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table of fourier transforms
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You would be very lucky to find any procedure that completed all functions.
Fourier and other transform methods are like algebra, integration and differentiation and many other techniques,
So we develop a 'table' of common standard results and also some rules for combining them to suit the expression of interest.
You need practise to become good at manipulating the table according to these rules.
Of course you also need to pay attention to the conditions of applicability which are, for Fourier integrals and transforms
1) F(t) must be a single valued function of a real variable t throught the range \( - \infty < t < \infty \), except for a finite number of discontinuities.
2) At a point of discontinuity the value of F(t) is defined as \(F({t_d}) = \frac{1}{2}\left[ {F\left( {{t_d} + 0} \right) + F\left( {{t_d} - 0} \right)} \right]\)
3)The integral \(\int\limits_{ - \infty }^\infty {|F(t)|dt} \) must exist.
A particular point to watch for when using tables of Fourier Transforms is that they are defined in two slightly different ways so the tables of transforms and inverse transforms must use the same definition.
The transform pair may either be defined by
\(F(t) = \int\limits_{ - \infty }^\infty {g(\omega ){e^{j\omega t}}dt} \) and \(g(\omega ) = \frac{1}{{2\pi }}\int\limits_{ - \infty }^\infty {F(t){e^{ - j\omega t}}dt} \)
or by
\(F(t) = \frac{1}{{\sqrt {2\pi } }}\int\limits_{ - \infty }^\infty {g(\omega ){e^{j\omega t}}dt} \) and \(g(\omega ) = \frac{1}{{\sqrt {2\pi } }}\int\limits_{ - \infty }^\infty {F(t){e^{ - j\omega t}}dt} \)
Where it can be seen that the factor of \(\frac{1}{{2\pi }}\) has been split between the transform and its inverse for symmetry.
Fourier and other transform methods are like algebra, integration and differentiation and many other techniques,
So we develop a 'table' of common standard results and also some rules for combining them to suit the expression of interest.
You need practise to become good at manipulating the table according to these rules.
Of course you also need to pay attention to the conditions of applicability which are, for Fourier integrals and transforms
1) F(t) must be a single valued function of a real variable t throught the range \( - \infty < t < \infty \), except for a finite number of discontinuities.
2) At a point of discontinuity the value of F(t) is defined as \(F({t_d}) = \frac{1}{2}\left[ {F\left( {{t_d} + 0} \right) + F\left( {{t_d} - 0} \right)} \right]\)
3)The integral \(\int\limits_{ - \infty }^\infty {|F(t)|dt} \) must exist.
A particular point to watch for when using tables of Fourier Transforms is that they are defined in two slightly different ways so the tables of transforms and inverse transforms must use the same definition.
The transform pair may either be defined by
\(F(t) = \int\limits_{ - \infty }^\infty {g(\omega ){e^{j\omega t}}dt} \) and \(g(\omega ) = \frac{1}{{2\pi }}\int\limits_{ - \infty }^\infty {F(t){e^{ - j\omega t}}dt} \)
or by
\(F(t) = \frac{1}{{\sqrt {2\pi } }}\int\limits_{ - \infty }^\infty {g(\omega ){e^{j\omega t}}dt} \) and \(g(\omega ) = \frac{1}{{\sqrt {2\pi } }}\int\limits_{ - \infty }^\infty {F(t){e^{ - j\omega t}}dt} \)
Where it can be seen that the factor of \(\frac{1}{{2\pi }}\) has been split between the transform and its inverse for symmetry.
Yes there are tables with a finite number of entries. Look at the CRC handbooks, or Rektorys, or Abramowitz and Stegun and other works that describe useful functions.
