We need some discussion to arrive at 23 Apr 19 EIE3001 Sig Sys Spring 2019 29

We need some discussion to arrive at 23 apr 19

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We need some discussion to arrive at: 23-Apr-19 EIE3001 Sig & Sys, Spring 2019 29
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Partial Fraction Expansion: Case 1 ¾ When the order of the numerator N ( s ) is less than that of the denominator D ( s ) and all the poles are of first order: Example 8: 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: 23-Apr-19 EIE3001 Sig & Sys, Spring 2019 30
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Partial Fraction Expansion: Case 2 ¾ If the numerator polynomial N ( s ) has an order no less than that of the denominator polynomial D ( s ), then long division can be used Example 9: find the time function x ( t ) which has the following Laplace transform We perform long division to obtain where the last term can be further expanded as which leads to 23-Apr-19 EIE3001 Sig & Sys, Spring 2019 31 and from Laplace transform property as we will see later
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Partial Fraction Expansion: Case 3 ¾ If the poles have multiple orders, then we need to use derivative Example 9: find x ( t ) which has the following Laplace transform First, we need to expend X(s) into the following form We already know how to compute A 1 and C 2 . To compute C 1 , notice that Take the derivative w.r.t. s , and substituting s = -1, we obtain 23-Apr-19 EIE3001 Sig & Sys, Spring 2019 32
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Partial Fraction Expansion: Case 3 ¾ If the poles have multiple orders, then we need to use derivative Example 9: find x ( t ) which has the following Laplace transform ---- (continued) Finally, we obtain and Together with the ROC, we obtain 23-Apr-19 EIE3001 Sig & Sys, Spring 2019 33
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*General Strategy for Partial Fraction Expansion Consider X ( s ) is rational and has the follow form: There are 4 cases: 23-Apr-19 EIE3001 Sig & Sys, Spring 2019 34
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*General Strategy for Partial Fraction Expansion 23-Apr-19 EIE3001 Sig & Sys, Spring 2019 35
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*General Strategy for Partial Fraction Expansion 23-Apr-19 EIE3001 Sig & Sys, Spring 2019 36
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Properties of Laplace Transform 23-Apr-19 EIE3001 Sig & Sys, Spring 2019 37
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Properties of Laplace Transform ¾ Caution: Applying the property will affect the ROC! ¾ Some property highlight Linearity where the ROC contains at least the intersection of the original two ROCs Time-shifting where the ROC is identical to that of X ( s ). Shifting in s -Domain where the ROC is identical to that of X ( s ). 23-Apr-19 EIE3001 Sig & Sys, Spring 2019 38
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Properties of Laplace Transform Time differentiation where the ROC includes the ROC of X ( s ) Differentiation in s -Domain where the ROC is identical to that of X ( s ). Convolution where the ROC contains at least the intersection of the original two ROCs 23-Apr-19 EIE3001 Sig & Sys, Spring 2019 39
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Examples In the previous Example 3, we used the linearity property to compute the Laplace transform of , and we concluded that the ROC is the union of the ROCs of each of the signals. Example 10: Consider , where the Laplace transforms of x 1 ( t ) and x 2 ( t ) are given by , and Determine X ( s ).
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