Maximal Invariants for Subspace Signal Detection in Unknown Interference With Multiple Observations

This article derives maximal invariants (MIs) for group invariant hypothesis tests to detect subspace signals in zero-mean complex Gaussian multivariate interference and noise with unknown covariance matrix given multiple observations. The signal is assumed to belong to a known
$M$
-dimensional subspace in
$\mathbb {C}^{N \times 1}$
. Given
$P$
independent observations of the test vector, we show that MIs are any two of three
$P \times P$
matrices constructed from the test matrix and the interference training data. The importance of MI is that the detection statistic of all constant false alarm rate (CFAR) tests are derived from MI and as such MI provides information on how input test matrix data and interference training data must be processed in order for the receiver to construct any one of the numerous CFAR tests.