16.322 Stochastic Estimation and Control, Fall 2004 Prof vander Velde Lecture 12 Non-zero power at zero frequency Non-zero power at non-zero frequency If R(r) includes a sinusoidal component corresponding to the component x()=Asin(o41+6) where 0 is uniformly distributed over 2t, A is random independent of 0, that component will be R(T)==A2 CoSOpt A coS O re erdt 121 J jer +e-jcur le-jordt o-eb) 1z[6(o-a)+6(+a) Page 1 of 816.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 1 of 8 Lecture 12 Non-zero power at zero frequency Non-zero power at non-zero frequency If ( ) Rxx τ includes a sinusoidal component corresponding to the component 0 xt A t ( ) sin( ) = + ω θ where θ is uniformly distributed over 2π , A is random, independent of θ , that component will be () () ( )( ) 0 0 0 0 2 0 2 0 2 2 2 0 0 1 ( ) cos 2 1 ( ) cos 2 1 1 2 2 1 1 2 2 1 2 xx j xx j j j j j R A S A ed A e e ed Ae e d A ωτ ωτ ωτ ωτ ωωτ ωω τ τ ωτ ω ωτ τ τ τ π δω ω δω ω ∞ − −∞ ∞ − − −∞ ∞ − −+ −∞ = = = + ⎡ ⎤ ⎣ ⎦ = + ⎡ ⎤ ⎣ ⎦ = −+ + ⎡ ⎤ ⎣ ⎦ ∫ ∫ ∫