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Stochastic processes

Lecture 8Ergodicty

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Random process

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Agenda (Lec. 8)

• Ergodicity• Central equations• Biomedical engineering example:

– Analysis of heart sound murmurs

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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.

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Ergodicity ilustrated

• statistics can be determined by time averaging of one realization

0 2 4 6 8 10-5

0

5Realization 1

t (s)

x(t)

0 2 4 6 8 10-5

0

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t (s)

Realization 2

x(t)

0 2 4 6 8 10-5

0

5

t (s)

Realization 3

x(t)

Estimate of E[X(x)]across Realizations

Estimate ofE[X(x)] from oneRealization overtime

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Ergodicity and stationarity

• Wide-sense stationary: Mean and Autocorrelation is constant over time

• Strictly stationary: All statistics is constant over time

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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

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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 ∞

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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

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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 ∞

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The time average autocorrelation(Discrete version)

N=12 𝑅𝑥𝑥 [𝑚 ]= ∑𝑛=0

𝑁−|𝑚|− 1

𝑥 [𝑛 ] 𝑥 [𝑛+𝑚]

-10 -5 0 5 10 15 20-2

-1

0

1

2

n

-10 -5 0 5 10 15 20-2

-1

0

1

2

n+m-15 -10 -5 0 5 10 15

-6

-4

-2

0

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8Autocorrelation

M=-10

-10 -5 0 5 10 15 20-2

-1

0

1

2

n

-10 -5 0 5 10 15 20-2

-1

0

1

2

n+m-15 -10 -5 0 5 10 15

-6

-4

-2

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8Autocorrelation

M=0

-10 -5 0 5 10 15 20-2

-1

0

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n

-10 -5 0 5 10 15 20-2

-1

0

1

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n+m-15 -10 -5 0 5 10 15

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-4

-2

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8Autocorrelation

M=4

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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?

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Example (2/2)

• The time averaged autocorrelation of the sample function

• ¿ lim𝑇→∞

𝐴2𝑇 ∫

−𝑇

𝑇

cos (2𝜋 𝑓 𝑐𝜏 )+¿ cos (4𝜋 𝑓 𝑐𝑡+2𝜋 𝑓 𝑐𝜏+𝜃 )¿

Thereby

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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 ∞

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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),

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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

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Essential equations

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Typical signals

• Dirac delta δ(t)

• Complex exponential functions

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Essential equationsDistribution and density functions

First-order distribution:

First-order density function:

2end order distribution

2end order density function

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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

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Essential equations Auto-correlations

• In the general case

– Thereby

• If X(t) is WSS

• If X(i) is Ergodic

– where

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Essential equations Cross-correlations

• In the general case

• In the case of WSS𝑅𝑥 𝑦 (𝑡 1 , 𝑡 2 )=𝐸 [𝑋 (𝑡 1 )𝑌 (𝑡 2 ) ]=𝑅𝑦𝑥∗(𝑡 2 , 𝑡 1)

𝑅𝑥 𝑦 (𝜏 )=𝑅𝑥𝑦 (𝑡+𝜏 ,𝑡 )=𝐸 [ 𝑋 (𝑡+𝜏 )𝑌 (𝑡)]

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Properties of autocorrelation and crosscorrelation

• Auto-correlation:Rxx(t1,t1)=E[|X(t)|2]When WSS:Rxx(0)=E[|X(t)|2]=σx

2+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;

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Essential equationsPSD

• Truncated Fourier transform of X(t):

• Power spectrum

• Or from the autocorrelation– The Fourier transform of the auto-correlation

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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

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Essential equationsLTI systems (2/4)

• Expected value (mean) of the output:

– If WSS:

• Expected Mean square value of the output

– If WSS:

𝑚𝑦=𝐸 [𝑌 (𝑡 ) ]=𝑚𝑥∫−∞

∞

h (𝛼 )𝑑𝛼

𝐸 [𝑌 (𝑡 ) ]=∫−∞

∞

𝐸 [ 𝑋 (𝑡−𝛼 ) ] h (𝛼 ) 𝑑𝛼=∫−∞

∞

𝑚𝑥 (𝑡−𝛼)h (𝛼 )𝑑𝛼

h𝑤 𝑒𝑟𝑒𝑚𝑥 (𝑡 ) 𝑖𝑠𝑚𝑒𝑎𝑛𝑜𝑓𝑋 (𝑡 )𝑎𝑠𝐸 [ 𝑋 (𝑡)]

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

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• Cross correlation function between input and output when WSS

• Autocorrelation of the output when WSS

Essential equationsLTI systems (3/4)

𝑅𝑦 𝑥 (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 )𝑑𝛼 𝑑𝛼

𝑅𝑦𝑦 (𝜏 )=𝑅𝑦𝑥 (𝜏 )∗h (−𝜏 )

𝑅𝑦𝑦 (𝜏 )=𝑅𝑥𝑥 (𝜏 )∗h (𝜏)∗h(−𝜏 )

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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

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A biomedical example on a stochastic process

• Analyze of Heart murmurs from Aortic valve stenosis using methods from stochastic process.

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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

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Aortic valve stenosis

• Narrowing of the Aortic valve

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Reflections of Aortic valve stenosis in the heart sound

• A clear diastolic murmur which is due to post stenotic turbulence

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Abnormal heart sounds

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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?

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exercise

• Chi meditation and autonomic nervous system