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Spectral Analysis Resolution and Study of the Uncertainty Relationship

机译:光谱分析分辨率和不确定度关系研究

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An uncertainty relationship between the width of a pulse frequency spectrum and its time duration and extension in space is investigated. Analysis of the uncertainty relationship is carried out for causal pulses with minimum-phase spectrum. Frequency spectra of elementary pulses are calculated using modified Fourier spectral analysis. Modification of the Fourier analysis is justified by the necessity of solving the paradox of ultralow frequencies in the Fourier spectra of unidirectional (nonosdilatory) pulses. It is established in the paper that the finite time of the measurement of signal spectra is the fundamental restriction for the determination of the frequency of harmonic components whose period is larger than the time of measurement. The occurrence of the physically nonobservable spectral components in Fourier spectra is caused by the varying resolution of Fourier analysis along the frequency axis. The modified Fourier spectral analysis has equal resolution along the frequency axis and excludes physically unobservable values from spectral and time concepts. It is proved that the Heaviside unit step function has an infinitely wide spectrum equal to one along the whole frequency range. The Dirac delta function has an infinitely wide spectrum within the scope of infinitely high frequencies. The difference between the propagations of wave and quasi-wave forms of energy motion is established from the analysis of the uncertainty relationship. The velocity of propagation of unidirectional pulses, or shock waves, depends on the relative width of the pulse spectra. The oscillating pulse propagation velocity is constant in a given nondispersive medium. The idea of the elementary wavelet is introduced. The hypothesis of the elementary wavelet is tested by the example of elastic and electromagnetic waves.
机译:研究了脉冲频谱的宽度与其持续时间和空间扩展之间的不确定性关系。对具有最小相位频谱的因果脉冲进行不确定性关系分析。基本脉冲的频谱是使用改进的傅立叶频谱分析计算的。傅里叶分析的修改是有必要解决单向(非振荡)脉冲的傅里叶频谱中超低频的矛盾。本文确定,信号频谱测量的有限时间是确定周期大于测量时间的谐波分量的频率的基本限制。傅立叶光谱中物理上不可观察到的光谱分量的出现是由于沿着频率轴进行的傅立叶分析的分辨率变化引起的。改进的傅立叶频谱分析在频率轴上具有相同的分辨率,并且从频谱和时间概念中排除了物理上不可观察的值。事实证明,Heaviside单位阶跃函数在整个频率范围内具有等于1的无限宽频谱。 Dirac delta函数在无限高的频率范围内具有无限广的频谱。通过对不确定性关系的分析,建立了能量运动的波和准波形式的传播之间的差异。单向脉冲或冲击波的传播速度取决于脉冲频谱的相对宽度。在给定的非分散介质中,振荡脉冲的传播速度是恒定的。介绍了基本小波的思想。基本小波的假设通过弹性波和电磁波的示例进行检验。

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