Question

3. Let the Laplace transforms of signals (t) and y(t) be X(s) and Y(s) with appropriate regions of convergence, respectively (a) Show that the Laplace transform of x(t) * y(t) is X(s)Y (s). What is the region of convergence? (b) Show that the Laplace transform of tx(t) is -dX(s)/ds with the same region of x(t) convergence as tn-1 1 for Re{sa} > 0. -at e (c) Show that the Laplace transform of 'u(t) is n 1)! (sa)" 1 for Refsa} < 0 (d) Show that the Laplace transform of e atu(-t) is s a

Answer 1

t 2ylt)xT (t) ylt-t) dT 1(a ) xt)y(t)edt x t) yit-)dt -St dt tet df dt (t-)) e dt &C (by chamginig the rdct t integrotion) CS Scanned with. CamScanner

x() ylt)(P+) dt du - -SC dc bJ P (t) eSt dt 3.e4 X(s) = x (t) e St. dt dittesemhatng / dxis) d s st t x(t) e dt dx (3) t -St dt dris) Scanned withs CamScanner

Stralaly for tf(t) ED d F(s) ds hore smu4ipied by t"- F()= 3ta at 6-)! Sta ds" n-2 dsn-2 1X S+a n-3 n-3 Sta at \e-1)! (Sra) +st dt (ats)t dt ta cS Scanned with CamScanner

1 Sta cs Scanned with CamScanner