{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,6]],"date-time":"2026-02-06T05:43:40Z","timestamp":1770356620165,"version":"3.49.0"},"reference-count":18,"publisher":"Springer Science and Business Media LLC","issue":"11","license":[{"start":{"date-parts":[[2010,10,22]],"date-time":"2010-10-22T00:00:00Z","timestamp":1287705600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sci. China Inf. Sci."],"published-print":{"date-parts":[[2010,11]]},"DOI":"10.1007\/s11432-010-4095-5","type":"journal-article","created":{"date-parts":[[2010,10,21]],"date-time":"2010-10-21T03:56:09Z","timestamp":1287633369000},"page":"2287-2299","source":"Crossref","is-referenced-by-count":20,"title":["The discrete multiple-parameter fractional Fourier transform"],"prefix":"10.1007","volume":"53","author":[{"given":"Jun","family":"Lang","sequence":"first","affiliation":[]},{"given":"Ran","family":"Tao","sequence":"additional","affiliation":[]},{"given":"Yue","family":"Wang","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2010,10,22]]},"reference":[{"key":"4095_CR1","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1093\/imamat\/25.3.241","volume":"25","author":"V. Namias","year":"1980","unstructured":"Namias V. The fractional Fourier transform and its application to quantum mechanics. J Inst Math Appl, 1980, 25: 241\u2013265","journal-title":"J Inst Math Appl"},{"key":"4095_CR2","doi-asserted-by":"crossref","first-page":"3084","DOI":"10.1109\/78.330368","volume":"42","author":"L. B. Almeida","year":"1994","unstructured":"Almeida L B. The fractional Fourier transform and time-frequency representations. IEEE Trans Signal Process, 1994, 42: 3084\u20133091","journal-title":"IEEE Trans Signal Process"},{"key":"4095_CR3","doi-asserted-by":"crossref","first-page":"495","DOI":"10.1016\/0030-4018(95)00268-D","volume":"48","author":"C. Shih","year":"1995","unstructured":"Shih C. Fractionalization of Fourier transform. Opt Comm, 1995, 48: 495\u2013498","journal-title":"Opt Comm"},{"key":"4095_CR4","doi-asserted-by":"crossref","first-page":"1360","DOI":"10.1007\/s11432-008-0123-0","volume":"51","author":"Q. Yang","year":"2008","unstructured":"Yang Q, Tao R, Wang Y. MIMO-OFDM system based on fractional Fourier transform and selecting algorithm for optimal order. Sci China Ser F-Inf Sci, 2008, 51: 1360\u20131371","journal-title":"Sci China Ser F-Inf Sci"},{"key":"4095_CR5","volume-title":"Introduction to Fractional Fourier Optics (in Chinese)","author":"Q. Ran","year":"2004","unstructured":"Ran Q, Tan L. Introduction to Fractional Fourier Optics (in Chinese). Beijing: Science Press, 2004"},{"key":"4095_CR6","doi-asserted-by":"crossref","first-page":"887","DOI":"10.1364\/OL.25.000887","volume":"25","author":"G. Unnikrishnan","year":"2000","unstructured":"Unnikrishnan G, Joseph J, Singh K. Optical encryption by double-random phase encoding in the fractional Fourier domain. Opt Lett, 2000, 25: 887\u2013889","journal-title":"Opt Lett"},{"key":"4095_CR7","doi-asserted-by":"crossref","first-page":"1159","DOI":"10.1364\/OL.25.001159","volume":"25","author":"B. Zhu","year":"2000","unstructured":"Zhu B, Liu S, Ran Q. Optical image encryption based on multifractional Fourier transforms. Opt Lett, 2000, 25: 1159\u20131161","journal-title":"Opt Lett"},{"key":"4095_CR8","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1364\/OL.28.000269","volume":"28","author":"B. Hennelly","year":"2003","unstructured":"Hennelly B, Sheridan J. Optical image encryption by random shifting in fractional Fourier domains. Opt Lett, 2003, 28: 269\u2013271","journal-title":"Opt Lett"},{"key":"4095_CR9","doi-asserted-by":"crossref","first-page":"1531","DOI":"10.1016\/j.optcom.2008.12.070","volume":"282","author":"R. Tao","year":"2009","unstructured":"Tao R, Lang J, Wang Y. The multiple-parameter discrete fractional Hadamard transform. Opt Comm, 2009, 282: 1531\u20131535","journal-title":"Opt Comm"},{"key":"4095_CR10","doi-asserted-by":"crossref","first-page":"581","DOI":"10.1364\/OL.33.000581","volume":"33","author":"R. Tao","year":"2008","unstructured":"Tao R, Lang J, Wang Y. Optical image encryption based on the multiple-parameter fractional Fourier transform. Opt Lett, 2008, 33: 581\u2013583","journal-title":"Opt Lett"},{"key":"4095_CR11","doi-asserted-by":"crossref","first-page":"2092","DOI":"10.1016\/j.optcom.2010.01.060","volume":"283","author":"J. Lang","year":"2010","unstructured":"Lang J, Tao R, Wang Y. Image encryption based on the multiple-parameter discrete fractional Fourier transform and chaos function. Opt Comm, 2010, 283: 2092\u20132096","journal-title":"Opt Comm"},{"key":"4095_CR12","doi-asserted-by":"crossref","first-page":"859","DOI":"10.1007\/s11432-008-0069-2","volume":"51","author":"R. Tao","year":"2008","unstructured":"Tao R, Zhang F, Wang Y. Research progress on discretization of fractional Fourier transform. Sci China Ser F-Inf Sci, 2008, 51: 859\u2013880","journal-title":"Sci China Ser F-Inf Sci"},{"key":"4095_CR13","doi-asserted-by":"crossref","unstructured":"Long J, Tao R, Wang Y. The generalized weighed fractional Fourier transform and its application to image encryption. In: The 2nd International Congress on Image and Signal Processing (CISP\u201909). Tianjin, 2009. 1\u20135","DOI":"10.1109\/CISP.2009.5301283"},{"key":"4095_CR14","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1109\/78.815493","volume":"48","author":"G. Cariolaro","year":"2000","unstructured":"Cariolaro G, Erseghe T, Kraniauskas P, et al. Multiplicity of fractional Fourier transforms and their relationships. IEEE Trans Signal Process, 2000, 48: 227\u2013241","journal-title":"IEEE Trans Signal Process"},{"key":"4095_CR15","doi-asserted-by":"crossref","first-page":"1010","DOI":"10.1007\/s11432-008-0073-6","volume":"51","author":"J. Lang","year":"2008","unstructured":"Lang J, Tao R, Wang Y, et al. The multiple-parameter fractional Fourier transform. Sci China Ser F-Inf Sci, 2008, 51: 1010\u20131024","journal-title":"Sci China Ser F-Inf Sci"},{"key":"4095_CR16","doi-asserted-by":"crossref","unstructured":"Ozaydin M, Nemati S, Yeary M, et al. Orthogonal projections and discrete fractional Fourier transforms. In: IEEE 12th-Signal Processing Education Workshop, Teton National Park, Wyoming, 2006. 429\u2013433","DOI":"10.1109\/DSPWS.2006.265461"},{"key":"4095_CR17","doi-asserted-by":"crossref","first-page":"1329","DOI":"10.1109\/78.839980","volume":"48","author":"C. Candan","year":"2000","unstructured":"Candan C, Kutay M, Ozaktas H. The discrete fractional Fourier transform. IEEE Trans Signal Process, 2000, 48: 1329\u20131337","journal-title":"IEEE Trans Signal Process"},{"key":"4095_CR18","doi-asserted-by":"crossref","first-page":"2141","DOI":"10.1109\/78.536672","volume":"44","author":"H. M. Ozaktas","year":"1996","unstructured":"Ozaktas H M, Arikan O, Kutay M A, et al. Digital computation of the fractional Fourier transform. IEEE Trans Signal Process, 1996, 44: 2141\u20132150","journal-title":"IEEE Trans Signal Process"}],"container-title":["Science China Information Sciences"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11432-010-4095-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11432-010-4095-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11432-010-4095-5","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,5]],"date-time":"2019-06-05T18:12:54Z","timestamp":1559758374000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11432-010-4095-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,10,22]]},"references-count":18,"journal-issue":{"issue":"11","published-print":{"date-parts":[[2010,11]]}},"alternative-id":["4095"],"URL":"https:\/\/doi.org\/10.1007\/s11432-010-4095-5","relation":{},"ISSN":["1674-733X","1869-1919"],"issn-type":[{"value":"1674-733X","type":"print"},{"value":"1869-1919","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,10,22]]}}}