the generalized correlation methods for estimation

THE GENERALIZED CORRELATION METHODS FOR ESTIMATION OF TIME DELAY WITH APPLICATION TO ELECTROMYOGRAPHY a

a

b

G.T. LUU , P. RAVIER , O. BUTTELI a

Images and Signals for Systems Team (ISS), Laboratory PRISME - University of Orleans, 12 rue de Blois 45067 Orléans Cedex. [email protected], [email protected] b Laboratoire Activité Motrice et Adaptation Physiologique (AMAPP), UFR STAPS - Université d’Orléans, 2 allée du Château BP 6237, 45067 Orléans Cedex 2, France [email protected]

Abstract Muscle fiber conduction velocity is based on the time delay estimation between electromyography recording channels. The aims of this study is to identify the best estimator of generalized correlation methods in the case where time delay is constant in order to extent these estimator to the time-varying delay case . The fractional part of time delay was calculated by using parabolic interpolation. The results indicate that Eckart filter and Hannan Thomson (HT) give the best results in the case where the signal to noise ratio (SNR) is 0 dB.

1. Introduction Muscle fiber conduction velocity (MFCV) is a relevant neuromuscular indicator because it reflects the functional state of the muscle through: • •

modifications in the motor unit (i.e. muscle functional unit) recruitment strategies of the motor control. modifications in the muscle fiber membrane properties.

MFCV is a useful tool to assess neuromuscular pathologies [8], fatigue [4], or pain [1]. MFCV can be estimated from intramuscular or surface EMG recordings [4]. In this work, we are only interested in surface EMG signals. However, the EMG signal suffers from several limitations, coming from anatomical problems and from modifications in the action potential volume conductor, that impact the conduction velocity estimation. This is particularly the case in dynamic muscle contraction conditions (the most applied daily conditions), in which both force and posture vary. In that case, three main factors affect the EMG signal: the nonstationarity property of the data; the change in conductivity properties of the tissues separating electrodes and muscle fibers; the relative shift of the electrodes with respect to the origin of the action potential. Answers to the last two points can be found by using a multi-channel device. The present work follows the thesis by Leclerc [7], which proposed a time-varying delay estimation in the case of two channels, taking into account the nonstationarity of EMG properties.

Using more than two surface EMG derivations along the fiber direction may significantly improve MFCV estimates in terms of variance of estimation , sensitivity of the measure to electrode location, and repeatability [5]. The purpose of this work is to investigate the use of multi-channel measurements in the time-varying MFCV estimation problem. First, surface EMG acquisitions induce various local signal-to-noise ratios (SNRs). Consequently, robust estimators must be developed to deal with spatial SNR variability. Thus, the study of performance has to be done as a function of the SNR. The first step of this work is to investigate the case where the time delay is considered as a constant. This is the focus of section 2. In a second step, the best estimators in this case will be extended in the case where the delay varies during the time. Second, methods for multichannel recordings must be specifically designed. We will work on spatial filtering techniques. Efficient strategies for combining multiple estimations will also be explored. In this study, we considered the surface EMG signal
 propagation between channel 1 and channel 2, a simple analytical model of two observed signals ଵ  and ଶ  in a discrete time domain, without shape differences, is the following: ଵ  =
 + ଵ  (1)   ଶ =
 +  + ଶ where

θ is the propagation delay between the two signals, and w1 ( n ) and w2 ( n) are assumed to

be independent, white, zero mean, additive Gaussian noises, of equal variance σ 2 . 2 Methods In the multi-channel case, the surface EMG signals are collected by a matrix of electrodes located at different positions of the skin. Due to the anisotropy of the membrane, the study has to be done assuming different SNRs depending on the channel. Robust algorithms have thus to be developed. In particular, the design of the delay estimators has to properly take into account the Power Spectral Density (PSD) shapes of the signals of interest. In this context, different methods of generalized correlation proposed in [2] were evaluated and tested for the EMG two-channel case. The signals are synthetic ones and are generated according to a Power Spectral Density (PSD) model described in [7]. Once the first channel has been generated, its delayed version is created thanks to the sinc-interpolator [7]. Finally, both channels are distorted by adding white noise at a given SNR level. In this study, the aims are firstly to identify the best estimators in the constant time delay case; secondly to extend the chosen estimators to the time-varying delay case; finally to investigate the multi-channel case. We present hereafter the theoretical estimators proposed in [2]. Simulation results are also given. Theoretical derivations To achieve a better resolution of the time delay estimation, the calculation of the crosscorrelation function R ୶భ ୶మ (τ) = TF ିଵ G୶భ ୶మ f is replaced by the following expression:

୶ ୶ (τ) = TF ିଵ ψ R భ మ ୶

భ ୶మ

୶ ୶ f f. G భ మ

(2)

୶ ୶ f is the estimated PSD. The functions Hଵ f and Hଶ f with ψ୶ ୶ f = Hଵ fHଶ∗ f, G భ మ భ మ are prefiltering operators aiming at transforming the two channel inputs such that the crosscorrelation function has better properties. In [2], the various estimation strategies proposed induce dedicated filters and produce different ψ୶ ୶ f functions (processors). భ మ

There are 6 processors. Their principle (shown in table 1) as well as their theoretical and estimated cross-correlation functions are given (table 2). The theoretical cross-correlation is obtained with the formula R ୶భ ୶మ (τ) = τ − D ∗ TF ିଵ Gୱୱ f. ψ୶ ୶ f. The cross-correlation భ మ

୶ ୶ f function as well as on the true PSDs of the function is based on the estimated G భ మ ଵ , ଶ  noise and
 EMG signal (that is assumed to be known). The time delay can be decomposed into an integer part plus a fractional part. The integer part is estimated by searching

୶ ୶ (τ).The fractional part is the maximum of the generalized cross-correlation function R భ మ calculated by using the parabolic interpolation [7] as:

+ d = D

− 0.5 τ = D ୖ

෡ ାଵ൧ିୖ౮ ౮ ൣୈ ෡ ୖ౮భ ౮మ ൣୈ భ మ ିଵ൧ ෡ ෡ ෡ ିଵ൧ ౮భ ౮మ ൣୈାଵ൧ିଶୖ౮భ ౮మ ൣୈ൧ାୖ౮భ ౮మ ൣୈ

(3)

are the estimators of the total delay, the integer and the fractional part Where τ, d and D respectively. Table 1: Comparative study of different processors PHAT Processor Simple Roth SCOT Eckart filter HT Cross(Smoothed (Phase (Hannan Coherence transform) correlation Thompson) Transfom) Principle

ψ࢞

૚ ࢞૛




Cross correlation

Wiener Filter

Function of Coherence

1

1 ௫భ௫భ


1 ௫భ௫భ
. ௫మ௫మ ()

PHAT (Phase transform) 1

௫భ௫మ


Maximization of SNR

௦௦
 | ଵଶ ()|ଶ 1 ௪భ௪భ
. ௪మ௪మ
 ௫ ௫
 1 − | ଵଶ ()| భ మ

Table 2 : Theoretical and estimated cross-correlation functions of different processor Processor Cross-correlation (theoretical) τ −  ∗ ௫భ ௫మ τ

Simple Crosscorrelation Roth

τ −  ∗ 

ାஶ

ିஶ

௦௦  .  ௝ଶగ௙τ .  ௦௦  + ௪భ ௪భ 

Maximum likelihood estimation

Cross-correlation (estimated)  ିଵ ௫భ ௫మ 

௫ ௫   ିଵ  భ మ  ௫భ ௫భ 

τ −  SCOT ାஶ (Smoothed ௦௦  Coherence ∗  ିஶ ௦௦  + ௪భ ௪భ  . ௦௦  + ௪మ ௪మ  Transfom) PHAT (Phase transform)

 ିଵ !

τ − 

ିஶ

where the coherence estimate function ଵଶ  =

# "

௫భ ௫మ . ௦௦   ିଵ   ௪భ ௪భ . ௪మ ௪మ 

௦௦ ଶ  τ −  ∗   ௝ଶగ௙τ .  ௪భ ௪భ . ௪మ ௪మ 

τ −  ଶ   ௦௦ ାஶ HT ௪భ ௪భ . ௪మ ௪మ  (Hannan ∗   ௝ଶగ௙τ .      ௦௦ ௦௦ Thompson) ିஶ 1 + + ௪భ ௪భ  ௪మ ௪మ 

௫భ ௫భ . ௫మ ௫మ 

௫ ௫   ିଵ  భ మ  $௫భ ௫మ $

ାஶ

Eckart filter

௫భ ௫మ 

௫ ௫  |ଵଶ ()|ଶ  ିଵ  భ మ  $௫భ ௫మ $ &1 − |ଵଶ ()|ଶ ' ீ෠ೣభೣమ ሺ௙ሻ

ටீೣభ ೣభ ሺ௙ሻ.ீೣమೣమ (௙)

3. Results Monte-Carlo simulations with 100 independent runs were performed for each SNR value in order to study the noise impact of these methods. In this work, two synthetic EMG signals have the same value of SNR. Duration of the signals is 1 second. The mean value and the standard deviation of the estimated time delays are presented in figure 1. As can be seen in figure 1(a), in the case where noise is strong (SNR=0dB), Roth, Scot, and PHAT give the worst results with respect to the Cross-correlation method. The Eckart filter gives the best result, followed by the HT processor. Roth and Scot processors give identical results because the power of noise is identical in this work ௪భ ௪భ  = ௪మ ௪మ  as predicted (figure 1(a) and 1(b)). The HT processor has the smallest bias and standard deviation among these processors (figure 1 (a)). When the noise is weak (20dB), the Eckart filter gives the best result (figure 1 (b)). The errors shown in figure 1 are synthesized by means of the mean square errors in table 2. As we can see, the Eckart filter and HT processors have the smallest MSE values in the case where noise is strong (0dB). When the noise is weak (20 dB), the generalized correlation methods did not improve the performance because the MSE values are nearly identical.

Table 2: Mean square error (MSE) for each value of SNR Methods

0 dB

20 dB

CC

0.1002

2.e-04

Roth

0.8881

0.0542

Scot

0.8882

0.0543

PHAT

0.7232

5 e-04

Eckart filter

0.0233

2.e-04

HT

0.0263

2.3e-04

(a)

(b) SNR=20 dB

4.6

Standard derivation T rue value =4.2

4.4 4.2 4 3.8 CC

ROTH SCOT EckartPHAT HT

Methods

Time delay (samples)

Time delay (samples)

SNR=0.dB 4.22

Standard derivation T rue value =4.2

4.21 4.2 4.19 4.18 CC

ROTH SCOT EckartPHAT HT

Methods

Figure 1: Time delay estimated values (mean ± standard deviation), see table 1 for the acronyms in abscissa. SNR is the Signal to Noise Ratio 4. Conclusion and perspectives The generalized correlation methods proposed in [2] were evaluated for two electromyography channels, the second one being a noisy delayed replica of the first one. To reach this goal, synthetic electromyography signals were generated and the performance was evaluated by Monte-Carlo simulations. The Eckart filter and HT processor were found to be the best time delay estimators in the case where the SNR is 0 dB. The results may also depend on the interpolation methods. Other interpolation methods should be applied to test whether the performance can be improved or not. This study will be continued by more precisely evaluating the role of other simulation parameters in the performance level (observation duration, SNR values, influence of EMG spectral shape,...). In this work, the power of noise and the power of signals are assumed known. In the case of real data, these parameters need to be estimated. This point will also be studied. The following work is the extension of these estimators to the time-varying delay case. Very few studies exist on this topic in the electromyography domain. To our knowledge, only one method has been published in [3] in the multi-channel case, concerning the MFCV

estimations during dynamic contractions. However, this approach does not use dedicated nonstationary tools such as time-frequency / time-scale representations or adaptive filtering. Nevertheless, time-frequency/time-scale coherency measures and adaptive filtering methods have already been proposed by Leclerc [7], but were limited to the two-channel case. Our contribution will permit this work to be pursued since the appropriate prefilters can now be included in time-frequency or time-scale tools for a better MFCV estimation. Based on these two channel MFCV estimation improvements, the multi-channel case will be investigated.

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