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hw03ab - ENEE 241 02 HOMEWORK ASSIGNMENT 3 Due Tue 02/17...

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ENEE 241 02* HOMEWORK ASSIGNMENT 3 Due Tue 02/17 Problem 3A Consider the discrete-time sinusoids x [ n ] = cos 7 πn 12 - 0 . 5 and y [ n ] = cos (1 . 83 n - 0 . 5) (i) (3 pts.) Which of the two sinusoids is periodic, and what is its fundamental period? (ii) (3 pts.) Use MATLAB to generate separate plots of x [ n ] and y [ n ] for n = 0 , . . . , 47. (iii) (4 pts.) Let N be the value of the period found in part (i). For what values of ω in [0 , π ] is cos( ωn ) periodic with fundamental period N ? (iv) (3 pts.) An equivalent form for x [ · ] is x [ n ] = cos ( ωn + φ ) where ω is between π and 2 π . What are the values of ω and φ ? (v) (2 pts.) The sequence v [ · ] is formed by taking every fourth sample in x [ · ], i.e., v [ n ] = x [4 n ] Write an equation for v [ n ]. What is the period of v [ · ]? (vi) (5 pts.) Using phasors, express y [ n ] - y [ n - 1] - 2 y [ n - 2] as a single real-valued sinusoid. Problem 3B 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 -3 -2 -1 0 1 2 3 Two periods of the sinusoid x ( t ) = A cos(Ω t + φ ) are plotted above. The value of x (0) equals - 3 sin( π/ 5).
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(i) (6 pts.) Determine A , Ω and φ . Express φ as an exact rational multiple of π in the range [0 , 2 π ).
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