Consider the convolution of a pulse x(t) = u(t + 0.5) − u(t − 0.5) with itself many times. Use MATLAB for the calculations and the plotting.

(a) Consider the result for N = 2 of these convolutions, i.e., y₂(t) = (x∗x) (t) Find Y₂(s) = L[y₂(t)] using the convolution property of the Laplace transform and find y₂(t)

(b) Consider the result for N = 3 of these convolutions, i.e., y₃(t) = (x∗x∗x) (t) Find Y₃(s) = L[y₃(t)] using the convolution property of the Laplace transform and find y₃(t)

(c) The signal x(t) can be considered the impulse response of an averager which ”smooths” out a signal. Letting y₁(t) = x(t), plot the three functions y_i(t) for i = 1, 2, and 3. Compare these signals on their smoothness and indicate their supports in time (for y₂(t) and y₃(t) how do their supports relate to the supports of the signals convolved?).