It can be easier to analyze systems using the frequency domain as a mathematical tool.
Suppose we have a transfer function:
Then, given an input of , in the time domain we would have:
where and is the output in response to the complex exponential .
If we are on the imaginary axis, such that , then is the system’s frequency response. Then, we have:
In other words, ==the frequency response is the steady-state response to a sinusoid (after time response disappears). ==
Thus, we have:
What this tell us is that given an input, it is amplified/attenuated by the magnitude of teh frequency response transfer function and there is a phase shift of due to the physics of the system.
If , we would have:
where .