Lecture 06 Laplace Transform meiling chen signals systems

Lecture #06 Laplace Transform meiling chen signals & systems 1

Eigenfunction A meiling chen signals & systems 2

meiling chen signals & systems 3

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

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 meiling chen signals & systems 5

Laplace transform Inverse Laplace transform meiling chen signals & systems 6

Unilateral Laplace transform for causal system meiling chen signals & systems 7

Laplace transform properties meiling chen signals & systems 8

Time convolution meiling chen signals & systems 9

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

Final Value Theorem Final-Value Theorem If and are Laplace transformable, if exists and if is analytic on the imaginary axis and in right half of the s-plane, then 1. No any pole on the imaginary axis or in right half of s-plane. 2. System is stable. meiling chen signals & systems 11

Example 2 Example 3 not exist meiling chen signals & systems 12

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

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

Example 6 or or or meiling chen signals & systems 15

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

Example 7 meiling chen signals & systems 17

Inverse Laplace transform Case III repeated root meiling chen signals & systems 18

Example 8 meiling chen signals & systems 19