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A novel high precision adaptive equalizer in digital coherent optical receivers
Optics Communications 353 (2015) 63–69
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
A novel high precision adaptive equalizer in digital coherent optical receivers Xiurong Ma a,b,c,n, Yujun Xu a,b,c, Xiao Wang a,b,c, Zhaocai Ding a,b,c a
The Department of Computer and Communication Engineering, Tianjin University of Technology, Tianjin 300384, China Engineering Research Center of Communication Devices and Technology, Ministry of Education, Tianjin 300384, China c Tianjin Key Laboratory of Film Electronic and Communication Devices, Tianjin 300384, China b
art ic l e i nf o
a b s t r a c t
Article history: Received 10 January 2015 Received in revised form 29 April 2015 Accepted 2 May 2015 Available online 4 May 2015
A novel high precision adaptive equalization method is introduced and applied to dynamic equalization for quadrature phase shift keying (QPSK) coherent optical communication system in this paper. A frequency-domain constant modulus algorithm (CMA) method is used to equalize the received signal roughly. Then, some non-ideal output signals will be picked out through the error measurement, and they will be equalized accurately further in a fixed time-domain CMA equalizer. This high precision equalization method can decrease the equalization error, then it can reduce the bit error ratio (BER) of coherent communication system. Simulation results show that there is a 6% decrease for computation complexity by proposed scheme when compared with time-domain CMA. Furthermore, compared with time-domain CMA and frequency-domain CMA, about 2 dB and 2.2 dB in OSNR improvement can be obtained by proposed scheme at the BER value of 1e 3, respectively. & 2015 Elsevier B.V. All rights reserved.
Keywords: Coherent optical communication system Constant modulus algorithm Error measurement
1. Introduction Polarization diversity optical coherent receivers, which combine a polarization-division multiplexing (PDM) and an M-ary modulation formats, were a promising solution for high-capacity optical communication systems. A superiority of coherent receiver was the equalization of linear distortions and the demultiplexing of the two PDM signals could be done in the electronic equalizer. The butterfly-structure multiple-input/multiple-output adaptive electronic equalizer was usually employed to accomplish the equalization procedure [1–3]. The sampling rate of received signals was always set as twice symbol rate for the sake of reduction in aliasing effect [4]. Under the condition of twofold oversampling, time-domain equalizer (TDE) with constant modulus algorithm (CMA) was adapted two samples per step in order to adjust the initial sampling phase [5,6]. The butterfly equalizer in optical coherent communication system was implemented in the time-domain generally. However, it was recently shown in [7] that, frequency-domain equalizers (FDE) could bring significant computational savings in the adaptive equalization than TDE. In [7,8], FDE was realized by using pilotsymbol sequence, but such approaches generally decreased the n Corresponding author at: The Department of Computer and Communication Engineering, Tianjin University of Technology, Tianjin 300384, China E-mail addresses: [email protected] (X. Ma), [email protected] (Y. Xu).
http://dx.doi.org/10.1016/j.optcom.2015.05.001 0030-4018/& 2015 Elsevier B.V. All rights reserved.
spectral efficiency of transmission systems. Another attractive solution for FDE was equalization in frequency domain based on the overlap-save technique (OS-FDE) [9], this scheme was apt to implement and any cyclic prefix were not required. A fractionally spaced overlap-save FDE was proposed in [10]. This implementation was mathematically equivalent to the blockadaptive TDE, but offered substantial complexity reduction. What’s more, Ref. [11] proposed another FDE using CMA, it could work on the double oversampling rate by introducing even and odd subequalizers. As a result, the polarization demultiplexing and sampling-phase adjustment could achieve together in FDE. However, only the sampling phase adjustment and reduction of the computation complexity were considered both in Ref. [10] and Ref. [11], no obvious improvement was shown in FDE performance compared with the TDE. A novel high precision adaptive equalizer (FAT) was proposed in this paper, which consisted of a FDE, an error measurement stage and a TDE. Firstly, the linear impairments could be equalized roughly through the FDE in frequency-domain, the sampling phase also would be adjusted to best position simultaneously. Then some undesirable equalized signals would be picked out by error measurement. The thresholds of error measurement are for computation complexity and final performance to decide. At last, a fixed tap weights TDE was proposed and implemented, giving the more precise equalization to the picked out signals. Simulation results showed that there was a 6% decrease for computation complexity by FAT when compared with TDE. Moreover, a remarkable
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decrease with 5.25 dB and 4.36 dB was founded for BER by the FAT compared with the FDE and the TDE.
2. System description The simulated QPSK coherent optical communication system is shown in Fig. 1. In the transmitter, two branches of QPSK signals were modulated by 40 Gb/s PRBS of length 215 1; the lasers of the transmitter and the local oscillator had a line-width which were both equal to 100 kHz. Then, two orthogonal branches were polarization-multiplexed by a polarization beam combiner (PBC) in transmission channel. In the SSMF channel, the setting of chromatic dispersion (CD) and first-order polarization-mode dispersion (PMD) was determined. Linear polarization controllers (PCs) were placed at 45° and 45° on both sides of SSMF. After detecting by the coherent receiver, the received signals were sampled by free-running analog-to-digital converters (ADCs) which operated at the twice symbol rate. Such oversampling significantly reduced the aliasing effect. Then, the sampled signal sequences were sent into the signal processing stage which consisted of an electric equalizer and a carrier recovery stage. At last, the signal sequence would be disposed in the data recovery stage to extract the original data and calculate the BER.
sub-equalizers and odd sub-equalizers. Four even sub-equalizers ‵e , Hxy ‵e , H‵yxe, H‵yye were connected in a two-by-two butterfly Hxx configuration. In the same way, four odd sub-equalizers ‵o , Hxy ‵o , H‵yxo, H‵yyo were placed in another two-by-two butterfly Hxx configuration. The input sequences were represented as [x(n),y (n)]T and the output signal sequence were represented as [x1(n), y1(n)]T. Before the process of equalizing, the input sequences [x(n), y(n)]T were divided into [xe(m),xo(m),ye(m),yo(m)]T by serial–parallel converter for sub-equalizers. Where, n ¼ 2m 1. After FFT transforming and overlapping, the input sequence of FDE could be expressed as [Xe(K),Xo(K),Ye(K),Yo(K)]T and the equalization output in frequency-domain could be expressed as [X1(K),Y1(K)]T. The equalization procedure was same as in Ref. [11]. The length of signal block was set as P, so the tap weights length of every subequalizer was 2P. Moreover, all the sub-equalizers were updated one symbol per step without down-sampling output sequence. The equalized signals can be expressed as [11] X ′(K ) = H′ e xx(K )⋅X e(K ) + H′ o xx(K )⋅X o(K ) + H′ e xy(K )⋅Y e(K ) + H′ o xy(K )⋅Y o(K ) Y ′(K ) = H′ e yx(K )⋅X e(K ) + H′ o yx(K )⋅X o(K ) + H′ e yy(K )⋅Y e(K ) + H′ o yy(K )⋅Y o(K )
where H′(K) represented the tap weights in frequency-domain. Thus, after IFFT transforming, the time-domain outputs are expressed by
( ) elements of IFFT (Y (K ))
x1(K ) = last p elements of IFFT X ′(K ) y1(K ) = last p
3. Electronic equalizer
(1)
′
(2) T
At last, the outputs of FDE [x1(n),y1(n)] could be achieved from [x1(K),y1(K)]T by parallel–serial converter.
3.1. Equalization by FDE The equalization scheme in FDE was same as in Ref. [11]. As shown in Fig. 2, the structure of FDE consisted of a CD compensation stage and an adaptive FDE. In the CD equalization stage, the CD was compensated in the frequency-domain firstly. Then, the PMD and the residual CD would be equalized in FDE roughly. In the FDE, eight frequency-domain filters consisted of even
3.2. Error measurement of FDE After equalization in FDE, a measurement for error function would be executed. Those signals still with larger error would be picked out and equalized refined in the TDE. The constellations of equalized signals by FDE and TDE are shown in Fig. 3. As shown in
Fig. 1. Setup for 80 Gbps QPSK coherent system.
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Fig. 2. Structure of frequency domain equalizer.
Fig. 3. Constellation of signals after equalization in (a) FDE and (b)TDE.
Fig. 3, the constellation of FDE was more disperse from the normalized constant modulus circle than TDE, this would affect the remapping of these signals. As the poor performance in the FDE, the relatively larger error signals need to be picked out and equalized accurately in the time-domain. This work could be accomplished by the error measurement. The QPSK constellation with thresholds is shown in Fig. 4. In Fig. 4, the standard mapping points of QPSK signals are expressed as the solid points; the normalized constant modulus of CMA is represented as solid circle; the two thresholds in the different directions are denoted by two dot dashed circles. The distance between thresholds and constant modulus were presented in the solid line Lx,y which represented the maximum of equalizer tolerance error value of FDE. To confirm the range of Lx,y, two questions needed to be considered. In the condition of small Lx,y, much more large error signals could be picked out and equalized accurately. In other words, the small Lx,y could ensure the precision of equalization. But the excessively little Lx,y would lead to some normal signals be judged as large error signals, which could increase the computation complexity of time-domain equalization. So it was necessary for equalization to select a appropriate thresholds. The appropriate thresholds could improve the performance of system and reduce the remarkably increasing of computation complexity in the equalization. Therefore, Lx,y were chosen as m times of the mathematical expectation of error function value. They are written as
Fig. 4. Figure of constellation with thresholds of error function of FDE. N
L x = m⋅ ∑ εx(n) /N ,
N
L y = m⋅ ∑ εy(n) /N
n= 1
(3)
n= 1
where N presented the length of signals and εx(n) and εy(n) are the error function of FDE. Thus, The thresholds range Rx and Ry of two branches can be written as
R x ∈ (1 − L x , 1 + L x ) ,
(
R y ∈ 1 − L y, 1 + L y
)
(4)
As a result, if the absolute value of FDE error function crossed
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3.4. Computational complexity analysis Computational complexity was always calculated in terms of the required number of complex multiplications per bit. The computational complexity of TDE can be expressed as [11]
CTDE =
6P + 2 log2(M )
(8)
where the delay spacing is T/2 and the length of signal sequence is N, the tap length is P and M is the order of modulation. Besides, the computational complexity of FDE can be expressed as [11]
CFDE =
12log2(P ) + 10 log2(M )
(9)
In the FAT scheme, the computation complexity of calculating absolute εx, y(n) value in Eq. (3) had been computed in the complexity of FDE. Thus, the error measurement stage had not computation complexity. Above all, the computation complexity was the sum of FDE and TDE which can be expressed as
CFAT = CFDE + Fig. 5. Structure of second stage equalizer by fixed TDE.
the border of thresholds range in any directions, these signals would be supposed to bad performance signals. Then, the accurate equalization in TDE of these signals would be done. 3.3. Equalization by fixed TDE The structure of the equalizer by fixed TDE still applied the butterfly-structure of FIR filters which is presented in Fig. 5. The picked up signals in the error function measurement could be represented as [x(in),y(in)]T. Then, these signals would be split to even and odd sequence [xe(in),xo(in),ye(in),yo(in)]T and equalized accurately in time-domain. Other than conventional TDE, the tap weights of the fixed TDE was fixed value. The tap weights in frequency-domain can be represented by
Hx′,ey, o(n) = FFT⎡⎣hx′,ey, o(n); 0L ⎤⎦
(5)
Thus, the time-domain tap weights of fixed TDE equalizer were determined by:
k⋅CTDE N
(10)
where k represented the number of signals which need to be equalized accurately in TDE. According to the above equation, the difference between CFDE and CFAT was determined by k, the value of k was affected by Lx,y, then the choice of Lx,y was for m to decide. That is to say, the selection of m was the key factor to increment of computation complexity between FAT and FDE. Fig. 6 shows the comparison of computational complexity between FDE, TDE and FAT (m ¼3) with different tap lengths. From Fig. 6, we could notice that when tap length was 16 or more, the advantage of computation complexity for FDE and FAT could be reflected. The BER versus OSNR line between FDE, TDE and FAT is shown in Fig. 7. In the simulation, the CD was 16.75 ps/ns km; the PMD was 0.4 ps/km; the noise figure of EDFA was 6 dB and the length of transmission was 400 km. From Fig. 7, when m was less than or equal to 3, the BER of FAT method were much lower than TDE. Fig. 8 shows the comparison of computational complexity between FDE, TDE and FAT with different m when the tap length was 16. From Fig. 8, when m was equal or greater than 2.5, the computational complexity of FAT was lower than TDE. Thus, m was chosen to be 3 to FAT equalization method. Under this condition, there was a 6% decrease for computation complexity by FAT when
′′ e, o = first L element of IFFT ⎡⎣Hxx ′ e, o(N )⎤⎦ hxx ′′ e, o = first L element of IFFT ⎡⎣Hxy ′ e, o(N )⎤⎦ hxy ′′ e, o = first L element of IFFT ⎡⎣Hyx ′ e, o(N )⎤⎦ hyx ′′ e, o = first L element of IFFT ⎡⎣Hyy ′ e, o(N )⎤⎦ hyy
(6)
where the length of tap weight is P, N is the signal length, H′e, (N) was the last tap weights of FDE sub-equalizer. According to the theory of CMA, H′e,o(N) could reflect the most accuracy information of channel. Thus, the output of fixed TDE can be written as
o
x2(in) = h′′ e xx⋅x e (in) + h′′ o xx⋅x o(in) + h′′ e xy⋅y e (in) + h′′ o xy⋅y o (in) y2 (in) = h′ e yx⋅x e (in) + h′′ o yx⋅x o(in) + h′′ e yy⋅y e (in) + h′′ o yy⋅y o (in)
(7)
As a result, when output signals of FDE were chosen to make the accurate equalization, the final equalization outputs of fixed TDE method were the x2(in) or y2(in) and the corresponding output of FDE x1(n) or y1(n) would be abandoned yet.
Fig. 6. Computation complexity comparison of TDE, FDE and FAT (m¼3) with different tap lengths.
X. Ma et al. / Optics Communications 353 (2015) 63–69
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Fig. 7. BER comparison of TDE, FDE, FAT and ideal scheme with different m.
Fig. 9. Whole flow chart of FAT.
4. Simulation results and discussion
Fig. 8. Computation complexity of TDE, FDE and FAT with different m.
compared with TDE. Moreover, from Fig. 7, the proposed scheme performed better than TDE and FDE, with about 2 dB and 2.2 dB in OSNR improvement at the BER value of 1e 3, respectively.
3.5. Description of equalization procedure Above all, the whole flow chart of proposed high precision equalization scheme is shown in Fig. 9. From Fig. 9, the PMD and the residual CD would be equalized in FDE roughly and [x1(n), y1(n)]T were their outputs. Then, a measurement for error function would be executed. Firstly, two thresholds Rx,y would be calculated by error function εx,y. Secondly, the original input of picked up signals would be named as [x(in),y(in)]T and equalized accurately in the TDE. [x2(in),y2(in)]T were outputs of TDE. When signals was opt for equalized in TDE, the corresponding outputs in FDE [x1(n), y1(n)]T would be replaced by [x2(in),y2(in)]T meanwhile. Then, carrier recovery and data recovery were performed. Above all, the recovered bit sequence were compared to the transmitted sequence and BER could be measured.
To analyze the novel constant modulus equalizer, a 80 Gb/s PMQPSK simulation system was built using MATLAB and Optisystem. As shown in Fig. 1, the optical communication link was simulated in Optisystem, and the electric equalizer and other signal processing block was executed in MATALB. The OSNR was controlled by the loaded ASE source and variable optical attenuator (VOA). The transmission fiber link consisted of recirculating loops which has 80-km-long SSMF with dispersion parameter of 16.75 ps/ns km, non-linearity with self-phase modulation, and erbium-doped fiber amplifier (EDFA) with 6-dB noise figure without optical dispersion compensation. The PMD was imposed in 0.4 ps/km and the length of transmission was 400 km. In the signal processing stage, the CD was compensated in frequency-domain, and the residual CD and PMD were equalized in the electric equalizer through FAT. In the carrier recovery stage, the frequency offset compensation was made by quadruplicate method [12] and phase recovery was executed by Viterbi–Viterbi (V–V) algorithm [13]. Under the condition of 16 dB OSNR, the equalized and carrier recovered constellations of signals are shown in Figs. 10 and 11, respectively. The BER versus OSNR line between FDE, FAT and ideal scheme is shown in Fig. 12. The tap length of equalizer was set as 16 and the m of FAT was 3. As shown in Figs. 10–12, it was apt to notice that FAT could give more precise equalization than FDE. Other than analyses of OSNR and equalization performance, there were some other criteria could influence the equalization performance of proposed algorithm. Firstly, the uppermost function of adaptive equalizer was the compensation of PMD. The BER versus PMD line is shown in Fig. 13, where the OSNR was set in 16 dB and 17 dB and the residual CD was 500 ps. From Fig. 13, during the various PMD from 20 ps to 140 ps, BER of the FDE and FAT were steady. This meant both FDE and FAT had a good tolerance of PMD increasing. Secondly, although almost CD impairments could be compensated by CD compensator, there were still residual CDs in the
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Fig. 10. Constellation of signals after equalization in (a) FAT and (b) FDE.
Fig. 11. Constellation of signals after carrier recovery in (a) FAT and (b) FDE.
Fig. 12. BER versus OSNR line in Ideal scheme, FDE and FAT (m¼3). Fig. 13. BER versus PMD line in FDE and FAT (m¼ 3).
received signals of electric equalizer. These linear impairment could also affect the equalization performance. The BER versus residual CD line is shown in Fig. 14, where the OSNR was set in 16 dB and 17 dB and the total amount of PMD was 50 ps. From Fig. 14, the residual CDs were varied from 500 ps/nm to 1500 ps/ nm, the BER performance could be steady when residual CD was below 1000 ps/nm in both the methods.
5. Conclusion In this paper, a novel high precision adaptive equalizer was introduced and applied to the dynamic equalization for QPSK coherent optical communication system. The proposed method
X. Ma et al. / Optics Communications 353 (2015) 63–69
[2] [3]
[4] [5]
[6]
[7] [8] [9] [10]
Fig. 14. BER versus residual CD line in FDE and FAT (m¼ 3).
could enhance the performance of equalization, and then the BER of coherent communication system reduced with it. In addition, the computation complexity of the novel method was more lower than that of the traditional time-domain CMA equalizer.
References [1] Zhao Ling, Wang Yeling, Hu Guijun, Cui Yunpeng, Shi Jian, Li Li, Polarization
[11] [12] [13]
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demultiplexing by recursive least squares constant modulus algorithm based on QR decomposition, Opt. Laser Technol. 49 (2013) 251–255. Seb J. Savory, Digital filters for coherent optical receivers, Opt. Express 16 (2) (2008) 804–817. Yan Han, Inwoong Kim, Guifang Li, Experimental demonstration of coherent optical polarization multiple-input–multiple-output, Opt. Express 13 (19) (2005) 7527–7534. Guifang Li, Recent advances in coherent optical communication, Adv. Opt. Photonics 1 (2009) 297–307. Yangyang Fan, Xue Chen, Weiqin Zhou, Xian Zhou, Hai Zhu, The comparison of CMA and LMS equalization algorithms in optical coherent receivers, in: Proceedings of the WICOM, 2010. K. Kikuchi, Clock recovering characteristics of adaptive finite-impulse-response filters in digital coherent optical receivers, Opt. Express 19 (6) (2011) 5611–5619. B. Spinnler, Equalizer design and complexity for digital coherent receivers, IEEE J. Sel. Top. Quantum Electron. 16 (5) (2010) 1180–1192. K. Ishihara, et al., Frequency-domain equalization for optical transmission systems, Electron. Lett. 44 (2008) 870–871. E. Ferrara, Fast implementation of LMS adaptive filters, IEEE Trans. Acoust. Speech Signal Process. 28 (4) (1980) 474–475. Omid Zia-Chahabi, Raphael Le Bidan, Efficient frequency-domain implementation of block-LMS/CMA fractionally spaced equalization for coherent optical communications, IEEE Photonics Technol. Lett. 23 (22) (2011) 1697–1699. M.S. Faruk, K. Kikuchi, Adaptive frequency-domain equalization in digital coherent optical receivers, Opt. Express 19 (13) (2011) 12789–12798. A. Leven, N. Kaneda, U.-V. Koc, Y.-K. Chen, Frequency estimation in intradyne reception, IEEE Photonics Technol. Lett. 19 (6) (2007) 366–368. A.J. Viterbi, A.M. Viterbi, Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission, IEEE Trans. Inf. Theory IT-29 (4) (1983) 543–551.
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
A novel high precision adaptive equalizer in digital coherent optical receivers Xiurong Ma a,b,c,n, Yujun Xu a,b,c, Xiao Wang a,b,c, Zhaocai Ding a,b,c a
The Department of Computer and Communication Engineering, Tianjin University of Technology, Tianjin 300384, China Engineering Research Center of Communication Devices and Technology, Ministry of Education, Tianjin 300384, China c Tianjin Key Laboratory of Film Electronic and Communication Devices, Tianjin 300384, China b
art ic l e i nf o
a b s t r a c t
Article history: Received 10 January 2015 Received in revised form 29 April 2015 Accepted 2 May 2015 Available online 4 May 2015
A novel high precision adaptive equalization method is introduced and applied to dynamic equalization for quadrature phase shift keying (QPSK) coherent optical communication system in this paper. A frequency-domain constant modulus algorithm (CMA) method is used to equalize the received signal roughly. Then, some non-ideal output signals will be picked out through the error measurement, and they will be equalized accurately further in a fixed time-domain CMA equalizer. This high precision equalization method can decrease the equalization error, then it can reduce the bit error ratio (BER) of coherent communication system. Simulation results show that there is a 6% decrease for computation complexity by proposed scheme when compared with time-domain CMA. Furthermore, compared with time-domain CMA and frequency-domain CMA, about 2 dB and 2.2 dB in OSNR improvement can be obtained by proposed scheme at the BER value of 1e 3, respectively. & 2015 Elsevier B.V. All rights reserved.
Keywords: Coherent optical communication system Constant modulus algorithm Error measurement
1. Introduction Polarization diversity optical coherent receivers, which combine a polarization-division multiplexing (PDM) and an M-ary modulation formats, were a promising solution for high-capacity optical communication systems. A superiority of coherent receiver was the equalization of linear distortions and the demultiplexing of the two PDM signals could be done in the electronic equalizer. The butterfly-structure multiple-input/multiple-output adaptive electronic equalizer was usually employed to accomplish the equalization procedure [1–3]. The sampling rate of received signals was always set as twice symbol rate for the sake of reduction in aliasing effect [4]. Under the condition of twofold oversampling, time-domain equalizer (TDE) with constant modulus algorithm (CMA) was adapted two samples per step in order to adjust the initial sampling phase [5,6]. The butterfly equalizer in optical coherent communication system was implemented in the time-domain generally. However, it was recently shown in [7] that, frequency-domain equalizers (FDE) could bring significant computational savings in the adaptive equalization than TDE. In [7,8], FDE was realized by using pilotsymbol sequence, but such approaches generally decreased the n Corresponding author at: The Department of Computer and Communication Engineering, Tianjin University of Technology, Tianjin 300384, China E-mail addresses: [email protected] (X. Ma), [email protected] (Y. Xu).
http://dx.doi.org/10.1016/j.optcom.2015.05.001 0030-4018/& 2015 Elsevier B.V. All rights reserved.
spectral efficiency of transmission systems. Another attractive solution for FDE was equalization in frequency domain based on the overlap-save technique (OS-FDE) [9], this scheme was apt to implement and any cyclic prefix were not required. A fractionally spaced overlap-save FDE was proposed in [10]. This implementation was mathematically equivalent to the blockadaptive TDE, but offered substantial complexity reduction. What’s more, Ref. [11] proposed another FDE using CMA, it could work on the double oversampling rate by introducing even and odd subequalizers. As a result, the polarization demultiplexing and sampling-phase adjustment could achieve together in FDE. However, only the sampling phase adjustment and reduction of the computation complexity were considered both in Ref. [10] and Ref. [11], no obvious improvement was shown in FDE performance compared with the TDE. A novel high precision adaptive equalizer (FAT) was proposed in this paper, which consisted of a FDE, an error measurement stage and a TDE. Firstly, the linear impairments could be equalized roughly through the FDE in frequency-domain, the sampling phase also would be adjusted to best position simultaneously. Then some undesirable equalized signals would be picked out by error measurement. The thresholds of error measurement are for computation complexity and final performance to decide. At last, a fixed tap weights TDE was proposed and implemented, giving the more precise equalization to the picked out signals. Simulation results showed that there was a 6% decrease for computation complexity by FAT when compared with TDE. Moreover, a remarkable
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X. Ma et al. / Optics Communications 353 (2015) 63–69
decrease with 5.25 dB and 4.36 dB was founded for BER by the FAT compared with the FDE and the TDE.
2. System description The simulated QPSK coherent optical communication system is shown in Fig. 1. In the transmitter, two branches of QPSK signals were modulated by 40 Gb/s PRBS of length 215 1; the lasers of the transmitter and the local oscillator had a line-width which were both equal to 100 kHz. Then, two orthogonal branches were polarization-multiplexed by a polarization beam combiner (PBC) in transmission channel. In the SSMF channel, the setting of chromatic dispersion (CD) and first-order polarization-mode dispersion (PMD) was determined. Linear polarization controllers (PCs) were placed at 45° and 45° on both sides of SSMF. After detecting by the coherent receiver, the received signals were sampled by free-running analog-to-digital converters (ADCs) which operated at the twice symbol rate. Such oversampling significantly reduced the aliasing effect. Then, the sampled signal sequences were sent into the signal processing stage which consisted of an electric equalizer and a carrier recovery stage. At last, the signal sequence would be disposed in the data recovery stage to extract the original data and calculate the BER.
sub-equalizers and odd sub-equalizers. Four even sub-equalizers ‵e , Hxy ‵e , H‵yxe, H‵yye were connected in a two-by-two butterfly Hxx configuration. In the same way, four odd sub-equalizers ‵o , Hxy ‵o , H‵yxo, H‵yyo were placed in another two-by-two butterfly Hxx configuration. The input sequences were represented as [x(n),y (n)]T and the output signal sequence were represented as [x1(n), y1(n)]T. Before the process of equalizing, the input sequences [x(n), y(n)]T were divided into [xe(m),xo(m),ye(m),yo(m)]T by serial–parallel converter for sub-equalizers. Where, n ¼ 2m 1. After FFT transforming and overlapping, the input sequence of FDE could be expressed as [Xe(K),Xo(K),Ye(K),Yo(K)]T and the equalization output in frequency-domain could be expressed as [X1(K),Y1(K)]T. The equalization procedure was same as in Ref. [11]. The length of signal block was set as P, so the tap weights length of every subequalizer was 2P. Moreover, all the sub-equalizers were updated one symbol per step without down-sampling output sequence. The equalized signals can be expressed as [11] X ′(K ) = H′ e xx(K )⋅X e(K ) + H′ o xx(K )⋅X o(K ) + H′ e xy(K )⋅Y e(K ) + H′ o xy(K )⋅Y o(K ) Y ′(K ) = H′ e yx(K )⋅X e(K ) + H′ o yx(K )⋅X o(K ) + H′ e yy(K )⋅Y e(K ) + H′ o yy(K )⋅Y o(K )
where H′(K) represented the tap weights in frequency-domain. Thus, after IFFT transforming, the time-domain outputs are expressed by
( ) elements of IFFT (Y (K ))
x1(K ) = last p elements of IFFT X ′(K ) y1(K ) = last p
3. Electronic equalizer
(1)
′
(2) T
At last, the outputs of FDE [x1(n),y1(n)] could be achieved from [x1(K),y1(K)]T by parallel–serial converter.
3.1. Equalization by FDE The equalization scheme in FDE was same as in Ref. [11]. As shown in Fig. 2, the structure of FDE consisted of a CD compensation stage and an adaptive FDE. In the CD equalization stage, the CD was compensated in the frequency-domain firstly. Then, the PMD and the residual CD would be equalized in FDE roughly. In the FDE, eight frequency-domain filters consisted of even
3.2. Error measurement of FDE After equalization in FDE, a measurement for error function would be executed. Those signals still with larger error would be picked out and equalized refined in the TDE. The constellations of equalized signals by FDE and TDE are shown in Fig. 3. As shown in
Fig. 1. Setup for 80 Gbps QPSK coherent system.
X. Ma et al. / Optics Communications 353 (2015) 63–69
65
Fig. 2. Structure of frequency domain equalizer.
Fig. 3. Constellation of signals after equalization in (a) FDE and (b)TDE.
Fig. 3, the constellation of FDE was more disperse from the normalized constant modulus circle than TDE, this would affect the remapping of these signals. As the poor performance in the FDE, the relatively larger error signals need to be picked out and equalized accurately in the time-domain. This work could be accomplished by the error measurement. The QPSK constellation with thresholds is shown in Fig. 4. In Fig. 4, the standard mapping points of QPSK signals are expressed as the solid points; the normalized constant modulus of CMA is represented as solid circle; the two thresholds in the different directions are denoted by two dot dashed circles. The distance between thresholds and constant modulus were presented in the solid line Lx,y which represented the maximum of equalizer tolerance error value of FDE. To confirm the range of Lx,y, two questions needed to be considered. In the condition of small Lx,y, much more large error signals could be picked out and equalized accurately. In other words, the small Lx,y could ensure the precision of equalization. But the excessively little Lx,y would lead to some normal signals be judged as large error signals, which could increase the computation complexity of time-domain equalization. So it was necessary for equalization to select a appropriate thresholds. The appropriate thresholds could improve the performance of system and reduce the remarkably increasing of computation complexity in the equalization. Therefore, Lx,y were chosen as m times of the mathematical expectation of error function value. They are written as
Fig. 4. Figure of constellation with thresholds of error function of FDE. N
L x = m⋅ ∑ εx(n) /N ,
N
L y = m⋅ ∑ εy(n) /N
n= 1
(3)
n= 1
where N presented the length of signals and εx(n) and εy(n) are the error function of FDE. Thus, The thresholds range Rx and Ry of two branches can be written as
R x ∈ (1 − L x , 1 + L x ) ,
(
R y ∈ 1 − L y, 1 + L y
)
(4)
As a result, if the absolute value of FDE error function crossed
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3.4. Computational complexity analysis Computational complexity was always calculated in terms of the required number of complex multiplications per bit. The computational complexity of TDE can be expressed as [11]
CTDE =
6P + 2 log2(M )
(8)
where the delay spacing is T/2 and the length of signal sequence is N, the tap length is P and M is the order of modulation. Besides, the computational complexity of FDE can be expressed as [11]
CFDE =
12log2(P ) + 10 log2(M )
(9)
In the FAT scheme, the computation complexity of calculating absolute εx, y(n) value in Eq. (3) had been computed in the complexity of FDE. Thus, the error measurement stage had not computation complexity. Above all, the computation complexity was the sum of FDE and TDE which can be expressed as
CFAT = CFDE + Fig. 5. Structure of second stage equalizer by fixed TDE.
the border of thresholds range in any directions, these signals would be supposed to bad performance signals. Then, the accurate equalization in TDE of these signals would be done. 3.3. Equalization by fixed TDE The structure of the equalizer by fixed TDE still applied the butterfly-structure of FIR filters which is presented in Fig. 5. The picked up signals in the error function measurement could be represented as [x(in),y(in)]T. Then, these signals would be split to even and odd sequence [xe(in),xo(in),ye(in),yo(in)]T and equalized accurately in time-domain. Other than conventional TDE, the tap weights of the fixed TDE was fixed value. The tap weights in frequency-domain can be represented by
Hx′,ey, o(n) = FFT⎡⎣hx′,ey, o(n); 0L ⎤⎦
(5)
Thus, the time-domain tap weights of fixed TDE equalizer were determined by:
k⋅CTDE N
(10)
where k represented the number of signals which need to be equalized accurately in TDE. According to the above equation, the difference between CFDE and CFAT was determined by k, the value of k was affected by Lx,y, then the choice of Lx,y was for m to decide. That is to say, the selection of m was the key factor to increment of computation complexity between FAT and FDE. Fig. 6 shows the comparison of computational complexity between FDE, TDE and FAT (m ¼3) with different tap lengths. From Fig. 6, we could notice that when tap length was 16 or more, the advantage of computation complexity for FDE and FAT could be reflected. The BER versus OSNR line between FDE, TDE and FAT is shown in Fig. 7. In the simulation, the CD was 16.75 ps/ns km; the PMD was 0.4 ps/km; the noise figure of EDFA was 6 dB and the length of transmission was 400 km. From Fig. 7, when m was less than or equal to 3, the BER of FAT method were much lower than TDE. Fig. 8 shows the comparison of computational complexity between FDE, TDE and FAT with different m when the tap length was 16. From Fig. 8, when m was equal or greater than 2.5, the computational complexity of FAT was lower than TDE. Thus, m was chosen to be 3 to FAT equalization method. Under this condition, there was a 6% decrease for computation complexity by FAT when
′′ e, o = first L element of IFFT ⎡⎣Hxx ′ e, o(N )⎤⎦ hxx ′′ e, o = first L element of IFFT ⎡⎣Hxy ′ e, o(N )⎤⎦ hxy ′′ e, o = first L element of IFFT ⎡⎣Hyx ′ e, o(N )⎤⎦ hyx ′′ e, o = first L element of IFFT ⎡⎣Hyy ′ e, o(N )⎤⎦ hyy
(6)
where the length of tap weight is P, N is the signal length, H′e, (N) was the last tap weights of FDE sub-equalizer. According to the theory of CMA, H′e,o(N) could reflect the most accuracy information of channel. Thus, the output of fixed TDE can be written as
o
x2(in) = h′′ e xx⋅x e (in) + h′′ o xx⋅x o(in) + h′′ e xy⋅y e (in) + h′′ o xy⋅y o (in) y2 (in) = h′ e yx⋅x e (in) + h′′ o yx⋅x o(in) + h′′ e yy⋅y e (in) + h′′ o yy⋅y o (in)
(7)
As a result, when output signals of FDE were chosen to make the accurate equalization, the final equalization outputs of fixed TDE method were the x2(in) or y2(in) and the corresponding output of FDE x1(n) or y1(n) would be abandoned yet.
Fig. 6. Computation complexity comparison of TDE, FDE and FAT (m¼3) with different tap lengths.
X. Ma et al. / Optics Communications 353 (2015) 63–69
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Fig. 7. BER comparison of TDE, FDE, FAT and ideal scheme with different m.
Fig. 9. Whole flow chart of FAT.
4. Simulation results and discussion
Fig. 8. Computation complexity of TDE, FDE and FAT with different m.
compared with TDE. Moreover, from Fig. 7, the proposed scheme performed better than TDE and FDE, with about 2 dB and 2.2 dB in OSNR improvement at the BER value of 1e 3, respectively.
3.5. Description of equalization procedure Above all, the whole flow chart of proposed high precision equalization scheme is shown in Fig. 9. From Fig. 9, the PMD and the residual CD would be equalized in FDE roughly and [x1(n), y1(n)]T were their outputs. Then, a measurement for error function would be executed. Firstly, two thresholds Rx,y would be calculated by error function εx,y. Secondly, the original input of picked up signals would be named as [x(in),y(in)]T and equalized accurately in the TDE. [x2(in),y2(in)]T were outputs of TDE. When signals was opt for equalized in TDE, the corresponding outputs in FDE [x1(n), y1(n)]T would be replaced by [x2(in),y2(in)]T meanwhile. Then, carrier recovery and data recovery were performed. Above all, the recovered bit sequence were compared to the transmitted sequence and BER could be measured.
To analyze the novel constant modulus equalizer, a 80 Gb/s PMQPSK simulation system was built using MATLAB and Optisystem. As shown in Fig. 1, the optical communication link was simulated in Optisystem, and the electric equalizer and other signal processing block was executed in MATALB. The OSNR was controlled by the loaded ASE source and variable optical attenuator (VOA). The transmission fiber link consisted of recirculating loops which has 80-km-long SSMF with dispersion parameter of 16.75 ps/ns km, non-linearity with self-phase modulation, and erbium-doped fiber amplifier (EDFA) with 6-dB noise figure without optical dispersion compensation. The PMD was imposed in 0.4 ps/km and the length of transmission was 400 km. In the signal processing stage, the CD was compensated in frequency-domain, and the residual CD and PMD were equalized in the electric equalizer through FAT. In the carrier recovery stage, the frequency offset compensation was made by quadruplicate method [12] and phase recovery was executed by Viterbi–Viterbi (V–V) algorithm [13]. Under the condition of 16 dB OSNR, the equalized and carrier recovered constellations of signals are shown in Figs. 10 and 11, respectively. The BER versus OSNR line between FDE, FAT and ideal scheme is shown in Fig. 12. The tap length of equalizer was set as 16 and the m of FAT was 3. As shown in Figs. 10–12, it was apt to notice that FAT could give more precise equalization than FDE. Other than analyses of OSNR and equalization performance, there were some other criteria could influence the equalization performance of proposed algorithm. Firstly, the uppermost function of adaptive equalizer was the compensation of PMD. The BER versus PMD line is shown in Fig. 13, where the OSNR was set in 16 dB and 17 dB and the residual CD was 500 ps. From Fig. 13, during the various PMD from 20 ps to 140 ps, BER of the FDE and FAT were steady. This meant both FDE and FAT had a good tolerance of PMD increasing. Secondly, although almost CD impairments could be compensated by CD compensator, there were still residual CDs in the
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Fig. 10. Constellation of signals after equalization in (a) FAT and (b) FDE.
Fig. 11. Constellation of signals after carrier recovery in (a) FAT and (b) FDE.
Fig. 12. BER versus OSNR line in Ideal scheme, FDE and FAT (m¼3). Fig. 13. BER versus PMD line in FDE and FAT (m¼ 3).
received signals of electric equalizer. These linear impairment could also affect the equalization performance. The BER versus residual CD line is shown in Fig. 14, where the OSNR was set in 16 dB and 17 dB and the total amount of PMD was 50 ps. From Fig. 14, the residual CDs were varied from 500 ps/nm to 1500 ps/ nm, the BER performance could be steady when residual CD was below 1000 ps/nm in both the methods.
5. Conclusion In this paper, a novel high precision adaptive equalizer was introduced and applied to the dynamic equalization for QPSK coherent optical communication system. The proposed method
X. Ma et al. / Optics Communications 353 (2015) 63–69
[2] [3]
[4] [5]
[6]
[7] [8] [9] [10]
Fig. 14. BER versus residual CD line in FDE and FAT (m¼ 3).
could enhance the performance of equalization, and then the BER of coherent communication system reduced with it. In addition, the computation complexity of the novel method was more lower than that of the traditional time-domain CMA equalizer.
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