Stochastic processes Lecture 8 Ergodicty
Agenda (Lec. 8) • Ergodicity • Central equations • Biomedical engineering example: • Analysis of heart sound murmurs
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.
Ergodicity ilustrated • statistics can be determined by time averaging of one realization
Ergodicity and stationarity • Wide-sense stationary: Mean and Autocorrelation is constant over time • Strictly stationary: All statistics is constant over time
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
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 ∞
Example • Ergodic in mean: X • Where: • is a random variable • a and θ are constant variables • Mean is impendent on the random variable • Not Ergodic in mean: X • Where: • and dcr are random variables • a and θ are constant variables • Mean is not impendent on the random variable
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∞
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?
Example (2/2) • The time averaged autocorrelation of the sample function Thereby
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 ∞
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),
Example • Ergodic in PSD: X • Where: • is a random variable • a and are constant variables • The PSD is impendent on the phase the random variable • Not Ergodic in PSD: X • Where: • are random variables • a and θ are constant variables • The PSD is not impendent on the random variable
Typical signals • Dirac delta δ(t) • Complex exponential functions
Essential equationsDistribution and density functions First-order distribution: First-order density function: 2end order distribution 2end order density function
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
Essential equations Auto-correlations • In the general case • Thereby • If X(t) is WSS • If X(i) is Ergodic • where
Essential equations Cross-correlations • In the general case • In the case of WSS
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) isindependent 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;
Essential equationsPSD • Truncated Fourier transform of X(t): • Power spectrum • Or from the autocorrelation • The Fourier transform of the auto-correlation
Essential equationsLTI 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
Essential equationsLTI systems (2/4) • Expected value (mean) of the output: • If WSS: • Expected Mean square value of the output • If WSS:
Essential equationsLTI systems (3/4) • Cross correlation function between input and output when WSS • Autocorrelation of the output when WSS
Essential equationsLTI systems (4/4) • PSD of the output • Where H(f) is the transfer function • Calculated as the four transform of the impulse response
A biomedical example on a stochastic process • Analyze of Heart murmurs from Aortic valve stenosis using methods from stochastic process.
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
Aortic valve stenosis • Narrowing of the Aortic valve
Reflections of Aortic valve stenosis in the heart sound • A clear diastolic murmur which is due to post stenoticturbulence
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?
exercise • Chi meditation and autonomic nervous system