Fourier Transform in Spectroscopy

8/8/2019 Fourier Transform in Spectroscopy

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Fourier Transform in

Spectroscopy

Fourier series representation

Fourier Transform Pair (FTP)

General properties of FTP Discrete Fourier Transform

Truncation and sampling of signal and

its effect

Fast Fourier Transform (FFT)

Naween Anand

Dept. of Physics

University of Florida


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Jean Baptiste Joseph Fourier (1768-1830)

Had crazy idea (1807):

Any periodic function can berewritten as a weighted sumofSines and Cosines ofdifferent frequencies.

Dont believe it? Neither did Lagrange,

Laplace, Poisson andother big wigs

Not translated intoEnglish until 1878!

But its true! called Fourier Series

Possibly the greatest tool

used in Engineering


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Fourier Series

Definition

Any periodic function, if it is

(1) piecewise continuous;

(2) square-integrable in one period,

it can be decomposed into a sum ofsinusoidal and

cosinoidal component functions---Fourier Series


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Fourier Series

f(t) is periodic with period T,

Here,

It is the nth harmonic (angular frequency) of the function f

!2

2cos

2 T

Tnn dtttf

Ta [

!

2

2sin

2 T

Tnn dtttf

Tb [

Tnn

T[

2!

? Ag

!

!1

0 sincos

2 nnnnn tbta

atf [[


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Fourier Series

In exponential form

where, the fourier coefficients are

Relationship between an,bn and cn

2nnn ibac !

g

g!

!n

ti

nnectf

[

!2

2

1 T

T

ti

n dtetfT

c n[


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Fourier Transform Pair (FTP)

p

p

ddd!

pp

ddd!

[[[

[[

dtitdtitftf

dfT

T

titdtitfT

tf

n

n

n

n

T

T

)exp(])exp()([2

1)(

givesThis

1andLet

)exp(})exp()(1

{)(

-

2/

2/


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Continue

We define

and constitute a Fourier Transform Pair.

Transform.FourierinversecalledisOperation

)}({)exp()()(

SimilarlyTransform.FouriercalledisOperation

)}({)exp()(2

1)(

1-

1

!!

!!

tfdttitfF

FdtiFtf

[[

[[[[

)(tf )([F


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The Fourier Transform and its

Inverse

Signalp

Spectrump( ) ( ) exp( )F f t i t dt [ [

g

g

! 1( ) ( ) exp( )

2 f t F i t d [ [ [

g

g

!


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0 0 . 2 0 . 4 0 .6 0 . 8 1 1 . 2 1 . 4 1 .6 1 . 8 2-2

-1

0

1

2

0 2 0 4 0 6 0 8 0 1 00 1 200

50

10 0

15 0

20 0

25 0

30 0

Famous Fourier Transforms

Sine wave

Delta function


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0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 00

0

1

0

2

0

3

0

4

0

5

0 5 0 1 00 1 50 2 00 2 500

1

2

3

4

5

6

Gaussian

Gaussian


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-1 -0 8 -0 6 -0 4 -0 2 0 0 2 0 4 0 6 0 8 1-0 5

0

0

5

1

1

5

-100 -5 0 0 5 0 1 000

1

2

3

4

5

6

Sinc function

Square wave


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Exponential

Lorentzian

0 5 0 1 00 1 50 2 00 2 500

5

10

15

20

25

30

0 0

2 0

4 0

6 0

8 1 1

2 1

4 1

6 1

8 20

0

2

0

4

0

6

0

8

1


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Properties of FT Scaling theorem

Addition theorem

Shift theorem

Derivative theorem

Modulation theorem

Parsevals theorem

Convolution theorem

Autocorrelation theorem


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Scale Theorem

The Fourier transformof a scaled function,f(at): { ( )} ( / ) / f at F a a[!F

{ ( )} ( ) exp( ) f at f at i t dt [

g

g

! F

{ ( )} ( ) exp( [ / ]) / f at f u i u a du a[g

g

! F

( ) exp( [ / ] ) / f u i a u du a[

g

g

! ( / ) / F a a[!

Ifa < 0, the limits flip when we change variables, introducing a

minus sign, hence the absolute value.

Assuming a > 0, change variables: u = at, so dt = du / a

Proof:


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The Scale

Theoremin action

f(t) F([)

Shortpulse

Medium-lengthpulse

Longpulse

The shorterthe pulse,

the broader

the spectrum!

This is the essenceof the UncertaintyPrinciple!

[

[

[

t

t

t


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The Fourier

Transform of a

sum of two

functions

{ ( ) ( )}

{ ( )} { ( )}

a f t bg t

a f t b g t !

F

F F

Also, constants factor out.

The Fourier transform is alinear function of functions.

f(t)

g(t)

t

t

t

[

[

[

F([)

G([)

f(t)+g(t)

F([)+G([)


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Shift Theorem

_ a

( ) :

( ) exp( )

( ) exp( [ ])

exp( ) ( ) exp( )

f t a

f t a f t a i t d t

u t a du dt

f u i u a du

i a f u i u du

[

[

[ [

g

g

g

g

g

g

!

! !

!

T ri r tr sf r f s ift f cti ,

r f :

ri l s : s

F

exp( ) ( )i a F[ [!

_ a( ) exp( ) ( ) f t a i a F [ [ ! F


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Application of the Shift Theorem

Suppose were measuring the spectrum ofE(t), but a small fraction

of its irradiance, say I, takes a longer path to the spectrometer.The extra light has the field, I E(ta), where a is the extra path.

The measured spectrum is:

2

( ) { ( ) ( )}S E t E t a[ I! F

2

( ) exp( ) ( )E i a E[ I [ [! % %

22( ) 1 exp( )E i a[ I [! %

22

( ) 1 cos( ) sin( )E a i a[ I [ I [! %

_ a2 22

( ) 1 cos( ) sin( )E

a a[I

[I

[

! - - %

E(t)

Spectro-

meter

E(t)

IE(t-a)

Performing the Fourier Transform andusing the Shift Theorem:


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Application of the Shift Theorem (contd)

NeglectingI

compared to I

and 1:

The contaminated spectrumwill have ripples with a periodof2T/a.

And these ripples will have asurprisingly large amplitude.

_ a2( ) 1 2 cos( )E a[ I [ I! %

_ a

2

( ) 1 2 cos( )E

a[ I [} %

_ a2

2 2

( ) 1 2 cos( ) cos ( ) sin ( )E a a a[ I [ I [ I [ ! - - %

IfI= 1% (a seemingly small amount), these ripples will have an amazing-ly large amplitude of2I= 20%! And peak-to-peak, theyre 40%!


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The Fourier Transform of the complex

conjugate of a function

_ a* *( ) ( ) f t F [! F

_ a* *

*

*

*

( ) ( ) exp( )

( ) exp( )

( ) exp( [ ] )

( )

f t f t i t dt

f t i t dt

f t i t dt

F

[

[

[

[

g

g

g

g

g

g

!

!

-

!

- !

F

Proof:


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Negative frequencies contain no

additional information for real functions.

*

( ) ( )F F[ [ !

If a function is real (e.g., a light wave!), thenf*(t)=f(t). So:

_ a _ a* ( ) ( ) f t f t !F F

Using the result we just proved:

So, at [, the real part of the Fourier transform offis the sameas at +[. And the imaginary part is just the negative of it.


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( ) [( ) exp( )]

( ) exp( )

(

f t i i t dt

i f t i t dt

i F

[ [

[ [

[ [

g

g

g

g

!

!

!

Derivative Theorem

The Fourier transform of a derivative of a function,f(t):

Proof:

Integrate by parts:

{ ( )} ( ) exp( ) f t f t i t dt[

g

g! F

{ ( )} ( ) f t i F[ [d !F

Remember that thefunction must be zero

at , so the otherterm, [f(t)exp(-i[t)]

vanishes.

+-

[ ] fg f g fg

f g fg fg

d d d!

d d !


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The Modulation Theorem:

The Fourier Transform ofE(t) cos([0 t)

_ a0 0( ) cos( ) ( ) cos( ) exp( )E t t E t t i t dt [ [ [g

g

! F

0 0

1( ) exp( ) exp( ) exp( )

2

E t i t i t i t dt [ [ [

g

g

! -

0 01 1( ) exp( [ ] ) ( ) exp( [ ] )2 2

E t i t dt E t i t dt [ [ [ [

g g

g g

!

_ a0 0 0

1 1( ) cos( ) ( ) ( )

2 2E t t E E[ [ [ [ [! % %F

[[0-[0

_ a0( ) cos( )E t t[FE(t)=exp(-t2)

t

Example: 0( ) cos( )E t t[


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Parsevals TheoremParsevals Theorem says that the

energy is the same, whether youintegrate over time or frequency:

Proof:

2 21

( ) ( )2f t dt F d[ [T

g g

g g!

2 ( ) ( ) * ( )

1 1( exp( ) * ( exp( )

2 2

1 1( ) * ( ) exp( [ ] )

2 2

1 1( ) * ( ) [2 )]

2 2

f t dt f t f t dt

F i t d F i t d dt

F F i t dt d d

F F d d

[ [ [ [ [ [T T

[ [ [ [ [ [T T

[ [ TH [ [ [T T

g g

g g

g g g

g g g

g g g

g g gg g

g g

! !

d d d ! - -

d d d !

- d d d!

21 1

( ) * ( ) ( )

2 2

F F d F d

[

[ [ [ [ [

T T

g g

g g

! !

Use [, not [, to avoid conflictsin integration variables.


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Time domain Frequency domainf(t)

|f(t)|2

F([)

|F([)|2

t [

t [

Parseval's Theorem in action

same.areareasT o


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( ) ( ) ( ) ( ) ( ) ( )f t g t f t g t f x g t x dxg

g

| |

The Convolution

The convolution allows one function to smear or broaden another.

changing variables:x t - x( ) ( )f t x g x dx

g

g

!

g f g

*

f

=

x tx


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The convolution

can be performedvisually:

rect rect

rect(t)* rect(t)=((t)

x

rect(x)

( ) ( ) ( ) ( )f t g t f t x g x dxg

g

! *

t

((t)


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Convolution with a delta

function

Convolution with a delta function simply centers the function on thedelta-function.

This convolution does not smear outf(t). Since a devices performancecan usually be described as a convolution of the quantity its trying tomeasure and some instrument response, a perfect device has a delta-function instrument response.

( ) ) ( ) ( )

( )

f t t f t x x dx

f t

H Hg

g

!

!

( ) ( ) f g f t x g x dx

g

g

!


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The Convolution Theorem

The Convolution Theorem turns a convolution into the inverse FT of

the product of the Fourier Transforms:

{ ( ) ( )}= ( ) ( )f t g t F G[ [ wF

{ ( ) ( )} ( ) ( ) exp( ) f t g t f x g t x dx i t dt [

g g

g g

! F( ) ( ) exp( )

( ){ ( exp( )}

( ) exp( ) ( ( (

f x g t x i t dt dx

f x G i x dx

f x i x dxG F G

[

[ [

[ [ [ [

g g

g g

g

g

g

g

!

!

! !

Proof:


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The Convolution Theorem in action

2

{ ( )}

sinc ( / 2)

t

[

( !F{rect( )}

sinc( / 2)

t

[

!F

rect( ) rect( ) ( )t t t ! (

2sinc( / 2) sinc( / 2) sinc ( / 2)[ [ [v !

We can show that the Fourier transform of((t) is sinc2.

0

1

1-1

rect(t)

t 0

( )t(

1

1-1 t

[0

2sinc ( / 2)[

1

[0

sinc( / 2)[

1

[0

sinc( / 2)[

1

-1 0

1

1

rect(t)

t


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The convolution of a functionf(x) with itself (the autoconvolution)is given by:

The Autocorrelation

( ) ( ) f f f x f t x dxg

g

!

Suppose that we dont negate any of the arguments, and we complex-conjugate the 2nd factor. Then we have the autocorrelation:

*( ) ( ) f f f x f x t dx

g

g|


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The Autocorrelation

Like the convolution, the autocorrelation also broadens the functionin time. For real functions, the autocorrelation is symmetrical (even).

As with the convolution, we can also perform the autocorrelationgraphically. Its similar to the convolution, but without the inversion.

f

x

f

x

=

t

f g


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The Autocorrelation TheoremThe Fourier Transform

of the autocorrelationis the spectrum!

Proof:

_ a2( ) *( ) ( ) f x f x t dx f t

g

g

!

F F

? A*

*

*

( ) *( ) exp( ) ( ) *( )

( ) exp( ) ( )

( ) exp( ) ( ) ( ) ( ) exp( )

( ) exp( ) *( ) ( ) *( ) (

f x f x t dx i t f x f x t dx dt

f x i t f x t dt dx

f x i t f x t dt dx f x F i x dx

f x i x dx F F F F

[

[

[ [ [

[ [ [ [ [

g g g

g g g

g g

g g

g g g

g g g

g

g

!

! -

d d d! ! -

! ! !

F

2)

t= t


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The Autocorrelation Theorem in action

_ a 2( ) sinc ( / 2)t [( !F

_ arect( ) sinc( / 2)t[

!F

2

sinc( / 2)

sinc( / 2)

sinc ( / 2)

[[

[

v!

rect( ) rect( )

( )

t t

t

!

(

0

1

1-1

rect(t)

t

-1 0

( )t(1

1 t [0

2sinc ( / 2)[1

[0

sinc( / 2)[1


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Sampling

If the signal to be analyzed is analog in nature then it must beconverted into digital form, as it is sampled, by an analog to digital(A/D) converter.

If delta is the time interval between consecutive samples, then thesampled time data can be represented as

Discrete and finite data points can not have as much information asa continuous signal can. This results into distortion of output.

...,,nnhhn 210)( ss!(!


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Discrete Fourier Transforms

Function sampled at Ndiscrete points sampling at evenly spaced intervals

Fourier transform estimated at discretevalues:

Almost the same symmetry properties asthe continuous Fourier transform

...3,2,1,0,1,2,3...,

)(

!

(!

n

nhhn

(!

N

nfn

2,...,

2

NNn !


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DFT formulas

? A ? A

? AN

iknh

tifhdttifthfH

N

kk

kn

N

k

knn

/2exp

2exp2exp)()(

1

0

1

0

T

TT

(

!

(}!

!

!

g

g

? A

!

|1

0

/2expN

k

kn NiknhH T nn HfH (})(

? A

!

!1

0

/2exp1 N

n

nk NiknHN

h T


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Frequency of Sampling

FT

x

k

every point

every second point


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FT

x

k

every point

every third point


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FT

x

k

every point

every fourth point


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Truncation

As sum covers finite segment (-T,T) of the signal, so this truncatedsignal could be represented as product of signal and boxcar or rect(t)function.

Fourier transform would be convolution of fourier transforms of rec(t)function and unmodified spectrum function.

)(*)}({)( ftrectFTfT !


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Effect of sampling

Spectrum of a signal sampled with the sampling interval (delta) isperiodic with the period (1/delta)

Because of loss of information, the final calculated spectrum is asuperposition of all the waves at frequency (f+n/delta).

Combined impact of sampling and truncation.


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Nyquist sampling

Critical sampling interval

Critical sampling frequency

max21

f!(

max21 ff

N!

(!


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FAST FOURIER TRANSFORM (FFT)

Uses mathematical identities to reduce the amount of computationthat is required.

The FFT is based on the Fourier Shift theorem.

The computational work increases as n log2n rather than as n2 whenyou use this trick which results in an enormous reduction of workwhen n is a large number.


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References

Introductory Fourier transform spectroscopy byR. J. Bell

Fourier transforms in spectroscopy by J. Kauppinen & J. Partanen

Fourier transform spectrometry by Sumner P. Davis

Advanced engineering mathematics by E. Kreyszig


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Thank you!!!

Documents

Fourier Transform in Spectroscopy