Post on 24-Feb-2016
description
1
Stochastic processes
Lecture 8Ergodicty
2
Random process
3
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
0 2 4 6 8 10-5
0
5Realization 1
t (s)
x(t)
0 2 4 6 8 10-5
0
5
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
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: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
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)
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
2
4
6
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
0
2
4
6
8Autocorrelation
M=0
-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
2
4
6
8Autocorrelation
M=4
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
• ¿ lim𝑇→∞
𝐴2𝑇 ∫
−𝑇
𝑇
cos (2𝜋 𝑓 𝑐𝜏 )+¿ cos (4𝜋 𝑓 𝑐𝑡+2𝜋 𝑓 𝑐𝜏+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: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
18
Essential equations
19
Typical signals
• Dirac delta δ(t)
• Complex exponential functions
20
Essential equationsDistribution 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
• 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
• If X(i) is Ergodic
– where
23
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)]
24
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;
25
Essential equationsPSD
• Truncated Fourier transform of X(t):
• Power spectrum
• Or from the autocorrelation– The Fourier transform of the auto-correlation
26
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
27
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
28
• 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 )
29
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
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