多维解析信号

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摘要翻译：
在本工作中，我们将解析信号理论推广到多维情形，当观测到在$d$正交方向上的振荡时。首先，给出了如何获得分离的相移分量，以及如何将它们组合成瞬时幅值和相位。其次，将适当的超复解析信号定义为一定上半空间边界上的全纯超复函数。其次，通过对超复傅里叶变换进行正频率限制，可以得到正确的相移分量。利用相应的Paley-Wiener定理，给出了利用全纯Fourier变换将超复解析信号解析推广到上超复半空间的充要条件。此外，还证明了对于D>2$，没有相应的非交换超复傅里叶变换（包括基于Clifford和Cayley-Dickson的变换）能够正确地恢复相移分量。
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英文标题：
《Analytic signal in many dimensions》
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作者：
Mikhail Tsitsvero, Pierre Borgnat, Paulo Gon\c{c}alves
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最新提交年份：
2019
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分类信息：

一级分类：Electrical Engineering and Systems Science        电气工程与系统科学
二级分类：Signal Processing        信号处理
分类描述：Theory, algorithms, performance analysis and applications of signal and data analysis, including physical modeling, processing, detection and parameter estimation, learning, mining, retrieval, and information extraction. The term "signal" includes speech, audio, sonar, radar, geophysical, physiological, (bio-) medical, image, video, and multimodal natural and man-made signals, including communication signals and data. Topics of interest include: statistical signal processing, spectral estimation and system identification; filter design, adaptive filtering / stochastic learning; (compressive) sampling, sensing, and transform-domain methods including fast algorithms; signal processing for machine learning and machine learning for signal processing applications; in-network and graph signal processing; convex and nonconvex optimization methods for signal processing applications; radar, sonar, and sensor array beamforming and direction finding; communications signal processing; low power, multi-core and system-on-chip signal processing; sensing, communication, analysis and optimization for cyber-physical systems such as power grids and the Internet of Things.
信号和数据分析的理论、算法、性能分析和应用，包括物理建模、处理、检测和参数估计、学习、挖掘、检索和信息提取。“信号”一词包括语音、音频、声纳、雷达、地球物理、生理、（生物）医学、图像、视频和多模态自然和人为信号，包括通信信号和数据。感兴趣的主题包括：统计信号处理、谱估计和系统辨识；滤波器设计；自适应滤波/随机学习；（压缩）采样、传感和变换域方法，包括快速算法；用于机器学习的信号处理和用于信号处理应用的机器学习；网络与图形信号处理；信号处理中的凸和非凸优化方法；雷达、声纳和传感器阵列波束形成和测向；通信信号处理；低功耗、多核、片上系统信号处理；信息物理系统的传感、通信、分析和优化，如电网和物联网。
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一级分类：Computer Science        计算机科学
二级分类：Information Theory        信息论
分类描述：Covers theoretical and experimental aspects of information theory and coding. Includes material in ACM Subject Class E.4 and intersects with H.1.1.
涵盖信息论和编码的理论和实验方面。包括ACM学科类E.4中的材料，并与H.1.1有交集。
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一级分类：Mathematics        数学
二级分类：Complex Variables        复变数
分类描述：Holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves
全纯函数，自守群作用与形式，伪凸性，复几何，解析空间，解析束
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一级分类：Mathematics        数学
二级分类：Information Theory        信息论
分类描述：math.IT is an alias for cs.IT. Covers theoretical and experimental aspects of information theory and coding.
它是cs.it的别名。涵盖信息论和编码的理论和实验方面。
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一级分类：Mathematics        数学
二级分类：Spectral Theory        光谱理论
分类描述：Schrodinger operators, operators on manifolds, general differential operators, numerical studies, integral operators, discrete models, resonances, non-self-adjoint operators, random operators/matrices
薛定谔算子，流形上的算子，一般微分算子，数值研究，积分算子，离散模型，共振，非自伴算子，随机算子/矩阵
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英文摘要：
  In this work we extend analytic signal theory to the multidimensional case when oscillations are observed in the $d$ orthogonal directions. First it is shown how to obtain separate phase-shifted components and how to combine them into instantaneous amplitude and phases. Second, the proper hypercomplex analytic signal is defined as holomorphic hypercomplex function on the boundary of certain upper half-space. Next it is shown that correct phase-shifted components can be obtained by positive frequency restriction of hypercomplex Fourier transform. Necessary and sufficient conditions for analytic extension of the hypercomplex analytic signal into the upper hypercomplex half-space by means of holomorphic Fourier transform are given by the corresponding Paley-Wiener theorem. Moreover it is demonstrated that for $d>2$ there is no corresponding non-commutative hypercomplex Fourier transform (including Clifford and Cayley-Dickson based) that allows to recover phase-shifted components correctly.
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PDF链接：
https://arxiv.org/pdf/1712.0935

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关键词：Hypercomplex Applications Experimental Optimization Differential analytic transform 正交 Fourier 给出