Generalized Eigenvalue Decomposition Research Articles

Hybrid Beamforming for Secure Transmission of Massive MIMO UAV Communication Networks

The powerful line-of-sight (LoS) characteristic between unmanned aerial vehicles (UAV) and terrestrial nodes could be leveraged by large-scale antenna arrays to significantly increase transmission rates, while the problem of secure transmission is emerging. Compared with fully digital beamforming, hybrid beamforming (HBF) could meet a compromise between cost and performance, which is precisely optimized in this article to maximize the secrecy rate of massive multiple-input–multiple-output (MIMO) UAV communication network. At first, we propose a secure HBF framework based on generalized eigenvalue decomposition and singular value decomposition. After that, considering the difficulty in directly solving the HBF optimization problem, an iterative algorithm based on successive convex approximation is developed for perfectly known channel state information (CSI). Furthermore, while CSI is imperfect, a neural network is constructed and trained to learn the statistical characteristics of wireless channels and complete the design of hybrid beamformer and combiner simultaneously. Performance validation and beampattern of the proposed HBF methods are given by extensive simulation results, which demonstrate the proposed secure HBF framework could considerably improve secrecy rate, and the deep learning-based HBF method possesses a positive average secrecy rate even with unknown eavesdropper's CSI.

Quantum canonical correlation analysis algorithm

Canonical correlation analysis (CCA) is a fundamental technique used to analyze data correlation in various fields, including video and medical data analysis. In this paper, we propose a quantum canonical correlation analysis (QCCA) algorithm. First, we introduce a combined density matrix representation method that transforms CCA into generalized eigenvalue decomposition. Moreover, to address the challenge of performing generalized eigenvalue decomposition in high-dimensional scenarios, we propose a quantum method for extracting the canonical principal axes. In this method, two sets of variables are transformed into a reduced density matrix, so that the product of variable matrices can be accelerated by phase estimation and controlled rotation. Complexity analysis shows that the QCCA algorithm achieves exponential acceleration in variable dimensions n, p and variable size m compared to classical algorithms. The QCCA algorithm serves as a foundation for the subsequent development of quantum algorithms for classification, regression, and other machine learning tasks.

An Improved Two-Stage Spherical Harmonic ESPRIT-Type Algorithm

Sensor arrays are gradually becoming a current research hotspot due to their flexible beam control, high signal gain, robustness against extreme interference, and high spatial resolution. Among them, spherical microphone arrays with complex rotational symmetry can capture more sound field information than planar arrays and can convert the collected multiple speech signals into the spherical harmonic domain for processing through spherical modal decomposition. The subspace class direction of arrival (DOA) estimation algorithm is sensitive to noise and reverberation, and its performance can be improved by introducing relative sound pressure and frequency-smoothing techniques. The introduction of the relative sound pressure can increase the difference between the eigenvalues corresponding to the signal subspace and the noise subspace, which is helpful to estimate the number of active sound sources. The eigenbeam estimation of signal parameters via the rotational invariance technique (EB-ESPRIT) is a well-known subspace-based algorithm for a spherical microphone array. The EB-ESPRIT cannot estimate the DOA when the elevation angle approaches 90°. Huang et al. proposed a two-step ESPRIT (TS-ESPRIT) algorithm to solve this problem. The TS-ESPRIT algorithm estimates the elevation and azimuth angles of the signal independently, so there is a problem with DOA parameter pairing. In this paper, the DOA parameter pairing problem of the TS-ESPRIT algorithm is solved by introducing generalized eigenvalue decomposition without increasing the computation of the algorithm. At the same time, the estimation of the elevation angle is given by the arctan function, which increases the estimation accuracy of the elevation angle of the algorithm. The robustness of the algorithm in a noisy environment is also enhanced by introducing the relative sound pressure into the algorithm. Finally, the simulation and field-testing results show that the proposed method not only solves the problem of DOA parameter pairing, but also outperforms the traditional methods in DOA estimation accuracy.

Double-Phase-Shifter Based Hybrid Beamforming for mmWave DFRC in the Presence of Extended Target and Clutters

In millimeter-wave (mmWave) dual-function radar-communication (DFRC) systems, hybrid beamforming (HBF) is recognized as a promising technique utilizing a limited number of radio frequency chains. In this work, in the presence of extended target and clutters, a HBF design based on the subarray connection architecture is proposed for a multiple-input multiple-output (MIMO) DFRC system. In this HBF, the double-phase-shifter (DPS) structure is embedded to further increase the design flexibility. We derive the communication spectral efficiency (SE) and radar signal-to-interference-plus-noise-ratio (SINR) with respect to the transmit HBF and radar receiver, and formulate the HBF design problem as the SE maximization subjecting to the radar SINR and power constraints. To solve the formulated nonconvex problem, the joinT Hybrid bRamforming and Radar rEceiver OptimizatioN (THEREON) is proposed, in which the radar receiver is optimized via the generalized eigenvalue decomposition, and the transmit HBF is updated with low complexity in a parallel manner using the consensus alternating direction method of multipliers (consensus-ADMM). Furthermore, we extend the proposed method to the multi-user multiple-input single-output (MU-MISO) scenario. Numerical simulations demonstrate the efficacy of the proposed algorithm and show that the solution provides a good trade-off between number of phase shifters and performance gain of the DPS HBF.

Filter Design and Power Allocation for Uplink Time-Reversal NOMA

The time-reversal division multiple access (TRDMA) system employs multipath channels to realize multi-user transmission via time-reversal filtering. The performance of TRDMA is affected by the correlation among users’ channels. When the correlation is high, inter-user interference (IUI) in the TRDMA system will be serious and the system capacity will be severely restricted. In this paper, we propose an uplink multiple access scheme time-reversal non-orthogonal multiple access (TR-NOMA), wherein TRDMA and power-domain non-orthogonal multiple access are combined. The base station employs a TR filter for each user to suppress IUI and applies successive interference cancellations in the detection process of users’ signals. Users’ transmit power and TR filters are jointly optimized to promote system capacity. The original optimization problem is non-convex. We decompose the optimization problem into two sub-problems: the design of TR filters and the allocation of users’ transmit power, which is solved by utilizing generalized eigenvalue decomposition and successive lower bound convex approximation, respectively. The solution to the original optimization problem is obtained by the iterative solution between the two sub-problems. Simulation results demonstrate that the sum rate of the optimized TR-NOMA system is significantly higher than that of the TRDMA system.

Pattern synthesis of nonuniform linear antenna array based on FFDM

To optimize the number, spacing and excitation of nonuniform linear array at the same time, a Fourier basis filter diagonalization method (FFDM) is proposed in this paper. By constructing a set of Hamiltonian operator functions that can be decomposed by orthogonal eigenvectors, the synthesis of nonuniform linear arrays is transformed into a generalized eigenvalue decomposition under Fourier basis. The introduction of Fourier basis can improve the computational rate. The least square method (LSM) is used to solve the array element excitation, which improves the matching accuracy. A number of numerical simulations shown that the proposed algorithm can improve the accuracy of the array factor (AF) pattern of many types of practical arrays reconstruction.

Transmit and receive hybrid beamforming design for OFDM dual-function radar-communication systems

This article investigates the design of hybrid digital and analog transmit/receive beamformers for orthogonal frequency division multiplexing dual-function radar-communication system. Specifically, we model the hybrid beamforming design by simultaneously optimizing the spectrum efficiency of communication and approximate output signal-to-interference noise ratio of radar sensing, through a flexible performance tradeoff. Due to the coupling relationship among the beamformers in the hybrid structure and the independence of signals in different frequency bands, the resultant optimization problem is NP-hard. Hence, a consensus alternating direction method of multipliers approach is designed to tackle the difficulty. The transmit hybrid beamformer is obtained by introducing auxiliary variables and exploiting the block continuous upper bound minimization method. Then, the receive beamformer can be optimized via generalized eigenvalue decomposition. Numerical simulations are provided to demonstrate the effectiveness of the proposed scheme.

Inclusion Methods for Multiplication of Three Point Matrices

This study considers inclusion methods for the multiplication of three point matrices. These methods are essential for verification methods such as for a matrix determinant, generalized eigenvalue decomposition, singular value decomposition, square roots and inverse square roots of a matrix, and Sylvester equations. We consider three methods that involve five matrix multiplications and compare the tightness of enclosures and numerical experiments. The proposed methods produce tighter results than the previous method with the same computational cost. We also show methods for obtaining the upper bound of the maximum norm of a product of three point matrices.

Low-Complexity Two-Step Optimization in Active-IRS-Assisted Uplink NOMA Communication

Recently, the active intelligent reflecting surface (IRS) has been proposed to adjust the phase and amplify the magnitude of the incident signal simultaneously. It has a more reasonable hardware overhead compared with the conventional relay. This letter aims to maximize the sum rate of multiple users in the active-IRS-assisted uplink non-orthogonal multiple access (NOMA) system. We propose a low-complexity two-step optimization algorithm to decompose the original non-convex problem into two sub-problems. First, the extreme low-complexity fixed point iteration (FPI) method is proposed to optimize the phase shifts. Then, two algorithms are proposed to solve the amplification optimization problem: the convergence-guaranteed quadratic transform (QT) and the low-complexity generalized eigenvalue decomposition (GEVD) algorithms. Simulation results show that the performance can be enhanced significantly compared with the orthogonal multiple access (OMA) and the passive-IRS scheme.

Sparse sliced inverse regression for high dimensional data analysis

BackgroundDimension reduction and variable selection play a critical role in the analysis of contemporary high-dimensional data. The semi-parametric multi-index model often serves as a reasonable model for analysis of such high-dimensional data. The sliced inverse regression (SIR) method, which can be formulated as a generalized eigenvalue decomposition problem, offers a model-free estimation approach for the indices in the semi-parametric multi-index model. Obtaining sparse estimates of the eigenvectors that constitute the basis matrix that is used to construct the indices is desirable to facilitate variable selection, which in turn facilitates interpretability and model parsimony.ResultsTo this end, we propose a group-Dantzig selector type formulation that induces row-sparsity to the sliced inverse regression dimension reduction vectors. Extensive simulation studies are carried out to assess the performance of the proposed method, and compare it with other state of the art methods in the literature.ConclusionThe proposed method is shown to yield competitive estimation, prediction, and variable selection performance. Three real data applications, including a metabolomics depression study, are presented to demonstrate the method’s effectiveness in practice.

Robust Joint Design of Transmit Waveform and Receive Filter for MIMO-STAP Radar Under Target and Clutter Uncertainties

This paper deals with the joint design problem of transmit waveform and receive filter for the robust detection of ground moving target with multiple-input multiple-output Space time adaptive processing (MIMO- STAP) radar in the presence of target and clutter uncertainties. With the prior knowledge of target and clutter statistics, the averaged signal-to-interference-plus-noise ratio (SINR) is formulated as a figure of merit to maximize. The robust joint design problems for the continuous and discrete phase cases, respectively, are formulated with the constant-modulus and similarity constraints. Then, an efficient iterative algorithm is developed for improving the SINR. Specifically, each iteration of the proposed algorithm involves the generalized eigenvalue decomposition to optimize the receive filter, while a nested iterative procedure involving Dinkelbach’s framework and alternating direction penalty method (ADPM) algorithm to design transmit waveform. The proposed algorithm can achieve higher SINR with reduced computational load compared to the recently developed ones. Numerical examples demonstrate the effectiveness of the proposed method for the robust detection of moving target in MIMO-STAP radar.

3D Multiple Sound Source Localization by Proposed T-Shaped Circular Distributed Microphone Arrays in Combination with GEVD and Adaptive GCC-PHAT/ML Algorithms.

Multiple simultaneous sound source localization (SSL) is one of the most important applications in the speech signal processing. The one-step algorithms with the advantage of low computational complexity (and low accuracy), and the two-step methods with high accuracy (and high computational complexity) are proposed for multiple SSL. In this article, a combination of one-step-based method based on the generalized eigenvalue decomposition (GEVD), and a two-step-based method based on the adaptive generalized cross-correlation (GCC) by using the phase transform/maximum likelihood (PHAT/ML) filters along with a novel T-shaped circular distributed microphone array (TCDMA) is proposed for 3D multiple simultaneous SSL. In addition, the low computational complexity advantage of the GCC algorithm is considered in combination with the high accuracy of the GEVD method by using the distributed microphone array to eliminate spatial aliasing and thus obtain more appropriate information. The proposed T-shaped circular distributed microphone array-based adaptive GEVD and GCC-PHAT/ML algorithms (TCDMA-AGGPM) is compared with hierarchical grid refinement (HiGRID), temporal extension of multiple response model of sparse Bayesian learning with spherical harmonic (SH) extension (SH-TMSBL), sound field morphological component analysis (SF-MCA), and time-frequency mixture weight Bayesian nonparametric acoustical holography beamforming (TF-MW-BNP-AHB) methods based on the mean absolute estimation error (MAEE) criteria in noisy and reverberant environments on simulated and real data. The superiority of the proposed method is presented by showing the high accuracy and low computational complexity for 3D multiple simultaneous SSL.

Canonical Polyadic Decomposition via the Generalized Schur Decomposition

The canonical decomposition (CPD) is a fundamental tensor decomposition which expresses a tensor as a sum of rank one tensors. In stark contrast to the matrix case, with light assumptions, the CPD of a low rank tensor is (essentially) unique. The essential uniqueness of CPD makes this decomposition a powerful tool in many applications as it allows for extraction of component information from a signal of interest. One popular algorithm for algebraic computation of a CPD is the generalized eigenvalue decomposition (GEVD) which selects a matrix subpencil of a tensor, then computes the generalized eigenvectors of the pencil. In this article, we present a simplification of GEVD which improves the accuracy of the algorithm. Surprisingly, the generalized eigenvector computation in GEVD is in fact unnecessary and can be replaced by a QZ decomposition which factors a pair of matrices as a product of unitary and upper triangular matrices. Computing a QZ decomposition is a standard first step when computing generalized eigenvectors, so our algorithm can been seen as a direct simplification of GEVD.

Distributed Combined Acoustic Echo Cancellation and Noise Reduction in Wireless Acoustic Sensor and Actuator Networks

The paper presents distributed algorithms for combined acoustic echo cancellation (AEC) and noise reduction (NR) in a wireless acoustic sensor and actuator network (WASAN) where each node may have multiple microphones and multiple loudspeakers, and where the desired signal is a speech signal. A centralized integrated AEC and NR algorithm, i.e., multichannel Wiener filter (MWF), is used as starting point where echo signals are viewed as background noise signals and loudspeaker signals are used as additional input signals to the algorithm. By including prior knowledge (PK), namely that the loudspeaker signals do not contain any desired signal component, an alternative centralized cascade algorithm (PK-MWF) is obtained with an AEC stage first followed by an MWF-based NR stage which has a lower computational complexity. Distributed algorithms can then be obtained from the MWF and PK-MWF algorithm, i.e., the generalized eigenvalue decomposition (GEVD)-based distributed adaptive node-specific signal estimation (DANSE) and PK-GEVD-DANSE algorithm, respectively. In the former, each node performs a reduced dimensional integrated AEC and NR algorithm and broadcasts only 1 fused signal (instead of all its signals) to the other nodes. In the PK-GEVD-DANSE algorithm, each node performs a reduced dimensional cascade AEC and NR algorithm and broadcasts only 2 fused signals (instead of all its signals) to the other nodes. The distributed algorithms achieve the same performance, upon convergence, as the corresponding centralized integrated (MWF) and centralized cascade (PK-MWF) algorithm. It is observed, however, that the communication cost in the PK-GEVD-DANSE algorithm can also be reduced, where each node then broadcasts only 1 fused signal (instead of 2 signals) to the other nodes. The resulting algorithm, referred to as the pruned PK-GEVD-DANSE (pPK-GEVD-DANSE) algorithm, then effectively combines the lowest possible communication cost (as low as in the GEVD-DANSE algorithm) with a lowest possible computational complexity in each node (further reduced from the PK-GEVD-DANSE computational complexity), within the class of algorithms considered in this paper.

MAXVAR-Based Distributed Correlation Estimation in a Wireless Sensor Network

The performance of most array signal processing tasks relies on the presence of correlation between sensor signals. In a wireless sensor network, where sensor nodes are spread out over a relatively large area, it is useful to identify nodes observing similar sensor signals and hence common phenomenons, for example to partition the network according to the observed latent signals and corresponding correlation structure. This can be achieved via the so-called MAXVAR formulation of generalized canonical correlation analysis, which finds a low-dimensional subspace that highlights correlated signal components between multiple nodes' observed signal subspaces. The classical procedure for computing the solutions of MAXVAR consists in performing a generalized eigenvalue decomposition after collecting all the sensors' signals at a fusion center. However, this typically incurs high communication and computational costs. In this paper, we describe a low communication and computational cost distributed algorithm that computes the solutions of MAXVAR without aggregating the nodes' observations at a central location. We show how a subset of those solutions can be used locally by each node to estimate the global correlation structure across all nodes in the network, thereby allowing any node to evaluate the presence of correlated signals at any other node, even if no direct link is shared. We prove the convergence of the algorithm and validate our method for estimating the correlation structure via simulations.

Joint Power Allocation and Hybrid Beamforming for Downlink mmWave-NOMA Systems

This paper investigates the application of nonorthogonal multiple access (NOMA) in millimeter-Wave (mmWave) communications to address resource allocation issues for multiuser downlink transmission. Inspired by the K-means clustering algorithm, a NOMA user grouping policy is first employed in accordance with the channel correlation between users. The first-stage decoding order for each NOMA group is then determined by merely utilizing the users’ channel gain to formulate a joint power allocation and hybrid beamforming optimization problem. Specifically, we use signal-to-leakage-plus-noise ratio (SLNR) as the performance index of the optimization problem to reduce the computational complexity, because it can decouple the design of the beamforming and power allocation issues so they can be executed iteratively. Under given beamforming matrix, the power allocation issue is expressed as a quadratic programming (QP) problem, and then transformed into a convex problem by introducing an auxiliary-positive real variable to solve through the Lagrange multiplier method. On the contrary, the beamforming optimization problem is non-convex and very difficult to solve due to the constant modulus constraints in the hybrid architecture. However, the optimum solution of the ideal full-digital beamforming can be acquired primarily by generalized eigenvalue decomposition (GED). In this condition, two hybrid beamforming algorithms are proposed, in which normalized phase matching and equal gain transmission are performed respectively in the analog domain. On this basis, a second-stage decoding order can be employed according to the effective channel gain of users to perform successive interference cancellation (SIC) successfully. Numerical results show that the proposed joint power allocation and hybrid beamforming algorithms are superior to the classical mmWave-NOMA and orthogonal multiple access (OMA) schemes not only in lower implementation complexity, but also in better spectrum efficiency and energy efficiency performance.

A Total Lp-Norm Optimization for Bearing-Only Source Localization in Impulsive Noise with SαS Distribution

This paper considers the problem of robust bearing-only source localization in impulsive noise with symmetric -stable distribution based on the Lp-norm minimization criterion. The existing Iteratively Reweighted Pseudolinear Least-Squares (IRPLS) method can be used to solve the least LP-norm optimization problem. However, the IRPLS algorithm cannot reduce the bias attributed to the correlation between system matrices and noise vectors. To reduce this kind of bias, a Total Lp-norm Optimization (TLPO) method is proposed by minimizing the errors in all elements of system matrix and data vector based on the minimum dispersion criterion. Subsequently, an equivalent form of TLPO is obtained, and two algorithms are developed to solve the TLPO problem by using Iterative Generalized Eigenvalue Decomposition (IGED) and Generalized Lagrange Multiplier (GLM), respectively. Numerical examples demonstrate the performance advantage of the IGED and GLM algorithms over the IRPLS algorithm.

Accelerated numerical modeling of RF circuits using network characteristic mode analysis

AbstractA fast numerical modeling approach based on network characteristic mode analysis (CMA) is presented and investigated for analyzing electrical layouts in miniature RF filters, such as surface acoustic wave filters. In this approach, a generalized eigenvalue decomposition is performed on the Z‐parameters of an electrical layout at one or two sampling frequencies that can be computed and extracted with any numerical full‐wave method. The obtained eigenvalues are used to extract modal resistance, inductance, and capacitance matrices for each eigenmode. The frequency dependence of the modal resistance matrix can be assumed a priori or determined automatically, and the modal inductance and capacitance matrices are assumed frequency independent. These modal matrices are then used to approximate the Z‐parameters at any other frequencies to provide the response of the electrical layout, which can then be combined with the frequency responses of other components, such as resonators, to yield the electrical response of an entire RF filter. Compared with the previously developed analytic extension of eigenvalues, this fast CMA‐based method is less affected by the frequency variation of eigenmodes since the frequency dependences of the eigenmodes are implicitly canceled out in its formulation. The accuracy of this approach is evaluated by comparing with results from full‐wave analyses. For RF circuits whose electrical sizes are small and whose frequency range of interest is relatively small, the proposed CMA‐based fast frequency sweep approach is found to be sufficiently accurate and highly practical for engineering applications.

A linearly extendible multi-artifact removal approach for improved upper extremity EEG-based motor imagery decoding.

Non-invasive multichannel Electroencephalography (EEG) recordings provide an alternative source of neural information from which motor imagery (MI) patterns associated with limb movement intent can be decoded for use as control inputs for rehabilitation robots. The presence of multiple inherent dynamic artifacts in EEG signals, however, poses processing challenges for brain-computer interface (BCI) systems. A large proportion of the existing EEG signal preprocessing methods focus on isolating single artifact per time from an ensemble of EEG trials and require calibration and/or reference electrodes, resulting in increased complexity of their application to MI-EEG controlled rehabilitation devices in practical settings. Also, a few existing multi-artifacts removal methods though explored in other domains, they have rarely been investigated in the space of MI-EEG signals for multiple artifacts cancellation in a simultaneous manner. Building on the premise of previous works, this study propose a semi-automatic EEG preprocessing method that combines Generalized Eigenvalue Decomposition driven by low-rank approximation and a Multi-channel Wiener Filter (GEVD-MWF) that employs a learning technique for simultaneous elimination of multiple artifacts from MI-EEG signals. The proposed method is applied to remove multiple artifacts from 64-channel EEG signals recorded from transhumeral amputees while they performed distinct classes of upper limb MI tasks before decoding their movement intent using a selection of features and machine learning algorithms. Experimental results show that the proposed GEVD-MWF method yields significant improvements in MI decoding accuracies, in the range of 13.23%-41.21% compared to four existing popular artifact removal algorithms. Further investigation revealed that the GEVD-MWF approach enabled accuracies in the range of 90.44% - 99.67% using "single trial" EEG recordings, which could eliminate the need to record and process large ensembles of EEG trials as commonly required in some existing approaches. Additionally, using a variant of the sequential forward floating selection algorithm, a subset of 9 channels was used to obtain a decoding accuracy of 93.73%±1.58%. Given its improved performance, reduced data requirements, and feasibility with few channels, the proposed GEVD-MWF could potentially spur the development of effective real-time control strategies for multi-degree of freedom EEG-based miniaturized rehabilitation robotic interfaces.

Securing Wireless Transmissions With RIS-Receiver Coordination: Passive Beamforming and Active Jamming

In this correspondence, we investigate the physical layer security issue with the assistance of a reconfigurable intelligent surface (RIS). In particular, we consider the single-input and multiple-output transmissions in the presence of a multi-antenna eavesdropper, where the legitimate receiver actively performs full-duplex jamming to confuse the eavesdropper. Meanwhile, the passive beamforming of the RIS is coordinated with the legitimate reception and jamming for security enhancement, leading to the joint optimization of receive beamforming, active jamming, and passive beamforming. The problem is solved under a block coordinate descent framework, where the receive beamforming is obtained through generalized eigenvalue decomposition, and the active jamming and passive beamforming are solved based on semidefinite programming. Simulation results show the combined effect of jamming and reflection upon the secrecy rate, also demonstrated is the performance superiority of the proposed scheme as compared with the conventional approaches in terms of wireless security.