信号的扩展傅里叶分析
Extended Fourier analysis of signals
AI总结：
本文提出扩展离散傅里叶变换，通过求解加权最小二乘估计器，在连续和离散域中推导出适应性频域表示，实现对不完整或非均匀采样数据的准确频谱估计。
摘要：
本文提出了扩展离散傅里叶变换（EDFT），该方法直接从傅里叶积分及其正交性质推导得出。通过在连续和离散域中求解加权最小二乘估计器，得到了一种自适应的频域表示，该表示与经典傅里叶框架完全一致。在均匀采样数据且频率网格大小相同时，EDFT恰好还原为经典离散傅里叶变换（DFT）。然而，当分析网格超过观测样本数量时，EDFT通过优化变换基底来规避传统零填充，从而实现对不完整或非均匀采样数据的准确频谱估计。因此，EDFT在强谱内容区域实现了增强的频率分辨率，同时保持全局分辨率的平衡，从而与不确定性原理保持一致。逆EDFT能够重构原始信号，并在频谱信息可用处生成插值或外推样本。EDFT无需显式分离确定性和随机性成分，能够同时准确捕捉宽带、瞬态和正弦波特征。仿真研究证实了其在非均匀采样、多个奈奎斯特区、缺失数据条件以及混合谱（包含线性和连续成分）下的鲁棒性。尽管EDFT的迭代计算相较于经典DFT具有更高的数值成本，但这一限制在1990年代显著存在，如今已通过现代计算资源得到很大程度的缓解，使EDFT在当代信号分析应用中变得实用。
英文摘要：
This summary of the doctoral thesis provides a comprehensive formulation of the Extended Discrete Fourier Transform (EDFT), derived directly from the Fourier integral and its orthogonality properties. The method is obtained by solving weighted least-squares estimators in both continuous and discrete domains, yielding an adaptive frequency-domain representation that remains fully consistent with the classical Fourier framework. In the special case of uniformly sampled data on a uniform frequency grid of the same size, the EDFT reduces exactly to the classical Discrete Fourier Transform (DFT). However, when the analysis grid exceeds the number of observed samples, EDFT circumvents conventional zero-padding by optimizing the transformation basis over the extended frequency set. This enables accurate spectral estimation from incomplete or nonuniformly sampled data. Consequently, the EDFT achieves enhanced frequency resolution in regions of strong spectral content while maintaining global resolution balance, thereby remaining consistent with the uncertainty principle. The inverse EDFT reconstructs the original signal and produces extrapolated or interpolated samples wherever spectral information is available. The EDFT requires no explicit separation of deterministic and stochastic components and accurately captures broadband, transient, and sinusoidal features simultaneously. Simulation studies confirm its robustness under nonuniform sampling, multiple Nyquist zones, missing-data conditions, and signals with mixed spectra comprising both line and continuous components. Although iterative computation of the EDFT entails higher numerical cost compared to the classical DFT, this limitation - significant in the 1990s - has been largely mitigated by modern computational resources, rendering the EDFT practical for contemporary signal analysis applications.
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