Electrical Engineering 20N - Spring 1999 - Final Exam

Electrical Engineering 20N - Spring 1999 - Final Exam - 1...

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1 EECS 20. Final Exam Solutions 15 May 1999 1. 15 points Answer these short questions and use the space below for your calculations. (a) The solutions of the equation e j 4 θ =1 are θ = Ans θ =0 ,π/ 2 ,π, 3 π/ 2 . (b) Express cos 3 θ and sin 3 θ in terms of cos θ and sin θ : cos 3 θ = sin 3 θ = Ans cos 3 θ + j sin 3 θ = e j 3 θ =[cos θ + j sin θ ] 3 =[ c o s 3 θ - 3cos θ sin 2 θ ]+ j [3 cos 2 θ sin θ - sin 3 θ So, cos 3 θ =c o s 3 θ - θ sin 2 θ sin 3 θ 3 c o s 2 θ sin θ - sin 3 θ ] (c) For what real-valued numbers ω is the function x periodic: n Ints, x ( n )=cos ωn and what is the period? Ans x is periodic with integer period p provided that ω ( n + p )= +2 πm ,o r ωp =2 ,or ω πm/p for some integer m . (d) The general form of the following matrix for n 0 is: ± 11 01 ² n = Ans ± ² n = ± 1 n ²
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2 + D + D D α z(n) z(n-1) z α zx y x+y 2 u(n) y(n) Figure 1: An LTI system can be built using unit delays, gains, and adders 2. 15 points A LTI system can be built using unit delay elements D ,gains α , and adders, shown on top of Figure 1 . (a) Express the relation between the input and output of the system in the lower part of the figure in the form: y ( n )= a 1 y ( n - 1) + ··· + a k y
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This note was uploaded on 05/17/2009 for the course EE 20N taught by Professor Ayazifar during the Spring '08 term at Berkeley.

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Electrical Engineering 20N - Spring 1999 - Final Exam - 1...

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