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Robert Newton
12-07-1994, 12:54 AM
I have received a further two excellent replies to my request for advice
on the optimal cutoff frequency for filtering noisy data. Unfortunately
they arrived too late to include in my original summary so I am
including them in this second posting.

Regards
Robert Newton
=================================================
I have a couple of points concerning the use of residuals for choosing
filter cut-off frequencies.

The residual method was I believe developed in order to filter noisy
data collected with a camera based system. The technique relies on the
residuals vs filter cut-off frequency plot becoming linear as the
increasing cut-off frequency of the filter attenuates the spectrally
flat white noise at the high end of the spectrum close to fs/2. If
there is not a section of white noise at the high frequency end then
the plot will not become linear. The Jackson method is a way of
detecting linearity by detecting when the second derivative becomes
zero.

Is it possible that your encoder doesnt actually have a significant
noise problem. If it is a digital encoder where the angle is converted
directly to a digital code within the encoder itself then there
shouldnt be noise from electrical inteference. Is there noise in the
data when the encoder is stationary ? There will of course be the
unavoidable (white) digitisation noise present in all A/D systems when
the encoder moves and the signal is varying.

I suspect that the residuals at high frequencies close to fs/2 are
insignificant compared to your signal power. In that case you could
choose a cut-off frequency at say the 99% power limit, ie the
frequency below which 99% of the power occurs ( Antonsson and Mann
1985). Alternatively if you can afford one you could buy a tachometer
and measure velocities directly (which would improve your angular
acceleration estimate also).

I havent seen any other references to the residual technique, but of
course there are a great many discussing optimal filtering of noisy
data by Woltring etc.

Antonsson, E.K. and Mann R.W. (1985) The Frequency Content of Gait,
J. Biomechanics, 18, 1, 39-47.

Best Wishes
Duncan Rand
================================================== =
After reading the responses to your request regarding the determination
of the optimal cutoff frequency for filtering noisy data sequences prior to
differentiation, I was surprised about the lack of information about existing
methods that prevails among biomechanists.

The problem has been dealt with in detail, and solved in principle, by Herman
Woltring and myself in the early 1980's. While Woltring used spline functions
(based on Utreras' work and computer programs),I developed an optimally
regularizing algorithm based on Fourier series approximation. In both cases,
the optimal filtering window (which determines the optimal cutoff frequency) is
computed automatically, not only for the data sequence itself, but also for the
(optimally filtered) first and second derivatives. The latter are computed
simultaneously with the optimally smoothed data sequences, thereby providing
an efficient and accurate means of rapidly processing noisy biomechanical data.
The user does not have to bother about estimating cutoff frequencies, as these
are automatically and optimally determined by the algorithm which performs
(automatically) a detailed statistical analysis of the information contained in
the
periodogram of the noise-contaminated data sequences. The only requirement
on the data is that their detrended version should represent a weakly stationary
stochastic process which requirement is fulfilled by most data sequences
occuring
biomechanical investigations. Details of the method can be found in the Journal
of Biomechanics, 14, pp. 13-18, 1981 (The use of optimally regularized Fourier
series for estimating higher-order derivatives of noisy biomechanical data),
which article also contains some of the shortcomings inherent in Jackson's
method.

Hope this hint is helpful.
H. Hatze
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