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NASA NTRS1981

Study of high speed complex number algorithms

A method of evaluating the radiation integral on the curved surface of a reflecting antenna is presented. A three dimensional Fourier transform approach is used to generate a two dimensional radiation cross-section along a planer cut at any angle phi through the far field pattern. Salient to the method is an algorithm for evaluating a subset of the total three dimensional discrete Fourier transform results. The subset elements are selectively evaluated to yield data along a geometric plane of constant. The algorithm is extremely efficient so that computation of the induced surface currents via the physical optics approximation dominates the computer time required to compute a radiation pattern. Application to paraboloid reflectors with off-focus feeds in presented, but the method is easily extended to offset antenna systems and reflectors of arbitrary shapes. Numerical results were computed for both gain and phase and are compared with other published work.

Heisler, R.↗

NASA NTRS1978

Fixed-point error analysis of Winograd Fourier transform algorithms

The quantization error introduced by the Winograd Fourier transform algorithm (WFTA) when implemented in fixed-point arithmetic is studied and compared with that of the fast Fourier transform (FFT). The effect of ordering the computational modules and the relative contributions of data quantization error and coefficient quantization error are determined. In addition, the quantization error introduced by the Good-Winograd (GW) algorithm, which uses Good's prime-factor decomposition for the discrete Fourier transform (DFT) together with Winograd's short length DFT algorithms, is studied. Error introduced by the WFTA is, in all cases, worse than that of the FFT. In general, the WFTA requires one or two more bits for data representation to give an error similar to that of the FFT. Error introduced by the GW algorithm is approximately the same as that of the FFT.

Patterson, R. W.↗

NASA NTRS1994

Adaptive model reduction for continuous systems via recursive rational interpolation

A method for adaptive identification of reduced-order models for continuous stable SISO and MIMO plants is presented. The method recursively finds a model whose transfer function (matrix) matches that of the plant on a set of frequencies chosen by the designer. The algorithm utilizes the Moving Discrete Fourier Transform (MDFT) to continuously monitor the frequency-domain profile of the system input and output signals. The MDFT is an efficient method of monitoring discrete points in the frequency domain of an evolving function of time. The model parameters are estimated from MDFT data using standard recursive parameter estimation techniques. The algorithm has been shown in simulations to be quite robust to additive noise in the inputs and outputs. A significant advantage of the method is that it enables a type of on-line model validation. This is accomplished by simultaneously identifying a number of models and comparing each with the plant in the frequency domain. Simulations of the method applied to an 8th-order SISO plant and a 10-state 2-input 2-output plant are presented. An example of on-line model validation applied to the SISO plant is also presented.

Lilly, John H.↗

NASA NTRS1976

CCD correlation techniques

Two CCD techniques were discussed for performing an N-point sampled data correlation between an input signal and an electronically programmable reference function. The design and experimental performance of an implementation of the direct time correlator utilizing two analog CCDs and MOS multipliers on a single IC were evaluated. The performance of a CCD implementation of the chirp z transform was described, and the design of a new CCD integrated circuit for performing correlation by multiplication in the frequency domain was presented. This chip provides a discrete Fourier transform (DFT) or inverse DFT, multipliers, and complete support circuitry for the CCD CZT. The two correlation techniques are compared.

Hewes, C. R.↗

NASA NTRS1993

Cramer-Rao Bound for Gaussian Random Processes and Applications to Radar Processing of Atmospheric Signals

Calculations of the exact Cramer-Rao Bound (CRB) for unbiased estimates of the mean frequency, signal power, and spectral width of Doppler radar/lidar signals (a Gaussian random process) are presented. Approximate CRB's are derived using the Discrete Fourier Transform (DFT). These approximate results are equal to the exact CRB when the DFT coefficients are mutually uncorrelated. Previous high SNR limits for CRB's are shown to be inaccurate because the discrete summations cannot be approximated with integration. The performance of an approximate maximum likelihood estimator for mean frequency approaches the exact CRB for moderate signal to noise ratio and moderate spectral width.

Frehlich, Rod↗

DOE OSTI2024

A FFT-based mesoscale continuum dislocation mechanics with defect energy: Applications to composites and polycrystals

A crystal plasticity elastoviscoplastic FFT (fast Fourier transform) formulation with a mesoscale continuum field dislocation mechanics model is presented, which incorporates a defect energy density that depends on GND densities and an associated material length scale. This allows to thermodynamically derive internal length scale dependent intra-crystalline backstress and Peach–Koehler force acting on GND densities. The model considers GND density evolution through a filtered numerical spectral approach, which is coupled with stress equilibrium through the elastoviscoplastic FFT algorithm. The discrete Fourier transform (DFT) method together with finite difference (FD) schemes is applied to solve both the backstress tensor and the Fourier–Green operator. Numerical results are first reported for two-phase laminate composites with plastic single crystal channels and elastic precipitates for shear loadings. Channel size effects are simulated and analyzed on the overall and local hardening behaviors during monotonous loadings. In addition, the evolutions of GND densities and the role of their associated backstress on size effects are examined during reversible shear loading. In a second part, the role of the defect energy internal length scale on polycrystal’s hardening during tension–compression is discussed. The results are compared to those obtained using FFT-based continuum field dislocation mechanics without defect energy.

36 MATERIALS SCIENCE↗

DOE OSTI2020

Non-Stationary Power System Forced Oscillation Analysis using Synchrosqueezing Transform

Non-stationary forced oscillations (FOs) have been observed in power system operations. However, most detection methods assume that the frequency of FOs is stationary. In this paper, we present a methodology for the analysis of nonstationary FOs. Firstly, Fourier synchrosqueezing transform (FSST) is used to provide a concentrated time-frequency representation of the signals that allows identification and retrieval of non-stationary signal components. To continue, the Dissipating Energy Flow (DEF) method is applied to the extracted components to locate the source of forced oscillations. The methodology is tested using simulated as well as real PMU data. In conclusion, the results show that the proposed FSST-based signal decomposition provides a systematic framework for the application of DEF Method to non-stationary FOs.

42 ENGINEERING↗

DOE OSTI2020

Lattice Dynamics of Sb 2 Se 3 from Inelastic Neutron and X-Ray Scattering

The lattice dynamics of orthorhombic Sb 2 Se 3 is studied by a combination of inelastic neutron and 121 Sb nuclear inelastic scattering giving access to the total and Sb partial density of phonon states (DPS). The Se partial DPS is determined from the difference between the total and Sb partial DPS. The total DPS is determined at 39, 150, and 300 K, and an analysis of the temperature-induced mode shifts in combination with low-temperature powder diffraction data is provided. Using an earlier reported theoretical approach, the corresponding total and partial DPS of Sb 2 Se 3 are calculated by first-principles calculations. Herein, a detailed analysis of the Grüneisen parameter, element-specific and bulk Debye temperatures, and the mean force constants as derived from the experimental data and discrete Fourier transform calculations is provided. In general, the calculations underestimate the strength of the covalent Sb-Se bonds.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2021

Experimental and theoretical analysis of carbon driven detonation waves in a heterogeneously premixed Rotating Detonation Engine

Coal dust explosions can be hazardous; however, they can also generate a significant rise in stagnation pressure if adequately harnessed. Rotating detonation combustors seek to take advantage of the stagnation pressure rise phenomenon in a more sustained and controlled manner via confinement to a physical annulus, leading to increased overall thermodynamic efficiency. Here this investigation presents an analysis of detonations fueled by Carbon Black, a solid particulate consisting of virtually pure carbon molecules and lean Hydrogen-Air mixtures. It is realized that with the addition of Carbon Black, an increase of lean mixture detonability and detonation velocities extending the operating limit over that of a pure hydrogen-air mixture is experienced. For all testing conditions, the total equivalence ratio is held at φ = 1, while the fuel mixture's carbon mass fraction is increased from 0 to 0.7 while the hydrogen is decreased. Detonation wave velocities are extracted from high-speed imaging through applying a Discrete Fourier Transform algorithm to determine changes to the wave speed as Carbon Black particles are introduced. As a result, due to the addition of Carbon Black as an auxiliary fuel source, detonations were formed instead of deflagrations in operating conditions where one would expect deflagrations at the same hydrogen-air equivalence ratios without Carbon Black addition. The detonation formation provides evidence that the coal particles are reacting within the detonation wave in a large enough capacity to support a detonation wave within the annulus. Furthermore, the wave speed is shown to increase with the additional of carbon particles. At a constant global equivalence ratio, the detonation wave velocities were found to decrease with hydrogen's incremental replacement with coal particles. Whereby, through a theoretical comparison of the heat of combustion as computed from the experimentally derived detonation wave velocities, a linear relationship of the two was shown to exist. Therefore, the heat of combustion has the potential to describe an operational limit to sustaining a detonation wave.

42 ENGINEERING↗

DOE OSTI2020

Machine learning surrogate models for Landau fluid closure

The first result of applying the machine/deep learning technique to the fluid closure problem is presented in this letter. As a start, three different types of neural networks (multilayer perceptron (MLP), convolutional neural network (CNN) and two-layer discrete Fourier transform (DFT) network) were constructed and trained to learn the well-known Hammett-Perkins Landau fluid closure in configuration space. We found that in order to train a well-preformed network, a minimum size of training data set is needed; MLP also requires a minimum number of neurons in the hidden layers equals to the degrees of freedom in Fourier space despite training data is fed in configuration space. Out of three models DFT performs the best for the clean data most likely due to the existence of nice Fourier expression for Hammett-Perkins closure but it is least robust with respect to input noise. Overall, with appropriate tuning and optimization, all three neural networks are able to accurately predict Hammett-Perkins closure and reproduce the inherit nonlocal feature, suggesting a promising path to calculate more sophisticated closures with the machine/deep learning technique.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2021

Multi-Dimensional Scaling on Groups

Leveraging the intrinsic symmetries in data for clear and efficient analysis is an important theme in signal processing and other data-driven sciences. A basic example of this is the ubiquity of the discrete Fourier transform which arises from translational symmetry (i.e. time-delay/phase-shift). Particularly important in this area is understanding how symmetries inform the algorithms that we apply to our data. In this paper we explore the behavior of the dimensionality reduction algorithm multi-dimensional scaling (MDS) in the presence of symmetry. We show that understanding the properties of the underlying symmetry group allows us to make strong statements about the output of MDS even before applying the algorithm itself. In analogy to Fourier theory, we show that in some cases only a handful of fundamental ``frequencies'' (irreducible representations derived from the corresponding group) contribute information for the MDS Euclidean embedding.

Dimensionality reduction, Representation theory, D↗

DOE OSTI2024

A Nonlinear Least Squares Phasor Estimation Algorithm with a Trust Metric

The paper presents a separable nonlinear least squares (NLLS)-based approach for estimation of fundamental-frequency phasors from sampled point-on-wave measurements. An analytical connection is established between the NLLS cost function and the discrete Fourier transform (DFT)-based periodogram of the input signal. This periodogram-based interpretation of the cost function offers an intuitive and easy-to-implement solution for the frequency estimate. Using the residual error of the NLLS-fit, the paper also presents an insightful measure for ascertaining the quality of the phasor estimates and their validity, especially for data windows containing signal transients.

Chatterjee, Kaustav↗

DOE OSTI2020

Validation of Numerical Models of the Offshore Wind Turbine From the Alpha Ventus Wind Farm Against Full-Scale Measurements Within OC5 Phase III

The main objective of the Offshore Code Comparison Collaboration Continuation, with Correlation (OC5) project is validation of aero-hydro-servo-elastic simulation tools for offshore wind turbines (OWTs) through comparison of simulated results to the response data of physical systems. Phase III of the OC5 project validates OWT models against the measurements recorded on a Senvion 5M wind turbine supported by the OWEC Quattropod from the alpha ventus offshore wind farm. The following operating conditions of the wind turbine were chosen for the validation: (1) idling below the cut-in wind speed, (2) rotor-nacelle assembly (RNA) rotation maneuver below the cut-in wind speed, (3) power production below and above the rated wind speed, and (4) shutdown. A number of validation load cases were defined based on these operating conditions. The following measurements were used for validation: (1) strains and accelerations recorded on the support structure and (2) pitch, yaw, and azimuth angles, generator speed, and electrical power recorded from the RNA. Strains were not directly available from the majority of the OWT simulation tools; therefore, strains were calculated based on out-of-plane bending moments, axial forces, and cross-sectional properties of the structural members. The simulation results and measurements were compared in terms of time series, discrete Fourier transforms, power spectral densities, and probability density functions of strains and accelerometers. A good match was achieved between the measurements and models setup by OC5 Phase III participants.

17 WIND ENERGY↗

DOE OSTI2026

Quantum Time-Space Tradeoffs for Matrix Problems

We consider the time and space required for quantum computers to solve a wide variety of problems involving matrices, many of which have only been analyzed classically in prior work. Our main results show that for a range of linear algebra problems—including matrix-vector product, matrix inversion, matrix multiplication and powering—existing classical time-space tradeoffs, several of which are tight for every space bound, also apply to quantum algorithms with at most a constant factor loss. For example, for almost all fixed matrices 𝐴, including the discrete Fourier transform matrix, we prove that quantum circuits with at most 𝑇 input queries and 𝑆 qubits of memory require 𝑇 = Ω⁢(𝑛 2 /𝑆) to compute matrix-vector product 𝐴⁢𝑥 for 𝑥 ∈{0,1 𝑛 . We similarly prove that matrix multiplication for 𝑛 ×𝑛 binary matrices requires 𝑇 = Ω⁢(𝑛 3 /$\sqrt{𝑆}$). Because many of our lower bounds are matched by deterministic algorithms with the same time and space complexity, our results show that quantum computers cannot provide any asymptotic advantage for these problems with any space bound. We obtain matching lower bounds for the stronger notion of quantum cumulative memory complexity—the sum of the space per layer of a circuit. We also consider Boolean (i.e., AND-OR) matrix multiplication and matrix-vector products, improving the previous quantum time-space tradeoff lower bounds for 𝑛 × 𝑛 Boolean matrix multiplication to 𝑇 = Ω⁢(𝑛 2.5 /𝑆 1/4 ) from 𝑇 = Ω⁢(𝑛 2.5 /𝑆 1/2 ). Our improved lower bound for Boolean matrix multiplication is based on a new coloring argument that extracts more from the strong direct product theorem that was the basis for prior work. To obtain our tight lower bounds for linear algebra problems, we require much stronger bounds than strong direct product theorems. We obtain these bounds by adding a new bucketing method to the quantum recording-query technique of Zhandry that lets us apply classical arguments to upper bound the success probability of quantum circuits.

lower bounds↗

DOE OSTI2026

Paradigm for universal quantum information processing with integrated acousto-optic frequency beamsplitters

Frequency-bin encoding offers tremendous potential in quantum photonic information processing, in which a single waveguide can support hundreds of lightpaths in a naturally phase-stable fashion. This stability, however, comes at a cost: arbitrary unitary operations can be realized by cascaded electro-optic phase modulators and pulse shapers, but require nontrivial numerical optimization for design and have thus far been limited to discrete tabletop components. In this article, we propose, formalize, and computationally evaluate a new paradigm for universal frequency-bin quantum information processing using acousto-optic scattering processes between distinct transverse modes. We show that controllable phase matching in intermodal processes enables 2 × 2 frequency beamsplitters and transverse-mode-dependent phase shifters, which together comprise cascadable FRequency-transverse-mODe Operations (FRODOs) that can synthesize any unitary via analytical decomposition procedures. Modeling the performance of both random gates and discrete Fourier transforms, we demonstrate the feasibility of high-fidelity quantum operations with existing integrated photonics technology, highlighting prospects of parallelizable operations achieving 100% bandwidth utilization. Our approach is realizable with CMOS technology, opening the door to scalable on-chip quantum information processing in the frequency domain.

Lukens, Joseph M. [Purdue Univ., West Lafayette, I↗

DOE OSTI2021

Open Source Fault-tolerant Grid Frequency Measurement for Solar Inverters

The Discrete Fourier transform (DFT) based measurement algorithms are one of the most common measurement algorithms for grid parameter estimation such as rms, phase angle, frequency. Over the past few years, many DFT based algorithms have been developed to enhance its measurement accuracy under steady-state and/or dynamic grid conditions. For example, an adaptive band-pass filter utilizing exponential modulation filter has been proposed to reduce measurement errors at the presence of large frequency deviations. Measurement accuracy of different algorithms including FIR filter, extended Kalman filtering (EKF), and enhanced DFT method have been compared in detail under different grid conditions. Two artificial signals that have 90-degree phase difference were constructed by the Clarke transformation to address the frequency spectrum leakage of DFT. A multi-module approach was developed to enhance both steady-state and dynamic measurement accuracies, in which each module was developed to eliminate some specific errors. Besides DFT-based measurement algorithms, some signal model-based algorithms have been developed to further improve the accuracy under dynamic conditions. However, a key drawback of the state-of-the-art algorithms is that they cannot perform measurements accurately during system transient faults. In the Blue Cut Fire event, there was a phase angle jump of about 26 degrees in the voltage waveform during the transient fault. The phase angle jump fault will cause waveform discontinuity, and these algorithms will fail to provide reliable measurements during this period because they typically assume the waveform to be measured is continuous, no matter what method (DFT, PLL, EKF, FIR, or Taylor WLS) is used for estimation. In fact, the measurement errors during the system transient faults like phase-jump is not required in the IEEE Standard. As a result, although a measurement instrument can pass the strict IEEE Standard, it could still be the source of the problem in the future if we have similar system transient faults, which could happen again. Therefore, developing the fault-tolerant measurement technology is the key to solve the problem.

14 SOLAR ENERGY↗

DOE OSTI2025

Determining the Axes of a Range-Doppler Image

Synthetic aperture radar (SAR) images formed with dechirp-on-receive data collection and rectangular format processing algorithm are the result of a two-dimensional discrete Fourier transform (DFT) applied to sampled data. There are several steps required to compute the range and Doppler values associated with each pixel in a SAR range-Doppler image. This memo walks readers through the process.

47 OTHER INSTRUMENTATION↗

NASA NTRS1972

Design study of a low bit rate transmitter and simulator

The results achieved toward developing a baseline communications channel model for studying low bit rate telemetry transmissions from Venus Planetary Explorer spacecraft are presented. The overall communications systems model, discrete channel model, input signal generation, and signal acquisition model are discussed. Discrete Fourier transform considerations, sampling routines, and flow charts are included.

Zusman, F. S.↗