solved03 - S 3.1 (P 1.19) _ (i) We know that w = W*T_s,...

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S 3.1 (P 1.19) ______________ (i) We know that w = W*T_s, i.e., w = 150*pi*(3/1000) = 0.45*pi (ii) w is a rational multiple of pi, therefore x[n] is periodic. The period N is determined by expressing w as (r/N)*2*pi where r and N have no common factors. In this case, w = (9/40)*2*pi so N=40. (iii) x[n] is constant for all n if w = 2*k*pi i.e., T_s = (2*k*pi)/(150*pi) = k*(13.3333. ..) ms or f_s = 1/T_s = 75/k samples/sec for some integer k. x[n] alternates in value between cos(q) and -cos(q) if w = pi + 2*k*pi i.e., T_s = (0.5 + k)*(13.3333. ..) ms or f_s = 1/T_s = 75/(k+0.5) samples/sec for some integer k. S 3.2 _____ (i) By inspection, the period T is given by 0.5 seconds. Thus W = 2*pi/T = 4*pi. Again by inspection, A=3.0, and since x(0) = 3*sqrt(2)/2, it follows that 3*cos(0*t + q) = 3*sqrt(2)/2 = sqrt(2)/2 i.e., cos(q) = -pi/4 or pi/4. Since the waveform is increasing at t=0, it follows that q = -pi/4. (ii) x[n] = 3*cos(W*T_s*n + q)
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This note was uploaded on 10/18/2011 for the course ENEE 241 taught by Professor Staff during the Spring '08 term at Maryland.

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solved03 - S 3.1 (P 1.19) _ (i) We know that w = W*T_s,...

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