**Digital Backward Propagation: A Technique to Compensate Fiber Dispersion and Non-Linear Impairments**

Rameez Asif, Chien-Yu Lin and Bernhard Schmauss

*Chair of Microwave Engineering and High Frequency Technology (LHFT), Erlangen Graduate School in Advanced Optical Technologies (SAOT), Friedrich-Alexander University of Erlangen-Nuremberg (FAU), Cauerstr. 9, (91058) Erlangen Germany*

#### **1. Introduction**

24 Applications of Digital Signal Processing

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> Recent numerical and experimental studies have shown that coherent optical QPSK (CO-QPSK) is the promising candidate for next-generation 100Gbit/s Ethernet (100 GbE) (Fludger et al., 2008). Coherent detection is considered efficient along with digital signal processing (DSP) to compensate many linear effects in fiber propagation i.e. chromatic dispersion (CD) and polarization-mode dispersion (PMD) and also offers low required optical signal-to-noise ratio (OSNR). Despite of fiber dispersion and non-linearities which are the major limiting factors, as illustrated in Fig. 1, optical transmission systems are employing higher order modulation formats in order to increase the spectral efficiency and thus fulfil the ever increasing demand of capacity requirements (Mitra et al., 2001). As a result of which compensation of dispersion and non-linearities (NL), i.e. self-phase modulation (SPM), cross-phase modulation (XPM) and four-wave mixing (FWM), is a point of high interest these days.

> Various methods of compensating fiber transmission impairments have been proposed in recent era by implementing all-optical signal processing. It is demonstrated that the fiber dispersion can be compensated by using the mid-link spectral inversion method (MLSI) (Feiste et al., 1998; Jansen et al., 2005). MLSI method is based on the principle of optical phase conjugation (OPC). In a system based on MLSI, no in-line dispersion compensation is needed. Instead in the middle of the link, an optical phase conjugator inverts the frequency spectrum and phase of the distorted signals caused by chromatic dispersion. As the signals propagate to the end of the link, the accumulated spectral phase distortions are reverted back to the value at the beginning of the link if perfect symmetry of the link is assured. In (Marazzi et al., 2009), this technique is demonstrated for real-time implementation in 100Gbit/s POLMUX-DQPSK transmission.

> Another all-optical method to compensate fiber transmission impairments is proposed in (Cvecek et al., 2008; Sponsel et al., 2008) by using the non-linear amplifying loop mirror (NALM). In this technique the incoming signal is split asymmetrically at the fiber coupler

Alternatively, infinite impulse response (IIR) filters can used (Goldfarb et al., 2007) to reduce

<sup>27</sup> Digital Backward Propagation:

A Technique to Compensate Fiber Dispersion and Non-Linear Impairments

However, with the use of higher order modulation formats, i.e QPSK and QAM, to meet the capacity requirements, it becomes vital to compensate non-linearities along with the fiber dispersion. Due to this non-linear threshold point (NLT) of the transmission system can be improved and more signal power can be injected in the system to have longer transmission distances. In (Geyer et al., 2010) a low complexity non-linear compensator scheme with automatic control loop is introduced. The proposed simple non-linear compensator requires considerably lower implementation complexity and can blindly adapt the required coefficients. In uncompensated links, the simple scheme is not able to improve performance, as the non-linear distortions are distributed over different amounts of CD-impairment. Nevertheless the scheme might still be useful to compensate possible non-linear distortions of the transmitter. In transmission links with full in-line compensation the compensator provides 1dB additional noise tolerance. This makes it useful in 10Gbit/s upgrade scenarios where optical CD compensation is still present. Another promising electronic method, investigated in higher bit-rate transmissions and for diverse dispersion mapping, is the digital backward propagation (DBP), which can jointly mitigate dispersion and non-linearities. The DBP algorithm can be implemented numerically by solving the inverse non-linear Schrödinger equation (NLSE) using split-step Fourier method (SSFM) (Ip et al., 2008). This technique is an off-line signal processing method. The limitation so far for its real-time implementation is the complexity of the algorithm (Yamazaki et al., 2011). The performance of the algorithm is dependent on the calculation steps (*h*), to estimate the transmission link parameters with

In this chapter we give a detailed overview on the advancements in DBP algorithm based on different types of mathematical models. We discuss the importance of optimized step-size

Pioneering concepts on backward propagation have been reported in articles of (Pare et al., 1996; Tsang et al., 2003). In (Tsang et al., 2003) backward propagation is demonstrated as a numerical technique for reversing femtosecond pulse propagation in an optical fiber, such that given any output pulse it is possible to obtain the input pulse shape by numerically undoing all dispersion and non-linear effects. Whereas, in (Pare et al., 1996) a dispersive medium with a negative non-linear refractive-index coefficient is demonstrated to compensate the dispersion and the non-linearities. Based on the fact that signal propagation can be interpreted by the non-linear Schrödinger equation (NLSE) (Agrawal, 2001). The inverse solution i.e. backward propagation, of this equation can numerically be solved by using split-step Fourier method (SSFM). So backward propagation can be implemented digitally at the receiver (see section 3.2 of this chapter). In digital domain, first important investigations (Ip et al., 2008; Li et al., 2008) are reported on compensation of transmission impairments by DBP with modern-age optical communication systems and coherent receivers. Coherent detection plays a vital role for DBP algorithm as it provides necessary information about the signal phase. In (Ip et al., 2008) 21.4Gbit/s RZ-QPSK transmission model over 2000km single mode fiber (SMF) is used to investigate the role of dispersion mapping, sampling ratio and multi-channel transmission. DBP is implemented by using a asymmetric split-step Fourier method (A-SSFM). In A-SSFM method each calculation step is solved by linear operator (*D*ˆ )

the complexity of the DSP circuit.

**2. State of the art**

accuracy, and on the knowledge of transmission link design.

selection for simplified and computationally efficient algorithm of DBP.

Fig. 1. Optical fiber transmission impairments.

into two counter-propagating signals. The weaker partial pulse passes first through the EDFA where it is amplified by about 20dB. It gains a significant phase shift due to self-phase modulation (Stephan et al., 2009) in the highly non-linear fiber (HNLF). The initially stronger pulse propagates through the fiber before it is amplified, so that the phase shift in the HNLF is marginal. At the output coupler the strong partial pulse with almost unchanged phase and the weak partial pulse with input-power-dependent phase shift interfere. The first, being much stronger, determines the phase of the output signal and therefore ensures negligible phase distortions.

Various investigations have been also been reported to examine the effect of optical link design (Lin et al., 2010a; Randhawa et al., 2010; Tonello et al., 2006) on the compensation of fiber impairments. However, the applications of all-optical methods are expensive, less flexible and less adaptive to different configurations of transmission. On the other hand with the development of proficient real time digital signal processing (DSP) techniques and coherent receivers, finite impulse response (FIR) filters become popular and have emerged as the promising techniques for long-haul optical data transmission. After coherent detection the signals, known in amplitude and phase, can be sampled and processed by DSP to compensate fiber transmission impairments.

DSP techniques are gaining increasing importance as they allow for robust long-haul transmission with compensation of fiber impairments at the receiver (Li, 2009; Savory et al., 2007). One major advantage of using DSP after sampling of the outputs from a phase-diversity receiver is that hardware optical phase locking can be avoided and only digital phase-tracking is needed (Noe, 2005; Taylor, 2004). DSP algorithms can also be used to compensate chromatic dispersion (CD) and polarization-mode dispersion (PMD) (Winters, 1990). It is depicted that for a symbol rate of *τ*, a *<sup>τ</sup>* <sup>2</sup> tap delay finite impulse response (FIR) filter may be used to reverse the effect of fiber chromatic dispersion (Savory et al., 2006). The number of FIR taps increases linearly with increasing accumulated dispersion i.e the number of taps required to compensate 1280 ps/nm of dispersion is approximately 5.8 (Goldfarb et al., 2007). At long propagation distances, the extra power consumption required for this task becomes significant. Moreover, a longer FIR filter introduces a longer delay and requires more area on a DSP circuitry. Alternatively, infinite impulse response (IIR) filters can used (Goldfarb et al., 2007) to reduce the complexity of the DSP circuit.

However, with the use of higher order modulation formats, i.e QPSK and QAM, to meet the capacity requirements, it becomes vital to compensate non-linearities along with the fiber dispersion. Due to this non-linear threshold point (NLT) of the transmission system can be improved and more signal power can be injected in the system to have longer transmission distances. In (Geyer et al., 2010) a low complexity non-linear compensator scheme with automatic control loop is introduced. The proposed simple non-linear compensator requires considerably lower implementation complexity and can blindly adapt the required coefficients. In uncompensated links, the simple scheme is not able to improve performance, as the non-linear distortions are distributed over different amounts of CD-impairment. Nevertheless the scheme might still be useful to compensate possible non-linear distortions of the transmitter. In transmission links with full in-line compensation the compensator provides 1dB additional noise tolerance. This makes it useful in 10Gbit/s upgrade scenarios where optical CD compensation is still present. Another promising electronic method, investigated in higher bit-rate transmissions and for diverse dispersion mapping, is the digital backward propagation (DBP), which can jointly mitigate dispersion and non-linearities. The DBP algorithm can be implemented numerically by solving the inverse non-linear Schrödinger equation (NLSE) using split-step Fourier method (SSFM) (Ip et al., 2008). This technique is an off-line signal processing method. The limitation so far for its real-time implementation is the complexity of the algorithm (Yamazaki et al., 2011). The performance of the algorithm is dependent on the calculation steps (*h*), to estimate the transmission link parameters with accuracy, and on the knowledge of transmission link design.

In this chapter we give a detailed overview on the advancements in DBP algorithm based on different types of mathematical models. We discuss the importance of optimized step-size selection for simplified and computationally efficient algorithm of DBP.
