meiling chensignals & systems1 Lecture #06 Laplace Transform

meiling chensignals & systems2 Eigenfunction A

meiling chensignals & systems3

meiling chensignals & systems4 LTI system h(t) is the impulse response of the LTI system According to the convolution: We define that

meiling chensignals & systems5 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 chensignals & systems6 Laplace transform Inverse Laplace transform

meiling chensignals & systems7 Unilateral Laplace transform for causal system

meiling chensignals & systems8 Laplace transform properties

meiling chensignals & systems9 Time convolution

meiling chensignals & systems10 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 chensignals & systems11 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 chensignals & systems12 Example 2 Example 3 not exist

meiling chensignals & systems13 Remark 1 Remark 2 If include impulse function at. Example 4 Example 5

meiling chensignals & systems14 Inverse Laplace transform F(s) is a strictly proper rational function Degree of denominator Case I simple root where

meiling chensignals & systems15 Example 6 or

meiling chensignals & systems16 Inverse Laplace transform Case II complex root let

meiling chensignals & systems17 Example 7

meiling chensignals & systems18 Inverse Laplace transform Case III repeated root

meiling chensignals & systems19 Example 8