Multidimensional Discrete Fourier Research Articles

Structural Damage Identification Based on Preprocessing Samples Mixed Training of Multi-Channel and Multi-Source Dynamic Response Data

Previous studies have shown that sampling processing and identification methods for neural networks damage identification in multi-channel and multi-source structural dynamic response data exhibit diversity, and the robustness and generalization issues of the model have not been effectively solved. This paper proposes a sample preprocessing technique and mixed training method suitable for multi-channel and multi-source dynamic response data to optimize the current structural damage identification methods based on neural networks. Through multi-dimensional discrete autocorrelation processing and Fourier transform, the preprocessed datasets from multiple sensors with multi-channel can access multi-channel CNN. Furthermore, the measured datasets from the actual structures are mixed with numerical simulation datasets before being used for neural network model training to address the model calibration and the imbalanced sample size under each classification label. The results of extensive model experiments and finite element verification of specimens show that the method performs outstandingly in detecting and identifying damage to simply supported beam structures using multi-channel CNN. The neural network model trained with preprocessed samples exhibits excellent robustness. The model using the mixed training method still performs well in identifying the accuracy of damage location and degree in simply supported beam structures after changing beam section and initial excitation. The method has certain generalization ability in detecting unknown damage in simply supported beam structures.

A New Harmonic Balance Approach Using Multidimensional Time

Abstract Over the past years, nonlinear frequency-domain methods have become a state-of-the-art technique for the numerical simulation of unsteady flow fields within multistage turbomachinery as they are capable of fully exploiting the given spatial and temporal periodicities, as well as modeling flow nonlinearities in a computationally efficient manner. Despite this success, it still remains a significant challenge to capture nonlinear interaction effects within the context of configurations with multiple fundamental frequencies. If all frequencies are integer multiples of a common fundamental frequency, the interval spanned by the sampling points typically resolves the period of the common base frequency. For configurations in which the common frequency is very low in relation to the frequencies of primary interest, many sampling points are required to resolve the highest harmonic of the common fundamental frequency and the method becomes inefficient. In addition, when a problem can no longer be described by harmonic perturbations that are integer multiples of one fundamental frequency, as it may occur in two-shaft configurations or when simulating the nonlinear interaction in the context of forced response or flutter, then the standard discrete Fourier transform is no longer suitable, and the basic harmonic balance (HB) method requires extension. One possible approach is to use almost periodic Fourier transforms with equidistant or nonequidistant time sampling. However, the definition of suitable sampling points that lead to well-conditioned Fourier transform matrices and small aliasing errors is an intricate issue and far from straightforward. To overcome the issues regarding multifrequency problems described earlier, a new harmonic balance approach based on multidimensional Fourier transforms in time is presented. The basic idea of the approach is that instead of defining common sampling points in a common time period, separate time domains, one for each base frequency, are spanned, and the sampling points are computed equidistantly within each base frequency's period. Since the sampling domain is now extended to a multidimensional time-domain, all time instant combinations covering the whole multidimensional domain are computed as the Cartesian product of the sampling points on the axes. In a similar fashion, the frequency-domain is extended to a multidimensional frequency-domain by the Cartesian product of the harmonics of each base frequency, so that every point defined by the Cartesian product is an integer linear combination of the occurring frequencies. In this way, the proposed method is capable of fully integrating the nonlinear coupling effects between higher harmonics of different fundamental frequencies by using multidimensional discrete Fourier transforms within the harmonic balance solution procedure. The aim of this paper is to introduce the multidimensional harmonic balance method in detail and demonstrate the capability of the approach to simultaneously capture unsteady disturbances with arbitrary excitation frequencies. Therefore, the well-established aeroelasticity testcase standard configuration 10 in the presence of an artificial inflow disturbance, which mimics an upstream blade wake, is investigated. The crucial aspect of the proposed testcase is that a small ratio of the frequency of the inflow disturbance and the blades vibration frequency is chosen. To demonstrate the advantages of the newly proposed multidimensional harmonic balance approach, the results are compared to unsteady simulations in the time-domain and to state-of-the-art frequency-domain methods based on one-dimensional discrete Fourier transforms.

Integer multiplication in time $O(n\mathrm{log}\, n)$

We present an algorithm that computes the product of two $n$-bit integers in $O(n \mathrm{log}\, n)$ bit operations, thus confirming a conjecture of Schönhage and Strassen from 1971. Our complexity analysis takes place in the multitape Turing machine model, with integers encoded in the usual binary representation. Central to the new algorithm is a novel ``Gaussian resampling" technique that enables us to reduce the integer multiplication problem to a collection of multidimensional discrete Fourier transforms over the complex numbers, whose dimensions are all powers of two. These transforms may then be evaluated rapidly by means of Nussbaumer's fast polynomial transforms.

Fast algorithms for the multi-dimensional Jacobi polynomial transform

We use the well-known observation that the solutions of Jacobi's differential equation can be represented via the non-oscillatory phase and amplitude functions to develop a fast algorithm for computing multi-dimensional Jacobi polynomial transforms. More explicitly, it follows from this observation that the matrix corresponding to the discrete Jacobi transform is the Hadamard product of a numerically low-rank matrix and a multi-dimensional discrete Fourier transform (DFT) matrix. The application of the Hadamard product can be carried out via r d fast Fourier transforms (FFTs), where r = O ( log ⁡ n log ⁡ log ⁡ n ) and d is the dimension, resulting in a nearly optimal algorithm to compute the multidimensional Jacobi polynomial transform.

A Flexible Framework for Multidimensional DFTs

Multidimensional discrete Fourier transforms (DFTs) are typically decomposed into multiple one-dimensional (1D) transforms. Hence, parallel implementations of any multidimentional DFT focus on para...

Radix‐4 3 based two‐dimensional FFT architecture with efficient data reordering scheme

Multi-dimensional Discrete Fourier Transforms (DFTs) play an important role in signal and image processing applications. Image reconstruction is a key component in signal processing applications like medical imaging, computer vision, face recognition etc. Two dimensional fast Fourier Transform (2D FFT) and Inverse FFT plays vital role in reconstruction. In this paper we present a fast 64 × 64 point 2D FFT architecture based on radix-43 algorithm using a parallel unrolled radix-4 3 FFT as the basic block. Our radix-4 3 architecture is a memory optimized parallel architecture which computes 64-point FFT, with least execution time. Proposed architecture produces reordered output of both 64-point one dimensional (1D) FFT and 64 × 64 point 2D FFT, without using any additional hardware for reordering. The proposed architecture has been implemented in UMC 40nm CMOS technology with clock frequency of 500 MHz and area of 0.841 mm 2 . The power consumption of proposed architecture is 358 mW at 500 MHz. Energy efficiency (FFTs computed per unit of energy) is 341 points/Joule. Computation time of 64 × 64 point FFT is 8.19 μs. ASIC implementation results shows better performance of proposed work in terms of computation time when compared with state-of-art implementation. Proposed architecture has also been implemented in Virtex-7 FPGA which gives comparable area.

Acoustic scattering for 3D multi-directional periodic structures using the boundary element method.

An efficient boundary element formulation is proposed to solve three-dimensional exterior acoustic scattering problems with multi-directional periodicity. The multi-directional periodic acoustic problem is represented as a multilevel block Toeplitz matrix. By exploiting the Toeplitz structure, the computational time and storage requirements to construct and to solve the linear system of equations arising from the boundary element formulation are significantly reduced. The generalized minimal residual method is implemented to solve the linear system of equations. To efficiently calculate the matrix-vector product in the iterative algorithm, the original matrix is embedded into a multilevel block circulant matrix. A multi-dimensional discrete Fourier transform is then employed to accelerate the matrix-vector product. The proposed approach is applicable to a periodic acoustic problem for any arbitrary shape of the structure in both full space and half space. Two case studies involving sonic crystal barriers are presented. In the first case study, a sonic crystal barrier comprising rigid cylindrical scatterers is modeled. To demonstrate the effectiveness of the proposed technique, periodicity in one, two, or three directions is examined. In the second case study, the acoustic performance of a sonic crystal barrier with locally resonant C-shaped scatterers is studied.

Multiple Beams From Planar Arrays

The paper describes a general theory of Multiple Beams from Planar Arrays establishing necessary and sufficient conditions for admitting a realizable lossless beamforming network. The developed theory applies concepts deriving from the Whittaker-Kotelnikov-Shannon (WKS) sampling theory, and from its multidimensional extension due to Miyakawa- Petersen- Middleton (MPM), to the problem of generating simultaneous multiple beams from planar phased arrays with generic periodicity and layout. It is proven that the multiple beams positions must satisfy certain reciprocal lattice conditions and that the phase steering vectors of the lossless beamforming network must correspond to the multi-dimensional discrete Fourier transform (MD-DFT). The nontrivial Hermitian orthogonality of the steering vectors, as well as of the multiple beams, is also proven.

Proof of unitarity of multidimensional discrete Fourier transform

The multidimensional discrete Fourier transform (MD-DFT) plays an important role in a growing number of signal processing applications. The fundamentals of its applicability as a unitary transform between discrete periodic sequences defined on multidimensional lattices stand on the Hermitian orthogonality of the vectors defining the MD-DFT matrix. A proof of the consistency of the MD-DFT formulation was first provided by Bernardini and Manduchi in 1994 using the Smith normal form theorem of integer matrices. In this reported work, a new proof is provided based on the nullity of the cardinal function on the nonzero cardinal points.

Conway matrices related to a non-transitive head-or-tail game with a q-sided die and their Hamming weight-spectra via DFT and the MacWilliams duality formula

We consider the well-known problem of string overlapping in connection with the so-called Penney Ante game with a q-sided die. We propose a simple and effective algorithm for calculation and visualization of Conway matrices Cn and the corresponding matrices Pn that give probabilities that player (1) will win for the chosen words w1 and w2 via the standard technique of the multidimensional discrete Fourier transform (DFT). We consider also the Hamming weight-spectra of these matrices and propose an algorithm for their computation and visualization.

Parallel FFT-based Poisson solver for isolated three-dimensional systems

We describe an implementation to solve Poissonʼs equation for an isolated system on a unigrid mesh using FFTs. The method solves the equation globally on mesh blocks distributed across multiple processes on a distributed-memory parallel computer. Test results to demonstrate the convergence and scaling properties of the implementation are presented. The solver is offered to interested users as the library PSPFFT. Program summaryProgram title: PSPFFTCatalogue identifier: AEJK_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEJK_v1_0.htmlProgram obtainable from: CPC Program Library, Queenʼs University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 110 243No. of bytes in distributed program, including test data, etc.: 16 332 181Distribution format: tar.gzProgramming language: Fortran 95Computer: Any architecture with a Fortran 95 compiler, distributed memory clustersOperating system: Linux, UnixHas the code been vectorized or parallelized?: Yes, using MPI. An arbitrary number of processors may be used (subject to some constraints). The program has been tested on from 1 up to ∼ 13 000 processors. RAM: Depends on the problem size, approximately 170 MBytes for 483 cells per process.Classification: 4.3, 6.5External routines: MPI (http://www.mcs.anl.gov/mpi/), FFTW (http://www.fftw.org), Silo (https://wci.llnl.gov/codes/silo/) (only necessary for running test problem).Nature of problem: Solving Poissonʼs equation globally on unigrid mesh distributed across multiple processes on distributed memory system.Solution method: Numerical solution using multidimensional discrete Fourier Transform in a parallel Fortran 95 code.Unusual features: This code can be compiled as a library to be readily linked and used as a blackbox Poisson solver with other codes.Running time: Depends on the size of the problem, but typically less than 1 second per solve.

Multidimensional DFT IP Generator for FPGA Platforms

Multidimensional (MD) discrete Fourier transform (DFT) is a key kernel algorithm in many signal processing applications. In this paper we describe an MD-DFT intellectual property (IP) generator and a bandwidth-efficient MD DFT IP for high performance implementations of 2-D and 3-D DFT on field-programmable gate array (FPGA) platforms. The proposed architecture is generated automatically and is based on a decomposition algorithm that takes into account FPGA resources and the characteristics of off-chip memory access, namely, the burst access pattern of the synchronous dynamic RAM (SDRAM). The IP generator has been integrated into an in-house FPGA development platform, AlgoFLEX, for easy verification and fast integration. The corresponding 2-D and 3-D DFT architectures have been ported onto the BEE3 board and their performance measured and analyzed. The results shows that the architecture can maintain the maximum memory bandwidth throughout the whole procedure while avoiding matrix transpose operations used in most other MD DFT implementations. To further enhance the performance, the proposed architecture is being ported onto the newly released Xilinx ML605 board. The simulation results show that 2 K × 2 K images with complex 64-bit precision can be processed in less than 27 ms.

Cluster expansion and the configurational theory of alloys

A rigorous mathematical foundation for the cluster-expansion method is presented. It is shown that the cluster basis developed by Sanchez et al. [Physica A 128, 334 (1984)] is a multidimensional discrete Fourier transform while the general formalism of Sanchez [Phys. Rev. B 48, 14013 (1993)] corresponds to a multidimensional discrete wavelet transform. For functions that depend nonlinearly on the concentration, it is shown that the cluster basis corresponding to a multidimensional discrete Fourier transform does not converge, as it is usually assumed, to a finite cluster expansion or to an Ising-type model representation of the energy of formation of alloys. The multidimensional wavelet transform, based on a variable basis cluster expansion, is shown to provide a satisfactory solution to the deficiencies of the discrete Fourier-transform approach. Several examples aimed at illustrating the main findings and conclusions of this work are given.

High resolution 4-D spectroscopy with sparse concentric shell sampling and FFT-CLEAN

Recent efforts to reduce the measurement time for multidimensional NMR experiments have fostered the development of a variety of new procedures for sampling and data processing. We recently described concentric ring sampling for 3-D NMR experiments, which is superior to radial sampling as input for processing by a multidimensional discrete Fourier transform. Here, we report the extension of this approach to 4-D spectroscopy as Randomized Concentric Shell Sampling (RCSS), where sampling points for the indirect dimensions are positioned on concentric shells, and where random rotations in the angular space are used to avoid coherent artifacts. With simulations, we show that RCSS produces a very low level of artifacts, even with a very limited number of sampling points. The RCSS sampling patterns can be adapted to fine rectangular grids to permit use of the Fast Fourier Transform in data processing, without an apparent increase in the artifact level. These artifacts can be further reduced to the noise level using the iterative CLEAN algorithm developed in radioastronomy. We demonstrate these methods on the high resolution 4-D HCCH-TOCSY spectrum of protein G's B1 domain, using only 1.2% of the sampling that would be needed conventionally for this resolution. The use of a multidimensional FFT instead of the slow DFT for initial data processing and for subsequent CLEAN significantly reduces the calculation time, yielding an artifact level that is on par with the level of the true spectral noise.

Determining the Weights of A Fourier Series Neural Network on the Basis of the Multidimensional Discrete Fourier Transform

Determining the Weights of A Fourier Series Neural Network on the Basis of the Multidimensional Discrete Fourier TransformThis paper presents a method for training a Fourier series neural network on the basis of the multidimensional discrete Fourier transform. The proposed method is characterized by low computational complexity. The article shows how the method can be used for modelling dynamic systems.

Fast algorithms for a multidimensional hypercomplex discrete Fourier transform with radix-3 decomposition

Combined algorithms for the multidimensional hypercomplex discrete Fourier transform (HDFT) of a real signal with data representation in the Hamilton-Eisenstein generalized codes are synthesized. The complexity of arithmetic operations in a commutative-associative hypercomplex algebra and its representation in generalized codes are obtained. It is shown that there exist only two essentially different commutative-associative hypercomplex algebras: the direct sums of real or complex algebras. The computational complexity of the algorithm synthesized is estimated.

Parallel algorithms for a hypercomplex discrete fourier transform

Methods for the parallel computation of a multidimensional hypercomplex discrete Fourier transform (HDFT) are considered. The basic idea consists in the application of the properties of the hypercomplex algebra in which this transform is performed. Additional possibilities for increasing the efficiency of the algorithm are provided by the natural parallelism of the multidimensional Cooley-Tukey scheme.

Vector coding algorithms for multidimensional discrete Fourier transform

A new fast algorithm is presented for the multidimensional discrete Fourier transform (DFT). This algorithm is derived using an interesting technique called “vector coding” (VC), and we call it the vector-coding fast Fourier transform (VC-FFT) algorithm. Since the VC-FFT is an extension of the Cooley–Tukey algorithm from 1-D to multidimensional form, the structure of the program is as simple as the Cooley–Tukey fast Fourier transform (FFT). The new algorithm significantly reduces the number of multiplications and recursive stages. The VC-FFT therefore comprehensively reduces the complexity of the algorithm as compared with other current multidimensional DFT algorithms.

Highly Parallel Computations: Algorithms and Applications

Section I: General-purpose parallel computing and metacomputing Cluster computing with message passing interface Scheduling UET grids with unit communication time delays into unbounded/fixed number of processors Metacomputing: technology and applications. Section II: Special parallel architectures and systolic application case studies Architectural design of array processors for multi-dimensional discrete fourier transform Massive parallel processing for matrix multiplication: a systolic approach Hexagonal systolic arrays for matrix multiplication Systolic block-Jacobi SVD algorithm for processor meshes An introduction to processor-time-optimal systolic arrays Hyper-systolic algorithms with applications in linear algebra and molecular dynamics Designing processor-time optimal systolic configurations. Section III: Neural algorithms and applications Word recognition system using neural networks Self-organised evolutionary neural networks: Algorithms and applications.

Multidimensional systolic arrays for the implementation of discrete Fourier transforms

This paper presents an efficient technique for using a multidimensional systolic array to perform the multidimensional discrete Fourier transform (DFT). Extensions of the multidimensional systolic array suitable for fast Fourier transform (FFT) computations such as the prime-factor computation or the 2/sup n/-point decomposed computation of the one-dimensional (1-D) discrete Fourier transform are also presented. The essence of our technique is to combine two distinct types of semisystolic arrays into one truly systolic array. The resulting systolic array accepts streams of input data (i.e., it does not require any preloading), and it produces output data streams at the boundary of the array. No networks for intermediate spectrum transposition between constituent transforms are required. The systolic array has regular processing elements that contain a complex multiplier accumulator and a few registers and multiplexers. Simple and regular connections are required between the PEs.