Low Sample Support Research Articles

A STAP Detection Scheme for Low Sample Support Maritime Environments

In airborne radar, reduced rank space-time adaptive processing techniques are employed when non-stationarity and non-homogeneity of the clutter causes insufficient sample support. There have been many approaches proposed to address this problem, including principal components, the cross-spectral metric and the multistage Wiener filter. This latter approach is superior to other reduced rank techniques in terms of computational efficiency, sample support and rank requirements. Regarding operation in heterogeneous environments, the single data set approach for clutter suppression has been proposed and operates solely on the cell under test to obtain a clutter covariance estimate. It is therefore highly effective in environments with limited training data that is homogeneous with the test data. In this paper, a single data set detection approach under the framework of the multistage Wiener filter is proposed and analysed to enhance clutter suppression capabilities. The target detection performance of the filter is evaluated using simulated maritime radar data.

Super-resolution Direction of Arrival Estimation Using a Minimum Mean-Square Error Framework

This paper develops a novel sparse direction-of-arrival (DOA) estimation technique that avoids the common requirement of hyperparameters, which are typically difficult to set suitably in practice. Using the presented minimum mean-square error (MMSE) estimation framework, we propose a computationally efficient super-resolution DOA estimator that is implemented using an alternate updating of the spatial power distribution of the signals and of the dictionary matrix using a ridge regression algorithm. The regularization parameter determining the sparsity of the solution is formed from the previous spatial power distribution estimates using a SPICE-based criteria. The method employs an adaptive gridding strategy to avoid the grid mismatch problem. The computational complexity is further reduced by the use of an integrated wideband dictionary, determing the active dictionary in an iterative manner. The method offers high resolution estimates of closely spaced sources even for very low sample support (single snapshot) without assuming prior knowledge of the number of sources. Our evaluations illustrate the preferable performance of the proposed estimator as compared to various state-of-the art estimators, also indicating that the method’s performance approaches the Cramér-Rao lower bound (CRB) for the examined problem as the signal-to-noise ratio (SNR) increases.

Physics-Aware Generative Adversarial Networks for Radar-Based Human Activity Recognition

Generative adversarial networks (GANs) have recently been proposed for the synthesis of RF micro-Doppler signatures to address the issue of low sample support and enable the training of deeper neural networks (DNNs) for enhanced RF signal classification. But GANs suffer from systemic kinematic inconsistencies that decrease performance when GAN-synthesized data is used for training DNNs in human activity recognition. As a solution to this problem, this paper proposes the design of a multi-branch GAN (MBGAN), which integrates domain knowledge into its architecture, and physics-aware metrics based on correlation and curve-matching in the loss function. The quality of the synthetic samples generated is evaluated via image quality metrics, the ability to synthesize data that reflects human physical properties and generalize to broader subject profiles, and the achieved classification accuracy. Our experimental results show the proposed approach generates synthetic data for training that more accurately matches target kinematics, resulting in an increase of 9% in classification accuracy when classifying 14 different ambulatory human activities.

Robust Minimum Variance Beamforming With Sidelobe-Level Control Using the Alternating Direction Method of Multipliers

Adaptive beamforming with sidelobe-level control in the presence of signal steering vector uncertainty is investigated. Unlike the traditional multiconstrained optimization strategy using the interior point method, iterative optimization algorithms with the aid of the alternating direction method of multipliers (ADMM) framework are proposed. The uncertainty set constraint and the sidelobe constraint are formulated into two optimization subproblems and handled with the Lagrange multiplier method. By introducing matrix decomposition techniques, subproblem 1 is transformed into a polynomial root-finding problem that can be solved with low computational complexity. For subproblem 2, a closed-form solution can be obtained directly. Furthermore, for the continuously receiving snapshots case, iterative gradient minimization is introduced and embedded into the ADMM iterations to give an approximate solution free from matrix decompositions. Theoretical analyses and simulations verify the low complexities and performance advantages of the proposed algorithms in the low sample support, steering vector mismatch, and real-time snapshot update scenarios.

Detection of a rank-one signal with limited training data

In this paper, we reconsider the problem of detecting a matrix-valued rank-one signal in unknown Gaussian noise, which was previously addressed for the case of sufficient training data. We relax the above assumption to the case of limited training data. We re-derive the corresponding generalized likelihood ratio test (GLRT) and two-step GLRT (2S–GLRT) based on certain unitary transformation on the test data. It is shown that the re-derived detectors can work with low sample support. Moreover, in sample-abundant environments the re-derived GLRT is the same as the previously proposed GLRT and the re-derived 2S–GLRT has better detection performance than the previously proposed 2S–GLRT. Numerical examples are provided to demonstrate the effectiveness of the re-derived detectors.

Space-Time Adaptive Detection at Low Sample Support

An important problem in space-time adaptive detection is the estimation of the large p-by-p interference covariance matrix from training signals. When the number of training signals n is greater than 2p, existing estimators are generally considered to be adequate, as demonstrated by fixed-dimensional asymptotics. But in the low-sample-support regime (n < 2p or even n < p) fixed-dimensional asymptotics are no longer applicable. The remedy undertaken in this paper is to consider the "large dimensional limit" in which n and p go to infinity together. In this asymptotic regime, a new type of estimator is defined (Definition 2), shown to exist (Theorem 1), and shown to be detection-theoretically ideal (Theorem 2). Further, asymptotic conditional detection and false-alarm rates of filters formed from this type of estimator are characterized (Theorems 3 and 4) and shown to depend only on data that is given, even for non-Gaussian interference statistics. The paper concludes with several Monte Carlo simulations that compare the performance of the estimator in Theorem 1 to the predictions of Theorems 2-4, showing in particular higher detection probability than Steiner and Gerlach's Fast Maximum Likelihood estimator.

Robust Two-Stage Reduced-Dimension Sparsity-Aware STAP for Airborne Radar With Coprime Arrays

Space-time adaptive processing (STAP) algorithms with coprime arrays can provide good clutter suppression potential with low cost in airborne radar systems as compared with their uniform linear arrays counterparts. However, the performance of these algorithms is limited by the training samples support in practical applications. To address this issue, a robust two-stage reduced-dimension (RD) sparsity-aware STAP algorithm is proposed in this work. In the first stage, an RD virtual snapshot is constructed using all spatial channels but only $m$ adjacent Doppler channels around the target Doppler frequency to reduce the slow-time dimension of the signal. In the second stage, an RD sparse measurement modeling is formulated based on the constructed RD virtual snapshot, where the sparsity of clutter and the prior knowledge of the clutter ridge are exploited to formulate an RD overcomplete dictionary. Moreover, an orthogonal matching pursuit (OMP)-like method is proposed to recover the clutter subspace. In order to set the stopping parameter of the OMP-like method, a robust clutter rank estimation approach is developed. Compared with recently developed sparsity-aware STAP algorithms, the size of the proposed sparse representation dictionary is much smaller, resulting in low complexity. Simulation results show that the proposed algorithm is robust to prior knowledge errors and can provide good clutter suppression performance in low sample support.

Subspace Averaging and Order Determination for Source Enumeration

In this paper, we address the problem of subspace averaging, with special emphasis placed on the question of estimating the dimension of the average. The results suggest that the enumeration of sources in a multi-sensor array, which is a problem of estimating the dimension of the array manifold, and as a consequence the number of radiating sources, may be cast as a problem of averaging subspaces. This point of view stands in contrast to conventional approaches, which cast the problem as one of identifiying covariance models in a factor model. We present a robust formulation of the proposed order fitting rule based on majorization–minimization algorithms. A key element of the proposed method is to construct a bootstrap procedure, based on a newly proposed discrete distribution on the manifold of projection matrices, for stochastically generating subspaces from a function of experimentally determined eigenvalues. In this way, the proposed subspace averaging (SA) technique determines the order based on the eigenvalues of an average projection matrix, rather than on the likelihood of a covariance model, penalized by functions of the model order. By means of simulation examples, we show that the proposed SA criterion is especially effective in high-dimensional scenarios with low sample support.

Detection of Gaussian Signal Using Adaptively Whitened Data

The adaptive matched filter, like many other adaptive detection schemes, uses in its test statistic the data under test whitened by the sample covariance matrix S of the training samples. Actually, it is a generalized likelihood ratio test (GLRT) based on the conditional (i.e., for given S) distribution of the adaptively whitened data. In this letter, we investigate detection of a Gaussian rank-one signal using the marginal (unconditional) distribution of the adaptively whitened data. A first contribution is to derive the latter and to show that it only depends on a scalar parameter, namely the signal to noise ratio. Then, a GLRT is formulated from this unconditional distribution and shown to have the constant false alarm rate property. We show that it bears close resemblance with the plain GLRT based on the whole data set (data under test and training samples). The new detector performs as well as the plain GLRT and even better with multiple cells under test and low training sample support.

Deep Neural Network Initialization Methods for Micro-Doppler Classification With Low Training Sample Support

Deep neural networks (DNNs) require large-scale labeled data sets to prevent overfitting while having good generalization. In radar applications, however, acquiring a measured data set of the order of thousands is challenging due to constraints on manpower, cost, and other resources. In this letter, the efficacy of two neural network initialization techniques—unsupervised pretraining and transfer learning—for dealing with training DNNs on small data sets is compared. Unsupervised pretraining is implemented through the design of a convolutional autoencoder (CAE), while transfer learning from two popular convolutional neural network architectures (VGGNet and GoogleNet) is used to augment measured RF data for training. A 12-class problem for discrimination of micro-Doppler signatures for indoor human activities is utilized to analyze activation maps, bottleneck features, class model, and classification accuracy with respect to training sample size. Results show that on meager data sets, transfer learning outperforms unsupervised pretraining and random initialization by 10% and 25%, respectively, but that when the sample size exceeds 650, unsupervised pretraining surpasses transfer learning and random initialization by 5% and 10%, respectively. Visualization of activation layers and learned models reveals how the CAE succeeds in representing the micro-Doppler signature.

Bayesian generalised likelihood ratio tests for distributed target detection in interference and noise

This study addresses the problem of detecting a distributed target in interference and noise with low sample support. The target signal and interference lie in two linearly independent subspaces and the spectral property of the noise is unknown. The number of available training data is too small to form a non-singular sample covariance matrix. To overcome the difficulty, the authors resort to the Bayesian framework and design two generalised likelihood ratio tests. Both detectors can reduce training requirement by utilising the a prior knowledge of the noise covariance matrix. Numerical results show that the proposed detectors provide better detection performance than the conventional solutions in training-limited scenarios.

A bias-compensated MUSIC for small number of samples

The multiple signal classification (MUSIC) method is known to be asymptotically efficient, yet with a small number of snapshots its performance degrades due to bias in MUSIC localization function. In this communication, starting from G-MUSIC which improves over MUSIC in low sample support, a high signal to noise ratio approximation of the G-MUSIC localization function is derived. This approximation results in closed-form expressions of the weights applied to each eigenvector of the sample covariance matrix. A new method which consists in minimizing this simplified G-MUSIC localization function is thus introduced, and referred to as sG-MUSIC. Interestingly enough, this sG-MUSIC criterion can be interpreted as a bias correction of the conventional MUSIC localization function. Numerical simulations indicate that sG-MUSIC incur only a marginal loss in terms of mean square error of the direction of arrival estimates, as compared to G-MUSIC, and performs better than MUSIC.

Adaptive Beamforming with Inadequate Snapshots

In array signal processing, the covariance matrix used to calculate the adaptive weights is often poor estimated when the snapshot number is inadequate. The prior environmental knowledge can be used to make the estimation more accuracy. In this paper, an alternative knowledge-aided adaptive beamforming approach that is robust to low sample support environment is proposed. In this algorithm the covariance matrix used to calculate the optimum weights is constructed by blending a sample covariance matrix and a priori structured covariance matrix. Numerical simulations demonstrate the proposed algorithm has the potential for substantial performance improvement.

Saddlepoint Approximations for Correlation Testing Among Multiple Gaussian Random Vectors

This letter considers the problem of threshold selection for a correlation test among multiple $({\geq}2)$ random vectors. The generalized likelihood ratio test (GLRT) for this problem uses a generalized Hadamard ratio to test for block diagonality in a composite covariance matrix. As the number of realizations used to estimate the composite covariance matrix grows large, the null distribution of the likelihood ratio statistic converges to a chi-squared distribution which can be used to prescribe thresholds needed to achieve a desired false alarm rate in high sample support situations. However, this asymptotic distribution can be slow to converge, making its use dubious in many practical scenarios. To address this problem, this letter uses saddlepoint approximations for the null distribution of the generalized Hadamard ratio. Simulations are provided to demonstrate the saddlepoint approximation’s ability to achieve a desired false alarm probability, even in situations with low sample support.

On the Expected Likelihood Approach for Assessment of Regularization Covariance Matrix

Regularization, which consists in shrinkage of the sample covariance matrix to a target matrix, is a commonly used and effective technique in low sample support covariance matrix estimation. Usually, a target matrix is chosen and optimization of the shrinkage factor is carried out, based on some relevant metric. In this letter, we rather address the choice of the target matrix. More precisely, we aim at evaluating, from observation of the data matrix, whether a given target matrix is a good regularizer. Towards this end, the expected likelihood (EL) approach is investigated. At a first step, we re-interpret the regularized covariance matrix estimate as the minimum mean-square error estimate in a Bayesian model where the target matrix serves as a prior. The likelihood function of the data is then derived, and the EL principle is subsequently applied. Over-sampled and under-sampled scenarios are considered.

Regularized &lt;formula formulatype="inline"&gt;&lt;tex Notation="TeX"&gt;$M$&lt;/tex&gt; &lt;/formula&gt;-Estimators of Scatter Matrix

In this paper, a general class of regularized $M$-estimators of scatter matrix are proposed which are suitable also for low or insufficient sample support (small $n$ and large $p$) problems. The considered class constitutes a natural generalization of $M$-estimators of scatter matrix (Maronna, 1976) and are defined as a solution to a penalized $M$-estimation cost function that depend on a pair $(\alpha,\beta)$ of regularization parameters. We derive general conditions for uniqueness of the solution using concept of geodesic convexity. Since these conditions do not include Tyler's $M$-estimator, necessary and sufficient conditions for uniqueness of the penalized Tyler's cost function are established separately. For the regularized Tyler's $M$-estimator, we also derive a simple, closed form and data dependent solution for choosing the regularization parameter based on shape matrix matching in the mean squared sense. An iterative algorithm that converges to the solution of the regularized $M$-estimating equation is also provided. Finally, some simulations studies illustrate the improved accuracy of the proposed regularized $M$-estimators of scatter compared to their non-regularized counterparts in low sample support problems. An example of radar detection using normalized matched filter (NMF) illustrate that an adaptive NMF detector based on regularized $M$-estimators are able to maintain accurately the preset CFAR level and at at the same time provide similar probability of detection as the (theoretical) NMF detector.

Approximate maximum likelihood estimation of two closely spaced sources

The performance of the majority of high resolution algorithms designed for either spectral analysis or Direction-of-Arrival (DoA) estimation drastically degrades when the amplitude sources are highly correlated or when the number of available snapshots is very small and possibly less than the number of sources. Under such circumstances, only Maximum Likelihood (ML) or ML-based techniques can still be effective. The main drawback of such optimal solutions lies in their high computational load. In this paper we propose a computationally efficient approximate ML estimator, in the case of two closely spaced signals, that can be used even in the single snapshot case. Our approach relies on Taylor series expansion of the projection onto the signal subspace and can be implemented through 1D Fourier transforms. Its effectiveness is illustrated in complicated scenarios with very low sample support and possibly correlated sources, where it is shown to outperform conventional estimators.

A Novel Robust Adaptive Beamformer using Diagonal Loading

beamfomer which utilizes MVDR beamformer along with SMI (sample matrix inversion), actual data is not available to calculate the covariance matrix. Instead, covariance matrix is estimated from the available data. It may result in bad conditioning. To avoid this, diagonal elements are introduced in the correlation matrix, which is called diagonal loading. Diagonal loading can be inserted by adding a scaled version of identity matrix to impart Robustness to the adaptive beamformer. This proves to be efficient against signal mismatch due to low sample support and helps to achieve desired sidelobe level and SINR improvement. A novel hybrid algorithm for MVDR(SMI beamformer with colored adaptive diagonal loading is proposed in this paper. The performance of the proposed method is compared with other methods such as Conventional, MVDR(SMI(Diagonal Loading, MVDR(SMI( Colored -DL, MVDR(SMI(Adaptive DL by conducting simulation experiments to prove its effectiveness in improving the directivity and SINR degradation in performance. The SINR which is a measure of performance of the beamformer degrades as sample support (the number of data) is low. The lower band on sidelobe levels of the beamformer when no interference sources are found at an angle is also calculated. Training issues like the presence of desired signal in the correlation matrix n i R + is also dealt with. The paper is organized as follows. In Section 2, Problem formulation and general model is presented. In Section 3 Adaptive beamforming with various beamforming methods are presented along with the Novel Hybrid algorithm (Adaptive colored diagonal loading. In Section 4 simulation experiments are presented. Section 5 contains Results and discussions. Section 6 presents the conclusions.

Robust DOA Estimation: The Reiterative Superresolution (RISR) Algorithm

A new approach for spatial direction-of-arrival (DOA) estimation, denoted as re-iterative superresolution (RISR), is developed based upon a recursive implementation of the minimum mean-square error (MMSE) framework. This recursive strategy alternates between updating an MMSE filter bank according to the previous receive spatial power distribution and then subsequently applying the new filter bank to the received data snapshots to obtain a new estimate of the receive spatial power distribution. Benefits of this approach include robustness to coherent sources such as can occur in multipath environments, operation with very low sample support to enable tracking of sources with rapidly changing DOA (e.g., bistatic pulse chasing), intrinsic determination of model order, and robustness to array modeling errors by exploiting approximate knowledge of array calibration tolerances. From an implementation perspective RISR belongs to a class of recursive algorithms that includes Interior Point methods, the minimum-norm-based FoCal underdetermined system solver (FOCUSS) algorithm, and the iterative reweighted least squares (IRLS) algorithm. However, the structure of RISR also enables the natural inclusion of spatial noise covariance information as well as a mechanism to account for array modeling errors which are known to induce degradation for existing superresolution methods. The inclusion of the latter is also found to facilitate an adaptive form of regularization that establishes a feasible (given model uncertainties) dynamic range for source estimates.

Adaptive coherence estimator based on the Krylov subspace technique for airborne radar

In this paper, we propose a novel adaptive detector for airborne radar space-time adaptive detection (STAD) in partially homogeneous environments. The novel detector combines the numerically stable Krylov subspace technique and diagonal loading technique, and it uses the framework of the adaptive coherence estimator (ACE). It can effectively detect a target with low sample support. Compared to its natural competitors, the novel detector has higher probability of detection (PD), especially when the number of the training data is low. Moreover, it is shown to be practical constant false alarm rate (CFAR).