Question

marhal A Lif ysi (a) Determine the system function H(z). (b) Find the impulse response of this system. (c) Determine the response of this system to the input x[n] 4(0.1)hu[n]. Hint: use the system function. (d) Determine the steady state response to the input xIn] 3 cos(0.1 Ttn). Hint: use the requency response.

Part D
Part D

Answer 1

y (n) - 0.3 y (n - 1) = 2 x (n) by applying transform on both sides we get y (2) [1 - 0.32^-1] = 2 lambda (2) y (2)/X (2) = 2/1 - 0.32^-1 Z = e^j w y (e^j w)/X (e^j w) = 2/1 - 0.3 e^j w y (e^j w)/x (e^j w) = 2 e^j w/2 e^j w - 0.3 = 2 [cos w + j sin w]/(cos w - 0.3) + j sin w Vertical-bar H (j w) Vertical-bar = 2/squareroot (cos w - 0.3)^2 + (sin w)^2 L H (e^j w) = tan^-1 (sin w/cos w) - tan^-1 (sin w/cos w - 0.3) Vertical-bar H e^(j w) Vertical-bar = 2/squareroot (cos (0.1 phi) - 0.3)^2 + (sin (0.1 phi))^2 Vertical-bar w = 0.1 phi L H (e^j w) = tan^-1 (sin 18/cos 18) - tan^-1 (sin 18/cos 18 - 0.3) = -7.39 (y (e^j w) = Vertical-bar X e^(j w) Vertical-bar Vertical-bar H (e^j w) Vertical-bar = 3 times 2.775 = 8.325 L Y (e^j w) Vertical-bar = (L X (e^j w) = 0 - 7.39 = 7.39 y (e^j w) = Vertical-bar y e^j w Vertical-bar L y (e^j w)