HW4_solutoin

HW4_solutoin - HW4_Solution 4.1-1 t(b x(t = te u(t X s = te...

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HW4_Solution 4.1-1 (b) () t x tt e u t = (1 ) (1 ) 00 0 1 1 ts t s t s t Xs t ee d t t e d t t d e s ∞∞ −− −+ === + ∫∫ (1 ) (1 ) 0 1 [| ] 0 1 st te e dt s =− + (1 ) 0 1 1 ed t s = + (1 ) (1 ) 222 0 111 || 0 (1 ) ) ) ee sss = = +++ (h) 2 cos(5 ) () t x te t u t θ =+ 2( 5 ) ( 5 ) 1 [] ( ) 2 tj t j t e u t θθ + (25) 1 ( ) 2 jt j jt j u t 0 1 [ ] 2 j s t X se e e e e d t 11 1 1 22 5 2 2 5 jj sj + + 2 1(2 5 ) (2 5 ) 2 ) 2 5 es je s j s ++ + +− = ++ 2 5 ) 5 ) 2 ) 2 5 j s = 2 1( 2) 2cos( ) (5 ) (2 sin( )) 2 ( 2) 25 j s +∗ + ∗ = 2 ( 2 ) c o s () 5 s i n 42 9 s s + = 4.1-3 (c) 2 ) ) 1 6( 3 ) ) 3 2 ss a b s s s s == = + + + + 2 3 ) |3 . 2 ) s s a s = + + ; 2 2 ) |0 . 2 (3 ) s s b s + = ; 32 3.2 0.2 () 1 () 3 .2 () 0 xt t eut e ut δ = + (i) 3 2 2 0.25 ) ( 25 ) 1(1 ) 25 sk A s B s s s s s + + + + +

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1 1 00 2 04 5 45 11 6 4 4 3 32 2 16 8 sk A B k B kA B A B →∞⇒ = + =⇒=−+ ⇒ + = + =⇒ = − + + + = Î 22 3/4 1/4 1 10 () 1( 1 ) 4 2 5 s Xs ss s s =− + + ++ + Use Table 4.1 10 (c) , A=1, B=-10 Î 1 5 100 20 11 1, 5, 2, 5.59, tan ( ) 70 51 4 o acb r θ == = = = = = 3 1 5.59 ( ) [( ) cos(2 70 )] ( ) 44 4 tt o x e e t u t −− + + 4.2-1 (e) ( ) te ut t e ut ττ τ + −=−+ e teu t e eu t = + Therefore [ ] ( 1) ( ( ( s s ee e e Xs e e e s s =+= + + + (g) 0 0 0 ( ) sin ( ) ( ) [sin( )cos( ) cos( )sin( )] ( ) xt t t t ω
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HW4_solutoin - HW4_Solution 4.1-1 t(b x(t = te u(t X s = te...

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