20101ee131A_1_ee131A_1_hw7sol-2010

20101ee131A_1_ee131A_1_hw7sol-2010 - 4.99 4.103 b x x x x 1...

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4.99 4.103
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b) 2 2 2 2 0 0 0 0 2 0 1 1 ( ) 2 ( ) 2 2 1 1 2 (1 ) 1 2 3 3 x x x x y y x x P Y X e dx ye dy e dx e e e dx < = = = = = 7.7 a) 2 10 20 ( ) ( ) ( ) 2 10 (2 )(10 ) Z X Y j j j j ω ω ω ω ω ω ω Φ = Φ Φ = = b) 20 5 1 1 ( ) (2 )(10 ) 2 2 10 5 2 1 10 4 2 4 10 Z j j j j j j ω ω ω ω ω ω ω Φ = = = Thus, by the linearity of Fourier transform, 2 10 2 10 5 1 5 5 ( ) 2 10 , 0 4 4 2 2 z z z z Z f z e e e e z = × × × × = .
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