{"id":"https://openalex.org/W4225701396","doi":"https://doi.org/10.1109/tsp.2022.3181342","title":"A Covariant, Discrete Time-Frequency Representation Tailored for Zero-Based Signal Detection","display_name":"A Covariant, Discrete Time-Frequency Representation Tailored for Zero-Based Signal Detection","publication_year":2022,"publication_date":"2022-01-01","ids":{"openalex":"https://openalex.org/W4225701396","doi":"https://doi.org/10.1109/tsp.2022.3181342"},"language":"en","primary_location":{"id":"doi:10.1109/tsp.2022.3181342","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tsp.2022.3181342","pdf_url":null,"source":{"id":"https://openalex.org/S168680287","display_name":"IEEE Transactions on Signal Processing","issn_l":"1053-587X","issn":["1053-587X","1941-0476"],"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 Signal Processing","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2202.03835","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5087385766","display_name":"Barbara Pascal","orcid":"https://orcid.org/0000-0003-2335-2693"},"institutions":[{"id":"https://openalex.org/I1294671590","display_name":"Centre National de la Recherche Scientifique","ror":"https://ror.org/02feahw73","country_code":"FR","type":"funder","lineage":["https://openalex.org/I1294671590"]},{"id":"https://openalex.org/I2279609970","display_name":"Universit\u00e9 de Lille","ror":"https://ror.org/02kzqn938","country_code":"FR","type":"education","lineage":["https://openalex.org/I2279609970"]},{"id":"https://openalex.org/I7454413","display_name":"\u00c9cole Centrale de Lille","ror":"https://ror.org/01x441g73","country_code":"FR","type":"education","lineage":["https://openalex.org/I7454413"]}],"countries":["FR"],"is_corresponding":true,"raw_author_name":"Barbara Pascal","raw_affiliation_strings":["University of Lille, CNRS, Centrale Lille, UMR 9189 CRIStAL, Lille, France"],"affiliations":[{"raw_affiliation_string":"University of Lille, CNRS, Centrale Lille, UMR 9189 CRIStAL, Lille, France","institution_ids":["https://openalex.org/I7454413","https://openalex.org/I2279609970","https://openalex.org/I1294671590"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5062461687","display_name":"R\u00e9mi Bardenet","orcid":"https://orcid.org/0000-0002-1094-9493"},"institutions":[{"id":"https://openalex.org/I2279609970","display_name":"Universit\u00e9 de Lille","ror":"https://ror.org/02kzqn938","country_code":"FR","type":"education","lineage":["https://openalex.org/I2279609970"]},{"id":"https://openalex.org/I7454413","display_name":"\u00c9cole Centrale de Lille","ror":"https://ror.org/01x441g73","country_code":"FR","type":"education","lineage":["https://openalex.org/I7454413"]},{"id":"https://openalex.org/I1294671590","display_name":"Centre National de la Recherche Scientifique","ror":"https://ror.org/02feahw73","country_code":"FR","type":"funder","lineage":["https://openalex.org/I1294671590"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Remi Bardenet","raw_affiliation_strings":["University of Lille, CNRS, Centrale Lille, UMR 9189 CRIStAL, Lille, France"],"affiliations":[{"raw_affiliation_string":"University of Lille, CNRS, Centrale Lille, UMR 9189 CRIStAL, Lille, France","institution_ids":["https://openalex.org/I7454413","https://openalex.org/I2279609970","https://openalex.org/I1294671590"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5087385766"],"corresponding_institution_ids":["https://openalex.org/I1294671590","https://openalex.org/I2279609970","https://openalex.org/I7454413"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.06413088,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"70","issue":null,"first_page":"2950","last_page":"2961"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11210","display_name":"Mathematical Analysis and Transform Methods","score":0.23119999468326569,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11210","display_name":"Mathematical Analysis and Transform Methods","score":0.23119999468326569,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.21690000593662262,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"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/T11038","display_name":"Advanced SAR Imaging Techniques","score":0.15539999306201935,"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/mathematics","display_name":"Mathematics","score":0.6223465800285339},{"id":"https://openalex.org/keywords/discrete-time-signal","display_name":"Discrete-time signal","score":0.5163460969924927},{"id":"https://openalex.org/keywords/covariance","display_name":"Covariance","score":0.5099118947982788},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.5092019438743591},{"id":"https://openalex.org/keywords/spectrogram","display_name":"Spectrogram","score":0.4929065704345703},{"id":"https://openalex.org/keywords/gaussian-noise","display_name":"Gaussian noise","score":0.474578857421875},{"id":"https://openalex.org/keywords/discretization","display_name":"Discretization","score":0.46859756112098694},{"id":"https://openalex.org/keywords/discrete-fourier-transform","display_name":"Discrete Fourier transform (general)","score":0.4574960172176361},{"id":"https://openalex.org/keywords/noise","display_name":"Noise (video)","score":0.44302308559417725},{"id":"https://openalex.org/keywords/gaussian","display_name":"Gaussian","score":0.4235418438911438},{"id":"https://openalex.org/keywords/fourier-transform","display_name":"Fourier transform","score":0.39501291513442993},{"id":"https://openalex.org/keywords/fractional-fourier-transform","display_name":"Fractional Fourier transform","score":0.34586071968078613},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.27148306369781494},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.26997190713882446},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.22506412863731384},{"id":"https://openalex.org/keywords/signal-transfer-function","display_name":"Signal transfer function","score":0.18596291542053223},{"id":"https://openalex.org/keywords/fourier-analysis","display_name":"Fourier analysis","score":0.17777323722839355},{"id":"https://openalex.org/keywords/speech-recognition","display_name":"Speech recognition","score":0.1615917682647705},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.1592124104499817},{"id":"https://openalex.org/keywords/analog-signal","display_name":"Analog signal","score":0.1276511549949646},{"id":"https://openalex.org/keywords/digital-signal-processing","display_name":"Digital signal processing","score":0.12762805819511414}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6223465800285339},{"id":"https://openalex.org/C146749787","wikidata":"https://www.wikidata.org/wiki/Q15963867","display_name":"Discrete-time signal","level":5,"score":0.5163460969924927},{"id":"https://openalex.org/C178650346","wikidata":"https://www.wikidata.org/wiki/Q201984","display_name":"Covariance","level":2,"score":0.5099118947982788},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.5092019438743591},{"id":"https://openalex.org/C45273575","wikidata":"https://www.wikidata.org/wiki/Q578970","display_name":"Spectrogram","level":2,"score":0.4929065704345703},{"id":"https://openalex.org/C4199805","wikidata":"https://www.wikidata.org/wiki/Q2725903","display_name":"Gaussian noise","level":2,"score":0.474578857421875},{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.46859756112098694},{"id":"https://openalex.org/C57733114","wikidata":"https://www.wikidata.org/wiki/Q1006032","display_name":"Discrete Fourier transform (general)","level":5,"score":0.4574960172176361},{"id":"https://openalex.org/C99498987","wikidata":"https://www.wikidata.org/wiki/Q2210247","display_name":"Noise (video)","level":3,"score":0.44302308559417725},{"id":"https://openalex.org/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.4235418438911438},{"id":"https://openalex.org/C102519508","wikidata":"https://www.wikidata.org/wiki/Q6520159","display_name":"Fourier transform","level":2,"score":0.39501291513442993},{"id":"https://openalex.org/C76563020","wikidata":"https://www.wikidata.org/wiki/Q4817582","display_name":"Fractional Fourier transform","level":4,"score":0.34586071968078613},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.27148306369781494},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.26997190713882446},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.22506412863731384},{"id":"https://openalex.org/C131021393","wikidata":"https://www.wikidata.org/wiki/Q7512759","display_name":"Signal transfer function","level":4,"score":0.18596291542053223},{"id":"https://openalex.org/C203024314","wikidata":"https://www.wikidata.org/wiki/Q1365258","display_name":"Fourier analysis","level":3,"score":0.17777323722839355},{"id":"https://openalex.org/C28490314","wikidata":"https://www.wikidata.org/wiki/Q189436","display_name":"Speech recognition","level":1,"score":0.1615917682647705},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.1592124104499817},{"id":"https://openalex.org/C13412647","wikidata":"https://www.wikidata.org/wiki/Q174948","display_name":"Analog signal","level":3,"score":0.1276511549949646},{"id":"https://openalex.org/C84462506","wikidata":"https://www.wikidata.org/wiki/Q173142","display_name":"Digital signal processing","level":2,"score":0.12762805819511414},{"id":"https://openalex.org/C115961682","wikidata":"https://www.wikidata.org/wiki/Q860623","display_name":"Image (mathematics)","level":2,"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/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}],"mesh":[],"locations_count":7,"locations":[{"id":"doi:10.1109/tsp.2022.3181342","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tsp.2022.3181342","pdf_url":null,"source":{"id":"https://openalex.org/S168680287","display_name":"IEEE Transactions on Signal Processing","issn_l":"1053-587X","issn":["1053-587X","1941-0476"],"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 Signal Processing","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:2202.03835","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2202.03835","pdf_url":"https://arxiv.org/pdf/2202.03835","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"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:HAL:hal-03553433v2","is_oa":true,"landing_page_url":"https://hal.science/hal-03553433","pdf_url":null,"source":{"id":"https://openalex.org/S4306402512","display_name":"HAL (Le Centre pour la Communication Scientifique Directe)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1294671590","host_organization_name":"Centre National de la Recherche Scientifique","host_organization_lineage":["https://openalex.org/I1294671590"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"ISSN: 1053-587X","raw_type":"info:eu-repo/semantics/article"},{"id":"pmh:oai:lilloa.univ-lille.fr:20.500.12210/59865","is_oa":true,"landing_page_url":"http://hdl.handle.net/20.500.12210/59865","pdf_url":null,"source":{"id":"https://openalex.org/S4306402203","display_name":"LillOA (Universit\u00e9 de Lille (University Of Lille))","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210123514","host_organization_name":"Centre d'Etudes en Civilisations, Langues et Litt\u00e9ratures Etrang\u00e8res","host_organization_lineage":["https://openalex.org/I4210123514"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"info:eu-repo/semantics/workingPaper"},{"id":"pmh:oai:lilloa.univ-lille.fr:20.500.12210/59932","is_oa":true,"landing_page_url":"http://hdl.handle.net/20.500.12210/59932","pdf_url":null,"source":{"id":"https://openalex.org/S4306402203","display_name":"LillOA (Universit\u00e9 de Lille (University Of Lille))","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210123514","host_organization_name":"Centre d'Etudes en Civilisations, Langues et Litt\u00e9ratures Etrang\u00e8res","host_organization_lineage":["https://openalex.org/I4210123514"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"info:eu-repo/semantics/workingPaper"},{"id":"pmh:oai:lilloa.univ-lille.fr:20.500.12210/79860","is_oa":true,"landing_page_url":"http://hdl.handle.net/20.500.12210/79860","pdf_url":null,"source":{"id":"https://openalex.org/S4306402203","display_name":"LillOA (Universit\u00e9 de Lille (University Of Lille))","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210123514","host_organization_name":"Centre d'Etudes en Civilisations, Langues et Litt\u00e9ratures Etrang\u00e8res","host_organization_lineage":["https://openalex.org/I4210123514"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"info:eu-repo/semantics/article"},{"id":"doi:10.48550/arxiv.2202.03835","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2202.03835","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"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":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article-journal"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2202.03835","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2202.03835","pdf_url":"https://arxiv.org/pdf/2202.03835","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"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":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":38,"referenced_works":["https://openalex.org/W852321735","https://openalex.org/W1519516901","https://openalex.org/W1605417594","https://openalex.org/W1654995442","https://openalex.org/W1983624841","https://openalex.org/W2032101051","https://openalex.org/W2045881429","https://openalex.org/W2054878302","https://openalex.org/W2058978315","https://openalex.org/W2073152266","https://openalex.org/W2087306836","https://openalex.org/W2094042860","https://openalex.org/W2106578604","https://openalex.org/W2120221444","https://openalex.org/W2139734935","https://openalex.org/W2139930049","https://openalex.org/W2142477181","https://openalex.org/W2150372634","https://openalex.org/W2154490308","https://openalex.org/W2168408847","https://openalex.org/W2222356937","https://openalex.org/W2252795400","https://openalex.org/W2582607838","https://openalex.org/W2962952170","https://openalex.org/W2963169422","https://openalex.org/W2963336936","https://openalex.org/W2980671223","https://openalex.org/W3115928068","https://openalex.org/W3152158897","https://openalex.org/W3158547810","https://openalex.org/W4206509365","https://openalex.org/W4211163609","https://openalex.org/W4213443160","https://openalex.org/W4249875616","https://openalex.org/W6634827438","https://openalex.org/W6752994227","https://openalex.org/W6800975862","https://openalex.org/W6846278635"],"related_works":["https://openalex.org/W55215308","https://openalex.org/W15853796","https://openalex.org/W44775234","https://openalex.org/W29006066","https://openalex.org/W61356396","https://openalex.org/W34872523","https://openalex.org/W49502152","https://openalex.org/W62701569","https://openalex.org/W27569920","https://openalex.org/W43459774"],"abstract_inverted_index":{"Recent":[0],"work":[1],"in":[2,23,80,98,210],"time-frequency":[3,82,110,148],"analysis":[4],"proposed":[5,128],"to":[6,53,125,134,155,187,276,284,288],"switch":[7],"the":[8,11,14,44,67,70,75,81,86,92,115,146,162,176,194,198,201,213,216,224,227,234,249,253,256,302,318,322],"focus":[9],"from":[10],"maxima":[12],"of":[13,21,46,48,63,69,88,91,145,160,179,197,200,204,215,226,233,252,255,262,296,305,321],"spectrogram":[15,50],"toward":[16],"its":[17],"zeros.":[18],"The":[19,60,140,259],"zeros":[20,47,199,214,228,254],"signals":[22],"white":[24,206],"Gaussian":[25,207,218],"noise":[26,208],"indeed":[27],"form":[28],"a":[29,34,49,108,119,126,242,306],"random":[30],"point":[31,195],"pattern":[32,45],"with":[33,185,212,274],"very":[35],"stable":[36],"structure.":[37],"Using":[38],"modern":[39],"spatial":[40,156,250],"statistics":[41,90,251],"tools":[42],"on":[43,237,248,317],"has":[51,165],"led":[52],"component":[54],"disentanglement":[55],"and":[56,74,121,152,183,271,293,309],"signal":[57,170,245,308,312],"detection":[58,246,279,313],"procedures.":[59,280],"major":[61],"bottlenecks":[62],"this":[64,238,263],"approach":[65],"are":[66,95],"discretization":[68],"Short-Time":[71],"Fourier":[72],"Transform":[73],"necessarily":[76],"bounded":[77],"observation":[78],"window":[79],"plane.":[83],"Both":[84],"impact":[85],"estimation":[87],"summary":[89],"zeros,":[93],"which":[94,112,173],"then":[96],"used":[97],"standard":[99],"statistical":[100,260],"tests.":[101],"To":[102],"circumvent":[103],"these":[104],"limitations,":[105],"we":[106,113,240],"propose":[107],"generalized":[109],"representation,":[111],"call":[114],"<i>Kravchuk":[116],"transform</i>,":[117],"providing":[118],"covariant":[120],"numerically":[122],"tractable":[123],"counterpart":[124],"recently":[127],"discrete":[129,205,307],"transform.":[130],"It":[131],"naturally":[132],"applies":[133],"<i>finite</i>":[135],"signals,":[136],"<i>i.e.</i>,":[137],"finite-dimensional":[138],"vectors.":[139],"corresponding":[141],"phase":[142],"space,":[143],"instead":[144],"whole":[147],"plane,":[149],"is":[150,229,265,314],"compact,":[151],"particularly":[153,286],"amenable":[154],"statistics.":[157],"On":[158],"top":[159],"this,":[161],"Kravchuk":[163,202,257,303],"transform":[164,203,304],"several":[166],"natural":[167],"properties":[168],"for":[169,244],"processing,":[171],"among":[172],"covariance":[174],"under":[175,231],"group":[177],"action":[178],"<inline-formula><tex-math":[180],"notation=\"LaTeX\">$\\text{SO}(3)$</tex-math></inline-formula>,":[181],"invertibility":[182],"symmetry":[184],"respect":[186,275],"complex":[188],"conjugation.":[189],"We":[190],"further":[191],"show":[192],"that":[193,223],"process":[196],"coincides":[209],"law":[211,225],"spherical":[217],"Analytic":[219],"Function.":[220],"This":[221],"implies":[222],"invariant":[230],"isometries":[232],"sphere.":[235],"Elaborating":[236],"theorem,":[239],"develop":[241],"procedure":[243,264],"based":[247],"spectrogram.":[258],"power":[261],"assessed":[266],"by":[267],"intensive":[268],"numerical":[269],"simulation,":[270],"compares":[272],"favorably":[273],"state-of-the-art":[277],"zeros-based":[278],"Furthermore":[281],"it":[282],"appears":[283],"be":[285],"robust":[287],"both":[289],"low":[290],"signal-to-noise":[291],"ratio":[292],"small":[294],"number":[295],"samples.":[297],"A":[298],"<small>Python</small>":[299],"toolbox":[300],"computing":[301],"implementing":[310],"zero-based":[311],"publicly":[315],"available":[316],"GitHub":[319],"page":[320],"first":[323],"author<xref":[324],"ref-type=\"fn\"":[325],"rid=\"fn1\"><sup>1</sup></xref><fn":[326],"id=\"fn1\"><label><sup>1</sup></label>":[327],"<uri>https://github.com/bpascal-fr/kravchuk-transform-and-its-zeros</uri>":[328],"</fn>.":[329]},"counts_by_year":[],"updated_date":"2026-03-20T23:20:44.827607","created_date":"2025-10-10T00:00:00"}
