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An introduction to fourier transforms, their mathematical definition, applications in various fields such as x-ray diffraction, electron microscopy, spectroscopy, image processing, and more. It also covers the properties of fourier transforms including the convolution theorem, correlation theorem, wiener-khinchin theorem, and parseval’s theorem.

Typology: Slides

2012/2013

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Download Fourier Transforms: Definition, Applications, and Properties and more Slides Electrical Engineering in PDF only on Docsity! Fourier Transform and its applications Docsity.com Fourier Transforms are used in • X-ray diffraction • Electron microscopy (and diffraction) • NMR spectroscopy • IR spectroscopy • Fluorescence spectroscopy • Image processing • etc. etc. etc. etc. Docsity.com Fourier Transforms • Different representation of a function – time vs. frequency – position (meters) vs. inverse wavelength • In our case: – electron density vs. diffraction pattern Docsity.com What is a Fourier transform? • A function can be described by a summation of waves with different amplitudes and phases. Docsity.com Docsity.com 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 −= Docsity.com DFT formulas [ ] [ ] [ ]Niknh tifhdttifthfH N k k nn N k knn /2exp 2exp2exp)()( 1 0 1 0 π ππ ∑ ∑∫ − = − = ∞ ∞− ∆= ∆≈= [ ]∑ − = ≡ 1 0 /2exp N k kn NiknhH π nn HfH ∆≈)( [ ]∑ − = −= 1 0 /2exp1 N n nk NiknHN h π Docsity.com Examples Docsity.com
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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 Docsity.com Special Convolutions Convolution with a Gauss function Gauss function: Fourier transform of a Gauss function: Docsity.com • Structure factor: ∑ = ⋅= n j jj if 1 ]2exp[)( SrSF π Docsity.com Correlation Theorem
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Docsity.com Calculation of the electron density [ ]∑ ⋅= j jj if SrSF π2exp)( [ ]dvi cell j∫ ⋅= SrrSF πρ 2exp)()( x,y and z are fractional coordinates in the unit cell 0 < x < 1 Docsity.com Calculation of the electron density [ ]∫ ∫ ∫ = = = ++= 1 0 1 0 1 0 )(2exp)()( x y z dxdydzlzkyhxixyzVhkl πρF [ ]dvi cell j∫ ⋅= SrrSF πρ 2exp)()( dxdydzVdv ⋅= yzklhx zyxzyx ++= ⋅⋅+⋅⋅+⋅⋅=⋅⋅+⋅+⋅=⋅ ScSbSaScbaSr )( Docsity.com Calculation of the electron density [ ]∫ ∫ ∫ = = = ++= 1 0 1 0 1 0 )(2exp)()( x y z dxdydzlzkyhxixyzVhkl πρF [ ])(2exp)(1)( lzkyhxihkl V xyz h k l ++−= ∑∑∑ πρ F This describes F(S), but we want the electron density We need Fourier transformation!!!!! F(hkl) is the Fourier transform of the electron density But the reverse is also true: Docsity.com