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Hsiang-simon Mong
07-20-1997, 07:05 PM
Reply Summary: Derivative Estimates of Noisy Signals in a Dynamic
Environment

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
E-mail: msmail5.clancye@tsod.lmig.com


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.
Pure
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),
1992.

Mallat S. G. A theory for multiresolution signal decomposition: the
wavelet
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
data
filtering. In 4th International Symposium on 3-D Analysis of Human
Movement,
Grenoble, 1996.

F. MAJID R.R. COIFMAN M.V. WICKERHAUSER. The Xwpl system Reference
Manual. Yale University Mathematics Department and FMA\&H, USA, December
1993.

M. BOURGES-SEVENIER. WaveLib 1.2 User's Guide. IRISA, Rennes, inria
edition,
1994.

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.