Invariant tests for rapid-fluctuating radar signal detection with unknown arrival time
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.
Section snippets
Rapid fluctuating radar detection problem
Assume that we have the data set of K range cells in a radar, and for each cell N signal samples are available. In the rapid fluctuating situation, the echoes are fluctuating such that no relation is assumed between the signal samples received from within a range cell. We assume that only in one of K range cells, there may be a target and the signal received from other cells are due to noise and clutter. If denotes the received vector for the th range cell, and denotes
Uniformly most powerful invariant test
In this section, we study the rapid-fluctuating radar signal detection (2) in AWGN. In particular, the vectors are Gaussian, zero-mean, independent and identically distributed random vectors with an unknown noise variance. The signal and the probable target cell are also assumed to be unknown. We first show that the UMPI test does exist for this problem only if the SNR is known. The UMPI test in known SNR is derived and its performance is used as a criterion for the evaluation of
GLRT for zero-mean additive white Gaussian noise interference
In this section, we use the GLRT approach to solve the binary hypothesis test (2) where the interference is a zero mean iid AWGN. We obtain the maximum likelihood estimates (MLEs) of the unknown parameters under each hypothesis and substitute them in the pdfs and construct the LR. Since the number of unknown parameters, is more than the number of observation samples , the GLRT cannot be obtained using just the primary data. Therefore, we derive the GLRT using both the
Relationship of the problem with detection of sinusoidal signal
In this section, we describe the relationship between the problem described in (1) and (2) and the detection of a single complex sinusoidal signal which is considered in [8]. In particular, we show how this paper extends the results of [8]. In order to better understand the relationship between the problem considered in this paper and the detection of a sinusoidal signal, we apply a discrete Fourier transform (DFT) to (1), as follows:where
Simulation examples
Simulation examples that illustrate the analytic relative performance results previously discussed are provided. In our simulations, we determined the threshold in each test experimentally, as follows: the decision statistics for independent trials in the absence of signal were sorted in ascending order and the threshold was chosen as the —percentile of the resulting data. For example, for , the threshold is chosen as the th ordered data; i.e., such that
Conclusion
In this paper, we studied the rapid-fluctuating radar signal detection in AWGN and clutter with unknown covariance matrix and proposed invariant detectors for each of them. In these detectors, instead of performing the test in individual range cells, the range cell index of the target is assumed as an unknown parameter and perform the test once for all the range cells data. We showed that in AWGN, the UMPI test does not exist unless if the SNR of the echo signal is known. We proposed GLR
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