16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 1 of 8 Lecture 13 Last time: () () 22 2 1 , () , a xx aT TR τ στ τ τ ? ?? +? ≤ ? ?? = ? ?? ? > ? () 22 22 0 2 2 2 2 22 () 1 2()21cos 2 2() 1cos sin 2 2() 2 T jj xx a T T a a a S aed ed T ad T aT T T aT T ωτ ωτ τ ω τσ τ τ π δω σ ωττ σ πδω ω ω ω πδω σ ω ∞ ?? ?∞ ? ?? =+? ?? ?? ?? =+? ?? ?? =+? ?? ?? ???? ?? =+ ?? ?? ?? ?? ∫∫ ∫ Amplitude of xx S falls off, but not very rapidly. 16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 2 of 8 Use error between early and late indicator to lock onto signal. Error is a linear function of shift, within the range (,)TT? . Return to the 1 st example process and take the case where the change points are Poisson distributed. 2 22 2 () a xx S λσ ω ω λ = + Take the limit of this as 2 a σ and λ become large in a particular relation: to establish the desired relation, replace 22 2 22 2 () () aa a xx k k kk S k σσ λλ λ σ ω ω λ → → = + and take the limit as k →∞. () 22 2 2 2 lim ( ) lim 2 a xx kk a k S k λσ ω λ σ λ →∞ →∞ = = Note this is independent of frequency. 16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 3 of 8 This is defined to be a “white noise” by analogy with white light, which is supposed to have equal participation by all wavelengths. Can shape ()x t to the correct spectrum so that it can be analyzed in this manner, by adding a shaping filter in the state-space formulation. Definition of a white noise process White means constant spectral density. 0 ( ) , constant xx SSω = 0 0 1 () 2 () j xx R Se d S τω τ ω π δτ ∞ ?∞ = = ∫ White noise processes only have a defined power density. The variance of a white noise process is not defined. If you start with almost any process ()x t , lim ( ) a ax at →∞ ? is a white noise 16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 4 of 8 So far we have looked only at the xx R and xx S of processes. We shall find that if we wish to determine only the yy R or yy S (thus the 2 y ) of outputs of linear systems, all we need to know about the inputs are their xx R or xx S . But what if we wanted to know the probability that the error in a dynamic system would exceed some bound? For this we need the first probability density function of the system error – an output. Very difficult in general. The pdf of the output ()yt satisfies the Fokker-Planck partial differential equation – also called the Kolmagorov forward equation. Applies to a continuous dynamic system driven by a white noise process. One case is easy: Gaussian process into a linear system, output is Gaussian. Gaussian processes are defined by the property that probability density functions of all order are normal functions. 11 2 2 ( , ; , ;... , ) -dimensional normal nnn fxtxt xt n= 1 1 2 2 1 () (2 ) T xM x n n fx e Mπ ? ? = M is the covariance matrix for 1 2 () () () n x t x t x x t ?? ?? ?? = ?? ?? ?? M Thus, () n f x for all n is determined by M - the covariance matrix for x . 16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 5 of 8 ()( ) (, ) ij i j xx i j M xt xt R tt = = Thus for a Gaussian process, the autocorrelation function completely defines all the statistical properties of the process since it defines the probability density functions of all order. This means: If ( ) (, ) yx i j xx j i R tt R t t=?, the process is stationary. If two processes (), ()x tyt are jointly Gaussian, and are uncorrelated () (, ) 0 xy i j Rtt= , they are independent processes. Most important: Gaussian input ? linear system ? Gaussian output. In this case all the statistical properties of the output are determined by the correlation function of the output – for which we shall require only the correlation functions for the inputs. Upcoming lectures will not cover several sections that deal with: Narrow band Gaussian processes Fast Fourier Transform Pseudorandom binary coded signals These are important topics for your general knowledge. Characteristics of Linear Systems Definition of linear system: If 11 22 () () () () ut yt ut yt → → Then 12 12 () () () ()au t bu t ay t by t+→+ 16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 6 of 8 By characterizing the response to a standard input, the response to any input can be constructed by superposing responses using the system’s linearity. (, )wtτ is the weighting function. 0 () lim (, ) ( ) (, ) ( ) i t iii i yt wt u wt u d τ τ ττ τττ ?→ ?∞ =?→ ∑ ∫ The central limit theorem says ()yt→normal if ()ut is white noise. Stable if every bounded input gives rise to a bounded output. ( , const. bounded for all wt d tττ ∞ ?∞ =<∞? ∫ Realizable if causal. ()(, ) 0, wt tτ τ=< State Space – An alternate characterization for a linear differential system 16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 7 of 8 If the input and output are related by an nth order linear differential equation, once can also relate input to output by a set of n linear first order differential equations. () () () () () () () () x tAtxtBtut yt Ctxt =+ = & The solution form is: 0 00 () () () () (, )( ) (,) ()() t t yt Ctxt x tttxt tBudτ τττ = =Φ + Φ ∫ where (, )t τΦ satisfies: ( , ) ( ) ( , ), ( , ) d tAtt I dt τττΦ=Φ Φ= Note that any system which can be cast in this form is not only mathematically realizable but practically realizable as well. Must add a gain times u to y to get as many zeroes as poles. For comparison with the weighting function description, take u and y to be scalars, and take 0 t =?∞. For stable systems, the transition from ?∞ to any finite time is zero. Specialize the state space model to single-input, single-output (SISO) and 0 t →?∞: (, ) 0 () () () () (, ) ( ) ( ) () () (, ) ( ) ( ) (, ) () (, ) ( ) T t t T T t yt Ct xt xt tbud yt Ct t b u d wt Ct t b ττττ τ τττ τττ ?∞ ?∞ Φ?∞= = =Φ =Φ =Φ ∫ ∫ which we recognize by comparison with the earlier expression for ()yt. 16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 8 of 8 For an invariant system, the shape of (, )wtτ is independent of the time the input was applied; the output depends only on the elapsed time since the application of the input. ( , ) ( ) ( ), 0 by conventionwt wt wtτ ττ→?= = Stability: () () const.wt d wt dtττ ∞∞ ?∞ ?∞ ?= =<∞ ∫∫ Note this implies ()wt is Fourier transformable. Realizability: ( ) 0, ( ) ( ) 0, ( 0) wt t wt t τ τ?= < =<