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procedure, for the smoothing process the precision was computed before each measurement after
the corresponding calibration took place.
B.5.
Denoise of COP Signal
B.5.1.
COP is Distorted with White Noise
The noise in the COP signal is the propagation noise arise from the combination of the components
used to compute it with function f . Generally, it is modeled as a wide-band additive, stationary,
zero-mean, and uncorrelated noise that contaminates the low-pass COP signal with noise variance
σ2 . However, even if the noise of the recorded GRF signals can be modeled as an additive zeromean “white noise”, the nonlinear transformation in COP computation destroys these properties to
some extend (Woltring, 1995). The noise in the COP signal becomes non-stationary (i.e., unequal
noise variance), except for the case where the Fz is constant (Fz = c). Therefore, noise stationarity
may be true for stabilometric studies and digital low-pass filter or smoothing techniques can be
used to remove high frequencies presented in the COP signal (Karlsson and Lanshammar, 1997;
Woltring, 1986). The optimal cut-off frequency for the low-pass digital filter can be found by
residual analysis (Winter and Patla, 1997) or by using the generalized, crossed-validation natural
splines smoothing algorithm (GCVSPL) (Woltring, 1986). Natural splines of m th order behave like
an m th order double Butterworth filter, where optimal cut-off frequency is the lowest frequency
for which the residual noise is white (Woltring et al., 1987).
Moreover, with sufficient oversampling is possible to retain significant signal components avoiding
aliasing errors, while reducing noise level (Furnée, 1989a). The noise variance presented in a
signal, or in its derivatives, after optimal smoothing depends on the band-limit of the signal and
is proportional to the sampling rate and the variance of the inherent band-limited “white noise”
presented in the raw data measurement (Lanshammar, 1982b), and is expressed as
σ2k = σ2 τ
ω2k+1
b
π(2k + 1)
(B.8)
where
σ2k is the noise variance in the estimate k th order derivative
σ2 is the noise variance in the raw measured data (additive “white noise”)
τ is the sampling interval (τ =
1
fs
=
1
2πω0
with ω0 ≥ 2ω b )
ω b is the band-limit of the signal
k is the order of the derivative
The term σ2 τ is known as spatiotemporal resolution criterion (Q ST ) and together with Shannon
sampling theorem can be regarded as sufficient criteria in order to choose the sampling frequency
ω0 (Furnée, 1989a,b; Woltring, 1984, 1995). When a quantizing data acquisition system with
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