Transfer Function

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

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

the entire signal can be written as a sum of contributions from each frequency,

$$f(t) = \int_{-\infty}^\infty f_\nu\,d\nu = \int_{-\infty}^\infty F(\nu)e^{2\pi i\nu t}\,d\nu.$$ (2)

If the signal is modified in some way, it will become

$$g_\nu(t) = \phi(\nu)f_\nu(t) = \phi(\nu)F(\nu)e^{2\pi i\nu t}$$ (3)

$$g(t) = \int_{-\infty}^\infty g_\nu(t)\,dt = \int_{-\infty}^\infty\phi(\nu)F(\nu)e^{2\pi i\nu t}\,d\nu,$$ (4)

where $\phi(\nu)$ is known as the ``transfer function.'' Fourier Transforming $\phi$ and $F$,

$$\phi(\nu) = \int_{-\infty}^\infty \Phi(t)e^{-2\pi i\nu t}\,dt$$ (5)

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

From the Convolution Theorem,

$$g(t) = f(t)*\Phi(t) = \int_{-\infty}^\infty f(t)\Phi(t-\tau)\,d\tau.$$ (7)

See also Convolution Theorem, Fourier Transform