Inverse Fourier transform

Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing

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 $\omega =2\pi f\Delta t{=2}\pi f$. The equation connecting the discrete-time integer n of the digital signal with the true time scale t of the continuous-time signal is $t=n\Delta t=n$. If we use equation 15 of Chapter 4, we see that the Nyquist frequency is $f_{n}={1/2}$ and the Nyquist angular frequency is ${\omega }_{n}=2\pi \left({1/2}\right)=\pi$. The discrete Fourier transform of the signal ($(b_{0})$, $b_{1}$, $b_{2}$ …, $b_{N}$) is

${\begin{aligned}B\left(\omega \right)=b_{0}+b_{l}e^{-i\omega }+b_{2}e^{-i\omega 2}+\dots +b_{N}e^{-i\omega N}.\end{aligned}}$

(32)

More generally, the discrete Fourier transform of the signal $x_{n}$, which can be infinitely long in both directions, is

${\begin{aligned}X\left(\omega \right)=\sum _{n=-}^{\infty }{x_{n}}e^{-i\omega n}={|}X\left(\omega \right){|}e^{-\rho \left(\omega \right)}.\end{aligned}}$

(33)

We see that the spectrum $X\left(\omega \right)$ is obtained from the geophysical Z-transform X(Z) by replacing Z with $e^{-i\omega }$. Use of the same symbol X for both the Z-transform and the spectrum is commonplace. The inverse Fourier transform is defined as

${\begin{aligned}x_{n}={\frac {l}{2\pi }}\int _{-\pi }^{\pi }{X}\left(\omega \right)e^{i\omega n}d\omega .\end{aligned}}$

(34)

To verify the assertion that equation 34 is correct, we substitute equation 33 into equation 34 and obtain

${\begin{aligned}x_{n}={\frac {1}{2\pi }}\int \limits _{-\pi }^{\pi }{\left({\sum \limits _{m=-\infty }^{\infty }{x_{m}e^{-i\omega m}}}\right)e^{i\omega \;n}\;d\omega =\sum \limits _{m=-\infty }^{\infty }{x_{m}\left({{\frac {1}{2\pi }}\int \limits _{-\pi }^{\pi }{e^{i\omega \;(n-m)}}d\omega }\right).}}\end{aligned}}$

(35)

The expression in parentheses on the right side of equation 35 is

${\begin{aligned}{\frac {1}{2\pi }}\int \limits _{-\pi }^{\pi }{e^{i\omega (n-m)}\;d\omega ={\frac {\sin \pi (n-m)}{\pi (n-m)}}=\delta _{n-m}=\left\langle {{\begin{array}{*{20}c}1&{\text{if}}&{n=m}\\0&{\text{if}}&{n\neq m}\\\end{array}}.}\right.}\end{aligned}}$

(36)

Thus, equation 35 becomes

${\begin{aligned}x_{n}=\sum \limits _{m=-\infty }^{\infty }{x_{m}\delta _{n-m}=x_{n},}\end{aligned}}$

(37)

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 $0\leq {|}\omega {|}\leq \Omega$ and stop band $\Omega {<}{|}\omega {|}\leq \pi$. The coefficients are given by the inverse Fourier transform as

${\begin{aligned}h_{n}={\frac {1}{2\pi }}\int \limits _{-\Omega }^{\Omega }{e^{i\omega n}}d\omega ={\frac {\sin \Omega n}{\pi n}},\;\;n=0,\pm 1,\;\pm 2,\;\ldots \end{aligned}}$

(38)

The ideal band-pass filter for passband ${\Omega }_{1}\leq {|}\omega {|}\leq {\Omega }_{2}$ and stop bands $0\leq {|}\omega {|}\leq {\Omega }_{l}$ and $\Omega _{2}\leq \;|\omega |\;\leq \;\pi$ is given as the difference between two low-pass filters. Thus, the coefficients are

${\begin{aligned}h_{n}={\frac {\mathrm {sin} \Omega _{2}n}{\pi n}}-{\frac {\mathrm {sin} \Omega _{1}n}{\pi n}},n=0,\pm {1,}\pm {2,}...\end{aligned}}$

(39)

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.

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Filtering	Wavelets
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Also in this chapter

• Frequency spectrum
• Magnitude spectrum and phase spectrum
• Fourier transform
• Minimum-phase spectrum
• Appendix F: Exercises

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