The transfer-function: $$G(s) = \frac{\beta s}{s^2 + \beta s + \omega_0^2}$$
is to be used in an application that requires the magnitude of the frequency response to be of the band-pass form.
- Show that this property is present by considering the cases when $\omega = 0, \omega = 1$ and show also that $\lvert G(j\omega_0)\rvert = 1$.
- Determine also the equation that defines the frequencies where the filter magnitude is $−3\textrm{ dB}$.
I use $G(s)=G(j\omega)|_{s=j\omega}$, then $$ \lvert G(j\omega_0)\rvert= \frac{j\beta \omega_0}{-\omega_0^2 + j\beta\omega_0 + \omega_0^2}=1 $$ when $\omega=0$, the numerator equal to $0$ so $\omega=0$ is be filtered out.
My questions are:
- Could I use the $G(s)=\lvert G(j\omega)\rvert$ directly or $G(s)=G(j\omega)=\lvert G(j\omega)\rvert e^{-jk\omega}$?
- I have no idea about the case when $\omega=+\infty$?
- The last question, I use $-20\log\lvert G(j\omega)\rvert=-3\textrm{ dB}$, so we can solve the $\omega$ from this equation, is that correct?