EE 20N - Spring 2002 - Varaiya - Final Solutions

# EE 20N - Spring 2002 - Varaiya - Final Solutions - EECS20...

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EECS20, Spring 2002 – Solutions to Final Exam 0. Typos/Corrections - Announced during exam. -All £ symbols stand for the letters f i . -For Problem 2, assume the system is causal. -Problem 4 is for 20 points, not 10 points. -In 4(c), the argument of y should be τ , not “ t ”. -In 4(d), take the input to be x ( n ) = δ ( n ) + 2 δ ( n - 2) + 3 δ ( n - 3) . 1. 20 points (a) By inspection, we obtain that w 0 = π/ 2 . A 1 = A 2 = A 6 = A 10 = 1 , A k = 0 otherwise. φ k = 0 for all k . (b) Using the relations X 0 = A 0 , X k = 0 . 5 A k e k , k = 1 , 2 , ··· X - k = X k * = 0 . 5 A k e - k , k = 1 , 2 , ··· we get X k = ( 0 . 5 , k = ± 1 , ± 2 , ± 6 , ± 10 0 , otherwise (c) y ( t ) = X k = -∞ X k H ( w 0 k ) e ikw 0 t = X 2 H ( π ) e iπt + X - 2 H ( - π ) e - iπt = - cos( πt ) . 1

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2. 15 points (a) s ( n ) = y ( n - 1) y ( n - 2) x ( n - 1) , A = 0 α 2 2 1 0 0 0 0 0 . , b
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EE 20N - Spring 2002 - Varaiya - Final Solutions - EECS20...

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