Question

PROBLEM # 1 Random signal I(), having a power spectrum is applied to a single pole low pass filter frequency response Hjo) . The filter response is denoted by Y(t). a) Find the average power in X() b) Find the autocorrelation and power spectrum of process Y(t). c) Find the average power in Y(). d) Repeat part (c) for H(jo)

Answer 1

S_x(w) = 2/w^2 + y Take Inverse fourier transform R_x(t) = Inverse F t {S_x (w)} R_x(t) = e^-0 Vertical-bar t Vertical-bar 2a/a^2 + w^2 R_x(t) = 1/2 e^-2 Vertical-bar t Vertical-bar Average power in x(t) = [R_x(t)]_t = 0 = 1/2 e^-2 Vertical-bar 0 Vertical-bar = 1/2 = 0.5 S_y(w) = Vertical-bar H Vertical-bar ^2 S_x(w) H(jw) = 1/jw + 1 Vertical-bar H(jw) Vertical-bar = 1/squareroot w^2 + 1 S_y(w) = S_x(w)/w^2 + 1 S_y(w) = 2/(w^2 + 1) (w^2 + 1) \r\n Using partial fraction S_y(w) Z = Z/(w^2 + 1) (w^2 + 4) = A/(w^2 + 1) + B/(w^2 + 4) w^2 = -1 2 = A(3) B = 2/3 w^2 = -4 B = -2/3 S_y(w) = 2/3 {1/w^2 + 1 - 1/w^2 + 4} R_y(t) = Inverse F T {S_y(w)} R_y(t) = 2/3{e^- Vertical-bar t Vertical-bar /2 - e^-2 Vertical-bar t Vertical-bar /4} R_y(t) = e^- Vertical-bar t Vertical-bar /3 - e^-2 Vertical-bar t Vertical-bar /6 Avg power of y(t) = R_y(0) = 1/3 - 1/6 = 1/2 Vertical-bar H(jw) Vertical-bar = 1/squareroot w^2 + 1 S_y(yw) = 4(2/(w^2 + 1) (w^2 + 4))