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 tω θ=+ where θ is uniformly distributed over 2π , A is random, independent of θ , that component will be () () ()() 00 00 2 0 2 0 2 2 2 00 1 () cos 2 1 () cos 2 11 22 11 22 1 2 xx j xx jjj jj RA SAed A eeed Ae e d A ωτ ωτ ωτ ωτ ωωτ ωωτ τωτ ωωττ τ τ πδωωδωω ∞ ? ?∞ ∞ ? ? ?∞ ∞ ??+ ?∞ = = ??=+ ?? ?? =+ ?? =?++ ∫ ∫ ∫ 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 () , where 2 j xx xx xx xx SRed xSd Sfdf f ωτ ωττ ωω π ω π ∞ ? ∞ ∞ ?∞ ∞ ?∞ = = == ∫ ∫ ∫ In this case, 2 ~ xx q S Hz Next most common: 2 1 () () 2 () j xx xx xx SRed xSd ωτ ω ττ π ωω ∞ ? ?∞ ∞ ?∞ = = ∫ ∫ In this case, 2 2 ~sec rad / sec xx q Sq= 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. 16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 3 of 8 Define (), ( ) () 0, elsewhere T x tTtT xt ?<< ? = ? ? () () () j TT T jt T X xte d x te dt ωτ ω ω τ ∞ ? ?∞ ? ? = = ∫ ∫ Then * 2 () () () lim 2 () lim 2 TT xx T T T XX 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 jt T xx R xtxt d T SRd xxtedt XX S T τ τ ω τ ττ τ ω τωττ ω ωω ω ? ? ? =+ ? = = = ∫ ∫ ∫ Standard deviation of xx S measured this way is approximately equal to mean. 16.322 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 () xx R τ from ()x t , it is faster to transform ()x t to ()X ω , form () xx S ω , and transform to get () xx R 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 ()yt are members of random ensembles, then we found before that () () () () () zz xx xy yx yy RRRRRτ ττττ=+++ so that () () () () () () j zz zz xx xy yx yy SRed SSSS ωτ ωττ ω ωωω ∞ ? ?∞ = =+++ ∫ where we have defined cross spectral density functions: () () () () j xy xy j yx yx SRed SRed ωτ ωτ ω ττ ω ττ ∞ ? ?∞ ∞ ? ?∞ = = ∫ ∫ This is equivalent to the definition 16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 5 of 8 * * () () ( ) lim 2 () () ( ) lim 2 TT xy T TT yx T XY S T YX S T ω ω ω ω ω ω →∞ →∞ = = We note from this that: * () () yx xy SSω ω= so that the sum of these two as they appear in () zz S ω is real. Also note that * () () () xy xy yx SSSω ωω?= = . Examples of Random Processes Analytically Defined Example: Random step function Amplitude n a independent, random Change points n t , Poisson-distributed with average density λ (points per second) () 1 () ! (0) k Pk e k Pe λτ λτ λτ ? ? = = [ ] () 22 22 22 () ()( ) (at least one change point in ) (no change point in ) 1 xx a R Extxt PaPa eaea ea λτ λτ λτ ττ ττ σ ?? ? =+ =? + =+ 16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 6 of 8 () () 22 0 22 2 0 0 22 2 0 22 2 2 22 () 2() 2() 11 2() 2 2() j xx a jj aa jj aa a a Saeed aeeded ae e a jj a λτ ωτ ωτ λτ ωτ λτ λωτ λωτ ωσ τ πδω σ τ σ τ σσ πδω λω λω πδωσ λωλω σλ πδω λω ∞ ? ? ?∞ ∞ ??? ?∞ ∞ ??+ ?∞ ?? =+ ?? =+ + =+ + ??? ?? =+ + ?? ?+ ?? =+ + ∫ ∫∫ Example: A signal random process Reference: Newton, G.C, L.A. Gould and J.F. Kaiser. Design of Linear Feedback Controls. John Wiley, 1961. p.100. Papoulis calls this the semirandom telegraph signal; p.288. A signal takes the values plus and minus 0 x only. It switches from one level to the other at “event points” which are Poisson distributed in time with constant average frequency λ . This is sometimes called a telegraph signal. 16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 7 of 8 Average rate of occurrence of change points λ= points per second () () ()() 2 0 2 0 22 00 0,2,4,... 1,3,5,... 2 0 () ()( ) (even number of change points in the interval ) (odd number of change points in the interval )( ) 11 !! 1 1 ! xx kk kk k k k RExtxt Px xexe ex k λτ λτ λτ ττ τ τ λτ λτ λτ ?? == ? =+?? ?? = +? =? =? ∑∑ () 0,1,2,... 2 0 0 22 0 ! k k xe k xe λτ λτ λτ = ∞ ? = ? ? = = ∑ ∑ You are generating higher harmonics in this case, as at each change point the amplitude changes sign. In the previous example, changes in amplitude at change point may be far smaller. () 2 0 2 2 2(2) () 2 xx x S λ ω ω λ = + Example: Binary function with an arbitrary amplitude distribution The problem considers a binary function with a more general amplitude distribution. The distribution of a is restricted to 1± with equal probability. That gives the pseudo-random binary code used by GPS. 16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 8 of 8 0 t is uniformly distributed over (0, )T Change points are periodic with period T Amplitudes independent with 2 ,aa [ ] 22 22 22 2 () ()( ) (1 or more change points in ) (No change point in ) 1 , 1 , () , xx a xx R Extxt PaPa aaT TT aT TR aT ττ ττ ττ τ τ στ τ τ =+ ?? =+? ≤ ?? ?? ? ?? ?+ ≤ ? ?? = ? ?? ? > ?