¾ Starting from this lecture to unify the notation we use to denote the Fourier

# ¾ starting from this lecture to unify the notation

• 28

This preview shows page 7 - 15 out of 28 pages.

¾ Starting from this lecture , to unify the notation, we use to denote the Fourier transform ¾ Note that, the Laplace transform is a mapping of a complex variable to complex amplitude. And, when s = j (purely imaginary), it is the Fourier transform. 19-Apr-19 EIE3001 Sig & Sys, Spring 2019 7 Our new notation of Fourier transform:

Subscribe to view the full document.

Connection with FT ¾ We have seen ¾ A more general connection is as follows. From we have The Laplace transform is the Fourier transform of the function x ( t ) multiplied by a real exponential function. 19-Apr-19 EIE3001 Sig & Sys, Spring 2019 8
LT May Resolve the Convergence Issue of FT ¾ Why is LT useful? Laplace transform may converge when Fourier transform doesn’t. The convergence issue of FT can be resolved by some choice of in LT. ¾ Example Find the LT of the step function u ( t ) Recall that we cannot apply the analysis equation to compute the FT of u ( t ), because u ( t ) is not absolutely integrable, and thus, FT does not converge. However, in LT: it is integrable under some choice of s . Caution: the algebraic expression is only valid for some s . (This is a key distinction from FT.) 19-Apr-19 EIE3001 Sig & Sys, Spring 2019 9

Subscribe to view the full document.

Examples ¾ Example 9.1 ¾ Example 9.2 19-Apr-19 EIE3001 Sig & Sys, Spring 2019 10
Examples – Graphical Illustration 19-Apr-19 EIE3001 Sig & Sys, Spring 2019 11 a > 0 a < 0 0 t 0 t 0 t 0 t FT: FT: doesn’t exist FT: FT: doesn’t exist LT: LT:

Subscribe to view the full document.

The Key Distinctions from the Previous Examples ¾ In FT, the existence of the FT X ( j ) depends on the signal itself ( e.g. , the parameter a of the signal . Once it exists, the transform is valid for all . ¾ In LT, the transform always converges ( i.e. , the algebraic expression of the transform does not depend on the signal x ( t )). However, the transform X ( s ) may be valid only for a limited range of s . ¾ This observation leads to a very important concept called Region of Convergence (ROC) of Laplace transform. In LT, not only do we need to specify the algebraic expression, but also we have to specify the region of convergence over which the expression is valid. 19-Apr-19 EIE3001 Sig & Sys, Spring 2019 12
ROC: Region of Convergence ¾ ROC: the set of s values over which Laplace transform exists. ¾ Example 9.3 19-Apr-19 EIE3001 Sig & Sys, Spring 2019 13 s -plane x x o -2 -3/2

Subscribe to view the full document.

ROC: Region of Convergence ¾ ROC: the set of s values over which the LT expression is valid.
• 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