Subspace-based Algorithms Research Articles

Experimental Evaluation of an SDR-Based UAV Localization System.

UAV communications have seen a rapid rise in the last few years. The drone class of UAV has particularly become more widespread around the world, and illicit behavior using drones has become a problem. Therefore, localization, tracking, and even taking control of drones have also gained interest. Knowing the frequency of a target signal, its position can be determined (as the angle of arrival with respect to a fixed receiver point) using radio frequency-based localization techniques. One such technique is represented by the subspace-based algorithms that offer highly accurate results. This paper presents the implementation of the MUSIC algorithm on an SDR-based system using a uniform circular antenna array and its experimental evaluation in relevant outdoor environments for drone localization. The results show the capability of the system to indicate the AoA of the target signal. The results are compared with the actual direction computed from the log files of the drone application and validated with a professional direction-finding solution (i.e., Narda SignalShark equipped with the automatic direction-finding antenna).

Super-resolution DOA estimation methods based on element multiplication and summation

In this article, we propose two direction-of-arrival (DOA) estimation methods to further improve the resolution and performance of subspace-based algorithms. While most subspace-based algorithms employ either noise or signal subspaces, we utilize the signal and noise subspaces jointly. First, we project the steering vectors to the estimated signal and noise subspaces respectively to obtain the projection vectors, then, by investigating the numerical characteristics behind the inner products between the steering vectors and the estimated eigenvectors of the sample covariance matrix, an initial estimator is proposed based on the multiplication of elements of the projection vectors instead of the summation. Though the initial estimator achieves higher resolution than most existing subspace-based methods, the appearance of pseudo peaks, especially in high SNR, may affect the estimation accuracy. By analyzing the main elements in the initial objective function that cause pseudo peaks, two improved algorithms incorporating element summation are proposed to balance the resolution capacity and estimation accuracy, which are especially suitable for small sample size, low SNR and closely spaced sources. The numerical simulations verify the effectiveness and superiority of the proposed algorithms.

High-Accuracy DOA Estimation Based on an Improved Sample Correlation Matrix

In a direction-finding process, high-resolution subspace-based algorithms are the most popular ones. It is well-known that their performance of direction of arrival estimation mainly depends on the accuracy of the signal subspace. However, the traditional methods of capturing the signal subspace do not mine the information hidden in the array output in depth, which may restrict their application to some extent. In this study, we elaborate on a novel scheme to extract the signal subspace through refinement of the correlation matrix of the array output. In the developed scheme, a collection of spatial temporal correlation matrices is firstly established. Then, we define a weighting vector for the correlation matrices, and take the weighted average of the correlation matrices as the covariance matrix of the array output. It is clear that this covariance matrix is more general than the traditional covariance matrix, and the signal subspace can be optimized through adjustment of the weighting vector. In this study, we present an optimal weighting vector by adopting the particle swarm optimization. Simulation results demonstrate that the proposed approach has better performance in root mean square error compared to the existing schemes.

A Comprehensive Analysis of the Impact of System Parameters on Subspace-based DoA Estimation Performance

This work provides an explanatory analysis of the influence of input parameters on the performance of subspace-based Direction of Arrival (DoA) estimation algorithms. The objective of this work is twofold. First, to drive a Steering Vector (SV) that works for arbitrary array configuration rather than just Uniform Linear Array (ULA) geometry. Second, to identify how the performance of the subspace-based algorithms is affected by tuning the input parameters. The later objective is crucial as it allows optimizing the algorithm through selecting optimum parameters to set an appropriate tradeoff between complexity and performance based on the intended applications. Toward that end, we firstly drive an SV for arbitrary array configuration followed by revealing the working principle of subspace based DoA techniques. Secondly, we evaluate the impact of several parameters namely Signal to Noise Ratio (SNR), number of snapshots, number of array elements, separation between array elements, number of available sources, and dependency between sources to conduct our analysis. Numerical simulations over a wide range of scenarios along with intensive Monte Carlo simulations are conducted to show the influence of these parameters on the resolution, accuracy, and complexity of the subspace based DoA estimation algorithm. As demonstrated by the obtained results, the performance of this class of DoA estimation method is determined mostly by the values of the input parameters. Furthermore, the simulation results show that tradeoff between performance and computational complexity needs to be considered when the system parameters are chosen for DoA estimation algorithms.

Performance of subspace-based algorithms associated with the sample sign covariance matrix

Complex-valued data in statistical signal processing applications have many advantages over their real-valued counterparts. It allows us to use the complete statistical information of the signal thanks to its statistical property of non-circularity. This paper presents a general framework for developing asymptotic theoretical results on the distribution-free sample sign covariance matrix (SSCM) under circular complex-valued elliptically symmetric (C-CES) and non-circular CES (NC-CES) multidimensional distributed data. It extends some partial asymptotic results on SSCM derived for real elliptically symmetric (RES) distributed data. In particular closed-form expressions of the first and second-order of the SSCM are derived for arbitrary spectra of eigenvalues for C-CES and NC-CES distributed data which facilitates the derivation of numerous statistical properties. Then, the asymptotic distributions of associated projectors are deduced, which are applied in the study of asymptotic performance analysis of SSCM-based subspace algorithms, followed by a comparison to the asymptotic results derived using Tyler's M estimate. However, a more in-depth analytical analysis of the efficiency of the SSCM relative to Tyler's M estimate is performed, yielding that the performances of the SSCM and Tyler's M estimate are close for a high-dimensional data and not too small dimension of the principal component space. We conclude therefore that, although the SSCM is inefficient relative to Tyler's M estimate, it is of great interest from the point of view of its lower computational complexity for high-dimensional data. Finally, numerical results illustrating the theoretical analysis are presented through the direction-of-arrival (DOA) estimation CES data models.

High-Accuracy DOA Estimation Algorithm at Low SNR Through Exploiting a Supervised Index

Performance of direction of arrival (DOA) estimation is one of the most important issues in array signal processing. Subspace-based algorithms provide a good compromise between the estimation accuracy and computational complexity. However, these methods are exposed to performance breakdown at low SNR scenarios. A major reason for such performance breakdown is the subspace swap phenomenon (intersubspace leakage). In this article, we elaborate on a novel modified signal subspace model through exploiting a supervised index for the estimation of DOA. With the developed model we refine the signal subspace so as to enhance the performance of the DOA estimation. In the proposed scheme, we define a fuzzy similarity matrix for the eigenvalues of the array output correlation matrix to capture the distribution of the eigenvalues. Then, we build up a transformation matrix between the fuzzy similarity matrix and the eigenspace of the correlation matrix, and construct a nonlinear transformation function to adjust the fuzzy similarity matrix. Subsequently, we define a supervised evaluation index named signal subspace reconstruction error for DOA estimation and construct a cost function of the signal subspace to develop a supervised model for the signal subspace. The signal subspace can be modified through adjusting the parameter in the nonlinear transformation function and optimizing the abovementioned cost function. Finally, the performance of DOA estimation can be enhanced with the modified signal subspace. The main characteristic of the proposed model is circularly applied feedback of the estimated DOA for refining the estimated subspace. It is a closed loop and supervised method not reported before. This article opens a specific way for improving the performance of the DOA estimation in array signal processing by a supervised index. However, the proposed method is still unsatisfying in some scopes of signal-to-noise ratio. We believe that exploiting a validity index for DOA estimation in array signal processing is still a general and interesting problem.

Low-Complexity 2D DOA Estimation and Self-Calibration for Uniform Rectangle Array with Gain-Phase Error

Most subspace-based algorithms need exact array manifold for direction of arrival (DOA) estimation, while, in practical applications, the gain-phases of different array elements are usually inconsistent, degrading their estimation performance. In this paper, a novel low-complexity 2D DOA and gain-phase error estimation algorithm is proposed by adding auxiliary array elements in a uniform rectangular array (URA). Firstly, the URA is modeled as the Kronecker product of two uniform linear arrays (ULAs) to decouple the 2D DOA estimation. Then, several well-calibrated auxiliary array elements are added in the two ULAs, based on which the rotation invariant factor of the URA destroyed by the gain-phase error is reconstructed by solving constrained optimization problems. Lastly, ESPRIT is used to estimate the 2-D DOA and the gain-phase error coefficients. The closed-form expressions of the estimation CRBs are also derived, providing insight into the impact of gain-phase error on DOA estimation. Simulation results are used to validate the effectiveness of the proposed algorithm and the correctness of the theoretical analysis.

High resolution DOA estimation of acoustic plane waves: An innovative comparison among Cuckoo search heuristics and subspace based algorithms

SONAR signal processing plays an indispensable role when it comes to parameter estimation of Direction of Arrival (DOA) of acoustic plane waves for closely spaced target exclusively under severe noisy environments. Resolution performance of classical MUSIC and ESPRIT algorithms and other subspace-based algorithms decreases under scenarios like low SNR, smaller number of snapshots and closely spaced targets. In this study, optimization strength of swarm intelligence of Cuckoo Search Algorithm (CSA) is accomplished for viable DOA estimation in different scenarios of underwater environment using a Uniform Linear Array (ULA). Higher resolution for closely spaced targets is achieved using smaller number of snapshots viably with CSA by investigating global minima of the highly nonlinear cost function of ULA. Performance analysis of CSA for different number of targets employing estimation accuracy, higher resolution, variance analysis, frequency distribution of RMSE over the monte Carlo runs and robustness against noise in the presence of additive-white Gaussian measurement noise is achieved. Comparative studies of CSA with Root MUSIC and ESPRIT along with Crammer Rao Bound analysis witnesses better results for estimating DOA parameters which are further endorsed from the results of Monte Carlo simulations.

Shift invariant sparse arrays and their optimal signal and noise subspaces

Many signal processing algorithms utilize singular vectors of the data received by a sensor array or eigenvectors of the sample covariance matrix. The performance of such subspace-based algorithms can be substantially improved by using the optimal signal or noise subspace estimate instead of using the subspaces spanned by the eigenvectors or the singular vectors. The optimal signal and noise subspaces are estimated by exploiting the shift-invariant structure of the sensor array. We develop a methodology to find the optimal subspaces in sparse arrays that possess shift-invariant structure and interpolate the optimal subspaces to match the subspaces of the fully populated arrays with an equal aperture. These shift-invariant sparse arrays’ direction of arrival estimation performance is superior compared to the methods that use eigenvectors or singular vectors directly. Our method also encompasses any symmetric array that is not shift-invariant, thus broadening the class of sparse arrays where our method can be applied.

FMCW Radar Estimation Algorithm with High Resolution and Low Complexity Based on Reduced Search Area.

We propose a frequency-modulated continuous wave (FMCW) radar estimation algorithm with high resolution and low complexity. The fast Fourier transform (FFT)-based algorithms and multiple signal classification (MUSIC) algorithms are used as algorithms for estimating target parameters in the FMCW radar systems. FFT-based and MUSIC algorithms have tradeoff characteristics between resolution performance and complexity. While FFT-based algorithms have the advantage of very low complexity, they have the disadvantage of a low-resolution performance; that is, estimating multiple targets with similar parameters as a single target. On the other hand, subspace-based algorithms have the advantage of a high-resolution performance, but have a problem of very high complexity. In this paper, we propose an algorithm with reduced complexity, while achieving the high-resolution performance of the subspace-based algorithm by utilizing the advantages of the two algorithms; namely, the low-complexity advantage of FFT-based algorithms and the high-resolution performance of the MUSIC algorithms. The proposed algorithm first reduces the amount of data used as input to the subspace-based algorithm by using the estimation results obtained by FFT. Secondly, it significantly reduces the range of search regions considered for pseudo-spectrum calculations in the subspace-based algorithm. The simulation and experiment results show that the proposed algorithm achieves a similar performance compared with the conventional and low complexity MUSIC algorithms, despite its considerably lower complexity.

Embedding the Localization and Imaging Functions in Mobile Systems: An Airport Surveillance Use Case

Driven by the extended applications and scarce spectrum resources, integrating the radio sensing functions into the future mobile system has been a consensus between the stakeholders. This paper demonstrates the feasibility of joint localization and imaging functions by exploiting the reference signal in the radio frame defined in the mobile system. Essentially, the subspace-based algorithms are adopted to jointly estimate the angle and distance and to enable the localization function for uplink and downlink signals and the derivation of theoretical performance boundaries. The vector antenna and virtual array concepts are introduced in the downlink scenario to enhance the angle estimation resolution. The uplink sounding reference signal is exploited to enable the imaging function as analog to synthetic-aperture radar (SAR). The joint localization and imaging performances are verified with a realistic ray-tracing channel based on the 3D ground and buildings model of Muret airport in France. The simulation results show that the reference signal can provide acceptable localization accuracy and target-distinguishing capability by adopting a virtual array and joint time-spatial smoothing in downlink and uplink, respectively. The joint localization and imaging results prove that the future mobile network is potentially a viable infrastructure to provide economic surveillance solutions for airports, particularly for secondary airports that are not well equipped with dedicated surveillance systems.

Joint Channel and AoA Estimation in OFDM Systems: One Channel Tap With Multiple AoAs Problem

Recently, a joint channel and angle-of-arrival (AoA) estimation scheme for pilot-assisted multiple-input multiple-output orthogonal frequency division multiplexing (MIMO-OFDM) systems with digital arrays has been proposed; in that scheme, compressive sensing-based time-domain channel estimation is applied. In the scheme, it is assumed that each channel tap has only one AoA. However, in reality, one channel tap may contain responses from multiple paths with different AoAs. In this letter, we propose an improved approach to solve the aforementioned problem. The underlying concept is to transmit multiple OFDM symbols with different beamforming vectors, rendering the time-invariant channel equivalently time-variant. Subsequently, the correlation matrix, imbedded with AoA information, can be estimated. Finally, subspace-based algorithms, such as multiple signal classification (MUSIC), can be used for AoA estimation. The advantage of the proposed method is that the number of detectable AoAs is much larger than that in conventional MUSIC. The performance of the proposed method was also theoretically analyzed. Simulation results demonstrated that with a small number of OFDM symbols, both channel gains and AoAs could be satisfactorily estimated and the theoretical analysis results were accurate.

A novel reduced-dimensional beamspace unitary ESPRIT algorithm for monostatic MIMO radar

In this paper, a novel direction of arrival (DOA) estimation algorithm is proposed for monostatic multiple-input multiple-output (MIMO) radar. Due to redundancy of the received data of monostatic MIMO radar, the computational complexity of the inversion of the covariance matrix is increased. In order to reduce the large computational complexity of existing subspace-based algorithms, we perform two reduced-dimensional operations. First, the high-dimensional received data is transformed the low-dimensional received data through a special transformation matrix. Second, the low-dimensional received data is transformed into beamspace. The receive beamspace filter is designed using convex optimization, and it can flexibly control the bandwidth and limit the sidelobe level. To ensure that the converted noise is Gaussian white noise, the prewhitening process is performed. Based on the above precondition, the reduced-dimensional beamspace technique can be effectively combined with the unitary ESPRIT model. Finally, the Cramer-Rao bound (CRB) on angle estimation in element space and reduced-dimensional beamspace is calculated for performance analysis. Numerical simulations verify the effectiveness of the proposed algorithm.

Nested MIMO Radar: Coarrays, Tensor Modeling, and Angle Estimation

This article addresses the problem of joint direction of departure (DOD) and direction of arrival (DOA) estimation with nested bistatic multiple input multiple output (MIMO) radar using tensor decomposition. We first employ the two-level nested transmit and receive arrays to develop the sum-difference coarray for constructing the Toeplitz and spatial smoothing matrices. We then generalize the three-way tensor model from DOD and DOA dimensions, and derive the optimized tensor by maximizing the number of detectable targets, where the existing COMFAC technique is exploited for angle estimation. We show that the proposed method can identify more targets and achieve better performance by enforcing the three-way structure information compared with the subspace-based algorithms. We also show that the conventional tensor model is just a special case. Finally, we derive the coarray Cramér–Rao Bound (CRB) for the nested MIMO radar, and also conduct a study for the conditions under which the CRB exists. Numerical simulations are provided to validate the theoretical analysis and demonstrate the performance improvement.

Robustness and performance analysis of subspace-based DOA estimation for rectilinear correlated sources in CES data model

This paper focuses on a theoretical performance analysis of subspace-based algorithms for the localization of spatially correlated rectilinear sources embedded in circular complex elliptically symmetric (C-CES) distributed noise model and also when the observations are non-circular CES (NC-CES) distributed with dependent scatter matrices on the direction of arrival (DOA) parameters. A perturbation analysis has been performed to derive closed-form expressions for the asymptotic covariance matrices of DOA estimates for non-circular subspace-based algorithms in two CES data models. Robustness of subspace-based algorithms is theoretical evaluated using robust covariance matrix estimators (instead of the sample covariance matrix (SCM)). We prove, for the first time, interpretable closed-form expressions of the asymptotic variance of the estimated DOA of two equi-power correlated sources, which allows us to derive a number of properties describing the DOA variance’s dependence on signals parameters and non-Gaussian distribution of the noise. Different robustness properties are theoretically analyzed. In particular, we prove in the framework of NC-CES distributed observations, that Tyler’s M-estimator enhances the performance for heavy-tailed distributions w.r.t. the SCM, with negligible loss in performance for circular Gaussian distributed observations. Finally, some Monte Carlo illustrations are given for quantifying this robustness and specifying the domain of validity of our theoretical asymptotic results.

Hierarchical Sparse Subspace Clustering (HESSC): An Automatic Approach for Hyperspectral Image Analysis

Hyperspectral imaging techniques are becoming one of the most important tools to remotely acquire fine spectral information on different objects. However, hyperspectral images (HSIs) require dedicated processing for most applications. Therefore, several machine learning techniques were proposed in the last decades. Among the proposed machine learning techniques, unsupervised learning techniques have become popular as they do not need any prior knowledge. Specifically, sparse subspace-based clustering algorithms have drawn special attention to cluster the HSI into meaningful groups since such algorithms are able to handle high dimensional and highly mixed data, as is the case in real-world applications. Nonetheless, sparse subspace-based clustering algorithms usually tend to demand high computational power and can be time-consuming. In addition, the number of clusters is usually predefined. In this paper, we propose a new hierarchical sparse subspace-based clustering algorithm (HESSC), which handles the aforementioned problems in a robust and fast manner and estimates the number of clusters automatically. In the experiment, HESSC is applied to three real drill-core samples and one well-known rural benchmark (i.e., Trento) HSI datasets. In order to evaluate the performance of HESSC, the performance of the new proposed algorithm is quantitatively and qualitatively compared to the state-of-the-art sparse subspace-based algorithms. In addition, in order to have a comparison with conventional clustering algorithms, HESSC’s performance is compared with K-means and FCM. The obtained clustering results demonstrate that HESSC performs well when clustering HSIs compared to the other applied clustering algorithms.

Direct Signal Space Construction Luenberger Tracker with Quadratic Least Square Regression for Computationally Efficient DoA Tracking

In this paper, we propose a computationally efficient direction-of-arrival (DoA) tracking scheme called the direct signal space construction Luenberger tracker (DSPCLT) with quadratic least square (QLS) regression. Also, we study analytically how the choice of observer gain affects the algorithm performance and illustrate how we can use the theoretical results in determining optimal observer gain value. The proposed scheme (DSPCLT) has several distinct features compared with existing algorithms. First, it requires only a fraction of computational complexity compared with other schemes. Secondly, it maintains robustness by treating separately the special case of object overlap in which subspace-based algorithms often suffer from lack of resolvability. Thirdly, the proposed scheme achieves enhanced performance by a method of delay compensation, which accounts for observation delay. Through numerical analysis, we show that DSPCLT achieves performance similar or superior to existing algorithms with only a fraction of computational requirement.

Efficiency of subspace-based estimators for elliptical symmetric distributions

Subspace-based algorithms that exploit the orthogonality between a sample subspace and a parameter-dependent subspace have proved very useful in many applications in signal processing. The purpose of this paper is to complement theoretical results already available on the asymptotic (in the number of measurements) performance of subspace-based estimators derived in the Gaussian context to real elliptical symmetric (RES), circular complex elliptical symmetric (C-CES) and non-circular CES (NC-CES) distributed observations in the same framework. First, the asymptotic distribution of M-estimates of the orthogonal projection matrix is derived from those of the M-estimates of the covariance matrix. This allows us to characterize the asymptotically minimum variance (AMV) estimator based on estimates of orthogonal projectors associated with different M-estimates of the covariance matrix. A closed-form expression is then given for the AMV bound on the parameter of interest characterized by the column subspace of the mixing matrix of general linear mixture models. We also specify the conditions under which the AMV bound based on Tyler’s M-estimate attains the stochastic Cramér-Rao bound (CRB) for the complex Student t and complex generalized Gaussian distributions. Finally, we prove that the AMV bound attains the stochastic CRB in the case of maximum likelihood (ML) M-estimate of the covariance matrix for RES, C-CES and NC-CES distributed observations, which is equal to the semiparametric CRB (SCRB) recently introduced.

Automated Harmonic Signal Removal Technique Using Stochastic Subspace-Based Image Feature Extraction.

This paper presents automated harmonic removal as a desirable solution to effectively identify and discard the harmonic influence over the output signal by neglecting any user-defined parameter at start-up and automatically reconstruct back to become a useful output signal prior to system identification. Stochastic subspace-based algorithms (SSI) methods are the most practical tool due to the consistency in modal parameters estimation. However, it will be problematic when applied to structures with rotating machines and the presence of harmonic excitations. Difficulties arise when automating this procedure without any human interaction and the problem is still unresolved because stochastic subspace-based algorithms (SSI) methods still require parameters (the maximum within-cluster distance) that are compulsory to be defined at start-up for each analysis of the dataset. Thus, the use of image-based feature extraction for clustering and classification of harmonic components and structural poles directly from a stabilization diagram and for modal system identification is the focus of the present paper. As a fundamental necessary condition, the algorithm has been assessed first from computed numerical responses and then applied to the experimental dataset with the presence of harmonic excitation. Results of the proposed approach for estimating modal parameters demonstrated very high accuracy and exhibited consistent results before and after removing harmonic components from the response signal.

Joint Angle–Frequency Estimation Using Nested Sampling with One-Bit Quantization

The problem of joint angle and frequency estimation is considered in this paper. An algorithm using spatial–temporal nested sampling and one-bit quantization is developed. It is shown that under moderate condition when the signal-to-noise ratio is low enough, the defined one-bit correlation matrix has linear relationship with the unquantized one, and therefore, subspace-based algorithms can be applied to this one-bit correlation matrix directly. The enhanced identification performance, offered by the nested sampling, can also be provided when one-bit quantization is used. Simulation results are provided to verify the performance of the presented algorithm.