Abstract
Smoothing and differentiation of noisy signals are common problems whenever it is difficult or impossible to obtain derivatives by direct measurement. In biomechanics body displacements are frequently assessed and these measurements are affected by noise. To avoid high-frequency noise magnification, data filtering before differentiation is needed. In the approach reported here an autoregressive model is fitted to the signal. This allows the evaluation of the filter bandwidth and the extrapolation of the data. The extrapolation also reduces edge effects. Low-pass filtering is performed in the frequency domain by a linear phase FIR filter and differentiation is performed in the frequency domain. The reported results illustrate the accuracy of the algorithm and its speed (mainly due to the use of the FFT algorithm). Automatic bandwidth selection also guarantees the homogeneity of the results.
Similar content being viewed by others
References
Akaike, M. (1969a) Fitting autoregressive models for prediction.Ann. Inst. Statist. Math.,21, 243–247.
Akaike, M. (1969b) Power spectrum estimation through autoregression model fitting.,21, 407–419.
Akaike, M. (1970a) On a semi-automatic spectrum estimation procedure. Proc. 3rd Haway Int. Conf. System Sci., Part 2, 974–977.
Akaike, M. (1970b) Statistical predictor identification.Ann. Inst. Statist. Math.,22, 203–217.
Akaike, M. (1971) Autoregressive model fitting for control.,23, 163–180.
Akaike, M. (1972) Use of an information theoretic quantity for statistical model identification. Proc. 5th Haway Int. Conf. System Sci., 249–250.
Akaike, M. (1974) A new look at the statistical model identification.IEEE Trans.,AC-19, 716–723.
Anderssen, R. S. andBloomfield, P. (1974) Numerical differentiation procedures for non-exact data.Numer. Math.,22, 157–182.
Blackman, R. B. andTukey, J. W. (1958)The measurement of power spectra from the point of view of communications engineering, Dover Publications Inc., New York.
Burg, S. P. (1967) Maximum entropy spectral analysis. Proc. 37th Meeting Society of Exploration Geophysicists, Oklahoma City, Oklahoma.
Cappozzo, A., Leo, T. andPedotti, A. (1975) A general computing method for the analysis of human locomotion.J. Biomech.,8, 307–344.
Cooley, J. W. andTukey, J. W. (1965) An algorithm for machine calculation of complex Fourier series.Math. Comput.,19, 297–301.
Craven, P. andWahba, G. (1979) Smoothing noisy data with spline functions.Numer. Math.,31, 377–403.
Cullum, J. (1971) Numerical differentiation and regularization.SIAM J. on Numer Computat.,8, 254–265.
Cullum, J. (1979) The effective choice of the smoothing norm in regularization.Math. of Computat.,33, 149–170.
D'Amico, M. andFerrigno, G. (in press) Estimation of velocities and accelerations from noisy kinematic data. Proc. ISBS 6th Int. Symp. of Biomech. in Sports, Bozeman, Montana, (in press).
Durbin, J. (1960) The fitting of time series models.Rev. Inst. Int. de Stat.,28, 233–244.
Ferrigno, G. andPedotti, A. (1985) ELITE: a digital dedicated hardware system for movement analysis via real-time TV signal processing.IEEE Trans.,BME-32, 943–950.
Gasser, T., Kohler, W., Jannen-Steinmetz, C. andSroka, L. (1986) The analysis of noisy signals by nonparametric smoothing and differentiation.,BME-32, 1129–1133.
Hadamard, J. (1902) Sur les problemes aux derivees partielles et leurs signification physiques.Bull. University of Princeton, 13.
Harris, F. J. (1978) On the use of windows for harmonic analysis with the discrete Fourier transform.Proc. IEEE,66, 51–83.
Jetto, L. (1985) Small-computer procedure for optimal filtering of haemodynamic data.Med. & Biol. Eng. & Comput.,23, 203–208.
Kay, S. N. andMarple, S. L. (1981) Spectrum analysis—a modern perspective.Proc. IEEE,69, 1380–1419.
Lanshammar, H. (1981) Precision limits on derivatives obtained from measurement data. InBiomechanics VII-A.Morecki, A., Fidelus, K., Kedzior, K. andWit, A. (Eds.), Polish Publishers' House and University Park Press, Baltimore.
Lanshammar, H. (1982a) On practical evaluation of differentiation techniques for human gait analysis.J. Biomech.,15, 99–105.
Lanshammar, H. (1982b) On precision limits for derivatives numerically calculated from noisy data.,15, 459–470.
Levinson, N. (1947) The Wiener (root mean square) error criterion in filter design and prediction.J. Math. Phys.,25, 261–278.
Makhoul, J. (1975) Linear prediction: a tutorial review.Proc. IEEE,63, 561–580.
Marple, S. L. (1987)Digital spectral analysis with applications. Prentice-Hall Inc., Englewood Cliffs, New Jersey.
Murphy, M. C. andMann, R. W. (1987) A comparison of smoothing and digital filtering/differentiation of kinematic data. 9th Ann. Conf. of IEEE Eng. in Med. & Biol. Soc., Boston, Massachusetts, 13th–16th Nov., 852–853.
Nuttall, A. H. (1976) Spectral analysis of a univariate process with bad data points, via maximum entropy, and linear predictive techniques. Naval Underwater System Center, Tech. Rep. 5303, New London, Connecticut.
Oppenheim, A. V. andSchafer, R. W. (1975)Digital signal processing. Prentice-Hall Inc., Englewood Cliffs, New Jersey.
Parzen, E. (1974) Some recent advances in time series modeling.IEEE Trans.,AC-19, 723–730.
Pezzack, J. C., Norman, R. W. andWinter, D. A. (1977) An assessment of derivative determining techniques used for motion analysis.J. Biomech.,10, 377–382.
Rabiner, L. R. andGold, B. (1975)Theory and application of digital signal processing. Prentice-Hall Inc., Englewood Cliffs, New Jersey.
Reinsch, C. H. (1967) Smoothing by spline functions.Numer. Math.,10, 177–183.
Reinsch, C. H. (1971) Smoothing by spline functions II.,16, 451–454.
Schuster, A. (1898) On the investigation of hidden periodicities with application to a supposed 26 day period of meteorological phenomena.Terrestrial Magnetism,3, 13–41.
Schuster, A. (1899) The periodogram of magnetic declination as obtained from the records of the Greenwich Observatory during the years 1871–1895.Trans. Cambridge Philosophical Soc.,18, 107–135.
Slepian D. (1976) On bandwidth.Proc. IEEE,64, 292–300.
Soudan, K. andDierckx, P. (1979) Calculation of derivatives and Fourier coefficients of human motion data, while using spline functions.J. Biomech.,12, 21–26.
Ulrych, T. J. andClayton, R. W. (1976) Time series modeling and maximum entropy.Phys. Earth Planetary Interior,12, 188–200.
Utreras, F. (1980) A package for smoothing noisy data with splines. Technical Report MA-80-B-209, Department of Mathematics, University of Chile, Santiago.
Vaughan, C. L. (1982) Smoothing and differentiation of displacement-time data: an application of splines and digital filtering.Int. J. Biomed. Comput.,2, 349–362.
Wiener, N. (1949)On the interpolation, smoothing and extrapolation of stationary time series. Wiley, New York.
Winter, D. A. (1987)The biomechanics and motor control of human gait. University of Waterloo Press, Dana Porter Library, Waterloo, Ontario, Canada.
Woltring, H. J. (1985) On optimal smoothing and derivative estimation from noisy displacement data in biomechanics. InHuman movement science, 4 (3), Elsevier Science Publishers B.V., North-Holland, 229–245.
Woltring, H. J. (1986) A FORTRAN package for generalized, cross-validatory spline smoothing and differentiation.Adv. Eng. Software,8, 104–113.
Wood, G. A. (1982) Data smoothing and differentiation procedures in biomechanics.Exercise & Sport Sci. Rev.,10, 308–362.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
D'Amico, M., Ferrigno, G. Technique for the evaluation of derivatives from noisy biomechanical displacement data using a model-based bandwidth-selection procedure. Med. Biol. Eng. Comput. 28, 407–415 (1990). https://doi.org/10.1007/BF02441963
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02441963