Coordinate Rotation Research Articles

Implementation of SVM-Based Low Power EEG Signal Classification Chip

EEG-based classifiers are being used in a growing number of applications, including assisting cognitive function and medical diagnosis. In this brief, a low power EEG signal detection chip is presented based on supporting vector machine (SVM) for classification, with the application example of epilepsy detection. This brief elaborates the detection algorithm and circuit implementation of the neural detection system. The feature extraction module is based on a 256-point FFT for band energies calculation. An exponential operation circuit with scaling unit is proposed to implement the kernel function in classifier. The exponential function operation circuit combines the coordinate rotation digital calculation method (CORDIC) and approximate circuits to obtain power-delay benefit. Fabricated in 130nm CMOS, the proposed chip achieves wide supply range, low power and high rate of correctness. The seizures detection is with an average accuracy of over 80% and sensitivity of 94.4%. The energy consumption of chip is 1.28uJ/class and the area is only 1.8* <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1.8mm^{2}$ </tex-math></inline-formula> .

A High-Accuracy Digital Implementation of the Morris–Lecar Neuron With Variable Physiological Parameters

A highly accurate digital implementation of the Morris-Lecar neuron model is presented with the intended application of hardware acceleration for neuroscience simulation. The novel implementation employs the COordinate Rotation DIgital Computer (CORDIC) algorithm to create a fixed-point implementation that is not only very accurate but requires low digital hardware resources. The accuracy exceeds that of the current state-of-the-art, requires fewer hardware resources to implement, and operates at a higher maximum clock frequency. The design is validated on FPGA and a normalized RMSE of 0.2039 is achieved at a maximum clock frequency of 378.07MHz.

A High-Accuracy and Energy-Efficient CORDIC Based Izhikevich Neuron With Error Suppression and Compensation.

Bio-inspired neuron models are the key building blocks of brain-like neural networks for brain-science exploration and neuromorphic engineering applications. The efficient hardware design of bio-inspired neuron models is one of the challenges to implement brain-like neural networks, as the balancing of model accuracy, energy consumption and hardware cost is very challenging. This paper proposes a high-accuracy and energy-efficient Fast-Convergence COordinate Rotation DIgital Computer (FC-CORDIC) based Izhikevich neuron design. For ensuring the model accuracy, an error propagation model of the Izhikevich neuron is presented for systematic error analysis and effective error reduction. Parameter-Tuning Error Compensation (PTEC) method and Bitwidth-Extension Error Suppression (BEES) method are proposed to reduce the error of Izhikevich neuron design effectively. In addition, by utilizing the FC-CORDIC instead of conventional CORDIC for square calculation in the Izhikevich model, the redundant CORDIC iterations are removed and therefore, both the accumulated errors and required computation are effectively reduced, which significantly improve the accuracy and energy efficiency. An optimized fixed-point design of FC-CORDIC is also proposed to save hardware overhead while ensuring the accuracy. FPGA implementation results exhibit that the proposed Izhikevich neuron design can achieve high accuracy and energy efficiency with an acceptable hardware overhead, among the state-of-the-art designs.

Vector Bessel beams: General classification and scattering simulations

Apart from a lot of fundamental interest, vector Bessel beams are widely used in optical manipulation, material processing, and imaging. However, the existing description of such beams remains fragmentary, especially when their scattering by small particles is considered. We propose a new general classification of all existing vortex Bessel beam types in an isotropic medium based on the superposition of transverse Hertz vector potentials. This theoretical framework contains duality and coordinate rotations as elementary matrix operations and naturally describes all relations between various beam types. This leads to various bases for Bessel beams and uncovers the novel beam type with circularly symmetric energy density. We also discuss quadratic functionals of the fields (such as the energy density and Poynting vector) and derive orthogonality relations between various beam types. Altogether, it provides a comprehensive reference of all properties of Bessel beams that may be relevant for applications. Next we generalize the formalism of the Mueller scattering matrices to arbitrary Bessel beams accounting for their vorticity. Finally, we implement these beams in the ADDA code - an open-source parallel implementation of the discrete dipole approximation. This enables easy and efficient simulation of Bessel-beams scattering by particles with arbitrary shape and internal structure.

FPGA implementation of feature detection and matching using ORB

Oriented FAST and Rotated BRIEF (ORB) is becoming the method of choice in real-time feature matching applications by offering performances comparable to other state-of-the-art descriptors at a fraction of the runtime and memory requirements. These characteristics have prompted multiple researchers to propose FPGA-based accelerators that allow for a significant increase in throughput compared to general-purpose processors. This increase is, however, achieved at the expense of the descriptor’s robustness as several approximations are necessary to map the algorithm efficiently into hardware which negatively affects its invariance. In this paper, we introduce an FPGA architecture capable of extracting ORB descriptors at high throughput with minimal impact to their invariance. Rotation of sample point coordinates is bypassed entirely, and methods for the accurate computation of the moments and descriptor generation in a single clock cycle are described. Moreover, a feature matching algorithm based on the cross-check criterion is proposed and implemented in hardware. The architecture achieved a throughput of 94 MPix/s using a 100 MHz clock while producing an almost identical performance compared to the reference implementation.

Performance analysis and trajectory planning of multi-locomotion mode ankle rehabilitation robot

A multi-locomotion mode ankle rehabilitation robot (MLMARR) based on the 2-UPU/RPU (U: universal; P: prismatic; R: revolute) parallel mechanism with actuators above the end effector is proposed. In addition to the rehabilitation training of the basic motion orientation of the ankle, the MLMARR enables up/down or back/forth traction rehabilitation training, ensuring the training of muscle groups and ligaments related to the ankle motion. First, degrees-of-freedom analysis is conducted based on the screw theory. Subsequently, using the closed-loop vector method and coordinate system rotation transformation, inverse position analysis is performed and the Jacobian matrix is described. In addition, three types of kinematic singularities are identified by analyzing the Jacobian matrix. Moreover, the workspace is determined by the limit boundary method. Three rehabilitation training modes are set and dynamic simulations are performed according to the ankle rehabilitation requirements; on this basis, the linear actuators can be selected reasonably. Finally, the effectiveness and accuracy of rehabilitation training are evaluated based on experimental data obtained using an MLMARR prototype. This research reveals the characteristics and superiority of the proposed MLMARR and offers the basis for the future improvement of the device.

CORDIC-Based FPGA Realization of a Spatially Rotating Translational Fractional-Order Multi-Scroll Grid Chaotic System

This paper proposes an algorithm and hardware realization of generalized chaotic systems using fractional calculus and rotation algorithms. Enhanced chaotic properties, flexibility, and controllability are achieved using fractional orders, a multi-scroll grid, a dynamic rotation angle(s) in two- and three-dimensional space, and translational parameters. The rotated system is successfully utilized as a Pseudo-Random Number Generator (PRNG) in an image encryption scheme. It preserves the chaotic dynamics and exhibits continuous chaotic behavior for all values of the rotation angle. The Coordinate Rotation Digital Computer (CORDIC) algorithm is used to implement rotation and the Grünwald–Letnikov (GL) technique is used for solving the fractional-order system. CORDIC enables complete control and dynamic spatial rotation by providing real-time computation of the sine and cosine functions. The proposed hardware architectures are realized on a Field-Programmable Gate Array (FPGA) using the Xilinx ISE 14.7 on Artix 7 XC7A100T kit. The Intellectual-Property (IP)-core-based implementation generates sine and cosine functions with a one-clock-cycle latency and provides a generic framework for rotating any chaotic system given its system of differential equations. The achieved throughputs are 821.92 Mbits/s and 520.768 Mbits/s for two- and three-dimensional rotating chaotic systems, respectively. Because it is amenable to digital realization, the proposed spatially rotating translational fractional-order multi-scroll grid chaotic system can fit various secure communication and motion control applications.

Low-Latency and Reconfigurable VLSI-Architectures for Computing Eigenvalues and Eigenvectors Using CORDIC-Based Parallel Jacobi Method

This article proposes a low-latency parallel Jacobi-method-based algorithm for computing eigenvalues and eigenvectors of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n\times n$ </tex-math></inline-formula> -sized real-symmetric matrix. It is a coordinate rotations digital-computer (CORDIC)-based iterative algorithm that comprises multiple rotations and hence the key contribution of our work is to reduce the time cost of each rotation. Thus, alleviating the total latency for computing eigenvalues and eigenvectors using the parallel Jacobi method. Based on this proposed algorithm and additional architectural optimizations, a new low-latency and highly accurate VLSI-architecture has been presented in this manuscript for computing eigenvalues and eigenvectors of real-symmetric matrix. Subsequently, this work proposes a reconfigurable algorithm and its VLSI-architecture for computing eigenvalues and eigenvectors of complex Hermitian (CH), complex skew-Hermitian (CSH), and real skew-symmetric (RSS) matrices. Performance analysis of the proposed architectures has demonstrated minimal error-percentage of 0.0106% which is adequate for the wide range of real-time applications. The proposed architectures are hardware implemented on Zynq Ultrascale+ field-programmable gate array (FPGA)-board that consumed short latency of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$9.377~\mu \text{s}$ </tex-math></inline-formula> while operating at maximum clock frequency of 172.75 MHz. Comparison of our implementation results with the reported works showed that the proposed architecture incurs 43.75% lower latency and 89.4% better accuracy than the state-of-the-art implementation.

A higher-order accurate VOF interface curvature computation scheme for 3D non-orthogonal structured meshes

This paper proposes a higher-order accurate curvature computation method for volume-of-fluid (VOF) interfaces on 3D non-orthogonal structured meshes. Analogous to the height-function (HF) method, a novel approach is introduced to identify columns of control volumes that straddle the interface. This is followed by a volume conserving piecewise-linear interface construction (PLIC). In the interest of efficiency and generality, the latter employs a novel sweep-plane algorithm with bracketing on convex-decomposed control volumes in each column. The emphasis is on the generalisation of a PLIC-based least squares polynomial surface fitting procedure owing to its invariance with respect to local coordinate rotations in 3D. The PLIC representation of the interface is then smoothed by iteratively refining the PLIC facet normals. The interface curvature is finally computed analytically by fitting either second- or fourth-order polynomial surfaces to local stencils of PLIC facets in a least squares manner. Formal second- and fourth-order accuracy of interface curvature is demonstrated by comparing numerical results with a variety of analytical interface definitions.

FPGA Implementation of Reconfigurable CORDIC Algorithm and a Memristive Chaotic System With Transcendental Nonlinearities

Coordinate Rotation Digital Computer (CORDIC) is a robust iterative algorithm that computes many transcendental mathematical functions. This paper proposes a reconfigurable CORDIC hardware design and FPGA realization that includes all possible configurations of the CORDIC algorithm. The proposed architecture is introduced in two approaches: multiplier-less and single multiplier approaches, each with its advantages. Compared to recent related works, the proposed implementation overpasses them in the included number of configurations. Additionally, it demonstrates efficient hardware utilization and suitability for potential applications. Furthermore, the proposed design is applied to a memristive chaotic system with different transcendental functions computed using the proposed reconfigurable block. The memristive system design is realized on the Artix-7 FPGA board, yielding throughputs of 0.4483 and 0.3972 Gbit/s for the two approaches of reconfigurable CORDIC.

Competition between collective and individual conical intersection dynamics in an optical cavity

Light-induced nonadiabatic phenomena arise when molecules or molecular ensembles are exposed to resonant external electromagnetic fields. The latter can either be classical laser or quantized cavity radiation fields, which can couple to either the electronic, nuclear or rotational degrees of freedom of the molecule. In the case of quantized radiation fields, the light–matter coupling results in the formation of two new hybrid light–matter states, namely the upper and lower ‘polaritons’. Light-induced avoided crossings and light-induced conical intersections (CIs) between polaritons exist as a function of the vibrational and rotational coordinates of single molecules. For ensembles of N molecules, the N − 1 dark states between the two optically active polaritons feature, additionally, so-called collective CIs, involving the coordinates of more than one molecule to form. Here, we study the competition between intramolecular and collective light-induced nonadiabatic phenomena by comparing the escape rate from the Franck–Condon region of a single molecule and of a molecular ensemble coupled to a cavity mode. In situations where the polaritonic gap would be large and the dark-state decay channels could not be reached effectively, the presence of a seam of light-induced CI between the polaritons facilitates again the participation of the dark manifold, resulting in a cooperative effect that determines the overall non-radiative decay rate from the upper into the lower polaritonic states.

Automatic detection of code-modulated visual evoked potentials using novel covariance estimators and short-time EEG signals

Background and Objective:Over the last years, code-modulated visual evoked potentials (cVEP)-based brain–computer interfaces (BCIs) have been developed as robust and non-invasive tools to construct high-speed communication systems. Recently, beamforming-based algorithms have extensively been used in cVEP-based BCI systems because of the need for short-time stimulation and less training data. One of the main drawbacks of the beamforming-based approaches is that their performance highly depends on estimating data covariance matrix and calculating activation patterns. Methods:In the present study, two novel covariance estimators (i.e., the modified convex combination (MCC) and the maximum likelihood (ML) techniques) are proposed to estimate a robust and more reliable covariance matrix. In the ML method, a new sparsity constraint is considered to express the specific eigendecomposition of the covariance matrix as a sparse matrix transform (SMT). Then, the SMT is calculated using the product of pairwise coordinate rotations. These rotations can be constructed by a cross-validation method. Two stimulation presentation rates of 60 and 120 Hz are used for the coding sequence. Results:Both of the suggested approaches (i.e., the MCC and SMT-based techniques) can efficiently improve the performance of the conventional spatiotemporal beamforming-based methods by providing a robust estimate of the covariance matrix in short stimulation times. Based on the experimental results, it can be concluded that the proposed SMT and MCC methods achieve the best results for the 60 and 120 Hz stimulus presentation rates, respectively. However, for both stimulus presentation rates, the proposed SMT and MCC-based methods remarkably outperform other state-of-the-art methods in cVEP-based BCI, such as conventional spatiotemporal beamforming and optimized support vector machines (SVM). Also, the results showed that the 120 Hz stimulus presentation rate provided faster communication. This procedure is performed by obtaining a maximal Information Transfer Rate (ITR) of 187.38 bits/minute. Conclusion:Finally, the present study suggested that the proposed MCC and SMT-based techniques could automatically detect the gazed targets. Also, these methods could be used as non-invasive alternatives over conventional methods.

Efficient and Accurate CORDIC Pipelined Architecture Chip Design Based on Binomial Approximation for Biped Robot

Recently, much research has focused on the design of biped robots with stable and smooth walking ability, identical to human beings, and thus, in the coming years, biped robots will accomplish rescue or exploration tasks in challenging environments. To achieve this goal, one of the important problems is to design a chip for real-time calculation of moving length and rotation angle of the biped robot. This paper presents an efficient and accurate coordinate rotation digital computer (CORDIC)-based efficient chip design to calculate the moving length and rotation angle for each step of the biped robot. In a previous work, the hardware cost of the accurate CORDIC-based algorithm of biped robots was primarily limited by the scale-factor architecture. To solve this problem, a binomial approximation was carefully employed for computing the scale-factor. In doing so, the CORDIC-based architecture can achieve similar accuracy but with fewer iterations, thus reducing hardware cost. Hence, incorporating CORDIC-based architecture with binomial approximation, pipelined architecture, and hardware sharing machines, this paper proposes a novel efficient and accurate CORDIC-based chip design by using an iterative pipelining architecture for biped robots. In this design, only low-complexity shift and add operators were used for realizing efficient hardware architecture and achieving the real-time computation of lengths and angles for biped robots. Compared with current designs, this work reduced hardware cost by 7.2%, decreased average errors by 94.5%, and improved average executing performance by 31.5%, when computing ten angles of biped robots.

Investigation of the Shielding and Attenuation Effects of a Dynamical High-Density Chaff Cloud on the Signal Based on Voxel Splitting

Chaff jamming is a widely used passive interference method. A chaff cloud diffusion model for the widespread chaff cloud was presented. The high-density chaff cloud aerodynamic model can rapidly predict the chaff elements’ time-varying spatial orientation, location, and overall spatial distribution. Basing on these pieces of information, the radar cross section (RCS) for the single chaff fibre can be obtained based on the method of moments (MoM). For high-density chaff clouds, the attenuation and shielding effects on the signal must be considered. This paper proposed a voxel splitting method in rotated coordinates, dividing the chaff cloud into rectangular cells with one side perpendicular to the direction of radar entry. The varied polarized RCS features of the chaff cloud were simulated and analysed on this foundation. By dividing the chaff cloud into dynamic voxels and considering the uneven density distribution of the chaff cloud and the signal attenuation in each segment caused by the continuous fluctuating shape, fine-grained simulations of the chaff cloud radar echoes can be performed.

Fracture prediction based on walkaround 3D3C vertical seismic profiling data: A case study from the Tarim Basin in China

Natural fractures are well developed in the Sangtamu carbonate formation, which is the primary oil and gas production unit in the Tarim Basin, China. The analysis of converted waves via S-wave splitting (SWS) is an effective tool for predicting natural fractures in carbonate units. Compared with surface seismic data measurements, vertical seismic profiling (VSP) is more advantageous for acquiring and imaging converted waves. At present, there is a lack of robust methods on using 3D3C VSP data for fracture prediction. We propose an innovative workflow for predicting the spatial fracture distribution with the first ever application of SWS analysis to a 3D3C VSP data set. Instead of traditional Cartesian coordinates, we generate image bins based on a polar coordinate system to obtain accurate P-to-S converted waves of different azimuth angles. This is followed by a two-step SWS analysis. First, we conduct a multidirectional SWS analysis to estimate the fracture-induced anisotropy in the upper layers. Then, the wavefield of the target layer is corrected using time compensation and coordinate rotation. Finally, we apply SWS analysis again to obtain the azimuth and spatial intensity of the fractures in the target layer. We find that there is an overall good agreement in the fracture densities derived from the VSP waveforms and well-log data. The areas with high fracture development, as indicated by the SWS and ant-tracking analyses, also are consistent. Our study indicates that azimuth processing of walkaround 3D3C VSP combined with SWS analysis can serve as a quantitative diagnostic tool for fractures in a carbonate reservoir.

How close can we get to the classical magnetotelluric sounding?

The classical magnetotelluric (MT) sounding can be defined as data that are invariant under rotation of coordinates -a feature we call centrality- and are affected solely by electromagnetic induction effects –a feature we call all-induction. The classical sounding can be realized only in the case of laterally isotropic media and perfectly horizontal layering. However, in general it is possible to process data from several sites to make an approximation. For instance, the electromagnetic array profiling (EMAP) method addresses the issue of all-induction by means of spatial filtering. The method is very effective but its application is not practical for long profiles because it requires too many contiguous dipoles. A more practical version exists that does not require contiguous dipoles but neither of them addresses the centrality issue; in the first case because the data are taken along a single direction, and in the other because it uses traditional orthogonal modes. In this work we improve over the last approach by introducing centrality to make the approximation closer to the classical sounding. We use the determinant of the impedance tensor, which is the only invariant among all that are known that can handle centrality, the galvanic distortions and also the isolation of all-induction effects through two-dimensional inversion. This approach to the classical MT sounding is illustrated using the synthetic dataset COPROD2S2 and the field dataset BC87 from British Columbia, Canada, which are commonly used for testing new ideas. We also apply it to a recent profile over the Colima Graben, México.

FPGA Implementation of Polyphase Digital Down Converter for WiMAX Systems

ABSTRACT This paper briefs the implementation of a reconfigurable digital down converter (DDC) on a field-programmablegate array (FPGA) development board to be used in worldwide interoperability for microwave access (WiMAX) applications. The proposed polyphase design includes a parallel number of coordinate rotation digital computer (CORDIC) processors and low-passfinite impulse response (FIR) filters. Furthermore, the single rate FIR filter works with complex data input using canonical implementation. This strategy reduces multiplications and computational complexity. Additionally, the sample rate factor can be dynamically reconfigured to support multistandard DDC architectures. The mathematical expressions of the proposed design are briefly described. Comparison results indicate that the proposed design saves the hardware resources to achieve a cost-effectivesolution for software-definedradio standards. The simulation and experimental results are used to verify the correctness and feasibility of the polyphase DDC structure.

Spectroscopic and theoretical study of the intramolecular π-type hydrogen bonding and conformations of 3-cyclohexen-1-ol

The conformations of 3-cyclohexen-1-ol (3CHOL) have been investigated by MP2/cc-pVTZ and CCSD/cc-pVTZ ab initio computations and by infrared spectroscopy. The six conformational minima correspond to specific values of the ring-twisting and OH internal rotation coordinates. The map of a two-dimensional potential energy surface (PES) that governs the conformational dynamics along the ring-twisting and the OH group internal rotation coordinates was generated. The relative energies of the conformational minima and the interconversion barriers for the ring twisting and internal rotations were computed. According to the MP2/cc-pVTZ computations, the conformer with intramolecular π-type hydrogen bonding has the lowest calculated energy by 108 cm−1. However, according to the CCSD/cc-pVTZ computations this conformer a tiny 10 cm−1 higher in energy than one without hydrogen bonding. The infrared spectrum in the OH stretching region agrees well with the predicted frequency differences between the conformers.

Pipeline Implementation of the Unified CORDIC Algorithm in FPGA

This work presents a pipeline implementation of the COordinate Rotation DIgital Computer (CORDIC) algorithm in VHDL. This implementation computes both the trigonometric (sine, cosine, and arctangent of two parameters) and hyperbolic (hyperbolic sine, hyperbolic cosine, and hyperbolic arctangent) fixed-point functions. The implementation was synthesized on a Xilinx ® Zynq UltraScale + ZCU102 FPGA evaluation board, reaching an operating frequency of 297.89 MHz. The results show the proper operation of the proposed architecture, achieving relative errors of less than 1.4% for a 16-bit fixed-point representation. The VHDL implementation is easily adjustable to meet the specific requirements of each system and is available as an open-source code under a Creative Commons license.

Nonlinear electro-elastic dynamics of a hub–cantilever bimorph rotor structure

Dynamics of a rotating mini hub–bimorph structure excited by periodic torque supplied to the hub is studied in this paper. The mathematical model of the system is formulated by means of the extended Hamilton’s principle. In the analysis the nonlinear constitutive behaviour of the piezoceramic material is adopted based on experimental results published in literature. Moreover, nonlinear inertia effects related to the system rotation are taken into account. The derived system of three integro-partial differential equations represents the electro-mechanical behaviour of the beam (transverse displacement and transducer output voltage) and the hub rotation coordinate. The state equations are discretized and solved numerically around the first resonance zone for the system excited by a periodic torque supplied to the hub. The obtained response curves reveal strong softening behaviour with additional unstable solutions identified out of the resonance zones. Within the framework of the performed numerical simulations the impact of torque excitation amplitude, torque mean value, hub inertia and electrical load resistance on system behaviour is discussed. Finally, the responses of the system with parallel vs series connected transducers are compared.