A Joint Frequency Offset and Phase Estimation Scheme Based on Cascaded EKF and LKF
1 引言
相干检测技术和数字信号处理(DSP)技术的结合广泛应用于现代光通信系统中。在相干传输系统中,载波频偏和载波相位噪声是相干传输系统中的主要损伤,DSP技术用于估计和补偿信号与本地振荡器之间的频偏(FO)以及载波相位噪声(PN)[1-3]。目前,已经提出频偏恢复算法[4-6]和载波相位恢复算法[7-9],其中常用的方法中,Viterbi-Viterbi相位估计算法(VVPE)仅适用于相移键控(PSK)调制格式的系统,不能有效地应用于正交振幅调制(QAM)信号;盲相位搜索(BPS)计算复杂度高。相比较而言,已经提出的卡尔曼滤波算法具有收敛速度快、计算复杂度低等优点。文献[
10]中提出基于并行运算的块状估计线性卡尔曼算法的载波频偏和相位的联合估计方案,有效地解决了因残余频偏而导致载波相位估计不够准确的问题。但其中块状处理的线性卡尔曼滤波对频偏估计范围小,需要提出一种简单有效的频偏初估方法快速地给出频偏范围,再利用线性卡尔曼滤波精确估计出残余频偏,最后进行精确的载波恢复。
本文将扩展卡尔曼滤波(EKF)与块状处理的线性卡尔曼滤波级联,应用于16QMA信号系统中实现频偏和载波相位的同时估计,解决了块状处理的线性卡尔曼滤波中频偏容忍度较低的问题。通过仿真分析了不同长度的块状数据和调优参数Q对估计性能的影响,同时给出此方案的线宽容忍度、频偏估计范围以及频偏漂移追踪。通过对实验数据的处理给出正交相移键控(QPSK)信号系统下的载波恢复性能。
2 工作原理
2.1 扩展卡尔曼滤波原理
首先利用EKF进行系统频偏的初估,状态参量为频偏fk。其测量方程为
式中Zk表示k时刻经过频偏干扰后观测到的信号,rk为k时刻接收到的信号,vk为观测噪声。
通过直接判决方式找到与其欧式距离最近的星座点,并且计算此次更新的测量余量dz。其基本过程如下:
式中左上角负号“—”代表预估计值,Q为过程噪声协方差,Pk表示后验协方差[10],R为观测噪声协方差,H为雅可比矩阵,可以表示为H=
,Kk为卡尔曼增益[11-12]。
2.2 块状处理的线性卡尔曼滤波
首先,把接收信号均分为长度为L的数据块。则第k个模块中L个符号的载波相位可以表示为
式中θk为第k个模块中点处符号的载波相位。基于块状卡尔曼滤波器的状态参量为Φk=[θk,fk]T,其状态更新方程和先验协方差方程可分别表示为
式中M为状态转移矩阵,M=
;Q代表过程噪声协方差。利用其测量方程对信号进行恢复后,通过直接判决找到与恢复信号欧式距离最近的理想星座点。最后利用极大似然函数来计算此时频偏和相位的测量残差,极大似然函数的基本表达式为
式中Re(·)代表取实部,r1表示接收到的信号,d1表示经过判决后的理想星座点,*表示取共轭。测量余量εk=[δθk,δfk]T,其具体算法见文献[
13]。
最后,利用测量余量来指导线性卡尔曼算法的下一个时刻的更新,基本方程为
提出的级联线性卡尔曼和扩展卡尔曼滤波器协同估计频偏和相位的方案如图1所示,图1中Z-1是指观测量(观测到的信号)的前一状态。
盲相位搜索算法用于相位恢复时不仅需要B个测试相位(或测试频率)对每个数据进行相位(频率)估计,还需要对这些测试相位(频率)进行判决以获得最佳估计值。相比盲相位搜索算法,块状处理卡尔曼算法大大减少了计算量。卡尔曼算法的调优参数Q用于决定其收敛速度和估计精度,下文将讨论块状数据的长度不同时所对应的最佳调优Q参量。
3 仿真与分析
利用仿真系统产生波特率为12 GBaud(Baud为波特率的单位,波特率指数据信号对载波的调制速率,它用单位时间内载波调制状态改变次数来表示)的16QAM信号,光信噪比(OSNR,ROSN)固定为17.6 dB。首先,设定激光线宽为100 kHz,频偏为1 GHz,分析不同块状数据的长度和Q参数下的滤波器性能。图2中给出不同块状数据的长度和不同Q参数下误码率(BER,RBE)的等高
图 1. EKF和LKF级联的算法框图
Fig. 1. Algorithm diagram of cascaded EKF and LKF
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图 2. 不同块状数据的长度和Q参数下的BER
Fig. 2. BER versus different block lengths and parameter Q
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线,图中横坐标是数据块的长度,纵坐标对应不同Q参数,BER反映在图中的曲线上。当数据块的长度b分别为8,16,32,64时,对应的最优Q参数分别为10-8,10-6,10-5,10-4,结果表明不同的块状数据长度对应不同的最优Q参数。并且在以上所有情况下误码率均不大于10-3,其中包括数据块长度较大的情况。这说明分块处理的数据块的长度取值较大时系统依然有较好的工作性能。
图3(a)中给出频偏固定在1 GHz时的不同线宽下BER 情况,可以看到在相位噪声为500 kHz范围内BER均在10-3附近,能够准确恢复出信号。图3(b)中给出线宽固定为100 kHz时算法对频偏的容忍度,可以看到,频偏从10 MHz变化到1.5 GHz时,BER基本无较大的波动,可以得出所提方案的频偏估计范围可以达到1.5 GHz,对频偏的容忍度较高。
实际情况中频偏会随时间变化而变化。仿真中频偏的变化则由线性变化的频偏漂移和初始的频偏相加组成[14]。图4中给出此方案对频偏漂移的追踪情况,频偏初始值为1 GHz,符号长度为215,波特率为12 Gbaud,即设置的数据传输速率为96 Gb/s。图4(a)中当频偏漂移速度为40 MHz/μs时能够成功追踪到频偏变化;图4(b)中给出频偏漂移速度达到320 MHz/μs时的频偏估计性能,能够看出此方案可精确快速地追踪到变化着的频偏。图5中给出此方案的频偏漂移容忍度,最大的频偏漂移可达到320 MHz/μs。
图 3. (a)在频偏为1 GHz时BER与线宽关系;(b)在线宽为100 kHz时BER与频偏关系
Fig. 3. (a) BER versus linewidth when frequency offset of 1 GHz; (b) BER versus FO when linewidth of 100 kHz
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图 4. 不同频偏漂移追踪情况。(a)频偏漂移速度为40 MHz/μs;(b)频偏漂移速度为320 MHz/μs
Fig. 4. Case of tracking different FO drifts. (a) FO drift velocity of 40 MHz/μs; (b) FO drift velocity of 320 MHz/μs
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图 5. BER与不同频偏漂移速度的关系
Fig. 5. BER versus frequency offset drift velocity
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4 实验分析
图6中给出12 Gbaud下的QPSK信号传输模型,激光线宽设置为100 kHz。光源(LD)经过偏振分束器(PBS)分为x和y偏振,在发射端将QPSK信号调制到激光的x和y偏振上,再通过偏振合束器(PBC)合束,并经过掺铒光纤放大器(EDFA)放大信号后背靠背传输。在接收端经带通滤波器(BPF)滤波后除去带外噪声,然后将光载波信号与本地振荡器(LO)产生光波进行相干混频,并利用光电探测器将光信号转变为电信号再通过模数转换器(ADC)转为数字信号,最后经过DSP进行相应的损耗补偿[12,15]。对于产生的12 Gbaud的QPSK信号,进行频偏估计性能分析,实验中设置4组不同频偏,并且真实频偏由快速傅里叶变换(FFT)算法给出。在OSNR为10 dB时,由FFT算法给出的4组真实频偏为0.0431,0.5457,1.0976,1.4916 GHz;在OSNR为16 dB时,4组真实频偏为0.1621,0.6158,1.1037,1.2998 GHz。利用此方案同时估计频偏和相位,频偏估计的结果与利用FFT进行频偏估计作为真实值的结果相比较。图7中给出频偏估计性能,图7(a)中给出在OSNR为10 dB时,LKF中不同数据块长度及Q参数下的频偏估计性能比较:进行了3种数据块长度下频偏估计的测试,分别为b=16,30,60,对应的Q参数为10-6,10-5,10-4。从图7中可以看出,3种情况下的频偏估计基本重合,这说明不同块状数据的长度在相对应的优化后的Q参数下,频偏估计性能基本不变。图7(b)是在OSNR分
图 6. 12 Gbaud双偏振QPSK光通信系统
Fig. 6. 12 Gbaud dual polarization QPSK signal system
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图 7. 频偏追踪。(a)不同块状数据长度下的频偏估计;(b)不同OSNR下的频偏估计
Fig. 7. Frequency offset tracking. (a) Frequency offset estimation with different block lengths; (b) frequency offset estimation with different OSNRs
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别为10 dB和16 dB时的频偏估计情况,可以看到2种情况都能准确追踪到频偏,且频偏的估计范围均较高。
5 结论
通过级联EKF和LKF成功地对载波信号的频偏和相位进行联合估计,解决了块状处理卡尔曼滤波对频偏估计容忍度较低的缺陷。该方案具有快速载波估计收敛能力和较高的估计精度,且频偏估计范围较大,能够达到符号速率的12.5%。另外,针对实际频偏会发生漂移的状况,通过仿真分析了对频偏漂移的追踪性能,结果显示,对频偏漂移的追踪速度可达到320 MHz/μs。算法对数据的块状处理使得计算复杂度低,缩短了计算时间,具有较高的实际应用性。最后利用实验对QPSK信号的频偏估计验证了该方案具有高估计精度、高频偏容忍度的性能。
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