soln5 - Solutions to Homework 5 EEL 5544 - Noise in Linear...

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Solutions to Homework 5 EEL 5544 - Noise in Linear Systems SS-1. 1. SS-2.
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SS-3. SS-4. 2. (a) In MATLAB: >> U=rand(1e5,1); >> X=sqrt(U); >> Y=U.ˆ2; >> mean(X) ans = 0.6666 >> sqrt(mean(U)) ans = 0.7070
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>> mean(Y) ans = 0.3332 >> mean(U)ˆ2 ans = 0.2498 Clearly, E [ f ( U )] is not equal to f ( E [ U ]) in general. (b) E [ U ] = 1 / 2 , so sqrtE [ U ] = 2 / 2 0 . 707 , and ( E [ U ]) 2 = 0 . 25 , which match up well with the estimates from the simulations. The derivations of E [ U ] and E [ U 2 ] are shown below, and again match up well with the simulation results: HW 5 Solutions
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3. HW 5 Solutions SS-5. E [ I A ( X )] = Z I A ( x ) f X ( x ) dx = Z A (1) f X ( x ) dx = P [ A ] 4. (a) Half-wave rectifier: g ( x ) = ± x, x 0 0 , x < 0 E [ Y 2 ] = E [ g 2 ( X )] = Z -∞ g 2 ( x ) f X ( x ) dx = Z 0 x 2 f X ( x ) dx By symmetry, Z 0 x 2 f X ( x ) dx = Z 0 -∞ x 2 f X ( x ) dx, so E [ Y 2 ] = 1 2 Z -∞ x 2 f X ( x ) dx = σ 2 2
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(b) Full-wave rectifier: h ( x ) = | x | Thus, E [ Y 2 ] = E [ h 2 ( X )] = Z -∞ h 2 ( x ) f X ( x ) dx = Z -∞ ( | x | ) 2 f X ( x ) dx = Z -∞ ( x ) 2 f X ( x ) dx = σ 2 5. Let g 1 ( X ) = g 2 ( X ) = X . Then
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This note was uploaded on 01/29/2012 for the course ECE 101 taught by Professor Wang during the Spring '11 term at Iowa State.

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soln5 - Solutions to Homework 5 EEL 5544 - Noise in Linear...

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