Page 227 - Acquisition and Processing of Marine Seismic Data

218 4. FUNDAMENTALS OF DATA PROCESSING

FIG. 4.5 Approximation to a box-car function using Fourier series expansion. (A) The box-car function given by Eq. (4.8),
and its Fourier series expansion for (B) 1 term (n ¼ 1), (C) first 2 terms (n ¼ 2), (D) first 6 terms (n ¼ 6), and (E) first 10 terms
(n ¼ 10) of Eq. (4.10). DC component, a 0 for n ¼ 0, is 1/2.

8
0 π < t π=2 Substituting Eq. (4.9) into Eq. (4.6), we get.
<
ftðÞ ¼ 1 π=2 < t π=2 (4.8)
0
: π=2 < t π 1 2 cos 3tðÞ cos 5tðÞ cos 7tðÞ
ftðÞ ¼ + cos tðÞ +
2 π 3 5 7
The period of this function is 2π and its angular

frequency is ω ¼ 2π/T ¼ 2π/2π ¼ 1. Since + cos 9tðÞ ⋯ (4.10)
f( t) ¼ f(t), box-car is an even function 9
(Fig. 4.5A), and therefore, all b n coefficients are Eq. (4.10) is actually an approximation to the
zero. Using Eq. (4.7), a 0 and a n can be obtained as periodic box-car function given in Eq. (4.8) by
a 0 ¼ 1=2 summing infinite number of sinusoids with dif-
8
0 n even ferent phase, amplitude and frequency values.

2 π < k
ð
a n ¼ sin n ¼ 2 1Þ In practice, the more terms are included in the
nπ 2 : n ¼ 2k +1, k ¼ 0,1,2,…
π 2k +1Þ summation (i.e., the higher the maximum n
ð
(4.9) value is incorporated), the closer the result to