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m08 - Properties of the Laplace Transform

Module by: C. Sidney Burrus

Summary: The Laplace Transform has properties similar to those of the Fourier Transform and Fourier series.

Properties of the Laplace Transform

Many of the properties of the Laplace transform are similar to those for Fourier transform [1] [2] , however, the basis functions for the Laplace transform are not orthogonal. Some of the more important ones are:

  1. Linear: ℒ { x + y } = ℒ { x } + ℒ { y } ℒ { x + y } = ℒ { x } + ℒ { y }
  2. Convolution: If y ( t ) = h ( t ) * x ( t ) = ∫ h ( t − τ ) x ( τ ) ⅆ τ y ( t ) = h ( t ) * x ( t ) = ∫ h ( t − τ ) x ( τ ) ⅆ τ then ℒ { h ( t ) * x ( t ) } = ℒ { h ( t ) } ℒ { x ( t ) } ℒ { h ( t ) * x ( t ) } = ℒ { h ( t ) } ℒ { x ( t ) }
  3. Derivative: ℒ { ⅆ x ⅆ t } = s ℒ { x ( t ) } ℒ { ⅆ x ⅆ t } = s ℒ { x ( t ) }
  4. Derivative (ULT): ℒ { ⅆ x ⅆ t } = s ℒ { x ( t ) } − x ( 0 ) ℒ { ⅆ x ⅆ t } = s ℒ { x ( t ) } − x ( 0 )
  5. Integral: ℒ { ∫ x ( t ) ⅆ t } = 1 s ℒ { x ( t ) } ℒ { ∫ x ( t ) ⅆ t } = 1 s ℒ { x ( t ) }
  6. Shift: ℒ { x ( t − T ) } = C ( k ) e − T s ℒ { x ( t − T ) } = C ( k ) e − T s
  7. Modulate: ℒ { x ( t ) e j ω 0 t } = X ( s − j ω 0 ) ℒ { x ( t ) e j ω 0 t } = X ( s − j ω 0 )

References

  1. R. N. Bracewell. (1985). The Fourier Transform and Its Applications. (Third). New York: McGraw-Hill.
  2. A. Papoulis. (1962). The Fourier Integral and Its Applications. McGraw-Hill.

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