SIGNAL PROCESSING - Biomechanics and Motor Control of Human Movement - page 39

RAW Y TOE MOTION, 9TH HARMONIC AND TOTAL

0.100

0.075

0.050

M
E
T
E
R
S

0.025

0.000

0.025

0.050

0.0 0.2 0.4 0.6

SECONDS

0.8 1.0 1.2

Figure 2.18 Fourier reconstruction of the vertical trajectory of a toe marker during
one walking stride. The actual trajectory is shown by the open square, the reconstruction
from the first nine harmonics is plotted with open triangles, and the contribution of the
ninth harmonic is plotted with open circles. The difference between the actual and the
reconstructed waveforms is the result of the lack of stationarity in the original signal.

errors in subsequent biomechanical analyses. This is because each harmonic
amplitude and phase values are average values, as we cautioned about in
Section 2.22, and this is especially true for a foot marker during gait, which
has high frequencies during swing and low frequencies during stance.

2.2.4.6 Fourier Analysis of White Noise. White noise was introduced in
Figure 2.4, where an autocorrelation showed that each point has zero corre-
lation with any points ahead or behind it in time. In a computer, white noise
can be simulated by a random number generator. The other characteristic
of white noise is in the frequency domain where the frequency spectrum is
equal across the whole range of the signal, and this is similar to some of the
noise apparent in kinematic data — cine, television, and optoelectric systems.
To demonstrate this frequency spectum characteristic we present an analysis
of the FFT of white noise. Figure 2.19 (a) is a white noise signal simulated
on Excel from a random number generator with amplitude 1 sampled at a

rate of 2048 samples/sec. Thus, the highest frequency present in this signal
is the Nyquist frequency of 1024 Hz, so our FFT will cover frequencies from
0 Hz to 1024 Hz. An FFT of the signal in Figure 2.19 (a) is presented in
Figure 2.19 (b). Note that the spectrum over that range appears “noisy,” but

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