A tunable detector for distributed targets when signal mismatch occurs

Aiming at the problem of distributed target detection when there is sig- nal mismatch, this paper proposes a tunable detector, which is charac-terised by a tunable parameter. The proposed detector can realise the ﬂexible detection of mismatch signals by adjusting the tunable parameter, and hence the directivity (robustness or selectivity to mismatched signals) of the tunable detector can be smoothly adjusted. In addition, the tunable detector can obtain approximately the same detection per- formance as the corresponding generalised likelihood ratio test in the absence of signal mismatch. The proposed tunable detector also pos- sesses the constant false alarm rate property. Numerical examples illus-trate the effectiveness of the proposed detector.

Introduction: Since its appearance, the functions of the radar have been continuously expanded and improved; however, target detection is always one of the most important functions. The problem of multichannel adaptive signal detection was first investigated by Kelly. Based on the generalised likelihood ratio test (GLRT) criterion, Kelly proposed his GLRT, denoted as Kelly's GLRT (KGLRT) [1]. Based on Kelly's research, a variety of detectors have been proposed e.g., adaptive matched filter (AMF) [2] and De Maio's Rao detector [3]. Recently, a book about multichannel adaptive signal detection was edited by De Maio and Greco in [4] and a literature review was given by Liu et al. in [5].
With the development of radar technology, radar resolution continues to improve. Hence, a single target may occupy multiple range resolution units. Conte et al. proposed several GLRT-based adaptive detectors for distributed targets [6]. However, the phenomenon of signal mismatch was not considered in [6]. In practice, a robust detector may be preferred if signal mismatch occur due to uncalibrated arrays, uncertainty of the target's direction of arrival (DOA), etc. In contrast, a selective detector may be needed if signal mismatch is due to sidelobe targets or jamming. A selective detector provides a probability of detection (PD), which diametrically reduces as the increase of amount of signal mismatch.
Two selective detectors were proposed in [7,8] for distributed target detection. However, these two detectors have only selectivity characteristics to mismatched signals, but lack robustness. A parametric adjustable detector for distributed target was proposed in [8] for partially homogeneous environment. However, this detector is not suitable for the homogeneous environment (HE). For this reason, this paper proposes a parametric adjustable detector suitable for mismatched signals of the distributed target in HE. By adjusting a parameter, the proposed detector can realise robust detection or selective rejection of the mismatch signal.
Detector Design: For an array radar containing N antennas, if a distributed target occupies K range bins, the data vector reflected by kth range bin within the distributed target can be denoted by an N × 1 vector x k , k = 1, 2, ..., K. The test data vector x k contains noise n k under the null hypothesis H 0 . In contrast, under the alternative hypothesis H 1 , x k includes signal s k and noise n k . The signal s k has the form s k = a*h, where a* is the signal amplitude, with ( · )* denoting conjugate, and h is the normalized signal steering vector. The noise n k is mutually independent and distributed as n k ∼ CN N (0 N×1 , R). That is to say, the mean of n k is 0 N × 1 , and the covariance matrix is R. In practical applications, R is usually unknown, so a certain number of training samples are required to estimate R. Suppose there are L training data, denoted as x e,l , l = 1, 2, ..., L, containing only noise n e,l ∼ CN N (0 N×1 , R). The detection problem can be expressed as where ( · ) H denotes conjugate transpose, X = [x 1 ,x 2 ,..., x K ], N= [n 1 ,n 2 ,...,n K ], a = [a 1 ,a 2 ,..., a K ] T , X L = [x e,1 ,x e,2 ,...,x e,L ], N L =[n e,1 , n e,2 , ..., n e,L ].
For the detection problem in (1), the GLRT and two-step GLRT are [4] t GLRT−HE = I K +X HX I K +X H P ⊥ sX (2) and respectively, whereX = S −1/2 X,s = S −1/2 s, S= L l=1 x l x H l is L times the sample covariance matrix. The detector in (3) was referred to as the generalised adaptive matched filter (GAMF) in [6]. Moreover, we refer the detector in (2) as the GLRT in HE (GLRT-HE) for convenience.
Note that the GLRT-HE in (2) and the GAMF in (3) were designed under the absence of signal mismatch. To devise a selective detector for mismatched signals, the detection problem in (1) were modified by adding a fictitious signal under hypothesis H 0 . Then, two selective detectors was designed in [7,8] according to the GLRT, described as which, for convenience, are denoted as the generalised ABORT in HE (G-ABORT-HE) and the generalised W-ABORT in HE (GW-ABORT-HE).
As mentioned above, a robust detector is preferred in many applications. However, none of the above detectors is designed with the purpose of robust detection. To manage the tradeoff between robustness and selectivity for mismatched signal, we introduce a tunable detector based on the GLRT-HE in (2).
For the GLRT-HE in (2) with a signal having fixed energy, when the amount of signal mismatch increases, the quantity |I K +X H P ⊥ sX | will increase too. Therefore, if the proportion of the quantity |I K +X H P ⊥ sX | increases, the detector will become more sensitive to mismatch. On the other hand, when the proportion of the quantity |I K +X H P ⊥ sX | reduces, the detector will become more robust. Based on the above analysis, we propose a parameter-tunable detector, describe as t T−GLRT−HE = I K +X HX where γ is the tunable parameter. The detector in (5) is called tunable GLRT in HE (T-GLRT-HE). It is easy to show that the larger γ is, the more sensitive the T-GLRT-HE is. In contrast, the smaller γ is, the more robust the T-GLRT-HE is. In particular, when γ = 1, the T-GLRT-HE degenerates into the GLRT-HE. When γ = 0, the detector T-GLRT-HE becomes which does not contain the orientation information of the target, so it has extremely robust characteristics. The T-GLRT-HE has the constant false alarm rate (CFAR) property, since the statistical properties of the quan-titiesX HX andX H P ⊥ sX do not depend on the noise covariance matrix R under hypothesis H 0 [9].
Numerical Examples: In this section, the detection performance of the T-GLRT-HE is evaluated. The PD is examined in two cases, namely, the case of signal mismatch and the case of no signal mismatch. The probability of false alarm (PFA) is set to be 10 -3 . The detection threshold and PD are obtained by 100/PFA and 10 4 Monte Carlo experiments, respectively. The (i, j) element of the R is ε |i − j| , i, j = 1, 2, ..., N. The signal-to-noise ratio (SNR) is defined as where h 0 is the actual signal steering vector. In all simulations, N=12, ε = 0.95. A commonly used quantity to measure signal mismatch is the cosine square of the angle between the actual signal steering vector h 0 and the presumed one h in whitening space, defined as The case of signal mismatch: Figures 1 and 2 show the detection performance of the detectors under different amounts of signal mismatches with the target expansion as a parameter. The results indicate that the T-GLRT-HE has the robust characteristics when 0.1 < γ < 1, while the T-GLRT-HE has the selectivity characteristics when γ > 1. In particular, the T-GLRT-HE has the most robust characteristics when γ = 0.5, while the T-GLRT-HE has the best selectivity characteristics when γ = 3. We can conclude that the directivity (selectivity and robustness) of the T-GLRT-HE can be smoothly change by adjusting the value of the tunable parameter.
The case of no signal mismatch: Figures 3 and 4 show the detection performance of the detectors under different SNRs in the absence of signal mismatch. The results indicate that the T-GLRT-HE can achieve roughly the same detection performance with the GLRT-HE. Moreover, the PDs of the detectors all decrease when the SNR is fixed and the target expansion becomes higher. An explanation is that the higher the target expansion, the larger the unknown parameter space. This fact leads to a decrease in detection performance. Figure 5 shows the detection performance of the T-GLRT-HE under different adjustable parameters without signal mismatch. It can be seen that as the adjustable parameter increases, the detection probability of the T-GLRT-HE first increases and then decreases. And the highest PD is equal to that of the GLRT-HE.

Conclusion:
In this paper, we proposed the T-GLRT-HE for distributed target detection in the presence of signal mismatch. With a large value of tunable parameter, the T-GLRT-HE can achieve selective detection by rejecting mismatched signals. In contrast, the T-GLRT-HE with a moderately small tunable parameter can provide robust detection by maintaining high PD for mismatched signals. In addition, when there is no signal mismatch, the T-GLRT-HE, under reasonable parameter settings, can provide a comparative PD with the GLRT-HE. Moreover, the T-GLRT-HE also has the CFAR property.