2.2j.4 Transfer Functions

Transfer Function

A transfer function is a mathematical representation that describes the input-output relationship of a system in the frequency domain, providing its frequency response. For a system $X(j\omega )$,
$H(j\omega )=\frac{X_{\text{out}}(j\omega )}{X_{\text{in}}(j\omega )}=|H(j\omega )|e^{j\omega \varphi },$
where by definition $|H(j\omega )|$ is the system’s gain (or magnitude), and $\varphi (j\omega )={\tan }^{-1}\frac{\mathrm{Im}\{H(\omega )\}}{\mathrm{Re}\{H(\omega )\}}$ is the system’s phase response.