Complex-valued Matrix Research Articles

VAN-DAMME: GPU-accelerated and symmetry-assisted quantum optimal control of multi-qubit systems

We present an open-source software package, VAN-DAMME (Versatile Approaches to Numerically Design, Accelerate, and Manipulate Magnetic Excitations), for massively-parallelized quantum optimal control (QOC) calculations of multi-qubit systems. To enable large QOC calculations, the VAN-DAMME software package utilizes symmetry-based techniques with custom GPU-enhanced algorithms. This combined approach allows for the simultaneous computation of hundreds of matrix exponential propagators that efficiently leverage the intra-GPU parallelism found in high-performance GPUs. In addition, to maximize the computational efficiency of the VAN-DAMME code, we carried out several extensive tests on data layout, computational complexity, memory requirements, and performance. These extensive analyses allowed us to develop computationally efficient approaches for evaluating complex-valued matrix exponential propagators based on Padé approximants. To assess the computational performance of our GPU-accelerated VAN-DAMME code, we carried out QOC calculations of systems containing 10 - 15 qubits, which showed that our GPU implementation is 18.4× faster than the corresponding CPU implementation. Our GPU-accelerated enhancements allow efficient calculations of multi-qubit systems, which can be used for the efficient implementation of QOC applications across multiple domains. Program summaryProgram Title: VAN-DAMMECPC Library link to program files::https://doi.org/10.17632/zcgw2n5bjf.1Licensing provisions: GNU General Public License 3Programming language: C++ and CUDANature of problem: The VAN-DAMME software package utilizes GPU-accelerated routines and new algorithmic improvements to compute optimized time-dependent magnetic fields that can drive a system from a known initial qubit configuration to a specified target state with a large (≈1) transition probability.Solution method: Quantum control, GPU acceleration, analytic gradients, matrix exponential, and gradient ascent optimization.

A reciprocity-aware, low-rank regularization for multidimensional deconvolution

A novel factorization-based, low-rank regularization method is introduced to solve multidimensional deconvolution (MDD) problems in the frequency domain. In this approach, each frequency component of the unknown wavefield is represented as a complex-valued square matrix and approximated as the product of a rectangular matrix and its transpose. The benefit of such a parameterization is twofold. First, each frequency matrix is guaranteed to be symmetric. From a physical standpoint, the recovered wavefield inherently adheres to the principle of reciprocity; in other words, the response to a receiver from a virtual source does not change when the two are swapped. Therefore, we refer to this low-rank parameterization as reciprocity-aware in that it respects the underlying physics of wave propagation associated with the MDD solution. Second, the size of the unknown matrix is greatly reduced compared with that of the original wavefield of interest (and halved compared with the existing factorization-based low-rank approximations). We further show that the associated objective function can be successfully optimized using the accelerated proximal gradient algorithm and present a robust strategy to define the initial guess of the solution. Numerical examples on synthetic and field data demonstrate the effectiveness of the proposed method in compressing the retrieved Green’s function while preserving its accuracy.

Reconstruction of electromagnetic parameters of the scattering from overfilled cavities using a complex-valued convolutional neural network

We reconstruct the electromagnetic parameters of the scattering from overfilled cavities by developing a complex-valued convolutional neural network (CV-CNN) method. Since the scattering field and electromagnetic parameters are usually complex values, the input, output and weight parameters of this method are set to complex values in order to capture the characteristics of the electromagnetic field and parameters more accurately. The complex-valued matrix of the input layer is composed of the scattering field intensity received by the receivers outside the overfilled cavity, which is solved by the Petrov–Galerkin finite element interface method based on uniform non-body-fitted meshes. By means of the received scattering field intensity outside the overfilled cavities, the proposed CV-CNN can accurately reconstruct the electromagnetic parameters of arbitrarily shaped overfilled cavities with homogeneous, inhomogeneous and anisotropic media. The experimental results show that the reconstruction accuracy of CV-CNN is significantly higher than real-valued convolutional neural network (RV-CNN) with the same architecture. Moreover, the appropriate optimizer and filter size have a positive influence on the reconstruction effects. Our study can provide a new development in the electromagnetic parameter reconstruction of the cavity scattering.

Chip-scale all-optical complex-valued matrix inverter

Matrix inversion is a fundamental and widely utilized linear algebraic operation but computationally expensive in digital-clock-based platforms. Optical computing is a new computing paradigm with high speed and energy efficiency, and the computation can be realized through light propagation. However, there is a scarcity of experimentally implemented matrix inverters that exhibit both high integration density and the capability to perform complex-valued operations in existing optical systems. For the first time, we experimentally demonstrated an iterative all-optical chip-scale processor to perform the computation of complex-valued matrix inversion using the Richardson method. Our chip-scale processor achieves an iteration speed of 10 GHz, which can facilitate ultra-fast matrix inversion with the assistance of high-speed Mach–Zehnder interferometer modulators. The convergence can be attained within 20 iterations, yielding an accuracy of 90%. The proposed chip-scale all-optical complex-valued matrix inverter represents a distinctive innovation in the field of all-optical recursive systems, offering significant potential for solving computationally intensive mathematical problems.

Complex Parameter Rao and Wald Tests for Assessing the Bandedness of a Complex-Valued Covariance Matrix

Banding the inverse of a covariance matrix has become a popular technique for estimating a covariance matrix from a limited number of samples. It is of interest to provide criteria to determine if a matrix is bandable, as well as to test the bandedness of a matrix. In this paper, we pose the bandedness testing problem as a hypothesis testing task in statistical signal processing. We then derive two detectors, namely the complex Rao test and the complex Wald test, to test the bandedness of a Cholesky-factor matrix of a covariance matrix’s inverse. Furthermore, in many signal processing fields, such as radar and communications, the covariance matrix and its parameters are often complex-valued; thus, it is of interest to focus on complex-valued cases. The first detector is based on the complex parameter Rao test theorem. It does not require the maximum likelihood estimates of unknown parameters under the alternative hypothesis. We also develop the complex parameter Wald test theorem for general cases and derive the complex Wald test statistic for the bandedness testing problem. Numerical examples and computer simulations are given to evaluate and compare the two detectors’ performance. In addition, we show that the two detectors and the generalized likelihood ratio test are equivalent for the important complex Gaussian linear models and provide an analysis of the root cause of the equivalence.

Efficient NVH Analysis of Electric Motors by Means of 2D Spectral Decomposition of Airgap Forces

The growing popularity of hybrid and electric vehicles has created a need for quieter electric motors. This has led to a focus on designing electric machines with reduced audible noise. Estimating the emitted noise requires complex calculations, but analyzing electromagnetic forces alone can provide valuable insights. Electric motors’ simple cylindrical structure allows for decoupling excitation forces and mode shapes, which are closely related. By representing the internal forces in a complex-valued matrix and calculating its 2D Fast Fourier Transform (FFT), we can obtain a comprehensive 2D spectrum that reveals temporal and spatial frequencies. The method enables individual analysis of unique harmonic components. This paper discusses the mathematical representation of forces, their spectra, and the relationship between force harmonics and mode shapes. Industrial examples of electric power-assisted steering systems demonstrate the practical application of this approach.

Complex-valued matrix-vector multiplication system for a large-scale optical FFT.

Recent advancements in optical convolutional neural networks (CNNs) and radar signal processing systems have brought an increasing need for the adoption of optical fast Fourier transform (OFFT). Presently, the fast Fourier transform (FFT) is executed using electronic means within prevailing architectures. However, this electronic approach faces limitations in terms of both speed and power consumption. Concurrently, existing OFFT systems struggle to balance the demands of large-scale processing and high precision simultaneously. In response, we introduce a novel, to the best of our knowledge, solution: a complex-valued matrix-vector system harnessed through wavelength selective switches (WSSs) for the realization of a 24-input optical FFT, achieving a high-accuracy level of 5.4 bits. This study capitalizes on the abundant wavelength resources available to present a feasible solution for an optical FFT system with a large N.

Shift-variant deconvolution beamforming using Implicit Neural Representation for near-field acoustic image measurement

The conventional beamforming (CBF) output can be expressed as a convolution of the source distribution and the array dependent point spread function (PSF), which is defined as the response of the beamformer to a point source. The resolution and accuracy of sound source localization can be improved by deconvolution of CBF. The shift-invariant PSF assumption is only a good approximation when the source region is small compared to the distance between the array and the source. However, the PSF of the near-field underwater acoustic map measurement is shift-variant, whose mismatch with the beam could lead to performance degradation. In this paper, we proposed an optimized deconvolution method by using the neural network called implicit neural representation (INR) to solve the beam mismatch problem, due to its strong performance in learning and reconstructing spatial scenes. Rather than recompute the PSF for each pixel in the scene, INR predicts the complex-valued weight matrix in two-dimensional space that encapsulates both the spatially varying properties of the PSF and the scatter distribution of the scene. Numerical simulations and experimental results verify the effectiveness of our method. And this work can be broadly applied to arrays with shift-variant PSFs.

2D-Unitary ESPRIT Based Multi-Target Joint Range and Velocity Estimation Algorithm for FMCW Radar

Millimeter-wave FMCW radar has been widely used in joint range-velocity estimation of multiple targets. However, most existing algorithms are unable to estimate the range-velocity information with high accuracy simultaneously and fail to discriminate the targets with either closely spaced ranges or closely spaced velocities in the 2D range-Doppler spectrum. In order to deal with these problems, this paper proposes a 2D-Unitary ESPRIT-based joint range and velocity estimation algorithm of multiple targets for FMCW radar. Firstly, The 1D-IF signal is constructed into a 2D virtual array signal, the virtual array signals are preprocessed by a 2D-spatial smoothing technique to generate a new matrix signal. Then, according to the 2D-Unitary ESPRIT algorithm, the 2D real-valued information of the target parameters is obtained from this matrix signal, and then a new complex-value matrix is constructed. Finally, the eigenvalue decomposition of this new complex-value matrix is performed, and the range-velocity estimates of multiple targets are, respectively, calculated from the real and imaginary parts of the eigenvalues, and paired automatically. The simulation results illustrate that the proposed algorithm not only provides highly accurate range-velocity estimates but also has high-resolution performance and achieves automatic pairing of the range-velocity estimates in multi-target scenarios, thus effectively improving the multi-target joint range and velocity estimation performance of FMCW radar.

A Novel Complex-Valued Gaussian Measurement Matrix for Image Compressed Sensing.

The measurement matrix used influences the performance of image reconstruction in compressed sensing. To enhance the performance of image reconstruction in compressed sensing, two different Gaussian random matrices were orthogonalized via Gram-Schmidt orthogonalization, respectively. Then, one was used as the real part and the other as the imaginary part to construct a complex-valued Gaussian matrix. Furthermore, we sparsified the proposed measurement matrix to reduce the storage space and computation. The experimental results show that the complex-valued Gaussian matrix after orthogonalization has better image reconstruction performance, and the peak signal-to-noise ratio and structural similarity under different compression ratios are better than the real-valued measurement matrix. Moreover, the sparse measurement matrix can effectively reduce the amount of calculation.

Fixed-time convergence integral-enhanced ZNN for calculating complex-valued flow matrix Drazin inverse

The complex-valued flow matrix Drazin inverse has recently attracted considerable interest from researchers due to its great academic value. In this paper, a fixed-time convergence integral-enhanced zeroing neural network (FTCIEZNN) model is proposed and investigated for calculating the Drazin inverse of complex-valued flow matrix. Since the FTCIEZNN model possesses fixed-time convergence, its upper limit of convergence time is irrelevant to initial conditions and can be adjusted by specified system parameters. Meanwhile, by adopting the newly designed reformed nonlinear activation function (RNAF) and variable parameters, the FTCIEZNN model converges rapidly in a relatively fast fixed-time and its robustness is dramatically strengthened. In addition, the upper limit of the convergence time in the absence of noise and the upper limit of the steady-state error in the presence of time-varying bounded noise are given by a scrupulous mathematical logic calculation. Furthermore, the outcomes of the numerical simulations demonstrate that the FTCIEZNN model outshines existing zeroing neural network models in calculating complex-valued flow matrix Drazin inverse. Finally, an application based on the FTCIEZNN model in image encryption fully illustrates the practical value of the FCIEZNN model.

Experimental Generation of Structured Light Beams through Highly Anisotropic Scattering Media with an Intensity Transmission Matrix Measurement

Structured light beams have played important roles in the fields of optical imaging and optical manipulation. However, light fields scatter when they encounter highly anisotropic scattering media, such as biological tissue, which destroys their original structured fields and turns them into speckle fields. To reconstruct structured light beams through highly anisotropic scattering media, we present a method based on intensity transmission matrix which only relates the input and output light intensity distributions. Compared with the conventional method which relies on the measurement of complex-valued transmission matrix, our scheme is easy to implement, fast and stable. With the assistance of spatial filters, three kinds of structured light beams, Bessel-like beams, vortex beams and cylindrical vector beams, were constructed experimentally through a ZnO scattering layer. The present method is expected to promote optical applications through highly anisotropic scattering media.

Acquisition of Visible Satellites Based on Antenna Arrays via Joint $\ell _{1,1}$-Norms Minimization

This work tackles the acquisition of visible satellites based on an antenna array in the presence of impulsive noise. First, the joint angle and number estimation of visible satellites is formulated as a joint <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _{1,1}$</tex-math></inline-formula> -norms minimization problem by exploiting the spatial sparsity of visible satellites. Then, in order to obtain a closed-form solution, a generalized conjugate function for complex-valued matrix variable is introduced and the joint <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _{1,1}$</tex-math></inline-formula> -norms minimization is transformed into a Lagrangian dual function maximization. Furthermore, the dual ascent framework and subgradient technique are employed to obtain a suitable iterative solution, where the convergence rate and steady-state value can be flexibly adjusted. Numerical results are presented to demonstrate the effectiveness of the proposed approach.

Energy-Efficient Power Allocation Maximization for MU-MIMO Multiple Access Channels

We propose an Iterative Water-Filling algorithm for Energy-Efficiency maximization problem of the Multi-User Multiple-Input Multiple-Output Multiple-Access-Channel (MU-MIMO-MAC) system, named as IWF-EE-MIMO. The algorithm can be regarded as two levels of operations. The inner level of IWF-EE-MIMO aims at computing the solution to each user of the family, while other users keeping their previous power allocation, in turn. The outer level of IWF-EE-MIMO aims at determining when to accept a good solution to the considered problem based on the results obtained by the inner level. For our designed algorithms in this paper, WF-EE-MIMO1, as the subordinate algorithm, is only used for the inner level. IWF-EE-MIMO, as the main algorithm, computes the solution to the considered problem. Note that IWF-EE-MIMO includes WF-EE-MIMO1, to avoid confusion. The considered problem is of non-linear fractional semi-definite optimization in complex-valued matrix optimization variables, beyond the range of standard (semi-definite) optimization theory. Thus, the optimality condition of the considered maximization problem needs to be created. Under this creation, for the considered problem, convergence or optimality of IWF-EE-MIMO is obtained with its efficiency.

Synchronization of oscillators not sharing a common ground

Networks of coupled LC oscillators that do not share a common ground node are studied. Both resistive coupling and inductive coupling are considered. For networks under resistive coupling, it is shown that the oscillator-coupler interconnection has to be bilayer if the oscillator voltages are to asymptotically synchronize. Also, for bilayer architecture (when both resistive and inductive couplers are present) a method is proposed to compute a complex-valued effective Laplacian matrix that represents the overall coupling. It is proved that the oscillators display synchronous behavior if and only if the effective Laplacian has a single eigenvalue on the imaginary axis.

Dynamic Synchrophasor Estimation Based on Weighted Real-Valued Sinc Interpolation Method

Phasor measurement units (PMUs) are the cornerstone of intelligent sensing and monitoring of power systems. The superiority of the sinc interpolation function-based least squares (SLS) method, a signal processing algorithm for PMUs, has been proven in harmonic phasor estimation. However, the SLS-based estimators suffer from spectral interferences and heavy computational burden. To this end, a real-valued sinc weighted least squares (SWLS-R) method is proposed to track the dynamic synchrophasor accurately and quickly. First, the weighted vector, i.e., window function, is introduced to enhance the SLS estimator’s behavior by mitigating the impact of spectral interference. Then, the complex-valued system matrix is transformed into the real-valued system matrix to lower the computational burden by avoiding complex matrix multiplication. Finally, an explicit and fast solution is designed to calculate the pseudoinverse matrix of the SWLS-R, which aims to further improve computational efficiency. The accuracy and effectiveness of the proposed method are validated by simulations. These are designed, under steady and dynamic conditions, according to the M-class requirements of the Standard IEC/IEEE 60255-118-1.

High-dimensional MVDR beamforming based on a second unitary transformation

Minimum variance distortion-less response (MVDR) beamforming is a popular method to spatial filtering among adaptive array signal processing. However, the inversion of sample covariance matrix, especially the high-dimensional complex-valued matrix’s inversion, is an inevitable obstacle to the real-time performance of the beamformer. In order to address this problem, a second unitary transformation (UT) technique is presented to transform the inversion of original high-dimensional (M-dimensional) complex-valued covariance matrix into two low-dimensional (M/2-dimensional) real-valued symmetrical sub-ones’ inversion. The proposed second unitary transformation minimum variance distortion-less response (SUT-MVDR) beamforming converts complex-valued computation to real-valued computation at the first UT stage, based on which the computational complexity of matrix’s inversion is reduced to about one fourth of the original at the second UT stage, depending on two half-size sub-matrices’ inversion. Numerical simulations show that the proposed beamformer has a higher computational efficiency and a better output signal-to-interference-plus-noise ratio (SINR) performance than other similar beamformers.

A novel eigenvalue-based iterative simulation method for multi-dimensional homogeneous non-Gaussian stochastic vector fields

The generation of sample functions of stochastic fields or processes is a prerequisite to use the Monte Carlo method in stochastic mechanics and structural reliability analysis. When it comes to non-Gaussian characteristics such as the spatial variability of material or structural properties, the simulation of multi-dimensional non-Gaussian stochastic vector fields becomes necessary. However, the conventional Cholesky decomposition-based iterative simulation method requires three times of decomposition operations at each iteration and a complex procedure of random shuffle, which inevitably augment the model complexity. To this end, this study develops a novel eigenvalue-based iterative simulation method which only requires one decomposition operation in the whole simulation process. Central to this method is to represent the cross power spectral density matrix with its eigenvalue matrix during the iteration procedure, which avoids the cumbersome Cholesky decomposition and random shuffle procedure. Meanwhile, it can yield unique convergence result and present a powerful ability in addressing cases with the complex-valued spectral matrix. Numerical example demonstrates that the proposed method has a similar accuracy as that of the Cholesky decomposition-based iterative simulation method. Meanwhile, the estimated cumulative distribution function from one simulated sample and the estimated ensemble correlation function from 1000 simulated samples are all consistent with their respective targets. In addition, the sensitive analysis is performed to demonstrate that the proposed method has a relatively wide applicability. Therefore, this method may have a great potential to simulate multi-dimensional homogeneous non-Gaussian stochastic vector fields in practice.

Iterative photonic processor for fast complex-valued matrix inversion

AnN×Niterative photonic processor is proposed for the first time, we believe, for fast computation of complex-valued matrix inversion, a fundamental but computationally expensive linear algebra operation. Compared to traditional digital electronic processing, optical signal processing has a few unparalleled features that could enable higher representational efficiency and faster computing speed. The proposed processor is based on photonic integration platforms–the inclusion of III-V gain blocks offers net neutral loss in the phase-sensitive loops. This is essential for the Richardson iteration method that is adopted in this paper for complex linear systems. Wavelength multiplexing can be used to significantly improve the processing efficiency, allowing the computation of multiple columns of the inverse matrix using a single processor core. Performances of the key building blocks are modeled and simulated, followed by a system-level analysis, which serves as a guideline for designing anN×NRichardson iteration processor. An inversion accuracy of&gt;98%can be predicted for a64×64photonic processor with a&gt;80times faster inversion rate than electronic processors. Including the power consumed by both active components and electronic circuits, the power efficiency of the proposed processor is estimated to be over an order of magnitude more energy-efficient than electronic processors. The proposed iterative photonic integrated processor provides a promising solution for future optical signal processing systems.

A complex-valued slow independent component analysis based incipient fault detection and diagnosis method with applications to wastewater treatment processes

Multivariate statistical process monitoring are the essential approaches to achieve better prognostics and health management (PHM) of process industries. However, incipient faults and complex behaviors (such as nonlinearity and dynamics) always render the traditional multivariate statistical process monitoring approaches inadequate. Thus, a complex-valued slow independent component analysis (CSICA) is proposed, which is able to extract optimized features from a complex-valued matrix containing both of raw data and their changing rates by resorting to a complex-valued independent component analysis operation and a batch of phase shifts. These features, named slow independent components (SICs), not only guarantee the statistical independence but also capture slowly-changing patterns, thus refining both dynamic and non-Gaussian information mostly related with incipient faults. The proposed algorithm together with novel statistics, Is2, If2 and SPE, as well as their control limits can sequentially detect incipient faults effectively. Then, together with the novel differential mapping reconstructed contribution plot (DM-RCP) and Granger causality analysis, the proposed method can accurately locate rooting causes of incipient faults. Finally, the proposed framework of process monitoring is validated through two data sets from a simulation platform and an oxidation-ditch-based wastewater treatment plant, respectively. The results demonstrate that the proposed method can achieve more accurate and efficient performances than conventional methods.