Real-valued Computations Research Articles

All-optical complex-valued convolution based on four-wave mixing

Optical complex-valued convolution can extract the feature of complex-valued data by processing both amplitude and phase information, enabling a wide range of future applications in artificial intelligence and high-speed optical computation. However, because optical signals at different wavelengths cannot interfere, optical systems based on wavelength multiplexing usually can only realize real-valued computation. Here, we experimentally demonstrate an all-optical computing scheme using Kerr-based optical four-wave mixing (FWM) that can perform complex-valued convolution of multi-wavelength signals. Specifically, this all-optical complex-valued convolution operation can be implemented based on the coherent superposition of converted light generated by multiple FWM processes. The computational throughput of this scheme can be expanded by increasing the number of optical wavelengths and the signal baud rate. To exemplify the application, we successfully applied this all-optical complex-valued convolution to four different orientations of image edge extraction. Our scheme can provide a basis for wavelength-parallel optical computing systems with the demanded complex-valued computation capability.

Accurate characterization of complex Bloch modes in optical chain waveguides using real-valued computations

This research presents a highly accurate and easy-to-implement method to characterize the complex Bloch modes propagating along optical chain waveguides with three-dimensional (3D) layered geometries and dispersive negative-epsilon material compositions. The technique combines commercial EM solver results with analytical post-processing to avoid iterative complex root estimation on the complex plane. The proposed methodology is based on the real-valued computations that yield the complex Bloch wavevector with superior accuracy even when both radiation and material losses are present. In addition, we introduce a single unit-cell technique to provide the possibility of dense meshing of 3D geometries when available computational resources are limited. To verify our results, two different plasmonic and dielectric case studies are discussed. The obtained results agree well with numerical and experimental results from the literature. Due to its generality, robustness, and high accuracy, the method is beneficial for studying a large variety of waveguide-based nanophotonic components.

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.

Accounting for viscoelastic effects in a multiscale fatigue model for the degradation of the dynamic stiffness of short-fiber reinforced thermoplastics

Under fatigue loading, the stiffness decrease in short-fiber reinforced polymers reflects the gradual degradation of the material. Thus, both measuring and modeling this stiffness is critical to investigate and understand the entire fatigue process. Besides evolving damage, viscoelastic effects within the polymer influence the measured dynamic stiffness. In this paper, we study the influence of a linear viscoelastic material model for the matrix on the obtained dynamic stiffness and extend an elastic multiscale fatigue-damage model to viscoelasticity. Our contribution is two-fold. First, we revisit the complex-valued elastic models known in the literature to predict the asymptotic periodic orbit of a viscoelastic material. For small phase shifts in an isotropic linear viscoelastic material, we show through numerical experiments that a real-valued computation of an “elastic” material is sufficient to approximate the dynamic stiffness of a microstructure with a generalized Maxwell material and equal Poisson’s ratios in every element as matrix, reinforced by elastic inclusions. This makes standard solvers applicable to fiber-reinforced thermoplastics. Secondly, we propose a viscoelastic fatigue-damage model for the thermoplastic matrix based on decoupling of the time scales where viscoelastic and fatigue-damage effects manifest. We demonstrate the capability of the multiscale model to predict the dynamic stiffness evolution under fatigue loading of short-fiber reinforced polybutylene terephthalate (PBT) by a validation with experimental results.

Efficient real-valued DOA estimation based on the trigonometry multiple angles transformation in monostatic MIMO radar

A novel real-valued polynomial rooting-based (RV-root-MUSIC) algorithm is proposed for efficient DOA estimation in monostatic MIMO radar. RV-root-MUSIC exploits the trigonometry multiple angles formula and variable substitution technique without transformation approximations to convert the original complex steering vector into a real one with respect to the cosine and sine functions. As such, RV-root-MUSIC implements the real-valued computation for both tasks of EVD and the polynomial rooting since the coefficients of polynomial are real. Numerical simulations are applied to demonstrate that with significantly reduced complexity, such trigonometry transformation is able to provide the similar performances as compared to the conventional root-MUSIC and U-root-MUSIC.

CRN++: Molecular programming language

Synthetic biology is a rapidly emerging research area, with expected wide-ranging impact in biology, nanofabrication, and medicine. A key technical challenge lies in embedding computation in molecular contexts where electronic micro-controllers cannot be inserted. This necessitates effective representation of computation using molecular components. While previous work established the Turing-completeness of chemical reactions, defining representations that are faithful, efficient, and practical remains challenging . This paper introduces CRN++, a new language for programming deterministic (mass-action) chemical kinetics to perform computation. We present its syntax and semantics, and build a compiler translating CRN++ programs into chemical reactions, thereby laying the foundation of a comprehensive framework for molecular programming. Our language addresses the key challenge of embedding familiar imperative constructs into a set of chemical reactions happening simultaneously and manipulating real-valued concentrations. Although some deviation from ideal output value cannot be avoided, we develop methods to minimize the error, and implement error analysis tools. We demonstrate the feasibility of using CRN++on a suite of well-known algorithms for discrete and real-valued computation. CRN++ can be easily extended to support new commands or chemical reaction implementations, and thus provides a foundation for developing more robust and practical molecular programs.

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.

A biologically plausible supervised learning method for spiking neural networks using the symmetric STDP rule

Spiking neural networks (SNNs) possess energy-efficient potential due to event-based computation. However, supervised training of SNNs remains a challenge as spike activities are non-differentiable. Previous SNNs training methods can be generally categorized into two basic classes, i.e., backpropagation-like training methods and plasticity-based learning methods. The former methods are dependent on energy-inefficient real-valued computation and non-local transmission, as also required in artificial neural networks (ANNs), whereas the latter are either considered to be biologically implausible or exhibit poor performance. Hence, biologically plausible (bio-plausible) high-performance supervised learning (SL) methods for SNNs remain deficient. In this paper, we proposed a novel bio-plausible SNN model for SL based on the symmetric spike-timing dependent plasticity (sym-STDP) rule found in neuroscience. By combining the sym-STDP rule with bio-plausible synaptic scaling and intrinsic plasticity of the dynamic threshold, our SNN model implemented SL well and achieved good performance in the benchmark recognition task (MNIST dataset). To reveal the underlying mechanism of our SL model, we visualized both layer-based activities and synaptic weights using the t-distributed stochastic neighbor embedding (t-SNE) method after training and found that they were well clustered, thereby demonstrating excellent classification ability. Furthermore, to verify the robustness of our model, we trained it on another more realistic dataset (Fashion-MNIST), which also showed good performance. As the learning rules were bio-plausible and based purely on local spike events, our model could be easily applied to neuromorphic hardware for online training and may be helpful for understanding SL information processing at the synaptic level in biological neural systems.

Real-Valued DOA Estimation for Non-Circular Sources via Reduced-Order Polynomial Rooting

A novel real-valued algorithm of the popular root multiple signal classification of non-circular sources (NC-root-MUSIC) is developed for low-complexity direction of arrival (DOA) estimation. The proposed real-valued NC-root-MUSIC (RV-NC-root-MUSIC) reduces the computational burden mainly in following three aspects. First, unlike the classical real-valued algorithms, such as unitary transform, the complex-valued operation is still implemented when acquiring DOAs. However, the proposed RV-NC-root-MUSIC implements real-valued received data. Therefore, both tasks of eigenvalue decomposition (EVD) (or singular value decomposition (SVD)) and polynomial rooting can be implemented completely by real-valued computations in the proposed RV-NC-root-MUSIC. Next, due to the conjugate expansion of the non-circular sources, the steering vector dimension is extended by a factor of two, and the real-valued signal- and noise-subspace can be calculated by two reduced-dimension sub-matrices. As such, large-scale EVD/SVD computation is avoided. Finally, a real coefficient rooting polynomial with a similar order as NC-root-MUSIC is constructed. A variable substitution technique is further proposed to reduce the degree of this polynomial by half without sacrificing accuracy, which allows fast coefficient computation and real-valued rooting. The real-valued computation and reduced-order polynomial rooting in the proposed method result in a significant reduction in computational cost as compared to state-of-the-art methods. Numerical computer simulations illustrate that with significantly reduced complexity, the fast RV-NC-root-MUSIC algorithm can provide similar root mean square errors (RMSE) close to the conventional methods.

Real-Valued Near-Field Localization of Partially Polarized Noncircular Sources With a Cross-Dipole Array

In this paper, a novel near-field localization algorithm is proposed for partially polarized noncircular sources. The algorithm is composed of two stages. In the first stage, based on the cross-dipole, the array observations are transformed into the real-valued domain. Then, an extended covariance matrix is constructed and the structure of the real-valued steering vector is utilized to factorize the direction of arrival (DOA) parameter from other parameters. Based on the rank reduction (RARE) principle, the DOA parameters can be obtained through the one-dimensional (1-D) spectral search. In the second stage, with the DOA estimates, the range parameters are generated via another RARE estimator, which also only requires a 1-D spectral search. Compared with the conventional second-order statistics (SOS)-based methods, the proposed algorithm is computationally more efficient since it only requires 1-D spectral search and real-valued computations. Moreover, it can resolve more sources and promote the DOA estimation accuracy. Numerical simulation results corroborate the effectiveness and efficiency of the proposed method.

Reduced-Complexity Direction of Arrival Estimation Using Real-Valued Computation with Arbitrary Array Configurations

A low-complexity algorithm is presented to dramatically reduce the complexity of the multiple signal classification (MUSIC) algorithm for direction of arrival (DOA) estimation, in which both tasks of eigenvalue decomposition (EVD) and spectral search are implemented with efficient real-valued computations, leading to about 75% complexity reduction as compared to the standard MUSIC. Furthermore, the proposed technique has no dependence on array configurations and is hence suitable for arbitrary array geometries, which shows a significant implementation advantage over most state-of-the-art unitary estimators including unitary MUSIC (U-MUSIC). Numerical simulations over a wide range of scenarios are conducted to show the performance of the new technique, which demonstrates that with a significantly reduced computational complexity, the new approach is able to provide a close accuracy to the standard MUSIC.

Efficient Real-Valued Rank Reduction Algorithm for DOA Estimation of Noncircular Sources Under Mutual Coupling

Noncircular sources are widely used in wireless communication array systems, which can offer more accurate estimates and detect more sources. However, in practical array systems, the direction-of-arrival (DOA) estimation performance may be severely degraded by mutual coupling effects. To solve this problem, we propose a real-valued DOA estimation algorithm for noncircular sources under unknown mutual coupling. Based on the sources’ noncircularity, an augmented real-valued covariance matrix is constructed. Then, utilizing the banded symmetric and Toeplitz property of the mutual coupling matrix, the middle subarray elements are considered as ideal ones, which have the same array gains. Finally, according to the subspace principle, a rank reduction-based virtual steering vector parameterizing method is derived, which extracts the DOAs from other nuisance parameters. Compared with conventional algorithms, the proposed one not only improves the estimation accuracy but also resolves more sources. Moreover, it is computationally efficient, since it only requires real-valued computations and 1-D spectral search. Numerical simulations demonstrate that the proposed method performs well under unknown mutual coupling and outperforms some of the existing approaches in resolution capability, estimation accuracy, and computational loads.

Reduced-complexity direction of arrival estimation with centro-symmetrical arrays and its performance analysis

A fast algorithm is proposed to dramatically reduce the computational complexity of the multiple signal classification (MUSIC) algorithm for direction-of-arrival (DOA) estimate using a centro-symmetrical array (CSA). The CSA is divided into two sub-arrays and a real matrix is constructed with the covariance matrices of the two sub-arrays and their cross-correlation ones. This real matrix is further regarded as the data covariance one observed by a virtual array which has a real array response, and a novel MUSIC-like cost function is derived accordingly. In the developed method, only real-valued computation is required and the spectral search is compressed into half of the total angular field-of-view. Furthermore, the dimensions of noise subspace and those of search vector are both reduced, leading to about 97% complexity reduction as compare to MUSIC. The non-asymptotic statistical performance of the new DOA estimator is analyzed and a closed-form expression is given to predict the mean square error (MSE) of DOA estimation by the new technique. The effectiveness of the presented approach as well as the theoretical analysis is verified through numerical computer simulations, and it is shown that the proposed method is able to provide good accuracy with low signal-to-noise ratio (SNR) and small numbers of snapshots.

MUSIC-like direction of arrival estimation based on virtual array transformation

In this paper, we propose a reduced-complexity algorithm to estimate the direction of arrivals (DOAs) of multiple uncorrelated narrow-band signals. We show that with an array of arbitrary configuration, the real part of the array covariance matrix can be equivalently reformulated as an entire array covariance matrix received by a virtual array with potential signal model available for DOA estimate. We provide in-depth insights into the eigenvalue decomposition (EVD) on the real part of the array covariance matrix, based on which we propose a novel virtual array transformation multiple signal classification (VAT-MUSIC) algorithm for DOA estimation with efficient real-valued computations. The new algorithm is able to reduce about 75% computational complexity and it has easy implementation advantages over state-of-the-art real-valued techniques. Furthermore, the performance of the proposed algorithm is theoretically analyzed and a closed-form expression is derived to predict the mean square error (MSE) of the new DOA estimator. Numerical simulations are conducted to demonstrate the advantages of the proposed algorithm and to verify the theoretical analysis.

LP-W- $\ell _{\infty }$ -SVD Algorithm for Direction-of-Arrival Estimation

This paper presents an effective direction-of-arrival (DOA) estimation method based on a new computation method of the $\ell _{\infty }$ -norm for a complex matrix. Making use of the new computation method of the $\ell _{\infty }$ -norm, the DOA estimation problem can be efficiently formulated as a linear programming (LP) problem. Thus, the proposed method can effectively give the DOA estimation by finding the sparse coefficients which are obtained by solving the LP problem. Except for the first singular value decomposition of the data matrix, the proposed method is effectively implemented in the light of the LP theory based on real-valued computation. It avoids solving the complicated second-order cone programming problem. Furthermore, it can primely suppress spurious peaks in DOA estimation. Simulation results demonstrate the efficiency of the presented approach.

Rigorous floating-point mixed-precision tuning

Virtually all real-valued computations are carried out using floating-point data types and operations. The precision of these data types must be set with the goals of reducing the overall round-off error, but also emphasizing performance improvements. Often, a mixed-precision allocation achieves this optimum; unfortunately, there are no techniques available to compute such allocations and conservatively meet a given error target across all program inputs. In this work, we present a rigorous approach to precision allocation based on formal analysis via Symbolic Taylor Expansions, and error analysis based on interval functions. This approach is implemented in an automated tool called FPTuner that generates and solves a quadratically constrained quadratic program to obtain a precision-annotated version of the given expression. FPTuner automatically introduces all the requisite precision up and down casting operations. It also allows users to flexibly control precision allocation using constraints to cap the number of high precision operators as well as group operators to allocate the same precision to facilitate vectorization. We evaluate FPTuner by tuning several benchmarks and measuring the proportion of lower precision operators allocated as we increase the error threshold. We also measure the reduction in energy consumption resulting from executing mixed-precision tuned code on a real hardware platform. We observe significant energy savings in response to mixed-precision tuning, but also observe situations where unexpected compiler behaviors thwart intended optimizations.

Subspace‐based direction‐of‐arrival estimation using centro‐symmetrical arrays

A fast algorithm is presented to dramatically reduce the complexity of the multiple signal classification (MUSIC) algorithm with centro-symmetrical arrays. The proposed method requires only real-valued computations with a limited spectral search over half of the angular field-of-view, and the dimension of the noise subspace and that of the search vector are reduced, leading to about 96.875% complexity reduction as compared with MUSIC. Simulations show the new estimator performs well at low signal-to-noise ratio and with a small number of snapshots.

Computationally efficient direction finding using polynomial rooting with reduced-order and real-valued computations

The root multiple signal classification (root-MUSIC) algorithm is one of the most important techniques for direction of arrival (DOA) estimation. Using a uniform linear array (ULA) composed of M sensors, this method usually estimates L signal DOAs by finding roots that lie closest to the unit circle of a (2M−1)-order polynomial, where L < M. A novel efficient root-MUSIC-based method for direction estimation is presented, in which the order of polynomial is efficiently reduced to 2L. Compared with the unitary root-MUSIC (U-root-MUSIC) approach which involves real-valued computations only in the subspace decomposition stage, both tasks of subspace decomposition and polynomial rooting are implemented with real-valued computations in the new technique, which hence shows a significant efficiency advantage over most state-of-the-art techniques. Numerical simulations are conducted to verify the correctness and efficiency of the new estimator.

LPWS Algorithm for Wideband Spectrum Sensing

Wideband or multi-band spectrum sensing is an essential functionality for cognitive radio (CR) networks. It enables a secondary user to identify spectral holes dynamically and transmit opportunistically so as not to interfere with cohabiting primary users over the same bands. This paper presents an effective wideband spectrum sensing method based on linear programming (named as LPWS algorithm) for a CR user equipped with a single receiving antenna. Firstly, the proposed method utilizes the temporal smoothing technique to form a virtual multi-antenna structure. Secondly, the wideband spectrum sensing problem is reformulated as a sparse reconstruction problem by exploiting a sparse representation of the virtual multi-antenna array covariance vector. Finally, making use of $$l_{\infty }$$lź-norm, the sparse reconstruction problem is modelled as a linear programming (LP) problem and hence can be solved efficiently. The presented method offers a number of advantages over other recently proposed methods. For examples, (1) it can reduce system overhead since single antenna is used instead of multiple antennas or sensing nodes. (2) The unknown noise variances can be eliminated effectively by a linear transformation. (3) It is computationally simpler since it is efficiently formulated in terms of the LP problem based on real-valued computation, etc. Simulation results are presented to verify the efficiency of the proposed method.

Unitary PUMA Algorithm for Estimating the Frequency of a Complex Sinusoid

One-dimensional (1-D) and two-dimensional (2-D) frequency estimation for a single complex sinusoid in white Gaussian noise is a classic signal processing problem with numerous applications. It is revisited here through a new unitary principal-singular-vector utilization modal analysis (PUMA) approach, which is realized in terms of real-valued computations. The 2-D unitary PUMA is first formulated as an iteratively weighted least squares optimization problem. Recognizing that only one iteration is sufficient when 2-D unitary PUMA is initialized using least squares, a computationally attractive closed-form solution is then obtained. A variant of 2-D unitary PUMA is also developed for the 1-D case. Due to the real-valued computations and closed-form expression for the frequency estimate, the unitary PUMA is more computationally efficient than a number of state-of-the-art methods. Furthermore, the asymptotic variances of 1-D and 2-D unitary PUMA estimators are theoretically derived, and numerical results are included to demonstrate the effectiveness of the proposed methods.