ECE 5345
Ergodic
Theorems
copyright R.J. Marks II
Ergodicity


In many cases, an ensemble of random
processes cannot be obtained.
If parameters of a random process can
be found from a single observation of
the process, the process is said to be
ergodic.


Mean ergodic
Distribution ergodic, etc.
copyright R.J. Marks II
Ergodicity


Mean Ergodic
Not Mean Ergodic Example

Battery Factory
copyright R.J. Marks II
Mean Ergodicity

Time Average
X (t )

 X (t )dt
T
If X(t) is WSS, the expected value is

E X (t )

T
1

2T
T
T

1

2T
T
 Xdt  X
T
Good so far!
copyright R.J. Marks II
Mean Ergodicity

E X (t )

We’d also like

var X (t )

T
 X
 0
T T 
If so, X(t) is mean ergodic!
copyright R.J. Marks II
Mean Ergodicity


Onward…
var X (t )
T
  E  X (t )
T
X

2
 1 T
 1 T

X (t )  X dt   X ( )  X d 
 E 

 2T T
 2T T

1 T T

E  X (t )  X  X ( )  X dtd
2 T T
4T
1 T T

C X (t   )dtd
2 T T
4T
copyright R.J. Marks II
Mean Ergodicity

Continuing:

var X (t )
1
 2
4T
T

1
 2
4T




 
T
T
T
T
 
C X (t   )dtd
 t    
C X (t   )  dtd
 2T   2T 
1

1
;
|
t
|


2

1
1
 t    ; | t |
2
2
1

0
;
|
t
|


2

Rectangle
Function
copyright R.J. Marks II
Mean Ergodicity
More:


var X (t )





T

1
4T
2

     t 
d 
dt
C X (t   )
 2T    2T 

  t 
copyright R.J. Marks II
Mean Ergodicity

var X (t )
T

1
4T 2


 

 t     t 
d  
dt
C X ( ) 
 2T    2T 

 
 t   t   

C ( )  
 
dt d
2  X
4T
   2T   2T  
1

 1    1   
  C X ( ) 

*
d

 2T  2T  2T  2T 
1 
  

C X ( )
 d

2T 
 2T 
 

copyright R.J. Marks II
   1 |  | 
 2
Ergodicity

var X (t )
T

1

2T
  
 C X ( ) 2T d

Thus, a WSS Random Process is mean ergodic if
1
lim
T  2T
  
 C X ( ) 2T d  0

Similar criteria for discrete time stochastic processes.
copyright R.J. Marks II
C X ( )  e
1
2T
1

T
2  | |
Telegraph Signal

  
 C X ( ) 2T d
2T  2 
 
0 e 1  2T d
1 2T 2
  e
d
T 0
 0 Ergodic.
T copyright R.J. Marks II
CX ( )  var( X )
Battery Factory
  
C
(

)

d



X

 2T 
1 T 
 
 var( X )  1 
d

T 0 
2T 
 var( X )  0
1
2T

T 
Not ergodic.
copyright R.J. Marks II
Autocorrelation Ergodic
When, for large T, can we approximate

RX ( )  X (t ) X (t  ) T ?

Define, for a fixed  ,
Y (t )  X (t ) X (t   )

X(t) is thus correlation ergodic
when Y(t) is mean ergodic for all .
copyright R.J. Marks II
Autocorrelation Ergodic

Note
CY  ( )  EY (t )Y (t   )
 EX ( ) X (  t ) X (  ) X (   t   )

Establishment of correlation
ergodicity thus requires
• fourth order moments of X(t).
• Better than WSS.
copyright R.J. Marks II