Fourier Transforms are used inX-ray diffractionElectron
microscopy (and diffraction)NMR spectroscopyIR
spectroscopyFluorescence spectroscopyImage processingetc. etc. etc.
etc.
Fourier TransformsDifferent representation of a function time
vs. frequencyposition (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 TransformIf h(t) is real:
Discrete Fourier TransformsFunction sampled at N discrete
pointssampling at evenly spaced intervalsFourier transform
estimated at discrete values:
e.g. ImagesAlmost the same symmetry properties as the continuous
Fourier transform
DFT formulas
Examples
Properties of Fourier TransformsConvolution TheoremCorrelation
TheoremWiener-Khinchin Theorem (autocorrelation)Parsevals
Theorem
ConvolutionAs a mathematical formula:Convolutions are
commutative:
Convolution illustrated
Convolution illustrated=
Convolution illustrated
Convolution TheoremThe Fourier transform of a convolution is the
product of the Fourier transformsThe Fourier transform of a product
is the convolution of the Fourier transforms
Special ConvolutionsConvolution with a Gauss functionGauss
function:Fourier transform of a Gauss function:
The Temperature Factor
Convolution with a delta functionThe delta function:The Fourier
Transform of a delta function
Structure factor:
Correlation Theorem
Autocorrelation
Calculation of the electron densityx,y and z are fractional
coordinates in the unit cell
0 < x < 1
Calculation of the electron density
Calculation of the electron densityThis 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 densityBecause F=|F|exp(ia):I(hkl)
is related to |F(hkl)| not the phase angle alpha===> The
crystallographic phase problem
Suggested
readinghttp://www.yorvic.york.ac.uk/~cowtan/fourier/fourier.html
and links thereinhttp://www.bfsc.leidenuniv.nl/ for the lecture
notes
