Part 6-Laplace Tansform - Laplace Transform Fourier...

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Fourier Analysis 4-1 Laplace Transform
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Fourier Analysis 4-2 Laplace Transform Laplace transform is a useful technique for analyzing systems described by differential equations. Two types of Laplace transform: 1. Bilateral Laplace transform 2. Unilateral Laplace transform a special case of the former one. it can only be used to analyze causal signals and causal systems. more convenient to use for causal signals and systems.
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Fourier Analysis 4-3 Unilateral Laplace Transform The unilateral Laplace transform of a signal x(t) is defined as Uniqueness property: For a given X(s), there is a unique inverse transform. It simplifies system analysis. Price to pay: It cannot be used to analyze non-causal systems or non-causal inputs. 0 ) ( ) ( dt e t x s X st
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Fourier Analysis 4-4 Region of Convergence The range of values of s for which X(s) converges is called the region of convergence (ROC) of the L.T. If for some real then for all dt t x | ) ( | e , 0 t - ) finite is ) ( i.e. ( 0 dt e t x st dt t x dt t x e s X s s st | ) ( | e | ) ( | | ) ( | , } Re{ , 0 t - 0
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Fourier Analysis 4-5 Example 1 Find the Laplace Transform of Solution: ) ( ) ( t u e t x at a s a s e a s dt e dt e t u e s X t s a t s a st at } { Re , 1 1 ) ( ) ( 0 ) ( 0 ) ( 0 a s a s t u e L at  } Re{ , 1 ) ( it is finite if Re{s} + a > 0. ( a is a real number)
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Fourier Analysis 4-6 s-Plane Example 1 Im(s ) Re(s) - a Re{ } s a   s-plane is commonly used to show the ROC of L.T.
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Fourier Analysis 4-7 Example 2
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Fourier Analysis 4-8 Example 3 Consider From Example 2.1, we have Combining, ) ( 2 ) ( 3 ) ( 2 t u e t u e t x t t . 2 Re , 2 1 ) ( , 1 Re , 1 1 ) ( 2   s s t u e s s t u e L t L t Both terms converge when Re{s} > 1. . 1 Re , 2 3 1 1 2 2 3 ) ( 2 ) ( 3 2 2  s s s s s s t u e t u e L t t
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Fourier Analysis 4-9 Inverse Laplace Transform x(t) can be obtained from X(s) via inverse Laplace transform: where c is a constant chosen to ensure the convergence of the above integral. Note: Computing the above integral is beyond the scope of this course. We will use another method to find the inverse. j c j c st ds e s X j t x ) ( 2 1 ) (
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Fourier Analysis 4-10 Inverse Laplace Transform Given the Laplace transform X(s), how to find the original signal x(t)? We use the following method, which includes two steps: 1. First, use partial-fraction expansion. 2. Then, use the transform-pair table to find x(t) from X(s).
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Fourier Analysis 4-11 Example 4 Find the causal signal x(t), whose Laplace transform is Solution: Expanding it into partial fraction: Since by the linearity of L.T., we have ) 2 )( 1 ( 1 ) ( s s s X 2 1 1 1 ) ( s s s X , 1 ) ( a s t u e L at  ). ( ) ( ) ( 2 t u e e t x t t
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