A discrete-time averager is represented by the input/output equation y[n] = (1/3)(x[n + 1] + x[n] + x[n − 1]), where x[n] is the input and y[n] the output.

(a) Determine whether this system is causal or not. Explain.

(b) Determine whether this system is BIBO stable or not. Explain.

(c) The input of the system is generated by sampling an analog signal x(t) = 2 cos (10t) using sampling periods T_s1 = 1 or T_s2 = πsec/sample. If we want the discrete-time signal x[n] = x(t)|_t=nTs to be periodic, which of the two sampling periods {T_si, i = 1, 2} would you use? For the chosen sampling period what would be the fundamental period of x[n]?

(d) If x[n] is periodic, would be the output of the averager be also periodic? If so, what would be the fundamental period of the output? Explain.