16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde Define x,(o) 0 eisewne X(o)=x,(0e e d x(ne dt Then X()X(o) 2T 2T in opposite order here than if the transform of the autocorrelation function is o Notice that the operations of transforming and averaging and product are dor calculated If one has only a finite record of a single random function, and Sa(o) is to be so calculated approximately under the ergodic hypothesis, it can be done either R(r)= x(o)x(t+ r)di T S(o)=2 R(r)cosordr x(o)= x(n)e -je dt X(o)X(o Standard deviation of Sxr measured this way is approximately equal to mean. 416.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 3 of 8 Define ( ), ( ) ( ) 0, elsewhere T x t TtT x t ⎧ − << = ⎨ ⎩ ( ) () ( ) j T T T j t T X x te d x t e dt ωτ ω ω τ ∞ − −∞ − − = = ∫ ∫ Then * 2 () () ( ) lim 2 ( ) lim 2 T T xx T T T X X S T X T ω ω ω ω →∞ →∞ = = Notice that the operations of transforming and averaging and product are done in opposite order here than if the transform of the autocorrelation function is calculated. If one has only a finite record of a single random function, and ( ) xx S ω is to be so calculated approximately under the ergodic hypothesis, it can be done either way. max 0 0 * 2 ( ) () ( ) ( ) 2 ( )cos ( ) () () () ( ) 2 T xx xx xx T j t T xx R xtxt d T S Rd x x t e dt X X S T τ τ ω τ τ τ τ ω τ ωτ τ ω ω ω ω − − − = + − = = = ∫ ∫ ∫ Standard deviation of xx S measured this way is approximately equal to mean