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4. (25 points) Laplace and LCCDE Systems Consider an LTI system with input-output relation described by the LCCDE: -2y(t) - y0) + 3x(t) + deco (O) = (a) (5 pts) Find the transfer function H(s) and write it in factored form. (b) (5 pt) Sketch the ROC corresponding to H(s) if it is known the system is causal. Mark the poles and zeros. (c) (5 pts) Sketch the ROC corresponding to H(s) if it is known the system is stable. Mark the poles and zeros. (d) (5 pts) Sketch the ROC corresponding to H(s) if it is known the system is left-sided. Mark the poles and zeros. (e) (5 pts) Given the input (1) = ((), calculate the output (1) for this interconnected LTI system.

Answer 1

Given, -2 y(t)-dyct) - dt +. 3x11) + d(1) d yn de Taking laplace Transform both sides -2. Y6 ) - S.763) + s’y15) - 3X(9) + S-X(5) (-2-s+s>) Y() = (3+5) xs) Y() - (5+3) 325-2 X(S) H(S) - YO) - (5+3) (S+) (5-2) HS)- MAS) = 841 poles Zeros located at 5=-1, 2 located at s=-3. right side of (6) For system Right most said to be causal Roc must be pole jw * Roc: 6 >+2 for causal. H(). © FOB HC) to be stable Roc must include ju' axis ROC: -1<642 for stable (5)

left sided Roc must be left side (d) For system said to be of left most pole. w 62-1 for lejlsided HCS) Ys X(S) - (s+3) [S+1) (5-2) Given x A) = u(t) X (5) Y() - (5+3) s(s+1)(5-2) Y60) = 3x + the end Taking Inverse Laplace Transform yct) = -Zult)+zetuct) +5etult)
kindly see the above images for the solution.