Introduction to Sports Biomechanics: Analysing Human Movement ...

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APPENDIX 4.1 DATA SMOOTHING AND FILTERING Digital low-pass filters Digital low-pass filters are widely used to remove, or filter, high-frequency noise from digital data. Butterworth filters (of order 2n where n is a positive integer) are often used in sports biomechanics, because they have a flat passband, the band of frequencies that is not affected by the filter (Figure 4.16). However, they have relatively shallow cut-offs. This can be improved by using higher-order filters, but round-off errors in computer calculations can then become a problem. They also introduce a phase shift, which must be removed by a second, reverse filtering, which increases the order of the filter and further reduces the cut-off frequency. Butterworth filters are recursive; that is, they use filtered values of previous data points as well as noisy data values to obtain filtered data values. This makes for faster computation but introduces problems at the ends of data sequences, at which filtered values must be estimated. This can mean that extra frames must be digitised at each end of the sequence and included in the data processing; these extra frames then have to be discarded after filtering. This can involve unwelcome extra work for the movement analyst; other solutions include various ways of padding the ends of the data sets. The main decision for the user, as with Fourier series truncation, is the choice of cut-off frequency (discussed on pages 133–7). The filtered data are not obtained in analytic form, so a separate numerical differentiation process must be used. Although Butterworth filtering appears to be very different from spline fitting (see below), the two are, in fact, closely linked. Figure 4.16 Low-pass filter frequency characteristics. QUANTITATIVE ANALYSIS OF MOVEMENT 153
INTRODUCTION TO SPORTS BIOMECHANICS Fourier series truncation The first step in Fourier series truncation is the transformation of the noisy data into the frequency domain by means of a Fourier transformation. This, in essence, replaces the familiar representation of displacement as a function of time (the time domain) as in Figure 4.17(a) by a series of sinusoidal waves of different frequencies. This ‘frequency domain’ representation of the data is then presented as amplitudes of the sinusoidal components at each frequency – the harmonic frequencies, as in Figure 4.17(b), or as a continuous curve. Figure 4.18(a) shows the frequency domain representation of the simplified data of Figure 4.17(a). The data are then filtered to remove high-frequency noise. This is done by reconstituting the data up to the chosen cut-off frequency and truncating the number of terms in the series from which it is made up. For the Figure 4.17 Displacement data represented in: (a) the time domain; (b) the frequency domain. 154
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Introduction to Sports Biomechanics
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First edition published 1997 This e
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Contents List of figures x List of
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Study tasks 216 Glossary of importa
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FIGURES 1.26 Underarm throw - femal
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FIGURES 5.9 Typical path of a swimm
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Tables 2.1 Examples of slowest sati
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Preface Why have I changed the cove
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Introduction MISSING TEXT The first
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INTRODUCTION principal movements in
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INTRODUCTION TO SPORTS BIOMECHANICS
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Figure 2.2 ‘Principles’ approac
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Figure 2.13 Identifying critical fe
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Figure 6.8 Main skeletal muscles: (
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INDEX 282 balls air flow past 173 b
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INDEX 284 energy 219 kinetic 219 mi
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INDEX 286 isokinetic contraction 24
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INDEX 288 muscle length-tension rel
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INDEX 290 three-dimensional 146-7,
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INDEX 292 validity 220 vantage poin
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