Application of the Family of Kravchenko-Rvachev Atomic Weight Functions (Windows) in Welch Method EEG Power Spectral Density Estimation

Application of atomic weight functions to the spectral analysis of electroencephalograms (EEG) is considered. In EEG signals several patterns of rhythmic or periodic activity are usually found: delta rhythm corresponds to deep-sleep (frequency less than 4Hz), theta rhythm appear at the beginning stages of sleep (frequency from 4 to 8Hz), alpha rhythm found in resting state (frequency from 8 to 13 Hz) and beta waves appear as background activity in tense and anxious state (frequency higher than 13 Hz). The discrepancy between the identified rhythmic activity and the patient's condition may be caused by various diseases. The function of the power spectral density (PSD) of the EEG signal allows to determine the presence of waves of a particular rhythm and, thus, carry out diagnostics. Welch method makes it possible to obtain consistent estimates of the power spectral density with decaying variance. The method consists of cutting out a part of the signal using the weight function, calculating the modified periodogram for each part, and finally averaging all the periodograms obtained. In this case, a compromise is usually sought between the spectral resolution and the decrease in the variance of the PSD estimate. High resolution is achieved by applying a weight function with a narrow main lobe. Variance can be reduced by using windows with low level of side lobes. Atomic functions are solutions of functional-differential equations of special form. They are infinitely differentiable and compactly supported. The spectra of atomic functions quickly decay at infinity. Listed properties allow to use atomic functions as effective windows. Atomic weight functions are superior to classical ones in such physical parameters as scalloping loss, worst case processing loss, highest side lobe level. Most atomic windows are parametric, which allows changing the parameter to choose the optimal window in a particular case. Combined weight functions that are atomic windows truncated with some classical window are also considered. This approach allows to decrease worst case processing loss preserving low side-lobe level. The efficiency of using weight windows based on atomic functions in the Welch method is confirmed by the results of a numerical experiment.