fourier transform
DESCRIPTION
Very good PPT on Fourier TransformTRANSCRIPT
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Fourier Transformand its applications
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Fourier Transforms are used inX-ray diffractionElectron microscopy (and diffraction)NMR spectroscopyIR spectroscopyFluorescence spectroscopyImage processingetc. etc. etc. etc.
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Fourier TransformsDifferent representation of a function time vs. frequencyposition (meters) vs. inverse wavelength
In our case:electron density vs. diffraction pattern
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What is a Fourier transform?A function can be described by a summation of waves with different amplitudes and phases.
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Fourier TransformIf h(t) is real:
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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
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DFT formulas
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Examples
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Properties of Fourier TransformsConvolution TheoremCorrelation TheoremWiener-Khinchin Theorem (autocorrelation)Parsevals Theorem
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ConvolutionAs a mathematical formula:Convolutions are commutative:
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Convolution illustrated
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Convolution illustrated=
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Convolution illustrated
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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
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Special ConvolutionsConvolution with a Gauss functionGauss function:Fourier transform of a Gauss function:
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The Temperature Factor
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Convolution with a delta functionThe delta function:The Fourier Transform of a delta function
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Structure factor:
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Correlation Theorem
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Autocorrelation
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Calculation of the electron densityx,y and z are fractional coordinates in the unit cell
0 < x < 1
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Calculation of the electron density
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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:
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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
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Suggested readinghttp://www.yorvic.york.ac.uk/~cowtan/fourier/fourier.html and links thereinhttp://www.bfsc.leidenuniv.nl/ for the lecture notes