{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,20]],"date-time":"2026-05-20T06:27:06Z","timestamp":1779258426093,"version":"3.51.4"},"reference-count":29,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2016,8,22]],"date-time":"2016-08-22T00:00:00Z","timestamp":1471824000000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"name":"Viet Nam NAFOSTED"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Wireless Pers Commun"],"published-print":{"date-parts":[[2017,1]]},"DOI":"10.1007\/s11277-016-3567-3","type":"journal-article","created":{"date-parts":[[2016,8,22]],"date-time":"2016-08-22T06:55:19Z","timestamp":1471848919000},"page":"623-637","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":32,"title":["Two New Convolutions for the Fractional Fourier Transform"],"prefix":"10.1007","volume":"92","author":[{"given":"P. K.","family":"Anh","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"L. P.","family":"Castro","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"P. T.","family":"Thao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"N. M.","family":"Tuan","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2016,8,22]]},"reference":[{"issue":"3","key":"3567_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. (1980). The fractional Fourier transform and its application to quantum mechanics. IMA Journal of Applied Mathematics, 25(3), 241\u2013265.","journal-title":"IMA Journal of Applied Mathematics"},{"issue":"2","key":"3567_CR2","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1093\/imamat\/39.2.159","volume":"39","author":"AC McBride","year":"1987","unstructured":"McBride, A. C., & Kerr, F. H. (1987). On Namias\u2019 fractional order Fourier transform. IMA Journal of Applied Mathematics, 39(2), 159\u2013175.","journal-title":"IMA Journal of Applied Mathematics"},{"issue":"5","key":"3567_CR3","doi-asserted-by":"crossref","first-page":"1037","DOI":"10.1080\/09500349414550971","volume":"41","author":"T Alieva","year":"1994","unstructured":"Alieva, T., Lopez, V., Agullo-Lopez, F., & Almeida, L. B. (1994). The fractional Fourier transform in optical propagation problems. Journal of Modern Optics, 41(5), 1037\u20131040.","journal-title":"Journal of Modern Optics"},{"key":"3567_CR4","doi-asserted-by":"crossref","first-page":"3084","DOI":"10.1109\/78.330368","volume":"42","author":"LB Almeida","year":"1994","unstructured":"Almeida, L. B. (1994). The fractional Fourier transform and time-frequency representation. IEEE Transactions on Signal Processing, 42, 3084\u20133091.","journal-title":"IEEE Transactions on Signal Processing"},{"key":"3567_CR5","doi-asserted-by":"crossref","first-page":"545","DOI":"10.4171\/ZAA\/1255","volume":"24","author":"LP Castro","year":"2005","unstructured":"Castro, L. P., & Zhang, B. (2005). Invertibility of convolution operators arising in problems of wave diffraction by a strip with reactance and Dirichlet conditions. Zeitschrift f\u00fcr Analysis und ihre Anwendungen, 24, 545\u2013560.","journal-title":"Zeitschrift f\u00fcr Analysis und ihre Anwendungen"},{"issue":"4","key":"3567_CR6","doi-asserted-by":"crossref","first-page":"1188","DOI":"10.1016\/j.sigpro.2009.10.002","volume":"90","author":"L Durak","year":"2010","unstructured":"Durak, L., & Aldirmaz, S. (2010). Adaptive fractional Fourier domain filtering. Signal Processing, 90(4), 1188\u20131196.","journal-title":"Signal Processing"},{"key":"3567_CR7","doi-asserted-by":"crossref","first-page":"1875","DOI":"10.1364\/JOSAA.10.001875","volume":"10","author":"D Mendlovic","year":"1993","unstructured":"Mendlovic, D., & Ozaktas, H. M. (1993). Fractional Fourier transforms and their optical implementation. Journal of the Optical Society of America A, 10, 1875\u20131881.","journal-title":"Journal of the Optical Society of America A"},{"key":"3567_CR8","doi-asserted-by":"crossref","first-page":"2141","DOI":"10.1109\/78.536672","volume":"44","author":"HM Ozaktas","year":"1996","unstructured":"Ozaktas, H. M., Ar\u0131kan, O., Kutay, M. A., & Bozda\u01e7\u0131, G. (1996). Digital computation of the fractional Fourier transform. IEEE Transactions on Signal Processing, 44, 2141\u20132150.","journal-title":"IEEE Transactions on Signal Processing"},{"key":"3567_CR9","doi-asserted-by":"crossref","first-page":"2522","DOI":"10.1364\/JOSAA.10.002522","volume":"10","author":"HM Ozaktas","year":"1993","unstructured":"Ozaktas, H. M., & Mendlovic, D. (1993). Fractional Fourier transforms and their optical implementation II. Journal of the Optical Society of America A, 10, 2522\u20132531.","journal-title":"Journal of the Optical Society of America A"},{"issue":"1","key":"3567_CR10","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1016\/S0893-6080(01)00120-4","volume":"15","author":"B Barshan","year":"2002","unstructured":"Barshan, B., & Ayrulu, B. (2002). Fractional Fourier transform pre-processing for neural networks and its application to object recognition. Neural Networks, 15(1), 131\u2013140.","journal-title":"Neural Networks"},{"key":"3567_CR11","doi-asserted-by":"crossref","unstructured":"Cui, D. (2009). Dual digital watermarking algorithm for image based on fractional Fourier transform. In Proceedings of the Second Pacific-Asia Conference on Web Mining and Web-based Application (WMWA 09), Wuhan, China, June 6\u20137, 51\u201354.","DOI":"10.1109\/WMWA.2009.60"},{"issue":"2","key":"3567_CR12","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1006\/jnca.2000.0128","volume":"24","author":"I Djurovic","year":"2001","unstructured":"Djurovic, I., Stankovic, S., & Pitas, I. (2001). Digital watermarking in the fractional Fourier transformation domain. Journal of Network and Computer Applications, 24(2), 167\u2013173.","journal-title":"Journal of Network and Computer Applications"},{"issue":"11","key":"3567_CR13","doi-asserted-by":"crossref","first-page":"1729","DOI":"10.1364\/OL.34.001729","volume":"34","author":"Q Ran","year":"2009","unstructured":"Ran, Q., Zhang, H., Zhang, J., Tang, L., & Ma, J. (2009). Deficiencies of the cryptography based on multiple-parameter fractional Fourier transform. Optics Letters, 34(11), 1729\u20131731.","journal-title":"Optics Letters"},{"key":"3567_CR14","doi-asserted-by":"crossref","unstructured":"Yu, F.Q., Zhang, Z.K., & Xu, M.H. (2006). A digital watermarking algorithm for image based on fractional Fourier transform. In Proceedings of the 2006 IEEE Conference on Industrial Electronics and Applications, Singapore, May 24\u201326, 1\u20135.","DOI":"10.1109\/ICIEA.2006.257372"},{"key":"3567_CR15","volume-title":"The fractional Fourier transform with applications in optics and signal processing","author":"HM Ozaktas","year":"2001","unstructured":"Ozaktas, H. M., Zalevsky, Z., & Kutay, M. A. (2001). The fractional Fourier transform with applications in optics and signal processing. New York: Wiley."},{"issue":"7","key":"3567_CR16","doi-asserted-by":"crossref","first-page":"941","DOI":"10.1109\/5.30749","volume":"77","author":"L Cohen","year":"1989","unstructured":"Cohen, L. (1989). Time-frequency distributions-A review. Proceedings of the IEEE, 77(7), 941\u2013981.","journal-title":"Proceedings of the IEEE"},{"issue":"2","key":"3567_CR17","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1109\/79.127284","volume":"9","author":"F Hlawatsch","year":"1992","unstructured":"Hlawatsch, F., & Bourdeaux-Bartels, F. G. (1992). Linear and quadratic time-frequency signal representations. IEEE Signal Processing Magazine, 9(2), 21\u201367.","journal-title":"IEEE Signal Processing Magazine"},{"key":"3567_CR18","unstructured":"Erden, M.F., Kutay, M.A., & Ozaktas, H.M. (1999). Applications of the fractional Fourier transform to filtering, estimation and restoration. In Proceedings of the IEEE-EURASIP Workshop on Nonlinear Signal and Image Processing (NSIP 99), Antalya, Turkey, 481\u2013485."},{"issue":"5","key":"3567_CR19","doi-asserted-by":"crossref","first-page":"1129","DOI":"10.1109\/78.575688","volume":"45","author":"MA Kutay","year":"1997","unstructured":"Kutay, M. A., Ozaktas, H. M., Arikan, O., & Onural, L. (1997). Optimal filtering in fractional Fourier domains. IEEE Transactions on Signal Processing, 45(5), 1129\u20131143.","journal-title":"IEEE Transactions on Signal Processing"},{"issue":"3","key":"3567_CR20","doi-asserted-by":"crossref","first-page":"459","DOI":"10.4213\/tmf6452","volume":"161","author":"S-G Liu","year":"2009","unstructured":"Liu, S.-G., & Fan, H.-Y. (2009). Convolution theorem for the three-dimensional entangled fractional Fourier transformation deduced from the tripartite entangled state representation. Teoreticheskaya i Matematicheskaya Fizika, 161(3), 459\u2013468.","journal-title":"Teoreticheskaya i Matematicheskaya Fizika"},{"issue":"1","key":"3567_CR21","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1007\/s10043-994-0015-5","volume":"1","author":"HM Ozaktas","year":"1994","unstructured":"Ozaktas, H. M., Barshan, B., & Mendlovic, D. (1994). Convolution and filtering in fractional Fourier domains. Optical Review, 1(1), 15\u201316.","journal-title":"Optical Review"},{"issue":"1","key":"3567_CR22","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1007\/s11277-011-0235-5","volume":"65","author":"AK Singh","year":"2012","unstructured":"Singh, A. K., & Saxena, R. (2012). On convolution and product theorems for FRFT. Wireless Personal Communications, 65(1), 189\u2013201.","journal-title":"Wireless Personal Communications"},{"issue":"10","key":"3567_CR23","doi-asserted-by":"crossref","first-page":"853","DOI":"10.1109\/LSP.2009.2026107","volume":"16","author":"D Wei","year":"2009","unstructured":"Wei, D., Ran, Q., Li, Y., Ma, J., & Tan, L. (2009). A convolution and product theorem for the linear canonical transform. IEEE Signal Processing Letters, 16(10), 853\u2013856.","journal-title":"IEEE Signal Processing Letters"},{"issue":"4","key":"3567_CR24","doi-asserted-by":"crossref","first-page":"102","DOI":"10.1109\/97.664179","volume":"5","author":"AI Zayed","year":"1998","unstructured":"Zayed, A. I. (1998). A convolution and product theorem for the fractional Fourier transform. IEEE Signal Processing Letters, 5(4), 102\u2013103.","journal-title":"IEEE Signal Processing Letters"},{"issue":"1","key":"3567_CR25","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1109\/97.551689","volume":"4","author":"LB Almeida","year":"1997","unstructured":"Almeida, L. B. (1997). Product and convolution theorems for the fractional Fourier transform. IEEE Transactions on Signal Processing Letters, 4(1), 15\u201317.","journal-title":"IEEE Transactions on Signal Processing Letters"},{"issue":"5","key":"3567_CR26","doi-asserted-by":"crossref","first-page":"592","DOI":"10.1007\/s11432-006-2016-4","volume":"49","author":"D Bing","year":"2006","unstructured":"Bing, D., Ran, T., & Yue, W. (2006). Convolution theorems for the linear canonical transform and their applications. Science in China Series F Information Sciences, 49(5), 592\u2013603.","journal-title":"Science in China Series F Information Sciences"},{"key":"3567_CR27","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1007\/s00020-009-1722-x","volume":"65","author":"BT Giang","year":"2009","unstructured":"Giang, B. T., Mau, N. V., & Tuan, N. M. (2009). Operational properties of two integral transforms of Fourier type and their convolutions. Integral Equations and Operator Theory, 65, 363\u2013386.","journal-title":"Integral Equations and Operator Theory"},{"key":"3567_CR28","doi-asserted-by":"crossref","first-page":"1758","DOI":"10.1002\/mana.200710192","volume":"283","author":"BT Giang","year":"2010","unstructured":"Giang, B. T., Mau, N. V., & Tuan, N. M. (2010). Convolutions for the Fourier transforms with geometric variables and applications. Mathematische Nachrichten, 283, 1758\u20131770.","journal-title":"Mathematische Nachrichten"},{"key":"3567_CR29","volume-title":"Functional analysis","author":"W Rudin","year":"1991","unstructured":"Rudin, W. (1991). Functional analysis. New York: McGraw-Hill."}],"container-title":["Wireless Personal Communications"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11277-016-3567-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11277-016-3567-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11277-016-3567-3","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11277-016-3567-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,6,24]],"date-time":"2017-06-24T21:05:47Z","timestamp":1498338347000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11277-016-3567-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,8,22]]},"references-count":29,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2017,1]]}},"alternative-id":["3567"],"URL":"https:\/\/doi.org\/10.1007\/s11277-016-3567-3","relation":{},"ISSN":["0929-6212","1572-834X"],"issn-type":[{"value":"0929-6212","type":"print"},{"value":"1572-834X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,8,22]]}}}