Consider a system described by the coupled difference equation

\[
\begin{aligned}
y(k+2)-v(k) & =0 \\
v(k+1)+y(k+1) & =u(k)
\end{aligned}
\]

where \(u(k)\) is the system input.

(a) Find a state-variable formulation for this system. Consider the outputs to be \(y(k+1)\) and \(v(k)\). Draw a simulation diagram first.

(b) Repeat part (a) with \(y(k)\) and \(v(k)\) as the outputs.

(c) Repeat part (a) with the single output \(v(k)\).

(d) Use (2-84) to calculate the system transfer function with \(v(k)\) as the system output, as in part (c); that is, find \(V(z) / U(z)\).

(e) Verify the transfer function \(V(z) / U(z)\) in part (d) by taking the \(z\)-transform of the given system difference equations and eliminating \(Y(z)\).

(f) Verify the transfer function \(V(z) / U(z)\) in part (d) by using Mason's gain formula on the simulation diagram of part (a).