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I(t) Q(t) A/D A/D FFE: feed-forward equalizer ▲Figure 1. Structure of the NLEE. By solving the nonlinear Schrödinger equation with first-order perturbation theory [27], the output can be viewed as having two parts: the linear solution μn and the nonlinear optical-field distortion, Δμn. In single-channel systems, only pulses with indices that satisfy the temporal matching condition l + m - k = n induce a noticeable distortion on the pulse μn[28]. This is equivalent to the phase-matching of four-wave mixing (FWM) in wavelength-division multiplexing (WDM). Intrachannel self-phase modulation (ISPM) and intrachannel cross-phase modulation (IXPM) are also included by containing the items l = m = k = n, l = n , or m = n (where l ≠ m). Assuming a nonlinear fiber system and Δμn, the following can be expressed according to the Volterra theory: N Fig. 1 shows the structure of the third-order NLEE, and the algorithm is detailed in [11]. The in-phase component of the received QPSK signals is I (t ), and the quadrature component of the received QPSK signals is Q (t ). The equalized signal for output-carrier phase estimation is ye(t ). The complex conjugation of the corresponding signals is ( )*. The symbol duration is T, and the sampling space is qT. The discrete coefficient is h. Intrachannel nonlinear distortion is hl,m,k , where the subscripts denote the optical pulse positions. The delay tap number, L = 2N + 1 , is equivalent to the nonlinear ISI length. The third-order items are defined as nonlinear terms, and the equalizer coefficients are adaptively determined using the recursive least square (RLS) algorithm. 3.2 Nonlinear Electrical Equalizer for Manakov Equation The NLEE in Fig. 1 can be extended to polarization-multiplexed systems by rewriting (11) according to the Manakov equation. We assume a transmission system has two polarizations, x and y. When compensating for polarization x nonlinearities, the distortions from polarization y should not be j yr(t ) • •• 0* 0* • • q T • q T • … … h 0 h 001 h 1 h 012 h 122 h 2 + FFE •• 0* • ye(t ) Δun =∑CpuP(n-p)+∑ ∑ ∑Cl,m,ku(n-l)u(n-m)u*(n-k )+... (11) P =-N N N N I =-N m =I k =m Data Data QAM or PSK De-Mapping QAM or PSK Mapping P/S NLEE S/P Slicer Clock Phase Correction LPF: low-pass filter NLEE: nonlinear electrical equalizer neglected. For polarization x, the adjusted third-order nonlinearities with nonlinear distortions from polarization y is expressed as [29] which can be expanded further as [30] where C l,m,k is the coefficient determined adaptively using the RLS algorithm, and ux and uy are the signals from the two orthogonal polarizations. For simplicity, we do not consider the intrachannel four-wave mixing from the orthogonal polarization. 4 Coherent Optical SCFDM System Fig. 2 shows the DSP block diagrams for the CO-SCFDM system. At the coder, the transmitted binary data is mapped into QAM or phase-shift keying (PSK) signals and then is grouped into blocks containing M symbols. The first step in the CO-SCFDM system is to perform an M -point DFT to produce a frequency-domain representation of the input symbols. Then, M-point DFT outputs are mapped to N (N ≥ M) orthogonal subcarriers. After the N-point inverse fast Fourier transform (IFFT), which transforms the subcarriers into a time domain signal, a cyclic prefix (CP) is inserted for each block before the data sequence is transmitted. At the decoder, the N -point FFT transforms the signals into the frequency domain, and channel equalization is performed. The equalized signal is transformed into the time domain by the M -point inverse discrete Fourier transform (IDFT) for decision. CO-SCFDM is very flexible, and its parameter design can have a high degree of commonality with OFDM. Nonlinear compensation algorithms of the BP and NLEE can be implemented at different positions (Fig. 2b). At the transmitter, the mapped quadrature phase-shift keying (QPSK) signals are first grouped into blocks of 1680 M-Point DFT M-Point IDFT Δux,n =∑∑∑C l,m,k[ux(l )ux(m)uz*(k ) Subcarrier Mapping l m k +ux (l )uy(m)uy*(k )] (12) Δu x,n =∑Cl [ux (l)( ux(l) 2 + uy(l ) 2 )] l +∑∑Cm,l[ux(l ) ( ux(m) 2 + uy (m) 2 )] N-Point IFFT Equalization N-Point FFT Remove CP CP: cyclic prefix IDFT: inverse discrete Fourier transform Back Propagation S/P LPF LPF: low-pass filter S/P: serial/parallel ▲Figure 2. DSP block diagram for (a) the CO-SCFDM coder and (b) the CO-SCFDM decoder. (a) (b) l m +∑∑Cl,m[ux(m)( ux(l ) 2 +ux (l )uy(m)uy*(l ) ] l m +∑∑∑C l,m,k [ux (l )ux(m)ux*(k )+ux(l )uy(m)uy*(k ) l m k P/S Add CP S pecial Topic Compensating for Nonlinear Effects in Coherent-Detection Optical Transmission Systems Fan Zhang LPF (13) Transmitted Signal Received Signal March 2012 Vol.10 No.1 <strong>ZTE</strong> COMMUNICATIONS 47
S pecial Topic Compensating for Nonlinear Effects in Coherent-Detection Optical Transmission Systems Fan Zhang ▼Table 1. Key CO-SCFDM design parameters FFT/IFFT (N -point) 2048 DFT/IDFT (M -point) 1712 Symbol Size 2168 Subcarrier Spacing 9.04 MHz FFT: fast Fourier transform IFFT: inverse fast Fourier transform Number of Pilot 32 Net Bit Rate 50 Gb/s Modulation PDM QPSK FEC Ethernet Redundancy DFT: discrete Fourier transform IDFT: inverse discrete Fourier transform symbols, and 32 pilots are uniformly time-multiplexed into each block. Then, the signal is transformed into the frequency domain by 1712-point DFT. In the frequency domain, 336 guard subcarriers are added and allocated at both sides of the band. These subcarriers provide about 20% oversampling so that the aliasing products can be spectrally separated with a low-pass filter. After subcarrier mapping, the signal is converted back to the time domain by 2048-point IFFT. The cyclic prefix and cyclic suffix, each 60 symbols long, are inserted into every data block before transmission. The duration of a CO-SCFDM symbol is 117.1 ns. The preamble is 2.44% and includes synchronization and training symbols (Table 1). Taking into account 12% redundancy for Ethernet overhead and forward-error correction, PDM QPSK is capable of 50 Gb/s net. The fiber link comprises ten spans of 80 km standard single-mode fiber, each with an average loss of 20 dB. The fiber dispersion is 17 ps/km/nm, and the fiber nonlinear coefficient is 1.32 km/W. An Erbium-doped fiber amplifier with 5 dB noise fully compensates for fiber attenuation. No in-line chromatic dispersion compensation is used. With an average launch power of 5 dBm, the signal suffers nonlinearity distortion during propagation. Nonlinearity is compensated for by either the NLEE or the BP. For NLEE, we choose NL [15]; for BP, we choose two different split-step sizes, one of which is equal to the span length. The other step is a full multistep approach determined by the maximum nonlinear phase shift in each step, which is the same 0.05 as the simulation of the transmission link. Fig. 3 shows the results of nonlinear compensation. The error vector magnitude (EVM) is the performance indicator. For Fig. 3(a) to (d), the EVM is 15.62 dB, 18.34 dB, 18.63 dB, and 18.75 dB, respectively. Although PDM signals are transmitted and equalized, we only present the results of x polarization for the sake of simplicity. The BP with one step per span improves EVM by about 3.0 dB, similar to the full multistep approach. The NLEE provides similar performance but with asymmetric constellations. 5 Conclusion In this paper, we review nonlinear compensation methods of digital BP propose an NLEE based on the Volterra series. In terms of computational complexity, the time-domain approach of NLEE requires one order of magnitude more than BP-SSF [11]. However, NLEE is much more convenient for real-time implementation because the pulses are processed 48 <strong>ZTE</strong> COMMUNICATIONS 12% March 2012 Vol.10 No.1 Symbol Duration 117.1 ns Preamble 2.44% in sequence rather than in a block. BP in [8] uses split-step FIR filtering and thus supports sequential processing. By comparison, the computational complexity of NLEE is one order of magnitude lower. NLEE has an important advantage over BP in that it does not require prior knowledge of the fiber-link parameters. The equalizer coefficients are adaptively determined, and this provides flexibility when signals are routed differently in optical networks. An NLEE based on the Volterra series can also be implemented in a frequency domain with low complexity [15], [31]. For both OFDM and SCFDM systems, Volterra series-based nonlinear equalization should be further studied for both time and frequency implementation. This paper is mainly focused on nonlinear compensation in single-channel operation. BP in WDM systems has been well covered in [10], [13], and [32]. Besides intrachannel nonlinearities, interchannel nonlinear interactions, such as XPM and FWM, should be considered. If A xm and A ym are, respectively, the envelopes of x and y polarization tributaries of the received mth WDM channel field, the reconstructed total optical field is A =∑[A xm·A ym ] T exp(im△ωt ) (14) m The channel spacing is Δf = Δω /2π. A set of local oscillators (LOs) is used for coherent detection and reconstruction of each WDM channel. To mitigate nonlinear distortions in WDM systems, the reconstructed optical field of Quadrature Quadrature 2 1 0 -1 -2 -2 2 1 0 -1 -2 -2 2 EVM=15.62 dB EVM=18.34 dB -1 -1 0 1 2 -2 -2 -1 0 In-Phase In-Phase (a) (b) 0 1 2 -2 -2 -1 0 In-Phase In-Phase (c) (d) EVM: error vector magnitude ▲Figure 3. Constellations of the recovery signals after ten-span transmission (a) without nonlinear compensation, (b) with nonlinear compensator as NL, (c) with one-step-per-span BP, (d) with multistep BP. Quadrature 2 EVM=18.63 dB EVM=18.75 dB Quadrature 1 0 -1 1 0 -1 1 1 2 2
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