Impulse Response

The impulse response is again the inverse Laplace transform of the transfer function. Expanding $H(s)$ into a sum of complex one-pole sections,

$$H(s) = 2\eta\cdot\frac{s}{s^2 + 2\eta s + \omega_0^2} = \frac{r_1}{s-p_1} + \frac{r_2}{s-p_2} = \frac{(r_1+r_2)s - (r_1p_2 + r_2p_1)}{s^2-(p_1 + p_2) + p_1p_2},$$

where $p_{1,2}\isdef -\eta \pm \sqrt{\eta^2 - \omega_0^2}$ . Equating numerator coefficients gives

$$\begin{eqnarray*} r_1+r_2 &=& 2\eta \;\mathrel{=}\; \frac{1}{RC}\\ r_1p_2 + r_2p_1 &=& 0. \end{eqnarray*}$$

This pair of equations in two unknowns may be solved for $r_1$ and $r_2$ . The impulse response is then

$$h(t) = r_1 e^{p_1 t} u(t) + r_2 e^{p_2 t} u(t).$$