Biomedical Engineering Reference
In-Depth Information
Both auto- and cross-correlations are calculated for various phase shifts
that a priori must be specified by the user, and this will have an impact on
the number of data points used in the formulae. If, for example,
x (n)
and
y(n)
are 1000 data points, and it is desired that
τ
100, then we can only
get 800 cross products and, therefore,
N
will be set to 800. Sometimes the
signals of interest are periodic (such as gait); then, we can wrap the signal on
itself and calculate the correlations using all the data points. Such an analysis
is known as a circular correlation.
=±
2.1.7 Application of Autocorrelations
As indicated in property #3 an autocorrelation indicates the frequency content
of
x (t )
. Figure 2.6 presents an EMG record and its autocorrelation. The upper
trace (a) is the raw EMG signal, which does not show any visible evidence of
hum, but the autocorrelation seen in the lower trace (b) is an even function
as predicted by property #2 and shows the presence of 60 Hz hum. Note
from Figure 2.3 that
R
xx
(τ )
for a sinusoidal wave has its first zero crossing
at
1
/
4
of a cycle of the sinusoidal frequency; thus, we can use that first zero
crossing of
R
xx
(τ )
to estimate the average frequency in the EMG. The first
zero crossing for this
R
xx
(τ )
occurred at about 3ms, representing an average
period of 12 ms, or an average frequency of about 83 Hz.
2.1.8 Applications of Cross-Correlations
2.1.8.1 Quantification of Cross-Talk in Surface Electromyography.
Cross-correlations quantify what is in common in the profiles of
x (t )
and
y(t )
but also any common signal present in both
x (t )
and
y(t )
. This may be
true in the recordings from surface electrodes that are close enough to be
subject to cross-talk. Because a knowledge of surface recording techniques
and the biophysical basis of the EMG signal is necessary to understand
cross-talk, the student is referred to Section 10.2.5 in Chapter 10 for a
detailed description of how
R
xy
(τ )
has been used to quantify cross-talk.
2.1.8.2 Measurement of Delay between Physiological Signals.
Experi-
mental research conducted to find the phase advance of one EMG signal
ahead of another has been used to advantage to find balance strategies in
walking (Prince et al., 1994). Balance of the head and trunk during gait
against large inertial forces is achieved by the paraspinal muscles. It was noted
that the head anterior/posterior (A/P) accelerations were severely attenuated
(0
.
48 m
/
s
2
) compared with hip accelerations (1
.
91 m
/
s
2
), and it was impor-
tant to determine how the activity of the paraspinal muscles contributed to
this reduced head acceleration. The EMG profiles at nine vertebral levels
from C7 down to L4 were analyzed to find the time delays between those
balance muscles. Figure 2.7 presents the ensemble average (see Section 2.3
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