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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^(2piinut),
(1)

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

 f(t)=int_(-infty)^inftyf_nudnu=int_(-infty)^inftyF(nu)e^(2piinut)dnu.
(2)

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

g_nu(t)=phi(nu)f_nu(t)
(3)
=phi(nu)F(nu)e^(2piinut)
(4)
g(t)=int_(-infty)^inftyg_nu(t)dt
(5)
=int_(-infty)^inftyphi(nu)F(nu)e^(2piinut)dnu,
(6)

where phi(nu) is known as the "transfer function." Fourier transforming phi and F,

 phi(nu)=int_(-infty)^inftyPhi(t)e^(-2piinut)dt
(7)
 F(nu)=int_(-infty)^inftyf(t)e^(-2piinut)dt.
(8)

From the convolution theorem,

 g(t)=f(t)*Phi(t)=int_(-infty)^inftyf(t)Phi(t-tau)dtau.
(9)

See also

Convolution Theorem, Fourier Transform

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Cite this as:

Weisstein, Eric W. "Transfer Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TransferFunction.html

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