Signal Processing

Take a complex harmonic signal with a sinusoidal component with amplitude <math>A_{in}</math>, angular frequency <math>\omega</math> and phase <math>p_{in}</math>

<math>x(t) = A_{in} e^{i(\omega t + p_{in})}</math>

(where i represents the imaginary unit) and use it as an input to a linear time-invariant system. The corresponding component in the output will match the following equation:

<math>x(t) = A_{out} e^{i(\omega t + p_{out})}</math>

Note that the fundamental frequency ω has not changed, only the amplitude and the phase of the response changed as it went through the system. The transfer function H(z) describes this change for every frequency ω in terms of 'Gain':

<math>\frac{A_{out}}{A_{in}} = | H(i\omega) |</math>

and 'Phase shift':

<math>p_{out} - p_{in} = \arg( H(i\omega))</math>.

The transfer function can also be derived by using the Fourier transform.

Control Engineering

In control engineering and control theory the transfer function is derived using the Laplace transform.