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

• 55

This preview shows page 18 - 29 out of 55 pages.

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.

Subscribe to view the full document.

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.
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

Subscribe to view the full document.

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
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

Subscribe to view the full document.

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
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

Subscribe to view the full document.

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
Inspection 23-Apr-19 EIE3001 Sig & Sys, Spring 2019 27

Subscribe to view the full document.

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
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.

Subscribe to view the full document.

• Fall '13

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern