Fourier Transform

and its applications

Fourier Transforms are used in

• X-ray diffraction

• Electron microscopy (and diffraction)

• NMR spectroscopy

• IR spectroscopy

• Fluorescence spectroscopy

• Image processing

• etc. etc. etc. etc.

Fourier Transforms

• Different representation of a function – time vs. frequency– position (meters) vs. inverse wavelength

• In our case:– electron density vs. diffraction pattern

What is a Fourier transform?

• A function can be described by a summation of waves with different amplitudes and phases.

Fourier Transform

dtiftthfH 2exp)()(

dfiftfHth

2exp)()(

)()( * fHfH If h(t) is real:

Discrete Fourier Transforms

• Function sampled at N discrete points– sampling at evenly spaced intervals– Fourier transform estimated at discrete values:

– e.g. Images

• Almost the same symmetry properties as the continuous Fourier transform

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

)(

n

nhhn

N

nfn2

,...,2

NNn

DFT formulas

Niknh

tifhdttifthfH

N

kk

nn

N

kknn

/2exp

2exp2exp)()(

1

0

1

0

1

0

/2expN

kkn NiknhH nn HfH )(

1

0

/2exp1 N

nnk NiknH

Nh

Examples

Properties of Fourier Transforms

• Convolution Theorem

• Correlation Theorem

• Wiener-Khinchin Theorem (autocorrelation)

• Parseval’s Theorem

Convolution

As a mathematical formula:

Convolutions are commutative:

Convolution illustrated

Convolution illustrated

=

Convolution illustrated

Convolution Theorem

•The Fourier transform of a convolution is the product of the Fourier transforms•The Fourier transform of a product is the convolution of the Fourier transforms

Special Convolutions

Convolution with a Gauss function

Gauss function:

Fourier transform of a Gauss function:

The Temperature Factor

2

2

2

2

1

4exp

sin2

4exp

sinexp)(

d

B

BBisoT

228 uB

Convolution with a delta functionThe delta function:

The Fourier Transform of a delta function

• Structure factor:

n

jjj if

1

]2exp[)( SrSF

Correlation Theorem

Autocorrelation

)()(),( * fGfGggC

2)(),( fGggC

Calculation of the electron density

j

jj if SrSF 2exp)(

dvicell

j SrrSF 2exp)()(

x,y and z are fractional coordinates in the unit cell

0 < x < 1

Calculation of the electron density

1

0

1

0

1

0

)(2exp)()(x y z

dxdydzlzkyhxixyzVhkl F

dv icell

j S r r S F 2 exp ) ( ) (

dxdydz V dv

yz kl hx

z y x z y x

S c S b S a S c b a S r) (

Calculation of the electron density

1

0

1

0

1

0

)(2exp)()(x y z

dxdydzlzkyhxixyzVhkl F

)(2exp)(1

)( lzkyhxihklV

xyzh k l

F

This describes F(S), but we want the electron densityWe need Fourier transformation!!!!!F(hkl) is the Fourier transform of the electron density

But the reverse is also true:

Calculation of the electron density

)(2exp)(1

)( lzkyhxihklV

xyzh k l

F

)()(2exp)(1

)( hklilzkyhxihklFV

xyzh k l

Because F=|F|exp(ia):

I(hkl) is related to |F(hkl)| not the phase angle alpha

===> The crystallographic phase problem

Suggested reading

• http://www.yorvic.york.ac.uk/~cowtan/fourier/fourier.html and links therein

• http://www.bfsc.leidenuniv.nl/ for the lecture notes