A LTI system is represented by the first-order difference equation

y[n] = x[n] − 0.5y[n − 1]                 n ≥ 0

where y[n] is the output and x[n] is the input.

(a) Find the Z-transform Y(z) in terms of X(z) and the initial condition y[−1].

(b) Find an input x[n] ≠ 0 and an initial condition y[−1] ≠ 0 so that the output is y[n] = 0 for n ≥ 0. Verify you get this result by solving the difference equation recursively.

(c) For a zero initial condition, find the input x[n]so that y[n] = δ[n] + 0.5δ[n − 1]