CUT-OFF FREQUENCY ESTIMATION OF KINEMATICAL DATA WITH DISCRETE FOURIER TRANSFORMATION AND REGRESSION ANALYSIS
AbstractINTRODUCTION This study addresses the problem of the differentiation process of human motion data. The central difficulty inherent'in any smoothing process is the correct cut-off frequency choice. A smoothing method, that utilizes the concepts of discrete Fourier transformation (DFT) and regression, has been assembled to analyze human motion and detect the optimum cut-off frequency. METHODS The transformation from the time domain to the frequency domain constructs orthogonal variables that correspond to the Fourier frequencies (Cappozzo, Leo, and Pedotti, 1975; Jackson, 1979). The unique variability of its variable frequency is examined with regression and the smoothed curve is reconstructed from the statistically significant frequencies. The maximum frequency accepted in the regression equation is considered the cut-off frequency. The algorithm was evaluated with four applications: a) the raw data from Pezzack, N o m and Winter (1977). b) data simulating a vertical jump, c) data simulating a counter motion, like a tennis drive, and d) data simulating a tracking movement. Pseudo-random error was superimposed to the synthetic data. The results were compared with the results from three other smoothing methods: a) b-splines, b) generalized cross validation (GCV), and c) power spectrum graph. The mean square error (MSE) and the mean signal to noise ratio (SNR) of the smoothed data were used as indexes for comparison. RESULTS AND CONCLUSIONS The DFT regression method showed good results with the Pezzack et al.'s (1977) data as its derivatives were very close with the real derivatives. B-splines and GCV had their best results with the non-periodic data. The DFT regression and the power spectrum graph methods failed to give a reasonable approximation for the jumping and counter movements. The least MSE and the highest SNR with the jumping movement data was those of the GCV method and for the counter movement data were those of the b-splines method. The DlT regression had the least MSE and highest SNR for the tracking motion. The DFT regression had excellent results with the periodic motions but not for the non-periodic ones where an oscillating Gibs phenomenon were observed.. The results suggests that DlT regression can be use with periodic motions, but further research should point to short term Fourier transform and wavelet analysis for non-periodic motions. REFERENCES Cappozzo, A., Leo, T,, & Pedotti, A. (1975). A generalized computing method for the analysis of human locomotion. Journal of biomechanics, 8,307-320. Jackson, K. M. (1979). Fitting of mathematical functions to biomechanical data. IEEE Transactions on Biomedical Engineering, BME-26(2), 122- 124. Pezzack, J. C., Norman, R. W., & Winters, D. A. (1977). An assessment of derivative determining techniques used for motion analysis. Journal of Biomechanics, 10, 377-382.
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