Читать книгу EEG Signal Processing and Machine Learning - Saeid Sanei - Страница 78

The EEG signals are subject to noise and artefacts. Electrocardiograms (ECGs), electro‐oculograms (EOG) or eye blinks affect the EEG signals. Any multimodal recording such as EEG–functional magnetic resonance imaging (fMRI) significantly disturbs the EEG signals because of both magnetic fields and the change in the blood oxygen level and sensitivity of oxygen molecule to the magnetic field (balisto‐cardiogram). Artefact removal from the EEGs will be explained in the related chapters. The noise in the EEGs, however, may be estimated and mitigated using adaptive and non‐adaptive filtering techniques.

The EEG signals contain neuronal information below 100 Hz (in many applications the information lies below 30 Hz). Any frequency component above these frequencies can be simply removed by using lowpass filters. In the cases where the EEG data acquisition system is unable to cancel out the 50 Hz line frequency (due to a fault in grounding or imperfect balancing of the inputs to the differential amplifiers associated with the EEG system) a notch filter is used to remove it.

The nonlinearities in the recording system related to the frequency response of the amplifiers, if known, are compensated by using equalizing filters. However, the characteristics of the internal and external noises affecting the EEG signals are often unknown. The noise may be characterized if the signal and noise subspaces can be accurately separated. Using principal component analysis (PCA) or independent component analysis (ICA) we are able to decompose the multichannel EEG observations to their constituent components such as the neural activities and noise. Combining these two together, the estimated noise components can be extracted, characterized, and separated from the actual EEGs. These concepts are explained in the following sections and their applications to the artefact and noise removal will be brought in the later chapters.

Adaptive noise cancellers used in communications, signal processing, and biomedical signal analysis can also be used for removing noise and artefacts from the EEG signals. An effective adaptive noise canceller however requires a reference signal. Figure 4.11 shows a general block diagram of an adaptive filter for noise cancellation. The reference signal carries significant information about the noise or artefact and its statistical properties. For example, in the removal of eye‐blinking artefacts (discussed in Chapter 16) a signature of the eye‐blink signal can be captured from the FP1 and FP2 EEG electrodes. In detection of the ERP signals, as another example, the reference signal can be obtained by averaging a number of ERP segments. There are many other examples such as ECG cancellation from EEGs and the removal of fMRI scanner artefacts from EEG‐fMRI simultaneous recordings where the reference signals can be provided.

Figure 4.11 An adaptive noise canceller.

Adaptive Wiener filters are probably the most fundamental type of adaptive filters. In Figure 4.11 the optimal weights for the filter, w(n), are calculated such that is the best estimate of the actual signal s(n) in the mean square sense. The Wiener filter minimizes the mean square value of the error defined as:

(4.91)

where w is the Wiener filter coefficient vector. Using the orthogonality principle [39] the final form of the mean squared error will be:

(4.92)

where E(.) represents statistical expectation:

(4.93)

and

(4.94)

By taking the gradient with respect to w and equating it to zero we have:

(4.95)

As R and p are usually unknown the above minimization is performed iteratively by substituting time averages for statistical averages. The adaptive filter in this case, decorrelates the output signals. The general update equation is in the form of:

(4.96)

where n is the iteration number which typically corresponds to discrete‐time index. Δ w (n) has to be computed such that E[e(n)]² reaches to a reasonable minimum. The simplest and most common way of calculating Δw(n) is by using gradient descent or steepest descent algorithm [39]. In both cases, a criterion is defined as a function of the squared error (often called a performance index) such as η (e(n)²), such that it monotonically decreases after each iteration and converges to a global minimum. This requires:

(4.97)

Assuming ΔW is very small, it is concluded that:

(4.98)

where, ∇_w (.)represents gradient with respect to w. This means that the above equation (Eq. 4.98) is satisfied by setting Δw = − μ∇_w (.), where μ is the learning rate or convergence parameter. Hence, the general update equation takes the form:

(4.99)

Using the least mean square (LMS) approach, ∇_w (η(w)) is replaced by an instantaneous gradient of the squared error signal, i.e.:

(4.100)

Therefore, the LMS‐based update equation is

(4.101)

Also, the convergence parameter, μ, must be positive and should satisfy the following:

(4.102)

where λ _max represents the maximum eigenvalue of the autocorrelation matrix R . The LMS algorithm is the most simple and computationally efficient algorithm. However, the speed of convergence can be slow especially for correlated signals. The recursive least‐squares (RLS) algorithm attempts to provide a high speed stable filter, but it is numerically unstable for real‐time applications [40, 41]. Defining the performance index as:

(4.103)

Then, by taking the derivative with respect to w we obtain

(4.104)

where 0 < γ ≤ 1 is the forgetting factor [40, 41]. Replacing for e(n) in the above equation (Eq. 4.104) and writing it in vector form gives:

(4.105)

where

(4.106)

and

(4.107)

From this equation:

(4.108)

The RLS algorithm performs the above operation recursively such that P and R are estimated at the current time n as:

(4.109)

(4.110)

In this case

(4.111)

where M represents the finite impulse response (FIR) filter order. Conversely:

(4.112)

which can be simplified using the matrix inversion lemma [42]:

(4.113)

and finally, the update equation can be written as:

(4.114)

where

(4.115)

and the error e(n) after each iteration is recalculated as:

(4.116)

The second term in the right‐hand side of the above equation is . Presence of R ⁻¹(n) in Eq. (4.115) is the major difference between RLS and LMS, but the RLS approach increases computation complexity by an order of magnitude.