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

Abstract

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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P_{e}</tex> , is a difficult problem. Because <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</tex> , the Hellinger integral, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H^{2}</tex> form an upper and a lower bound for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P_{e}</tex> , we minimize <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</tex> instead. We find that the set of observables that minimizes <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</tex> is a set of coefficients of the simultaneously orthogonal expansions of the two signals. The same set of observables maximizes the Hájek <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">J</tex> -divergence as well.