Lecture #06

1. Lecture #06 Laplace Transform signals & systems

2. Eigenfunction A signals & systems

3. signals & systems

4. LTI system h(t) is the impulse response of the LTI system According to the convolution: We define that signals & systems

5. We identify as an eigenfunction of the LTI system and H(s) as the corresponding eigenvalue. In which s is a complex frequency Is the Fourier transform of signals & systems

6. Laplace transform Inverse Laplace transform signals & systems

7. Unilateral Laplace transform for causal system signals & systems

8. Laplace transform properties signals & systems

9. Time convolution signals & systems

10. If is continuous at and may different and if is not impulse function or derivative of impulse function, then Initial Value Theorem Initial-Value Theorem Example 1 signals & systems

11. If and are Laplace transformable, if exists and if is analytic on the imaginary axis and in right half of the s-plane, then Final Value Theorem Final-Value Theorem • No any pole on the imaginary axis or in right half of s-plane. • System is stable. signals & systems

12. Example 2 Example 3 not exist signals & systems

13. If include impulse function at . Remark 1 Example 4 Remark 2 Example 5 signals & systems

14. Case I simple root where Inverse Laplace transform F(s) is a strictly proper rational function Degree of denominator signals & systems

15. Example 6 or or or signals & systems

16. Case II complex root Inverse Laplace transform let signals & systems

17. Example 7 signals & systems

18. Case III repeated root Inverse Laplace transform signals & systems

19. Example 8 signals & systems