09-01-2014, 04:20 PM
Fourier series
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Fourier Transform
The Fourier transform is a generalization of the complex Fourier series in the limit
Fourier analysis = frequency domain analysis
Low frequency: sin(nx),cos(nx) with a small n
High frequency: sin(nx),cos(nx) with a large n
Note that sine and cosine waves are infinitely long – this is a shortcoming of Fourier analysis, which explains why a more advanced tool, wavelet analysis, is more appropriate for certain signals
What is a Fourier transform?
A function can be described by a summation of waves with different amplitudes and phases.
Fourier Transforms are used in
X-ray diffraction
Electron microscopy (and diffraction)
NMR spectroscopy
IR spectroscopy
Fluorescence spectroscopy
Image processing
etc. etc. etc. etc.
Applications of Fourier Transform
Physics
Solve linear PDEs (heat conduction, Laplace, wave propagation)
Antenna design
Seismic arrays, side scan sonar, GPS, SAR
Signal processing
1D: speech analysis, enhancement …
2D: image restoration, enhancement …
Applications
In image processing:
Instead of time domain: spatial domain (normal image space)
frequency domain: space in which each image value at image position F represents the amount that the intensity values in image I vary over a specific distance related to F