A second-order system is represented by the difference equation y[n] = 0.25y[n − 2] + x[n] where x[n] is the input and y[n] the output.

(a) For the zero-input case, i.e., when x[n] = 0, find the initial conditions y[ − 1] and y[ − 2] so that y[n] = 0.5ⁿ u[n].

(b) Suppose the input is x[n] = u[n], without solving the difference equation can you find the corresponding steady state y_ss[n]? Explain how and give the steady-state output.

(c) Find the input x[n] so that for zero initial conditions, the output is given as y[n] = 0.5ⁿ u[n].

(d) If x[n] = δ[n] + 0.5δ[n − 1] is the input to the above difference equation, find the impulse response h[n]of the system.