The output of a discrete-time system is y[n] = w[n] x[n] where x[n] is the input, and w[n] = u[n] − u[n − 5] is a rectangular window.

(a) The input is x[n] = 4 sin (πn/2), −∞ < n < ∞, determine if x[n] is periodic, and if so indicate its fundamental period N₀.

(b) Let z[n] = w[n] x[n − N] for what values of N is y[n] = z[n]? According to this, is the system time-invariant? Explain.

(c) Suppose that we let x₁[n] = x[n] u[n], i.e., x₁[n] is the causal component of x[n]. Determine the outputs due to x₁[n] and to x₁[n − 4]. Again according to this, is the system time-invariant? Explain.