Finite Fourier Series Expansion Research Articles

Integer factorization and finite Fourier series expansion

Given two integers a and k > 0, the number of factorizations of a (mod k) is the number of ordered pairs (s, t) ∈ {0, 1, . . . , k − 1}2 satisfying s · t ≡ a (mod k). This number is known to be expressible by a formula involving the greatest common divisor function. Motivated by such a formula, we derive several formulae counting the number of factorizations of a (mod k) subject to certain other natural restrictions. Some of these formulae are obtained as consequences of finite Fourier series expansions of the greatest common divisor function, whereas some are shown to be closely connected with the notion of unitary divisors.

Adaptive Transmit Waveform Design Using Multitone Sinusoidal Frequency Modulation

This paper presents an adaptive waveform design method using Multi-Tone Sinusoidal Frequency Modulation (MTSFM). The MTSFM waveform's modulation function is represented as a finite Fourier series expansion. The Fourier coefficients are utilized as a discrete set of design parameters that may be modified to adapt the waveform's properties. The MTSFM's design parameters are adjusted to shape the spectrum, Auto-Correlation Function (ACF), and Ambiguity Function (AF) shapes of the waveform. The MTSFM waveform model naturally possesses the constant envelope and spectrally compact waveforms that make it well suited for transmission on practical radar/sonar transmitters which utilize high power amplifiers. The MTSFM has an exact mathematical definition for its time-series using Generalized Bessel Functions which allow for deriving closed-form analytical expressions for its spectrum, AF, and ACF. These expressions allow for establishing well-defined optimization problems that finely tune the MTSFM's properties. This adaptive waveform design model is demonstrated by optimizing MTSFM waveforms that initially possess a "thumbtack-like" AF shape. The resulting optimized designs possess substantially improved sidelobe levels over specified regions in the range-Doppler plane without increasing the Time-Bandwidth Product (TBP) that the initialized waveforms possessed. Simulations additionally demonstrate that the optimized thumbtack-like MTSFM waveforms are competitive with thumbtack-like phase-coded waveforms derived from design algorithms available in the published literature.

Construction of weighted crystallographic orientations capturing a given orientation density function

To be useful in numerical simulations of e.g. deformation processes, EBSD datasets of crystallographic orientations have to be downsized by several orders of magnitude yet preserving the orientation density function approximately. The objective is either to preserve the overall shape of the initially kernel estimated orientation density function and in particular its non-negativity, or to preserve the unbiased estimates of the first Fourier coefficients up to a given finite order. Methods are presented how to construct a much smaller set of weighted orientations such that their kernel density estimate approximates the initial estimate. To preserve its overall shape the de la Vallee Poussin kernel is applied as it is the only known non-negative kernel with a finite Fourier series expansion avoiding truncation errors. If the first Fourier coefficients are to be preserved the Dirichlet kernel applies as it is the only kernel providing unbiased estimates of the Fourier coefficients up to any given finite order. The weights are determined numerically by resolving a least squares or a maximum likelihood problem. Due to the linearity of kernel density estimation and the Fourier transform the approaches in spatial and spectral domain are related to each other in a unique complementary way. For an exemplary practical application we use a large EBSD dataset of about 80.000 orientations from a recrystallized low alloyed Zirconium sheet. Our methods reduce the size of the dataset by about \(99.75\,\%\) to the order of 200 weighted orientations supporting a secondary approximate distribution with a volume portion of crystallites oriented differently than initially of less than \(10\,\%\).

Fast and accurate thermal modeling and simulation of manycore processors and workloads

Today's microprocessors require careful analysis of their thermal behavior both at design time as well as at runtime. However, accurate prediction of thermal behavior is possible only through simulations due to the complexity and the high dynamicity of their operation. Unfortunately, accurate simulations of such complex systems are computationally intensive, therefore time consuming, while on the other hand, simplified models often cause mispredictions leading to overdesign and lowered performance. In our work, we propose a new discrete-time method of assessment of the run-time temperature of a processor based on an approximation of its instructions-per-cycle (IPC) by a finite Fourier series expansion. Our method is AC-based, and shows significant increase of accuracy in comparison to well-known DC (average)-based models. Furthermore, we provide a new Dynamic Voltage and Frequency Scaling (DVFS) model based on our estimations.

Novel adaptive learning control of linear systems with completely unknown time delays

A novel output-feedback adaptive learning control approach is developed for a class of linear time-delay systems. Three kinds of uncertainties: time delays, number of time delays, and system parameters are all assumed to be completely unknown, which is different from the previous work. The design procedure includes two steps. First, according to the given periodic desired reference output and the allowed bound of tracking error, a suitable finite Fourier series expansion (FSE) is chosen as a practical reference output to be tracked. Second, by expressing the delayed practical reference output as a known time-varying vector multiplied by an unknown constant vector, we combine three kinds of uncertainties into an unknown constant vector and then estimate the vector by designing an adaptive law. By constructing a Lyapunov-Krasovskii functional, it is proved that the system output can asymptotically track the practical reference signal. An example is provided to illustrate the effectiveness of the control scheme developed in this paper.

OPTIMAL INPUT DESIGN FOR IDENTIFICATION OF CONTINUOUS-TIME SYSTEMS

In this paper, we present an optimal input design method for the identification of single input single output continuous-time transfer functions. In the proposed algorithm, we show that this optimal input design problem can be rewritten as that for discrete-time systems proposed by Antoulas et al.(Antoulas and Anderson, 1999; Antoulas, 1997; Antoulas and Astolfi, 1998), if the input is approximated by the finite Fourier series expansion. The derivative signals of input/output are required to design the optimal input, and filters are used to obtain the derivative signals. We also numerically consider the effect of the filters to the optimality. Through numerical examples, its effectiveness is verified.

Natural vibration analysis of coaxial shells coupled with fluid

Investigated in this study are the natural vibration characteristics of the coaxial cylindrical shells coupled with a fluid. Theoretical method is developed to find the natural frequencies of the shell using the finite Fourier series expansion, and their results are compared with those of finite element method to verify the validation of the method developed. The effect of the fluid-filled annulus and the boundary conditions on the modal characteristics of the coaxial shells is investigated using a finite element modeling.

FREE VIBRATION OF A CIRCULAR CYLINDRICAL SHELL FILLED WITH BOUNDED COMPRESSIBLE FLUID

This paper presents an analytical method for evaluating the linear free vibration of a circular cylindrical shell filled with bounded compressible fluid. The analytical method was developed by means of the finite Fourier series expansion method. The motion of compressible fluid coupled with the shell was determined by means of the linearized velocity potential flow theory. To clarify the validity of the analytical method, the coupled natural frequencies of a water-filled circular cylindrical shell with a clamped–clamped boundary condition were obtained by the analytical method and finite element analysis. Excellent agreement on the natural frequencies of the liquid-coupled shell structure was found. The results show that the natural frequencies decrease with an increase in fluid compressibility. However, the compressibility effect on the lower circumferential mode frequencies was found to be relatively significant. Additionally, it was found that the rigid bounding of fluid at the ends of the shell decreases the natural frequencies for the lower circumferential modes when compared with the sloshing fluid case.

Error analysis of the finite‐strip method for parabolic equations

AbstractThe finite‐strip method (FSM) is a hybrid technique which combines spectral and finite‐element methods. Finite‐element approximations are made for each mode of a finite Fourier series expansion. The Galerkin formulated method is set apart from other weighted‐residual techniques by the selection of two types of basis functions, a piecewise linear interpolating function and a trigonometric function. The efficiency of the FSM is due in part to the orthogonality of the complex exponential basis: The linear system which results from the weak formulation is decoupled into several smaller systems, each of which may be solved independently. An error analysis for the FSM applied to time‐dependent, parabolic partial differential equations indicates the numerical solution error is O(h2 + M−r). M represents the Fourier truncation mode number and h represents the finite‐element grid mesh. The exponent r ≥ 2 increases with the exact solution smoothness in the respective dimension. This error estimate is verified computationally. Extending the result to the finite‐layer method, where a two‐dimensional trigonometric basis is used, the numerical solution error is O(h2 + M−r + N−q). The N and q represent the trucation mode number and degree of exact solution smoothness in the additional dimension. © 1993 John Wiley & Sons, Inc.

A finite surface element model for two concentric nearly circular rings in partial contact

A numerical model is derived for the plane-stress elastic deformation of two concentric hollow cylinders with acircular contact surfaces. Given the detailed undeformed section geometry, the model computes the circumferential distributions of interfacial pressure and gap, and the distortion of the two free surfaces. The interface is partitioned into axial strip elements with radial displacement nodes at their boundaries and a piecewise linear pressure distribution from node to node. Influence coefficients linearly relating nodal displacements to nodal pressures are computed using a finite Fourier series expansion of the pressure distribution in conjuction with exact expressions for the deformation of a sinusoidally loaded thick ring. The iterative solution of the nonlinear system of node equations entails computing the currently nonzero nodal pressures by matrix inversion and then updating the contact/gap configuration. The model is demonstrated for a 400 node, multiple-contact interface between a machined shaft and sleeve with measured acircularity profiles.

Meridional Rib-Stiffened Shells

A semianalytical technique is proposed for the analysis of a meridional rib stiffened shell of revolution which accounts for the discrete stiffness of the rib in a rational manner. The method uses discrete field concepts, finite Fourier series expansion of the dependent variables in the circumferential direction and Galerkin FEM in the meridional direction to analyse such a class of shells. In the process the two-dimensional physical problem is reduced to a one-dimensional mathematical model, thereby increasing computation efficiency. This scheme can be employed to study the size and effect of the ribs on the shell behavior more efficiently than a full-blown FEM. Through the use of a numerical example, the effect of the ribs in a typical hyperbolic shell of revolution is illustrated. Shell deflection and stress resultants are compared for both stiffened and unstiffened shell subjected to wind pressure. The analysis is applicable to such linear elastic models where small deflection theory is valid.

Interferometric Spectroscopy in the Far Infrared

Michelson’s visibility technique for obtaining spectral distributions interferometrically in the visible region is compared with the new far-infrared interferogram method. The scanning function for the lamellar-grating type interferometric modulator that is used for the far-infrared region has been derived. Its theoretical resolution limit is compared with the spectrum estimate obtained from experimental results with a microwave source. Discrepancies between the derived spectrum estimate, obtained by a finite Fourier series expansion, and the true spectrum are discussed. A simple technique, due to Rayleigh, is invoked to avoid distortion of the spectrum due to the so-called Gibbs phenomenon. A spectrum of H2O extending from 8 to 60 cm−1 (λ = 1250 to 167 μ) illustrates the application of Rayleigh’s method.