ENEE241hw02

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

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Unformatted text preview: 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(Ωt + 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 Ω 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 ￿ ￿ 5π n π x[n] = cos + 8 4 and y [n] = cos ￿ 10π n 2π − 7 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 Nx and Ny are the two fundamental periods found in part (i), show that u[n] = x[n] + y [n] is also periodic with period Nu = Nx Ny . Is Nu 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[2n] Write an equation for v [n]. What is the period of v [ · ]? (vi) (5 pts.) Using phasors, express x[n] − 2x[n − 1] + 3x[n − 2] as a single real-valued sinusoid. Problem 2C Two periods of the sinusoid x(t) = A cos(Ωt + φ) are plotted below. The value of x(0) equals 4 sin(2π /7). 4 3 2 1 0 −1 −2 −3 −4 0 0.01 0.02 0.03 0.04 0.05 0.06 (i) (4 pts.) Determine A, Ω and φ. Express φ as an exact rational multiple of π in the range [0, 2π ). (You may want to verify your answers by plotting the resulting sinusoid—it should be identical to the one in the figure.) In what follows: A, Ω and φ are as found in part (i) and x[n] = x(nTs ), where Ts is a suitably chosen sampling period. (ii) (4 pts.) Using phasors, express y (t) = x(t) + 2x(t − (π /4Ω)) as a single sinusoid. (iii) (3 pts.) Determine all values of Ts such that x[n] is proportional to constant for all n. (iv) (3 pts.) Determine all values of Ts such that x[n] = A cos((π /12)n + φ). (v) (3 pts.) Determine all values of Ts such that x[n] = x[n + 4] for all n (Note: The fundamental period of x[ · ] will equal 2 or 4.) (vi) (3 pts.) Determine all values of Ts such that x[n] = A cos((5π /6)n − φ). Solved Examples S 2.1 (P 1.11 in textbook). Consider the continuous-time sinusoid x(t) = 5 cos(500π t + 0.25) where t is in seconds. (i) What is the first value of t greater than 0 such that x(t) = 0? (ii) Consider the following MATLAB script which generates a discrete approximation to x(t): t = 0 : 0.0001 : 0.01 ; x = 5*cos(500*pi*t + 0.25) ; For which values of n, if any, is x(n) zero? S 2.2 (P 1.12 in textbook). The value of the continuous-time sinusoid x(t) = A cos(Ωt + φ) (where A > 0 and 0 ≤ φ < 2π ) is between −2.0 and +2.0 for 70% of its period. (i) What is the value of A? (ii) If it takes 300 ms for the value of x(t) to rise from −2.0 to +2.0, what is the value of Ω? (iii) If t = 40 ms is the first positive time for which x(t) = −2.0 and x￿ (t) (the first derivative) is negative, what is the value of φ? S 2.3 (P 1.13 in textbook). The input voltage v (t) to a light-emitting diode circuit is given by A cos(Ωt + φ), where A > 0, Ω > 0 and φ are unknown parameters. The circuit is designed in such a way that the diode turns on at the moment the input voltage exceeds A/2, and turns off when the voltage falls below A/2. (i) What percentage of the time is the diode on? (ii) Suppose the voltage v (t) is applied to the diode at time t = 0. The diode turns on instantly, turns off at t = 1.5 ms, then turns on again at t = 9.5 ms. Based on this information, determine Ω and φ. S 2.4. Find all frequencies ω in [0, π ] for which the discrete-time sinusoid x[n] = cos ω n is periodic with fundamental period N = 16. S 2.5 (P 1.15 in textbook). (i) For exactly one value of ω in [0, π ], the discrete-time sinusoid v [n] = A cos(ω n + φ) is periodic with period equal to N = 4 time units. What is that value of ω ? (ii) For that value of ω , suppose the first period of v [n] is given by v [0] = 1, v [1] = 1, v [2] = −1 and v [3] = −1 What are the values of A > 0 and φ? S 2.6 (P 1.17 in textbook). (i) Use MATLAB to plot four periods of the discrete-time sinusoid ￿ ￿ 7π n π x1 [n] = cos + 9 6 (ii) Show that the product x2 [n] = x1 [n] · cos(π n) is also a (real-valued) discrete-time sinusoid. Express it in the form A cos(ω n + φ), where A > 0, ω ∈ [0, π ] and φ ∈ (0, 2π ). S 2.7 (P 1.16 in textbook; more difficult). (i) Use the trigonometric identity cos(α + β ) = cos α cos β − sin α sin β to show that (ii) Suppose cos(ω (n + 1) + φ) + cos(ω (n − 1) + φ) = 2 cos(ω n + φ) cos ω x[1] = 1.7740, x[2] = 3.1251 and x[3] = 0.4908 are three consecutive values of the discrete-time sinusoid x[n] = A cos(ω n + φ), where A > 0, ω ∈ [0, π ] and φ ∈ [0, 2π ]. Use the equation derived in (i) to evaluate ω . Then use the ratio x[2]/x[1] together with the given identity for cos(α + β ) to evaluate tan φ and hence φ. Finally, determine A. S 2.8 (P 1.19 in textbook). The continuous-time sinusoid x(t) = cos(150π t + φ) is sampled every Ts = 3.0 ms starting at t = 0. The resulting discrete-time sinusoid is x[n] = x(nTs ) (i) Express x[n] in the form x[n] = cos(ω n + φ) i.e., determine the value of ω . (ii) Is the discrete-time sinusoid x[n] periodic? If so, what is its period? (iii) Suppose that the sampling rate fs = 1/Ts is variable. For what values of fs is x[n] constant for all n? For what values of fs does x[n] alternate in value between − cos φ and cos φ? √ S 2.9. The first period of the sinusoid x(t) = A cos(Ωt + φ) is plotted below, where x(0) = 3 2/2. 3 2 x(t) 1 0 −1 −2 −3 0 0.1 0.2 0.3 t (seconds) 0.4 0.5 (i) Determine the values of A, Ω and φ. (ii) The sinusoid is sampled every Ts = 0.05 seconds starting at t = 0 to produce x[n] = x(nTs ) Write an equation for x[n]. Is x[n] periodic, and if so, what is its period? (iii) What is the relationship between the vector (x[0], . . . , x[4]) and the vector (x[5], . . . , x[9])? S 2.10 (this problem has a somewhat different flavor). The continuous time sinusoid x(t) = cos(1200π t) is sampled every Ts seconds starting at time t = 0. The value of Ts is chosen so that every zero of x( · ) is also a (zero-valued) sample in x[ · ]. (i) What are the possible values of Ts ? (ii) What is the maximum value of Ts (less than one-quarter period of x( · )) such that the difference between two consecutive samples does not exceed 0.01 (in absolute value)? ...
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This note was uploaded on 09/27/2011 for the course ENEE 241 taught by Professor Staff during the Spring '08 term at Maryland.

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