Conference on Information Sciences and Systems The Johns Hopkins University March   Adaptive Notch Filter for EEG Signals Based on the LMS Algorithm with Variable StepSize Parameter Daniel Olgu n Ol

Conference on Information Sciences and Systems The Johns Hopkins University March Adaptive Notch Filter for EEG Signals Based on the LMS Algorithm with Variable StepSize Parameter Daniel Olgu n Ol - Description

mx Frantz Bouchereau email fblitesmmx Sergio Mart57524305nez email smartitesmmx Abstract This paper presents the use of an adaptive noise canceler ANC with variable stepsize parameter for the elimination of power line interference in the recording of ID: 26054 Download Pdf

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Conference on Information Sciences and Systems The Johns Hopkins University March Adaptive Notch Filter for EEG Signals Based on the LMS Algorithm with Variable StepSize Parameter Daniel Olgu n Ol

mx Frantz Bouchereau email fblitesmmx Sergio Mart57524305nez email smartitesmmx Abstract This paper presents the use of an adaptive noise canceler ANC with variable stepsize parameter for the elimination of power line interference in the recording of

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Conference on Information Sciences and Systems The Johns Hopkins University March Adaptive Notch Filter for EEG Signals Based on the LMS Algorithm with Variable StepSize Parameter Daniel Olgu n Ol




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2005 Conference on Information Sciences and Systems, The Johns Hopkins University, March 16–18, 2005 Adaptive Notch Filter for EEG Signals Based on the LMS Algorithm with Variable Step-Size Parameter Daniel Olgu ın Olguın e-mail: A00347811@itesm.mx Frantz Bouchereau e-mail: fbl@itesm.mx Sergio Martınez e-mail: 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 unex- plored

gamma-band (35- 100 Hz). The use of an adap- tive step-size parameter offers a balance in terms of convergence, misadjustment, and rejection bandwidth optimization. Simulation results are presented to sup- port the proposed algorithm and compare its perfor- mance with fixed step-size ANC schemes. It will be shown that the proposed algorithm outperforms clas- sical 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 trans-

mission lines in the recording of physiological signals of elec- trical nature has been an active topic of research for the last few decades [1], [2], [3]. The majority of electrophysiological recordings unavoidably contain an undesired level of interfer- ence 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 sim- ple subtractive filter from being completely effective. A fixed notch filter may eliminate the noise when its distri- bution is

centered exactly at the frequency for which the filter was designed [2]. However, the frequency of the power-line noise is not constant at exactly 60 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 elim- inating 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 (0.1-30 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 predic- tion and epilepsy treatment through several therapeutic meth- ods. The results presented in [4] suggest the need to design a

notch filter with an optimal rejection bandwidth that effec- tively eliminates the time-varying noise introduced by power All three authors are with the Department of Electrical Engi- neering, Tecnologico de Monterrey, Campus Monterrey, Mexico. The authors acknowledge the finantial support provided by the Consejo de Ciencia y Tecnologıa del Estado de Nuevo Leon (CO- CYTENL). transmission lines. The proposed filter should have an opti- mum 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 ra- tio 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 line- frequency 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 min- imize 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 tech- niques [2], [3]. In applications where the information of in- terest is contained within the classical EEG bands: delta (0-4 Hz), theta (4-7 Hz),

alpha (7-13 Hz) and beta (13-35 Hz); it is of common practice to use a 60/50 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 50 Hz to assure the integrity of the data. In [2], three different adap- tive 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 band- width conditions. In this paper, we propose an adaptive notch filter to elim- inate the interference introduced

by power transmission lines in the recording of EEG signals within the gamma-band (35- 100 Hz), based on an adaptive noise canceling scheme imple- mented with a variable step-size LMS algorithm. The pro- posed algorithm avoids the cumbersome trial and error pro- cess needed to choose an adequate value for the step-size pa- rameter 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 algo-

rithm with the classical fixed step-size adaptive noise cancel- ing scheme. We will consider the case where the frequency of
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the noise is varying around the nominal frequency of 60 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 sig- nals. The classical adaptive noise canceler (ANC) is intro- duced in Section III where the proposed algorithm for the step-size parameter adaptation is also presented. Section IV presents simulation results.

Here, several comparisons be- tween 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 )= )+ (1) where ) is the EEG signal of interest and ) is an additive time-varying sinusoidal interference. Several mathematical models have been developed to de- scribe EEG signals. Some examples include AR models, matching pursuit method-based models, Kalman filters, and Markov processes. Since the EEG signal is highly non- stationary its statistical properties are difficult to

model ac- curately. In this paper we use a Markov process amplitude model originally developed by Nakamura et al. [6]. The artifi- cially generated EEG signal is formed by a linear combination of different oscillations =1 , ..., K ) as given by )= =1 )= =1 ) sin(2 πm (2) where ) is the model’s amplitude obtained from a first- order Gauss-Markov process, is the dominant -th fre- quency, and is the initial phase. The ( + 1)-th value of the model’s amplitude is defined as +1)= )+ (3) where ) is a random increment of Gaussian distribution with zero mean and variance

ξ,k , and is the coefficient of the first-order Markov process which must satisfy the condi- tion 0 < 1 for stability. The power line noise interference is a frequency-varying si- nusoidal with a center frequency of 60 Hz. In this work this interference will be modeled as )= cos )] (4) Here, is assumed to be a constant and deterministic ampli- tude, is the central frequency with a value of 60 Hz, the initial phase is a random variable uniformly distributed over [0 ], and ) is a slowly varying random frequency which is assumed to be a steady state realization of a zero-mean

Gauss-Markov process given by ( +1 )= ρf ( )+ ( (5) where is the time interval index for the process which might be larger than the signal sampling interval is the coefficient of the first-order Markov process, and ( )isa random increment of Gaussian distribution with zero mean and variance III. Adaptive Noise Canceling System with Variable Step-Size Parameter The transfer function for a 2nd-order FIR notch filter is given by )=1 2 cos(2 πf (6) with a null at frequency . 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 character- istics of the filter we may consider a 2nd-order IIR notch filter with transfer function )= 2 cos(2 πf cos(2 πf (7) where 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 sig- nal. 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 sinu- soid as closely as possible, allowing them to then be subtracted out. The system is shown in Figure 1. Figure 1: Adaptive noise canceling system. Applying this scheme to the problem of filtering a noisy EEG signal, the primary input ) of the system corre- sponds to the clean EEG signal ) corrupted by power line noise ). These

signals are assumed to be uncorrelated, =0, n,k ). The reference input ) is a sinu- soidal signal with frequency and zero phase. The value of is set to the noise interference center frequency . This reference signal given by )= cos(2 πf ), is applied to an -stage tapped delay line. Here is a constant deter- ministic amplitude usually different from (the interference amplitude). The values at the taps at time form the reference -vector )=[ 1) ...r + 1)] The output of the filter ) is estimated to match the noise ) in the primary input. The noise and the reference signals are

assumed to be correlated, }6 =0. If we define the -dimensional filter coefficient vector as )=[ ...w )] , then the equations that describe the adaptation of the system based on the LMS al- gorithm with fixed step-size are given by )= (8) )= (9) +1)= )+ e (10)
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Let ) denote the desired signal, which in this case is equivalent to the primary input ). The error signal )is defined as the difference between the desired signal ) and the filter’s output signal )= ), then )= )= )+ )= (11) Clearly ) 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 (MSE) given by {| {| It is well known that the ANC transfer function from to ),is given by [8] )= cos (2 πf C cos (2 πf C (12) By comparing equations (12) and (7), we observe that =1 MC MC (13) If is small enough, this can be approximated to =1 MC . Then, it is clear that (12) is the transfer function of a 2nd-order digital IIR notch filter with a null centered at the reference frequency . It can be shown that the 3-dB bandwidth of the

null is given by BW dB MC πT Hz (14) It is clear from equations (12) and (14) that the position of the filter poles and the notch bandwidth are directly affected by the step-size parameter . The pole locations become closer to the unit circle and the rejection bandwidth becomes smaller as decreases. Hence, the choice of the step-size parameter in the ANC algorithm represents a tradeoff between misadjust- ment, speed of convergence, tracking, notch attenuation and rejection bandwidth. For the application of interest it is de- sired to have the smallest notch

bandwidth, however, it is not possible to minimize this bandwidth by making 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], [10]. In [10] the recursion to obtain ) is based on an estimator of ∂e / ), however, the complexity of the algorithm and its requirement of the independence condition =0, n,k makes

it not suitable for our application. The objective is to ensure large ) when the algorithm is far from the optimum, and decreasing ) as we approach the optimum hence decreasing the notch bandwidth and in- creasing 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 ) is as follows +1)= )+ γe (15) The constant is a forgetting factor with values between <α< 1 and γ> 0

is the step-size parameter for the adaptation of . Substituting the variable step-size )in (10) the update equation for the filter coefficients becomes +1)= )+ (16) The initial step-size (0) is usually set to max and this max- imum 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 effec- tiveness by analyzing the rate of convergence, misadjustment, rejection bandwidth and

tracking capabilities. Figure 2 shows the power spectral density (PSD) of the EEG signal generated by using the Markov process amplitude model described in Section II with = 2 dominant frequen- cies located at 3 Hz and 12.5 Hz, corresponding to an EEG recording during baseline. To create the frequency peaks we selected =0 98, =0 99, and ξ, ξ, =0 01. 10 20 30 40 50 60 70 80 90 100 −60 −55 −50 −45 −40 −35 −30 −25 −20 PSD of artificially generated EEG signal s(n) Frequency (Hz) dB Figure 2: PSD of EEG signal generated by using a

Markov process amplitude model with dominant frequencies at 3 Hz and 12.5 Hz. Let us present four experiments to compare the perfor- mance of the fixed step-size ANC algorithm and the varying step-size ANC algorithm. The sampling frequency used along the experiments was set to = 400 Hz and the number of filter coefficients was set to =8. Let ) and ˆs )be -dimensional vectors of the noise-free and estimated signal at the -th experiment realization. Then, the ensemble average MSE presented in the following results is obtained as MSE =1 ˆs (17) For the experiments presented in

this section, = 16384 samples and = 200 trials of the experiment. Note that this MSE is with respect to the noise free and estimated signal which is different to the MSE defined in Section III for the output error ). Experiment 1 The amplitude of the generated signal ) was normalized to [ 1] and a power line noise signal ) with amplitude
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40 45 50 55 60 65 70 75 80 −60 −55 −50 −45 −40 −35 −30 −25 −20 PSD of x(n) in the gamma−band Frequency (Hz) dB 40 45 50 55 60 65 70 75 80 −60 −55 −50

−45 −40 −35 −30 −25 −20 PSD of s(n) in the gamma−band Frequency (Hz) dB Figure 3: (Top) PSD of signal plus noise, (Bottom) PSD of orig- inal EEG signal 2000 4000 6000 8000 10000 12000 14000 16000 10 −6 10 −5 10 −4 10 −3 Variable step−size =0.02 ↓µ =0.5 Mean Square Error Number of iterations MSE Figure 4: Comparison of MSE using a variable step-size param- eter (n), and two different fixed step-size parameters =0.02, and =0.5 for Experiment 1. =0 1 and constant frequency = 60 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 vari- able step-size parameter ). 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 MSE curves for three different cases of step-size parameter selection. For the first case the value of step-size parame- ter was fixed at =0 02. This value is below the optimum value found when using a

variable step-size parameter and therefore the algorithm converges slowly after approximately 14,000 iterations. For the second case the step-size parameter was fixed at =0 5, near its maximum allowable value max We can observe that the algorithm converges very fast, after approximately 500 iterations, but with the disadvantage of a large misadjustment. For the third case the noisy EEG sig- nal 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 parame- ter when using equation (15). The initial step-size parameter was set to max and this parameter converged to its average optimum value after approximately 500 iterations. Note that after the step-size has reached its average optimum value, which in this case was found to be opt =0 05, it 2000 4000 6000 8000 10000 12000 14000 16000 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Step−size parameter variation over time Number of iterations (n) Figure 5: Convergence behavior of the step-size parameter for Experiment 1.

2000 4000 6000 8000 10000 12000 14000 16000 10 −5 10 −4 10 −3 Variable Step−Size =0.02 ↓µ =0.5 Mean Square Error Number of iterations MSE Figure 6: MSE behavior when a step-change of frequency is ap- plied to the noise signal. continues to vary around this value following the changes of the estimated EEG signal )= ). It is clear that the adaptive step-size algorithm will minimize the value of 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 2 To analyze the tracking capabilities of the algorithm we set the noise signal to have a frequency step-change of 10 Hz. This means that the center frequency of the interference varied from = 60 Hz to = 50 Hz. Obviously, the reference signal frequency was kept fixed at = 60 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 MSE when the change in frequency was applied. We can see from the figure that the MSE 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 pa- rameter. It is interesting to note that even with the large
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2000 4000 6000 8000 10000 12000 14000 16000 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Step−size parameter variation over time

Number of iterations (n) Figure 7: Convergence behavior of the step-size parameter for Experiment 2. 10 20 30 40 50 55 60 65 70 Time (sec) Time−frequency variation of noise power spectral density Frequency (Hz) PSD Figure 8: Time-frequency magnitude plot of the noise PSD for Experiment 3. 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 MSE 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 = 60 Hz, following the Gauss-Markov model described in Section II. We defined = 2 seconds, =0 99, and =0 1 and generated a power line noise signal whose frequency variation with time is shown in Figure 8. Figure 9 shows the MSE 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 10 shows the PSD of the signal estimate ) 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 2000 4000 6000 8000 10000 12000 14000

16000 10 −6 10 −5 10 −4 10 −3 Variable Step−Size =0.02 =0.5 Mean Square Error Number of iterations MSE Figure 9: MSE behavior when the noise frequency is varying every 2 seconds. 40 45 50 55 60 65 70 75 80 −70 −65 −60 −55 −50 −45 −40 −35 −30 PSD of s(n) in the gamma−band Frequency (Hz) dB 40 45 50 55 60 65 70 75 80 −70 −65 −60 −55 −50 −45 −40 −35 −30 PSD of estimate of s(n) in the gamma−band when using variable Frequency (Hz) dB 40 45 50 55 60 65

70 75 80 −70 −65 −60 −55 −50 −45 −40 −35 −30 PSD of estimate of s(n) in the gamma−band when using =0.02 Frequency (Hz) dB 40 45 50 55 60 65 70 75 80 −70 −65 −60 −55 −50 −45 −40 −35 −30 PSD of estimate of s(n) in the gamma−band when using =0.5 Frequency (Hz) dB Figure 10: Comparison of power spectrum in the gamma-band of the noise free signal and its estimates. (Top) Spectrum of noise free signal ). (Second) ) obtained with adaptive , (Third) obtained with =0.02, and (Bottom)

) obtained with =0.5.
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20.5 20.6 20.7 20.8 20.9 21 21.1 21.2 21.3 21.4 −0.1 −0.05 0.05 0.1 0.15 Original EEG signal + noise in the gamma−band Time (sec) x(n) 20.5 20.6 20.7 20.8 20.9 21 21.1 21.2 21.3 21.4 −0.1 −0.05 0.05 0.1 0.15 Original EEG signal in the gamma−band Time (sec) s(n) 20.5 20.6 20.7 20.8 20.9 21 21.1 21.2 21.3 21.4 −0.1 −0.05 0.05 0.1 0.15 Signal estimate when using variable step−size Time (sec) Estimate of s(n) Figure 11: Gamma-band signals: (Top) Signal plus interference, (Center) Original EEG signal s(n),

(Bottom) Estimated signal (n) using the variable step-size parameter algorithm. 2000 4000 6000 8000 10000 12000 14000 16000 10 −4 10 −3 Variable step−size =0.02 ↑µ =0.5 Mean Square Error Number of iterations MSE Figure 12: MSE behavior in the presence of a periodic non- sinusoidal 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 =0 02. On the other hand, when the step-size was set to =0 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 60 Hz for this scenario. Figure 11 shows the EEG signal plus noise in the gamma-band (40-80 Hz), the original noise-free EEG signal, and the signal estimate ) 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 non- sinusoidal noise on the ANC algorithm. Figure 12 shows the MSE when the interference was set to be a sinusoidal trun- cated at values of 50% 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 sat- uration may occur and in cases where the signal is windowed in time. V. Conclusions When dealing with EEG signals in the gamma-band (35- 100 Hz), it

is desirable to have a notch filter with a small re- jection 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 minimiza- tion 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 monitor- ing. The choice of the step-size parameter in the adaptation al- gorithm 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 re- jection bandwidth required to effectively eliminate the time- varying 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 10th Annual International Conference , p. 1168, 1988. [2] 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 Engi- neering , vol. 41, no. 6, pp. 529–536, 1994. [3] M. V. Dragosevic, and S. S. Stankovic “An Adaptive Notch Filter with Improved

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IEEE Workshop on Digital Signal Processing , pp. 33–36, 1997.