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Transfer Function

The engineering terminology for one use of Fourier Transforms. By breaking up a wave pulse into its frequency spectrum

f_\nu = F(\nu)e^{2\pi i\nu t},
\end{displaymath} (1)

the entire signal can be written as a sum of contributions from each frequency,
f(t) = \int_{-\infty}^\infty f_\nu\,d\nu = \int_{-\infty}^\infty F(\nu)e^{2\pi i\nu t}\,d\nu.
\end{displaymath} (2)

If the signal is modified in some way, it will become
$\displaystyle g_\nu(t)$ $\textstyle =$ $\displaystyle \phi(\nu)f_\nu(t) = \phi(\nu)F(\nu)e^{2\pi i\nu t}$ (3)
$\displaystyle g(t)$ $\textstyle =$ $\displaystyle \int_{-\infty}^\infty g_\nu(t)\,dt = \int_{-\infty}^\infty\phi(\nu)F(\nu)e^{2\pi i\nu t}\,d\nu,$  

where $\phi(\nu)$ is known as the ``transfer function.'' Fourier Transforming $\phi$ and $F$,
\phi(\nu) = \int_{-\infty}^\infty \Phi(t)e^{-2\pi i\nu t}\,dt
\end{displaymath} (5)

F(\nu) = \int_{-\infty}^\infty f(t)e^{-2\pi i\nu t}\,dt.
\end{displaymath} (6)

From the Convolution Theorem,
g(t) = f(t)*\Phi(t) = \int_{-\infty}^\infty f(t)\Phi(t-\tau)\,d\tau.
\end{displaymath} (7)

See also Convolution Theorem, Fourier Transform

© 1996-9 Eric W. Weisstein