A state-space representation of the scalar difference equation system
$$
\begin{aligned}
y(k+n) & +a_1(k) y(k+n-1)+\cdots+a_n(k) y(k) \\
& =b_0(k) u(k+n)+b_1(k) u(k+n-1)+\cdots+b_n(k) u(k)
\end{aligned}
$$
where $k=0,1,2, \ldots$, may be given by
$$
\begin{aligned}
{\left[\begin{array}{c}
x_1(k+1) \\
x_2(k+1) \\
\vdots \\
x_{n-1}(k+1) \\
x_n(k+1)
\end{array}\right] } & =\left[\begin{array}{ccccc}
0 & 1 & \cdots & 0 & 0 \\
0 & 0 & \cdots & 0 & 0 \\
\vdots & \vdots & & \vdots & \vdots \\
0 & 0 & \cdots & 0 & 1 \\
-a_n(k) & -a_{n-1}(k) & \cdots & -a_2(k) & -a_1(k)
\end{array}\right]\left[\begin{array}{c}
x_1(k) \\
x_2(k) \\
\vdots \\
x_{n-1}(k) \\
x_n(k)
\end{array}\right]+\left[\begin{array}{c}
h_1(k) \\
h_2(k) \\
\vdots \\
h_{n-1}(k) \\
h_m(k)
\end{array}\right] u(. \\
y(k) & =x_1(k)+b_0(k-n) u(k)
\end{aligned}
$$
Determine $h_1(k), h_2(k), \ldots, h_s(k)$ in terms of $a_i(k)$ and $b_f(k)$, where $i=1,2$, . and $j=0,1, \ldots, n$. Determine also the initial values of the state variables $x_1(t$ $x_2(0), \ldots, x_m(0)$ in terms of the input sequence $u(0), u(1), \quad, u(n-1)$ and the outp sequence $y(0), y(1), \ldots, y(n-1)$