hw1 - EE 562a Homework Set 1 Due Monday, 22 Jan 2007 1 (The...

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Unformatted text preview: EE 562a Homework Set 1 Due Monday, 22 Jan 2007 1 (The following well-defined problems come from different sources, and the notation used may vary. Don’t let that bother you!) 1. A function is called concave (or equivalently convex ∩ (cap) or convex upward ) if g ( αx + (1- α ) y ) ≥ αg ( x ) + (1- α ) g ( y ) for all x,y ∈ R . Show that if g ( x ( u )) is a concave function of a real random variable x ( u ), then E { g ( x ( u )) } ≤ g ( E { x ( u ) } ) . 2. Define two random processes x ( u,t ) and y ( u,t ) with the index set J = { 1 , 2 , 3 ,... } . Let x ( u,t ) be an i.i.d. random process with mean value function E { x ( u,t ) } = 0 and correlation function E { x ( u,t 1 ) x ( u,t 2 ) } = δ t 1 t 2 . The process y ( u,t ) is defined by y ( u, 1) = (1- p ) 1 / 2 x ( u, 1) , and y ( u,t ) = p 1 / 2 y ( u,t- 1) + (1- p ) 1 / 2 x ( u,t ) , t ≥ 2 , where 0 < p < 1. Find the mean value function m y ( t ) and the correlation function R y ( t 1 ,t 2 ). 3. Amplitude and Phase Modulation: In this problem you are asked to find the second moment description of two random processes (models for communication signals). You maymoment description of two random processes (models for communication signals)....
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This note was uploaded on 05/06/2008 for the course EE 562a taught by Professor Toddbrun during the Spring '07 term at USC.

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hw1 - EE 562a Homework Set 1 Due Monday, 22 Jan 2007 1 (The...

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