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
Time Shift a(t-r)esX(s), same ROC as X(s) Exampl 3. +2 9e{}>-24 ST s+2 Re{s}>-24→c2u(t) t→→t-T ↓T=-3 3s s+2 e{s}>-2←c-2(+32(t+3)
Time Shift
Time-Domain Differentiation + dar( t + sX(se ds da sX(s); with ROC containing the ROC of X ( s) ROC could be bigger than the roc of X(s), if there is pole-zero cancellation. E. g 9te{s}>0 dt 8(t)+1=s. ROC= entire s-plane S-Domain Differentiation dx ta(t) ds, with same RoC as X(s)(Derivation is similar to d dt E te-(t ds s+ 9e{s}>-a S+a
Time-Domain Differentiation ROC could be bigger than the ROC of X( s), if there is pol e-zero cancellation. E.g., s -Domain Differentiation
Convolution Property h() +y(t)=b(t)米(t) Forx(t)←X(s),y(t)←→Y(s),b(t)←→H(s) Then Y(s)=H(s)·X(s) ROC of Y(s)=H(s)X(s: at least the overlap of the roCs of H(s)& X(s) ROC could be empty if there is no overlap between the two ROCs E. g x(t=eu(t and h(t)=-e u(t) ROC could be larger than the overlap of the two. e. g a()米b(t)=6(t)
Convolution Property For Then • ROC of Y(s) = H(s)X(s): at least the overlap of the ROCs of H(s) & X(s) • ROC could be empty if there is no overlap between the two ROCs E.g. • ROC could be larger than the overlap of the two. E.g. x(t ) e u(t ), h(t ) e u( t ) t t = = − − − and
The system Function of an LTI System h(t)<) H(s)-the system function The system function characterizes the system System properties correspond to properties of H(s)and its roc a first example System is stable +/ h(t) dt < oo ROC of H(s) includes the jw axis
The System Function of an LTI System The system function charact erizes the system ⇓ System properties correspond to properties of H( s) and its ROC A first example:
Geometric Evaluation of Rational laplace Transforms Example #1: X1(s) 8- A first-order zero Graphic evaluation of X1 s= s-plane Can reason about vector length S-a ∠X1(s angle w/ real axis ∠Ⅹ(s1 aa e
Geometric Evaluation of Rational Laplace Transforms Example #1: A first-order zero Graphic evaluation of Can reason about - vector length - angle w/ real axis