I don't understand the follow step in my notes:

It is about the filter roll-off.

$H(w)={1\over 1+j(w/w_c)}$

$\Rightarrow$ $|H(w)|= {1\over \sqrt(1+(w/w_c)^2)}$

where $H(w) = {v_o \over v_i}$

$w_c$ = cut-off frequency


For any arbitrary rational complex function $H(\omega) = \frac{N(\omega)}{D(\omega)}$,

$$ |H(\omega)| = \frac{|N(\omega)|}{|D(\omega)|} $$

This follows straightforwardly from the properties of the polar (i.e. magnitude/phase) representation of complex numbers. So, for your example:

$$ |H(\omega)| = \frac{|1|}{|1 + j \frac{\omega}{\omega_c}|} $$

The magnitude of an arbitrary complex number is:

$$ |a + jb| = \sqrt{a^2+b^2} $$

So for your example:

$$ \begin{align} |H(\omega)| &= \frac{1}{\sqrt{1^2 + \left(\frac{\omega}{\omega_c}\right)^2}} \\ &= \frac{1}{\sqrt{1 + \left(\frac{\omega}{\omega_c}\right)^2}} \end{align} $$


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