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lecture12 - 16.322 Stochastic Estimation and Control Fall...

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16.322 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 () xx R τ includes a sinusoidal component corresponding to the component 0 ( ) sin( ) xt A t ω θ =+ where is uniformly distributed over 2 π , A is random, independent of , that component will be 00 2 0 2 0 2 2 2 1 c o s 2 1 c o s 2 11 22 1 2 xx j xx jj j RA SA e d A eee d Ae e d A ωτ ωωτ τω ωω πδ ωδ −∞ −∞ −− + −∞ = = ⎡⎤ ⎣⎦ =− + +

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16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 2 of 8 Units of xx S Mean squared value per unit frequency interval. Usually: 2 () 1 2 () , w h e r e 2 j xx xx xx xx SR e d xS d Sf d f f ωτ τ ωω π ω −∞ −∞ = = == In this case, 2 ~ xx q S Hz Next most common: 2 1 2 j xx xx xx e d d ττ −∞ −∞ = = In this case, 2 2 ~s e c rad /sec xx q Sq = There is an alternate form of the power spectral density function. Since xx S is a measure of the power density of the harmonic components of x t , one should be able to get xx S also from the Fourier Transform of x t which is a direct decomposition of x t into its infinitesimal harmonic components. This is true, and is the approach taken in the text. One difficulty is that the Fourier Transform does not converge for members of stationary ensembles. The mathematics are handled by a limiting process.
16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 3 of 8 Define () , ( ) 0, elsewhere T x tT t T xt −<< = ( ) j TT T jt T X xte d t ωτ ω τ −∞

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lecture12 - 16.322 Stochastic Estimation and Control Fall...

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