The [[Frequency Response Function]] of some [[Signal Filter]] is a [[Function]] [[Invariant]] of the [[Time Domain]] $H(\omega)$. $\huge \begin{align} y_{t} &= H(\omega) e^{i\omega t} \\ H(\omega) &= r_{\omega}e^{i\theta_{\omega}} \\ \end{align} $ > This page uses the second example from [[Finite Impulse Response Filters]]. ## Magnitude and Phase Response The [[Frequency Response Function|magnitude response]] is defined as the [[Absolute Value]] of $H(\omega)$ $\huge \begin{align} H(\omega) &= \underbrace{\left| H(\omega) \right|}_{\text{Magnitude Response}}e^{i \overbrace{\theta \omega}^{\text{Phase Response}}} \\ \end{align} $ Where $\theta$ is the [[Phase]]. $H(\omega)$ is a function of a [[Real Numbers|Real Number]] *or* can be "embedded" into the [[Complex Plane]] on the [[Unit Circle]], ie with inputs $e^{i\omega}$ with $0\leq\omega\leq{2}\pi$. ## Filters with Phasors How we get $H(\omega)$![[Frequency Response .excalidraw.svg]] by inserting $e^{i\omega t}$ into the filter as $x_{t}$, and get $y_{t}$ as $H(\omega) e^{i\omega t}$. ie. $\huge \begin{align} y_t &=e^{i\omega t}+e^{i \omega(t-1)} \\ &= e^{i \omega t} \underbrace{\left( 1+e^{-i\omega} \right)}_{H(\omega)} \end{align} $ So the frequency response of a [[Signal Filter|filter]] is the [[Complex Numbers|Complex Number]], algebraically dependent of $\omega$, which multiples $e^{i \omega t}$ when this [[Phasor]] is input into the [[Signal Filter|filter]].