Study of Sparsity-Aware Reduced-Dimension Beam-Doppler Space-Time Adaptive Processing
aa r X i v : . [ ee ss . SP ] M a r STUDY OF SPARSITY-AWARE REDUCED-DIMENSION BEAM-DOPPLER SPACE-TIMEADAPTIVE PROCESSING
Zhaocheng Yang , and Rodrigo C. de Lamare College of Information Engineering, Shenzhen University, Shenzhen, Guangdong, 518060, China Communications Research Group, University of York, York YO10 5DD, United [email protected] and [email protected]
ABSTRACT
Existing reduced-dimension beam-Doppler space-time adap-tive processing (RD-BD-STAP) algorithms are confined tothe beam-Doppler cells used for adaptation, which oftenleads to some performance degradation. In this work, a novelsparsity-aware RD-BD-STAP algorithm, denoted SparseConstraint on Beam-Doppler Selection Reduced-DimensionSpace-Time Adaptive Processing (SCBDS-RD-STAP), isproposed can adaptively selects the best beam-Doppler cellsfor adaptation. The proposed SCBDS-RD-STAP approachformulates the filter design as a sparse representation prob-lem and enforcing most of the elements in the weight vectorto be zero (or sufficiently small in amplitude). Simulationresults illustrate that the proposed SCBDS-RD-STAP algo-rithm outperforms the traditional RD-BD-STAP approacheswith fixed beam-Doppler localized processing.
Index Terms — Space-time adaptive processing, reduced-dimension, sparsity-aware, clutter suppression.
1. INTRODUCTION
Space-time adaptive processing (STAP) is a leading tech-nology candidate for improving detection performance ofphased-array airborne radar [1] and other related approaches.However, STAP techniques often suffer from the lack ofsnapshots for training the receive filter, especially in nonho-mogeneous environments, which is a crucial concern in thedevelopment of STAP algorithms [1, 2, 3].In the past decades, many related works have been in-vestigated to improve the clutter mitigation performance inscenarios with a number of snapshots (see [1, 2, 3, 4, 5, 6, 7, 8,9, 13, 12, 10, 11, 63] and the references therein). For instance,the auxiliary channel receiver (ACR) [4], the joint domain lo-calized approach (JDL) [5, 6], the space-time multiple-beam(STMB) [7] are three kinds of effective reduced-dimension(RD) algorithms in the beam-Doppler domain. However,the filter design in [4, 5, 6, 7] relies on fixed beam-Dopplercells and cannot provide optimal selection, suffering signifi-cant performance degradation in the presence of sensor array errors. To overcome this issue, the studies in [8] and [9] pro-posed sequential methods that reduce the required partiallyadaptive dimension in the transformed domain.Motivated by the rank deficiency in clutter suppression,sparsity-aware beamformers have been proposed to improvethe convergence by exploiting the sparsity of the receiveddata and filter weights [10, 11]. The studies in [12] and [13]developed a Min-Max STAP strategy based on the selectionof an optimum subset of antenna-pulse pairs that maximizesthe separation between the target and the clutter trajectory.Both the sparsity-aware beamformers and the Min-Max STAPstrategy are in the antenna-pulse domain. The former is adata-dependent strategy and the latter is a data-independentstrategy which requires prior knowledge of the clutter ridge.By drawing inspiration from compressive sensing, recentlyreported sparsity-based STAP algorithms have formulated theSTAP problem as a sparse representation that exploits thesparsity of the entire observing scene in the whole angle-Doppler plane [63]. However, this kind of approach suffersfrom high computational complexity due to the large dimen-sion of the discretized angle-Doppler plane. Previous worksimply that the degrees of freedom (DoFs) used for STAP fil-ters required to mitigate the clutter are much smaller than thefull dimension, and different selection strategies have resultedin various levels of performance.In this work, we introduce the idea of sparse selection inthe beam-Doppler domain and formulate the STAP filter de-sign as a sparse representation problem. Unlike the sparsity-based STAP [63], the proposed Sparse Constraint on Beam-Doppler Selection Reduced-Dimension STAP (SCBDS-RD-STAP) algorithm does not discretize the angle-Doppler planeinto a large number of grids, but only transforms the receiveddata into a same size beam-Doppler domain. Differently fromthe sparsity-aware beamformers [10, 11] or the Min-MaxSTAP strategy [12, 13], the proposed SCBDS-RD-STAP al-gorithm designs the filter in the beam-Doppler domain andautomatically selects the best beam-Doppler cells used foradaptation by solving a sparse representation problem. Inaddition, an analysis of the complexity is performed for theproposed algorithm. Simulation results show the effective-ess of the proposed algorithm.This paper is structured as follows: Section 2 describesthe signal model of a pulse Doppler side-looking airborne sys-tem and states the problem. Section 3 details the proposedSCBDS-RD-STAP algorithm along with approximate solu-tions and their computational complexity. Section 4 presentsand discusses simulation results while Section 5 provides theconcluding remarks.
2. SIGNAL MODEL AND PROBLEMFORMULATION
In this section we describe the signal model of a pulseDoppler side-looking airborne radar system and state theproblem of designing a beam-Doppler STAP.
Considering a pulse Doppler side-looking airborne radar witha uniform linear array (ULA) consisting of M elements. Theradar transmits a coherent burst of N pulses at a constantpulse repetition frequency (PRF) f r . Generally, for a rangebin with the space-time snapshot x , target detection can beformulated as a binary hypothesis problem and expressed as H : x = x u H : x = α t s + x u , (1)where H and H denote the disturbance only and the tar-get plus disturbance hypotheses, respectively, α t is a complexgain, s is the N M × target space-time steering vector and x u denotes the clutter-plus-noise vector which encompassesthe clutter and the thermal noise [1].The STAP filter based on a minimum variance distortion-less response (MVDR) approach by minimizing the clutter-plus-noise output power while constraining a unitary gain inthe direction of the desired target signal is expressed as [3] w opt = R − ss H R − s . (2)where R = E (cid:2) x u x Hu (cid:3) denotes the clutter-plus-noise co-variance matrix. Approaches to compute the beamformingweights include [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25,26, 37, 27, 28, 29, 30, 31, 32, 33, 34, 39, 36, 38, 39, 40, 41,42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57,58, 59, 60, 61, 62] The beam-Doppler STAP approaches firstly transform thedata x in the antenna-pulse domain to the beam-Dopplerdomain, denoted as ˜ x , where “ ˜ ” above x signifies the beam-Doppler domain. This procedure can be represented by ˜ x = T H LP x , (3) where T LP denotes the transformation matrix. The commonidea under the beam-Doppler STAP approaches is to choose alocalized processing (LP) region, or equivalently, the matrix T LP , corresponding to a set of beam-Doppler responses, foradaptive processing. The optimal beam-Doppler STAP filtercan be represented by ˜ w opt = ˜ R − ˜ s ˜ s H ˜ R − ˜ s , (4)where ˜ R = T H LP RT LP and ˜ s = T H LP s . f d f s Target Clutter (a) ACR f d f s Target Clutter (b) JDL f d f s TargetClutter (c) STMB
Fig. 1 . The LP region selections in different beam-DopplerLP approaches. ◦ denotes the selected beam-Doppler cell,and × denotes the target beam-Doppler cell.Observing (4), the key challenge is how to efficiently se-lect the LP region. The ACR method [4] suggests to selectthe LP region placed along the clutter ridge, as shown inFig.1(a). The JDL method [5, 6] chooses the beam-Dopplercells around the target cell, which turns out to be a rectangularshape, as shown in Fig.1(b). Unlike ACR and JDL, STMB [7]chooses the beam-Doppler cells with a “cross” shape centeredat the target cell, as shown in Fig.1(c). All these approachescan reduce the STAP filter dimension, resulting in improvedconvergence and steady-state performance in a small train-ing data set. However, the ACR requires the knowledge ofthe clutter ridge, and there is no rule to determine the op-timum size of the chosen beam-Doppler LP region for JDLand STMB. The optimum choice of the beam-Doppler regionshould be related to the scenario or the data rather than justfixed.
3. PROPOSED SCBDS-RD-STAP ALGORITHM
In this section, we detail the proposed SCBDS-RD-STAP al-gorithm, how to design the receive filter and discuss the com-putational complexity.
The core idea of the proposed SCBDS-RD-STAP scheme isbased on a transformation matrix and a filter with sparse con-straints. The received space-time data vector x is first mappedby an N M × N M transformation matrix T into an N M × beam-Doppler domain data vector. Here, T can be con-structed as T = (cid:2) s T aux (cid:3) , (5)where T aux is an N M × ( N M − matrix, given by T aux = (cid:0) v d ( f d,t ) ⊗ v s ( f s,t + M ) (cid:1) T ... (cid:0) v d ( f d,t ) ⊗ v s ( f s,t + M − M ) (cid:1) T (cid:0) v d ( f d,t + N ) ⊗ v s ( f s,t ) (cid:1) T ... (cid:0) v d ( f d,t + N ) ⊗ v s ( f s,t + M − M ) (cid:1) T ... (cid:0) v d ( f d,t + N − N ) ⊗ v s ( f s,t + M − M ) (cid:1) T T . (6)Denoting d = s H x and ˜ x = T H aux x , we note that d is thecomponent at the target beam-Doppler cell (also called mainchannel), and elements of ˜ x are the components from oth-erwise beam-Doppler cells (also called auxiliary channels).Following the concept of GSC, we can expect to reduce theclutter in d by employing a filter on the auxiliary channel data ˜ x . Furthermore, based on the first three observations ana-lyzed above, we do not need to use all auxiliary channel databut only a few of them. In order to realize this idea, we per-form a sparse constraint on the STAP filter weight vector ˜ w .Precisely, we design the filter ˜ w by solving the following op-timization problem min ˜ w E h(cid:12)(cid:12) d − ˜ w H ˜ x (cid:12)(cid:12) i + κ k ˜ w k , (7)where κ is the regularization parameter that controls the bal-ance between the sparsity and total squared error. Theoreti-cally, the optimum choice can be determined by an algorithmthat is properly designed for the task. To show an intuitiveobservation of the above idea, we will provide examples bysimulations later on. I am not sure about the above but itwould be useful to include a table with the pseudo-code ofthe SCBDS-RD-STAP algorithm here. Since the sparse regularization function is l -norm, it leads toan NP-hard problem. In the following, we use the relaxationpenalty l p -norm (where < p ≤ ) instead of the l -normand rewrite (7) as min ˜ w E h(cid:12)(cid:12) d − ˜ w H ˜ x (cid:12)(cid:12) i + κ k ˜ w k p . (8)In practice, since the expectation in (8) cannot be ob-tained, we now modify (8) based on a least-squares type costfunction. Let X = [ x , x , · · · , x L ] denote the space-time data matrix formed by L training snapshots, and let d = X T s ∗ , and ˜ X = T H aux X , then the least-squares type costfunction is described by min ˜ w (cid:13)(cid:13)(cid:13) d ∗ − ˜ X H ˜ w (cid:13)(cid:13)(cid:13) + κ k ˜ w k p . (9)Note that (9) is a standard sparse representation problem andcan be solved by the regularized focal underdetermined sys-tem solution (R-FOCUSS) algorithm.It should be noted that the sensing matrix or the dictio-nary ˜ X H of the optimization problem (9) of the proposedSCBDS-RD-STAP scheme is formed by the received data(i.e., snapshots from the beam-Doppler domain), and is dif-ferent from that of the sparsity-based STAP approaches [63],which is composed of known space-time steering vectorsfrom the discretized angle-Doppler plane. Furthermore, un-like the ACR, JDL, and STMB, which are performed withfixed beam-Doppler LP region, the proposed SCBDS-RD-STAP scheme provides an iterative approach to automati-cally select the beam-Doppler LP region aided by a sparseconstraint. Additionally, the auxiliary channel data are for-mulated by a standard 2-D discrete Fourier transform withexplicit physical meaning in the proposed SCBDS-RD-STAPscheme, whereas the auxiliary channel data are formulatedby a signal blocking matrix in the sparsity-aware beamformer[10]. We detail the computational complexity of the proposedSCBDS-RD-STAP algorithm, sparsity-aware beamformer[10], and JDL [6]/STMB [7], as shown in Table 1. Here, forthe proposed algorithm, K foc is the total iteration numberand D foc ,q is the number of elements above the preset thresh-old at the q th iteration, which is decided by the sparsity; forthe JDL/STMB, D is the number of selected beam-Dopplerelements. From the table, we see that the computationalcomplexity of the proposed algorithm is comparable or evenlower than that of the sparsity-aware beamformer, and higherthan those of the JDL and STMB. This is because the numberof snapshots L used in the proposed SCBDS-RD-STAP algo-rithm is much smaller than N M (which can be seen in thesimulations), the value of D foc ,q after several iterations willalso be much smaller than N M , and the pseudo-inversioncan be calculated by the conjugate gradient approach, whichhas low complexity [64].
4. SIMULATIONS
In this section, we assess the performance of the proposedSCBDS-RD-STAP algorithm and compare it with other exist-ing algorithms, namely, the JDL ( × ) [6], STMB ( ) In our experiments, the average time used for the proposed algorithmwith a fixed p and the sparsity-aware beamformer are . second and . second, respectively. able 1 . Computational ComplexityAlgorithm ComplexitySparsity-aware beamformer O (cid:0) L ( N M ) (cid:1) JDL/STMB O (cid:0) LD + D (cid:1) Proposed SCBDS-RD-STAP O (cid:16)P K foc q =1 D foc ,q L (cid:17) [7], and sparsity-aware beamformer [10] in terms of the out-put signal-to-clutter-plus-noise-ratio (SCNR) loss [1], whichis defined as SCNR loss = σ (cid:12)(cid:12) ˆ w H s (cid:12)(cid:12) N M ˆ w H R ˆ w , (10)where ˆ w = s − T aux ˜ w is the corresponding filter weight vec-tor in the original domain. We consider a side-looking ULA(half-wavelength inter-element spacing) airborne radar withthe following parameters: uniform transmit pattern, M = 12 , N = 12 , carrier frequency . GHz, f r = 2 kHz, platformvelocity m/s, platform altitude km, clutter-to-noise ratio(CNR) dB. For the following examples: in the sparsity-aware beamformer, we set parameters as those in [10]; in theproposed SCBDS-RD-STAP algorithm, we set the regulariza-tion parameter to , the maximum iteration number is ,and the stopping criterion is decided by the preset limit rel-ative change of the solution between two adjacent iterations − .
10 20 30 40 50 60 70 80 90 100
Snapshots -12-10-8-6-4-20 S CNR Lo ss ( d B ) Sparsity-Aware BeamformerJDL(3 3)STMB(8+4+1)Proposed SCBDS-RD-STAP: p=0.4Proposed Switched-SCBDS-RD-STAPOptimum
Fig. 2 . The SCNR loss against the number of snapshots fortraining.In the first example, we examine the convergence perfor-mance (signal-to-clutter-plus-noise ratio (SCNR) loss againstthe number of snapshots) of the proposed SCBDS-RD-STAPalgorithm, as shown in Fig.2. The true target is supposedto be boresight aligned with normalized Doppler frequency − . . The curves show that the proposed Switched-SCBDS-RD-STAP algorithm converges to a higher SCNR loss with much fewer training snapshots compared to all theconsidered algorithms. -0.4 -0.2 0 0.2 0.4 Normalized Doppler frequency -0.5-0.4-0.3-0.2-0.100.10.20.30.4 N o r m a li z ed S pa t i a l f r equen cy Fig. 3 . 2-D view of the weight vector of the Switched-SCBDS-RD-STAPFig.3 illustrates the 2-D view of the weight vector, specif-ically, each element in the weight vector is represented byone grid point, and its amplitude is depicted by the grayscaleof the grid. Note that, each element in the weight vector isassociated to one auxiliary channel in the GSC, and a zeroamplitude implies the associated auxiliary channel is not in-volved in the adaption. Apparently, most of the elementsin the weight vector have zero amplitudes, which impliesthat the Switched-SCBDS-RD-STAP selects very few beam-Doppler cells for adaptation. -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4
Normalized Doppler frequency -12-10-8-6-4-20 S CNR Lo ss ( d B ) Sparsity-Aware BeamformerJDL(3 3)STMB(8+4+1)Proposed SCBDS-RD-STAP: p=0.4Proposed Switched-SCBDS-RD-STAPOptimum
Fig. 4 . The SCNR loss versus different target Doppler fre-quencies.In the third example, we assess the performance of theproposed SCBDS-RD-STAP algorithm under different targetDoppler frequencies, as depicted in Fig.4. Here, we set thenumber of snapshots for training used in the JDL ( ), STMB ), proposed SCBDS-RD-STAP ( ), and sparsity-awarebeamformer ( ). The curves illustrates that the proposedSCBDS-RD-STAP algorithm provides much better perfor-mance than other algorithms for small target Doppler fre-quencies. That is to say, the proposed SCBDS-RD-STAPalgorithm is suitable for slow moving target detection. Al-though the performance of the SCBDS-RD-STAP algorithmis slightly lower than that of the sparsity-aware beamformerfor large target Doppler frequencies, the number of snapshotsused in the SCBDS-RD-STAP algorithm is much less.
5. CONCLUSIONS
This paper has proposed a novel STAP algorithm based onthe beam-Doppler selection for clutter mitigation for airborneradar with small sample support. The SCBDS-RD-STAP al-gorithm transforms the received data into beam-Doppler do-main, employs a sparse constraint on the filter weight forsparse beam-Doppler selection and formulates this selectionas a sparse representation problem, where the sensing ma-trix is formed by the data matrix. Simulations have demon-strated the effectiveness of the proposed SCBDS-RD-STAPalgorithm and shown its improvement in target detection overthe existing algorithms, such as the JDL, STMB, and sparsity-aware beamformer both in absence and presence of array er-rors.
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