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 , is a difficult problem. Because , the Hellinger integral, and form an upper and a lower bound for , we minimize instead. We find that the set of observables that minimizes is a set of coefficients of the simultaneously orthogonal expansions of the two signals. The same set of observables maximizes the Hájek -divergence as well.