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Fourier series

About: Fourier series is a research topic. Over the lifetime, 16548 publications have been published within this topic receiving 322486 citations.


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09 Jan 1997
TL;DR: In this article, the Fourier Transform was used to detect the presence of complex tones in the frequency domain and to detect and detect modulation in a noisy environment, where the amplitude and pitch of Sine Tones were measured.
Abstract: Preface.- 1. Pure Tones.- 2. Complex Representation.- 3. Power, Intensity, and Decibels.- 4. Intensity and Loudness.- 5. Fourier Series.- 6. Perception of Periodic Complex Tones.- 7. Delta Functions.- 8. Fourier Integral.- 9. Filters.- 10. Auditory Filters.- 11. Musical Measures of Frequency.- 12. Pitch of Sine Tones.- 13. Applications of the Fourier Transform.- 14. Correlation Functions and Spectra.- 15. Delay-and-Add Filtering.- 16. Probability Density Functions.- 17. Beats and Amplitude Modulation.- 18. The Envelope.- 19. Frequency Modulation.- 20. Modulation Detection and Perception.- 21. Sampled Signals.- 22. Nonlinear Distortion.- 23. Noise.- 24. Signal Detection Theory.- Appendices A - K.- References.- Index.

411 citations

28 Feb 2010
TL;DR: In this article, the authors introduce the concepts of participant triangularity and triangular flow in heavy-ion collisions, analogous to the definitions of participant eccentricity and elliptic flow, and show that triangular flow is present in data.
Abstract: We introduce the concepts of participant triangularity and triangular flow in heavy-ion collisions, analogous to the definitions of participant eccentricity and elliptic flow. The participant triangularity characterizes the triangular anisotropy of the initial nuclear overlap geometry and arises from event-by-event fluctuations in the participant-nucleon collision points. In studies using a multiphase transport model (AMPT), a triangular flow signal is observed that is proportional to the participant triangularity and corresponds to a large third Fourier coefficient in two-particle azimuthal correlation functions. Using two-particle azimuthal correlations at large pseudorapidity separations measured by the PHOBOS and STAR experiments, we show that this Fourier component is also present in data. Ratios of the second and third Fourier coefficients in data exhibit similar trends as a function of centrality and transverse momentum as in AMPT calculations. These findings suggest a significant contribution of triangular flow to the ridge and broad away-side features observed in data. Triangular flow provides a new handle on the initial collision geometry and collective expansion dynamics in heavy-ion collisions.

406 citations

Journal ArticleDOI
TL;DR: In this paper, a new method employing periodic orthogonal polynomials to fit the observations and the analysis of variance (ANOVA) statistic to evaluate the quality of the fit is presented.
Abstract: The classical methods for searching for a periodicity in uneven sampled observations suffer from a poor match of the model and true signals and/or use of a statistic with poor properties. We present a new method employing periodic orthogonal polynomials to fit the observations and the analysis of variance (ANOVA) statistic to evaluate the quality of the fit. The orthogonal polynomials constitute a flexible and numerically efficient model of the observations. Among all popular statistics, ANOVA has optimum detection properties as the uniformly most powerful test. Our recurrence algorithm for expansion of the observations into the orthogonal polynomials is fast and numerically stable. The expansion is equivalent to an expansion into Fourier series. Aside from its use of an inefficient statistic, the Lomb-Scargle power spectrum can be considered a special case of our method. Tests of our new method on simulated and real light curves of nonsinusoidal pulsators demonstrate its excellent performance. In particular, dramatic improvements are gained in detection sensitivity and in the damping of alias periods.

399 citations

Journal ArticleDOI
TL;DR: In this paper, an approximation to the exact evolution equation for the Fourier coefficients of the disturbance is proposed and it is shown, by an asymptotic analysis valid at large times, that the solution of the approximate equations develops a singularity at a critical time.
Abstract: The evolution of a small amplitude initial disturbance to a straight uniform vortex sheet is described by the Fourier coefficients of the disturbance. An approximation to the exact evolution equation for these coefficients is proposed and it is shown, by an asymptotic analysis valid at large times, that the solution of the approximate equations develops a singularity at a critical time. The critical time is proportional to ln $(\epsilon ^{-1})$, where $\epsilon $ is the initial amplitude of the disturbance and the singularity itself is such that the nth Fourier coefficient decays like $n^{-2.5}$ instead of exponentially. Evidence-not conclusive, however-is present to show that the approximation used is adequate. It is concluded that the class of vortex layer motions correctly modelled by replacing the vortex layer by a vortex sheet is very restricted; the vortex sheet is an inadequate approximation unless it is everywhere undergoing rapid stretching.

389 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
2023270
2022702
2021511
2020510
2019589
2018580