Nuclear magnetic resonance spectroscopy originated in physics and quickly found versatile applications of paramount importance in other sciences, including chemistry. Signal processing in this methodology is a key to data analysis and interpretation. Herein, one of the most powerful tools from mathematical theory of approximations, known as rational polynomials, is the prime example of reliable handling of the two stumbling blocks that hamper further progress: noise suppression and resolution improvement. Within this realm resides the fast Padé transform (FPT), which simultaneously solves both these problems. It has a self-correcting procedure, which is automatically built in rational polynomials through noise suppression by pole-zero cancellations in spectra. Moreover, by solving the quantification problem (called spectral analysis in mathematics), the FPT can unequivocally separate overlapped peaks and thereby improve resolution. Further, lineshape estimations are provided by both non-parametric and parametric signal processing in the FPT. Since the FPT includes singularities (poles) of the expanded function, it achieves exponental convergence exp(-N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\exp {(-N)}$$\end{document} (the so-named spectral resolution) with respect to the size N of the basis set. This is contrasted to merely the inverse-power-law convergence 1 / N in the fast Fourier transform because its basis functions do not describe the singularities of the expanded function. The present investigation reports on practical aspects of all these critical features and gives several representative illustrations for measured time signals heavily contaminated with noise.

中文翻译：

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磁共振波谱信号处理中的自动自校正：降噪、分辨率提高和分裂重叠峰

核磁共振波谱起源于物理学，并很快发现了在其他科学（包括化学）中至关重要的多功能应用。这种方法中的信号处理是数据分析和解释的关键。在此，近似数学理论中最强大的工具之一（称为有理多项式）是可靠处理阻碍进一步发展的两个绊脚石的主要示例：噪声抑制和分辨率改进。在这个领域内存在快速 Padé 变换 (FPT)，它同时解决了这两个问题。它具有自校正程序，通过频谱中的零极点消除来抑制噪声，自动在有理多项式中构建。而且，通过解决量化问题（数学中称为光谱分析），FPT 可以明确分离重叠峰，从而提高分辨率。此外，线形估计由 FPT 中的非参数和参数信号处理提供。由于 FPT 包括扩展函数的奇点（极点），它实现指数收敛 exp(-N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\exp {(-N)}$$\end{document} （所以-命名光谱分辨率）相对于基组的大小 N。这与快速傅立叶变换中仅反幂律收敛 1 / N 形成对比，因为其基函数不描述扩展函数的奇点。目前的调查报告了所有这些关键特征的实际方面，并给出了一些受噪声严重污染的测量时间信号的代表性说明。