Question

For each of the cases below, sketch the signals r(t) and h(t) and compute and sketch the signal y(t) -r(t) * h(t) where * denotes the convolution integral (1+1+3-5 points for each part) (anln+1] -uln -1] and h[n]un-un-2. (1+1+3-5 points) (b) n un-un-2 and hn ()un. (1+1+3-5 points) 0, else h[n+1] - un -1. (1+1+3-5 points)

Please solve and show work! Thank you in advance!

Answer 1

(a) c[n] = u[n + 1 - u[n - 1] = {1, n = -1, 0, 1 0, else h[x] = u[n] - u[n - 2] = [1, n = 0, 1, 2 0, else x[n] * h[n] = Sum_k = -infinity h[k] x[n - k] = h[0] x[n - 0] + h[1] x[n - 1] + h[2] x[n - 2] + 0 Ellipsis other terms vanish {h[0] = h[1] = h[2] = 1} x[n] * h[n] = x[n] + x[n - 1] + x[n - 2] \r\n y[n] = x[n] * h[n] = x[n] + c[n - 1] + x[n - 2] = {0 n < 0 1 n = 0 2 n = 1 3 n = 2 3 n = 3 3 n = 4 Ellipsis y[n] = [0 0 0 0 1 2 3 3 3 Ellipsis] \r\n (b) x[n] = u[n] - u[n - 2] {1: n = 0, 1, 2 0: Elsewhere h[n] = {(1/2)^n: n greaterthanorequalto 0 0: n < 0 y[n] = Sum_k = -infinity^infinity = h[k] x[n - k] = Sum_k = 0^infinity (1/2)^k x[n - k] y[n] = (1/2)^0 x[n] + (1/2) x[n - 1] + (1/2)^2 x[n - 2]