# Ee301 Hw1

Due: October 4th , 2007, Thursday

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1. A discrete-time signal x[n] is shown in the ﬁgure above. Sketch and label carefully each of the following signals. (a) x[n − 4] (e) x[n]u[3 − n] (b) x[3 − n] (f) x[n − 2]δ(n − 2) (c) x[3n] 1 (g) 2 x[n] + 1 (−1)n x[n] 2 (d) x[3n + 1] (h) x[(n − 1)2 ]

2. Determine whether or not each of the following signals is periodic. If the signal is periodic, determine its fundamental period. (a) x(t) = jej10t (c) x(t) = [cos(2t − π )]2 3 (e) x(t) = ∞ e−(2t−n) n=−∞ 3 (g) x[n] = 3e(j 5 (n+1/2)) (i) x[n] = cos( π n2 ) 8 (b) x(t) = ej(πt−1) (d) x(t) = Ev{cos(4πt)u(t)} (f) x[n] = ej7πn (h) x[n] = cos( π n) cos( π n) 3 6 (j) x[n] = 2 cos( π n) + sin( π n) − 2 cos( π n + π ) 4 8 2 6

1 3. (a) Show that δ(2t) = 2 δ(t).

(b) Let x[n] be an arbitrary signal with even and oﬀ parts denoted by xe [n] = Ev{x[n]} and xo [n] = Od{x[n]}. Show that +∞ n=−∞

x2 [n] =

+∞ n=−∞

x2 [n] + e

+∞ n=−∞

x2 [n] o

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