$y(t)=\displaystyle\int_{-\infty}^{\infty}d\tau h(\tau)x(t-\tau)$

y(t)=\int_{-\infty}^{\infty}d\tau h(\tau)x(t-\tau)

$y(\omega)=H(\omega)x(\omega)$

y(\omega)=H(\omega)x(\omega)

$Y(\omega)=\displaystyle\int_{-\infty}^{\infty}y(t)e^{-i\omega t}dt$

Y(\omega)=\displaystyle\int_{-\infty}^{\infty}y(t)e^{-i\omega t}dt

$H(\omega)=\displaystyle\int_{-\infty}^{\infty}h(t)e^{-i\omega t}dt$

H(\omega)=\int_{-\infty}^{\infty}h(t)e^{-i\omega t}dt

$m\displaystyle\frac{d^2y}{dt^2}+c\displaystyle\frac{dy}{dt}+ky=k_0x(t)$

m\displaystyle\frac{d^2y}{dt^2}+c\displaystyle\frac{dy}{dt}+ky=k_0x(t)

$\omega_0=\sqrt{\displaystyle\frac{k}{m}}$

\omega_0=\sqrt{\displaystyle\frac{k}{m}}

$\zeta=\displaystyle\frac{c}{2\sqrt{km}}$

\zeta=\displaystyle\frac{c}{2\sqrt{km}}

$|H(\omega)|^2=\displaystyle\frac{1}{k\sqrt{(1-\omega^2/\omega_0^2)^2+4\zeta^2\omega^2/\omega_0^2}}$

|H(\omega)|^2=\displaystyle\frac{1}{k\sqrt{(1-\omega^2/\omega_0^2)^2+4\zeta^2\omega^2/\omega_0^2}}

$\phi(\omega)=\arctan\left(\displaystyle\frac{2\zeta\omega/\omega_0}{1-\omega^2/\omega_0^2}\right)$

\phi(\omega)=\arctan\left(\displaystyle\frac{2\zeta\omega/\omega_0}{1-\omega^2/\omega_0^2}\right)