Question

Determine whether the following discrete-time signals are periodic or not? For the periodic ones, find their fundamental period. 2. (5 points each) a. x1 [n] =sin(0.5n +z/4) c. x1n] = cos (tn/3) + sin(nn/5)-2cos(tn/10) d. x4[n]- sin(tn/12) cos(tn/3) (Hint: use trigonometric identities to write the signal as a sum of sinusoids)

Answer 1

Given that A discrete time signals are periodic x_1 [n] = sin (0.5 n + pi/4) x_1 [n] = sin (0.5 n + pi/4) where w = 0.5: pi/4 = phi x_1 [n] = sin (w_n + phi) w/2 pi = 0.5/2 pi = 0.5/2(3.14) = 1/4 pi Hence not a rational number x_1 [n] = sin (0.5 n + pi/4) is a periodic x_2 [n] = e^j n pi/2 = e^j n . w where w = pi/2 \r\n w/2 pi = pi/2/2 pi = (3.14/2)/2(3.14) = 1.57/6.28 = 1/4 rightarrow Rational number rightarrow periodic w/2 pi = M/N = 1/4 period = N = 4 x_3 [n] = cos (pi n/3) + sin (pi n/s) - 2 cos (pi n/10) cos (pi n/3) + sin (pi n/5) - 2 cos (pi n/10) cos (pi n/3) = w_1 rightarrow pi/3 sin (pi n/5) = w_2 rightarrow pi/5 2 cos (n pi/10) = w_3 rightarrow pi/10 \r\n w_1 = pi/3: w_2 = pi/5: w_3 = pi/10