Fourier Series and the Fourier Transform of Discrete Signals

2016 ◽  
pp. 141-190
Author(s):  
Taan S. ElAli
Keyword(s):  
Fourier Transform ◽  
Fourier Series ◽  
Discrete Signals ◽  
The Fourier Transform

A Central Crack in a Finite Rectangular Plate With Four Hinged Edges Under the Opening Mode

1990 ◽  
Vol 57 (4) ◽  
pp. 1079-1081
Author(s):  
S. W. Ma ◽  
Y. G. Tsuei
Keyword(s):  
Stress Intensity Factor ◽  
Integral Equation ◽  
Fourier Transform ◽  
Fourier Series ◽  
Rectangular Plate ◽  
Uniform Loading ◽  
Central Crack ◽  
Opening Mode ◽  
Loading Case ◽  
The Fourier Transform

By combining linearly the Fourier transform and Fourier series, the stress intensity factor of a central crack in a finite rectangular plate with four hinged edges under the opening mode is expressed as the Fredholm integral equation of the second kind. The uniform loading case is considered in detail. The numerical results include the predictions by Koiter and Fichter as limiting cases.

scholarly journals Proper Definitions and Demonstration of Dirichlet Conditions

2021 ◽  
Author(s):  
Pushpendra Singh ◽  
Amit Singhal ◽  
Binish Fatimah ◽  
Anubha Gupta ◽  
Shiv Dutt Joshi
Keyword(s):  
Signal Processing ◽  
Fourier Transform ◽  
Fourier Series ◽  
Finite Number ◽  
Large Body ◽  
Dirichlet Condition ◽  
Countable Number ◽  
Dirichlet Conditions ◽  
Time Period ◽  
The Fourier Transform

<div>Fourier theory is the backbone of the area of Signal Processing (SP) and Communication Engineering. However, Fourier series (FS) or Fourier transform (FT) do not exist for some signals that fail to fulfill a predefined set of Dirichlet conditions (DCs). We note a subtle gap in the explanation of these conditions as available in the popular signal processing literature. They lack a certain degree of explanation essential for the proper understanding of the same. For example, </div><div>the original second Dirichlet condition is the requirement of bounded variations over one time period for the convergence of Fourier Series, where there can be at most infinite but countable number of maxima and minima, and at most infinite but countable number of discontinuities of finite magnitude. However, a large body of the literature replaces this statement with the requirements of finite number of maxima and minima over one time period, and finite number of discontinuities. The latter incorrectly disqualifies some signals from having valid FS representation. Similar problem holds in the description of DCs for the Fourier transform. Likewise, while it is easy to relate the first DC with the finite value of FS or FT coefficients, it is hard to relate the second and third DCs as specified in the signal processing literature with the Fourier representation as to how the failure to satisfy these conditions disqualifies those signals from having valid FS or FT representation. <br></div><div><br></div>

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