1. Stochastic processes
Lecture 8
Ergodicty

2. Random process

4. Agenda (Lec. 8) Ergodicity Central equations
Biomedical engineering example:
Analysis of heart sound murmurs

5. 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.

6. Ergodicity ilustrated
statistics can be determined by time averaging of one realization

7. Ergodicity and stationarity
Wide-sense stationary: Mean and Autocorrelation is constant over time
Strictly stationary: All statistics is constant over time

8. 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

9. 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 ∞

10. Example Ergodic in mean: Not Ergodic in mean: X 𝑡 =a sin⁡(2𝜋 𝜔 𝑟 +𝜃)
Where:
𝜔 𝑟 is a random variable
a and θ are constant variables
Mean is impendent on the random variable 𝜔 𝑟
Not Ergodic in mean:
X 𝑡 =𝑎 sin 2𝜋𝜔𝑟+𝜃 + 𝑑𝑐 𝑟
𝜔 𝑟 and dcr are random variables
Mean is not impendent on the random variable 𝑑𝑐 𝑟

11. 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 ∞

12. The time average autocorrelation (Discrete version)
𝑅𝑥𝑥 𝑚 = 𝑛=0 𝑁− 𝑚 −1 𝑥 𝑛 𝑥[𝑛+𝑚]

13. 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?

14. Example (2/2) The time averaged autocorrelation of the sample function
= lim 𝑇→∞ 𝐴 2𝑇 −𝑇 𝑇 cos 2𝜋 𝑓 𝑐 𝜏 + cos 4𝜋 𝑓 𝑐 𝑡+2𝜋 𝑓 𝑐 𝜏+𝜃
Thereby

15. 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 ∞

16. 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),

17. Example Ergodic in PSD: Not Ergodic in PSD: X 𝑡 =a sin⁡(2𝜋 𝜔 + 𝜃 𝑟 )
Where:
θ 𝑟 is a random variable
a and 𝜔 are constant variables
The PSD is impendent on the phase the random variable 𝜃 𝑟
Not Ergodic in PSD:
X 𝑡 =𝑎 sin 2𝜋 𝜔 𝑟 +𝜃
𝜔 𝑟 are random variables
a and θ are constant variables
The PSD is not impendent on the random variable 𝜔 𝑟

18. Essential equations

19. Typical signals Dirac delta δ(t) Complex exponential functions
𝛿 𝑡 = ∞ 𝑡=0 0 𝑒𝑙𝑠𝑒
−∞ ∞ 𝛿 𝑡 𝑑𝑡=0
Complex exponential functions
𝑒 𝑗𝑡 = cos 𝑡 +𝑗𝑠𝑖𝑛(𝑡)

20. Essential equations Distribution and density functions
First-order distribution: First-order density function: 2end order distribution 2end order density function

21. Essential equations Expected value 1st order (Mean)
Expected value (Mean)
In the case of WSS
𝑚𝑥=𝐸[𝑋(𝑡)]
In the case of ergodicity
Where<> denotes time averaging such as

22. Essential equations Auto-correlations
In the general case
Thereby
If X(t) is WSS
𝑅𝑥𝑥 𝜏 =𝑅𝑥𝑥 𝑡+𝜏,𝑡 =𝐸[𝑋 𝑡+𝜏 𝑋(𝑡)]
If X(i) is Ergodic
where

23. Essential equations Cross-correlations
In the general case
In the case of WSS
𝑅𝑥𝑦 𝑡1,𝑡2 =𝐸 𝑋 𝑡1 𝑌 𝑡2 =𝑅𝑦𝑥∗(𝑡2,𝑡1)
𝑅𝑥𝑦 𝜏 =𝑅𝑥𝑦 𝑡+𝜏,𝑡 =𝐸[𝑋 𝑡+𝜏 𝑌(𝑡)]

24. Properties of autocorrelation and crosscorrelation
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;

25. Essential equations PSD
Truncated Fourier transform of X(t):
Power spectrum
Or from the autocorrelation
The Fourier transform of the auto-correlation

26. 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

27. Essential equations LTI systems (2/4)
Expected value (mean) of the output:
If WSS:
Expected Mean square value of the output
𝐸 𝑌 𝑡 = −∞ ∞ 𝐸 𝑋 𝑡−𝛼 ℎ 𝛼 𝑑𝛼 = −∞ ∞ 𝑚𝑥(𝑡−𝛼)ℎ 𝛼 𝑑𝛼
𝑤ℎ𝑒𝑟𝑒 𝑚𝑥 𝑡 𝑖𝑠 𝑚𝑒𝑎𝑛 𝑜𝑓𝑋 𝑡 𝑎𝑠 𝐸[𝑋(𝑡)]
𝑚𝑦=𝐸 𝑌 𝑡 =𝑚𝑥 −∞ ∞ ℎ 𝛼 𝑑𝛼
𝐸 𝑌 𝑡 2 = −∞ ∞ −∞ ∞ 𝑅𝑥𝑥(𝑡−𝛼,𝑡−𝛽)ℎ 𝛼 ℎ 𝛽 𝑑𝛼1𝑑𝛼2
𝐸 𝑌 𝑡 2 = −∞ ∞ −∞ ∞ 𝑅𝑥𝑥(𝛼−𝛽) ℎ 𝛼 ℎ 𝛽 𝑑𝛼1𝑑𝛼2

28. Essential equations LTI systems (3/4)
Cross correlation function between input and output when WSS
Autocorrelation of the output when WSS
𝑅𝑦𝑥 𝜏 = −∞ ∞ 𝑅𝑥𝑥 𝜏−𝛼 ℎ 𝛼 𝑑𝛼=𝑅𝑥𝑥 𝜏 ∗ℎ(𝜏)
𝑅𝑦𝑦 𝜏 = −∞ ∞ −∞ ∞ 𝐸[𝑋 𝑡+𝜏−𝛼 𝑋 𝑡+𝛼 ]ℎ 𝛼 ℎ −𝑎 𝑑𝛼𝑑𝛼
𝑅𝑦𝑦 𝜏 =𝑅𝑦𝑥 𝜏 ∗ℎ(−𝜏)
𝑅𝑦𝑦 𝜏 =𝑅𝑥𝑥 𝜏 ∗ℎ(𝜏)∗ℎ(−𝜏)

29. Essential equations LTI systems (4/4)
PSD of the output
Where H(f) is the transfer function
Calculated as the four transform of the impulse response
𝑆𝑦𝑦 𝑓 =𝑆𝑥𝑥 𝑓 𝐻 𝑓 𝐻 ∗ (𝑓)
𝑆𝑦𝑦 𝑓 =𝑆𝑥𝑥 𝑓 |𝐻 𝑓 | 2

30. A biomedical example on a stochastic process
Analyze of Heart murmurs from Aortic valve stenosis using methods from stochastic process.

31. 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

32. Aortic valve stenosis
Narrowing of the Aortic valve

33. Reflections of Aortic valve stenosis in the heart sound
A clear diastolic murmur which is due to post stenotic turbulence

34. Abnormal heart sounds

35. 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?

36. exercise
Chi meditation and autonomic nervous system