scholarly journals 量子力学、香农采样定理与数理统计学的数学联系

2021 ◽  
Author(s):  
Zheng-Ling YANG
Keyword(s):  
Sampling Theorem ◽  
Shannon Sampling Theorem

对于原子的基本结构，目前物理学认为电子围绕原子核运动，其不同轨道之间通常是分离的。对波函数、坍塌的解释有不同的观点。从信号处理的角度看：不妨假设电子围绕原子核做连续的复杂运动；利用奈奎斯特-香农采样定理（Nyquist-Shannon sampling theorem），可以把高速运动的微观粒子的连续行为的长期观察结果，采样成不连续的概率方式，即量子力学。“标准”的量子力学建立时（1926年前后），数学里的概率论和数理统计学还没有发展成熟。数学工具的不足，是引起波函数与坍塌解释不同看法的数学原因。

scholarly journals Nyquist-Shannon sampling theorem applied to refinements of the atomic pair distribution function

2011 ◽  
Vol 84 (13) ◽  
Author(s):  
Christopher L. Farrow ◽  
Margaret Shaw ◽  
Hyunjeong Kim ◽  
Pavol Juhás ◽  
Simon J. L. Billinge
Keyword(s):  
Distribution Function ◽  
Pair Distribution Function ◽  
Sampling Theorem ◽  
Atomic Pair Distribution Function ◽  
Atomic Pair ◽  
Shannon Sampling Theorem

scholarly journals Shannon Wavelets for the Solution of Integrodifferential Equations

2010 ◽  
Vol 2010 ◽  
pp. 1-22 ◽  
Author(s):  
Carlo Cattani
Keyword(s):  
Hypergeometric Series ◽  
Sampling Theorem ◽  
Integrodifferential Equations ◽  
Series Connection ◽  
Wavelet Representation ◽  
Connection Coefficients ◽  
Shannon Sampling Theorem ◽  
Shannon Wavelet ◽  
Definition Of ◽  
Shannon Wavelets

Shannon wavelets are used to define a method for the solution of integrodifferential equations. This method is based on (1) the Galerking method, (2) the Shannon wavelet representation, (3) the decorrelation of the generalized Shannon sampling theorem, and (4) the definition of connection coefficients. The Shannon sampling theorem is considered in a more general approach suitable for analysing functions ranging in multifrequency bands. This generalization coincides with the Shannon wavelet reconstruction ofL2(ℝ)functions. Shannon wavelets areC∞-functions and their any order derivatives can be analytically defined by some kind of a finite hypergeometric series (connection coefficients).

scholarly journals A note on Whittaker's cardinal series in harmonic analysis

1986 ◽  
Vol 29 (3) ◽  
pp. 349-357 ◽  
Author(s):  
M. M. Dodson ◽  
A. M. Silva ◽  
V. Soucek
Keyword(s):  
Control Theory ◽  
Data Processing ◽  
Harmonic Analysis ◽  
Real Function ◽  
Sampling Theorem ◽  
Communication Engineering ◽  
Shannon Sampling Theorem ◽  
Band Limited ◽  
Image Position ◽  
Sampling Set

The sampling theorem, often referred to as the Shannon or Whittaker-Kotel'nikov- Shannon sampling theorem, is of considerable importance in many fields, including communication engineering, electronics, control theory and data processing, and has appeared frequently in various forms in engineering literature (a comprehensive account of its numerous extensions and applications is given in [3]). The result states that a band-limited signal, i.e. a real function f of the formwhere w>0, is under reasonable conditions on the even function F, determined by its values on the sampling set (l/2w)ℤ and can be reconstructed from the samples f(k/2w), k∈ℤ, by the series

Study on Fault Diagnosis of Gear Transmission which is Based on ADAMS

2012 ◽  
Vol 215-216 ◽  
pp. 812-816
Author(s):  
Shi Ming Wang ◽  
Xian Zhu Ai ◽  
Chao Lv ◽  
Li Na Ma
Keyword(s):  
Time Domain ◽  
Mechanical Energy ◽  
Electric Energy ◽  
Image Data ◽  
Real Data ◽  
Sampling Theorem ◽  
Drive Gear ◽  
Ocean Wave Energy ◽  
Shannon Sampling Theorem ◽  
Simulation Parameters

Introduced a transmission system of a new oscillation buoy ocean wave energy generation device, the system can transform the mechanical energy into electric energy. A pair of gear model was built by SOLIDWORKS, the parameter is just the same as the real data, then imported the model into ADAMS. Under the same simulation parameters, two experiments were done, one engaged without failure, the other engaged with one broken tooth of drive wheels. Calculated TIME and STEPS by Shannon sampling theorem, simulated the marker point’s acceleration of the drive gear, then obtain image data of time domain and frequency domain, after analyzed, found this method has a significant meaning to practice.

scholarly journals On the uncertainty inequality as applied to discrete signals

2006 ◽  
Vol 2006 ◽  
pp. 1-22 ◽  
Author(s):  
Y. V. Venkatesh ◽  
S. Kumar Raja ◽  
G. Vidyasagar
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Sampling Theorem ◽  
Discrete Signals ◽  
Bandlimited Signal ◽  
Time Signals ◽  
Shannon Sampling Theorem ◽  
Heisenberg Uncertainty ◽  
Comparison Of The Results ◽  
Uncertainty Inequality

Given a continuous-time bandlimited signal, the Shannon sampling theorem provides an interpolation scheme forexactly reconstructingit from its discrete samples. We analyze the relationship between concentration (orcompactness) in thetemporal/spectral domainsof the (i) continuous-time and (ii) discrete-time signals. The former is governed by the Heisenberg uncertainty inequality which prescribes a lower bound on the product ofeffectivetemporal and spectral spreads of the signal. On the other hand, the discrete-time counterpart seems to exhibit some strange properties, and this provides motivation for the present paper. We consider the following problem:for a bandlimited signal, can the uncertainty inequality be expressed in terms of the samples, using thestandard definitions of the temporal and spectral spreads of the signal?In contrast with the results of the literature, we present a new approach to solve this problem. We also present a comparison of the results obtained using the proposed definitions with those available in the literature.

scholarly journals Frequency Aliasing-Based Spatial-Wavenumber Filter for Online Damage Monitoring

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Bin Liu ◽  
Tingzhang Liu ◽  
Jianfei Zhao ◽  
Dan Hang
Keyword(s):  
Lamb Wave ◽  
Sensor Array ◽  
Laminate Plate ◽  
Sampling Theorem ◽  
Spatial Sampling ◽  
Filter Method ◽  
Damage Monitoring ◽  
Shannon Sampling Theorem ◽  
Damage Imaging ◽  
Frequency Aliasing

The spatial-wavenumber filter method can extract the specific mode of the Lamb wave, thereby distinguishing the incident wave and the damage reflection wave. This method has been widely studied for damage imaging. However, the diameter of piezoelectric transducer (PZT) sensor limits the spatial sampling wavenumber of the linear PZT sensor array, which limits the application of this method because of the Nyquist–Shannon sampling theorem. Therefore, the wavenumber filtering range of spatial-wavenumber filter should be less than half of the spatial sampling wavenumber. In this paper, a frequency aliasing based spatial-wavenumber filter for online damage monitoring is proposed. In this method, the wavenumber filtering range is extended to the spatial sampling wavenumber, and two wavenumber results will be calculated as for the frequency aliasing. Subsequently, the wavenumber of the received Lamb wave signal can be obtained according to the average arrival time difference between the two adjacent sensors in the linear PZT sensor array. Finally, the damage is localized using the spatial-wavenumber filter and cruciform PZT sensor array. This method was validated on an epoxy laminate plate. The maximum damage localization errors are less than 2 cm. It is indicated that this method can extend the spatial-wavenumber filtering range to the spatial sampling wavenumber and the application of spatial-wavenumber filter-based online damage monitoring.

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