Invariance Theory for Radar Detection in Disturbance With Kronecker Product Covariance Structure—Part I: Gaussian Environment

This two-part paper addresses maximally invariant detection of range-spread targets embedded in disturbance characterized by an unknown Kronecker product-structured covariance matrix. Part I focuses on Gaussian interference, whereas Part II extends the study to compound-Gaussian, clutter-dominated environments. Leveraging the principle of invariance, this part identifies a suitable transformation group that effectively compresses the nuisance parameter space, ensuring the constant false alarm rate (CFAR) property (with respect to the Kronecker-structured covariance matrix) for all invariant detectors. A maximal invariant and an induced maximal invariant are subsequently derived, serving as powerful tools to guide the design of CFAR detectors. Some existing two-step CFAR detectors for this structured situation are expressed as functions of the derived maximal invariant. Furthermore, two novel detectors (whose CFARity holds true under some mild technical conditions) are devised: the former employs a pseudo-missing strategy by treating elements possibly contaminated by target signals as missing and utilizes an Expectation-Maximization algorithm to perform the covariance matrix estimation; the latter is based on the one-step generalized likelihood ratio test criterion and is implemented via an alternate optimization algorithm. Finally, their CFAR behavior and detection performance are assessed through numerical examples, demonstrating their superiority with respect to some conventional decision rules.