","doi":"https://doi.org/10.1109/tac.2015.2446851","title":"On the Factorization of Rational Discrete-Time Spectral Densities","display_name":"On the Factorization of Rational Discrete-Time Spectral Densities","publication_year":2015,"publication_date":"2015-06-18","ids":{"openalex":"https://openalex.org/W2249626609","doi":"https://doi.org/10.1109/tac.2015.2446851","mag":"2249626609"},"language":"en","primary_location":{"id":"doi:10.1109/tac.2015.2446851","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tac.2015.2446851","pdf_url":null,"source":{"id":"https://openalex.org/S184954342","display_name":"IEEE Transactions on Automatic Control","issn_l":"0018-9286","issn":["0018-9286","1558-2523","2334-3303"],"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 Transactions on Automatic Control","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1410.0765","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":null,"display_name":"Giacomo Baggio","orcid":null},"institutions":[{"id":"https://openalex.org/I138689650","display_name":"University of Padua","ror":"https://ror.org/00240q980","country_code":"IT","type":"education","lineage":["https://openalex.org/I138689650"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Giacomo Baggio","raw_affiliation_strings":["Dipartimento di Ingegneria dell'Informazione, Universit\u00e0 di Padova, Padova, Italy"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Dipartimento di Ingegneria dell'Informazione, Universit\u00e0 di Padova, Padova, Italy","institution_ids":["https://openalex.org/I138689650"]}]},{"author_position":"last","author":{"id":null,"display_name":"Augusto Ferrante","orcid":null},"institutions":[{"id":"https://openalex.org/I138689650","display_name":"University of Padua","ror":"https://ror.org/00240q980","country_code":"IT","type":"education","lineage":["https://openalex.org/I138689650"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Augusto Ferrante","raw_affiliation_strings":["Dipartimento di Ingegneria dell'Informazione, Universit\u00e0 di Padova, Padova, Italy"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Dipartimento di Ingegneria dell'Informazione, Universit\u00e0 di Padova, Padova, Italy","institution_ids":["https://openalex.org/I138689650"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I138689650"],"apc_list":null,"apc_paid":null,"fwci":2.0057,"has_fulltext":false,"cited_by_count":22,"citation_normalized_percentile":{"value":0.87816753,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":94,"max":98},"biblio":{"volume":"61","issue":"4","first_page":"969","last_page":"981"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.49570000171661377,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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/T10792","display_name":"Matrix Theory and Algorithms","score":0.49570000171661377,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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/T10046","display_name":"Stability and Control of Uncertain Systems","score":0.14159999787807465,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems 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/T11022","display_name":"Spectral Theory in Mathematical Physics","score":0.08739999681711197,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/factorization","display_name":"Factorization","score":0.7789000272750854},{"id":"https://openalex.org/keywords/corollary","display_name":"Corollary","score":0.7777000069618225},{"id":"https://openalex.org/keywords/constructive","display_name":"Constructive","score":0.7573000192642212},{"id":"https://openalex.org/keywords/matrix-decomposition","display_name":"Matrix decomposition","score":0.6050000190734863},{"id":"https://openalex.org/keywords/constraint","display_name":"Constraint (computer-aided design)","score":0.541700005531311},{"id":"https://openalex.org/keywords/inverse","display_name":"Inverse","score":0.5067999958992004},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.42640000581741333},{"id":"https://openalex.org/keywords/constructive-proof","display_name":"Constructive proof","score":0.4230000078678131},{"id":"https://openalex.org/keywords/spectral-theorem","display_name":"Spectral theorem","score":0.4172999858856201}],"concepts":[{"id":"https://openalex.org/C187834632","wikidata":"https://www.wikidata.org/wiki/Q188804","display_name":"Factorization","level":2,"score":0.7789000272750854},{"id":"https://openalex.org/C2780012671","wikidata":"https://www.wikidata.org/wiki/Q1343870","display_name":"Corollary","level":2,"score":0.7777000069618225},{"id":"https://openalex.org/C2778701210","wikidata":"https://www.wikidata.org/wiki/Q28130034","display_name":"Constructive","level":3,"score":0.7573000192642212},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7346000075340271},{"id":"https://openalex.org/C42355184","wikidata":"https://www.wikidata.org/wiki/Q1361088","display_name":"Matrix decomposition","level":3,"score":0.6050000190734863},{"id":"https://openalex.org/C2776036281","wikidata":"https://www.wikidata.org/wiki/Q48769818","display_name":"Constraint (computer-aided design)","level":2,"score":0.541700005531311},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.5067999958992004},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.42899999022483826},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.42640000581741333},{"id":"https://openalex.org/C202854965","wikidata":"https://www.wikidata.org/wiki/Q3044470","display_name":"Constructive proof","level":2,"score":0.4230000078678131},{"id":"https://openalex.org/C35958075","wikidata":"https://www.wikidata.org/wiki/Q1425077","display_name":"Spectral theorem","level":3,"score":0.4172999858856201},{"id":"https://openalex.org/C75190567","wikidata":"https://www.wikidata.org/wiki/Q41237","display_name":"Rational function","level":2,"score":0.4115000069141388},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3926999866962433},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.37459999322891235},{"id":"https://openalex.org/C18762648","wikidata":"https://www.wikidata.org/wiki/Q42213","display_name":"Work (physics)","level":2,"score":0.36570000648498535},{"id":"https://openalex.org/C135252773","wikidata":"https://www.wikidata.org/wiki/Q1567213","display_name":"Inverse problem","level":2,"score":0.34360000491142273},{"id":"https://openalex.org/C2985906921","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Spectral properties","level":2,"score":0.3353999853134155},{"id":"https://openalex.org/C6802819","wikidata":"https://www.wikidata.org/wiki/Q1072174","display_name":"Linear system","level":2,"score":0.2937000095844269},{"id":"https://openalex.org/C124681953","wikidata":"https://www.wikidata.org/wiki/Q339062","display_name":"Decomposition","level":2,"score":0.2892000079154968},{"id":"https://openalex.org/C2988995629","wikidata":"https://www.wikidata.org/wiki/Q2915729","display_name":"Matrix algebra","level":3,"score":0.2824000120162964},{"id":"https://openalex.org/C43656286","wikidata":"https://www.wikidata.org/wiki/Q2100961","display_name":"Pole\u2013zero plot","level":3,"score":0.27059999108314514},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.2703999876976013},{"id":"https://openalex.org/C137559481","wikidata":"https://www.wikidata.org/wiki/Q1231787","display_name":"Dixon's factorization method","level":5,"score":0.26190000772476196},{"id":"https://openalex.org/C2781089630","wikidata":"https://www.wikidata.org/wiki/Q21856745","display_name":"Realization (probability)","level":2,"score":0.25459998846054077},{"id":"https://openalex.org/C152671427","wikidata":"https://www.wikidata.org/wiki/Q10843505","display_name":"Non-negative matrix factorization","level":4,"score":0.25429999828338623},{"id":"https://openalex.org/C168110828","wikidata":"https://www.wikidata.org/wiki/Q1331626","display_name":"Spectral density","level":2,"score":0.25029999017715454}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1109/tac.2015.2446851","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tac.2015.2446851","pdf_url":null,"source":{"id":"https://openalex.org/S184954342","display_name":"IEEE Transactions on Automatic Control","issn_l":"0018-9286","issn":["0018-9286","1558-2523","2334-3303"],"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 Transactions on Automatic Control","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1410.0765","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1410.0765","pdf_url":"https://arxiv.org/pdf/1410.0765","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:www.research.unipd.it:11577/3196663","is_oa":false,"landing_page_url":"http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=7128353","pdf_url":null,"source":{"id":"https://openalex.org/S4306402547","display_name":"Padua Research Archive (University of Padova)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I138689650","host_organization_name":"University of Padua","host_organization_lineage":["https://openalex.org/I138689650"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1410.0765","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1410.0765","pdf_url":"https://arxiv.org/pdf/1410.0765","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":38,"referenced_works":["https://openalex.org/W178677418","https://openalex.org/W1972481868","https://openalex.org/W1981842601","https://openalex.org/W1993408859","https://openalex.org/W1997129429","https://openalex.org/W2002357820","https://openalex.org/W2005276415","https://openalex.org/W2014111265","https://openalex.org/W2023723226","https://openalex.org/W2024132874","https://openalex.org/W2028812282","https://openalex.org/W2029382655","https://openalex.org/W2050457751","https://openalex.org/W2063084473","https://openalex.org/W2063841006","https://openalex.org/W2065097322","https://openalex.org/W2068901771","https://openalex.org/W2079628196","https://openalex.org/W2089959648","https://openalex.org/W2100434865","https://openalex.org/W2107300011","https://openalex.org/W2107453951","https://openalex.org/W2117470445","https://openalex.org/W2126700203","https://openalex.org/W2129288433","https://openalex.org/W2132658950","https://openalex.org/W2137466941","https://openalex.org/W2163005618","https://openalex.org/W2163101083","https://openalex.org/W2171352244","https://openalex.org/W2172506337","https://openalex.org/W2339524658","https://openalex.org/W3022209387","https://openalex.org/W4241280266","https://openalex.org/W4248816015","https://openalex.org/W4252222815","https://openalex.org/W6602369057","https://openalex.org/W6637743642"],"related_works":[],"abstract_inverted_index":{"In":[0],"this":[1,63],"paper,":[2],"we":[3],"consider":[4],"an":[5],"arbitrary":[6],"matrix-valued,":[7],"rational":[8],"spectral":[9],"density":[10],"\u03a6(z).":[11],"We":[12],"show":[13],"with":[14,47],"a":[15,21,69],"constructive":[16],"proof":[17],"that":[18,43,51,71],"\u03a6(z)":[19],"admits":[20],"factorization":[22,82],"of":[23,79],"the":[24,48,60,76,80],"form":[25],"\u03a6(z)=WT(z-1)":[26],"W(z),":[27],"where":[28],"W(z)":[29,34],"is":[30],"stochastically":[31],"minimal.":[32],"Moreover,":[33],"and":[35],"its":[36],"right":[37],"inverse":[38],"are":[39],"analytic":[40],"in":[41,87],"regions":[42],"may":[44,72],"be":[45,73],"selected":[46],"only":[49],"constraint":[50],"they":[52],"satisfy":[53],"some":[54],"symplectic-type":[55],"conditions.":[56],"By":[57],"suitably":[58],"selecting":[59],"analyticity":[61],"regions,":[62],"extremely":[64],"general":[65],"result":[66],"particularizes":[67],"into":[68],"corollary":[70],"viewed":[74],"as":[75],"discrete-time":[77],"counterpart":[78],"matrix":[81],"method":[83],"devised":[84],"by":[85],"Youla":[86],"his":[88],"celebrated":[89],"work":[90],"[48].":[91]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2024,"cited_by_count":4},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":6},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":4},{"year":2017,"cited_by_count":2}],"updated_date":"2026-08-05T07:39:15.569665","created_date":"2016-06-24T00:00:00"}