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
P_{e}
, is a difficult problem. Because
H
, the Hellinger integral, and
H^{2}
form an upper and a lower bound for
P_{e}
, 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.