Question

Problem 8.3.1 Determine the Laplace transform of the following signals using Laplace Transform table and the time-shifting property. In other words, represent each signal using functions with known Laplace transforms, and then apply time-shifting property to find Laplace transform of the signals. thre (e) Optional: find the Laplace transforms and the ROC for the above signals using direct integration. Problem 8.3.2 Find the Laplace transforms of the following functions using Laplace Transform table and the time-shifting property (if needed) of the unilateral Laplace transform: (a) x(t)=sin(@(t-1))u(t ) 6) X(t)=sin(0,(t-1))u(t) C) x(t)=sin((())u(t-t.)

Answer 1

8.3.1 L.T(x(+1) = x(s) = (x (t)est dat -60 (BL:T] - set estat o CULT) & (t) = t. octai othuwise X.15) - Inici est Xils) = 2, (4) can be written as X, (t) = 8(t)- 24-1) - utt) rtt ut L{xi (t) } XLS) = L[vct)) - Lot-1)] - Lust-1)]

7 S2 westfo2_509 S2 42 Fi 2 (t) = ? 31 y, it x, (t) = sint 4CA) 4, 6 c Sịp (+-) u (t-1) 12 (t) = x, (t) t 9. (t) H₂ (t) = sint utt) a sin (t-a) uta)

x2 = sint ut) + sin(A-7) uct-7) TS tets - + e- X2(5) sti srt s²+1 70(4) | 13:19 = items RS 1te X (5) st 22ty canelout 2,0) + yit) Los 1,t (uci) - (04-)) - 144-1) ezt (eid)-ud-)) + utt) têt, ua) nt eet-1)-1 utt) = e. tuc) - (t u (t-1) - 1 ult-1) + el - z 'l duct) - uct-1) = 44-1 +244-1? 4 4007 sti X90) = 2(727 st

① integration method dt 83 (t) = det octa let ist.co. 2,64) = Steve dhe -sztést att pratadt Xyks) C + 3cs+de ja (s tot dit S+1) 11 St sti sti

@ x (t) = sin wolt-to) u (t-to) of it) = sinuot u(t) L f (t)} = F(S) = wo 5 + woh Let t- to = alt= to ty x H) = Sinwor. UP) XL) = wo Stor Time shifting proped X (s) = ē stowo stwo

@ a(t) = sin wot. u t-to) Let t-to =T t = tott a) = sin wo( toti)UCT) - [sinwoto.coswort coswoto.sinwor. Ut Let Y=t - (S in woto.coswot & Coswoto.sin wot) ucts y (s) = -stossinwo to two coswo to e stwor