Performance Comparison of Two Subspace-Based GLRTs for Rank-1 Signal Detection in Unknown Interference With Multiple Observations

This article compares the receiver operating characteristics (ROC) of two subspace-based generalized likelihood ratio tests (GLRTs) referred to in this article as the unconstrained and constrained GLRTs. The tests are derived for slightly different assumptions and are applied to detect an unknown rank-1 signal that belongs to a known
$M$
-dimensional subspace in
$\mathbb {C}^{N \times 1}$
. The receiver is given multiple (say
$P$
) statistically independent observations of unknown multivariate zero-mean complex Gaussian interference-plus-noise that may contain a rank-1 signal as the test matrix and signal-free training data to estimate the interference-plus-noise covariance matrix. The two detectors are shown to coincide when
$M = 1$
and/or
$P = 1$
. New analytical expressions for the probability of false alarm and probability of detection are derived for the unconstrained GLRT and to the best of our knowledge, no such analytical expressions are available for evaluating the performance of the constrained GLRT in the general case, where
$M > 1$
and
$P > 1$
. The ROC of the unconstrained GLRT is a lower bound on the performance of the constrained GLRT. For small signal-to-interference-plus-noise ratios (SINRs) and
$M/N \ll 1$
, the lower bound is tight, which makes the derived analytical results both interesting and useful.