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Unformatted text preview: Laplace Transform • Laplace Transform of Signals: Signals are represented by func tions of time t . For a signal f ( t ), its onesided Laplace transform is defined by F ( s ) := L [ f ( t )] = Z ∞ f ( t ) e st dt. where s = σ + jω is a complex number and σ is sufficiently large so that the integral converges. can be simply replaced by 0 if f ( t ) does not contain impulse at t = 0. • Inverse Laplace Transform: Given F ( s ), f ( t ) can be obtained as f ( t ) = 1 2 πj Z σ c + j ∞ σ c j ∞ F ( s ) e st ds where σ c is large enough to include all p k , poles of F ( s ), on the left of straight line s = σ c . Thus f ( t ) can be computed using residue theorem: f ( t ) = X Res p k ‰ F ( s ) e st = X Res p k { F ( s ) } e p k t . 1 Transfer Function • Systems: Systems process signals in a certain way. y ( t ) u ( t ) H For a linear timeinvariant system, it can be represented by trans fer function defined by H ( s ) = L [ y ( t )] L [ u ( t )] fl fl fl fl fl fl fl zero initial condition = Y ( s ) U ( s ) . Thus input and output have a simpler relation in sdomain: Y ( s ) = H ( s ) U ( s ) Laplace transform is a tool to analyze signals and systems. 2 Example 3.5: Find the Laplace Transform of the step function a 1( t ) and ramp function bt 1( t ). Solution: By the Laplace transform integral L [ a 1( t )] = Z ∞ a 1( t ) e st dt = ae st s fl fl fl fl fl fl fl ∞ = a s Note that e st → 0 as t → ∞ for s = σ + jω with σ > 0. Similarly, L [ bt 1( t )] = Z ∞ bt 1( t ) e st dt =  bte st s be st s 2 ∞ = b s 2 Example 3.6: Find the Laplace Transform of the unitimpulse function δ ( t ). Solution: By the Laplace transform integral L [ δ ( t )] = Z ∞ δ ( t ) e st dt = Z + δ ( t ) e st dt = 1 . Example 3.7: Find the Laplace Transform of the sinusoid function. Solution: By the Laplace transform integral L [sin ωt ] = Z ∞ sin ωt e st dt = Z ∞ e jωt e jωt 2 j e st dt = Z ∞ e ( jω s ) t e ( jω s ) t 2 j dt = 1 2 j 1 s jω 1 s + jω = ω s 2 + ω 2 3 Properties of Laplace Transforms 1. Superposition: L [ α 1 f 1 ( t ) + α 2 f 2 ( t )] = α 1 F 1 ( s ) + α 2 F 2 ( s ). 2. Time delay: L [ f ( t T )] = e sT F ( s ). L [ f ( t T )] = Z ∞ f ( t T ) e st dt = Z ∞ T f ( τ ) e s ( τ + T ) dτ = e sT Z ∞ f ( τ ) e sτ dτ = e sT F ( s ) 3. Timescaling: L [ f ( at )] = 1 a F ( s a ) , a > 0. L [ f ( at )] = Z ∞ f ( at ) e st dt = 1 a Z ∞ f ( at ) e ( s/a )( at )) d ( at ) = F ( s/a ) a 4. Shift in frequency: L [ e at f ( t )] = F ( s + a ). L [ e at f ( t )] = Z ∞ e at f ( t ) e st dt = Z ∞ f ( t ) e ( s + a ) t dt = F ( s + a ) 5. Differentiation: L [ ˙ f ( t )] = sF ( s ) f (0 ), and L • f ( m ) ( t ) ‚ = s m F ( s ) s m 1 f (0 ) s m 2 ˙ f (0 ) ···  f ( m 1) (0 ) ....
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This note was uploaded on 02/03/2012 for the course EE 3530 taught by Professor Chen during the Fall '07 term at LSU.
 Fall '07
 Chen

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