Publicado en por Edison Del Rosario

s3Eva_IIT2017_T5 Sistema lineal

Z(t)=X(t)−Y(t) Z(t) = X(t) -Y(t) Y(t)=h(t)∗X(t) Y(t) = h(t)*X(t)

a) autocorrelación de Z(t)

E[Z(t1)Z(t2)]=E[(X(t1)−Y(t1))(X(t2)−Y(t2))] E[Z(t_1)Z(t_2)] = E[(X(t_1) - Y(t_1))(X(t_2) -Y(t_2))] =E[X(t1)(X(t2)−X(t1)Y(t2)−Y(t1)X(t2)+Y(t1)Y(t2)] = E[X(t_1)(X(t_2) - X(t_1)Y(t_2) - Y(t_1)X(t_2) + Y(t_1)Y(t_2)] =E[X(t1)(X(t2)]−E[X(t1)Y(t2)]−E[Y(t1)X(t2)]+E[Y(t1)Y(t2)] = E[X(t_1)(X(t_2)] - E[X(t_1)Y(t_2)] - E[Y(t_1)X(t_2)] + E[Y(t_1)Y(t_2)] =RX(τ)−RXY(τ)−RYX(τ)+RY(τ) = R_X(\tau) - R_{XY}(\tau) - R_{YX}(\tau) + R_Y(\tau)

b) Densidad espectral de potencia de Z(t)

Sz(f)=F[RX(τ)]−F[RXY(τ)]−F[RYX(τ)]+F[RY(τ)] Sz(f) = F[R_X(\tau)] - F[R_{XY}(\tau)] - F[R_{YX}(\tau)] + F[R_Y(\tau)] =SX(f)−SXY(f)−SYX(f)+SY(f) = S_X(f) - S_{XY}(f) - S_{YX}(f) + S_Y(f) =SX(f)−SXY(f)−SXY∗(f)+SY(f) = S_X(f) - S_{XY}(f) - S_{XY}^* (f) + S_Y(f) =SX(f)−H(f)SX(f)−H∗(f)SX(f)+SY(f) = S_X(f) - H(f) S_{X}(f) - H^* (f) S_{X}(f) + S_Y(f) =[1−H(f)−H∗(f)+∣H(f)∣2]SX(f) = [1 - H(f) - H^* (f) + |H(f)|^2] S_X(f) =[1−2Re[H(f)]+∣H(f)∣2]SX(f) = [1 - 2Re[H(f)] + |H(f)|^2 ] S_X(f) Sz(f)=∣1−H(f)∣2SX(f) Sz(f) = |1 - H(f)|^2 S_X(f)