Adaptive Notch Filter for EEG Signals Based on the LMS Algorithm with Variable Step-Size Parameter

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1 5 Conference on Information Sciences and Systems, The Johns Hopkins University, March 16 18, 5 Adaptive Notch Filter for EEG Signals Based on the LMS Algorithm with Variable Step-Size Parameter Daniel Olguín Olguín 1 A347811@itesm.mx Frantz Bouchereau 1 fbl@itesm.mx Sergio Martínez 1 smart@itesm.mx Abstract This paper presents the use of an adaptive noise canceler (ANC) with variable step-size parameter for the elimination of power line interference in the recording of EEG signals within the relatively unexplored gamma-band (35-1 Hz). The use of an adaptive step-size parameter offers a balance in terms of convergence, misadjustment, and rejection bandwidth optimization. Simulation results are presented to support the proposed algorithm and compare its performance with fixed step-size ANC schemes. It will be shown that the proposed algorithm outperforms classical fixed step-size ANC algorithms and eliminates the cumbersome trial and error process needed to choose an adequate value for such parameter. I. Introduction The elimination of the interference caused by power transmission lines in the recording of physiological signals of electrical nature has been an active topic of research for the last few decades [1], [], [3]. The majority of electrophysiological recordings unavoidably contain an undesired level of interference deriving from the power transmission lines. Moreover, the line-frequency contamination is not of constant amplitude, phase, or even frequency [1]. Such variability prevents a simple subtractive filter from being completely effective. A fixed notch filter may eliminate the noise when its distribution is centered exactly at the frequency for which the filter was designed []. However, the frequency of the power-line noise is not constant at exactly 6 Hz. The importance of the work presented in this paper relies on the fact that there is existence of epileptiform oscillations with frequencies nearby the power line interference frequency which have been ignored because of the lack of an effective notch filter capable of eliminating the noise components without affecting the original electroencephalographic (EEG) signal. Worrell et al. [4] have found that currently available clinical EEG systems and EEG analysis methods utilize a dynamic range (.1-3 Hz) that discards clinically important information. Their results show that the dynamic range utilized in current clinical practice largely ignores fundamental oscillations that are signatures of an epileptogenic brain. A finer study of high-frequency EEG oscillations may open a new possibility for patients who are poor candidates to epilepsy surgery, allowing seizure prediction and epilepsy treatment through several therapeutic methods. The results presented in [4] suggest the need to design a notch filter with an optimal rejection bandwidth that effectively eliminates the time-varying noise introduced by power 1 All three authors are with the Department of Electrical Engineering, Tecnológico de Monterrey, Campus Monterrey, Mexico. The authors acknowledge the finantial support provided by the Consejo de Ciencia y Tecnología del Estado de Nuevo León (CO- CYTENL). transmission lines. The proposed filter should have an optimum speed of convergence and allow the minimization of loss of information and distortion of the signal of interest. Power line interference recorded in EEG generally results from poor electrode application on the scalp. This noise is often due to high-impedance electrodes that, when connected to the recording device, affect the common mode rejection ratio of the amplifier, which changes when the impedances of electrodes and scalp are not matched. Ensuring impedance measurements of less than 5 kω will usually reduce such linefrequency noise [5]. The standard practice now is to measure all potentials relative to a common electrode which is isolated from ground. This improves subject safety and reduces power line noise. However, taking all the existing measures to minimize the interference is not enough when we are interested in measuring signals with frequency components that are very close to those of the interference, and with amplitudes which are one or more orders of magnitude smaller than the noise. There have been several attempts of eliminating power line interference by using digital signal processing techniques [], [3]. In applications where the information of interest is contained within the classical EEG bands: delta (-4 Hz), theta (4-7 Hz), alpha (7-13 Hz) and beta (13-35 Hz); it is of common practice to use a 6/5 Hz notch filter with a fixed null in its frequency response characteristic to remove the noise from the data. Sometimes the EEG signal is further low-pass filtered with a cut-off frequency of less than 5 Hz to assure the integrity of the data. In [], three different adaptive notch filters are considered: an FIR second-order filter, a second-order IIR filter with fixed zeros and varying poles, and a second-order IIR filter with varying zeros and poles. The three filters were designed using a constrained LMS algorithm with fixed step-size. It was observed that only the non-fixed pole-zero IIR filter was able to track the frequency variation with a variable bandwidth. However, a difficult design issue that arised from this filtering scheme was that of choosing an adequate step-size parameter to adjust the filter s coefficients and obtain optimal convergence, tracking and rejection bandwidth conditions. In this paper, we propose an adaptive notch filter to eliminate the interference introduced by power transmission lines in the recording of EEG signals within the gamma-band (35-1 Hz), based on an adaptive noise canceling scheme implemented with a variable step-size LMS algorithm. The proposed algorithm avoids the cumbersome trial and error process needed to choose an adequate value for the step-size parameter and will minimize the rejection bandwidth required to effectively eliminate the time-varying interference while, at the same time, preserving optimal convergence, tracking and misadjustment conditions. Numerical results will be presented to compare the algorithm with the classical fixed step-size adaptive noise canceling scheme. We will consider the case where the frequency of

2 the noise is varying around the nominal frequency of 6 Hz, following a first-order Gauss-Markov process model. The paper is organized as follows. Section II describes the model utilized for the EEG and time-varying interference signals. The classical adaptive noise canceler (ANC) is introduced in Section III where the proposed algorithm for the step-size parameter adaptation is also presented. Section IV presents simulation results. Here, several comparisons between the fixed step-size ANC and our proposed algorithm are presented. Finally, Section V presents conclusions. II. Signal Model The signal observation model is given by x(n) =s(n)+p(n), (1) where s(n) is the EEG signal of interest and p(n) is an additive time-varying sinusoidal interference. Several mathematical models have been developed to describe EEG signals. Some examples include AR models, matching pursuit method-based models, Kalman filters, and Markov processes. Since the EEG signal is highly nonstationary its statistical properties are difficult to model accurately. In this paper we use a Markov process amplitude model originally developed by Nakamura et al. [6]. The artificially generated EEG signal is formed by a linear combination of K different oscillations x k (k =1,,..., K) as given by K K s(n) = a k(n)x k(n) = a k(n) sin(πm kn + φ k), () k=1 k=1 III. Adaptive Noise Canceling System with Variable Step-Size Parameter The transfer function for a nd-order FIR notch filter is given by H(z)=1 cos(πf )z 1 + z, (6) with a null at frequency f. The problem with this filter is that the notch has a relatively large bandwidth, which means that other frequency components around the desired null are severely attenuated [7]. To improve the frequency characteristics of the filter we may consider a nd-order IIR notch filter with transfer function H(z)= 1 cos(πf)z 1 + z, (7) 1 ζ cos(πf )z 1 + ζ z where ζ is a constant that defines the location of the poles in the unit circle. A very narrow notch is usually desired in order to filter out a sinusoidal interference without distorting the original signal. However, if the interference is not precisely known, and if the notch is very narrow, the center of the notch may not fall exactly over the interference. When a reference for the interference is available, the adaptive noise canceling method originally proposed in [8] may be used. In this method, the interference is adaptively filtered to match the interfering sinusoid as closely as possible, allowing them to then be subtracted out. The system is shown in Figure 1. where a k(n) is the model s amplitude obtained from a firstorder Gauss-Markov process, m k is the dominant k-th frequency, and φ k is the initial phase. The (n + 1)-th value of the model s amplitude is defined as a k(n +1)=γ ka k(n)+ξ k(n), (3) where ξ k(n) is a random increment of Gaussian distribution with zero mean and variance σ ξ,k, and γ k is the coefficient of the first-order Markov process which must satisfy the condition <γ k < 1 for stability. The power line noise interference is a frequency-varying sinusoidal with a center frequency of 6 Hz. In this work this interference will be modeled as p(n) =A cos{π[f + f v(n)]n + ψ}. (4) Here, A is assumed to be a constant and deterministic amplitude, f is the central frequency with a value of 6 Hz, the initial phase ψ is a random variable uniformly distributed over [, π], and f v(n) is a slowly varying random frequency which is assumed to be a steady state realization of a zero-mean Gauss-Markov process given by f v( k+1) =ρf v( k)+η( k), (5) where k is the time interval index for the process which might be larger than the signal sampling interval T, ρ is the coefficient of the first-order Markov process, and η( k)isa random increment of Gaussian distribution with zero mean and variance ση. Figure 1: Adaptive noise canceling system. Applying this scheme to the problem of filtering a noisy EEG signal, the primary input x(n) of the system corresponds to the clean EEG signal s(n) corrupted by power line noise p(n). These signals are assumed to be uncorrelated, E{s(n)p(k)} =, (n, k). The reference input r(n) is a sinusoidal signal with frequency f r and zero phase. The value of f r is set to the noise interference center frequency f. This reference signal given by r(n) =C cos(πf rn), is applied to an M-stage tapped delay line. Here C is a constant deterministic amplitude usually different from A (the interference amplitude). The values at the M taps at time n form the reference M-vector r(n) =[r(n) r(n 1)... r(n M + 1)] T. The output of the filter y(n) is estimated to match the noise p(n) in the primary input. The noise and the reference signals are assumed to be correlated, E{p(n)r(k)}. If we define the M-dimensional filter coefficient vector as w(n) =[w (n) w 1(n)... w M 1(n)] T, then the equations that describe the adaptation of the system based on the LMS algorithm with fixed step-size are given by y(n) =w T (n)r(n), (8) e(n) =d(n) y(n), (9) w(n +1) =w(n) +µe(n)r(n). (1)

3 Let d(n) denote the desired signal, which in this case is equivalent to the primary input x(n). The error signal e(n) is defined as the difference between the desired signal d(n) and the filter s output signal y(n) = ˆp(n), then e(n) =x(n) y(n), e(n) =s(n) +p(n) ˆp(n), e(n) =ŝ(n). (11) Clearly ŝ(n) is an estimate of the noise-free EEG signal and the LMS algorithm is designed to minimize an instantaneous version of the mean square error () given by E{ e(n) } = E{ ŝ(n) }. It is well known that the ANC transfer function from d(n) to e(n),is given by [8] H(z) = ( 1 1 MµC 4 1 cos (πf r) z 1 + z ) ( cos(πf r)z MµC By comparing equations (1) and (7), we observe that ζ =1 MµC ) z, (1) ( ) MµC +. (13) 4 If µ is small enough, this can be approximated to ζ =1 MµC. Then, it is clear that (1) is the transfer function of a nd-order digital IIR notch filter with a null centered at the reference frequency f r. It can be shown that the 3- bandwidth of the null is given by BW 3 = MµC (Hz). (14) 4πT It is clear from equations (1) and (14) that the position of the filter poles and the notch bandwidth are directly affected by the step-size parameter µ. The pole locations become closer to the unit circle and the rejection bandwidth becomes smaller as µ decreases. Hence, the choice of the step-size parameter in the ANC algorithm represents a tradeoff between misadjustment, speed of convergence, tracking, notch attenuation and rejection bandwidth. For the application of interest it is desired to have the smallest notch bandwidth, however, it is not possible to minimize this bandwidth by making µ arbitrarily small. Instead, to ensure an optimal equilibrium between all the desired filter characteristics it becomes necessary to find a method to chose the step-size parameter in an optimal way at every iteration of the algorithm. Several variable step-size parameter algorithms have been proposed in the literature [9], [1]. In [1] the recursion to obtain µ(n) is based on an estimator of e(n)/ µ(n), however, the complexity of the algorithm and its requirement of the independence condition E{r(n)r(k)} =, n, k makes it not suitable for our application. The objective is to ensure large µ(n) when the algorithm is far from the optimum, and decreasing µ(n) as we approach the optimum hence decreasing the notch bandwidth and increasing noise attenuation. The step-size adjustment proposed in [9] is controlled by the square of the prediction error. The simplicity of the algorithm and its sensibility to changes in the error signal allowed us to implement it in the ANC scheme. The algorithm for updating µ(n) is as follows µ(n +1)=αµ(n)+γe (n). (15) The constant α is a forgetting factor with values between < α < 1 and γ > is the step-size parameter for the adaptation of µ. Substituting the variable step-size µ(n) in (1) the update equation for the filter coefficients becomes w(n +1)=w(n)+µ(n)e(n)r(n). (16) The initial step-size µ() is usually set to µ max and this maximum value is chosen to ensure stability of the algorithm. IV. Simulation Results We applied the algorithm described by equations (8), (9), (15) and (16) to artificially generated EEG signals corrupted with time-varying power line noise and evaluated its effectiveness by analyzing the rate of convergence, misadjustment, rejection bandwidth and tracking capabilities. Figure shows the power spectral density (PSD) of the EEG signal generated by using the Markov process amplitude model described in Section II with K = dominant frequencies located at 3 Hz and 1.5 Hz, corresponding to an EEG recording during baseline. To create the frequency peaks we selected γ 1 =.98, γ =.99, and σ ξ,1 = σ ξ, =.1. 5 PSD of artificially generated EEG signal s(n) Figure : PSD of EEG signal generated by using a Markov process amplitude model with dominant frequencies at 3 Hz and 1.5 Hz. Let us present four experiments to compare the performance of the fixed step-size ANC algorithm and the varying step-size ANC algorithm. The sampling frequency used along the experiments was set to f s = 4 Hz and the number of filter coefficients was set to M =8. Let s(i) and ŝ(i) be N-dimensional vectors of the noise-free and estimated signal at the i-th experiment realization. Then, the ensemble average presented in the following results is obtained as = 1 Q Q s(i) ŝ (i). (17) i=1 For the experiments presented in this section, N = samples and Q = trials of the experiment. Note that this is with respect to the noise free and estimated signal which is different to the defined in Section III for the output error e(n). Experiment 1 The amplitude of the generated signal s(n) was normalized to [ 1, 1] and a power line noise signal p(n) with amplitude

4 PSD of x(n) in the gamma band.1 Step size parameter variation over time.9.8 PSD of s(n) in the gamma band 5 µ(n) Figure 3: (Top) PSD of signal plus noise, (Bottom) PSD of original EEG signal Figure 5: Convergence behavior of the step-size parameter for Experiment µ=. 1 3 µ=. Variable Step Size 1 4 Variable step size µ= µ= Figure 4: Comparison of using a variable step-size parameter µ(n), and two different fixed step-size parameters µ=., and µ=.5 for Experiment 1. A =.1 and constant frequency f = 6 Hz was generated in order to analyze the rate of convergence of the LMS algorithm that adapts the weights of the ANC when using two different fixed values for the step-size parameter and when using a variable step-size parameter µ(n). We can see in Figure 3 that the power of the noise signal is significantly superior to the power of the EEG signal in the gamma-band. Figure 4 shows the curves for three different cases of step-size parameter selection. For the first case the value of step-size parameter was fixed at µ =.. This value is below the optimum value found when using a variable step-size parameter and therefore the algorithm converges slowly after approximately 14, iterations. For the second case the step-size parameter was fixed at µ =.5, near its maximum allowable value µ max. We can observe that the algorithm converges very fast, after approximately 5 iterations, but with the disadvantage of a large misadjustment. For the third case the noisy EEG signal was filtered using the variable step-size ANC algorithm. Clearly this algorithm maintains an equilibrium between fast convergence and small misadjustment. Figure 5 shows the ensemble average (over two hundred realizations) for the adaptation curve of the step-size parameter when using equation (15). The initial step-size parameter was set to µ = µ max and this parameter converged to its average optimum value after approximately 5 iterations. Note that after the step-size has reached its average optimum value, which in this case was found to be µ opt =.5, it Figure 6: behavior when a step-change of frequency is applied to the noise signal. continues to vary around this value following the changes of the estimated EEG signal e(n) = ŝ(n). It is clear that the adaptive step-size algorithm will minimize the value of µ after convergence to minimize misadjustment. Recalling equation (14), this means that the algorithm will optimize the rejection bandwidth while keeping excellent convergence properties as well as tracking capabilities as will be shown next. Experiment To analyze the tracking capabilities of the algorithm we set the noise signal to have a frequency step-change of 1 Hz. This means that the center frequency of the interference varied from f = 6 Hz to f = 5 Hz. Obviously, the reference signal frequency was kept fixed at f r = 6 Hz. The abrupt change proposed in this example would never occur in reality since the power line frequency must be robust enough as to drift only in small quantities. However, this test was used for analysis purposes only. Figure 6 shows the behavior of the when the change in frequency was applied. We can see from the figure that the increases when the change in frequency occurs, and then it converges again to a minimum value for each of the different cases of step-size parameter selection. Figure 7 shows the ensemble average (over two hundred realizations) for the adaptation curve of the step-size parameter. It is interesting to note that even with the large

5 .1 Step size parameter variation over time µ(n) µ= Variable Step Size µ= Figure 7: Convergence behavior of the step-size parameter for Experiment. 1 6 Time frequency variation of noise power spectral density Figure 9: behavior when the noise frequency is varying every seconds. 4 6 PSD PSD of s(n) in the gamma band Figure 8: Time-frequency magnitude plot of the noise PSD for Experiment frequency step-change, the step-size values did not change considerably. This means that the rejection bandwidth of the filter remains fairly constant even in the presence of large unstationarities. We can conclude once more that the variable step-size algorithm minimizes the while preserving good tracking capabilities, optimal step-size values and small rejection bandwidths. Experiment 3 Finally we consider the case when the power line frequency is constantly varying around its nominal value f = 6 Hz, following the Gauss-Markov model described in Section II. We defined k = seconds, ρ =.99, and σ η =.1 and generated a power line noise signal whose frequency variation with time is shown in Figure 8. Figure 9 shows the for the three different cases of step-size parameter selection. We can readily identify the instants in which the noise drifted in frequency. It is important to notice that in all cases the filters were able to track the frequency changes. However, the variable step-size algorithm was able to track the frequency changes while maintaining a fast convergence rate, a small misadjustment and an optimum step-size value and hence a minimum rejection bandwidth. Figure 1 shows the PSD of the signal estimate ŝ(n) for the three different selections of step-size parameter considered in this experiment. By comparing these plots with the PSD of the original EEG signal we can see that when the step-size PSD of estimate of s(n) in the gamma band when using variable µ PSD of estimate of s(n) in the gamma band when using µ= PSD of estimate of s(n) in the gamma band when using µ= Figure 1: Comparison of power spectrum in the gamma-band of the noise free signal and its estimates. (Top) Spectrum of noise free signal s(n). (Second) ŝ(n) obtained with adaptive µ, (Third) ŝ(n) obtained with µ=., and (Bottom) ŝ(n) obtained with µ=.5.

6 x(n) s(n) Estimate of s(n) Original EEG signal + noise in the gamma band Original EEG signal in the gamma band Signal estimate when using variable step size Figure 11: Gamma-band signals: (Top) Signal plus interference, (Center) Original EEG signal s(n), (Bottom) Estimated signal ŝ(n) using the variable step-size parameter algorithm Variable step size µ=. µ=.5 Figure 1: behavior in the presence of a periodic nonsinusoidal noise signal. parameter is adequately chosen, there is no distortion of the spectral content in the filtered signal. This is true for the varying step-size parameter as well as for the case when it was kept constant with µ =.. On the other hand, when the step-size was set to µ =.5 the rejection bandwidth of the filter became too large and the distortion of the spectral content became apparent. One can clearly observe a null attenuating several frequencies around 6 Hz for this scenario. Figure 11 shows the EEG signal plus noise in the gamma-band (4-8 Hz), the original noise-free EEG signal, and the signal estimate ŝ(n) obtained with the varying step-size parameter algorithm. We can appreciate that the original signal is completely masked by the noise signal, and how well it is reconstructed by the ANC system with varying step-size parameter proposed in this paper. Experiment 4 In this experiment we analyze the effects of periodic nonsinusoidal noise on the ANC algorithm. Figure 1 shows the when the interference was set to be a sinusoidal truncated at values of ±5% of its peak value. It is clear that although the convergence time of the algorithm increased, it was still able to eliminate the interference using a sinusoidal reference. This result is relevant in cases where amplifier saturation may occur and in cases where the signal is windowed in time. V. Conclusions When dealing with EEG signals in the gamma-band (35-1 Hz), it is desirable to have a notch filter with a small rejection bandwidth that effectively eliminates the time-varying noise introduced by power transmission lines. The proposed ANC system based on a variable step-size LMS algorithm is able to find an optimum speed of convergence which is of great importance in real-time applications and allows the minimization of information loss and signal distortion by keeping the notch bandwidth as small as possible. The proposed filters could be implemented in existing EEG recording devices or in new devices intended for real-time ambulatory EEG monitoring. The choice of the step-size parameter in the adaptation algorithm plays an important role in the rate of convergence, stability, tracking capabilities and rejection bandwidth of the filters. The proposed variable step-size method may overcome the cumbersome trial and error process needed to choose an adequate value for such parameter and will minimize the rejection bandwidth required to effectively eliminate the timevarying interference introduced by power transmission lines. This last property is of great importance since, as mentioned in the introductory paragraphs, valuable signal information is found around the interference frequency band. References [1] K. J. Eriksen Non-Distorting Post-Acquisition Line-Frequency for Evoked Potentials, Proceedings of the IEEE EMBS 1th Annual International Conference, p. 1168, [] M. Ferdjallah and R. E. Barr Adaptive Digital Notch Filter Design on the Unit Circle for the Removal of Powerline Noise from Biomedical Signals, IEEE Trans. on Biomedical Engineering, vol. 41, no. 6, pp , [3] M. V. Dragosevic, and S. S. Stankovic An Adaptive Notch Filter with Improved Tracking Properties, IEEE Trans. Signal Processing, vol. 43, no. 9, pp , [4] G. A. Worrell, S. D. Cranstoun, R. Jonas, G. Baltuch and B. Litt High-frequency oscillations and seizure generation in neocortical epilepsy, Brain, vol. 17, no. 7, pp , 4. [5] J. S. Ebersole, and T. A. Pedley Current practice of clinical electroencephalography, Lippincott Williams and Wilkins, 3rd Ed., USA, p. 81, 3. [6] O. Bai, M. Nakamura, A. Ikeda, and H. Shibasaki Nonlinear Markov Process Amplitude EEG Model for Nonlinear Coupling Interaction of Spontaneous EEG, IEEE Trans. on Biomedical Engineering, vol. 47, no. 9, pp ,. [7] J. G. Proakis, and D. G. Manolakis Digital Signal Processing, Prentice Hall, 3rd Ed., pp , USA, [8] J. R. Glover Adaptive Noise Canceling Applied to Sinusoidal Interferences, IEEE Trans. on Acoustics, Speech, and Signal Processing, vol. ASSP-5, no. 6, pp , [9] R. H. Kwong, and E. W. Johnston A Variable Step Size LMS Algorithm, IEEE Trans. Signal Processing, vol. 4, no. 7, pp , 199. [1] A. M. Kuzminskiy A Robust Step Size Adaptation Scheme for LMS Adaptive Filters, IEEE Workshop on Digital Signal Processing, pp , 1997.