ps9soln - MIT OpenCourseWare http://ocw.mit.edu 2.161...

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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 2.161 Signal Processing: Continuous and Discrete Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.161 Signal Processing - Continuous and Discrete Fall Term 2008 Solution of Problem Set 9 Assigned: November 20, 2008 Due: December 4, 2008 Problem 1: + ∞ + ∞ 1 1 = xp ( x ) dx = x ( λe − λx ) dx = e − λx ( − x − ) + ∞ = µ x λ λ −∞ 0 + ∞ + ∞ 1 2 σ x 2 = µ x − µ 2 x = x 2 p ( x ) dx − µ 2 x = x 2 ( λe − λx ) dx − λ 2 −∞ 0 1 1 1 2 1 1 σ 2 = e − λx ( − x 2 + 2 ( − x − )) + ∞ − = − = x λ λ λ 2 λ 2 λ 2 λ 2 1 σ x = λ Problem 2: ff ( j ω ) gg ( j ) H ( j ) c c- 1 ω ω ω ω Φ Φ Input f is a white noise with flat spectrum and hence we assume that Φ ff ( jω ) = 1. The output can be computed from: Φ gg ( jω ) = | H ( jω ) | 2 Φ ff ( jω ) = | H ( jω ) | 2 . 1 = | H ( jω ) | 2 + ∞ ω c Φ gg ( τ ) = F − 1 (Φ gg ( jω )) = 1 | H ( jω ) | 2 e jwτ dω = 1 1 .e jωτ dω = ω c sinc( ω c τ ) = sin( ω c τ ) 2 π 2 π π πτ −∞ − ω c 1 Problem 3: The inverse filter for this high-precision problem can be implemented as the exact inverse of the input filter. We will revisit this problem in the 2nd quiz where additive noise of...
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This note was uploaded on 02/27/2012 for the course MECHANICAL 2.161 taught by Professor Derekrowell during the Fall '08 term at MIT.

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ps9soln - MIT OpenCourseWare http://ocw.mit.edu 2.161...

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