Which of the following discrete-time state-space models $(A,B,C,D)$ of the form
$x(t+1)=Ax(t)+Bu(t), \quad y(t)=Cx(t)+Du(t), \quad t\in \mathbb{N}$
with $A$ in jordan form has its impulse response given by
$h(t)=\left\{ \begin{array}{ll} \delta(t-1)+2^{t-1} \ \text{for} \ t = 1,2,3,...\\ 0 \ \text{for} \ t = 0,\\ \end{array} \right.$ $\quad$ where $\delta = 0^t = \left\{ \begin{array}{ll} 1 \ \text{for} \ t = 0\\ 0 \ \text{for} \ t = 1,2,3,...\\ \end{array} \right.$
Answer:
$A) \quad \left[ \begin{array}{c|c} A&B \\ \hline C&D \end{array} \right] = \left[ \begin{array}{cc|c} 2&0 &1 \\ 0&0&1 \\ \hline 1&1&0 \end{array} \right]$
$B) \quad \left[ \begin{array}{c|c} A&B \\ \hline C&D \end{array} \right] = \left[ \begin{array}{cc|c} 1&1&1 \\ 1&1&0 \\ \hline 2&0&0 \end{array} \right]$
$C) \quad \left[ \begin{array}{c|c} A&B \\ \hline C&D \end{array} \right] = \left[ \begin{array}{c|c} 2&1 \\ \hline 1&0 \end{array} \right]$
$D) \quad \left[ \begin{array}{c|c} A&B \\ \hline C&D \end{array} \right] = \left[ \begin{array}{cc|c} 2&0&0 \\ 0&0&1 \\ \hline 1&0&0 \end{array} \right]$
$E) \quad \text{None of the above}$
I know that the DT impulse response output equals: $y(t) = CA^{t-1}B$
so $h(t) = CA^{t-1}B$.
Entering a few values of $t$ into $h(t)$ gives: $h(t) = 1 \ \text{for} \ t = 1, $ $ \ h(t) = 2 \ \text{for} \ t = 2, $ $h(t) = 4 \ \text{for} \ t = 3, $ All answers fit this patern.
I know that $2^{t-1}$ means that the jordan form has at least a $2$ in it so $B$ is incorrect.
$A$ is the right answer, but I don't know how to get there.