On simultaneously orthogonal expansions and choosing observables for discriminating two Gaussian signals

Relations among four simultaneously orthogonal expansions (SOE), two of them previously known, are obtained. It is shown with the help of information theory that any of these SOE´s can be derived from any other. The coefficients of the SOE´s are employed to investigate the problem of choosing observables for discriminating two Gaussian signals under the probability of error criterion. The best single observable is found. It is also shown that the best n independent observables are not necessarily those having the smallest n individual error probabilities. The difficulties in picking the best observables are pointed out. As a byproduct, it is proved that even when the most important features of one signal are the least important of the other and vice versa; still, feature extraction and signal discrimination are not related in the sense that the best set for the former is not necessarily the best for the latter.