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 Shannon Wavelets Theory

2008 ◽  
Vol 2008 ◽  
pp. 1-24 ◽  
Author(s):  
Carlo Cattani
Keyword(s):  
Hypergeometric Series ◽  
Sampling Theorem ◽  
Connection Coefficients ◽  
Shannon Sampling Theorem ◽  
Shannon Wavelet ◽  
Differential Properties ◽  
Derivatives Of ◽  
Shannon Wavelets

Shannon wavelets are studied together with their differential properties (known as connection coefficients). It is shown that the Shannon sampling theorem can be considered in a more general approach suitable for analyzing functions ranging in multifrequency bands. This generalization coincides with the Shannon wavelet reconstruction ofL2(ℝ)functions. The differential properties of Shannon wavelets are also studied through the connection coefficients. It is shown that Shannon wavelets areC∞-functions and their any order derivatives can be analytically defined by some kind of a finite hypergeometric series. These coefficients make it possible to define the wavelet reconstruction of the derivatives of theCℓ-functions.

Shannon Wavelet Approach to Sub-Band Coding

2003 ◽  
Vol 01 (02) ◽  
pp. 233-242 ◽  
Author(s):  
Charles K. Chui ◽  
Jianzhong Wang
Keyword(s):  
Continuous Time ◽  
Wavelet Packet ◽  
Sampling Rate ◽  
Sampling Theorem ◽  
Wavelet Theory ◽  
Bandlimited Signals ◽  
Bandlimited Signal ◽  
Time Signals ◽  
Shannon Sampling Theorem ◽  
Shannon Wavelet

It is well known that the Shannon Sampling Theorem allows us to fully recover a continuous-time bandlimited signal from its digital samples, as long as the sampling rate to be chosen is not smaller than the Nyquist frequency. This theory applies to all bandlimited signals, which may or may not occupy the entire frequency band. Hence, it is intuitively convincing that for continuous-time signals, such as those in speech, that do not fully utilize the entire frequency intervals, less digital samples are required for their full recovery. Current techniques in sub-band coding are used for achieving this goal. The objective of this paper is to present a wavelet theory for establishing the mathematical foundation of this sub-band coding approach. A wavelet packet decomposition of the signal provides the optimal sub-band coding bit-rate by using the Shannon wavelet library introduced in this paper.

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

Signal Recovery

2009 ◽  
Author(s):  
Robert J Marks II
Keyword(s):  
Interpolation Problem ◽  
Super Resolution ◽  
Sampling Theorem ◽  
Signal Recovery ◽  
Continuous Sampling ◽  
Bandlimited Signals ◽  
Special Cases ◽  
Extrapolation Problem ◽  
Definition Of

The literature on the recovery of signals and images is vast (e.g., [23, 110, 112, 257, 391, 439, 791, 795, 933, 934, 937, 945, 956, 1104, 1324, 1494, 1495, 1551]). In this Chapter, the specific problem of recovering lost signal intervals from the remaining known portion of the signal is considered. Signal recovery is also a topic of Chapter 11 on POCS. To this point, sampling has been discrete. Bandlimited signals, we will show, can also be recovered from continuous samples. Our definition of continuous sampling is best presented by illustration.Asignal, f (t), is shown in Figure 10.1a, along with some possible continuous samples. Regaining f (t) from knowledge of ge(t) = f (t)Π(t/T) in Figure 10.1b is the extrapolation problem which has applications in a number of fields. In optics, for example, extrapolation in the frequency domain is termed super resolution [2, 40, 367, 444, 500, 523, 641, 720, 864, 1016, 1099, 1117]. Reconstructing f (t) from its tails [i.e., gi(t) = f (t){1 − Π(t/T)}] is the interval interpolation problem. Prediction, shown in Figure 10.1d, is the problem of recovering a signal with knowledge of that signal only for negative time. Lastly, illustrated in Figure 10.1e, is periodic continuous sampling. Here, the signal is known in sections periodically spaced at intervals of T. The duty cycle is α. Reconstruction of f (t) from this data includes a number of important reconstruction problems as special cases. (a) By keeping αT constant, we can approach the extrapolation problem by letting T go to ∞. (b) Redefine the origin in Figure 10.1e to be centered in a zero interval. Under the same assumption as (a), we can similarly approach the interpolation problem. (c) Redefine the origin as in (b). Then the interpolation problem can be solved by discarding data to make it periodically sampled. (d) Keep T constant and let α → 0. The result is reconstructing f (t) from discrete samples as discussed in Chapter 5. Indeed, this model has been used to derive the sampling theorem [246]. Figures 10.1b-e all illustrate continuously sampled versions of f (t).

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.

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