Removal of non-white noise from single trial event-related EEG signals using soft-thresholding

Infrequent stimulation of a subject generates event-related potential (ERP) signals masked by background EEG activity. It is generally assumed that this background activity is normal white noise. Another assumption made is that the underlying evoked signals are deterministic and they do not vary from trial to trial. The signal, thus, is modeled as y_i^(j)=x_i+σ _i^(j), where y_i^(j) is the jth trial and z_i is a white noise. The signal x_i is recovered by averaging the observations y_i^(j). In reality, the background activity is “colored” and not always Gaussian. The time samples of the background activity are generally correlated. In addition, the signal x_i varies across observations. The purpose of this study is to extract single trial ERPs from the EEG. The authors are interested in the trial to trial variation of the ERPs and their clinical applicability. They have investigated the wavelet soft-thresholding method to remove the background noise and separate out the single trial response. The non-white background activity, after wavelet transformation, may concentrate in certain resolution levels. The authors determined these levels by testing the Gaussianity of the wavelet coefficients, using both the χ² and Kolmogorov-Smirnov goodness of fit tests. In the resolution levels at which the null hypothesis was not rejected, the noise level was estimated. The de-noising threshold was then calculated using a level dependent rule T_j,N=√(2 log(N))·MAD(C_j,k)/0.6745, where C_j,k are the wavelet coefficients and MAD(C_j,k)=Median(|C_j,k| ) is an estimator of the noise level. The resolution levels at which the null hypothesis was rejected were not thresholded