Question

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6. Find the impulse response of an LTI system, if the output y[n] and the input x[n] are shown in the figures below. (15 points) y[n] 4 x[n] 2 3 2 1 →n -1 0 1 -1 0 1 2 3 -1.5 0 -1

Answer 1

{ 2, -1,5, 1} a an Y(2) = { yon 6 given anla In) = {4, 1, 3,-1, 2} apply z-transform to x(n) = {2, -1.5, 13 X (E)= { am z = x(-1) 2' + a(o)r + x(i) z' X(Z) = 27-1,5 +/z to y m) = {4, 1, 3,-1, 2} -) Y (-1) 2'+9 (0) 2 + y)2 + y(2) 2 + y)23 hordo apply z-transfora - proje 42+1 +2 -12 + 2 / 2 Impulge response A (Z) = Y(2) XL) H (2) = 42+1 + 3 2 2 2 2 2 2 + 2 / 2 27 - 1,5 + /

H(A) = -3 +22 22-1,5+z 22 27-1,5+{")428+1+32 -4 °+2222+24+22 y=-3+22) W+'-42 M-324 222 If it 471_3 2² +22 42'_32²+ 223 (-) : | HG)= 2 +22² +222. Now apply Inverse zo transform, H2) = x(m) zn H(z)= x(0) 2 + (DE + x (272 2. Compare ① and ② equation, h(n). 9, a } 9.