Problem III $(20\%).$ Bode's gain$-$phase relationship states that for any stable minimum$-$phase transfer function $G(j),$ the phase of $G(j)$ is uniquely related to the magnitude of $G(j),$ by

$\frac{?}{{?}_G(j)}=\frac{1}{}{\displaystyle \underset{-}{\overset{+}{}}}\frac{dM}{du}W(u)du(in$ radians$),$

where

$M=log$ magnitude $=ln|G(j)|,$

$u=$ normalized frequency $=ln(\frac{}{{}_0})$

$d\frac{M}{d}u$~~ slope $n,$ the slope of $G(j)$ in units of decade of amplitude per decade of frequency,

$(|)/(2|)$

Use it to explain why it is desirable to have $n=-1$ for $$ around ${}_c,$ the crossover frequency for about a decade. Give a numerical example.