论文标题
多频离散的真实和复杂信号的确切分解
Exact Decomposition of Multifrequency Discrete Real and Complex Signals
论文作者
论文摘要
数十年来，教科书中的标准内容之一已成为“窗口和纠察围栏效应的光谱泄漏”。光谱泄漏和纠察栅栏效应将导致信号的幅度，频率和相位的扭曲，这些幅度一直引起人们的关注，并尝试解决它们。本文提出了两个新颖的分解定理，这些定理可以完全消除光谱泄漏和纠察栅栏效应，并可以扩大信号处理的知识。首先，为多频离散的真实信号和复杂信号构建了两个广义特征值方程。然后证明两个分解定理。在这些碱基上，提出了针对真实和复杂信号的精确分解方法。对于具有M正弦组件的无噪声多频实际信号，只需仅使用4M-1离散值及其二阶导数即可精确计算每个组件的频率，振幅和相位。对于多频复杂信号，仅需要2M-1离散值及其一阶导数。数值实验表明，所提出的方法具有很高的分辨率，并且采样率不一定遵守奈奎斯特采样定理。借助嘈杂的信号，所提出的方法具有非凡的精度。
'The spectral leakage from windowing and the picket fence effect from discretization' have been among the standard contents in textbooks for many decades. The spectral leakage and picket fence effect would cause the distortions in amplitude, frequency, and phase of signals, which have always been of concern, and attempts have been made to solve them. This paper proposes two novel decomposition theorems that can totally eliminate the spectral leakage and picket fence effect, and could broaden the knowledge of signal processing. First, two generalized eigenvalue equations are constructed for multifrequency discrete real signals and complex signals. The two decomposition theorems are then proved. On these bases, exact decomposition methods for real and complex signals are proposed. For a noise-free multifrequency real signal with m sinusoidal components, the frequency, amplitude, and phase of each component can be exactly calculated by using just 4m-1 discrete values and its second-order derivatives. For a multifrequency complex signal, only 2m-1 discrete values and its first-order derivatives are needed. The numerical experiments show that the proposed methods have very high resolution, and the sampling rate does not necessarily obey the Nyquist sampling theorem. With noisy signals, the proposed methods have extraordinary accuracy.
