- Research Article
- Open Access

# Parametric Adaptive Radar Detector with Enhanced Mismatched Signals Rejection Capabilities

- Chengpeng Hao
^{1}Email author, - Bin Liu
^{2}, - Shefeng Yan
^{1}and - Long Cai
^{1}

**2010**:375136

https://doi.org/10.1155/2010/375136

© Chengpeng Hao et al. 2010

**Received:**12 August 2010**Accepted:**2 November 2010**Published:**8 November 2010

## Abstract

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.

## Keywords

- Side Lobe
- Constant False Alarm Rate
- Noncentrality Parameter
- Mismatched Signal
- Rejection Capability

## 1. Introduction

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 [1] derives a constant false alarm rate (CFAR) test for detecting target signals known up to a scaling factor; Robey et al. [2] 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 [7]. In addition, considering the presence of mutual coupling and near-field effects, De Maio et al. [8] 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) [9] 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 [18], 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 [24], Kalson devises a parametric detector obtained by merging the statistics of Kelly's GLRT and of the AMF, whereas in [25], 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 [26]. 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 [18]. We also give another two-stage detector with enhanced robustness, which is obtained by cascading the GLRT-based Subspace Detector (SD) [27] 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).

- (i)
where denotes expectation and conjugate transposition;

- (ii)
is the unit-norm steering vector of main lobe target echo, which is possibly different from that of the nominal steering vector ;

- (iii)
is an unknown deterministic factor which accounts for both target reflectivity and channel effects.

is the decision statistic of Kelly's GLRT.

where is the design parameter.

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 [26]. As a consequence, it ensures the CFAR property with respect to the unknown covariance matrix of the noise.

## 3. Performance Assessment

### 3.1. of the KRAO

- (i)
given , is ruled by the complex central F-distribution with 1, degrees of freedom, namely, ;

- (ii)
is a complex central beta distribution random variable (rv) with , degrees of freedom, namely, .

- (i)
- (ii)
- (iii)
- (iv)

### 3.2. of the KRAO

- (i)given , is ruled by the complex noncentral F-distribution with 1, degrees of freedom and noncentrality parameter
namely, , where is the total available signal-to-noise ratio;

- (ii)is a complex noncentral beita distribution rv with , degrees of freedom and noncentrality parameter
namely, .

where is the pdf of the rv , and then, given , is the cdf of the rv .

- (i)
- (ii)
- (iii)
- (iv)

### 3.3. Performance Analysis

In this subsection, we present numerical examples to illustrate the performance of the KRAO. The curves are obtained by numerical integration and the probability of false alarm is set to .

## 4. Two-Stage Detector Based on the KRAO

- (i)
namely, the two-stage detector achieves the same performance as that of the KRAO test;

- (ii)

where is the pdf of the rv , and is the cdf of the rv , given .

## 5. Improved Two-Stage Detector Based on the KRAO

where is the radar operating wavelength, is the interelement spacing, and denotes transposition.

- (i)
given and , is ruled by the complex central F-distribution with 1, degrees of freedom, namely, ;

- (ii)
is a complex central F-distribution random variable (rv) with , degrees of freedom, namely, ;

- (iii)
obeys the complex central F-distribution with , degrees of freedom, namely, ;

- (iv)
and are statistically independent rv's.

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
namely, ;

- (ii)is a complex noncentral F-distribution rv with , degrees of freedom and noncentrality parameter
namely, ;

- (iii)given , obeys the complex noncentral F-distribution with , degrees of freedom and noncentrality parameter
namely, .

where is the pdf of the rv , is the pdf of the rv , given , and is the cdf of , given and .

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.

## 6. Conclusions

- (i)
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.

- (ii)
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.

- (iii)
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.

- (iv)
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.

## Declarations

### Acknowledgments

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.

## Authors’ Affiliations

## References

- Kelly EJ: An adaptive detection algorithm.
*IEEE Transactions on Aerospace and Electronic Systems*1986, 22(2):115-127.View ArticleGoogle Scholar - Robey FC, Fuhrmann DR, Kelly EJ, Nitzberg R: A CFAR adaptive matched filter detector.
*IEEE Transactions on Aerospace and Electronic Systems*1992, 28(1):208-216. 10.1109/7.135446View ArticleGoogle Scholar - Greco M, Gini F, Diani M: Robust CFAR detection of random signals in compound-Gaussian clutter plus thermal noise.
*IEE Proceedings: Radar, Sonar and Navigation*2001, 148(4):227-232. 10.1049/ip-rsn:20010475Google Scholar - Younsi A, Greco M, Gini F, Zoubir AM: Performance of the adaptive generalised matched subspace constant false alarm rate detector in non-Gaussian noise: an experimental analysis.
*IET Radar, Sonar and Navigation*2009, 3(3):195-202. 10.1049/iet-rsn:20080101View ArticleGoogle Scholar - de Maio A, Alfano G, Conte E: Polarization diversity detection in compound-Gaussian clutter.
*IEEE Transactions on Aerospace and Electronic Systems*2004, 40(1):114-131. 10.1109/TAES.2004.1292147View ArticleGoogle Scholar - Shuai X, Kong L, Yang J: Performance analysis of GLRT-based adaptive detector for distributed targets in compound-Gaussian clutter.
*Signal Processing*2010, 90(1):16-23. 10.1016/j.sigpro.2009.05.008View ArticleMATHGoogle Scholar - Conte E, de Maio A, Ricci G: GLRT-based adaptive detection algorithms for range-spread targets.
*IEEE Transactions on Signal Processing*2001, 49(7):1336-1348. 10.1109/78.928688View ArticleGoogle Scholar - de Maio A, Landi L, Farina A: Adaptive radar detection in the presence of mutual coupling and near-field effects.
*IET Radar, Sonar and Navigation*2008, 2(1):17-24. 10.1049/iet-rsn:20060077View ArticleGoogle Scholar - Pulsone NB, Rader CM: Adaptive beamformer orthogonal rejection test.
*IEEE Transactions on Signal Processing*2001, 49(3):521-529. 10.1109/78.905870View ArticleGoogle Scholar - Conte E, Lops M, Ricci G: Asymptotically optimum radar detection in compound-Gaussian clutter.
*IEEE Transactions on Aerospace and Electronic Systems*1995, 31(2):617-625.View ArticleGoogle Scholar - Kraut S, Scharf LL: The CFAR adaptive subspace detector is a scale-invariant GLRT.
*IEEE Transactions on Signal Processing*1999, 47(9):2538-2541. 10.1109/78.782198View ArticleGoogle Scholar - Bandiera F, Besson O, Ricci G: An ABORT-like detector with improved mismatched signals rejection capabilities.
*IEEE Transactions on Signal Processing*2008, 56(1):14-25.MathSciNetView ArticleGoogle Scholar - Bandiera F, Besson O, Orlando D, Ricci G: Theoretical performance analysis of the W-ABORT detector.
*IEEE Transactions on Signal Processing*2008, 56(5):2117-2121.MathSciNetView ArticleGoogle Scholar - Greco M, Gini F, Farina A: Radar detection and classification of jamming signals belonging to a cone class.
*IEEE Transactions on Signal Processing*2008, 56(5):1984-1993.MathSciNetView ArticleGoogle Scholar - de Maio A: Robust adaptive radar detection in the presence of steering vector mismatches.
*IEEE Transactions on Aerospace and Electronic Systems*2005, 41(4):1322-1337. 10.1109/TAES.2005.1561887View ArticleGoogle Scholar - Besson O: Detection of a signal in linear subspace with bounded mismatch.
*IEEE Transactions on Aerospace and Electronic Systems*2006, 42(3):1131-1139.MathSciNetView ArticleGoogle Scholar - Bandiera F, de Maio A, Ricci G: Adaptive CFAR radar detection with conic rejection.
*IEEE Transactions on Signal Processing*2007, 55(6):2533-2541.MathSciNetView ArticleGoogle Scholar - de Maio A: Rao test for adaptive detection in Gaussian interference with unknown covariance matrix.
*IEEE Transactions on Signal Processing*2007, 55(7):3577-3584.MathSciNetView ArticleGoogle Scholar - Richmond CD: Performance of a class of adaptive detection algorithms in nonhomogeneous environments.
*IEEE Transactions on Signal Processing*2000, 48(5):1248-1262. 10.1109/78.839973MathSciNetView ArticleGoogle Scholar - Richmond CD: Performance of the adaptive sidelobe blanker detection algorithm in homogeneous environments.
*IEEE Transactions on Signal Processing*2000, 48(5):1235-1247. 10.1109/78.839972MathSciNetView ArticleGoogle Scholar - Bandiera F, Orlando D, Ricci G: A subspace-based adaptive sidelobe blanker.
*IEEE Transactions on Signal Processing*2008, 56(9):4141-4151.MathSciNetView ArticleGoogle Scholar - Bandiera F, Besson O, Orlando D, Ricci G: A two-stage detector with improved acceptance/rejection capabilities.
*Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP '08), April 2008, Las Vegas, Nev, USA*2301-2304.Google Scholar - Bandiera F, Besson O, Orlando D, Ricci G: An improved adaptive sidelobe blanker.
*IEEE Transactions on Signal Processing*2008, 56(9):4152-4161.MathSciNetView ArticleGoogle Scholar - Kalson SZ: An adaptive array detector with mismatched signal rejection.
*IEEE Transactions on Aerospace and Electronic Systems*1992, 28(1):195-207. 10.1109/7.135445View ArticleGoogle Scholar - Bandiera F, Orlando D, Ricci G: One- and two-stage tunable receivers.
*IEEE Transactions on Signal Processing*2009, 57(6):2064-2073.MathSciNetView ArticleGoogle Scholar - Bose S, Steinhardt AO: Maximal invariant framework for adaptive detection with structured and unstructured covariance matrices.
*IEEE Transactions on Signal Processing*1995, 43(9):2164-2175. 10.1109/78.414779View ArticleGoogle Scholar - Kraut S, Scharf LL, McWhorter LT: Adaptive subspace detectors.
*IEEE Transactions on Signal Processing*2001, 49(1):1-16. 10.1109/78.890324View ArticleGoogle Scholar - Kelly EJ:
*Adaptive detection in non-stationary interference—part III.*MIT, Lincoln Laboratory, Lexington, Mass, USA; August 1987.Google Scholar

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