Question

Please be neat and show all work. Thank you.

1. Determine the z-transform of the signal x(n)s {100 elsewhere 1} x(n)=( 0; elsewhere

Answer 1

X(n) = {1 , 0lessthanorequalto n lessthanorequaltoN - 1 0, elsewhere X(z) = sigma^infinity_-infinity x(n) z^-n X(z) = x(0) z^-0 + sigma^N - 1_n = 1 x(n) z^-n X(z) = x(0) + sigma^N - 1_n = 1 x(n) z^-n X(z) = 1 + sigma^N - 1/n = 1 z^-n X(z) = 1 + [z^-1 + z^-2 + ....+ z^-(N - 1)], + Vertical-bar z Vertical-bar > 1 above series in G.P therefore In G.P . Sum of N - finite numbers, = 1/1 - r Here, a = first term = 1 r = z^-1 therefore X(z) = 1/ 1 - z^-1 X(z) = 1/1 - Z^-1 Vertical-bar z Vertical-bar > 1