# hw5 - The homework has to be stapled PSTAT 174/274...

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Unformatted text preview: - The homework has to be stapled. PSTAT 174/274. Homework #5 Name: Due: 11/22/06 at the beginning of the class Score: /10 Credit: People you worked with: Sources consulted(Reference): For Question 1 - 3, assume ω k = 2 πk n and n is an even number. Useful formulas e iω = cos ω + i sin ω 2 cos a cos b = cos( a + b ) + cos( a- b ) 2 sin a sin b = cos( a- b )- cos( a + b ) 2 sin a cos b = sin( a + b ) + sin( a- b ) 1. (3 points) Prove that { sin( ω k t ) , cos( ω k t ) : k = 0 , 1 , 2 , ··· ,n/ 2 } is a collection of orthogonal func- tions. Hint > To prove φ k ( t ) and φ j ( t ) ( t ∈ D ) are orthogonal functions, you need to prove X t ∈ D φ k ( t ) φ j ( t ) = 0 , k 6 = j 6 = 0 , k = j 2. (1 point each) With the proof from Question 1, X t ( t = 1 , 2 , ··· ,n ) can be written as X t = ∑ n/ 2 k =0 { a k sin( ω k t ) + b k cos( ω k t ) } . Show that (a) a k = 1 n ∑ n t =1 X t cos( ω k t ) , k = 0 ,n/ 2 2 n ∑ n t =1 X t cos( ω k t ) , k...
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## This note was uploaded on 05/01/2009 for the course PSTAT 120A taught by Professor Mackgalloway during the Spring '08 term at UCSB.

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hw5 - The homework has to be stapled PSTAT 174/274...

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