EE3530-3.1b

# EE3530-3.1b - Laplace Transform • Laplace Transform of Signals Signals are represented by func tions of time t For a signal f t its one-sided

This preview shows pages 1–5. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Laplace Transform • Laplace Transform of Signals: Signals are represented by func- tions of time t . For a signal f ( t ), its one-sided 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 time-invariant 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 s-domain: 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 unit-impulse 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. Time-scaling: 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- ) ....
View Full Document

## This note was uploaded on 02/03/2012 for the course EE 3530 taught by Professor Chen during the Fall '07 term at LSU.

### Page1 / 21

EE3530-3.1b - Laplace Transform • Laplace Transform of Signals Signals are represented by func tions of time t For a signal f t its one-sided

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online