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Unformatted text preview: The Laplace Transform Do not distribute these notes 220 Prof. R.T. M’Closkey, UCLA The Bilateral Laplace Transform The Laplace transform comes in two flavors. The most common version is the unilateral Laplace transform , denoted L , which “operates" on functions defined on the time interval [0 , ∞ ) , L ( f ) = Z ∞ f ( t ) e st dt ( domain of f is [0 , ∞ ) ) The lower limit is used to explicitly indicate that the rapid transitions near t = 0 in “abrupt" inputs, like impulses or steps, are included in the analysis. Before studying the properties and uses of the unilateral Laplace transform, the bilateral Laplace transform is introduced. The bilateral Laplace transform , denoted L , operates on functions that are defined on the bilateral interval (∞ , ∞ ) , L ( f ) = Z ∞∞ f ( t ) e st dt ( domain of f is (∞ , ∞ )) (87) The variable s ∈ C is restricted to a region of the complex plane where the integral is welldefined and exists. This region of convergence (ROC) is always an open halfplane with boundary parallel to the imaginary axis. The ROC must be specified along with the function of s that is computed from (87). The idea behind the bilateral Laplace transform is to multiply f by e σt for some σ ∈ R so that the Fourier transform of f ( t ) e σt exists. This permits us to study functions that are of engineering relevance but that cannot be analyzed with the Fourier transform. Do not distribute these notes 221 Prof. R.T. M’Closkey, UCLA Examples. 1. Consider the unit step function, μ , L ( μ ) = Z ∞∞ μ ( t ) e st dt = Z ∞ e st dt = 1 s assuming R e ( s ) > . Thus, the bilateral Laplace transform is 1 s with ROC R e ( s ) > . ï 10 ï 5 5 10 ï 5 ï 4 ï 3 ï 2 ï 1 1 2 3 4 5 seconds unit step ï 10 ï 5 5 10 ï 10 ï 8 ï 6 ï 4 ï 2 2 4 6 8 10 real imag ROC 1 s + μ ( t ) NOTE: we will adopt the same compact notation to represent the Laplace transform of a signal that was used in the Fourier transform analysis. In other words, we will use the “ ˆ · " notation to denote the Laplace transform of a signal: ˆ u = L ( u ) . We will explicitly state which transformation is being used if there is any ambiguity. Do not distribute these notes 222 Prof. R.T. M’Closkey, UCLA 2. Now compute L ( f ) where f ( t ) = e t μ ( t ) , ˆ f ( s ) = L ( f ) = Z ∞∞ e t μ ( t ) e st dt = Z ∞ e (1+ s ) t dt = 1 s + 1 assuming R e ( s ) > 1 . ï 5 5 ï 10 ï 8 ï 6 ï 4 ï 2 2 4 6 8 10 real imag ROC + ï 5 5 ï 5 ï 4 ï 3 ï 2 ï 1 1 2 3 4 5 seconds signal 1 s + 1 e − t μ ( t ) Do not distribute these notes 223 Prof. R.T. M’Closkey, UCLA 3. Compute L ( f ) where f ( t ) = e t μ ( t ) , ˆ f ( s ) = L ( f ) = Z ∞∞ e t μ ( t ) e st dt = Z∞ e (1+ s ) t dt = 1 s + 1 assuming R e ( s ) < 1 ....
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This note was uploaded on 08/08/2011 for the course MAE 107 taught by Professor Tsao during the Spring '06 term at UCLA.
 Spring '06
 TSAO
 Laplace

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