Development in Signal Processing Based on Correntropy and Cyclic Correntropy
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摘要:
在无线电监测和目标定位等应用中，接收信号经常会受到脉冲噪声和同频带干扰等复杂电磁环境的影响，传统的基于2阶统计量的信号处理方法往往不能正常工作，基于分数低阶统计量的信号处理方法也由于对信号噪声统计先验知识的依赖性而遇到困难。近年来提出并受到信号处理领域普遍关注的相关熵和循环相关熵信号处理理论与方法，是解决复杂电磁环境下信号分析处理、参数估计、目标定位和其他应用问题的有效技术手段，有力促进了非高斯、非平稳信号处理理论方法和应用的发展。该文系统性地综述了相关熵和循环相关熵信号处理的基本理论和基本方法，包括相关熵与循环相关熵的起源背景、定义概念、性质特点，以及所包含的数学物理意义。该文还介绍了相关熵与循环相关熵信号处理在多个领域的应用问题，希望对非高斯、非平稳统计信号处理的研究和应用有所裨益。
Abstract:In radio monitoring and target location applications, the received signals are often affected by complex electromagnetic environment, such as impulsive noise and cochannel interference. Traditional signal processing methods based on second-order statistics often fail to work properly. The signal processing methods based on fractional lower order statistics also encounter difficulties due to their dependence on prior knowledge of signals and noises. In recent years, the theory and method of correntropy and cyclic correntropy signal processing, which are widely concerned in the field of signal processing, are put forward. They are effective technical means to solve the problems of signal analysis and processing, parameter estimation, target location and other applications to complex electromagnetic environment. They promote greatly the development of the theory and application of non-Gaussian and non-stationary signal processing. This paper reviews systematically the basic theory and methods of correntropy and cyclic correntropy signal processing, including the background, definition, properties and characteristics of correntropy and cyclic correntropy, as well as their mathematical and physical meanings. This paper introduces also the applications of correntropy and cyclic correntropy signal processing to many fields, hoping to benefit the research and application of non-Gaussian and non-stationary statistical signal processing.
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Key words:
- Signal processing /
- Correntropy /
- Cyclic correntropy /
- Non-Gaussian /
- Non-stationary
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1. 引言
雷电是一种发生在大气中的天气现象，发生时会产生瞬态大电流、高电压和强电磁脉冲辐射等[1]。根据卫星监测和气象部门的数据统计，全球每秒大约有2000 多个雷电发生[2]。近年来，由于电子、通信设备发展迅速，导致雷电灾害由以往的森林火灾和人员伤亡形式逐渐转向对电子设备以及大型通讯网络设备的干扰与破坏，对军事、铁路、航天航空等领域造成严重影响。因此开展雷电探测研究，提供雷电预警，对防雷减灾的工作具有十分重要的指导意义[3-6]。直至今日，人类对雷电探测技术的研究已经发展了200多年，从20世纪20年代起，美国开始研究利用磁环天线来对雷电位置进行精确探测，成为雷电探测技术发展史上的一个里程碑，代表了雷电探测技术逐渐走向成熟[7-11]。
目前雷电定位系统(Lightning Location System, LLS)主要采用正交磁环天线(Orthogonal Magnetic Loop Antenna, OMLA)作为核心探测器件[12]。为提高测向精度，以往对OMLA的研究多注重于减小其测向角度误差，如对由雷电回击通道和地面不垂直造成的极化误差研究[13]、由雷电测向设备附近地形地势造成的场地误差研究[14]、由天线磁环非正交性造成的误差研究[15]以及OMLA自身结构误差的研究等[16]，而对采用新型磁环天线结构设计提高天线测向精度的研究则相对较少。随着LLS探测精度要求的不断提高，采用正交磁环天线探测雷电方向的测量精度达到瓶颈。具体而言，在排除外界条件造成天线的测角误差后，虽然可以通过引入修正矩阵来降低磁环的非正交性误差[15]，或引入补偿系数来矫正正交磁环天线的一致性误差[16]，在一定程度上提高测向精度，但经矫正后的正交磁环天线仍存在一定的结构误差角，导致测角误差随雷电方向角呈波浪形变化。即当雷电信号方向平行于其中一个磁环时，测角误差达到最大，最大值约等于天线结构误差角的值。因此研究如何设计新型磁环天线结构以减轻结构误差角对测角的影响，对于提升雷电测向精度具有一定的意义。
本文提出一种由3个磁环两两呈60°组成的新型三环天线，由于其特殊结构，可以有效降低结构误差角对测角的影响，提高天线测向精度。首先根据三磁环天线特殊结构推导了低频信号情况下三环天线的测角公式，建立了结构误差角与测角误差的仿真模型；然后通过实验比对三环天线与同尺寸正交磁环天线测角精度。本文最后部分给出分析结论。
2. 三磁环天线结构及测向原理
雷电与地面的回击通道可以被看作一个垂直的电偶极子，当地面为理想导体时，雷电电磁脉冲仅激发横磁波[17]。三磁环天线结构如图1所示，假设H为垂直地面的闪电回击通道，3个磁环两两呈60°组成天线来进行雷电源方位角的探测，中轴线O与回击通道H平行。
设HO平面(即雷电来向)与A环、B环、C环的夹角分别为
θ , 60°–θ , 120°–θ ，3个磁环受雷电电磁脉冲激发产生的自感电动势峰值分别为EA ,EB ,EC 。再假设磁环中电动势、电流沿逆时针为正方向，且磁环自感磁链的正方向和电流的正方向之间符合右手螺旋关系，根据法拉第电磁感应定律，A环感应电动势计算公式为EA=2πλBNSμrsinθ (1) 式中，λ为信号波长；B为信号磁感应强度；N为天线匝数；S为天线面积；
μr 为磁环的相对磁导率；θ 为雷电信号与A环的夹角，同理将B环、C环与雷电来向夹角代入式(1)可得其各自感应电动势EB ,EC 。考虑3个磁环间还存在互感电动势：3个磁环受雷电电磁脉冲激发产生自感电动势后，每个磁环内变化的感应电流会在其他两个磁环上激发互感电动势。设
VA ,VB ,VC 分别为磁环A, B, C两端实际电动势；eAB 为A环电流在B环激发的感应电动势，同理eBC ,eAC 分别为B环电流在C环激发的感应电动势以及A环电流在C环激发的感应电动势，则有VA=EA+eBA+eCAVB=EB+eAB+eCBVC=EC+eAC+eBC} (2) 考虑雷电信号一部分处于200 kHz以下的低频频段，在此情况下，磁环间互感电动势很小，因此3个磁环间互感可忽略不计。根据欧姆定律将3个磁环的感应电动势两两组合进行计算，分别可得3组雷电测向角度
θ1 ,θ2 ,θ3 θ1 = arctan(√3EAEA−2EB)θ2 = arctan(√3EAEA+2EC)θ3 = arctan(√3(EC−EB)EC+EB)} (3) 对求得的3个角度求算数平均值作为最终测向结果。此外由于三角函数的周期性特征，由式(3)所计算得到的方向角
θ 存在180°的二义性，可使用垂直极化的电场天线判断雷电信号极性，进一步确定方向角符号。3. 三磁环天线测角误差仿真
为验证三磁环天线结构误差角对测角误差的影响，建立测角误差关于结构误差角的仿真模型。结构误差角由天线磁环夹角不理想造成，理论上天线的3个磁环保持两两呈60°，但实际情况下由于加工、磁环扭曲等问题，使得磁环间存在等效结构误差角
σ 。首先推导只有单个磁环存在结构误差角的特殊情况时测角误差计算式，进而拓展到一般情况。如图2所示，设A环准确东西摆放，角度自西向东逆时针由–180°～180°递进。C环准确朝向120°，B环与理想方向60°之间的夹角σ 即为等效结构误差角。定义雷电来向与正东方向实际夹角为
θ ，通过式(3)得到的计算角度为θ′ 。根据几何关系，理想情况下B环感应电动势E′B 与实际情况下感应电动势EB 的比值为E′BEB=sin(60∘−θ)sin(60∘−(θ−σ))=K(σ,θ) (4) 根据A环、B环电压关系可进一步求得计算角度
θ′1 θ′1=arctan(√3E′AE′A−2E′B/2E′BK(σ,θ)K(σ,θ)) (5) 代入
E′A 消去E′B 即可化简得到计算角度θ′1 关于结构误差角σ 和实际角度θ 的函数关系θ′1=arctan(√3tanθ(1+√3sinσ−cosσ)tanθ+√3cosσ+sinσ) (6) 同理可得
θ′3 θ′3=arctan(−√3(√3sinσ−cosσ−1)tanθ + √3cosσ + sinσ−√3(√3sinσ−cosσ + 1)tanθ + √3cosσ + sinσ + √3) (7) 由于C环不存在结构误差角，所以
θ′2 =θ 。定义计算角度θ′ 与实际角度θ 的差为测角误差δ ，则有δ=θ′1+θ′3−2θ3 (8) 为更直观地体现结构误差角
σ 对测角误差δ 的影响，仿真测角误差δ 关于实际角度θ 的变化情况，如图3所示。由图3可知，在B环结构误差角度
σ 一定的情况下，测角误差δ 随实际角度θ 呈波浪形变化，当θ 在60°和–120°附近时测角误差最大，最大值约等于0.7倍的结构误差角σ 。进一步讨论一般情况，当B环、C环均存在结构误差角时，如图4所示，B环、C环与60°,120°的夹角分别为
σ1 ,σ2 ，规定当磁环沿逆时针偏离理想方向时，其结构误差角为正，反之为负。定义σ1 ,σ2 中绝对值较大的为三磁环天线结构误差角σ 。为便于分析，先设定σ1 ,σ2 大小相等。类似单磁环存在结构误差角时求解方法，可分别求得两种情况时测角误差
δ 与三磁环天线结构误差角σ 的函数关系，进一步仿真测角误差δ 关于实际角度θ 的变化情况，如图5所示。由图5可知，当三磁环天线结构误差角
σ 一定时，测角误差δ 总体呈波浪形变化趋势。分析图5(a)可知，当σ1 ,σ2 同号时，测角误差最大值出现在90°和–90°附近，最大值约为0.75σ 。当σ1 ,σ2 大小不相等时，保持σ1 值不变，改变σ2 的大小在0～σ1 之间变化，通过仿真可得，测角误差最大值在0.7σ1 ～0.75σ1 之间变化，当σ2 为0时，测角误差最大值为0.7σ1 。因此在σ1 ,σ2 同号的情况下，测角误差最大值为0.75σ 。分析图5(b)可知，当
σ1 ,σ2 异号时，测角误差最大值出现在±60°和±120°附近，最大值的绝对值约等于0.6σ 。当σ1 ,σ2 大小不相等时，保持σ1 值不变，改变σ2 的大小在0～σ1 变化，通过仿真可得，测角误差最大值在0.6σ1 ～0.7σ1 之间变化，当σ2 为0时，测角误差最大值为0.7σ1 。因此这种情况下三磁环天线测角误差最大值为0.7σ 。上述推导需满足在同等电磁场条件下，3个磁环的感应电动势峰值相同。然而在实际情况中由于制作工艺等原因会造成磁环的面积、阻抗不同，进而导致各磁环感应电动势峰值不同，由此引发的误差可称为一致性误差。一致性误差可通过引入补偿系数进行矫正，设三磁环感应电动势峰值分别为
VA ,VB ,VC ，选定A环感应电压值为标准值，对B环、C环电压值进行补偿，则B环补偿系数ηB 可表示为ηB=VBVA (9) 同理可得C环补偿系数
ηC 。在多个电磁场强度下，分别测量三磁环在相同角度时感应电动势，对磁环A,B和磁环A,C两组电压数据分别作拟合直线，两直线斜率即为B环和C环的补偿系数。通过上述方法能够基本矫正磁环的一致性误差。考虑正交磁环天线磁环间夹角90°与三磁环天线夹角60°比值为1.5，因此在同等结构误差角条件下，正交磁环天线结构误差角约为三磁环天线的1.5倍，即正交磁环天线结构误差角约为1.5
σ 。已知正交磁环天线测角误差最大值约等于其结构误差角[16]，仿真结果如图6所示。综上所述，不考虑一致性误差的影响，理论上采用三磁环天线进行雷电测向，测角误差能够降低约50%。4. 三磁环天线测向误差实验
以下通过对照实验对比三磁环天线与同尺寸正交磁环天线测角精度。实验选取半径为7.5 cm的磁环，根据关于雷电测向的正交磁环天线的接特性研究[18]，可确定该磁环天线在最佳接收效益时的线圈匝数为60匝。设计参数如表1所示的三磁环天线，实物如图7所示。
表 1 磁环天线参数磁环半径r(cm) 线圈宽度d(cm) 导线直径
Φ(mm)磁环匝数
(N)7.5 1.5 0.3 60 利用信号发生器产生200 kHz的正弦信号，示波器连接信号发射线圈，发射线圈匝数为30匝，半径为40 cm，面积远大于天线线圈。将天线固定在位于发射线圈正前方10 cm的转台上，通过转动天线来改变信号方向角。将天线3个磁环分别接入终端阻抗为1 MΩ、耦合方式为直流的MDO-3052型数字示波器的3个通道，记录天线各磁环的响应电压。
首先对三磁环天线的一致性误差进行检验和矫正，利用发射线圈从0°～90°间隔10°分别对3个磁环进行响应测试并记录3组电压值数据，采用第3节所述方法可求得B环、C环的补偿系数
ηB ,ηC 分别为1.03, 1.02。接着利用发射线圈从–180°～180°间隔10°对天线进行响应测试，将3个磁环的响应电压值通过补偿系数修正后代入式(3)求得方向角计算值
θ′ ，与实际方向角θ 相减得到测角误差δ 。用相同参数的磁环制作正交磁环天线，先对其一致性误差进行矫正，然后通过对照响应试验求得不同角度时正交磁环天线的测角误差δ′ 。实验结果表明，相比于正交磁环天线，三磁环天线测角误差明显降低，误差最大值由3.3°下降为1.19°，降低65%，测角误差平均值从1.67°下降为0.49°，降低70.6%，标准差由1.01°下降为0.32°，降低68.3%。为更直观体现优化程度，将两种天线测角误差取模并作多项式拟合，得出曲线图如图8所示。由图8可知，相比于正交磁环天线，采用三磁环天线对信号源进行方位角探测的测角误差显著降低，整体误差降低约50%。进一步分析，正交磁环天线测角误差峰值分布在 ±90°附近，误差峰值约为3°；三磁环天线的误差角度峰值分布在 ±120°和±60°附近，误差峰值约为1°，结合第3节仿真可知三磁环天线两结构误差角
σ1 ,σ2 为异号，测角误差变化趋势与仿真结果基本相符。5. 结论
本文分析了正交磁环天线由于结构误差角造成的测角误差较大的问题，提出一种改进的三磁环天线，该天线通过两两呈60°的磁环进行测向。通过建立测角误差仿真模型分析了三磁环天线测角误差优化效果，对比实验表明，在同等制作工艺下，相较于正交磁环天线，三磁环天线测角误差整体降低约50%，提高了测向精度。
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