¾ Apr 19 EIE3001 Sig Sys Spring 2019 14 s plane x x o 2 32

# ¾ apr 19 eie3001 sig sys spring 2019 14 s plane x x

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¾ Example 9.3 19-Apr-19 EIE3001 Sig & Sys, Spring 2019 14 s -plane x x o -2 -3/2 Important Terminologies Zeros : roots of N ( s ), i.e. , s * such that N ( s *) = 0. Poles : roots of D ( s ) If X ( s ) is a rational polynomial:

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ROC Properties ¾ Property 1: The ROC of X ( s ) consists of strips parallel to the j - axis of the s -plane. ¾ Property 2: For rational Laplace transforms, the ROC does not contain any poles. ¾ Property*: The ROC is a connected region. 19-Apr-19 EIE3001 Sig & Sys, Spring 2019 15 s -plane x x o -2 -3/2
Example: Find Eligible ROCs using Properties 1 and 2 ¾ Given the algebraic expression of the Laplace transform X ( s ), there could be several possibilities of the ROC. There are only 3 possibilities of ROC associated with this expression. 19-Apr-19 EIE3001 Sig & Sys, Spring 2019 16

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ROC Properties ¾ Property 3: If x ( t ) is of finite duration and is absolutely integrable, then the ROC is the entire s -plane. 19-Apr-19 EIE3001 Sig & Sys, Spring 2019 18 - Recall that the existence of the Laplace transform may relates to - Also the existence associates with the existence of the Fourier transform:
ROC Properties ¾ Property 4: If x ( t ) is right -sided, and if the line is in the ROC, then all values of s for which will be in the ROC. 19-Apr-19 EIE3001 Sig & Sys, Spring 2019 19 Implication: if x ( t ) is right -sided and X ( s ) is rational, then the ROC is to the right of the rightmost pole.

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ROC Properties ¾ Property 5: If x ( t ) is left -sided, and if the line is in the ROC, then all values of s for which will be in the ROC. 19-Apr-19 EIE3001 Sig & Sys, Spring 2019 20 Implication: if x ( t ) is left -sided and X ( s ) is rational, then the ROC is to the left of the leftmost pole.
ROC Properties ¾ Properties 6: If x ( t ) is two- sided, and if the line is in the ROC, then the ROC will consist of a strip in the s -plane that includes the line . 19-Apr-19 EIE3001 Sig & Sys, Spring 2019 21

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ROC Properties ¾ To summarize, for any signal with a Laplace transform, there are only 4 possibilities its ROCs: The entire s-plane (for finite duration signals) A left-half plane (for left-sided signals) A right-half plane (for right-sided signals) A single strip (for two-sided signals) ¾ Using these properties, we can infer the ROC from the time function. 19-Apr-19 EIE3001 Sig & Sys, Spring 2019 22
Examples ¾ Determine the Laplace transform of the following signals a) , where the parameters a and b are real b) c) Discuss whether the Fourier transform exist.

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