solution6 - EL630: Homework6 Solutions 1) a) ( 29 ( 29 ( 29...

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Unformatted text preview: EL630: Homework6 Solutions 1) a) ( 29 ( 29 ( 29 X a j j aX j Y j Y e E e e E e E = = = + ) ( ) ( . Now, the second factor here is just the characteristic function of ) , ( 2 N with replaced by a . Thus, 2 / 2 / 2 2 2 2 2 2 ) ( a j a j Y e e e-- = = . Note that this is the characteristic function of ) , ( 2 2 a N . So we conclude that Y is a normal r.v. with mean and variance 2 2 a . b) ( 29 - -- -- = = = dx e dx e e e E x j x x j X j Z 2 2 2 2 2 2 2 1 2 / 2 1 2 1 ) ( Now we use the following integral formula. A dx e x A = -- 2 Which holds for any complex number A with positive real part. Note that if 2 2 1 = A then this formula just says that the integral of the normal density is 1. Using this formula we get 2 2 2 1 1 2 1 2 1 ) ( j j Z- =- = ....
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This note was uploaded on 04/24/2011 for the course ECE 6303 taught by Professor Voltz during the Spring '11 term at NYU Poly.

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solution6 - EL630: Homework6 Solutions 1) a) ( 29 ( 29 ( 29...

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