16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde The second approach is faster In fact, with the advent of the Fast Fourier Transform(initiated by Cooley and Tukey), even if one wanted to calculate Ra(r) from x(o), it is faster to transform x(o)to X(o), form S(o), and transform to get R(r)than to integrate x(O)x(t+r) directly for all desired values of t The Fast Fourier Transform is an amazingly efficient procedure for digital calculation of finite fourier transforms References: Full issue-IEEE Transactions on audio and electroacoustics Vol. au-15, No. 2 June 1967 Tutorial article: Brighton, E O and Morrow, R.E.: The Fast Fourier Transform, IEEE Spectrum; Dec 1967. Cross spectral density In dealing with more than one random process, the cross power spectral densit arises naturally. For example, if 二(1)=x(1)+y(1) where x(o) and y(o are members of random ensembles, then we found before R(r=R(t)+r(t)+r(r)+r(r) so that S(o)=R_(r)e dr SH(o)+S,(o)+S(o)+Sy(o) where we have defined cross spectral density functions S2(o)=∫R(r)e-da This is equivalent to the definition Page 4 of 816.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 4 of 8 The second approach is faster. In fact, with the advent of the Fast Fourier Transform (initiated by Cooley and Tukey), even if one wanted to calculate ( ) Rxx τ from x( )t , it is faster to transform x( )t to X ( ) ω , form ( ) xx S ω , and transform to get ( ) Rxx τ than to integrate xtxt () ( ) +τ directly for all desired values of τ . The Fast Fourier Transform is an amazingly efficient procedure for digital calculation of finite Fourier Transforms. References: Full issue – IEEE Transactions on Audio and Electroacoustics, Vol. AU-15, No.2; June 1967. Tutorial article: Brighton, E.O. and Morrow, R.E.: The Fast Fourier Transform, IEEE Spectrum; Dec. 1967. Cross spectral density In dealing with more than one random process, the cross power spectral density arises naturally. For example, if zt xt yt () () () = + where x( )t and y t( ) are members of random ensembles, then we found before that () () () () () RRRRR zz xx xy yx yy τ =+++ ττττ so that ( ) () () () () () j zz zz xx xy yx yy S R ed SSSS ωτ ω ττ ω ωωω ∞ − −∞ = =+++ ∫ where we have defined cross spectral density functions: ( ) () ( ) () j xy xy j yx yx S R ed S R ed ωτ ωτ ω τ τ ω τ τ ∞ − −∞ ∞ − −∞ = = ∫ ∫ This is equivalent to the definition