The engineering terminology for one use of Fourier Transforms. By breaking up a wave pulse into
its frequency spectrum
![\begin{displaymath}
f_\nu = F(\nu)e^{2\pi i\nu t},
\end{displaymath}](t_1130.gif) |
(1) |
the entire signal can be written as a sum of contributions from each frequency,
![\begin{displaymath}
f(t) = \int_{-\infty}^\infty f_\nu\,d\nu = \int_{-\infty}^\infty F(\nu)e^{2\pi i\nu t}\,d\nu.
\end{displaymath}](t_1131.gif) |
(2) |
If the signal is modified in some way, it will become
where
is known as the ``transfer function.'' Fourier Transforming
and
,
![\begin{displaymath}
\phi(\nu) = \int_{-\infty}^\infty \Phi(t)e^{-2\pi i\nu t}\,dt
\end{displaymath}](t_1137.gif) |
(5) |
![\begin{displaymath}
F(\nu) = \int_{-\infty}^\infty f(t)e^{-2\pi i\nu t}\,dt.
\end{displaymath}](t_1138.gif) |
(6) |
From the Convolution Theorem,
![\begin{displaymath}
g(t) = f(t)*\Phi(t) = \int_{-\infty}^\infty f(t)\Phi(t-\tau)\,d\tau.
\end{displaymath}](t_1139.gif) |
(7) |
See also Convolution Theorem, Fourier Transform
© 1996-9 Eric W. Weisstein
1999-05-26