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Stochastic processes Lecture 8 Ergodicty 1 Random process 2 3 Agenda (Lec. 8) • Ergodicity • Central equations • Biomedical engineering example: – Analysis of heart sound murmurs 4 Ergodicity • A random process X(t) is ergodic if all of its statistics can be determined from a sample function of the process • That is, the ensemble averages equal the corresponding time averages with probability one. 5 Ergodicity ilustrated • statistics can be determined by time averaging of one realization Realization 1 x(t) 5 0 -5 0 2 4 6 t (s) Realization 2 8 10 0 2 4 8 10 x(t) 5 0 -5 6 t (s) Realization 3 x(t) 5 Estimate of E[X(x)] from one Realization over time 0 -5 0 2 4 Estimate of E[X(x)] across Realizations 6 8 10 t (s) 6 Ergodicity and stationarity • Wide-sense stationary: Mean and Autocorrelation is constant over time • Strictly stationary: All statistics is constant over time 7 Weak forms of ergodicity • The complete statistics is often difficult to estimate so we are often only interested in: – Ergodicity in the Mean – Ergodicity in the Autocorrelation 8 Ergodicity in the Mean • A random process is ergodic in mean if E(X(t)) equals the time average of sample function (Realization) • Where the <> denotes time averaging • Necessary and sufficient condition: X(t+τ) and X(t) must become independent as τ approaches ∞ 9 Example • Ergodic in mean: X = a sin(2 + ) • Where: – is a random variable – a and θ are constant variables • Mean is impendent on the random variable • Not Ergodic in mean: X = sin 2 + + – Where: – and dcr are random variables – a and θ are constant variables • Mean is not impendent on the random variable 10 Ergodicity in the Autocorrelation • Ergodic in the autocorrelation mean that the autocorrelation can be found by time averaging a single realization • Where • Necessary and sufficient condition: X(t+τ) X(t) and X(t+τ+a) X(t+a) must become independent as a approaches ∞ 11 The time average autocorrelation (Discrete version) N=12 − −1 = [ + ] =0 Autocorrelation Autocorrelation M=-10 M=0 M=4 222 888 111 666 000 444 -1-1 -1 -2-2 -2 -10 -10 -10 -5 -5 -5 000 555 nnn 10 10 10 15 15 15 20 20 20 222 000 222 111 -2 -2 -2 000 -4 -4 -4 -1-1 -1 -2-2 -2 -10 -10 -10 -5 -5 -5 000 555 n+m n+m n+m 10 10 10 15 15 15 20 20 20 -6 -6 -6 -15 -15 -15 -10 -10 -10 -5 -5 0 5 10 15 12 Example (1/2) Autocorrelation • A random process – where A and fc are constants, and Θ is a random variable uniformly distributed over the interval [0, 2π] – The Autocorraltion of of X(t) is: – What is the autocorrelation of a sample function? 13 Example (2/2) • The time averaged autocorrelation of the sample function • = lim →∞ 2 − cos 2 + cos 4 + 2 + Thereby 14 Ergodicity of the First-Order Distribution • If an process is ergodic the first-Order Distribution can be determined by inputting x(t) in a system Y(t) • And the integrating the system • Necessary and sufficient condition: X(t+τ) and X(t) must become independent as τ approaches ∞ 15 Ergodicity of Power Spectral Density • A wide-sense stationary process X(t) is ergodic in power spectral density if, for any sample function x(t), 16 Example • Ergodic in PSD: X = a sin(2 + ) • Where: – θ is a random variable – a and are constant variables • The PSD is impendent on the phase the random variable • Not Ergodic in PSD: X = sin 2 + – Where: – are random variables – a and θ are constant variables • The PSD is not impendent on the random variable 17 Essential equations 18 Typical signals • Dirac delta δ(t) = ∞ 0 =0 ∞ = 0 −∞ • Complex exponential functions = cos + () 19 Essential equations Distribution and density functions First-order distribution: First-order density function: 2end order distribution 2end order density function 20 Essential equations Expected value 1st order (Mean) • Expected value (Mean) • In the case of WSS = [()] • In the case of ergodicity Where<> denotes time averaging such as 21 Essential equations Auto-correlations • In the general case – Thereby • If X(t) is WSS = + , = [ + ()] • If X(i) is Ergodic – where 22 Essential equations Cross-correlations • In the general case 1, 2 = 1 2 ∗ = (2, 1) • In the case of WSS = + , = [ + ()] 23 Properties of autocorrelation and crosscorrelation • Auto-correlation: Rxx(t1,t1)=E[|X(t)|2] When WSS: Rxx(0)=E[|X(t)|2]=σx2+mx2 • Cross-correlation: – If Y(t) and X(t) is independent Rxy(t1,t2)=E[X(t)Y(t)]=E[X(t)]E[Y(t)] – If Y(t) and X(t) is orthogonal Rxy(t1,t2)=E[X(t)Y(t)]=E[X(t)]E[Y(t)]=0; 24 Essential equations PSD • Truncated Fourier transform of X(t): • Power spectrum • Or from the autocorrelation – The Fourier transform of the auto-correlation 25 Essential equations LTI systems (1/4) • Convolution in time domain: Where h(t) is the impulse response Frequency domain: Where X(f) and H(f) is the Fourier transformed signal and impluse response 26 Essential equations LTI systems (2/4) • Expected value (mean) of the output: ∞ = ∞ − ℎ = −∞ ( − )ℎ −∞ – If WSS: ℎ [()] ∞ = = ℎ −∞ • Expected Mean square value of the output ∞ 2 ∞ = ( − , − )ℎ ℎ 12 −∞ −∞ – If WSS: ∞ 2 = ∞ ( − ) ℎ ℎ 12 −∞ −∞ 27 Essential equations LTI systems (3/4) • Cross correlation function between input and output when WSS ∞ = − ℎ = ∗ ℎ() −∞ • Autocorrelation of the output when WSS ∞ = ∞ [ + − + ]ℎ ℎ − −∞ −∞ = ∗ ℎ(−) = ∗ ℎ() ∗ ℎ(−) 28 Essential equations LTI systems (4/4) • PSD of the output = ∗ () = | |2 • Where H(f) is the transfer function – Calculated as the four transform of the impulse response 29 A biomedical example on a stochastic process • Analyze of Heart murmurs from Aortic valve stenosis using methods from stochastic process. 30 Introduction to heart sounds • The main sounds is S1 and S2 – S1 the first heart sound • Closure of the AV valves – S2 the second heart sound • Closure of the semilunar valves 31 Aortic valve stenosis • Narrowing of the Aortic valve 32 Reflections of Aortic valve stenosis in the heart sound • A clear diastolic murmur which is due to post stenotic turbulence 33 Abnormal heart sounds 34 Signals analyze for algorithm specification • Is heart sound stationary, quasi-stationary or non-stationary? • What is the frequency characteristic of systolic Murmurs versus a normal systolic period? 35 exercise • Chi meditation and autonomic nervous system 36