Maximal Invariants and Performance of Some Invariant Hypothesis Tests for an Adaptive Detection Problem

Maximal invariants for adaptive detection of a signal in unknown interference from multiple observations is derived. Given coherent samples from
$P$
sets of observations, it is shown that a maximal invariant statistic for the detection problem is a
$2P \times 1$
-dimensional vector comprising the eigenvalues of two Hermitian positive definite matrices obtained from the data set. Two invariant detectors, well known for
$P=1$
, are generalized for the case of multiple observations and closed form expressions for the probability of detection and probability of false alarm are derived along with the distributions of the signal-to-interference-plus-noise loss factors. Several novel invariant detectors are constructed from the maximal invariants and the receiver operating characteristics of the detectors compared.