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NASA NTRS1981

Some aspects of algorithm performance and modeling in transient thermal analysis of structures

The status of an effort to increase the efficiency of calculating transient temperature fields in complex aerospace vehicle structures is described. The advantages and disadvantages of explicit and implicit algorithms are discussed. A promising set of implicit algorithms with variable time steps, known as the GEAR package is described. Four test problems, used for evaluating and comparing various algorithms, have been selected and finite-element models of the configurations are described. These problems include a Space Shuttle frame component, an insulated cylinder, a metallic panel for a thermal protection system, and a model of the Space Shuttle Orbiter wing. Results generally indicate a preference for implicit over explicit algorithms for solution of transient structural heat transfer problems when the governing equations are 'stiff' (typical of many practical problems such as insulated metal structures).

Adelman, H. M.↗
NASA NTRS2003

Subgrid Combustion Modeling for the Next Generation National Combustion Code

In the first year of this research, a subgrid turbulent mixing and combustion methodology developed earlier at Georgia Tech has been provided to researchers at NASA/GRC for incorporation into the next generation National Combustion Code (called NCCLES hereafter). A key feature of this approach is that scalar mixing and combustion processes are simulated within the LES grid using a stochastic 1D model. The subgrid simulation approach recovers locally molecular diffusion and reaction kinetics exactly without requiring closure and thus, provides an attractive feature to simulate complex, highly turbulent reacting flows of interest. Data acquisition algorithms and statistical analysis strategies and routines to analyze NCCLES results have also been provided to NASA/GRC. The overall goal of this research is to systematically develop and implement LES capability into the current NCC. For this purpose, issues regarding initialization and running LES are also addressed in the collaborative effort. In parallel to this technology transfer effort (that is continuously on going), research has also been underway at Georgia Tech to enhance the LES capability to tackle more complex flows. In particular, subgrid scalar mixing and combustion method has been evaluated in three distinctly different flow field in order to demonstrate its generality: (a) Flame-Turbulence Interactions using premixed combustion, (b) Spatially evolving supersonic mixing layers, and (c) Temporal single and two-phase mixing layers. The configurations chosen are such that they can be implemented in NCCLES and used to evaluate the ability of the new code. Future development and validation will be in spray combustion in gas turbine engine and supersonic scalar mixing.

Menon, Suresh↗
NASA NTRS2019

Application of High-Dimensional Fuzzy K-Means Cluster Analysis to CALIOP/CALIPSO Version 4.1 Cloud-Aerosol Discrimination

This study applies fuzzy k-means (FKM) cluster analyses to a subset of the parameters reported in the CALIPSO lidar level 2 data products in order to classify the layers detected as either clouds or aerosols. The results obtained are used to assess the reliability of the cloud–aerosol discrimination (CAD) scores reported in the version 4.1 release of the CALIPSO data products. FKM is an unsupervised learning algorithm, whereas the CALIPSO operational CAD algorithm (COCA) takes a highly supervised approach. Despite these substantial computational and architectural differences, our statistical analyses show that the FKM classifications agree with the COCA classifications for more than 94 % of the cases in the troposphere. This high degree of similarity is achieved because the lidar-measured signatures of the majority of the clouds and the aerosols are naturally distinct, and hence objective methods can independently and effectively separate the two classes in most cases. Classification differences most often occur in complex scenes (e.g., evaporating water cloud filaments embedded in dense aerosol) or when observing diffuse features that occur only intermittently (e.g., volcanic ash in the tropical tropopause layer). The two methods examined in this study establish overall classification correctness boundaries due to their differing algorithm uncertainties. In addition to comparing the outputs from the two algorithms, analysis of sampling, data training, performance measurements, fuzzy linear discriminants, defuzzification, error propagation, and key parameters in feature type discrimination with the FKM method are further discussed in order to better understand the utility and limits of the application of clustering algorithms to space lidar measurements. In general, we find that both FKM and COCA classification uncertainties are only minimally affected by noise in the CALIPSO measurements, though both algorithms can be challenged by especially complex scenes containing mixtures of discrete layer types. Our analysis results show that attenuated backscatter and color ratio are the driving factors that separate water clouds from aerosols; backscatter intensity, depolarization, and mid-layer altitude are most useful in discriminating between aerosols and ice clouds; and the joint distribution of backscatter intensity and depolarization ratio is critically important for distinguishing ice clouds from water clouds.

Zeng, Shan↗
NASA NTRS2006

Probabilistic Classification Using Elemental Abundance Distributions and Lossless Image Compression in Apollo 17 Lunar Dust Samples from Mare Serenitatis

We have previously outlined a strategy for the detection of fossils [Storrie-Lombardi and Hoover, 2004] and extant microbial life [Storrie-Lombaudi and Hoover, 20051 during robotic missions to Mars using co-registered structural and chemical signatures. Data inputs included image lossless compression indices to estimate relative textural complexity and elemental abundance distributions. Two exploratory classification algorithms (principal component analysis and hierarchical cluster analysis) provide an initial tentative classification of all targets. Nonlinear stochastic neural networks are then trained to produce a Bayesian estimate of algorithm classification accuracy. The strategy previously has been successful in distinguishing regions of biotic and abiotic alteration of basalt glass from unaltered samples. [Storrie-Lombardi and Fisk, 2004; Storrie-Lombardi and Fisk, 2004] Such investigations of abiotic versus biotic alteration of terrestrial mineralogy on Earth are compromised by .the difficulty finding mineralogy completely unaffected by the ubiquitous presence of microbial life on the planet. The renewed interest in lunar exploration offers an opportunity to investigate geological materials that may exhibit signs of aqueous alteration, but are highly unlikely to contain contaminating biological weathering signatures. We here present an extension of our earlier data set to include lunar dust samples obtained during the Apollo 17 mission. Apollo 17 landed in the Taurus-Littrow Valley in Mare Serenitatis. Most of the rock samples from this region of the lunar highlands are basalts comprised primarily of plagioclase and pyroxene and selected examples of orange and black volcanic glass. SEM images and elemental abundances (C6, N7, O8, Na11, Mg12, Al13, Si14, P15, S16, Cll7, K19, Ca20, Fe26) for a series of targets in the lunar dust samples are compared to the extant cyanobacteria, fossil trilobites, Orgueil meteorite, and terrestrial basalt targets previously discussed. The data set provides a first step in producing a quantitative probabilistic methodology for geobiological analysis of returned lunar samples or in situ exploration.

Storrie-Lombardi, Michael C.↗
NASA NTRSDate not supplied

Architecture Options for Navigation in Cislunar Space for Human Landing System Vehicles

As part of architecture studies and insight analysis focused on requirements development into Human Landing System lunar architecture designs, multiple studies are underway to understand the sensitivities and options for achieving high precision landing on the lunar surface. The baseline approach utilizes a combination of multiple sensors to capture autonomous state observations of the lander with respect to the lunar surface. These systems are typically constrained in terms of operational altitudes by parameters such as onboard map size, camera focus, or sensor transmitted power (for altimeter observations). While these sensor suites do enable high precision landing, they are typically very complex and expensive. For a human-rated vehicle, fault detection algorithms are needed in addition to redundant sensors drive additional design complexity. Conversely, for these early missions, mass performance is key, so extended analysis is required to identify numbers of sensors, their ideal placement, and integration algorithms. A key part of this analysis is to help identify key sensor suites and options to help alleviate this design tension. An alternate approach is to take advantage and build out in-situ assets to allow for GPS-like navigation within the lunar regime through the use of navigation references or beacons. This can be achieved through the integration of navigation services into potential relays and pre-placed lunar surface assets. This research focuses on the capability of this infrastructure to support navigation in all areas of cislunar space such as: approach to the moon, in orbit around the moon, and ascent/descent operations to the surface. An augmented state linear covariance analysis (LinCov) and navigation state covariance analysis (NavCov) tools were used to assess a variety of navigation reference locations and how they can support vehicle operations through both understanding of state uncertainties and trajectory dispersions. This research helps to supplement existing studies focused on communication link analysis by providing additional insight into specific vehicle operational scenarios that are tied closely to potential Human Landing System scenarios. Key aspect of this analysis focus on the sensitivity to state knowledge of the references, the accuracy of inter-asset measurements, and placement in support of the various scenarios.

Evan J Anzalone↗
NASA NTRS2019

MARGInS Model-Based Analysis of Realizable Goals in Systems

The high complexity of modern aircraft and spacecraft requires elaborate Verification and Validation (V&V) approaches to make sure that such complex systems work properly and reliably. MARGInS is a framework for the analysis, understanding, and prediction of the behavior of a complex, hybrid system. MARGInS contains a set of machine learning and statistical algorithms for multivariate clustering, treatment learning, critical factor determination, time-series analysis, event prediction, and safety-boundary detection and characterization. The framework supports system testing and can be configured to find novel features in test suites, determine classes of behavior, propose new experiments that can efficiently explore and characterize the boundaries between classes of system behavior, and to create visualizations and reports.

He, Yuning↗
NASA NTRS2015

Use of Generalized Fluid System Simulation Program (GFSSP) for Teaching and Performing Senior Design Projects at the Educational Institutions

This paper describes the experience of the authors in using the Generalized Fluid System Simulation Program (GFSSP) in teaching Design of Thermal Systems class at University of Alabama in Huntsville. GFSSP is a finite volume based thermo-fluid system network analysis code, developed at NASA/Marshall Space Flight Center, and is extensively used in NASA, Department of Defense, and aerospace industries for propulsion system design, analysis, and performance evaluation. The educational version of GFSSP is freely available to all US higher education institutions. The main purpose of the paper is to illustrate the utilization of this user-friendly code for the thermal systems design and fluid engineering courses and to encourage the instructors to utilize the code for the class assignments as well as senior design projects. The need for a generalized computer program for thermofluid analysis in a flow network has been felt for a long time in aerospace industries. Designers of thermofluid systems often need to know pressures, temperatures, flow rates, concentrations, and heat transfer rates at different parts of a flow circuit for steady state or transient conditions. Such applications occur in propulsion systems for tank pressurization, internal flow analysis of rocket engine turbopumps, chilldown of cryogenic tanks and transfer lines, and many other applications of gas-liquid systems involving fluid transients and conjugate heat and mass transfer. Computer resource requirements to perform time-dependent, three-dimensional Navier-Stokes computational fluid dynamic (CFD) analysis of such systems are prohibitive and therefore are not practical. Available commercial codes are generally suitable for steady state, single-phase incompressible flow. Because of the proprietary nature of such codes, it is not possible to extend their capability to satisfy the above-mentioned needs. Therefore, the Generalized Fluid System Simulation Program (GFSSP1) has been developed at NASA Marshall Space Flight Center (MSFC) as a general fluid flow system solver capable of handling phase changes, compressibility, mixture thermodynamics and transient operations. It also includes the capability to model external body forces such as gravity and centrifugal effects in a complex flow network. The objectives of GFSSP development are: a) to develop a robust and efficient numerical algorithm to solve a system of equations describing a flow network containing phase changes, mixing, and rotation; and b) to implement the algorithm in a structured, easy-to-use computer program. The analysis of thermofluid dynamics in a complex network requires resolution of the system into fluid nodes and branches, and solid nodes and conductors as shown in Figure 1. Figure 1 shows a schematic and GFSSP flow circuit of a counter-flow heat exchanger. Hot nitrogen gas is flowing through a pipe, colder nitrogen is flowing counter to the hot stream in the annulus pipe and heat transfer occurs through metal tubes. The problem considered is to calculate flowrates and temperature distributions in both streams. GFSSP has a unique data structure, as shown in Figure 2, that allows constructing all possible arrangements of a flow network with no limit on the number of elements. The elements of a flow network are boundary nodes where pressure and temperature are specified, internal nodes where pressure and temperature are calculated, and branches where flowrates are calculated. For conjugate heat transfer problems, there are three additional elements: solid node, ambient node, and conductor. The solid and fluid nodes are connected with solid-fluid conductors. GFSSP solves the conservation equations of mass and energy, and equation of state in internal nodes to calculate pressure, temperature and resident mass. The momentum conservation equation is solved in branches to calculate flowrate. It also solves for energy conservation equations to calculate temperatures of solid nodes. The equations are coupled and nonlinear; therefore, they are solved by an iterative numerical scheme. GFSSP employs a unique numerical scheme known as simultaneous adjustment with successive substitution (SASS), which is a combination of Newton-Raphson and successive substitution methods. The mass and momentum conservation equations and the equation of state are solved by the Newton-Raphson method while the conservation of energy and species are solved by the successive substitution method. GFSSP is linked with two thermodynamic property programs, GASP2 and WASP3 and GASPAK4, that provide thermodynamic and thermophysical properties of selected fluids. Both programs cover a range of pressure and temperature that allows fluid properties to be evaluated for liquid, liquid-vapor (saturation), and vapor region. GASP and WASP provide properties of 12 fluids. GASPAK includes a library of 36 fluids. GFSSP has three major parts. The first part is the graphical user interface (GUI), visual thermofluid analyzer of systems and components (VTASC). VTASC allows users to create a flow circuit by a 'point and click' paradigm. It creates the GFSSP input file after the completion of the model building process. GFSSP's GUI provides the users a platform to build and run their models. It also allows post-processing of results. The network flow circuit is first built using three basic elements: boundary node, internal node, and branch.

Majumdar, A. K.↗
NASA NTRS1993

Adaptive fuzzy system for 3-D vision

An adaptive fuzzy system using the concept of the Adaptive Resonance Theory (ART) type neural network architecture and incorporating fuzzy c-means (FCM) system equations for reclassification of cluster centers was developed. The Adaptive Fuzzy Leader Clustering (AFLC) architecture is a hybrid neural-fuzzy system which learns on-line in a stable and efficient manner. The system uses a control structure similar to that found in the Adaptive Resonance Theory (ART-1) network to identify the cluster centers initially. The initial classification of an input takes place in a two stage process; a simple competitive stage and a distance metric comparison stage. The cluster prototypes are then incrementally updated by relocating the centroid positions from Fuzzy c-Means (FCM) system equations for the centroids and the membership values. The operational characteristics of AFLC and the critical parameters involved in its operation are discussed. The performance of the AFLC algorithm is presented through application of the algorithm to the Anderson Iris data, and laser-luminescent fingerprint image data. The AFLC algorithm successfully classifies features extracted from real data, discrete or continuous, indicating the potential strength of this new clustering algorithm in analyzing complex data sets. The hybrid neuro-fuzzy AFLC algorithm will enhance analysis of a number of difficult recognition and control problems involved with Tethered Satellite Systems and on-orbit space shuttle attitude controller.

Mitra, Sunanda↗
NASA NTRS2002

Dynamics of Quantum Adiabatic Evolution Algorithm for Number Partitioning

We have developed a general technique to study the dynamics of the quantum adiabatic evolution algorithm applied to random combinatorial optimization problems in the asymptotic limit of large problem size n. We use as an example the NP-complete Number Partitioning problem and map the algorithm dynamics to that of an auxiliary quantum spin glass system with the slowly varying Hamiltonian. We use a Green function method to obtain the adiabatic eigenstates and the minimum exitation gap, gmin = O(n2(sup -n/2)), corresponding to the exponential complexity of the algorithm for Number Partitioning. The key element of the analysis is the conditional energy distribution computed for the set of all spin configurations generated from a given (ancestor) configuration by simultaneous flipping of a fixed number of spins. For the problem in question this distribution is shown to depend on the ancestor spin configuration only via a certain parameter related to the energy of the configuration. As the result, the algorithm dynamics can be described in terms of one-dimensional quantum diffusion in the energy space. This effect provides a general limitation of a quantum adiabatic computation in random optimization problems. Analytical results are in agreement with the numerical simulation of the algorithm.

Smelyanskiy, Vadius↗
NASA NTRS2002

Dynamics of Quantum Adiabatic Evolution Algorithm for Number Partitioning

We have developed a general technique to study the dynamics of the quantum adiabatic evolution algorithm applied to random combinatorial optimization problems in the asymptotic limit of large problem size n. We use as an example the NP-complete Number Partitioning problem and map the algorithm dynamics to that of an auxiliary quantum spin glass system with the slowly varying Hamiltonian. We use a Green function method to obtain the adiabatic eigenstates and the minimum excitation gap. g min, = O(n 2(exp -n/2), corresponding to the exponential complexity of the algorithm for Number Partitioning. The key element of the analysis is the conditional energy distribution computed for the set of all spin configurations generated from a given (ancestor) configuration by simultaneous flipping of a fixed number of spins. For the problem in question this distribution is shown to depend on the ancestor spin configuration only via a certain parameter related to 'the energy of the configuration. As the result, the algorithm dynamics can be described in terms of one-dimensional quantum diffusion in the energy space. This effect provides a general limitation of a quantum adiabatic computation in random optimization problems. Analytical results are in agreement with the numerical simulation of the algorithm.

Smelyanskiy, V. N.↗
NASA NTRS1986

COFS 1: Beam dynamics and control technology overview

The Control of Flexible Structures (COFS) 1 Project provides the invaluable opportunity to test, validate, and measure the effectiveness of theories, structural concepts, control systems, and flight certification processes for future missions through a research program focusing on multiple issues in large flexible structures, dynamics, and controls. The COFS 1 Project consists of a series of ground and flight activities building progressively from modeling and dynamic characterization of large space systems to the more complex issues of flexible-body control. The program objectives are to: determine the degree to which theory and ground testing can predict flight performance of next-generation low-frequency structures; evaluate structural fidelity of representative next-generation large deployable precision structure; assess math modeling requirements for large lightweight complex systems on which ground test results are questionable; determine degree to which scale model analysis and tests can be correlated to full-scale performance; evaluate system identification and state estimation algorithms on complex lightweight structures in the space environment; evaluate and verify controls/structures modeling capability; evaluate control laws and control systems; and evaluate damping effects in micro-g environment.

Allen, John L.↗
NASA NTRS2001

Development of an Unstructured Mesh Code for Flows About Complete Vehicles

This report describes the research work undertaken at the Massachusetts Institute of Technology, under NASA Research Grant NAG4-157. The aim of this research is to identify effective algorithms and methodologies for the efficient and routine solution of flow simulations about complete vehicle configurations. For over ten years we have received support from NASA to develop unstructured mesh methods for Computational Fluid Dynamics. As a result of this effort a methodology based on the use of unstructured adapted meshes of tetrahedra and finite volume flow solvers has been developed. A number of gridding algorithms, flow solvers, and adaptive strategies have been proposed. The most successful algorithms developed from the basis of the unstructured mesh system FELISA. The FELISA system has been extensively for the analysis of transonic and hypersonic flows about complete vehicle configurations. The system is highly automatic and allows for the routine aerodynamic analysis of complex configurations starting from CAD data. The code has been parallelized and utilizes efficient solution algorithms. For hypersonic flows, a version of the code which incorporates real gas effects, has been produced. The FELISA system is also a component of the STARS aeroservoelastic system developed at NASA Dryden. One of the latest developments before the start of this grant was to extend the system to include viscous effects. This required the development of viscous generators, capable of generating the anisotropic grids required to represent boundary layers, and viscous flow solvers. We show some sample hypersonic viscous computations using the developed viscous generators and solvers. Although this initial results were encouraging it became apparent that in order to develop a fully functional capability for viscous flows, several advances in solution accuracy, robustness and efficiency were required. In this grant we set out to investigate some novel methodologies that could lead to the required improvements. In particular we focused on two fronts: (1) finite element methods and (2) iterative algebraic multigrid solution techniques.

Peraire, Jaime↗
NASA NTRS2001

Unstructured Mesh Methods for the Simulation of Hypersonic Flows

This report describes the research work undertaken at the Massachusetts Institute of Technology. The aim of this research is to identify effective algorithms and methodologies for the efficient and routine solution of hypersonic viscous flows about re-entry vehicles. For over ten years we have received support from NASA to develop unstructured mesh methods for Computational Fluid Dynamics. As a result of this effort a methodology based on the use, of unstructured adapted meshes of tetrahedra and finite volume flow solvers has been developed. A number of gridding algorithms flow solvers, and adaptive strategies have been proposed. The most successful algorithms developed from the basis of the unstructured mesh system FELISA. The FELISA system has been extensively for the analysis of transonic and hypersonic flows about complete vehicle configurations. The system is highly automatic and allows for the routine aerodynamic analysis of complex configurations starting from CAD data. The code has been parallelized and utilizes efficient solution algorithms. For hypersonic flows, a version of the, code which incorporates real gas effects, has been produced. One of the latest developments before the start of this grant was to extend the system to include viscous effects. This required the development of viscous generators, capable of generating the anisotropic grids required to represent boundary layers, and viscous flow solvers. In figures I and 2, we show some sample hypersonic viscous computations using the developed viscous generators and solvers. Although these initial results were encouraging, it became apparent that in order to develop a fully functional capability for viscous flows, several advances in gridding, solution accuracy, robustness and efficiency were required. As part of this research we have developed: 1) automatic meshing techniques and the corresponding computer codes have been delivered to NASA and implemented into the GridEx system, 2) a finite element algorithm for the solution of the viscous compressible flow equations which can solve flows all the way down to the incompressible limit and that can use higher order (quadratic) approximations leading to highly accurate answers, and 3) and iterative algebraic multigrid solution techniques.

Peraire, Jaime↗
NASA NTRSDate not supplied

Joint ESA-NASA Multi-Mission Algorithm and Analysis Platform (MAAP)

The scientific community is faced with a need for greatly improved data sharing, analysis, visualization and advanced collaboration based firmly on open science principles. Recent and upcoming launches of new satellite missions with more complex and voluminous data, as well as the ever more urgent need to better understand the global carbon budget and related ecological processes, provided the immediate rational for the ESA-NASA Multi-mission Algorithm and Analysis Platform (MAAP). This highly collaborative joint project of ESA and NASA established a framework between ESA and NASA to share data, science algorithms and compute resources in order to foster and accelerate scientific research conducted by ESA and NASA EO data users. Presented to the public in October 2021, the current version of MAAP provides a common cloud-based platform with computing capabilities co-located with the data, a collaborative coding and analysis environment, and a set of interoperable tools and algorithms developed to support the estimation and visualization of global above-ground biomass. Data from the Global Ecosystem Dynamics Investigation (GEDI) mission on the International Space Station and the Ice, Cloud, and Land Elevation Satellite-2 (ICESat-2) have been instrumental in the first products of MAAP including the first comprehensive map of Boreal above-ground Biomass and a current Global Biomass Harmonization Activity, but the platform is also being specifically designed to support the forthcoming ESA Biomass mission and incorporate data from the upcoming NASA-ISRO SAR (NISAR) mission. While these missions and the corresponding research which includes airborne, field, and calibration/validation data collection and analyses, provide a wealth of data and information relating to global biomass estimation, they also present data storing, processing and sharing challenges. The NISAR mission alone will produce about 80TB/day. These large data volumes present a challenge that would otherwise place accessibility limits on the scientific community and impact scientific progress. Other challenges being addressed by MAAP include: 1) Enabling researchers to easily discover, process, visualize and analyze large volumes of data from both agencies; 2) Providing a wide variety of data in the same coordinate reference frame to enable comparison, analysis, data evaluation, and data generation; 3) Providing a version-controlled science algorithm development environment that supports tools, co-located data and processing resources; and 4) Addressing intellectual property and sharing challenges related to collaborative algorithm development and sharing of data and algorithms. MAAP products can be explored on the MAAP Dashboard at https://earthdata.nasa.gov/maap-biomass or the joint platform entrance at scimaap.net. MAAP also can be accessed through individual NASA (https://maap-project.org) and ESA (https://esa-maap.org/) landing pages.

cloud computing↗
NASA NTRS2019

Cluster Analysis of IRIS Spectroscopic Line Profiles and SDO/AIA EUV Emission in Observations and RMHD Simulations of the Solar Atmosphere

Spatially-resolved observations from the IRIS and SDO/AIA satellites, especially when coupled with realistic 3D RMHD simulations, are a powerful tool for analysis of processes in the solar chromosphere, transition region, and corona. However, the complexity of the data makes understanding the observations and modeling results difficult. In this work, we apply unsupervised clustering algorithms for analysis of observational and synthetic chromospheric Mg II h&k 2796Å&2803Å and transition region C II 1334Å&1335Å line profiles observed by IRIS, and extreme ultraviolet (EUV) emission observed by SDO/AIA, for various types of problems. The synthetic line profiles are computed for simulations of the quiescent solar atmosphere (using the StellarBox and RH1.5 codes). The K-Means clustering algorithm is applied, and the selection of an optimal number of clusters is supported by the average silhouette width technique. We discuss applications of the line profile clustering method to 1) visualization of computational and observational spectroscopic imaging data; 2) understanding of evolutionary trends and behavior patterns of quiet Sun emission and during solar flares; and 3) recognition of heating events and shock waves.

Sadykov, Viacheslav↗
NASA NTRS1992

A VLSI architecture for simplified arithmetic Fourier transform algorithm

The arithmetic Fourier transform (AFT) is a number-theoretic approach to Fourier analysis which has been shown to perform competitively with the classical FFT in terms of accuracy, complexity, and speed. Theorems developed in a previous paper for the AFT algorithm are used here to derive the original AFT algorithm which Bruns found in 1903. This is shown to yield an algorithm of less complexity and of improved performance over certain recent AFT algorithms. A VLSI architecture is suggested for this simplified AFT algorithm. This architecture uses a butterfly structure which reduces the number of additions by 25 percent of that used in the direct method.

Reed, Irving S.↗
NASA NTRS2009

Discrete Fourier Transform Analysis in a Complex Vector Space

Alternative computational strategies for the Discrete Fourier Transform (DFT) have been developed using analysis of geometric manifolds. This approach provides a general framework for performing DFT calculations, and suggests a more efficient implementation of the DFT for applications using iterative transform methods, particularly phase retrieval. The DFT can thus be implemented using fewer operations when compared to the usual DFT counterpart. The software decreases the run time of the DFT in certain applications such as phase retrieval that iteratively call the DFT function. The algorithm exploits a special computational approach based on analysis of the DFT as a transformation in a complex vector space. As such, this approach has the potential to realize a DFT computation that approaches N operations versus Nlog(N) operations for the equivalent Fast Fourier Transform (FFT) calculation.

Dean, Bruce H.↗