Biomedical Engineering Reference
In-Depth Information
R
xx
(
τ
)
1
τ
−
1
Figure 2.3
Autocorrelation of a sine or cosine signal. Note that this is an even function
and the repetitive nature of
R
xx
(τ )
at the frequency of the sine and cosine wave.
R
xx
(
τ
)
1
τ
−
1
Figure 2.4
0, indicat-
ing that each data point has 0 correlation with all other data points ahead and behind
it in time.
Autocorrelation of white noise. Note that
R
xx
(τ )
=
0atall
τ
=
2.1.4 Three Properties of the Cross-Correlation Function
Property #1.
The cross-correlation of
x (t )
and
y(t )
is not an even function.
Because the two signals are completely different, the phase shifting in the
+
ve
direction will not result in the same “cross products” as shifting in the
- ve direction. Thus,
R
xy
(τ )
=
R
xy
(
−
τ)
.
Property #2.
The maximum value of
R
xy
(τ )
is not necessarily at
τ
=
0.
The maximum
ve
or negative peak of
R
xy
(τ )
will occur when the two
signals are most in phase or most out of phase. For example, if
x (t )
is a
sine wave and
y(t )
is a cosine wave of the same frequency at
τ
+
0, the
signals are 90
◦
out of phase with each other, and the cross products over one
=
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