23 Apr 19 EIE3001 Sig Sys Spring 2019 19 Implication if x t is right sided and

23 apr 19 eie3001 sig sys spring 2019 19 implication

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23-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. 23-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.
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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 . 23-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. 23-Apr-19 EIE3001 Sig & Sys, Spring 2019 22
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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. Remember: Laplace transform consists of The the algebraic expression X ( s ), and The ROC 23-Apr-19 EIE3001 Sig & Sys, Spring 2019 23
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Existence of the Fourier Transform from the ROC of the Laplace Transform ¾ In general, exists if its ROC includes the imaginary axis ( j -axis). Example 4: 23-Apr-19 EIE3001 Sig & Sys, Spring 2019 24 a > 0, FT exists a < 0, FT doesn’t exist
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Existence of the Fourier Transform from the ROC of the Laplace Transform ¾ In general, exists if its ROC includes the imaginary axis ( j -axis). Example 5: 23-Apr-19 EIE3001 Sig & Sys, Spring 2019 25 a > 0, FT doesn’t exist a < 0, FT exists
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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 23-Apr-19 EIE3001 Sig & Sys, Spring 2019 26
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Inspection 23-Apr-19 EIE3001 Sig & Sys, Spring 2019 27
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Partial Fraction Expansion ¾ Example 6. Consider Step 1: Factorize the denominator Step 2: Convert to an expansion Step 3: Inspection Finally 23-Apr-19 EIE3001 Sig & Sys, Spring 2019 28
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Note: Need to Pay Attention to the ROC ¾ Example 7: 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.
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