scholarly journals Expansions of Functions Based on Rational Orthogonal Basis with Nonnegative Instantaneous Frequencies

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Xiaona Cui ◽  
Suxia Yao
Keyword(s):  
Fourier Transform ◽  
Fast Fourier Transform ◽  
Fast Algorithm ◽  
Orthogonal Basis ◽  
Basis Functions ◽  
Almost Everywhere ◽  
Integrable Functions ◽  
Instantaneous Frequencies ◽  
Square Integrable Functions

We consider in this paper expansions of functions based on the rational orthogonal basis for the space of square integrable functions. The basis functions have nonnegative instantaneous frequencies so that the expansions make physical sense. We discuss the almost everywhere convergence of the expansions and develop a fast algorithm for computing the coefficients arising in the expansions by combining the characterization of the coefficients with the fast Fourier transform.

scholarly journals On bicomplex Fourier–Wigner transforms

2020 ◽  
Vol 18 (03) ◽  
pp. 2050008
Author(s):  
A. El Gourari ◽  
A. Ghanmi ◽  
K. Zine
Keyword(s):  
Orthogonal Basis ◽  
Hermite Functions ◽  
Window Function ◽  
Specific Class ◽  
Wigner Transform ◽  
Basic Properties ◽  
Bicomplex Numbers ◽  
Integrable Functions ◽  
Square Integrable Functions

We consider the [Formula: see text]d and [Formula: see text]d bicomplex analogues of the classical Fourier–Wigner transform. Their basic properties, including Moyal’s identity and characterization of their ranges giving rise to new bicomplex–polyanalytic functional spaces are discussed. Details concerning a special window function are developed explicitly. An orthogonal basis for the space of bicomplex-valued square integrable functions on the bicomplex numbers is constructed by means of a specific class of bicomplex Hermite functions.

scholarly journals Hermite Functions and Fourier Series

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 853
Author(s):  
Enrico Celeghini ◽  
Manuel Gadella ◽  
Mariano del Olmo
Keyword(s):  
Fourier Transform ◽  
Hermite Functions ◽  
Quantum Oscillator ◽  
Linear Combinations ◽  
Ladder Operators ◽  
Orthonormal Bases ◽  
Integrable Functions ◽  
The Fourier Transform ◽  
The One ◽  
Square Integrable Functions

Using normalized Hermite functions, we construct bases in the space of square integrable functions on the unit circle (L2(C)) and in l2(Z), which are related to each other by means of the Fourier transform and the discrete Fourier transform. These relations are unitary. The construction of orthonormal bases requires the use of the Gramm–Schmidt method. On both spaces, we have provided ladder operators with the same properties as the ladder operators for the one-dimensional quantum oscillator. These operators are linear combinations of some multiplication- and differentiation-like operators that, when applied to periodic functions, preserve periodicity. Finally, we have constructed riggings for both L2(C) and l2(Z), so that all the mentioned operators are continuous.

scholarly journals Caracterización en línea de la dinámica temporal de señales para el monitoreo del sistema eléctrico de potencia

2020 ◽  
pp. 26-34
Author(s):  
Francisco Román Lezama-Zárraga ◽  
Jorge de Jesús Chan-González ◽  
Meng Yen Shih ◽  
Roberto Carlos Canto-Canul
Keyword(s):  
Nonlinear Oscillations ◽  
Electrical Power ◽  
Orthogonal Basis ◽  
Basis Functions ◽  
Electrical Power System ◽  
Transient Phenomena ◽  
On Line ◽  
Measurement Units ◽  
Dynamic Phenomena

The characterization of dynamic phenomena is essential for monitoring the Electrical Power System subject to disturbances. This article proposes an On-line time systematic approach to analyze and characterize the temporal evolution of transient and nonlinear oscillations in these systems. Two methods are used; the first method is based on a local decomposition of the signal under study into orthogonal basis functions to obtain the dynamics of transient oscillations. Next, a second method is applied to those orthogonal basis functions to obtain analytical signals and characterize the instantaneous amplitude, phase and frequency attributes of the oscillations and determine a physical interpretation of the system’s behavior. The proposed methodology is a time-frequencyenergy analysis which can be applied to the timesynchronized Phasor Measurement Units measurements. The results demonstrate that the proposed methodology provide an accurate characterization of transient phenomena with non-stationary effects.

scholarly journals Resolution of the Identity of the Classical Hardy Space by Means of Barut-Girardello Coherent States

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Zouhaïr Mouayn
Keyword(s):  
Fourier Transform ◽  
Hardy Space ◽  
Real Line ◽  
Coherent States ◽  
Positive Real ◽  
Integrable Functions ◽  
Resolution Of The Identity ◽  
Square Integrable Functions ◽  
Complex Valued ◽  
Classical Hardy Space

We construct a one-parameter family of coherent states of Barut-Girdrardello type performing a resolution of the identity of the classical Hardy space of complex-valued square integrable functions on the real line, whose Fourier transform is supported by the positive real semiaxis.

Characterization of fatigued Al lines by means of SThM and XRD: Analysis using fast Fourier transform

2012 ◽  
Vol 52 (4) ◽  
pp. 711-717 ◽  
Author(s):  
R.F. Szeloch ◽  
P. Janus ◽  
J. Serafińczuk ◽  
P.M. Szecówka ◽  
G. Jóźwiak
Keyword(s):  
Fourier Transform ◽  
Fast Fourier Transform ◽  
Xrd Analysis

scholarly journals Generalized Orthogonal Discrete W Transform and Its Fast Algorithm

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Jichao Sun ◽  
Zhengping Zhang
Keyword(s):  
Signal Processing ◽  
Fourier Transform ◽  
Fast Algorithm ◽  
Basis Functions ◽  
Real Sequence ◽  
Communication Signal ◽  
Numerical Computing ◽  
Orthogonal Transform ◽  
Generalized Orthogonal ◽  
And Storage

Based on the generalized discrete Fourier transform, the generalized orthogonal discrete W transform and its fast algorithm are proposed and derived in this paper. The orthogonal discrete W transform proposed by Zhongde Wang has only four types. However, the generalized orthogonal discrete W transform proposed by us has infinite types and subsumes a family of symmetric transforms. The generalized orthogonal discrete W transform is a real-valued orthogonal transform, and the real-valued orthogonal transform of a real sequence has the advantages of simple operation and facilitated transmission and storage. The generalized orthogonal discrete W transforms provide more basis functions with new frequencies and phases and hence lead to more powerful analysis and processing tools for communication, signal processing, and numerical computing.

scholarly journals Sparse Time-Frequency Decomposition for Multiple Signals with Same Frequencies

2017 ◽  
Vol 09 (04) ◽  
pp. 1750010
Author(s):  
Thomas Y. Hou ◽  
Zuoqiang Shi
Keyword(s):  
Fourier Transform ◽  
Fast Fourier Transform ◽  
Frequency Analysis ◽  
Special Structure ◽  
Data Driven ◽  
Multiple Signals ◽  
Time Frequency ◽  
Frequency Decomposition ◽  
Engineering Problems ◽  
Instantaneous Frequencies

In this paper, we consider multiple signals sharing the same instantaneous frequencies. This kind of data is very common in scientific and engineering problems. To take advantage of this special structure, we modify our data-driven time-frequency analysis by updating the instantaneous frequencies simultaneously. Moreover, based on the simultaneous sparsity approximation and the Fast Fourier Transform, we develop several efficient algorithms to solve this problem. Since the information of multiple signals is used, this method is very robust to the perturbation of noise and it is applicable to the general nonperiodic signals even with missing samples or outliers. Several synthetic and real signals are used to demonstrate the robustness of this method. The performances of this method seems quite promising.

Sign in / Sign up

Export Citation Format

Share Document