# Exam1a - -Oregon Institute of TechnologyDigital Signal...

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-Oregon Institute of Technology- Digital Signal Processing Dr. Mateo Aboy EET 471 Practice Midterm Exam Instructions This is a closed book, closed notes exam. You are allowed to use calculator. 120 minutes. 1 Sinusoids 1. Given the x ( t ) defined below, (a) make a plot of x ( t ) for - 30 t 30, (b) label the plot with the time location of the first peak t 1 ( t 1 > 0) in seconds, and (c) determine the amplitude ( A ), phase ( φ ), and frequency ( f 0 ) in Hz. x ( t ) = 5 cos π 6 t - π 5 2. Use complex exponentials to prove the following trigonometric identity, cos(2 θ ) = cos 2 ( θ ) - sin 2 ( θ ) 3. Given the x ( t ) defined below, express x ( t ) in the form x ( t ) = A cos( wt + φ ), x ( t ) = 3 X k =1 A k cos(2 πf k t + φ k ) where A 1 = 5, A 2 = 9, A 3 = 2, φ 1 = 1 4 π , φ 2 = 1 6 π , φ 3 = 0, and f 1 = f 2 = f 3 . Show all your steps. 4. Given x ( t ) = A cos(2 πf 0 t + π ) = A cos(2 πf 0 ( t - t 1 )), and T 0 = 9 s, (a) determine time shift in seconds t 1 corresponding to a phase shift of φ = 2 π 2 , (b) determine the phase shift φ corresponding to a time shift t 1 of 7 seconds.

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