Digital Signal Processing Reference
In-Depth Information
Fig. 6.7
Slow and fast processes
X
and
Y
The process
X
(
t
) is assumed to be a WS stationary and, as a such, has the first
order density function of:
f
X
ðx; tÞ¼f
X
ðxÞ:
(6.41)
The process
Y
(
t
) is obtained from the process
X
(
t
) by accelerating it. Therefore,
both processes will have the same amplitude characteristics, i.e., both will have the
same density function:
f
X
ðxÞ¼f
Y
ðyÞ:
(6.42)
However, even though both processes have the same statistical characteristic,
that of a first order PDF, they are obviously different. The process
X
(
t
) changes in
amplitude slowly over time, while the process
Y
(
t
) changes in amplitude very
quickly. As a result, amplitudes of the process
X
(
t
) will be more similar to each
other in the time instants
t
1
and
t
2
, than will the amplitudes of the process
Y
(
t
) in the
same time instants. In order to measure a similarity of amplitudes in time instants
t
1
and
t
2
, thus obtaining a measure of a change of the process in time, we need an
autocorrelation function.
The autocorrelation functions are given in the following equations and are
shown in Fig.
6.8
.
1
R
XX
ðtÞ¼XðtÞXðt þ tÞ¼
x
1
x
2
f
X
1
X
2
ðx
1
; x
2
;
tÞ
d
x
1
d
x
2
;
(6.43)
1
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