Doppler Spectrum Estimation by Ramanujan Fourier Transforms

The Doppler spectrum estimation of a weather radar signal in a classic way can be made by two methods, temporal one based in the autocorrelation of the successful signals, whereas the other one uses the estimation of the power spectral density PSD by using Fourier transforms. We introduces a new tool of signal processing based on Ramanujan sums cq(n), adapted to the analysis of arithmetical sequences with several resonances p/q. These sums are almost periodic according to time n of resonances and aperiodic according to the order q of resonances. New results will be supplied by the use of Ramanujan Fourier Transform (RFT) for the estimation of the Doppler spectrum for the weather radar signal

A fast analysis-based discrete Hankel transform using asymptotic expansions

A fast and numerically stable algorithm is described for computing the discrete Hankel transform of order $0$ as well as evaluating Schl\"{o}milch and Fourier--Bessel expansions in $\mathcal{O}(N(\log N)^2/\log\!\log N)$ operations. The algorithm is based on an asymptotic expansion for Bessel functions of large arguments, the fast Fourier transform, and the Neumann addition formula. All the algorithmic parameters are selected from error bounds to achieve a near-optimal computational cost for any accuracy goal. Numerical results demonstrate the efficiency of the resulting algorithm.Comment: 22 page

Detection of variable frequency signals using a fast chirp transform

The detection of signals with varying frequency is important in many areas of physics and astrophysics. The current work was motivated by a desire to detect gravitational waves from the binary inspiral of neutron stars and black holes, a topic of significant interest for the new generation of interferometric gravitational wave detectors such as LIGO. However, this work has significant generality beyond gravitational wave signal detection. We define a Fast Chirp Transform (FCT) analogous to the Fast Fourier Transform (FFT). Use of the FCT provides a simple and powerful formalism for detection of signals with variable frequency just as Fourier transform techniques provide a formalism for the detection of signals of constant frequency. In particular, use of the FCT can alleviate the requirement of generating complicated families of filter functions typically required in the conventional matched filtering process. We briefly discuss the application of the FCT to several signal detection problems of current interest

Comparison of five methods of computing the Dirichlet-Neumann operator for the water wave problem

We compare the effectiveness of solving Dirichlet-Neumann problems via the Craig-Sulem (CS) expansion, the Ablowitz-Fokas-Musslimani (AFM) implicit formulation, the dual AFM formulation (AFM*), a boundary integral collocation method (BIM), and the transformed field expansion (TFE) method. The first three methods involve highly ill-conditioned intermediate calculations that we show can be overcome using multiple-precision arithmetic. The latter two methods avoid catastrophic cancellation of digits in intermediate results, and are much better suited to numerical computation. For the Craig-Sulem expansion, we explore the cancellation of terms at each order (up to 150th) for three types of wave profiles, namely band-limited, real-analytic, or smooth. For the AFM and AFM* methods, we present an example in which representing the Dirichlet or Neumann data as a series using the AFM basis functions is impossible, causing the methods to fail. The example involves band-limited wave profiles of arbitrarily small amplitude, with analytic Dirichlet data. We then show how to regularize the AFM and AFM* methods by over-sampling the basis functions and using the singular value decomposition or QR-factorization to orthogonalize them. Two additional examples are used to compare all five methods in the context of water waves, namely a large-amplitude standing wave in deep water, and a pair of interacting traveling waves in finite depth.Comment: 31 pages, 18 figures. (change from version 1: corrected error in table on page 12

Development of a vibration measurement device based on a MEMS accelerometer

© 2017 by SCITEPRESS. Published under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International licence (CC BY-NC-ND 4.0: https://creativecommons.org/licenses/by-nc-nd/4.0/)This paper proposes a portable and low cost vibration detection device. Enhanced vibration calculation, reduction of error and low storage memory are complementary accomplishments of this research. The device consists of a MEMS capacitive accelerometer sensor and microcontroller unit, which operates based on a novel algorithm designed to obtained vibration velocity, bypassing the usual time-based integration process. The proposed algorithm can detect vibrations within 15Hz-1000Hz frequencies. Vibration in this frequency range cannot be easily and accurately evaluated with conventional low cost digital sensors. The proposed technique is assessed and validated by comparing results with an industrial grade vibration meter

Higher-order splitting algorithms for solving the nonlinear Schr\"odinger equation and their instabilities

Since the kinetic and the potential energy term of the real time nonlinear Schr\"odinger equation can each be solved exactly, the entire equation can be solved to any order via splitting algorithms. We verified the fourth-order convergence of some well known algorithms by solving the Gross-Pitaevskii equation numerically. All such splitting algorithms suffer from a latent numerical instability even when the total energy is very well conserved. A detail error analysis reveals that the noise, or elementary excitations of the nonlinear Schr\"odinger, obeys the Bogoliubov spectrum and the instability is due to the exponential growth of high wave number noises caused by the splitting process. For a continuum wave function, this instability is unavoidable no matter how small the time step. For a discrete wave function, the instability can be avoided only for \dt k_{max}^2{<\atop\sim}2 \pi, where $k_{max}=\pi/\Delta x$.Comment: 10 pages, 8 figures, submitted to Phys. Rev.