Question 11.7: Find a closed-form expression for the impulse response of th......

The transfer function of a system is given by

H(z)=(C(z I-A)^{-1}B+D)

H(z)={\left[\begin{array}{l l}{1-2}\end{array}\right]}\left [ \begin{matrix} \frac{z}{z^{2}-2z+3} -{\frac{3}{z^{2}-2z+3}} \\ {\frac{1}{z^{2}-2z+3}} \ \ \ {\frac{z-2}{z^{2}-2z+3}} \end{matrix} \right ] \left [ \begin{matrix} 1 \\ 0 \end{matrix} \right ]+2=\frac{z-2}{(z^{2}-2z+3)}+2

Expanding into partial fractions, we get

H(z)=2+\frac{0.5+j\frac{1}{2\sqrt{2}}}{z-1-j\sqrt{2}}+\frac{0.5-j\frac{1}{2\sqrt{2}}}{z-1+j\sqrt{2}}

Finding the inverse z-transform and simplifying, we get

h(n)=2\delta(n)+((\sqrt{3})^{n-1}\cos((\tan^{-1}(\sqrt{2}))(n-1))

-\;\frac{1}{\sqrt{2}}(\sqrt{3})^{n-1}\sin((\tan^{-1}(\sqrt{2}))(n-1)))u(n-1),\;n=0,1,2,\ldots