^{1}

^{1, 2}

^{1}

^{3}

^{1}

^{2}

^{3}

We propose a subspace-tracking-based space-time adaptive processing technique for airborne radar applications. By applying a modified approximated power iteration subspace tracing algorithm, the principal subspace in which the clutter-plus-interference reside is estimated. Therefore, the moving targets are detected by projecting the data on the minor subspace which is orthogonal to the principal subspace. The proposed approach overcomes the shortcomings of the existing methods and has satisfactory performance. Simulation results confirm that the performance improvement is achieved at very small secondary sample support, a feature that is particularly attractive for applications in heterogeneous environments.

The primary goal of airborne ground moving target indicator (GMTI) radar is to detect and estimate the physical location of moving targets. Therefore, the clutter mitigation approach, that is, space-time adaptive processing (STAP) technique, must be employed to separate moving targets from stationary clutter. In essence, STAP combines spatial and temporal degrees of freedom to design a bidimension filter. It is recognized as central to effectively eliminate interference since the publication of Brennan and Reed [

The optimal STAP processor derives a data-dependent weighting vector which can offer a significant increasing in output signal-to-interference-plus-noise ratio (SINR). However, this calculation requires the knowledge of the space-time covariance matrix at the cell under test (CUT). Two major reasons that have restrained its application in practice are the high computational cost and the substantial amount of stationary sample support (also called secondary data). These issues have motivated the development of suboptimal methodology such as reduced-rank STAP [

Moreover, the STAP algorithm which is based on the use of subspace projection has been shown to be an attractive approach for rank reduction and can achieve good performance with very little sample support, since only the principal subspace needs to be estimated. The estimation of the principal subspace is commonly based on the traditional eigenvalue decomposition (EVD) or singular value decomposition (SVD). However, the main drawback of these decomposition approaches is that they are inherently computationally expensive. Therefore, a large number of algorithms have been proposed for performing the fast subspace tracking task. See [

We present here a new post-Doppler subspace-based STAP approach which is performed in two stages. The first step is used to recursively calculate the principal subspace. Once the signal subspace has been calculated, the adaptive weight vector can be achieved with the use of subspace projection in the last stage. This algorithm presents several advantages such as the following: an orthonormal subspace basis is computed at each updating step, the computational cost can be significantly reduced, the dimension of the dominant subspace can be adaptively determined, and convergence comparable with subspace estimation using the EVD can be achieved.

The paper is organized as follows. In Section

The radar transmitter emits a pulse and collects samples of the received signal until the next pulse is emitted. Each of these samples corresponding to a given range is commonly referred to as “range bin”. For each range bin, the data at the output of the array is arranged into a vector

The weight vector of the linearly constrained minimum variance (LCMV) is given by

Note that the clutter-plus-jammer component exists in a relatively low dimensional subspace of the

Since we chose

Thanks to the rank-reduction nature of the space-time correlation matrix, the subspace-based methods are known to converge faster than SMI with very little sample support. Popularly, the size of the secondary data sample set of the principal component canceller approach can be reduced to

The complexity of the dominant subspace estimation with EVD procedure can be reduced by adopting subspace tracking method. These subspace trackers are based on the minimization of the mean square error according to the following cost function:

In spite of the fact that different subspace tracking methods can be derived, here, we are more particularly interested in the FAPI algorithm [

The key of this algorithm is the assumption of

Once the principal subspace was estimated, the subspace-based weight vector is then

Two factors affect the direct use of FAPI for the fully adaptive STAP application. First, since the FAPI algorithm updates all dominant eigenvectors per iteration, it propagates the error of eigenbasis derivation. This error propagation consequently decreases the convergence rate of the dominant subspace. Second, either adopting subspace methods or not, all the characteristics of the received data, such as clutter locus, are not taken into consideration in fully adaptive STAP.

In fact, an improved performance of convergence rate can be achieved if the aforementioned error propagation effect can be avoided. Moreover, using the partially adaptive architecture, the performance of SINR loss can be improved. Note that the post-Doppler approaches, which provides a good tradeoff between detection performance and computational burden, are indeed the most popular among others. That is why only a modified FAPI method for post-Doppler STAP will be considered in here.

Let

Representations of space snapshot for given range gate

Then, instead of updating all dominant eigenvectors per iteration, we will estimate principal eigenvectors sequentially utilizing all training data. Let

Once the first eigenvector

We clarify that our goal here is not to justify whether the partially adaptive STAP approaches with FAPI are of appropriateness or not. An answer to the question can be found in [

Perhaps the biggest difficulty associated with adopting a principal subspace tracker for STAP is determining what subspace dimension results in the best performance. The assumption that the subspace dimension is known is only used to simplify our presentation. In practice, adopting the well-known selection techniques, such as the Akaike information criterion (AIC) [

We would like to point out that the orthonormality between the estimated eigenvectors cannot be guaranteed by adopting only the clearing operation due to a numerical stability problem and noise, especially when the dimension of the estimated subspace is larger than that of the true dominant subspace and/or the train data set is small. However, our m-FAPI algorithm can guarantee the orthonormality between the estimated eigenvectors at each iteration.

In this section, two simulations are provided to illustrate the effectiveness of our proposed approach. Firstly, the performance of the subspace estimation is analyzed according to the following estimation error cost function:

The estimation error

equal SNR {20 dB 20 dB 20 dB 20 dB}

nonequal SNR {30 dB 20 dB 10 dB 20 dB}

Secondly, the performance of clutter cancellation is analyzed based on the SINR loss, which is defined as the loss in SINR compared with a matched filter in the presence of white noise only. We consider a side-looking airborne radar, operating at

SINR loss versus sample number.

SINR loss versus normalized Doppler.

STAP technique is the bidimensional adaptive processing in space and time which is employed for the purposes of clutter mitigation to enable the detection of slow moving targets. In this paper, a new method of partially adaptive STAP was proposed. The approach is based on the use of subspace projection, where the principal subspace has been tracked with m-FAPI algorithm. Both in the terms of estimation error and SINR loss, the method is proven able to outperform FAPI and OPAST algorithm. Furthermore, it is shown that m-FAPI algorithm converges quickly, and the method achieves clutter mitigation performance quite comparable with the EVD method. Finally, the proposed subspace tracking algorithm can be applied as the starting point of a real-time STAP processor, where the expensive EVD procedure in eigenvector computation can be avoided.

This work was supported by the National Natural Science Foundation of China (60901066), the New Teacher Foundation of Ministry of Education (20090203120006), and the Fundamental Research Funds for the Central University.