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

This preview shows pages 1–6. Sign up to view the full content.

Solutions to Homework 5 EEL 5544 - Noise in Linear Systems SS-1. 1. SS-2.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
>> 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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
(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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## 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.

### Page1 / 21

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

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online