Biomedical Engineering Reference
In-Depth Information
9.5.1 Properties of the Cross-Correlation Function
The cross-correlation function possesses several properties which are valuable in appli-
cations to signal and data processing:
1.
R
xy
(
τ
)
=
R
xy
(
−
τ
) The cross-correlation is not an “Even Function.”
R
xy
(
)
≤
2.
τ
R
xy
(0)
R
yx
(0)
=
x
2
∗
y
2
(the product of the RMS values of
x
and
y
)
3.
If
x
and
y
are independent variables, then
R
xy
(0)
=
R
yx
(0)
=
∗
y
(the product of the mean values of
x
and
y
)
x
4.
If
x
and
y
are both periodic with period
=
T
0
, then
R
xy
and
R
yx
are also periodic
with period
=
T
0
.
5.
The cross-power spectrum of
x
and
y
is computed as the Fourier Transform of
the cross-correlation function.
9.5.2
Applications of the Cross-Correlation Function
1.
An indication of the predictive power of
x
to
y
(or vice versa) is given by the
magnitude of the correlation function. Delays of the effects of
x
on
y
are indicated
by the value of
t
at which
R
xy
reaches a maximum.
2.
The RMS and mean values of
x
or
y
can be estimated through the use of
properties 2 and 3 given above, provided that the values of one function are
known.
3.
Periodicities in
x
and
y
can be detected in the cross-correlation function (Prop-
erty 4).
4.
The cross-power spectrum, given by the Fourier transform of the cross-
correlation, is used in the calculation of the coherence of the functions.
Search WWH ::
Custom Search