Parametric detector in the situation of mismatched signals - IEEE Xplore

www.ietdl.org Published in IET Radar, Sonar and Navigation Received on 30th January 2013 Revised on 9th June 2013 Accepted on 25th June 2013 doi: 10.1049/iet-rsn.2013.0044

ISSN 1751-8784

Parametric detector in the situation of mismatched signals Weijian Liu1,2, Wenchong Xie2, Yongliang Wang2 1

College of Electronic Science and Engineering, National University of Defense Technology, Changsha 410073, People’s Republic of China 2 Key Research Laboratory, Wuhan Radar Academy, Wuhan 430019, People’s Republic of China E-mail: [email protected]

Abstract: This study deals with the problem of adaptive detection in the presence of signal mismatch. An effective parametric detector is proposed, which encompasses Kelly’s generalised likelihood ratio test (GLRT), adaptive matched filter (AMF) and adaptive beamformer orthogonal rejection test (ABORT) as its three special cases. The novel detector can control the degree to which the mismatched signals are rejected. A remarkable feature is that the novel detector can achieve slightly higher probability of detection, for matched signals than those of Kelly’s GLRT, AMF and ABORT. Furthermore, it can also provide both improved robustness to the mismatched signals and enhanced rejection of the mismatched signals than its natural competitors.

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Introduction

A considerable amount of work has been directed towards the problem of detecting a multichannel signal, known up to a scaling factor in Gaussian or non-Gaussian disturbance with unknown covariance matrix. A large number of approaches have been proposed under various scenarios, among which Kelly’s generalised likelihood ratio test (GLRT) [1], adaptive matched filter (AMF) [2] and adaptive coherence estimator [3] are most famous. Other techniques have been successfully used to detect targets in environments with unknown parameters. Some of them are based on: component analysis in the frequency domain [4, 5], artificial neural networks [6–8] and genetic algorithms [7, 9]. However, the possibility of the signal mismatch is not taken into account at the design stage of the detectors mentioned above. In practice, some real effects, such as imperfect array calibration, pointing errors and wavefront distortions, often cause a deviation of the actual signal steering vector from the nominal one. This phenomenon is the so-called signal mismatch. An approach to cope with the signal mismatch is modifying the hypothesis test by adding a fictitious signal under the null hypothesis. This fictitious signal is assumed to be orthogonal to the presumed one in some manner. When there is no target in the presumed direction but one in another direction, the detector will incline towards the null hypothesis. This is the rationale of the adaptive beamformer orthogonal rejection test (ABORT) [10], whitened ABORT (W-ABORT) [11] and two Bayesian detectors [12]. They can achieve improved mismatched signals rejection, but have poor robustness to slightly mismatched signals. In [13–16], two-stage detectors, that is, the adaptive sidelobe blanker 48

& The Institution of Engineering and Technology 2013

(ASB) and its improved versions, are introduced. These detectors are formed by cascading two detectors with converse capabilities of mismatched signals rejection. One declares the presence of a target only when data survive both detection thresholds. However, the capabilities of rejection of and robustness to mismatched signals of the ASB-like detectors are confined by these capabilities of their individual detectors. In [17–21], the signal is modelled as a vector belonging to a proper cone or it is assumed that the angle of the signal is totally or partially uncertain. The corresponding detectors can achieve a visible performance improvement in terms of target sensitivity (robustness) but at the price of poor mismatched signals rejection. Additionally, there are usually no analytical solutions. Another scheme is that a constrained noise-like interference is assumed to be existent only in the cell under test (primary data). According to this model, the so-called double-normalised AMF (DN-AMF) is proposed in [22, 23]. However, it is found that the DN-AMF exhibits enhanced mismatched signal rejection, but with poor robustness to the slightly mismatched signals. Other effective approaches are the parametric (or tunable) detectors [24–26]. By tuning a scalar parameter, they can control the level to which the mismatched signals are rejected. However, the parametric detector in [24] has limited capabilities of mismatched signal rejection, whereas the parametric detectors in [25, 26] have limited robustness. In this paper, we introduce a novel parametric detector. Unlike the parametric detectors in [24–26], this new detector is parameterised by two positive scalars, referred to as the tunable parameters. It encompasses Kelly’s GLRT, AMF and ABORT as its three special cases. Compared with the existing parametric detectors, the novel detector IET Radar Sonar Navig., 2014, Vol. 8, Iss. 1, pp. 48–53 doi: 10.1049/iet-rsn.2013.0044

www.ietdl.org has an improved flexibility of controlling the degree to which the mismatched signals are rejected.

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fact that the AMF is CFAR, and the quantity −1 bf = 1 + xH S −1 x − tAMF

Problem formulation and design issues

We assume that there is a linear array with Na antennas and each antenna collects Np samples. The primary data are denoted by an N-dimensional complex vector x, with N = NaNp. We want to know if there exists a useful signal in x. Hence, the detection problem to be solved can be formulated in terms of the following binary hypothesis test

H0 : x = n H1 : x = asm + n

(1)

where sm is the actual direction of the mainlobe target echo, which is not necessarily aligned with the nominal steering vector s. a is an unknown complex scalar, standing for the signal amplitude. n is the disturbance, including the clutter and noise, which is modelled as a zero-mean complex Gaussian vector with an unknown positive-definite covariance matrix E[nnH] = R. This essentially necessitates the use of adaptive methods. As customary, we suppose that a secondary data set of L samples X = [x1, x2, ..., xL] is available. It is assumed that each secondary sample xl contains only noise and clutter, and shares the same covariance as the primary data, that is, E xt xH = R. For t convenience, let S = XXH, which is L times the sample covariance matrix (SCM). Under both hypotheses, it is assumed that L > N. This ensures that the SCM is non-singularity with probability 1 [1]. For the problem shown in (1), the detection statistic of the AMF is [2] tAMF

2 = sH S −1 x /sH S −1 s

(3)

which has the following equivalent form tK W tK / 1 − tK = tAMF / 1 + xH S −1 x − tAMF

(4)

Moreover, the detection statistic of the ABORT is found to be [10] tABORT

= 1 + tAMF / 1 + xH S −1 x − tAMF

(6)

which is denoted as KMABORT. The first two letters in the acronym KMABORT stand for Kelly’s GLRT and the AMF, respectively, while the last five letters denotes the ABORT. The terms α and γ are named as the tunable parameters. Note that when α = γ = 1, (6) degenerates into the ABORT, while when α = 0 and γ = 1, it is statistically equivalent to Kelly’s GLRT. Moreover, with γ = 0, (6) is tantamount to the AMF regardless of the value of α. A quick comment on the KMABORT is that it possesses the constant false alarm rate (CFAR) property, due to the IET Radar Sonar Navig., 2014, Vol. 8, Iss. 1, pp. 48–53 doi: 10.1049/iet-rsn.2013.0044

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Performance assessment

In this section, we analytically evaluate the detection performance of the KMABORT both in terms of probability of false alarm (Pfa) and probability of detection (Pd). Precisely, we consider the general case, in which the actual signal sm in the primary data x does not need to be aligned with the nominal steering vector s. The generalised cosine angle square between sm and s is defined as [10] H −1 H −1 −1 2 cos2 f = sH m R s / s m R sm s R s

(8)

which is a central quantity for the detection performance for mismatched signals. It is found that under hypothesis H1, the quantity tK , shown in (4), conditioned on the quantity βφ, is ruled by the non-central complex F distribution, with 1 and L − N + 1 degrees of freedom (DOFs), and a non-central parameter ρβ = SCNR·cos2φ·βφ [13], where SCNR = |a|2sHR − 1s is defined as the signal-to-clutter-plus-noise ratio (SCNR). It is written symbolically as

tK |bf CF 1,L−N +1 rb

(9)

(5)

By comparing (2), (4) and (5), we introduce the following novel parametric detector, with the detection statistic: g tKMABORT = a + tAMF / 1 + xH S −1 x − tAMF

is essentially not related to the covariance matrix (see, e.g. [13]). Note that tAMF can be regarded as the energy of the quasi-whitened primary data projected onto the subspace spanned by the quasi-whitened nominal signal, whereas the quantity xHS − 1x − tAMF can be taken as the energy of the quasi-whitened primary data projected onto the orthogonal complement of the quasi-whitened nominal signal subspace. Hence, we can reasonably conjecture that with the increase in the value of γ, the ability of rejection of mismatched signals becomes stronger. Besides, increasing the value of α will weaken the effect of the term tAMF. Consequently, this still result in enhanced mismatched signals rejection, provided the value of γ is not very small. In fact, this is indeed the case; see Section 4.

(2)

while that of Kelly’s GLRT is given by [1] tK = tAMF / 1 + xH S −1 x

(7)

where the notation ‘|’ stands for ‘be conditioned on’. Moreover, one can easily verify the following relationship tAMF = tK /bf The quantity βφ, under hypothesis H1, is subject to the non-central complex Beta distribution with L − N + 2 and N−1 DOFs, and a non-central parameter 2 2 ˜ db = SCNR · sin f, denoted as bf CbL−N +2,N −1 d2b ; see, for example, [24]. Gathering the results in (6), (7) and (9), after some algebra, we have tK bgf−1 tKMABORT = abfg +

(10) 49


www.ietdl.org Hence, the Pd of the KMABORT is Pd = Pr ttKMABORT . h; H1 = Pr tK . hbf1−g − abf ; H1

(11)

where η is the threshold. Equation (11) can be recast as

1

g Pd = 1 − P1 hb1− − abf f1 bf dbf f

(12)

0

where P1(·) is the cumulative distribution function (CDF) of tK , with βφ fixed, under hypothesis H1 [13] L−N h P 1 (h ) = (1 + h)L−N +1 k=0

rb L−N +1 k h · IGk+1 1+k 1+h

Fig. 1 Pd of the KMABORT, under different α and γ, for the matched signals Na = 4, Np = 6, L = 48 and SCNR = 16 dB. The plane is the Pd of Kelly’s GLRT

(13) Gamma function, with in which IGm + 1(a) is the incomplete k the expression IGm+1 (a) = e−a m a /k!. k=0 The term f1(βφ) in (12) is the probability density function (PDF) of βφ under hypothesis H1, given by f1 (bf ) = 2

e−db bf

L−N +2 m=0

L−N +2 m

L! d2m f b (L + m)! b L−N +2,N +m−1 f (14)

where fm,n(βφ) is the PDF of the central complex Beta distribution with m and n DOF’s. As a consequence, the Pfa is g tK . hb1− − abf ; H0 Pfa = Pr f =

1

−(L−N +1)

g 1 + hb1− − ab f0 bf dbf f f

(15)

0

where we have used the identity 1 − P0(x) = (1 + x) − (L − N + 1) [25], where, in turn, P0(·) is the CDF of tK , with βφ fixed, under hypothesis H0; f0(βφ) is the PDF of βφ under H0, which can be obtained by setting d2b = 0 in (14). Note that if no signal mismatch exists, the distribution of βφ is the same under both hypotheses, that is, f1(βφ) = f0(βφ). Before closing this section, we would like to point out that the most time consuming term of the KMABORT, Kelly’s GLRT, AMF and ABORT is the matrix inversion operation, which is nearly the same for these detectors. Hence, the four detectors have roughly the same computational complexity.

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Numerical examples

In this section, we illustrate the detection performance of the KMABORT and compare it with Kelly’s GLRT, AMF and ABORT both for matched and mismatched signals. For all the figures, Pfa = 10−4. Fig. 1 depicts the Pd of the KMABORT with different tunable parameters for matched signals. For convenience, the values of the Pd of the KMABORT less than 0.9 are set to be 0.9. It is shown that there is about one-fifth of the 50

number of the couples of γ and α, with which the Pd of the KMABORT is slightly higher than that of Kelly’s GLRT. Furthermore, for fixed γ with small value, increasing or decreasing the value of α does not significantly affect the Pd of the KMABORT, while when the value of α is fixed, increasing the value of γ results in lower Pd. Moreover, simultaneously increasing the values of α and γ leads to lower Pd, too. However, this will bring enhanced rejection of mismatched signals; see Figs. 6 and 7 for details. Fig. 2 displays the Pd s of the detectors against SCNR for matched signals, that is, cos2φ = 1. The values of α and γ of the KMABORT, are set to be 1 and 0.25, respectively. For independent confirmation, we also show the Pd obtained by Monte Carlo trials, which is labelled ‘KMABORT (MC)’. In order to evaluate the PD and the thresholds necessary to ensure the prescribed Pfa, we resort to 2 × 104 and 200/Pfa independent data realisations, respectively. It is seen that, for the specific setting, the KMABORT achieves detection performance comparable to that of Kelly’s GLRT. Precisely, the KMABORT suffers a slightly detection loss, compared with Kelly’s GLRT, when the SCNR is low or moderate. In contrast, if the SCNR is high enough (say,


Fig. 2 Pd s of the detectors for the matched signals. Na = 4, Np = 6 and L = 48 IET Radar Sonar Navig., 2014, Vol. 8, Iss. 1, pp. 48–53 doi: 10.1049/iet-rsn.2013.0044

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Fig. 3 Pd s of the detectors, for the matched signals, against the number of the secondary data. Na = 8 and Np = 1

SCNR > 16 dB), the KMABORT can provide slightly higher Pd than those of Kelly’s GLRT, AMF and ABORT. It is worth noting that the Pd of the KMABORT is higher than those of the AMF and ABORT for nearly all SCNR in the interval from 5 to 17 dB. Fig. 3 plots the Pd of the KMABORT under different number of the secondary data for the matched signals, also in comparison with those of the AMF, Kelly’s GLRT and ABORT. For the KMABORT, the values of α and γ are set to be 1 and 0.25, respectively. It is seen that with the increase of the number of the secondary data, the Pd s of the detectors also increase. This is because of the improvement of the estimation precision of the covariance matrix. Moreover, when the number of the secondary data is not very large, the Pd s of Kelly’s GLRT and ABORT are slightly higher than that of the KMABORT. In contrast, when the number of the secondary data is large, the Pd of the KMABORT is commensurate to or even higher than those of Kelly’s GLRT, ABORT and AMF. Furthermore, it seems that if the SCNR is not high enough, no matter how large is the number of the secondary data, the Pd s of the detectors can never approach unity.

Fig. 4 shows the Pd of the KMABORT with fixed α and different γ, or with fixed γ and different α. Note that the figures are two slices of Fig. 1. It is seen that, from Fig. 4a, fixing the value of α, increasing the value of γ from zero to a relatively small quantity, the Pd increases; when the value of γ continues to increase, then the Pd decreases. Similarly, Fig. 4b reveals that, for fixed value of γ not equal to zero, with the increase of the value of α, the Pd first increases then decreases. Note that when γ = 0, the value of α does not affect the Pd; this is due to the fact that the KMABORT is equivalent to the AMF in this setting. An interesting feature is that if the value of γ (α) is relatively small, the greater value of α (γ), the higher Pd it provides. In contrast, increasing α and γ from certain values greater than zero results in the decrease of the Pd for matched signals, yet this gains enhanced mismatched signals rejection; see Figs. 6 and 7. The detection performance for mismatched signals is shown in Figs. 5–7. Particularly, Fig. 5 displays the Pd of the KMABORT for mismatched signals under different parameters. The results imply that only increasing the value of α or γ can realise mismatched signals rejection. Besides, when the values of SCNR and/or cos2φ are low enough, the Pd of the KMABORT is very low, irrespective of the values of other parameters. In Fig. 6, the contours of constant Pd are represented as a function of SCNR and cos2φ. (That is to say, the numbers 0.01, 0.1, 0.5, 0.9 and 0.99, given in the figure, are the certain values of the Pd for the corresponding detector.) Such plots were first introduced in [10], where they are referred to as mesh-plots. For the KMABORT, the tunable parameters are set to be α = 1 and γ = 2. It is seen that the KMABORT offers better performance in terms of mismatched signal rejection than that of the ABORT, which in turn, is more selective than Kelly’s GLRT. The AMF has the least mismatch discrimination capabilities. Fig. 7 characterises the Pd of the KMABORT for the mismatched signals against cos2φ. For comparison purposes, the Pd s of Kelly’s GLRT, AMF and ABORT, are also shown. It is seen that increasing the value of α or γ, or both, brings enhanced capabilities of mismatched signals rejection. In particular, with α = γ = 0, the KMABORT is reduced to the AMF, and it has the least capabilities in terms of mismatched signal rejection. Or equivalently, the

Fig. 4 Pd of the KMABORT, for matched signals, against α or γ, SCNR = 15 dB. Na = 24, Np = 1 and L = 48 a Different α b Different γ IET Radar Sonar Navig., 2014, Vol. 8, Iss. 1, pp. 48–53 doi: 10.1049/iet-rsn.2013.0044

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Fig. 5 Pd of the KMABORT against SNCR, α, γ and cos2φ. Na = 24, Np = 1 and L = 48 a SCNR = 20 dB, γ = 1 b SCNR = 20 dB, α = 0.5 c SCNR = 16 dB, cos2φ = 0.9 d α = 0.5, γ = 1.2 e α = 1, cos2φ = 0.9 f γ = 1, cos2φ = 0.9

Fig. 6 Contours of constant Pd s for the AMF (subplot a), Kelly’s GLRT (subplot b), ABORT (subplot c) and KMABORT (subplot d). Na = 24, Np = 1 and L = 48 52


IET Radar Sonar Navig., 2014, Vol. 8, Iss. 1, pp. 48–53 doi: 10.1049/iet-rsn.2013.0044

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Fig. 7 Pd s of the detectors against cos2φ. Na = 24, Np = 1, L = 48 and SCNR = 20 dB

AMF is most robust to the mismatched signals. The KMABORT, with α = 3 and γ = 1, exhibits the strongest capabilities of mismatched signals rejection.

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Conclusions

In this paper, we have considered the problem of adaptive detection in the presence of signal mismatch. A novel doubly parameterised detector has been proposed, namely, the KMABORT. It can govern the level to which the mismatched signals are rejected (or detected). Moreover, the KMABORT, with proper tunable parameters, can provide slightly higher Pd for matched signals than those of Kelly’s GLRT, AMF and ABORT, provided the SCNR is relatively high. It can also achieve enhanced mismatched signal rejection and improved robustness than its natural competitors. As for the selection of the tunable parameters, a general criterion is that if one wants enhanced mismatched signal rejection, the values of the tunable parameters should be set relatively large. An example is that the radar works in tracking mode. In contrast, if one wants the robustness to the mismatched signals, the values of the tunable parameters should be chosen as small as possible. An instance is that the radar operates in searching mode.

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Acknowledgment

This work was supported in part by the National Nature Science Foundation of China under contract numbers 61102169 and 60925005.

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