1 Introduction
Spacetime adaptive processing (STAP) is a leading technology candidate for improving detection performance of phasedarray airborne radar [1] and other related approaches. However, STAP techniques often suffer from the lack of snapshots for training the receive filter, especially in nonhomogeneous environments, which is a crucial concern in the development of STAP algorithms [1, 2, 3].
In the past decades, many related works have been investigated to improve the clutter mitigation performance in scenarios 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 localized approach (JDL) [5, 6], the spacetime multiplebeam (STMB) [7] are three kinds of effective reduceddimension (RD) algorithms in the beamDoppler domain. However, the filter design in [4, 5, 6, 7] relies on fixed beamDoppler cells and cannot provide optimal selection, suffering significant performance degradation in the presence of sensor array errors. To overcome this issue, the studies in [8] and [9] proposed sequential methods that reduce the required partially adaptive dimension in the transformed domain.
Motivated by the rank deficiency in clutter suppression, sparsityaware beamformers have been proposed to improve the convergence by exploiting the sparsity of the received data and filter weights [10, 11]. The studies in [12] and [13] developed a MinMax STAP strategy based on the selection of an optimum subset of antennapulse pairs that maximizes the separation between the target and the clutter trajectory. Both the sparsityaware beamformers and the MinMax STAP strategy are in the antennapulse domain. The former is a datadependent strategy and the latter is a dataindependent strategy which requires prior knowledge of the clutter ridge. By drawing inspiration from compressive sensing, recently reported sparsitybased STAP algorithms have formulated the STAP problem as a sparse representation that exploits the sparsity of the entire observing scene in the whole angleDoppler plane [63]
. However, this kind of approach suffers from high computational complexity due to the large dimension of the discretized angleDoppler plane. Previous works imply that the degrees of freedom (DoFs) used for STAP filters required to mitigate the clutter are much smaller than the full dimension, and different selection strategies have resulted in various levels of performance.
In this work, we introduce the idea of sparse selection in the beamDoppler domain and formulate the STAP filter design as a sparse representation problem. Unlike the sparsitybased STAP [63], the proposed Sparse Constraint on BeamDoppler Selection ReducedDimension STAP (SCBDSRDSTAP) algorithm does not discretize the angleDoppler plane into a large number of grids, but only transforms the received data into a same size beamDoppler domain. Differently from the sparsityaware beamformers [10, 11] or the MinMax STAP strategy [12, 13], the proposed SCBDSRDSTAP algorithm designs the filter in the beamDoppler domain and automatically selects the best beamDoppler cells used for adaptation by solving a sparse representation problem. In addition, an analysis of the complexity is performed for the proposed algorithm. Simulation results show the effectiveness of the proposed algorithm.
This paper is structured as follows: Section 2 describes the signal model of a pulse Doppler sidelooking airborne system and states the problem. Section 3 details the proposed SCBDSRDSTAP algorithm along with approximate solutions and their computational complexity. Section 4 presents and discusses simulation results while Section 5 provides the concluding remarks.
2 Signal Model and Problem Formulation
In this section we describe the signal model of a pulse Doppler sidelooking airborne radar system and state the problem of designing a beamDoppler STAP.
2.1 Signal Model
Considering a pulse Doppler sidelooking airborne radar with a uniform linear array (ULA) consisting of elements. The radar transmits a coherent burst of pulses at a constant pulse repetition frequency (PRF) . Generally, for a range bin with the spacetime snapshot , target detection can be formulated as a binary hypothesis problem and expressed as
(1) 
where and denote the disturbance only and the target plus disturbance hypotheses, respectively, is a complex gain, is the target spacetime steering vector and denotes the clutterplusnoise vector which encompasses the clutter and the thermal noise [1].
The STAP filter based on a minimum variance distortionless response (MVDR) approach by minimizing the clutterplusnoise output power while constraining a unitary gain in the direction of the desired target signal is expressed as
[3](2) 
where denotes the clutterplusnoise covariance matrix. Approaches to compute the beamforming weights 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]
2.2 The BeamDoppler STAP Approaches
The beamDoppler STAP approaches firstly transform the data in the antennapulse domain to the beamDoppler domain, denoted as , where “ ” above signifies the beamDoppler domain. This procedure can be represented by
(3) 
where denotes the transformation matrix. The common idea under the beamDoppler STAP approaches is to choose a localized processing (LP) region, or equivalently, the matrix , corresponding to a set of beamDoppler responses, for adaptive processing. The optimal beamDoppler STAP filter can be represented by
(4) 
where and .
Observing (4), the key challenge is how to efficiently select the LP region. The ACR method [4] suggests to select the LP region placed along the clutter ridge, as shown in Fig.1(a). The JDL method [5, 6] chooses the beamDoppler cells around the target cell, which turns out to be a rectangular shape, as shown in Fig.1(b). Unlike ACR and JDL, STMB [7] chooses the beamDoppler cells with a “cross” shape centered at the target cell, as shown in Fig.1(c). All these approaches can reduce the STAP filter dimension, resulting in improved convergence and steadystate performance in a small training data set. However, the ACR requires the knowledge of the clutter ridge, and there is no rule to determine the optimum size of the chosen beamDoppler LP region for JDL and STMB. The optimum choice of the beamDoppler region should be related to the scenario or the data rather than just fixed.
3 Proposed SCBDSRDSTAP Algorithm
In this section, we detail the proposed SCBDSRDSTAP algorithm, how to design the receive filter and discuss the computational complexity.
3.1 Proposed SCBDSRDSTAP Scheme
The core idea of the proposed SCBDSRDSTAP scheme is based on a transformation matrix and a filter with sparse constraints. The received spacetime data vector is first mapped by an transformation matrix into an beamDoppler domain data vector. Here, can be constructed as
(5) 
where is an matrix, given by
(6) 
Denoting and , we note that is the component at the target beamDoppler cell (also called main channel), and elements of are the components from otherwise beamDoppler cells (also called auxiliary channels). Following the concept of GSC, we can expect to reduce the clutter in by employing a filter on the auxiliary channel data . Furthermore, based on the first three observations analyzed above, we do not need to use all auxiliary channel data but only a few of them. In order to realize this idea, we perform a sparse constraint on the STAP filter weight vector . Precisely, we design the filter by solving the following optimization problem
(7) 
where is the regularization parameter that controls the balance between the sparsity and total squared error. Theoretically, the optimum choice can be determined by an algorithm that is properly designed for the task. To show an intuitive observation of the above idea, we will provide examples by simulations later on. I am not sure about the above but it would be useful to include a table with the pseudocode of the SCBDSRDSTAP algorithm here.
3.2 Approximate Solutions
Since the sparse regularization function is norm, it leads to an NPhard problem. In the following, we use the relaxation penalty norm (where ) instead of the norm and rewrite (7) as
(8) 
In practice, since the expectation in (8) cannot be obtained, we now modify (8) based on a leastsquares type cost function. Let denote the spacetime data matrix formed by training snapshots, and let , and , then the leastsquares type cost function is described by
(9) 
Note that (9) is a standard sparse representation problem and can be solved by the regularized focal underdetermined system solution (RFOCUSS) algorithm.
It should be noted that the sensing matrix or the dictionary of the optimization problem (9) of the proposed SCBDSRDSTAP scheme is formed by the received data (i.e., snapshots from the beamDoppler domain), and is different from that of the sparsitybased STAP approaches [63]
, which is composed of known spacetime steering vectors from the discretized angleDoppler plane. Furthermore, unlike the ACR, JDL, and STMB, which are performed with fixed beamDoppler LP region, the proposed SCBDSRDSTAP scheme provides an iterative approach to automatically select the beamDoppler LP region aided by a sparse constraint. Additionally, the auxiliary channel data are formulated by a standard 2D discrete Fourier transform with explicit physical meaning in the proposed SCBDSRDSTAP scheme, whereas the auxiliary channel data are formulated by a signal blocking matrix in the sparsityaware beamformer
[10].3.3 Computational Complexity
We detail the computational complexity of the proposed SCBDSRDSTAP algorithm, sparsityaware beamformer [10], and JDL [6]/STMB [7], as shown in Table 1. Here, for the proposed algorithm, is the total iteration number and is the number of elements above the preset threshold at the th iteration, which is decided by the sparsity; for the JDL/STMB, is the number of selected beamDoppler elements. From the table, we see that the computational complexity of the proposed algorithm is comparable or even lower^{1}^{1}1In our experiments, the average time used for the proposed algorithm with a fixed and the sparsityaware beamformer are second and second, respectively. than that of the sparsityaware beamformer, and higher than those of the JDL and STMB. This is because the number of snapshots used in the proposed SCBDSRDSTAP algorithm is much smaller than (which can be seen in the simulations), the value of after several iterations will also be much smaller than , and the pseudoinversion can be calculated by the conjugate gradient approach, which has low complexity [64].
Algorithm  Complexity 

Sparsityaware beamformer  
JDL/STMB  
Proposed SCBDSRDSTAP 
4 Simulations
In this section, we assess the performance of the proposed SCBDSRDSTAP algorithm and compare it with other existing algorithms, namely, the JDL () [6], STMB () [7], and sparsityaware beamformer [10] in terms of the output signaltoclutterplusnoiseratio (SCNR) loss [1], which is defined as
(10) 
where is the corresponding filter weight vector in the original domain. We consider a sidelooking ULA (halfwavelength interelement spacing) airborne radar with the following parameters: uniform transmit pattern, , , carrier frequency GHz, kHz, platform velocity m/s, platform altitude km, cluttertonoise ratio (CNR) dB. For the following examples: in the sparsityaware beamformer, we set parameters as those in [10]; in the proposed SCBDSRDSTAP algorithm, we set the regularization parameter to , the maximum iteration number is , and the stopping criterion is decided by the preset limit relative change of the solution between two adjacent iterations .
In the first example, we examine the convergence performance (signaltoclutterplusnoise ratio (SCNR) loss against the number of snapshots) of the proposed SCBDSRDSTAP algorithm, as shown in Fig.2. The true target is supposed to be boresight aligned with normalized Doppler frequency . The curves show that the proposed SwitchedSCBDSRDSTAP algorithm converges to a higher SCNR loss with much fewer training snapshots compared to all the considered algorithms.
Fig.3 illustrates the 2D view of the weight vector, specifically, each element in the weight vector is represented by one grid point, and its amplitude is depicted by the grayscale of the grid. Note that, each element in the weight vector is associated to one auxiliary channel in the GSC, and a zero amplitude implies the associated auxiliary channel is not involved in the adaption. Apparently, most of the elements in the weight vector have zero amplitudes, which implies that the SwitchedSCBDSRDSTAP selects very few beamDoppler cells for adaptation.
In the third example, we assess the performance of the proposed SCBDSRDSTAP algorithm under different target Doppler frequencies, as depicted in Fig.4. Here, we set the number of snapshots for training used in the JDL (), STMB (), proposed SCBDSRDSTAP (), and sparsityaware beamformer (). The curves illustrates that the proposed SCBDSRDSTAP algorithm provides much better performance than other algorithms for small target Doppler frequencies. That is to say, the proposed SCBDSRDSTAP algorithm is suitable for slow moving target detection. Although the performance of the SCBDSRDSTAP algorithm is slightly lower than that of the sparsityaware beamformer for large target Doppler frequencies, the number of snapshots used in the SCBDSRDSTAP algorithm is much less.
5 Conclusions
This paper has proposed a novel STAP algorithm based on the beamDoppler selection for clutter mitigation for airborne radar with small sample support. The SCBDSRDSTAP algorithm transforms the received data into beamDoppler domain, employs a sparse constraint on the filter weight for sparse beamDoppler selection and formulates this selection as a sparse representation problem, where the sensing matrix is formed by the data matrix. Simulations have demonstrated the effectiveness of the proposed SCBDSRDSTAP algorithm and shown its improvement in target detection over the existing algorithms, such as the JDL, STMB, and sparsityaware beamformer both in absence and presence of array errors.
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