Transformada de Fourier

From SEG Wiki
Jump to navigation Jump to search
This page is a translated version of the page Fourier transform - book and the translation is 53% complete.
ADVERTISEMENT
Other languages:
Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
DigitalImaging.png
Series Geophysical References Series
Title Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
Author Enders A. Robinson and Sven Treitel
Chapter 7
DOI http://dx.doi.org/10.1190/1.9781560801610
ISBN 9781560801481
Store SEG Online Store

What is the Fourier transform (or spectrum) of a wavelet? Recall that in Chapter 6, we learned about the Fourier transform and its many applications for using a system’s time function to determine the system’s frequency function. Now we will extend this discussion and give more applications. The time functions in question will be Wavelets.

Geophysicists follow the same convention that electrical engineers use with respect to the Fourier transform. This practice is wise because electrical engineers write most of the books on digital filtering. Specifically, electrical engineers choose to use Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle e^{-2\pi if\Delta tn} } (which has a negative sign) in their definition of the Fourier transform and not Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle e^{+2\pi if\Delta tn}} (which has a positive sign), as mathematicians usually do. The variable f represents cyclical frequency in units of hertz (i.e., cycles per second). The electrical engineering version of the Fourier transform (or spectrum) of a two-sided wavelet b is


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} B\left(f\right)=\sum^{\infty }_{n=-\infty }{b_n}e^{-2\pi jf\Delta tn}=A\left(f\right)e^{i\psi \left(f\right)}\ =\ A\left(f\right)e^{-i\phi \left(f\right)}, \end{align}} (8)

where A(f) is the amplitude spectrum, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \psi (f) } is the phase-lead spectrum and where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi \left(f\right) } is the phase-lag spectrum. The phase-lag spectrum is the negative of the phase-lead spectrum. Either one can be called the phase spectrum. However, because the exponential Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle e^{-2\pi i f\Delta tn} } in the Fourier transform carries a negative sign, the choice of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle e^{-i\phi \left(f\right)} } , which also has a negative sign in the exponential, seems to be favored. For that reason, we choose to let the word phase refer to the phase lag Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi \left(f\right) } and not to the phase lead Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \psi (f)} . For example, the Fourier transform of the unit delay Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\delta }_{n-{\rm l}} } (which is 1 for n = 1 and 0 otherwise) is simply


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \sum^{\infty }_{n=-\infty}{{\delta }_{n-{\rm l}} }e^{-2\pi if\Delta tn}=e^{-2\pi if\Delta t}=A\left(f\right)e^{-i\ \phi \left(f\right)}, \end{align}} (9)

so the phase spectrum (i.e., the phase-lag spectrum) is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi \left(f\right)=2\pi f\Delta t} .

The sampling frequency is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_s={\rm 1/}\Delta {\rm t} } . Thus,


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} e^{-2\pi if_{{\rm s}} \Delta tn}\ =\ e^{-2\pi in}=1. \end{align}} (10)

Hence,


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} &B\left(f+f_s\right)=\sum^{\infty }_{n=-\infty}{b_n}e^{-2\pi i\left(f+f_s\right)\Delta tn}=B\left(f\right) , \end{align}} (11)

which shows that the spectrum is periodic with a period equal to the sampling frequency. Often, it is more convenient to compute the spectrum over the range Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 0\le f<f_s } instead of the more usual Nyquist range Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle -f_s/2\le f<f_s{/2} } , to which we were introduced in Chapter 4.

If we define the time unit not as 1 s but as the sampling interval Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Delta t } (so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Delta t=1 } ), then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 2\pi f\Delta t=2\pi f=\omega } . The variable Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \omega} represents angular frequency in units of cycles per radian. As a result, the Fourier transform becomes


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} B\left(\omega \right)=\sum^{\infty }_{n=-\infty}{b_n}e^{-i\omega n}=A\left(\omega \right)e^{-i\phi \left(\omega \right)}, \end{align}} (12)

where the Nyquist range is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\rm is}-\pi \le \omega <\pi } .

What is the Fourier transform (or spectrum) of the zero-point reverse wavelet? The spectrum of the zero-point reverse of the wavelet b is


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} B^{ZPR}\left(f\right)=\sum^{\infty }_{k=-\infty }{b^*_{-k}} e^{-2\pi if\Delta {\rm t}k}=\sum^{\infty }_{n=-}{b^{*}_n}e^{2\pi if\Delta m}=B^*\left(f\right)=A\left(f\right)e^{i\phi \left(f\right)} \end{align}} (13)

Thus, the zero-point reverse wavelet has an amplitude spectrum that is the same as the wavelet’s amplitude spectrum, and it has a phase spectrum that is the negative the wavelet’s phase spectrum.

What is the discrete Fourier transform? Often, we deal with a finite wavelet Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle b=\ (b_0, b_{1}, b_{2},\dots, b_{{\rm N}})} of length N + 1, whose coefficients may be complex. Its Fourier transform reduces to the discrete Fourier transform (DFT)


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} B\left(k\right)=\sum^{N-1}_{n=0}{b_n}w^{nk}_N, \end{align}} (14)

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle w_N=e^{-2\pi {i/}N} } for k 0, 1, …, N – 1. The inverse discrete Fourier transform (IDFT) is


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} b_n=\frac{1}{N}\sum^{N-1}_{k=0}{B}\left(k\right)w^{-'{\rm t}k}_N \end{align}} (15)

for n = 0, 1, …, N – 1. The fast Fourier transform (FFT) (Cooley and Tukey, 1965[1]) gives a computationally efficient way of computing the DFT and its inverse.


Referencias

  1. Cooley, J. W., and J. W. Tukey, 1965, An algorithm for the machine calculation of complex Fourier series: Mathematics of Computation, 19, 297–301.

Sigue leyendo

Sección previa Siguiente sección
Ondículas Transformada Z
Capítulo previo Siguiente capítulo
Frecuencia Sintéticos

Tabla de contenido


También en este capítulo


Vínculos externos

find literature about
Fourier transform - book/es
SEG button search.png Datapages button.png GeoScienceWorld button.png OnePetro button.png Schlumberger button.png Google button.png AGI button.png