Eigenvalue Decomposition Of Matrix Research Articles

Robust multicomponent LFM signals synthesis algorithm based on masked ambiguity function

Wigner distribution (WD) based multiple components linear frequency modulation (LFM) signal synthesis method (SSM) is often adversely affected by cross-terms. Masked WD (MWD) is one of the widely used cross-term suppression techniques in practice due to its simplicity and efficiency. However, the cross-terms are hardly masked out when auto-terms and cross-terms are overlapped (Case I) or the components are very close to each other in the time–frequency (TF) plane (Case II). To solve these problems, we present a robust ambiguity function (AF) based approach for multicomponent signals synthesis. This algorithm consists of two stages. First, a SSM from the AF is proposed according to matrix rearrangement and eigenvalue decomposition. However, the existence of cross-term makes the signal synthesis entirely erroneous. To settle this issue, we present a masked AF (MAF) algorithm based on Radon and its inverse transforms in the second stage. Applying the presented algorithm, multicomponent signals can be synthesized efficiently even in Case I and Case II. Simulation results demonstrate the effectiveness and feasibility of the proposed algorithm.

Efficient Two-Dimensional Direction Finding via Auxiliary-Variable Manifold Separation Technique for Arbitrary Array Structure

A polynomial rooting direction of arrival (DOA) algorithm for multiple plane waves incident on an arbitrary array structure that combines the multipolynomial resultants and matrix computations is proposed in this paper. Firstly, a new auxiliary-variable manifold separation technique (AV-MST) is used to model the steering vector of arbitrary array structure as the product of a sampling matrix (dependent only on the array structure) and two Vandermonde-structured wavefield coefficient vectors (dependent on the wavefield). Then the propagator operator is calculated and used to form a system of bivariate polynomial equations. Finally, the automatically paired azimuth and elevation estimates are derived by polynomial rooting. The presented algorithm employs the concept of auxiliary-variable manifold separation technique which requires no sector by sector array interpolation and thus does not suffer from any mapping errors. In addition, the new algorithm does not need any eigenvalue decomposition of the covariance matrix and exhausted search over the two-dimensional parameter space. Moreover, the algorithm gives automatically paired estimates, thus avoiding the complex pairing procedure. Therefore, the proposed algorithm shows low computational complexity and high robustness performance. Simulation results are shown to validate the effectiveness of the proposed method.

DISPERSION ANALYSIS OF DOUBLE-SIDED OPEN PERIODIC MEDIA USING INHOMOGENEOUS PLANE WAVE EXCITATION

Double-sided open periodic structures are analyzed using inhomogeneous plane wave scattering. The leaky and surface wave modes of several unit cells of different structures are computed using the poles of generalized reflection and transmission coefficients of inhomogeneous plane waves in the spectral domain. It is shown that the reflection and transmission coefficients of the zeroth order Floquet mode contain the poles of the Green's function of the complex stratified periodic structure. The properties of evanescent mode amplification as well as super resolution near field imaging in a wire medium are addressed. A balanced leaky wave antenna unit cell with double-sided radiation feature is introduced and it is shown that, in contrast to grounded structures, total absorption in lossless non- chiral double-sided open unit cells is not feasible as long as the behavior of the unit cell is well described by its fundamental mode. In homogeneous open dielectric structures, the spectral representation of the Green's function leads to a series of propagating discrete surface wave modes, together with a continuous integral representation indicating the branch cut in the Riemann sheet (1). By introducing possibly inhomogeneous periodicities into these structures, the analysis of the whole configuration can be reduced to the analysis of one period known as unit cell. This unit cell has open boundaries at the unbounded sides, and periodic boundary conditions are assumed at its side walls, see Fig. 1. In homogeneous layered dielectric media, the spectral domain Green's functions have closed form (2). However, inhomogeneous complex objects do not have any closed form Green's function. Therefore, numerical methods are employed to compute the eigenvalues of these structures. In the numerical computation of open boundary structures, often perfectly matched layers (PML) are chosen as absorbing boundary condition (3). As a result, the spectral domain Green's function can be expressed in terms of the summation of three different groups of modes namely Berenger modes, leaky modes and surface wave modes (4). Among these modes, Berenger modes are complex modes which are more concentrated in the PML layer, while leaky modes are complex modes more concentrated in the guiding structure (5). For the analysis of complex objects immersed in or placed on top of a layered medium, the computation of leaky complex modes is possible with several methods. The most common approach is based on eigenvalue decomposition of the system matrix resulting from a numerical model (6-8). Another robust method is trying to find the stationary points of stored energy derived from physical observations. Since eigenvalue decomposition is a numerically expensive procedure, a physics based method for the computation of complex and surface wave modes of planar periodic configurations was addressed in (9, 10). This method was based on the scattering of inhomogeneous plane waves from grounded unit cells and the eigenmodes were explored by exciting the poles of the reflection coefficient.

Two-Dimensional DOA Estimation for Uniform Rectangular Array Using Reduced-Dimension Propagator Method

A novel algorithm is proposed for two-dimensional direction of arrival (2D-DOA) estimation with uniform rectangular array using reduced-dimension propagator method (RD-PM). The proposed algorithm requires no eigenvalue decomposition of the covariance matrix of the receive data and simplifies two-dimensional global searching in two-dimensional PM (2D-PM) to one-dimensional local searching. The complexity of the proposed algorithm is much lower than that of 2D-PM. The angle estimation performance of the proposed algorithm is better than that of estimation of signal parameters via rotational invariance techniques (ESPRIT) algorithm and conventional PM algorithms, also very close to 2D-PM. The angle estimation error and Cramér-Rao bound (CRB) are derived in this paper. Furthermore, the proposed algorithm can achieve automatically paired 2D-DOA estimation. The simulation results verify the effectiveness of the algorithm.

DOA and Noncircular Phase Estimation of Noncircular Signal via an Improved Noncircular Rotational Invariance Propagator Method

We consider the computationally efficient direction-of-arrival (DOA) and noncircular (NC) phase estimation problem of noncircular signal for uniform linear array. The key idea is to apply the noncircular propagator method (NC-PM) which does not require eigenvalue decomposition (EVD) of the covariance matrix or singular value decomposition (SVD) of the received data. Noncircular rotational invariance propagator method (NC-RI-PM) avoids spectral peak searching in PM and can obtain the closed-form solution of DOA, so it has lower computational complexity. Animproved NC-RI-PMalgorithm of noncircular signal for uniform linear array is proposed to estimate the elevation angles and noncircular phases with automatic pairing. We reconstruct the extended array output by combining the array output and its conjugated counterpart. Our algorithm fully uses the extended array elements in the improved propagator matrix to estimate the elevation angles and noncircular phases by utilizing the rotational invariance property between subarrays. Compared with NC-RI-PM, the proposed algorithm has better angle estimation performance and much lower computational load. The computational complexity of the proposed algorithm is analyzed. We also derive the variance of estimation error and Cramer-Rao bound (CRB) of noncircular signal for uniform linear array. Finally, simulation results are presented to demonstrate the effectiveness of our algorithm.

Sequential Matrix Diagonalization Algorithms for Polynomial EVD of Parahermitian Matrices

For parahermitian polynomial matrices, which can be used, for example, to characterize space-time covariance in broadband array processing, the conventional eigenvalue decomposition (EVD) can be generalized to a polynomial matrix EVD (PEVD). In this paper, a new iterative PEVD algorithm based on sequential matrix diagonalization (SMD) is introduced. At every step the SMD algorithm shifts the dominant column or row of the polynomial matrix to the zero lag position and eliminates the resulting instantaneous correlation. A proof of convergence is provided, and it is demonstrated that SMD establishes diagonalization faster and with lower order operations than existing PEVD algorithms.

Direction Finding and Information Detection Algorithm with an L-Shaped CCD Array

ABSTRACTThis paper proposes a quaternion-ESPRIT (estimate signal parameters via rotation invariance) algorithm for estimating frequency, direction of arrival (DOA) and polarization, using a collocated crossed dipole (CCD) L-shaped array configuration. First, two sets of synchronous sampling data are used to construct the data covariance matrix, the array steering vector of the whole array is estimated using a quaternion eigenvalue decomposition of the data covariance matrix. Second, the direction cosine of the x-axis and y-axis are acquired by applying block operation to the spatial steering vectors, hence the estimates of two-dimensional DOA are obtained. Finally, the polarization parameters are estimated using the relationship between the dipole sub-array steering vectors of the x-axis and y-axis. The closed-form solutions for each parameter are obtained and parameters can automatically match. The simulation results verify that the performance of quaternion method is obviously better than that of the long-vector method.

Study on the Improvement of MUSIC Algorithm Fire Wireless Communication

Music algorithm has good spatial resolution, provides the possibility to further improve the performance of fire radio communication system, but the algorithm in the target range rapidly changing circumstances poor stability. Aiming at this problem, this paper proposes a MUSIC algorithm based on time domain analytical signals (TAMUSIC, Time-domain Analysis MUSIC). The TAMUISC algorithm first constructs analytical time-domain signal; then the time domain analytical signal covariance matrix; finally the covariance matrix eigenvalue decomposition, the noise subspace estimation results of spatial spectrum. The simulation results show that, TAMUSIC algorithm in target azimuth change quickly, compared with the conventional MUSIC algorithm, need a short observation time, observation has smaller variance.

Distributed adaptive generalized eigenvector estimation of a sensor signal covariance matrix pair in a fully connected sensor network

The generalized eigenvalue decomposition (GEVD) of a pair of matrices generalizes the concept of the eigenvalue decomposition (EVD) of a single matrix. It is a widely used tool in signal processing applications, in particular in a context of spatial filtering and subspace estimation. In this paper, we describe a distributed adaptive algorithm to estimate generalized eigenvectors (GEVCs) of a pair of sensor signal covariance matrices in a fully connected wireless sensor network. The algorithm computes these GEVCs in an iterative fashion without explicitly constructing the full network-wide covariance matrices. Instead, the nodes only exchange compressed sensor signal observations, providing a significant reduction in per-node communication and computational cost compared to the scenario where all the raw sensor signal observations are collected and processed in a fusion center.

Extending networks of protected areas to optimize connectivity and population growth rate

Protected areas (PAs) are recognized as the flagship tool to offset biodiversity loss on Earth. Spatial conservation planning seeks optimal designs of PAs that meet multiple targets such as biodiversity representation and population persistence. Since connectivity between PAs is a fundamental requirement for population persistence, several methods have been developed to include connectivity into PA design algorithms. Among these, the eigenvalue decomposition of the connectivity matrix allows for identifying clusters of strongly connected sites and selecting the sites contributing the most to population persistence. So far, this method was only suited to optimize an entire network of PAs without considering existing PAs in the new design. However, a more cost‐effective and realistic approach is to optimize the design of an extended network to improve its connectivity and thus population persistence. Here, we develop a flexible algorithm based on eigenvalue decomposition of connectivity matrices to extend existing networks of PAs while optimizing connectivity and population growth rate. We also include a splitting algorithm to improve cluster identification. The new algorithm accounts for the change in connectivity due to the increased biological productivity often observed in existing PAs. We illustrate the potential of our algorithm by proposing an extension of the network of ∼100 Mediterranean marine PAs to reach the targeted 10% surface area protection from the current 1.8%. We identify differences between the clean slate scenario, where all sites are available for protection, irrespective of their current protection status, and the scenario where existing PAs are forced to be included into the optimized solution. By integrating this algorithm to existing multi‐objective and multi‐specific algorithms of PA selection, the demographic effects of connectivity can be explicitly included into conservation planning.

Linear and Instantaneous Mixtures Following Eigenvalue Decomposition

The physical signiflcance of independent component analysis is not clear, and it does not have only one solution. Aiming at the disadvantages, a novel linear and instantaneous mixture algorithm method based on eigenvalue decomposition of observation signals autocorrelation matrix is proposed. Assumptions of this new method are difierent from the traditional blind source separation model and independent component analysis algorithm. For difierent types of mixed matrix, this new method gets difierent information of source signals. The numerical simulation results show that, it can separate multiple unrelated source signals in despite of Gauss measurement noise and can also achieve noise reduction efiect after separation and reconstruction of observation signals.

Approximation of Spatio-Temporal Random Processes Using Tensor Decomposition

AbstractA new representation of spatio-temporal random processes is proposed in this work. In practical applications, such processes are used to model velocity fields, temperature distributions, response of vibrating systems, to name a few. Finding an efficient representation for any random process leads to encapsulation of information which makes it more convenient for a practical implementations, for instance, in a computational mechanics problem. For a single-parameter process such as spatial or temporal process, the eigenvalue decomposition of the covariance matrix leads to the well-known Karhunen-Loève (KL) decomposition. However, for multiparameter processes such as a spatio-temporal process, the covariance function itself can be defined in multiple ways. Here the process is assumed to be measured at a finite set of spatial locations and a finite number of time instants. Then the spatial covariance matrix at different time instants are considered to define the covariance of the process. This set of square, symmetric, positive semi-definite matrices is then represented as a third-order tensor. A suitable decomposition of this tensor can identify the dominant components of the process, and these components are then used to define a closed-form representation of the process. The procedure is analogous to the KL decomposition for a single-parameter process, however, the decompositions and interpretations vary significantly. The tensor decompositions are successfully applied on (i) a heat conduction problem, (ii) a vibration problem, and (iii) a covariance function taken from the literature that was fitted to model a measured wind velocity data. It is observed that the proposed representation provides an efficient approximation to some processes. Furthermore, a comparison with KL decomposition showed that the proposed method is computationally cheaper than the KL, both in terms of computer memory and execution time.

Using The Matrix Ridge Approximation to Speedup Determinantal Point Processes Sampling Algorithms

Determinantal point process (DPP) is an important probabilistic model that has extensive applications in artificial intelligence. The exact sampling algorithm of DPP requires the full eigenvalue decomposition of the kernel matrix which has high time and space complexities. This prohibits the applications of DPP from large-scale datasets. Previous work has applied the Nystrom method to speedup the sampling algorithm of DPP, and error bounds have been established for the approximation. In this paper we employ the matrix ridge approximation (MRA) to speedup the sampling algorithm of DPP, showing that our approach MRA-DPP has stronger error bound than the Nystrom-DPP. In certain circumstances our MRA-DPP is provably exact, whereas the Nystrom-DPP is far from the ground truth. Finally, experiments on several real-world datasets show that our MRA-DPP is more accurate than the other approximation approaches.

Kernel Based CSP Algorithm for Classifying Multi-class EEG Data by Combining K-means Clustering and Nystrom Approximation

Because Common Spatial Pattern (CSP) algorithm in brain-computer interface assumes that the relation between the observed EEG signal and the brain source signal is strictly linear and very limited nonlinear kernel functions exist, both linear CSP and nonlinear kernel CSP can not characterize the brain pattern accurately. In this paper, a novel hybrid kernel CSP algorithm is proposed in order to improve the performance of pattern classification, which is based on the combination of linear and nonlinear kernel function. K-means clustering and Nystrom approximation were used to relieve the computational complexity and the requirement for memory in eigenvalue decomposition of kernel matrix. The band pass filtered EEG data were clustered for dimensionality reduction and then Nystrom approximation was performed to ensure the validity of low rank decomposition of kernel matrix in a high dimensional feature space. The performance of the algorithm was tested on a four-class dataset and compared with that of linear and kernel CSP. Using a pairwise classification strategy, the proposed algorithm achieved superior performance in terms of classification accuracy.

A stabilization technique for the regularization of nearly singular extended finite elements

In this contribution a simple, robust and efficient stabilization technique for extended finite element (XFEM) simulations is presented. It is useful for arbitrary crack geometries in two or three dimensions that may lead to very bad condition numbers of the global stiffness matrix or even ill-conditioning of the equation system. The method is based on an eigenvalue decomposition of the element stiffness matrix of elements that only possess enriched nodes. Physically meaningful zero eigenmodes as well as enrichment scheme dependent numerically reasonable zero eigenmodes are filtered out. The remaining subspace is stabilized depending on the magnitude of the respective eigenvalues. One of the main advantages is the fact that neither the equation solvers need to be changed nor the solution method is restricted. The efficiency and robustness of the method is demonstrated in numerous examples for 2D and 3D fracture mechanics.

Novel Field-Programmable Gate Array Architecture for Computing the Eigenvalue Decomposition of Para-Hermitian Polynomial Matrices

In this paper, we introduce a field-programmable gate array (FPGA) hardware architecture for the realization of an algorithm for computing the eigenvalue decomposition (EVD) of para-Hermitian polynomial matrices. Specifically, we develop a parallelized version of the second-order sequential best rotation (SBR2) algorithm for polynomial matrix EVD (PEVD). The proposed algorithm is an extension of the parallel Jacobi method to para-Hermitian polynomial matrices, and as such, it is the first architecture devoted to PEVD. Hardware implementation of the algorithm is achieved via a highly pipelined, nonsystolic FPGA architecture. The efficient hardware solution is accomplished by using the coordinate rotation digital computer algorithm to calculate the trigonometric functions and vector multiplications that are performed by the algorithm. The architecture, which is scalable in terms of the size of the input para-Hermitian matrix, has been designed using the Xilinx system generator tool. We verify the algorithmic convergence of the architecture and demonstrate its decomposition performance through computer simulations and FPGA-in-the-loop hardware co-simulations. Results confirm that the proposed solution gives low execution times while reducing the number of resources required from the FPGA.

A feature selection method using fixed-point algorithm for DNA microarray gene expression data

As the performance of hardware is limited, the focus has been to develop objective, optimized and computationally efficient algorithms for a given task. To this extent, fixed-point and approximate algorithms have been developed and successfully applied in many areas of research. In this paper we propose a feature selection method based on fixed-point algorithm and show its application in the field of human cancer classification using DNA microarray gene expression data. In the fixed-point algorithm, we utilize between-class scatter matrix to compute the leading eigenvector. This eigenvector has been used to select genes. In the computation of the eigenvector, the eigenvalue decomposition of the scatter matrix is not required which significantly reduces its computational complexity and memory requirement.

Nonlinear Dimensionality Reduction via the ENH-LTSA Method for Hyperspectral Image Classification

The problems of neglecting spatial features in hyperspectral imagery (HSI) and the high complexity of Local Tangent Space Alignment (LTSA) still exist in the nonlinear dimensionality reduction with LTSA for classification. Therefore, this paper proposes an innovative ENH-LTSA (Enhanced-Local Tangent Space Alignment) method to solve the two problems. First, random projection is introduced to preliminarily reduce the dimension of HSI data. It aims to improve the speed of neighbor searching and the local tangent space construction. Then, the new method presents the similarity measure via the adaptive weighted summation kernel (AWSK) distance. The AWSK distance considers both spectral and spatial features in HSI data, and attempts to ameliorate the k-nearest neighbors (KNNs) of each pixel. Furthermore, the adaptive spatial window is proposed to automatically estimate the proper window size for the description of spatial features. After that, fast approximate KNNs graph construction via Recursive Lanczos Bisection is incorporated into the new method to reduce the complexity of KNNs searching. When finishing constructing each local tangent space, the new method uses a fast low-rank approximate singular value decomposition to speed up eigenvalue decomposition of the global alignment matrix that is constituted with local manifold coordinates. Five groups of experiments with two different HSI datasets are designed to completely analyze and testify the ENH-LTSA method. Experimental results show that ENH-LTSA outperforms LTSA, both in classification results and in computational speed.

UL-Isomap based nonlinear dimensionality reduction for hyperspectral imagery classification

The paper proposes an upgraded landmark-Isometric mapping (UL-Isomap) method to solve the two problems of landmark selection and computational complexity in dimensionality reduction using landmark Isometric mapping (LIsomap) for hyperspectral imagery (HSI) classification. First, the vector quantization method is introduced to select proper landmarks for HSI data. The approach considers the variations in local density of pixels in the spectral space. It locates the unique landmarks representing the geometric structures of HSI data. Then, random projections are used to reduce the bands of HSI data. After that, the new method incorporates the Recursive Lanczos Bisection (RLB) algorithm to construct the fast approximate k-nearest neighbor graph. The RLB algorithm accompanied with random projections improves the speed of neighbor searching in UL-Isomap. After constructing the geodesic distance graph between landmarks and all pixels, the method uses a fast randomized low-rank approximate method to speed up the eigenvalue decomposition of the inner-product matrix in multidimensional scaling. Manifold coordinates of landmarks are then computed. Manifold coordinates of non-landmarks are computed through the pseudo inverse transformation of landmark coordinates. Five experiments on two different HSI datasets are run to test the new UL-Isomap method. Experimental results show that UL-Isomap surpasses LIsomap, both in the overall classification accuracy (OCA) and in computational speed, with a speed over 5 times faster. Moreover, the UL-Isomap method, when compared against the Isometric mapping (Isomap) method, obtains only slightly lower OCAs.

Joint Estimation of DOA and Polarization with CLD Pair Cylindrical Array Based on Quaternion Model

This paper proposes a quaternion-ESPRIT algorithm for closed-form estimation of direction of arrival (DOA) and polarization, using a cylindrical conformal array configuration. First, the array steering vector and the elevation are estimated using a quaternion eigenvalue decomposition of the data covariance matrix. Second, the azimuth is estimated. Finally, the polarization parameters are estimated using the relationships between the dipoles and the loops. These estimates are automatically matched. Simulation results show that the performance of quaternion method is obviously better than that of the long-vector method.