KINEMATICS - Biomechanics and Motor Control of Human Movement - page 73

us that the noise content, mainly introduced by the human digitizing process,
is the same for all markers. This regression line has an intercept of 1.8 mm,
which indicates that the rms of the noise is 1.8 mm. In this case, the cine
camera was 5 m from the subject and the image was 2 m high by 3 m wide.
Thus, the rms noise is less than one part in 1000.
Also, we see distinct differences in the frequency content of different
markers. The residual shows the more rapidly moving markers on the heel
and ball to have power up to about 6 Hz, while the vertical displacements
of the rib and hip markers were limited to about 3 Hz. Thus, through this
selection technique, we could have different cutoff frequencies specified for
each marker displacement.

3.4.4.4 Optimal Cutoff Frequency. The residual analysis technique
described in the previous section suggested the choice of a frequency
where the signal distortion was equal to the residual noise. This optimal
applies to displacement data only. However, this may not be the optimum
frequency for all amplitudes of signal and noise, all sampling frequencies,
and all levels of differentiation: velocities versus accelerations. Giakas and
Baltzopoulos (1997) showed that the optimal cutoff frequencies depended
on noise level and whether displacements, velocities, or accelerations were
being considered. Unfortunately, their reference displacement signal was
reconstituted from a harmonic analysis, and in Section 2.2.4.5 this technique
was shown to have major problems because of lack of stationarity of each
harmonics amplitude and phase. Yu et al. (1999) carried out a detailed anal-
ysis to estimate the optimum cutoff frequency for higher-order derivatives,
especially accelerations. They found the optimum cutoff frequencies to be
somewhat higher than those estimated for the displacement residual analysis.
This is not suprising when we consider that the acceleration increases as
the square of the frequency (Section 3.4.3); thus, the higher-frequency
noise in the acceleration waveform will increase far more rapidly than the
signal itself. Also, when the sampling frequency, f s , increases, the sampling
period, t 1 /f s , decreases, and thus the noise as calculated by finite

differences increases [see Equations (3.17) and (3.18c)]. Thus, Yu et al.
(1999) estimated that the optimum cutoff frequency was not only a function
of the residual between the filtered and unfiltered data but also a function of
f s . Their estimated optimal cutoff frequency, f c. 2 ,was:

=

0 . 000022 f s f c ,2 = 0 . 06 f s − + 5 . 95 / ε (3.10)

where f s is the sampling frequency and ε is the relative mean residual between
X i and X i [terms defined in Equation (3.9)]. These authors present example
acceleration curves (see Figure 3.4 in Yu et al.,1999) that shows a reasonable
match between the accelerometer data and the filtered film data, except that
the lag of the filtered data suggests that a second-order low-pass filter was
used rather than the desired fourth-order zero-lag filter.