电磁干扰要素的数学表征及其方法研究

Abstract：The high-dimensional, strongly non-linear correlation and non-stationary characteristics of electromagnetic interference factors (EIFs) in modern electronic equipments lead to the unavailability of existing measurement and mathematical simulation methods. In this proposal, we will focus on mathematical modeling methods, i.e., mathematical representation, of the high-dimensional, strongly non-linear and non-stationary electromagnetic interference factors, especially in the background of national information systems. With the summarization of previous approaches, we believe it is a good solution for modeling EIFs to formulate the mathematical representation task in the following three steps: signal separation, signal decomposition and typical EIFs (or EIF basis) construction. By signal separation, source EIFs could be recovered from the correlated mixture signals, which could be further decomposed into a set of basic signals through signal decomposition. By removing the redundancy and noise of the signals set, we finally could construct a finite approximately complete EIFs basis, which then could be directly used for EIF modeling. The contents of the research of this proposal include: a blind source separation based signal separation that can address the strongly non-linear correlation; an improved wavelet analysis based signal decomposition that could address the non-stationary; a high dimensional clustering and low-rank decomposition based optimization method that could remove the noise and redundancy of a EIFs set. Our final aim is to propose a systematic mathematical representation framework for EIFs, fill the blank of the mathematical theory in the research on EIFs, and provide theoretic support for system design, prediction and assessment techniques of electromagnetic comparability.

现代电子设备中的电磁干扰要素呈现高维、强非线性和非平稳特征，导致对电磁干扰要素分析的精度、准确度都存在较大偏差和不确定性，现有的数学仿真模型不适用。因此本课题以国家信息系统中的电子系统为背景，研究高维、强非线性和非平稳电磁干扰要素的数学表征及方法：首先基于盲源分离等理论，对强非线性混合信号开展分离方法研究；然后基于小波分析等方法，研究非平稳电磁源信号的分解方法，得到电磁干扰要素基函数，初步构建相关基函数系；通过研究高维聚类分析、低秩分解等方法，对基函数系进行优化，构建有限近似完备基函数库。最终建立一套电磁干扰要素的数学表征理论，提出电磁干扰要素的数学表征方法，弥补国内外电磁干扰要素数学理论方面研究的空白，为我国电磁兼容总体设计技术、预测技术和评估技术提供相关的数学理论支持和方法，达到国内领先、国际先进水平。

本项目对电磁干扰要素的数学表征及其方法进行了深入研究，取得了较大进展。提出了电磁干扰要素的频谱特征；研究了高维强非线性电磁干扰要素的信号去噪、分离、分解、聚类及分类方法，并对其表征基函数构建和优化方法进行了探索；对相关数学表征理论进行了研究，提出了基于特殊矩阵计算的高效数学表征方法，增强方法的实用性和计算稳定性。项目执行期间共发表论文24篇，其中在Applied Mathematics and Computation, Journal of Computational and Applied Mathematics等国际权威学术期刊上发表SCI检索论文11篇，在国际学术会议上发表论文6篇，申请专利2项。代表性成果有：. 1. 研究电磁干扰要素的频谱特性和干扰信息的整体特性，构建了适用于多种典型电磁干扰要素的局部和全局特征，增强了现有电磁干扰要素表征的可区分性。. 2. 提出了基于经验模态分解和智能算法的强非线性电磁干扰要素的混合信号分离方法：不同于以往的方法对输入信号的线性混合和多通道假设，该方法利用经验模态分解可处理单通道信号，并通过量子遗传等智能算法进行优化，避免陷入局部最优。提出了基于B样条、可调三次三角Cardinal样条插值经验模态分解的电磁信号分解方法，解决了过冲和欠冲问题，分解精度有较大提升。. 3. 提出了基于人工免疫演化等智能算法的高维强非线性非平稳电磁干扰要素的聚类和分类方法。在此基础上，提出了基于Toeplitz矩阵表征和奇异值分解的电磁干扰要素基函数空间构造及重构方法。. 4. 提出了基于几类特殊矩阵计算理论的电磁干扰要素高效数学表征方法：考虑到电磁干扰要素表征中的诸多问题可归结为Toeplitz矩阵、三对角矩阵、稀疏矩阵等特殊矩阵的计算问题，因此利用这些矩阵的特殊代数结构，基于共轭梯度法、谱分析、预处理矩阵等设计了高效的矩阵最小二乘求解、线性方程组求解算法，提高了算法的收敛速度，增强了电磁干扰要素的数学表征方法的实用性和计算稳定性。. 本项目按计划完成研究内容，构建了可应用于复杂电磁环境下电磁干扰要素的数学表征理论及算法，完成了论文发表。期间共培养博士研究生7名，硕士研究生17名。