Biomedical Engineering Reference
In-Depth Information
17.3 AUTOCORRELATION
The autocorrelation function gives an average measure of the time-domain properties of
a signal waveform. It is defined as (17.1):
T
0
/
2
lim
1
T
0
R
xx
(
τ
)
=
f
(
t
)
f
(
t
+
τ
)
dt
(17.1)
T
0
→∞
−
T
0
/
2
This is the average product of the signal,
f
(
t
), and a time-shifted version of itself,
). The expression above applies to the case of a continuous signal of infinite
duration. In practice, the intervals must be finite and it is necessary to use a modified
version as given by (17.2).
f
(
t
+
τ
∞
R
xx
(
τ
)
=
f
1
(
t
)
f
1
(
t
+
τ
)
dt
(17.2)
−∞
The autocorrelation function may be applied to deterministic as well as random
signals. Each of the frequency components in the signal
f
(
t
) produces a corresponding
term in the autocorrelation function having the same period in the time-shifted variable,
τ
, as the original component has in the time variable,
t
. The amplitude is equal to half of
the squared value of the original. The phase shifts of each signal produce a single cosine
term in the autocorrelation as shown in Fig. 17.2.
f(t)
r
xy
(
t
)
t
t
t=
0
t
=
0
(a)
(b)
FIGURE 17.2
:
Autocorrelation. Trace (b) is the autocorrelation of the signal shown in (a)
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