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摘要: 现实世界中, 所获得的信号大部分都是非平稳和非线性的, 将此类复杂信号分解为多个简单的子信号是重要的信号处理方法. 1998年, 提出希尔伯特–黄变换(Hilbert-Huang transform, HHT)以来, 历经20余年的发展, 信号分解已经成为信号处理领域相对独立又具有创新性的重要内容. 特别是近10年, 多元/多变量/多通道信号分解理论方法方兴未艾, 在诸多领域得到了成功应用, 但目前尚未见到相关综述报道. 为填补这个空缺, 从单变量和多变量两个方面系统综述了国内/外学者对主要信号分解方法的研究现状, 对这些方法的时频表达性能进行分析和比较, 指出这些分解方法的优势和存在的问题. 最后, 对信号分解研究进行总结和展望.Abstract: Most signals obtained in the real world are non-stationary and nonlinear, decomposing such complex signals into several simple sub-signals is an important signal processing method. Since the Hilbert-Huang transform (HHT) was proposed in 1998, after more than 20 years of development, signal decomposition has become a relatively independent and innovative important content in the field of signal processing. Especially in the past decade, multivariate signal decomposition methods and theoretical research are in the ascendant, which have been successfully applied in many fields. However, there is no relevant overview report at present. Therefore, this paper systematically summarizes the development of signal decomposition theory and methods from both univariate and multivariate aspects. This work analyzes and compares the time-frequency expression performance of these methods, and points out the advantages and issues. Finally, the future research of signal decomposition is prospected and summarized.
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图 10 多元/多通道/多变量信号分解领域术语的图形化解释
Fig. 10 Graphical interpretation of terms in multivariate signal decomposition
图 11 单变量ICMD分解多变量信号的结果
Fig. 11 The decomposition results of multivariate signals by the univariate ICMD
图 12 多变量ICMD分解多变量信号的结果
Fig. 12 The decomposition results of multivariate signals by the multivariate ICMD
图 14 BEMD的两种二维包络的均值计算示意图
Fig. 14 Schematic diagram of the calculation of the mean value of two envelopes of two-dimensional signal for BEMD
表 1 常见单变量信号分解方法归类总结
Table 1 Classification and summary of common univariate signal decomposition methods
方法名称 作用域 优点 局限性 FT 频域 经典方法, 理论完备, 简单高效 仅适用于线性平稳信号 STFT 时频域 经典方法, 简单高效 窗函数选取问题, 分辨率固定 WVD 时频域 经典方法, 理论完备 不能处理交叉频率和多分量情况 WT 时频域 经典方法, 理论完备 母小波和尺度需人为指定 EMD 时域 自适应性强, 适用于非线性和非平稳信号, 应用场景广泛 噪声敏感, 模态混叠和端点效应问题严重, 缺乏理论基础 EEMD 时域 自适应性强, 对信号间歇性鲁棒 计算效率低, 重构误差大, 受辅助噪声参数影响大 CEEMD 时域 对信号间歇性鲁棒, 计算效率和重构误差优于EEMD 辅助噪声的参数会影响分解结果 MEEMD 时域 对信号间歇性鲁棒, 噪声鲁棒性好, 模态分裂概率低 计算效率低于EEMD MCEEMD 时域 噪声鲁棒性和分解完备性好、模态分裂概率低 计算效率低于CEEMD LMD 时域 能处理非平稳信号 噪声敏感、参数影响大 ITD 时域 计算效率优于EMD, 易于实施在线计算 噪声敏感、模态提取能力劣于EMD SST 时频域 能有效表征非平稳信号的时变调频特征 在处理强、变信号时, 会产生较大误差且无法处理时频面
交叉和重叠信号EWT 频域 数据驱动自适应划分频段 噪声鲁棒性弱, 分辨率有限 VMD 频域 噪声鲁棒性和采样频率鲁棒性好, 数学理论完善 局限于处理窄带信号, 参数影响大 NCMD 时频域 数学理论完善, 宽带信号处理能力强 需要提前指定参数 ICMD 时频域 宽带信号处理能力强, 计算效率高, 能处理交叉瞬时频率 需要提前指定参数 
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表 2 多元信号分解方法归类总结
Table 2 Classification and summary of multivariate signal decomposition methods
方法名称 拓展方式 优点 局限性 CEMD 复数性质 可处理复数信号 实部虚部模态数量可能不一致 RCEMD 复数空间
旋转概念复数空间中极值定义明确, 实部虚部模态一致 局限于处理复数信号 BEMD 单位圆投影向量 可分解双变量信号 局限于处理双变量信号 TEMD 球面投影 可分解三变量信号 局限于处理三变量信号 QEMD 超球面投影 可分解四变量信号 局限于处理四变量信号 MEMD 高维空间投影 适用于双变量及多变量信号分解 投影向量的数量和方向敏感, 抗噪声能力差, 计算效率低 FMEMD 高维空间投影 大幅提高了MEMD的计算效率 投影向量的数量和方向敏感, 噪声鲁棒性略低于MEMD IMITD 高维空间投影 局部特征处理效果好, 计算效率高于MEMD 投影向量的数量和方向会影响到基线提取 DMITD 高维空间投影 投影向量鲁棒性优于IMITD 运算效率低于IMITD MSST 多变量振荡 时频谱清晰, 适用于探索性数据分析 不能直接重构模态 MEWT 多变量振荡 可以重构模态 需要有效的频谱分割, 来显示构造自适应小波滤波器组 CVMD 复数性质 噪声和采样频率鲁棒性好 局限于处理复数窄带信号, 参数影响大 MVMD 多变量调制振荡 噪声和采样频率鲁棒性好, 模态之间信息泄露
少, 自适应多变量最优维纳滤波器局限于处理窄带多变量信号, 参数影响大 MNCMD 多变量调制振荡 可对时变多元信号进行分解 复杂度高, 需要预估信号中的噪声水平和调整参数 MICMD 多变量调制振荡 适用于宽带多元信号分解与时频分析, 计算复杂度低,
参数鲁棒性好, 模态正交性强, 信息泄露少在强噪声条件下, 分解性能下降, 零频分量波动较明显
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表 3 常见多元信号分解方法的适用场景
Table 3 Applicable scenarios of common multivariate signal decomposition methods
方法 适用场景 CEMD 只适用于复数信号 BEMD 只适用于双变量信号 MEMD 适合分析信噪比高, 实时性要求低, 采样频率足够高, 模态频率间隔两倍以上的多元信号, 可以作为有效的探索性分析方法 FMEMD 适合分析信噪比高, 实时性要求高, 采样频率足够高, 模态频率间隔两倍以上的信号; 数据量大时, 建议采用FMEMD, 不采用MEMD IMITD 适合分析局部特征明显, 实时性要求高, 采样频率足够高的多元信号 DMITD 适合分析通道间差异大, 实时性要求低, 采样频率足够高的多元信号 CVMD 只适用于具有窄带性质的复数信号 MVMD 适用于分量频率范围不重叠的窄带多元信号, 处理宽带信号效果非常有限 MNCMD 适用于宽带多元信号, 但计算复杂度较高 MICMD 适用于宽带多元信号和时频曲线有交叉的多元信号, 计算复杂度较低
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