A periodic discrete-time signal x[n] with a fundamental period N = 3 is passed through a filter with impulse response  h[n] = (1/3) (u[n] − u[n − 3]). Let y[n] be the filter output. We begin the  filtering at n = 0, i.e., we are interested in y[n], n ≥ 0 and could assume y[n] = 0, n < 0. For a period, x[n] is x[0] = 1, x[1] = − 2, x[2] = 1.

(a) Determine the values of y[0], y[1], and y[2]. Because x[n] is periodic you can assume its values for n < 0 are given.

(b) Use the convolution sum to calculate y[n], n ≥ 0. What is the  steady state output? When is the steady state of y[n] attained?

(c) Compute the output of the filter using the DTFT.