# ps9soln - MIT OpenCourseWare http/ocw.mit.edu 2.161 Signal...

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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 .

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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 0 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 0 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 ( ) = 1. The output can be computed from: Φ gg ( ) = | H ( ) | 2 Φ ff ( ) = | H ( ) | 2 . 1 = | H ( ) | 2 + ω c Φ gg ( τ ) = F 1 gg ( )) = 1 | H ( ) | 2 e jwτ = 1 1 .e jωτ = ω c sinc( ω c τ ) = sin( ω c τ ) 2 π 2 π π πτ −∞ ω c 1
Problem 3:

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