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Robust STAP algorithms using prior knowledge for airborne radar applications
Signal Processing 79 (1999) 273}287
Robust STAP algorithms using prior knowledge for airborne radar applicationsq X. Lin, R.S. Blum* EECS Department, Packard Lab, Lehigh University, Bethlehem, PA 18015-3084, USA Received 12 April 1999; received in revised form 29 June 1999
Abstract Space}time adaptive processing (STAP) schemes have shown promise for airborne radar applications. However, the majority of schemes develop an estimate of the covariance matrix for the test cell by averaging over surrounding range cells, called reference data. This method is only guaranteed to ensure good performance when a large set of homogeneous reference data is available with the same statistics as the test cell. In this paper, a new methodology is proposed for obtaining a covariance matrix that uses a priori knowledge. The approach is useful for detecting weak signals in cases with discretes in some range cells which do not appear in other range cells. The focus is on the use of a simple model for ground clutter that incorporates our prior knowledge on the structure of the ground clutter. The new methodology can be applied to most existing STAP schemes. This is illustrated by applying the methodology to three speci"c existing schemes. The modi"ed schemes are generally shown to outperform the existing schemes in non-stationary measured data cases. ( 1999 Elsevier Science B.V. All rights reserved. Zusammenfassung Verfahren der adaptiven Raum}Zeit-Verarbeitung (STAP) haben sich als vielversprechend fuK r Anwendungen auf dem Gebiet des Luft-Radar erwiesen. Jedoch wird beim Gro{teil der Methoden die Kovarianzmatrix fuK r die Testzelle durch Mittelung uK ber umliegende Entfernungszellen (sogenannte Referenzdaten) geschaK tzt. Diese Methode garantiert eine gute LeistungsfaK higkeit nur dann, wenn eine gro{e Menge homogener Referenzdaten mit der gleichen Statistik wie jene der Testzelle zur VerfuK gung stehen. In diesem Artikel wird eine neue, a-priori-Wissen verwendende Methodik zur Bestimmung der Kovarianzmatrix vorgeschlagen. Dieser Ansatz ist nuK tzlich zur Detektion schwacher Signale in jenen FaK llen, in denen diskrete Objekte in einigen Entfernungszellen existieren, jedoch nicht in anderen Entfernungszellen. Der Schwerpunkt liegt auf der Verwendung eines einfachen Modells fuK r Bodenechos, welches unser a-priori-Wissen uK ber die Struktur der Bodenechos einbezieht. Die neue Methodik kann auf die meisten existierenden STAP-Verfahren angewandt werden. Dies wird durch die Anwendung der Methodik auf drei spezi"sche existierende Verfahren veranschaulicht. Es wird gezeigt, da{ in FaK llen mit nichtstationaK ren gemessenen Daten die LeistungsfaK higkeit der modi"zierten Verfahren jene der existierenden Verfahren im allgemeinen uK bertri!t. ( 1999 Elsevier Science B.V. All rights reserved.
q This paper was supported by the Air Force Research Laboratory, Sensors Directorate and the Air Force O$ce of Scienti"c Research, Air Force Material Command under contracts F30602-97-C-0065-01 and F49620-97-C-0065-01. * Corresponding author. Tel.: #1-610-758-4070; fax: #1-610-758-6279. E-mail address: [email protected] (R.S. Blum)
0165-1684/99/$ - see front matter ( 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 5 - 1 6 8 4 ( 9 9 ) 0 0 1 0 1 - 2
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X. Lin, R.S. Blum / Signal Processing 79 (1999) 273}287
Re2 sume2 Les meH thodes de traitement adaptatif espace}temps (STAP) se sont aveH reH es prometteuses pour les applications de radar aeH rien embarqueH . Toutefois dans la majoriteH des approches on deH veloppe une estimeH e de la matrice de covariance pour la cellule de test, en moyennant sur les cellules l'entourant, appeleH es donneH es de reH feH rence. Cette meH thode est garantie assurer de bonnes performances uniquement lorsqu'un ensemble eH tendu de donneH es de reH feH rence homoge`nes ayant les me( mes statistiques que la cellule de test est disponible. Une meH thodologie nouvelle d'obtention de la matrice de covariance, utilisant les connaissances a priori, est proposeH e dans cet article. Cette approche est utile pour la deH tection de signaux faibles dans les cas ou` ceux-ci sont preH sents dans certaines cellules impliqueH es mais pas dans d'autres. Nous nous focalisons sur l'utilisation d'un mode`le simple du fouillis du( au sol, incorporant nos connaissances a priori sur sa structure. Cette meH thodologie nouvelle peut e( tre appliqueH e a` la plupart des techniques STAP existantes. Ceci est illustreH par une application de cette meH thodologie a` trois techniques existantes speH ci"ques. Les techniques modi"eH es sont montreH es surpasser en geH neH ral les techniques existantes dans des cas de donneH es non stationnaires mesureH es. ( 1999 Elsevier Science B.V. All rights reserved. Keywords: Space}time adaptive processing; STAP; Adaptive antenna arrays; Airborne radar; Non-homogeneous clutter
1. Introduction Space}time adaptive processing (STAP) schemes for airborne radar have received signi"cant attention recently [2,16,17]. Typically these schemes involve using observations from neighboring range cells, called reference data, to estimate the covariance matrix of the clutter-plus-interference in the test range cell. The estimated covariance matrix is then used to de"ne a weight vector, which is used in a linear processing scheme. The weight vector is chosen with the hope that it will suppress the clutter-plus-interference in the test range cell. If the estimated covariance matrix accurately represents the true covariance matrix for the test cell, then most STAP schemes perform well. However, this may not occur in practice. The data taken from surrounding range cells can have di!erent statistics than data taken from the test cell. This type of behavior has been observed in the measured data [7]. The reason for this type of behavior can also be motivated. Consider ground clutter, for example. The clutter part of the radar returns corresponding to di!erent range cells are produced by re#ections from di!erent portions of the ground. If di!erent types of objects reside on these di!erent portions of ground, it is reasonable to expect the clutter returns from these di!erent range cells will be di!erent. In particular, if Gaussian
clutter1 is assumed, the covariance matrices which characterize the clutter returns from the di!erent range cells will be di!erent. As a consequence of this nonstationarity behavior across range cells, the amount of reference data available for use in the estimation is usually quite limited. In this research, a new methodology for constructing robust STAP schemes is proposed for use in highly nonstationary environments. Our approach is to signi"cantly reduce the number of parameters to be estimated. In fact, this idea is similar to what has been used to justify localized or beam-space (more generally reduced dimension) STAP schemes [2,7,16,17], but our reduction is achieved by exploiting the structure of clutter. This idea is new and would appear to have signi"cant potential. The purpose of this paper is to investigate this potential. In order to do this we use a simple model for simple cases with ground clutter only. In the cases we consider here, we assume a clutter ridge which is generally well modeled as being linear with no aliasing. This is similar to the assumption in [3]. Such a model would be obtained in the side looking radar case with a linear array as considered in [16,17]. It is possible to
1 Here we assume Gaussian clutter, but extensions to nonGaussian clutter are possible as we discuss in the conclusions.
X. Lin, R.S. Blum / Signal Processing 79 (1999) 273}287
handle other cases with an altered version of the model we consider, but this will not be addressed here. Thus deviations from this model are not considered. More complicated clutter models could also be employed, however, we show that the resulting scheme can typically provide good performance in simulations or tests we tried with measured radar data. For example, the approach might be extended for jamming. This will be discussed in the conclusions. In Section 2, we introduce the new approach and describe its application to a large set of STAP algorithms. In Section 3, we generally describe our method for extracting necessary parameters from real data. In Section 4, we describe the measured data set used to test performance in this paper. We follow the procedures described in Section 3 to estimate the necessary parameters for this data set. In Section 5, three speci"c STAP algorithms are modi"ed using our approach and their performance is tested using measured data and Monte Carlo simulations. Conclusions are given in Section 6.
2. General approach for modi5ed scheme We now generally discuss some traditional STAP schemes and then describe the modi"cations we suggest based on a simple clutter model proposed in [16]. Three example schemes, the sample matrix inversion algorithm (SMI) [16], the extended factored approach (EFA) [2,17], and the joint-domain localized (JDL) approach [16], are considered. These schemes are described in Appendix A. 2.1. Traditional schemes Assume that the observations to be processed are taken from M di!erent pulse returns from the kth range cell which are received by N antenna elements. Each return is assumed to contain a possible signal in additive noise-plus-clutter. Denote the observation corresponding to the ith pulse at the jth antenna element as x . Each observation is a comij plex number corresponding to the in-phase and quadrature components of the received matched
275
Fig. 1. Processing #ow for a general STAP scheme.
"ltered waveform. The observations to be processed are ordered as X "(x , x ,2, x , x ,2, x )T, (1) k 11 21 N1 12 NM where xT denotes the transpose of the vector x. In (1) the subscript k reminds us this data comes from the kth range cell. A reasonably large set of STAP processing schemes can be described using the general description shown in Fig. 1. The description is broken into three parts: pre-processing, adaptive processing and post-processing. The pre-processing can be described by (2) XI (p)"(A ?B )HX , p"0,1,2,2, P!1, k p p k where X is given in (1). In (2) ? stands for k a Kronecker product2 and A and B are schemep p dependent matrices (speci"c examples are given in Appendix A). The operations in (2) generate P vectors from the single observation vector. Each vector is produced by a possibly di!erent coordinate transformation and selection operation on the observed vector. A simple example is where the observed vector is divided into several equal size vectors each of which corresponds to di!erent groups of pulses which are non-overlapping. The
2 The Kronecker product is best illustrated by an example:
C D 1 2 1 2
C DC D 1 1 1 1
?
1 2
3 4 3 4
"
3 4
1 2 1 2 3 4 3 4
.
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adaptive processing is applied to each of the P vectors as in [17] to produce y (p)"SHR~1(p)XI (p)/U , k k p k p"0,1,2,2, P!1,
(3)
which is a set of P complex numbers. In (3) (see examples in Appendix A) S"S ?S , (4) 5 4 where S is the scheme-dependent temporal steering 5 vector and S is the scheme-dependent spatial steer4 ing vector [16]. The quantity U is the normalizp ation needed to provide a constant false alarm rate (CFAR) in homogeneous clutter [1] and is given by U "JSHR~1(p)S. (5) k p R (p) is an estimated covariance matrix. The postk processing is not necessary if y (p), for a particular k p, is the "nal output of interest. In this case, Dy (p)D k is compared to a threshold directly to decide if target is present. If y (p) is not the "nal output of k interest, then we assemble the complex outputs produced by (3) into a vector > "[y (0), k k y (1),2, y (p!1)]T and compute k k z "f H> . (6) k k In (6), f is typically a particular column of a P]P "lter matrix F (a DFT matrix is one example), and z is the "nal output whose magnitude is compared k to a threshold to make a decision. The subscript k of the "nal output reminds us we are processing data from the kth range cell. In the traditional schemes we considered, the estimated covariance matrix is calculated as k`Q@2`1 + XI (p)XI (p)H (7) i i i/k~Q@2~1,iEk,k~1,k`1 by averaging over Q range cells (assuming Q is even) surrounding the kth range cell, excluding the test cell and the two closest range cells. Most of the STAP algorithms discussed in [17] can be described using (2)}(6). Some important examples are provided in Appendix A. We note that the schemes in Appendix A had been previously described [2,7,16,17], but their description in the general framework of (2)}(6) is apparently new. 1 R (p)" k Q
2.2. Modixed schemes In order to use our prior knowledge of the nature of ground clutter, namely the known structure of the clutter ridge in angle-Doppler space, we use a simple model for ground clutter proposed in [16]. Assume that the clutter portion of the reference samples is Gaussian distributed with the two-dimensional power spectral density (psd) described in [16]
C A
L p2 ( f !f )2 #,d #5,d P ( f , f )" + exp ! 5 # 5 4 2pp p 2p2 f5,d f4,d f5,d d/1 ( f !f )2 #4,d # 4 , (8) 2p2 f4,d which is a function of normalized Doppler frequency f [17] and spatial frequency f [17]. The 5 4 psd in (8) consists of ¸ Gaussian-shaped humps, the dth of which is centered at ( f , f )"( f , f ) and 5 4 #5,d #4,d has amplitude controlled by p2 and a spread in #,d angle and Doppler controlled by p2 and p2 . f4,d f5,d Our approach will assume the clutter samples are described by the psd in (8). Then the data will be used to estimate the parameters in (8). Once the estimates for the parameters in (8) are found, the covariance matrix for clutter described by (8) is given by [16]
BD
L R " + p2 C ?C , (9) # #,d 5,d 4,d d/1 where C and C are Toeplitz matrices speci"ed 5,d 4,d by C "ToeplitzM[1, e~2(ppf5,d )2`*2pf#5,d , 2 , 5,d e~2(ppf5,d (M~1))2`*(M~1)2pf#5,d ]N (10) and C "ToeplitzM[1, e~2(ppf4,d )2`*2pf#4,d , 2 , 4,d e~2(ppf4,d (M~1))2`*(M~1)2pf#4,d ]N, (11) respectively. Now, we can modify the traditional schemes by using the covariance matrix in (9), possibly with slight modi"cation, to replace the covariance matrix estimate of (7).
X. Lin, R.S. Blum / Signal Processing 79 (1999) 273}287
For our modi"ed schemes, the covariance matrix is derived using (2) as R (p)"EM(XI (p)!EMXI (p)N) k k k ](XI (p)!EMXI (p)N)HN k k "(A ?B )HR (A ?B ) p p # p p L "(A ?B )H + p2 C ?C #,d 5,d 4,d p p d/1 ](A ?B ) p p L (12) " + p2 (AHC A )?(BHC B ), p 4,d p #,d p 5,d p d/1 where (9) was used along with Theorems 8.8.2 and 8.8.6 in [5]. C and C are given in (10) and (11). 5,d 4,d A discussion of methods for choosing proper values for the parameters follows.
A
B
3. Estimating the model parameters The most important parameter of the model in (8) would appear to be the slope of the clutter ridge along which we center the Gaussian functions. Since the slope of clutter ridge is related to airplane's velocity [17], this parameter can be obtained directly from instruments on the aircraft and can be updated frequently. For simplicity, we "x the amplitude of the humps, p2 , d"1,2, ¸, at #,d a common value. Without loss of generality, the common value can be "xed at unity, since this will just introduce a scalar multiplier in (3) and (6) that will be compensated for by the threshold of the test, which will be selected to yield a given false alarm probability. For the other parameters, we just attempt to get reasonably accurate values and we attempt to update them as often as possible. This would allow the parameters to change as the clutter environment changes drastically. Of course, in order to allow frequent updates, it is important to keep the procedures used to estimate the parameters simple. While it is theoretically possible to obtain maximum-likelihood estimates of the parameters based on the observed data, we found the complexity of such an approach was very high. Thus, we felt this would defeat the goal of simplicity that we hoped for. We wanted to demonstrate that our approach
277
works well without complicated estimation procedures. Further, we know the clutter model we employ is simplistic and so the advantage of very accurate parameter estimates is questionable. Finally, we expect that good performance can be achieved without extremely accurate estimates, as long as the model roughly approximates the most important characteristics of the clutter. For simplicity, we "x the number of humps used at a particular value ¸ and place them uniformly along a line in the ( f , f ) plane that passes through 5 4 the origin and has slope set by the airplane velocity. Clearly if the number of humps is too small, the model might not be able to represent clutter well. On the other hand, if the number of humps is too large, computational complexity increases with little performance improvement. The parameters which control the width of the clutter ridge, were also taken to be identical so by p2 "p2 "p2 f5,d f4,d for d"1,2, ¸. De"ne R as the (i, j)th component ij of the R"X XH where X is the vector from the k k k test cell which may contain a weak signal (with power smaller than or on the order of the clutter). De"ne R (p, ¸) as the (i, j)th component of R in #,ij # (9). Then we choose p and ¸ which minimize D(p, ¸)"+ DR (p, ¸)!R D2. (13) #,ij ij i,j In order to limit complexity, the calculation can be carried out with a relatively small dimension vector, possibly formed by selecting only some of the components of the true observation vector X . The k estimated width and the number of humps can still be applied to the model for the larger vector X . In k practice, one would use the largest-dimension vector possible with the available computing power, but tests with reasonably small dimensions (3 or 4) appeared to give estimates that worked well.
4. Testing with MCARM data We use measured monostatic airborne radar data that comes from the multi-channel airborne radar measurements (MCARM) program [4,12]. The goal of this program was to accelerate the development of STAP technology through the use of a common set of data. The MCARM array is
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installed on a British Aerospace BAC-11 commercial aircraft, operated by Westinghouse Electric Corporation to serve as an airborne radar testbed. The radar operates at L band. The array is mounted to be sidelooking from a location which is forward from the left wing of the aircraft. The MCARM array employs 22 independent channels and 128 pulses comprise a single coherent processing interval. The number of independent channels sets the maximum size of N, while the number of pulses in a coherent processing interval sets the maximum value of M. The combination of N and M determines the maximum size of the vector X from (1). Here we use data from #ight 5, acquisik tion 575. These data were acquired from a #ight over the Delmarva Peninsula on the east coast of the United States. For more details about MCARM we refer the reader to [4,12]. In our tests with MCARM data, we generally selected the model parameters as described in Section 3. After attempting to select p and ¸ to minimize D(p, ¸), we found "ve humps and p"0.08 provided a near-optimum choice in all of our investigations with MCARM data. This is illus-
trated in Fig. 2, which demonstrates the relationship between D(p, ¸) and the hump width p for a typical ¸"5 case. We note that D(p, ¸) is fairly #at for 0.08(p(0.16 so we selected the smallest value p"0.08 to allow the least nulling of targets close to the clutter ridge while still allowing approximately the same good "t in (9). Fig. 3 shows D(p, ¸) versus ¸ for p"0.08. We choose ¸"5 since D(p, ¸) is fairly #at over a range of ¸ starting near ¸"5 so increasing ¸ further, which increases complexity, yields little decrease in D(p, ¸). Figs. 2 and 3 are for full vector lengths and for range cell 415, but the results for smaller vector lengths and the other cells we considered were so similar, we used these values throughout our study. This implies one may be able to update these parameters infrequently in some case.
5. Performance comparisons To illustrate the modi"ed approach described in Section 2, three typical STAP schemes: SMI, EFA and JDL are used as examples. We call the new
Fig. 2. Normalized D versus the width of humps for data from range cell 415 when the number of humps is set to ¸"5.
X. Lin, R.S. Blum / Signal Processing 79 (1999) 273}287
279
Fig. 3. Normalized D versus ¸ for data from range cell 415 when the hump width is p"0.08.
schemes, obtained using (2)}(6), modi"ed SMI, modi"ed EFA and modi"ed JDL, respectively. For the modi"ed schemes, we do not develop our covariance matrix estimate by performing averages over the adjacent range cells. Instead the covariance matrix is determined using the model parameters in (9) which are obtained from the test cell. Thus mismatch between the statistics of the reference data and the data from the test cells is avoided, which can be a problem for traditional schemes in a highly non-homogeneous environment. To overcome problems from the small amount of data we use prior knowledge on the structure of the interference as shown in (9). In the traditional SMI, EFA and JDL schemes used in our comparisons, we use a set of reference samples on either side of the test cell. In our examples, Q"66 cells (33 on either side of the test cell) are used for SMI and EFA. Q"10 cells are used for JDL ("ve on each side of the test cell). A covariance matrix is formed using (7). We are generally interested in cases with covariance mismatch as we describe next.
If there is a mismatch between the covariance matrix describing the test cell and the covariance matrix describing a particular reference cell, this can occur in two general ways. Either the two covariance matrices di!er only in scale (so that the psds have the same shape) or the psds may have di!erent shapes (and about the same average levels). We call the "rst case scale mismatch and the second shape mismatch. The "rst is due to a change in power level only and is equivalent to incorrectly setting the threshold of the test from (3) and (6). Thus in cases with scale mismatch the quantity in (5) will not provide CFAR. Obviously this can degrade performance. This is really a CFAR problem, which is beyond the scope of this paper.3 The second type of mismatch is due to discretes (or
3 We comment that there are promising techniques to handle this problem, which can be employed in conjunction with our approach. For example, it may be possible to get several &looks' at the same range cell and obtain an averaged estimate of the correct power from the multiple looks.
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some extended interference) which are in one cell but which are not in adjacent cells. There are two possibilities. The "rst case is where the discretes are in the reference cells, but not in the test cell. These cases can cause the target to be nulled if a discrete occupies a location near the target location. The other case is that where the discretes appear only in the test cell. If these discretes are not suppressed by the adaptive pattern, the performance of the traditional schemes might be negatively a!ected. To construct an example to illustrate the "rst kind of shape mismatch, we include range cells 390}395 as part of our reference cells. There is a discrete in range cells 390}395 at Doppler frequency 0.0625 and spatial frequency 0.125 which does not exist in the surrounding range cells. Thus suppose we attempt to "nd a target which is inserted in range bin 380 or 400. If we use range cells 390}395 as part of the reference cells, the adaptive processing in the traditional schemes will try to produce a null at the discrete position. If the target in the test cell happens to be close to this position, the target can be seriously attenuated. To examine this, we generated a set of 4 test cases as given in Table 1. We did not "nd representative examples of the second kind of shape mismatch in the measured data set. However, we performed Monte Carlo simulations to explore the performance degradation in such cases. 5.1. Numerical results of measured data To evaluate the performance in the tests using measured data, we compute an approximation to the signal-to-noise ratio (SNR)4 after the adaptive processing, which is denoted by SNR "10 log 1045 10
A
B
Dz D2!Dz D2 k,1 k,0 , Dz D2 k,0
(14)
where Dz D and Dz D are both de"ned by (6), but k,1 k,0 Dz D is for the case where a target is inserted in the k,1 test cell and Dz D is for the case where it is not. k,0 4 When we use the term SNR we really mean signal-to-noiseplus-clutter ratio. We consider the clutter as part of the noise unless otherwise stated.
Since we inserted the target we are able to compute both cases. To interpret results easily, the amplitude of the target is set as a"0.02 for SMI, and at a"0.005 for EFA and JDL. Previous studies have shown [2,16,17] the performance of EFA and JDL is much better than the performance of SMI in nonhomogeneous cases. Listed in Table 2 is a crude estimate of the average SNR before processing for the range cells where the target is inserted in our test for each of the cases of interest. This crude estimate gives some idea of the power level of the signal as compared to the power level of the noise and is computed as a2 a2 P" " , (15) DX D2 + Dx D2 k i,j ij where the notation from (1) has been employed. 5.1.1. SMI From Table 1, the modi"ed scheme is always better than the traditional scheme for the nonhomogeneous cases considered. This is consistent with previous results which demonstrate that SMI can perform poorly in nonhomogeneous environments [7]. 5.1.2. EFA In all of the cases shown in Table 1, the modi"ed scheme performs better than the traditional one. Generally, the improvement for SMI is larger than EFA. This is reasonable since SMI is a fully adaptive scheme [16,17] and so it would generally be expected to be the most sensitive to poor covariance matrix estimates. 5.1.3. JDL The modi"ed scheme performs better than the traditional one in all cases shown in Table 1. The performance improvement is larger than that for EFA. This is reasonable in the cases we consider since the target location is close to the location of the discrete. Recall for the examples in Table 1, the discrete is located only in the reference data. JDL uses only a few angle-Doppler bins near the target and in the cases we consider the discrete falls into the few angle-Doppler bins used. Thus, all of the
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281
Table 1 Performance comparison between modi"ed scheme and traditional scheme of SMI, EFA and JDL when a target is inserted in di!erent locations SMI (dB) No.
1 2 3 4
EFA (dB)
JDL (dB)
Range cell
Doppler frequency
Space frequency
Traditional
Modi"ed
Traditional
Modi"ed
Traditional
Modi"ed
380 380 400 400
0.0469 0.0859 0.0625 0.0859
0.1250 0.1250 0.0859 0.1641
1.88 0.60 3.81 0.96
10.95 16.29 11.29 19.65
4.26 18.97 7.49 6.61
5.97 20.06 8.74 9.82
!16.09 20.63 10.16 9.74
5.61 30.39 21.63 16.53
Table 2 Average SNR before processing for di!erent range cells under test Average SNR before processing (dB) Target amplitude
Range bin 380
Range bin 400
Range bin 415
0.005 0.02
!27.95 !15.91
!15.79 !3.75
!14.93 !2.89
highly localized energy of the discrete is felt by JDL. We should note that if the discrete is far from the target, JDL works very well, much as in the homogeneous case [16]. In homogeneous cases, using only a few angle-Doppler bins can help avoid the discrete. This is illustrated in our tests using simulations. 5.2. Monte Carlo simulation results To better understand the impact of shape mismatch on performance, we also performed Monte Carlo simulations. We generated the Gaussian clutter data using a covariance matrix R calculated # from a hump model as described in (9). The parameters for the particular hump model used are shown in Table 3 for d"1,2,5. To simulate the two kinds of shape mismatch mentioned above, we added one extra hump, modeling a discrete, in the reference cells or the test cell. The parameters for this discrete is given in Table 3 for d"6. For each kind of mismatch, two cases are considered. In one case, called case (a), the target position (with Doppler frequency set as 0.0469 and angle frequency set as 0.1641) is close to the discrete position. In the
Table 3 Parameters of assumed hump model with the sixth hump simulating a discrete d
p #,d
p f5,d
p f4,d
f #5,d
f #4,d
1 2 3 4 5 6
2 2 8 2 2 6
0.08 0.08 0.08 0.08 0.08 0.02
0.08 0.08 0.08 0.08 0.08 0.02
!0.333 !0.167 0 0.167 0.333 0.086
!0.333 !0.167 0 0.167 0.333 0.086
other case, called case (b), the target position (with Doppler frequency set as !0.1563 and angle frequency set as !0.0313) is far away from the discrete position. By examining these two cases, we studied how the relative position of a target to a discrete a!ects the performance for di!erent schemes. For a "xed false alarm probability (set at 0.015), we compared the probability of detection of the traditional scheme and the modi"ed scheme as a function of SNR calculated in full dimension space as SNR"10 log (sHR~1s), where s is the # 10 target observed in zero-mean clutter with covariance matrix R . In our Monte Carlo simula# tions, 10 000 runs were used. 5.2.1. SMI When a discrete is in the reference cells and not in the test cell, as in the case considered in Fig. 4,
5 To improve the accuracy in our simulations, this value is set higher than the practical values, which are typically in the range of 10~6 or less. Note that this does not alter the trends observed.
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Fig. 4. Performance comparison for modi"ed and traditional SMI when a discrete is in the reference cells (Monte Carlo simulation).
the modi"ed scheme is always better than the traditional scheme regardless of the relative position of the target and the discrete. This is expected since SMI is a fully adaptive scheme and sensitive to incorrect covariance estimates. Further, it is reasonable that more performance improvement is obtained in case (a). Fig. 4 also shows that the two curves for modi"ed SMI representing cases (a) and (b) almost overlap each other. This is reasonable because in this case the estimated covariance matrix is not in#uenced by the discrete since the discrete is in the reference data and these data are not used to estimate the covariance matrix. Thus no matter where the target is, correct covariance estimates are obtained and similar performance should be achieved in both cases (a) and (b). When a discrete is in the test cell and not in the reference cells, the modi"ed scheme is still fairly insensitive to the discrete as shown in Fig. 5. This time the reason is that the discrete is properly nulled by the modi"ed scheme. 5.2.2. EFA Similar results are observed for both kinds of shape mismatch as illustrated in Figs. 6 and 7. We
have already explained the advantage of modi"ed EFA over traditional EFA after presenting the measured data results. However, the performance improvement observed in simulations are even larger than that which were observed in Table 1. The reason is that only some of reference cells had discretes in them in the measured data cases, but in the simulation cases all the reference cells contain discretes. Once again the modi"ed scheme provides large improvements for case (a). There is a slight performance improvement in case (b) when using the modi"ed scheme instead of the traditional scheme. This might be expected since EFA is an element}space scheme. Thus it does not perform the adaptive processing completely in the angleDoppler domain so it is di$cult for EFA to completely exploit the occurance of the target being far from the discrete. This is in contrast to JDL as we shall see. 5.2.3. JDL As expected, modi"ed JDL performs much better than traditional JDL in the case when the target is close to the discrete. When the target is far from the
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Fig. 5. Performance comparison for modi"ed and traditional SMI when a discrete is in test cell (Monte Carlo simulation).
Fig. 6. Performance comparison for modi"ed and traditional EFA when a discrete is in the reference cells (Monte Carlo simulation).
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Fig. 7. Performance comparison for modi"ed and traditional EFA when a discrete is in test cell (Monte Carlo simulation).
discrete, modi"ed JDL has similar performance to traditional JDL. This is illustrated in Figs. 8 and 9. Since JDL performs adaptive processing using only a few angle-Doppler bins near the target location, this makes sense.
6. Conclusions We have suggested the use of some modi"ed STAP schemes for airborne radar applications, which use knowledge of the structure of the interference. The speci"c numerical results given use a model that only assumes ground clutter, but extensions are possible. Further our approach works well for a measured radar data set that includes other interference. From the comparisons, it appears that the modi"ed scheme can provide good performance for cases with shape mismatch. This is signi"cant since these cases are traditionally considered to be di$cult and have been observed in measured radar data. The improvements are especially signi"cant for schemes that are sensitive to poor covariance
estimates. The modi"ed schemes generally perform better in non-homogeneous cases when compared to schemes that average over range to estimate the required covariance matrix. In practical situations it would be useful to determine whether the clutter is non-homogeneous in order to determine if the modi"ed schemes must be used. Recent research [8] suggests some methods to make such determinations. These methods could be useful in our application. In [8] the authors discuss three methods for assessing the relative homogeneity among a set of available secondary data. The data are assumed to be Gaussian distributed, the same assumption made in this paper. The authors demonstrate that their nonhomogeneity detectors do work. In fact, they use their method to provide improved STAP performance for cases where one has a reasonably large set of reference data, some of which is homogeneous. The methods we propose here would be suitable for cases where a reasonably large set of homogeneous data is not available. In this paper we considered only cases with ground clutter in formulating our method. It might
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285
Fig. 8. Performance comparison for modi"ed and traditional JDL when a discrete is in the reference cells (Monte Carlo simulation).
Fig. 9. Performance comparison for modi"ed and traditional JDL when a discrete is in the test cell (Monte Carlo simulation).
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be possible to also handle other types of interference and, in particular, jamming with a slight extension. If the location of a jammer or another target (other than the one currently being searched for) is known from a previous scan then this knowledge could be employed by locating a null in the adaptive pattern at the appropriate angle and Doppler. On the other hand, if there is signi"cant interference that cannot be easily modeled due to lack of knowledge, then a di!erent approach might be justi"ed. If the interference is assumed to be independent of the ground clutter, a reasonable assumption for jamming for example, then the total covariance matrix is
Gaussian clutter, the spherically invariant random vector (SIRV) model [11]. Under some fairly unrestrictive conditions, a canonical form for the optimal nonlinear receiver structure for this case is presented in [13]. As shown in Fig. 10, this form incorporates the conventional Gaussian receiver ¹ (r)"sHR~1r which is the optimum for Gaussian ' clutter. Here r is the vector of the received data, s corresponds to the signal vector which may be present in r, and R is the covariance matrix of r. As illustrated in Fig. 10, the optimum test also incorporates the quadratic form ¹ (r)"rHR~1r. The 2 optimum Neyman Pearson test under the fairly unrestrictive conditions [11,13] can be expressed as
R "R #R , (16) 5 # I where R comes from (9) and R , the covariance # I matrix of the interference, could be estimated as in (7). This approach still insures the structure of ground clutter is included, but allows some arbitrary adaption. Also notice that, in the presence of jamming, the estimations of p and ¸ might be a!ected and deviate from their &true' values when (13) is applied. We suggest performing the estimations in the angle-Doppler domain by using robust image-processing techniques to limit this somewhat. In a slight extension, R and R can each be # I weighted by scalars which incorporate how con"dent we are that only ground clutter is present. We can also imagine developing methods for obtaining these scalar weights adaptively from the observed data. In this paper we considered only Gaussian clutter. However, our method can be extended to some cases with non-Gaussian and, in particular, impulsive clutter. Consider a common model for non-
¹(r)"g [¹ (r), ¹ (r)];HH1 g, (17) NL ' q :0 where g [),)] is the optimal memoryless non-linNL earity which can be determined for any SIRV clutter distribution [10,13]. Using our technique to produce the covariance matrix R allows us to be able to handle non-Gaussian clutter cases with SIRV clutter distributions as shown in Fig. 10. Another popular model for non-Gaussian clutter, the sub-Gaussian alpha stable random vector model [15], can also employ our covariance matrix. The application is the same as for SIRVs, since the sub-Gaussian alpha stable random vector model is a special case of an SIRV model. In obtaining the parameters of our model, one could employ robust estimation techniques [6]. These techniques generally employ limiting so that large impulsive samples do not overly bias the estimates. A recent application using simple robust estimates for cases with alpha stable noise appears in [14]. For further reading, see [9].
Fig. 10. Canonical forms of the optimum SIRV processor.
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Appendix A SMI
EFA
JDL
In our tests of SMI we use M"22 pulses and N"22 antennas in the adaptive processing. For the SMI scheme, we have A "I , B "I in (2), where I , I are p M p N M N M]M and N]N identity matrices, respectively. S and S are the target temporal and 5 4 spatial response vectors, respectively [16]. Post-processing is not necessary for SMI. For the EFA scheme, the pre-processing is as in (2) with A "[f ,f ,f ], B "I , (A.1) p p~1 p p`1 p N where f ,f ,f are three adjacent Dopp~1 p p`1 pler "lters and f corresponds to the norp malized Doppler frequency of target. The adaptive processing uses S "[0 1 0]T and 5 S set to the target spatial response vector. 4 Only processing for a single p must be performed and so post-processing is not needed. For JDL, the matrices in (2) should be taken as A "[f , f , f ], B "[g , g , g ], p p~1 p p`1 p q~1 q q`1 (A.2) where f ,f ,f are three Doppler "lters p~1 p p`1 and f is that "lter which corresponds to the p normalized Doppler frequency of target. f and f are Doppler "lters correp~1 p`1 sponding to neighboring Doppler frequencies. g ,g ,g are three spatial "lters q~1 q q`1 and g is that "lter which corresponds the q spatial steering vector for the target. g and g are spatial "lters correspondq~1 q`1 ing to neighboring spatial frequencies. The steering vector S in the adaptive processing is S"S ?S "[0 1 0]?[0 1 0]. (A.3) 5 s As for JDL, processing must be performed for only one p so post-processing is not needed.
References [1] W.S. Chen, I.S. Reed, A new CFAR detection test for radar, Digital Signal Process. 4 (October 1991) 198}214.
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[2] R.C. Dipietro, Extended factored space-time processing techniques for airborne radar, in: Proceedings of the 25th Asilomar Conference, Paci"c Grove, PA, October 1992, pp. 425}430. [3] A. Farina, P. Lombardo, M. Pirri, Nonlinear nonadaptive space-time processing for airborne early warning radar, in: Proceedings of IEE on Radar, Sonar and Navigation, Vol. 145, No. 1, February 1998, pp. 9}18. [4] D.K. Fenner, W.F. Hoover, Test results of a space-time adaptive processing system for airborne early warning radar, in: Proceedings of the IEEE National Radar Conference, Ann Arbor, MI, May 1996, pp. 88}93. [5] F.A. Fraybill, Introduction to Matrices with Applications in Statistics, Wadsworth Publishing, Belmont, CA, 1969, especially pp. 201}204. [6] S.A. Kassam, H.V. Poor, Robust techniques for signal processing: a survey, Proc. IEEE 73 (3) (March 1985) 433}481. [7] W.L. Melvin, B. Himed, Comparative analysis of spacetime adaptive algorithms with measured airborne data, Presented at the Seventh International Conference on Signal Processing Applications and Technology, 7}10 October 1996, pp. 1479}1483. [8] W.L. Melvin, M.C. Wicks, Improving practical space-time adaptive radar, in: Proceedings of the IEEE National Radar Conference, Syracuse, NY, May 1997, pp. 48}53. [9] W.L. Melvin, M.C. Wicks, R.D. Brown, Assessment of multichannel airborne radar measurements for analysis and design of space-time processing architectures and algorithms, in: Proceedings of the IEEE National Radar Conference, Ann Arbor, MI, May 1996, pp. 130}135. [10] A. Ozturk, An application of the distributed identi"cation algorithm to signal detection problems, in: Twenty-seventh Asilomar Conference on Signals, Systems, and Computers, Vol. 2, November 1993, pp. 248}252. [11] M. Rangaswamy, D. Weiner, A. Ozturk, Non-Gaussian random vector identi"cation using spherically invariant random processes, IEEE Trans. Aerospace Electron. Systems AES-29 (January 1993) 111}124. [12] D. Sloper et al., MCARM Final Report, Rome Lab tech. Rept., RL-TR-96-49, April 1996. [13] D.L. Stadelman, D.D. Weiner, Detection of weak signals with random parameters in non-Gaussian clutter, in: Proceedings of SPIE } The International Society for Optical Engineering, Vol. 2561, Bellingham, WA, 1995, pp. 31}42. [14] A. Swami, B. Sadler, Parameter estimation for linear alpha-stable processes, IEEE Signal Process. Lett. 5 (February 1998) 48}50. [15] G.A. Tsihrintzis, C.L. Nikias, Data-adaptive algorithms for signal detection in sub-Gaussian impulsive interference, IEEE Trans. Signal Process. (July 1997) 1873}1878. [16] H. Wang, L. Cai, On adaptive spatial-temporal processing for airborne surveillance radar systems, IEEE Trans. Aerospace Electron. Systems 30 (July 1994) 660}669. [17] J. Ward, Space-time adaptive processing for airborne radar, Technical Report 1015, Lincoln Laboratory, 1995.
Robust STAP algorithms using prior knowledge for airborne radar applicationsq X. Lin, R.S. Blum* EECS Department, Packard Lab, Lehigh University, Bethlehem, PA 18015-3084, USA Received 12 April 1999; received in revised form 29 June 1999
Abstract Space}time adaptive processing (STAP) schemes have shown promise for airborne radar applications. However, the majority of schemes develop an estimate of the covariance matrix for the test cell by averaging over surrounding range cells, called reference data. This method is only guaranteed to ensure good performance when a large set of homogeneous reference data is available with the same statistics as the test cell. In this paper, a new methodology is proposed for obtaining a covariance matrix that uses a priori knowledge. The approach is useful for detecting weak signals in cases with discretes in some range cells which do not appear in other range cells. The focus is on the use of a simple model for ground clutter that incorporates our prior knowledge on the structure of the ground clutter. The new methodology can be applied to most existing STAP schemes. This is illustrated by applying the methodology to three speci"c existing schemes. The modi"ed schemes are generally shown to outperform the existing schemes in non-stationary measured data cases. ( 1999 Elsevier Science B.V. All rights reserved. Zusammenfassung Verfahren der adaptiven Raum}Zeit-Verarbeitung (STAP) haben sich als vielversprechend fuK r Anwendungen auf dem Gebiet des Luft-Radar erwiesen. Jedoch wird beim Gro{teil der Methoden die Kovarianzmatrix fuK r die Testzelle durch Mittelung uK ber umliegende Entfernungszellen (sogenannte Referenzdaten) geschaK tzt. Diese Methode garantiert eine gute LeistungsfaK higkeit nur dann, wenn eine gro{e Menge homogener Referenzdaten mit der gleichen Statistik wie jene der Testzelle zur VerfuK gung stehen. In diesem Artikel wird eine neue, a-priori-Wissen verwendende Methodik zur Bestimmung der Kovarianzmatrix vorgeschlagen. Dieser Ansatz ist nuK tzlich zur Detektion schwacher Signale in jenen FaK llen, in denen diskrete Objekte in einigen Entfernungszellen existieren, jedoch nicht in anderen Entfernungszellen. Der Schwerpunkt liegt auf der Verwendung eines einfachen Modells fuK r Bodenechos, welches unser a-priori-Wissen uK ber die Struktur der Bodenechos einbezieht. Die neue Methodik kann auf die meisten existierenden STAP-Verfahren angewandt werden. Dies wird durch die Anwendung der Methodik auf drei spezi"sche existierende Verfahren veranschaulicht. Es wird gezeigt, da{ in FaK llen mit nichtstationaK ren gemessenen Daten die LeistungsfaK higkeit der modi"zierten Verfahren jene der existierenden Verfahren im allgemeinen uK bertri!t. ( 1999 Elsevier Science B.V. All rights reserved.
q This paper was supported by the Air Force Research Laboratory, Sensors Directorate and the Air Force O$ce of Scienti"c Research, Air Force Material Command under contracts F30602-97-C-0065-01 and F49620-97-C-0065-01. * Corresponding author. Tel.: #1-610-758-4070; fax: #1-610-758-6279. E-mail address: [email protected] (R.S. Blum)
0165-1684/99/$ - see front matter ( 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 5 - 1 6 8 4 ( 9 9 ) 0 0 1 0 1 - 2
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Re2 sume2 Les meH thodes de traitement adaptatif espace}temps (STAP) se sont aveH reH es prometteuses pour les applications de radar aeH rien embarqueH . Toutefois dans la majoriteH des approches on deH veloppe une estimeH e de la matrice de covariance pour la cellule de test, en moyennant sur les cellules l'entourant, appeleH es donneH es de reH feH rence. Cette meH thode est garantie assurer de bonnes performances uniquement lorsqu'un ensemble eH tendu de donneH es de reH feH rence homoge`nes ayant les me( mes statistiques que la cellule de test est disponible. Une meH thodologie nouvelle d'obtention de la matrice de covariance, utilisant les connaissances a priori, est proposeH e dans cet article. Cette approche est utile pour la deH tection de signaux faibles dans les cas ou` ceux-ci sont preH sents dans certaines cellules impliqueH es mais pas dans d'autres. Nous nous focalisons sur l'utilisation d'un mode`le simple du fouillis du( au sol, incorporant nos connaissances a priori sur sa structure. Cette meH thodologie nouvelle peut e( tre appliqueH e a` la plupart des techniques STAP existantes. Ceci est illustreH par une application de cette meH thodologie a` trois techniques existantes speH ci"ques. Les techniques modi"eH es sont montreH es surpasser en geH neH ral les techniques existantes dans des cas de donneH es non stationnaires mesureH es. ( 1999 Elsevier Science B.V. All rights reserved. Keywords: Space}time adaptive processing; STAP; Adaptive antenna arrays; Airborne radar; Non-homogeneous clutter
1. Introduction Space}time adaptive processing (STAP) schemes for airborne radar have received signi"cant attention recently [2,16,17]. Typically these schemes involve using observations from neighboring range cells, called reference data, to estimate the covariance matrix of the clutter-plus-interference in the test range cell. The estimated covariance matrix is then used to de"ne a weight vector, which is used in a linear processing scheme. The weight vector is chosen with the hope that it will suppress the clutter-plus-interference in the test range cell. If the estimated covariance matrix accurately represents the true covariance matrix for the test cell, then most STAP schemes perform well. However, this may not occur in practice. The data taken from surrounding range cells can have di!erent statistics than data taken from the test cell. This type of behavior has been observed in the measured data [7]. The reason for this type of behavior can also be motivated. Consider ground clutter, for example. The clutter part of the radar returns corresponding to di!erent range cells are produced by re#ections from di!erent portions of the ground. If di!erent types of objects reside on these di!erent portions of ground, it is reasonable to expect the clutter returns from these di!erent range cells will be di!erent. In particular, if Gaussian
clutter1 is assumed, the covariance matrices which characterize the clutter returns from the di!erent range cells will be di!erent. As a consequence of this nonstationarity behavior across range cells, the amount of reference data available for use in the estimation is usually quite limited. In this research, a new methodology for constructing robust STAP schemes is proposed for use in highly nonstationary environments. Our approach is to signi"cantly reduce the number of parameters to be estimated. In fact, this idea is similar to what has been used to justify localized or beam-space (more generally reduced dimension) STAP schemes [2,7,16,17], but our reduction is achieved by exploiting the structure of clutter. This idea is new and would appear to have signi"cant potential. The purpose of this paper is to investigate this potential. In order to do this we use a simple model for simple cases with ground clutter only. In the cases we consider here, we assume a clutter ridge which is generally well modeled as being linear with no aliasing. This is similar to the assumption in [3]. Such a model would be obtained in the side looking radar case with a linear array as considered in [16,17]. It is possible to
1 Here we assume Gaussian clutter, but extensions to nonGaussian clutter are possible as we discuss in the conclusions.
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handle other cases with an altered version of the model we consider, but this will not be addressed here. Thus deviations from this model are not considered. More complicated clutter models could also be employed, however, we show that the resulting scheme can typically provide good performance in simulations or tests we tried with measured radar data. For example, the approach might be extended for jamming. This will be discussed in the conclusions. In Section 2, we introduce the new approach and describe its application to a large set of STAP algorithms. In Section 3, we generally describe our method for extracting necessary parameters from real data. In Section 4, we describe the measured data set used to test performance in this paper. We follow the procedures described in Section 3 to estimate the necessary parameters for this data set. In Section 5, three speci"c STAP algorithms are modi"ed using our approach and their performance is tested using measured data and Monte Carlo simulations. Conclusions are given in Section 6.
2. General approach for modi5ed scheme We now generally discuss some traditional STAP schemes and then describe the modi"cations we suggest based on a simple clutter model proposed in [16]. Three example schemes, the sample matrix inversion algorithm (SMI) [16], the extended factored approach (EFA) [2,17], and the joint-domain localized (JDL) approach [16], are considered. These schemes are described in Appendix A. 2.1. Traditional schemes Assume that the observations to be processed are taken from M di!erent pulse returns from the kth range cell which are received by N antenna elements. Each return is assumed to contain a possible signal in additive noise-plus-clutter. Denote the observation corresponding to the ith pulse at the jth antenna element as x . Each observation is a comij plex number corresponding to the in-phase and quadrature components of the received matched
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Fig. 1. Processing #ow for a general STAP scheme.
"ltered waveform. The observations to be processed are ordered as X "(x , x ,2, x , x ,2, x )T, (1) k 11 21 N1 12 NM where xT denotes the transpose of the vector x. In (1) the subscript k reminds us this data comes from the kth range cell. A reasonably large set of STAP processing schemes can be described using the general description shown in Fig. 1. The description is broken into three parts: pre-processing, adaptive processing and post-processing. The pre-processing can be described by (2) XI (p)"(A ?B )HX , p"0,1,2,2, P!1, k p p k where X is given in (1). In (2) ? stands for k a Kronecker product2 and A and B are schemep p dependent matrices (speci"c examples are given in Appendix A). The operations in (2) generate P vectors from the single observation vector. Each vector is produced by a possibly di!erent coordinate transformation and selection operation on the observed vector. A simple example is where the observed vector is divided into several equal size vectors each of which corresponds to di!erent groups of pulses which are non-overlapping. The
2 The Kronecker product is best illustrated by an example:
C D 1 2 1 2
C DC D 1 1 1 1
?
1 2
3 4 3 4
"
3 4
1 2 1 2 3 4 3 4
.
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adaptive processing is applied to each of the P vectors as in [17] to produce y (p)"SHR~1(p)XI (p)/U , k k p k p"0,1,2,2, P!1,
(3)
which is a set of P complex numbers. In (3) (see examples in Appendix A) S"S ?S , (4) 5 4 where S is the scheme-dependent temporal steering 5 vector and S is the scheme-dependent spatial steer4 ing vector [16]. The quantity U is the normalizp ation needed to provide a constant false alarm rate (CFAR) in homogeneous clutter [1] and is given by U "JSHR~1(p)S. (5) k p R (p) is an estimated covariance matrix. The postk processing is not necessary if y (p), for a particular k p, is the "nal output of interest. In this case, Dy (p)D k is compared to a threshold directly to decide if target is present. If y (p) is not the "nal output of k interest, then we assemble the complex outputs produced by (3) into a vector > "[y (0), k k y (1),2, y (p!1)]T and compute k k z "f H> . (6) k k In (6), f is typically a particular column of a P]P "lter matrix F (a DFT matrix is one example), and z is the "nal output whose magnitude is compared k to a threshold to make a decision. The subscript k of the "nal output reminds us we are processing data from the kth range cell. In the traditional schemes we considered, the estimated covariance matrix is calculated as k`Q@2`1 + XI (p)XI (p)H (7) i i i/k~Q@2~1,iEk,k~1,k`1 by averaging over Q range cells (assuming Q is even) surrounding the kth range cell, excluding the test cell and the two closest range cells. Most of the STAP algorithms discussed in [17] can be described using (2)}(6). Some important examples are provided in Appendix A. We note that the schemes in Appendix A had been previously described [2,7,16,17], but their description in the general framework of (2)}(6) is apparently new. 1 R (p)" k Q
2.2. Modixed schemes In order to use our prior knowledge of the nature of ground clutter, namely the known structure of the clutter ridge in angle-Doppler space, we use a simple model for ground clutter proposed in [16]. Assume that the clutter portion of the reference samples is Gaussian distributed with the two-dimensional power spectral density (psd) described in [16]
C A
L p2 ( f !f )2 #,d #5,d P ( f , f )" + exp ! 5 # 5 4 2pp p 2p2 f5,d f4,d f5,d d/1 ( f !f )2 #4,d # 4 , (8) 2p2 f4,d which is a function of normalized Doppler frequency f [17] and spatial frequency f [17]. The 5 4 psd in (8) consists of ¸ Gaussian-shaped humps, the dth of which is centered at ( f , f )"( f , f ) and 5 4 #5,d #4,d has amplitude controlled by p2 and a spread in #,d angle and Doppler controlled by p2 and p2 . f4,d f5,d Our approach will assume the clutter samples are described by the psd in (8). Then the data will be used to estimate the parameters in (8). Once the estimates for the parameters in (8) are found, the covariance matrix for clutter described by (8) is given by [16]
BD
L R " + p2 C ?C , (9) # #,d 5,d 4,d d/1 where C and C are Toeplitz matrices speci"ed 5,d 4,d by C "ToeplitzM[1, e~2(ppf5,d )2`*2pf#5,d , 2 , 5,d e~2(ppf5,d (M~1))2`*(M~1)2pf#5,d ]N (10) and C "ToeplitzM[1, e~2(ppf4,d )2`*2pf#4,d , 2 , 4,d e~2(ppf4,d (M~1))2`*(M~1)2pf#4,d ]N, (11) respectively. Now, we can modify the traditional schemes by using the covariance matrix in (9), possibly with slight modi"cation, to replace the covariance matrix estimate of (7).
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For our modi"ed schemes, the covariance matrix is derived using (2) as R (p)"EM(XI (p)!EMXI (p)N) k k k ](XI (p)!EMXI (p)N)HN k k "(A ?B )HR (A ?B ) p p # p p L "(A ?B )H + p2 C ?C #,d 5,d 4,d p p d/1 ](A ?B ) p p L (12) " + p2 (AHC A )?(BHC B ), p 4,d p #,d p 5,d p d/1 where (9) was used along with Theorems 8.8.2 and 8.8.6 in [5]. C and C are given in (10) and (11). 5,d 4,d A discussion of methods for choosing proper values for the parameters follows.
A
B
3. Estimating the model parameters The most important parameter of the model in (8) would appear to be the slope of the clutter ridge along which we center the Gaussian functions. Since the slope of clutter ridge is related to airplane's velocity [17], this parameter can be obtained directly from instruments on the aircraft and can be updated frequently. For simplicity, we "x the amplitude of the humps, p2 , d"1,2, ¸, at #,d a common value. Without loss of generality, the common value can be "xed at unity, since this will just introduce a scalar multiplier in (3) and (6) that will be compensated for by the threshold of the test, which will be selected to yield a given false alarm probability. For the other parameters, we just attempt to get reasonably accurate values and we attempt to update them as often as possible. This would allow the parameters to change as the clutter environment changes drastically. Of course, in order to allow frequent updates, it is important to keep the procedures used to estimate the parameters simple. While it is theoretically possible to obtain maximum-likelihood estimates of the parameters based on the observed data, we found the complexity of such an approach was very high. Thus, we felt this would defeat the goal of simplicity that we hoped for. We wanted to demonstrate that our approach
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works well without complicated estimation procedures. Further, we know the clutter model we employ is simplistic and so the advantage of very accurate parameter estimates is questionable. Finally, we expect that good performance can be achieved without extremely accurate estimates, as long as the model roughly approximates the most important characteristics of the clutter. For simplicity, we "x the number of humps used at a particular value ¸ and place them uniformly along a line in the ( f , f ) plane that passes through 5 4 the origin and has slope set by the airplane velocity. Clearly if the number of humps is too small, the model might not be able to represent clutter well. On the other hand, if the number of humps is too large, computational complexity increases with little performance improvement. The parameters which control the width of the clutter ridge, were also taken to be identical so by p2 "p2 "p2 f5,d f4,d for d"1,2, ¸. De"ne R as the (i, j)th component ij of the R"X XH where X is the vector from the k k k test cell which may contain a weak signal (with power smaller than or on the order of the clutter). De"ne R (p, ¸) as the (i, j)th component of R in #,ij # (9). Then we choose p and ¸ which minimize D(p, ¸)"+ DR (p, ¸)!R D2. (13) #,ij ij i,j In order to limit complexity, the calculation can be carried out with a relatively small dimension vector, possibly formed by selecting only some of the components of the true observation vector X . The k estimated width and the number of humps can still be applied to the model for the larger vector X . In k practice, one would use the largest-dimension vector possible with the available computing power, but tests with reasonably small dimensions (3 or 4) appeared to give estimates that worked well.
4. Testing with MCARM data We use measured monostatic airborne radar data that comes from the multi-channel airborne radar measurements (MCARM) program [4,12]. The goal of this program was to accelerate the development of STAP technology through the use of a common set of data. The MCARM array is
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installed on a British Aerospace BAC-11 commercial aircraft, operated by Westinghouse Electric Corporation to serve as an airborne radar testbed. The radar operates at L band. The array is mounted to be sidelooking from a location which is forward from the left wing of the aircraft. The MCARM array employs 22 independent channels and 128 pulses comprise a single coherent processing interval. The number of independent channels sets the maximum size of N, while the number of pulses in a coherent processing interval sets the maximum value of M. The combination of N and M determines the maximum size of the vector X from (1). Here we use data from #ight 5, acquisik tion 575. These data were acquired from a #ight over the Delmarva Peninsula on the east coast of the United States. For more details about MCARM we refer the reader to [4,12]. In our tests with MCARM data, we generally selected the model parameters as described in Section 3. After attempting to select p and ¸ to minimize D(p, ¸), we found "ve humps and p"0.08 provided a near-optimum choice in all of our investigations with MCARM data. This is illus-
trated in Fig. 2, which demonstrates the relationship between D(p, ¸) and the hump width p for a typical ¸"5 case. We note that D(p, ¸) is fairly #at for 0.08(p(0.16 so we selected the smallest value p"0.08 to allow the least nulling of targets close to the clutter ridge while still allowing approximately the same good "t in (9). Fig. 3 shows D(p, ¸) versus ¸ for p"0.08. We choose ¸"5 since D(p, ¸) is fairly #at over a range of ¸ starting near ¸"5 so increasing ¸ further, which increases complexity, yields little decrease in D(p, ¸). Figs. 2 and 3 are for full vector lengths and for range cell 415, but the results for smaller vector lengths and the other cells we considered were so similar, we used these values throughout our study. This implies one may be able to update these parameters infrequently in some case.
5. Performance comparisons To illustrate the modi"ed approach described in Section 2, three typical STAP schemes: SMI, EFA and JDL are used as examples. We call the new
Fig. 2. Normalized D versus the width of humps for data from range cell 415 when the number of humps is set to ¸"5.
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279
Fig. 3. Normalized D versus ¸ for data from range cell 415 when the hump width is p"0.08.
schemes, obtained using (2)}(6), modi"ed SMI, modi"ed EFA and modi"ed JDL, respectively. For the modi"ed schemes, we do not develop our covariance matrix estimate by performing averages over the adjacent range cells. Instead the covariance matrix is determined using the model parameters in (9) which are obtained from the test cell. Thus mismatch between the statistics of the reference data and the data from the test cells is avoided, which can be a problem for traditional schemes in a highly non-homogeneous environment. To overcome problems from the small amount of data we use prior knowledge on the structure of the interference as shown in (9). In the traditional SMI, EFA and JDL schemes used in our comparisons, we use a set of reference samples on either side of the test cell. In our examples, Q"66 cells (33 on either side of the test cell) are used for SMI and EFA. Q"10 cells are used for JDL ("ve on each side of the test cell). A covariance matrix is formed using (7). We are generally interested in cases with covariance mismatch as we describe next.
If there is a mismatch between the covariance matrix describing the test cell and the covariance matrix describing a particular reference cell, this can occur in two general ways. Either the two covariance matrices di!er only in scale (so that the psds have the same shape) or the psds may have di!erent shapes (and about the same average levels). We call the "rst case scale mismatch and the second shape mismatch. The "rst is due to a change in power level only and is equivalent to incorrectly setting the threshold of the test from (3) and (6). Thus in cases with scale mismatch the quantity in (5) will not provide CFAR. Obviously this can degrade performance. This is really a CFAR problem, which is beyond the scope of this paper.3 The second type of mismatch is due to discretes (or
3 We comment that there are promising techniques to handle this problem, which can be employed in conjunction with our approach. For example, it may be possible to get several &looks' at the same range cell and obtain an averaged estimate of the correct power from the multiple looks.
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some extended interference) which are in one cell but which are not in adjacent cells. There are two possibilities. The "rst case is where the discretes are in the reference cells, but not in the test cell. These cases can cause the target to be nulled if a discrete occupies a location near the target location. The other case is that where the discretes appear only in the test cell. If these discretes are not suppressed by the adaptive pattern, the performance of the traditional schemes might be negatively a!ected. To construct an example to illustrate the "rst kind of shape mismatch, we include range cells 390}395 as part of our reference cells. There is a discrete in range cells 390}395 at Doppler frequency 0.0625 and spatial frequency 0.125 which does not exist in the surrounding range cells. Thus suppose we attempt to "nd a target which is inserted in range bin 380 or 400. If we use range cells 390}395 as part of the reference cells, the adaptive processing in the traditional schemes will try to produce a null at the discrete position. If the target in the test cell happens to be close to this position, the target can be seriously attenuated. To examine this, we generated a set of 4 test cases as given in Table 1. We did not "nd representative examples of the second kind of shape mismatch in the measured data set. However, we performed Monte Carlo simulations to explore the performance degradation in such cases. 5.1. Numerical results of measured data To evaluate the performance in the tests using measured data, we compute an approximation to the signal-to-noise ratio (SNR)4 after the adaptive processing, which is denoted by SNR "10 log 1045 10
A
B
Dz D2!Dz D2 k,1 k,0 , Dz D2 k,0
(14)
where Dz D and Dz D are both de"ned by (6), but k,1 k,0 Dz D is for the case where a target is inserted in the k,1 test cell and Dz D is for the case where it is not. k,0 4 When we use the term SNR we really mean signal-to-noiseplus-clutter ratio. We consider the clutter as part of the noise unless otherwise stated.
Since we inserted the target we are able to compute both cases. To interpret results easily, the amplitude of the target is set as a"0.02 for SMI, and at a"0.005 for EFA and JDL. Previous studies have shown [2,16,17] the performance of EFA and JDL is much better than the performance of SMI in nonhomogeneous cases. Listed in Table 2 is a crude estimate of the average SNR before processing for the range cells where the target is inserted in our test for each of the cases of interest. This crude estimate gives some idea of the power level of the signal as compared to the power level of the noise and is computed as a2 a2 P" " , (15) DX D2 + Dx D2 k i,j ij where the notation from (1) has been employed. 5.1.1. SMI From Table 1, the modi"ed scheme is always better than the traditional scheme for the nonhomogeneous cases considered. This is consistent with previous results which demonstrate that SMI can perform poorly in nonhomogeneous environments [7]. 5.1.2. EFA In all of the cases shown in Table 1, the modi"ed scheme performs better than the traditional one. Generally, the improvement for SMI is larger than EFA. This is reasonable since SMI is a fully adaptive scheme [16,17] and so it would generally be expected to be the most sensitive to poor covariance matrix estimates. 5.1.3. JDL The modi"ed scheme performs better than the traditional one in all cases shown in Table 1. The performance improvement is larger than that for EFA. This is reasonable in the cases we consider since the target location is close to the location of the discrete. Recall for the examples in Table 1, the discrete is located only in the reference data. JDL uses only a few angle-Doppler bins near the target and in the cases we consider the discrete falls into the few angle-Doppler bins used. Thus, all of the
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Table 1 Performance comparison between modi"ed scheme and traditional scheme of SMI, EFA and JDL when a target is inserted in di!erent locations SMI (dB) No.
1 2 3 4
EFA (dB)
JDL (dB)
Range cell
Doppler frequency
Space frequency
Traditional
Modi"ed
Traditional
Modi"ed
Traditional
Modi"ed
380 380 400 400
0.0469 0.0859 0.0625 0.0859
0.1250 0.1250 0.0859 0.1641
1.88 0.60 3.81 0.96
10.95 16.29 11.29 19.65
4.26 18.97 7.49 6.61
5.97 20.06 8.74 9.82
!16.09 20.63 10.16 9.74
5.61 30.39 21.63 16.53
Table 2 Average SNR before processing for di!erent range cells under test Average SNR before processing (dB) Target amplitude
Range bin 380
Range bin 400
Range bin 415
0.005 0.02
!27.95 !15.91
!15.79 !3.75
!14.93 !2.89
highly localized energy of the discrete is felt by JDL. We should note that if the discrete is far from the target, JDL works very well, much as in the homogeneous case [16]. In homogeneous cases, using only a few angle-Doppler bins can help avoid the discrete. This is illustrated in our tests using simulations. 5.2. Monte Carlo simulation results To better understand the impact of shape mismatch on performance, we also performed Monte Carlo simulations. We generated the Gaussian clutter data using a covariance matrix R calculated # from a hump model as described in (9). The parameters for the particular hump model used are shown in Table 3 for d"1,2,5. To simulate the two kinds of shape mismatch mentioned above, we added one extra hump, modeling a discrete, in the reference cells or the test cell. The parameters for this discrete is given in Table 3 for d"6. For each kind of mismatch, two cases are considered. In one case, called case (a), the target position (with Doppler frequency set as 0.0469 and angle frequency set as 0.1641) is close to the discrete position. In the
Table 3 Parameters of assumed hump model with the sixth hump simulating a discrete d
p #,d
p f5,d
p f4,d
f #5,d
f #4,d
1 2 3 4 5 6
2 2 8 2 2 6
0.08 0.08 0.08 0.08 0.08 0.02
0.08 0.08 0.08 0.08 0.08 0.02
!0.333 !0.167 0 0.167 0.333 0.086
!0.333 !0.167 0 0.167 0.333 0.086
other case, called case (b), the target position (with Doppler frequency set as !0.1563 and angle frequency set as !0.0313) is far away from the discrete position. By examining these two cases, we studied how the relative position of a target to a discrete a!ects the performance for di!erent schemes. For a "xed false alarm probability (set at 0.015), we compared the probability of detection of the traditional scheme and the modi"ed scheme as a function of SNR calculated in full dimension space as SNR"10 log (sHR~1s), where s is the # 10 target observed in zero-mean clutter with covariance matrix R . In our Monte Carlo simula# tions, 10 000 runs were used. 5.2.1. SMI When a discrete is in the reference cells and not in the test cell, as in the case considered in Fig. 4,
5 To improve the accuracy in our simulations, this value is set higher than the practical values, which are typically in the range of 10~6 or less. Note that this does not alter the trends observed.
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Fig. 4. Performance comparison for modi"ed and traditional SMI when a discrete is in the reference cells (Monte Carlo simulation).
the modi"ed scheme is always better than the traditional scheme regardless of the relative position of the target and the discrete. This is expected since SMI is a fully adaptive scheme and sensitive to incorrect covariance estimates. Further, it is reasonable that more performance improvement is obtained in case (a). Fig. 4 also shows that the two curves for modi"ed SMI representing cases (a) and (b) almost overlap each other. This is reasonable because in this case the estimated covariance matrix is not in#uenced by the discrete since the discrete is in the reference data and these data are not used to estimate the covariance matrix. Thus no matter where the target is, correct covariance estimates are obtained and similar performance should be achieved in both cases (a) and (b). When a discrete is in the test cell and not in the reference cells, the modi"ed scheme is still fairly insensitive to the discrete as shown in Fig. 5. This time the reason is that the discrete is properly nulled by the modi"ed scheme. 5.2.2. EFA Similar results are observed for both kinds of shape mismatch as illustrated in Figs. 6 and 7. We
have already explained the advantage of modi"ed EFA over traditional EFA after presenting the measured data results. However, the performance improvement observed in simulations are even larger than that which were observed in Table 1. The reason is that only some of reference cells had discretes in them in the measured data cases, but in the simulation cases all the reference cells contain discretes. Once again the modi"ed scheme provides large improvements for case (a). There is a slight performance improvement in case (b) when using the modi"ed scheme instead of the traditional scheme. This might be expected since EFA is an element}space scheme. Thus it does not perform the adaptive processing completely in the angleDoppler domain so it is di$cult for EFA to completely exploit the occurance of the target being far from the discrete. This is in contrast to JDL as we shall see. 5.2.3. JDL As expected, modi"ed JDL performs much better than traditional JDL in the case when the target is close to the discrete. When the target is far from the
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Fig. 5. Performance comparison for modi"ed and traditional SMI when a discrete is in test cell (Monte Carlo simulation).
Fig. 6. Performance comparison for modi"ed and traditional EFA when a discrete is in the reference cells (Monte Carlo simulation).
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Fig. 7. Performance comparison for modi"ed and traditional EFA when a discrete is in test cell (Monte Carlo simulation).
discrete, modi"ed JDL has similar performance to traditional JDL. This is illustrated in Figs. 8 and 9. Since JDL performs adaptive processing using only a few angle-Doppler bins near the target location, this makes sense.
6. Conclusions We have suggested the use of some modi"ed STAP schemes for airborne radar applications, which use knowledge of the structure of the interference. The speci"c numerical results given use a model that only assumes ground clutter, but extensions are possible. Further our approach works well for a measured radar data set that includes other interference. From the comparisons, it appears that the modi"ed scheme can provide good performance for cases with shape mismatch. This is signi"cant since these cases are traditionally considered to be di$cult and have been observed in measured radar data. The improvements are especially signi"cant for schemes that are sensitive to poor covariance
estimates. The modi"ed schemes generally perform better in non-homogeneous cases when compared to schemes that average over range to estimate the required covariance matrix. In practical situations it would be useful to determine whether the clutter is non-homogeneous in order to determine if the modi"ed schemes must be used. Recent research [8] suggests some methods to make such determinations. These methods could be useful in our application. In [8] the authors discuss three methods for assessing the relative homogeneity among a set of available secondary data. The data are assumed to be Gaussian distributed, the same assumption made in this paper. The authors demonstrate that their nonhomogeneity detectors do work. In fact, they use their method to provide improved STAP performance for cases where one has a reasonably large set of reference data, some of which is homogeneous. The methods we propose here would be suitable for cases where a reasonably large set of homogeneous data is not available. In this paper we considered only cases with ground clutter in formulating our method. It might
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Fig. 8. Performance comparison for modi"ed and traditional JDL when a discrete is in the reference cells (Monte Carlo simulation).
Fig. 9. Performance comparison for modi"ed and traditional JDL when a discrete is in the test cell (Monte Carlo simulation).
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be possible to also handle other types of interference and, in particular, jamming with a slight extension. If the location of a jammer or another target (other than the one currently being searched for) is known from a previous scan then this knowledge could be employed by locating a null in the adaptive pattern at the appropriate angle and Doppler. On the other hand, if there is signi"cant interference that cannot be easily modeled due to lack of knowledge, then a di!erent approach might be justi"ed. If the interference is assumed to be independent of the ground clutter, a reasonable assumption for jamming for example, then the total covariance matrix is
Gaussian clutter, the spherically invariant random vector (SIRV) model [11]. Under some fairly unrestrictive conditions, a canonical form for the optimal nonlinear receiver structure for this case is presented in [13]. As shown in Fig. 10, this form incorporates the conventional Gaussian receiver ¹ (r)"sHR~1r which is the optimum for Gaussian ' clutter. Here r is the vector of the received data, s corresponds to the signal vector which may be present in r, and R is the covariance matrix of r. As illustrated in Fig. 10, the optimum test also incorporates the quadratic form ¹ (r)"rHR~1r. The 2 optimum Neyman Pearson test under the fairly unrestrictive conditions [11,13] can be expressed as
R "R #R , (16) 5 # I where R comes from (9) and R , the covariance # I matrix of the interference, could be estimated as in (7). This approach still insures the structure of ground clutter is included, but allows some arbitrary adaption. Also notice that, in the presence of jamming, the estimations of p and ¸ might be a!ected and deviate from their &true' values when (13) is applied. We suggest performing the estimations in the angle-Doppler domain by using robust image-processing techniques to limit this somewhat. In a slight extension, R and R can each be # I weighted by scalars which incorporate how con"dent we are that only ground clutter is present. We can also imagine developing methods for obtaining these scalar weights adaptively from the observed data. In this paper we considered only Gaussian clutter. However, our method can be extended to some cases with non-Gaussian and, in particular, impulsive clutter. Consider a common model for non-
¹(r)"g [¹ (r), ¹ (r)];HH1 g, (17) NL ' q :0 where g [),)] is the optimal memoryless non-linNL earity which can be determined for any SIRV clutter distribution [10,13]. Using our technique to produce the covariance matrix R allows us to be able to handle non-Gaussian clutter cases with SIRV clutter distributions as shown in Fig. 10. Another popular model for non-Gaussian clutter, the sub-Gaussian alpha stable random vector model [15], can also employ our covariance matrix. The application is the same as for SIRVs, since the sub-Gaussian alpha stable random vector model is a special case of an SIRV model. In obtaining the parameters of our model, one could employ robust estimation techniques [6]. These techniques generally employ limiting so that large impulsive samples do not overly bias the estimates. A recent application using simple robust estimates for cases with alpha stable noise appears in [14]. For further reading, see [9].
Fig. 10. Canonical forms of the optimum SIRV processor.
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Appendix A SMI
EFA
JDL
In our tests of SMI we use M"22 pulses and N"22 antennas in the adaptive processing. For the SMI scheme, we have A "I , B "I in (2), where I , I are p M p N M N M]M and N]N identity matrices, respectively. S and S are the target temporal and 5 4 spatial response vectors, respectively [16]. Post-processing is not necessary for SMI. For the EFA scheme, the pre-processing is as in (2) with A "[f ,f ,f ], B "I , (A.1) p p~1 p p`1 p N where f ,f ,f are three adjacent Dopp~1 p p`1 pler "lters and f corresponds to the norp malized Doppler frequency of target. The adaptive processing uses S "[0 1 0]T and 5 S set to the target spatial response vector. 4 Only processing for a single p must be performed and so post-processing is not needed. For JDL, the matrices in (2) should be taken as A "[f , f , f ], B "[g , g , g ], p p~1 p p`1 p q~1 q q`1 (A.2) where f ,f ,f are three Doppler "lters p~1 p p`1 and f is that "lter which corresponds to the p normalized Doppler frequency of target. f and f are Doppler "lters correp~1 p`1 sponding to neighboring Doppler frequencies. g ,g ,g are three spatial "lters q~1 q q`1 and g is that "lter which corresponds the q spatial steering vector for the target. g and g are spatial "lters correspondq~1 q`1 ing to neighboring spatial frequencies. The steering vector S in the adaptive processing is S"S ?S "[0 1 0]?[0 1 0]. (A.3) 5 s As for JDL, processing must be performed for only one p so post-processing is not needed.
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[2] R.C. Dipietro, Extended factored space-time processing techniques for airborne radar, in: Proceedings of the 25th Asilomar Conference, Paci"c Grove, PA, October 1992, pp. 425}430. [3] A. Farina, P. Lombardo, M. Pirri, Nonlinear nonadaptive space-time processing for airborne early warning radar, in: Proceedings of IEE on Radar, Sonar and Navigation, Vol. 145, No. 1, February 1998, pp. 9}18. [4] D.K. Fenner, W.F. Hoover, Test results of a space-time adaptive processing system for airborne early warning radar, in: Proceedings of the IEEE National Radar Conference, Ann Arbor, MI, May 1996, pp. 88}93. [5] F.A. Fraybill, Introduction to Matrices with Applications in Statistics, Wadsworth Publishing, Belmont, CA, 1969, especially pp. 201}204. [6] S.A. Kassam, H.V. Poor, Robust techniques for signal processing: a survey, Proc. IEEE 73 (3) (March 1985) 433}481. [7] W.L. Melvin, B. Himed, Comparative analysis of spacetime adaptive algorithms with measured airborne data, Presented at the Seventh International Conference on Signal Processing Applications and Technology, 7}10 October 1996, pp. 1479}1483. [8] W.L. Melvin, M.C. Wicks, Improving practical space-time adaptive radar, in: Proceedings of the IEEE National Radar Conference, Syracuse, NY, May 1997, pp. 48}53. [9] W.L. Melvin, M.C. Wicks, R.D. Brown, Assessment of multichannel airborne radar measurements for analysis and design of space-time processing architectures and algorithms, in: Proceedings of the IEEE National Radar Conference, Ann Arbor, MI, May 1996, pp. 130}135. [10] A. Ozturk, An application of the distributed identi"cation algorithm to signal detection problems, in: Twenty-seventh Asilomar Conference on Signals, Systems, and Computers, Vol. 2, November 1993, pp. 248}252. [11] M. Rangaswamy, D. Weiner, A. Ozturk, Non-Gaussian random vector identi"cation using spherically invariant random processes, IEEE Trans. Aerospace Electron. Systems AES-29 (January 1993) 111}124. [12] D. Sloper et al., MCARM Final Report, Rome Lab tech. Rept., RL-TR-96-49, April 1996. [13] D.L. Stadelman, D.D. Weiner, Detection of weak signals with random parameters in non-Gaussian clutter, in: Proceedings of SPIE } The International Society for Optical Engineering, Vol. 2561, Bellingham, WA, 1995, pp. 31}42. [14] A. Swami, B. Sadler, Parameter estimation for linear alpha-stable processes, IEEE Signal Process. Lett. 5 (February 1998) 48}50. [15] G.A. Tsihrintzis, C.L. Nikias, Data-adaptive algorithms for signal detection in sub-Gaussian impulsive interference, IEEE Trans. Signal Process. (July 1997) 1873}1878. [16] H. Wang, L. Cai, On adaptive spatial-temporal processing for airborne surveillance radar systems, IEEE Trans. Aerospace Electron. Systems 30 (July 1994) 660}669. [17] J. Ward, Space-time adaptive processing for airborne radar, Technical Report 1015, Lincoln Laboratory, 1995.