{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,26]],"date-time":"2026-03-26T06:31:55Z","timestamp":1774506715601,"version":"3.50.1"},"reference-count":33,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2015,12,9]],"date-time":"2015-12-09T00:00:00Z","timestamp":1449619200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"name":"Italian MIUR FIRB project COTONE"},{"name":"EU FP-7 G\u00c9ANT project COFFEE"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Photon Netw Commun"],"published-print":{"date-parts":[[2016,6]]},"DOI":"10.1007\/s11107-015-0586-z","type":"journal-article","created":{"date-parts":[[2015,12,9]],"date-time":"2015-12-09T11:23:31Z","timestamp":1449660211000},"page":"493-502","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":37,"title":["Single-step digital backpropagation for nonlinearity mitigation"],"prefix":"10.1007","volume":"31","author":[{"given":"Marco","family":"Secondini","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Simon","family":"Rommel","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gianluca","family":"Meloni","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Francesco","family":"Fresi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Enrico","family":"Forestieri","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Luca","family":"Pot\u00ec","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2015,12,9]]},"reference":[{"issue":"22","key":"586_CR1","doi-asserted-by":"crossref","first-page":"22796","DOI":"10.1364\/OE.18.022796","volume":"18","author":"R Asif","year":"2010","unstructured":"Asif, R., Lin, C.Y., Holtmannspoetter, M., Schmauss, B.: Optimized digital backward propagation for phase modulated signals in mixed-optical fiber transmission link. Opt. Express 18(22), 22796\u201322807 (2010)","journal-title":"Opt. Express"},{"issue":"1","key":"586_CR2","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1109\/LPT.2004.838152","volume":"17","author":"E Ciaramella","year":"2005","unstructured":"Ciaramella, E., Forestieri, E.: Analytical approximation of nonlinear distortions. IEEE Photonics Technol. Lett. 17(1), 91\u201393 (2005)","journal-title":"IEEE Photonics Technol. Lett."},{"issue":"18","key":"586_CR3","doi-asserted-by":"crossref","first-page":"2790","DOI":"10.1109\/JLT.2011.2164237","volume":"29","author":"G Colavolpe","year":"2011","unstructured":"Colavolpe, G., Foggi, T., Forestieri, E., Secondini, M.: Impact of phase noise and compensation techniques in coherent optical systems. J. Lightwave Technol. 29(18), 2790\u20132800 (2011)","journal-title":"J. Lightwave Technol."},{"issue":"1","key":"586_CR4","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1109\/JLT.2012.2225600","volume":"31","author":"F Cugini","year":"2013","unstructured":"Cugini, F., Paolucci, F., Meloni, G., Berrettini, G., Secondini, M., Fresi, F., Sambo, N., Poti, L., Castoldi, P.: Push-pull defragmentation without traffic disruption in flexible grid optical networks. J. Lightwave Technol. 31(1), 125\u2013133 (2013)","journal-title":"J. Lightwave Technol."},{"key":"586_CR5","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1090\/S0025-5718-1965-0178586-1","volume":"19","author":"JW Cooley","year":"1965","unstructured":"Cooley, J.W., Tukey, J.W.: An algorithm for the machine calculation of complex Fourier series. Math. Comp. 19, 297\u2013301 (1965)","journal-title":"Math. Comp."},{"issue":"16","key":"586_CR6","doi-asserted-by":"crossref","first-page":"17075","DOI":"10.1364\/OE.18.017075","volume":"18","author":"LB Du","year":"2010","unstructured":"Du, L.B., Lowery, A.J.: Improved single channel backpropagation for intra-channel fiber nonlinearity compensation in long-haul optical communication systems. Opt. Express 18(16), 17075\u201317088 (2010)","journal-title":"Opt. Express"},{"key":"586_CR7","first-page":"3","volume-title":"Optical Communication Theory and Techniques","author":"E Forestieri","year":"2004","unstructured":"Forestieri, E., Secondini, M.: Solving the nonlinear Schr\u00f6dinger equation. In: Forestieri, E. (ed.) Optical Communication Theory and Techniques, pp. 3\u201311. Springer, New York (2004)"},{"issue":"11","key":"586_CR8","doi-asserted-by":"crossref","first-page":"8815","DOI":"10.1364\/OE.17.008815","volume":"17","author":"G Goldfarb","year":"2009","unstructured":"Goldfarb, G., Li, G.: Efficient backward-propagation using wavelet-based filtering for fiber backward-propagation. Opt. Express 17(11), 8815\u20138821 (2009)","journal-title":"Opt. Express"},{"key":"586_CR9","first-page":"423","volume":"15","author":"RH Hardin","year":"1973","unstructured":"Hardin, R.H., Tappert, F.D.: Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations. SIAM Rev. 15, 423 (1973)","journal-title":"SIAM Rev."},{"key":"586_CR10","doi-asserted-by":"crossref","unstructured":"Ip, E., Bai, N., Wang, T.: Complexity versus performance tradeoff for fiber nonlinearity compensation using frequency-shaped, multi-subband backpropagation. In: Optical Fiber Communications Conference (OFC), p. OThF4 (2011)","DOI":"10.1364\/OFC.2011.OThF4"},{"issue":"20","key":"586_CR11","doi-asserted-by":"crossref","first-page":"3416","DOI":"10.1109\/JLT.2008.927791","volume":"26","author":"E Ip","year":"2008","unstructured":"Ip, E., Kahn, J.M.: Compensation of dispersion and nonlinear impairments using digital backpropagation. J. Lightwave Technol. 26(20), 3416\u20133425 (2008)","journal-title":"J. Lightwave Technol."},{"key":"586_CR12","doi-asserted-by":"crossref","unstructured":"Irukulapati, N.V., Marsella, D., Johannisson, P., Secondini, M., Wymeersch, H., Agrell, E., Forestieri, E.: On maximum likelihood sequence detectors for single-channel coherent optical communications. In: European Conference on Optical Communications (ECOC), p. P.3.19 (2014)","DOI":"10.1109\/ECOC.2014.6964134"},{"issue":"11","key":"586_CR13","doi-asserted-by":"crossref","first-page":"3956","DOI":"10.1109\/TCOMM.2014.2362534","volume":"62","author":"NV Irukulapati","year":"2014","unstructured":"Irukulapati, N.V., Wymeersch, H., Johannisson, P., Agrell, E.: Stochastic digital backpropagation. IEEE Trans. Commun. 62(11), 3956\u20133968 (2014)","journal-title":"IEEE Trans. Commun."},{"issue":"16","key":"586_CR14","doi-asserted-by":"crossref","first-page":"3614","DOI":"10.1109\/JLT.2009.2024963","volume":"27","author":"M Kuschnerov","year":"2009","unstructured":"Kuschnerov, M., Hauske, F.N., Piyawanno, K., Spinnler, B., Alfiad, M.S., Napoli, A., Lankl, B.: DSP for coherent single-carrier receivers. J. Lightwave Technol. 27(16), 3614\u20133622 (2009)","journal-title":"J. Lightwave Technol."},{"key":"586_CR15","doi-asserted-by":"crossref","unstructured":"Li, L., Tao, Z., Dou, L., Yan, W., Oda, S., Tanimura, T., Hoshida, T., Rasmussen, J.C.: Implementation efficient nonlinear equalizer based on correlated digital backpropagation. In: Optical Fiber Communications Conference (OFC), p. OWW3 (2011)","DOI":"10.1364\/OFC.2011.OWW3"},{"issue":"2","key":"586_CR16","doi-asserted-by":"crossref","first-page":"880","DOI":"10.1364\/OE.16.000880","volume":"16","author":"X Li","year":"2008","unstructured":"Li, X., Chen, X., Goldfarb, G., Mateo, E., Kim, I., Yaman, F., Li, G.: Electronic post-compensation of WDM transmission impairments using coherent detection and digital signal processing. Opt. Express 16(2), 880\u2013888 (2008)","journal-title":"Opt. Express"},{"issue":"11","key":"586_CR17","doi-asserted-by":"crossref","first-page":"14655","DOI":"10.1364\/OE.23.014655","volume":"23","author":"X Liang","year":"2015","unstructured":"Liang, X., Kumar, S.: Correlated digital back propagation based on perturbation theory. Opt. Express 23(11), 14655\u201314665 (2015)","journal-title":"Opt. Express"},{"key":"586_CR18","doi-asserted-by":"crossref","unstructured":"Maher, R., Xu, T., Galdino, L., Sato, M., Alvarado, A., Shi, K., Savory, S.J., Thomsen, B.C., Killey, R.I., Bayvel, P.: Spectrally shaped DP-16QAM super-channel transmission with multi-channel digital back-propagation. Sci. Rep. 5, 8214 (2015). doi: 10.1038\/srep08214","DOI":"10.1038\/srep08214"},{"issue":"5","key":"586_CR19","doi-asserted-by":"crossref","first-page":"908","DOI":"10.1109\/JLT.2013.2294457","volume":"32","author":"D Marsella","year":"2014","unstructured":"Marsella, D., Secondini, M., Forestieri, E.: Maximum likelihood sequence detection for mitigating nonlinear effects. J. Lightwave Technol. 32(5), 908\u2013916 (2014)","journal-title":"J. Lightwave Technol."},{"issue":"7","key":"586_CR20","doi-asserted-by":"crossref","first-page":"2806","DOI":"10.1109\/JLT.2006.875953","volume":"24","author":"C Menyuk","year":"2006","unstructured":"Menyuk, C., Marks, B.: Interaction of polarization mode dispersion and nonlinearity in optical fiber transmission systems. J. Lightwave Technol. 24(7), 2806\u20132826 (2006)","journal-title":"J. Lightwave Technol."},{"issue":"7","key":"586_CR21","doi-asserted-by":"crossref","first-page":"1351","DOI":"10.1109\/JLT.2014.2301492","volume":"32","author":"A Napoli","year":"2014","unstructured":"Napoli, A., Maalej, Z., Sleiffer, V.A., Kuschnerov, M., Rafique, D., Timmers, E., Spinnler, B., Rahman, T., Coelho, L.D., Hanik, N.: Reduced complexity digital back-propagation methods for optical communication systems. J. Lightwave Technol. 32(7), 1351\u20131362 (2014)","journal-title":"J. Lightwave Technol."},{"key":"586_CR22","volume-title":"Discrete-Time Signal Processing","author":"AV Oppenheim","year":"1999","unstructured":"Oppenheim, A.V., Schafer, R.W.: Discrete-Time Signal Processing. Prentice Hall, Upper Saddle River (1999)"},{"key":"586_CR23","doi-asserted-by":"crossref","unstructured":"Peddanarappagari, K.V., Brandt-Pearce, M.: Volterra series transfer function of single-mode fibers. J. Lightwave Technol. 15, 2232\u20132241 (1997)","DOI":"10.1109\/50.643545"},{"issue":"10","key":"586_CR24","doi-asserted-by":"crossref","first-page":"9453","DOI":"10.1364\/OE.19.009453","volume":"19","author":"D Rafique","year":"2011","unstructured":"Rafique, D., Mussolin, M., Forzati, M., M\u00e5rtensson, J., Chugtai, M.N., Ellis, A.D.: Compensation of intra-channel nonlinear fibre impairments using simplified digital back-propagation algorithm. Opt. Express 19(10), 9453\u20139460 (2011)","journal-title":"Opt. Express"},{"issue":"22","key":"586_CR25","doi-asserted-by":"crossref","first-page":"2016","DOI":"10.1109\/LPT.2012.2217952","volume":"24","author":"M Secondini","year":"2012","unstructured":"Secondini, M., Forestieri, E.: Analytical fiber-optic channel model in the presence of cross-phase modulation. IEEE Photonics Technol. Lett. 24(22), 2016\u20132019 (2012)","journal-title":"IEEE Photonics Technol. Lett."},{"issue":"22","key":"586_CR26","doi-asserted-by":"crossref","first-page":"2252","DOI":"10.1109\/LPT.2014.2353217","volume":"26","author":"M Secondini","year":"2014","unstructured":"Secondini, M., Forestieri, E.: On XPM mitigation in WDM fiber-optic systems. IEEE Photonics Technol. Lett. 26(22), 2252\u20132255 (2014)","journal-title":"IEEE Photonics Technol. Lett."},{"issue":"23","key":"586_CR27","doi-asserted-by":"crossref","first-page":"3839","DOI":"10.1109\/JLT.2013.2288677","volume":"31","author":"M Secondini","year":"2013","unstructured":"Secondini, M., Forestieri, E., Prati, G.: Achievable information rate in nonlinear WDM fiber-optic systems with arbitrary modulation formats and dispersion maps. J. Lightwave Technol. 31(23), 3839\u20133852 (2013)","journal-title":"J. Lightwave Technol."},{"key":"586_CR28","doi-asserted-by":"crossref","unstructured":"Secondini, M., Marsella, D., Forestieri, E.: Enhanced split-step fourier method for digital backpropagation. In: Proceedings of the European Conference on Optical Communications (ECOC), p. We.3.3.5 (2014)","DOI":"10.1109\/ECOC.2014.6964122"},{"key":"586_CR29","unstructured":"Secondini, M., Rommel, S., Fresi, F., Forestieri, E., Meloni, G., Pot\u00ec, L.: Coherent 100G nonlinear compensation with single-step digital backpropagation. In: Optical Network Design and Modeling (ONDM), 2015, pp. 63\u201367. IEEE (2015)"},{"key":"586_CR30","doi-asserted-by":"crossref","unstructured":"Stockham, T.G., Jr.: High-speed convolution and correlation. In: Proceedings of the 1966 Spring Joint Computer Conference, AFIPS, vol.\u00a028, pp. 229\u2013233 (1966)","DOI":"10.1145\/1464182.1464209"},{"key":"586_CR31","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1016\/0021-9991(84)90003-2","volume":"5","author":"TR Taha","year":"1984","unstructured":"Taha, T.R., Ablowitz, M.J.: Analytical and numerical aspects of certain nonlinear evolution equation, II, numerical, nonlinear Schroedinger equation. J. Comput. Phys. 5, 203\u2013230 (1984)","journal-title":"J. Comput. Phys."},{"key":"586_CR32","doi-asserted-by":"crossref","unstructured":"Taylor, M.G.: Coherent detection method using DSP for demodulation of signal and subsequent equalization of propagation impairments. IEEE Photonics Technol. Lett. 16(2), 674\u2013676 (2004)","DOI":"10.1109\/LPT.2003.823106"},{"issue":"7","key":"586_CR33","doi-asserted-by":"crossref","first-page":"1102","DOI":"10.1109\/JLT.2002.800376","volume":"20","author":"A Vannucci","year":"2002","unstructured":"Vannucci, A., Serena, P., Bononi, A.: The RP method: a new tool for the iterative solution of the nonlinear Schr\u00f6dinger equation. J. Lightwave Technol. 20(7), 1102\u20131112 (2002)","journal-title":"J. Lightwave Technol."}],"container-title":["Photonic Network Communications"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11107-015-0586-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11107-015-0586-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11107-015-0586-z","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,9,2]],"date-time":"2019-09-02T04:47:49Z","timestamp":1567399669000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11107-015-0586-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,12,9]]},"references-count":33,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2016,6]]}},"alternative-id":["586"],"URL":"https:\/\/doi.org\/10.1007\/s11107-015-0586-z","relation":{},"ISSN":["1387-974X","1572-8188"],"issn-type":[{"value":"1387-974X","type":"print"},{"value":"1572-8188","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,12,9]]}}}