Biomedical Engineering Reference
In-Depth Information
If our data contained no signal, just random noise, the residual plot would
be a straight line decreasing from an intercept at 0 Hz to an intercept on the
abscissa at the Nyquist frequency (0
.
5
f
s
). The line
de
represents our best
estimate of that noise residual. The intercept
a
on the ordinate (at 0 Hz) is
nothing more than the rms value of the noise, because
X
i
for a 0-Hz filter is
nothing more than the mean of the noise over the
N
samples. When the data
consist of true signal plus noise, the residual will be seen to rise above the
straight (dashed) line as the cutoff frequency is reduced. This rise above the
dashed line represents the signal distortion that is taking place as the cutoff
is reduced more and more.
The final decision is where
f
c
should be chosen. The compromise is always
a balance between the signal distortion and the amount of noise allowed
through. If we decide that both should be equal, then we simply project a
line horizontally from
a
to intersect the residual line at
b
. The frequency
chosen is
f
c
, and at this frequency the signal distortion is represented by
bc
. This is also an estimate of the noise that is passed through the filter.
Figure 3.21 is a plot of the residual of four markers from one stride of gait
data, and both vertical and horizontal coordinates were analyzed (Wells and
Winter, 1980). As can be seen, the straight regression line that represents the
noise is essentially the same for both coordinates on all markers. This tells
25
Motion
horizontal vertical
Marker
20
Rib
Hip
Heel
Ball
15
10
Best fit linear regression for harmonics 11 to 20:
Residual (mm)
1.8
0.05n, r
0.79
=
−
=
5
0
0
2
4
6
8
1
harmonic n
12
14
16
18
20
0
2
4
6
8
10
12
14
16
18
Frequency Hz
Figure 3.21
Plot of the residual of four markers from a walking trial; both vertical
and horizontal displacement data. Data were digitized from movie film with the camera
5 m from the subject.
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