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ELE3410 Random Process and DSP ELE3410 Random Process & DSP Tutorial #5 Ergodic Processes Autocorrelation and Autocovariance Function Ergodicity: If a process is said to be ergodic, its ensemble averages equal appropriate time averages. (Note that this is NOT the definition of ergodicity but the property!) M1η(t ) = x(t , S)∑0 0Ensemble average: M s =1− The average value of M sample at a give time t . 0− Dependent on t . 0− Natural way to estimate η(t ). 0T1lim x(t, S)dt∫Time average: T → ∞ 2T −T− The average value in a long enough period for a specified outcome S. − Dependent on the outcomes S ⇒ Different S should have different time average. − If the process is stationary and E[|x(t, s)|] is finite, then the limit should exist for almost every S. Tutorial 6 [1/5] ELE3410 Random Process and DSP Definition: A random process x(t) is said to be ergodic if all its statistics can be determined from a single of the process. (Compare the difference among Ergodic, Stationary and Statistically Determined.) thThat means: The n -order pdf of the process can be deduced by examining either: 1) one member of the process over a long time, or 2) x(t ), …, x(t ), the process at t , …, t many times. 1 n 1 n x(t) = cos( ϖ ⋅ t + θ )Example 1) if , where θ is a random variable uniformly distributed in [- π, π], is said to be ergodic, show that its time average and its ensemble average are equal. Solution: M1η(t ) = x(t , S)∑0 ...
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ELE3410
Random Process and DSP
Tutorial 6
[1/5]
ELE3410
Random Process & DSP
Tutorial #5
Ergodic Processes
Autocorrelation and Autocovariance Function
Ergodicity:
If a process is said to be
ergodic,
its
ensemble averages
equal appropriate
time
averages
. (Note that this is
NOT
the definition of ergodicity but the property!)
Ensemble average:
∑
=
=
M
s
S
t
x
M
t
1
0
0
)
,
(
1
)
(
η
−
The average value of
M
sample at a give time
t
0
.
−
Dependent on
t
0
.
−
Natural way to estimate
η
(
t
0
).
Time average:
∫
−
∞
→
T
T
T
dt
S
t
x
T
)
,
(
2
1
lim
−
The average value in a long enough period for a specified outcome
S
.
−
Dependent on the outcomes
S
⇒
Different
S
should have different time
average.
−
If the process is stationary and E[|
x
(
t
,
s
)|] is finite, then the limit should exist
for almost every
S
.
ELE3410
Random Process and DSP
Tutorial 6
[2/5]
Definition: A random process
x
(
t
) is said to be ergodic if
all
its statistics can be
determined from a
single
of the process. (Compare the difference among
Ergodic
,
Stationary
and
Statistically Determined
.)
That means: The
n
th
-order
pdf
of the process can be deduced by examining either:
1)
one member of the process over a long time, or
2)
x
(
t
1
), …,
x
(
t
n
), the process at
t
1
, …,
t
n
many times.
Example 1) if
)
cos(
)
(
θ
ϖ
+
⋅
=
t
t
x
, where
θ
is a random variable uniformly
distributed in [-
π
,
π
], is said to be ergodic, show that its time average and its
ensemble average are equal.
Solution:
Ensemble average:
∑
=
=
M
s
S
t
x
M
t
1
0
0
)
,
(
1
)
(
η
0
)
(
)
cos(
)]
[cos(
)]
(
[
=
+
=
+
=
∫
−
θ
θ
θ
θ
π
π
d
p
wt
wt
E
t
x
E
Time average:
∫
−
∞
→
T
T
T
dt
S
t
x
T
)
,
(
2
1
lim
0
)]
sin(
)
[sin(
1
2
1
lim
)
cos(
2
1
lim
)
,
(
2
1
lim
0
0
0
=
+
−
−
+
=
+
=
∞
→
−
∞
→
−
∞
→
∫
∫
θ
θ
θ
wT
wT
w
T
dt
wt
T
dt
S
t
x
T
T
T
T
T
T
T
T
So they are equal.
ELE3410
Random Process and DSP
Tutorial 6
[3/5]
Autocorrelation and Autocovariance:
Autocorrelation:
2
1
2
1
2
1
2
1
2
1
2
1
)
,
;
,
(
)]
(
)
(
[
)
,
(
dx
dx
t
t
x
x
p
x
x
t
x
t
x
E
t
t
R
∫
∫
∞
∞
−
∞
∞
−
=
•
=
The ensemble average of the product
x
(
t
1
) and
x
(
t
2
) of a random process
x
(
t
).
Autocovariance:
(
)
(
)
(
)
(
)
(
)
(
)
(
)
[
]
(
)
(
)
(
)
2
1
2
1
2
2
1
1
2
1
,
,
t
t
t
t
R
t
t
x
t
t
x
E
t
t
C
µ
µ
µ
µ
−
=
−
−
=
(Compare with covariance
µ
11
=
(
)(
)
[
]
y
y
x
x
E
−
−
.)
Variance:
(
)
(
)
(
)
(
)
(
)
(
)
t
t
C
dx
t
x
p
t
t
x
t
,
,
2
2
=
−
=
∫
∞
∞
−
µ
σ
special case of
(
)
2
1
,
t
t
C
For a
stationary
random process
x
(
t
) of mean
η
：
)]
)
(
)(
)
(
[(
)
(
η
τ
η
τ
−
+
−
=
Φ
t
x
t
x
E
Normalized Autocovariance:
]
1
,
1
[
)
(
)
(
2
−
∈
Φ
=
Φ
σ
τ
τ
n
For a
stationary
random process
x
(
t
) of variance
σ
2
. Why?
Note the difference between their definitions. Remember the relationship between
Autocorrelation and Autocovariance (including the proof).
Autocorrelation:
(Note the relationship with
power spectrum
in later lectures)
If
x
(
t
) is ergodic
⇒
p
(
x
1
,
x
2
;
t
1
,
t
2
) =
p
(
x
1
,
x
2
;
τ
)
⇒
)
(
)
(
)
(
2
1
lim
)
,
(
2
1
τ
τ
R
dt
t
x
t
x
T
t
t
R
T
T
T
=
+
=
∫
−
∞
→
, where
τ
=
t
2
-
t
1
.
Properties: If
x
(
t
),
y
(
t
) and
z
(
t
) are ergodic random processes with autocorrelation
functions
R
x
(
τ
) ,
R
y
(
τ
) and
R
z
(
τ
) (Remember these properties and the proofs):
1)
R
(0)
≥
0
2)
R
(
τ
) =
R
(-
τ
)
3)
R
(0)
≥
R
(
τ
)
∀
τ
4)
If
dt
t
dx
t
z
)
(
)
(
=
,
2
2
)
(
)
(
dt
R
d
R
z
τ
τ
=
ELE3410
Random Process and DSP
Tutorial 6
[4/5]
5)
If
∫
∞
−
=
t
dt
t
x
t
z
)
(
)
(
,
∫
∫
∞
−
∞
−
−
=
τ
τ
v
z
dudv
u
R
R
)
(
)
(
6)
If
z
(
t
) =
x
(
t
) +
y
(
t
), where
x
(
t
) and
y
(
t
) are
uncorrelated
processes,
R
z
(
τ
) =
R
x
(
τ
) +
R
y
(
τ
).
7)
If
z
(
t
) =
x
(
t
)
y
(
t
), where
x
(
t
) and
y
(
t
) are
statistically independent
processes,
R
z
(
τ
)
=
R
x
(
τ
)
R
y
(
τ
).
Example 2) If
)
cos(
)
(
θ
ϖ
+
⋅
=
t
t
x
, where
θ
is a random variable uniformly
distributed in [-
π
,
π
], is an ergodic random process, Find its mean,
autocorrelation function and autocovariance function.
Solution:
Method(1)
0
)
(
)
cos(
)]
[cos(
)]
(
[
=
+
=
+
=
∫
−
θ
θ
θ
θ
π
π
d
p
wt
wt
E
t
x
E
2
cos
]
2
cos
)
2
2
cos(
[
)]
(
cos(
)
[cos(
)]
(
cos(
)
[cos(
)]
(
)
(
[
)
(
τ
τ
θ
τ
θ
τ
θ
θ
τ
θ
τ
τ
w
w
w
wt
E
t
w
wt
E
t
w
wt
E
t
x
t
x
E
R
=
+
+
+
=
+
+
+
=
+
+
+
=
+
=
）
）
2
cos
)]
)
(
)(
)
(
[(
)
(
τ
η
τ
η
τ
w
t
x
t
x
E
=
−
+
−
=
Φ
ELE3410
Random Process and DSP
Tutorial 6
[5/5]
Method(2)
2
cos
)
cos
2
)
2
2
sin(
)
2
2
sin(
(
2
1
lim
2
cos
)
2
2
cos(
2
1
lim
)
(
)
(
2
1
lim
)
(
τ
τ
θ
τ
θ
τ
τ
θ
τ
τ
τ
w
w
T
w
w
wT
w
wT
T
dt
w
w
wt
T
dt
t
x
t
x
T
R
T
T
T
T
T
T
T
=
+
+
+
−
−
+
+
=
+
+
+
=
+
=
∞
→
−
∞
→
−
∞
→
∫
∫
2
cos
)
(
)
(
τ
τ
τ
w
R
=
=
Φ
