Digital Signal Processing Reference
In-Depth Information
Fig. 6.12
Stationarity and
ergodicity
Since the particular realizations of ergodic process are generated by equal
independent sources, we can expect that, with enough time, all time functions
(realizations) will pass the same number of times the same amplitude values, but
with a different order of appearance. This is shown in Fig.
6.13a
, where time
realizations are divided into small intervals [LEE 60].
If all amplitudes from Fig.
6.13a
are ordered using a criterion of decreasing
amplitude and not as they appear, we will obtain the same realizations as shown in
Fig.
6.13b
.
In another words, if we imagine that each realization is divided into small
intervals of infinitesimal duration
Dt
, then we can expect a very large set of those
intervals with the amplitudes
...
; x
i
ð
0
Þ; x
i
ðDtÞ; x
i
ð
2
DtÞ;
...
(6.139)
will be equal for all realizations, as shown in Fig.
6.13b
.
On the other hand, due to stationarity, the amplitudes of the intervals in
kDt
...
; x
i
1
ðkDtÞ; x
i
ðkDtÞ; x
iþ
1
ðkDtÞ;
...
(6.140)
will be equal to the corresponding set for any other time interval
jDt
,
...
; x
i
1
ðjDtÞ; x
i
ðjDtÞ; x
iþ
1
ðjDtÞ;
...
(6.141)
Note that set (
6.140
) contains one element of the set (
6.139
), taken from all
realizations, as shown in Fig.
6.13a
. In this way, we can conclude that there is
equivalence between set (
6.139
) taken from any realization (Fig.
6.13b
) and the set
(
6.140
) obtained in arbitrary time instant
kDt
(Fig.
6.13c
).
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