16.322 Stochastic Estimation and Control, Fall 2004 Prof vander Velde Rx(12)=x(1)x(L2) ∫a4∫dea)fsi(omn+)si(mn+) 2 sin Asin B=5[cos(A-B)-cos(A+B) R2(42)=5∫,-o(-4)-coso(4+)+20]db l2-1 So the autocorrelation function is sinusoidal with the same frequenc This periodic property is true in general. If all members of a stationary ensemble are periodic, x((+nT)=x(o) R(t +nT)=x(ox(+r+nT) x(1)x(t+r) Identification of a periodic signal in noise We have recorded a signal from an experimental device which looks like just It is of interest to know if there are periodic components contained in it Conside x(1)=R(1)+P() where P(t) is any periodic function of period T and R(t)is a random process16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 5 of 9 ( )( ) 12 1 2 2 2 1 2 0 0 "" "" (, ) ()( ) 1 ( ) sin sin 2 xx B A R t t xt xt dA d f a A t t π θ ωθ ωθ π ∞ = = ++ ∫ ∫ 14243 14243 ()() 1 sin sin cos cos 2 A B AB AB = −− + ⎡ ⎤ ⎣ ⎦ { } () () 2 2 12 2 1 1 2 0 2 2 1 1 1 ( , ) cos cos 2 2 2 1 cos , 2 Rxx tt A t t t t d A tt π ω ω θθ π ωτ τ = − − ++ ⎡ ⎤⎡ ⎤ ⎣ ⎦⎣ ⎦ = =− ∫ So the autocorrelation function is sinusoidal with the same frequency. This periodic property is true in general. If all members of a stationary ensemble are periodic, x( ) () t nT x t + = ( ) () ( ) () ( ) ( ) xx xx R nT x t x t nT xtxt R τ τ τ τ + = ++ = + = Identification of a periodic signal in noise We have recorded a signal from an experimental device which looks like just hash. It is of interest to know if there are periodic components contained in it. Consider: x() () () t Rt Pt = + where P t( ) is any periodic function of period T and R( )t is a random process independent of P