Testing the Energy of Random Signals in a Known Subspace: An Optimal Invariant Approach

We consider the problem of testing whether the energy of a random signal projected onto a known subspace exceeds some specified value
$\tau \geq 0$
. The probability distribution of the signal is assumed to be unknown and this signal is observed in additive and independent white Gaussian noise with known variance. The proposed theoretical framework relies on the invariance of the problem and the resulting test is shown to be uniformly most powerful invariant in an extended sense suitable for random signals. This work extends Scharf and Friedlander’s matched subspace detector.