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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)
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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
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 ∞
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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
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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
•

= lim
→∞ 2

−
cos 2  + cos 4  + 2  + 
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 ∞
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),
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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 
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Essential equations
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Typical signals
• Dirac delta δ(t)
  =
∞
0
=0

∞
   = 0
−∞
• Complex exponential functions
  = cos  + ()
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Essential equations
Distribution 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
 1, 2 =   1  2
∗
=  (2, 1)
• In the case of WSS
  =   + ,  = [  +  ()]
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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]=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;
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Essential equations
PSD
• Truncated Fourier transform of X(t):
• Power spectrum
• Or from the autocorrelation
– The Fourier transform of the auto-correlation
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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
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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
−∞ −∞
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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
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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?
35
exercise
• Chi meditation and autonomic nervous system
36

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