Fourier series

Problems in Exploration Seismology and their Solutions

Series	Geophysical References Series
Title	Problems in Exploration Seismology and their Solutions
Author	Lloyd P. Geldart and Robert E. Sheriff
Chapter	9
Pages	295 - 366
DOI	http://dx.doi.org/10.1190/1.9781560801733
ISBN	ISBN 9781560801153
Store	SEG Online Store

Problem 9.1a

Show that the Fourier series coefficients, $a_{n}$ and $b_{n}$ in equation (9.1a), are given by equations (9.1d,e).

Background

If $g(t)$ is a periodic function, that is, one that repeats exactly at intervals $T$ (called the period) and has a finite number of maxima, minima, and discontinuities in each interval $T$, then it can be expressed in a Fourier series in any one of the three following equivalent forms, $n$ being an integer:

${\begin{aligned}g(t)={\frac {1}{2}}a_{0}+\sum \limits _{n=1}^{+\infty }(a_{n}\cos \omega _{n}t+b_{n}\sin \omega _{n}t)\end{aligned}}$

(9.1a)

${\begin{aligned}={\frac {1}{2}}c_{0}+\sum \limits _{n=1}^{+\infty }c_{n}\cos(\omega _{n}t-\gamma _{n})\end{aligned}}$

(9.1b)

${\begin{aligned}=\sum \limits _{n=-\infty }^{+\infty }\alpha _{n}e^{{\hbox{j}}\omega _{n}t},\end{aligned}}$

(9.1c)

where $\omega _{0}=2\pi f_{0}=2\pi /T=$ fundamental angular frequency, $\omega _{n}=n\omega _{0}=$ harmonic frequency, and the amplitudes and phases are given by the following equations where $0<n\leq +\infty$, except for equation (9.1h) where $-\infty \leq n\leq +\infty$:

${\begin{aligned}a_{n}={\frac {2}{T}}\int _{-T/2}^{+T/2}g(t)\cos \,\omega _{n}t\,\mathrm {d} t,\end{aligned}}$

(9.1d)

${\begin{aligned}b_{n}={\frac {2}{T}}\int _{-T/2}^{+T/2}g(t)\sin \,\omega _{n}t\,\mathrm {d} t,\end{aligned}}$

(9.1e)

${\begin{aligned}c_{n}={\frac {2}{T}}\int _{-T/2}^{+T/2}g(t)\cos \,(\omega _{n}t-\gamma _{n})\,\mathrm {d} t,\end{aligned}}$

(9.1f)

${\begin{aligned}\gamma _{n}=\tan ^{-1}(b_{n}/a_{n})=\mathrm {phase} ,\gamma _{0}=0,\end{aligned}}$

(9.1g)

${\begin{aligned}\alpha _{n}={\frac {1}{T}}\int _{-T/2}^{+T/2}g(t)e^{-{\hbox{j}}\omega _{n}t}\mathrm {d} t.\end{aligned}}$

(9.1h)

Solution

We multiply both sides of equation (9.1a) by ${\rm {\;cos\;}}\omega _{r}t$, $r$ being any nonzero positive integer, and integrate between the limits $-T/2$ and $+T/2$. (We omit the limits on the integrals in the following proofs since they are all the same.) Equation (9.1a) now becomes

${\begin{aligned}\int g(t)\cos \omega _{r}t\mathrm {d} t&=\sum \limits _{n=1}^{+\infty }a_{n}\int \cos \,\omega _{r}t\cos \omega _{n}t\,\mathrm {d} t\\&+\sum \limits _{n=1}^{+\infty }b_{n}\int \cos \omega _{r}t\sin \omega _{n}t\,\mathrm {d} t.\end{aligned}}$

By using the identities

${\begin{aligned}\cos x\cos y={\frac {1}{2}}[\cos(x+y)+\cos(x-y)]\end{aligned}}$

${\begin{aligned}{\hbox{and}}\quad \quad \quad \cos x\sin y={\frac {1}{2}}[\sin(x+y)-\sin(x-y)],\end{aligned}}$

the first integrand reduces to the sum of two cosines of the form $\cos(\omega _{r}\pm \omega _{n})t$ while the second integrand becomes the difference between two sines of the form $\sin(\omega _{r}\pm \omega _{n})t.$ If $r\neq n$, the integrals are zero, because at the limits the arguments of the cosines and sines are equal to $2p\omega _{0}T/2=2p\pi$, $p$ integral. When $r=n$, the first integrand becomes $\cos ^{2}\omega _{n}t={\frac {1}{2}}(1+\cos \,2\omega _{n}t)$ and the right-hand side equals $a_{n}T/2$. When $r=n$, the second integrand becomes $\cos \omega _{n}t\sin \omega _{n}t\,\mathrm {d} t={\frac {1}{2}}\sin \,2\omega _{n}t\,\mathrm {d} t$, which again gives zero upon integration.

Thus we are left with

${\begin{aligned}\int g(t)\cos \omega _{n}t\,\mathrm {d} t=a_{n}T/2,\quad n=0,1,2,\ldots .\end{aligned}}$

Solving for $a_{n}$ gives equation (9.1d).

To verify equation (9.1e), we multiply both sides of equation (9.1a) by $\sin \omega _{r}t$ and integrate, all integrals on the right side of the equation giving zero except for the one with the integrand $b_{n}\sin ^{2}\omega _{n}t=(b_{n}/2)(1-\cos \,2\omega _{n}t)$. Proceeding as before, we arrive at equation (9.1e).

When we set $n=0$ in equations (9.1d,e) we get $b_{0}=0$ and

${\begin{aligned}a_{0}/2={\frac {1}{T}}\int g(t)\,\mathrm {d} t=\mathrm {average\ value\ of} \,g(t).\end{aligned}}$

Problem 9.1b

Verify equations (9.1f,g) for the Fourier series coefficients $c_{n}$ and $\gamma _{n}$.

Solution

To verify equation (9.1f) for $c_{n}$, we multiply both sides of equation (9.1b) by $\cos(\omega _{r}t-\gamma _{r})$, $r\neq 0$, and proceed as in part (a). All integrals vanish except when $r=n$, and the result is equation (9.1f).

When $r=0$, we have

${\begin{aligned}\int g(t)\,\mathrm {d} t={\frac {c_{0}}{2}}\int \mathrm {d} t={\frac {c_{0}T}{2}},\\{\hbox{so}}\quad \quad \quad {\frac {c_{0}}{2}}={\frac {1}{T}}\int g(t)\,\mathrm {d} t=\mathrm {average\ value\ of} \ g(t).\end{aligned}}$

To verify equation (9.1g) for $\gamma _{n}$, we compare terms in equations (9.1a) and (9.1b) that involve $\omega _{n}t$, $n\neq 0$. The result is

${\begin{aligned}a_{n}\cos \omega _{n}t+b_{n}\sin \omega _{n}t&=c_{n}\cos(\omega _{n}t-\gamma _{n})\\&=c_{n}(\cos \omega _{n}t\,\cos \,\gamma _{n}+\sin \omega _{n}t\,\sin \,\gamma _{n}).\end{aligned}}$

Equating coefficients of $\cos \omega _{n}t$ and $\sin \omega _{n}t$, we get

${\begin{aligned}a_{n}=c_{n}\cos \gamma _{n},\quad b_{n}=c_{n}\sin \gamma _{n},\quad c_{n}^{2}=a_{n}^{2}+b_{n}^{2},\quad \tan \gamma _{n}=b_{n}/a_{n}.\end{aligned}}$

When $n=0$, $a_{0}=c_{0}$, $b_{0}=0$, $\gamma _{0}=0$.

Problem 9.1c

Verify the coefficients for the exponential form, equation (9.1h).

Solution

To derive equation (9.1h), we use Euler’s equations (Sheriff and Geldart, 1995, problem 15.12a), namely,

${\begin{aligned}\cos x={\frac {1}{2}}(e^{{\hbox{j}}x}+e^{-{\hbox{j}}x}),\quad \sin \,x={\frac {1}{2{\hbox{j}}}}(e^{{\hbox{j}}x}-e^{-{\hbox{j}}x}),\\{\hbox{or}}\quad \quad \quad e^{{\hbox{j}}x}=(\cos x+{\hbox{j}}\sin x),\quad e^{-{\hbox{j}}x}=(\cos x-{\hbox{j}}\sin x).\end{aligned}}$

Multiplying both sides of equation (9.1c) by $e^{-{\hbox{j}}\omega _{r}t}$, $r\neq 0$, and integrating, we get a series of integrals with integrands such as $e^{{\hbox{j}}(\omega _{n}-\omega _{r})t}$. Using Euler’s formula, $e^{{\hbox{j}}(\omega _{n}-\omega _{r})t}={\hbox{cos}}(\omega _{n}-\omega _{r})t+{\hbox{j}}\ {\hbox{sin}}(\omega _{n}-\omega _{r})t$, so the integrals vanish for all values of $n$ except when $n=r$; for this value we get

${\begin{aligned}\int g(t)e^{-{\hbox{j}}\omega _{n}t}\,\mathrm {\rm {d}} t=\alpha _{n}\int \mathrm {d} t=\alpha _{n}T,\end{aligned}}$

which is equation (9.1h), $n\neq 0$. For $n=0$,

${\begin{aligned}\alpha _{0}={\frac {1}{T}}\int g(t)\,\mathrm {d} t=\mathrm {average\ value\ of} \,g(t).\end{aligned}}$

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Table of Contents (book)

Also in this chapter

• Fourier series
• Space-domain convolution and vibroseis acquisition
• Fourier transforms of the unit impulse and boxcar
• Extension of the sampling theorem
• Alias filters
• The convolutional model
• Water reverberation filter
• Calculating crosscorrelation and autocorrelation
• Digital calculations
• Semblance
• Convolution and correlation calculations
• Properties of minimum-phase wavelets
• Phase of composite wavelets
• Tuning and waveshape
• Making a wavelet minimum-phase
• Zero-phase filtering of a minimum-phase wavelet
• Deconvolution methods
• Calculation of inverse filters
• Inverse filter to remove ghosting and recursive filtering
• Ghosting as a notch filter
• Autocorrelation
• Wiener (least-squares) inverse filters
• Interpreting stacking velocity
• Effect of local high-velocity body
• Apparent-velocity filtering
• Complex-trace analysis
• Kirchhoff migration
• Using an upward-traveling coordinate system
• Finite-difference migration
• Effect of migration on fault interpretation
• Derivative and integral operators
• Effects of normal-moveout (NMO) removal
• Weighted least-squares

External links

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