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经过CMA均衡后的输出信号可能存在一定的相位误差,单独使用时,需要加入相位恢复模块。MCMA可以修正CMA造成的相位旋转,同时能在误比特率方面得到一定改善。本文作者在基于强耦合少模光纤的模分复用系统中比较了CMA和MCMA的均衡效果。接下来分别对两种算法的原理进行介绍。
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CMA的基本思想[17]如下:恒模信号通过理想光纤信道传输后仍具有恒模特性,但实际的光纤信道存在损耗、色散等损伤,需要使用算法更新均衡器的抽头系数,使信号重新具有恒模特性。
图 1所示是CMA的示意图[18]。图中,x(n)表示发送信号,h(n)表示传输信道的脉冲响应,n(n)表示加性高斯白噪声,y(n)表示经过信道传输后的接收信号,即均衡器的输入信号,w(n)表示均衡器的抽头系数,z(n)表示均衡器的输出信号,(n)表示判决器的输出信号。
CMA的代价函数为[19]:
$ J(n)=E\left[\left(|z(n)|^2-R_2\right)^2\right] $
(1) 式中:E[·]是期望函数;R2为CMA模值,是一个常数。R2可表示为:
$ R_2=\frac{E\left[|x(n)|^4\right]}{E\left[|x(n)|^2\right]} $
(2) 由式(1)可知,代价函数只包含了接收信号的幅度信息,没有包含相位信息。因此,发送信号通过复数信道后会出现相位偏移。
误差函数表示为:
$ e(n)=z(n)\left(|z(n)|^2-R_2\right) $
(3) 根据梯度下降法,可得均衡器w(n)的权向量迭代过程如下:
$ \begin{gathered} w(n+1)=w(n)-\mu \cdot \nabla J(n)= \\ w(n)-\mu \cdot z(n)\left(|z(n)|^2-R_2\right) \cdot y^*(n)= \\ w(n)-\mu \cdot e(n) \cdot y^*(n) \end{gathered} $
(4) 式中:μ表示迭代步长; y*(n)表示对y(n)进行共轭运算。CMA的迭代步长μ是常数,其大小决定算法的收敛速度。
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MCMA算法的原理是把式(1)的代价函数的实虚部分开,再分别计算[20]:
$ J(n)=\operatorname{Re}(J(n))+\operatorname{Im}(J(n)) $
(5) 其中:
$ \operatorname{Re}(J(n))=E\left[\left(|\operatorname{Re}(z(n))|^2-\operatorname{Re}\left(R_2\right)\right)^2\right] $
(6) $ \operatorname{Im}(J(n))=E\left[\left(|\operatorname{Im}(z(n))|^2-\operatorname{Im}\left(R_2\right)\right)^2\right] $
(7) 式中:Re(·)和Im(·)分别表示实部、虚部。Re(R2)、Im(R2)可以表示为:
$ \operatorname{Re}\left(R_2\right)=\frac{E\left[|\operatorname{Re}(x(n))|^4\right]}{E\left[|\operatorname{Re}(x(n))|^2\right]} $
(8) $ \operatorname{Im}\left(R_2\right)=\frac{E\left[|\operatorname{Im}(x(n))|^4\right]}{E\left[|\operatorname{Im}(x(n))|^2\right]} $
(9) 误差函数定义如下:
$ e(n)=\operatorname{Re}(e(n))+\operatorname{Im}(e(n)) $
(10) $ \begin{gathered} \operatorname{Re}(e(n))= \\ \operatorname{Re}(z(n))\left[|\operatorname{Re}(z(n))|^2-\operatorname{Re}\left(R_2\right)\right] \end{gathered} $
(11) $ \begin{gathered} \operatorname{Im}(e(n))= \\ \operatorname{Im}(z(n))\left[|\operatorname{Im}(z(n))|^2-\operatorname{Im}\left(R_2\right)\right] \end{gathered} $
(12) MCMA的代价函数同时包含信号的幅度和相位信息,能够有效修正信号经过复信道后引起的相位旋转。
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收敛速度和均方误差(mean square error,MSE)是盲均衡算法的两个主要性能指标[21],这两个指标都可以从MSE曲线得到。通常情况下,MSE被定义为期望数据和均衡后数据对应点误差的平方和的均值,用σ表示,表达式如下:
$ \sigma(n)=E\left[e^2(n)\right]=E\left\{[\hat{z}(n)-z(n)]^2\right\} $
(13) 在CMA和MCMA算法中,σ(n)一般定义为e(n)。
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为了分析CMA和MCMA的性能,利用VPI仿真软件,基于功率耦合理论搭建了一个3模模分复用系统,系统结构如图 2所示。在仿真中,6×56 Gbit/s的DP-QPSK信号通过模式复用器在具有LP01、LP11, a和LP11, b光纤模式的3模光纤中传输,3种模式之间只发生强耦合。该系统的传输距离为80 km,光信号在传输过程中还受到色度色散、偏振模色散和模式群时延的影响,具体参数值如表 1所示。最后,信号经过一个放大器和一个模式解复用器,接收机得到受损信号。在本次仿真中,激光器线宽被设置为100 kHz。
表 1 参数设置
Table 1. Parameter settings
parameter name parameter value bit rate 56 Gbit/s bit sequence length 32768 total fiber length 80 km fiber dispersion coefficient 20 ps/(nm·km) polarization mode dispesion 0.05 ps/(km)1/2 DMGD (LP11, a, LP11, b-LP01) 130 ps/km 对于传输后的受损信号需要进行一系列数字信号处理(digital signal processing,DSP),才能恢复原始信号,主要包括模数转换(analog-to-digital converter,ADC)、色散补偿(CD compensation)、MIMO均衡和载波相位估计(carrier phase estimation)。DSP的实现流程如图 3所示。其中,MIMO均衡是本文中的研究重点。
模分复用系统中盲均衡算法的均衡性能研究
Research on equalization performance of blind equalization algorithms in mode-division multiplexing system
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摘要: 恒模算法(CMA)是一种广泛应用于模分复用系统的算法, 可对系统中模式耦合、差分模式群时延和色散等损伤因素进行均衡补偿, 进而得到理想信号。为了研究CMA算法在强耦合模分复用系统中的均衡性能, 采用功率耦合理论搭建6×6模分复用系统模型, 并在接收端使用CMA和修正的恒定模数算法(MCMA)对系统输出信号进行均衡, 获得了星座图、均方根误差(RMSE)值和误比特率(BER)。结果表明, 在星座图方面, MCMA可以减少散点, 使星座点更紧凑; 在RMSE方面, MCMA均衡后的信号的RMSE值小于CMA均衡后得到的RMSE值, 说明MCMA均衡后的数据离散程度较低; 在BER方面, 当BER为10-3时, MCMA要求的光信噪比比CMA低1.0 dB, 因此, MCMA的均衡效果优于CMA。该研究结果为强耦合模分复用系统中的均衡算法提供了一些参考。Abstract: The constant modulus algorithm (CMA) is a popular algorithm for mode-division multiplexing systems to equalize and compensate for impairments such as mode coupling, differential mode group delay, and dispersion in the system to obtain the desired signal. In order to study the equalization performance of the CMA in the strong coupling mode-division multiplexing system, the power coupling theory was used to build a 6×6 mode-division multiplexing system model and use the CMA and modified constant modulus algorithm (MCMA) at the receiving end to equalize the system output signal and obtain the constellation diagrams, root mean square error (RMSE) values and bit error rate (BER). The results show that in terms of the constellation diagram, MCMA can reduce scatter points and make constellation points more compact; in terms of RMSE, the RMSE value of the signal after MCMA equalization is smaller than the RMSE obtained after CMA equalization, indicating that the data dispersion level after MCMA equalization is low; in terms of BER, when BER is 10-3, the optical signal-to-noise ratio required by MCMA is 1.0 dB lower than that of CMA, therefore, MCMA equalization outperforms CMA. The results of this study provide some references for the equalization algorithm in the strong coupling mode-division multiplexing system.
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Key words:
- optical communication /
- blind equalization /
- mean square error /
- bit error rate
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表 1 参数设置
Table 1. Parameter settings
parameter name parameter value bit rate 56 Gbit/s bit sequence length 32768 total fiber length 80 km fiber dispersion coefficient 20 ps/(nm·km) polarization mode dispesion 0.05 ps/(km)1/2 DMGD (LP11, a, LP11, b-LP01) 130 ps/km -
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