16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde Lecture 11 Last time: Ergodic processes An ergodic process is necessarily stationary Example: Binary process t+t t+2T At each time step the signal may switch polarity or stay the same. Both +xo and -xo are equally likely Is it stationary and is it ergodic? For this distribution, we expect most of the members of the ensemble to have a hange point near t=0
16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 1 of 9 Lecture 11 Last time: Ergodic processes An ergodic process is necessarily stationary. Example: Binary process At each time step the signal may switch polarity or stay the same. Both 0 +x and 0 −x are equally likely. Is it stationary and is it ergodic? For this distribution, we expect most of the members of the ensemble to have a change point near t=0
16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde Rn(1,l2)=E[x(4)x(2) ≈x(1)x(2) 0 Rx(2,14)≈x(13) 14-13=12-1=x2 Chance of spanning a change point is the same over each regular interval, so the process is stationary. Is it ergodic Some ensemble members possess properties which are not representative of the ensemble as a whole. As an infinite set, the probability that any such member of the ensemble occurs is zero Page 2 of 9
16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 2 of 9 12 1 2 [ ] 1 2 2 34 3 2 43 21 0 (, ) ()( ) ()( ) 0 (, ) () xx xx R t t E xt xt xt xt R t t xt ttttx = ≈ = ≈ −=−= Chance of spanning a change point is the same over each regular interval, so the process is stationary. Is it ergodic? Some ensemble members possess properties which are not representative of the ensemble as a whole. As an infinite set, the probability that any such member of the ensemble occurs is zero
16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde x(t) t+t t+2T (t) 0 t+t t+2T t+T countable set of infinity which constitute a set of zero measure. The ey are a All of these exceptional points are associated with rational points. Th complementary set of processes are an uncountable infinity associated with irrational numbers which constitute a set of measure one
16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 3 of 9 All of these exceptional points are associated with rational points. They are a countable set of infinity which constitute a set of zero measure. The complementary set of processes are an uncountable infinity associated with irrational numbers which constitute a set of measure one
16.322 Stochastic Estimation and Control, Fall 2004 Prof. VanderⅤelde For ergodic processes E[x(o]=limx(dt x2=E[x()]=lim[x(dt R(r)=E[(ox(+r]=lim x(Ox(t+r)dt →2T R,()=E[(On(+D)]=lim ,T ]x(O)y (+rdr a time invariant system may be defined as one such that any translation in time le input affects the output only by an equal translation in time System This system will be considered time invariant if for every r, the input u(t+r) causes the output v((+r). note that the system may be either linear or non- linear It is proved directly that if u(t) is a stationary random process having the ergodic property and the system is time invariant, then y(() is a stationary random process having the ergodic property, in the steady state. This requires the system to be stable, so a defined steady state exists, and to have been operating in the presence of the input for all past time Example: Calculation of an autocorrelation function semble: x(0=Asin(at +0) B, A are independent random variable 0 is uniformly distributed over 0, 2T This process is stationary(the uniform distribution of 0 hints at this)but not ergodic. Unless we are certain of stationarity, we should calculate: Page 4 of 9
16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 4 of 9 For ergodic processes: [ ] [ ] [ ] 22 2 1 ( ) lim ( ) 2 1 [ ( ) ] lim ( ) 2 1 ( ) ( ) ( ) lim ( ) ( ) 2 1 ( ) ( ) ( ) lim ( ) ( ) 2 T T T T T T T xx T T T xy T T x E x t x t dt T x E x t x t dt T R E x t x t x t x t dt T R E x t y t x t y t dt T ττ τ τ τ τ →∞ − →∞ − →∞ − →∞ − = = = = = += + = += + ∫ ∫ ∫ ∫ A time invariant system may be defined as one such that any translation in time of the input affects the output only by an equal translation in time. This system will be considered time invariant if for every τ , the input u t( ) +τ causes the output y t( ) +τ . Note that the system may be either linear or nonlinear. It is proved directly that if u t( ) is a stationary random process having the ergodic property and the system is time invariant, then y t( ) is a stationary random process having the ergodic property, in the steady state. This requires the system to be stable, so a defined steady state exists, and to have been operating in the presence of the input for all past time. Example: Calculation of an autocorrelation function Ensemble: xt A t ( ) sin( ) = + ω θ θ, Aare independent random variables θ is uniformly distributed over 0,2π This process is stationary (the uniform distribution of θ hints at this) but not ergodic. Unless we are certain of stationarity, we should calculate:
16.322 Stochastic Estimation and Control, Fall 2004 Prof vander Velde Rx(12)=x(1)x(L2) ∫a4∫dea)fsi(omn+)si(mn+) 2 sin Asin B=5[cos(A-B)-cos(A+B) R2(42)=5∫,-o(-4)-coso(4+)+20]db l2-1 So the autocorrelation function is sinusoidal with the same frequenc This periodic property is true in general. If all members of a stationary ensemble are periodic, x((+nT)=x(o) R(t +nT)=x(ox(+r+nT) x(1)x(t+r) Identification of a periodic signal in noise We have recorded a signal from an experimental device which looks like just It is of interest to know if there are periodic components contained in it Conside x(1)=R(1)+P() where P(t) is any periodic function of period T and R(t)is a random process
16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 5 of 9 ( )( ) 12 1 2 2 2 1 2 0 0 "" "" (, ) ()( ) 1 ( ) sin sin 2 xx B A R t t xt xt dA d f a A t t π θ ωθ ωθ π ∞ = = ++ ∫ ∫ 14243 14243 ()() 1 sin sin cos cos 2 A B AB AB = −− + ⎡ ⎤ ⎣ ⎦ { } () () 2 2 12 2 1 1 2 0 2 2 1 1 1 ( , ) cos cos 2 2 2 1 cos , 2 Rxx tt A t t t t d A tt π ω ω θθ π ωτ τ = − − ++ ⎡ ⎤⎡ ⎤ ⎣ ⎦⎣ ⎦ = =− ∫ So the autocorrelation function is sinusoidal with the same frequency. This periodic property is true in general. If all members of a stationary ensemble are periodic, x( ) () t nT x t + = ( ) () ( ) () ( ) ( ) xx xx R nT x t x t nT xtxt R τ τ τ τ + = ++ = + = Identification of a periodic signal in noise We have recorded a signal from an experimental device which looks like just hash. It is of interest to know if there are periodic components contained in it. Consider: x() () () t Rt Pt = + where P t( ) is any periodic function of period T and R( )t is a random process independent of P