Fourier transform - book

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	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 $e^{-2\pi if\Delta tn}$ (which has a negative sign) in their definition of the Fourier transform and not $e^{+2\pi if\Delta tn}$ (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

${\begin{aligned}B\left(f\right)=\sum _{n=-\infty }^{\infty }{b_{n}}e^{-2\pi jf\Delta tn}=A\left(f\right)e^{i\psi \left(f\right)}\ =\ A\left(f\right)e^{-i\phi \left(f\right)},\end{aligned}}$

(8)

where A(f) is the amplitude spectrum, where $\psi (f)$ is the phase-lead spectrum and where $\phi \left(f\right)$ 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 $e^{-2\pi if\Delta tn}$ in the Fourier transform carries a negative sign, the choice of $e^{-i\phi \left(f\right)}$, 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 $\phi \left(f\right)$ and not to the phase lead $\psi (f)$. For example, the Fourier transform of the unit delay ${\delta }_{n-{\rm {l}}}$ (which is 1 for n = 1 and 0 otherwise) is simply

${\begin{aligned}\sum _{n=-\infty }^{\infty }{{\delta }_{n-{\rm {l}}}}e^{-2\pi if\Delta tn}=e^{-2\pi if\Delta t}=A\left(f\right)e^{-i\ \phi \left(f\right)},\end{aligned}}$

(9)

so the phase spectrum (i.e., the phase-lag spectrum) is $\phi \left(f\right)=2\pi f\Delta t$.

The sampling frequency is $f_{s}={\rm {1/}}\Delta {\rm {t}}$. Thus,

${\begin{aligned}e^{-2\pi if_{\rm {s}}\Delta tn}\ =\ e^{-2\pi in}=1.\end{aligned}}$

(10)

Hence,

${\begin{aligned}&B\left(f+f_{s}\right)=\sum _{n=-\infty }^{\infty }{b_{n}}e^{-2\pi i\left(f+f_{s}\right)\Delta tn}=B\left(f\right),\end{aligned}}$

(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 $0\leq f<f_{s}$ instead of the more usual Nyquist range $-f_{s}/2\leq f<f_{s}{/2}$, to which we were introduced in Chapter 4.

If we define the time unit not as 1 s but as the sampling interval $\Delta t$ (so $\Delta t=1$), then $2\pi f\Delta t=2\pi f=\omega$. The variable $\omega$ represents angular frequency in units of cycles per radian. As a result, the Fourier transform becomes

${\begin{aligned}B\left(\omega \right)=\sum _{n=-\infty }^{\infty }{b_{n}}e^{-i\omega n}=A\left(\omega \right)e^{-i\phi \left(\omega \right)},\end{aligned}}$

(12)

where the Nyquist range is ${\rm {is}}-\pi \leq \omega <\pi$.

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

${\begin{aligned}B^{ZPR}\left(f\right)=\sum _{k=-\infty }^{\infty }{b_{-k}^{*}}e^{-2\pi if\Delta {\rm {t}}k}=\sum _{n=-}^{\infty }{b_{n}^{*}}e^{2\pi if\Delta m}=B^{*}\left(f\right)=A\left(f\right)e^{i\phi \left(f\right)}\end{aligned}}$

(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 $b=\ (b_{0},b_{1},b_{2},\dots ,b_{\rm {N}})$ of length N + 1, whose coefficients may be complex. Its Fourier transform reduces to the discrete Fourier transform (DFT)

${\begin{aligned}B\left(k\right)=\sum _{n=0}^{N-1}{b_{n}}w_{N}^{nk},\end{aligned}}$

(14)

where $w_{N}=e^{-2\pi {i/}N}$ for k 0, 1, …, N – 1. The inverse discrete Fourier transform (IDFT) is

${\begin{aligned}b_{n}={\frac {1}{N}}\sum _{k=0}^{N-1}{B}\left(k\right)w_{N}^{-'{\rm {t}}k}\end{aligned}}$

(15)

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

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Wavelets	Z-transform
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Frequency	Synthetics
Table of Contents (book)

Also in this chapter

• Wavelets
• Z-transform
• Delay: Minimum, mixed, and maximum
• Two-length wavelets
• Illustrations of spectra
• Delay in general
• Energy
• Autocorrelation
• Canonical representation
• Zero-phase wavelets
• Symmetric wavelets
• Ricker wavelet
• Appendix G: Exercises

External links

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