Time-frequency analysis of band-limited EEG with BMFLC and Kalman filter for BCI applications
© Wang et al.; licensee BioMed Central Ltd. 2013
Received: 22 May 2013
Accepted: 19 November 2013
Published: 25 November 2013
Time-Frequency analysis of electroencephalogram (EEG) during different mental tasks received significant attention. As EEG is non-stationary, time-frequency analysis is essential to analyze brain states during different mental tasks. Further, the time-frequency information of EEG signal can be used as a feature for classification in brain-computer interface (BCI) applications.
To accurately model the EEG, band-limited multiple Fourier linear combiner (BMFLC), a linear combination of truncated multiple Fourier series models is employed. A state-space model for BMFLC in combination with Kalman filter/smoother is developed to obtain accurate adaptive estimation. By virtue of construction, BMFLC with Kalman filter/smoother provides accurate time-frequency decomposition of the bandlimited signal.
The proposed method is computationally fast and is suitable for real-time BCI applications. To evaluate the proposed algorithm, a comparison with short-time Fourier transform (STFT) and continuous wavelet transform (CWT) for both synthesized and real EEG data is performed in this paper. The proposed method is applied to BCI Competition data IV for ERD detection in comparison with existing methods.
Results show that the proposed algorithm can provide optimal time-frequency resolution as compared to STFT and CWT. For ERD detection, BMFLC-KF outperforms STFT and BMFLC-KS in real-time applicability with low computational requirement.
EEG uses electrodes to record electrical brain activity that originates from the post-synaptic potentials, aggregates at the cortex, and transfers through the skull to the scalp. The EEG signal is the reflection of brains neuronal oscillations. These oscillations with similar frequency and energy lead to the separation of frequency bands . Numerous studies have tried to identify the relation between the frequency bands and brain states and it still remains as a hotspot of ongoing neuroscience research [2, 3]. EEG activity related with voluntary movements has been the center of interest, as it is applicable to brain-computer interfaces (BCI) [4–8].
Several BCI systems rely on an amplitude attenuation phenomenon, namely event-related desynchronization (ERD) that can be voluntarily controlled by movement imagery. It was shown in  that during both planning and execution of hand movements, the ERD can be detected in most of the subjects within the band of μ-rhythm (6-14 Hz). By utilizing this amplitude attenuation phenomenon, an alternative communication pathway can be built directly from human brain to the computer . The accuracy of this class of methods was examined in . In recent years, this type of BCIs has been applied for limb function recovery  and robotic system control , which can improve the quality of life of the subject with severe motor function impairment. The energy decrease in ERD usually occurs in a specific frequency band for a subject. When the frequency characteristics of signal are required, the fast Fourier transform (FFT) is often used. For BCI applications, the ERD in the EEG signal is considered as a percentage change of the signal amplitude with respect to a experiment cue [9, 11–13]. Since the FFT-based methods cannot provide time-frequency information, the time-frequency representation (TFR) of EEG signal is extremely important for ERD analysis.
As the performance of EEG-based BCI systems rely on the time-domain and frequency-domain features, a variety of EEG features (such as power spectrum within a pre-defined frequency band  and phase lock value ) that reflects ongoing brain states have been attempted for designing BCI systems. In general, the feature extraction algorithm requires accumulation of sufficient number of samples for generating control commands. In , an EEG-based BCI for three-dimensional movement control has been implemented where the power spectrum was calculated for every 50 ms with a 16-order autoregressive(AR) model. In a recent research , BCI with a noninvasive functional electrical stimulation has been studied. In the on-line processing phase, the BCI aggregates 500 ms EEG signal and then FFT is applied to obtain power estimates for classification. It is clear that the response time of BCIs mainly depends on the time required by the feature extraction algorithm to store and process a sufficient number of samples. Since power spectrum is a frequency domain feature, the time-frequency representation(TFR) methods that can provide amplitude variation along time axis, can be directly applied to BCI systems. Furthermore, the accuracy of BCIs can be improved by employing a narrow subject-specific frequency band [9, 18, 19]. In , an adaptive filtering approach was employed for identifying the subject-specific frequency band to improve the classification accuracy.
The TFR methods can be categorized into two types, namely non-parametric and parametric methods. The non-parametric TFRs such as band-pass filtering, short time Fourier transform (STFT) and continuous wavelet transform (CWT) were successfully applied in time-frequency analysis of EEG [20, 21]. However, all the traditional methods have pros and cons in temporal and spectral resolutions. In band-pass filtering, the temporal and spectral resolution is highly dependent on the filter type, center frequency of the filter and its order. The temporal and spectral resolution of STFT is determined by the window length. The CWT can be considered as the best TFR technique among the available methods. However, it still suffers with the tradeoff between temporal and spectral resolution as STFT. The computational requirement of CWT remains as a major barrier for real-time BCI applications. A performance comparison of all the TFR methods for EEG time-frequency analysis can be found in .
Recently, a new method band-limited multiple Fourier linear combiner (BMFLC) was developed for estimating band-limited signals within a pre-defined frequency band . By incorporating the idea of linear time varying model with a fixed frequency band, BMFLC can provide an alternative spectral estimation method for band-limited signals. The original BMFLC adopted a truncated Fourier series as the model and estimated the Fourier coefficients by least mean squares (LMS) algorithm [23, 24]. As LMS requires time to converge to the steady-state, the algorithm accuracy cannot be guaranteed for small data segments. Especially, LMS algorithm is not suitable when the main objective is to accurately track the amplitude changes of a band-limited signal. To improve the accuracy of the time-frequency decomposition and the tracking ability of the existing BMFLC for real-time applications, Kalman Filter is employed. The proposed method is designed to extract time-frequency amplitude distribution that is more suitable for BCI applications. The performance of proposed method is evaluated in comparison with STFT, CWT and the existing BMFLC-LMS method.
This section first reviews the existing methods for time-frequency representation and later presents the proposed methods.
Classical time-frequency methods
As the traditional fast Fourier transform (FFT) does not provide time-domain information, the intuitive way to overcome this is to isolate the signal in time domain by multiplying with a window function and compute the Fourier coefficients in that time interval, then shift the time window through the time line to capture the entire time-frequency information of the signal .
It can be illustrated as a box centered at (t,ω), with the length equal to △t in time domain and the width equal to △ω in the frequency domain, to sweep the whole time-frequency domain to extract the time-frequency information. Since this product remains constant, the increase of one quantity will cause degradation in the other. This degradation (leakage effect) is also due to this constant product. Thus the information that can be extracted via STFT is actually the information within that box. If the window function has a Gaussian envelope, the STFT can achieve the lower bound defined in (1) . The STFT with a Gaussian window function is commonly referred as Gabor transform.
In STFT, after fixing the length of the window function, its time-frequency resolution remains constant for the entire time-frequency domain. The continuous wavelet transform (CWT) solves this problem by adopting a dilated and translated versions of the same function namely, the mother wavelet [26, 27]. The dilated version of the mother wavelet is controlled by a scalar parameter a in CWT, where the corresponding time-frequency resolution can be provided as . Although the time-frequency resolution is still bounded by Heisenberg uncertainty, but the length and width are scaled by the parameter a. Therefore, for CWT the time-frequency resolution can be adjusted by the parameter a compared to a fixed time-frequency resolution in STFT.
where f and a represent the Fourier frequency and wavelets scale respectively.
The STFT and CWT belong to the analytic time-frequency representation methods, where the time-frequency resolution has a lower bound that is determined by the Heisenberg uncertainty. To apply the CWT and STFT to a signal, an appropriate length of the signal is required, as all the samples within the window function should be used at each time-frequency decomposition. The power spectrum defined in STFT and CWT is relative to the true signal spectrum that can be obtained from FFT. For some applications where the exact measurement of amplitude or power at a specific frequency is required, the CWT and STFT are not suitable.
Band-limited multiple Fourier linear combiner
The Fourier linear combiner (FLC) proposed in  works by adaptively estimating the Fourier coefficients of a known base frequency together with its harmonics with the help of the least mean squares (LMS) algorithm. In [18, 23], to estimate the unknown band-limited signal, a pre-defined frequency band [ ω1-ω n ] is considered and divided into ‘n’ finite number of divisions. Then n-FLC’s are combined to form the BMFLC to estimate bandlimited signals. The frequency resolution of BMFLC, (i.e.), is the frequency gap between two adjacent frequency components. The selection of frequency gap is a balance between signal characteristics and analysis requirement.
where ε k is the error between modeling and measurement and μ is adaptive gain of LMS. As LMS algorithm relies on gradient based method for error minimization, the accuracy of the algorithm can be affected by the dynamic changes in the characteristics of the signal. LMS algorithm has only a single adjustable parameter for controlling the convergence rate, namely, the adaptive gain μ. Selection of proper μ is very important for stability and convergence of the algorithm. For difference choices of frequency gap (△f), different values of μ are required to ensure stability and convergence . As shown in , BMFLC-LMS requires around 5 seconds for the initial weights to track the amplitude changes in a 25s EEG signal. For the practical BCI systems, the length of EEG signal is generally in a few seconds after the experiment cue. To improve the performance, we employ Kalman Filter (KF).
BMFLC with Kalman filter (BMFLC-KF) for real-time estimation
where η k is the state error.
where R and Q are measurement error covariance and state error covariance respectively. By employing the Kalman filter, the optimal estimation of the state of the dynamical system at time instant k can be obtained with the measurement sequence Y1: k-1=[y1,y2,⋯,yk-1] where y k is the output defined in (4). Although the premise of the noise may not hold for the signal to be analyzed, it was shown in [35, 36] that Kalman filter can also provide the minimum mean-squared error estimation within the class of linear estimators. The adaptive algorithm together with the BMFLC is shown in Figure 1.
with initial condition , P0. K k is the Kalman gain updated at each time instant. The BMFLC-KF does not require the matrix inverse as it only involves a scalar observation. The proposed BMFLC-KF is computationally fast and is well suited for real-time applications.
BMFLC with Kalman Smoother (BMFLC-KS) for off-line analysis
where and are estimated through recursion with the Kalman filter. The smoother estimation is then obtained by running the stored estimates backward in time. This procedure is suitable for off-line analysis.
The convergence properties of the proposed method are determined by the Kalman filter. The convergence analysis for autoregressive(AR) model with Kalman filter was well documented in [37–40]. It was shown in  that the Kalman filter is uniformly exponentially stable, if the output matrix sequence and the covariance of state error are bounded. In proposed method, the output matrix in (9) is a combination of user-defined frequency components. As sine and cosine functions are bounded, the output matrix is bounded. The covariance of state error is user-defined and is bounded. Hence the convergence of proposed algorithm can be established similar to . Further the convergence rate can be quantified as in .
Time-frequency decomposition with BMFLC
where the operator ⊙ represents the element by element multiplication of the matrix. The D matrix provide the time-domain information of all individual frequency component and it can be directly used for time-frequency representation.
Comparing with the LMS based BMFLC in [18, 32], by adopting the Kalman filter combined with the smoother procedure, an accurate weights adaption process in BMFLC can be achieved. Hence it provides an accurate time-frequency decomposition. The usage of smoother is optional and it depends on the purpose of analysis. For the off-line analysis, if an accurate time-frequency mapping is required, the smoother procedure can be employed.
where e is the estimation error.
For estimation of synthesized signals and EEG, we set the following parameters for BMFLC based algorithms: f1=6 Hz, ω1=2π f1,f n =14 Hz, ω n =2π f n . Frequency spacing is set to be △f=0.5 Hz . The weights are initialized with 0, i.e. w0=0. For the BMFLC-LMS algorithm, the adaptive gain μ=0.035 is chosen for optimal performance for the corresponding frequency spacing △f=0.5 Hz . The co-relation between the parameter μ and step-size is discussed in .
To further justify the selection of parameters, experiments are conducted with the EEG data. EEG data of all trials of subject 1 are selected. Similar to the earlier, R is estimated in the algorithm, and the q is selected based on the RMS error. Based on the RMS error obtained for all selections of q and for all trials of the subject #1, the 95% confidence interval (CI) is estimated. To obtain reliable estimation of CI, Bootstrap method  with 2000 re-sampling is employed and the results are shown in Figure 4(d)-(e). Hence, we select q=0.01 and R=0.01 for optimal performance.
where L denotes the length of the window function, F s is the sampling frequency and △f=△ω/2π. The step size of STFT is 1/F s . For the wavelet-based TFR method, the Morlet wavelet is employed with ω0=6 to offer good trade-off between temporal and spectral resolutions . A total of 17 scales are calculated in this paper, which are equally spaced within the range of 6H z to 14H z with the same frequency gap employed in BMFLC and STFT. The wavelet scale is transformed to Fourier frequency with (3).
In this section, we provide comprehensive analysis of all the five methods, BMFLC-LMS, BMFLC-KF, BMFLC-KS, STFT and CWT for synthetic and EEG data sets discussed in earlier section.
To compare the estimation accuracy of BMFLC based methods, the estimated signal together with the estimation error for synthesized signals (23, 24) and single trial EEG data (C3 electrode for right hand movement imagery) are shown in Figure 2.
In Figure 2(a2) and Figure 2(b2), error can be observed at the transition points for BMFLC-KF. By comparing the results of BMFLC-LMS in Figure 2(a4) and BMFLC-KF/KS in Figure 2(a2) and (a3), it is clear that the estimation accuracy depends on the adaptive algorithm. For the EEG data, BMFLC-KF/KS performed better compared to BMFLC-LMS as shown in Figure 2(c4).
For BMFLC-KS, the error is backward averaged to obtain the smoothed estimation. Since the system transition matrix is modelled as identity matrix, the accuracy of the algorithm cannot be improved with Kalman-smoother . However, we can observe from Figure 2(b2) and Figure 2(b3) that the transient performance can be improved.
Estimation accuracy of BMFLC based methods
Estimation accuracy for different frequency gaps
Frequency gap in BMFLC
Temporal and spectral resolution: comparison of all five methods
BMFLC-KF provides accurate spectral estimation as shown in Figure 5(a2) and (b2). However, when there is a sudden change in the frequency, BMFLC-KF requires an adapting period for tracking the spectral changes in the signal. Although the estimation accuracy is high, there is some disturbance at the frequency transition at 5 sec as seen in Figure 5(a2) and (b2). As the amplitude weights of BMFLC-KF are initialized with zero, BMFLC-KF requires an initial adaptation period. This is mainly due to the random walk model employed for the state transition in BMFLC. By comparing Figure 5(a2) with Figure 5(a3) and Figure 5(b2) with Figure 5(b3), the estimation of smoother relays on the information provided by the Kalman filter, and operates backward to smooth the errors in the estimation. It is clear that the BMFLC-KS provides improved spectral estimation.
In order to study the real-time applicability, the computational complexity of the TFR methods is presented. The difference between BMFLC-KF and non-parametric methods (CWT and STFT) is that they require sufficient length of input samples to be stored in order to provide the time-frequency mapping. Therefore the computational complexity of these methods relies on the length of the input data.
O(N1l o g2N1)
O(N l o g2N1)
O(3N2l o g2N2)
In BCIs, the feature required for generating a control command is extracted every 500 ms [16, 45]. Let us consider the sampling frequency as F s =512 Hz and frequency resolution as △f=0.5 Hz . For the given △f, the order of BMFLC-KF can be obtained as . In order to maintain the same frequency resolution in STFT and FFT with BMFLC, the data is padded with zeros in order to have sufficient data length (28) for analysis. For quantitative comparison, the operations required for all the methods are for a given data length is also provided in Table 3. The computational complexity of FFT  is also discussed in the Table. This comparison quantifies the computational complexity of the algorithms for real-time implementation. BMFLC-KF has comparatively lower computational requirement compared to other methods.
Computational complexity of BMFLC-KF
△f=0. 2 Hz
△f=0. 4 Hz
△f=0. 5 Hz
where are the estimated weights from time-frequency decomposition algorithm at j th sample of the i th trial of frequency f, is the mean of weights over all trials and N is the number of trials/subject. R f is average power in the pre-defined reference period [ r0,r0+k]. The reference period is generally the period before the cue is onset . In line with earlier works [9, 32], we select the reference period as 0 to 1.5 s before the cue onset (cue at 2 sec) for calculation of ERD. The time-frequency map for ERD is obtained by combining all the row vectors , with m being the number of data samples in a given trial. The obtained ERD % is the average over all trials of a subject.
To statistically validate the performance, bootstrap with 2000 times re-sampling is used to estimate the 95% confidence interval for obtained ERD%. As stated in , if both confidence values of an ERD show same sign then it can be considered as significant. The bootstrap test shows that the obtained ERD mapping is significant for all methods.
By comparing the results in Figure 11(b2)-(b5), ERD can be observed in the frequency range of 10 Hz- 12 Hz in all the methods except BMFLC-LMS. Within these methods, STFT (Figure 11(b4)) and CWT (Figure 11(b5)) offer the best temporal accuracy. On the other hand, BMFLC-KF/KS also provides narrower band similar to CWT. By comparing the results of Figure 11(b1) and (b2), the adaptation period in Kalman filter can be decreased with the smoother procedure. Also, the frequency transitions in the spectral domain can be accurately estimated in the time-frequency mapping for BCI applications with the proposed method.
Although the proposed method ensures high accuracy for all choices of (Δ f), an optimal selection of (Δ f) is required for real-time implementation. For EEG, a frequency gap of 0.5 Hz is optimal  and ensures accurate spectral estimation. However, for off-line analysis a small (Δ f) can be selected. The comparative study conducted on computational complexity of the TFRs shows that the proposed method has lower computational demand for a bandlimited signal. Hence, the proposed method is more suitable for accurate spectral estimation in a narrow frequency band.
While the proposed method is only discussed for a narrow frequency band in this paper, the method can still be applied for a wide frequency band. Multiple BMFLCs can be implemented in parallel by dividing the wide band into small narrow bands to ensure stability and accuracy for the algorithm. However, if the band of interests is too wide or the frequency resolution for each sub-band is too high, traditional methods would be more appropriate. A wide frequency band increases the computational requirement in BMFLC. As we confine our study to a narrow frequency band, the CWT could not provide better performance compared to STFT as the frequency resolution in CWT is scaled in the narrow band.
Heisenberg uncertainty exists in all the methods. Even though the estimation accuracy is high, the proposed method requires an additional time for the frequency weights to settle, as both amplitude and frequency cannot be estimated at the same time. The uncertainty on frequency can be clearly seen at the sudden frequency or amplitude transition. For EEG signals, an additional 0.5 s is required for the frequency weights to settle. However, it only occurs at the start of the estimation process. This initial estimation delay is inherent for all adaptive based methods and can be improved with proper initialization. The lower computational complexity of the proposed method can offset for the delay caused in the estimation compared to traditional methods. Comparatively, the CWT has the best performance in ERD detection. However, the implementation of CWT method can be a problem for real-time applications.
Although only the ERD detection is considered in this paper as application, the proposed algorithm is applicable for any bandlimited signal estimation. As ERD and ERS (event related synchronization) lies in a specific frequency bands, the proposed method can be applied for ERD and ERS detection simultaneously by employing multiple BMFLC’s in parallel. Since the weights in the BMFLC are directly related to the real amplitudes of the individual frequency components, the algorithm can utilized where an accurate amplitude estimation of a specific frequency component is required.
In this research, the performance of existing BMFLC-LMS is improved by incorporating a Kalman filter. A comparison study of the BMFLC based methods with STFT and CWT is performed with both synthetic and real EEG data. The results indicate that the BMFLC-KF/KS can be used as an alternative time-frequency analysis methods for band-limited signals. As most of the frequency-based BCI applications rely on amplitude features in a fixed frequency band (μ rhythm) for classification, BMFLC-KF can be directly applied to most existing BCI systems. With the linear model employed, optimal estimation can be obtained with the Kalman filter. Thus the proposed method can provide an accurate time-frequency mapping with less computational complexity as compared to STFT and CWT for real-time applications. The results also show that the BMFLC-KS can provide more accurate time-frequency representation for off-line analysis.
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant No. 2011-0023999).
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