Filters are mathemematical operations running on a timeserie as a whole. They can be used as a preprocessing step, applied to the raw measurment data, at the end of the processing as a postprocessing step or „temporarily“ for visualisation in a plot sheet.

There is a default set of filters implemented. The available filters are explained in the following chapters.

Every filter can have attributes. Some attributes are used from most of the filters. For some of these attributes default values are defined:

stepWidth        =1/120
samplingFrequency="120"
windowWidth      ="12"
noisyData        ="true"
cutOffFrequency  ="5"
zeroPhase        ="true"
sectionMinLength ="10" 

The filter types and characteristics can be defined in the XML markerset/labelset files which define the which data is loaded for processing or which parts of the processed data should be written in an output file.

Further filters can be added by an plugin architecture.

The filter-elements defining the filter charatericstics can be used at different positions in the hierarchie of a labelset:

<?xml version="1.0" encoding="UTF-8"?>
<LabelSet>
 <Filter name="butter2" implName="default" cutOffFrequency="20"
         windowWidth="" stepWidth="" noisyData="true" samplingFrequency="1211"
         zeroPhase="true" sectionMinLength="3"/>  
 <Labels fileSuffix="c3d" mimeType="application/c3d-vicon">
    <Filter name="butter2" implName="default" cutOffFrequency="20"
            windowWidth="" stepWidth="" noisyData="true" samplingFrequency="1211"
            zeroPhase="true" sectionMinLength="3"/>  
    <Label name="ABC" comment="" flag="" firstDerivative=""
           secondDerivative="" thirdDerivative="" differentiations=""
           optional="" unchangeable="" mathType="">
       <Filter name="butter2" implName="default" cutOffFrequency="20"
               windowWidth="" stepWidth="" noisyData="true" samplingFrequency="1211"
               zeroPhase="true" sectionMinLength="3"/>  
    </Label>
 </Labels>
</LabelSet>

Defined as the first child of the labelset-element it is applied to all timeseries, as a child of an label-element it is only applied to the timeserie the label corresponds to.

The filter-elements defining the filter charatericstics, can be used also as children of the label-elements in a plot sheet definition.

<?xml version="1.0" encoding="iso-8859-1" ?>
<Sheet name="Bilateral shoulder abdcution">
  <Pages>
    <Page name="Mir 100"  rows="4" columns="3">
    ...
       <Diagram name="Height" 
           column="1"
           row="1"
           xunit="%"
           yunit="mm"
           diagramDescriptionDown="down"
           diagramDescriptionUp="up"
           plane="" 
           diagramType="xt" 
           diagramTitle="Height">
 
           <Label name="Height"
              label="Height">
              <Filter name="dft" implName="default"/>
           </Label>
       </Diagram>  

• Till now, only in xt()- and xy()-plots timeseries are filterd.

A differentiation method, that simultaneously smoothes the data by application of a 4-th order interpolation polynom (5 points). Minimum trial length is four.

• Hildebrand, F.B. (1974). Introduction to Numerical Analysis, 2nd Edition, pp.111
• Kreyszig, Erwin (1983). Advanced Engineering Mathematics, 5th Edition, pp.793
• Yakowitz Sydney and Szidarovsky, Ferenc (1989). An Introduction to Numerical Computations, 2nd Edition, pp.18

A differentiation method that simultaneously smoothes the data. This method is based on a sliding window technique. The data is approximated within the window with a quadratic polynomial. The coefficients of the quadratic polynomial were then used for calculating the derivative at the window's center (Rand and Stelmach, 2005).

• The original C++ implementation is from Dr. Yuri Shimansky.
• Thanks for Laetitia Fradet for the idea to use this filter and to have a look in a Matlab implementation of this filter.

Calculate the mean of a chunk of a timeserie.

The timeserie length must be a multiple of the windows size. The phase of the result is shifted by (window size) /2 frames to the front. The result timeserie length = (original timeserie length)/(window size). This reduction of the timeserie length makes this filter useless in most of the cases.

Simple Low pass digital filter.

result[i] = (data[i-2] + 3d*data[i-1] + 4d*data[i] + 3d*data[i+1] + data[i+2])/12d

This filter determines the derivative based on a 4-th order interpolation polynom.

Hildebrand, F.B. (1974). Introduction to Numerical Analysis, 2nd Edition, pp.111 Kreyszig, Erwin (1983). Advanced Engineering Mathematics, 5th Edition, pp.793 Yakowitz, Sydney and Szidarovsky, Ferenc (1989). An Introduction to Numerical Computations, 2nd Edition, pp.18

Zero-phase forward and reverse butterworth filter of 2. order.

Computes 1D Discrete Fourier Transform (DFT). Each DFT output value S[k] is represents the amplitude at frequency 2πkN. The multiplication by the exponential function effectively shifts s[n] down in frequency by 2πkN. The sum over N time samples can then be thought of as applying a decimating lowpass filter. So, effectively, the N outputs of the DFT represent the results of applying a bank of equally-spaced filters across s[n]'s frequency band, thus measuring „how much“ energy is present in various frequency bands in the input signal.

Assuming that a large-enough sample rate was used during the conversion to discrete-time, the result of the DFT will look a lot like the Fourier transform of the original signal s(t).The DFT is only applicable to finite-length signals, so in order to apply it, one must truncate the discrete-time signal s[n] to some finite length N, resulting in a potentially-shorter signal sN[n].

The result array contains the power specturm based on the dft() without the leading DC component for k=0:

spectrum[k]=Math.sqrt(Math.pow(data[k*2],2.0)+Math.pow(data[k*2+1],2.0));

The center frequencies for the bins are available, if the needed attributes are set:

• The result array with the power spectrum has half size as the input array. Thatś why this filter can not be used in the input- or output-labelsets. It is assumed, that this filter is usefull only in the plotsheet functionality. If there are new applications which needs this filter to be defined in input- or output-labelset the needed resampling code can be added easy.

Natural B-spline data smoothing, using the Generalized Cross-Validation and Mean-Squared Prediction Error Criteria of Craven & Wahba (1979).

The following parameters are determined and can be printed in the log-file:

CutOffFrequency, if it is autodetected GeneralizedCrossValidationValue (WK(1) ) MeanSquaredResidual (WK(2) ) ResidualSumSquaresDimension (WK(3)) = Estimate of the number of degrees of freedom of the residual sum of squares per dataset, with 0.lt.WK(3).lt.N-M. SmoothingParameter (WK(4)) = Smoothing parameter p, multiplicative with the splines' derivative constraint. TrueMeanSquareError (WK(5) ) = Estimate of the true mean squared error (different formula for |MD| = 3). GaussMarkovErrorVariance (WK(6))

MEMO: GCVSPL software package (C) COPYRIGHT 1985, 1986: H.J. Woltring

For large datasets (N » 0) and negligible boundary artefacts, the behaviour of a natural spline approximates that of a periodic spline. For the latter case, the frequency characteristic in the equidistantly sampled, uniformly weighted case is that of a double, phase-symmetric Butterworth filter, with transfer function H(w) = [1 + (w/wo)^2M]^-1, where w is the frequency, wo = (p*T)^(-0.5/M) the filter's cut-off frequency, p the smoothing parameter, T the sampling interval, and 2M the order of the spline. If T is expressed in seconds, the frequen- cies are expressed in radians/second.

It has been found empirically, that the effective number of estimated spline parameters Np is related to the Butterworth cut-off frequency wo as Np ~ M/2 + KM * wo * N * T, where Np ranges between M and N, and where KM is the integral over x from 0 to infinity of (1 + x^2M)^- divided by PI. For large M, KM approaches 1/PI from above; values for small M are: K1 = 1/2, K2 = 1/V8, K3 = 1/3. This relation has also been found to apply for uniformly weighted data which are sampled slightly anequidistantly, with T taken as the average sampling inter-val. For large Np, the relation with wo * N * T becomes nonlinear.