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Derivative Estimates of Noisy Signals in a Dynamic Environme

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  • Derivative Estimates of Noisy Signals in a Dynamic Environme

    Reply Summary: Derivative Estimates of Noisy Signals in a Dynamic

    Recently, I posted a request for help in taking derivatives (first
    and second) of noisy signals when the bandwidth of the signal changes
    over time. My application is in electromyography (EMG), in which I am
    interested in the first and second derivative of the AMPLITUDE of the
    EMG. The signal to be differentiated is quite noisy (the noise is
    roughly as large as the signal). I wondered if techniques existed which
    would adapt their smoothing of the derivative to the local character of
    the signal (when the signal is slowly varying, a long smoothing window
    is desired; when the signal is rapidly varying, a short smoothing
    window is desired).

    I received several replies, the suggested techniques being
    summarized below. My kind thanks to all for lending a hand.

    Ted Clancy
    Liberty Mutual Research Center for Safety and Health
    71 Frankland Road
    Hopkinton, MA 01748
    Tel. (508) 435-9061 x206
    Fax. (508) 435-8136

    1) Automated techniques already exist to evaluate a COMPLETE signal
    recording and select an appropriate amount of smoothing for derivative
    estimates of the COMPLETE recording. [Note that a posting of freeware
    (QuickSAND) for several such techniques, which included literature
    references, was recently made to this server by Jeff Walker
    (approximately June 27, 1997). For brevity, I won't repeat that
    material here.] However, a technique which adapts to the local
    character of a signal could be formed by breaking a complete signal into
    several (perhaps overlapping) segments. Then, the existing techniques
    could independently be applied to each segment, with the results
    combined into one derivative estimate.

    2) Wavelet Transforms and Wavelet Networks. The wavelet transform
    permits the synthesis of signals in terms of dilations and translations
    of a generating function. It can be used to analyze transitional and
    brief phenomenon and/or signals with local time behavior. Hence,
    filters based on the wavelet transform might be appropriate for this
    problem. Some useful references include:

    Daubechie I. Orthonormal bases of Compactly Supported Wavelets. Comm.
    Appl. Math., 40:909--996, 1988.

    Daubechie I. Ten Lectures on Wavelets. CBMS-NSF Regional Conferences
    Series in
    Applied Mathematics, SIAM (Capital City Press, Montpellier, Vermont),

    Mallat S. G. A theory for multiresolution signal decomposition: the
    representation. IEEE Transactions on Pattern Analysis and Machine
    Intelligence, 11(7):674--693, 1989.

    Mallat S. G. Multiresolution approximations and wavelet orthonormal
    bases of
    L_2(R). Transaction Am. Math. Soc., 315:69--87, 1989.

    O. COUSSI F. BREMAND G. BESSONNET. Wavelet transform and biomechanical
    filtering. In 4th International Symposium on 3-D Analysis of Human
    Grenoble, 1996.

    F. MAJID R.R. COIFMAN M.V. WICKERHAUSER. The Xwpl system Reference
    Manual. Yale University Mathematics Department and FMA\&H, USA, December

    M. BOURGES-SEVENIER. WaveLib 1.2 User's Guide. IRISA, Rennes, inria

    Q. ZHANG. Regressor selection an wavelet network construction. Technical
    Report 1967, INRIA, Rennes, April 1993.

    3) Savitzky-Golay filtering. Savitzky-Golay filtering is a fast
    technique for implementing polynomial smoothing filters in the form of a
    linear digital filter. Generally, this technique does not adapt to the
    local character of the signal, however certain adaptations may be
    possible. Some useful references include:

    Press WH et al. (1992) "Numerical Recipes in Fortran", Cambridge Univ.
    Press, pp. 644-649.

    Savitzky A, Golay MJE (1964) Anal. Chem. 36: 1627-1639.

    Or any advanced book on digital filters.

    4) Another suggestion is a "common sense" approach in which the
    adaptation is manually tuned to the problem at hand. For EMG
    processing, an initial EMG AMPLITUDE estimate would be made (e.g.
    rectify and smooth). This stage would be followed by an adaptive
    derivative estimator (say an adaptive linear filter, or some other
    simple non-linear element) in which the adapted parameter is the cut-off
    frequency. Adaptation would be performed manually, perhaps tuned to the
    particular application of interest.