chapter6.pdf - Chapter 6 Laplace Transform 1 Definition of...

This preview shows page 1 - 6 out of 16 pages.

ELEC242: CT Signals and Systems Chapter 6 Laplace Transform 1. Definition of Laplace Transform 2. Unilateral vs. Bilateral Laplace Transform 3. Inverse Laplace Transform 4. Properties of Laplace Transform 5. Poles and zeros 6. LTIC System Analysis Sections: 6.1 – 6.10
Image of page 1

Subscribe to view the full document.

ELEC242: CT Signals and Systems Laplace Transform: Definition Activity 1: Derive the following Laplace transform pair. ( ) ( ) = dt e t x s X st σ + σ π = j j st j ds e s X t x ) ( ) ( 2 1 Analysis equation: Synthesis equation: 1 ( ) with ROC: Re{ } L at e u t s a s a ←ί→ > +
Image of page 2
ELEC242: CT Signals and Systems Computing DTFT: Exponential t 0 x 1 ( t ) Activity 1 (contd): Compute the Laplace transform of the decaying exponential function x 1 ( t ) = exp( at ) u ( t ), a ~ R + . Solution: Re{ s } Im{ s } a 0 ( ) { } { } Ϊ Ϊ Ω Ϊ Ϊ Ψ Χ > + = . Re for undefined Re for ) ( 1 a s a s a s s X
Image of page 3

Subscribe to view the full document.

ELEC242: CT Signals and Systems Computing DTFT: Exponential Activity 2: Compute the Laplace transform of the decaying exponential function x ( t ) = exp( at ) u ( t ), a ~ R + . Solution: t 0 x ( t ) = e at u ( t ) ( ) { } { } Ϊ Ϊ Ω Ϊ Ϊ Ψ Χ < + = . Re for undefined Re for ) ( 1 a s a s a s s X Re{ s } 0 Im{ s } a
Image of page 4
ELEC242: CT Signals and Systems Properties of ROC 1. The ROC consists of 2D strips parallel to the j -axis. 2. For a right sided function, the ROC takes the form Re{ s } > σ 0 consisting of the RHS of the s-plane. 3. For a left sided function, the ROC takes the form Re{ s } < σ 0 consisting of the LHS of the s-plane. 4. For a finite duration function, the ROC consists of the entire s-plane except for the possible deletion of the point s = 0. 5. For a double sided function, the ROC takes the form σ 1 < Re{s} < σ 2 and is a strip within the s-plane.
Image of page 5

Subscribe to view the full document.

Image of page 6
  • Fall '19
  • ω0t, σ0 consisting of the RHS of the s-plane

What students are saying

  • Left Quote Icon

    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.

    Student Picture

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

  • Left Quote Icon

    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.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    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.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern

Ask Expert Tutors You can ask 0 bonus questions You can ask 0 questions (0 expire soon) You can ask 0 questions (will expire )
Answers in as fast as 15 minutes