Beam-Steering in Switched 4D Arrays Based on the Discrete Walsh Transform

The continuous Fourier transformation of I(h3) (h3 = 0,1) is proposed as a method for the analysis of the stacking sequence in close-packed structures. By this method, the diffraction pattern is directly translated into autocorrelation parameters P0n, P+n, P−n of the stacking sequence. It applies to periodic stacking sequences, for which the Fourier coefficients can be analysed in terms of the Zdhanov symbol, as well as to faulted stacking sequences, where the Fourier coefficients can be analysed in terms of faulting probabilities. In the work reported, the method proposed is applied to powder diffraction data from β′1 Cu–Al martensite. The stacking sequence in the martensite investigated can be dealt with as statistical, with Reichweite s = 3, and faulting parameters α = 0.18, β = 0.54. There is reasonably good agreement between the experimentally measured Fourier coefficients and the Fourier coefficients calculated from the above mentioned α and β values.

Application of the least-squares inversion method: Fourier series versus waveform inversion

A novel representation of a quasi-periodic modified von Mangoldt function L(n) on prime numbers and its decomposition into Fourier series has been investigated. We focus on some particular quantities characterizing the modified von Mangoldt function. The results indicate that prime number progression can be decomposed into periodic sequences. The main approach is to decompose it into sin or cosine function. Basically, it is applied to extract hidden periodicities in seemingly quasi periodic prime function. Numerical evidences were provided to confirm the periodic distribution of primes.

and that the corresponding limit at x = e(O), where 0 is not congruent (mod 1) to an integral multiple of a, is zero. P. Sziisz has pointed out that for the proof of (1) and (2) Davenport's conditions can be reduced to the following: (a) limN_. EInIN ane(nt) exist for each real t, and (b) I E jnI,N a.e(nt) I <K < co uniformly in N and t. That is, the convergence of Ean and condition (6) in Davenport's paper can be replaced by (a). The purpose of this note is to generalize the above results in a couple of directions. It is to be noticed that the power series coefficients in (2), A,=g(va+y), constitute in some general sense an almost periodic sequence with the Fourier expansion

We consider a class of generalized trigonometric functions (distributions) given by the divergent trigonometric series; coefficients of these series have a certain order of growth. These functions have no values in usual sense and manifest themselves only as convolution with the test functions. Classes of test functions are formed by even periodical functions that are either defined by chosen Fourier coefficients or have certain differential properties. Therefore generalized trigonometric functions are analogous to well known Dirac -functions. The operations of differentiation and integration of these generalized trigonometric functions are considered. The class of generalized trigonometric functions of zero order with coefficients forming -periodical sequences is investigated in details. Such functions under certain conditions that are received in this paper have interpolating properties. The results of calculations for the test example are given; these results are well correlated to those predicted by theory.

Hitherto, most clothings have been empirically, designed and produced by garment manufactures. This study aims to produce clothings by a scientiffic method “deductive reasoning” considering what clothings should be from the work design's point of view. For this purpose, the fundamental terms expressing visual impressions have been investigated and free curves have been analyzed by one group of the subjects and attempts have been made to generate clothing figures for women on the basis of these curves.Another group of the subjects judged whether the figures generated above can be recognized as clothings. Fourier's coefficients were calculated to apply the Fourier's series equation to the figures recognized as clothings. Next, periodogram were figured on the basis of those Fourier's coefficients. Then, it should bc possible to reappear the clothing figures generated above by using dominat periodic components in that periodogram.The computeaided models thus obained include these coefficients as characteristic values with the equation of Fourier's series. Thus, the method has been developed to produce any clothing figure expressing visual impressions.

Computations of Fourier coefficients and related integrals of the associated Legendre functions with a new method along with their application to spherical harmonics analysis and synthesis are presented. The method incorporates a stable three-step recursion equation that can be processed separately for each colatitudinal Fourier wavenumber. Recursion equations for the zonal and sectorial modes are derived in explicit single-term formulas to provide accurate initial condition. Stable computations of the Fourier coefficients as well as the integrals needed for the projection of Legendre functions are demonstrated for the ultra-high degree of 10,800 corresponding to the resolution of one arcmin. Fourier coefficients, computed in double precision, are found to be accurate to 15 significant digits, indicating that the normalized error is close to the machine round-off error. The orthonormality, evaluated with Fourier coefficients and related integrals, is shown to be accurate to O(10−15) for degrees and orders up to 10,800. The Legendre function of degree 10,800 and order 5,000, synthesized from Fourier coefficients, is accurate to the machine round-off error. Further extension of the method to even higher degrees seems to be realizable without significant deterioration of accuracy. The Fourier series is applied to the projection of Legendre functions to the high-resolution global relief data of the National Geophysical Data Center of the National Oceanic and Atmospheric Administration, and the spherical harmonic degree variance (power spectrum) of global relief data is discussed.

A fundamental challenge in learning is the presence of nonlinear redundancies and dependencies in the data. To address this, we propose a Fourier-based approach to characterize feature redundancies, in unsupervised learning, and feature-label dependencies, in the supervised variant of the problem. We first develop a novel Fourier expansion for functions (more generally stochastic mappings) of correlated binary random variables. This is a generalization of the standard Fourier expansion on the Boolean cube beyond product probability spaces. As an important application of this analysis, we investigate learning with feature subset selection. In the unsupervised variant of this problem, we characterize feature redundancies via the Shannon entropy and group the features into sufficiently informative and redundant. Then, we make a connection to the proposed Fourier expansion and derive an upper bound on the joint entropy. Based on that, we propose a measure to quantify feature redundancies and present an unsupervised learning algorithm. We test our method on various real-world and synthetic datasets and demonstrate improvements on conventional unsupervised feature selection techniques. Then, we investigate the supervised feature subset selection and reformulate it in the Fourier domain. Bridging the Bayesian error rate with the Fourier coefficients, we demonstrate that the Fourier expansion provides a powerful tool to characterize nonlinear feature-label dependencies. Further, we introduce a computationally efficient measure for selecting relevant features. Via a theoretical analysis, we show that our proposed measure finds provably <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">asymptotically optimal</i> feature subsets. Lastly, we present an algorithm based on this measure and via numerical experiments demonstrate its improvements on various supervised feature selection algorithms.

The volume velocity function of the glottis can be computed from a pitch-synchronous spectral analysis of the speech pressure wave. The method consists of computing the Fourier coefficients of a pitch period, locating the formant frequencies, removing the formant poles from the spectrum, and regenerating the glottal waveform from the residual by Fourier synthesis. To obtain accurate results it is necessary to record the original speech without phase distortion while maintaining at least 55–60 db signal-to-noise ratio. The data reported here were derived from recordings originating with a WE-640-AA condenser microphone in a free-space room feeding a digital tape recorder. Two male speakers uttered six vowel sounds between h and d at different pitches and intensities. The glottal waveforms were examined to determine both shape and phase with respect to the speech wave as a function of the sound, the pitch and the intensity. Sequences of periods were studied for variations from period to period.

Sufficiency conditions for cone-beam data are well known for the case of continuous data collection along a cone-vertex curve with continuous detectors. These continuous conditions are inadequate for real-world data where discrete vertex geometries and discrete detector arrays are used. The authors present a theoretical formulation of cone-beam tomography with arbitrary discrete arrays of detectors and vertices. The theory models the imaging system as a linear continuous-to-discrete mapping and represents the continuous object exactly as a Fourier series. The reconstruction problem is posed as the estimation of some subset of the Fourier coefficients. The main goal of the theory is to determine which Fourier coefficients can be reliably determined from the data delivered by a specific discrete design. A Fourier component will be well determined by the data if it satisfies two conditions: it makes a strong contribution to the data, and this contribution is relatively independent of the contribution of other Fourier components. To make these considerations precise, the authors introduce a concept called the cross-talk matrix. A diagonal element of this matrix measures the strength of a Fourier component in the data, while an off-diagonal element quantifies the dependence or aliasing of two different components. One reasonable approach to system design is to attempt to make the diagonal elements of this matrix large and the off-diagonal elements small for some set of Fourier components. If this goal can be achieved, simple linear reconstruction algorithms are available for estimating the Fourier coefficients. To illustrate the usefulness of this approach, numerical results on the cross-talk matrix are presented for different discrete geometries derived from a continuous helical vertex orbit, and simulated images reconstructed with two linear algorithms are presented.

Currently time-marching methods for unsteady turbomachinery flow calculations suffer from the inability to deal with multiple perturbations in a single-passage solution. A new periodic boundary condition method has been proposed, in which one can perform a single-passage solution with multiple perturbations. At the periodic boundaries, all of the perturbations are identified by their own phase-shifted periodicities and approximated by Fourier series. A combined parameter is chosen for the Fourier transform so that the Fourier coefficients can be efficiently updated. Corrections on both the variables at a current time level and on the Fourier coefficients are carried out repeatedly until a periodic state is reached. Numerical examples for oscillating cascade flows subject to inlet/outlet perturbations have demonstrated the validity of the present method and its ability to deal with situations of practical interest.

Local stress in periodic composites via the Riesz summability method

The line profile of the hth order reflection from a set of lattice planes is known to be given by a Fourier series with coefficients A(h,m)=N(m)J(h,m). The function N(m) depends on the particle shapes and sizes, and J(h,m) depends on the state of strain Neither can be completely determined from the experimental Fourier coefficients for a single value of h. They have, however, different behaviours for small m, and A(h,m) has an unambiguous interpretation in this region only.: its initial slope gives the mean particle thickness, and its initial curvature sets a lower limit to the mean-square strain. The practical value of this interpretation is subject to two limitations: (1) the initial shape of the A(h,m) curve depends on the `tail' of the line profile, (11) if the crystals are large enough to contain several regions of compression and extension, probability theory shows that J(m) is likely to be approximately proportional to exp {-const.|m|} for m large. Extrapolation to small m could then tend to give fictitiously small values for both particle size and strain

This paper studies the Fourier expansion of Hecke–Maass eigenforms for GL(2,ℚ) of arbitrary weight, level, and character at various cusps. Translating well known results in the theory of adelic automorphic representations into classical language, a multiplicative expression for the Fourier coefficients at any cusp is derived. In general, this expression involves Fourier coefficients at several different cusps. A sufficient condition for the existence of multiplicative relations among Fourier coefficients at a single cusp is given. It is shown that if the level is 4 times (or in some cases 8 times) an odd squarefree number then there are multiplicative relations at every cusp. We also show that a local representation of GL(2,ℚ p ) which is isomorphic to a local factor of a global cuspidal automorphic representation generated by the adelic lift of a newform of arbitrary weight, level N, and character χ (mod N) cannot be supercuspidal if χ is primitive. Furthermore, it is supercuspidal if and only if at every cusp (of width m and cusp parameter=0) the mp l Fourier coefficient, at that cusp, vanishes for all sufficiently large positive integers l. In the last part of this paper, a three term identity involving the Fourier expansion at three different cusps is derived.

II - Fourier Series and Fourier Transforms