## Abstract

Surface electromyography is used in research, to estimate the activity of muscle, in prosthetic design, to provide a control signal, and in biofeedback, to provide subjects with a visual or auditory indication of muscle contraction. Unfortunately, successful applications are limited by the variability in the signal and the consequent poor quality of estimates. I propose to use a nonlinear recursive filter based on Bayesian estimation. The desired filtered signal is modeled as a combined diffusion and jump process and the measured electromyographic (EMG) signal is modeled as a random process with a density in the exponential family and rate given by the desired signal. The rate is estimated on-line by calculating the full conditional density given all past measurements from a single electrode. The Bayesian estimate gives the filtered signal that best describes the observed EMG signal. This estimate yields results with very low short-time variability but also with the capability of very rapid response to change. The estimate approximates isometric joint torque with lower error and higher signal-to-noise ratio than current linear methods. Use of the nonlinear filter significantly reduces noise compared with current algorithms, and it may therefore permit more effective use of the EMG signal for prosthetic control, biofeedback, and neurophysiology research.

## INTRODUCTION

Surface electromyography has been studied for >50 yr as a noninvasive measure of the activity in voluntary muscles. There are currently at least three major areas of application: *1*) interpretation of neural control signals for research (Merletti et al. 1999), *2*) extraction of a voluntary command signal for control of prosthetic or robotic devices (Dipietro et al. 2005; Light and Chappell 2000; Light et al. 2002; Park and Meek 1995) or stimulation of functionally denervated muscles (Hefftner and Jaros 1988; Hefftner et al. 1988), and *3*) biofeedback to subjects to change patterns of voluntary muscle contraction (Ince et al. 1984, 1985). Each of these applications requires filtering of the electromyographic (EMG) signal to extract a smoothed signal related to muscle force or voluntary drive to muscle.

Many current methods for filtering are based on a model of the EMG signal as amplitude-modulated band-limited noise (Clancy et al. 2001; D'Alessio and Conforto 2001). Algorithms for interpretation of the EMG signal attempt to demodulate the signal to recover the amplitude envelope (Hogan 1976). Over 25 yr ago, Hogan showed that the maximum likelihood estimator for the envelope can be extracted by squaring the EMG signal at each point in time, followed by low-pass filtering to smooth the resulting estimate (Hogan and Mann 1980a,b). In most current applications, this procedure is slightly modified so that the EMG signal is full-wave rectified rather than squared before filtering (Evans et al. 1984), and some journals consider this the “standard” method of analysis (ISEK 1996). It can be shown that this procedure is an optimal estimator under slight modifications of the original assumptions (Clancy and Hogan 1999; Clancy et al. 2001).

Unfortunately, the low-pass filter smooths not only the undesired variability in the EMG signal, but also any intentional rapid changes (St-Amant et al. 1998). It therefore limits the ability to detect rapid onset or offset of EMG activity. When used as an on-line estimator, the low-pass filter must be causal and this will introduce a delay that can be hundreds of milliseconds long. Increasing the bandwidth of the filter improves responsiveness but at the expense of greater variability in the estimated signal (Meek and Fetherston 1992). Recent innovations including the use of multiple electrodes or electrode arrays (Clancy and Hogan 1994, 1995; Clancy et al. 2006; Staudenmann et al. 2005, 2006) and the use of whitening filters before rectification can significantly improve the quality of estimates (Clancy and Farry 2000; Clancy and Hogan 1997; Clancy et al. 2002; Prakash et al. 2005), but there remains a trade-off between smoothness of estimates and the ability to make rapid changes. Other linear or quasi-linear methods show promise for discrimination and classification of EMG signals, including hidden Markov models (Chan and Englehart 2005; Chan et al. 2002), wavelet processing (Al-Assaf and Al-Nashash 2005; Englehart et al. 2001), Gaussian mixture models (Huang et al. 2005), and neural network estimators (Karlik et al. 2003), although efficient implementation of these methods may be difficult. Several other methods use the EMG signal to classify movement commands rather than estimate a continuous underlying explanatory signal (Englehart and Hudgins 2003; Englehart et al. 2001; Huang et al. 2005). Bayesian estimators were previously used to identify rapid changes in the EMG signal and to estimate its variance (Johnson et al. 2003), but these estimators do not appear to have a recursive implementation and thus cannot be used for real-time filtering.

I propose a new recursive algorithm for on-line Bayesian filtering of the surface EMG signal. The purpose of this algorithm is to allow the filtered signal to remain constant with low variability during periods of constant drive, but also to permit very rapid “step” changes. I model the EMG signal as a filtered random process with random rate and I show that the likelihood function for the rate evolves in time according to a Fokker–Planck partial differential equation. I provide a causal on-line algorithm for propagation of the Fokker–Planck equation and I demonstrate its performance on recordings of biceps and triceps activity from single electrodes during isometric contraction.

## METHODS

### Model of the EMG signal

The surface EMG signal results from propagation of action potentials along muscle fibers that may be close to the skin or buried deep within the muscle. The signal will depend on the relative timing of potentials, the particular size and shape of the muscle and muscle fiber, the placement of the recording electrode relative to fibers, and the impedance at the skin–electrode interface relative to the amplifier input impedance. I do not attempt to derive a detailed model for the surface EMG signal, but instead propose that the purpose of filtering is to extract a signal that describes the measured surface EMG signal. I will refer to this signal as the “driving signal,” although it may not be directly related to the actual neural drive to the muscle.

The instantaneous relation between the latent driving signal x and the resulting EMG signal can be described by a conditional probability density P(EMG|x). Because the driving signal cannot be measured directly, I suggest three alternative models for the conditional density and test them in the estimation procedure. In particular, I suggest models for the conditional probability of the rectified EMG signal, which can be written as emg = |EMG| because the rectified EMG signal is commonly used in current estimation algorithms.

Under the assumption that the rectified EMG signal results from random depolarization events of multiple muscle fibers, then the average amplitude in a small time window will be proportional to the number of depolarization events during that time. The number of events *n* can be modeled as a Poisson process so that one can write where x is the average rate of events for the whole muscle, and thus represents the unknown driving signal, and *n* is an integer that is proportional to the magnitude of the rectified EMG signal. This model ignores the considerable complexity arising from filtering properties of the soft tissues, electrode placement, action potential shape, and many other physical effects. I will refer to this model as the “Poisson measurement model.” The Poisson model with rate described by a stochastic differential equation (see following text) was previously shown to be a useful model of communications and other nonstationary processes (Boel and Benes 1980).

Empirical observation of the EMG signal has led to the common assumption that it can be described as amplitude-modulated zero-mean Gaussian noise (Hogan and Mann 1980). In this case For the rectified EMG signal this function is only defined for values of emg > 0, so I refer to this model as the “Half-Gaussian measurement model.”

Close observation of the EMG signal suggests that the density may be better approximated by a Laplacian density (Clancy and Hogan 1999) which for the rectified EMG signal is given by Because this is an exponential distribution (with rate 1/x) I refer to this model as the “Exponential measurement model.”

It should be noted that all three models are intended to describe the empirically measured signal; they do not provide a detailed model for the physiological processes responsible for generating the EMG signal. For all three models, the expected value of the rectified EMG signal given x is E[emg|x] = x and the average value of the rectified EMG signal is thus an unbiased estimator of the driving signal x. If the driving signal is constant, then the law of large numbers states that the average of the rectified EMG signal will approach the true value of x if it is averaged over sufficiently long periods of time.

However, when the driving signal x is time varying one cannot use the law of large numbers because estimates must be rapid enough to keep up with changes in x. This is the justification for using a low-pass linear filter to estimate x from the rectified EMG signal. However, low-pass filters have an undesirable property: rapid response to change requires a high bandwidth, but high bandwidth will not remove rapid variations in the EMG signal that are unrelated to the driving signal x. Therefore an improved estimator for x will have both rapid response and adequate rejection of irrelevant components of the EMG signal.

### Model of the driving signal

Although it is often not stated in the derivation of EMG signal estimators, all estimators imply a presumed model of the statistics of the driving signal. One obtains very different results depending on assumptions about the bandwidth of the driving signal. For example, estimation of muscle force by low-pass filtering the rectified EMG signal <20 Hz corresponds to the reasonable assumption that force does not change more rapidly than 20 Hz.

Here, I take the different approach of specifying the driving signal in terms of a stochastic model. Because the resulting estimate will fit this model, the model should be selected based on the intended purpose of the estimate. For instance, if the purpose is to estimate joint torque then a model that fits the temporal statistics of torque should be used. If the purpose is to control a mechanical device, then a model that matches the allowable variation in the control signal should be used. I choose a flexible model that is intended to capture two properties useful for both control and biofeedback: *1*) changes in the driving signal are almost always smooth and *2*) large jumps may occur at unpredictable but infrequent intervals. Although the model will be implemented in discrete time, it is convenient to write it as a continuous-time stochastic differential equation where the equation is to be interpreted in the Ito sense, dW is the differential of standard Brownian motion, dN_{β} is the differential of a counting process with rate β events per unit time, and U is a random variable uniformly distributed on [0, 1]. Most of the time, dN_{β} = 0 so that this reduces to dx = α(dW), which is a random walk. At intervals determined by the rate constant β, the value of x(t) will be replaced by U(t), which corresponds to a jump to a new randomly selected value of x.

To maintain x(t) within the interval [0, 1], we further assume where δ(t) is the Dirac delta function. This equation forces x(t) to remain within the interval [0, 1]. Because of this requirement, the Fokker–Planck equation for the evolution of the density of x(t) will not have a closed-form solution. However, an approximate solution is given by The term ∂^{2}p(x, t)/∂x^{2} is the well-known density evolution for a diffusion process. β indicates the probability of a jump to an arbitrary value of x. As time progresses, β causes p(x, t) to converge exponentially to 1 as the chance of not having jumped to an arbitrary value of x becomes vanishingly small.

### Recursive density propagation

The recursive algorithm and discrete-time approximation here are closely related to an algorithm developed for nonlinear state estimation (Challa and Bar-Shalom 2000).

Given a single measurement of emg(t) (rectified EMG signal) at time t, the function P[emg(t)|x(t)] specifies the likelihood of each possible value of x(t) given that particular measurement. Because x(t) is a random variable, Bayes's rule gives the posterior density where P[x(t)] is the probability density for x(t) immediately before the measurement of emg(t). The maximum a posteriori (MAP) estimate of x is found by maximizing P(x|emg) and it is common practice to ignore the denominator term because it is independent of x(t). It is sometimes helpful to include an additional term where the small constant γ indicates the probability that the measurement model or the measurement itself is incorrect and the constant C is chosen to make the density integrate to 1.

In general, the prior P[x(t)] will depend on the entire past history of measurements, so one can make this dependency explicit by writing P[x(t)|emg(s); s < t]. Estimation of P[x(t)] can be performed using a recursive algorithm based on discrete-time measurements, where t is an integer. In discrete time, the goal is to calculate P[x(t)|emg(t), emg(t − 1),…] and using Bayes's rule where C is a constant chosen so that the density integrates to 1, P[emg(t)|x(t)] is given by one of the measurement models presented earlier, and the standard assumption has been made that successive measurements of the EMG signal are conditionally independent given x(t).

From the previous step of the recursion, one obtains an estimate of P[x(t − 1)|emg(t − 1), emg(t − 2),…], which can be written as p(x, t − 1). It is now necessary to estimate the conditional density of x at time t, but immediately before the measurement of emg(t). To simplify notation, write this density as p(x, t−), which is calculated by propagating the density of p(x, t − 1) forward in time by one sampling interval. This can be accomplished by numerical integration of the Fokker–Planck equation. In discrete time, this equation is approximated by (The original values of α and β must be multiplied by the sampling interval Δt and the density p(x, t) must be renormalized to have integral 1 after each step.) When x is discretized into bins of width ε [so that p(x, t−) is represented as a histogram with *n* = max (x)/ε elements], the first term can be approximated using second differences (suppressing the time index for clarity) so that now where α is the diffusion “drift” rate and β is the Poisson “jump” rate for the driving signal model. As soon as a new measurement emg(t) is available, one can calculate At the first step, p(x, 0) can be initialized to a constant (uniform density). At each step, the denominator constant C is chosen to maintain normalization of the density so ∫ p(x, t)dx = 1.

The MAP (Bayesian) estimate of x(t) at each step is

### Recursive algorithm

The full algorithm can be summarized as follows:

)Initialize p(x, 0) = 1

)Forward propagate p(x, t−) ≈ αp(x − ε, t − 1) + (1 − 2α)p(x, t − 1) + αp(x + ε, t − 1) + β + (1 − β)p(x, t − 1)

)Measure the rectified EMG signal

)Calculate the posterior likelihood function P(x, t) ≈ P(emg|x)p(x, t−)

)Output the signal estimate MAP[x(t)] = argmax P(x, t)

)Divide p(x, t) by a constant C so that ∫ p(x, t)dx = 1

)Repeat from step 2

The measurement model P(emg|x) can be chosen to be one of the three models proposed earlier. For example, for the exponential model step 4 is given by There are four free parameters: α, β, γ, and ε. α specifies the expected rate of gradual drift in the signal, in signal units per unit time; β specifies the expected rate of sudden shifts in the signal, in number of expected shifts per unit time; γ specifies the measurement uncertainty, as probability of error per single measurement; and ε = m/N specifies the bin width for discretization of the estimate x, where m is the maximum value of x. In the results shown later, α = 0.001Δt, β = 10^{−24}Δt, γ = 0, ε = m/50, and the sample rate is 1,000 Hz (Δt = 0.001 s). These values were selected empirically by testing on a different data set. The very low value of β represents the assumption that large shifts in the estimate *independent of all prior measurements* are extremely unlikely. However, β should not be zero because it allows shifts to occur when prior measurements make these likely. (For efficient implementation it is helpful to clip the rectified input signal so that rare high values do not lead to an inappropriately large value of m; otherwise, the algorithm will attempt to estimate the conditional density for these rare values, but it is unlikely that there will be enough data to do so accurately. In the examples, clipping has been chosen to eliminate samples >3 SDs from zero.) Sample Matlab code and data can be downloaded from http://www.kidsmove.org/bayesemgdemo.html.

### Subjects and data collection

Five healthy right-handed adult subjects between 22 and 26 yr of age (four male, one female) were recruited and all signed written informed consent to participate as well as US Health Information Portability and Accountability Act (HIPAA) authorization for use of medical and research records, according to approval of the Stanford University Institutional Review Board.

Subjects were seated comfortably and positioned with straps in an adjustable chair (Biodex Medical Systems, Shirley, NY), and the right arm was strapped into a custom-built aluminum frame. The arm was positioned with the elbow at 90° and placed at shoulder height, 45° from the sagittal plane. The forearm was vertical and attached using Velcro straps to a load cell (model 1500ASK-50; Interface, Scottsdale, AZ). Surface EMG electrodes (model DE2.3; Delsys, Boston, MA) were attached over the belly of the biceps and triceps muscle. These electrodes have a gain of 1,000 and band-pass frequency response from 20 to 450 Hz. Force and EMG data were digitized simultaneously at 1,000 Hz (model Power 1401; Cambridge Electronic Design, Cambridge, UK) and data were stored off-line for subsequent analysis.

The force during maximum voluntary elbow flexion [maximal voluntary contraction (MVC)] was measured for each subject as the maximum value over three attempts of ≥5 s each. Subsequently, each subject performed 30 trials of 6 s each. On each trial, subjects initially were required to maintain relaxation (confirmed by EMG signal and force measurement). They were then asked to attempt to flex the elbow (against the isometric constraint) to match a desired force level. Most subjects required <1 s to achieve a steady-state force. The target level was displayed on a monitor by a horizontal line and the actually achieved force level was also displayed, so that subjects could match the two lines. Target forces of 2.5, 5, 10, 15, 20, and 25% of MVC were used in pseudorandom order with five trials at each force level. Low forces were chosen to avoid fatigue and 1–2 min of rest were given between trials. Subjects were asked to state whether they felt fatigued, but this did not occur.

### Statistical analysis

Computations were performed using Matlab (The MathWorks, Natick, MA) and statistical analyses were performed using R (R project group). Performance was compared with two standard linear algorithms. The first algorithm was a low-pass filter applied to the rectified EMG signal, with cutoff frequencies of 5, 1, or 0.1 Hz. Low-pass filters of order 1,000 were calculated using the fir1 function in Matlab and filtering was applied separately to the biceps and triceps EMG signal. The second algorithm was the optimal linear estimator of torque given the rectified EMG signal from both biceps and triceps. For each subject and each trial, the optimal 100th-order finite-impulse-response (FIR) linear filter was calculated using the Steiglitz–McBride method (stmcb function in Matlab).

Bayesian decoders were implemented with the Poisson, Half-Gaussian, and Exponential measurement models. Each of the Bayesian decoders produced a separate estimate for biceps and triceps activation levels. To estimate torque, the best linear estimate of torque from the two activation levels was calculated using linear regression for each task condition for each subject and for each of the algorithms. Linear regression was performed time point by time point, so that there were only two scalar regression coefficients (for biceps and triceps) for each subject and each task condition.

The ability to estimate torque was used as the primary outcome measure because the subjects had been instructed to produce a specified torque level. Therefore torque was considered to be a surrogate measure of the intended muscle activation level.

The root mean square error (RMSE) for estimated torque was calculated for each of the algorithms, for each task and each subject. Signal-to-noise ratio (SNR) was calculated as the average squared torque divided by the mean squared error. The correlation coefficient (*r*^{2}) between measured torque and the estimated torque was also calculated. Values of SNR, RMSE, and *r*^{2} were compared between algorithms using a pairwise two-tailed Student's *t*-test. Results for the primary outcome (SNR) were reported as significant for *P* ≤ 0.05.

## RESULTS

The relation between the mean and variance of the raw EMG signal can be used to assess the relative goodness of fit of the different EMG signal models. Figure 1 shows a plot of the SD versus the mean of the raw biceps EMG signal over all subjects and all MVC levels; *r*^{2} = 0.97. The quality of the linear fit can be assessed by regression of log (variance) on log (mean), which produces a slope of 1.98 (confidence interval: 1.91–2.06) consistent with a linear relationship between the mean activity and the square root of the variance (the SD). Note that the SD is proportional to the mean for an exponential distribution or a half-Gaussian distribution, but not a Poisson distribution (for Poisson, the variance is proportional to the mean). This suggests that the Poisson model is a poor fit to these data, but that either the Exponential of Half-Gaussian models could be appropriate estimators.

Figure 2 shows sample plots of the raw EMG signal, the measured torque, and the estimates from several different algorithms. The mean values of SNR, RMSE, and *r*^{2} are shown in Table 1. The Bayesian algorithm assuming an exponential or half-Gaussian distribution on the input performed significantly better (Student's *t*-test) in all three measures compared with all other algorithms. The Bayesian algorithm with exponential distribution was significantly different from the half-Gaussian distribution only for SNR.

Figure 3 shows the relative performance of the Bayesian algorithm with exponential input distribution for varying values of the parameters α and β. The SNR does not change significantly over many orders of magnitude and SNR is significantly different from the best linear algorithm (*P* < 0.0001 compared with 1-Hz filter by paired *t*-test) for all cases except β = 1 (*P* = 0.074).

Figure 4 shows the relative performance of the Bayesian and linear (1-Hz) algorithms with α = 0.001, β = 10^{−24} for each of the different levels of target force. The SNR for the Bayesian algorithm is significantly higher (*P* < 0.001) for all values of force except 2.5% (*P* = 0.11).

## DISCUSSION

The relation between the SD and the mean of the EMG signal (Fig. 1) is most consistent with either an Exponential or a Half-Gaussian model for the measurement. This may explain the relatively better performance of the Bayesian algorithm when using either of these measurement models compared with the Poisson measurement model. In all cases, the Bayesian algorithm outperforms the linear algorithms (Table 1). For example, the Bayesian algorithm with the exponential measurement model improves the SNR of the estimate by a factor of 3 compared with the best linear model. Note that the “optimal” linear estimator did not perform as well as the 1-Hz low-pass filter, and this is most likely explained by the 100-point time window used by the optimal estimator (vs. the 1,000-point time window of the low-pass filter). An estimator with a longer time window may perform better, but estimation of the parameters of a 1,000th-order optimal finite impulse response filter would require a much larger amount of data per trial than was available.

Comparison of the Bayesian algorithm with linear methods in Fig. 2 shows that the Bayesian algorithm has more rapid response to EMG signal onset and offset. In most cases, changes at the output of the Bayesian filter precede changes in torque because the Bayesian filter has the ability to produce changes in output that are more rapid than the mechanical response. Thus the rate of rise of the torque estimate can be more rapid than the rate of rise of the measured torque.

Figure 3 shows that the signal-to-noise ratio of the Bayesian algorithm is relatively insensitive to the particular values of its parameters. However, changes in the parameters will affect the smoothness of the filtered result, so these parameters should be chosen to match the intended purpose of the output. In particular, for estimation of torque they should be chosen to match the statistics of torques that actually occur. For control of a motor they should be chosen to match the capabilities of the motor.

Figure 4 shows that both the Bayesian and linear (1-Hz) algorithms perform best at target force between 10 and 20% of maximum voluntary force. One reason for this may be that the exponential model for the EMG signal provides a better fit in this range. Another reason may be that the higher energy in the signal provides more information per unit time for estimation. Further exploration of the applicability of the specific electromyography model will require data from many different isometric and nonisometric conditions.

The Bayesian estimation algorithm is a new algorithm for recursive estimation of the driving signal of surface EMG signal. It is very flexible and can be adapted to use different models of the statistical distribution of the EMG signal and different assumptions on the statistics of the driving signal. The primary advantage of the new algorithm is the possibility for very smooth output signals without eliminating the possibility of sudden and large changes in value. Therefore the output signal may be more useful for prosthetic control or biofeedback and it may better indicate the volitional drive to muscles.

Another important advantage of the algorithm is the flexibility in implementation. As derived earlier, there are four parameters that can control the drift, jumps, measurement uncertainty, and quantization. Increasing the measurement uncertainty will require a larger number of consistent data samples before a change in output will occur, but will also lead to a higher ultimate likelihood. So this provides a straightforward method for trading off uncertainty against response time, with the assurance that the resulting filter will have the minimum possible response time.

The algorithm belongs to a class of nonlinear filters that was the focus of a number of studies (Smith and Brown 2003; Twum-Danso and Brockett 2001). To reduce computational requirements, forward propagation of the conditional density p(x, t) can be implemented using the “Particle Filter” algorithm (Brockwell et al. 2004; Brown et al. 2001), although particle filters remain very time consuming to implement. Because of the specific assumptions about the form of the stochastic differential equation and simplifying assumptions used in calculating the discrete-time approximation, a much more efficient closed-form approximate solution can be found (Challa and Bar-Shalom 2000). This significantly reduces the computational burden and allows the algorithm to be implemented in portable hardware. For example, with the algorithm running under Matlab on a 1.2-GHz PowerPC computer (Apple Computer), filtering a 10-s signal requires about 3 s. A compiled implementation of the algorithm on the same computer can perform estimates in real time at 100 Hz (producing one estimate for every 10 input samples at 1,000 Hz, ε = m/256). An implementation on a 41-MHz ARM7 microcontroller without a floating-point unit (Aduc7020, Analog Devices) can generate estimates in real time at 30 Hz (producing one estimate for every 33 input samples at 1,000 Hz, ε = m/64).

The derivation given here makes several simplifying assumptions about the form of the driving signal. The signal is assumed to be time varying, but its relation to the surface EMG signal is not. Therefore I do not account for effects of fatigue, muscle length, velocity of shortening, temperature, or changes in skin conductance on the relation between driving signal and surface EMG signal. The statistical model of the driving signal is a Markov process and is thus unable to model signals whose properties change over time or depend on random events in the remote past. There is no attempt to include sophisticated models of the relationship between force and the EMG signal (Lawrence and De Luca 1983; Zhou and Rymer 2004), although such models could certainly be included when estimation of force is the goal.

Further experiments will be needed to assess the performance of the Bayesian algorithm under nonisometric conditions, as well as under unusual or pathologic conditions in which there may be rapid bursting, large motor units, abnormal recruitment patterns, fatigue, or other changes in the EMG signal. However, note that under such conditions there is no “correct” answer to which the filtered estimate can be compared. Here, I have used isometric contractions to match a known force that is <25% of maximum to: *1*) provide a known “correct” answer that is related to both the muscle force and the voluntary activation of the muscle, *2*) reduce potential effects of fatigue, and *3*) eliminate effects of passive muscle and limb properties on the relation between torque and the EMG signal. For example, in a nonisometric task the relation between the EMG signal and the joint torque will be determined by many factors including limb inertia, the muscle length–tension and velocity–tension curves, and the geometry of the muscle–tendon attachments to bone. Because many of these relations are incompletely known for humans, nonisometric conditions cannot at this time be used to validate the filtering algorithm. However, one might hope that in the future the use of this algorithm to find an “equivalent isometric driving signal” during nonisometric tasks could be helpful to determine the relationship between the EMG signal and joint angle, angular velocity, and torque.

Successful use of surface electromyography for research, control, and biofeedback depends on the quality, immediacy, and stability of the estimated signals. Given its assumptions, the Bayesian algorithm presented here will maximize the quality, immediacy, and stability over all other algorithms with similar assumptions. The algorithm produces adequate estimates from only a single electrode. Further work is needed to confirm its utility for control and biofeedback applications and to include compensation for fatigue and other slow changes in the surface EMG signal (Park and Meek 1993). Further work is also needed to investigate possible extensions to multielectrode systems or to the estimation of other bioelectric signals. The ultimate goal is to provide an easily implementable on-line estimation algorithm for bioelectric signals that can be used for research, medical treatment, and rehabilitation.

## GRANTS

This work was supported by National Institute of Neurological Disorders and Stroke Grant NS-041243 and by generous additional support from the Crowley Carter Foundation.

## Acknowledgments

I am grateful to V. Chu for collecting the sample data and to S. Sherman-Levine for clinical research coordination.

## Footnotes

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- Copyright © 2007 by the American Physiological Society