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>
(where i represents the imaginary unit) and use it as an input to a linear timeinvariant system. The corresponding component in the output will match the following equation:
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':
and 'Phase shift':
The transfer function can also be derived by using the Fourier transform.
In control engineering and control theory the transfer function is derived using the Laplace transform.
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