16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde units of s Mean squared value per unit frequency interval Usually S2(o)=∫R2(r)emdr 12x s(o)do S()df, where f=o In this case e, Sx HE Next most common S(o)=-「R(r) e- Jerde x2=s(o)do In this case, s There is an alternate form of the power spectral density function Since S(o) is a measure of the power density of the harmonic components of x(O), one should be able to get S(o) 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 x(t) Page 2 of 816.322 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 ( ) , where 2 j xx xx xx xx S R ed x Sd S f df f ωτ ω ττ ω ω π ω π ∞ − ∞ ∞ −∞ ∞ −∞ = = = = ∫ ∫ ∫ In this case, 2 xx ~ q S Hz Next most common: 2 1 ( ) () 2 ( ) j xx xx xx S R ed x Sd ωτ ω τ τ π ω ω ∞ − −∞ ∞ −∞ = = ∫ ∫ In this case, 2 2 ~ sec rad / sec xx q S q = There is an alternate form of the power spectral density function. Since ( ) xx S ω is a measure of the power density of the harmonic components of x( )t , one should be able to get ( ) xx S ω 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