Parametric Adaptive Radar Detector with Enhanced Mismatched Signals Rejection Capabilities
© Chengpeng Hao et al. 2010
Received: 12 August 2010
Accepted: 2 November 2010
Published: 8 November 2010
We consider the problem of adaptive signal detection in the presence of Gaussian noise with unknown covariance matrix. We propose a parametric radar detector by introducing a design parameter to trade off the target sensitivity with sidelobes energy rejection. The resulting detector merges the statistics of Kelly's GLRT and of the Rao test and so covers Kelly's GLRT and the Rao test as special cases. Both invariance properties and constant false alarm rate (CFAR) behavior for this detector are studied. At the analysis stage, the performance of the new receiver is assessed and compared with several traditional adaptive detectors. The results highlight better rejection capabilities of this proposed detector for mismatched signals. Further, we develop two two-stage detectors, one of which consists of an adaptive matched filter (AMF) followed by the aforementioned detector, and the other is obtained by cascading a GLRT-based Subspace Detector (SD) and the proposed adaptive detector. We show that the former two-stage detector outperforms traditional two-stage detectors in terms of selectivity, and the latter yields more robustness.
Adaptive detection of signals embedded in Gaussian or non-Gaussian disturbance with unknown covariance matrix has been an active research field in the last few decades. Several generalized likelihood ratio test- (GLRT-) based methods are proposed, which utilize secondary (training) data, that is, data vectors sharing the same spectral properties, to form an estimate of the disturbance covariance. In particular, Kelly  derives a constant false alarm rate (CFAR) test for detecting target signals known up to a scaling factor; Robey et al.  develops a two-step GLRT design procedure, called adaptive matched filter (AMF). Based on the above methods, some improved approaches have been proposed, for example, the non-Gaussian version of Robey's adaptive strategy in [3–6] and the extended targets version of Kelly's adaptive detection strategy in . In addition, considering the presence of mutual coupling and near-field effects, De Maio et al.  redevises Kelly's GLRT detector and the AMF.
Most of the above methods work well, provided that the exact knowledge of the signal array response vector is available; however, they may experience a performance degradation in practice when the actual steering vector is not aligned with the nominal one. A side lobe mismatched signal may appear subject to several causes, such as calibration and pointing errors, imperfect antenna shape, and wavefront distortions. To handle such mismatched signals, the Adaptive Beamformer Orthogonal Rejection Test (ABORT)  is proposed, which takes the rejection capabilities into account at the design stage, introducing a tradeoff between the detection performance for main lobe signals and rejection capabilities for side lobe ones. The directivity of this detector is in between that of the Kelly's GLRT and the Adaptive Coherence Estimator (ACE) [10, 11]. A Whitened ABORT (W-ABORT) [12, 13] is proposed to address adaptive detection of distributed targets embedded in homogeneous disturbance via GLRT and the useful and fictitious signals orthogonal in the whitened space, which has an enhanced rejection capability for side lobe signals. Some alternative approaches are devised [14–17], which basically depend on constraining the actual signature to span a cone, whose axis coincides with its nominal value. Moreover, in , a detector based on the Rao test criterion is introduced and assessed. It is worth noting that the Rao test exhibits discrimination capabilities of mismatched signals better than those of the ABORT, although it does not consider a possible spatial signature mismatch at the design stage.
From another point of view, increased robustness to mismatch signals can be obtained by two-stage tunable receivers that are formed by cascading two detectors (usually with opposite behaviors), in which case, only data vectors exceeding both detection thresholds will be declared as the target bearings [19–23]. Remarkably, such solutions can adjust directivity by proper selection of the two thresholds to trade good rejection capabilities of side lobe signals for an acceptable detection loss for matched signals. An alternative approach to design tunable receivers relies on the parametric adaptive detectors, which allow us to trade off target sensitivity with side lobes energy rejection via tuning a design parameter [24, 25]. In particular, in , Kalson devises a parametric detector obtained by merging the statistics of Kelly's GLRT and of the AMF, whereas in , Bandiera et al. propose another parametric adaptive detector, which is obtained by mixing the statistic of Kelly's GLRT with that of the W-ABORT.
In this paper, we attempt to increase the rejection capabilities of tunable receivers and develop a novel adaptive parametric detector, which is obtained by merging the statistics of the Kelly's GLRT and of the Rao test. We show that the proposed detector is invariant under the group of transformations defined in . As a consequence, it ensures the CFAR property with respect to the unknown covariance matrix of the noise. The performance assessment, conducted analytically for matched and mismatched signals, highlights that specified with a appropriate design parameter the new detector has better rejection capabilities for side lobe targets than existing decision schemes. However, if the value of the design parameter is bigger than or equals to unity, this new detector leads to worse detection performance than Kelly's receiver. To circumvent this drawback, a two-stage detector is proposed, which consists of the AMF followed by the proposed parametric adaptive detector and can be taken as an improved alternative of the two-stage detector in . We also give another two-stage detector with enhanced robustness, which is obtained by cascading the GLRT-based Subspace Detector (SD)  and the proposed parametric adaptive receiver.
The paper is organized as follows. In the next section, we formulate the problem and then propose the adaptive parametric detector. In Section 3, we analyze the performance of the proposed receiver. We present two newly proposed two-stage tunable detectors, respectively, in Sections 4 and 5. Section 6 contains conclusions and avenues for further research. Finally, some analytical derivations are given in the Appendix.
2. Problem Formulation and Design Issues
We assume that data are collected from sensors and denote by the complex vector of the samples where the presence of the useful signal is sought (primary data). As customary, we also suppose that a secondary data set , , is available ( ), that each of such snapshots does not contain any useful target echo and exhibits the same covariance matrix as the primary data (homogeneous environment).
is the decision statistic of Kelly's GLRT.
It is clear that our detector covers Kelly's GLRT and the Rao test as special cases, respectively, when and . Moreover, since can be expressed in terms of the maximal invariant statistic ( , ), it is invariant with respect to the transformations defined in . As a consequence, it ensures the CFAR property with respect to the unknown covariance matrix of the noise.
3. Performance Assessment
3.3. Performance Analysis
4. Two-Stage Detector Based on the KRAO
namely, the two-stage detector achieves the same performance as that of the KRAO test;
5. Improved Two-Stage Detector Based on the KRAO
where , is the pdf of the rv , is the pdf of the rv , and is the cdf of the rv , given and . As can be seen from (41), the of the SKRAO-ASB depends on the threshold pairs ( ) and the design parameter , as a consequence of which, the SKRAO-ASB possesses the constant false alarm rate (CFAR) property with respect to the disturbance covariance matrix .
- (i)given and , is ruled by the complex noncentral F-distribution with 1, degrees of freedom and noncentrality parameter
- (ii)is a complex noncentral F-distribution rv with , degrees of freedom and noncentrality parameter
- (iii)given , obeys the complex noncentral F-distribution with , degrees of freedom and noncentrality parameter
Finally, we compare the SKRAO-ASB and the KRAO-ASB in terms of computational complexity. We focus on the first stage of each detector, since the second stage of each detector is to be computed only if the fist stage declares a detection. Observe that the AMF does not require the on-line inversion of the matrix ( ) and the computation of the extra term , which are necessary to implement the SD decision statistic. It is thus apparent that the KRAO-ASB is faster to implement than the SKRAO-ASB. Anyway, resorting to the usual Landau notation, the SKRAO-ASB involves floating-point operations (flops), whereas the KRAO-ASB requires flops.
We propose a new parametric radar detector, KRAO, by merging the statistics of the Kelly's GLRT test and of the Rao test. We discuss its invariance and CFAR property. We derive the closed-form expressions for the probability of false alarm and the probability of detection in matched and mismatched cases.
We demonstrate performance of KRAO via simulations. Numerical results show that, with a properly selected value for the design parameter, the proposed KRAO can yield better rejection capabilities of mismatched signals than its counterparts. However, when the sensitivity parameter is greater than or equal to unity, it has a nonnegligible loss for matched signals compared with Kelly's GLRT.
To compensate the matched detection performance of the KRAO, we propose a two-stage detector consisting of an adaptive matched filter followed by the KRAO. We show that such a two-stage detector has desirable property in terms of selectivity. Its invariance and CFAR property have been studied.
To increase the robustness of the aforementioned two-stage detector, we introduce another two-stage detector by cascading a GLRT-based subspace detector and the KRAO. It possesses the CFAR property with respect to the unknown covariance matrix of the noise and it can guarantee a wider range of directivity values with respect to aforementioned two-stage detector.
Further work will involve the analysis of the proposed tunable receivers in a partially homogeneous (Gaussian) environment scenario, that is, when the noise covariance matrices of the primary and the secondary data have the same structure but are at different power levels. It is also needed to investigate these tunable receivers in a clutter-dominated non-Gaussian scenario.
The authors are very grateful to the anonymous referees for their many helpful comments and constructive suggestions on improving the exposition of this paper. This work was supported by the National Natural Science Foundation of China under Grant no. 60802072.
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