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# m08 - Properties of the Laplace Transform

Module by: C. Sidney Burrus. E-mail the author

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