Question

(a) Let x(t) be a continuous-time signal known to have a first derivative ct) that is a smooth, continuous function over all t in (-00,00). Then the integral [ [x(t) – e(t – 7)][8(t – 3) + 6(t – 10)] dt evaluates to which of the following expressions: 1. x(t)8(t – 3) 2. x(3) 3. x(3) – č(-4) + x(10) – č(3) 4. x(3) – 3(-4) (b) A continuous-time dynamic system is described by the differential equation dyſt) + 4y(t) = da{t) 04 – 3) dt dt where x(t) and y(t) denote the system input and output, respectively. Which of the following statements is true about this system? 1. The system is linear and time-invariant. 2. The system is nonlinear and time-invariant. 3. The system is linear and time-varying. 4. The system is nonlinear and time-varying. 5. None of the above statements are true.

Answer 1

0 Given that { [ xH - ult-)] [$(4-3) + &lt-10)] at - = xy.dlt-3) art / x H x (t-10) at I & t-7) 214-3/dt 070 - J&t=7) 511-10) dt . O = J x H I (t =3) at the Vrt-7) 2(4-3) dt 0 because they are The in remaining equation zero Out oft range Limits. - X (3) WIT S xH alt-to) = X(to) = x(3) - X (3-7) EX (3) ~ * (-4)) option 4' is conect.

6 044 + 444 = dah,x(4-3) at For the input x, (t. dy & + 44,C4 - d xy(t) oxılt-3) dt at for the anput x2(ts. dyalti + 4%, 4 = I x ₂ (t) X₂ (+-3) at for both Xift & X2 (1) give simultanesonly. dyit + 4 4 t + dy(t) + 4 4, (+) = axit X, (+3) td X (H) x₂ 143 dt dt at d[ its + Y, CH) + 4 T 4 ts + bf xt] = 2 x 6 x 1 (43) + d x2 (4) X2 (4-3). dt Both are not equal at not satisfy the SPT : so it is non Linear. at dy-to) +4 4t - to) = 2 x t-toox (t-to-3). output due to delayed rul CH t at xlt-1-3) Glt, T = 24 (t-T) +44(E-T) = d xHatlo delay delayed by To d 4(4-1) +48 (t-1) = ax (+-7] ox (4-7-3) ct CH Ans - The system is non Linear & Time -encanant option 2 is comect.