Question

3. As indicated in Section 9.5, many of the properties of the Laplace transform and their derivation are analogous to corresponding properties of the Fourier transform and their derivation, as developed in Chapter 4. In this problem, you are asked to outline the derivation of a number of the Laplace transform properties. Observing the derivation for the corresponding property in Chapter 4 for the Fourier transform, derive each of the following Laplace transform properties. Your derivation must include a consideration of the region of convergence. (a) Time shifting (Section 9.5.2) (b) Shifting in the s-domain (Section 9.5.3) (c) Time scaling (Section 9.5.4) (d) Convolution property (Section 9.5.6)

Answer 1

Solm Lay Let at). be a signal xls) a signal whose Laplace transform is With Roc R le LT M14) Xls) with Ruc=R 2(t-to) il est xls) ; Roc = R Prook. -st alt-tore adt Put t-to=2 thus at=de -s(etto) Xirie indt -sto o fuos:e" sa -sto E xls) proved the ē Sto term will not affact Roc. because it is not providing any pole.. and we know pole location can determine Roc. Poles of Xes) and xs) are Seene-

b) Shifting in s domain let Xit) LT X(s) ; Roc=R Sot e LIT X(+) ♡ XISso); Roc=rthes sob, Proof LITY & Sot Sot -st Rets}= e alt).e odt -(S-sot altre .dt x(5-c.) [Preved ) Roc. suppose X(S) has pole at S=b then X(S-so) will have a pole at Sosob Sabt so Now, 9t is clear to us. that If Roc to Xis) is Ressy > then of XC Saso) is Refs} > b + Resso} ie Roc= R + Resso}

taime scaling 3 If HH) LT then → X(S) ; Roc=R. X lat) Luty to x (%); front X (S/); Rue= Ra LT {Klats? est alat.e rdt Put at= ir This dta de a LT{xcatly alelie a de -Sat xt.e .de a . As a I X(/). [proved is real Constant mang be positive or negative so gewerializing it L.T &x cat) = The X (%) Roc If Roc of X(S) is R lic Refs}=R xifa) is Rie Ress) = Ra Sol

Convolution multe plication in s domini. eng XS) - Roc; Ressal S-1 Now X54) 2654-1) Res972 2 J S-2 Roc; X Convolution in time X in time domain corresponds to Il 1,14 LIT î Yils) ; Roc = Ry and N₂14) Lit 7 42 (s) ; Roca R2 then Ailth & Malt) La & Xals? Kale) ; Rocio Rn R₂ Proof * {2, ) * vg141) 142,43 * 2,4est de Ivonautinn st 1,12). (tre dr.dto é eft

on Rearranging We get. LITS X (H) **, Hly = facce se --514-7) H-2) e at 5 Xi (s). 1 % H-?).e dt put Jautrefois bruozes 3 to the -40. t-rou Thus .dt=du -su {aaluse ridu = X1 (). X2 [5[Praved] Roc let Xilt) - e ult) -24 Roca Ress} 7-2 canal 3t N₂ (t) = e ult) ; . Roc = Re&s}}-3 Now, Litpalto 1(A) * 921+13 X16S). X215) 1 (5+2) (5+3) Jw 3 alt bi

G Xi (59. X2 is) is We can say Roc Refsyy-2n Retf32-3