Electrical Engineering 20N - Spring 2003 - Varaiya - Mid 2

Electrical Engineering 20N - Spring 2003 - Varaiya - Mid 2...

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EECS 20. Midterm No. 2 Solution April 11, 2003. 1. 15 points For the following hybrid system sketch (a) 10 points the state trajectory (both the mode and the continuous state) and (b) 5 points the output signal for 0 t 3 . { s ( t ) = 1}/ ping s ( t ): = 1 { s ( t ) = 0}/ pong s ( t ):=0 { ping , pong, absent } ping-pong s (0) := 0 s ( t ) = 1 . mode1 s ( t ) = - 2 . mode2 0123 mode t s ( t ) t y ( t ) 01 2 3 y ( t ) t 1 1.5 absent ping pong 1.5 1.5 2.5 2.5 2.5 mode2 mode1 2. 15 points, 5 points each part Give the units of period and frequency below (a) Consider the discrete-time signal x given by n Integers, x ( n )=cos ωn. For what values of ω is x periodic, and what is the period? x is periodic with period p samples if ωp is a multiple of 2 π , i.e. if ω =2 mπ/p rad/sample for integers m, p . To get the smallest period p , m, p must be coprime. 1
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(b) Consider the discrete-time signal x given by n Integers, x ( n )=1+cos(4 πn/ 9) . What is its period p and what is its fundamental frequency? The period is p =9 samples , and the fundamental frequency is ω 0 =2 π/ 9 rads/sample .
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Electrical Engineering 20N - Spring 2003 - Varaiya - Mid 2...

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