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20075ee102_1_SolutionsMidtermII

# 20075ee102_1_SolutionsMidtermII - EE 102 – Solutions for...

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Unformatted text preview: EE 102 – Solutions for Midterm II 11/21/07 Duration: 1 hour and 30 minutes The midterm is closed book and closed lecture notes. You can use a single page of handwritten notes. Please carefully justify all your answers. Problem 1: Consider the following signals: ∀ t ∈ Reals x ( t ) = 5 sin(2 π 27 t ) + 3 cos(2 π 3 t ) ∀ t ∈ Reals y ( t ) = sin( t ) + 2 t and the system S ( z ) = w defined by the following differential equation: 2 d 2 dt 2 w + 5 w = z 1. Is signal x or signal y periodic? If so, determine the corresponding period, the funda- mental frequency and its units. Signal x is periodic if there exists p ∈ Reals such that x ( t + p ) = x ( t ), that is: 5 sin(2 π 27 t + 2 π 27 p ) + 3 cos(2 π 3 t + 2 π 3 p ) = 5 sin(2 π 27 t ) + 3 cos(2 π 3 t ) Since sin and cos are periodic functions with period 2 π we conclude that the following must hold: 2 π 27 p = α 2 π 2 π 3 p = β 2 π with α, β ∈ Integers . Moreover, the period p should be the smallest rational solution to the above equations. We thus conclude that p = 1 3 and thus f = 1 p = 3 Hz or ω = 2 π p = 6 π rad/s....
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20075ee102_1_SolutionsMidtermII - EE 102 – Solutions for...

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