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Fourier transformation is a tool which generates the spectrum of a
signal yielding a frequency-domain representation.
Since this transformation is unambiguous the original signal can be
reconstructed from its spectrum by an inverse transformation.
The Fourier transform
of a 1D function
is defined as:
![\begin{displaymath}
F(u)=\int_{-\infty }^{\infty }f(x)\cdot e^{2\pi ifx}dx,
\end{displaymath}](img31.gif) |
(6) |
where
is a value in the frequency domain. The inverse Fourier
transformation for reconstructing
from
is defined as:
![\begin{displaymath}
f(x)=\int_{-\infty }^{\infty }F(u)\cdot e^{-2\pi ifu}du
\end{displaymath}](img33.gif) |
(7) |
which is rather similar, except that the exponential term has the
opposite sign. In the 3D case, the Fourier transform of a
function
is defined as follows:
![\begin{displaymath}
F(u,v,w)=\int \int \int f(x,y,z)\cdot e^{2\pi i(ux+vy+wz)}dxdydz
\end{displaymath}](img35.gif) |
(8) |
The inverse transformation is analogous to the 1D case.
Subsections
Ivan Viola, Matej Mlejnek
2001-03-22