H AA Signals and Systems Fall 2003 Lecture #18 6 November 2003 Inverse Laplace Transforms Laplace transform Properties The System Function of an lti System Geometric Evaluation of laplace transforms and Frequency responses
Signals and Systems Fall 2003 Lecture #18 6 November 2003 • Inverse Laplace Transforms • Laplace Transform Properti e s • The System Function of an LTI System • Geometric Evaluation of Laplace Transforms and Frequency Responses
Inverse Laplace transform tes dt, s=0+jwE ROC Fa(te oj Fix o E ROC and apply the inverse Fourier transform ate o X(o+ju)e X(o+jw)e(a+ju )t dw But s=0+jo(o fixed)= ds=jda X(sesd 丌 0-1
Inverse Laplace Transform But s = σ + j ω (σ fixed) ⇒ ds = jdω Fix σ ∈ ROC and apply the inverse Fourier transform
Inverse laplace transforms Via Partial fraction Expansion and Properties Example: s+3 A B A B Three possible roc's- corresponding to three different signals gm ne Recall 9e e u(t) left-sided sta 3n,{8>=一c“) right-sided
Inverse Laplace Transforms Via Partial Fraction Expansion and Properties Example: Three possible ROC’s — corresponding to three different signals Recall
ROC I ft-sided signal Aeu(t)-Beu(t l(-t) Diverges as t→-0 ROC II Two-sided signal, has Fourier Transform Ae u(t)- Beu(t) 2e-t)+c2(-) 0ast→±∝ ROC III: Right-Sided signal Ae u(t)+ Beu(t) () Diverges as t→+∞
ROC I: — Left-sided signal. ROC III:— Right-sided signal. ROC II: — Two-sided signal, has Fourier Transform
Properties of laplace Transforms Many parallel properties of the Ctft, but for Laplace transforms we need to determine implications for the roc For example Inear an1(t)+b2(t)aX1(s)+bX2 (s) ROC at least the intersection of ROCs of X(s)and X2(s) ROC can be bigger(due to pole-zero cancellation) E. g and a Then a.1(t)+bm2()=0→→X(s)=0 → RoC entire s- plane
Properties of Laplace Transforms • For example: Lineari t y ROC at least the intersection of ROCs of X1( s) and X2 ( s) ROC can be bigger (due to pole-zero cancellation) • Many parallel properties of the CTFT, but for Laplace transforms we need to determine implications for the ROC ⇒ ROC entir e s-plane