preserves all information contained in the original real-. Special thanks to CNPq/Brazil and the Wishbone Trust for the financial support of this work.
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A Novel Spectral Representation of Electromyographic Signals 1
A. O. Andrade1, P. J. Kyberd2, S. D. Taffler3
Department of Cybernetics, University of Reading, Reading, UK Department of Cybernetics, University of Reading, Reading, UK 3 Oxford Orthopaedic Engineering Centre, University of Oxford, Oxford, UK 2
Abstract—Time/frequency and temporal analyses have been widely used in biomedical signal processing. These methods represent important characteristics of a signal in both time and frequency domain. In this way, essential features of the signal can be viewed and analysed in order to understand or model the physiological system. Historically, Fourier spectral analyses have provided a general method for examining the global energy/frequency distributions. However, an assumption inherent to these methods is the stationarity of the signal. As a result, Fourier methods are not generally an appropriate approach in the investigation of signals with transient components. This work presents the application of a new signal processing technique, empirical mode decomposition and the Hilbert spectrum, in the analysis of electromyographic signals. The results show that this method may provide not only an increase in the spectral resolution but also an insight into the underlying process of the muscle contraction. Keywords—Electromyography, empirical mode decomposition, Hilbert spectrum
I. INTRODUCTION Time/frequency and temporal analysis methods have been widely used in electromyographic (EMG) signal processing. These methods represent the temporal characteristics of the EMG signal by its spectral components in the frequency domain. Historically, Fourier spectral analysis has provided a general method for examining the global energy/frequency distributions. As a result, the term ‘spectrum’ has become almost synonymous with the Fourier transform of the data [1]. Fourier methods relate the temporal characteristics of a signal to its frequencyrepresenting spectrum, by the signal as an infinite summation of weighted sine and cosine waves of increasing frequencies, multiples of a fundamental frequency. However, an underlying assumption in these methods is the stationarity of the original signal. As a result, they are not generally an appropriate approach in the analysis of signals with transient components [2],[3]. Nonetheless, Fourier methods have been largely used in the analysis of electromyographic signals, which are nonstationary and nonlinear. To satisfy the stationarity condition and to extend the applicability of Fourier transform, it is common to divide the EMG signal into blocks of short duration, which is basically to place a window process over the input signal. In this way, the signal is divided into a number of segments inside each it is stationary. Akay [2] points out two potential disadvantages
of this method: firstly, it depends on the choice of the window function and secondly, in order to achieve a good time resolution, one should use a short window. However, a short window has poor frequency resolution since the number of samples inside the window is limited and this compromises the assumption of stationarity within the window. In order to overcome drawbacks of Fourier methods this work presents the application of a signal processing technique, empirical mode decomposition and the Hilbert spectrum in the analysis of EMG signals. Results from the analyses of these signals show that this technique may reveal physiological phenomena associated to the muscle contraction and an increase in the spectral resolution. II. METHODOLOGY The Hilbert spectral analysis method is based on the assumption that any signal consists of a set of different simple intrinsic mode oscillations. Since electromyographic signals may be considered as the summation of many motor unit action potentials and this sum yields a nonlinear and nonstationary time-series the method is thoroughly appropriate for the analysis of such signals. The method requires two steps in analysing the data. The first step is to pre-process them by the empirical mode decomposition (EMD) method with which the data are decomposed into a number of intrinsic mode functions (IMFs). This will identify the appropriate time-scales that will reveal the physical characteristics of the system and expand the data on a basis derived from them. In practice can be demonstrated that this decomposition process is complete, orthogonal, adaptive and local. The second step is to apply the Hilbert transform to decomposed IMFs and construct the energy and frequency against time distribution, designated as the Hilbert spectrum, from which the time localities of events will be preserved, in other words, the instantaneous frequency and energy rather than the global frequency and energy as described by the Fourier spectral analysis. A. The notion of instantaneous frequency and the Hilbert Transform The notion of instantaneous frequency can be described if an analytical signal, Z(t), is created from an arbitrary time-series, S(t). Z(t) is a complex-valued signal that preserves all information contained in the original real-
Special thanks to CNPq/Brazil and the Wishbone Trust for the financial support of this work.
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valued signal S(t) [4]. Theoretically, there are many ways of defining the imaginary part of Z(t) (e.g. quadrature methods), but the Hilbert transform, HT, provides a simple way of defining the imaginary part so that the result is an analytic function, as shown in (1).
Z (t ) = S (t ) + iHT {S (t )} = a(t )e iθ (t )
(1)
given time, the data may include more than one oscillatory mode. That is why the simple Hilbert transform cannot provide the full description of the frequency content for the general data and for this reason the data should be decomposed into IMF components. An arbitrary time-series S(t) can be described as a summation of IMFs in addition to a residue or trend function, r(t), as indicated by (7).
Where HT{S(t)} can be estimated as HT {S (t )} =
n
S (t ) = å imf i (t ) + r (t ) i =1
¥
S(t) 1 Pò dt , π - ¥ t '-t
(2)
and P is the Cauchy principal value defined as ¥
Pò
-¥
R S (t ) S (t ) dt = lim ò dt , R ® ¥ - R t '-t t '-t
(3)
From (1) it is possible to define the instantaneous amplitude, a(t), as
[
]
a (t ) = S 2 (t ) + HT 2 {S (t )}
1/2
,
(4)
and the instantaneous phase, θ(t), as æ HT {S (t )} ö ÷÷ . q (t ) = arctançç è S (t ) ø
dq (t ) dt
a)
c) d) e) f)
(5)
Thus, the notion of instantaneous frequency, ω, can be obtained differentiating the instantaneous phase as shown by (6). v=
Where imfi(t) is an IMF component, n is the total number of components and t is time. The steps presented below can be carried out in order to decompose S(t) into its constituent IMFs:
b)
where both, t and t’, represent a continuous time [5].
(6)
B. The concept of intrinsic mode functions Huang et al. [1] have defined IMFs as a class of functions that satisfy two conditions: 1) In the whole data set, the number of extrema and the number of zero-crossings must be either equal or differ at most by one; 2) At any point, the mean value of the envelope defined by the local maxima and the envelope defined by the local minima is zero. C. Empirical mode decomposition method: the sifting process Unfortunately, most of the data are not IMFs. At any
(7)
g)
h)
X (a dummy variable) is set to the signal to be analyzed. Splines are fitted to the positive extrema (Ex+) and the negative extrema (Ex-). The average envelope, Aenv, is calculated as the arithmetic mean between Ex+ and Ex-. A possible IMF, imfp, is estimated as the difference between X and Aenv. If imfp is not an IMF than its value is assigned to the dummy variable X and the steps of a to e are repeated. If imfp is an IMF it is saved as imfi, where i is the ith component. The mean squared error, mse, between two consecutives IMFs, imfi-1 and imfi, is calculated and this value is compared to an ad hoc stop condition (usually a very small value, i.e. 10-5). If the stopping condition (step g) is reached then the sifting process is finalized and the final residue r can be estimated as the difference between S(t) and the sum of all IMFs obtained. Otherwise, the partial residue, rn, is estimated as the difference between a previous partial residue rn-1 and imfn, and its content is assigned to the dummy variable X and the steps of a to e are repeated. It is important to observe that r-1 is the input time-series S(t).
D. The Hilbert spectrum and the Hilbert marginal spectrum The five steps below are necessary in order to generate the Hilbert spectrum, H(w,t), where w is the frequency and t time [6]. They are as follows: a) Decompose the input signal, S(t), into IMFs. b) Calculate the Hilbert transform of each IMF. c) Generate analytical signals IMF + jHT{IMF}. d) Calculate the instantaneous attributes of the analytic signals.
3 of 4 e) The instantaneous frequency and the instantaneous amplitudes are functions of the time and they can be represented in a three-dimensional plot, in which the amplitude can be contoured on the frequency-time plane. This frequency-time distribution of the amplitude is designated as the Hilbert spectrum. Since the Hilbert spectrum is defined, it is also possible to define the Hilbert marginal spectrum (MHS), h(w), as shown by (7). h( w) = ò0T H ( w, t )dt
(7)
Where w is frequency, t is time and T is the period. The Hilbert marginal spectrum offers a measure of total amplitude (or energy) contribution from each frequency value. It represents the cumulated amplitude over the time/ data span in a probabilistic sense. III. RESULTS A. Hilbert spectrum calibration In order to calibrate the Hilbert spectrum, a quadratic swept-frequency cosine was simulated and its Hilbert spectrum and Fourier spectrogram were calculated. Fig. 1 shows that the energy in the Hilbert spectrum is concentrated only on the frequencies of the signal whereas in the Fourier spectrogram (Fig. 2) the dispersion of energy is clear and unavoidable.
B. Analysis of simulated electromyographic signals With the aim of evaluating the Hilbert spectrum and its potential, the analysis of simulated EMG signals was performed. An electromyographic signal (Fig. 3) was simulated as a summation of the individual action potentials of five active motor units, as described in [7],[8],[9]. The Hilbert spectrum of that signal was estimated, as shown in Fig. 4. The crosses are not part of the spectrum and include the firing time of one of those active motor units (chosen arbitrarily). A comparison between that time and the time when the local maxima of the instantaneous frequencies occur, suggests that it is possible to detect the instant when that motor unit is firing. This is possible because the technique (EMD) decomposes the original data into components with distinct time-scales. Therefore transitory events in the time domain (e.g. firing of motor units) may be detected by short-time scale components whereas slow events may be detected by large-time scale components. C. Analysis of surface electromyographic signals Following the theoretical analysis a practical experiment was accomplished. A pair of passive-surface electrodes was placed on the posterior deltoid of a subject and signals were collected at 4000 Hz. Four bursts of EMG signals were detected and the first one (Fig. 5) was selected, by using a rectangular window, for investigation.
Fig. 3. Simulated EMG signal.
Fig. 1. The Hilbert spectrum of a quadratic swept-frequency cosine with frequencies varying from 100 Hz to 500 Hz.
Fig. 4. Hilbert spectrum of the simulated EMG signal. Fig. 2. Fourier spectrogram of the same quadratic swept-frequency cosine with frequencies varying from 100 Hz to 500 Hz.
Fig. 5. Selected burst of EMG activity.
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Fig. 7. Power spectrum estimation using MHS and Fourier analyses.
The results in the present work have also shown that this mathematical tool may offer a new way of investigating the underlying process associated to the muscle contraction, i.e. the firing time of some active motor units. V. CONCLUSION Fig. 6. Intrinsic mode functions obtained from the decomposition of the EMG signal in analysis. Time-scales vary from the shortest one (top) to the largest one (bottom).
The selected EMG signal was decomposed into its intrinsic mode functions as shown in Fig. 6. The top component represents the shortest time-scale whereas the last one represents the largest time-scale. This implies that transitory events in the time domain may be located using the finest scale. The obtained IMFs were then used to estimate the Hilbert spectrum and from which was possible to calculate the Marginal Hilbert spectrum (Fig. 7). A power spectrum obtained using conventional Fourier technique is included for comparison. It is apparent that the MHS shows a concentration of the energy of the signal into its component frequencies. IV. DISCUSSION Traditional digital signal processing techniques that are based on Fourier analysis are not very suitable for the analysis of electromyographic signals. Those types of analyses do not take into account the nonlinearnonstationary nature of the signal. For this reason inaccuracies may occur. Consider, for instance, the investigation of muscle fatigue. Shifts in the median frequency obtained from the power spectrum function are always associated with this phenomenon. But since the PSD function is not accurately estimated, especially due to lack of resolution, it is not possible to quantify such shifts effectively. From the results presented in this work, the superiority of the empirical mode decomposition and the Hilbert spectrum is evident in representing time-frequency distributions, when compared with Fourier analysis. Huang et al. [1] showed also comparisons with Wavelets in the analysis of seismic data and also proved the superiority of the method.
This work has introduced a novel technique for analyzing EMG signals. The method is adaptive and based on local characteristics of the signal. From the analysis of the intrinsic mode functions it is possible to conclude that the technique is best suited to cases when signal time-scales are distinct from the noise time-scales. The results have shown that this technique may provide not only an increase in the spectral resolution but also an insight into the basic process of the muscle contraction. REFERENCES [1] N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N.-C. Yen, C. C. Tung, and H. H. Liu, "The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis," Procedures of Royal Society of London, vol. 454, pp. 903-995, 1998. [2] M. Akay, Detection and estimation methods for biomedical signals. San Diego - California - USA: Academic Press, 1999. [3] M. Akay, Biomedical signal processing, 1 ed. Sandiego, California: Academic Press, Inc, 1994. [4] S. Lawrence Marple, Jr, "Computing the discrete-time ‘analytic’ signal via FFT," IEEE Transactions on Signal Processing, vol. 47, pp. 26002603, 1999. [5] L. Debnath and P. Mikusinski, Introduction to Hilbert spaces with applications, Second ed. San Diego, California, USA: Academic Press, 1999. [6] S. Taffler, "The use of the Hilbert Spectrum in the analysis of electromyographic signal and its application in the development of myoelectric prosthesis controllers," Ph.D. dissertation in Department of Engineering Science. Oxford: The University of Oxford, 2001, pp. 183. [7] P. Rosenfalck, "Intra- and extracellular potential fields of active nerve and muscle fibres: a physico-mathematical analysis of different models," Acta Physiologica Scandinavica, vol. Supplementum 321, pp. 1-166, 1969. [8] K. C. McGill, Z. C. Lateva, and S. Xiao, "A model of the muscle action potential for describing the leading edge, terminal wave, and slow afterwave," IEEE Transactions on Biomedical Engineering, vol. 48, pp. 1357 - 1365, 2001. [9] D. Stashuk, "EMG signal decomposition: how it is performed and how it can be used?," Journal of Electromyography and Kinesiology, vol. 11, pp. 151-173, 2001.
