Invariant tests for rapid-fluctuating radar signal detection with unknown arrival time

Abstract

We study the rapid-fluctuating radar signal detection, where the arrival time of the received signal, i.e., the range cell index of the target, as well as the complex amplitude of the signal and the noise are unknown. We show that even in additive white Gaussian noise (AWGN) environment, uniformly most powerful invariant (UMPI) test does not exist. Instead, the UMPI detector in known SNR is used as an upper performance bound for performance evaluation of any invariant detector performance. In addition, we derive generalized likelihood ratio (GLR) detectors for this signal in AWGN with unknown noise variance and also in clutter with unknown covariance matrix. The GLR test statistic for AWGN represents the ratio of the maximum power over all cells to the total power. The GLRT for a clutter environment is also a power ratio which is the maximum over all range cells of the Euclidean norms of the spatially whitened observed sequence. Simulation results demonstrate that the performance of the proposed GLR test in AWGN is very close to the upper bound performance, i.e., to that of the UMPI test in known SNR.

Introduction

A radar detects the existence of a target where some parameters of the target and/or environment are unknown. Marcum in [1] proposed a solution for the radar signal detection in additive white Gaussian noise (AWGN). In his work, the target is modeled as a known signal and is considered as a point target. Swerling [2] introduced four models for the target namely non-fluctuating, slowly fluctuating, moderate-fluctuating and rapid-fluctuating, where the non-fluctuating model is assumed in Marcum's work. Brennan et al. in [3] and Nayebi et al. in [4] proposed detectors for non-fluctuating and slowly fluctuating radar signals, respectively. Modarres-Hashemi et al. in [5] developed an average likelihood ratio (ALR) solution for the coherent detection of the rapid-fluctuating targets. In particular, they derived a locally ALR detector for small signal-to-noise ratio (SNR). The noise variance is considered as a known parameter in these references.

In [6], [7], [8], the coherent detection of slowly fluctuating radar signal is considered where the Doppler frequency and the noise variance are unknown. This problem is modeled as the detection of a sinusoid signal with unknown amplitude, phase and frequency in unknown Gaussian noise environment. Kay and Gabriel in [8] derived a generalized likelihood ratio (GLR) detector for the detection of a sinusoid in complex AWGN environment where the noise variance and the signal amplitude, phase and frequency are all unknown. Using invariance principles, they showed that this problem is invariant under the composition of two groups of scale and modulation and found that the uniformly most powerful invariant (UMPI) test does not exist. Instead, they derived a UMPI detector in known SNR that gives an upper performance bound for any invariant test (such as GLR test) for this problem.

The problem of radar signal detection in the presence of noise and clutter is modeled as the detection of a signal in the presence of an interference with unknown correlation statistics [9], [10]. Kelly developed the GLR test for the detection of a known signal with unknown multiplicative constant in zero-mean complex Gaussian N-vector additive interference. This method requires a set of secondary data (a set of mutually independent N-vectors) in order to estimate the unknown covariance matrix. The statistical properties of the secondary data must be similar to those of the interference in the primary data, to be used in the absence of the knowledge about the covariance matrix. Raghavan et al. in [11] considered the detection of an unknown N-dimensional complex vector whose correlation properties are unknown. They derived the GLR test for the rapid-fluctuating radar signal detection in clutter and noise and showed that their proposed GLR test is equivalent to the UMPI test [11].

In practice, since the arrival time of the echo signal is unknown, the existing radar receivers divide the range into multiple cells and apply hypothesis tests to signals received from within each of the individual range cells; i.e. for each range cell a test is performed to detect the presence of a probable target. One disadvantage of this method is that the same test must be repeated for all range cells. Another disadvantage is that such a detector is not of constant false alarm rate (CFAR). In this method, in order to test the existence of the target in each range cell, the detector employs the data samples from other range cells as the secondary data to estimate the noise variance or the covariance matrix of the interference. The test results for those cells without target is impacted (not only by the noise but also) by the signal of a possible target in other range cells and therefore the probability of false alarm depends on the power of the possible target in other cells, i.e., this detector is not CFAR. In order to improve the detection performance, an alternative approach is to consider the target location, i.e., the arrival time delay, as an unknown parameter in the detection problem. In this approach the data from different range cells are used simultaneously in the detection process; i.e., the unknown target location is also estimated along with the detection procedure. Kay [12] used this method for the derivation of the GLR detector for a slowly fluctuating radar signal in known variance white Gaussian noise.

In this paper, we consider the detection of a target using a rapid-fluctuating radar in AWGN assuming the target location as an unknown parameter (i.e., the arrival time of the received signal is to be estimated as well). We show that the problem is invariant under the composition of three groups of orthogonal transformation, permutation and scale. We show that for the case of unknown noise variance, the UMPI test does not exist unless the SNR is known. The performance of this test in known SNR gives an upper bound performance for evaluation of invariant tests. In addition, we propose a GLR detector which is invariant, since for invariant problems, the GLR test (GLRT) is always invariant [13]. Computer simulation examples indicate that GLRT performance is very close to the UMPI upper-bound performance. We finally derive a GLR test for the case of unknown clutter and evaluate its performance via simulations.

The paper is organized as follows: Section 2 describes the signal model. In Section 3, the UMPI test is presented. The GLR method used in both white and colored noise are derived in Section 4. In Section 6 simulation details and results are described. Concluding remarks are added in Section 7.