A CFAR Adaptive Subspace Detector for Second-Order ... - IEEE Xplore
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 3, MARCH 2005
871
A CFAR Adaptive Subspace Detector for Second-Order Gaussian Signals Yuanwei Jin, Member, IEEE, and Benjamin Friedlander, Fellow, IEEE
Abstract—In this paper, we study the problem of detecting subspace signals described by the Second-Order Gaussian (SOG) model in the presence of noise whose covariance structure and level are both unknown. Such a detection problem is often called Gauss–Gauss problem in that both the signal and the noise are assumed to have Gaussian distributions. We propose adaptive detectors for the SOG model signals based on a single observation and multiple observations. With a single observation, the detector can be derived in a manner similar to that of the generalized likelihood ratio test (GLRT), but the unknown covariance structure is replaced by sample covariance matrix based on training data. The proposed detectors are constant false alarm rate (CFAR) detectors. As a comparison, we also derive adaptive detectors for the First-Order Gaussian (FOG) model based on multiple observations under the same noise condition as for the SOG model. With a single observation, the seemingly ad hoc CFAR detector for the SOG model is a true GLRT in that it has the same form as the GLRT CFAR detector for the FOG model. We give an approximate closed form of the probability of detection and false alarm in this case. Furthermore, we study the proposed CFAR detectors and compute the performance curves. Index Terms—Adaptive detection, CFAR detector, first-order Gaussian model, Generalized Likelihood Ratio Test (GLRT), second-order Gaussian model.
NOMENCLATURE
Tr diag
Transpose. Hermitian transpose. Expected value. Trace. Diagonal matrix whose diagonal is the vector . Determinant of matrix . Vector (matrices) Frobenius norm. Identity matrix of order . I. INTRODUCTION
D
ETECTING subspace signals in the presence of noise is a common problem in multidimensional signal processing. By a low-rank subspace signal, we mean that each observation of the signal waveform can be modeled as some linear combination of basis vectors or “modes,” where is the subspace rank. Manuscript received June 24, 2003; revised Apr. 1, 2004. This work was supported by the Office of Naval Research under Grant N00014-01-1-0075. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Carlos H. Muravchik. Y. Jin is with the Department of Electrical and Computer Engineering, Carnegie Mellon Uiversity, Pittsburgh, PA 15213 USA (e-mail: ywjin@andrew. cmu.edu). B. Friedlander is with the Department of Electrical Engineering, University of California, Santa Cruz, CA 95060 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TSP.2004.842196
When we say that the signal obeys the linear subspace model , we are saying that the vector actually lies in a -dimensional subspace of , which we denote . The subis the range of the transformation . It is spanned by space the columns of the matrix . These columns comprise a basis are for the subspace, and the elements of the coordinates of with respect to this basis. Depending on the statistical property of the coordinate vector , we have different signal models. For a large class of detection problems, the signal of interest is modeled as the Second-Order Gaussian (SOG) model, of which the signal coordinate vector has a Gaussian distribution, i.e., (1) Such a problem is usually called Gauss–Gauss problem in that both the signal and the noise are assumed to have Gaussian distributions. One example of this type of model is the underwater acoustic source generated by, for instance, ships or submarines. The acoustic signal is random in nature and is spread in frequency and space. For another type of detection problem, the signal of interest is modeled as the First-Order Gaussian (FOG) model, which means that the signal coordinate vector is deterministic but unknown. This kind of signal model typically arises in radar, where the received radar echoes are described as unknown but deterministic signals corrupted by noise (see, e.g., [20], [28], [29], and [31]). For the FOG model, various forms of detection problem have been discussed in the statistics literature and in the signal processing field. The commonly used generalized likelihood ratio (GLR) detector has been extensively studied by many authors (see, e.g., Steinhardt [3], Burgess [5], Kelly [19], Kirsteins [20], and Trees [32]). Among others, Kraut [22], Scharf [30], and Friedlander [31] have investigated low-rank properties of the subspace signals and designed the nonadaptive and adaptive matched subspace filters. In addition, the theory of invariance in hypothesis testing is exploited by many authors (see, e.g., Bose and Steinhardt [3] and Scharf [30]), which leads to detectors that share a natural set of invariance with respect to scaling, transformation, and rotation. The detection problem of this type of model has a wide application in active radar and sonar, space-time adaptive processing (STAP), mobile communication systems, and many other multisensor or time series applications (see, e.g., [6], [7], [12], [26], and [31]). Some radar applications of the SOG model have be addressed by several authors (see, e.g., Gini [10] and Raghavan [26], [27]).
1053-587X/$20.00 © 2005 IEEE
872
Nevertheless, unlike the well-studied FOG models, the detection problem for the SOG model does not seem to be well understood. Recently, a general second-order signal model is presented by McWhorter, et al. (see, e.g., [24] and [25]). They study the matched subspace detectors (MSDs) in the presence of white noise based on a single observation. They discover that the GLRT for the detection of a FOG model signal is identical to the GLRT for the SOG model signal. The fact that the MSD for the FOG model has the same form as that for SOG model does not occur by coincidence. The difference between the FOG model and the SOG model lies in different statistical descriptions of the unknown coordinate vector in a known signal subspace. Intuitively, with a single observation, there is insufficient information to estimate the unknown coordinate vector ; therefore, it is a natural result that the MSD detectors for the two models are identical. However, the situation may change when multiple observations are used. Detection of signals on the basis of multiple observations is of interest in many applications, such as adaptive radar (see [8] and [16]). In general, the detector based on multiple observations is not a straightforward extension of that based on a single observation. Hence, further investigation on this issue is needed. In addition, prior work on nonadaptive subspace detectors motivates us to study adaptive subspace detectors for the SOG model and the FOG model and to investigate the connections between the adaptive detectors for these two models. We study in this paper the adaptive detectors of a subspace signal based on a single and multiple observations. We begin with the SOG signals. The detectors are derived based on the GLRT principle assuming that the covariance is known. After the test statistics are derived, the maximum likelihood (ML) estimate of the covariance matrix based on the secondary data is inserted in place of the known covariance. The resulting test statistic are CFAR detectors. In order to gain insights into the SOG detection problem, we also derive adaptive detectors for the FOG model based on multiple observations under the same noise condition as for the SOG model. The proposed CFAR adaptive detectors for the SOG and the FOG model signals based on single and multiple observations, to best of our knowledge, appear to be new. One important piece of work in this paper is that, for the SOG model, we are able to derive a closed-form expression of the approximate distribution function of the detection statistic and the detection probability for the false alarm rate . This form of approximation, which is verified by Monte Carlo simulation, appears to have quite good accuracy. This approximate closed form simplifies the computation of the detection threshold. In addition, the closed form reveals that the test statistic derived for the SOG has a central -distribution. By comparing the detectors for the SOG model and the FOG model, we observe that the derived CFAR detectors for both models have the same form under a single observation assumption. In fact, the detector for the FOG model on the basis of a single observation is proven to be a GLRT detector (see Kraut [21]). This observation suggests that, in this case, the derived adaptive detector for the SOG model is the true GLRT detector, although it is not proven mathematically. Our results extend the results on nonadaptive matched subspace detectors (see, e.g.,
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 3, MARCH 2005
[25]) to adaptive subspace detectors and expand the range of applications of adaptive subspace detectors for the FOG model to that for the SOG model. The remainder of paper is organized as follows. Section II introduces the subspace detection problem for the SOG model and the FOG model. Section III computes the CFAR adaptive detector for the SOG model and provides analytical and quantitative performance results. Section IV discusses the derived detectors for the SOG and the FOG models and their connections. We conclude the paper in Section V. II. FORMULATION OF THE DETECTION PROBLEM be a sequence of statistically independent, Let stationary, complex Gaussian distributed random data vectors . In an array processing application, typfor ically represents a snapshot (sample of the sensor outputs at time ) collected from an array of sensors, and it is assumed that is the superposition of the signal of interest and the noise. We define the random data matrix as a concatenation of all available data. We consider the as the primary data for detection. data The general signal model we consider in this paper is described as follows: (2) where distributed as dimension is given as
is the known signal subspace. is . is a positive definite matrix of . In other words, the signal covariance matrix
(3) is the noise data vector and is distributed as , where denotes the noise level. The CFAR requirement of a candidate algorithm ensures that the false alarm rate may be prescribed at a given value independent of the correlation properties between the various noise components. In the nomenclature of the nonadaptive detection literature, “CFAR” is with respect to noise level or variance in the test data. In the adaptive detection literature (see, e.g., [29]), “CFAR” is, with respect to the noise covariance matrix , assumed to be uniform over test and training data. However, if we allow the noise level to vary between training and test data and , respectively, we then mean with covariance matrix “CFAR” with respect to both the shared noise covariance matrix structure and independent scaling of the noise in the test data. This generalizes the meaning of “CFAR” in both the nonadaptive and adaptive detection literature, where “CFAR” is respect to a shared covariance matrix or presumed gain factor between test data and training data, respectively [22]. Our goal is to determine the existence of a signal in the received data matrix . Posing the problem as a hypothesis test, we let the null hypothesis be that the data is signal free and the alternative hypothesis be that the data contains a signal. We also assume that one has access to secondary data, or the “signal . Hence, the free” data and
JIN AND FRIEDLANDER: CFAR ADAPTIVE SUBSPACE DETECTOR FOR SECOND-ORDER GAUSSIAN SIGNALS
873
TABLE I CFAR SUBSPACE DETECTORS FOR SOG MODEL
SOG detection problem we consider in the paper can be formulated as
(4)
is assumed. A possible way to cope with the aforementioned a priori uncertainty is to resort to the GLRT, which is tantamount to replacing the unknown parameters by their ML estimates under each hypothesis [23]. In other words, the GLRT is to be derived from (9)
and
(5) Notice that is the scale factor accounting for the power mismatch between the primary and secondary data. The arbitrary scaling between the test data and training data is of significance can be considered as an idealized in practical situation. condition. In fact, we hope that the false alarm rate is insensitive to when it deviates from unity [21]. As a comparison, we also study the detection problem for the FOG signals. Without knowing the distribution of the unknown vector , we assume that this vector at each sample time is deterministic but unknown. The detection problem for the FOG model is formulated as follows:
(6) and
(7) In the next section, we will derive adaptive detectors for the SOG signals. III. ADAPTIVE SUBSPACE DETECTOR FOR SOG MODEL WITH OBSERVATIONS According to the Neyman–Pearson criterion, the optimum solution to the above hypothesis testing problem (4) and (5) is the likelihood ratio test. However, for the case under consideration, it cannot be employed since the total ignorance of the parameters (8)
is the joint densities under and . where Unfortunately, it has been well known that when both the interference and signal covariance are unknown, the GLRT detector is intractable [3], [24], [26]. In order to circumvent this drawback, we resort to an ad hoc two-step design procedure: First, we derive the GLRT detector assuming the covariance is known. After the test statistic is derived, the ML estimate of the covariance matrix based on the secondary data is inserted in place of the known covariance matrix. The resulting detectors have the desirable CFAR property. A. Derivation of the CFAR Adaptive Detectors , Before we proceed, we define a prewhitening filter is the Cholesky factorization of [13]. Then, where and the disturbance the prewhitened measurement becomes white noise . In reality, we will replace this filter by its estimate from the training data by means of either ML estimate or some forms of reduced rank processing. Let us define several notations before we begin the derivation of the GLRT detector. We use the notation to represent the whitened signal covariance matrix. Thus, the detection problem can be formulated as the following simple hypothesis test:
(10) The proposed CFAR subspace detectors are summarized in single data snapshot, the GLRT is given as Table I. With (see Appendix I-B for details) (11) where , and is the projection operator on the subspace while is its nulling projector. This detector is also called generalized energy detector in that the detection statistic is basically the ratio of signal power projected onto the signal subspace to the noise power projected onto its
874
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 3, MARCH 2005
null space. Taking into account the prewhitening filter, we obtain the following test:
and Similarly, when follows:
,
. , the corresponding detector is given as
(12) (18) data snapshots, the proposed CFAR detector is With given as below (see Appendix I-C): (13) or taking into account the prewhitening filter (14) It should be noted that this is not a GLRT detector in a strict sense. In fact, the GLRT when is given by (see Appendix I-A) Tr
(15)
Tr where
, and . When , it is easy to see that the detector (13) and (15) are equivalent. However, the proposed CFAR detector (13) generally outperforms the GLRT (15). In fact, it can be shown that the proposed CFAR detector can be obtained through a maximization based on a loosened condition (see Appendix I-C). We also see that for the SOG signal of rank , the GLRT detectors take different forms with different . This is because the detection statistic depends on signal power distribution along each dimension for the SOG model. When data snapshots, the total signal power can be there are resolved onto each dimension of the signal subspace, and the signal-to-noise powers along each dimension of the signal subdata snapshot and space are accounted for. With only without a priori knowledge of , resolving signal onto each dimension of the signal subspace is intractable; hence, only the total signal power projected onto the whole signal subspace is accounted for. If the noise covariance matrix were known, then we would use the detector described by (12) and (14). In general, the covariance matrix is unknown and must be estimated by using adaptive techniques. In this paper, we use an ad hoc procedure by substituting the unknown covariance matrix with its ML estimate based on the secondary data. The resulting detector when is given as follows: (16) where is the ML estimate of the noise covariance matrix from the secondary data, i.e., (17)
So far, we have not specified the structure of the covariance matrix . The adaptive detector uses the covariance matrix estimate to calculate the adaptive weights and then filter the primary data. Additional noise residue exists at the detector output be. This residue degrades the detection performance. cause It is well known that (see, e.g., [28]) the number of data sam. ples to obtain sufficient estimation accuracy requires Certainly, if is a structured covariance matrix, for instance, , where is a low-rank matrix of rank , we will use the ML estimate that incorporates the structure of (see, e.g., [2] and [20]) or other reduced-rank processing schemes (see, e.g., Gau [9], Goldstein [12], Guerci [14], Haimovich [15], and Kirsteins [20]). With a modification based on Anderson’s [1] early work, the ML estimate of the structured for the case can be obtained. when the number of secondary samples We skip the derivation for purposes of brevity. Let the eigen-dediag . The ML esticomposition of be mate of is given as follows: diag
(19)
where the noise estimate is given as follows: (20) are the largest eigenvalues of and is where the estimate of the rank of interference, or it can also be interpreted as the upper bound of the rank . is the ML estimate of the eigenvector matrix of the sample covariance matrix . Certainly, an a priori knowledge of the structure of loosens the requirement for the estimation of . However, a sample size thorough study of the impact of on performance when has a particular structure is out of the scope of this paper. Equivalently, the CFAR detector (16) can also be written as (21) due to the fact that
is a monotonic function of (16), or (22)
We immediately recognize that this detector has the same form as the GLRT CFAR adaptive subspace detector for the FOG model [21]. Hence, we have a conjecture that this seemingly ad hoc CFAR detector is a true GLRT. B. Decision Thresholds A closed form of the test’s probability of false alarm and detection is usually difficult to obtain. However, we find that when the signal subspace is known, an approximate closed form of
JIN AND FRIEDLANDER: CFAR ADAPTIVE SUBSPACE DETECTOR FOR SECOND-ORDER GAUSSIAN SIGNALS
875
the and is achievable for the proposed CFAR detectors (11) and (13) (see Appendix II). This approximate closed form reveals an insight as how the designed detector differs from the one designed for the FOG model although the two bear the same structure. That is, the test statistic for the SOG model has a central -distribution, whereas the test statistic for the FOG model has a noncentral -distribution [31]. We write to denote the central -distribution with degrees of freedom of , where the detection statistics under and are given as follows: (23) (24) where , , , and are defined in (100). Equation (23) is a function of the signal subspace rank , , and the number of observations . It the total dimension and level . This clearly is independent of noise structure indicates that the proposed detector (11) and (13) are CFAR detectors.
A
Fig. 1. P versus SNR for Gaussian random signals confined in subspace with p = [1; 3; 5] dimension. P = 0:01 and K = 1. The symbols square ( ), cross (+), and star (3) denote the Monte Carlo trial results for p = 1, p = 3, and p = 5, respectively, whereas the lines denote the corresponding analytical results.
C. Performance Results In this section, the computer simulation is conducted to verify the analytical result presented in the previous subsection. The simulation setting is chosen to be the underwater acoustic scenario where the change of the signal subspace rank is caused by a change of the signal spatial angular spread (see [17] for details). Nevertheless, this particular choice of simulation setup can be extended to other scenarios in principle. We use a unisensors with half-wavelength unit form linear array of apart. We consider the detector performance when the number is made of training samples is sufficient. The choice since this condition provides a reasonable accuracy for estimation of the covariance matrix of noise . We will study the perand snapshots, formance of the detector with respectively, which represents the cases of single snapshot and multiple snapshots detection. In Figs. 1–3, the detection performance versus the SNR value for different dimension of signal subspace is depicted. We caland snapculate the detection probability under and , reshots for false alarm rate of spectively. In these figures, the symbols denote the Monte Carlo trial results, whereas the lines denote the theoretical results. The three trial cases show that the theoretical results match with the Monte Carlo results quite well. However, caution should be taken when using this approximation if a precise detection probability is required. Figs. 1 and 2 show that for a single snapshot detection, there is a cross-over point on the receiver operating characteristic (ROC) curves. It demonstrates that within a certain SNR range, the increase of the dimension of subspace improves the detection performance. However, when multiple snapshots are used for detection, as is shown in Fig. 3, the increase of the rank of subspace reduces the detection performance. The results are consistent with the results reported in [17] and [26]. The explanation is as follows. The performance of a detector depends on both SNR and the shape of probability density functions. The degrees of
A
Fig. 2. P versus SNR for Gaussian random signals confined in subspace with p = [1; 3; 5] dimension. P = 0:001 and K = 1. The symbols square ( ), cross (+), and star (3) denote the Monte Carlo trial results for p = 1, p = 3, and p = 5, respectively, whereas the lines denote the corresponding analytical results.
freedom (DOF) of the pdfs depend on the subspace signal rank and the number of test data snapshots , as shown in (23) and (24). When the degrees of freedom are small and with a certain SNR range, the increase of DOF improves the detection performance due to a favorable change of the shape of probability density function. However, further increase of the DOF brings down the detection performance, as is shown in Fig. 3, where large number of observations is used. It is clear that the performance decreases when the signal subspace rank increases. It is interesting to see that the derived CFAR detector (or ) and becomes with . This is the test statistic of the conventional minimum variance distortionless response
876
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 3, MARCH 2005
A
Fig. 3. P versus SNR for Gaussian random signals confined in subspace with p = [1; 3; 5] dimension. P = 0:001 and K = 20. The symbols square ( ), star (3), and triangle ( ) denote the Monte Carlo trial results for p = 1, p = 3, and p = 5, respectively, whereas the lines denote the corresponding analytical results.
Fig. 5.
Performance comparison of CFAR detector (13) and (14). K = 20. The rank p is set to be p = [1; 3; 5].
P
=
0:001 and
ambiguity between the subspaces spanned by the signal components and the noise components. We discuss in Fig. 6 and 7 the detection curves under mismatched conditions. Here, the , and the estimated subspace true signal subspace rank is . When , signal components caprank tured by the adaptive detector are lost; on the other hand, when , more noise components are projected onto the signal subspace. The performance dropoff for the mismatched signals is evident in both scenarios. IV. DISCUSSIONS In this section, we compare nonadaptive and adaptive detectors for the FOG model and the SOG model based on and observations. The detectors for FOG models are listed in Table II (see Appendix III for details). In [25], it has been shown that GLRT for the FOG model is identical to the GLRT for the SOG model for the nonadaptive case ( is known). In our notation, the GLRT detector has the form
Fig. 4. Performance of the CFAR detector and the MVDR beamformer detector. P versus SNR for Gaussian random signals confined in subspace with p = [1; 3; 5] dimension. P = 0:001 and K = 1.
A
(MVDR) beamformer for subspace signal with scaled by . Fig. 4 depicts the performance of , , and the conventhe derived CFAR detector with tional MVDR beamformer. It is expected that as increases, the performance of the conventional MVDR beamformer degrades very quickly. This is due to an increasing loss of signal-to-noise ratio as the signal subspace opens up and the MVDR fails to capture the signal energy with a single main beam. , the proposed CFAR In Fig. 5, we show that when detector outperforms the GLRT detector under several testing scenarios. No claim to optimality is made for the GLRT. This is one example of a technique that is superior. Finally, we study the performance of the CFAR detectors to the mismatched signals when there exists signal subspace rank
(25) For the adaptive detector, it has been shown that when , the adaptive subspace detector (ASD) is the sample-matrix version of the CFAR matched subspace detector (MSD) [22]. The resulting detector takes the form as (26) This observation leads to a conjecture that CFAR detector (22) may be GLRT although it has not been proven mathematically. The GLRTs for the SOG and the FOG models when is . This is understandable in known take different form for that different statistical characteristics of the data for two models generally lead to different detector structures.
JIN AND FRIEDLANDER: CFAR ADAPTIVE SUBSPACE DETECTOR FOR SECOND-ORDER GAUSSIAN SIGNALS
Fig. 6. CFAR test with mismatch. P versus SNR for Gaussian random with p = 5. P = 0:001, and K = 1. The signals confined in subspace estimate of p is given as p = [1; 3; 5; 7; 9].
A
877
This leads us to a conjecture, although not proven mathematically, that the proposed CFAR detector is the true GLRT detector. With multiple observations, the GLRT test is not optimal in the Neyman–Pearson sense. The proposed CFAR test has a probability of detection that is higher than that of the GLRT for the test scenarios. There are several open questions that deserve further investigation. For instance, the detection performance is affected by the estimation errors of the noise covariance. The two dominant factors in the estimation errors are the number of secondary data samples and the knowledge of principal eigenvalues and eigenvectors. The unavailability of sufficient number of secondary data for large aperture arrays and the unknown effective rank of the noise covariance structure in practical situations have profound impact on detection performance. Furthermore, the problem becomes more complicated when the noise level is time varying. The understanding of the optimal or various suboptimal detector structures and their sensitivities to different kinds of noise covariances is of great importance from both theoretical and practical standpoint. APPENDIX I DERIVATION OF GLRT WHEN IS KNOWN FOR SOG MODELS In this Appendix, the GLRT detector, when is given, is and denote the derived. Let prewhitened signal subspace and test data, respectively. We derive the GLRT test for the following binary hypothesis test: (27) and (28) The GLRT test takes the form of (29)
Fig. 7. CFAR test with mismatch. P versus SNR for Gaussian random signals confined in subspace with p = 5. P = 0:001 and K = 20. The estimate of p is given as p = [1; 3; 5; 7; 9].
A
where is the whitened data matrix. Next, we would follow McWhorter and Clark’s [24] derivation to derive the GLRT test of this hypothesis problem. To proceed, we present the signal model in a different form. Let (30)
TABLE II CFAR SUBSPACE DETECTORS FOR FOG MODEL SIGNALS
where
is the unitary matrix, diagonal matrix, and
diag
is the
(31)
V. CONCLUSION We have proposed the adaptive subspace detectors for the Second-Order and the First-Order Gaussian models based on multiple observations. The proposed tests are CFAR tests. They are invariant to arbitrary scaling of the training data and the test data. It is interesting that, with a single observation, the proposed CFAR detector for the Second-Order Gaussian model has the same form as the GLRT for the First-Order Gaussian model.
It is straightforward to see that the columns of are a set of orthonormal vectors. Furthermore, let denote the null space of dimension , and it is assumed that each of is orthonormal. Thus, we have column of (32) The covariance matrix
can then be written as (33)
878
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 3, MARCH 2005
It follows that
and the log-likelihood as
(42)
(34) The ML estimate of tion. Toward that end
where
is obtained by maximizing this func-
(35) into After decomposing equivalent GLRT as follows:
and
(43)
, we then obtain an This implies that (36) (44)
Under hypothesis
, the likelihood function is given as Plugging (44) into (42), we obtain that
(45) (37) or equivalently
or its log-likelihood function as (38) It is straightforward to calculate the ML estimate of
(46) (39) Let
The compressed likelihood function under
denote
is then (47) (40)
The likelihood function under
can be written as
If we call an artificial noise, then it is reasonable to call the signal-to-noise ratio resolved onto the one-dimensional (1-D) , and subspace (48) is the signal-to-noise ratio resolved onto the -dimensional sub. space over leads to Maximizing (49) or
(41)
(50)
JIN AND FRIEDLANDER: CFAR ADAPTIVE SUBSPACE DETECTOR FOR SECOND-ORDER GAUSSIAN SIGNALS
Hence, the compressed likelihood function becomes
879
is a set of orthog-
Due to the constraint that onal basis, we conclude that Tr
(60)
(51) Hence, the compressed the likelihood function has the form where
of
denotes (52)
Tr
and (53)
(61)
Hence, the GLRT detection statistic is given as
Equations (48) and (53) are (52) and (48) with being replaced by its ML estimate in (50), respectively. Up to this point, the remaining question is how to maximize or, equivalently, . We conthe term sider the following three cases: 1) large number of data snap, 2) single snapshot , and 3) a postulated shots . solution for
Tr (62) where
A. Large Data Record
Tr
In this case, a large number of test data samples are used. . Notice that (52) can be further Let written as follows:
Tr
Tr
Tr Tr
Tr
(63)
Tr
Tr (54)
Tr
Notice that plification of
is a monotonic function of . Further simby ignoring the constant term yields
where
Tr
, will be a full rank matrix of size When eigendecomposition of yields the following: diag
. An
(64)
Tr
(55)
It is easy to see that this is a CFAR detector. Taking into account the prewhitening filter, we obtain the following test: Tr
(56) Tr
Asymptotically,
(65) (57) B.
Hence (58) Assuming
, it is straightforward to see that (59)
It should be noted that with large number of data sample, ; in we are able to estimate the signal subspace matrix other words, the energy distribution within the signal subspace is tractable. However, with a single data snapshot , we are no longer able to do that. Interestingly, in this case, the GLRT takes another form.
880
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 3, MARCH 2005
To proceed, based on (44), we notice that (66)
or , respectively. Thus, we prove this case. Next, ; then, for we assume that the argument is true for , we have
which leads to the following: (67) and
(74) (68)
The above equation means that given such that the product of all the find
, we would like to is minimized.
Here, and should not be confused with these defined in (47) and (48). Similarly (75) (69) and (70) Equations (67) and (68) imply that one can rotate each indisuch that (67) is always met, whereas the sum convidual straint simply implies that the signal-to-noise ratios in the individual subspaces much account for all of the signal-to-noise ratios in the overall subspace. denote (51) with . If the maximization Let over is equivalent to maximizing of with the constraint of (70), we obtain (71) of is equal to 1, and the The equality occurs where remaining is . This constrained maximization can be obtained by the simple although tedious inductive reasoning. The proof is provided as follows. Theorem 1: For real numbers , with a constant constraint , then (72) of is equal to 1, and the The equality occurs where . remaining is Proof: We conduct the proof by utilizing inductive rea, we have , and ; soning. For thus,
The above function is a parabolic function with its maximum . To obtain the minimum, we at only need to check the following numbers or . In fact, and produce the , we have same function value. When (76) or when
, we have
(77) Both sequences satisfy the conditions set by the theorem. We thus complete the proof. This result gives rise to the ML estimator of . Geometrically, the subspace is rotated so that the all the signal energy is resolved onto 1-D subspace , whereas the artificial . noise power is resolved evenly onto subspace The remaining question is whether a corresponding choice of exists and why the ML estimator of the subspace should work like this. The answer is that with a single observation, the may not be resolved into the signal subtotal energy space along each dimension explicitly. Thus, the ML estimation places all the signal energy and one unit of noise in one dimenand the remaining units of noise power in the sion coordinates based on theorem 1 (see also [25]). remaining , where the total This is different from the case in which signal energy can be resolved along each dimension of the signal subspace. Hence, the compressed likelihood function takes the form
(78) The GLRT detection statistic is then given as follows:
(73) is a parabolic with the maximum at . Since . Hence, we know that the minimum value or . In other words, occurs at either
(79)
JIN AND FRIEDLANDER: CFAR ADAPTIVE SUBSPACE DETECTOR FOR SECOND-ORDER GAUSSIAN SIGNALS
Furthermore, taking the derivative of log-function of leads to
(80) The last line is due to the fact that . Thus, function (79) is a monotonic function of , and an equivalent detection statistic is given as foillows:
881
the signal subspace and its null space. It is assumed that the target signal is independent of noise, and therefore, the signal component and the residue noise component projected onto the signal subspace are independent of the noise component in the null space. The numerical simulations (see, e.g., Figs. 1–3) show that the accuracy of the simple approximation is quite good. This method may be usefully employed to supplement the accurate (but less suggestive) exact methods. to denote the central ChiIn what follows, we write degrees of freedom. Similarly, we squared distribution with to denote the central -distribution with degrees write of freedom of and to be the probability density function (PDF) of a central F-distribution with degrees . of freedom The false alarm rate is computed as follows:
(81) (85) Taking into account the prewhitening filter, we obtain the detector
Let
; then, under
,
, we have
(82) (86)
or equivalently (83)
C. Postulated CFAR Detector for
Observations
A postulated CFAR detector utilizing described as follows:
observations can be
(84) In fact, this detector can be obtained if we are allowed to loosen the condition (52) to be , (53) to be , and to utilize Theorem 1. Consequently, the maximization results in a greater likelihood function value in (51) than that given in (61). Not surprisingly, it turns out that this detector has better performance than the GLRT (64) proposed earlier. APPENDIX II CALCULATION OF PROBABILITY OF FALSE ALARM AND DETECTION In this Appendix, we derive the probability of false alarm and is known. As detection for the detector (84), assuming that we can see, the detection statistic is the ratio of two non-negative quadratic forms. In [17], we have shown that the approximation to the distribution of a positive quadratic form by a scaled Chisquared distribution is fairly satisfactory. We then attempt to approximate the ratio of two independent quadratic forms by Chi-squared distributions in the numerator and the denominator. Caution should be taken to ensure the mutual independence of the numerator and the denominator. In fact, after the interference nulling filter, the received signal is separated into two subspaces:
is . It is shown The rank of the projection matrix is central in [30] that the quadratic form Chi-squared distributed with degrees of freedom if and is idempotent, i.e., . only if is a projection matrix; therefore, this In our case, . Obviously, condition is satisfied, and . Taking the independent and identically distributed snapshots into account, the detection has the following distribution: statistic under (87) is whereas the probability density function under by the Jacobian transformation, where the density function
(88) Similarly, the detection probability is computed as follows:
(89) The PDF of detection probability is more complicated. We will employ the following theorem, which is a straightforward extension of [4, Th. 2]. The theorem is given as follows. Theorem 2: If denotes a column vector of random varihaving expectation zero and distributed in a ables
882
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 3, MARCH 2005
complex multinormal distribution with variance-covariin any real quadratic form of ance matrix , and if , then is distributed like a quantity rank (90) where each variate is distributed independently of every other, and ’s are the real nonzero eigenvalues of the matrix . Proof: This theorem is a straightforward extension of [4, Th. 2.1] from real numbers to complex numbers. Based on this theorem, we know that the numerator of is distributed as a weighted sum of a chi-squared distribution, i.e., (91)
Theorem 3: (96) is distributed approximately as
, where (97)
Based on the above theorem, we have (98) and (99) where
are the eigenvalues of the matrix . Let denote the eigenvalues of the matrix , and notice that where the
(100) Hence the detection statistic under imate distribution:
has the following approx-
(92)
(101)
In the above, we use the fact that, given two matrices and , the eigenvalues of matrix are identical to those (see, e.g., [18]). Therefore, the eigenvalues of of matrix are identical to those of (which are ). also equal to the eigenvalues of the matrix It is worth noticing that the eigenvalues of are essential for the Gauss–Gauss problem we are studying here. , the random variable Given a signal vector represents the signal-to-interference-plus-noise ratio (SINR). Hence
whereas the probability density function takes the form by Jacobian transformation. Hence, the probability of detection and false alarm can be approximated as follows:
Tr
(102)
(103)
(93)
denote the eigenvalues of the Specifically, if , then is the average SINR matrix confined within the signal subspace. Carrying out the similar operations on the denominator, we have (94) The eigenvalue set
(95) Next we notice that a weighted chi-squared distribution can be approximated by a scaled chi-squared distribution given by the following theorem [4].
where
, and
.
APPENDIX III ADAPTIVE DETECTORS FOR FIRST-ORDER MODEL WITH OBSERVATIONS The detection problem is formulated in (6) and (7). In general, the GLRT detector is to be derived from (104) where is the joint densities under and , respectively, for the above detection problem. is However, we soon find out that the GLRT for general . Hence, we utilize an ad intractable, except for the case hoc approach, assuming is known. Prefiltering on data
JIN AND FRIEDLANDER: CFAR ADAPTIVE SUBSPACE DETECTOR FOR SECOND-ORDER GAUSSIAN SIGNALS
and subspace , we have the following ; thus, the detection problem becomes
and
883
or equivalently (116)
(105)
Taking the prewhitening filter into consideration, we have
and (106) Under
(117)
, the joint density is given as Replacing
with its ML estimate
from training data, we have (118)
Particularly when
, we have (119)
(107) The ML estimate of
ACKNOWLEDGMENT
is given as (108)
Hence, the compressed density function under
is given as
REFERENCES
(109) It is easy to verify that the density under
is
(110) The intermediate GLRT is then
(111) or equivalently (112) Finally, maximizing over the unknown vector equivalent to minimizing the quadratic form , yielding
The authors would like to thank the anonymous reviewers for their helpful suggestions that considerably improved the quality of this paper.
is
(113) Hence (114) We then obtain that (115)
[1] T. W. Anderson, An Introduction to Multivariate Statistical Analysis, 2nd ed. New York: Wiley, 1971, p. 58. [2] G. Bienvenu and L. Kopp, “Optimality of high resolution array processing using the eigen-system approach,” IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-31, no. 5, pp. 1235–1248, Oct. 1983. [3] S. Bose and A. O. Steinhardt, “A maximal invariant framework for adaptive detection with structured and unstructured covariance matrices,” IEEE Trans. Signal Process., vol. 43, no. 9, pp. 2164–2175, Sep. 1995. [4] G. E. P. Box, “Some theorems on quadratic forms applied in the study of analysis of variance problems, I. Effect of inequality of variance in the one-way classification,” Ann. Math. Statist., vol. 25, no. 2, pp. 290–302, Jun. 1954. [5] K. A. Burgess and B. D. Van Veen, “Subspace-based adaptive generalized likelihood ratio detection,” IEEE Trans. Signal Process., vol. 44, no. 4, pp. 912–927, Apr. 1996. [6] E. Conte, A. De Maio, and G. Ricci, “GLRT-based adaptive detection algorithms for range-spread targets,” IEEE Trans. Signal Process., vol. 49, no. 7, pp. 1336–1348, Jul. 2001. , “CFAR detection of distributed targets in non-Gaussian distur[7] bance,” IEEE Trans. Aerosp. Electron. Syst., vol. 38, no. 2, pp. 612–621, Apr. 2002. [8] J. V. DiFranco and W. L. Rubin, Radar Detection. Englewood Cliffs, NJ: Prentice-Hall, 1968. [9] Y. Gau and I. S. Reed, “An improved reduced-rank CFAR space-time adaptive radar detection algorithm,” IEEE Trans. Signal Process., vol. 46, no. 8, pp. 2139–2146, Jul. 1998. [10] R. Gini and A. Farina, “Matched subspace CFAR detection of hovering helicopters,” IEEE Trans. Aerosp. Electron. Syst., vol. 35, no. 4, pp. 1293–1305, Oct. 1997. [11] C. H. Gierull and B. Balaji, “Minimal sample support space-time adaptive processing with fast subspace techniques,” Proc. Inst. Elect. Eng.–Radar Sonar Navig., vol. 149, no. 5, pp. 209–220, Oct. 2002. [12] J. S. Goldstein and I. S. Reed, “Reduced rank adaptive filtering,” IEEE Trans. Signal Process., vol. 45, no. 2, pp. 492–496, Feb. 1997. [13] G. H. Golub and C. F. Van Loan, Matrix Computations, 2nd ed. Baltimore, MA: Johns Hopkins Univ. Press, 1989. [14] J. R. Guerci, J. S. Goldstein, and I. S. Reed, “Optimal and adaptive reduced rank STAP,” IEEE Trans. Aerosp. Electron. Syst., vol. 36, no. 2, pp. 647–663, Apr. 2000. [15] A. M. Haimovich, “The eigencaceler: adaptive radar by eigenanalysis methods,” IEEE Trans. Aerosp. Electron. Syst., vol. 32, pp. 532–542, Apr. 1996.
884
[16] C. W. Helstrom, Satistical Theory of Signal Detection, 2nd ed. New York: Pergamon, 1968. [17] Y. Jin and B. Friedlander, “Detection of distributed sources using sensor arrays,” IEEE Trans. Signal Processing, vol. 52, no. 6, pp. 1537–1548, Jun. 2004. [18] T. Kailath, Linear System. Englewood Cliffs, NJ: Prentice-Hall, 1980. [19] E. J. Kelly, “An adaptive detection algorithm,” IEEE Trans. Aerosp. Electron. Syst., vol. AES-22, no. 1, pp. 115–127, Mar. 1986. [20] I. P. Kirsteins and D. W. Tufts, “Adaptive detection using low rank approximation to a data matrix,” IEEE Trans. Aerosp. Electron. Syst., vol. 30, no. 1, pp. 55–67, Jan. 1994. [21] S. Kraut and L. L. Scharf, “The CFAR adaptive subspace detector is a scale-invariant GLRT,” IEEE Trans. Signal Processing, vol. 47, no. 9, pp. 2538–2541, Sep. 1999. [22] S. Kraut, L. L. Scharf, and L. T. McWhorter, “Adaptive subspace detectors,” IEEE Trans. Signal Process., vol. 49, no. 1, pp. 1–16, Jan. 2001. [23] E. Lehmann, Testing Statistical Hypotheses, 2nd ed. New York: WileyInterscience, 1986, pp. 284–286. [24] L. T. McWhorter and M. Clark, “Matched Subspace Detectors and Classifiers,”, Mission Research Tech. Rep. MRC/MRY-R-073, Aug. 2001. [25] L. T. McWhorter and L. L. Scharf, “Matched subspace detectors for stochastic signals,” unpublished, Sep. 2001 (private communication). [26] R. S. Raghavan, N. Pulsone, and D. J. McLaughlin, “Performance of the GLRT for adaptive vector subspace detection,” IEEE Trans. Aerosp. Electron. Syst., vol. 32, no. 4, pp. 1473–1487, Oct. 1996. [27] R. S. Raghavan, H. E. Qiu, and D. J. McLaughlin, “CFAR detection in clutter with unknown correlation properties,” IEEE Trans. Aerosp. Electron. Syst., vol. 31, no. 2, pp. 647–657, Apr. 1995. [28] I. Reed, J. Mallet, and L. Brennan, “Rapid convergence rate in adaptive arrays,” IEEE Trans. Aerosp. Electron. Syst., vol. AES-10, pp. 853–863, Nov. 1974. [29] F. Robey, D. R. Fuhrmann, E. J. Kelly, and R. Nitzberg, “A CFAR adaptive matched filter detector,” IEEE Trans. Aerosp. Electron. Syst., vol. 28, no. 1, pp. 208–216, Jan. 1992. [30] L. L. Scharf, Statistical Signal Processing. Reading, MA: AddisonWesley, 1991. [31] L. L. Scharf and B. Friedlander, “Matched subspace detection,” IEEE Trans. Signal Processing, vol. 42, no. 8, pp. 2146–2157, Aug. 1994. [32] H. L. Van Trees, Part I: Detection, Estimation, and Modulation Theory. New York: Wiley, 1968.
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 3, MARCH 2005
Yuanwei Jin (S’98–M’03) received the B.Sc. and the M.Sc. degrees from East China Normal University, Shanghai, China, in 1993 and 1996, respectively, and the Ph.D. degree in electrical engineering from the University of California, Davis (UC Davis), in 2003. He was a lecturer with the Department of Mathematics, Nanjing University of Science and Technology, Nanjing, China, from 1996 to 1997. In the summer of 1999, he was with Sensys Instruments Corporation, Santa Clara, CA. He was also with the University of California, Santa Cruz. He is now with Carnegie Mellon University, Pittsburgh, PA. His research interests are in signal processing, array processing, estimation, and detection with applications in digital communications and sonar/radar. Dr. Jin was a recipient of the Earle C. Anthony Fellowship from UC Davis from 1997 to 1999.
Benjamin Friedlander (S’74–M’76–SM’82–F’87) received the B.Sc. and M.Sc. degrees in electrical engineering from the Technion—Israel Institute of Technology, Haifa, Israel, in 1968 and 1972, respectively, and the Ph.D. degree in electrical engineering and the M.Sc. degree in statistics from Stanford University, Stanford, CA, in 1976. From 1976 to 1985, he was with Systems Control Technology, Inc., Palo Alto, CA. From November 1985 to July 1988, he was with Saxpy Computer Corporation, Sunnyvale, CA. From 1989 to 1999, he was with the University of California, Davis. Currently, he is a professor of electrical engineering with the University of California, Santa Cruz. Dr. Friedlander received the 1983 ASSP Senior Award, the 1985 Award for the Best Paper of the Year from the European Association for Signal Processing (EURASIP), the 1989 Technical Achievement Award of the Signal Processing Society, and the IEEE Third Millennium Medal.
871
A CFAR Adaptive Subspace Detector for Second-Order Gaussian Signals Yuanwei Jin, Member, IEEE, and Benjamin Friedlander, Fellow, IEEE
Abstract—In this paper, we study the problem of detecting subspace signals described by the Second-Order Gaussian (SOG) model in the presence of noise whose covariance structure and level are both unknown. Such a detection problem is often called Gauss–Gauss problem in that both the signal and the noise are assumed to have Gaussian distributions. We propose adaptive detectors for the SOG model signals based on a single observation and multiple observations. With a single observation, the detector can be derived in a manner similar to that of the generalized likelihood ratio test (GLRT), but the unknown covariance structure is replaced by sample covariance matrix based on training data. The proposed detectors are constant false alarm rate (CFAR) detectors. As a comparison, we also derive adaptive detectors for the First-Order Gaussian (FOG) model based on multiple observations under the same noise condition as for the SOG model. With a single observation, the seemingly ad hoc CFAR detector for the SOG model is a true GLRT in that it has the same form as the GLRT CFAR detector for the FOG model. We give an approximate closed form of the probability of detection and false alarm in this case. Furthermore, we study the proposed CFAR detectors and compute the performance curves. Index Terms—Adaptive detection, CFAR detector, first-order Gaussian model, Generalized Likelihood Ratio Test (GLRT), second-order Gaussian model.
NOMENCLATURE
Tr diag
Transpose. Hermitian transpose. Expected value. Trace. Diagonal matrix whose diagonal is the vector . Determinant of matrix . Vector (matrices) Frobenius norm. Identity matrix of order . I. INTRODUCTION
D
ETECTING subspace signals in the presence of noise is a common problem in multidimensional signal processing. By a low-rank subspace signal, we mean that each observation of the signal waveform can be modeled as some linear combination of basis vectors or “modes,” where is the subspace rank. Manuscript received June 24, 2003; revised Apr. 1, 2004. This work was supported by the Office of Naval Research under Grant N00014-01-1-0075. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Carlos H. Muravchik. Y. Jin is with the Department of Electrical and Computer Engineering, Carnegie Mellon Uiversity, Pittsburgh, PA 15213 USA (e-mail: ywjin@andrew. cmu.edu). B. Friedlander is with the Department of Electrical Engineering, University of California, Santa Cruz, CA 95060 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TSP.2004.842196
When we say that the signal obeys the linear subspace model , we are saying that the vector actually lies in a -dimensional subspace of , which we denote . The subis the range of the transformation . It is spanned by space the columns of the matrix . These columns comprise a basis are for the subspace, and the elements of the coordinates of with respect to this basis. Depending on the statistical property of the coordinate vector , we have different signal models. For a large class of detection problems, the signal of interest is modeled as the Second-Order Gaussian (SOG) model, of which the signal coordinate vector has a Gaussian distribution, i.e., (1) Such a problem is usually called Gauss–Gauss problem in that both the signal and the noise are assumed to have Gaussian distributions. One example of this type of model is the underwater acoustic source generated by, for instance, ships or submarines. The acoustic signal is random in nature and is spread in frequency and space. For another type of detection problem, the signal of interest is modeled as the First-Order Gaussian (FOG) model, which means that the signal coordinate vector is deterministic but unknown. This kind of signal model typically arises in radar, where the received radar echoes are described as unknown but deterministic signals corrupted by noise (see, e.g., [20], [28], [29], and [31]). For the FOG model, various forms of detection problem have been discussed in the statistics literature and in the signal processing field. The commonly used generalized likelihood ratio (GLR) detector has been extensively studied by many authors (see, e.g., Steinhardt [3], Burgess [5], Kelly [19], Kirsteins [20], and Trees [32]). Among others, Kraut [22], Scharf [30], and Friedlander [31] have investigated low-rank properties of the subspace signals and designed the nonadaptive and adaptive matched subspace filters. In addition, the theory of invariance in hypothesis testing is exploited by many authors (see, e.g., Bose and Steinhardt [3] and Scharf [30]), which leads to detectors that share a natural set of invariance with respect to scaling, transformation, and rotation. The detection problem of this type of model has a wide application in active radar and sonar, space-time adaptive processing (STAP), mobile communication systems, and many other multisensor or time series applications (see, e.g., [6], [7], [12], [26], and [31]). Some radar applications of the SOG model have be addressed by several authors (see, e.g., Gini [10] and Raghavan [26], [27]).
1053-587X/$20.00 © 2005 IEEE
872
Nevertheless, unlike the well-studied FOG models, the detection problem for the SOG model does not seem to be well understood. Recently, a general second-order signal model is presented by McWhorter, et al. (see, e.g., [24] and [25]). They study the matched subspace detectors (MSDs) in the presence of white noise based on a single observation. They discover that the GLRT for the detection of a FOG model signal is identical to the GLRT for the SOG model signal. The fact that the MSD for the FOG model has the same form as that for SOG model does not occur by coincidence. The difference between the FOG model and the SOG model lies in different statistical descriptions of the unknown coordinate vector in a known signal subspace. Intuitively, with a single observation, there is insufficient information to estimate the unknown coordinate vector ; therefore, it is a natural result that the MSD detectors for the two models are identical. However, the situation may change when multiple observations are used. Detection of signals on the basis of multiple observations is of interest in many applications, such as adaptive radar (see [8] and [16]). In general, the detector based on multiple observations is not a straightforward extension of that based on a single observation. Hence, further investigation on this issue is needed. In addition, prior work on nonadaptive subspace detectors motivates us to study adaptive subspace detectors for the SOG model and the FOG model and to investigate the connections between the adaptive detectors for these two models. We study in this paper the adaptive detectors of a subspace signal based on a single and multiple observations. We begin with the SOG signals. The detectors are derived based on the GLRT principle assuming that the covariance is known. After the test statistics are derived, the maximum likelihood (ML) estimate of the covariance matrix based on the secondary data is inserted in place of the known covariance. The resulting test statistic are CFAR detectors. In order to gain insights into the SOG detection problem, we also derive adaptive detectors for the FOG model based on multiple observations under the same noise condition as for the SOG model. The proposed CFAR adaptive detectors for the SOG and the FOG model signals based on single and multiple observations, to best of our knowledge, appear to be new. One important piece of work in this paper is that, for the SOG model, we are able to derive a closed-form expression of the approximate distribution function of the detection statistic and the detection probability for the false alarm rate . This form of approximation, which is verified by Monte Carlo simulation, appears to have quite good accuracy. This approximate closed form simplifies the computation of the detection threshold. In addition, the closed form reveals that the test statistic derived for the SOG has a central -distribution. By comparing the detectors for the SOG model and the FOG model, we observe that the derived CFAR detectors for both models have the same form under a single observation assumption. In fact, the detector for the FOG model on the basis of a single observation is proven to be a GLRT detector (see Kraut [21]). This observation suggests that, in this case, the derived adaptive detector for the SOG model is the true GLRT detector, although it is not proven mathematically. Our results extend the results on nonadaptive matched subspace detectors (see, e.g.,
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 3, MARCH 2005
[25]) to adaptive subspace detectors and expand the range of applications of adaptive subspace detectors for the FOG model to that for the SOG model. The remainder of paper is organized as follows. Section II introduces the subspace detection problem for the SOG model and the FOG model. Section III computes the CFAR adaptive detector for the SOG model and provides analytical and quantitative performance results. Section IV discusses the derived detectors for the SOG and the FOG models and their connections. We conclude the paper in Section V. II. FORMULATION OF THE DETECTION PROBLEM be a sequence of statistically independent, Let stationary, complex Gaussian distributed random data vectors . In an array processing application, typfor ically represents a snapshot (sample of the sensor outputs at time ) collected from an array of sensors, and it is assumed that is the superposition of the signal of interest and the noise. We define the random data matrix as a concatenation of all available data. We consider the as the primary data for detection. data The general signal model we consider in this paper is described as follows: (2) where distributed as dimension is given as
is the known signal subspace. is . is a positive definite matrix of . In other words, the signal covariance matrix
(3) is the noise data vector and is distributed as , where denotes the noise level. The CFAR requirement of a candidate algorithm ensures that the false alarm rate may be prescribed at a given value independent of the correlation properties between the various noise components. In the nomenclature of the nonadaptive detection literature, “CFAR” is with respect to noise level or variance in the test data. In the adaptive detection literature (see, e.g., [29]), “CFAR” is, with respect to the noise covariance matrix , assumed to be uniform over test and training data. However, if we allow the noise level to vary between training and test data and , respectively, we then mean with covariance matrix “CFAR” with respect to both the shared noise covariance matrix structure and independent scaling of the noise in the test data. This generalizes the meaning of “CFAR” in both the nonadaptive and adaptive detection literature, where “CFAR” is respect to a shared covariance matrix or presumed gain factor between test data and training data, respectively [22]. Our goal is to determine the existence of a signal in the received data matrix . Posing the problem as a hypothesis test, we let the null hypothesis be that the data is signal free and the alternative hypothesis be that the data contains a signal. We also assume that one has access to secondary data, or the “signal . Hence, the free” data and
JIN AND FRIEDLANDER: CFAR ADAPTIVE SUBSPACE DETECTOR FOR SECOND-ORDER GAUSSIAN SIGNALS
873
TABLE I CFAR SUBSPACE DETECTORS FOR SOG MODEL
SOG detection problem we consider in the paper can be formulated as
(4)
is assumed. A possible way to cope with the aforementioned a priori uncertainty is to resort to the GLRT, which is tantamount to replacing the unknown parameters by their ML estimates under each hypothesis [23]. In other words, the GLRT is to be derived from (9)
and
(5) Notice that is the scale factor accounting for the power mismatch between the primary and secondary data. The arbitrary scaling between the test data and training data is of significance can be considered as an idealized in practical situation. condition. In fact, we hope that the false alarm rate is insensitive to when it deviates from unity [21]. As a comparison, we also study the detection problem for the FOG signals. Without knowing the distribution of the unknown vector , we assume that this vector at each sample time is deterministic but unknown. The detection problem for the FOG model is formulated as follows:
(6) and
(7) In the next section, we will derive adaptive detectors for the SOG signals. III. ADAPTIVE SUBSPACE DETECTOR FOR SOG MODEL WITH OBSERVATIONS According to the Neyman–Pearson criterion, the optimum solution to the above hypothesis testing problem (4) and (5) is the likelihood ratio test. However, for the case under consideration, it cannot be employed since the total ignorance of the parameters (8)
is the joint densities under and . where Unfortunately, it has been well known that when both the interference and signal covariance are unknown, the GLRT detector is intractable [3], [24], [26]. In order to circumvent this drawback, we resort to an ad hoc two-step design procedure: First, we derive the GLRT detector assuming the covariance is known. After the test statistic is derived, the ML estimate of the covariance matrix based on the secondary data is inserted in place of the known covariance matrix. The resulting detectors have the desirable CFAR property. A. Derivation of the CFAR Adaptive Detectors , Before we proceed, we define a prewhitening filter is the Cholesky factorization of [13]. Then, where and the disturbance the prewhitened measurement becomes white noise . In reality, we will replace this filter by its estimate from the training data by means of either ML estimate or some forms of reduced rank processing. Let us define several notations before we begin the derivation of the GLRT detector. We use the notation to represent the whitened signal covariance matrix. Thus, the detection problem can be formulated as the following simple hypothesis test:
(10) The proposed CFAR subspace detectors are summarized in single data snapshot, the GLRT is given as Table I. With (see Appendix I-B for details) (11) where , and is the projection operator on the subspace while is its nulling projector. This detector is also called generalized energy detector in that the detection statistic is basically the ratio of signal power projected onto the signal subspace to the noise power projected onto its
874
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 3, MARCH 2005
null space. Taking into account the prewhitening filter, we obtain the following test:
and Similarly, when follows:
,
. , the corresponding detector is given as
(12) (18) data snapshots, the proposed CFAR detector is With given as below (see Appendix I-C): (13) or taking into account the prewhitening filter (14) It should be noted that this is not a GLRT detector in a strict sense. In fact, the GLRT when is given by (see Appendix I-A) Tr
(15)
Tr where
, and . When , it is easy to see that the detector (13) and (15) are equivalent. However, the proposed CFAR detector (13) generally outperforms the GLRT (15). In fact, it can be shown that the proposed CFAR detector can be obtained through a maximization based on a loosened condition (see Appendix I-C). We also see that for the SOG signal of rank , the GLRT detectors take different forms with different . This is because the detection statistic depends on signal power distribution along each dimension for the SOG model. When data snapshots, the total signal power can be there are resolved onto each dimension of the signal subspace, and the signal-to-noise powers along each dimension of the signal subdata snapshot and space are accounted for. With only without a priori knowledge of , resolving signal onto each dimension of the signal subspace is intractable; hence, only the total signal power projected onto the whole signal subspace is accounted for. If the noise covariance matrix were known, then we would use the detector described by (12) and (14). In general, the covariance matrix is unknown and must be estimated by using adaptive techniques. In this paper, we use an ad hoc procedure by substituting the unknown covariance matrix with its ML estimate based on the secondary data. The resulting detector when is given as follows: (16) where is the ML estimate of the noise covariance matrix from the secondary data, i.e., (17)
So far, we have not specified the structure of the covariance matrix . The adaptive detector uses the covariance matrix estimate to calculate the adaptive weights and then filter the primary data. Additional noise residue exists at the detector output be. This residue degrades the detection performance. cause It is well known that (see, e.g., [28]) the number of data sam. ples to obtain sufficient estimation accuracy requires Certainly, if is a structured covariance matrix, for instance, , where is a low-rank matrix of rank , we will use the ML estimate that incorporates the structure of (see, e.g., [2] and [20]) or other reduced-rank processing schemes (see, e.g., Gau [9], Goldstein [12], Guerci [14], Haimovich [15], and Kirsteins [20]). With a modification based on Anderson’s [1] early work, the ML estimate of the structured for the case can be obtained. when the number of secondary samples We skip the derivation for purposes of brevity. Let the eigen-dediag . The ML esticomposition of be mate of is given as follows: diag
(19)
where the noise estimate is given as follows: (20) are the largest eigenvalues of and is where the estimate of the rank of interference, or it can also be interpreted as the upper bound of the rank . is the ML estimate of the eigenvector matrix of the sample covariance matrix . Certainly, an a priori knowledge of the structure of loosens the requirement for the estimation of . However, a sample size thorough study of the impact of on performance when has a particular structure is out of the scope of this paper. Equivalently, the CFAR detector (16) can also be written as (21) due to the fact that
is a monotonic function of (16), or (22)
We immediately recognize that this detector has the same form as the GLRT CFAR adaptive subspace detector for the FOG model [21]. Hence, we have a conjecture that this seemingly ad hoc CFAR detector is a true GLRT. B. Decision Thresholds A closed form of the test’s probability of false alarm and detection is usually difficult to obtain. However, we find that when the signal subspace is known, an approximate closed form of
JIN AND FRIEDLANDER: CFAR ADAPTIVE SUBSPACE DETECTOR FOR SECOND-ORDER GAUSSIAN SIGNALS
875
the and is achievable for the proposed CFAR detectors (11) and (13) (see Appendix II). This approximate closed form reveals an insight as how the designed detector differs from the one designed for the FOG model although the two bear the same structure. That is, the test statistic for the SOG model has a central -distribution, whereas the test statistic for the FOG model has a noncentral -distribution [31]. We write to denote the central -distribution with degrees of freedom of , where the detection statistics under and are given as follows: (23) (24) where , , , and are defined in (100). Equation (23) is a function of the signal subspace rank , , and the number of observations . It the total dimension and level . This clearly is independent of noise structure indicates that the proposed detector (11) and (13) are CFAR detectors.
A
Fig. 1. P versus SNR for Gaussian random signals confined in subspace with p = [1; 3; 5] dimension. P = 0:01 and K = 1. The symbols square ( ), cross (+), and star (3) denote the Monte Carlo trial results for p = 1, p = 3, and p = 5, respectively, whereas the lines denote the corresponding analytical results.
C. Performance Results In this section, the computer simulation is conducted to verify the analytical result presented in the previous subsection. The simulation setting is chosen to be the underwater acoustic scenario where the change of the signal subspace rank is caused by a change of the signal spatial angular spread (see [17] for details). Nevertheless, this particular choice of simulation setup can be extended to other scenarios in principle. We use a unisensors with half-wavelength unit form linear array of apart. We consider the detector performance when the number is made of training samples is sufficient. The choice since this condition provides a reasonable accuracy for estimation of the covariance matrix of noise . We will study the perand snapshots, formance of the detector with respectively, which represents the cases of single snapshot and multiple snapshots detection. In Figs. 1–3, the detection performance versus the SNR value for different dimension of signal subspace is depicted. We caland snapculate the detection probability under and , reshots for false alarm rate of spectively. In these figures, the symbols denote the Monte Carlo trial results, whereas the lines denote the theoretical results. The three trial cases show that the theoretical results match with the Monte Carlo results quite well. However, caution should be taken when using this approximation if a precise detection probability is required. Figs. 1 and 2 show that for a single snapshot detection, there is a cross-over point on the receiver operating characteristic (ROC) curves. It demonstrates that within a certain SNR range, the increase of the dimension of subspace improves the detection performance. However, when multiple snapshots are used for detection, as is shown in Fig. 3, the increase of the rank of subspace reduces the detection performance. The results are consistent with the results reported in [17] and [26]. The explanation is as follows. The performance of a detector depends on both SNR and the shape of probability density functions. The degrees of
A
Fig. 2. P versus SNR for Gaussian random signals confined in subspace with p = [1; 3; 5] dimension. P = 0:001 and K = 1. The symbols square ( ), cross (+), and star (3) denote the Monte Carlo trial results for p = 1, p = 3, and p = 5, respectively, whereas the lines denote the corresponding analytical results.
freedom (DOF) of the pdfs depend on the subspace signal rank and the number of test data snapshots , as shown in (23) and (24). When the degrees of freedom are small and with a certain SNR range, the increase of DOF improves the detection performance due to a favorable change of the shape of probability density function. However, further increase of the DOF brings down the detection performance, as is shown in Fig. 3, where large number of observations is used. It is clear that the performance decreases when the signal subspace rank increases. It is interesting to see that the derived CFAR detector (or ) and becomes with . This is the test statistic of the conventional minimum variance distortionless response
876
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 3, MARCH 2005
A
Fig. 3. P versus SNR for Gaussian random signals confined in subspace with p = [1; 3; 5] dimension. P = 0:001 and K = 20. The symbols square ( ), star (3), and triangle ( ) denote the Monte Carlo trial results for p = 1, p = 3, and p = 5, respectively, whereas the lines denote the corresponding analytical results.
Fig. 5.
Performance comparison of CFAR detector (13) and (14). K = 20. The rank p is set to be p = [1; 3; 5].
P
=
0:001 and
ambiguity between the subspaces spanned by the signal components and the noise components. We discuss in Fig. 6 and 7 the detection curves under mismatched conditions. Here, the , and the estimated subspace true signal subspace rank is . When , signal components caprank tured by the adaptive detector are lost; on the other hand, when , more noise components are projected onto the signal subspace. The performance dropoff for the mismatched signals is evident in both scenarios. IV. DISCUSSIONS In this section, we compare nonadaptive and adaptive detectors for the FOG model and the SOG model based on and observations. The detectors for FOG models are listed in Table II (see Appendix III for details). In [25], it has been shown that GLRT for the FOG model is identical to the GLRT for the SOG model for the nonadaptive case ( is known). In our notation, the GLRT detector has the form
Fig. 4. Performance of the CFAR detector and the MVDR beamformer detector. P versus SNR for Gaussian random signals confined in subspace with p = [1; 3; 5] dimension. P = 0:001 and K = 1.
A
(MVDR) beamformer for subspace signal with scaled by . Fig. 4 depicts the performance of , , and the conventhe derived CFAR detector with tional MVDR beamformer. It is expected that as increases, the performance of the conventional MVDR beamformer degrades very quickly. This is due to an increasing loss of signal-to-noise ratio as the signal subspace opens up and the MVDR fails to capture the signal energy with a single main beam. , the proposed CFAR In Fig. 5, we show that when detector outperforms the GLRT detector under several testing scenarios. No claim to optimality is made for the GLRT. This is one example of a technique that is superior. Finally, we study the performance of the CFAR detectors to the mismatched signals when there exists signal subspace rank
(25) For the adaptive detector, it has been shown that when , the adaptive subspace detector (ASD) is the sample-matrix version of the CFAR matched subspace detector (MSD) [22]. The resulting detector takes the form as (26) This observation leads to a conjecture that CFAR detector (22) may be GLRT although it has not been proven mathematically. The GLRTs for the SOG and the FOG models when is . This is understandable in known take different form for that different statistical characteristics of the data for two models generally lead to different detector structures.
JIN AND FRIEDLANDER: CFAR ADAPTIVE SUBSPACE DETECTOR FOR SECOND-ORDER GAUSSIAN SIGNALS
Fig. 6. CFAR test with mismatch. P versus SNR for Gaussian random with p = 5. P = 0:001, and K = 1. The signals confined in subspace estimate of p is given as p = [1; 3; 5; 7; 9].
A
877
This leads us to a conjecture, although not proven mathematically, that the proposed CFAR detector is the true GLRT detector. With multiple observations, the GLRT test is not optimal in the Neyman–Pearson sense. The proposed CFAR test has a probability of detection that is higher than that of the GLRT for the test scenarios. There are several open questions that deserve further investigation. For instance, the detection performance is affected by the estimation errors of the noise covariance. The two dominant factors in the estimation errors are the number of secondary data samples and the knowledge of principal eigenvalues and eigenvectors. The unavailability of sufficient number of secondary data for large aperture arrays and the unknown effective rank of the noise covariance structure in practical situations have profound impact on detection performance. Furthermore, the problem becomes more complicated when the noise level is time varying. The understanding of the optimal or various suboptimal detector structures and their sensitivities to different kinds of noise covariances is of great importance from both theoretical and practical standpoint. APPENDIX I DERIVATION OF GLRT WHEN IS KNOWN FOR SOG MODELS In this Appendix, the GLRT detector, when is given, is and denote the derived. Let prewhitened signal subspace and test data, respectively. We derive the GLRT test for the following binary hypothesis test: (27) and (28) The GLRT test takes the form of (29)
Fig. 7. CFAR test with mismatch. P versus SNR for Gaussian random signals confined in subspace with p = 5. P = 0:001 and K = 20. The estimate of p is given as p = [1; 3; 5; 7; 9].
A
where is the whitened data matrix. Next, we would follow McWhorter and Clark’s [24] derivation to derive the GLRT test of this hypothesis problem. To proceed, we present the signal model in a different form. Let (30)
TABLE II CFAR SUBSPACE DETECTORS FOR FOG MODEL SIGNALS
where
is the unitary matrix, diagonal matrix, and
diag
is the
(31)
V. CONCLUSION We have proposed the adaptive subspace detectors for the Second-Order and the First-Order Gaussian models based on multiple observations. The proposed tests are CFAR tests. They are invariant to arbitrary scaling of the training data and the test data. It is interesting that, with a single observation, the proposed CFAR detector for the Second-Order Gaussian model has the same form as the GLRT for the First-Order Gaussian model.
It is straightforward to see that the columns of are a set of orthonormal vectors. Furthermore, let denote the null space of dimension , and it is assumed that each of is orthonormal. Thus, we have column of (32) The covariance matrix
can then be written as (33)
878
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 3, MARCH 2005
It follows that
and the log-likelihood as
(42)
(34) The ML estimate of tion. Toward that end
where
is obtained by maximizing this func-
(35) into After decomposing equivalent GLRT as follows:
and
(43)
, we then obtain an This implies that (36) (44)
Under hypothesis
, the likelihood function is given as Plugging (44) into (42), we obtain that
(45) (37) or equivalently
or its log-likelihood function as (38) It is straightforward to calculate the ML estimate of
(46) (39) Let
The compressed likelihood function under
denote
is then (47) (40)
The likelihood function under
can be written as
If we call an artificial noise, then it is reasonable to call the signal-to-noise ratio resolved onto the one-dimensional (1-D) , and subspace (48) is the signal-to-noise ratio resolved onto the -dimensional sub. space over leads to Maximizing (49) or
(41)
(50)
JIN AND FRIEDLANDER: CFAR ADAPTIVE SUBSPACE DETECTOR FOR SECOND-ORDER GAUSSIAN SIGNALS
Hence, the compressed likelihood function becomes
879
is a set of orthog-
Due to the constraint that onal basis, we conclude that Tr
(60)
(51) Hence, the compressed the likelihood function has the form where
of
denotes (52)
Tr
and (53)
(61)
Hence, the GLRT detection statistic is given as
Equations (48) and (53) are (52) and (48) with being replaced by its ML estimate in (50), respectively. Up to this point, the remaining question is how to maximize or, equivalently, . We conthe term sider the following three cases: 1) large number of data snap, 2) single snapshot , and 3) a postulated shots . solution for
Tr (62) where
A. Large Data Record
Tr
In this case, a large number of test data samples are used. . Notice that (52) can be further Let written as follows:
Tr
Tr
Tr Tr
Tr
(63)
Tr
Tr (54)
Tr
Notice that plification of
is a monotonic function of . Further simby ignoring the constant term yields
where
Tr
, will be a full rank matrix of size When eigendecomposition of yields the following: diag
. An
(64)
Tr
(55)
It is easy to see that this is a CFAR detector. Taking into account the prewhitening filter, we obtain the following test: Tr
(56) Tr
Asymptotically,
(65) (57) B.
Hence (58) Assuming
, it is straightforward to see that (59)
It should be noted that with large number of data sample, ; in we are able to estimate the signal subspace matrix other words, the energy distribution within the signal subspace is tractable. However, with a single data snapshot , we are no longer able to do that. Interestingly, in this case, the GLRT takes another form.
880
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 3, MARCH 2005
To proceed, based on (44), we notice that (66)
or , respectively. Thus, we prove this case. Next, ; then, for we assume that the argument is true for , we have
which leads to the following: (67) and
(74) (68)
The above equation means that given such that the product of all the find
, we would like to is minimized.
Here, and should not be confused with these defined in (47) and (48). Similarly (75) (69) and (70) Equations (67) and (68) imply that one can rotate each indisuch that (67) is always met, whereas the sum convidual straint simply implies that the signal-to-noise ratios in the individual subspaces much account for all of the signal-to-noise ratios in the overall subspace. denote (51) with . If the maximization Let over is equivalent to maximizing of with the constraint of (70), we obtain (71) of is equal to 1, and the The equality occurs where remaining is . This constrained maximization can be obtained by the simple although tedious inductive reasoning. The proof is provided as follows. Theorem 1: For real numbers , with a constant constraint , then (72) of is equal to 1, and the The equality occurs where . remaining is Proof: We conduct the proof by utilizing inductive rea, we have , and ; soning. For thus,
The above function is a parabolic function with its maximum . To obtain the minimum, we at only need to check the following numbers or . In fact, and produce the , we have same function value. When (76) or when
, we have
(77) Both sequences satisfy the conditions set by the theorem. We thus complete the proof. This result gives rise to the ML estimator of . Geometrically, the subspace is rotated so that the all the signal energy is resolved onto 1-D subspace , whereas the artificial . noise power is resolved evenly onto subspace The remaining question is whether a corresponding choice of exists and why the ML estimator of the subspace should work like this. The answer is that with a single observation, the may not be resolved into the signal subtotal energy space along each dimension explicitly. Thus, the ML estimation places all the signal energy and one unit of noise in one dimenand the remaining units of noise power in the sion coordinates based on theorem 1 (see also [25]). remaining , where the total This is different from the case in which signal energy can be resolved along each dimension of the signal subspace. Hence, the compressed likelihood function takes the form
(78) The GLRT detection statistic is then given as follows:
(73) is a parabolic with the maximum at . Since . Hence, we know that the minimum value or . In other words, occurs at either
(79)
JIN AND FRIEDLANDER: CFAR ADAPTIVE SUBSPACE DETECTOR FOR SECOND-ORDER GAUSSIAN SIGNALS
Furthermore, taking the derivative of log-function of leads to
(80) The last line is due to the fact that . Thus, function (79) is a monotonic function of , and an equivalent detection statistic is given as foillows:
881
the signal subspace and its null space. It is assumed that the target signal is independent of noise, and therefore, the signal component and the residue noise component projected onto the signal subspace are independent of the noise component in the null space. The numerical simulations (see, e.g., Figs. 1–3) show that the accuracy of the simple approximation is quite good. This method may be usefully employed to supplement the accurate (but less suggestive) exact methods. to denote the central ChiIn what follows, we write degrees of freedom. Similarly, we squared distribution with to denote the central -distribution with degrees write of freedom of and to be the probability density function (PDF) of a central F-distribution with degrees . of freedom The false alarm rate is computed as follows:
(81) (85) Taking into account the prewhitening filter, we obtain the detector
Let
; then, under
,
, we have
(82) (86)
or equivalently (83)
C. Postulated CFAR Detector for
Observations
A postulated CFAR detector utilizing described as follows:
observations can be
(84) In fact, this detector can be obtained if we are allowed to loosen the condition (52) to be , (53) to be , and to utilize Theorem 1. Consequently, the maximization results in a greater likelihood function value in (51) than that given in (61). Not surprisingly, it turns out that this detector has better performance than the GLRT (64) proposed earlier. APPENDIX II CALCULATION OF PROBABILITY OF FALSE ALARM AND DETECTION In this Appendix, we derive the probability of false alarm and is known. As detection for the detector (84), assuming that we can see, the detection statistic is the ratio of two non-negative quadratic forms. In [17], we have shown that the approximation to the distribution of a positive quadratic form by a scaled Chisquared distribution is fairly satisfactory. We then attempt to approximate the ratio of two independent quadratic forms by Chi-squared distributions in the numerator and the denominator. Caution should be taken to ensure the mutual independence of the numerator and the denominator. In fact, after the interference nulling filter, the received signal is separated into two subspaces:
is . It is shown The rank of the projection matrix is central in [30] that the quadratic form Chi-squared distributed with degrees of freedom if and is idempotent, i.e., . only if is a projection matrix; therefore, this In our case, . Obviously, condition is satisfied, and . Taking the independent and identically distributed snapshots into account, the detection has the following distribution: statistic under (87) is whereas the probability density function under by the Jacobian transformation, where the density function
(88) Similarly, the detection probability is computed as follows:
(89) The PDF of detection probability is more complicated. We will employ the following theorem, which is a straightforward extension of [4, Th. 2]. The theorem is given as follows. Theorem 2: If denotes a column vector of random varihaving expectation zero and distributed in a ables
882
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 3, MARCH 2005
complex multinormal distribution with variance-covariin any real quadratic form of ance matrix , and if , then is distributed like a quantity rank (90) where each variate is distributed independently of every other, and ’s are the real nonzero eigenvalues of the matrix . Proof: This theorem is a straightforward extension of [4, Th. 2.1] from real numbers to complex numbers. Based on this theorem, we know that the numerator of is distributed as a weighted sum of a chi-squared distribution, i.e., (91)
Theorem 3: (96) is distributed approximately as
, where (97)
Based on the above theorem, we have (98) and (99) where
are the eigenvalues of the matrix . Let denote the eigenvalues of the matrix , and notice that where the
(100) Hence the detection statistic under imate distribution:
has the following approx-
(92)
(101)
In the above, we use the fact that, given two matrices and , the eigenvalues of matrix are identical to those (see, e.g., [18]). Therefore, the eigenvalues of of matrix are identical to those of (which are ). also equal to the eigenvalues of the matrix It is worth noticing that the eigenvalues of are essential for the Gauss–Gauss problem we are studying here. , the random variable Given a signal vector represents the signal-to-interference-plus-noise ratio (SINR). Hence
whereas the probability density function takes the form by Jacobian transformation. Hence, the probability of detection and false alarm can be approximated as follows:
Tr
(102)
(103)
(93)
denote the eigenvalues of the Specifically, if , then is the average SINR matrix confined within the signal subspace. Carrying out the similar operations on the denominator, we have (94) The eigenvalue set
(95) Next we notice that a weighted chi-squared distribution can be approximated by a scaled chi-squared distribution given by the following theorem [4].
where
, and
.
APPENDIX III ADAPTIVE DETECTORS FOR FIRST-ORDER MODEL WITH OBSERVATIONS The detection problem is formulated in (6) and (7). In general, the GLRT detector is to be derived from (104) where is the joint densities under and , respectively, for the above detection problem. is However, we soon find out that the GLRT for general . Hence, we utilize an ad intractable, except for the case hoc approach, assuming is known. Prefiltering on data
JIN AND FRIEDLANDER: CFAR ADAPTIVE SUBSPACE DETECTOR FOR SECOND-ORDER GAUSSIAN SIGNALS
and subspace , we have the following ; thus, the detection problem becomes
and
883
or equivalently (116)
(105)
Taking the prewhitening filter into consideration, we have
and (106) Under
(117)
, the joint density is given as Replacing
with its ML estimate
from training data, we have (118)
Particularly when
, we have (119)
(107) The ML estimate of
ACKNOWLEDGMENT
is given as (108)
Hence, the compressed density function under
is given as
REFERENCES
(109) It is easy to verify that the density under
is
(110) The intermediate GLRT is then
(111) or equivalently (112) Finally, maximizing over the unknown vector equivalent to minimizing the quadratic form , yielding
The authors would like to thank the anonymous reviewers for their helpful suggestions that considerably improved the quality of this paper.
is
(113) Hence (114) We then obtain that (115)
[1] T. W. Anderson, An Introduction to Multivariate Statistical Analysis, 2nd ed. New York: Wiley, 1971, p. 58. [2] G. Bienvenu and L. Kopp, “Optimality of high resolution array processing using the eigen-system approach,” IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-31, no. 5, pp. 1235–1248, Oct. 1983. [3] S. Bose and A. O. Steinhardt, “A maximal invariant framework for adaptive detection with structured and unstructured covariance matrices,” IEEE Trans. Signal Process., vol. 43, no. 9, pp. 2164–2175, Sep. 1995. [4] G. E. P. Box, “Some theorems on quadratic forms applied in the study of analysis of variance problems, I. Effect of inequality of variance in the one-way classification,” Ann. Math. Statist., vol. 25, no. 2, pp. 290–302, Jun. 1954. [5] K. A. Burgess and B. D. Van Veen, “Subspace-based adaptive generalized likelihood ratio detection,” IEEE Trans. Signal Process., vol. 44, no. 4, pp. 912–927, Apr. 1996. [6] E. Conte, A. De Maio, and G. Ricci, “GLRT-based adaptive detection algorithms for range-spread targets,” IEEE Trans. Signal Process., vol. 49, no. 7, pp. 1336–1348, Jul. 2001. , “CFAR detection of distributed targets in non-Gaussian distur[7] bance,” IEEE Trans. Aerosp. Electron. Syst., vol. 38, no. 2, pp. 612–621, Apr. 2002. [8] J. V. DiFranco and W. L. Rubin, Radar Detection. Englewood Cliffs, NJ: Prentice-Hall, 1968. [9] Y. Gau and I. S. Reed, “An improved reduced-rank CFAR space-time adaptive radar detection algorithm,” IEEE Trans. Signal Process., vol. 46, no. 8, pp. 2139–2146, Jul. 1998. [10] R. Gini and A. Farina, “Matched subspace CFAR detection of hovering helicopters,” IEEE Trans. Aerosp. Electron. Syst., vol. 35, no. 4, pp. 1293–1305, Oct. 1997. [11] C. H. Gierull and B. Balaji, “Minimal sample support space-time adaptive processing with fast subspace techniques,” Proc. Inst. Elect. Eng.–Radar Sonar Navig., vol. 149, no. 5, pp. 209–220, Oct. 2002. [12] J. S. Goldstein and I. S. Reed, “Reduced rank adaptive filtering,” IEEE Trans. Signal Process., vol. 45, no. 2, pp. 492–496, Feb. 1997. [13] G. H. Golub and C. F. Van Loan, Matrix Computations, 2nd ed. Baltimore, MA: Johns Hopkins Univ. Press, 1989. [14] J. R. Guerci, J. S. Goldstein, and I. S. Reed, “Optimal and adaptive reduced rank STAP,” IEEE Trans. Aerosp. Electron. Syst., vol. 36, no. 2, pp. 647–663, Apr. 2000. [15] A. M. Haimovich, “The eigencaceler: adaptive radar by eigenanalysis methods,” IEEE Trans. Aerosp. Electron. Syst., vol. 32, pp. 532–542, Apr. 1996.
884
[16] C. W. Helstrom, Satistical Theory of Signal Detection, 2nd ed. New York: Pergamon, 1968. [17] Y. Jin and B. Friedlander, “Detection of distributed sources using sensor arrays,” IEEE Trans. Signal Processing, vol. 52, no. 6, pp. 1537–1548, Jun. 2004. [18] T. Kailath, Linear System. Englewood Cliffs, NJ: Prentice-Hall, 1980. [19] E. J. Kelly, “An adaptive detection algorithm,” IEEE Trans. Aerosp. Electron. Syst., vol. AES-22, no. 1, pp. 115–127, Mar. 1986. [20] I. P. Kirsteins and D. W. Tufts, “Adaptive detection using low rank approximation to a data matrix,” IEEE Trans. Aerosp. Electron. Syst., vol. 30, no. 1, pp. 55–67, Jan. 1994. [21] S. Kraut and L. L. Scharf, “The CFAR adaptive subspace detector is a scale-invariant GLRT,” IEEE Trans. Signal Processing, vol. 47, no. 9, pp. 2538–2541, Sep. 1999. [22] S. Kraut, L. L. Scharf, and L. T. McWhorter, “Adaptive subspace detectors,” IEEE Trans. Signal Process., vol. 49, no. 1, pp. 1–16, Jan. 2001. [23] E. Lehmann, Testing Statistical Hypotheses, 2nd ed. New York: WileyInterscience, 1986, pp. 284–286. [24] L. T. McWhorter and M. Clark, “Matched Subspace Detectors and Classifiers,”, Mission Research Tech. Rep. MRC/MRY-R-073, Aug. 2001. [25] L. T. McWhorter and L. L. Scharf, “Matched subspace detectors for stochastic signals,” unpublished, Sep. 2001 (private communication). [26] R. S. Raghavan, N. Pulsone, and D. J. McLaughlin, “Performance of the GLRT for adaptive vector subspace detection,” IEEE Trans. Aerosp. Electron. Syst., vol. 32, no. 4, pp. 1473–1487, Oct. 1996. [27] R. S. Raghavan, H. E. Qiu, and D. J. McLaughlin, “CFAR detection in clutter with unknown correlation properties,” IEEE Trans. Aerosp. Electron. Syst., vol. 31, no. 2, pp. 647–657, Apr. 1995. [28] I. Reed, J. Mallet, and L. Brennan, “Rapid convergence rate in adaptive arrays,” IEEE Trans. Aerosp. Electron. Syst., vol. AES-10, pp. 853–863, Nov. 1974. [29] F. Robey, D. R. Fuhrmann, E. J. Kelly, and R. Nitzberg, “A CFAR adaptive matched filter detector,” IEEE Trans. Aerosp. Electron. Syst., vol. 28, no. 1, pp. 208–216, Jan. 1992. [30] L. L. Scharf, Statistical Signal Processing. Reading, MA: AddisonWesley, 1991. [31] L. L. Scharf and B. Friedlander, “Matched subspace detection,” IEEE Trans. Signal Processing, vol. 42, no. 8, pp. 2146–2157, Aug. 1994. [32] H. L. Van Trees, Part I: Detection, Estimation, and Modulation Theory. New York: Wiley, 1968.
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 3, MARCH 2005
Yuanwei Jin (S’98–M’03) received the B.Sc. and the M.Sc. degrees from East China Normal University, Shanghai, China, in 1993 and 1996, respectively, and the Ph.D. degree in electrical engineering from the University of California, Davis (UC Davis), in 2003. He was a lecturer with the Department of Mathematics, Nanjing University of Science and Technology, Nanjing, China, from 1996 to 1997. In the summer of 1999, he was with Sensys Instruments Corporation, Santa Clara, CA. He was also with the University of California, Santa Cruz. He is now with Carnegie Mellon University, Pittsburgh, PA. His research interests are in signal processing, array processing, estimation, and detection with applications in digital communications and sonar/radar. Dr. Jin was a recipient of the Earle C. Anthony Fellowship from UC Davis from 1997 to 1999.
Benjamin Friedlander (S’74–M’76–SM’82–F’87) received the B.Sc. and M.Sc. degrees in electrical engineering from the Technion—Israel Institute of Technology, Haifa, Israel, in 1968 and 1972, respectively, and the Ph.D. degree in electrical engineering and the M.Sc. degree in statistics from Stanford University, Stanford, CA, in 1976. From 1976 to 1985, he was with Systems Control Technology, Inc., Palo Alto, CA. From November 1985 to July 1988, he was with Saxpy Computer Corporation, Sunnyvale, CA. From 1989 to 1999, he was with the University of California, Davis. Currently, he is a professor of electrical engineering with the University of California, Santa Cruz. Dr. Friedlander received the 1983 ASSP Senior Award, the 1985 Award for the Best Paper of the Year from the European Association for Signal Processing (EURASIP), the 1989 Technical Achievement Award of the Signal Processing Society, and the IEEE Third Millennium Medal.
