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ENEE 241 02* HOMEWORK ASSIGNMENT 3 Due Tue 02/ 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 2π. 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[4n] 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
−3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.
−
−
0
1
2
3
Two periods of the sinusoid x(t) = A cos(Ωt + φ) are plotted above. The value of x(0) equals −3 sin(π/5).
(i) (6 pts.) Determine A, Ω and φ. Express φ as an exact rational multiple of π in the range [0, 2 π). In what follows: A and φ are as found in part (i) and x[n] = x(nTs), where Ts is a suitably chosen sampling period. (ii) (3 pts.) For what values of Ts is it true that x[n] is constant in n? (iii) (3 pts.) For what values of Ts is it true that x[n] = −x[n − 1] for all n? (iv) (4 pts.) Determine all values of Ts such that x[n] = A cos((5π/8)n + φ). (v) (4 pts.) Determine all values of Ts such that x[n] = A cos((π/6)n − φ).
Solved Examples S 3.1. Find all frequencies ω in [0, π] for which the discrete-time sinusoid
x[n] = cos ωn is periodic with fundamental period N = 16. S 3.2 (P 1.15 in textbook). (i) For exactly one value of ω in [0, π], the discrete-time sinusoid v[n] = A cos(ωn + 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 3.3 (P 1.17 in textbook). (i) Use MATLAB to plot four periods of the discrete-time sinusoid
x 1 [n] = cos
( (^7) πn 9 +^
π 6
(ii) Show that the product x 2 [n] = x 1 [n] · cos(πn) is also a (real-valued) discrete-time sinusoid. Express it in the form A cos(ωn + φ), where A > 0, ω ∈ [0, π] and φ ∈ (0, 2 π). S 3.4 (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 cos(ω(n + 1) + φ) + cos(ω(n − 1) + φ) = 2 cos(ωn + φ) cos ω (ii) Suppose x[1] = 1. 7740 , x[2] = 3. 1251 and x[3] = 0. 4908
