Sparsity-Aware STAP Algorithms Using L 1 -norm Regularization For Radar Systems
aa r X i v : . [ c s . I T ] A p r Sparsity-Aware STAP Algorithms Using L -normRegularization For Radar Systems Zhaocheng Yang, Rodrigo C. de Lamare,
Senior Member, IEEE and Xiang Li,
Member IEEE
Abstract —This article proposes novel sparsity-aware space-time adaptive processing (SA-STAP) algorithms with l -normregularization for airborne phased-array radar applications.The proposed SA-STAP algorithms suppose that a number ofsamples of the full-rank STAP data cube are not meaningfulfor processing and the optimal full-rank STAP filter weightvector is sparse, or nearly sparse. The core idea of the proposedmethod is imposing a sparse regularization ( l -norm type) tothe minimum variance (MV) STAP cost function. Under somereasonable assumptions, we firstly propose a l -based samplematrix inversion (SMI) to compute the optimal filter weight vec-tor. However, it is impractical due to its matrix inversion, whichrequires a high computational cost when in a large phased-arrayantenna. Then, we devise lower complexity algorithms based onconjugate gradient (CG) techniques. A computational complexitycomparison with the existing algorithms and an analysis of theproposed algorithms are conducted. Simulation results with bothsimulated and the Mountain Top data demonstrate that fastsignal-to-interference-plus-noise-ratio (SINR) convergence andgood performance of the proposed algorithms are achieved. Index Terms — l regularization, Sparsity-aware Space-timeadaptive processing, Conjugate gradient techniques, Airborneradar, Mountain Top data. I. I
NTRODUCTION
Space-time adaptive processing (STAP) is an efficient toolfor detection of slow targets by airborne or spaceborne radarsystems in serious environments, such as strong clutter andlots of jammers [1]–[4]. However, the full-rank adaptive STAPbased on linearly constrained minimum variance (LCMV)criterion gives rise to two of the major limitations in practicalapplications of radar [2], [4]. First, the computational loadrequired to solve the interference matrix inversion is quiteintense. In addition, the number of training data samplesrequired for an accurate estimate of the interference covariancematrix can become impractical for high-dimensional problems,particularly in heterogeneous environments. It is thereforedesirable to develop STAP techniques with low computationalcomplexity and that can provide high performance in small-sample support situations.The diagonal loading sample matrix inversion (LSMI) tech-nique is considered to be a simple and robust approach for bothhomogeneous and heterogeneous environments [5], but has ahigh computational cost. Reduced-rank techniques have been
Z. Yang and X. Li are with Research Institute of Space Electronics,Electronics Science and Engineering School, National University of DefenseTechnology, Changsha, 410073, China. e-mail: [email protected],[email protected]. C. de Lamare is with Communications Research Group, De-partment of Electronics, University of York, YO10 5DD, UK. e-mail:[email protected] investigated for solving the previously discussed problems inthe last decades [6]–[12], [14]–[16], [20]–[22]. One of themost important reduced-rank techniques is the class of theKrylov subspace methods, which includes the auxiliary-vectorfilters (AVF) [7], [8], the multistage Wiener filter (MWF) [9]–[12] and the conjugate gradient (CG) algorithm [14]–[16],[20]. These methods project the observation data onto a lower-dimensional Krylov subspace and can obtain an improvedconvergence and tracking performance. The main differencesamongst them lie on the computational cost, structure of adap-tation and ease of implementation. Knowledge-aided (KA)STAP techniques have currently gained significant attention asan effective STAP algorithm to mitigate the effects of the het-erogeneity in the secondary data by exploiting a priori knowl-edge [21]–[24]. However, the exact form of prior knowledge isstill problem-dependent and hard to be derived. More recently,several authors have considered sparse recovery (SR) ideas formoving target indication (MTI) and STAP problems [25]–[31].These work based on SR techniques relys on the recovery ofthe clutter power in the angle-Doppler plane, which is usuallycarried out via two steps: first, recovering the clutter angle-Doppler profile by some SR algorithms; second, estimatingthe covariance matrix based on the result obtained in thefirst step, and computing the Capon’s optimal filter. Althoughsome fast sparse recovery algorithms are proposed, e.g., thefast iterated shrinkage/thresholding (FISTA) algorithm [27],and the focal underdetermined system solution (FOCUSS)based algorithm [30], it is more computationally expensivethan conventional STAP because of the Capon’s optimal filterrequiring matrix inversion, and the recovery procedure beingan additive computational burden.In airborne radar systems, most interference suppressionproblems are rank deficient in nature [2]–[4], that is theyrequire less adaptive degrees of freedom (DOF) than the fullDOF provided by the array. In this case, the total adaptiveDOF provided by the array will be much great than thenumber that needed to suppress the interference. Motivatedby this, the authors in [32] proposed an sequential approachthat gave a sparse solution for the transformation matrix toselect the ”best” DOF to be retained in a partially adaptivebeamformer. Moreover, the property described above can beseen that there is a high degree of sparsity of the filter weightvector. Hence, in our prior work, an l type regularizationto the generalized sidelobe canceler (GSC) STAP processorusing the l -based online coordinate gradient (OCD) method[33] and the l -based recursive least squares method [34] isintroduced to exploit the sparsity of the received data and filterweights, resulting in an improvement in both convergence rate and steady-state signal-to-interference-plus-noise ratio (SINR)performance. In this paper, we extend the work presentedin [33] and [34] to the direct filter STAP processor (DFP).By adding the sparsity constraint ( l -norm regularization) tothe MV cost function, we derive the l -regularized optimalfilter weight vector under some reasonable assumptions, andthen propose a sparsity-aware (SA) adaptive STAP strategyfor airborne radar systems. One direct way is to use the l -based SMI recursion algorithm to compute the filter weights.However, it requires the matrix inversion operation, whichprevents its use in practice. The CG method has a lowcomputational complexity and is the simplest Krylov subspacemethod since it only needs the forward stage, unlike the MWFthat requires both forward and backward stages. Therefore,low complexity l -based CG type algorithms are devised. Thesimulations are conducted using both simulated and measureddata, which show that the proposed algorithms exhibit im-proved performance as compared to existing techniques.This paper is organized as follows. Section II introducesthe STAP signal model for airborne radar. In Section III, wefirst introduce the strategy of the SA-STAP algorithm. Then l -based SMI and l -based CG type algorithms are developed andtheir computational complexity is also shown. Furthermore,we conduct an analysis of the proposed algorithms. In SectionIV, some examples of performance of the proposed algorithmswith both simulated and the Mountain Top data are exhibited.Finally, the conclusions are given in Section V.Notation: In this paper, scalar quantities are denoted withitalic typeface. Lowercase boldface quantities denote vectorsand uppercase boldface quantities denote matrices. The op-erations of transposition, complex conjugation, and conjugatetransposition are denoted by superscripts T , ∗ , and H , re-spectively. The symbols ⊗ represents the Kronecker productand ⊙ denotes the Hadamard matrix product. Finally, thesymbol E {·} denotes the expected value of a random quantity,operator ℜ [ · ] selects the real part of argument, and the symbol k · k p denotes the l p -norm operation of a vector.II. S IGNAL M ODEL AND P ROBLEM S TATEMENT
The system under consideration is a pulsed Doppler radarresiding on an airborne platform. The radar antenna is auniformly linear spaced array (ULA) which consists of M elements. The platform is at altitude h p and moving withconstant velocity v p . The chosen coordinate system is shownin Fig. ?? . The angle variables φ and θ refer to elevation andazimuth. The radar transmits a coherent burst of pulses ata constant pulse repetition frequency (PRF) f r = 1 /T r ,where T r is the pulse repetition interval (PRI). The transmittercarrier frequency is f c = c/λ c , where c is the propagationvelocity and λ c is the wavelength. The coherent processinginterval (CPI) length is equal to N T r . For each PRI, K time samples are collected to cover the range interval. Aftermatched filtering to the radar returns from each pulse, thereceived data set for one CPI comprises KN M complexbaseband samples, which is referred to as the radar datacubeshown in Fig. ?? . The data are then processed at one rangeof interest, which corresponds to a slice of the CPI datacube. The slice is an M × N matrix which consists of M × spatialsnapshots for pulses at the range of interest. It is convenient tostack the matrix column-wise to form the N M × vector x [ k ] ,termed a space-time snapshot, where k is the range sampleindex and ≤ k ≤ K [2]–[4].Target detection in airborne radar systems can be formulatedinto a binary hypothesis problem, where the hypothesis H corresponds to target absence and the hypothesis H corre-sponds to target presence, given as H : x = x u H : x = α s s + x u , (1)where α s is a complex gain and the vector s , which is the N M × normalized space-time steering vector in the space-time look-direction, defined as s = s t ( f d ) ⊗ s s ( f s ) k s t ( f d ) ⊗ s s ( f s ) k , (2)where s t ( f d ) denotes the N × temporal steering vectorat the target Doppler frequency f d and s s ( f s ) denotes the M × spatial steering vector in the direction provided by thetarget frequency f s . The vector x u encompasses any undesiredinterference or noise component of the data including clutter x c , jamming x j and thermal noise x n . Generally, we assumethe thermal noise is spatially and temporally uncorrelated, andthe jamming is temporally uncorrelated but spatially stronglycorrelated. As for the clutter, a general model for the clutterspace-time snapshot is given by [35] x c [ k ] = N r X m =1 N c X n =1 σ c/k ; m,n (cid:0) α t ( k ; m, n ) ⊙ s t ( f d/k ; m,n ) (cid:1) ⊗ (cid:0) α s ( k ; m, n ) ⊙ s s ( f s/k ; m,n ) (cid:1) , (3)where N r is the number of range ambiguities, N c is the num-ber of independent clutter patches that are evenly distributedin azimuth about the radar, α t ( k ; m, n ) is a vector describingthe normalized pulse-to-pulse voltages, and α s ( k ; m, n ) ac-counts for spatial decorrelation. σ c/k ; m,n describes the averagevoltage for the mn th clutter patch and k th range. The clutter-jammer-noise (for short, calling interference in the followingpart) covariance matrix R can be expressed as R = E (cid:8) x u x Hu (cid:9) = R c + R j + R n , (4)where R c = E (cid:8) x c x Hc (cid:9) , R j = E (cid:8) x j x Hj (cid:9) and R n = E (cid:8) x n x Hn (cid:9) , denote clutter, jammer and thermal noise covari-ance matrix, respectively.Generally, the space-time processor linearly combines theelements of the data snapshot, yielding the scalar output [4] y = w H x , (5)where w is the N M × weight vector. The idea behindLCMV approach is to minimize the STAP output power whilstconstraining the gain in the direction of the desired signal. Thisleads to the following power minimization with constraints min w J ( w ) = E (cid:8) k w H x k (cid:9) s.t. w H s = 1 . (6) Using the method of Lagrange multipliers, the optimal full-rank LCMV STAP weights are given by [1] w LCMV = R − ss H R − s . (7)III. SA-STAP WITH L - NORM R EGULARIZATION
In this section, we detail the design of the proposed SA-STAP strategy, derive the l -based SMI recursion algorithmand the l -based CG type algorithms and detail their complex-ity. Finally, the analysis of the proposed SA-STAP algorithmsis shown. A. SA-STAP Strategy
In airborne radar systems, most interference suppressionproblems are rank deficient in nature, that is they require lessadaptive DOF than are offered by the array, the additionalDOF that are not required can be discarded so that onlythose that are important are retained, which is termed aspartially STAP technique [32]. Furthermore, full DOF willlead to slow convergence, i.e. requiring many snapshots totraining the filter, which is difficult to obtain especially innon-homogeneous clutter environments. As a result, the totaladaptive DOF provided by the array will be much great thanthe number that needed to suppress the interference. In anotherword, there is a high degree of sparsity of the filter weightvector. However, in the practical, it is not easy to estimatethe required DOF, related with the sparsity, and to decidewhich DOF are the most important ones. Herein, the authorsin [33], [34] proposed an l regularized STAP algorithm forGSC structure to exploit the sparsity of the received data andfilter weights. In this paper, we extend this work to a moregeneral framework for airborne radar systems, by employingthe sparse regularization to the MV STAP cost function, whichis described as the following optimization problem min w E n(cid:13)(cid:13) w H x (cid:13)(cid:13) o + 2 λ Γ ( w ) s.t. w H s = 1 , (8)where λ is a positive scalar which provides a trade-off betweenthe sparsity and the output interference power. The largerthe chosen λ , the more components are shrunk to zero [36].The sparse regularization is usually conducted by the l -normconstraint [37]–[39]. However, since this kind of optimizationproblem is known to be NP-hard, one of the approximationalgorithms, called l -norm, is considered for the convexity andsimple complexity [38]. In the following, we adopt the l -norm regularization, i.e., Γ ( w ) = k w k . Now, the questionthat arises is how to effectively solve the l regularizedMV STAP. Albeit convex, the cost function J ( w ) is non-differentiable which leads to difficulty with the use of themethod of Lagrange multipliers directly. Thus, we proposean approximation to the regularization term, which is givenby Γ ( w ) = k w k ≈ w H Λ w , (9)where Λ = diag (cid:26) | w | + ǫ , | w | + ǫ , · · · , | w NM | + ǫ (cid:27) , (10) where ǫ is a small positive constant (e.g., ǫ = 0 . isacceptable),and w i , i = 1 , · · · , M N are the entries of thefilter weight vector w . Thus the regularization term w H Λ w has a quadratic structure, if we assume the diagonal matrix Λ to be fixed. Minimization can be done iteratively by assumingthat the term Λ is fixed, being computed with the currentsolution w [37]. So, fixing the term Λ , we take the differentialterm with respect to w ∗ of (9), which is given as follows ∂ k w k ∂ w ∗ ≈ ∂ (cid:0) w H Λw (cid:1) ∂ w ∗ = Λ w . (11)The above constrained optimization problem described by(8) can be transformed into an unconstrained optimizationproblem by the method of Lagrange multipliers, whose costfunction becomes L = E n(cid:13)(cid:13) w H x (cid:13)(cid:13) o + 2 λ k w k + 2 ℜ (cid:8) κ ∗ (cid:0) w H s − (cid:1)(cid:9) , (12)where κ is a complex Lagrange multiplier. Computing thegradient terms of (12) with respect to w ∗ and κ ∗ , we get ∇L w ∗ = Rw + λ Λw + κ ∗ s ∇L κ ∗ = w H s − . (13)By equating the above gradient terms to zero, we obtain thefilter weight vector w = [ R + λ Λ ] − ss H [ R + λ Λ ] − s . (14)By inspecting (14), we verify that there is an additionalterm λ Λ in the inverse of the interference covariance matrix R , which is due to the l -norm regularization. One should notethat the filter weight vector expression in (14) is not a closed-form solution since Λ is a function of w . Thus it is necessaryto develop an iterative procedure to compute the filter weightvector, which will be shown in the following parts. B. L -Based SMI Recursion Algorithm In practice, because the interference covariance is unknownto us, it is most common to compute the interference covari-ance matrix estimate as [2]–[4] ˆ R = 1 L L X n =1 x [ n ] x H [ n ] , (15)where { x [ n ] } Ln =1 are known as the secondary or training data.In our following derivation, to develop an iterative procedure,we add an exponential weighting factor to the interferencecovariance matrix, which may allow the STAP algorithms toaccommodate possible non-stationarities in the input. We writethe ˆ R [ k ] as ˆ R [ k ] = i X n =1 β k − n x [ n ] x H [ n ] = β ˆ R [ k −
1] + x [ i ] x H [ i ] , (16)where β is the forgetting factor, and ˆ R [0] = δ I , where δ isa small positive quantity and I is the identity matrix. Since Λ [ k ] is a function of w [ k ] , we assume that the filter weightvalues do not change significantly in a single snapshot step,which is reasonable because we want the instantaneous error of the filter weight vector to change slowly [40]. Hence, Λ [ k ] can be approximated by Λ [ k ] ≈ Λ [ k −
1] = diag (cid:26) | w [ k − | + ǫ , · · · , | w NM [ k − | + ǫ (cid:27) . (17)However, we note that the computational complexity ofthe l -based SMI recursion algorithm is proportional to O (cid:16) ( N M ) (cid:17) , which is not practical, especially in large sizeof phased-array antenna. In the next section, we will developsome low complexity algorithms. C. L -Based CG Algorithms In order to reduce the computational complexity of the l -based SMI recursion algorithm, we introduce low complexityadaptive algorithms based on CG techniques to iterativelycompute the filter weights. There are two different basic strate-gies for using the CG method. One is the conventional CG(CCG) [15], [16], which executes several iterations per sampleand runs the reset periodically for convergence. The otheris the modified CG (MCG) [14], [16], [20], which operatesonly one iteration per sample. CCG has a faster convergencethan MCG, but a higher computational complexity. In thefollowing, we detail the derivation of the l -based SA-STAPalgorithms based on these two strategies, called l -based CCGalgorithm and l -based MCG algorithm. For simplicity, wefirstly introduce an auxiliary vector given by v [ k ] = h ˆ R [ k ] + λ Λ [ k ] i − r t . (18)Then the STAP filter weight vector can be described as w [ k ] = v [ k ] / ( sv [ k ]) . The solution of v [ k ] described by (18) is alsothe solution of the following minimal optimization problem[16]: min J ( v ) = v H h ˆ R + λ Λ i v − ℜ (cid:8) v H s (cid:9) . (19)Then the CG-based weight vector is expressed by v [ k ] = v [ k −
1] + α [ k ] p [ k ] , (20)where p [ k ] is the direction vector, α [ k ] is the correspondingadaptive step size.For the l -based CCG algorithm, the iteration procedurefor the CG-based weight vector v is executed per sample.For the k th sample, it assumes constant ˆ R [ k ] + λ Λ [ k ] withinthe internal iterations, and D internal iterations are performedper input data sample. The main difference between the l -based CCG algorithm and the existing CCG algorithm afterthe derivation is that we add an additional term λ Λ [ k ] to theestimated interference covariance matrix ˆ R [ k ] . A summary ofthe algorithm is shown in Table I.The l -based CCG algorithm operates multiple iterationsper sample and runs the reset periodically for convergence,which increases the computational load in the sample-by-sample update. In the following, we detail the derivations ofthe l -based MCG algorithm with one iteration per sample forSTAP. From [14], one way to realize the conjugate gradientmethod with one iteration per snapshot is the application of the degenerated scheme, which means that the residualvector g [ k ] will not be completely orthogonal to the subspacespanned by the direction vectors { p [0] , p [1] , · · · , p [ k − } .Under this condition, the adaptive step size α [ k ] has to fulfillthe convergence bound given by ≤ (cid:12)(cid:12) p H [ k ] g [ k ] (cid:12)(cid:12) ≤ . (cid:12)(cid:12) p H [ k ] g [ k − (cid:12)(cid:12) , (21)where g [ k ] is the negative gradient vector of J ( v ) in (19).Thus, g [ k ] can be written as g [ k ] = −∇J ( v ) v ∗ = − h ˆ R [ k ] + λ Λ [ k ] i v [ k ] + s , (22)which can be calculated recursively by g [ k ] = (1 − β ) s + β g [ k − − α [ k ] h ˆ R [ k ] + λ Λ [ k ] i p [ k ] − (cid:8) λ [1 − β ] Λ [ k ] + x [ k ] x H [ k ] (cid:9) v [ k − . (23)In the previous equation, we use the approximation that Λ [ k − ≈ Λ [ k ] . Premultiplying (23) by p H [ k ] , taking theexpectation of both sides and considering p [ k ] uncorrelatedwith s , x [ k ] and v [ k − [14], we obtain E (cid:2) p H [ k ] g [ k ] (cid:3) ≈ βE (cid:2) p H [ k ] g [ k − (cid:3) − βE (cid:2) p H [ k ] s (cid:3) − E h α [ k ] p H [ k ] (cid:16) ˆ R [ k ] + λ Λ [ k ] (cid:17) p [ k ] i . (24)Here, it is assumed that the algorithm convergeswith the assumption that E [ v [ k − − v opt ] ≈ , E (cid:2) x [ k ] r H [ k ] v [ k − (cid:3) ≈ s , and E [ λ [1 − β ] Λ [ k ] v [ k − ≈ . Making a rearrangement of (25) and following theconvergence bound (21), we obtain α [ k ] = h p H [ k ] (cid:16) ˆ R [ k ] + λ Λ [ k ] (cid:17) p [ k ] i − (cid:8) β (cid:2) p H [ k ] g [ k − − p H [ k ] s (cid:3) − µ p H [ k ] g [ k − (cid:9) . (25)where ≤ µ ≤ . . The direction vector is a linear com-bination from the previous direction vector and the negativegradient, which is described as p [ k ] = g [ k −
1] + ν [ k ] p [ k ] , (26)where ν [ k ] is computed for avoiding the reset procedure byemploying the Polak-Ribiere approach, which should have animproved performance [14], [16], and is stated as ν [ k ] = [ g [ k ] − g [ k − H g [ k ] g H [ k − g [ k − . (27)The proposed l -based MCG STAP algorithm is summarizedin Table II.From above discussions, two aspects should be noted that:First, the performance of our proposed algorithms (both l -based SMI and l -based CG-type algorithms) depends onregularization parameter λ . An approach to choose λ isintroduced in [34], which can be easy to extend to ourproposed algorithms, but not discussed in this paper for savingspace. Second, the convergence analysis in [16] is suitable toour proposed CG-type algorithms, where the convergence isgoverned by k ς i +1 [ k ] k G [ k ] ≤ p τ max /τ min − τ max /τ min + 1 ! i k ς [ k ] k G [ k ] , (28) where ς i [ k ] = v opt [ k ] − v i [ k ] is the CG-based weight vectorerror at the i th iteration for the k th snapshot, v opt [ k ] is theoptimal solution at the k th snapshot, τ max and τ min are themaximal and minimal eigenvalues with respect to G [ k ] =ˆ R [ k ] + λ Λ [ k ] , and k ς i [ k ] k G [ k ] = ς Hi [ k ] G [ k ] ς i [ k ] . From theabove equation, we note that the convergence behavior of theproposed algorithms is related to the CG-based weight vectorerror ς [ k ] and the condition number τ max /τ min . D. Complexity Analysis
In this section, we detail the computational complexity interms of complex additions and complex multiplications ofthe proposed l -based SMI, l -based CG type algorithms, andother existing STAP algorithms, namely the LSMI, the AVF,the MWF and the conventional CG type algorithms, as shownin Table III. One aspect should be noted that, the rank D maynot equal to the clutter rank, and can be smaller than that.This is because the principle of the Krylov subspace approachis different from that of the eigen-decomposition approach.An eigen-decomposition approach would usually require anSVD on the full-rank covariance matrix and the selection ofthe D eigenvectors associated with the D largest eigenvalues,which is high related to the clutter rank. In contrast to that, theKrylov-based approach does not require eigen-decompositionand selects the D basis vectors which minimize the desiredcost function and will form the projection matrix, where D can be decreased without significantly degrading the SINR[12]. In the table, D is the rank for CCG type, AVF andMWF algorithms, and L = N M is the system size. Seenfrom the table, the computational complexity of l -based SMIis similar to the conventional LSMI algorithm, both requiringone to calculate the matrix inversion. With respect to theproposed l -based CG type algorithms, the computationalcomplexity is nearly the same as the conventional CG typealgorithms. Note that the complexity of CCG type, AVF andMWF algorithms is dependent on the rank D . This is atradeoff between complexity and performance. We found thatthe rank of the proposed l -based CCG algorithm with D = 7 works well (while the best rank for AVF and MWF is muchlarger), as will be verified in the following simulations. Thelow-rank characteristic will bring computational savings. Thecomputational complexity of all algorithms is shown in Fig.1,where we use the best rank obtained from the simulations forthese algorithms ( D = 7 for CCG type, D = 18 for AVFand D = 14 for MWF). We see that the proposed CG typealgorithms have much lower complexity than AVF and MWFalgorithms.Furthermore, it requires to compute the filter weights repeat-edly for target detection in airborne radar systems, especiallyin heterogeneous environment. In this case, our proposedalgorithms can work in an iterative way and do not need torecompute all the filter weights, which can lead to significantcomputational savings. Usually, secondary data of the slidingwindow are used in detection procedures, where the parameterthat defines the length of the sliding window is K . Assume R i [ K ] denotes the estimated interference covariance matrixaccording to (16) and w i [ K ] denotes the filter weight vector at the cell under test (CUT) of the i th range bin, respectively.Consider the case of the i + 1 th CUT, we first remove theimpact of i + 1 th CUT, given by β R i +1 [ K −
1] = R i [ K ] − x [ i + 1] x H [ i + 1] . (29)Since an exponentially decaying data window is used, we donot need to remove the first snapshot used to compute thefilter weights. Then, similarly, we consider the case of addingsnapshots. Two snapshots, one is at the primary i th CUT andanother is the new snapshot x new which was not included inthe sliding window before, should be added to the i +1 th CUTsecondary data. The procedure can be written as R i +1 [ K ] = β R i +1 [ K − β x new x H new + x [ i + 1] x H [ i + 1] . (30)As for the filter weight vector w i +1 [ K ] at the i + 1 th CUT, itcan be updated using the new interference covariance matrixand the filter weight vector w i [ K − .In addition, the proposed algorithms adopt an adaptive filter-ing approaches, which can obtain a near optimum interferencerejection at a low cost [41]. The advantage of this approachis that filtering can be accomplished in a pipeline mode asthe echo pulses come in. The required number of calculationsfor filtering can be realized easily with nowadays digitaltechnology [42]. E. Analysis of the SA-STAP Algorithm
At this point, we have finished the derivation of the SA-STAP algorithms. The following simulation results will showthat the proposed SA-STAP algorithms have a faster SINRconvergence speed and better SINR steady-state performancethan the conventional algorithms. This translates into a supe-rior detection performance. However, why do the SA-STAPalgorithms work is an interesting question. This section willtry to explain that from two points of view.First, to understand the behavior, we write the filter weightvector using the eigenvalue decomposition (EVD) of ˆ R . Weassume that the eigenvalues of the estimated interferencecovariance matrix are ˆ γ n with the corresponding eigenvectorsdenoted by u n , n = 1 , , · · · , N M . The eigenvalues areordered as, ˆ γ ≥ ˆ γ ≥ · · · ≥ ˆ γ NM = ˆ γ min . (31)Thus, through the EVD, the estimated interference covariancematrix can be written as ˆ R = NM X n =1 ˆ γ n u n u Hn . (32)Substituting (32) into (14), the filter weight vector of the SA-STAP algorithm can be written as w SA = ς SA ( r t − NM X n =1 ˆ γ n + ∆ n − ˆ γ min ˆ γ n + δ SAmin + ∆ n (cid:0) u Hn r t (cid:1) u n ) , (33)where δ SAmin = min (cid:16) λ | w n | + ǫ (cid:17) , n = 1 , , · · · , N M , ∆ n isthe difference between λ | w n | + ǫ and δ SAmin , and ς SA is a scalarquantity, which does not affect the SINR. By inspecting (33), we observe that the SA-STAP belongsto the class of diagonal loading STAP techniques in a sense.Moreover, it is equivalent to an adaptive diagonal loading tech-nique, which will apply to each eigenbeam of the interferencecovariance matrix different weights and exploit the sparsity ofthe filter weights and the received data.Second, we will investigate the relationship between theSINR performance and the l -norm-sum quantity of the filterweights. Assume the scene is the same as the one withhomogeneous environment introduced in the next section. Wecompute the SINR loss and the l -norm-sum quantity of thefilter weights against the number of snapshots using the SMIalgorithm. The results are plotted in Fig.2. From the figure,we find that the better the SINR performance, the smaller the l -norm-sum quantity of the filter weights. From this point ofview, a constraint on the l -norm-sum quantity of the filterweights can lead to fast convergence, which in fact exploitsthe sparsity of the received data and filter weights.IV. P ERFORMANCE A SSESSMENT
In this section, we assess the proposed SA-STAP algorithmsusing both simulated and measured data and compare themwith the existing algorithms, such as the conventional CCG,MCG, MWF, AVF and LSMI algorithms. We measure theSINR, the SINR loss and the probability of detection curves,where the SINR and the SINR loss are defined as follows [4],respectively. SINR = (cid:12)(cid:12) ˆ w H x (cid:12)(cid:12) | ˆ w H R ˆ w | , (34)SINR loss = (cid:12)(cid:12) ˆ w H x (cid:12)(cid:12) | ˆ w H R ˆ w | | s H R − s | , (35)where R is the exact interference covariance matrix at thedetection range bin and ˆ w is the estimated filter weights usingthe neighbor secondary data. A. Simulated Data
Consider a monostatic sidelooking radar with M = 10 antenna elements and N = 8 pulses in one CPI, giving aspace-time steering vector of length L = 80 . We assumea simulated scenario with the following parameters: half-wavelength spaced antennas, uniform transmit pattern, carrierfrequency MHz, PRF set to
Hz, platform velocity of m/s and height of m, the clutter uniformly distributedfrom azimuth − π/ to π/ with clutter-to-noise-ratio (CNR)of dB, two jammer located at − and with jammer-to-noise-ratio (JNR) of dB, the target located at ◦ azimuthwith Doppler frequency of Hz and signal-to-noise-ratio(SNR) of dB, and the thermal noise power is . W. Weconsider the inner clutter motion (ICM) in simulated data.One common model, referred to as the Billingsley model, wasdeveloped by Billingsley of MIT Lincoln Laboratory [3]. Theonly parameters required to specify the clutter Doppler powerspectrum are essentially the shape parameter b and the windspeed parameter ω . In this paper, we assume b = 3 . and ω = 51 . miles per hour (mph). All presented results areaverages over 100 independent Monte Carlo runs.In our first example, we consider the SINR performanceversus the rank D of the proposed l -based CCG algorithm,the conventional CCG algorithm, the AVF algorithm and theMWF algorithm. A total of K = 160 snapshots are considered.The results in Fig.3 show that our proposed l -based CCGalgorithm can obtain its best performance when the rank islarger than D = 7 . It is much lower rank to obtain its bestperformance than that of AVF ( D = 18 ) and MWF ( D = 14 )algorithms. The low-rank characteristic will bring considerablecomputational savings, which is very important for STAP inradar systems. One should note that the performance of theconventional CCG algorithm will degrade when the rank is toolarge, while our proposed l -based CCG can always keep goodperformance resulting in further robustness. Since the SINRperformance is much worse when the rank is lower than thebest rank, thus, we will use D = 7 for CCG type algorithms, D = 18 for the AVF algorithm and D = 14 for the MWFalgorithm in the following examples.In the next example, we evaluate the SINR loss performanceagainst the number of snapshots K = 320 of the proposedalgorithms with the existing algorithms, as depicted in Fig.4.The curves show that: (1) the SINR performance of the pro-posed l -based SMI algorithm is a suboptimal algorithm, butexhibits the best performance compared with other algorithms.(2) l -based CG type algorithms outperform conventional CGtype algorithms in terms of convergence rate and steady-state performance; (3) the SINR performance of the l -basedCCG algorithm is better than AVF and MWF algorithms.(4) Although the l -based MCG algorithm shows slowerSINR convergence than the MWF algorithm, we can obtaina better SINR performance when the number of snapshots islarger than . One should note that the proposed CG typealgorithms have a much lower computational complexity thanLSMI, AVF and MWF algorithms.In the third example, we present the probability of detection P d versus SNR with the target injected at the azimuth of ◦ and Doppler frequency Hz in Fig.5. We assume the falsealarm rate P fa is set to − and the number of the secondarydata is K = 110 . The plots illustrate a similar trend to theSINR loss performance in the second example. Note that weobtain a performance gain of about dB in terms of SNR for l -based CG type algorithms, as compared with conventionalCG type algorithms.Fig.6 shows the SINR performance against the targetDoppler frequency at the azimuth of ◦ with a total of K = 160 snapshots. Here, we suppose the potential Dopplerfrequency space is from − Hz to
Hz. The parametersof all algorithms are the same as the second example. Thecurves in the figure demonstrate a similar trend to the results ofprevious examples. Additionally, the l -based SMI algorithmdisplays much better performance to the slow targets than otheralgorithms. B. Measured Data
In this section, we apply the proposed algorithms to theMountain-Top data set. This data was collected from com- manding sites (mountain tops) and radar motion is emulatedusing a technique developed at Lincoln Laboratories [6], [43].The sensor consists of elements and the data are organizedin CPIs of pulses. Here, we use the data file t pre v CPI , which could be obtained from the internet [44]. Thepulse PRF was Hz and the instance bandwidth after pulsecompression was kHz. There are independent rangesamples available for the training data support. The clutterwas located around ◦ azimuth and the target was at ◦ with a Doppler frequency Hz. All the data processedfollowing are through pulse compression firstly. Note that theclutter and target have the same Doppler frequency, henceseparation is impossible in the Doppler domain but possible inthe spatial domain. The estimated angle-Doppler profile usingall samples is given in Fig. ?? , which shows a seriousheterogeneity.Fig. and Fig. display the STAP output power of all algo-rithms in the range of - km. Here, the interferencecovariance matrix estimated using a symmetric sliding windowwith a total of snapshots for Fig. and snapshots for Fig..For each CUT, the snapshots do not include the snapshotsaround the CUT. In the figures, we also give the unadaptedweight vector, which equals the steering vector w = s . Wesee that the target is clearly not detectable without adaptiveprocessing. To have a clear comparison amongst differentalgorithms, we show the differences between the output powerat km and the next highest power peak in Table IV,where ”-” presents the target not detectable. Here, range binsaround the rang bin of the target is not used for comparisonsince they are the guide cells. Seen from the table, we findthat: (1) the proposed l -based SMI algorithm obtains thebest detection performance in both situations, which is thesame conclusion as that using simulated data; (2) the proposed l -based CG type algorithms obtain better performance thanthe conventional CG type algorithms (although the proposed l -based MCG algorithm has a pseudo target at the range km, when the secondary data record is snapshots, theconventional MCG algorithm can not detect the target atall.); (3) the proposed l -based CCG algorithm outperformsAVF and MWF algorithms in both situations. Hence, wecan conclude that our proposed algorithms show a robustperformance in heterogeneous environments.V. C ONCLUSIONS
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CCG A
LGORITHM
Initialization: ˆ R [0] = δ I , v [0] = s , η CCG , Recursion: For each snapshot k = 1 , · · · , L STEP 1: Start: ˆ R [ k ] = β ˆ R [ k −
1] + x [ k ] x H [ k ] , Λ [ k ] = diag n | w [ k − | + ǫ , · · · , | w NM [ k − | + ǫ o , G [ k ] = ˆ R [ k ] + λ Λ [ k ] , g [ k ] = s − G [ k ] v [ k ] , p [ k ] = g [ k ] , ρ [ k ] = g H [ k ] g [ k ] , STEP 2: For d = 1 , · · · , D and ρ d − [ k ] > η CCG z d [ k ] = G [ k ] p d [ k ] , α d [ k ] = (cid:2) p Hd [ k ] z d [ k ] (cid:3) − ρ k − [ k ] , v d [ k ] = v d − [ k −
1] + α d [ k ] p d [ k ] , g d [ k ] = g d − [ k ] − α d [ k ] z d [ k ] , ρ d [ k ] = g Hd [ k ] g d [ k ] , ν d [ k ] = ρ d [ k ] ρ d − [ k ] , p d +1 [ k ] = g d [ i ] + ν d [ k ] p d [ k ] , STEP 3: After end STEP 2 v [ k + 1] = v d [ k ] , w [ k ] = v [ k ] s H v [ k ] , Final output: y [ k ] = w H [ k ] x [ k ] . TABLE IIT HE l - BASED
MCG A
LGORITHM
Initialization: ˆ R [0] = δ I , v [0] = s , w [0] = s , g [0] = s , p [1] = s , Recursion: For each snapshot k = 1 , · · · , L ˆ R [ k ] = β ˆ R [ k −
1] + x [ k ] x H [ k ] , Λ [ k ] = diag n | w [ k − | + ǫ , · · · , | w NM [ k − | + ǫ o , G [ k ] = ˆ R [ k ] + λ Λ [ k ] , α [ k ] = (cid:2) p H [ k ] G [ i ] p [ k ] (cid:3) − × (cid:8) β (cid:2) p H [ k ] g [ k − − p H [ k ] s (cid:3) − µ p H [ k ] g [ k − (cid:9) , v [ k ] = v [ i −
1] + α [ i ] p [ i ] , g [ k ] = (1 − β ) s + β g [ k − − α [ k ] G [ k ] p [ k ] − (cid:8) λ [1 − β ] Λ [ k ] + x [ k ] x H [ k ] (cid:9) v [ k − , ν [ k ] = [ g [ k ] − g [ k − H g [ k ] g H [ k − g [ k − , p [ k + 1] = g [ k −
1] + ν [ k ] p [ k ] , w [ k ] = v [ k ] s H v [ k ] , Output: y [ k ] = w H [ k ] x [ k ] . TABLE IIIC
OMPARISON OF THE C OMPUTATIONAL C OMPLEXITY
Algorithm Additions MultiplicationsLSMI O (cid:0) L (cid:1) + O (cid:0) L (cid:1) O (cid:0) L (cid:1) + O (cid:0) L (cid:1) l -based SMI O (cid:0) L (cid:1) + O (cid:0) L (cid:1) O (cid:0) L (cid:1) + O (cid:0) L (cid:1) MWF DL + (4 D − D ) L DL + (5 D − D ) L + D − D − D + D − D + D AVF (2 D + 1) L + (4 D D + 1) L +1) L − D − DL + L CCG ( D + 2) L + (4 D ( D + 3) L +2) L − D − DL + 3 L MCG L + 10 L − L + 13 L + 2 l -based CCG ( D + 3) L + (4 D + ( D + 3) L L − D − DL + Ll -based MCG L + 11 L − L + 13 L + 2 TABLE IVR
ESULTS OF M OUNTAIN T OP DATA
Algorithms snapshots snapshotsunadapted - -LSMI . dB . dB l -based SMI . dB . dBMWF . dB . dBAVF . dB . dBCCG . dB . dBMCG - . dB l -based CCG . dB . dB l -based MCG dB . dB
20 40 60 80 100 12010 L N u m b e r o f M u lti p li ca ti on s
20 40 60 80 100 12010 L N u m b e r o f A dd iti on s LSMIL1−based SMICCG D=7MCGL1−based CCG D=7L1−based MCGAVF D=18MWF D=14
Fig. 1. The computational complexity per snapshot.
100 150 200 250 300 350 400 450−10−8−6−4−20 Snapshots S I NR Lo ss ( d B )
100 150 200 250 300 350 400 45010 Snapshots L1 − no r m − s u m Fig. 2. The relationship between the SINR performance and the l -norm-sum quantity of the filter weights. S I N R ( d B ) CCGL1−based CCGAVFMWFOptimum
Fig. 3. The SINR performance versus the rank D . Parameters: the diagonalloading factor for AVF and MWF algorithms is dB to the thermal noisepower; β = 0 . , η CCG = 10 − and R [0] = 0 . I for CCG typealgorithms; λ = 2 for l -based CCG algorithms.
50 100 150 200 250 300−14−12−10−8−6−4−20 Snapshot S I NR Lo ss ( d B ) LSMIL1−based SMICCG D=7MCGL1−based CCG D=7L1−based MCGAVF D=18MWF D=14
Fig. 4. The SINR loss performance against the number of snapshots K = 320 . Parameters: the diagonal loading factor for LSMI, AVF andMWF algorithms is dB to the thermal noise power; β = 0 . and R [0] = 0 . I for CG type algorithms; λ = 1 for the l -based SMI, λ = 2 for the l -based CCG algorithm and λ = 1 for the l -based MCG algorithm; η CCG = 10 − . −12 −10 −8 −6 −4 −2 0 2 4 6 800.10.20.30.40.50.60.70.80.91 SNR (dB) P D Probability of Detection for snapshots:110
LSMIL1−based SMICCG D=7MCGL1−based CCG D=7L1−based MCGAVF D=18MWF D=14Optimum
Fig. 5. Probability of detection performance versus SNR with K = 110 snapshots. P fa = 10 − and the other parameters are the same as the secondexample. −100 −80 −60 −40 −20 0 20 40 60 80 10002468101214161820 Doppler (Hz) S I NR ( d B ) LSMIL1−based SMICCG D=7MCGL1−based CCG D=7L1−based MCGAVF D=18MWF D=14Optimum
Fig. 6. SINR performance against Doppler frequency with K = 160 snapshots and Doppler frequency space from − to100