Complex-valued Matrix Research Articles

A Neural Network for Moore—Penrose Inverse of Time-Varying Complex-Valued Matrices

The Moore—Penrose inverse of a matrix plays a very important role in practical applications. In general, it is not easy to immediately solve the Moore—Penrose inverse of a matrix, especially for solving the Moore—Penrose inverse of a complex-valued matrix in time-varying situations. To solve this problem conveniently, in this paper, a novel Zhang neural network (ZNN) with time-varying parameter that accelerates convergence is proposed, which can solve Moore—Penrose inverse of a matrix over complex field in real time. Analysis results show that the state solutions of the proposed model can achieve super convergence in finite time with weighted sign-bi-power activation function (WSBP) and the upper bound of the convergence time is calculated. A related noise-tolerance model which possesses finite-time convergence property is proved to be more efficient in noise suppression. At last, numerical simulation illustrates the performance of the proposed model as well.

Compressed Arrays and Hybrid Channel Sensing: A Cramér-Rao Bound Based Analysis

This letter provides a Cramer-Rao Bound (CRB) based analysis of compressive arrays with applications in mmWave channel sensing, where the measurements at the output of an antenna array are further compressed using a complex-valued compression matrix, in order to reduce the system complexity and power consumption. While necessary conditions for the existence of CRB for compressed arrays have been recently derived, currently no sufficient conditions exist that can guarantee the existence of the CRB in different compressive regimes, and therefore can be used to guide the design of the overall system. We overcome this drawback by deriving tight sufficient conditions (that agree with the necessary conditions) for almost all choices of the compression matrix. Our results decisively demonstrate the additional benefit gained by using sparse arrays (such as nested array) even when a compression matrix is deployed.

A Fully Complex-Valued and Robust ZNN Model for Dynamic Complex Matrix Inversion Under External Noises

Dynamic complex-valued matrix inversion is often used in the field of mathematics and engineering. Over the past years, many recurrent neural network models have been designed and researched to analyse the process of matrix inversion without noises interference. However, there are many types of uncertain noises in actual model design and analysis. In this article, a novel fully complex-valued and robust Zeroing neural network (CVRZNN) is firstly proposed for calculating the dynamic complex matrix inversion under the interference of external noise environment, and its robustness is analysed and demonstrated in the presence of various types of external noises. Compared with the previous zeroing neural network (ZNN) and the gradient neural network (GNN) for dynamic complex matrix inversion, this novel CVRZNN model has good robustness under three kinds of external noises. Besides, the theoretical analysis shows that the CVRZNN model can globally converge to zero under constant noise. Through comparative simulation results, the excellent performance of the proposed CVRZNN model is obviously demonstrated, which is much better than that of the previous GNN and ZNN models.

Adaptive Discontinuous Galerkin Modeling of Intrinsic Attenuation Anisotropy for Fluid-Saturated Porous Media

An hp- and memory-adaptive discontinuous Galerkin time-domain algorithm is presented to efficiently model wave propagation in poroelastic media, with the incorporation of 3-D fully anisotropic intrinsic attenuation. From the perspective of physics, the attenuation from the triclinic rock frame, the loss in the pore fluid, and the friction for their interaction are all considered. From the perspective of implementation, a new frequency-domain constitutive equation is introduced, involving complex-valued poroelasticity matrix. A new $Q$ value is defined as the ratio between its real and imaginary parts for every entry. Then, a generalized Maxwell body is adopted to approximate this frequency-dependent $Q$ in the time domain. Mathematically speaking, the hyperbolicity is preserved for the new viscous poroelastic system. Our results corroborate that the intrinsic attenuation anisotropy makes tangible effects in fluid-saturated porous formations.

Sampling of entire functions of several complex variables on a lattice and multivariate Gabor frames

ABSTRACT We give a general construction of entire functions in d complex variables that vanish on a lattice of the form for an invertible complex-valued matrix. As an application, we exhibit a class of lattices of density >1 that fail to be a sampling set for the Bargmann–Fock space in . By using an equivalent real-variable formulation, we show that these lattices fail to generate a Gabor frame.

A Noise-Tolerant Zeroing Neural Network for Time-Dependent Complex Matrix Inversion Under Various Kinds of Noises

Complex-valued time-dependent matrix inversion (TDMI) is extensively exploited in practical industrial and engineering fields. Many current neural models are presented to find the inverse of a matrix in an ideal noise-free environment. However, the outer interferences are normally believed to be ubiquitous and avoidable in practice. If these neural models are applied to complex-valued TDMI in a noise environment, they need to take a lot of precious time to deal with outer noise disturbances in advance. Thus, a noise-suppression model is urgent to be proposed to address this problem. In this article, a complex-valued noise-tolerant zeroing neural network (CVNTZNN) on the basis of an integral-type design formula is established and investigated for finding complex-valued TDMI under a wide variety of noises. Furthermore, both convergence and robustness of the CVNTZNN model are carefully analyzed and rigorously proved. For comparison and verification purposes, the existing zeroing neural network (ZNN) and gradient neural network (GNN) have been presented to address the same problem under the same conditions. Numerical simulation consequences demonstrate the effectiveness and excellence of the proposed CVNTZNN model for complex-valued TDMI under various kinds of noises, by comparing the existing ZNN and GNN models.

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.

Global µ-stability of neutral-type impulsive complex-valued BAM neural networks with leakage delay and unbounded time-varying delays

This paper investigates the existence and uniqueness of the equilibrium point and its global µ-stability for neutral-type impulsive complex-valued bidirectional associative memory neural networks with leakage delay and unbounded time-varying delays, by considering two types of Lipschitz conditions to be satisfied by the activation functions. The homeomorphism lemma is used to obtain a sufficient condition for the existence and uniqueness of the equilibrium point. Sufficient delay-dependent conditions both in terms of complex-valued and real-valued linear matrix inequalities which ensure the global µ-stability of the equilibrium point are obtained by constructing appropriate Lyapunov–Krasovskii functionals with simple, double, and triple integral terms, and using the free weighting matrix method, simple and double complex-valued Jensen inequalities, the complex-valued reciprocally convex combination inequality, and the complex-valued Wirtiger-based integral inequality. Lastly, two numerical examples are provided to illustrate the effectiveness of the obtained theoretical results.

Global asymptotic stability analysis for neutral-type complex-valued neural networks with random time-varying delays

ABSTRACTThis paper investigates the global asymptotic stability problem for a class of neutral-type complex-valued neural networks with random time-varying delays. By introducing a stochastic variable with Bernoulli distribution, the information of time-varying delay is assumed to be random time-varying delays. By constructing an appropriate Lyapunov–Krasovskii functional and employing inequality technique, several sufficient conditions are obtained to ensure the global asymptotically stability of equilibrium point for the considered neural networks. The obtained stability criterion is expressed in terms of complex-valued linear matrix inequalities, which can be simply solved by effective YALMIP toolbox in MATLAB. Finally, three numerical examples are given to demonstrate the efficiency of the proposed main results.

Synchronization of two nonidentical complex-valued neural networks with leakage delay and time-varying delays

This paper investigates the global asymptotical synchronization of two nonidentical complex-valued neural networks (CVNNs) with leakage delay and time-varying delays via integral sliding mode control approach. A suitable sliding surface and an appropriate Lyapunov–Krasovskii functional are constructed. By using free-weighting matrix method combined with inequality technique, a sliding mode controller is designed to ensure the synchronization for the considered CVNNs. The obtained criterion is presented in the form of complex-valued linear matrix inequalities (LMIs), which could be calculated through YALMIP with solver of SDPT3 in MATLAB. An example is provided to demonstrate the obtained theoretical result.

Global dissipativity of a class of quaternion-valued BAM neural networks with time delay

This paper is concerned with the problem of the global dissipativity of a class of quaternion-valued BAM neural networks with time delay. Two different types of activation functions are considered, including general bounded and Lipschitz-type activation functions. Based on the plural decomposition method of quaternion, the quaternion-valued BAM neural network is separated into two complex-valued systems. Using Lyapunov second method and inequality techniques, some sufficient conditions in complex-valued linear matrix inequality form are derived to ensure the global dissipativity of the discussed network. Moreover, the framework of the globally attractive sets are given out as well. Here, the existence and the uniqueness of the equilibrium point needs not to be considered. Finally, two examples are presented and analyzed to illustrate the effectiveness of our theoretical results.

Sleep stages classification from EEG signal based on Stockwell transform

Sleep has great effect on physical health and quality of life. Electroencephalogram (EEG) signal is used in studying sleep process and recently, time–frequency transforms are increasingly utilised in EEG signal analysis. This study proposes an efficient method for sleep stages classification based on a time–frequency transform, namely Stockwell transform. In the introduced method, at first, the Stockwell transform is used to map each 30 s epoch of EEG signal into the time–frequency domains, which results in a complex-valued matrix. Then, the frequency domain is divided into different non-overlapping segments, leading to several matrices. After that, entropy features are extracted from the obtained matrices. In order to determine the sleep stage of each epoch, the computed features are applied to classifier. Support vector machine, weighted K -nearest neighbour, and ensemble bagged tree classifiers are considered. The Pz–Oz and Fpz–Cz channels of EEG signal from Sleep-EDF data set and C3–A2 channel from ISRUC-Sleep data set are used in this research. The results indicate that the proposed method outperforms the recently introduced methods.

Single stationary domain equivalent inverter admittance for three‐phase grid‐inverter system considering the interaction between grid and inverter

Admittance matrix has been used to represent the characteristics of three-phase inverter for grid-inverter system stability evaluation in both stationary, sequence, and dq domain. For the dq domain admittance matrix, the PLL and Park transformation introduce additional dynamics during the admittance measurement that may influence the application of dq model. For the stationary and sequence domain admittance matrix, the model measurement is not affected by the above additional parts but may also encounter other difficulties in actual measurement. First, current frequency response analyser (FRA) is not able to perform two-frequency measurement required by the admittance matrix. Second, practical limitations exist in the course of solving complex-valued admittance matrix equations. Here, a single stationary domain equivalent admittance of three-phase inverter for grid-inverter system with considering the interaction between grid and inverter is defined so that the above two difficulties can be avoided. Then, an equivalent admittance model of prototype inverter is constructed for stability analysis of the grid-inverter system using Nyquist stability criterion, instead of the generalised Nyquist criterion (GNC). The effectiveness of the proposed equivalent inverter admittance model is validated through simulation and experiment verification.

Leakage delay-dependent stability analysis for complex-valued neural networks with discrete and distributed time-varying delays

This paper investigates the leakage delay-dependent global asymptotic stability problem for a class of complex-valued neural networks (CVNNs) with discrete and distributed time-varying delays. In order to handle this issue easily, an appropriate Lyapunov–Krasovskii functional (LKF) is constructed with some augmented delay-dependent terms. By employing integral inequalities, several delay-dependent sufficient conditions are derived that ensure the global asymptotic stability of the considered system model. Moreover, the results obtained in this paper have expressed in terms of complex-valued linear matrix inequalities (LMIs), whose feasible solutions can be easily verified by effective YALMIP control toolbox in MATLAB LMI. Finally, two benchmark illustrative examples are given to show the effectiveness and advantages of the proposed results.

Unitary Matrix Completion-Based DOA Estimation of Noncircular Signals in Nonuniform Noise

In this paper, a novel direction-of-arrival (DOA) estimation algorithm is proposed for noncircular signals with nonuniform noise by using the unitary matrix completion (UMC) technique. First, the proposed method utilizes the noncircular property of signals to design a virtual array for approximately doubling the array aperture. Then, the virtual complex-valued covariance matrix with the unknown nonuniform noise is transformed into the real-valued one by utilizing the unitary transformation to improve the computational efficiency. Next, a novel UMC method is formulated for the DOA estimation to remove the influence of nonuniform noise. Finally, the DOA without the influence of the unknown noncircularity phase is obtained by using the modified estimation of signal parameters via rotational invariance technique (ESPRIT). Especially, for handling the coherent sources, the forward–backward spatial smoothing technique is utilized to reconstruct a full-rank covariance matrix so that the signal subspace and the noise subspace can be correctly separated. Due to utilizing the extended array aperture and the unitary transformation, the proposed method can identify more sources than the number of physical sensors and provides higher angular resolution and better estimation performance. Compared with the existing DOA estimation algorithms for noncircular signals, the proposed one can effectively suppress the influence of the nonuniform noise. The simulation results are provided to verify the effectiveness and superiority of the proposed method.

Underdetermined Independent Component Analysis Based on First- and Second-Order Statistics

This paper proposes a class of new algorithms based on first- and second-order statistics for independent source extraction of circular signals in underdetermined complex-valued mixture. The complex-valued mixing matrix is estimated by two extremely cost-effective novel methods based on the conditional mean of the mixtures which require some prior knowledge of the positive support of the real and/or imaginary parts of the sources. And the sources are recovered by combining the conventional minimum mean-squared error-based beamforming approach with the acquired prior knowledge. Based on how much prior knowledge is got, we propose several new algorithms. The complexity analysis about the proposed algorithms suggests that the algorithms which employ more prior knowledge have higher complexity, but their computational cost is significantly low. Two examples are provided for showing the possible applications of these proposed algorithms. Simulation results validate the effectiveness and reliability of all presented methods.

Complex-Domain Super MDS: A New Framework for Wireless Localization With Hybrid Information

We revisit the super multidimensional scaling (SMDS) wireless localization algorithm first proposed a decade ago, recasting it onto the complex-domain. Under this new formulation, the edge kernel, which carries both angle and distance information simultaneously and plays a central role in the SMDS algorithm, becomes a complex-valued rank-one matrix, resulting in a new complex-domain SMDS framework, which yields several advantages over the original, including the elimination of redundancy, the enhancement of conditions to handle information erasure, and the possibility of designing several algorithmic variations that offer different complexity/performance improvements. To cite some concrete results, it is shown, for instance, that a distance-based localization system with 20 targets employing one of the new algorithms dubbed complex-domain SMDS (CD-SDMDS) outperforms the original SMDS at a tenth of the computational cost, and that the handling of missing information via matrix completion is superior in CD-SMDS compared to SMDS. If the same network collects also anchor-to-target angle information, it is furthermore shown that localization is achieved with nearly $20\times $ lower complexity and still higher accuracy using another new algorithm dubbed Turbo MRC-SMDS, and over $25\times $ faster using yet another method dubbed Iterative MRC-SMDS, with only a slight degradation.

Multichannel Blind Sound Source Separation Using Spatial Covariance Model With Level and Time Differences and Nonnegative Matrix Factorization

This paper presents an algorithm for multichannel sound source separation using explicit modeling of level and time differences in source spatial covariance matrices (SCM). We propose a novel SCM model in which the spatial properties are modeled by the weighted sum of direction of arrival (DOA) kernels. DOA kernels are obtained as the combination of phase and level difference covariance matrices representing both time and level differences between microphones for a grid of predefined source directions. The proposed SCM model is combined with the NMF model for the magnitude spectrograms. Opposite to other SCM models in the literature, in this work, source localization is implicitly defined in the model and estimated during the signal factorization. Therefore, no localization preprocessing is required. Parameters are estimated using complex-valued nonnegative matrix factorization with both Euclidean distance and Itakura–Saito divergence. Separation performance of the proposed system is evaluated using the two-channel SiSEC development dataset and four channels signals recorded in a regular room with moderate reverberation. Finally, a comparison to other state-of-the-art methods is performed, showing better achieved separation performance in terms of SIR and perceptual measures.

Architectural Optimizations for a High-Throughput Sorted QR Decomposition Circuit in MIMO Communication Systems

This paper presents the VLSI architecture of a low-latency and high-throughput sorted-QR decomposition (SQRD) engine for multiple-input multiple-output (MIMO) communication systems. In order to achieve a high processing throughput, the proposed design is architected based on a novel pipelined Givens rotation (GR) structure comprising of multi-dimension COordinate rotation DIgital computer (CORDIC) (MD-CORDIC) processing elements (PEs). Moreover, this design delivers the vector norm and conducts the sorting operation as a by-product of the vectoring operation on the execution flow of the CORDIC process. Therefore, excessive overheads for norm-calculation and sorting are excluded, and thus the latency is greatly reduced and throughput is enhanced. In addition, the proposed SQRD engine is operating directly on the complex-valued channel matrix to avoid the matrix augmentation caused by the real-valued decomposition of the channel matrix. This design has been synthesized, placed and routed, and the post-layout estimation results have shown that the processing throughput of the proposed SQRD architecture achieves an approximately 2[Formula: see text] improvement compared to the prior arts.

A Fast Converging Channel Estimation Algorithm for Wireless Sensor Networks

A set-membership affine projection algorithm is proposed that can estimate a complex-valued channel matrix using a set of complex-valued pilots in the presence of additive white Gaussian noise. It is shown that the algorithm converges faster than the well-known set-membership normalized least mean square algorithm (SM-NLMS) while it resolves the high steady-state error value and the complexity issues in the regular affine projection algorithm. The fast convergence of the proposed algorithm means that a shorter training sequence inside each data block is required, which in turn improves the effective bit rate. This fast convergence is more pronounced when the pilot vectors are highly correlated. We incorporated the set-membership filtering framework into our study to reduce the computational complexity of the algorithm and preserve energy in the WSNs. Other studies have shown the superiority of the adaptive filtering algorithms, and in particular the NLMS algorithm, over other alternatives in various signal processing areas in WSNs, therefore, our proposed algorithm is a powerful substitute for a variety of algorithms. In other words, the implementation of various signal processing algorithms for different purposes can be replaced with the implementation of the proposed multipurpose algorithm. In this paper, we combine the results of our previous studies and prove the convergence of the algorithm. Furthermore, the steady-state analysis in the output mean square error is presented for two cases of pilot signals, and in the conducted simulations, the MSE performance of the algorithm is compared with the regular affine projection algorithm and the SM-NLMS algorithm.