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LT Tables_rev3

LT Tables_rev3 - s X s Convolution in Time t h t x s H s X...

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Laplace Transform Table Time Signal Laplace Transform ) ( t u s / 1 0 ), ( ) ( > c c t u t u 0 , / ) 1 ( > c s e cs K , 3 , 2 , 1 ), ( = N t u t N K , 3 , 2 , 1 , ! 1 = + N s N N ) ( t δ 1 real ), ( c c t δ real , c e cs complex or real ), ( b t u e bt complex or real , 1 b b s + K , 3 , 2 , 1 ), ( = N t u e t bt N K , 3 , 2 , 1 , ) ( ! 1 = + + N b s N N ) ( ) cos( t u t o ω 2 2 o s s ω + ) ( ) sin( t u t o ω 2 2 o o s ω ω + ) ( ) ( cos 2 t u t o ω ) 4 ( 2 2 2 2 2 o o s s s ω ω + + ) ( ) ( sin 2 t u t o ω ) 4 ( 2 2 2 2 o o s s ω ω + ) ( ) cos( t u t e o bt ω 2 2 ) ( o b s b s ω + + + ) ( ) sin( t u t e o bt ω 2 2 ) ( o o b s ω ω + + ) ( ) cos( t u t t o ω 2 2 2 2 2 ) ( o o s s ω ω + ) ( ) sin( t u t t o ω 2 2 2 ) ( 2 o o s s ω ω + ( ) [ ] 2 2 1 : where ) ( 1 sin ζ ω α ζ ω ζω = n n t A t u t Ae n 2 2 2 n n s s ω ζω α + + ( ) [ ] ( ) = + = + n n n n n t A t u t Ae n ζω α ζ ω φ ζ ω ζω α β φ ζ ω ζω 2 1 2 2 2 2 1 tan 1 ) 1 ( ) ( 1 sin 2 2 2 n n s s s ω ζω α β + + + ) ( ) cos( t u t te o bt ω 2 2 2 2 2 ) ) (( ) ( o o b s b s ω ω + + + ) ( ) sin( t u t te o bt ω 2 2 2 ) ) (( ) ( 2 o o b s b s ω ω + + +

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Laplace Transform Properties Property Name Property Linearity ) ( ) ( t bv t ax + ) ( ) ( s bV s aX + Right Time Shift ( Causal Signal) 0 ), ( > c c t x ) ( s X e cs Time Scaling 0 ), ( > a at x 0 ), / ( 1 > a a s X a Multiply by t n K , 3 , 2 , 1 ), ( = n t x t n K , 3 , 2 , 1 ), ( ) 1 ( = n s X ds d n n n Multiply by Exponential complex or real ), ( a t x e at complex or real ), ( a a s X Multiply by Sine ) ( ) sin( t x t o ω [ ] ) ( ) ( 2 o o j s X j s X j ω ω + Multiply by Cosine ) ( ) cos( t x t o ω [ ] ) ( ) ( 2 1 o o j s X j s X ω ω + + ) ( t x & Time Differentiation 2 nd Derivative n th Derivative ) ( t x & & ) ( ) ( t x N ) 0 ( ) ( x s sX ) 0 ( ) 0 ( ) ( 2 x sx s X s & ) 0 ( ) 0 (
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Unformatted text preview: s X s Convolution in Time ) ( * ) ( t h t x ) ( ) ( s H s X [ ] ) ( lim ) ( s sX x s ∞ → = Initial-Value Theorem [ ] ) ( ) ( lim ) ( 2 sx s X s x s − = ∞ → & [ ] ) ( ) ( ) ( ) ( lim ) ( ) 1 ( 1 1 ) ( − − + ∞ → − − − − = N N N N s N sx x s x s s X s x L & Final-Value Theorem ) ( lim ) ( lim then exists, ) ( lim If s sX t x t x s t t → ∞ → ∞ → =...
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LT Tables_rev3 - s X s Convolution in Time t h t x s H s X...

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