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ENEE241hw02

# ENEE241hw02 - ENEE 241 02 HOMEWORK ASSIGNMENT 2 Due Tue...

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ENEE 241 02 HOMEWORK ASSIGNMENT 2 Due Tue 02/15 Problem 2A Use your calculator for algebraic calculations only. Solutions based on trial and error, inspection of numerical plots, etc., are not acceptable. Consider the sinusoid x ( t ) = A cos( t + φ ), where A > 0 and φ ( π , π ]. Time t is in seconds. It is known that x ( t ) 2 . 4 for exactly 18% of each period; it takes 0.123 seconds for the value of the sinusoid to drop from 2 . 4 to the next minimum (“valley”); the first zero of the sinusoid in positive time occurs at t = 0 . 040 (seconds). (i) (5 pts.) Determine the amplitude A . (ii) (5 pts.) Determine the period and angular frequency of x ( t ). (iii) (5 pts.) Determine the initial phase φ of x ( t ) as a fraction of π . (iv) (5 pts.) Write simple MATLAB code which computes and plots two periods of x ( t ) starting at t = 0, using 150 uniformly spaced samples per period (i.e., a total of 300 samples). Attach a printout of the code and a plot of the result. Problem 2B Consider the discrete-time sinusoids x [ n ] = cos 5 π n 8 + π 4 and y [ n ] = cos 10 π n 7 2 π 3 (i) (3 pts.) What is the fundamental period of each sinusoid? (ii) (4 pts.) Use MATLAB to generate separate plots of x [ n ] and y [ n ] for n = 0 , . . . , 111. (iii) (3 pts.) If N x and N y are the two fundamental periods found in part (i), show that u [ n ] = x [ n ] + y [ n ] is also periodic with period N u = N x N y . Is N u the fundamental period of u [ n ]? (iv) (3 pts.) An equivalent form for y [ · ] is y [ n ] = cos ( ω n + φ ) where ω is between 0 and π . What are the values of ω and φ ? (v) (2 pts.) The sequence v [ · ] is formed by taking every other sample in x [ · ], i.e., v [ n ] = x [2 n ] Write an equation for v [ n ]. What is the period of v [ · ]?

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ENEE241hw02 - ENEE 241 02 HOMEWORK ASSIGNMENT 2 Due Tue...

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