Averaging of spectral power and phase via vector diagram best fits without reference electrode or reference channel

Abstract

A method is proposed to average results of Fourier analysis over epochs without assigning zero amplitude values to any electrode or zero phase values to any channel. The Fourier sine and cosine values of each electrode define its entry location into a vector diagram for the frequency component. The entry constellations thus formed are translated so that the mean of all entries (the virtual average reference which is the location of best fit) is at the centre of each constellation, and then they are rotated around the means into angles of best fit so that the sum of the squared distances between corresponding entries is minimal. The method leads to non-ambiguous mean results, with variance information for all recording points. Approaches for the assessment of the constellations, their transformation into field maps, and the problems with conventional phase measurements vs. real electrodes are discussed. The virtual average reference (spatial high pass filter) is proposed for assessing the time sequence (phase differences) between entry points, thus describing the internal structure of the constellations without bias and without reference to a zero phase point. Comparison of voltage and current density result constellations and consideration of reference-independent descriptors of the scalp field potential maps support the validity of the average reference for the approach.