02-02-2013, 04:52 PM
Fourier transform application
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Analysis of differential equations
Fourier transforms and the closely related Laplace transforms are widely used in solving differential equations. The Fourier transform is compatible with differentiation in the following sense: if ƒ(x) is a differentiable function with Fourier transform ƒ̂(ξ), then the Fourier transform of its derivative is given by 2πiξ ƒ̂(ξ). This can be used to transform differential equations into algebraic equations. This technique only applies to problems whose domain is the whole set of real numbers. By extending the Fourier transform to functions of several variables partial differential equations with domain Rn can also be translated into algebraic equations.
Fourier transform spectroscopy
The Fourier transform is also used in nuclear magnetic resonance (NMR) and in other kinds of spectroscopy, e.g. infrared (FTIR). In NMR an exponentially shaped free induction decay (FID) signal is acquired in the time domain and Fourier-transformed to a Lorentzian line-shape in the frequency domain. The Fourier transform is also used in magnetic resonance imaging (MRI) and mass spectrometry.
Quantum mechanics and signal processing
In quantum mechanics, Fourier transforms of solutions to the Schrödinger equation are known as momentum space (or k space) wave functions. They display the amplitudes for momenta. Their absolute square is the probabilities of momenta. This is valid also for classical waves treated in signal processing, such as in swept frequency radar where data is taken in frequency domain and transformed to time domain, yielding range. The absolute square is then the power