Using EMD-FrFT filtering to mitigate high power interference in chirp tracking radars

This letter presents a new signal processing subsystem for conventional monopulse tracking radars that offers an improved solution to the problem of dealing with manmade high power interference (jamming). It is based on the hybrid use of empirical mode decomposition (EMD) and fractional Fourier transform (FrFT). EMD-FrFT filtering is carried out for complex noisy radar chirp signals to decrease the signal's noisy components. An improvement in the signal-to-noise ratio (SNR) of up to 18 dB for different target SNRs is achieved using the proposed EMD-FrFT algorithm

ShearLab: A Rational Design of a Digital Parabolic Scaling Algorithm

Multivariate problems are typically governed by anisotropic features such as edges in images. A common bracket of most of the various directional representation systems which have been proposed to deliver sparse approximations of such features is the utilization of parabolic scaling. One prominent example is the shearlet system. Our objective in this paper is three-fold: We firstly develop a digital shearlet theory which is rationally designed in the sense that it is the digitization of the existing shearlet theory for continuous data. This implicates that shearlet theory provides a unified treatment of both the continuum and digital realm. Secondly, we analyze the utilization of pseudo-polar grids and the pseudo-polar Fourier transform for digital implementations of parabolic scaling algorithms. We derive an isometric pseudo-polar Fourier transform by careful weighting of the pseudo-polar grid, allowing exploitation of its adjoint for the inverse transform. This leads to a digital implementation of the shearlet transform; an accompanying Matlab toolbox called ShearLab is provided. And, thirdly, we introduce various quantitative measures for digital parabolic scaling algorithms in general, allowing one to tune parameters and objectively improve the implementation as well as compare different directional transform implementations. The usefulness of such measures is exemplarily demonstrated for the digital shearlet transform.Comment: submitted to SIAM J. Multiscale Model. Simu

An Algorithm for Precise Aperture Photometry of Critically Sampled Images

We present an algorithm for performing precise aperture photometry on critically sampled astrophysical images. The method is intended to overcome the small-aperture limitations imposed by point-sampling. Aperture fluxes are numerically integrated over the desired aperture, with sinc-interpolation used to reconstruct values between pixel centers. Direct integration over the aperture is computationally intensive, but the integrals in question are shown to be convolution integrals and can be computed ~10000x faster as products in the wave-number domain. The method works equally well for annular and elliptical apertures and could be adapted for any geometry. A sample of code is provided to demonstrate the method.Comment: Accepted MNRA

Modern Methods of Time-Frequency Warping of Sound Signals

Tato práce se zabývá reprezentací nestacionárních harmonických signálů s časově proměnnými komponentami. Primárně je zaměřena na Harmonickou transformaci a jeji variantu se subkvadratickou výpočetní složitostí, Rychlou harmonickou transformaci. V této práci jsou prezentovány dva algoritmy využívající Rychlou harmonickou transformaci. Prvni používá jako metodu odhadu změny základního kmitočtu sbírané logaritmické spektrum a druhá používá metodu analýzy syntézou. Oba algoritmy jsou použity k analýze řečového segmentu pro porovnání vystupů. Nakonec je algoritmus využívající metody analýzy syntézou použit na reálné zvukové signály, aby bylo možné změřit zlepšení reprezentace kmitočtově modulovaných signálů za použití Harmonické transformace.This thesis deals with representation of non-stationary harmonic signals with time-varying components. Its main focus is aimed at Harmonic Transform and its variant with subquadratic computational complexity, the Fast Harmonic Transform. Two algorithms using the Fast Harmonic Transform are presented. The first uses the gathered log-spectrum as fundamental frequency change estimation method, the second uses analysis-by-synthesis approach. Both algorithms are used on a speech segment to compare its output. Further the analysis-by-synthesis algorithm is applied on several real sound signals to measure the increase in the ability to represent real frequency-modulated signals using the Harmonic Transform.

Multigrid waveform relaxation for the time-fractional heat equation

In this work, we propose an efficient and robust multigrid method for solving the time-fractional heat equation. Due to the nonlocal property of fractional differential operators, numerical methods usually generate systems of equations for which the coefficient matrix is dense. Therefore, the design of efficient solvers for the numerical simulation of these problems is a difficult task. We develop a parallel-in-time multigrid algorithm based on the waveform relaxation approach, whose application to time-fractional problems seems very natural due to the fact that the fractional derivative at each spatial point depends on the values of the function at this point at all earlier times. Exploiting the Toeplitz-like structure of the coefficient matrix, the proposed multigrid waveform relaxation method has a computational cost of $O(N M \log(M))$ operations, where $M$ is the number of time steps and $N$ is the number of spatial grid points. A semi-algebraic mode analysis is also developed to theoretically confirm the good results obtained. Several numerical experiments, including examples with non-smooth solutions and a nonlinear problem with applications in porous media, are presented

Efficient numerical solution of the time fractional diffusion equation by mapping from its Brownian counterpart

The solution of a Caputo time fractional diffusion equation of order $0<\alpha<1$ is expressed in terms of the solution of a corresponding integer order diffusion equation. We demonstrate a linear time mapping between these solutions that allows for accelerated computation of the solution of the fractional order problem. In the context of an $N$-point finite difference time discretisation, the mapping allows for an improvement in time computational complexity from $O\left(N^2\right)$ to $O\left(N^\alpha\right)$, given a precomputation of $O\left(N^{1+\alpha}\ln N\right)$. The mapping is applied successfully to the least-squares fitting of a fractional advection diffusion model for the current in a time-of-flight experiment, resulting in a computational speed up in the range of one to three orders of magnitude for realistic problem sizes.Comment: 9 pages, 5 figures; added references for section