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.

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.