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.