Difference between revisions of "Dictionary:Fourier transform"

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{{#category_index:F|Fourier transform}}
 
{{#category_index:F|Fourier transform}}
 
<translate>
 
<translate>
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<!--T:1-->
 
[[File:Segf19.jpg|thumb|F-19. Fourier transform pairs. The time functions on the left are Fourier transforms of the frequency functions on the right and vice-versa. Many more transform pairs could be shown. The above are all even functions and hence have zero phase. Transforms for real odd functions are imaginary, i.e., they have a phase shift of +&#x03C0;/2. Transforms of functions that are neither odd nor even involve variations of phase with frequency. Note ''f''=1/''t''.]]
 
[[File:Segf19.jpg|thumb|F-19. Fourier transform pairs. The time functions on the left are Fourier transforms of the frequency functions on the right and vice-versa. Many more transform pairs could be shown. The above are all even functions and hence have zero phase. Transforms for real odd functions are imaginary, i.e., they have a phase shift of +&#x03C0;/2. Transforms of functions that are neither odd nor even involve variations of phase with frequency. Note ''f''=1/''t''.]]
  
 +
<!--T:2-->
 
Formulas that convert a time function ''g''(''t'') (waveform, seismic record trace, etc.) into its frequency-domain representation ''G''(''f'') and vice versa. ''G''(''f'') and ''g''(''t'') constitute a [[Special:MyLanguage/Dictionary:Fourier-transform pair|Fourier-transform pair]]; see Figure [[Special:MyLanguage/Dictionary:Fig_F-19|F-19]] below. An example is  
 
Formulas that convert a time function ''g''(''t'') (waveform, seismic record trace, etc.) into its frequency-domain representation ''G''(''f'') and vice versa. ''G''(''f'') and ''g''(''t'') constitute a [[Special:MyLanguage/Dictionary:Fourier-transform pair|Fourier-transform pair]]; see Figure [[Special:MyLanguage/Dictionary:Fig_F-19|F-19]] below. An example is  
  
 +
<!--T:3-->
 
<center><math>g(t) \leftrightarrow G(f) = \int\limits_{- \infty}^{\infty} g(t)e^{-j2\pi ft} dt = \int\limits_{- \infty}^{\infty} g(t) \cos(j2\pi ft) dt - j \int\limits_{- \infty}^{\infty} g(t) \sin(j2\pi ft) dt</math></center>
 
<center><math>g(t) \leftrightarrow G(f) = \int\limits_{- \infty}^{\infty} g(t)e^{-j2\pi ft} dt = \int\limits_{- \infty}^{\infty} g(t) \cos(j2\pi ft) dt - j \int\limits_{- \infty}^{\infty} g(t) \sin(j2\pi ft) dt</math></center>
  
 +
<!--T:4-->
 
The [[Special:MyLanguage/Dictionary:inverse transform|inverse transform]] is  
 
The [[Special:MyLanguage/Dictionary:inverse transform|inverse transform]] is  
  
 +
<!--T:5-->
 
<center><math>g(t) = \int\limits_{- \infty}^{\infty} G(f)e^{j2\pi ft} dt = \int\limits_{- \infty}^{\infty} G(f) \cos(j2\pi ft) dt + j \int\limits_{- \infty}^{\infty} G(f) \sin(j2\pi ft) dt</math></center>
 
<center><math>g(t) = \int\limits_{- \infty}^{\infty} G(f)e^{j2\pi ft} dt = \int\limits_{- \infty}^{\infty} G(f) \cos(j2\pi ft) dt + j \int\limits_{- \infty}^{\infty} G(f) \sin(j2\pi ft) dt</math></center>
  
 +
<!--T:6-->
 
Finding ''G''(''f'') from ''g''(''t'') is called [[Special:MyLanguage/Dictionary:Fourier analysis|Fourier analysis]] and finding ''g''(''t'') from ''G''(''f'') is called [[Special:MyLanguage/Dictionary:Fourier synthesis|Fourier synthesis]]. ''G''(''f'') is the [[Special:MyLanguage/Dictionary:complex spectrum|complex spectrum]], the real part being the [[Special:MyLanguage/Dictionary:cosine transform|cosine transform]] and the imaginary part the [[Special:MyLanguage/Dictionary:sine transform|sine transform]] whenever ''g''(''t'') is real. Another expression for ''G''(''f'') is  
 
Finding ''G''(''f'') from ''g''(''t'') is called [[Special:MyLanguage/Dictionary:Fourier analysis|Fourier analysis]] and finding ''g''(''t'') from ''G''(''f'') is called [[Special:MyLanguage/Dictionary:Fourier synthesis|Fourier synthesis]]. ''G''(''f'') is the [[Special:MyLanguage/Dictionary:complex spectrum|complex spectrum]], the real part being the [[Special:MyLanguage/Dictionary:cosine transform|cosine transform]] and the imaginary part the [[Special:MyLanguage/Dictionary:sine transform|sine transform]] whenever ''g''(''t'') is real. Another expression for ''G''(''f'') is  
  
 +
<!--T:7-->
 
<center><math>G(f) = \left\vert A(f) \right\vert e^{j\gamma (f)}</math></center>
 
<center><math>G(f) = \left\vert A(f) \right\vert e^{j\gamma (f)}</math></center>
  
 +
<!--T:8-->
 
where the functions ''A''(''f'') and &#x03B3;(''f'') are real. They are, respectively, the [[Special:MyLanguage/Dictionary:amplitude spectrum|amplitude spectrum]] and the [[Special:MyLanguage/Dictionary:phase spectrum|phase spectrum]] of ''g''(''t''):  
 
where the functions ''A''(''f'') and &#x03B3;(''f'') are real. They are, respectively, the [[Special:MyLanguage/Dictionary:amplitude spectrum|amplitude spectrum]] and the [[Special:MyLanguage/Dictionary:phase spectrum|phase spectrum]] of ''g''(''t''):  
  
 +
<!--T:9-->
 
<center><math>A(f)=\sqrt{[\text{real part of } G(f)]^2 + [\text{imaginary part of } G(f)]^2}</math></center>
 
<center><math>A(f)=\sqrt{[\text{real part of } G(f)]^2 + [\text{imaginary part of } G(f)]^2}</math></center>
  
 +
<!--T:10-->
 
<center><math>\gamma (f) = \arctan \bigg(\frac{\text{imaginary part of } G(f)}{\text{real part of } G(f)}\bigg) </math></center>
 
<center><math>\gamma (f) = \arctan \bigg(\frac{\text{imaginary part of } G(f)}{\text{real part of } G(f)}\bigg) </math></center>
  
 +
<!--T:11-->
 
&#x03B3;(''f'') is in the first or second quadrant if the imaginary part is positive, in the first or fourth quadrant if the real part is positive. A record trace ''h''(''t'') that extends only from 0 to ''T'' may be assumed to be repeated indefinitely and so expanded in a [[Special:MyLanguage/Dictionary:Fourier series|Fourier series]] of period ''T'':  
 
&#x03B3;(''f'') is in the first or second quadrant if the imaginary part is positive, in the first or fourth quadrant if the real part is positive. A record trace ''h''(''t'') that extends only from 0 to ''T'' may be assumed to be repeated indefinitely and so expanded in a [[Special:MyLanguage/Dictionary:Fourier series|Fourier series]] of period ''T'':  
  
 +
<!--T:12-->
 
<center><math>h(t) = \sum_{n=0}^{\infty} a_n \cos(2\pi n ( \frac{t}{T}) ) + \sum_{n=1}^{\infty} b_n \sin(2\pi n ( \frac{t}{T}) )</math></center>
 
<center><math>h(t) = \sum_{n=0}^{\infty} a_n \cos(2\pi n ( \frac{t}{T}) ) + \sum_{n=1}^{\infty} b_n \sin(2\pi n ( \frac{t}{T}) )</math></center>
  
 +
<!--T:13-->
 
where  
 
where  
  
 +
<!--T:14-->
 
<center><math>a_n = \frac{2}{T} \int\limits_{0}^{T} h(t) \cos(2\pi n ( \frac{t}{T} ) ) dt,</math></center>
 
<center><math>a_n = \frac{2}{T} \int\limits_{0}^{T} h(t) \cos(2\pi n ( \frac{t}{T} ) ) dt,</math></center>
  
 +
<!--T:15-->
 
<center><math>b_n = \frac{2}{T} \int\limits_{0}^{T} h(t) \sin(2\pi n ( \frac{t}{T} ) )dt,</math></center>
 
<center><math>b_n = \frac{2}{T} \int\limits_{0}^{T} h(t) \sin(2\pi n ( \frac{t}{T} ) )dt,</math></center>
  
 +
<!--T:16-->
 
and  
 
and  
  
 +
<!--T:17-->
 
<center><math>h(t) \leftrightarrow H_n = \left\vert A_n \right\vert e^{j\gamma_n}</math></center>
 
<center><math>h(t) \leftrightarrow H_n = \left\vert A_n \right\vert e^{j\gamma_n}</math></center>
  
 +
<!--T:18-->
 
<center><math>A_n = \sqrt{a_n^2 + b_n^2}, \gamma_n=\arctan \Bigg(\frac{b_n}{a_n}\Bigg) </math></center>
 
<center><math>A_n = \sqrt{a_n^2 + b_n^2}, \gamma_n=\arctan \Bigg(\frac{b_n}{a_n}\Bigg) </math></center>
  
 +
<!--T:19-->
 
The same rules for quadrants apply to &#x03B3;<sub>''n''</sub> as expressed for &#x03B3;(''f''); ''a''<sub>0</sub> is the [[Special:MyLanguage/Dictionary:zero-frequency component|zero-frequency component]] (or [[Special:MyLanguage/Dictionary:dc shift|dc shift]]). The frequency spectrum is discrete if the function is periodic. If ''h''<sub>''t''</sub> is a sampled time series sampled at intervals of time ''t''<sub>2</sub>, then we can stop summing when ''n''&#x003E;2''T''/''t''<sub>2</sub> (see [[Special:MyLanguage/Dictionary:sampling_theorem|sampling theorem]]). In this case ''a''<sub>''n''</sub> and ''b''<sub>''n''</sub> can be expressed as sums:
 
The same rules for quadrants apply to &#x03B3;<sub>''n''</sub> as expressed for &#x03B3;(''f''); ''a''<sub>0</sub> is the [[Special:MyLanguage/Dictionary:zero-frequency component|zero-frequency component]] (or [[Special:MyLanguage/Dictionary:dc shift|dc shift]]). The frequency spectrum is discrete if the function is periodic. If ''h''<sub>''t''</sub> is a sampled time series sampled at intervals of time ''t''<sub>2</sub>, then we can stop summing when ''n''&#x003E;2''T''/''t''<sub>2</sub> (see [[Special:MyLanguage/Dictionary:sampling_theorem|sampling theorem]]). In this case ''a''<sub>''n''</sub> and ''b''<sub>''n''</sub> can be expressed as sums:
  
 +
<!--T:20-->
 
<center><math>a_n = \frac{2}{T} \sum_{n=0}^{2\frac{T}{t_2}} h_t \cos(2\pi n ( \frac{t}{T}) )</math></center>
 
<center><math>a_n = \frac{2}{T} \sum_{n=0}^{2\frac{T}{t_2}} h_t \cos(2\pi n ( \frac{t}{T}) )</math></center>
  
 +
<!--T:21-->
 
and  
 
and  
  
 +
<!--T:22-->
 
<center><math>b_n = \frac{2}{T} \sum_{n=1}^{2\frac{T}{t_2}} h_t \sin(2\pi n ( \frac{t}{T}) )</math></center>
 
<center><math>b_n = \frac{2}{T} \sum_{n=1}^{2\frac{T}{t_2}} h_t \sin(2\pi n ( \frac{t}{T}) )</math></center>
  
 +
<!--T:23-->
 
Also see [[Special:MyLanguage/Dictionary:phase_response|phase response]] and [[Special:MyLanguage/Dictionary:fast_Fourier_transform_(FFT)|fast Fourier transform]].
 
Also see [[Special:MyLanguage/Dictionary:phase_response|phase response]] and [[Special:MyLanguage/Dictionary:fast_Fourier_transform_(FFT)|fast Fourier transform]].
  
 +
<!--T:24-->
 
[[File:Segf20.jpg|thumb|F-20. Equivalence of Fourier transform operations. Doing the time operation is equivalent to doing the frequency operation on the transform of the data. Note: ''g''(''t'')&#x2194;''G''(''f'') and ''h''(''t'')&#x2194;''H''(''f'').]]
 
[[File:Segf20.jpg|thumb|F-20. Equivalence of Fourier transform operations. Doing the time operation is equivalent to doing the frequency operation on the transform of the data. Note: ''g''(''t'')&#x2194;''G''(''f'') and ''h''(''t'')&#x2194;''H''(''f'').]]
  
 +
<!--T:25-->
 
Operations in one domain have equivalent operations in the transform domain (see Figure [[Special:MyLanguage/Dictionary:Fig_F-20|F-20]]). Computations can sometimes be carried out more economically in one domain than the other and Fourier transforms provide a means of accomplishing this. The Fourier-transform relations can be generalized for more than one dimension (see Figure [[Special:MyLanguage/Dictionary:Fig_F-21|F-21]]). For example,  
 
Operations in one domain have equivalent operations in the transform domain (see Figure [[Special:MyLanguage/Dictionary:Fig_F-20|F-20]]). Computations can sometimes be carried out more economically in one domain than the other and Fourier transforms provide a means of accomplishing this. The Fourier-transform relations can be generalized for more than one dimension (see Figure [[Special:MyLanguage/Dictionary:Fig_F-21|F-21]]). For example,  
  
 +
<!--T:26-->
 
<center><math>G(k,\omega) = \int\limits_{- \infty}^{\infty} \int\limits_{- \infty}^{\infty} g(x,t)e^{-j(k x+\omega t)} dx dt</math></center>
 
<center><math>G(k,\omega) = \int\limits_{- \infty}^{\infty} \int\limits_{- \infty}^{\infty} g(x,t)e^{-j(k x+\omega t)} dx dt</math></center>
  
 +
<!--T:27-->
 
and  
 
and  
  
 +
<!--T:28-->
 
<center><math>g(x,t) = \frac{1}{\pi} \int\limits_{- \infty}^{\infty} \int\limits_{- \infty}^{\infty} G(k,\omega)e^{j(k x+\omega t)} d\omega dk</math></center>
 
<center><math>g(x,t) = \frac{1}{\pi} \int\limits_{- \infty}^{\infty} \int\limits_{- \infty}^{\infty} G(k,\omega)e^{j(k x+\omega t)} d\omega dk</math></center>
  
 +
<!--T:29-->
 
The 1/4&pi; factor is sometimes distributed between the two integrals; where calculations involve an arbitrary scaling factor, the 1/4&pi; factor may be dropped entirely.
 
The 1/4&pi; factor is sometimes distributed between the two integrals; where calculations involve an arbitrary scaling factor, the 1/4&pi; factor may be dropped entirely.
  
 +
<!--T:30-->
 
Fourier transforms are discussed in Sheriff and Geldart (1995, 277, 532–533)<ref>Sheriff, R. E. and Geldart, L. P., 1995, Exploration Seismology, 2nd Ed., Cambridge Univ. Press.</ref>. Theorems relating to Fourier transforms are shown in Figure [[Special:MyLanguage/Dictionary:Fig_F-22|F-22]].
 
Fourier transforms are discussed in Sheriff and Geldart (1995, 277, 532–533)<ref>Sheriff, R. E. and Geldart, L. P., 1995, Exploration Seismology, 2nd Ed., Cambridge Univ. Press.</ref>. Theorems relating to Fourier transforms are shown in Figure [[Special:MyLanguage/Dictionary:Fig_F-22|F-22]].
  
  
  
 +
<!--T:31-->
 
[[File:Segf22.jpg|center|thumb|500px|F-22. Fourier transform theorems.]]
 
[[File:Segf22.jpg|center|thumb|500px|F-22. Fourier transform theorems.]]
  
== Notations and sign conventions ==
+
== Notations and sign conventions == <!--T:32-->
  
 +
<!--T:33-->
 
The notations and sign conventions used above are common in the electrical engineering world. Many exploration geophysicists may be familiar, however, with the conventions
 
The notations and sign conventions used above are common in the electrical engineering world. Many exploration geophysicists may be familiar, however, with the conventions
 
used by physicists and mathematicians. These differ subtly from those used above. Many geophysicists whose education may in those fields, or who draw
 
used by physicists and mathematicians. These differ subtly from those used above. Many geophysicists whose education may in those fields, or who draw
Line 75: Line 107:
 
discussion of these conventions.
 
discussion of these conventions.
  
 +
<!--T:34-->
 
It is more common in the world of mathematics and theoretical physics for the convention of <math> i = \sqrt{-1} </math> rather than <math> j </math>.  
 
It is more common in the world of mathematics and theoretical physics for the convention of <math> i = \sqrt{-1} </math> rather than <math> j </math>.  
 
Electrical engineers prefer to use the letter <math> j </math> because the letter <math> i </math> is reserved for current. Mathematicians and mathematical
 
Electrical engineers prefer to use the letter <math> j </math> because the letter <math> i </math> is reserved for current. Mathematicians and mathematical
Line 83: Line 116:
  
  
== 1D transforms ==
+
== 1D transforms == <!--T:35-->
  
 +
<!--T:36-->
 
Putting all of these together, we obtain a common notational convention for the ''forward'' Fourier transform in time as
 
Putting all of these together, we obtain a common notational convention for the ''forward'' Fourier transform in time as
  
 +
<!--T:37-->
 
<center> <math>  \tilde{f}(\omega) = \int_{-\infty}^{\infty} f(t) e^{i \omega t } \; dt </math> </center>
 
<center> <math>  \tilde{f}(\omega) = \int_{-\infty}^{\infty} f(t) e^{i \omega t } \; dt </math> </center>
  
 +
<!--T:38-->
 
and the ''inverse'' Fourier transform
 
and the ''inverse'' Fourier transform
  
 +
<!--T:39-->
 
<center> <math>  f(t) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} \tilde{f}(\omega) e^{- i \omega t } \; d\omega . </math> </center>
 
<center> <math>  f(t) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} \tilde{f}(\omega) e^{- i \omega t } \; d\omega . </math> </center>
  
 +
<!--T:40-->
 
The 1D spatial forward Fourier transform differs from the temporal transform in that the integration is on infinite limits and the sign of the exponent in
 
The 1D spatial forward Fourier transform differs from the temporal transform in that the integration is on infinite limits and the sign of the exponent in
 
the exponential is negative
 
the exponential is negative
  
 +
<!--T:41-->
 
<center> <math>  \tilde{f}(k) = \int_{-\infty}^{\infty} f(x) e^{- i k x } \; dx </math> </center>
 
<center> <math>  \tilde{f}(k) = \int_{-\infty}^{\infty} f(x) e^{- i k x } \; dx </math> </center>
  
 +
<!--T:42-->
 
and the ''inverse'' spatial Fourier transform is similarly different
 
and the ''inverse'' spatial Fourier transform is similarly different
  
 +
<!--T:43-->
 
<center> <math>  f(t) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} \tilde{f}(k) e^{i k x} \; dk . </math> </center>
 
<center> <math>  f(t) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} \tilde{f}(k) e^{i k x} \; dk . </math> </center>
  
  
== Transforms in <math> n </math> spatial dimensions ==
+
== Transforms in <math> n </math> spatial dimensions == <!--T:44-->
  
 +
<!--T:45-->
 
In <math> n</math> dimensions, the spatial transforms become
 
In <math> n</math> dimensions, the spatial transforms become
  
 +
<!--T:46-->
 
<center> <math>  \tilde{f}(\mathbf{k}) = \int_{-\infty}^{\infty} ... \mbox{n total integrations}... \int_{-\infty}^{\infty} f(x) e^{- i \mathbf{k} \cdot  \mathbf {x} } \; d\mathbf{x} </math> </center>
 
<center> <math>  \tilde{f}(\mathbf{k}) = \int_{-\infty}^{\infty} ... \mbox{n total integrations}... \int_{-\infty}^{\infty} f(x) e^{- i \mathbf{k} \cdot  \mathbf {x} } \; d\mathbf{x} </math> </center>
  
 +
<!--T:47-->
 
and the ''inverse'' spatial Fourier transform is similarly different
 
and the ''inverse'' spatial Fourier transform is similarly different
  
 +
<!--T:48-->
 
<center> <math>  f(\mathbf{x}) = \frac{1}{(2 \pi)^n} \int_{-\infty}^{\infty} ... \mbox{n total integrations}... \int_{-\infty}^{\infty}  \tilde{f}(\mathbf{k}) e^{i \mathbf{k} \cdot \mathbf{x} } \; d\mathbf{k} . </math> </center>
 
<center> <math>  f(\mathbf{x}) = \frac{1}{(2 \pi)^n} \int_{-\infty}^{\infty} ... \mbox{n total integrations}... \int_{-\infty}^{\infty}  \tilde{f}(\mathbf{k}) e^{i \mathbf{k} \cdot \mathbf{x} } \; d\mathbf{k} . </math> </center>
  
 +
<!--T:49-->
 
Here we have used the conventions that <math> \mathbf{x} = (x_1,x_2,...,x_n),  </math> <math> d\mathbf{x} = dx_1 dx_2...dx_n,  </math> <math> \mathbf{k} = (k_1,k_2,...,k_n),  </math> <math> d\mathbf{k} = dk_1 dk_2...dk_n,  </math> and <math> \mathbf{k} \cdot \mathbf{x} = k_1 x_1 + k_2 x_2 + ... + k_n x_n. </math>  
 
Here we have used the conventions that <math> \mathbf{x} = (x_1,x_2,...,x_n),  </math> <math> d\mathbf{x} = dx_1 dx_2...dx_n,  </math> <math> \mathbf{k} = (k_1,k_2,...,k_n),  </math> <math> d\mathbf{k} = dk_1 dk_2...dk_n,  </math> and <math> \mathbf{k} \cdot \mathbf{x} = k_1 x_1 + k_2 x_2 + ... + k_n x_n. </math>  
  
  
== Transforms in 1 temporal and 3 spatial dimensions ==
+
== Transforms in 1 temporal and 3 spatial dimensions == <!--T:50-->
  
 
   
 
   
 +
<!--T:51-->
 
In 3 dimensions of space and 1 dimension of time, as is encountered in problems dealing with the wave equation, we have the forward Fourier transform from
 
In 3 dimensions of space and 1 dimension of time, as is encountered in problems dealing with the wave equation, we have the forward Fourier transform from
 
<math> (\mathbf{x}, t) \rightarrow (\mathbf{k},\omega) </math>
 
<math> (\mathbf{x}, t) \rightarrow (\mathbf{k},\omega) </math>
  
 +
<!--T:52-->
 
<center> <math>  \tilde{F}(\mathbf{k}, \omega) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}  \int_{0}^{\infty} f(\mathbf{x},t) e^{-i(\mathbf{k}\cdot\mathbf{x} - \omega t)} \; dt \, d^3 x </math> </center>
 
<center> <math>  \tilde{F}(\mathbf{k}, \omega) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}  \int_{0}^{\infty} f(\mathbf{x},t) e^{-i(\mathbf{k}\cdot\mathbf{x} - \omega t)} \; dt \, d^3 x </math> </center>
  
 +
<!--T:53-->
 
and the corresponding inverse Fourier transform
 
and the corresponding inverse Fourier transform
 
<center> <math>  f(\mathbf{x},t) = \frac{1}{(2\pi)^{n+1}} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}  \int_{\infty}^{\infty}
 
<center> <math>  f(\mathbf{x},t) = \frac{1}{(2\pi)^{n+1}} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}  \int_{\infty}^{\infty}
 
\tilde{F}(\mathbf{k},\omega) e^{i(\mathbf{k}\cdot\mathbf{x} - \omega t)} \; d\omega \, d^3 k. </math> </center>
 
\tilde{F}(\mathbf{k},\omega) e^{i(\mathbf{k}\cdot\mathbf{x} - \omega t)} \; d\omega \, d^3 k. </math> </center>
  
 +
<!--T:54-->
 
Here, <math> d^3 x = dx_1 dx_2 dx_3 </math> and <math> d^3 k = dk_1 dk_2 dk_3. </math>
 
Here, <math> d^3 x = dx_1 dx_2 dx_3 </math> and <math> d^3 k = dk_1 dk_2 dk_3. </math>
  
  
==References==
+
==References== <!--T:55-->
  
 +
<!--T:56-->
 
<references />
 
<references />
 
</translate>
 
</translate>

Latest revision as of 05:57, 15 June 2017

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F-19. Fourier transform pairs. The time functions on the left are Fourier transforms of the frequency functions on the right and vice-versa. Many more transform pairs could be shown. The above are all even functions and hence have zero phase. Transforms for real odd functions are imaginary, i.e., they have a phase shift of +π/2. Transforms of functions that are neither odd nor even involve variations of phase with frequency. Note f=1/t.

Formulas that convert a time function g(t) (waveform, seismic record trace, etc.) into its frequency-domain representation G(f) and vice versa. G(f) and g(t) constitute a Fourier-transform pair; see Figure F-19 below. An example is

The inverse transform is

Finding G(f) from g(t) is called Fourier analysis and finding g(t) from G(f) is called Fourier synthesis. G(f) is the complex spectrum, the real part being the cosine transform and the imaginary part the sine transform whenever g(t) is real. Another expression for G(f) is

where the functions A(f) and γ(f) are real. They are, respectively, the amplitude spectrum and the phase spectrum of g(t):

γ(f) is in the first or second quadrant if the imaginary part is positive, in the first or fourth quadrant if the real part is positive. A record trace h(t) that extends only from 0 to T may be assumed to be repeated indefinitely and so expanded in a Fourier series of period T:

where

and

The same rules for quadrants apply to γn as expressed for γ(f); a0 is the zero-frequency component (or dc shift). The frequency spectrum is discrete if the function is periodic. If ht is a sampled time series sampled at intervals of time t2, then we can stop summing when n>2T/t2 (see sampling theorem). In this case an and bn can be expressed as sums:

and

Also see phase response and fast Fourier transform.

F-20. Equivalence of Fourier transform operations. Doing the time operation is equivalent to doing the frequency operation on the transform of the data. Note: g(t)↔G(f) and h(t)↔H(f).

Operations in one domain have equivalent operations in the transform domain (see Figure F-20). Computations can sometimes be carried out more economically in one domain than the other and Fourier transforms provide a means of accomplishing this. The Fourier-transform relations can be generalized for more than one dimension (see Figure F-21). For example,

and

The 1/4π factor is sometimes distributed between the two integrals; where calculations involve an arbitrary scaling factor, the 1/4π factor may be dropped entirely.

Fourier transforms are discussed in Sheriff and Geldart (1995, 277, 532–533)[1]. Theorems relating to Fourier transforms are shown in Figure F-22.


F-22. Fourier transform theorems.

Notations and sign conventions

The notations and sign conventions used above are common in the electrical engineering world. Many exploration geophysicists may be familiar, however, with the conventions used by physicists and mathematicians. These differ subtly from those used above. Many geophysicists whose education may in those fields, or who draw from the scientific results from those fields prefer conventions different from those of electrical engineers. It is, therefore, important to include a short discussion of these conventions.

It is more common in the world of mathematics and theoretical physics for the convention of rather than . Electrical engineers prefer to use the letter because the letter is reserved for current. Mathematicians and mathematical physicists prefer using the angular frequency where the units are radians per time, rather than cycles per time. This means that there may be a factor of discrepancy between computations using the differing conventions. Finally, there may be different sign conventions on the exponent of the exponentials in the Fourier transform definitions. Because signals recorded in the space-time domain are causal, meaning that there are no arrivals before time the forward temporal transform integration will start at rather than .


1D transforms

Putting all of these together, we obtain a common notational convention for the forward Fourier transform in time as

and the inverse Fourier transform

The 1D spatial forward Fourier transform differs from the temporal transform in that the integration is on infinite limits and the sign of the exponent in the exponential is negative

and the inverse spatial Fourier transform is similarly different


Transforms in spatial dimensions

In dimensions, the spatial transforms become

and the inverse spatial Fourier transform is similarly different

Here we have used the conventions that and


Transforms in 1 temporal and 3 spatial dimensions

In 3 dimensions of space and 1 dimension of time, as is encountered in problems dealing with the wave equation, we have the forward Fourier transform from

and the corresponding inverse Fourier transform

Here, and


References

  1. Sheriff, R. E. and Geldart, L. P., 1995, Exploration Seismology, 2nd Ed., Cambridge Univ. Press.