Trigonometric Transformation Research Articles

A New Transceiver for OFDM Systems Using Smooth Local Trigonometric Transforms

This letter proposes a new transceiver for OFDM systems based on Smooth Local Trigonometric Transform (LTT). In our transceiver, the transmitter is realized by firstmodulating the original serial data using a constellation mapper, then feeding the results into the inverse LTT modulator. Unlike the conventional DFT-OFDM system, which always uses the roll cosine function as its window function, the proposed system needs no additional window function for the reason that LTT transform includes a bell-shaped window function by itself. Moreover, each LTT-OFDM symbol has a much more rapid attenuation rate outside of the spectral bandwidth and better spectrum convergence. In the receiver, the original data is recovered by demodulating the received data using forward LTT. Comparative simulation results from the conventional DFT-OFDM system, the system we proposed, and the recently proposed DCT based OFDM system are discussed in terms of bit error rate (BER).

Evaluation of rock-mass permeability tensor and prediction of tunnel inflows by means of geostructural surveys and finite element seepage analysis

In this paper, a new practical technical approach for the evaluation of hydraulic conductivity and tunnel water inflow in complex fractured rock masses is presented. This study was performed in order to evaluate water flow into tunnels planned along the new highway project from Firenze Nord gate to Barberino di Mugello (Tuscany). The results are based on detailed and comprehensive geostructural characterization of rock masses, by means of field surveys, geological and hydrogeological studies. Starting from discontinuities properties surveyed in the field, the permeability K tensor was calculated using the Kiraly equation, integrated with the introduction of the effective hydraulic opening of fissures ( e). Principal directions of tensor K were calculated for each geostructural survey station using an automated software script especially developed for this purpose. Available K data from Lugeon tests were also collected and analyzed with results that do not fully represent the rock mass due to its structural (hydraulic) variability. In order to evaluate water flow into tunnels planned for excavation under the water table (for a total number of eight), a finite elements seepage analysis was performed on 38 representative geological sections transverse to tunnel paths ( α-planes). Each section referred to the nearest and geologically most compatible geostructural station. Principal directions of K tensors were projected on the α-planes by means of trigonometric transformations. Unitary water inflows were then evaluated for long-term steady-state, as well as for initial state immediately after tunnel excavation. Inflow values calculated for each unitary section were extended to geologically homogeneous lengths of the tunnel, according to the variability of the water head above excavation, and then summed up for the whole length of each tunnel. Inflow values obtained with FE seepage analysis were also compared to other inflow evaluation methods.

A Superfast Algorithm for Toeplitz Systems of Linear Equations

In this paper we develop a new superfast solver for Toeplitz systems of linear equations. To solve Toeplitz systems many people use displacement equation methods. With displacement structures, Toeplitz matrices can be transformed into Cauchy-like matrices using the FFT or other trigonometric transformations. These Cauchy-like matrices have a special property, that is, their off-diagonal blocks have small numerical ranks. This low-rank property plays a central role in our superfast Toeplitz solver. It enables us to quickly approximate the Cauchy-like matrices by structured matrices called sequentially semiseparable (SSS) matrices. The major work of the constructions of these SSS forms can be done in precomputations (independent of the Toeplitz matrix entries). These SSS representations are compact because of the low-rank property. The SSS Cauchy-like systems can be solved in linear time with linear storage. Excluding precomputations the main operations are the FFT and SSS system solve, which are both very efficient. Our new Toeplitz solver is stable in practice. Numerical examples are presented to illustrate the efficiency and the practical stability.

Évaluation de la stimulation du noyau sous-thalamique utilisant un cadre de Fischer adapté, un guidage par fusion IRM-scanner et un contrôle radiographique stéréotaxique peropératoire, dans la maladie de Parkinson

Background We present the method and results of an original technique to implant electrodes in the subthalamic nucleus (STN) to treat Parkinson's disease, based on adaptations of the Fisher ZD stereotactic frame. Methods Targets coordinates were calculated after fusion of stereotactic CT-scan and MRI images. STN was localized by its theoretical coordinates according to AC–PC and by its direct visualization on T2 images. Electrodes were implanted after local anesthesia, using peroperative multicanal microrecordings and test stimulation. Electrodes location was checked by peroperative perpendicular radiographs. To avoid projection of the frame arm on the area of interest on anteroposterior and lateral radiographs, the arm was fixed at 45° from the usual 90° position. This original fixation needed a trigonometric transformation of the X and Y stereotactic coordinates. Radiopaque markers, fixed on the frame, were identified on the radiographs, allowing the calculation of the stereotactic coordinates of the electrode tip, which were then entered in the stereotactic MRI, to check its location from the defined target. Results No problem due to adaptations of the frame occurred in the 60 patients. In all cases, peroperative radiographs allowed to confirm the correct location of electrodes. Six months after surgery, UPDRS III score without medication was decreased by 52% with stimulation “on”. UPDRS IV items 32, 33 and 39 scores were decreased by 75,7, 79,5 and 72%. Daily dopa-equivalent dose was decreased by 71%. One asymptomatic thalamic hematoma and two wound infections occurred. Conclusion This method was efficient and safe to implant deep electrodes.

Transformations for complex discrete linear Hamiltonian and symplectic systems

This paper is concerned with transformations for complex discrete linear Hamiltonian systems and complex discrete linear symplectic systems. A general complex discrete trigonometric system is studied and a criterion for it is established. Based on these results, the Prüfer and trigonometric transformations for complex discrete linear Hamiltonian systems and complex discrete linear symplectic systems are formulated. The relative existing results in real cases are extended.

Automated and readable simplification of trigonometric expressions

Automated simplification of trigonometric expressions is an important problem that hasn’t been completely solved by current computer algebra systems. This paper presents a number of unique prescriptions for the ordering of some trigonometric transformation rules, which have been derived by observing how human experts follow their intuitive rules. We have implemented the procedure in Lisp because of its suitability for formula manipulations and rule-based reasoning systems. Consequently, it can simplify many trigonometric expressions which are even difficult to do by hand, and it achieves much better results for many hard problems than any of Maple, Mathematica, and Maxima do.

Technical note: Out‐of‐plane angular correction based on a trigonometric function for use in two‐dimensional kinematic studies

In two-dimensional (2D) kinematic studies, limb positions in three-dimensional (3D) space observed in lateral view are projected onto a 2D film plane. Elbow and knee-joint angles that are less than 20 degrees out-of-plane of lateral-view cameras generally exhibit very little measurable difference from their 3D counterparts (Plagenhoef 1979 Environment, Behavior, and Morphology; New York: Gustav Fisher, p. 95-118). However, when limb segment angles are more than 20 degrees out-of-plane, as is often the case in locomotor studies of arboreal primates, elbow and knee angles can appear significantly more extended than they actually are. For this reason, a methodology is described that corrects 2D out-of-plane angular estimates using a series of trigonometric transformations.

A superfast solver for real symmetric Toeplitz systems using real trigonometric transformations

AbstractA new superfast O(n log2 n) complexity direct solver for real symmetric Toeplitz systems is presented. The algorithm is based on 1. the reduction to symmetric right‐hand sides, 2. a polynomial interpretation in terms of Chebyshev polynomials, 3. an inversion formula involving real trigonometric transformations and 4. an interpretation of the equations as a tangential interpolation problem. The tangential interpolation problem is solved via a divide and conquer strategy and fast DCT. Copyright © 2005 John Wiley & Sons, Ltd.

Analytic signals and harmonic measures

We prove that a sufficient and necessary condition for H e i Θ ( s ) = − i e i Θ ( s ) , where H is Hilbert transformation, Θ is a continuous and strictly increasing function with | Θ ( R ) | = 2 π , is that d Θ ( s ) is a harmonic measure on the line. The counterpart result for the periodic case is also established. The study is motivated by, and has significant impact to time–frequency analysis, especially to aspects of analytic signals inducing instantaneous amplitude and frequency. As a by-product we introduce the theory of Hardy-space-preserving weighted trigonometric series and Fourier transformations induced by harmonic measures in the respective contexts.

Characterizing Uncertain Outcomes with the Restricted HT Transformation

Restrictions on the hyperbolic trigonometric (HT) transformation are imposed to guarantee that a probability density function is obtained from maximum likelihood estimation. Performance of the restricted HT transformation using data generated from normal, beta, gamma, logistic, log-normal, Pareto, Weibull, order statistic, and bimodal populations is investigated via sampling experiments. Results suggest that the restricted HT transformation is sufficiently flexible to compete with the actual population distributions in most cases. Application of the restricted HT transformation is illustrated by characterizing uncertain net income per acre for community-supported agriculture farms in the northeastern United States.

A necessary and sufficient condition for oscillation of the Sturm–Liouville dynamic equation on time scales

In this paper, we investigate oscillatory properties of the second order Sturm–Liouville dynamic equation on a time scale (∗) (r(t)x Δ ) Δ +c(t)x σ=0. Using the so-called trigonometric transformation we establish a necessary and sufficient condition for oscillation of (∗). This condition is then used to derive an explicit oscillation criterion for this equation.

Asymptotic Behavior of Trigonometric Series with O-Regularly Varying Quasimonotone Coefficients, II

In this paper, the asymptotic behavior of a trigonometric series with O-regularly varying quasimonotone coefficients is investigated. Our result generalizes the corresponding ones given in Boas's book (1967, “Integrability Theorems for Trigonometric Transforms,” Springer-Verlag, Berlin/New York) and Nurcombe's paper (1993, J. Math. Anal. Appl.178, 63–69).

Representations of Toeplitz-plus-Hankel matrices using trigonometric transformations with application to fast matrix-vector multiplication

Representations of real Toeplitz and Toeplitz-plus-Hankel matrices are presented that involve real trigonometric transformations (DCT, DST, DHT) and diagonal matrices. These representations can be used for fast matrix-vector multiplication. In particular, it is shown that the multiplication of an n × n Toeplitz-plus-Hankel matrix by a vector requires only 4 transformations of length n plus O( n) operations.

Mode matches and their locations in the hydrophobic free energy sequences of peptide ligands and their receptor eigenfunctions.

Patterns in sequences of amino acid hydrophobic free energies predict secondary structures in proteins. In protein folding, matches in hydrophobic free energy statistical wavelengths appear to contribute to selective aggregation of secondary structures in "hydrophobic zippers." In a similar setting, the use of Fourier analysis to characterize the dominant statistical wavelengths of peptide ligands' and receptor proteins' hydrophobic modes to predict such matches has been limited by the aliasing and end effects of short peptide lengths, as well as the broad-band, mode multiplicity of many of their frequency (power) spectra. In addition, the sequence locations of the matching modes are lost in this transformation. We make new use of three techniques to address these difficulties: (i) eigenfunction construction from the linear decomposition of the lagged covariance matrices of the ligands and receptors as hydrophobic free energy sequences; (ii) maximum entropy, complex poles power spectra, which select the dominant modes of the hydrophobic free energy sequences or their eigenfunctions; and (iii) discrete, best bases, trigonometric wavelet transformations, which confirm the dominant spectral frequencies of the eigenfunctions and locate them as (absolute valued) moduli in the peptide or receptor sequence. The leading eigenfunction of the covariance matrix of a transmembrane receptor sequence locates the same transmembrane segments seen in n-block-averaged hydropathy plots while leaving the remaining hydrophobic modes unsmoothed and available for further analyses as secondary eigenfunctions. In these receptor eigenfunctions, we find a set of statistical wavelength matches between peptide ligands and their G-protein and tyrosine kinase coupled receptors, ranging across examples from 13.10 amino acids in acid fibroblast growth factor to 2.18 residues in corticotropin releasing factor. We find that the wavelet-located receptor modes in the extracellular loops are compatible with studies of receptor chimeric exchanges and point mutations. A nonbinding corticotropin-releasing factor receptor mutant is shown to have lost the signatory mode common to the normal receptor and its ligand. Hydrophobic free energy eigenfunctions and their transformations offer new quantitative physical homologies in database searches for peptide-receptor matches.

Transformation techniques for Toeplitz and Toeplitz-plus-Hankel matrices. I. Transformations

Transformations of the form C are investigated that transform Toeplitz and Toeplitz-plus-Hankel matrices into generalized Cauchy matrices. C and C are matrices related to the discrete Fourier transformation or to various real trigonometric transformations. Combining these results with pivoting techniques, in paper II algorithms for Toeplitz and Toeplitz-plus-Hankel systems will be presented that are more stable than classical algorithms.

Error expansions for multidimensional trapezoidal rules with Sidi transformations

In 1993, Sidi introduced a set of trigonometric transformations x = ψ(t) that improve the effectiveness of the one-dimensional trapezoidal quadrature rule for a finite interval. In this paper, we extend Sidi's approach to product multidimensional quadrature over [0,1] N . We establish the Euler–Maclaurin expansion for this rule, both in the case of a regular integrand function f(x) and in the cases when f(x) has homogeneous singularities confined to vertices.

Domains of Trigonometric Transforms

Domains of Trigonometric Transforms

On the Use of Certain Matrix Algebras Associated with Discrete Trigonometric Transforms in Matrix Displacement Decomposition

The authors extend some recent results of Di Fiore and Zellini [Linear Algebra Appl., to appear], obtaining new classes of formulas for the displacement operator-based decomposition of matrices. It is shown how an arbitrary matrix can be expressed as the sum of products of matrices belonging to matrix algebras associated with certain versions of sine and cosine transforms. Applications to the representation of the inverse of a Toeplitz and a Toeplitz plus Hankel matrix, with and without symmetry, are presented. Implications on the computation of the product of these matrices by a vector are discussed.

Fast Fourier wavelet packet transformation

The wavelet basis generated by Shannon's sampling theorem is presented. Periodical finite dimensional wavelet and Fourier wavelet packets are suggested. A link of constructed bases with complex trigonometric series and discrete Fourier transformation is considered. The Fourier wavelet packet may be used as widely as discrete Fourier transformation.

De Jericho au bang sonique. Les précurseurs des arts et sciences acoustiques

After Adam, Cain, Henoch,... «Yubal was the gather of all which as handle the harp and organ» (Bible, Genesis, IV. 21). From the Bible, and through Babylonian, Chinese, Egyptian, Arabs civilisations, the Middle-ages and after, we found many precursive ideas or realisations of our acoustic theories or instruments. Some examples are given: musical scales, vibrating wires and pipes, nature of sound, its propagation and velocity, echo and thunder, sound recording, room acoustics, sound detection, noise annoyance, even trigonometric series and Fourier Transform, shock waves and sonic boom