Remember Laplace transform consists of The the algebraic expression X s and The

# Remember laplace transform consists of the the

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Remember: Laplace transform consists of The the algebraic expression X ( s ), and The ROC 19-Apr-19 EIE3001 Sig & Sys, Spring 2019 23

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The Inverse Laplace Transform ¾ The Laplace transform and inverse Laplace transform are one-to- one mapping provided that ROC is given. (Similar to the Fourier transform case, when the Fourier transform converges.) ¾ Three commonly used techniques for inverse Laplace transform: Inspection Partial fraction expansion Contour integration (will not be discussed in this course) * For the integration technique, one can perform inverse Fourier transform based on the relationship for some that the Fourier transform exists. Specifically, one can compute 19-Apr-19 EIE3001 Sig & Sys, Spring 2019 24
Inspection 19-Apr-19 EIE3001 Sig & Sys, Spring 2019 25

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Partial Fraction Expansion ¾ Let’s begin with an example Step 1: Factorize the denominator Step 2: Convert to an expansion Step 3: Inspection Finally 19-Apr-19 EIE3001 Sig & Sys, Spring 2019 26
Note: Need to Pay Attention to the ROC ¾ An example with the same algebraic expression but different ROC: Step 1 and 2: After factorization and expansion Step 3: Inspection. The ROC implies that the time function consists of a left-sided function and a right-sided function. We need some discussion to arrive at: 19-Apr-19 EIE3001 Sig & Sys, Spring 2019 27

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A General Strategy for the Factorization ¾ Find the time function x ( t ) for the following Laplace transform We can write the series as where the coefficients can be found as Together with the ROC, we obtain: 19-Apr-19 EIE3001 Sig & Sys, Spring 2019 28
Summary and Remark ¾ Expected learning outcome Be able to explain the connection between Laplace transform and Fourier transform Be able to associate the ROC with the property of the time function Be able to find the inverse Laplace transform using inspection and partial series expansion techniques ¾ Suggested reading: Sections 9.1—9.3. 19-Apr-19 EIE3001 Sig & Sys, Spring 2019 29
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