## 1 Introduction

It is part of common knowledge that excessive bodily exercise will lead to muscle fatigue, easily assessed as the muscles’ inability to continuously deliver a constant force. Both physiotherapy and Functional Electro-Stimulation (FES), the noninvasive stimulation of de-innervated muscles, have gone to great lengths to develop stimulation patterns to avoid early muscle fatigue, yet to provide optimal therapeutic outcome. Insofar, it is of great interest to objectively assess muscular fatigue and adjust training accordingly. Muscular fatigue can be detected by subtle changes in electromyographic recordings from the affected muscle. An overview of known fatigue detection algorithms can be found in Gonzalez-Izal et al. [5]. Our investigation compared the performance of different reported fatigue detection algorithms depending on the sampling rate of the input sEMG (surface EMG) signal. We considered algorithms based on mean (MNF) and median (MDF) frequencies as well as the spectral moment ratio (SMR), sample (SampEn) and fuzzy approximate entropy (fApEn). These algorithm’s performance was evaluated based on sEMG signals with technically most relevant sampling rates between 256 and 1024 Hz.

### 2 Methods

The investigations are based on sEMG recordings from upper arm’s biceps muscles while pulling against a constant force. The experiment was conducted with 12 healthy volunteers (age between 17 and 56, 6 female and 6 male). Written consent to take part in the experiment was obtained from the volunteers prior to the start of the experiment.

### 2.1 Experimental setup

Each subject was requested to pull a rope attached to his/her wrist with a bent arm and the elbow leaning flat on a support. The experiment was conducted with the subject sitting upright in front of a table fitted with pulleys to enable isotonic lifting of a defined weight (see Fig. 1 a). The block and pulley construction exerted a constant force on the biceps muscles, once the prepared weight was lifted off the ground.

For calibration purposes the volunteer was asked to pull against a spring scale with his/her maximal achievable effort. That way the maximum voluntary contraction (MVC) was determined. Afterwards the subjects were granted a resting period of at least 30 minutes.

Then the subject was requested to lift a weight of 20% of his/her previously determined MVC for 3 minutes. This trial was intended to produce a non-fatiguing sEMG recording. In this case all subjects persevered the maximum duration of 3 minutes.

After another resting period of at least 30 minutes the subject was requested to pull with 60% of his/her MVC to produce a fatiguing sEMG recording. Each volunteer was told to be free to let go the load and relax when feeling exhausted. Subjects were able to sustain the load between 60 seconds until the end of the trial after 3 minutes. The average duration of the 60% MVC trial was 135 seconds.

sEMG was recorded from eight self-adhesive electrodes (Covidien Kendall, H124SG) placed on the upper arm on top of the biceps (see Fig. 1 b). They were placed in two rows (first row electrodes 1-4, second row electrodes 5-8). Electrodes 3 and 7 were placed above the thickest point of the biceps and electrodes 1 and 4 were placed on the distal end of the upper arm. Electrodes were attached next to each other without gap in between. The centers of adjacent electrodes were thus (in accordance with their diameter) 24 mm apart.

Recordings were made with a Porti amplifier (TMSI, Oldenzaal, Netherlands). The amplifier included a 22 bit analog digital converter and was operated with a sampling rate of 1024 Hz. The common electrode of the amplifier was connected on the back side of the upper arm, opposite to the other electrodes, on top of the triceps muscle.

### 2.2 Signal processing

Raw values provided by the amplifier were imported into a PC for offline analysis. All further steps of signal processing were conducted in R [3]. In a first step 28 differential channels representing all possible combinations of two from eight electrodes were calculated. To remove a possible baseline offset, all differential channels were filtered with a high pass filter (3rd order Butterworth IIR filter with 2Hz corner frequency).

Downsampled sEMG signals were derived from the original 1024 Hz signal to investigate the effect of sampling rate limitations. The resampled sEMG signals had sampling rates of 256, 384, 512, 640, 768 and 896 Hz. The re-sampling was carried out by bandlimited interpolation [8] as implemented by the resample method from the R signal package.

Fatigue was quantified based on the high pass filtered sEMG signal as well as the derived resampled sEMG signals by the following, briefly described fatigue evaluation algorithms. The evaluation was performed epoch-wise with an epoch size of one second. A new epoch was started every 128 ms, resulting in eight epochs overlapping at each point in time and in fatigue tracings with 8 Hz sampling rate.

### 2.2.1 Frequency based fatigue algorithms

Frequency based fatigue algorithms quantify the spectral compression that is connected to the reduction of the conduction velocity occurring in fatiguing muscles as explained by Lindström et al. [6]. These algorithms are based on the epoch-wise estimated power spectral density (PSD). The PSD was estimated by the FFT based Welch method with eight sub segments, no overlapping and the Hamming window function.

An alternative PSD estimation was obtained by the Burg algorithm [2]. The Burg algorithm is a parametric PSD estimation method. In a first step parametric PSD estimators determine a linear filter. This filter is designed in a way that it would transform white noise into a signal with the same spectral shape as the original signal to be analyzed. In case of the Burg algorithm filter coefficients of an autoregressive filter and a gain are calculated. The order of the filter was adaptively determined by the Akaike Information Criterion as implemented by the R function ar.burg(). In a second step the PSD estimation was calculated from the autoregressive filter coefficients.

The mean (also known as central) frequency

Furthermore the PSD estimation was evaluated with the spectral moment ratio (SMR) reported as new spectral index in [4]. Instead of using fixed frequency bands with fixed boundaries as with the old H/L ratio [9], Dimitrov and colleagues used the ratio of the spectral moments of the orders -1 and 5. The spectral moment of order k is defined as

### 2.2.2 Entropy based fatigue algorithms

Entropy estimation is a nonlinear method that might be used as a fatigue indicator. One possible entropy calculation method is the sample entropy (SampEn) as introduced by Richman and Moorman [7]. Xie et al. [10] suggested an improved algorithm named fuzzy approximate entropy (fApEn) as a fatigue indicator.

Sample entropy is defined by Richman and Moorman [7] as negative natural logarithm of the conditional probability that two sequences similar for *m* points remain similar at the next point *m*+1. Two vectors are considered similar if their distance according to the uniform norm is below a threshold *r*. The sample entropy is calculated epoch-wise. Each epoch is first normalized to have zero mean and a standard deviation of one. The sample entropy was calculated for vector length *m* = 1, 2, 3, 4, 5, 6, 7 and thresholds of *r* = 0.3, 0.4, 0.5, 0.6, 0.7. A value of vector length of one in combination with *r* = 0.3 and a vector length of two in combination with *r* = 0.7 proved satisfactory. The sample entropy was calculated using code available from http://www.physionet.org/physiotools/sampen/c/.

The fuzzy approximate entropy is calculated in a similar manner. In contrast to the sample entropy the mean of each part vector is subtracted prior to comparison under the uniform norm. Moreover instead of the simplification to a binary state threshold, a fuzzy function *d*. In case of the fuzzy approximate entropy a vector length of two and a threshold of *r* = 0.6 were used.

### 2.3 Evaluation of quality

The performance of the fatigue methods were compared with respect to two criteria. The first criterion of a good fatigue detection algorithm is to demonstrate little noise in case of a fatigue producing load. The second criterion is that it will provide a good discrimination of different degrees of fatigue, assuming that these are resulting from the different load levels.

To account for the first criterion the fatigue index’ noise was quantified by calculating the coefficient of determination *R*^{2} following the suggestion of Xie et al. [10]. The coefficient of determination states how much of a signal can be described by a line with constant slope. The closer *R*^{2} is to one, the better. This analysis is based on the first 60 s of the 60% MVC load case. Figure 2 illustrates an example. An average *R*^{2} value was evaluated for each combination of fatigue algorithm and sampling rate across all subjects and electrode channels.

To investigate how well different degrees of fatigue can be discriminated the results of the different load levels were analyzed. This analysis is based on the assumption that different load levels will lead to different degrees of fatigue. A best fit linear slope was calculated for the first 60 s of all fatigue signals for both the 20% and 60% MVC load cases. See Figure 2 for an example. The slope of this line corresponds to the change of the underlying fatigue signal in the first 60 s of the contraction. For each combination of sampling rate and fatigue algorithm a separabilty value

### 3 Results

Figure 3a shows the average coefficient of determination for the 60% MVC load case. In general the coefficient of determination rises with larger sampling rates. In case of the frequency based methods the spectral moment ratio shows the largest *R*^{2} values followed by mean and median frequency. All three PSD evaluation methods show larger coefficients of determination if they are based on Burg spectral estimation compared to the classical FFT based Welch algorithm. Both entropy algorithms suffer a strong decrease of the average coefficient of determination for small sampling rates.

Figure 3b shows the values quantifying the separability of the different load levels. In the field of frequency based algorithms the spectral moment ratio lies closely above the mean frequency followed by the median frequency. In contrast to the evaluation of the coefficient of determination the results of the Burg spectral estimation are very close to those based on the Welch method. The results of the entropy based methods are again more sensitive to smaller sampling rates than the frequency based methods.

### 4 Conclusion

The results show that the fatigue detection quality of all tested algorithms is sensitive to the sampling rate of the input sEMG signal. At the three tested frequency based methods the median frequency yielded the worst results. Above a sampling rate of 640 Hz the results are not improving for both investigated spectral estimations and in respect to both *R*^{2} and. It seems quite reasonable that this effect is connected to the characteristic of the median to be robust against outliers. Small portions of the frequency compression in higher bands are attenuated similar to outliers. In contrast the results of the mean frequency are slightly improving for sampling rates larger than 640 Hz. The SENIAM recommendations [1] suggest a sampling rate of 1 kHz or larger for general sEMG applications. The results show that this recommendation seems reasonable for mean and median frequency based fatigue detection.

At low sampling rates the results of the spectral moment ratio are comparable to the mean frequency. Towards higher sampling rates the advantage of the spectral moments method above the mean frequency becomes more and more obvious. It seems that the intention of the smooth weighting function to take the spectral compression also for higher frequencies into account is effective and that the algorithm needs a high sampling rate to display this strength. This is in accordance with the fact that Dimitrov et al. [4] used a sampling rate of even 2 kHz.

Kammeyer and Kroschel [2] showed that the Burg algorithm can yield a very good spectral estimation even for small epoch sizes if the autoregressive model is suitable to the data. The results suggest that this is the case for the investigated sEMG data as the Burg algorithm leads to higher *R*^{2} values in case of all three PSD based evaluation methods possibly caused by Burg’s lower estimation variance.

The results also show that the entropy based fatigue methods are more sensitive against a reduction of the sampling rate than the frequency based methods. In case of the sample entropy the choice of the best parameters strongly depends on the sampling rate whereas this is not the case with the fuzzy approximate entropy. The fatigue detection quality based on fApEn showed also best results at a sampling rate of 1024 Hz. This is in accordance with the 1 kHz sampling rate used by Xie et al. [10].

**Funding**: This work was funded by Drägerwerk AG & Co. KGaA, Lübeck.

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**Conflict of interest**: Authors state no conflict of interest. Material and Methods: Informed consent: Informed consent has been obtained from all individuals included in this study. Ethical approval: The research related to human use has been complied with all the relevant national regulations, institutional policies and in accordance the tenets of the Helsinki Declaration, and has been approved by the authors’ institutional review board or equivalent committee.

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