Many stationary random processes are also
Ergodic. For an Ergodic Random Process we can
exchange Ensemble Averages for Time
Averages. This is equivalent to assuming that our
ensemble of random signals is just composed of all possible time
shifts of a single signal
Xt
X
t
.
Recall from our previous discussion of Expectation that the
expectation of a function of a random variable is given by
EgX=∫gx
fX
xdx
g
X
x
g
x
fX
x
(1)
This result also applies if we have a
random
function
g.
g
.
of a
deterministic variable such as
tt. Hence
Egt=∫gt
fT
tdt
g
t
t
g
t
fT
t
(2)
Because
tt is linearly increasing,
the pdf
fT
t
fT
t
is uniform over our measurement interval, say
−T
T
to
TT, and will be
12T
1
2
T
to make the pdf valid (integral = 1). Hence
Egt=∫−TTgt12Tdt=12T∫−TTgtdt
g
t
t
T
T
g
t
1
2
T
1
2
T
t
T
T
g
t
(3)
If we wish to measure over all time, then we take the limit as
T→∞
T
.
This leads to the following results for Ergodic WSS random
processes:
-
Mean Ergodic:
EXt=∫−∞∞x
f
X
(
t
)
xdx=limit T→
∞
12T∫−TTXtdt
X
t
x
x
f
X
(
t
)
x
T
1
2
T
t
T
T
X
t
(4) -
Correlation Ergodic:
r
X
X
τ=EXtXt+τ=∫−∞∞∫−∞∞
x1
x2
f
X
(
t
)
,
X
(
t
+
τ
)
x1
x2
d
x1
d
x2
=limit T→
∞
12T∫−TTXtXt+τdt
r
X
X
τ
X
t
X
t
τ
x2
x1
x1
x2
f
X
(
t
)
,
X
(
t
+
τ
)
x1
x2
T
1
2
T
t
T
T
X
t
X
t
τ
(5)
and it is often assumed when estimating
moments (or correlations) for such processes.
In almost all practical situations, processes are stationary
only over some limited time interval (say
T1
T1
to
T2
T2
) rather than over all time. In that case we
deliberately keep the limits of the integral finite and adjust
f
X
(
t
)
f
X
(
t
)
accordingly. For example the autocorrelation function
is then measured using
r
X
X
τ=1
T2
−
T1
∫
T1
T2
XtXt+τdt
r
X
X
τ
1
T2
T1
t
T1
T2
X
t
X
t
τ
(6)
This avoids including samples of
XX
which have incorrect statistics, but it can suffer from errors
due to limited sample size.
