Real-valued Matrix Research Articles

A High-Efficiency Beam Sweeping Algorithm for DOA Estimation in the Hybrid Analog-Digital Structure

The spatial covariance matrix, as the core of subspace direction of arrival (DOA) estimation, is unavailable in the hybrid analog-digital structure, which is widely used in large-scale transceivers to reduce the number of radio frequency chains. In previous studies, the beam sweeping algorithm (BSA) has been used for effectively reconstructing the spatial covariance matrix. However, the BSA is not practical owning to the intractable computational cost due to the high-dimensional matrix operations. To address this problem, a high-efficiency BSA (HeBSA) is proposed. The main motivation behind the HeBSA is to avoid the high-dimensional matrix multiplication and inversion. By appropriately adjusting weights connected to antennas, the real-valued spatial covariance matrix can be reconstructed according to low-dimensional matrix multiplication. Then the classical U-root multiple signal classification can be adopted to estimate the DOA angle of each signal. Finally, simulation experiments are conducted to verify the performance of the HeBSA. The results indicate that the HeBSA can achieve DOA estimation performance comparable with that of BSA even at considerably lower computational cost.

Outage Performance Analysis of Widely Linear Receivers in Uplink Multi-User MIMO Systems

This paper considers the application of widely linear (WL) receivers in an uplink multi-user system using real-valued modulation schemes, where the cellular base station (BS) with multiple antennas provides connectivity for randomly deployed single-antenna users. The targeted use case is massive machine type communication (mMTC) with grant-free access in the uplink, where the network is required to host a large number of low data rate devices transmitting in an uncoordinated fashion. Four types of WL receivers are investigated, namely the WL zero-forcing (ZF) and the WL minimum mean-squared error (MMSE) receivers, along with their enhanced versions employing successive interference cancellation (SIC) with channel-dependent ordering, i.e., the WL-ZF-SIC and WL-MMSE-SIC receivers. The outage performances of these receivers are analytically characterized in the high signal-to-noise ratio (SNR) regime and compared to those of conventional linear (CL) receivers using complex-valued modulation schemes. For the non-SIC receivers, we show that, when compared to the CL counterparts, the WL receivers yield a higher diversity gain when decoding the same number of users and have the same diversity gain but a decreased coding gain when the number of users is nearly doubled. The outage performance analysis of WL-SIC receivers is facilitated by the marginal distribution of ordered eigenvalues of a real-valued Wishart matrix. It is shown that the SIC operation with channel-dependent ordering brings no additional diversity gain to the WL receivers but instead increases the coding gain. Moreover, the coding gain of WL-SIC receivers grows as the number of users increases and even exceeds that of CL-SIC receivers under suitable conditions. For the mMTC scenario with grant-free transmission, it is demonstrated that the WL receivers outperform their CL counterparts in terms of offering a lower outage (and packet drop) probability and a higher system throughput for a given packet drop probability.

Multi-view clustering by joint spectral embedding and spectral rotation

Discrete Spectral clustering is an effective tool for directly getting discrete labels. However, existing spectral clustering carry out spectral embedding and spectral rotation separately, which may limit their clustering performance. Seamlessly connecting above two processes is challenging. In this paper, we propose a new multi-view clustering framework, namely Multi-view Clustering by Joint Spectral Embedding and Spectral Rotation. In the framework, the differences of Laplacian matrices from different views are learned adaptively. Moreover, the real-valued cluster indicator matrix is approximated by continuous orthogonalization of the discrete clustering index matrix. By doing so, our method has better convergence, which is also strictly mathematically proven. Extensive experiments indicate that our method is superior to several state-of-the-art methods.

Weighted multiwindow discrete Gabor transform

According to the Heisenberg uncertainty principle, an analysis window with a high time resolution in time domain will result in a low frequency resolution in frequency domain, and vice versa. To obtain the Gabor spectrum with high time-frequency resolution and concentration, weighted multiwindow discrete Gabor transform (M-DGT) using weights and the biorthogonal analysis method for analyzing long (or infinite) sequences is proposed in this paper, in which the combined Gabor coefficients constructed by a combination of M-DGT coefficients can be adaptively changed according to the time-frequency distributions of an analyzed signal containing multiple time-varying frequencies. To obtain the weights of the weighted M-DGT, the M-DGT is converted into a sparse problem with ℓ1-ℓ2 regularization, then an efficient iterative algorithm for solving the weights in terms of real-valued matrix and real-valued vector is derived. The convergence of the iterative algorithm is proved by optimization theory. The experimental results demonstrate that the proposed method is an effective and efficient tool for nonstationary time-frequency analysis of signals.

Focusing light through multimode fibres using a digital micromirror device: a comparison study of non-holographic approaches

Focusing light through a multimode fibre (MMF) has attracted significant research interest, mainly driven by the need for miniature endoscopes in biomedicine. In recent years, digital micromirror devices (DMD) have become increasingly popular as a high-speed alternative to liquid-crystal spatial light modulators for light focusing via wavefront shaping based on binary amplitude modulations. To exploit the potentials and limitations of the state-of-the-art DMD-based wavefront shaping methods, in this study, for the first time, we compared four representative, non-holographic and DMD-based methods that are reported so far in literature with the same experimental and simulation conditions, including a real-valued intensity transmission matrix (RVITM)-based algorithm, a complex-valued transmission matrix (TM)-based algorithm, a conditional probability algorithm and a genetic algorithm. We investigated the maximum achievable peak-to-background ratio (PBR) in comparison to theoretical expectations, and further improved the performance of the RVITM-based method. With both numerical simulations and experiments, we found that the genetic algorithm offered the highest PBR but suffered from the lowest focusing speed, while the RVITM-based algorithm provided a comparable PBR to that of the genetic algorithm, and the highest focusing speed.

Joint angle and range estimation for bistatic FDA-MIMO radar via real-valued subspace decomposition

The frequency diverse array (FDA) is mainly applied to achieving target localization under the complicated electromagnetic jamming condition. The bistatic multiple-input-multiple-output (MIMO) radar with FDA has gained comprehensive attention in recent years. An innovative method covering the transmitting subarray scheme and the unitary estimation of signal parameters via rotational invariance technology (ESPRIT) is proposed for joint direction of departure (DOD), direction of arrival (DOA), and range estimation. First, a non-overlapping subarray scheme is designed for the transmitting array to decouple DOD and range. For the purpose of enhancing the estimation accuracy and reducing the computational complexity, the real-valued rotational invariance matrix is obtained via applying the unitary transformation to the extended data. The removal method is proposed to avoid the periodic phase ambiguity. The pairing method is put forward to achieving the match of three-dimensional parameter (DOD, DOA and range) for multiple targets. The computational complexity of the proposed algorithm and conventional ESPRIT algorithm are also provided. Finally, extensive experiment results indicate that the proposed algorithm shows the superior performance than the ESPRIT algorithm.

High-speed photoacoustic-guided wavefront shaping for focusing light in scattering media.

Wavefront shaping is becoming increasingly attractive as it promises to enable various biomedical applications by breaking through the optical diffusion limit that prevents light focusing at depths larger than in biological tissue. However, despite recent advancements in wavefront shaping technology, such as those exploiting non-invasive photoacoustic-guidance, in vivo demonstrations remain challenging mainly due to rapid tissue speckle decorrelation. In this work, we report a high-speed photoacoustic-guided wavefront shaping method with a relatively simple experimental setup, based on the characterization of a scattering medium with a real-valued intensity transmission matrix. We demonstrated light focusing through an optical diffuser by optimizing 4096 binary amplitude modulation modes of a digital micromirror device within , leading to a system runtime of 75 µs per input mode, which is 3 orders of magnitude smaller than the smallest runtime reported in literature so far using photoacoustic-guided wavefront shaping. Thus, our method is a solid step forward toward in vivo applications of wavefront shaping.

Matrix Completion with Deterministic Sampling: Theories and Methods.

In some significant applications such as data forecasting, the locations of missing entries cannot obey any non-degenerate distributions, questioning the validity of the prevalent assumption that the missing data is randomly chosen according to some probabilistic model. To break through the limits of random sampling, we explore in this paper the problem of real-valued matrix completion under the setup of deterministic sampling. We propose two conditions, isomeric condition and relative well-conditionedness, for guaranteeing an arbitrary matrix to be recoverable from a sampling of the matrix entries. It is provable that the proposed conditions are weaker than the assumption of uniform sampling and, most importantly, it is also provable that the isomeric condition is necessary for the completions of any partial matrices to be identifiable. Equipped with these new tools, we prove a collection of theorems for missing data recovery as well as convex/nonconvex matrix completion. Among other things, we study in detail a Schatten quasi-norm induced method termed isomeric dictionary pursuit (IsoDP), and we show that IsoDP exhibits some distinct behaviors absent in the traditional bilinear programs.

Toeplitz-block sensing matrix based on bipolar chaotic sequence

Compressed sensing is a revolutionary signal processing technique, which allows the signals of interest to be acquired at a sub-Nyquist rate, meanwhile still permitting the signals from highly incomplete measurements to be reconstructed perfectly. As is well known, the construction of sensing matrix is one of the key technologies to promote compressed sensing from theory to application. Because the Toeplitz sensing matrix can support fast algorithm and corresponds to discrete convolution operation, it has essential research significance. However, the conventional random Toeplitz sensing matrix, due to the uncertainty of its elements, is subject to many limitations in practical applications, such as high memory consumption and difficulty of hardware implementation. To avoid these limitations, we propose a bipolar Toeplitz block-based chaotic sensing matrix (Bi-TpCM) by combining the intrinsic advantages of Toeplitz matrix and bipolar chaotic sequence. Firstly, the generation of bipolar chaotic sequence is introduced and its statistical characteristics are analyzed, showing that the generated bipolar chaotic sequence is an independent and identically distributed Rademacher sequence, which makes it possible to construct the sensing matrix. Secondly, the proposed Bi-TpCM is constructed, and it is proved that Bi-TpCM has almost optimal theoretical guarantees in terms of the coherence, and also satisfies the restricted isometry condition. Finally, the measurement performances on one-dimensional signals and images by using the proposed Bi-TpCM are investigated and compared with those of its counterparts, including random matrix, random Toeplitz matrix, real-valued chaotic matrix, and chaotic circulant sensing matrix. The results show that Bi-TpCM not only has better performance for these testing signals, but also possesses considerable advantages in terms of the memory cost, computational complexity, and hardware realization. In particular, the proposed Bi-TpCM is extremely suitable for the compressed sensing measurement of linear time-invariant (LTI) systems with multiple inputs and single output, such as the joint parameter and time-delay estimation for finite impulse response. Moreover, the construction framework of the proposed Bi-TpCM can be extended to different chaotic systems, such as Logistic or Cat chaotic systems, and it is also possible for the proposed Bi-TpCM to derive the Hankel blocks, additional stacking of blocks, partial circulant blocks sensing matrices. With these block-based sensing architectures, we can more easily implement compressed sensing for various compressed measurement problems of LTI systems.

The complex step approximation to the higher order Fréchet derivatives of a matrix function

The k th Frechet derivative of a matrix function f is a multilinear operator from a cartesian product of k subsets of the space $\mathbb {C}^{n\times n}$ into itself. We show that the k th Frechet derivative of a real-valued matrix function f at a real matrix A in real direction matrices E1, E2, $\dots $ , Ek can be computed using the complex step approximation. We exploit the algorithm of Higham and Relton (SIAM J. Matrix Anal. Appl. 35(3):1019–1037, 2014) with the complex step approximation and mixed derivative of complex step and central finite difference scheme. Comparing with their approach, our cost analysis and numerical experiment reveal that half and seven-eighths of the computational cost can be saved for the complex step and mixed derivative, respectively. When f has an algorithm that computes its action on a vector, the computational cost drops down significantly as the dimension of the problem and k increase.

Abstract 5486: Time-dependent mathematical modeling of genetic alterations in acute myeloid leukemia

Abstract Acute myeloid leukemia (AML) is a complex, aggressive type of neoplasia characterized by mutations in myeloid stem cells. Here we perform a hierarchical and time-dependent classification of genomic and transcriptomic events that drive AML using data from The Cancer Genome Atlas. The problem of mathematically modeling the interplay between somatic mutations, translocations, and gene expression changes during AML evolution is formulated as a mixed integer linear program. We define a connectivity index between driver genes and genes with expression changes that determines the assignment of abnormally expressed genes. Input data consists of a binary matrix with 46 columns (known driver genes) and a real-valued expression matrix with 210 columns (genes implicated in cancer development, chosen by overlapping the KEGG Pathways in Cancer set with our dataset). Gene expression data is preprocessed using Python v3.7.4 scripts so as to generate a binary matrix. The output of the model is a set of phases of cancer progression, each of which contains driver genes and genes with expression changes. The model was solved using CPLEX v12.6 with default parameters. A total of 10 drivers, including DNMT3A, U2AF1, ASXL1 and translocations t(15;17) and t(8;21) were assigned to the initial phase of our model. This phase is characterized by point mutations in epigenetic modifier genes known to play a role in clonal hematopoiesis condition and pre-leukemic stages. The intermediate phase of our model comprised a total of 21 driver genes, including NPM1, RUNX1 and CEBPA, which are commonly used for AML risk stratification. The final phase contained a number of drivers typically associated with late stages of AML, such as FLT3 and TP53. Our three-phase model also revealed a gradual progression in gene expression changes. The majority of genes with altered expression, independently of the phase of progression, were linked to cell proliferation, the main affected pathways being PI3K/AKT activation and Wnt signaling. The initial phase included genes that code for transcription factors, cytoskeleton, GTPases, and cyclins, the majority of which are involved in cell proliferation and differentiation. The intermediate phase included genes involved in apoptosis, necroptosis, other specific pathways of myeloid differentiation, DNA damage repair and glucose homeostasis for energy metabolism. The final phase contained tumor suppressor genes, proto oncogenes activation, immune modulators and developmental genes. Taken together, our results suggest that, at first, early epigenetic events lead to cell proliferation; then additional mutations impair myeloid differentiation, activate death mechanisms, DNA repair and metabolism processes, and further increase cellular capacity for proliferation; finally, new mutations in tumor suppressor genes establish malignant transformation, and immune pathways are modulated in AML cells. Citation Format: Julia L. Fleck, Matheus O. Meirim, Ana B. Pavel, Luciana M. Gutiyama, Ilana R. Zalcberg. Time-dependent mathematical modeling of genetic alterations in acute myeloid leukemia [abstract]. In: Proceedings of the Annual Meeting of the American Association for Cancer Research 2020; 2020 Apr 27-28 and Jun 22-24. Philadelphia (PA): AACR; Cancer Res 2020;80(16 Suppl):Abstract nr 5486.

Fluid pressure diffusion effects on the excess compliance matrix of porous rocks containing aligned fractures

SUMMARYThe presence of sets of open fractures is common in most reservoirs, and they exert important controls on the reservoir permeability as fractures act as preferential pathways for fluid flow. Therefore, the correct characterization of fracture sets in fluid-saturated rocks is of great practical importance. In this context, the inversion of fracture characteristics from seismic data is promising since their signatures are sensitive to a wide range of pertinent fracture parameters, such as density, orientation and fluid infill. The most commonly used inversion schemes are based on the classical linear slip theory (LST), in which the effects of the fractures are represented by a real-valued diagonal excess compliance matrix. To account for the effects of wave-induced fluid pressure diffusion (FPD) between fractures and their embedding background, several authors have shown that this matrix should be complex-valued and frequency-dependent. However, these approaches neglect the effects of FPD on the coupling between orthogonal deformations of the rock. With this motivation, we considered a fracture model based on a sequence of alternating poroelastic layers of finite thickness representing the background and the fractures, and derived analytical expressions for the corresponding excess compliance matrix. We evaluated this matrix for a wide range of background parameters to quantify the magnitude of its coefficients not accounted for by the classical LST and to determine how they are affected by FPD. We estimated the relative errors in the computation of anisotropic seismic velocity and attenuation associated with the LST approach. Our analysis showed that, in some cases, considering the simplified excess compliance matrix may lead to an incorrect representation of the anisotropic response of the probed fractured rock.

Dissipativity of impulsive matrix-valued neural networks with leakage delay and mixed delays

A generalization of real-, complex-, and quaternion-valued neural networks is represented by matrix-valued neural networks (MVNNs), for which the states and weights are matrices. The dissipativity of impulsive MVNNs with leakage delay and mixed delays is studied in this paper, by giving sufficient criteria expressed in terms of real-valued linear matrix inequalities. After decomposing the MVNNs into real-valued systems, Lyapunov–Krasovskii functionals with double, triple, and quadruple integral terms are formulated. Also, the free weighting matrix method, simple, double, and triple Jensen inequalities, the reciprocally convex combination inequality, and the Wirtiger-based integral inequality are used to establish the sufficient criteria. Two numerical examples illustrate the feasibility and correctness of the proposed theoretical results.

A Real-Valued Toeplitz Matrix Method for DOA Estimation

Based on the optimized coprime array (OpCA), we propose a real-valued Toeplitz matrix method. This article analyzes the feasibility of converting the manifold matrix of a virtual domain constructed from a physical array into a real-valued matrix, and the unitary estimation of signal parameters via rotational invariance technique is introduced to transform the equivalent received signal rearranged Toeplitz matrix into a real-valued matrix and obtain directions of arrival (DOAs). Moreover, the proposed method offers higher estimation accuracy. Simulations are demonstrated to prove the validity of this method.

A Novel Real-Valued DOA Algorithm Based on Eigenvalue.

To solve the high complexity of the subspace-based direction-of-arrival (DOA) estimation algorithm, a super-resolution DOA algorithm is built in this paper. However, in this method, matrix decomposition is required for each search angle. Therefore, in this paper, real-valued processing is used to reduce the scanning range by half, which is less effective in algorithm complexity. The super-resolution algorithm mainly uses the conservation of energy. By exploring the relationship between the covariance matrix and its complex conjugate, we constructed the real-valued matrix and introduced a real-valued searching source to make the operation of the matrix real-valued. Finally, the simulation experiments show that the proposed algorithm not only reduces the spectral search range by half but also has a higher angular resolution than the traditional algorithm.

Efficient Localization Algorithm for Near-Field Noncircular Sources via Dual-Polarization Sensor Array

Noncircular signals are widely used in the area of radar, sonar, and wireless communication array systems, which can offer more accurate estimates and detect more sources. In this paper, the noncircular signals are employed to improve source localization accuracy and identifiability. Firstly, an extended real-valued covariance matrix is constructed to transform complex-valued computation into real-valued computation. Based on the property of noncircular signals and symmetric uniform linear array (SULA) which consist of dual-polarization sensors, the array steering vectors can be separated into the source position parameters and the nuisance parameter. Therefore, the rank reduction (RARE) estimators are adopted to estimate the source localization parameters in sequence. By utilizing polarization information of sources and real-valued computation, the maximum number of resolvable sources, estimation accuracy, and resolution can be improved. Numerical simulations demonstrate that the proposed method outperforms the existing methods in both resolution and estimation accuracy.

Real‐valued off‐grid DOA estimation based on fourth‐order cumulants using sparse Bayesian learning in spatial coloured noise

In this study, the authors address the problem of off-grid sparsity-inducing direction-of-arrival (DOA) estimation in the context of real-valued fourth-order cumulants (FOC) in the presence of spatially coloured noise. Firstly, a selection matrix is constructed to eliminate the redundant data of FOC and rearrange the data in the de-redundant FOC matrix to facilitate real processing. Then a new virtual overcomplete dictionary is constructed with coupling symmetric property by linear transform with the selection matrix. Next, the FOC matrix is transformed into a real-valued matrix via a unitary transformation which can be sparsely represented by a real-valued virtual overcomplete dictionary. The real-valued sparse model is vectorised for transforming to a single measurement vector (SMV) model, and the redundant data in the vector model is further removed by another selection matrix. Finally, an off-grid sparse model based on the real-valued SMV is established and solved by utilising the SBL strategy. The proposed method not only reduces the computational complexity but also obtains an extended-aperture array with increased degrees of freedom which yields high resolution, and provides superiority in performance and robustness against coloured noise. The simulation results demonstrate the effectiveness of the proposed method.

Multiple-image encryption based on chaotic phase mask and equal modulus decomposition in quaternion gyrator domain

In order to transmit multiple-image synchronously, this paper introduces an encryption algorithm by combining equal modulus decomposition with quaternion gyrator transform. Firstly, each color image is encoded into a quaternion-valued matrix for a holistic processing. With the chaotic random phase mask, the quaternion gyrator spectrum is obtained. It is subsequently split into a complex-valued interim matrix and the equal modulus decomposition is performed to enhance the security, where the spectrum is divided into two complex-valued masks. Thereafter, the two set of phase masks are respectively superimposed and followed by gyrator transforms. Finally, a real-valued matrix is constructed as the final ciphertext image by splicing the real and imaginary parts together. The phase masks that are generated using the chaotic and real-valued ciphertext are convenient to storage and transmission. Moreover, the initial conditions are closely related with the plaintext images, which makes the cryptosystem achieve high security. Numerical simulations are made to demonstrate the reliability of the proposed cryptosystem.

Compensation of Current Transformers’ Nonlinearities by Tensor Linearization

With the proliferation of harmonic sources in power systems, current transformers (CTs), which are the most widely used current transducers in power systems, will have to measure and monitor signals with increasing levels of distortion. The accuracy of the measurement can be affected by the nonlinear response of the CT, which is dependent on the distortion in the measured signal, but nonlinear mathematical models can be used to compensate the measured value. In this paper, the complex ratio of the CT is modeled using a frequency domain model based on tensor linearization, obtaining a real-valued compensation matrix. An accurate measurement setup has been built to characterize the CTs performance in distorted conditions. Experimental results using the proposed compensation technique are presented and discussed for two commercial CTs of accuracy class 0.5 and 1. It is demonstrated that the proposed technique can accurately reconstruct the harmonic components (up to several kilohertz) of the CT primary current from the measured CT distorted secondary current.

Orthogonal and symplectic Harish-Chandra integrals and matrix product ensembles

In this paper, we highlight the role played by orthogonal and symplectic Harish-Chandra integrals in the study of real-valued matrix product ensembles. By making use of these integrals and the matrix-valued Fourier-Laplace transform, we find the explicit eigenvalue distributions for particular Hermitian anti-symmetric matrices and Hermitian anti-self dual matrices, involving both sums and products. As a consequence of these results, the eigenvalue probability density function of the random product structure [Formula: see text], where each [Formula: see text] is a standard real Gaussian matrix, and [Formula: see text] is a real anti-symmetric matrix can be determined. For [Formula: see text] and [Formula: see text] the bidiagonal anti-symmetric matrix with 1’s above the diagonal, this reclaims results of Defosseux. For general [Formula: see text], and this choice of [Formula: see text], or [Formula: see text] itself a standard Gaussian anti-symmetric matrix, the eigenvalue distribution is shown to coincide with that of the squared singular values for the product of certain complex Gaussian matrices first studied by Akemann et al. As a point of independent interest, we also include a self-contained diffusion equation derivation of the orthogonal and symplectic Harish-Chandra integrals.