$(25$ pt$)$ For a system whose continuous$-$time transfer function is given as:

$G(s)=\frac{K({}_{0}s+1)}{({}_{1}s+1)({}_{2}s+1)},$

show that the discrete$-$time transfer function with sample time $T_S$ is given by

$G(z)=\frac{b_1{z}^{-1}+b_2{z}^{-2}}{1+a_1{z}^{-1}+a_2{z}^{-2}}$

where, if we define ${}_1=$exp$(-\frac{T_S}{{}_1}),{}_2=$exp$(-\frac{T_S}{{}_2}),$ then the indicated coefficients are

given by

$a_1=-({}_1+{}_2),a_2={}_1{}_2$

$b_1=K[1-\frac{{}_1-{}_0}{{}_1-{}_2}{}_1-\frac{{}_2-{}_0}{{}_2-{}_1}{}_2],b_2=K[{}_1{}_2-\frac{{}_1-{}_0}{{}_1-{}_2}{}_2-\frac{{}_2-{}_0}{{}_2-{}_1}{}_1].$

$[$Hint: Use partial fraction expansion first, and then convert first$-$order systems.$]$

please solve it not just instructions