HW 09_examples - 1 ME360 - Solved Examples for Homework #9...

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1 ME360 - Solved Examples for Homework #9 Problem 1: Let x ( t )=2 rect ( t +1) and h ( t t rect ( t 1) . (a) Find the autocorrelation r xx ( t ) ; (b) Find the autocorrelation r hh ( t ) ; (c) Find the cross-correlation r hx ( t ) ; (d) Find the cross-correlation r xh ( t ) . -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x(t) = 2rect(t+1) Time (sec) x(t)
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2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 h(t) = 2 t rect(t-1) Time (sec) h(t) (a) Find the autocorrelation r xx ( t ) . Solution: x ( t )=2 rect ( t +1)=2 rect ( t ( 1)) x ( λ t rect ( λ t ( 1)) Definition of Autocorrelation: r xx ( t )= x ( t ) BBx ( t R −∞ x ( λ ) x ( λ t ) There are four regions over which the evaluation of the integral can be separated: -1.5 -1 -.5 x( ) λ λ x( t) λ− λ t-1.5 t-0.5 -1.5 -1 -.5
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3 Region 1: t 0 . 5 < 1 . 5 t< 1 No overlap: r xx =0 Region 2: 1 . 5 t 0 . 5 ≤− 0 . 5 ⇒− 1 t 0 r xx = R t 0 . 5 1 . 5 x ( λ ) x ( λ t ) = R t 0 . 5 1 . 5 2 · 2 =4 λ | t 0 . 5 1 . 5 =4( t ( 1)) Region 3: 1 . 5 t 1 . 5 0 . 5 0 t 1 r xx = R 0 . 5 t 1 . 5 2 · 2 λ | 0 . 5 t 1 . 5 0 . 5 ( t 1 . 5)) = 4( t 1) Region 4: t 1 . 5 > 0 . 5 t> 1 No overlap: r xx So, r xx ( t )= 0 ,t< 1 t ( 1)) , 1 t 0 t 1) , 0 t 1 0 ,t> 1 -4 -3 -2 -1 0 1 2 3 4 0 0.5 1 1.5 2 2.5 3 3.5 4 Autocorrelation Function Rxx(t) x(t) = 2 rect(t+1) Time (sec) Rxx(t) (b) Find the autocorrelation r hh ( t ) . Solution: h ( t )=2 t rect ( t 1) h ( λ t t rect ( λ t 1) There are four regions over which the evaluation of the integral can be separated: Region 1: t +1 . 5 < 0 . 5 1 r hh Region 2: 0 . 5 t . 5 1 . 5 1 t 0 r hh = R t +1 . 5 0 . 5 h ( λ ) h ( λ t R t +1 . 5 0 . 5 2 λ · 2( λ t ) = R t +1 . 5 0 . 5 ¡ 4 λ 2 4 λt ¢ = £ 4 3 λ 3 2 λ 2 t ¤ t +1 . 5 0 . 5 = 4 3 ( t . 5) 3 2 t ( t . 5) 2 + 1 2 t 1 6 Region 3: 0 . 5 t +0 . 5 1 . 5 0 t 1
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4 r hh = R 1 . 5 t +0 . 5 ¡ 4 λ 2 4 λt ¢ = £ 4 3 λ 3 2 λ 2 t ¤ 1 . 5 t +0 . 5 = 4 3 ( t +0 . 5) 3 +2 t ( t . 5) 2 +4 . 5 4 . 5 t Region 4: t> 1 r hh =0 So, r xx ( t )= 0 ,t< 1 4 3 ( t +1 . 5) 3 2 t ( t . 5) 2 + 1 2 t 1 6 , 1 t 0 4 3 ( t . 5) 3 t ( t . 5) 2 . 5 4 . 5 t, 0 <t 1 0 ,t> 1 -4 -3 -2 -1 0 1 2 3 4 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Autocorrelation of h(t) Rhh(t) Time (sec) Rhh(t) (c) Find the cross-correlation r hx ( t ) .
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HW 09_examples - 1 ME360 - Solved Examples for Homework #9...

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