Fourier Transform and its applications Fourier Transforms are
- Slides: 49
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.
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 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
DFT formulas
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: Fourier transform of a Gauss function:
The Temperature Factor
Convolution with a delta function The delta function: The Fourier Transform of a delta function
• Structure factor:
Correlation Theorem
Autocorrelation
Calculation of the electron density x, y and z are fractional coordinates in the unit cell 0<x<1
Calculation of the electron density
Calculation of the electron density 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:
Calculation of the electron density 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/fouri er/fourier. html and links therein • http: //www. bfsc. leidenuniv. nl/ for the lecture notes
- The fourier transform and its applications
- Insidan region jh
- Short time fourier transform
- Complex fourier transform
- Inverse dtfs
- Fourier transform amplitude and phase
- Fourier
- Relationship between laplace and fourier transform
- Laplace tranform table
- Fourier transform of product of two functions
- Fourier transform delta function
- Ctft
- Fourier transform table
- Parseval's identity for fourier transform
- Fourier transform properties table
- Synthesis equation fourier series
- Matlab ramp function
- Fourier transform mri
- Fourier transform properties solved examples
- Fourier transform of unit step function
- Difference of gaussians
- Fourier transform of 1
- Fourier time shift
- Properties of fourier transform in digital image processing
- Sin 2pift
- Inverse fourier transform formula
- Fourier transform of an integral
- Short time fourier transform
- Polar fourier series
- Fourier transform of product of two functions
- Discrete fourier transform
- Sinc fourier transform
- Fourier transform of impulse train
- Discrete fourier transform
- Fourier transform of impulse train
- Circ function fourier transform
- Rect t/2
- Fourier transform is defined for
- Fourier series formulas
- Integral of unit step
- Filter
- Sine fourier transform
- Duality of fourier transform
- Fourier series half range
- Windowed fourier transform
- Fourier transform
- Continuous fourier transform formula
- Introduction to fast fourier transform
- Fourier transform solver
- Fourier transform of a periodic function