fourier transform transformations

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Introduction to Fourier Transform



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Transforms:

Mathematical transformations are applied to signals to obtain afurther information from that signal that is not readily available

in the raw signal.

There are number of transformations that can be applied,among which the Fourier transforms are probably by far the

most popular.



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Fourier transform:

Jean B. Joseph Fourier

(1768-1830)

³An arbitrary function, continuous or with

discontinuities, defined in a finite interval by an

arbitrarily capricious graph can always be expressed

as a sum of sinusoids´

J.B.J. Fourier

December, 21, 1807



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Fourier transform:

Original Signal Constituent Sinusoids of different frequencies



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Fourier transform:

For example the following signal

x(t)=cos(2*pi*10*t)+cos(2*pi*25*t)+cos(2*pi*50*t)+cos(2*pi*100*t)

is a stationary signal, because it has frequencies of 10, 25,50, and 100 Hz at any given time instant.

This signal is plotted below:



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Complex Signal Representation:



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FrequencyC

omponents:



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Fourier transform: If f(x) is a continuous function of a real variable x, then the

Fourier Transform of f(x), denoted by , is defined by theequation:

Given F(u), f(x) can be obtained by using the inverse FourierTransform:

? A ´g

g

!! dxe x f u F x f ux j T 2)()()(

? A)()( 1 u F x f !

1

2

? A´g

g

! duux ju F T 2exp)(

Eq (1) and (2) are collectively called as Fourier Transform Pair

? A)( x f



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Fourier transform: As Fourier transform is Complex function, it can be expressed

as:

F(u) = R(u) + j I(u)

Where R(u) and I(u) are, respectively, the real and imaginarycomponents of F(u).

3



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Fourier transform: It is often convenient to express eq: 3 in exponential

form(polar coordinates ):

The magnitude function is called the Fourier Spectrum of f(x) and J(u) its phase angle or phase spectrum of thetransform.

? A1/222)()()( u I u Ru !

¼½

»

¬-

«

!

)(

)(

tan)(1

u

u I

uJ

)(u F

5

6

Where:

and:

)()()( u jeuu J ! 4



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Fourier transform: The square of the spectrum, is often referred as ´Power

Spectrum´ of f(x).

Another common term used is ³Spectral Density´

2)()( u F u P !

= R 2(u) + I2(u)



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Fourier transform: The variable ³u´ appearing in the Fourier transform is often

called as ³Frequency Variable´.

This arises from the fact, that, If we expand the exponential

term by using Euler¶s formula, it is:

If we interpret the Integral in Eq:(1) as a limit-summation of

discrete terms, it is evident that F(u) is composed of an Infinitesum of Sine & Cosine terms, and that each value of ³u´ determines the frequency of its corresponding sine-cosine pair.

ux je T 2 =Cos(2ux) ± jSin(2ux)



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Fourier transform:

Fig (a) Fig (b)



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Fourier transform: Consider the function as shown in the Fig. (a) Its Fourier

transform is obtained from Eq. (1) as follows:

? A´

! d xux j x f u T 2exp)()(

? A´ ! X

dxux j A0

2exp T

? A ? A122

2

0

2 !

! uX j X ux j eu j

eu j

T T

T T

For simplification Multiply & Divide byu X j

eT



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Fourier transform:

? Au X

ju X

ju X

j

eeeu j

AT T T T ! 2

? A uX juX juX j e jeeu

T T T

T ! 2/)(

? A u X ju X ju X j eeeu j

A T T T

T

!

2

uX jeuX u

u T T T

! )sin()(? A UUU sin2/)( ! iee iiAs ===>



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Fourier transform: So we have obtained the Fourier transform, that is

Frequency Domain representation :

As F(u) is a complex term, we can find out the Fourier

Spectrum by:

uX jeuX u

u T T T

! )sin()(

uX

uX X

T T )sin(

!

A Plot of F(u) is shown in Fig (b) above.



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2-D Fourier transform: The Fourier transform can be easily extended to a

function f (x, y) of two variables.

If f (x, y) is continuous and F (u, v) is integrable, wehave that the following Fourier Transform pair :

? A

´

¢

¢́ !

!

d udvvyux j

evu y x f

vu y x f

)](2[),(),(

),(1

),(

T

? A

´ ´

!

!

d xdyvyux j

y x f vu

y x f vu

)](2[exp),(),(

),(),(

T 7

8



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2-D Fourier transform: As in the one ±dimensional case. , the Fourier spectrum,

phase, and power spectrum, respectively, are:

? A1/2

),(

2

),(

2

),( vu I vuvu F !

¼½

»¬-

«!

),(

),(tan),( 1

vu

vu I vuJ

),(2

),(22

),(),( vu I vuvu F vu P !! 11

10

9



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2-D Fourier transform:

Figure.2 (a) A two-dimensional function. (b) It¶s aFourier spectrum and (c) the spectrum displayed as anintensity function.



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2-D Fourier transform: Consider the function as shown in the Fig. 2(a) Its Fourier

transform is obtained from Eq. (1) as follows:

´ ´

! d xdyvyux j y x f vu )](2[exp),(),( T

? A ? A´ ´ !X

yd yvy jdxux j

0 02exp2exp T T

1

2



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2-D Fourier transform:Y

vy j X

ux j

u j

e

u j

e

0

2

0

2

22

¼

½

»¬

-

«

¼

½

»¬

-

«

!

T T

T T

]1[2

1]1[

2

1 22

! uY ju X j eu j

eu j

A T T

T T

¼½

»¬-

«¼½

»¬-

«!

uY

euY

u X

eu X A X Y

u X ju X j

T T

T T T T 22

)sin()sin(

3

4

5



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2-D Fourier transform:

¼½

»¬-

«¼½

»¬-

«!uY

uY

uX

uX X Y vu

T T

T T )sin()sin(

),( 6

As F(u,v) is a complex term, we can find out the FourierSpectrum by:



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fourier transform transformations

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