Square Wave and Truncated Fourier Series

2015 ◽  
Author(s):  
J. Scott Tyo ◽  
Andrey Alenin
Keyword(s):  
Fourier Series ◽  
Square Wave ◽  
Truncated Fourier Series

Investigation of discontinuous periodic signals using the measuring labview complex

2020 ◽  
pp. 72-79
Author(s):  
E.B. Solovyeva ◽  
◽  
Yu.M. Inshakov ◽  
Keyword(s):  
Fourier Series ◽  
Laboratory Experiment ◽  
Gibbs Phenomenon ◽  
Electrical Signals ◽  
Measuring Complex ◽  
Universal Measuring ◽  
Depth Study ◽  
Periodic Signals ◽  
Truncated Fourier Series

General approaches to the analysis of the Gibbs phenomenon for discontinuous periodic signals approximated by the truncated Fourier series are considered. Methods for smoothing the truncated Fourier series and improving its convergence are discussed. The software means for modeling is a universal measuring complex LabVIEW, which possesses a convenient environment for analyzing electrical signals, on the basis of this complex a laboratory experiment is carried out. The advantages of the measuring LabVIEW complex and its capabilities for in-depth study of discontinuous periodic signals are noted.

scholarly journals Dynamics of Fourier Modes in Torus Generative Adversarial Networks

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 325
Author(s):  
Ángel González-Prieto ◽  
Alberto Mozo ◽  
Edgar Talavera ◽  
Sandra Gómez-Canaval
Keyword(s):  
Fourier Series ◽  
Generative Adversarial Networks ◽  
Learning Models ◽  
Training Process ◽  
Small Perturbations ◽  
Adversarial Networks ◽  
Novel Method ◽  
Truncated Fourier Series ◽  
Real Flow ◽  
Machine Learning Models

Generative Adversarial Networks (GANs) are powerful machine learning models capable of generating fully synthetic samples of a desired phenomenon with a high resolution. Despite their success, the training process of a GAN is highly unstable, and typically, it is necessary to implement several accessory heuristics to the networks to reach acceptable convergence of the model. In this paper, we introduce a novel method to analyze the convergence and stability in the training of generative adversarial networks. For this purpose, we propose to decompose the objective function of the adversary min–max game defining a periodic GAN into its Fourier series. By studying the dynamics of the truncated Fourier series for the continuous alternating gradient descend algorithm, we are able to approximate the real flow and to identify the main features of the convergence of GAN. This approach is confirmed empirically by studying the training flow in a 2-parametric GAN, aiming to generate an unknown exponential distribution. As a by-product, we show that convergent orbits in GANs are small perturbations of periodic orbits so the Nash equillibria are spiral attractors. This theoretically justifies the slow and unstable training observed in GANs.

scholarly journals Gibbs-Wilbraham Phenomenon in Square Wave Function

2018 ◽  
Vol 14 (1) ◽  
pp. 206-215
Author(s):  
Bhupendra Budha ◽  
Man Bahadur Subedi
Keyword(s):  
Wave Function ◽  
Fourier Series ◽  
Spline Interpolation ◽  
Cubic Spline ◽  
Cubic Spline Interpolation ◽  
Smooth Curves ◽  
Square Wave ◽  
Mathematical Relation

The aim of this paper is to show the fluctuation on overshoots of particular term due to change in magnitude of discontinuity in the square wave function represented in Fourier series and also is to address the mathematical relation between those parameters. Along with that, some graphical plots of different terms are given to illustrate the results and to interpolate data with smooth curves using Cubic spline interpolation in MATLAB.Journal of the Institute of Engineering, 2018, 14(1): 206-215

Truncated Fourier series formulation for bipedal walking balance control

Robotica ◽  
2009 ◽  
Vol 28 (1) ◽  
pp. 81-96 ◽  
Author(s):  
Lin Yang ◽  
Chee-Meng Chew ◽  
Yu Zheng ◽  
Aun-Neow Poo
Keyword(s):  
Fourier Series ◽  
Control Method ◽  
Balance Control ◽  
Bipedal Walking ◽  
Reference Trajectory ◽  
Dynamic Feedback ◽  
Long Distance ◽  
Time Dynamic ◽  
Truncated Fourier Series ◽  
Walking Balance

SUMMARYThis paper studies the parameters contained in the truncated Fourier series (TFS) formulation for bipedal walking balance control. Using the TFS generated lateral motion reference, 3D bipedal walking can be directly achieved without any parameter adjustment. Furthermore, the potential of this TFS formulation for motion balance control has also been investigated. One more motion balance strategy is developed through the reinforcement learning, which adjusts the motion's reference trajectory according to the selected dynamic feedback in real time. Dynamic simulation results of the presented balance control method show that the resulting motion can be constrained periodical and long-distance 3D bipedal walking motions are achievable.

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