JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 32, NO. 7, APRIL 1, 2014 1351 Reduced Complexity Digital Back-Propagation Methods for Optical Communication Systems Antonio Napoli, Zied Maalej, Vincent A. J. M. Sleiffer, Student Member, IEEE, Maxim Kuschnerov, Danish Rafique, Erik Timmers, Bernhard Spinnler, Talha Rahman, Student Member, IEEE, Leonardo Didier Coelho, Member, IEEE, and Norbert Hanik, Member, IEEE Abstract—Next-generation optical communication systems will continue to push the (bandwidth · distance) product towards its physical limit. To address this enormous demand, the usage of dig- ital signal processing together with advanced modulation formats and coherent detection has been proposed to enable data-rates as high as 400 Gb/s per channel over distances in the order of 1000 km. These technological breakthroughs have been made possible by full compensation of linear fiber impairments using digital equaliza- tion algorithms. While linear equalization techniques have already matured over the last decade, the next logical focus is to explore so- lutions enabling the mitigation of the Kerr effect induced nonlinear channel impairments. One of the most promising methods to com- pensate for fiber nonlinearities is digital back-propagation (DBP), which has recently been acknowledged as a universal compensator for fiber propagation impairments, albeit with high computational requirements. In this paper, we discuss two proposals to reduce the hardware complexity required by DBP. The first confirms and ex- tends published results for non-dispersion managed link, while the second introduces a novel method applicable to dispersion man- aged links, showing complexity reductions in the order of 50% and up to 85%, respectively. The proposed techniques are vali- dated by comparing results obtained through post-processing of simulated and experimental data, employing single channel and WDM configurations, with advanced modulation formats, such as quadrature phase shift keying (QPSK) and 16-ary quadrature am- plitude modulation (16-QAM). The considered net symbol rate for all cases is 25 GSymbol/s. Our post-processing results show that we can significantly reduce the hardware complexity without af- fecting the system performance. Finally, a detailed analysis of the obtained reduction is presented for the case of dispersion managed link in terms of number of required complex multiplications per transmitted bit. Manuscript received September 14, 2013; revised December 4, 2013 and December 23, 2013; accepted January 1, 2014. Date of publication January 19, 2014; date of current version March 3, 2014. A. Napoli, M. Kuschnerov, D. Rafique, B. Spinnler, and L. D. Coelho are with Coriant R&D GmbH, Munich 81541, Germany (e-mail: antonio.napoli@ coriant.com; maxim.kuschnerov@coriant.com; danish.rafique@corian.com; bernhard.spinnler@coriant.com; leonardo.coelho@ieee.org). Z. Maalej was with Technische Universit¨ at M¨ unchen, Munich 85748, Germany. Now he is with Vodafone Deutschland, Munich (e-mail: zied.maalej@vodafone.com). V. A. J. M. Sleiffer and T. Rahman are with the Eindhoven University of Technology, Eindhoven, The Netherlands (e-mail: v.a.j.m.sleiffer@tue.nl; t.rahman@tue.nl). E. Timmers was with Eindhoven University of Technology, Eindhoven, The Netherlands. He is now with easyGIS, Eindhoven, The Netherlands (e-mail: etimmers@gmail.com). N. Hanik is with the Institute for Communications Engineering, Tech- nische Universit¨ at M¨ unchen, Munich 80290, Germany (e-mail: norbert. hanik@tum.de). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JLT.2014.2301492 Index Terms—Advanced modulation formats, coherent detec- tion, digital signal processing(DSP), nonlinear mitigation. I. INTRODUCTION O VER the last decade, communication traffic showed an exponential growth [1]. This trend has lately been sup- ported by the introduction of new multimedia services, such as cloud computing, video streaming, and social networks, essen- tially pushing currently available transmission capacity to its limits [2]. Due to the aforementioned stringent capacity demands, the legacy networks employing 10 Gb/s on-off keying (OOK) traffic are not able to fuel the needs of state-of-the-art multimedia applications. Consequently, over the past few years research has intensified on optical systems based on transceivers employing coherent detection and digital signal processing (DSP), together with phase and amplitude modulated formats such as binary phase shift keying, quadrature phase shift keying (QPSK) or quadrature amplitude modulation (e.g., 16-QAM) [3]–[5]. In particular, the recent application of DSP to optical com- munications has moved the balance of system complexity from the transmission link to the end nodes. With current techno- logical advancements, modern DSP-based coherent receivers (DSP-CRX) 1 are able to fully compensate for linear channel impairments, such as chromatic dispersion (CD) and polariza- tion mode dispersion (PMD) [4]. Consequently, the transmis- sion performance remains ultimately limited by nonlinear fiber effects [6]. For this purpose different nonlinear compensation methods have been proposed and investigated to reduce the impact of Kerr induced nonlinearities. Among them Volterra series based equalizer [7], radio-frequency (RF)-pilot tone tech- nique [8], [9], phase-conjugated twin waves [10] and eventually the digital back-propagation algorithm (DBP) [11] are worth to be mentioned. DBP has recently gained a momentum [11]–[14], and it is now commonly acknowledged as one of the most suit- able candidates for joint compensation of linear and nonlinear effects. In particular, fiber impairments such as self-phase mod- ulation (SPM), cross-phase modulation (XPM) and four-wave mixing represent the bottle-neck for future optical transport networks, capable to transmit data-rates as high as 400 Gb/s per channel with a high spectral efficiency over distances of about 1000 km [5], [15]. Theoretically, DBP can compensate for any deterministic nonlinear effect, however practically, due to 1 Hereafter, only intra-dyne coherent receivers are considered.