Seismic Data ProcessingLecture 4

Fourier Series and Fourier TransformPrepared by

Dr. Amin E. KhalilSchool of Physics, USM, Malaysia

Today's Agenda

• Examples on Fourier Series

• Definition of Fourier transform

•Examples on Fourier transform

Examples on Fourier Series

Example: 1

Solution:

The function f(x) is an odd function, thus the a- terms vanishes and the transform will be:

Increasing the number of terms we arrive at better approximation.

Another Example

The function is even function and thus:

Fourier-Discrete Functions

iN

xi2

.. the so-defined Fourier polynomial is the unique interpolating function to the function f(xj) with N=2m

it turns out that in this particular case the coefficients are given by

,...3,2,1,)sin()(2

,...2,1,0,)cos()(2

1

*

1

*

kkxxfN

b

kkxxfN

a

N

jjj

N

jjj

k

k

)cos(2

1)sin()cos(

2

1)( *

1

1

****0 kxakxbkxaaxg m

m

km kk

... what happens if we know our function f(x) only at the points

Fourier Spectrum

)(

)(arctan)(arg)(

)()()()(

)()()()(

22

)(

R

IF

IRFA

eAiIRF i

)(

)(

A Amplitude spectrum

Phase spectrum

In most application it is the amplitude (or the power) spectrum that is of interest.

Remember here that we used the properties of complex numbers.

When does the Fourier transform work?

Gdttf )(

Conditions that the integral transforms work:

f(t) has a finite number of jumps and the limits exist from both sides

f(t) is integrable, i.e.

Properties of the Fourier transform for special functions:

Function f(t) Fouriertransform F(w)

even even

odd odd

real hermitian

imaginary antihermitian

hermitian real

Some properties of the Fourier Transform

Defining as the FT: )()( Ftf

Linearity

Symmetry

Time shifting

Time differentiation

)()()()( 2121 bFaFtbftaf

)(2)( Ftf

)()( Fettf ti

)()()( Fi

t

tf nn

n

Time differentiation )()()( Fi

t

tf nn

n

Examples on Fourier Transform

Graphically the spectrum is:

Important applications of FT

• Convolution and Deconvolution

• Filtering

• Sampling of Seismic time series

Thank you