On the best finite set of linear observables for discriminating two Gaussian signals

Consider the problem of discriminating two Gaussian signals by using only a finite number of linear observables. How to choose the set of n observables to minimize the error probability P₄, is a difficult problem. Because H, the Hellinger integral, and H^2 form an upper and a lower bound for P₄, we minimize H instead. We find that the set of observables that minimizes H is a set of coefficients of the simultaneously orthogonal expansions of the two signals. The same set of observables maximizes the Hájek J -divergence as well.