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HW6 - ECE 301 Homework#6 due date...

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ECE 301, Homework #6, due date: 10/6/2010 http://cobweb.ecn.purdue.edu/ chihw/10ECE301F/10ECE301F.html Question 1: [Basic] Review of linear algebra: Consider row vectors of dimension 3. Let x 1 = ( 2 / 2 , - 2 / 2 , 0), x 2 = ( 3 / 3 , 3 / 3 , 3 / 3), and x 3 = ( 6 / 6 , 6 / 6 , - 2 6 / 6)) Show that { x 1 , x 2 , x 3 } is an orthonormal basis. Namely, show that | x i | 2 = 1 for all i = 1 , 2 , 3, and show that the inner product x i · x j = 0 for i 6 = j . If we know that x = 0 . 7 x 1 + 0 . 3 x 2 + 0 . 4 x 3 , find x . If we know that x 0 = (0 . 7 , 0 . 3 , 0 . 4), find α 1 , α 2 , α 3 such that x 0 = α 1 x 1 + α 2 x 2 + α 3 x 3 . Why are we interested in rewriting x 0 = α 1 x 1 + α 2 x 2 + α 3 x 3 ? Note: There is a simple formula of solving α 1 , α 2 , α 3 when x 1 , x 2 , and x 3 being orthonor- mal . Please refer to any linear algebra textbook or website, or come to the office hours if you are not familiar with that formula. It might take too much time for you to re-derive existing results. Question 2: [Basic] Consider a LTI system with impulse response h ( t ) = 3 - t U ( t ). What is the output y ( t
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