A finite impulse response (FIR) filter has an input/output relation y[n] = x[n] − x[n − 5] where x[n] is the input and y[n] the output.

(a) Find the impulse response h[n] of this filter. Plot h[n] as a function of n, and indicate if the filter is causal and BIBO stable or not.

(b) Suppose the input is x[n] = u[n], find the corresponding output y[n] and carefully plot it. Are x[n] and y[n] finite-energy signals?

(c) If x[n] = sin (2πn/5)u[n], find its corresponding output y[n]. Determine the energies of the input x[n] and of y[n]. Are both finite energy?

(d) Determine the frequencies {ω₀} of the input x[n] = sin (ω₀n) u[n] for which the corresponding output y[n] is finite energy. If you choose a frequency different from these frequencies, is the output finite energy?