Question

Problem 4. Given the input/output system represented by t-1 y(t) = 2 ( x(y - 3) dy where x(t) is the input and y(t) is the output, a) Determine whether the system is linear or non-linear. b) Determine the impulse response h(t, to) of the system by setting x(t)= 8(t–to). c) Determine whether the system is time invariant or time variant. d) Determine whether the system is causal or non-causal.

Answer 1

t-1 2 Y(+) x(7-3) dr Concept-: Linear / non-linear for the tube systemt linear, it should satisfy 2 Property, Wi-ty, and H2(+) Y.(t). Additivity ey, 4t Y (H) then H, (+)+ Hz (+) »crH) → cyt → Homogenity ax, (H) + b22 H) ay, (t)t by (t) super position Time invariant : A system is T.I. if the input and output characteristics donot change with time, for ult) yıt a t-to) yy (tato) Causal - A causal is whose present system Responce does not depend on the future value of + the input- CS Scanned with CamScanner

t-1 a Here linear © is x (+) Y(+1=25&(8-3) dr Integration is given so it must be a system. s(t-to) 2() s[r-Yo] X(7-3) = $[Y-3-Y] y (+=21 8 [Y-3-Y.] dr 7-8 Y3t-1 y (H)= 2 s s[r-13+)] dr y=-00 if +-1 > 3+Y. t> 4+80 then, í s udt=2] y (t) = 2 A 24+ro of Folte > 1-7 then, ly (t)=0 8(++3) dt =. CS Scanned with CamScanner

رو) t-1 y (t) = ? . x(8-3) dr shiftin input t-1 Y, 2 sa[8-3 3-6.] dr Let y-to = dr=dh When, - Y=-a when r=+-1, dt-1-to t-1-to y, (H) 2 s x (2-3 dd 0 t-tool shistin output, u sa(8-3)*) dr Y, (7-to) = we see from Equation ① and 6 (11 G so itis Time invariant- CS Scanned with CamScanner

t-1 ② YH = 25 x 18-30 dr . Let 2-3=d dr= dd - YE- tz-00 = t-1-3 = t-4 ہ rat- 7-4 y (+= 2s x(Add Here the present output is depends on the input so, Iris Causal past systems CS Scanned with CamScanner