Consider the original coefficients of a $2^{nd}$ order filter:

$\{\begin{array}{c}\text{:}{A}_0^D=A_0(\frac{2}{T}{)}^2+A_1\frac{2}{T}+A_2\end{array}$

$\{\begin{array}{c}\text{:}{A}_1^D=-2A_0(\frac{2}{T}{)}^2+2A_2\end{array}$

$\{\begin{array}{c}\text{:}{A}_2^D=A_0(\frac{2}{T}{)}^2-A_1\frac{2}{T}+A_2\end{array}$

A$.$ Modify the equation with the consideration of frequency pre$-$warping, which may prevent the filter from frequency mismatch. $(5\%)$

B$.$ Under what kind of situation, a serious frequency mismatch will be induced? $(5\%)$ Hint: Frequency Pre$-$warping, Bilinear Transform,