Multichannel adaptive signal detection: Basic theory and literature review
Weijian Liu, Jun Liu, Chengpeng Hao, Yongchan Gao, Yong-Liang Wang
aa r X i v : . [ s t a t . A P ] F e b SCIENCE CHINA
Information Sciences . RESEARCH PAPER .Multichannel adaptive signal detection: Basic theoryand literature review
Weijian LIU , Jun LIU , Chengpeng HAO , Yongchan GAO & Yong-Liang WANG Wuhan Electronic Information Institute, Wuhan , China; Department of Electronic Engineering and Information Science, University of Science and Technology of China, Hefei , China; State Key Laboratory of Acoustics, Institute of Acoustics, Chinese Academy of Sciences, Beijing , China; Xidian University, Xi’an , China
Abstract
Multichannel adaptive signal detection jointly uses the test and training data to form an adap-tive detector, and then make a decision on whether a target exists or not. Remarkably, the resulting adaptivedetectors usually possess the constant false alarm rate (CFAR) properties, and hence no additional CFARprocessing is needed. Filtering is not needed as a processing procedure either, since the function of filteringis embedded in the adaptive detector. Moreover, adaptive detection usually exhibits better detection perfor-mance than the filtering-then-CFAR detection technique. Multichannel adaptive signal detection has beenmore than 30 years since the first multichannel adaptive detector was proposed by Kelly in 1986. However,there are fewer overview articles on this topic. In this paper we give a tutorial overview of multichanneladaptive signal detection, with emphasis on Gaussian background. We present the main deign criteria foradaptive detectors, investigate the relationship between adaptive detection and filtering-then-CFAR detec-tion, relationship between adaptive detectors and adaptive filters, summarize typical adaptive detectors, shownumerical examples, give comprehensive literature review, and discuss some possible further research tracks.
Keywords
Constant false alarm rate, multichannel signal, signal mismatch, statistical distribution, sub-space signal.
Citation
Liu W, Liu J, Hao C, et al. Multichannel adaptive signal detection: Basic theory and literature review.Sci China Inf Sci, for review
Signal detection in noise is a fundamental problem in various areas, such as radar, sonar, communications,optical image, hyperspectral imagery, remote sensing, medical imaging, subsurface prospecting, and so on.Taking the radar system for example, the received data for early radar systems are of single channel, andhence, the data are scalar-valued. In contrast, with the applications of pulsed Doppler techniques and/ormultiple transmit/receive (T/R) modules, along with the increase in computation power and advances inhardware design, the received data for modern radar systems are usually multichannel, namely, vector-valued or even matrix-valued. Moreover, the frequency diversity, polarization diversity, or waveformdiversity can also lead to the multichannel form of the received data. The multichannel data containmore information, compared with the single-channel data. On one hand, using the multichannel data,we have more degrees of freedom (DOFs) to design adaptive processors. On the other hand, using themultichannel data model, it is more convenient to characterize the correlated properties between data indifferent channels. Using these correlated properties, one can design a filter, whose output signal-to-noise(SNR) is often higher than that for a single-channel data. Similarly, utilizing the data correlation, onecan devise a detector, which has superior detection performance to a detector for single-channel data.Remarkably, noise is ubiquitous, which, in a general sense, usually includes thermal noise and clutter.For multichannel data in the cell under test (also called primary data), the noise covariance matrixis unknown and needs to be estimated. A common strategy is using the training data (also calledsecondary data) to form appropriate estimator. It is pointed out in [1] that modern strategy for radar * Corresponding author (email: [email protected])iu W , et al. Sci China Inf Sci detection should include the following three features: 1) being adaptive to the noise spectral density orits probability density function (PDF), 2) maintaining constant false alarm rate (CFAR) property, and3) having a relatively simple processing scheme. Multichannel adaptive signal detection is a kind of thisstrategy. It jointly utilizes the test data and training data to design adaptive detectors, which usuallypossess the CFAR property. The resulting adaptive detector is then compared with a certain detectionthreshold, set to ensure a fixed probability of false alarm (PFA). Finally, a target is declared to be present(absent) if the threshold is exceeded (not exceeded).Two points are worth to be emphasized. One is that the word “adaptive” in the first feature aboveindicates that the spectrum character of the noise is unknown in advance or is changing in the operationalenvironment, and hence adaptive techniques are needed. The other is that the CFAR property or theCFARness , which, for single-channel signal, means that the detection threshold of a detector is indepen-dent of the noise power. Equivalently, the statistical property of the detector is functionally independentof the noise power under the signal-absence hypothesis. In contrast, for multichannel signal detection,the CFARness means that the statistical property of the detector is also functionally independent of thestructure of the noise covariance matrix under the signal-absence hypothesis. This kind of CFARness isreferred to as the matrix CFAR in [2] and covariance matrix-CFAR in [3].Multichannel adaptive signal detection was first investigated by Kelly in 1986. In the seminal paper [4],Kelly proposed the famous detector, i.e., Kelly’s GLRT (KGLRT) for detecting a rank-one signal in ho-mogeneous environment (HE). The rank-one signal has a known steering vector but unknown amplitude.For the HE model, the noise in the training and test data is both subject to mean-zero circularly complexGaussian distribution, with the same covariance matrix.There is more than three decades since Kelly proposed the famous KGLRT in 1986. Multichanneladaptive signal detection has been adopted in various areas. Based on different design criteria, numerousdetectors have been proposed for different problems. Recently, an important book is edited by De Maioand Greco [5]. However, there are seldom survey papers on multichannel signal detection. In particular,references [6] and [7] gave overview of signal detection in compound-Gaussian clutter for subspace signalsand rank-one signals, respectively. These two references are mainly on known clutter or known noisecovariance matrix. Moreover, the target is point-like and no signal mismatch is considered. Differentfrom the above two references, in this paper we give a review of multichannel adaptive signal detectionin unknown noise, with emphasis on Gaussian background.In this paper, we give a tutorial on multichannel adaptive signal detection, and present a brief surveyof the state of the art. For brevity, “adaptive detection” always means “multichannel adaptive signaldetection” in the following. In Section 2, we present the basic theory for adaptive detection, includingdata model, main detector design criteria, relationship between adaptive detection and filter-then-CFARdetection, and relationship between adaptive detection and adaptive filtering. In Section 3, we givecomprehensive literature review. In Section 4, we analyse and compare the detection performance ofsome typical adaptive detectors. Finally, Section 5 summarizes this paper and gives some further researchtracks in adaptive detection. The GLRT, Rao test, and Wald test are three main detector design criteria . These three criteria arereferred to as “the Holy Trinity” in statistical inference [26]. Before listing these criteria, we need toformulate a binary hypothesis mathematically. A binary hypothesis has two possible cases, namely, thenull (signal-absence) hypothesis and alternative (signal-presence) hypothesis. Hence, a binary hypothesistest can be written as ( H : x = n , x e ,l = n e ,l , l = 1 , , · · · , L, H : x = s + n , x e ,l = n e ,l , l = 1 , , · · · , L, (1)
1) CFARness is an important property required by an effective detector in practice, because the PFA may be dramatically raisedto an unaffordable value if a detector does not maintain CFARness and the noise changes severely.2) There are also some other often used criteria, such as the gradient test [8], Durbin test [9], test based on maximal invariantstatistic [10], multifamily likelihood ratio test [11], and other modifications of the likelihood ratio test [11], which are utilized foradaptive detector design, e.g., [12–25].iu W , et al. Sci China Inf Sci where H denotes the null hypothesis, H denotes the alternative hypothesis, x is the test data, s is thesignal to be detected, n is the noise in the test data, whose covariance matrix, denoted as R , is generallyunknown, { x e ,l } Ll =1 are L training data, used to estimate the unknown R .For the detection problem in (1), the GLRT is [27] t GLRT = max Θ f ( x , X L )max Θ f ( x , X L ) , (2)where Θ and Θ denote the unknown parameters under hypotheses H and H , respectively, f ( x , X L )and f ( x , X L ) are the joint PDFs of the test data x and training data X L = [ x e,1 , x e,2 , · · · , x e ,L ] underhypotheses H and H , respectively.To derive the Rao and Wald tests, we need the Fisher information matrix (FIM), which, for circularlysymmetric random parameters, is defined as [28] I ( Θ ) = E (cid:20) ∂ ln f ( x , X L ) ∂ Θ ∗ ∂ ln f ( x , X L ) ∂ Θ T (cid:21) . (3)For convenience, the FIM is usually partitioned as I ( Θ ) = " I Θ r , Θ r ( Θ ) I Θ r , Θ s ( Θ ) I Θ s , Θ r ( Θ ) I Θ s , Θ s ( Θ ) , (4)where Θ = [ Θ T r , Θ T s ] T , (5) I Θ r , Θ r ( Θ ) = E (cid:20) ∂ ln f ( x , X L ) ∂ Θ ∗ r ∂ ln f ( x , X L ) ∂ Θ T r (cid:21) , (6a) I Θ r , Θ s ( Θ ) = E (cid:20) ∂ ln f ( x , X L ) ∂ Θ ∗ r ∂ ln f ( x , X L ) ∂ Θ T s (cid:21) , (6b) I Θ s , Θ r ( Θ ) = E (cid:20) ∂ ln f ( x , X L ) ∂ Θ ∗ s ∂ ln f ( x , X L ) ∂ Θ T r (cid:21) , (6c) I Θ s , Θ s ( Θ ) = E (cid:20) ∂ ln f ( x , X L ) ∂ Θ ∗ s ∂ ln f ( x , X L ) ∂ Θ T s (cid:21) . (6d)In (5), Θ r is the relevant parameter, such as the signal amplitude, Θ s is the nuisance parameter, e.g., thenoise covariance matrix. Note that if ln f ( x , X L ) is twice differential with respect to Θ , then the FIM in(3), under the regularity condition, can be calculated by [29] I ( Θ ) = − E (cid:20) ∂ ln f ( x , X L ) ∂ Θ ∗ ∂ Θ T (cid:21) , (7)which is often more easier to be derived.Then, the Rao and Wald tests for complex-valued signals are [29] t Rao = ∂ ln f ( x , X L ) ∂ Θ r (cid:12)(cid:12)(cid:12)(cid:12) T Θ = ˆ Θ [ I − ( ˆ Θ )] Θ r , Θ r ∂ ln f ( x , X L ) ∂ Θ ∗ r (cid:12)(cid:12)(cid:12)(cid:12) Θ = ˆ Θ , (8)and t Wald = ( ˆ Θ r − Θ r ) H n [ I − ( ˆ Θ )] Θ r , Θ r o − ( ˆ Θ r − Θ r ) , (9)respectively, where ˆ Θ and ˆ Θ are the maximum likelihood estimates (MLEs) of Θ under hypotheses H and H , respectively, ˆ Θ r is the MLE of Θ r under hypothesis H , Θ r is the value of Θ r under hypothesisH , and (cid:8) [ I − ( Θ )] Θ r , Θ r (cid:9) − is the Schur complement of I Θ s , Θ s ( Θ ), namely, (cid:8) [ I − ( Θ )] Θ r , Θ r (cid:9) − = I Θ r , Θ r ( Θ ) − I Θ r , Θ s ( Θ ) I − Θ s , Θ s ( Θ ) I Θ s , Θ r ( Θ ) . (10)
3) The complex-valued Rao test is also given in [30] which is a generalization of the one in (8) and suitable of non-circularlysymmetric random parameters.iu W , et al. Sci China Inf Sci In some cases the relevant parameter Θ r and/or the nuisance parameter Θ s may be known. Obviously,in these cases we use these true values, and do not need to derive their MLEs.It is worthy pointing out that the two-step variations of the three design criteria are also adopted.Precisely, the GLRT, Rao test, or Wald test is first derived under the assumption that the noise covariancematrix is known or its structure is known. Then the noise covariance matrix in the corresponding detectoris replaced by a proper estimate by using the training data. For example, the two-step GLRT (2S-GLRT)can be mathematically expressed as t = max Θ ′ f ( x , X L )max Θ ′ f ( x , X L ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R = ˆ R , (11)where Θ ′ and Θ ′ denote the unknown parameters except for R under hypotheses H and H , respectively,and ˆ R is an appropriate estimation of R .From the three detector design criteria in (2), (8) and (9), we know that one of the key point to how tofind the derivatives of scalar real-valued functions, such as the PDFs, with respect to a complex-valuedscalar, vector, or matrix. One of the most useful book on this topic may be the one by Hjørungnes [31],which is written in engineering-oriented manner. The theory of finding complex-valued derivatives in [31]is based on the complex differential of the objective function. Using the complex differential is muchmore easier to find a derivative than using the component-wise approach, such as the famous book byMagnus and Neudecker [32], which mainly focuses on real-valued derivatives.It is worthy pointing out that the following fact is often used in deriving a detector with simplifieddetection statistic or in a form whose statistical distribution is easy to be derived. Precisely, if a detectorcan be expressed as a monotonically increasing function of another one, then these two detectors areequivalent. We try to find a related reference. However, it is not found. Hence, we summarize the abovefact in the following theorem. Theorem 1:
Let t and t are two detectors, and t = g ( t ) (12)monotonically increases with t . Then t are t have the same detection performance such that they havethe identical probability of detection (PD) under the same PFA. Proof:
Let the PFAs of t and t be PFA and PFA , respectively. ThenPFA = Pr[ t > η ; H ] , (13)PFA = Pr[ t > η ; H ] , (14)where η and η are detection thresholds of t and t , respectively. According to (12), (14) can berewritten as PFA = Pr[ g ( t ) > η ; H ] = Pr[ t > g − ( η ); H ] , (15)where the second equality is owing to the fact that g ( t ) is a momotonically increasing function of t , and g − ( · ) denotes the inverse function of g ( · ). Comparing (13) and (15), and using PFA = PFA , we have η = g − ( η ) . (16)The PDs of t and t can be expressed asPD = Pr[ t > η ; H ] (17)and PD = Pr[ t > η ; H ] , (18)respectively. Since t = g ( t ) is a monotonically increasing function of t , (18) can be recast asPD = Pr[ t > g − ( η ); H ] = Pr[ t > η ; H ] = PD , (19)where the second equality is obtained according to (16). This completes the proof. (cid:4) Adaptive detection is different from filtering-then-CFAR detection, which is widely adopted in mostradar systems. Moreover, adaptive detection is highly related with adaptive filtering, although theirpurposes are different. In the following two subsections, we investigate the relationship between them. iu W , et al. Sci China Inf Sci Nowadays, the mainly used detection scheme in most radar systems is the filtering-then-CFAR approach.Precisely, the test data are first filtered and then processed by the CFAR techniques. The CFAR pro-cessing is a technique which makes the detection threshold of a detector independent of noise covariancematrix. Or, equivalently, through CFAR processing, the statistical characteristics of the detector doesnot depend on the noise covariance matrix under the signal-absence hypothesis. There are many CFARtechnologies, such as cell-averaging CFAR (CA-CFAR), greatest-of-selection CFAR (GO-CFAR), orderedstatistic CFAR (OS-CFAR), and so on [33, 34]. It seems that the filtering-then-CFAR detection schemeis a natural approach for detecting a target in noise, since adaptive filtering can obtain high output SNR,which benefits the detection process.The theoretical basis behind the filtering-then-CFAR detection scheme for multichannel data can betraced back to the classic paper [35]. Precisely, for airborne radar space-time two-dimensional signalprocessing, the test data, if containing the target signal, can be written as x = a s + n , (20)where x is an N a N p × N a is the number of the antennas, N p is the number of thepulses received by each antenna, s = s p ⊗ s a is an N a N p × s p and s a being an N p × N a × ⊗ denotesthe Kronecker product, and n is the noise, including clutter and thermal noise, distributed as circularlycomplex Gaussian distribution with covariance matrix R .In [35], to detect the target in (20), the test data vector x is first filtered by an N a N p × w . Hence, the output of the filter can be expressed as y = w H x . (21)For the filtered data y , the optimum detector, in the Neyman-Pearson sense, is the likelihood ratio test,given by t LRT = f ( y | x = a s + n ) f ( y | x = n ) , (22)where f ( · ) and f ( · ) are the PDFs under signal-presence and signal-absence hypotheses, respectively.The optimum filter weight w can be obtained by maximizing (22), written symbolically as w opt = max w f ( y | x = a s + n ) f ( y | x = n ) , (23)which is shown to be equivalent to [35] w opt = max w | w H s | w H Rw , (24)and the solution to (24) is w opt = µ R − s , (25)where µ is an arbitrary non-zero constant.A well-known equivalent solution to (24) is the minimum variance distortionless response (MVDR),which is mathematically formed as [36] min w w H Rw , s.t. w H s = 1 , (26)and the corresponding solution is w MVDR = R − ss H R − s . (27)Taking (27) into (21) and performing the norm-squared operation leads to t MF = | s H R − x | ( s H R − s ) . (28) iu W , et al. Sci China Inf Sci Gathering the above results indicates that the optimum detection in (22) is equivalent to the optimumfiltering in (24), and the optimum filter weight is given in (25). Based on the above results, a techniquecalled space-time adaptive processing (STAP) came into being, which is regarded as one of most effectivetechnology for airborne radar clutter cancellation, and numerous achievements have been obtained [37–39]. Note that STAP is a filtering technique , whose aim is to maximize the output SNR. To realize thefinal target detection, CFAR processing is needed.It is worth pointing out that the above equivalence between optimum detection and optimum filteringholds under certain processing flow and certain assumptions . The specific processing flow is filtering-then-detection. Precisely, the multichannel test data vector x is first filtered by the weight vector w , resultingin the scalar-valued data y . Then, a detector is devised based on the filtered data y . The assumptionis that the noise n in the test data is Gaussian distributed, and its covariance matrix R is known inadvance. Unfortunately, the above assumption is usually not satisfied in practice, since radar systemworks in varying environment. When the noise covariance matrix R is unknown, it is usually replacedby the sample covariance matrix (SCM), formed by using the training data received in the vicinity ofthe test data. Then the optimum filter in (25) becomes the sub-optimum filter of the sample covarianceinversion (SMI) [42]. To complete target detection, it also needs appropriate CFAR processing.Note that the above filtering-then-CFAR detection scheme adopts adaptive filtering. However, thereis another filtering-then-CFAR detection scheme, which performs non-adaptive filtering, such as movingtarget indication (MTI) and moving target detection (MTD) and pulse Doppler processing. The keypoint in the MTD and pulse Doppler processing is Doppler filtering using multiple pulses. However, thenumber of the pulses used in the MTD is much smaller than that used in the pulse Doppler processing.Moreover, the MTD is often used by ground-based radar, while the pulse Doppler processing is mainlyused by airborne radar. This non-adaptive filtering-then-CFAR detection scheme usually has lower com-plexity compared with the adaptive filtering-then-CFAR detection scheme, however, suffers from certainperformance loss, since its filtering performance is limited.For unknown noise, if the test and training data are directly utilized to devise multichannel adaptivedetectors, then better detection performance can be obtained, compared with the above filtering-then-CFAR detection scheme. Adaptive detection is just a kind of this detection scheme. Precisely, for adaptivedetection, the test and training data are jointly utilized to design an adaptive detector, and then it iscompared with a detection threshold, set according to a pre-assigned PFA. If the value of a detector isgreater than the threshold, a target is claimed. Otherwise, no target is claimed.The block diagrams of filtering-then-CFAR detection and adaptive detection are summarized in Figure1. It can be concluded that the filtering-then-CFAR detection approach (adaptive or non-adaptive)needs two independent processing procedures, as its name indicates, i.e., filtering and CFAR processing.In contrast, independent filtering processing is not needed for adaptive detection, which achieves thefunction of filtering and CFAR processing simultaneously, both embedded in the detection statistic of theadaptive detection. As explained above, adaptive filters and adaptive detectors have different purposes, since the formertries to maximize the output SNR, while the latter tries to maximize the PD with a fixed PFA. However,adaptive filters and adaptive detectors have some common feature. They both adopt adaptivity. Precisely,they use training data to adaptively estimate the unknown noise covariance matrix. This is the essentialpoint in adaptive processors. Moreover, adaptive detectors have the function of adaptive filtering, which,however, is not achieved in an independent procedure, as pointed above.As an example, Figure 2 shows the block diagrams of one adaptive filter, namely, the SMI [42], andthree adaptive detectors, namely, the KGLRT [4], adaptive matched filter (AMF) [43, 44], and De Maio’sRao (DMRao) [45] . The SMI can be obtained by replacing R with the SCM S in (28), resulting in t SMI = ˜ x H P ˜ s ˜ x ˜ s H ˜ s . (29)
4) Strictly speaking, the STAP technique is much less than its literal meaning. Precisely, STAP is a filtering technique to rejectthe clutter and jammer (if present) for airborne radar [40, 41].5) The SMI is proposed based on the idea of filtering-then-CFAR detection. Mathematically, it can be written as (cid:20) max w | w H s | w H Rw (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) R = 1 L S . The KGLRT, AMF and DMRao are proposed for the detection problem in (1) according to the crite-ria of GLRT, 2S-GLRT and Rao test, respectively. The AMF can also be obtained according to the Wald test.iu W , et al. 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(cid:38)(cid:41)(cid:36)(cid:53)(cid:3)(cid:51)(cid:85)(cid:82)(cid:70)(cid:72)(cid:86)(cid:86)(cid:76)(cid:81)(cid:74) (cid:36)(cid:79)(cid:74)(cid:82)(cid:85)(cid:76)(cid:87)(cid:75)(cid:80)(cid:86) (cid:40)(cid:86)(cid:87)(cid:76)(cid:80)(cid:68)(cid:87)(cid:72)(cid:71)(cid:3)(cid:49)(cid:82)(cid:76)(cid:86)(cid:72)(cid:3)(cid:51)(cid:82)(cid:90)(cid:72)(cid:85)(cid:3) (cid:49)(cid:82)(cid:81)(cid:16)(cid:36)(cid:71)(cid:68)(cid:83)(cid:87)(cid:76)(cid:89)(cid:72)(cid:3)(cid:41)(cid:76)(cid:79)(cid:87)(cid:72)(cid:85)(cid:76)(cid:81)(cid:74) (cid:39)(cid:72)(cid:87)(cid:72)(cid:70)(cid:87)(cid:82)(cid:85)(cid:3)(cid:39)(cid:72)(cid:86)(cid:76)(cid:74)(cid:81)(cid:55)(cid:72)(cid:86)(cid:87)(cid:3)(cid:39)(cid:68)(cid:87)(cid:68)(cid:55)(cid:85)(cid:68)(cid:76)(cid:81)(cid:76)(cid:81)(cid:74)(cid:3)(cid:39)(cid:68)(cid:87)(cid:68) (cid:39)(cid:72)(cid:87)(cid:72)(cid:70)(cid:87)(cid:76)(cid:82)(cid:81)(cid:3)(cid:54)(cid:87)(cid:68)(cid:87)(cid:76)(cid:86)(cid:87)(cid:76)(cid:70) (cid:20)(cid:19) (cid:43)(cid:43) (cid:33)(cid:31) (cid:55)(cid:75)(cid:85)(cid:72)(cid:86)(cid:75)(cid:82)(cid:79)(cid:71) (cid:39)(cid:72)(cid:70)(cid:76)(cid:86)(cid:76)(cid:82)(cid:81)(cid:51)(cid:85)(cid:72)(cid:16)(cid:68)(cid:86)(cid:86)(cid:76)(cid:74)(cid:81)(cid:72)(cid:71)(cid:3)(cid:51)(cid:41)(cid:36)(cid:58)(cid:75)(cid:76)(cid:87)(cid:72)(cid:81)(cid:72)(cid:71)(cid:3)(cid:48)(cid:68)(cid:87)(cid:70)(cid:75)(cid:72)(cid:71)(cid:3)(cid:41)(cid:76)(cid:79)(cid:87)(cid:72)(cid:85)(cid:76)(cid:81)(cid:74)(cid:55)(cid:72)(cid:86)(cid:87)(cid:3)(cid:39)(cid:68)(cid:87)(cid:68) (cid:39)(cid:72)(cid:87)(cid:72)(cid:70)(cid:87)(cid:76)(cid:82)(cid:81)(cid:3)(cid:54)(cid:87)(cid:68)(cid:87)(cid:76)(cid:86)(cid:87)(cid:76)(cid:70) (cid:20)(cid:19) (cid:43)(cid:43) (cid:33)(cid:31) (cid:55)(cid:75)(cid:85)(cid:72)(cid:86)(cid:75)(cid:82)(cid:79)(cid:71) (cid:39)(cid:72)(cid:70)(cid:76)(cid:86)(cid:76)(cid:82)(cid:81)(cid:55)(cid:85)(cid:68)(cid:76)(cid:81)(cid:76)(cid:81)(cid:74)(cid:3)(cid:39)(cid:68)(cid:87)(cid:68) (cid:238) (cid:54)(cid:70)(cid:68)(cid:79)(cid:76)(cid:81)(cid:74)(cid:3)(cid:41)(cid:68)(cid:70)(cid:87)(cid:82)(cid:85) (cid:51)(cid:85)(cid:72)(cid:16)(cid:68)(cid:86)(cid:86)(cid:76)(cid:74)(cid:81)(cid:72)(cid:71)(cid:3)(cid:51)(cid:41)(cid:36) (cid:38)(cid:41)(cid:36)(cid:53)(cid:3)(cid:51)(cid:85)(cid:82)(cid:70)(cid:72)(cid:86)(cid:86)(cid:76)(cid:81)(cid:74) (cid:36)(cid:79)(cid:74)(cid:82)(cid:85)(cid:76)(cid:87)(cid:75)(cid:80)(cid:86) (cid:40)(cid:86)(cid:87)(cid:76)(cid:80)(cid:68)(cid:87)(cid:72)(cid:71)(cid:3)(cid:49)(cid:82)(cid:76)(cid:86)(cid:72)(cid:3)(cid:51)(cid:82)(cid:90)(cid:72)(cid:85)(cid:3) (cid:36)(cid:71)(cid:68)(cid:83)(cid:87)(cid:76)(cid:89)(cid:72)(cid:3)(cid:41)(cid:76)(cid:79)(cid:87)(cid:72)(cid:85)(cid:76)(cid:81)(cid:74) (cid:55)(cid:85)(cid:68)(cid:76)(cid:81)(cid:76)(cid:81)(cid:74)(cid:3)(cid:39)(cid:68)(cid:87)(cid:68) (cid:54)(cid:38)(cid:48)(cid:29) (cid:54)(cid:91)(cid:59) (cid:58)(cid:75)(cid:76)(cid:87)(cid:72)(cid:81) (cid:86) (cid:58)(cid:75)(cid:76)(cid:87)(cid:72)(cid:81) (cid:91) (cid:4) (cid:86) (cid:4) (cid:86) (cid:51) (cid:4) (cid:54)(cid:48)(cid:44)(cid:36)(cid:48)(cid:41) (cid:36)(cid:38)(cid:40)(cid:46)(cid:42)(cid:47)(cid:53)(cid:55) (a) Non-adaptive filtering-then-CFAR (cid:48)(cid:68)(cid:87)(cid:70)(cid:75)(cid:72)(cid:71)(cid:3)(cid:41)(cid:76)(cid:79)(cid:87)(cid:72)(cid:85)(cid:76)(cid:81)(cid:74)(cid:55)(cid:72)(cid:86)(cid:87)(cid:3)(cid:39)(cid:68)(cid:87)(cid:68) (cid:39)(cid:72)(cid:87)(cid:72)(cid:70)(cid:87)(cid:76)(cid:82)(cid:81)(cid:3)(cid:54)(cid:87)(cid:68)(cid:87)(cid:76)(cid:86)(cid:87)(cid:76)(cid:70) (cid:20)(cid:19) (cid:43)(cid:43) (cid:33)(cid:31) (cid:55)(cid:75)(cid:85)(cid:72)(cid:86)(cid:75)(cid:82)(cid:79)(cid:71) (cid:39)(cid:72)(cid:70)(cid:76)(cid:86)(cid:76)(cid:82)(cid:81)(cid:55)(cid:85)(cid:68)(cid:76)(cid:81)(cid:76)(cid:81)(cid:74)(cid:3)(cid:39)(cid:68)(cid:87)(cid:68) (cid:238) (cid:54)(cid:70)(cid:68)(cid:79)(cid:76)(cid:81)(cid:74)(cid:3)(cid:41)(cid:68)(cid:70)(cid:87)(cid:82)(cid:85) (cid:51)(cid:85)(cid:72)(cid:16)(cid:68)(cid:86)(cid:86)(cid:76)(cid:74)(cid:81)(cid:72)(cid:71)(cid:3)(cid:51)(cid:41)(cid:36) (cid:38)(cid:41)(cid:36)(cid:53)(cid:3)(cid:51)(cid:85)(cid:82)(cid:70)(cid:72)(cid:86)(cid:86)(cid:76)(cid:81)(cid:74) (cid:36)(cid:79)(cid:74)(cid:82)(cid:85)(cid:76)(cid:87)(cid:75)(cid:80)(cid:86) (cid:40)(cid:86)(cid:87)(cid:76)(cid:80)(cid:68)(cid:87)(cid:72)(cid:71)(cid:3)(cid:49)(cid:82)(cid:76)(cid:86)(cid:72)(cid:3)(cid:51)(cid:82)(cid:90)(cid:72)(cid:85)(cid:3) (cid:49)(cid:82)(cid:81)(cid:16)(cid:36)(cid:71)(cid:68)(cid:83)(cid:87)(cid:76)(cid:89)(cid:72)(cid:3)(cid:41)(cid:76)(cid:79)(cid:87)(cid:72)(cid:85)(cid:76)(cid:81)(cid:74) (cid:39)(cid:72)(cid:87)(cid:72)(cid:70)(cid:87)(cid:82)(cid:85)(cid:3)(cid:39)(cid:72)(cid:86)(cid:76)(cid:74)(cid:81)(cid:55)(cid:72)(cid:86)(cid:87)(cid:3)(cid:39)(cid:68)(cid:87)(cid:68)(cid:55)(cid:85)(cid:68)(cid:76)(cid:81)(cid:76)(cid:81)(cid:74)(cid:3)(cid:39)(cid:68)(cid:87)(cid:68) (cid:39)(cid:72)(cid:87)(cid:72)(cid:70)(cid:87)(cid:76)(cid:82)(cid:81)(cid:3)(cid:54)(cid:87)(cid:68)(cid:87)(cid:76)(cid:86)(cid:87)(cid:76)(cid:70) (cid:20)(cid:19) (cid:43)(cid:43) (cid:33)(cid:31) (cid:55)(cid:75)(cid:85)(cid:72)(cid:86)(cid:75)(cid:82)(cid:79)(cid:71) (cid:39)(cid:72)(cid:70)(cid:76)(cid:86)(cid:76)(cid:82)(cid:81)(cid:51)(cid:85)(cid:72)(cid:16)(cid:68)(cid:86)(cid:86)(cid:76)(cid:74)(cid:81)(cid:72)(cid:71)(cid:3)(cid:51)(cid:41)(cid:36)(cid:58)(cid:75)(cid:76)(cid:87)(cid:72)(cid:81)(cid:72)(cid:71)(cid:3)(cid:48)(cid:68)(cid:87)(cid:70)(cid:75)(cid:72)(cid:71)(cid:3)(cid:41)(cid:76)(cid:79)(cid:87)(cid:72)(cid:85)(cid:76)(cid:81)(cid:74)(cid:55)(cid:72)(cid:86)(cid:87)(cid:3)(cid:39)(cid:68)(cid:87)(cid:68) (cid:39)(cid:72)(cid:87)(cid:72)(cid:70)(cid:87)(cid:76)(cid:82)(cid:81)(cid:3)(cid:54)(cid:87)(cid:68)(cid:87)(cid:76)(cid:86)(cid:87)(cid:76)(cid:70) (cid:20)(cid:19) (cid:43)(cid:43) (cid:33)(cid:31) (cid:55)(cid:75)(cid:85)(cid:72)(cid:86)(cid:75)(cid:82)(cid:79)(cid:71) (cid:39)(cid:72)(cid:70)(cid:76)(cid:86)(cid:76)(cid:82)(cid:81)(cid:55)(cid:85)(cid:68)(cid:76)(cid:81)(cid:76)(cid:81)(cid:74)(cid:3)(cid:39)(cid:68)(cid:87)(cid:68) (cid:238) (cid:54)(cid:70)(cid:68)(cid:79)(cid:76)(cid:81)(cid:74)(cid:3)(cid:41)(cid:68)(cid:70)(cid:87)(cid:82)(cid:85) (cid:51)(cid:85)(cid:72)(cid:16)(cid:68)(cid:86)(cid:86)(cid:76)(cid:74)(cid:81)(cid:72)(cid:71)(cid:3)(cid:51)(cid:41)(cid:36) (cid:38)(cid:41)(cid:36)(cid:53)(cid:3)(cid:51)(cid:85)(cid:82)(cid:70)(cid:72)(cid:86)(cid:86)(cid:76)(cid:81)(cid:74) (cid:36)(cid:79)(cid:74)(cid:82)(cid:85)(cid:76)(cid:87)(cid:75)(cid:80)(cid:86) (cid:40)(cid:86)(cid:87)(cid:76)(cid:80)(cid:68)(cid:87)(cid:72)(cid:71)(cid:3)(cid:49)(cid:82)(cid:76)(cid:86)(cid:72)(cid:3)(cid:51)(cid:82)(cid:90)(cid:72)(cid:85)(cid:3) (cid:36)(cid:71)(cid:68)(cid:83)(cid:87)(cid:76)(cid:89)(cid:72)(cid:3)(cid:41)(cid:76)(cid:79)(cid:87)(cid:72)(cid:85)(cid:76)(cid:81)(cid:74) (cid:55)(cid:85)(cid:68)(cid:76)(cid:81)(cid:76)(cid:81)(cid:74)(cid:3)(cid:39)(cid:68)(cid:87)(cid:68) (cid:54)(cid:38)(cid:48)(cid:29) (cid:54)(cid:91)(cid:59) (cid:58)(cid:75)(cid:76)(cid:87)(cid:72)(cid:81) (cid:86) (cid:58)(cid:75)(cid:76)(cid:87)(cid:72)(cid:81) (cid:91) (cid:4) (cid:86) (cid:4) (cid:86) (cid:51) (cid:4) (cid:54)(cid:48)(cid:44)(cid:36)(cid:48)(cid:41) (cid:36)(cid:38)(cid:40)(cid:46)(cid:42)(cid:47)(cid:53)(cid:55) (b) Adaptive filtering-then-CFAR (cid:48)(cid:68)(cid:87)(cid:70)(cid:75)(cid:72)(cid:71)(cid:3)(cid:41)(cid:76)(cid:79)(cid:87)(cid:72)(cid:85)(cid:76)(cid:81)(cid:74)(cid:55)(cid:72)(cid:86)(cid:87)(cid:3)(cid:39)(cid:68)(cid:87)(cid:68) (cid:39)(cid:72)(cid:87)(cid:72)(cid:70)(cid:87)(cid:76)(cid:82)(cid:81)(cid:3)(cid:54)(cid:87)(cid:68)(cid:87)(cid:76)(cid:86)(cid:87)(cid:76)(cid:70) (cid:20)(cid:19) (cid:43)(cid:43) (cid:33)(cid:31) (cid:55)(cid:75)(cid:85)(cid:72)(cid:86)(cid:75)(cid:82)(cid:79)(cid:71) (cid:39)(cid:72)(cid:70)(cid:76)(cid:86)(cid:76)(cid:82)(cid:81)(cid:55)(cid:85)(cid:68)(cid:76)(cid:81)(cid:76)(cid:81)(cid:74)(cid:3)(cid:39)(cid:68)(cid:87)(cid:68) (cid:238) (cid:54)(cid:70)(cid:68)(cid:79)(cid:76)(cid:81)(cid:74)(cid:3)(cid:41)(cid:68)(cid:70)(cid:87)(cid:82)(cid:85) (cid:51)(cid:85)(cid:72)(cid:16)(cid:68)(cid:86)(cid:86)(cid:76)(cid:74)(cid:81)(cid:72)(cid:71)(cid:3)(cid:51)(cid:41)(cid:36) (cid:38)(cid:41)(cid:36)(cid:53)(cid:3)(cid:51)(cid:85)(cid:82)(cid:70)(cid:72)(cid:86)(cid:86)(cid:76)(cid:81)(cid:74) (cid:36)(cid:79)(cid:74)(cid:82)(cid:85)(cid:76)(cid:87)(cid:75)(cid:80)(cid:86) (cid:40)(cid:86)(cid:87)(cid:76)(cid:80)(cid:68)(cid:87)(cid:72)(cid:71)(cid:3)(cid:49)(cid:82)(cid:76)(cid:86)(cid:72)(cid:3)(cid:51)(cid:82)(cid:90)(cid:72)(cid:85)(cid:3) (cid:49)(cid:82)(cid:81)(cid:16)(cid:36)(cid:71)(cid:68)(cid:83)(cid:87)(cid:76)(cid:89)(cid:72)(cid:3)(cid:41)(cid:76)(cid:79)(cid:87)(cid:72)(cid:85)(cid:76)(cid:81)(cid:74) (cid:39)(cid:72)(cid:87)(cid:72)(cid:70)(cid:87)(cid:82)(cid:85)(cid:3)(cid:39)(cid:72)(cid:86)(cid:76)(cid:74)(cid:81)(cid:55)(cid:72)(cid:86)(cid:87)(cid:3)(cid:39)(cid:68)(cid:87)(cid:68)(cid:55)(cid:85)(cid:68)(cid:76)(cid:81)(cid:76)(cid:81)(cid:74)(cid:3)(cid:39)(cid:68)(cid:87)(cid:68) (cid:39)(cid:72)(cid:87)(cid:72)(cid:70)(cid:87)(cid:76)(cid:82)(cid:81)(cid:3)(cid:54)(cid:87)(cid:68)(cid:87)(cid:76)(cid:86)(cid:87)(cid:76)(cid:70) (cid:20)(cid:19) (cid:43)(cid:43) (cid:33)(cid:31) (cid:55)(cid:75)(cid:85)(cid:72)(cid:86)(cid:75)(cid:82)(cid:79)(cid:71) (cid:39)(cid:72)(cid:70)(cid:76)(cid:86)(cid:76)(cid:82)(cid:81)(cid:51)(cid:85)(cid:72)(cid:16)(cid:68)(cid:86)(cid:86)(cid:76)(cid:74)(cid:81)(cid:72)(cid:71)(cid:3)(cid:51)(cid:41)(cid:36)(cid:58)(cid:75)(cid:76)(cid:87)(cid:72)(cid:81)(cid:72)(cid:71)(cid:3)(cid:48)(cid:68)(cid:87)(cid:70)(cid:75)(cid:72)(cid:71)(cid:3)(cid:41)(cid:76)(cid:79)(cid:87)(cid:72)(cid:85)(cid:76)(cid:81)(cid:74)(cid:55)(cid:72)(cid:86)(cid:87)(cid:3)(cid:39)(cid:68)(cid:87)(cid:68) (cid:39)(cid:72)(cid:87)(cid:72)(cid:70)(cid:87)(cid:76)(cid:82)(cid:81)(cid:3)(cid:54)(cid:87)(cid:68)(cid:87)(cid:76)(cid:86)(cid:87)(cid:76)(cid:70) (cid:20)(cid:19) (cid:43)(cid:43) (cid:33)(cid:31) (cid:55)(cid:75)(cid:85)(cid:72)(cid:86)(cid:75)(cid:82)(cid:79)(cid:71) (cid:39)(cid:72)(cid:70)(cid:76)(cid:86)(cid:76)(cid:82)(cid:81)(cid:55)(cid:85)(cid:68)(cid:76)(cid:81)(cid:76)(cid:81)(cid:74)(cid:3)(cid:39)(cid:68)(cid:87)(cid:68) (cid:238) (cid:54)(cid:70)(cid:68)(cid:79)(cid:76)(cid:81)(cid:74)(cid:3)(cid:41)(cid:68)(cid:70)(cid:87)(cid:82)(cid:85) (cid:51)(cid:85)(cid:72)(cid:16)(cid:68)(cid:86)(cid:86)(cid:76)(cid:74)(cid:81)(cid:72)(cid:71)(cid:3)(cid:51)(cid:41)(cid:36) (cid:38)(cid:41)(cid:36)(cid:53)(cid:3)(cid:51)(cid:85)(cid:82)(cid:70)(cid:72)(cid:86)(cid:86)(cid:76)(cid:81)(cid:74) (cid:36)(cid:79)(cid:74)(cid:82)(cid:85)(cid:76)(cid:87)(cid:75)(cid:80)(cid:86) (cid:40)(cid:86)(cid:87)(cid:76)(cid:80)(cid:68)(cid:87)(cid:72)(cid:71)(cid:3)(cid:49)(cid:82)(cid:76)(cid:86)(cid:72)(cid:3)(cid:51)(cid:82)(cid:90)(cid:72)(cid:85)(cid:3) (cid:36)(cid:71)(cid:68)(cid:83)(cid:87)(cid:76)(cid:89)(cid:72)(cid:3)(cid:41)(cid:76)(cid:79)(cid:87)(cid:72)(cid:85)(cid:76)(cid:81)(cid:74) (cid:55)(cid:85)(cid:68)(cid:76)(cid:81)(cid:76)(cid:81)(cid:74)(cid:3)(cid:39)(cid:68)(cid:87)(cid:68) (cid:54)(cid:38)(cid:48)(cid:29) (cid:54)(cid:91)(cid:59) (cid:58)(cid:75)(cid:76)(cid:87)(cid:72)(cid:81) (cid:86) (cid:58)(cid:75)(cid:76)(cid:87)(cid:72)(cid:81) (cid:91) (cid:4) (cid:86) (cid:4) (cid:86) (cid:51) (cid:4) (cid:54)(cid:48)(cid:44)(cid:36)(cid:48)(cid:41) (cid:36)(cid:38)(cid:40)(cid:46)(cid:42)(cid:47)(cid:53)(cid:55) (c) Adaptive detection Figure 1
Block diagrams for filtering-then-CFAR detection and adaptive detection
Moreover, the detection statistics of the KGLRT, AMF, and DMRao are t KGLRT = ˜ x H P ˜ s ˜ x x H ˜ x − ˜ x H P ˜ s ˜ x , (30) t AMF = ˜ x H P ˜ s ˜ x , (31)and t DMRao = ˜ x H P ˜ s ˜ x (1 + ˜ x H ˜ x )(1 + ˜ x H ˜ x − ˜ x H P ˜ s ˜ x ) , (32)respectively, where ˜ x = S − x , ˜ s = S − s , x is the test data vector, s is the signal steering vector, S = X L X HL is the SCM , and P ˜ s = ˜ s ˜ s H ˜ s H ˜ s is the orthogonal projection matrix of ˜ s .The SMI and AMF can be taken as the outputs of certain adaptive filters, and then their correspondingweight vectors are w SMI = S − ss H S − s (33)and w AMF = S − s √ s H S − s , (34)respectively. However, the KGLRT and DMRao cannot be expressed as the output of a filter.Two key functions of adaptive filtering are clutter rejection and signal integration. The former isachieved by the “whiten” model, accomplished by the matrix S − , while the latter is achieved by theorthogonal projection matrix “ P ˜ s ”. It is seen from Figure 2, along with (29)-(32), that the SMI, KGLRT,AMF, and DMRao all have the function of adaptive filtering. Moreover, the AMF and SMI have thesame filtering performance, since they have the same output SNR. This can be verified by substituting
6) A more common SCM in adaptive filtering is defined as S ′ = L X L X HL . However, for adaptive detection it is usually moreconvenient to use the SCM defined as S = X L X HL .7) Note that the SMI weight in (33) satisfies the constraint w H SMI s = 1.iu W , et al. Sci China Inf Sci (cid:48)(cid:68)(cid:87)(cid:70)(cid:75)(cid:72)(cid:71)(cid:3)(cid:41)(cid:76)(cid:79)(cid:87)(cid:72)(cid:85)(cid:76)(cid:81)(cid:74)(cid:55)(cid:72)(cid:86)(cid:87)(cid:3)(cid:39)(cid:68)(cid:87)(cid:68) (cid:39)(cid:72)(cid:87)(cid:72)(cid:70)(cid:87)(cid:76)(cid:82)(cid:81)(cid:3)(cid:54)(cid:87)(cid:68)(cid:87)(cid:76)(cid:86)(cid:87)(cid:76)(cid:70) (cid:20)(cid:19) (cid:43)(cid:43) (cid:33)(cid:31) (cid:55)(cid:75)(cid:85)(cid:72)(cid:86)(cid:75)(cid:82)(cid:79)(cid:71) (cid:39)(cid:72)(cid:70)(cid:76)(cid:86)(cid:76)(cid:82)(cid:81)(cid:55)(cid:85)(cid:68)(cid:76)(cid:81)(cid:76)(cid:81)(cid:74)(cid:3)(cid:39)(cid:68)(cid:87)(cid:68) (cid:238) (cid:54)(cid:70)(cid:68)(cid:79)(cid:76)(cid:81)(cid:74)(cid:3)(cid:41)(cid:68)(cid:70)(cid:87)(cid:82)(cid:85) (cid:51)(cid:85)(cid:72)(cid:16)(cid:68)(cid:86)(cid:86)(cid:76)(cid:74)(cid:81)(cid:72)(cid:71)(cid:3)(cid:51)(cid:41)(cid:36) (cid:38)(cid:41)(cid:36)(cid:53)(cid:3)(cid:51)(cid:85)(cid:82)(cid:70)(cid:72)(cid:86)(cid:86)(cid:76)(cid:81)(cid:74) (cid:36)(cid:79)(cid:74)(cid:82)(cid:85)(cid:76)(cid:87)(cid:75)(cid:80)(cid:86) (cid:40)(cid:86)(cid:87)(cid:76)(cid:80)(cid:68)(cid:87)(cid:72)(cid:71)(cid:3)(cid:49)(cid:82)(cid:76)(cid:86)(cid:72)(cid:3)(cid:51)(cid:82)(cid:90)(cid:72)(cid:85)(cid:3) (cid:49)(cid:82)(cid:81)(cid:16)(cid:36)(cid:71)(cid:68)(cid:83)(cid:87)(cid:76)(cid:89)(cid:72)(cid:3)(cid:41)(cid:76)(cid:79)(cid:87)(cid:72)(cid:85)(cid:76)(cid:81)(cid:74) (cid:39)(cid:72)(cid:87)(cid:72)(cid:70)(cid:87)(cid:82)(cid:85)(cid:3)(cid:39)(cid:72)(cid:86)(cid:76)(cid:74)(cid:81)(cid:55)(cid:72)(cid:86)(cid:87)(cid:3)(cid:39)(cid:68)(cid:87)(cid:68)(cid:55)(cid:85)(cid:68)(cid:76)(cid:81)(cid:76)(cid:81)(cid:74)(cid:3)(cid:39)(cid:68)(cid:87)(cid:68) (cid:39)(cid:72)(cid:87)(cid:72)(cid:70)(cid:87)(cid:76)(cid:82)(cid:81)(cid:3)(cid:54)(cid:87)(cid:68)(cid:87)(cid:76)(cid:86)(cid:87)(cid:76)(cid:70) (cid:20)(cid:19) (cid:43)(cid:43) (cid:33)(cid:31) (cid:55)(cid:75)(cid:85)(cid:72)(cid:86)(cid:75)(cid:82)(cid:79)(cid:71) (cid:39)(cid:72)(cid:70)(cid:76)(cid:86)(cid:76)(cid:82)(cid:81)(cid:51)(cid:85)(cid:72)(cid:16)(cid:68)(cid:86)(cid:86)(cid:76)(cid:74)(cid:81)(cid:72)(cid:71)(cid:3)(cid:51)(cid:41)(cid:36)(cid:58)(cid:75)(cid:76)(cid:87)(cid:72)(cid:81)(cid:72)(cid:71)(cid:3)(cid:48)(cid:68)(cid:87)(cid:70)(cid:75)(cid:72)(cid:71)(cid:3)(cid:41)(cid:76)(cid:79)(cid:87)(cid:72)(cid:85)(cid:76)(cid:81)(cid:74)(cid:55)(cid:72)(cid:86)(cid:87)(cid:3)(cid:39)(cid:68)(cid:87)(cid:68) (cid:39)(cid:72)(cid:87)(cid:72)(cid:70)(cid:87)(cid:76)(cid:82)(cid:81)(cid:3)(cid:54)(cid:87)(cid:68)(cid:87)(cid:76)(cid:86)(cid:87)(cid:76)(cid:70) (cid:20)(cid:19) (cid:43)(cid:43) (cid:33)(cid:31) (cid:55)(cid:75)(cid:85)(cid:72)(cid:86)(cid:75)(cid:82)(cid:79)(cid:71) (cid:39)(cid:72)(cid:70)(cid:76)(cid:86)(cid:76)(cid:82)(cid:81)(cid:55)(cid:85)(cid:68)(cid:76)(cid:81)(cid:76)(cid:81)(cid:74)(cid:3)(cid:39)(cid:68)(cid:87)(cid:68) (cid:238) (cid:54)(cid:70)(cid:68)(cid:79)(cid:76)(cid:81)(cid:74)(cid:3)(cid:41)(cid:68)(cid:70)(cid:87)(cid:82)(cid:85) (cid:51)(cid:85)(cid:72)(cid:16)(cid:68)(cid:86)(cid:86)(cid:76)(cid:74)(cid:81)(cid:72)(cid:71)(cid:3)(cid:51)(cid:41)(cid:36) (cid:38)(cid:41)(cid:36)(cid:53)(cid:3)(cid:51)(cid:85)(cid:82)(cid:70)(cid:72)(cid:86)(cid:86)(cid:76)(cid:81)(cid:74) (cid:36)(cid:79)(cid:74)(cid:82)(cid:85)(cid:76)(cid:87)(cid:75)(cid:80)(cid:86) (cid:40)(cid:86)(cid:87)(cid:76)(cid:80)(cid:68)(cid:87)(cid:72)(cid:71)(cid:3)(cid:49)(cid:82)(cid:76)(cid:86)(cid:72)(cid:3)(cid:51)(cid:82)(cid:90)(cid:72)(cid:85)(cid:3) (cid:36)(cid:71)(cid:68)(cid:83)(cid:87)(cid:76)(cid:89)(cid:72)(cid:3)(cid:41)(cid:76)(cid:79)(cid:87)(cid:72)(cid:85)(cid:76)(cid:81)(cid:74) (cid:55)(cid:85)(cid:68)(cid:76)(cid:81)(cid:76)(cid:81)(cid:74)(cid:3)(cid:39)(cid:68)(cid:87)(cid:68) (cid:54)(cid:38)(cid:48)(cid:29) (cid:54)(cid:91)(cid:59) L (cid:58)(cid:75)(cid:76)(cid:87)(cid:72)(cid:81) (cid:86) (cid:58)(cid:75)(cid:76)(cid:87)(cid:72)(cid:81) (cid:91) (cid:5) (cid:86) (cid:5) (cid:86) (cid:51) (cid:5) (cid:54)(cid:48)(cid:44)(cid:36)(cid:48)(cid:41) (cid:39)(cid:48)(cid:53)(cid:68)(cid:82)(cid:46)(cid:42)(cid:47)(cid:53)(cid:55) Figure 2
Block diagrams for one adaptive filter and three adaptive detectors (33) and (34) into the quantity to be maximized in the right-hand side of (24). However, their detectionperformance is different, since the AMF has the CFAR property, whereas the SMI does not .In summary, adaptive detectors use the test and training data to form specific structures, which areCFAR and have the function of filtering, embedded in the detection statistics. According to different criteria, the problem of adaptive detection can be sorted into different types.For example, according to the extension of a target, adaptive detection can be sorted into point targetdetection and distributed (spread) target detection; according to the fact that whether the signal ismismatched or not, adaptive detection can be sorted as detection in the absence of signal mismatchand detection in the presence of signal mismatch; according to statistical property of the noise, adaptivedetection can be sorted into detection in Gaussian noise and detection in non-Gaussian noise; accordingto the characters of the test and training data, adaptive detection can be sorted into detection in HEand detection in non-homogeneous (heterogeneous) environment; etc. However, the above classificationsare too rough. Hence, we review the literature in the following six categories . For convenience, in eachsubsection we summarize the corresponding taxonomies in a table. Table 1
Related Taxonomy in Subsection 3.1Taxonomy MeaningHE A scenario that test and training data have the same noise covariance matrix.PHE A scenario that test and training data have the same noise covariance matrix upon to unknownscaling factor.Nonhomogeneity A scenario that the data in the collection of test and training data do not have the same noisecovariance matrix.Compound-Gaussian process A random process which is in the form of a product of two components. One is the is the squareroot of a positive scalar random process (called texture, accounting for local power change),while the other is a complex Gaussian process (called speckle, accounting for local scattering).Rank-one signal A kind of signal, modelled by the product of a known vector and an unknown scaling factor.Subspace signal A kind of signal, modelled by the product of a known matrix and an unknown vector. That isto say, a subspace signal lies in a known subspace but with unknown coordinates.
In the seminal paper [4], Kelly considered the detection problem for a point target in HE. Precisely, thepoint target has a known signal steering vector, embedded in Gaussian noise with unknown covariancematrix. To estimate the unknown noise covariance matrix, a set of IID training data was used, whichis signal-free and shares the same noise covariance matrix with the test data. Then Kelly proposed thefamous KGLRT. According to the 2S-GLRT, Chen et al. [43] and Robey et al. [44] independently derivedthe well-known AMF, which has small complexity compared with the KGLRT. The corresponding Raotest was obtained by De Maio [45], i.e., the DMRao, which has lower PD than the KGLRT and AMF.However, the DMRao has better performance in terms of rejecting mismatched signals. The corresponding
8) The statistical performance analysis of the multi-band generalization of the SMI, called the modified SMI (MSMI), in [46]indicates that the detection threshold of the SMI depends on the noise covariance matrix R .9) We are sorry to any researcher whose work is overlooked or otherwise not discussed.iu W , et al. Sci China Inf Sci Wald test was also derived by De Maio [47], which coincides with the AMF. Noticeably, in 1994, Gerlachproposed the nonconcurrent mean level adaptive detector (N-MLAD) [48] and concurrent mean leveladaptive detector (C-MLAD) [49]. The N-MLAD and C-MLAD are essentially the AMF and DMRao,respectively; see also [50, 51]. Moreover, the AMF was utilized in [52] for simultaneous detection andparameter estimation (i.e., target’s Doppler and bearing).The three detector KGLRT, DMRao, and AMF were all devised under the assumption of the HE.However, the data may have different statistical properties, owing to rapidly changed environmental fac-tors or instrumental factors, such as adaptation of conformal array, bistatic radar, or multisite radar.Partially homogeneous environment (PHE) is a widely used nonhomogeneity model, which well charac-terizes the environment for airborne radars with low number of training data [53] and also suitable forwireless communications with fades over multiple sources of interference [54]. The GLRT for point targetdetection in PHE was derived by Kraut et al. , denoted as the adaptive coherent estimator (ACE) [55]. Itwas found in [56] that the Rao and Wald tests in PHE coincide with the ACE. In [57], a simple approachfor the threshold setting of ACE, as well as the AMF, was provided. An invariance property of theACE was given in [58], and it was shown to be uniformly most powerful invariant (UMPI) in [54]. Morerecently, it was shown in [59] that the ACE using the fixed-point covariance estimate [60] coincides witha maximal invariant component . It is worth to pointing out that the ACE is effective in two kinds ofnon-homogeneous environment. One is spherically invariant noise [64] or compound-Gaussian noise [61].The other is Bayesian heterogeneity. Precisely, the covariance matrix of the training data is subject toinverse complex Wishart distribution, and is proportional to the covariance matrix in the test data [65].Moreover, the ACE is also called the adaptive normalized matched filter (ANMF) [64, 66] or normalizedAMF (NAMF) [67]. In [68] the CFAR behavior using experimentally measured data was investigatedfor the KGLRT, AMF, and two variations of the ACE, namely, recursive ANMF (R-ANMF) [69] andpersymmetric (RP-ANMF) [70]. It was shown in [68] that all these detectors exhibit a false alarm ratehigher than the preassigned value, and the RP-ANMF is most robust among them. More recently, Theproblem of target separation detection (TSD) was considered in [71], where TSD tests were designedaccording to the GLRT. It was shown therein that the TSD tests can effectively monitor the event oftarget separation.The above detectors are for rank-one signals, which have a known steering vector. However, a signalmay naturally lie in a subspace, but with unknown coordinates, such as polarimetric target detection[72–76]. This type of signal is called subspace signal, which can be mathematically expressed as theproduct of a full-column-rank matrix and a vector. Under the background of polarimetric target detection,references [77] and [78] generalized the KGLRT and AMF to the case of 2-dimensional subspace. Then,references [79, 80] generalized the KGLRT to the case of subspace with dimension greater than 2, andthe detector can be named as the subspace-based GLRT (SGLRT). Similarly, the AMF was generalizedto the case of subspace with dimension greater than 2 in [81], and the detector was referred to asthe subspace-based AMF (SAMF). The subspace versions of the DMRao and ACE were given in [82]and [83], respectively, and the resulting detectors can be denoted as the subspace-based Rao (SRao) testand adaptive subspace detector (ASD), respectively. The statistical properties of the SGLRT was givenin [79, 84], the statistical properties of the SAMF was given in [81], the statistical properties of the ASDwas given in [85, 86], and the statistical properties of the SRao was given in [87].
For a high-resolution radar (HRR), a target may be spread in range, especially a big target, such as a largeship. It was shown in [88] that a properly designed HRR can provide improved detection performance.This is mainly due to two factors. One is that increasing the capability of range resolution of the radar canreduce the amount of energy per range bin backscattered by the clutter. The other is that a distributedtarget is usually less fluctuated than an unresolved point target.It was assumed in [53] that the echoes reflected by the distributed target all came from the samedirection, and the GLRT and 2S-GLRT for distributed target detection in HE and PHE were derived.The corresponding Rao and Wald tests in HE were derived in [89], while the Rao and Wald test in PHEwere given in [90]. The 2S-GLRT in HE in [53] was known as the generalized AMF (GAMF). Similarly,
10) It is observed that the upper-bound performance of the ACE is provide by the normalized matched filter (NMF), which wasgiven in [61, 62]. Moreover, the NMF was shown in [63] to be the UMPI detector in spherically invariant random vector (SIRV)disturbance with a specific texture.iu W , et al. Sci China Inf Sci Table 2
Related Taxonomy in Subsection 3.2Taxonomy MeaningDistributed target A target which occupies more than one range bins for a radar system.DD A detection problem, for which the received echoes all come from the same direction.However, the corresponding signal steering vector is only known to lie in a given subspace.GDD A detection problem, for which both the column and row components of a rank-one matrix-valuedsignal are constrained to lie in known subspaces, but with unknown coordinates.DOS signal A kind of signal, which is matrix-valued and its row and column elements both lie in known subspacesbut with unknown coordinates. we can name the GLRT in HE in [53] as generalized KGLRT (GKGLRT), since it is a generalization ofthe KGLRT. It is observed that the GLRT in HE proposed in [53] shares the same form as the multibandGLR (MBGLR) in [91] .Reference [92] investigated the problem of detecting a distributed target, whose signal steering vectorwas unknown. The GLRT, 2S-GLRT, modified 2S-GLRT (M2S-GLRT), and spectral norm test (SNT)were proposed. It was shown in [93] that the 2S-GLRT and M2S-GLRT can be obtained according to theWald test and Rao test, respectively. Some intuitive interpretations about the detectors were also givenin [93]. Recently, reference [94] considered the case when the test data matrix was of rank two, and ageneralization of ACE was proposed and its analytical performance was given.In [95] it was assumed that the echoes backscattered by the distributed target all came from the samedirection. However, the corresponding signal steering vector was only known to lie in a given subspace.This correspond detection problem was referred to as the direction detection (DD) therein, and the so-called generalized adaptive direction detector (GADD) was proposed according to the 2S-GLRT in PHE.From a mathematical point of view, for the problem of direction detection, the matrix-valued signal to bedetected is of rank one, and its column components are constrained to a known subspace, while its rowcomponents are completely unknown. A more general signal model was adapted in [96], where both thecolumn and row components of a rank-one matrix-valued signal are constrained to lie in known subspaces,but with unknown coordinates. This kind of problem can be taken as a generalized direction detection(GDD). However, it did not use the training data in [96]. Instead, it was assumed that the dimension ofthe test data satisfied certain constraint. Then a set of virtual training data can be obtained by usinga unitary matrix transformation to the test data. As a consequence, the row structure of the signal waslost. Then the corresponding GLRT and 2S-GLRT were proposed therein. Essentially, the data modelin [96] was equivalent to that in [95], but the environments were homogeneous. The Wald test for theDD in HE was proposed in [97], and it was shown that there is no reasonable Rao test for the problem ofdirection detection. The problem of GDD in HE was exploited in [98], where the corresponding GLRTand 2S-GLRT were proposed. Moreover, the 2S-GLRT in PHE for GDD was given in [99].For the problem of detecting a distributed target, a systematic and comprehensive investigation wasthe report by Kelly and Forsythe in 1989 [100], where the solid mathematical background for adaptivesignal detection was given. In [100] the signal to be detected is matrix-valued and its row and columnelements both lie in known subspaces but with unknown coordinates. This kind of signal model is referredto as the double subspace (DOS) signal in [82, 101]. The DOS signal model is very general and includesmany types of point targets and distributed targets as the special cases. In [100], no training data setwas utilized. In contrast, a dimension constraint was posed on the test data. Then after a unitary matrixtransformation on the test data, a set of virtual training data was obtained. Unfortunately, the rowstructure of the DOS signal is lost after the unitary matrix transformation. The problem of detectinga DOS signal was generalized in [82, 101], where true training data were assumed available, and manydetectors were proposed and compared.Compared with the detectors for point targets, the statistical performance of the detectors designedfor distributed targets is difficult to be derived. In particular, the statistical performance of the GLRTand 2S-GLRT for distributed target in HE, proposed in [53], was given in [91] and [102], respectively.Moreover, the result in [91] was generalized in [103] to the case of signal mismatch. Signal mismatch willbe explained detailed in the next subsection. iu W , et al. Sci China Inf Sci Table 3
Related Taxonomy in Subsection 3.3Taxonomy MeaningSignal mismatch The phenomenon that the actual signal steering vector is not aligned with the nominal one adopted bythe radar system.Robustness A property that the detection performance of a detector does not decrease severely with the increase of signalmismatch.Selectivity A property that the detection performance of a detector decreases rapidly with the increase of signal mismatch.Directivity The property (including robustness and selectivity) of a detector when detecting a mismatch signal.Tunable detector A kind of detector, which is parameterized by one or more positive scaling factors, called the tunableparameters. By adjusting the tunable parameters, the directivity property of the detector can be changed.Cascaded detector A kind of detector, formed by cascading a robust detector and a selective detector.Weighted detector A kind of detector, formed by weighting a robust detector and a selective detector.
In practice, there often exists signal mismatch [104]. Precisely, the actual signal steering vector is notaligned with the nominal one adopted by the radar system. The statistical performance analysis foradaptive detectors in the presence of signal mismatch was first dealt with in [105], where it is shownthat a key quantity controlling the detection performance of the KGLRT with mismatched signals isthe generalized cosine-squared between the actual signal and the nominal signal in the whitened space.Based on the result in [105], the statistical performance of the AMF and ACE was given in [106], whilethe performance of the DMRao was dealt with in [45]. The statistical performance of the subspace-baseddetectors was addressed in [107] for the case of mismatched subspace signals, which is a generalization ofthe rank-one signal.Signal mismatch can be caused by array error or target maneuvering. Moreover, signal mismatch canalso be caused by jamming signals coming from the radar sidelobe, due to electronic countermeasures(ECM). For different sources of signal mismatch, different types of detectors are needed. For the firstcase, a robust detector is preferred, which achieves satisfied detection performance when signal mismatchoccurs. In contrast, for the second case, a selective detector is preferred, whose detection performancedecreases rapidly with the increase of signal mismatch.One method to design a robust detector for mismatched signals is adopting subspace signal model(for rank-one signals) [79] or enlarging the signal subspace (for subspace signals) [108, 109]. Anothermethod is constraining the actual angle or Doppler frequency lie in an compact interval [110, 111]. Then,maximization of the concentrated likelihood function over the actual angle or Doppler can be formulatedas a semidefinite programming (SDP) convex problem, and hence easily solved. A third method is toassume that the actual signal lies in a convex cone, whose axes coincide with the nominal signal steeringvector. Then a robust detector is designed by using second-order cone (SOC) programming [112–116]. Afourth method is to adding a random component in the test data under the signal-presence hypothesis.This makes the hypothesis more plausible when signal mismatch happens [117].To design a selective detector, one approach is to modify the original hypothesis test by adding adeterminant unknown fictitious signal (or jammer) under the null hypothesis. The fictitious signal satisfiescertain constraints. A useful constraint is that the fictitious signal is orthogonal to the nominal signal inthe quasi-whitened space [118] or whitened space [119]. Then the resulting detector will be inclined tochoose the null hypothesis when there is no target in the nominal direction but in other directions. Underthis idea, many selective detectors have been proposed, such as the adaptive beamformer orthogonalrejection test (ABORT) [118], whitened ABORT (W-ABORT) [119], their Bayesian variations [120],and other modifications [121–123]. The proposed selective detectors in the aforementioned referenceswere mainly under the assumption of the HE. In contrast, a selective detector was proposed in [124] fordistributed target detection in PHE. However, the selectivity property of the proposed detector is limited.In [125] a detector with improved selectivity was proposed for distributed target detection in PHE.Another approach to design selective detector is adding a random unknown fictitious signal under boththe null and alternative hypotheses. An intuitive interpretation may be lack. However, it works in certainparameter setting, such as the double-normalized AMF (DN-AMF) [126].Note that the directivity (robustness or selectivity) of the above detectors cannot be adjusted. Inother words, for a given detector, it either works as a robust detector or a selective detector, not both.
11) The MBGLR was proposed for point target detection when a radar system has multiple frequency bands.iu W , et al. Sci China Inf Sci This limits the flexibility of the detectors in detecting mismatched signals. Tunable detectors, cascadeddetectors, weighted detectors, as well as their combinations, can overcome the above limitation.Tunable detectors are mainly obtained by comparing the similarities in the detection statistics oftwo or more detectors with different directivity properties, and they, with specific tunable parameters,usually contain conventional detectors as their special cases. Directivity property of a tunable detectorfor mismatched signals can be smoothly changed by adjusting one or two parameters, called tunableparameters. The first tunable detector was proposed by Kalson in 1992 [127], which contains the KGLRTand AMF as two special cases. However, the selectivity of this tunable detector cannot exceed theKGLRT. Another tunable detector was proposed by Hao et al. in [128], termed as KRAO, which containsthe KGLRT and DMRao as two special cases. The KRAO has enhanced selectivity but its robustness islimited. In [129] a tunable detector termed as KMABORT, was proposed, which contains the KGLRT,AMF, and ABORT as three special cases. The KMABORT is characterized by two tunable parameters,and hence it has more freedoms in detecting mismatched signals. However, its best robust propertyfor mismatched signals is tantamount to that of the AMF. Fortunately, the AMF is very robust formismatched signals, although it is not designed specially for robust detection of mismatched signals.A tunable detector, called KWA, was proposed in [130], which contains the KGLRT, W-ABORT, andadaptive energy detector (AED) [131] as its special cases. The KWA can provide even more robustproperty than the AMF. As a special case of the KWA, the AED does not need the nominal signalsteering vector, instead, it only tests whether there exists a signal with sufficient energy. In other words,it does not differentiate between matched signals and mismatched signals. As a result, the AED is mostrobust. There are some other tunable detectors, such as the ones in [132–135].A cascaded detector is forming by cascading a robust detector and a selective detector, and hence ithas numerous pairs of detection thresholds. By changing the pair of detection thresholds, it can changethe directivity property for mismatched signals. This type of cascaded detector is also called two-stagedetector. A two-stage detector, referred to as 2SGLRT, cascading the KGLRT and AMF was proposedin [136]. In [106], a two-stage detector, called adaptive sidelobe blanker (ASB), was proposed, whichcascades the AMF and ACE. In [45], a two-stage detector, denoted as AMF-Rao, which cascades theAMF and DMRao. In [137], a two-stage detector, called WAS-ASB was proposed, which cascades theSGLRT and W-ABORT. In [138], a two-stage detector, called S-ASB was proposed, which cascadesthe SGLRT and ACE. In [130], a two-stage detector called KWAS-ASB was proposed, which cascadesthe KWA and SGLRT. In [128] two two-stage detectors were proposed, named as the KRAO-ASB andSKRAO-ASB. The former cascades the AMF and KRAO, while the latter cascades the SGLRT andKRAO. In [139], a two-stage detector, called SD-RAO was proposed, which cascades the SGLRT andDMRao. The above two-stage detectors were all designed for rank-one signals. In contrast, a two-stagedetector, named AESD, was proposed in [140] for mismatched subspace signal by cascading the AED andASD. The useful lecture [141] summarized the selective detectors ABORT and W-ABORT, the tunabledetector KWA, the two-stage detectors ASB, AMF-Rao, S-ASB and WAS-ASB. Recently, a survey onthe two-stage detector was given in [142].A weighted detector is constructed by weighting a robust detector and a selective detector. By adjustingthe weight, the directivity can be smoothly changed. A weighted detector, called SAMF-ASD, wasproposed in [143].All the tunable detectors, two-stage detectors, and weighted detectors above are designed for pointtarget in HE. The ABORT was generalized in [124] for the distributed target detection both in HE andPHE. For distributed target detection, the W-ABORT was generalized in [144] and [125] in HE and PHE,respectively. Moreover, a tunable detector for distributed target detection in PHE was proposed in [125],called tunable GLRT in PHE (T-GLRT-PHE).Note that the capabilities of robustness or selectivity of the two-stage detector and weighted detectorcannot exceed their corresponding cascaded detectors and weighted detectors, respectively. In contrast,the tunable detector usually has much more freedoms to change the directivity for mismatched signals.
Most of the aforementioned detectors are designed without taking into account the presence of interfer-ence. In practice, however, there usually exists interference, besides noise and possible signal of interest.Interference can be caused by the intentional ECM or unintentional industrial production.Masking and deception are two main effects of interference on radar system. Noise interference has iu W , et al. Sci China Inf Sci Table 4
Related Taxonomy in Subsection 3.4Taxonomy MeaningNoise interference A type of random interference, having the effect of thermal noise or clutter.Coherent interference A type of interference, having the effect of deceiving the radar system,which only lies in a direction and occupies a Doppler bin.Subspace interference A type of coherent interference, which can be modelled by a subspace model.Orthogonal interference A type of coherent interference, which is orthogonal to the signal in some manner. the effect of masking the radar system, while coherent interference has the effect of deceiving the radarsystem. Noise interference plays the role of thermal noise or clutter. Hence, it raises the level of the noise.As a result, in order to maintain CFAR property, the radar system has to raise the detection threshold,which reduces the radar sensitivity for target detection [145, pp. 114-115]. Coherent interference usuallyimitates a real target, and hence it can deceive the radar system. This requires the interference workscoherent to the radar system. Coherent interference can also be called false-target interference, includingfalse-range interference, false-velocity interference, and false-direction interference.From the point of view of data model, coherent interference is usually constrained to lie in a knownsubspace, and hence is often referred to as subspace interference in the field of adaptive detection. Muchwork was done by Scharf et al. [146–148] for detecting a multichannel signal in subspace interference andthermal noise (or colored noise with known covariance matrix). Some other relative work in subspaceinterference and colored noise with known covariance matrix was given in [149–152].In practical applications, the noise covariance matrix is usually unknown, and needed to be estimated.For distributed target detection in subspace interference, it was assumed in [153] that the noise covariancematrix was unknown. To estimate the noise covariance matrix, a set of sufficient training data was used.The GLRT and 2S-GLRT were derived both in HE and PHE therein. The PFA of the GLRT in HEwas given in [154]. The corresponding Rao test and two-step Rao (2S-Rao) tests in HE and PHE werederived in [155]. The Wald test and two-step Wald (2S-Wald) tests for point target detection in subspaceinterference were derived in [156]. Moreover, a modified Rao test was given in [157], which took both thesignal coordinate matrix and interference coordinate matrix as the relative parameter. It is shown in [156]that in HE the 2S-GLRT, 2S-Rao, and Wald test (the other detectors all strongly related with these threedetectors) whiten the noise (or equivalently reject the clutter) in the same manner. However, they rejectthe subspace interference in different manners. Recently, the statistical performance of the GLRT forsubspace interference was analyzed in [158] for the case that the signal was of rank one. Moreover, thestatistical performance of the GLRT-based detectors for point target detection in subspace interferencewas analysed in [159] for the case of signal mismatch, including the signal match as a special case. It wasshown in [159] that the coherent interference and signal mismatch affect the detection performance ofthe GLRT-based detectors through two generalized angles. One is the angle between the whitened actualsignal and the whitened interference subspace. The other is the angle of the actual signal and nominalsignal matrix after they are both projected onto the interference-orthogonalized subspace. Reference [160]investigated the detection problem in subspace interference when signal mismatch happens. Two selectivedetectors and a tunable detector were proposed, and their statistical performance was also given therein.The detection problem in subspace interference was addressed in [161–165] in the framework of invarianceprinciple. When the subspace interference lies in both the test and training data, it was pointed thatin [23] that there is no effective GLRT, and a modified GLRT was proposed based on the method ofsieves therein.For the DD problem in the presence of subspace interference in HE, the GLRT and 2S-GLRT weredeveloped in [166], while the Wald test and 2S-Wald test were obtained in [167]. The corresponding2S-GLRT and 2S-Wald tests in PHE were derived in [168].In the above references, sufficient information about the coherent interference is assumed available.However, this is not always the case in practice. It was assumed in [169] that the interference subspacewas unknown except for its dimension, and a GLRT-like detector was proposed therein. In [87], itwas assumed that the coherent interference was unknown but it was orthogonal to the signal in thewhitened space. This type of interference was called orthogonal interference therein . Then three
12) The orthogonal interference satisfies the generalized eigenrelation (GER) defined in [170], which can be approximately metin practice, especially for the out-of-mainbeam interference [171]. It is pointed out in [171] that using secondary data selectionstrategies, e.g., the power selected training [172], results in the orthogonality of the signal and interference in the whitened space.iu W , et al. Sci China Inf Sci detectors were proposed, according to the criteria of GLRT, Rao test, and Wald test. Remarkably, theresulting three detectors share the same forms as the SGLRT, SRao, and SAMF, respectively. However,statistical performance analysis indicated that the orthogonal interference can degrade the detectionperformance [87]. Moreover, it was assumed in [173–175] that there were uncertainties in signal andcoherent interference. To account for these uncertainties, the signal and interference were constrained tocertain proper cones. Then effective detectors were proposed by using convex optimization.The adaptive detection in completely unknown coherent interference was dealt with in [176]. At thestage of detector design, the unknown interference was assumed to lie in a subspace orthogonal to thesignal. According to the GLRT and Wald test, two detectors were proposed, and the detector derivedaccording to the GLRT was called adaptive orthogonal rejection detector (AORD). It was shown thatthe AORD has better detection performance than others in completely unknown interference. Anotherdistinctive feature of the AORD is that it can even provide significantly performance improvement,compared with the KGLRT and AMF in the absence of interference. This was shown in [177], where thestatistical performance of the AORD was also given.The above references mainly deal with coherent interference. It was assumed in [178] that therewas a completely unknown noise interference, and the corresponding GLRT for rank-one signals wasshown to be equivalent to the ACE. The corresponding Rao test was given in [126], i.e., the DN-AMF,mainly adopted for mismatched signal detection, as explained in Subsection 3.3. The above resultswere generalized in [179] when there existed additional coherent interference, and the GLRT, Rao test,and Wald test were derived for subspace signals. In [180] the noise interference was constrained by theGER, and the GLRT was shown to be the same as the KGLRT. Moreover, the corresponding Rao andWald tests were shown to be the DMRao and AMF, respectively [181]. The results in [180, 181] weregeneralized in [182] for subspace signals. It was assumed in [183] that the noise interference lies in asubspace orthogonal to the signal subspace, and a detector was proposed according to the 2S-Rao test,named as two-step orthogonal SAMF (2S-OSAMF). Numerical examples shew that the 2S-OSAMF hasbetter detection performance than its competitors even the noise interference is completely unknown.In [184] the authors considered the problem of determining whether the test data contained a noiseinterference or not. This problem was solved by formulating the problem as a binary hypothesis test,and a detector was designed according to GLRT criterion. In [185] the authors considered the problem ofdetecting a signal in the presence of noise interference, which only occupied parts of training data. TwoGLRT-related detectors were proposed, which were shown to have better performance than the existingdetectors. In [186] the authors considered two scenarios for the signal detection problem in interference.One was that only noise interference existed, and the other is that both noise interference and coherentinterference existed. For the first scenario, an effective estimate for the interference covariance wasproposed and then utilized in the AMF, which can mitigate the deleterious effects of the noise interference.For the second scenario, a compressive sensing-based GLRT was proposed. Some other detection problemsinvolved in noise interference were given in [187–189]. Table 5
Related Taxonomy in Subsection 3.5Taxonomy MeaningLow-rank structure Noise covariance matrix is a sum of a scaled identity matrix and a low-rank matrix, with eigenvaluesmuch greater than unity.Persymmetry Noise covariance matrix is persymmetric about its cross diagonal and Hermitian about its diagonal.Spectral symmetry Ground clutter has a symmetric PSD centred around the zero-Doppler frequency.
For adaptive processing, e.g., adaptive detection or adaptive filtering, it usually needs sufficient trainingdata to estimate the unknown noise covariance matrix. In particular, it was shown in [42] that theadaptive filter SMI needs at least 2 N − N being the dimension of the test data.This is known as the Reed-Mallett-Brenann (RMB) rule [42] . However, this requirement may not be
13) Recently, a simple proof of the RMB rule has been given in [190]. It is worth pointing out for adaptive detection, more than2 N − , et al. Sci China Inf Sci always satisfied in practice. Taking an example of the STAP filtering for airborne radar, if the numberof antenna elements is 30, the number of pulses is 40, and the system bandwidth is 10 MHz, in order tomeet the requirement of the RMB rule, each filter needs received data within a range of roughly 36 km.The IID assumption usually cannot be guaranteed in such a wide range . A priori information-based method and dimension reduction are two main kinds of approach to alleviatethe requirement of sufficient IID training data.3.5.1
A priori information-based methodsA priori information-based method includes several sub-kinds, namely, Bayesian methods, parametricmethods, special structure-based methods, etc.For Bayesian methods , the noise covariance matrix is ruled by a certain statistical distribution[196], and the distribution parameters can be obtained by using limited training data. In [197] thenoise covariance matrix was assumed to be subject to a given inverse Wishart distribution, and theBayesian one-step GLRT (B1S-GLRT) and Bayesian 2S-GLRT (B2S-GLRT) were proposed. It wasshown by simulated and experiment data that these two Bayesian detectors can provide better detectionperformance than the conventional ones with low sample support. Notice that the B1S–GLRT and B2S–GLRT can be taken as the Bayesian generalizations of the KGLRT and AMF, respectively. The Bayesianversion of the ACE was derived in [198, 199]. The Bayesian method was also adopted in [120, 200] todevise selective detectors with limited training data. Noticeably, the Bayesian method can be used evenno training data are available [201].Parametric (or model-based) method approximates the interference spectrum with a low-order mul-tichannel autoregressive (AR) model [202]. In other words, the noise covariance matrix can be wellcharacterized by using only a few parameters. Hence, this method largely reduces the required trainingdata. At the same time, it also reduces the computational complexity. In [202], the parametric AMF(PAMF) was proposed. The PAMF was shown to be equivalent to the parametric Rao test in [203],where the asymptotic (in the case of large sample case) statistical distribution was also derived. Thecorresponding parametric GLRT was obtained in [204], which was shown to have better detection per-formance than the PAMF. In [205], the nonstationary PAMF (NS-PAMF) and nonstationary normalizedPAMF (NS-PAMF) were proposed for adaptive signal detection in hyperspectral imaging. There aremany other parametric detectors, e.g., [206–221].The “structure” for special structure-based methods is for the noise covariance matrix, which mayhave different kinds of special structures for different antenna configurations or different radar operatingenvironments. The special structures for the noise covariance matrix include low-rank structure, Toeplitz[222], Kronecker [223, 224], persymmetry, spectral symmetry, etc.For the low-rank structure, which is data-dependent, the noise covariance matrix is a sum of a scaledidentity matrix (corresponding to weak thermal noise) and a low-rank matrix (corresponding to strongclutter), with eigenvalues much greater than unity. Then, with limited training data, the principalcomponent approximation of the SCM is usually a better estimation for the noise covariance matrixthan the SCM itself [225]. Under this guideline, many reduced-rank approaches have been developed.Precisely, the reduced-rank versions of the KGLRT, AMF, and ACE were exploited in [226] for theproblem of space-time adaptive detection (STAD) in airborne radar with the data received by multiplesensors under different pulses. There are many other well-known reduced-rank detectors or filters, such asthe principal component analysis (PCA) [225], cross-spectral metric (CSM) [227], multistage Wiener filter(MWF) [228, 229], auxiliary-vector filter (AVF) [230], joint iterative optimization (JIO) [231], conjugategradient (CG)-based AMF (CG-AMF) [232], and some others [239–243]. Moreover, the diagonallyloaded versions of the KGLRT, AMF and ACE were investigated in [12, 244]. Diagonal loading can beoften taken as a kind of reduced-rank method, since it uses the low-rank structure information of thenoise covariance matrix.
14) In other words, in many applications only a few number of data are IID. There are many approaches to choose qualified data,such as reiterative censored fast maximum likelihood (CFML) [191], generalized inner product (GIP) [192], approximate maximumlikelihood (AML) [193], etc.15) The Bayesian methods were also used to model the detection problem in non-homogeneous environment, e.g., [194, 195].16) It is worthy pointing out that the MWF, AVF, and CG are equivalent to each other [233, 234], and they all belong to theKrylov subspace technique, which was originally used in numerical calculation [235] and have been recently successfully used insignal processing [236]. Remarkably, the Krylov subspace technique needs neither matrix inversion nor eigenvalue decomposition(EVD), and it can provide better performance than the EVD-based methods [237, 238].iu W , et al. Sci China Inf Sci Persymmetry is another useful structure for the covariance matrix estimation with low sample support.For the persymmetric covariance matrix, it is persymmetric about its cross diagonal and Hermitian aboutits diagonal. This structure exists when symmetrically spaced linear arrays and/or pulse trains are used.In addition, persymmetry can be found in other situations, e.g., standard rectangular arrays, uniformcylindrical arrays (with an even number of elements), and some standard exagonal arrays [245]. In [246]and [247], the maximum likelihood estimates of persymmetric covariance matrices were provided in theabsence and presence of white noise, respectively. It has been proven in [248, 249] that the exploitationof persymmetry is tantamount to doubling the number of training data in adaptive processing. Fortarget detection by exploiting the persymmetry, references [250–254] considered the case of point targetwith a single observation in HE, while references [255–258] considered the case of distributed target orpoint target with multiple observations/multi-bands in HE. Persymmetric detectors in HE with improvedrejection capabilities were given in [259]. In the PHE, several persymmetric detection algorithms weredesigned in [260–265]. The above references exploiting persymmetry mainly focus on rank-one signaldetection. In contrast, persymmetric detection of subspace signals was considered in [266–271]. Moreover,persymmetry can also be used in non-Gaussian noise [70, 250, 272–275] or multiple-input multiple-output(MIMO) radars [276–279].Spectral symmetry exists in ground clutter, when it is observed by a stationary monostatic radarsystem. Precisely, the ground clutter has a symmetric power spectral density (PSD) centred around thezero-Doppler frequency. This special structure is confirmed by real data in [280, 281]. Other situationswhere spectral symmetry exists were discussed in details in [245]. Exploiting the spectral symmetry,adaptive detectors were proposed for the HE [245, 282] and PHE [283]. Simulation results indicate thatutilizing the spectral symmetry is equivalent to doubling the number of the training data.The above special structures can be combined together to further improve the performance, for instance,Bayesian method plus parametric method [284], parametric method plus persymmetry [285,286], low-rankstructure plus persymmetry [287], persymmetry plus spectral symmetry [288–290].3.5.2
Dimension reduction methods
The method of utilizing a priori information may suffer from significantly performance loss, if the priorinformation greatly departs from the actual one. Another approach to alleviate the requirement ofsufficient IID training data is dimension reduction, which is data-independent. To this end, a reduced-dimension transformation is applied to the test and training data before adaptive processing. This has theeffect of projecting the noise covariance matrix onto a low-dimension subspace. As a result, the requirednumber of IID training data can be considerably reduced, and the computational complexity is reduced aswell. Various reduced-dimension approaches have been proposed, such as auxiliary channel receiver (ACR)[291], extended factor approach (EFA) [292], space-time multiple-beam (STMB) [293], sum-differenceSTAP (Σ∆–STAP) [294], best channel method (BCM) [295], alternating low-rank decomposition (ALRD)[296], among others [297].The aforementioned approaches were proposed for filtering. In contrast, the joint-domain localizedGLR (JDL-GLR) detector was proposed in [298] for airborne radar target detection. The JDL-GLRfirst transforms the test and training data into a reduced-dimension space, and then uses the KGLRTstructure to form the final detector. Another similar reduced-dimension GLRT was proposed in [299].In [52] two reduced-dimension detectors were proposed, which adopted the AMF structure. In [300], areduced-dimension detector was proposed by using subarray processing. Recently, a random matrix-basedreduced-dimension detector was given in [301]. The detector also uses the KGLRT structure. However,the reduced-dimension matrix is chosen in a different manner. Precisely, one column of the reduced-dimension matrix is aligned with signal steering vector, while the other columns are chosen randomly inthe subspace orthogonal to the signal steering vector.In [302, 303] the test and training data were first projected on the one-dimensional signal subspace,resulting in scalar data. Then using the resultant scalar data, two reduced-dimension detectors weredesigned. The above reduced-dimension detectors are mainly for rank-one signals. In contrast, areduced-dimension detector for subspace signal detection was proposed in [304], referred to as subspacetransformation-based detector (STBD). It is shown in [304] that the STBD, which can also serve as afilter, can provide improved detection and filtering performance even in some sample-abundant scenarios,besides the case of limited training data.Besides the
A priori information-based method and dimension reduction method, there may be some iu W , et al. Sci China Inf Sci other technologies to alleviate (or even not need) the requirement of training data. For example, in [305]the authors considered the problem of detecting a multichannel spatial signal in unknown noise withouttraining data. To estimate the unknown noise covariance matrix, a number of echo signals reflected bythe test data were utilized. Table 6
Related Taxonomy in Subsection 3.6Taxonomy MeaningDistributed MIMO radar MIMO radar with widely separate antennas.Colocated MIMO radar MIMO radar with closely spaced antennas.Spatial diversity The transmit antennas are far enough from each other, and hence the target radar cross sectionscan be taken as independent random variables for different transmit-receive paths. With a spatiallydiverse set of “looks”, each set of received data carries independent information about the target.
A MIMO radar adopts multiple elements at both transmit and receive antennas. The transmittedwaveforms are linearly independent or orthogonal [306]. According to the antenna configuration, thereare two basic categories for MIMO radar. One is distributed MIMO radar, whose antennas are far fromeach other [307], while the other is colocated MIMO radar, whose antennas are closely spaced [308].Strictly speaking, the review of MIMO radar target detection can also be carried out from above fiveaspects, or be included in the above five aspects. However, as an emerging research area, MIMO radarhas received considerable attention. Hence, we would like to review MIMO radar target detection in anindependent subsection from the following three aspects: adaptive detection for distributed MIMO radar,adaptive detection for colocated MIMO radar, and adaptive detection for other types of MIMO radar.3.6.1
Adaptive detection for distributed MIMO radar
For the distributed MIMO radar detection, it was shown in [309] that the distributed MIMO radar canprovide better detection performance than traditional phased-array radar in high SNR regions. Thisimprovement is due to the fact that spatial diversity can alleviate the impact of target scintillation, andspatial diversity gain is higher than the coherent processing gain of phased-array radar. Based on theresults in [309], the expressions for the PD of the GLRT was derived in [310] when the target consists ofa finite number of small scatterers. Reference [311] considered the problem of joint target detection andparameter estimation, and it was shown that distributed MIMO radars provide significant improvementover phased-array radars for distributed targets. Reference [312] dealt with the MIMO radar detectionproblem when phase synchronization mismatch arose between the transmit and receive antennas. Thephase error was modelled as the von Mises distribution, and the corresponding GLRT was derived.Polarimetric MIMO radar detection in Gaussian noise was investigated in [313], and it was shown thatoptimal design of the antenna polarizations leads to better detection performance than MIMO radarstransmitting fixed polarized waveforms over all antennas.In the above references, the target’s movement feature was not taken into consideration. When thetransmitted waveform was orthogonal and Doppler processing was adopted, the GLRT and 2S-GLRTwere derived in [314] for moving target detection in Gaussian background, and the expression for thePFA of the GLRT was given in [315]. It was assumed in [276] that the noise covariance matrix had thepersymmetric property, then the GLRT, as well as its statistical property, was derived for distributedMIMO radar which transmitted orthogonal waveforms and adopted Doppler processing. Under the sameantenna configuration, as well as adopting the Doppler processing, the 2S-GLRT was given in [316] forcompound-Gaussian clutter, while the corresponding 2S-Rao and 2S-Wald tests were derived in [317].When the distributed MIMO radar transmitted orthogonal waveforms and adopted Doppler processing,reference [318] derived the GLRT for polarimetric moving target detection in the Gaussian noise. Thedetection problem in [314] was generalized in [319] by assuming that the target velocity was unknown, andit was shown that distributed MIMO radar has better detection performance than the phased-array radarwhen detecting a target with small radial velocities and environment is homogeneous. The distributedMIMO radar detection in non-homogeneous clutter was considered in [320], where the correspondingGLRT was derived and analytically evaluated. It was also shown that the GLRT in [320] has betterdetection performance than the detector in [319], as well as the corresponding phased-array detector. iu W , et al. Sci China Inf Sci Note that in the above references orthogonal waveforms are adopted. Under the assumption of whiteGaussian noise, it is shown in [321] that a detector will suffer from certain detection performance loss ifthe orthogonality property of the waveforms transmitted by different antennas is not satisfied. However,the above result may not suitable for colored noise. Reference [322] derived the GLRT for distributedMIMO radar with arbitrary transmitted waveforms and arbitrary time-correlation of the noise, and itwas shown that there is an inherent trade-off between diversity and integration, and that no uniformlyoptimum waveform design strategy exists. In [323], the GLRT was derived for distributed MIMO radar,with arbitrary transmit waveform and adopting Doppler processing. It was assumed that all transmit-receive pairs share the same known covariance matrix, then the expressions for the PD of the GLRT wasgiven [323], according to which the optimum transmit waveform was given. Reference [324] generalized thedata model in [323] to the case that different transmit-receive pairs have different but known covariancematrices. Then the statistical performance of the corresponding GLRT was given for Swerling I target.The signal model in [323] was also adopted in [325–329], however, the noise was assumed to becompound-Gaussian. Precisely, the 2S-Rao and 2S-Wald tests for distributed MIMO radar were givenin [325], while several Bayesian 2S-GLRTs were derived in [326, 327]. The 2S-Rao and 2S-Wald testsin [325] was generalized to polarimetric distributed MIMO radar detection in [330] for point targets andin [331] for distributed targets. Moreover, reference [328] derived the 2S-GLRT for point target detectionwith polarimetric distributed MIMO radar in compound-Gaussian clutter, and it was generalized in [329]for the case of distributed target detection.3.6.2
Adaptive detection for colocated MIMO radar
For the colocated MIMO radar detection, reference [332] derived the GLRT and its asymptotic statisticaldistribution for colocated MIMO radar after beamforming in white Gaussian noise. Reference [333]proposed three 2S-GLRTs for colocated MIMO radar with randomly distributed arrays in compound-Gaussian clutter, and it was shown that the configuration of randomly distributed arrays achieve detectionperformance improvement at the directions with strong clutter.Remarkably, it was shown in [334] that colocated MIMO radars make it possible that detecting atarget or estimating its parameters does not need training data or even range compression. Without thetraining data, the problem of parameter estimation for colocated MIMO radar was addressed in [335],where the GLRT was derived to suppress the false peak induced by strong jammer. The correspondingRao and Wald tests, as well as their statistical properties, were given in [336]. When signal mismatchoccurs, a tunable MIMO radar detector was proposed in [135], which includes the Rao and Wald testsin [336] as special cases. The proposed tunable detector in [135] has flexibility in controlling the directionproperty, selectivity or robustness, for mismatched signals. Two robust detectors were proposed in [337]for mismatched signals by assuming the actual signal lying in certain subspaces. The GLRT in [335] wasgeneralized in [277] when the persymmetry of noise covariance matrix was exploitation. It was shownby simulation and experimental data that by utilization the persymmetry, the proposed detector in [277]can achieve better detection performance. The correspond persymmetric Rao test Wald test were givenin [279] and [278], and in [279] a two-stage detector was also given for mismatched signal detection bycascading the above persymmetric Rao and Wald tests.More recently, for a colocated MIMO radar, a robust Wald-type test were proposed in [338]. Perfor-mance analysis showed that there always exists a sufficient number of (virtual) antennas such that therequired performance are satisfied, without prior knowledge of the noise statistical property. This typeof MIMO radar was referred to as the massive MIMO radar therein. Moreover, in [339] three adaptiveGLRTs were proposed for colocated MIMO radar equipped with frequency diverse array (FDA).3.6.3
Adaptive detection for other types of MIMO radar
There are several types of variations of the distributed MIMO radar and colocated MIMO radar, such asphased MIMO (Phased-MIMO) radar [340], hybrid MIMO phased array radar (HMPAR) [341], transmitsubaperturing MIMO (TS-MIMO) radar [342], and multi-site radar system MIMO (MSRS-MIMO) [343].The MSRS-MIMO radar has multiple widely separate sub-arrays, and each sub-arrays has multiple colo-cated antennas. According to the waveforms, the MSRS-MIMO radar can be classified as two kinds.One is that the waveforms are different or orthogonal in different transmit antennas [344]. The otheris that the waveforms transmitted by the antennas are scaled versions of a single waveform [345]. For iu W , et al. Sci China Inf Sci convenience, the first type of MSRS-MIMO radar is referred to as the distributed-colocated MIMO radar,while the latter one is referred to as the distributed-phased MIMO radar .When the waveform are orthogonal, the GLRT for distributed-colocated MIMO radar was obtained innon-Gaussian environment in [2], and the expression for the PFA was given therein under the constraintthat the product of the number of transmit elements and receive elements is the same for each pair oftransmit-receive sub-array. The results for the PFA in [2] was generalized in [348] by eliminating theabove constraint. For distributed-colocated MIMO radar with non-orthogonal waveform, the GLRT inGaussian noise was derived in [349], while the two-step Rao and Wald tests were given in [350]. Innon-Gaussian background, the 2S-GLRT, Rao test, and Wald test were exploited in [351] for distributed-colocated MIMO radar with non-orthogonal waveform. Moreover, reference [352] considered the problemof detecting a mismatched signal in distributed-phased MIMO radar, and proposed three selective detec-tors.Before closing this section, we summarize important progress in Table I. Table 7
Important Progress in Multichannel Adaptive Signal DetectionYear Important Progress Author(s) Ref.1986 first paper on adaptive detection Kelly [4]1989 solid mathematical background foradaptive signal detection Kelly and Forsythe [100]19911992 well-known detector for pointtargets: AMF Chen, Fobey, et al. [43][44]1992 tunable detector for mismatchedsignals Kalson [127]1992 persymmetry structure based detect-or with limited training data Cai and Wang [256]1995/1999 well-known detector for pointtargets: ACE Conte, Lops,Kraut, Scharf, et al. [61][55]1996 subspace-based signal detection Raghavan, Pulsone, et al. [79]1996 direction detection Bose and Steinhardt [96]1997 distributed target detection Gerlach, Steiner, et al. [353]2000 two-stage detector for mismatchedsignals Pulsone and Zatman [136]2000 low-rank structure based detectorwith limited training data Ayoub andHaimovich [299]2000/2006 parametric detector withlimited training data Roman, Rangaswamy,Li, Michels, et al. [202][205]2001 selective detector for mismatchedsignals Pulsone and Rader [118]2003 adatpive detectors based on Raoand Wald tests Conte and De Maio [272]2004 multiple target detection Gini, Bordoni, et al. [354]2005 adaptive detection based onconvex optimization De Maio [112]2007 adaptive detection in subspaceinterference Bandiera, De Maio, et al. [153]2007 Bayesian detector in heterogeneousenvironment Besson, Tourneret, et al. [194]2008 MIMO radar detection in unknownnoise Xu, Li, et al. [335]2010 spectral symmetry based detectorwith limited training data De Maio, Orlando, et al. [245]2014 Rao and Wald tests for complex-valued signals with circularlysymmetric random parameters Liu, Wang, et al. [29]2014 double subspace signal detection Liu, Xie, et al. [82, 101]2016 Rao test for complex-valued signalswith circularly or non-circularlysymmetric random parameters Kay and Zhu [30]2016 adaptive detection based oncovariance structure classification Carotenuto, De Maio, et al. [355]17) It is worth pointing out that the data model of the distributed-phased MIMO radar is the same as the conventionaldistributed MIMO radar which adopts coherent pulse processing in each sub-array, such as [320, 346, 347].iu W , et al. Sci China Inf Sci Multichannel adaptive signal detection was first investigated for a point target in 1986 by Kelly [4].Based on Kelly’s work, all kinds of problems were dealt with, and numerous detectors were proposed. Inthis section, we first summarize the statistical properties of many well-known detectors for point targets,since the statistical properties are the primary tool to evaluate the detection performance of the detectors.Then, we generalize the case of point target detection to distributed target detection and signal detectionin the presence of interference . Note that subspace signal model is more general than the rank-one signal model adopted in (20). It ispointed out in [146] that the matched subspace detector is the general building block of signal processing,and it contains the rank-one matched filter or detector as a special case. Hence, in this subsection thedetectors for point targets are all based on subspace signal model.For the detection problem in (1), if the signal s lies in a known subspace spanned by an N × p full-column-rank matrix H , then we have s = H θ , with θ being a p × x is the same as that in the training data x e ,l . Then,for the detection problem in (1) with s being replaced by H θ , the GLRT [84], Rao test [82], and Waldtest [82] are t SGLRT = ˜ x H P ˜ H ˜ x x H ˜ x − ˜ x H P ˜ H ˜ x , (35) t SRao = ˜ x H P ˜ H ˜ x (1 + ˜ x H ˜ x )(1 + ˜ x H ˜ x − ˜ x H P ˜ H ˜ x ) , (36)and t SAMF = ˜ x H P ˜ H ˜ x , (37)respectively, where ˜ H = S − H and P ˜ H = ˜ H ( ˜ H H ˜ H ) − ˜ H H . The detectors in (35)-(37) are referred asthe SGLRT, SRao, and SAMF, respectively.In the PHE, the noise covariance matrices in the test and training data can be modified as R e = σ R and R , respectively, with σ being an unknown positive scaling factor, standing for the power mismatchbetween the test and training data. In the PHE, the GLRT, Rao test, and Wald test coincide with eachother, and are found to be [101] t ASD = ˜ x H P ˜ H ˜ x ˜ x H ˜ x , (38)which is named as ASD in [83]. Note that the SGLRT, SRao, SAMF, and ASD are the subspacegeneralizations of the KGLRT, DMRao, AMF, and ACE, respectively.The above four detectors are designed without taking into account the possibility of signal mismatch.On the one hand, signal mismatch may be caused by antenna error, mutual coupling, or target maneu-vering. On the other hand, signal mismatch can also be caused by a strong target or a jamming signallocated in the radar sidelobe, generated by the ECM. For different sources of signal mismatch, differentdirectivity properties (the capability of selectivity or robustness to signal mismatch) of the detector arepreferred. For the first case, a robust detector is needed, which maintains good detection performance inthe presence of signal mismatch. In contrast, for the second case, a selective detector is preferred, whosedetection performance decreases rapidly with the increase of signal mismatch.To design selective detectors in the case of signal mismatch, an effective approach is adding artificiallydeterminant factitious jammer under hypothesis H [118]. Then, the detection problem in (1) can bemodified to be ( H : x = n + q , x e ,l = n e ,l , l = 1 , , · · · , L, H : x = H θ + n , x e ,l = n e ,l , l = 1 , , · · · , L, (39)where the N × q denotes the artificially injected determinant factitious jammer. Re-markably, the injection of the factitious jammer q makes the resulting detector tend to choose hypothesis
18) We choose the above three cases because they are representative in the field of adaptive detection and extensively studiedin the literature.iu W , et al. Sci China Inf Sci H if signal mismatch occurs. When q is constrained to be orthogonal to the signal subspace in thequasi-whitened space, i.e., H H S − q = p × , (40)the GLRT for the detection problem in (39) is t SABORT = 1 + ˜ x H P ˜ H ˜ x x H ˜ x − ˜ x H P ˜ H ˜ x , (41)which is a special case of the adaptive direction detector with mismatched signal rejection of type 2(ADD-MSR2) in [123], and a subspace generalization of the ABORT, proposed in [118]. Hence, forconvenience, the detector in (41) is referred to as the subspace-based ABORT (SABORT).If the determinant factitious jammer in (40) is modified as H H R − q = p × . (42)That is to say, the factitious jammer is orthogonal to the signal subspace in the truely whitened space.Then the GLRT for the detection problem in (39) becomes t W-SABORT = 1 + ˜ x H ˜ x (1 + ˜ x H ˜ x − ˜ x H P ˜ H ˜ x ) , (43)which is a special case of the adaptive direction detector with mismatched signal rejection of type 1(ADD-MSR1) in [123], and a subspace generalization of the W-ABORT in [119]. The detector in (43) isdenoted as the whitened SABORT (W-SABORT) for convenience.Moreover, another approach to devise a selective detector is injecting an unknown, rank-one, noise-like,fictitious jammer v under both hypotheses. As a results, the noise covariance matrix in the test databecomes R = R e + vv H . Then, the selective detector derived according to the Rao test is t DN-SAMF = ˜ x H P ˜ H ˜ x ˜ x H ˜ x (1 + ˜ x H ˜ x − ˜ x H P ˜ H ˜ x ) , (44)which is a special case of the Rao test in [179] and a subspace generalization of the DN-AMF in [126].For convenience, the detector in (44) is denoted as the doubly normalized SAMF (DN-SAMF).Different from the above devised selective detectors, a robust detector to the signal mismatch may bepreferred in many applications. A robust detector can be designed by assuming the desired signal tobe detected is completely unknown. In other words, the signal s in (1) is unknown, or equivalently, thedimension of the signal matrix H is N × N . Then the corresponding GLRT is given by [131] t AED = ˜ x H ˜ x , (45)which can be denoted as AED. It is shown in [82] that the Rao test and Wald test are both equivalentto the GLRT, i.e., the AED in (45).Since the case of signal match can be taken as a special case of signal mismatch (i.e., the mismatchedangle is zero), we only summarize the statistical properties of the above detectors in the presence ofsignal mismatch. As mentioned above, the statical performance of the detectors in the presence of signalmismatch was first dealt with by Kelly in [105] for the KGLRT in the case of rank-one signal. Basedon this result, the statistical performance of the SGLRT, SAMF, and ASD was given in [107]. In thefollowing, we summarize the statical properties of the above eight detectors, some of which were notfound in the open literature.To obtain the statical distributions of the detectors, it is convenient to introduce the following quantity β = 11 + ˜ x H ˜ x − ˜ x H P ˜ H ˜ x , (46)which can be taken as a loss factor.If signal mismatch happens, the actual signal, denoted as s , may not completely lie in the signalsubspace spanned by the columns of H . Then, it is shown in [107] that the statistical distribution of iu W , et al. Sci China Inf Sci the SGLRT in (35), with β given, under hypothesis H is complex noncentral F-distribution, with p and L − N + 1 DOF and a noncentrality parameter βρ cos φ , written symbolically as t SGLRT | [ β, H ] ∼ CF p,L − N +1 (cid:0) βρ pnt cos φ (cid:1) , (47)where ρ pnt is the output SNR, defined as ρ pnt = s H R − s , (48)cos φ = s H R − H ( H H R − H ) − H H R − s s H R − s , (49)and the notation “ | [ β, H ]” denotes the fact that the above statistical distribution holds under hypothesisH on the condition that β is given. Equation (49) can be rewritten ascos φ = ¯ s H P ¯ H ¯ s ¯ s H ¯ s , (50)where s = R − s , ¯ H = R − H , P ¯ H = ¯ H ( ¯ H H ¯ H ) − ¯ H H . It follows from (50) that the quantity cos φ measures cosine-squared of the angle between the whitened actual signal ¯ s and the whitened nomi-nal signal subspace spanned by the columns of ¯ H . cos φ plays a key role in controlling the detectionperformance of a detector in the presence of signal mismatch. This is numerically shown in the nextsection.Moreover, it is shown in [107] that the statistical distribution of the loss factor β in (46) under hypoth-esis H is a complex noncentral Beta distribution, with L − N + p + 1 and N − p DOFs and a noncentralityparameter δ , written symbolically as β | H ∼ CB L − N + p +1 ,N − p ( δ ) , (51)where δ = ρ pnt sin φ, (52)and sin φ = 1 − cos φ .In contrast, under hypothesis H , the statistical distributions of the SGLRT in (35) and the loss factor β in (46) become t SGLRT | [ β, H ] ∼ CF p,L − N +1 , (53)and β | H ∼ CB L − N + p +1 ,N − p , (54)respectively.The analytical expressions for the PDF and cumulative distribution function (CDF) of the complexnoncentral F-distribution and complex noncentral Bea distribution were exploited in detail in Kelly andForsythe’s classic report [100], also summarized in [106, 141]. One can use these CDFs and PDFs toderive the expressions for the PDs and PFAs of the above detectors.It is straightforward to verify that the following seven equations hold t SAMF = t SGLRT β , (55) t ASD = t SGLRT − β , (56) t SRao = βt SGLRT t SGLRT , (57) t SABORT = β + t SGLRT , (58) t W–SABORT = (1 + t SGLRT ) β, (59) t DN–SAMF = βt SGLRT (1 − β )(1 − β + t SGLRT ) , (60) iu W , et al. Sci China Inf Sci t AED = 1 − β + t SGLRT β . (61)Based on the conditional distribution of the SGLRT in (47) and the statistical distribution of the lossfactor β in (51), along with the statistical dependences in (55)-(61), one can readily obtain analyticalexpressions for the PDs and PFAs of the detectors. Interested readers can refer to [107] for examples.Note that one can obtain the expressions for the PD and PFA of the AED in a more direct manner byderiving the statistical distribution of the AED [356]. Precisely, according to Theorem 3.2.13 in [357, p.98]or Theorem 5.2.2 in [358, p.176], the statistical distribution of the AED in (45) under hypotheses H andH are t AED | H ∼ CF N,L − N +1 ( ρ pnt ) (62)and t AED | H ∼ CF N,L − N +1 , (63)respectively.In order to evaluate the detection performance of the detectors under different numbers of trainingdata, we consider the detector with known noise covariance matrix. Precisely, when R is known, theGLRT for the detection problem in (1) with s being replaced by H θ is t SMF = x H R − H ( H H R − H ) − H H R − x , (64)which is referred to as the subspace-based matched filter (SMF). It can also be obtained by the criteriaof GLRT, Rao and Wald tests. The statistical distribution of the SMF in (64) under hypothesis H is acomplex noncentral Chi-square distribution with p DOFs and a noncentrality parameter ρ [87], writtensymbolically as t SMF | H ∼ C χ p ( ρ pnt ) . (65)Under hypothesis H , the above distribution becomes central, i.e., t SMF | H ∼ C χ p (66) In this subsection we compare the detection performance of the detectors with numerical examples, andonly focus on the case of HE. Two cases are considered, namely, the case of no signal mismatch and thecase of signal mismatch. The PD curves of all detectors are obtained by using the theoretical results, andconfirmed by Monte Carlo simulations, which are not shown for a clear display.Figure 3 compares the detection performance of the adaptive detectors under different SNRs in theabsence of signal mismatch. For comparison purpose, the result for the SMF is also reported. The resultsindicate that, for the chosen parameters, the SGLRT, among the eight adaptive detectors, has the highestPD and slightly better than the SAMF and SABORT, the DN-SAMF has the lowest PD, and the PDs ofthe ASD, W-SABORT, SRao, and AED are in between. Moreover, the detection performance loss of theSGLRT in terms of SNR is roughly 4 dB when PD = 0 .
9, compared with the SMF. This is quite differentfrom adaptive filtering, since it is well-known from the RMB rule [42] that 2 N independent identicallydistributed (IID) training data can maintain 3 dB SNR loss, compared with the optimum filter. Theabove detection loss is owing to two factors [4]. One is the effective SNR loss factor (similar to adaptivefiltering), and the other is the CFAR loss of the adaptive detectors. The effective SNR loss factor dependsroughly on the ratio of L to N , while the CFAR loss depends solely on L , whose increase results in thedecrease of the CFAR loss.Figure 4 shows the detection performance of the adaptive detectors under different amount of signalmismatch. As expected, the AED is the most robust and its PD does not vary with the change of cos φ .However, its PD cannot attain unity for the chosen parameters. The robustness of the SAMF, SGLRT,SABORT, ASD, W-SABORT, SRao, and DN-SAMF reduces in sequence.Another method to illustrate the detection performance for mismatched signals is showing the contoursof PDs as functions of SNR and cos φ , first introduced in [118] and named as mesa plot. This is displayedin Figure 5 for the above detectors. The directivities of the detectors are the same as those in Figure4. However, more information can be inferred from Figure 5. Taking the SAMF for example, it is veryrobust to signal mismatch. It can provide a PD as high as 0.9 as long as the SNR is high enough, even the iu W , et al. Sci China Inf Sci (cid:1008)(cid:3)(cid:282)(cid:17) Figure 3
PD versus SNR. N = 12, p = 2, L = 2 N , and PFA = 10 − . whitened actual signal is orthogonal to the whitened nominal signal subspace, i.e., the case of cos φ = 0.In contrast, for a selective detector, such as the SABORT, it does not achieve a PD higher than 0.5when cos φ < .
55, no matter how high the SNR is. It is worth pointing out for the chosen parameters,the SAMF and SABORT have comparable PDs for matched signals as shown in Figure 3. Hence, if aselective detector is needed, the SABORT is a better candidate than the SAMF.Before closing this section, we would like to give the following three remarks. First, it is knownfrom (53), (54), and (55)-(61) that all the adaptive detectors exploited above have the CFAR propertywith respect to the noise covariance matrix R . Second, only the ASD, among the above eight adaptivedetectors, possesses the CFAR property in PHE, although the ASD has lower PD than some otherdetectors in HE. Third, the DN-SAMF can behave quite well when the number of system dimension N is large enough, as shown in [126]. The detection problem in (1) has been generalized in many aspects, and hence, many other adaptivedetectors have been proposed besides the ones shown in the above subsection. Distributed target detection(without interference) and signal detection in interference are two import generations, which will be shownbelow.4.3.1
Adaptive detectors for distributed targets
A large target usually occupies multiple range bins, especially for high-resolution radar system [359]. Inthis case, the detection problem in (1) should be modified as ( H : X = N , x e ,l = n e ,l , l = 1 , , · · · , L, H : X = sa H + N , x e ,l = n e ,l , l = 1 , , · · · , L, (67)where X is an N × K matrix denoting the test data, with N being the number of system channels and K being the number of range bins occupied by the distributed target, N is the noise in the test data, s is the signal steering vector, a is the coordinate vector of the signal, x e ,l is the l th training data vector,and n e ,l is the noise in x e ,l . The columns of N are IID, having the noise covariance matrix R t . Denotethe noise covariance matrix of n e ,l as R . Then, in HE, R t = R , while in PHE R t = σ R , with σ beingthe unknown power mismatch between the test data and training data. iu W , et al. Sci China Inf Sci cos φ P r obab ili t y o f D e t e c t i on SGLRTSAMFSRaoASDSABORTW-SABORTDN-SAMFAED
Figure 4
PD versus cos φ . N = 12, p = 2, L = 2 N , SNR = 18 dB, and PFA = 10 − . For the detection problem in (67), the GLRT and its two-step variation for the HE and PHE were allproposed in [53]. Precisely, for the HE, the GLRT and 2S-GLRT are t GKGLRT = ˜ s H ˜ X ( I K + ˜ X H ˜ X ) − ˜ X H ˜ s ˜ s H ˜ s − ˜ s H ˜ X ( I K + ˜ X H ˜ X ) − ˜ X H ˜ s (68)and t GAMF = ˜ s H ˜ X ˜ X H ˜ s ˜ s H ˜ s , (69)respectively. Moreover, for the PHE, the GLRT and 2S-GLRT are t GLRT–PHE = (ˆ σ ) NKL + K (cid:12)(cid:12)(cid:12)(cid:12) I K + 1ˆ σ ˜ X H ˜ X (cid:12)(cid:12)(cid:12)(cid:12) (ˆ σ ) NKL + K (cid:12)(cid:12)(cid:12)(cid:12) I K + 1ˆ σ ˜ X H P ⊥ ˜ s ˜ X (cid:12)(cid:12)(cid:12)(cid:12) (70)and t GASD = ˜ s H ˜ X ˜ X H ˜ s ˜ s H ˜ s tr( ˜ X H ˜ X ) , (71)respectively. In (70), ˆ σ and ˆ σ are the sole solutions of r X k =1 λ k λ k + σ = N KL + K (72)and r X k =1 ξ k ξ k + σ = N KL + K , (73)respectively, where r = min( N, K ), r = min( N − , K ), λ k is the k th non-zero eigenvalue of ˜ X H ˜ X , k = 1 , , · · · , r , and ξ k is the k th non-zero eigenvalue of ˜ X H P ⊥ ˜ s ˜ X , k = 1 , , · · · , r .The detectors in (69) and (71) were referred to as generalized AMF (GAMF) and generalized adaptivesubspace detector (GASD), respectively in [53]. For convenience, the detector in (68) is denoted asGKGLRT in this paper. iu W , et al. Sci China Inf Sci . . . . . . . SNR (dB) c o s φ . . . . . SNR (dB) c o s φ (a) SMF (b) SGLRT . . . . . . SNR (dB) c o s φ . . . . SNR (dB) c o s φ (c) SAMF (d) SRao . . . . SNR (dB) c o s φ . . . . SNR (dB) c o s φ (e) ASD (f) SABORT . . . . SNR (dB) c o s φ . . . SNR (dB) c o s φ (g) W-SABORT (h) DN-SAMF . . . . . . SNR (dB) c o s φ (i) AED Figure 5
Contours of the PDs versus SNR and cos φ . N = 12, p = 2, L = 2 N , and PFA = 10 − .iu W , et al. Sci China Inf Sci Moreover, for the detection problem in (67) in HE, the Wald test is the same as the GAMF, while theRao test was proposed in [89], described as t Rao-HE = s H ( S + XX H ) − XX H ( S + XX H ) − ss H ( S + XX H ) − s . (74)For the detection problem in (67) in PHE, the Rao test and Wald test were proposed in [90], given by t Rao–PHE = 1ˆ σ tr h X H ˆ R − H ( H H ˆ R − H ) − H H ˆ R − X i (75)and t Wald–PHE = 1ˆ σ tr h X H ˆ R − H ( H H ˆ R − H ) − H H ˆ R − X i , (76)respectively, where ˆ σ and ˆ σ are the sole solutions of (72) and (73), respectively,ˆ R = 1 L + K (cid:18) S + 1ˆ σ XX H (cid:19) (77)and ˆ R = 1 L + K S (cid:18) I N + 1ˆ σ P ⊥ ˜ s ˜ X ˜ X H P ⊥ ˜ s (cid:19) S . (78)Different from the case of point target, it is more difficult to derive the statistical performance for thedistributed-target-based detectors. At presence, only the statistical performance of the GKGLRT andGAMF is known. The statistical distribution of the GKGLRT was first proposed in [91] for the case of nosignal mismatch, and then generalized to the case of signal mismatch in [103]. The statistical distributionof the GAMF was given in [102]. Precisely, under hypothesis H , the conditional distribution of theGKGLRT in (68) is t GKGLRT | H ∼ CF K,L − N +1 (cid:0) ρ dstr cos φ rk1 β GKGLRT (cid:1) , (79)where ρ dstr = a H a · s H R − s , (80)can be taken as the output SNR, with s being the actual signal steering vector,cos φ rk1 = | s H R − s | s H R − s s H R − s (81)is generalized cosine-squared between the actual signal s and the nominal signal s in the whitened space,and β GKGLRT is a loss factor for the GKGLRT, haveing the statistical distribution β GKGLRT | H ∼ CB L + K − N +1 ,N − ( ρ sin φ rk1 ) , (82)with sin φ rk1 = 1 − cos φ rk1 . Under hypothesis H , equations (79) and (82) become t GKGLRT | H ∼ CF K,L − N +1 (83)and β GKGLRT | H ∼ CB L + K − N +1 ,N − , (84)respectively. Moreover, under hypothesis H the conditional distribution of the GAMF in (69) is β GAMF t GAMF | H ∼ CF K,L − N +1 ( β GAMF ρ dstr ) , (85)where β GAMF is a loss factor for the GAMF, with the statistical distribution β GAMF | [H and H ] ∼ CB L − N +2 ,N − . (86)
19) Using matrix inversion lemma, it is easy to show that equation (74) can be recast as t Rao-HE = ˜ s H ˜ X ( I K + ˜ X H P ⊥ ˜ s ˜ X ) − I K + ˜ X H ˜ X ) − X H ˜ s ˜ s H ˜ s .iu W , et al. Sci China Inf Sci Under hypothesis H , equation (85) turns to be β GAMF t GAMF | H ∼ CF K,L − N +1 . (87)There are two kinds of further generalizations of the detection problem in (67). One is that the signalsteering vector s lies in a given subspace spanned by an N × p full-column matrix H . Hence, s can beexpressed as s = H θ , with θ being p × ( H : X = N , x e ,l = n e ,l , l = 1 , , · · · , L, H : X = H θ a H + N , x e ,l = n e ,l , l = 1 , , · · · , L, (88)The GLRT and 2S-GLRT in HE were proposed in [96], described as t GLRDD = λ max h ˜ X H P ˜ H ˜ X ( I K + ˜ X H ˜ X ) − i (89)and t AMDD = λ max (cid:16) ˜ X H P ˜ H ˜ X (cid:17) , (90)respectively, where λ max ( · ) denotes the maximum eigenvalue of the matrix argument. It was shown in [97]that there is no reasonable Rao test for the detection problem in (88), the 2S-Wald test is the same asthe detector in (90), and the Wald test is given by t SNRDD = θ H max ˜ H H ˜ X ˜ X H ˜ H θ max θ H max ˜ H H ˜ H θ max , (91)where θ max is a principal eigenvector (the eigenvector corresponding to the maximum eigenvalue) of thematrix ( ˜ H H ˜ H ) − ˜ H H ˜ X ( I K + ˜ X H ˜ X ) − ˜ X H ˜ H . The detectors in (89), (90), and (91) are referred to as GLR-based direction detector (GLRDD), adaptive matched direction detector (AMDD), SNR-based directiondetector (SNRDD) in [97]. The 2S-GLRT for the detection problem in (88) in PHE was proposed in [95],given by t GADD = λ max (cid:16) ˜ X H P ˜ H ˜ X (cid:17) tr (cid:16) ˜ X H ˜ X (cid:17) , (92)which was denoted as GADD therein.It follows from (89)-(92) that the detectors choose a direction among the subspace spanned by thecolumns of H . In other words, the detection problem in (88) tantamount to finding a direction with thelargest possibility in a gvien subspace, and hence it is called direction detection in [95].The problem of direction detection can be further generalized when both the column component androw component of the signal to be detected lie in given subspaces. To be precise, the test data underhypothesis H becomes X = H θα H C + N , with C a given M × K full-row-rank matrix and α an M × X hasa slightly different steering vector in the sense that these steering vectors are different but all come fromthe same subspace. Hence, the detection model in (67) can be modified as ( H : X = N , x e ,l = n e ,l , l = 1 , , · · · , L, H : X = HΦ + N , x e ,l = n e ,l , l = 1 , , · · · , L, (93)where H is an N × p full-column-rank matrix, and Φ is a p × K matrix standing for the coordinates. TheGLRT in HE was proposed in Kelly and Forsythe’s classic report [100], while the Rao tes and Wald testin HE can be obtained according to the results in [82]. Precisely, the GLRT, Rao test, and Wald test aregiven by t GLRT = (cid:12)(cid:12) I K + X H S − X (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) I K + X H S − X − X H S − H ( H H S − H ) − H H S − X (cid:12)(cid:12)(cid:12) , (94)
20) The GADD in (92) can be also derived according to 2S-Wald test.iu W , et al. Sci China Inf Sci t Rao = tr (cid:26) X H ( S + XX H ) − H h H H ( S + XX H ) − H i − H H ( S + XX H ) − X (cid:27) , (95)and t Wald = tr h X H S − H ( H H S − H ) − H H S − X i , (96)respectively.Note that when p = N in the detection problem in (93), the signal steering vectors lie in the wholeobservation space. Or, equivalently, the steering vectors are completely unknown. The correspond GLRT,2S-GLRT, and a modified 2S-GLRT (M2S-GLRT) in HE were proposed in [92]. It was also shown in [93]that the M2S-GLRT is essentially the corresponding Rao test, while the 2S-GLRT can also be derivedaccording to the Wald test. Moreover, in [82, 101] the test data in (93) were generalized to the case X = HΦC + N , with Φ being a p × M unknown matrix, C being an M × K known full-row-rank matrix.The corresponding signal model was called double subspace (DOS) model in [82, 101], where manyadaptive detectors were proposed.4.3.2 Adaptive detectors in the presence of interference
Most of the above detectors are designed without taking into account the possibility of interference,which usually exists in practice. Interference can be caused intentionally (jamming due to the ECM)or unintentionally (communication signals or radar signals transmitted by other radar systems). In thiscase, the detection problem in (1) can be modified as ( H : x = n , x e ,l = n e ,l , l = 1 , , · · · , L, H : x = s + j + n , x e ,l = n e ,l , l = 1 , , · · · , L, (97)where j stands for the interference. Roughly speaking, there are two main kinds of interference. One iscoherent interference, while the other is noise interference. For the former, it works like a real target,which usually lies in a certain spatially direction and/or occupies a Doppler bin. Hence, the coherentinterference can be modelled by a subspace model. For the latter, it works like thermal noise or clutter.As a result, the noise interference changes the noise covariance matrix of the test data.Based on the above analysis, the coherent interference can be modelled as j = J φ , where the N × q full-column-rank matrix spans the subspace where the interference lies, and the q × φ denotesthe unknown coordinates. For coherent interference and subspace signals (i.e., the signal in (97) can beexpressed as s = H θ ), the GLRT and 2S-GLRT in HE and PHE for the detection problem in (97) were allproposed in [153], and the GLRT and 2S-GLRT in PHE coincide with each other. Precisely, the GLRTand 2S-GLRT in HE are t GLRT-HE-I = ˜ x H P P ⊥ ˜ J ˜ H ˜ x x H P ⊥ ˜ J ˜ x − ˜ x H P P ⊥ ˜ J ˜ H ˜ x (98)and t = ˜ x H P P ⊥ ˜ J ˜ H ˜ x , (99)respectively, while the GLRT in PHE is t GLRT-PHE-I = ˜ x H P P ⊥ ˜ J ˜ H ˜ x ˜ x H P ⊥ ˜ J ˜ x , (100)where P P ⊥ ˜ J ˜ H = P ⊥ ˜ J ˜ H ( ˜ H H P ⊥ ˜ J ˜ H ) − ˜ H H P ⊥ ˜ J , P ⊥ ˜ J = I N − P ˜ J , and P ˜ J = ˜ J (˜ J H ˜ J ) − ˜ J H . For convenience, thedetectors in (98), (99), and (100) are referred to the GLRT in HE with interference rejection (GLRT-HE-I), 2S-GLRT in HE with interference rejection (2S-GLRT-HE-I), and GLRT in PHE with interferencerejection (GLRT-PHE-I), respectively.
21) The DOS signal model was first introduced in [100]. However, it was assumed in [100] that no training data were available.Instead, it was assumed K > M + N . This constraint ensures the existence of a set of virtual training data, generated by a certainunitary matrix to the test data.iu W , et al. Sci China Inf Sci For coherent interference, the Rao test and 2S-Rao test were proposed in [155], while the Wald testand 2S-Wald test were derived in [156]. Precisely, in HE the Rao test and 2S-Rao test are t Rao-HE-I = ˜ x H P ⊥ ˜ J P ˜ H P ⊥ ˜ J ˜ x (1 + ˜ x H P ⊥ ˜ J ˜ x )(1 + ˜ x H P ⊥ ˜ J P ⊥ ˜ H P ⊥ ˜ J ˜ x ) (101)and t = ˜ x H P ⊥ ˜ J P ˜ H P ⊥ ˜ J ˜ x , (102)respectively, while in PHE the Rao test is the same as the 2S-Rao test, given by t Rao-PHE-I = ˜ x H P ⊥ ˜ J P ˜ H P ⊥ ˜ J ˜ x ˜ x H P ⊥ ˜ J ˜ x . (103)The Wald test is the same as the 2S-Wald test both in HE and PHE, given by t Wald-HE-I = ˜ x H P H ˜ H | ˜ J P ˜ H | ˜ J ˜ x (104)and t Wald-PHE-I = ˜ x H P H ˜ H | ˜ J P ˜ H | ˜ J ˜ x ˜ x H P ⊥ ˜ B ˜ x , (105)respectively, where P ˜ H | ˜ J = ˜ H ( ˜H H P ⊥ ˜ J ˜ H ) − ˜ H H P ⊥ ˜J is the oblique projection matrix onto the subspacespanned by ˜ H along the subspace spanned by ˜ J . Detailed analysis and comparison of the above detectorscan be found in [156].At present, only the GLRT-HE-I, 2S-GLRT-HE-I, and GLRT-PHE-I have known statistical properties,given in [159]. Precisely, the conditional distribution of the GLRT-HE-I in (98) with a fixed β I underhypothesis H , is t GLRT–HE–I | [ β I , H ] ∼ CF p,L − N + q +1 ( ρ eff β I ) , (106)where ρ eff = ¯ s H P ⊥ ¯ J P P ⊥ ˜ J ˜ H P ⊥ ¯ J ¯ s (107)is defined as the effective SNR (eSNR), and β I is loss factor defined as β I = 11 + ˜ x H P ⊥ ˜ J ˜ x − ˜ x H P P ⊥ ˜ J ˜ H ˜ x . (108)The statistical distribution of β I under hypothesis H is β I | H ∼ CB L − N + p + q +1 ,N − p − q ( δ ) , (109)where δ = ¯ s H P ⊥ ¯ J P ⊥ P ⊥ ¯ J ¯ H P ⊥ ¯ J ¯ s , (110)with P ⊥ P ⊥ ˜ J ˜ H = I N − P P ⊥ ˜ J ˜ H . Under hypothesis H , (106) and (109) reduce to t GLRT–HE–I | [ β I , H ] ∼ CF p,L − N + q +1 (111)and β I | H ∼ CB L − N + p + q +1 ,N − p − q , (112)respectively.More geometric interpretation about the eSNR in (107) can be found in [159]. Moreover, the followingtwo equations can be easily verified t = t GLRT–HE–I β I , (113) t GLRT–PHE–I = t GLRT–HE–I − β I . (114) iu W , et al. Sci China Inf Sci Using (113) and (114), along with (106), (109), (111) and (112), we can obtain the analytical expressionsfor the PDs and PFAs of the 2S-GLRT-HE-I and GLRT-PHE-I.For completely unknown noise interference, it was shown in [178] that the GLRT for rank-one signalsis equivalent to the ACE. In The corresponding Rao test was derived in [126], i.e., the DN-AMF, origi-nally adopted for mismatched signal detection. The results in [126, 178] were generalized in [179] whenadditional coherent interference existed. In [180] the noise interference was assumed to be orthogonal tothe signal of interest in the whitened space, and it was shown that the GLRT coincides with the KGLRT.Moreover, it was shown in [181] that the corresponding Rao and Wald tests are the same as the DMRaoand AMF, respectively. The results in [180,181] were generalized in [182] for the case of subspace signals.Some other generalizations for noise interference can be found in [183, 185–189].
In this paper, we investigated the detector design criteria for adaptive detection, analysed the relationshipbetween adaptive detection and the filtering-then-CFAR detection approach, as well as the relationshipbetween adaptive detectors and adaptive filters, gave a comprehensive review, summarized and comparedtypical adaptive detectors. Adaptive detection jointly uses the test and training data to form an adap-tive detector. Compared with the filtering-then-CFAR detection approach (adaptive or non-adaptive),adaptive detection has many distinct features. It achieves the function of filtering and CFAR processingsimultaneously, and hence, it has simple detection procedure. Moreover, it can provide better detectionperformance.We hope that this paper will stimulate new researches on adaptive detection. Some possible furtherresearch tracks are listed below. 1) The statistical performance of many adaptive detectors are neededto be studies, such as the Rao and Wald tests in subspace interference [155, 156], the 2S-GLRT in HEin the presence of signal mismatch [53]. Obtaining these results can reveal how the signal mismatchand/or interference affect the detection performance. 2) Nowadays, multichannel signal detection hasbeen combined with compressive sensing or sparse representation, which is an emerging signal processingtechnique for efficiently acquiring and reconstructing a compressible signal, by using much fewer samples.Several compressive sensing-based detectors were proposed, such as [361–367] and the references therein.However, most proposed detectors based on compressive sensing are for known noise or white Gaussiannoise with unknown variance. Much more challenging task is for colored noise with unknown covariancematrix. 3) Most existing adaptive detectors were designed under specific assumptions on the noise, eitherhomogeneous, partially homogeneous, compound-Gaussian, or structure nonhomogeneity. However, theactual noise may be different from the assumed one, due to system and environment uncertainties.As a consequence, the designed detectors may suffer from significant performance loss. Therefore, itis necessary to devise fully adaptive detection approaches which can adjust the detection strategy toaccommodate the changing environments. Recently, some preliminary analysis on classification of noisecovariance structure in Gaussian background was proposed in [290, 355, 368, 369]. 4) Recently, somepreliminary results of machine learning were utilized in adaptive detection [370–372]. However, it was notfully addressed the fundamental problem that why and how the detection performance can be improvedby using machine learning technologies.In this paper, we mainly focused on the Gaussian background. In practice, the environment may exhibitnon-Gaussian character [60, 61, 373–375]. Interesting readers can refer to a recently overview paper [7]on compound-Gaussian clutter, for the case that the relevant properties of the clutter are assumed to beknown in advance.
Acknowledgements
This work was supported in part by National Natural Science Foundation of China (Grant Nos. 62071482and 61871469), in part by the National Natural Science Foundation of China and Civil Aviation Administration of China underGrant U1733116, in part by the Youth Innovation Promotion Association CAS (Grant CX2100060053), in part by the National KeyResearch and Development Program of China (Grant 2018YFB1801105), and by China Postdoctoral Science Foundation (Grant2020T130493).
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