Abstract

We propose a novel detection scheme called Stokes vector direct detection (SV-DD) to realize high electrical spectral efficiency and cost-effective optical communication for short and medium reach. With SV-DD, the signal is modulated in only one polarization and combined with the carrier in the orthogonal polarization for fiber transmission. At reception, the combined signal is detected in Stokes space by three or four photo-detectors. Compared with conventional DD technique, SV-DD is resilient to both chromatic dispersion and signal-to-signal beat noise. Furthermore, SV-DD does not require polarization tracking or narrow band optical filtering for carrier extraction. In this paper, we present for the first time the numerical analysis and experimental demonstration of single-carrier SV-DD. We report 62.5-Gb/s data rate single-carrier SV-DD transmission over 160-km SSMF using 12.5-Gbaud 32-QAM modulation.

© 2014 Optical Society of America

1. Introduction

The emergence of many Internet-centric applications such as cloud computing has created a dire need for the high-speed optical transport. For short or medium reach applications, a data rate of 40 Gb/s and 100 Gb/s has been actively studied [1] and will soon become a reality. For these applications, systems at low cost and low complexity such as direct detection (DD) [25] are preferred. Compared with coherent detection, DD is a suitable option which requires neither polarization diversity nor local oscillator (LO). The digital signal processing (DSP) is also much simpler because there is no laser frequency offset or phase noise. However, there is a fundamental limit for 40 Gb/s and beyond DD systems using double sideband (DSB) modulation due to chromatic dispersion (CD) induced fading. Single sideband (SSB) modulation is an efficient way of avoiding CD induced fading [2,3], but it halves the electrical spectral efficiency (SE) of transmitter. Additionally, DD system suffers from signal-to-signal beat noise (SSBN) due to the power law detection of photodiode (PD). Various techniques for the cancellation of SSBN have been proposed but with the sacrifice of SE [3], or incur additional DSP complexity [4]. Recently we proposed a DSB modulation DD scheme called block-wise phase switching (BPS) [6,7] which eliminates the CD induced fading by introducing phase diversity. Since two blocks are transmitted for a complete complex signal, the SE of BPS is only 50%. A higher SE can be achieved by using a narrow bandwidth optical band-pass filter (OBPF) to extract the carrier [8]. However, this OBPF could be potentially expensive and difficult to operate. In this paper, we propose a novel DD technique called Stokes vector direct detection (SV-DD) where the signal is detected in Stokes space. With SV-DD, spectrally efficient high speed optical transmission is demonstrated, achieving 100% SE with reference to the coherent detection of single polarization. Compared with polarization multiplexed modulation with coherent detection, SV-DD only requires single polarization modulation and does not need an LO, which lowers the cost and reduces the system complexity. We report 62.5-Gb/s SV-DD transmission over 160-km standard single-mode fiber (SSMF) using 32-QAM modulation, which achieves a net electrical SE of 9.34 bit/s/Hz after 7% forward error correction (FEC).

2. Concept of Stokes vector direct detection

It is known that an optical signal in arbitrary polarization state can be represented in either Jones or Stokes space. In the SV-DD scheme, we launch signal (ES) and carrier (Ec) into orthogonal polarizations which are combined together with a polarization beam combiner (PBC). The Jones vector (JV) of transmitted signal is T = [Es, Ec]T where superscript ‘T’ stands for transpose. After fiber transmission, the received is detected in Stokes space. The JV can be converted to the Stokes vector (SV) S = [S0, S1, S2, S3]T defined as

S0=ExEx*+EyEy*=|Ex|2+|Ey|2,S1=ExEx*EyEy*=|Ex|2|Ey|2S2=ExEy*+EyEx*=2Re(ExEy*),S3=i(ExEy*EyEx*)=2Im(ExEy*)

The 0th component of SV (S0) shows the intensity of the entire optical signal, which is not related to the polarization rotation. Following Eq. (1), we obtain the SV of the transmitted signal, T=[S1,S2,S3]T=[|Es|2|EC|2,2Re(EsEC*),2Im(EsEC*)]T where Re(.) and Im(.) denote the real and imaginary part of a complex signal. The SV of received signal can be expressed as R=[S1,S2,S3]T=[|Ex|2|Ey|2,2Re(ExEy*),2Im(ExEy*)]Twhere the prime stands for the signals at receive. The SV of received signal and transmitted signal are related by the following equation R=HT, where H is a 3 × 3 rotation matrix in Stokes space. The SV of received signal can be detected as follows: the optical signal is first split into two orthogonal polarizations by a polarization beam splitter (PBS). Both polarizations are split again using two 3-dB couplers, generating two identical sets of signal in x- and y- polarizations (Ex and Ey). One pair of signals are respectively fed into two single-end PDs, with the two outputs denoted as O1 and O4. The other pair of signals are fed into a 90° optical hybrid. The outputs of the optical hybrid are detected by two balanced PDs with the outputs denoted as O2 and O3, as shown in Fig. 1(a). Ignoring some constants, the four outputs of SV-DD receiver areO1=|Ex|2,O2=2Re(ExEy*), O3=2Im(ExEy*), and O4=|Ey|2. Using Eq. (1), the SV of received signal can be obtained from the four outputs O1~O4. Alternatively, the SV can be also detected with 3 balanced PDs as shown in Fig. 1(b), or 4 single-end PDs as shown Fig. 1(c). The outputs of these 3- or 4 PD detection schemes are a linear superposition of all four SV components.

 figure: Fig. 1

Fig. 1 Receiver schemes for Stokes vector detection. (a) 2 single-end PDs and 2 balanced PDs, (b) 3 balanced PDs, and (c) 4 single-end PDs. PBS: polarization beam splitter, PD: photodiode.

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To estimate the SV rotation matrix, we send three orthogonal training symbols in the JV form of [0, 1]T, [1,1]T and [i, 1]T, as shown in Fig. 2(a). These training symbols correspond to the SV of [-1, 0, 0]T, [0, 1, 0]T and [0, 0, 1]T in Stokes space, as shown in Fig. 2(b). The SV rotation matrix Hcan thus be estimated by H=RT1 where T and R stand for the transmitted and received SV of training symbols in 3 × 3 matrix form, as shown in Fig. 2(c). By inverting H and multiplying it by the received SV, we can recover the SV of received signal back to the transmitted polarization state T=H1R. After equalization, we combine the 2nd and 3rd SV components of Tto form a complex signal Er=S1+iS2=2EsEC*. It can be seen that the recovered complex signal has the full phase diversity of the transmitted signal. Therefore there is no CD induced fading. Furthermore, the 2nd order nonlinear terms of signal are all included in the 0th and 1st components of SV. Therefore SSBN does not affect the signal recovered from 2nd and 3rd SV components.

 figure: Fig. 2

Fig. 2 Stokes space training algorithm. (a) Signal structure, (b) training symbols in Jones and Stokes vector form, and (c) channel estimation in Stokes space.

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3. Digital signal processing for SV-DD receivers

As shown in Fig. 3(a), the receiver DSP of SV-DD is as follows, (1) low-pass filtering (5th order Bessel), (2) CD compensation in the three SV components [S1,S2,S3]T using three T/2-spaced fixed time-domain finite impulse response (FIR) equalizers (EQ1), (3) clock recovery, (4) Stokes space adaptive equalization (SS-AEQ), (5) constellation reconstruction and decision. Here we propose two algorithms for the SS-AEQ which is training symbol (TS) based and blind. For the TS based AEQ (TS-AEQ) as shown in Fig. 3(b), we perform the following operations to equalize the received signal R, (i) TS based rotation matrix estimation and inversion using the method as proposed in Section 2. After obtaining H1, we equalize the signal R with three one-tap filters (EQ2) for the polarization recovery and complex signal forming. The following relationship applies between the filter coefficients (w1, w2 and w3) and H1: w1 = h21 + ih31, w2 = h22 + ih32, and w3 = h23 + ih33, where hjk denotes the matrix element ofH1at j-th row and k-th column; and (ii) an 64-tap adaptive filter (EQ3) with filter coefficient h for the compensation of residual CD and other linear distortion, using decision-directed normalize least mean square (DD-NLMS) [9] algorithm with step size of 0.32 or recursive least square (RLS) with forget factor = 1. For the blind AEQ (B-AEQ) as shown in Fig. 3(c), we perform the following operations to equalize the received signal R: three 64-tap adaptive filters (EQ4) for the polarization recovery, complex signal forming, and compensation of residual CD and other linear distortion. The adaptive algorithm of EQ4 is a modified multi-input-single-output (MISO) DD-NLMS that contains the following procedures to update the filter coefficients: (i) initialize the filter coefficients at time k, denoted as wm(k)(m = 1, 2 or 3) with all tap weights set to zero except the central tap set to unitary; (ii) equalize the received SVs using the initialize filter coefficients, y(k)=w(k)R(k)where w(k)=[w1(k),w2(k),w3(k)],R(k)is the received 3 SVs of signalR=at time k; (iii) calculate the instantaneous error by e(k)=d(k)y(k), where d(k)is the symbol closest to y(k); and (iv) update the filter coefficients by

wm(k+1)=wm(k)+μe(k)Sm(k)/R(k)2
where μ is the step size of the adaptive filter, ||·||2 denotes the L2 norm of braced vector. The algorithm minimizes the instantaneous squared error at new time k + 1 and the deviation of filter coefficients from the prior estimate at time k,

 figure: Fig. 3

Fig. 3 Conceptual diagram of (a) digital signal processing at the SV-DD receiver, (b) Training symbol based SS-AEQ, and (c) blind SS-AEQ. LPF: low-pass filter.

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minw(k+1){e(k+1)2+(1/μ1)R(k+1)2w(k+1)w(k)2}

Similar to that in coherent detection, B-AEQ may have steady state error without pre-convergence. TS-AEQ does not have steady state error, but will introduce additional overheads. To overcome this problem, we apply a partial TS-based SS-AEQ (PTS-AEQ) algorithm which performs the following two-stage operations to update the filter coefficients: (1) at the initial equalization stage, we use TS based SS-AEQ until EQ3 reaches a steady state for pre-convergence; and (2) once EQ3 converges, we switch to the B-AEQ for the subsequent data symbols by initializing the filter coefficients of EQ4 as wm(k)=wmh, where wm is the filter coefficients of EQ2 (polarization recovery) and his the filter coefficients of EQ3 after convergence. The advantage of PTS-AEQ is that the TS only need to be sent once so the overhead is almost negligible.

To demonstrate the feasibility of SV-DD and SS-AEQ algorithm, we first numerically evaluate the system performance of SV-DD through simulations. The simulation parameters are as follows: the signal is a pseudo random binary sequence (PRBS) with a length of 215-1 that is mapped into 32 QAM, preceded by 8 sets of three orthogonal Stokes training symbols with a length of 128 sample points per symbol. The signal baud rate is 12.5-Gbaud or 25-Gbaud (raw data rate of 62.5-Gb/s or 125-Gb/s). The fiber link is modeled as 2-spans of 80-km SSMF. The fiber parameter is as follows: loss α = 0.2 dB/km, chromatic dispersion D = 17 ps/nm/km, nonlinear coefficient γ = 1.3 W−1* km−1. The fiber loss in each span is fully compensated by an erbium-doped fiber amplifier (EDFA) with noise figure of 6 dB. The receiver sampling rate is 50 GSa/s and the electrical bandwidth (BW) is 25 GHz. For each scheme, the Monte Carlo simulation is performed 100 times and Q factors or BERs are obtained by averaging over all instances. We first search the optimum launch power for 160-km SSMF transmission. The result is shown in Fig. 4(a). We find out that the optimum launch power is about 2.5 dBm for 12.5-Gbaud and 3 dBm for 25-Gbaud scheme. We then analyze the optical signal-to-noise ratio (OSNR) sensitivity of SV-DD for back-to-back (B2B) and after 160-km SSMF transmission. The OSNR is defined as the ratio of the total signal power to the noise power within 0.1 nm bandwidth, and the OSNR sensitivity is defined as the required OSNR at the receiver to achieve BER of 3.8 × 10−3 (7% FEC). The result is illustrated in Fig. 4(b). It can be seen that the required OSNR is about 21.5 dB (B2B) and 21.9 dB (160 km) for 12.5-Gbaud SV-DD. This value increases to 24.5 dB (B2B) and 25.2 dB (160 km) for 25-Gbaud SV-DD. The performance is similar for all three proposed AEQ algorithms (B-, TS- and PTS-) when the algorithms converge. Furthermore, the performance of SV-DD with respect to the carrier-to-signal power ratio (CSPR) is also analyzed. It is found that the best performance is achieved when CSPR = 0 dB, confirming that the 2nd nonlinearity is completely removed. Finally, we analyze the impact of polarization mode dispersion (PMD) on the performance of SV-DD. Simulation results show that for 1-dB OSNR penalty, the differential group delay (DGD) tolerance is 2.8 ps for 62.5-Gb/s SV-DD and 1.4 ps for 125-Gb/s SV-DD, respectively.

 figure: Fig. 4

Fig. 4 Simulated system performance for 12.5- and 25-Gbaud SV-DD. (a) Q factor vs. optical launch power after 160-km SSMF transmission, (b) BER vs. OSNR at back-to-back and after 160-km SSMF transmission.

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4. Experiment and results

The system setup is shown in Fig. 5. We use an external-cavity laser (ECL) with a fixed wavelength at 1550-nm as the light source. The continuous wave (CW) from ECL is first split into two branches by a 3-dB coupler. The CW light in upper branch passes through an IQ modulator driven by a Tektronix arbitrary waveform generator (AWG) for signal modulation. The sampling rate of AWG is 12.5 GSa/s. The RF signal incorporating the Stokes training symbols and 32-QAM data symbols is generated offline with a Matlab program and loaded onto AWG. The RF signal pattern is shown in inset (i) of Fig. 5. The lower branch just simply passes through to provide main carrier. The fiber length in the upper and lower paths is matched to minimize the impact of laser phase noise, and the CSPR is maintained at 0 dB for optimal operation. The signal and carrier are then combined together with a PBC, and launched into a fiber link consisting of two spans of 80-km SSMF. At the SV-DD receiver, we elect to use the scheme of 2 two single-end PDs and 2 balanced PDs to measure SV, as shown in Fig. 1(a). The electrical signal of 4 outputs is sampled by a 4-channel Tektronix oscilloscope at a sampling rate of 50GSa/s. The bandwidth of oscilloscope is 20 GHz. The acquired signal is processed offline using the same DSP as in simulations. The raw data rate of received signal is 62.5 Gb/s and the net data rate is 58.4 Gb/s (7% FEC) after counting all the overheads. A total of 3.14 × 106 data bits are collected for the measurement of BER. Inset (iv) in Fig. 5 shows the electrical spectra of received signal before and after SV rotation. A strong 2nd order nonlinearity component is observed in the spectrum before SV rotation, which has been successfully removed after SV rotation indicating that our DSP algorithm is effective. We first analyze the optimum launch power for 160-km SSMF transmission. Figure 6(b) shows the Q-factor of the system against optical launch power. We can see that the optimum launch power is 2 dBm. By fixing the launch power at 2 dBm, we then measure the OSNR sensitivity of the SV-DD system by optically load noise at the receiver. The result is shown in Fig. 6(b). The simulated BER curve for SV-DD and coherent detection is also plotted for comparison. We find that at 7% FEC threshold, the required OSNR for back-to-back (B2B) is 22.6 dB and all three AEQ algorithms have similar performance. After 160-km SSMF transmission, the required OSNR increases slightly to 24.4 dB (PTS-AEQ) and 26.2 dB (TS-AEQ), respectively. The B-AEQ does not always have stable convergence and thus not illustrated. Using PTS-AEQ algorithm, the required OSNR is only 2.8 dB higher than the simulated result of SV-DD (21.6 dB), or is 8.8 dB higher than the simulated result of coherent detection (15.6 dB). If a 20% FEC (BER = 2 × 10−2) is used, the required OSNR is 19.2 dB (B2B), 20.2 dB (160-km PTS-AEQ) and 20.9 dB (160 km TS-AEQ). The maximum achievable OSNR of received signal is more than 31 dB [Fig. 6(b)] after 160-km transmission using commercial available EDFAs, showing that the SV-DD system has more than 6.7 dB (7% FEC) or 10.8 dB (20% FEC) OSNR margin thus the signal can potentially be received at a longer reach or a higher data rate.

 figure: Fig. 5

Fig. 5 Experimental setup for 62.5-Gb/s 32QAM SV-DD. Insets: (i) RF signal loaded onto AWG, (ii) optical spectrum of transmitted signal, (iii) optical spectrum of received signal, and (iv) electrical spectra of received signal before and after SV rotation.

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 figure: Fig. 6

Fig. 6 (a) Q factor as a function of fiber launch power, and (b) BER as a function of OSNR for 62.5 Gb/s 32-QAM SV-DD. Dashed line in (b): 7% FEC (black) and 20% FEC (orange). Insets: recovered signal constellation for (i) launch power of 2 dBm (optimum), (ii) launch power of 10 dBm, (iii) back-to-back, and (iv) after 160-km transmission. SV: SV-DD, Co: coherent detection.

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References and links

1. O. Ishida, “40/100GbE Technologies and Related Activities of IEEE Standardization,” in Proc. Optical Fiber Communication Conference (OFC), 2009, paper OWR5. [CrossRef]  

2. A. J. Lowery and J. Armstrong, “Orthogonal-frequency-division multiplexing for dispersion compensation of long-haul optical systems,” Opt. Express 14(6), 2079–2084 (2006). [CrossRef]   [PubMed]  

3. W.-R. Peng, X. Wu, V. R. Arbab, K.-M. Feng, B. Shamee, L. C. Christen, J.-Y. Yang, A. E. Willner, and S. Chi, “Theoretical and Experimental Investigations of Direct-Detected RF-Tone-Assisted Optical OFDM Systems,” J. Lightwave Technol. 27(10), 1332–1339 (2009). [CrossRef]  

4. W.-R. Peng, B. Zhang, K.-M. Feng, X. Wu, A. E. Willner, and S. Chi, “Spectrally Efficient Direct-Detected OFDM Transmission Incorporating a Tunable Frequency Gap and an Iterative Detection Techniques,” J. Lightwave Technol. 27(24), 5723–5735 (2009). [CrossRef]  

5. M. Nazarathy and A. Agmon, “Doubling direct-detection data rate by polarization multiplexing of 16-QAM without active polarization control,” Opt. Express 21(26), 31998–32012 (2013). [CrossRef]   [PubMed]  

6. X. Chen, A. Li, D. Che, Q. Hu, Y. Wang, J. He, and W. Shieh, “Block-wise phase switching for double-sideband direct detected optical OFDM signals,” Opt. Express 21(11), 13436–13441 (2013). [CrossRef]   [PubMed]  

7. A. Li, D. Che, X. Chen, Q. Hu, Y. Wang, and W. Shieh, “61 Gbits/s direct-detection optical OFDM based on blockwise signal phase switching with signal-to-signal beat noise cancellation,” Opt. Lett. 38(14), 2614–2616 (2013). [CrossRef]   [PubMed]  

8. L. Xu, J. Hu, D. Qian, and T. Wang, “Coherent Optical OFDM Systems Using Self-Optical Carrier Extraction,” in Proc. Optical Fiber Commun. Conf. (OFC), 2008, paper OMU4. [CrossRef]  

9. S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express 16(2), 804–817 (2008). [CrossRef]   [PubMed]  

References

  • View by:

  1. O. Ishida, “40/100GbE Technologies and Related Activities of IEEE Standardization,” in Proc. Optical Fiber Communication Conference (OFC), 2009, paper OWR5.
    [Crossref]
  2. A. J. Lowery and J. Armstrong, “Orthogonal-frequency-division multiplexing for dispersion compensation of long-haul optical systems,” Opt. Express 14(6), 2079–2084 (2006).
    [Crossref] [PubMed]
  3. W.-R. Peng, X. Wu, V. R. Arbab, K.-M. Feng, B. Shamee, L. C. Christen, J.-Y. Yang, A. E. Willner, and S. Chi, “Theoretical and Experimental Investigations of Direct-Detected RF-Tone-Assisted Optical OFDM Systems,” J. Lightwave Technol. 27(10), 1332–1339 (2009).
    [Crossref]
  4. W.-R. Peng, B. Zhang, K.-M. Feng, X. Wu, A. E. Willner, and S. Chi, “Spectrally Efficient Direct-Detected OFDM Transmission Incorporating a Tunable Frequency Gap and an Iterative Detection Techniques,” J. Lightwave Technol. 27(24), 5723–5735 (2009).
    [Crossref]
  5. M. Nazarathy and A. Agmon, “Doubling direct-detection data rate by polarization multiplexing of 16-QAM without active polarization control,” Opt. Express 21(26), 31998–32012 (2013).
    [Crossref] [PubMed]
  6. X. Chen, A. Li, D. Che, Q. Hu, Y. Wang, J. He, and W. Shieh, “Block-wise phase switching for double-sideband direct detected optical OFDM signals,” Opt. Express 21(11), 13436–13441 (2013).
    [Crossref] [PubMed]
  7. A. Li, D. Che, X. Chen, Q. Hu, Y. Wang, and W. Shieh, “61 Gbits/s direct-detection optical OFDM based on blockwise signal phase switching with signal-to-signal beat noise cancellation,” Opt. Lett. 38(14), 2614–2616 (2013).
    [Crossref] [PubMed]
  8. L. Xu, J. Hu, D. Qian, and T. Wang, “Coherent Optical OFDM Systems Using Self-Optical Carrier Extraction,” in Proc. Optical Fiber Commun. Conf. (OFC), 2008, paper OMU4.
    [Crossref]
  9. S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express 16(2), 804–817 (2008).
    [Crossref] [PubMed]

2013 (3)

2009 (2)

2008 (1)

2006 (1)

Agmon, A.

Arbab, V. R.

Armstrong, J.

Che, D.

Chen, X.

Chi, S.

Christen, L. C.

Feng, K.-M.

He, J.

Hu, Q.

Li, A.

Lowery, A. J.

Nazarathy, M.

Peng, W.-R.

Savory, S. J.

Shamee, B.

Shieh, W.

Wang, Y.

Willner, A. E.

Wu, X.

Yang, J.-Y.

Zhang, B.

J. Lightwave Technol. (2)

Opt. Express (4)

Opt. Lett. (1)

Other (2)

L. Xu, J. Hu, D. Qian, and T. Wang, “Coherent Optical OFDM Systems Using Self-Optical Carrier Extraction,” in Proc. Optical Fiber Commun. Conf. (OFC), 2008, paper OMU4.
[Crossref]

O. Ishida, “40/100GbE Technologies and Related Activities of IEEE Standardization,” in Proc. Optical Fiber Communication Conference (OFC), 2009, paper OWR5.
[Crossref]

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Figures (6)

Fig. 1
Fig. 1 Receiver schemes for Stokes vector detection. (a) 2 single-end PDs and 2 balanced PDs, (b) 3 balanced PDs, and (c) 4 single-end PDs. PBS: polarization beam splitter, PD: photodiode.
Fig. 2
Fig. 2 Stokes space training algorithm. (a) Signal structure, (b) training symbols in Jones and Stokes vector form, and (c) channel estimation in Stokes space.
Fig. 3
Fig. 3 Conceptual diagram of (a) digital signal processing at the SV-DD receiver, (b) Training symbol based SS-AEQ, and (c) blind SS-AEQ. LPF: low-pass filter.
Fig. 4
Fig. 4 Simulated system performance for 12.5- and 25-Gbaud SV-DD. (a) Q factor vs. optical launch power after 160-km SSMF transmission, (b) BER vs. OSNR at back-to-back and after 160-km SSMF transmission.
Fig. 5
Fig. 5 Experimental setup for 62.5-Gb/s 32QAM SV-DD. Insets: (i) RF signal loaded onto AWG, (ii) optical spectrum of transmitted signal, (iii) optical spectrum of received signal, and (iv) electrical spectra of received signal before and after SV rotation.
Fig. 6
Fig. 6 (a) Q factor as a function of fiber launch power, and (b) BER as a function of OSNR for 62.5 Gb/s 32-QAM SV-DD. Dashed line in (b): 7% FEC (black) and 20% FEC (orange). Insets: recovered signal constellation for (i) launch power of 2 dBm (optimum), (ii) launch power of 10 dBm, (iii) back-to-back, and (iv) after 160-km transmission. SV: SV-DD, Co: coherent detection.

Equations (3)

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S 0 = E x E x * + E y E y * = | E x | 2 + | E y | 2 , S 1 = E x E x * E y E y * = | E x | 2 | E y | 2 S 2 = E x E y * + E y E x * =2Re( E x E y * ), S 3 =i( E x E y * E y E x * )=2Im( E x E y * )
w m (k+1)= w m (k)+μe(k) S m (k)/ R(k) 2
min w(k+1) { e(k+1) 2 +( 1/μ1 ) R(k+1) 2 w (k+1) w (k) 2 }

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