Rao tests for distributed target detection in interference and noise

Highlights

• We propose two detectors for distributed target detection in interference and noise.

• The proposed detectors have the CFAR property.

• They have better detection performance than the existing ones.

Abstract

This paper deals with the problem of detecting a distributed target in interference and noise. The target signal and interference are assumed to lie in two linearly independent subspaces, and their coordinates are unknown. The noise is Gaussian distributed, with an unknown covariance matrix. To estimate the covariance matrix, a set of training data is supposed available. We derive the Rao test and its two-step variant both in homogeneous and partially homogeneous environments. All of the proposed detectors exhibit a desirable constant false alarm rate. Numerical examples show that the proposed detectors can provide better detection performance than their natural counterparts in some scenarios.

Introduction

Detection of a distributed target in unknown disturbance is an interesting research topic. Up to now a large number of approaches have been proposed to solve the problem in various scenarios. Particularly, in [1] a distributed target is embedded in unknown Gaussian noise, and the echoes reflected by the distributed target are assumed to have the same direction. To estimate the unknown covariance matrix, a set of signal-free training (secondary) data is required. When the test (primary) data and secondary data have the identical covariance matrix, it is usually referred to as the homogeneous environment (HE) [1]. In contrast, when the primary and secondary data share the same structure of the covariance matrix, but with unknown power mismatch, it is denoted as the partial HE (PHE) [1]. Based on the generalized likelihood ratio test (GLRT) criterion, several detectors for the HE and PHE are proposed in [1]. The Rao and Wald tests are derived in [2] for the HE and in [3] for the PHE. Remarkably, a milestone in the area of multichannel signal detection in unknown Gaussian noise is the report in [4], where the model is very general and the GLRT-based detectors are proposed and detailed analyzed. More recently, the detection model in [4] is further generalized in [5], [6] and many detectors are devised. Some other related work is exploited in [7], [8], [9], [10], [11], [12], [13], [14] and the references therein.

It is worth noting that interference is not taken into account in the references mentioned above. However, in many practical applications there usually exists interference due to electronic countermeasure (ECM) systems or civil broadcasting systems. The detection problem is addressed in [15], [16], [17], [18], [19], [20] in the presence of deterministic interference under the assumption of white Gaussian noise or colored Gaussian noise with known covariance matrix. Moreover, the GLRT is derived in [21] for detecting a point-like target in undernulled noise-like interference and unknown Gaussian noise. The resulting GLRT is shown to be equivalent to the adaptive coherence estimator (ACE). In [22], the problem of detecting a point-like target in deterministic interference and unknown Gaussian noise is dealt with. The interference lies in the primary and secondary data, and it is confined to a known subspace. The detection problem is solved by the method of sieves. For the distributed target detection, in [23] the echo signals reflected from the target are all assumed to come from the same direction, and the signal steering vector lies in a known subspace with an unknown coordinate. The GLRT and two-step GLRT (2S-GLRT) are proposed in the presence of deterministic interference for the HE. The detection problem in [23] is extended in [24], where the deterministic interference is assumed to lie in an unknown subspace except for the knowledge of the interference subspace dimension. Additionally, in [25] the interference and target signal are assumed to lie in two linearly independent subspaces, and the GLRT and 2S-GLRT are proposed for the HE and PHE.

Note that there exists no uniformly most powerful (UMP) test for the detection problem in [25], since the noise covariance matrix and the coordinates of the signal and interference are unknown. Hence, it may be reasonable to adopt approaches different from the GLRT and 2S-GLRT employed therein to devise detectors. Besides the GLRT criterion, the other two criteria widely used for detector design are the Rao test and Wald test, e.g., [26], [27], [28], [29], [30]. In this paper we adopt the Rao test, including the one-step and two-step versions, to design detectors for the detection problem in [25], since the Rao test can achieve best detection performance in some scenarios¹. It is shown that all the proposed detectors are constant false alarm rate (CFAR) with respect to (w.r.t.) the unknown covariance matrix. Moreover, the proposed detectors can achieve better detection performance than the existing ones in some situations.

The rest of the paper is organized as follows. Section 2 formulates the detection problem to be solved. Section 3 derives the one-step Rao test and two-step Rao (2S-Rao) test in the HE and PHE. Section 4 compares the detection performance of the proposed detectors with the existing ones by Monte Carlo simulations. Finally, some concluding remarks are given in Section 5.

Notations: Matrices, vectors, scalars, are denoted by bold-face upper case letters, bold-face lower case letters, and light-face lower case letters, respectively. The superscripts ${(\cdot )}^{⁎}$, ${(\cdot )}^T$, and ${(\cdot )}^H$ stands for the conjugate, transpose, and conjugate transpose of a vector or matrix, respectively. The notation $\otimes$ is the Kronecker product. For a complex number $a$, $abs(a)$ stands for its modulus. For an $N\times K$ matrix $\mathit{A}$, $E\left[\mathit{A}\right]$ denotes the statistical expectation of $\mathit{A}$, $vec(\mathit{A})$ vectorizes $\mathit{A}$ by stacking its columns, ${\mathit{P}}_{\mathit{A}}$ is the orthogonal projector (projection matrix) onto the subspace spanned by the columns of $\mathit{A}$, i.e., ${\mathit{P}}_{\mathit{A}}=\mathit{A}{({\mathit{A}}^H\mathit{A})}^{-1}{\mathit{A}}^H$, and ${\mathit{P}}_{\mathit{A}}^{\perp }={\mathit{I}}_N-{\mathit{P}}_{\mathit{A}}$. When $\mathit{A}$ becomes a square matrix, $\left|\mathit{A}\right|$ and $tr(\mathit{A})$ denote its determinant and trace, respectively. Further, when $\mathit{A}$ turns into a positive definite matrix, ${\mathit{A}}^{1/2}$ denotes its square-root matrix (positive definite and satisfying ${\mathit{A}}^{1/2}{\mathit{A}}^{1/2}=\mathit{A}$), and ${\mathit{A}}^{-1/2}$ is the inverse of ${\mathit{A}}^{1/2}$. $\min (a,b)$ denotes the minimum value of $a$ and $b$. $lnf$ stands for the natural logarithm of the scalar function $f$, and $\partial f/\partial \mathit{\Xi }$ is the partial derivative of $f$ wr.t. $\mathit{\Xi }$, with $\mathit{\Xi }$ being a vector or matrix. Moreover, ${\widehat{\mathit{\Xi }}}_i$ is the maximum likelihood estimate (MLE) of $\mathit{\Xi }$ under hypothesis $H_i$, $i=0,1$. Finally, ${\mathit{I}}_N$ is an $N\times N$ identity matrix and ${\mathbf{0}}_{p\times q}$ is a $p\times q$ null matrix.