Inverse Fourier transform
 | |
| Series | Geophysical References Series |
|---|---|
| Title | Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing |
| Author | Enders A. Robinson and Sven Treitel |
| Chapter | 6 |
| DOI | http://dx.doi.org/10.1190/1.9781560801610 |
| ISBN | 9781560801481 |
| Store | SEG Online Store |
The Fourier equations can be simplified by choosing the discrete time spacing to define one unit of time - that is, by choosing $ \Delta t={1} $. Then the angular frequency 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/":): \omega=2\pi f\Delta t{=2}\pi f . The equation connecting the discrete-time integer n of the digital signal with the true time scale t of the continuous-time signal 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/":): t=n\Delta t=n . If we use equation 15 of Chapter 4, we see that the Nyquist 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/":): f_n={1/2} and the Nyquist angular 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/":): {\omega }_n=2\pi \left({1/2}\right)=\pi . The discrete Fourier transform of the signal (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/":): (b_0) , 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/":): b_{{1}} , 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/":): b_{2} …, 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/":): b_N ) is
$ {\begin{aligned}B\left(\omega \right)=b_{0}+b_{l}e^{-i\omega }+b_{2}e^{-i\omega 2}+\dots +b_{N}e^{-i\omega N}.\end{aligned}} $ ()
More generally, the discrete Fourier transform of the signal 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/":): x_n , which can be infinitely long in both directions, 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/":): \begin{align} X\left(\omega \right)=\sum^{\infty }_{n=-}{x_n}e^{-i\omega n}={|}X\left(\omega \right){|}e^{-\rho \left(\omega \right)}. \end{align} ()
We see that the spectrum 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/":): X\left(\omega \right) is obtained from the geophysical Z-transform X(Z) by replacing Z with 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/":): e^{-i\omega } . Use of the same symbol X for both the Z-transform and the spectrum is commonplace. The inverse Fourier transform is defined as
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/":): \begin{align} x_n=\frac{{l}} {2\pi }\int^{\pi }_{-\pi }{X}\left(\omega \right)e^{i\omega n}d\omega . \end{align} ()
To verify the assertion that equation 34 is correct, we substitute equation 33 into equation 34 and obtain
$ {\begin{aligned}x_{n}={\frac {1}{2\pi }}\int \limits _{-\pi }^{\pi }{\left({\sum \limits _{m=-\infty }^{\infty }{x_{m}e^{-i\omega m}}}\right)e^{i\omega \;n}\;d\omega =\sum \limits _{m=-\infty }^{\infty }{x_{m}\left({{\frac {1}{2\pi }}\int \limits _{-\pi }^{\pi }{e^{i\omega \;(n-m)}}d\omega }\right).}}\end{aligned}} $ ()
The expression in parentheses on the right side of equation 35 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/":): \begin{align} \frac{1} {{2\pi }} \int\limits_{ - \pi }^\pi {e^{i\omega (n - m)} \;d\omega = \frac{{\sin \pi (n - m)}} {{\pi (n - m)}} = \delta _{n - m} = \left\langle {\begin{array}{*{20}c} 1 & {\text{if}} & {n = m} \\ 0 & {\text{if}} & {n \ne m} \\ \end{array} .} \right.} \end{align} ()
Thus, equation 35 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/":): \begin{align} x_n = \sum\limits_{m = - \infty }^\infty {x_m \delta _{n - m} = x_n ,} \end{align} ()
which takes us full circle and thereby establishes the assertion.
Let us use the inverse Fourier transform to find the coefficients of an ideal low-pass filter with passband 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/":): 0\le{|}\omega {|}\le \Omega and stop band 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/":): \Omega {<}{|}\omega {|}\le \pi . The coefficients are given by the inverse Fourier transform as
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/":): \begin{align} h_n = \frac{1} {{2\pi }} \int\limits_{ - \Omega }^\Omega {e^{i\omega n} } d\omega = \frac{{\sin \Omega n}} {{\pi n}} ,\;\;n = 0, \pm 1,\; \pm 2,\; \ldots \end{align} ()
The ideal band-pass filter for passband 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/":): {\Omega }_{{1}}\le {|}\omega {|}\le {\Omega }_{2} and stop bands 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/":): 0\le {|}\omega {|}\le {\Omega }_{{l}} and 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/":): \Omega _2 \le \;|\omega |\; \le \;\pi is given as the difference between two low-pass filters. Thus, the coefficients are
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/":): \begin{align} h_n=\frac{\mathrm{sin}\Omega_{2}n}{\pi n}-\frac{\mathrm{sin}\Omega_{1}n}{\pi n}, n =0, \pm {1,} \pm {2,}... \end{align} ()
Our present discussion has been concerned primarily with the underlying principles of digital filtering. We have shown in some detail how causal FIR filters operate on a discrete input to yield a discrete output. We have found that the mechanics of this process can be visualized with greatest ease in the time domain, but usually it is necessary to think of filtering in terms of both the time domain and the frequency domain. We have attempted to present a thorough although heuristic description of filter behavior in both these domains. Finally, we have introduced the minimum-phase concept for classifying digital filters.
Continue reading
| Previous section | Next section |
|---|---|
| Minimum-phase spectrum | Appendix F: Exercises |
| Previous chapter | Next chapter |
| Filtering | Wavelets |
Also in this chapter
- Frequency spectrum
- Magnitude spectrum and phase spectrum
- Fourier transform
- Minimum-phase spectrum
- Appendix F: Exercises
External links
| find literature about Inverse Fourier transform/en |
|
|
|
|
|
|
|