Page 223 - Acquisition and Processing of Marine Seismic Data

214 4. FUNDAMENTALS OF DATA PROCESSING

TABLE 4.1 Mathematical and Graphical Expressions and Important Properties of Generalized Functions Used in
Signal Processing, Where * Denotes Convolution, ℑ{} is Fourier Transform, a n and b n are Fourier Series Coefficients, ω is
Angular Frequency, T is Period, and i is ( 1) 1/2

Function Graphic Specifications
Even functions T=2 T=2
R R
f(t)¼f( t) ftðÞdt ¼ 2 ftðÞdt
T=2 0
In Fourier series summation b n ¼ 0

Odd functions T=2
R
f( t)¼ f(t) ftðÞdt ¼ 0
T=2
f(0) ¼ 0
In Fourier series summation a n ¼ 0

Dirac Delta Function R ∞
δ tðÞdt ¼ 1
1 t ¼ 0 ∞
δ tðÞ ¼
0 t 6¼ 0 δ( t)¼δ(t)
∗
f(t) δ(t)¼f(t)
f(t)δ(t)¼f(0)δ(t)

Unit step function R ∞ R ∞
utðÞϕ tðÞdt ¼ ϕ tðÞdt
1 t 0 ∞ 0
utðÞ ¼
0 t < 0 du tðÞ
dt ¼ δ tðÞ

Box-Car function ℑ{w(t)}¼sinc(ω)

10 t < t 0
wtðÞ ¼
0 other

Sinc function ℑ{sinc(t)}¼w(ω)
sin tðÞ sinc(0) ¼ 1
sinc tðÞ ¼
t

Signum function sign( t)¼ sign(t)
2
1 t 0 ℑ sign tðÞ ¼
1 iω
sign tðÞ ¼ ¼
t jj 1 t > 0